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Evaluation of wavefunctions by electron momentum spectroscopy Bawagan, Alexis Delano Ortiz 1987

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EVALUATION  OF WAVEFUNCTIONS BY E L E C T R O N  MOMENTUM  SPECTROSCOPY by A L E X I S D E L A N O ORTIZ B.Sc.(cum laude), University of the M.Sc, A THESIS  BAWAGAN Philippines,  University of Houston,  1979  1982  SUBMITTED IN PARTIAL F U L F I L M E N T OF  T H E REQUIREMENTS FOR THE DEGREE OF DOCTOR O F  PHILOSOPHY  in T H E F A C U L T Y OF G R A D U A T E STUDIES Chemistry Department We accept this thesis as to the  conforming  required standard  T H E U N I V E R S I T Y O F BRITISH C O L U M B I A August, c  1987  Alexis Delano Ortiz Bawagan,  1987  In presenting  this thesis in partial fulfilment  of the  requirements for an advanced  degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department  or  by  his  or  her  representatives.  It  is  understood  that  copying  or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  DE-6(3/81)  'Sa aking mga Magulang' (To my Mother and Father)  i  ABSTRACT Electron  momentum  spectroscopy  molecular electronic structure and the  (EMS)  provides  experimental  atomic  information in terms of the binding energy  experimental momentum  spectrum  profile (XMP), which is a direct probe  of the  electron momentum distribution in specific molecular orbitals. The measured permit a detailed quantitative quantum  and  XMPs  evaluation of theoretical ab initio wavefunctions in  chemistry and also provide a means to investigate  traditional  concepts  in chemical reactivity at the fundamental electronic level.  This thesis reports high momentum resolution E M S measurements of the orbitals of H 0 , D 0 , N H 2  2  and H C O  3  obtained using an E M S spectrometer of  2  the symmetric, non-coplanar type operated measured  experimental  molecule have stringent now  been  momentum  at an impact energy of 1200eV. The  profiles  for  the  placed on a common intensity  quantitative  comparison between  valence  experimental  and  experiment  theoretical  orbitals  scale, which and  momentum  has  theory.  confirm earlier preliminary investigations that suggested  between  valence  of  each  allowed a  These  studies  serious  discrepancies  distributions.  Exhaustive  consideration of possible rationalizations of these discrepancies indicate that double zeta quality and even near Hartree-Fock quality wavefunctions describing the outermost  valence orbitals of H 0  for  that  H CO 2  describing  also indicate the  outermost  efforts have therefore  2b  2  3  Hartree-Fock wavefunctions  orbital.  Interactive  and  are  collaborative  incapable of theoretical  led to the development of new Hartree-Fock limit and also  highly correlated (CI) wavefunctions for H 0 , N H 2  highly extended  and N H . Preliminary results  2  near  are insufficient in  basis sets including diffuse  ii  3  and H C O .  functions  2  and the  It is found that adequate inclusion  of correlation and  relaxation effects  are  necessary  accurate prediction of  in the  experimental momentum profiles as measured by electron momentum  New E M S measurements are also reported  for the outermost  NF ,  and  NH CH ,  3  2  NH (CH ) ,  3  exploratory  3  studies  2  have  N (CH ) 3  illustrated  3  useful  revealed  chemical  calculations. These the  trends  which  chemical  pair'  in  are  consistent  with  experimental  arguments  trends,  qualitatively  quality  E M S measurements  of para-dichlorobenzene  orbitals  were  3  show  based  both  in the  predicted  by  wavefunctions.  However  on case  more  of  with  E M S . In  molecular  electron density amines  the  different  of the  orbital  accurate  and  3  the 7T  and  2  profiles  effects.  These  para-dichlorobenzene,  calculations prediction  IT  momentum  resonance  amines  orbital  derealization of  non-degenerate  and  amines  in comparison to  experimental  inductive  molecular  of  applications  consistent  calculations suggest extensive  NH .  These  orbitals of the methylated  so-called nitrogen 'lone pair' in the methylated  Tone  valence orbitals of  para-dichlorobenzene.  particular, E M S measurements of the outermost have  spectroscopy.  of  using the  double  zeta  experimental  momentum profiles of these molecules will need more extended basis sets and the inclusion of correlation and relaxation effects  as  suggested  by the  studies  based  on the smaller molecules.  An also  integrated been  computer  developed  package  based  (HEMS)  for  on improvements  studies testing a new prototype  momentum to  existing programs.  multichannel (in the  are described.  iii  space calculations  has  Development  0 plane) E M S spectrometer  T A B L E  O F  C O N T E N T S  ABSTRACT  "  T A B L E OF CONTENTS  *  LIST O F F I G U R E S  v  v i i  LIST O F T A B L E S  x i  LIST O F A B B R E V I A T I O N S  xii  ACKNOWLEDGEMENTS  xiv  Chapter  1. I N T R O D U C T I O N 1.1. Historical Remarks 1.2. Binding Energy Spectrum (BES): A Physical Observable 1.3. Experimental Momentum Profile (XMP): A Physical Observable 1.4. Scope of Thesis  1 1 3 5 9  Chapter 2. T H E O R E T I C A L M E T H O D 11 2.1. Theory of Electron Momentum Spectroscopy (EMS) 11 2.1.1. Introductory Remarks 11 2.1.2. Plane Wave Impulse Approximation (PWIA) .. 14 2.1.3. Target Hartree-Fock Approximation (THFA) ... 20 2.1.4. Full Ion-Neutral Overlap Amplitude 24 2.2. Momentum Space Chemistry 25 2.3. Basis Sets for Ab initio Wavefunctions 29 2.4. Electron Correlation 32 2.5. Green's Function Methods and Ionization Spectra 34 2.6. H E M S : A Computer Package for Momentum-Space Calculations 36 1  Chapter 3. E X P E R I M E N T A L M E T H O D 3.1. Electron Momentum Spectrometer 3.1.1. Description of Spectrometer 3.1.2. Coincidence Detection, Event Processing and Control 3.2. Modelling the Effects of Finite Momentum Resolution 3.2.1. Planar Grid Method 3.2.2. Analytic Gaussian Function Method 3.2.3. Defining the Optimum p-Value  44 44 44  Chapter 4. W A T E R : P A R T I 4.1. Overview 4.2. Binding Energy Spectra of Water  71 71 73  iv  51 57 59 63 66  4.3. Momentum Distributions of Water 4.4. Comparison of Experimental and Theoretical Momentum Distributions 4.5. Orbital Density Maps and Surfaces 4.6. Wide Range Momentum Density Maps Chapter 5. WATER: 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8.  80 82 95 97  PART II Overview Extended Basis Sets for Water Experimental Details: Normalization of Data Vibrational Effects Basis Set Effects Correlation and Relaxation Effects Calculated Properties Near the H F and CI Limits Summary  100 100 108 Ill 117 119 129 144 149  Chapter 6. AMMONIA 152 6.1. Overview 152 6.2. Basis Sets for Literature SCF Wavefunctions 153 6.2.1. A 126-GTO Extended Basis Set for N H ... 158 6.2.2. Inclusion of Correlation: Calculation of the Ion-Neutral Overlap Distribution 161 6.3. Binding Energy Spectra 165 6.4. Comparison of Experimental Momentum Profiles with Theoretical Predictions 174 6.5. Ion-Neutral Overlap Distributions (OVDs) 184 6.6. Position-Space and Momentum-Space Density Maps 189 6.7. Exterior Electron Distribution (EED) Ratios and XMPs 193 6.8. Summary 195 3  Chapter 7. FORMALDEHYDE 7.1. Overview 7.2. Basis Sets for SCF Wavefunctions 7.3. Binding Energy Spectra 7.4. Comparison of Experimental Momentum Profiles with Theoretical Predictions 7.5. Summary  196 196 197 200  Chapter 8. PARA-DICHLOROBENZENE 8.1. Overview 8.2. Experimental Momentum Profiles 8.3. Calculated Momentum Distributions 8.4. Summary  222 222 225 228 234  Chapter 9. M E T H Y L A T E D AMINES AND N F 9.1. Overview 9.2. Basis Sets for SCF Wavefunctions 9.3. Measured and Calculated Momentum distributions 9.4. The Methyl Inductive Effect  236 236 238 239 254  3  v  207 220  9.5. Summary Chapter  256  10. T H E H A L O G E N S : A T H E O R E T I C A L S T U D Y 10.1. Overview 10.2. Molecular Chlorine ( C l ) 10.3. Molecular Bromine ( B r ) 10.4. Molecular Iodine ( l ) 10.5. General Trends  258 258 259 270 276 283  11. M U L T I C H A N N E L E M S : P R O S P E C T S A N D D E V E L O P M E N T S .. 11.1. Multichannel Electron Momentum Spectrometer (MEMS) 11.1.1. Accurate Alignment 11.1.2. Suppression of Background Secondary Electrons 11.1.3. C A M A C Interface 11.1.4. Modes of Operation 11.2. Preliminary Results 11.2.1. Helium 11.2.2. Argon  286  2  2  2  Chapter  286 287 292 292 296 299 299 299  CONCLUDING REMARKS  304  REFERENCES  308  vi  List of Figures Fig. 2.1 The (e,2e) reaction  12  F i g . 2.2 Layout of the H E M S computer package  38  Fig. 2.3 Input structure  40  of H E M S package  Fig. 2.4 Sample output of H E M S package Fig. 2.5 Density difference  41  (CI - SCF) maps for H  42  2  Fig. 3.1 Schematic of Electron Momentum Spectrometer  46  Fig. 3.2 Design parameters of 135° sector C M A  49  Fig. 3.3 Timing coincidence electronics of E M S spectrometer  52  Fig. 3.4 Collision volume defined by spectrometer  60  apertures and beam size  Fig. 3.5 Sample results using planar grid method  62  Fig. 3.6 Sample results using analytic gaussian function method  65  Fig. 3.7 P-value histogram at different values of <p  68  Fig. 4.1 Binding energy spectra of H O at 0 = 0 °  74  0  2  and 8°  Fig. 4.2 Comparison of inner valence (2a, ) binding energy with previous experimental and theoretical work  spectrum 77  Fig. 4.3 Spherically averaged momentum distribution and distribution difference for the l b , orbital of H 0  91  Fig. 4.4 Spherically averaged momentum distribution and distribution difference for the 3a, orbital of H 0  92  Fig. 4.5 Spherically averaged momentum distribution and distribution difference for the l b orbital of H 0  93  Fig. 4.6 Spherically averaged momentum distribution and distribution difference for the 2a, orbital of H 0  94  Fig. 4.7 Wide range momentum density contour maps for the valence orbitals of H 0  98  2  2  2  2  2  2  Fig. 5.1 Detailed comparison of the X M P s of the and D O with theoretical calculations 2  vii  lb,  orbital of H O 2  113  Fig. 5.2 Detailed comparison of the X M P s of the 3a, and D O with theoretical calculations  orbital of H O  Fig. 5.3 Detailed comparison of the X M P s of the and D O with theoretical calculations  orbital of H O  2  114  2  lb  2  2  115  2  Fig. 5.4 Detailed comparison of the X M P s of the 2a, and D O with theoretical calculations  orbital of H O 2  116  2  Fig. 5.5 Comparison of the calculated valence M D s using the best Slater and Gaussian basis sets Fig. 5.6 Correlation effects in the calculated M D s of the of water  130 lb,  orbital 134  Fig. 5.7 Correlation effects in the calculated M D s of the 3a, of water  orbital  Fig. 5.8 Correlation effects in the calculated M D s of the  orbital  lb  2  135  of water  136  Fig. 5.9 Binding energy spectrum of water in the inner valence region  141  Fig. 5.10 Two-dimensional density difference (CI-THFA) plots in momentum space and position space  145  Fig. 5.11 Convergence of calculated properties Fig. 6.1 Binding energy spectra of N H  3  146  at 0 = 0 ° and </» = 8°  Fig. 6.2 Comparison of the inner valence binding energy spectrum with theoretical predictions Fig. 6.3 Comparison of valence X M P s of N H with M D s calculated from various wavefunctions 3  167  169 176  Fig. 6.4 Comparison of valence X M P s of N H with ion-neutral overlaps (OVDs) calculated from correlated wavefunctions  179  Fig. 6.5 Comparison of valence X M P s of N H from target natural orbitals (TNOs)  190  3  3  with M D s calculated  Fig. 6.6 Position space and momentum space density contour maps for N H  192  3  Fig. 7.1 Binding energy spectra of H C O measured at 0 = 0° and 0 = 6°  202  Fig. 7.2 Inner valence binding energy spectra of H CO  205  2  2  viii  Fig. 7.3 Binding energy scans in the region 14-22eV as a function of azimuthal angles, <> / Fig.  208  7.4 Comparison of the 2b experimental momentum profile of H C O with calculated M D s  212  7.5 Comparison of the l b , experimental momentum profile of H C O with calculated M D s  213  7.6 Comparison of the 5a, experimental momentum profile of H C O with calculated MDs  215  7.7 Comparison of the l b experimental momentum profile o f , H C O with calculated MDs  216  Fig. 7.8 Comparison of the 4a, experimental momentum profile of H C O with calculated M D s  218  2  2  Fig.  2  Fig.  2  Fig.  2  2  2  Fig.  7.9 Comparison of the 3a, of H C O 2  experimental momentum profile  with calculated MDs  219  Fig. 8.1 Binding energy spectrum of para-dichlorobenzene  227  Fig. 8.2 Measured momentum profiles for the ir and ir orbitals of para-dichlorobenzene Fig. 8.3 Calculated momentum distributions for para-diclorobenzene and benzene , 3  2  229 231  Fig. 8.4 Schematic representation of the wavefunction amplitudes for the IT orbitals of para-dichlorobenzene and benzene  232  Fig. 9.1 Comparison of the experimental momentum profiles of the outermost valence orbitals of NH , NH CH , NH ( C H ) N(CH ) and N F 3  3  3  2  3  3  2  , 241  3  Fig. 9.2 Two-dimensional contour plots of the position space and momentum space densities of the outermost valence orbitals of N H , NH CH , NH(CH ) , N(CH ) and N F 3  2  3  3  2  3  3  3  244  Fig. 9.3 Two-dimensional density difference contour maps in both position space and momentum space  248  Fig.  10.1 Experimental and theoretical M D s for 2ir^ orbital of C l  2  263  Fig.  10.2 Experimental and theoretical M D s for 2ir  orbital of C l  2  264  Fig.  10.3 Experimental and theoretical M D s for 5o^ orbital of C l  2  265  ix  u  ^  Fig.  10.4 Experimental and theoretical MDs for 4o"  orbital of C l  2  266  Fig.  10.5 Experimental and theoretical MDs for 4ag orbital of C l  2  267  Fig.  10.6 Effect of basis set polarization on calculated MDs of C l  2  269  Fig.  10.7 Experimental and theoretical MDs for 4ir^ orbital of B r  2  271  Fig.  10.8 Experimental and theoretical MDs for 4TT  orbital of B r  2  272  Fig.  10.9 Experimental and theoretical MDs for 8a^ orbital of B r  2  273  Fig.  10.10  Experimental and theoretical MDs for 7 a  Fig. 10.11  Experimental and theoretical MDs for la  Fig.  10.12  Experimental and theoretical MDs for 67Tg orbital of I  2  277  Fig.  10.13  Experimental and theoretical MDs for 67T  2  278  Fig.  10.14  Experimental and theoretical MDs for H ^ g orbital of I  2  279  Fig.  10.15  Experimental and theoretical MDs for 10a  orbital of I  2  280  Fig.  10.16  Experimental and theoretical MDs for 10<7g orbital of I  2  281  Fig.  11.1 Schematic of Multichannel EMS Spectrometer  288  Fig.  11.2 Comparison of observed elastic signal  293  Fig.  11.3 MEMS modes of operation  294  Fig.  11.4 He Is binding energy spectrum  300  Fig.  11.5 Ar 3p experimental momentum profile  302  u  x  u  u  orbital of B r  2  274  orbital of Br  2  275  orbital of I  U  u  List of Tables Table 1.1 Highlights in Electron Momentum Spectroscopy  2  Table 2.1 Subprograms in the H E M S package  37  Table 3.1 Components of E M S timing coincidence electronics  53  Table 4.1 Orbital ionization energies and peak intensities of water  78  Table 4.2 Comparison of theoretical S C F wavefunctions  83  Table 5.1 Properties of theoretical S C F and CI wavefunctions  109  Table 5.2 Characteristics of calculated orbital M D s and experimental momentum profiles  126  Table 5.3 CI calculations of the ground and final ion states of H 0  133  Table 5.4 Binding energy spectrum of water in the inner valence region  137  Table 6.1 Properties of Theoretical S C F and CI wavefunctions for N H  154  2  Table 6.2 CI calculations of the ground and final ion states of N H  3  3  Table 6.3 CI calculations of the pole strengths and energies in the binding energy spectrum of N H Table 6.4 Experimental and calculated energies and relative pole strengths 3  for the ionization of the 2a,  orbital of N H  Table 7.1 Properties of theoretical wavefunctions  162  163 170  3  for H C O  198  2  Table 7.2 Binding energies and relative ionization intensities for H CO  203  Table 9.1 Charateristics of theoretical S C F wavefunctions  240  Table 10.1 Wavefunctions for C l  261  2  2  Table 11.1 Design parameters of 360° C M A  289  Table 11.2 Configuration of C A M A C system  295  xi  LIST OF A B B R E V I A T I O N S ADC  Analog-to-Digital Converter  ADC(n)  Algebraic Diagrammatic  Construction  to nth  order  au  atomic  units  BES  Binding Energy  CI  Configuration  CGTO  Contracted Gaussian Type Orbital  CMA  Cylindrical Mirror Analyser  D  Debye  DAC  Digital-to-Analog Converter  DZ  Double Zeta  EMS  Electron Momentum  eV  electron Volt  FOA  Frozen Orbital Approximation  fwhm  full width at half maximum  GTO  Gaussian Type Orbital  HEMS  H-compiler optimized programs  Spectrum  Interaction  Spectroscopy  for E M S HF  Hartree-Fock  IP  Ionization Potential  ISR  Interrupt Service Routine  LCAO  Linear Combination of Atomic Orbitals  MBPT  Many Body Perturbation  MBS  Minimal Basis Set  xii  Theory  MCSCF  Multiconfiguration Self Consistent Field  MD  theoretical Momentum Distribution  MO  Molecular Orbital  MRSDCI  Multireference Singles and Doubles Configuration  Interaction  OVD  Ion-Neutral Overlap Distribution  PES  Photoelectron  PWIA  Plane Wave Impulse Approximation  R  base Resolution  SAC-CI  Symmetry Adapted Cluster Configuration  Spectroscopy  Interaction  SCF  Self Consistent Field  SDCI  Singles and Doubles Configuration  STO  Slater Type Orbital  TAC  Time-to-Amplitude Converter  THFA  Target Hartree-Fock Approximation  XMP  Experimental Momentum Profile  XPS  X-ray Photoelectron  2ph-TDA  2 particle-hole Tamm Dancoff Approximation  xiii  Interaction  Spectroscopy  ACKNOWLEDGEMENTS First and foremost, I would like to thank my research supervisor, Prof. Chris E . Brion, for his unwavering support,  both academically and personally,  throughout  the course of this work. It has been a great privilege to share with him many long discussions from which I have definitely learned much and, most of all, to have been a witness to his infectious enthusiasm for scientific research.  I would also like to thank Dr. Tong Leung for his invaluable assistance the  early  phase of this  work. Special  thanks  are  also extended  to the  during many  colleagues with whom I have interacted and had discussions in the course of this work, in particular (1) Prof. Ernest R. Davidson,  Dr. David  Feller  and Caroline Boyle  extended basis set (SCF and CI) calculations for H 0 , N H 2  (2) Prof. Mike Coplan (3)  3  -  for  the  and H C O ; 2  for his collaboration in the work on para-dichlorobenzene;  Dr. L . Frost, Dr. A . M . Grisogono, Prof.  C. Brion,  Prof. W. von Niessen,  Prof. A . Sgamellotti, Prof. E . Weigold, M r . R. Pascual and Dr. P . K . Mukherjee -  for the data obtained in the work on the halogens;  (4) Prof. Erich Weigold (5) Prof. Larry  for assistance with the multichannel E M S spectrometer;  Weiler, Prof. Del Chong and Dr. Mark  Casida -for discussions  pertaining to work on the amines;  Special acknowledgements  are  also due to the many support staff at U B C who  'stretched that extra arm' to enable the continued optimal operation of the E M S spectrometer.  In particular, I wish to thank  E d Gomm, Bill Henderson, Brunius  Snapkauskas, Brian Greene and Philip Carpendale.  xiv  I also gratefully  acknowledge the  University of British Columbia for receipt of a  University Graduate Fellowship (UGF) during the period  Finally,  I  would  like to  thank  Adelaida  for  specially, for her sincere and loving support.  xv  typing  the  1983-87.  bibliography  and,  most  C H A P T E R 1. INTRODUCTION  1.1. HISTORICAL  REMARKS  Electron momentum  spectroscopy  spectroscopic et  al.  technique  [AE69],  (EMS)t  following  Camilloni  et  the  al.  has emerged  pioneering  [CG72] and  as  a unique  experimental Weigold  and powerful  studies  et al.  of  [WH73]  Amaldi and  the  insightful theoretical investigations of Neudatchin et al. [NN69] and Glassgold and Ialongo [GI68]. The novel idea of obtaining electronic structure information using the  (e,2e) coincidence experiments  knockout  (p,2p)  developed  for  reaction (p,2p)  [BM60,  reactions,  Initial speculations  potential  (e,2e)  distributions  (MDs) as  molecules and thin experiments then  several  reactions well  films.  around the reviews  at  for as  from the  SN66].  was  experiments. of  came  The  quickly that  the  theoretical  applied  time  probing  to  nuclear framework,  the  [GI68, N N 6 9 ,  individual  investigation  These speculations  analogous  orbital  analysis  quasi-elastic originally of  (e,2e)  L72] indicated electron  of correlation effects  the  momentum in  atoms,  were soon to be verified in several  world [CG72, W H 7 3 , H H 7 6 , M C 7 8 , N L 8 1 , RD84]. Since  [MW76,  WM78,  GF81, N L 8 1 , MT82,  B86, MW87]  have  been published. The important highlights in E M S for the past decade and a half are  listed  applications  in of  Table  1.1.  E M S . For  The a  list more  illustrates extensive  the  various  survey  of  the  capabilities  and  literature  the  bibliography published by Leung and Brion [LB85] should be consulted.  t Electron momentum spectroscopy is also known as binary (e,2e) spectroscopy. 1  2  Table 1.1. Highlights in Experimental Electron Momentum Spectroscopy. Amaldi et [AE69]  Demonstrated feasibility of (e,2e) reactions for obtaining the binding energy spectra of K-shell electrons in carbon film. E = 14.6keV, A E = 1 5 0 e V .  al.  0  Camilloni et [CG72]  al.  Obtained the first K-shell and L-shell momentum distributions of carbon film using the coplanar symmetric (e,2e) reaction. E = 9 k e V , A E = 45eV. 0  Weigold et [WH73]  al.  Obtained the first resolved 3p and 3s valence shell momentum distributions of argon using the non-coplanar symmetric (e,2e) reaction. Evidence for inner valence (3s""b correlation states were also reported. E = 4 0 0 e V , A E = 5eV. 0  Dixon et [DM76]  Hood et [HH76]  Moore et [MC78]  al.  al.  al.  Lohmann and Weigold [LW81]  Observed population of n = 2 states in He illustrating potential of E M S for observing ground state correlations. Serious disagreement between the experimental and theoretical momentum distributions of water were observed in low momentum resolution E M S experiments. Inadequacies in the ab initio wavefunctions for water were postulated. Introduced multichannel E M S spectrometer in the 0-plane. Reported the first experimental verification of the exact solution of the Schro dinger equation for the H-atom. E M S measurements are reported for E = 400eV, 800eV, 1200eV. 0  Leung and Brion [LB83a, LB84a]  Obtained an experimental estimation of the (spherically averaged) chemical bond in H .  Cook et [CM84]  Introduced the multichannel microchannel plate E M S spectrometer in the energy dispersive plane.  al.  2  3 1.2.  BINDING  ENERGY  SPECTRUM  (BES):  A  PHYSICAL  OBSERVABLE  The intimate relation between experiment and theory is well illustrated by the development  of  quantum  theory.  Heisenberg  [H30]  points  out  the  classic  experiments of 1900-1930 which marked the genesis of the fundamental concepts in quantum theory. One example is the series of collision experiments of Franck and Hertz [FH13, H23] which demonstrated that atoms can only assume discrete energy values. These findings gave valuable support to Bohr's ideas concerning the 'stationary states of an atom'.  It became apparent that the energy (E) of an atom in its ground, excited or ionized state is a well-defined physical observable defined by, <*|H|*> H  is  the  = E  [1.1]  linear Hamiltonian  operator for  the  system  and 4" represents  the  particular state (ground, excited or ionized) wavefunction. Quantum theory [D57] provides a prescription  for calculating the energy observable, E (see  Eqn. 1.1)  which we then identify with the state wavefunction, «P.  The study of energies of the different possible states (energy spectra) of atoms or molecules has been the focus of various types of experiments. Electron energy loss spectroscopy (EELS) [TR70, WB72] and photoabsorption [WM77b] experiments probe the energies of, the excited states of the system. Photoelectron spectroscopy (PES) [SN69, TB70, KK81], dipole (e,2e) spectroscopy [BH81, B82] and Penning ionization electron spectroscopy (PIES) [OM84] have been useful in the study of  4  the energies of the different  ionic states, t  Electron momentum  spectroscopy  also  has the capability of obtaining the energy spectra of the different ionic states of an atom or molecule. Since it is not energy-limited by resonance  light sources  (e.g. H e l limit is 21.2eV), E M S is capable of observing spectra over the complete binding  energy  range  including both  the  outer  and  the  inner  valence  regions.  E M S and P E S studies in the inner valence region have shown extra structure to what  one  would  structures  are  expect  from  the  now attributed  correlation  (many-body)  transition  probabilities is  generally  semi-quantitative  single-particle picture  to  states  and  correlation states. The understanding  as  fairly  of ionization  well  recent  agreement  as  the  [MS76, obtained  accurate  CD77, between  the  of these  prediction  CD86].  these  of  their  Nevertheless,  the  experimental binding  energy spectra obtained in E M S and the calculated spectra is quite impressive. It is  clear  from  single-particle  the  studies  picture  conducted  of ionization,  so  far  that  specially in the  the  inner  breakdown valence  of  the  region, is  a  general phenomenon [CD86].  Although  E M S and  P E S both  give  some  information regarding  state (satellite) intensities, the two techniques It  should  be  (p = 0 - 3 a ~ ) 1  0  components only in the PES  pointed whereas  (e.g. case  out  that  higher  energy  Al Ka, p=lOa " ). 1  0  of states within  (synchroton and X-ray  sources)  the  correlation  are of an entirely different nature.  E M S probes  photon  the  of the  the  low  momentum  P E S probes target  the  wavefunction.  same symmetry manifold  might be expected  high  that  components momentum Thus  it is  E M S and  to give similar  satellite  t Strictly speaking the observed excitation energies or ionization potentials refer to the energy difference between the initial neutral state and the final neutral or ionic state.  5  relative intensity distributions [M85, BB87d, K K 8 7 ,  SH87]. Accurate experimental  and theoretical data on correlation state intensities are few and future  studies in  this area will clearly feature even higher energy resolution E M S studies.  1.3.  EXPERIMENTAL  MOMENTUM  PROFILE  (XMP):  A  PHYSICAL  and the  position-space  OBSERVABLE  The correspondence representation  of the  in quantum  momentum-space theory  representation  [D57] is illustrated by the  Fourier  transform  relationship,  *<p) = ( 2 T T ) "  3 / 2  J e "  1  ^  Although the position-space representation new  insights  to  momentum-space study  of  different  chemical  and  representation.  the  properties  atoms  and  [1.2]  <Mr) d r  is more intuitive and easy to visualize,  physical  problems  are  available  in  the  Coulson and Duncanson [CD41] first initiated the  of  momentum-space  molecules.  Since  then  wavefunctions  significant  and  theoretical  densities work  in  [SW65,  ET77, N T 8 1 , R83, DS84, RB84, K K 8 6 ] has developed into what could be called the  field  of  exploited the DS84] have  momentum-space  chemistry.  Theoretical  advantages of working in momentum showed that the  be solved easier  space.  Hartree-Fock equations  and more directly in momentum  quantum  chemists  Recent reports  have [NT81,  for molecular systems  space because the  may  singularities  in the potential (due to the multi-center nature of molecules) are reduced to one singularity  at  p = 0. Others  [HL80,  DJ84] have  investigated  the  use  of optimal  basis sets obtained from extensive analysis of the momentum space Hartree-Fock wavefunctions.  6 Epstein  and  Tanner  [ET77]  outline  chemistry and how it applies to the  the  major  interpretation  principles  in  momentum-space  of molecular  momentum-space  density distributions. These principles or properties can be summarized as  [ET77,  CB82], (i) Inverse spatial weighting; (ii) Preservation of molecular symmetry; (iii) Addition of inversion symmetry; (iv) Bond oscillation; (v) Bond directional reversal; These  Fourier  interpretation  transform  properties  of momentum-space  momentum-space  and  or  principles have  experimental  position-space  density  been  quantities  maps.  invaluable  and  Further  the  in  the  corresponding  discussion  of  these  properties will be made in the following chapter.  Experimental momentum-space techniques. Compton  chemistry involves a wide variety of experimental  Electron momentum spectroscopy [MW76, WM78, L B 8 5 , B86, MW87], scattering  [W77,  L77,  C85], high  energy  electron  scattering  [BF74,  BW77] and positron annihilation [W73, B77] are some of the techniques used to measure the experimental momentum distribution. In E M S the angular correlation spectra  obtained  for  a  particular  energy-selected  state  are  momentum profiles (XMPs).  The term X M P is the preferred  previous  as  confusion  designations between  such the  theoretical quantity (MD).  purely  experimental experimental  momentum quantity  called  experimental  name as opposed to  distributions. This avoids (XMP) and  the  purely  7 Like  the  total  energy  (Sec.  1.2)  the  XMP  is also a  observable within the limits defined by the theory of EMS  well-defined physical and is proportional to  the spherically averaged orbital momentum distribution, p(p)  =  fdQ|tf(p) |  [1.3]  2  \^(p) is the single-particle (momentum-space) orbital from which the electron was ionized. Energy-selected electron distribution  XMPs therefore provide a very  in atoms, molecules  and  detailed probe of the  solids. In fact, EMS  has  been  referred to in the literature as an orbital imaging [BB87a] or as a wavefunction mapping  [MW87] technique. Since EMS  is orbital selective it has  advantage compared to Compton scattering and  a distinct  high energy electron scattering  which measure a quantity (the Compton profile) equal to an integral over the total electron momentum distribution. Similarly the two-photon coincidence rate in positron annihilation gives information on the total momentum distribution [B77].  With the current improvements in momentum resolution, EMS now  providing a  sensitive  test  of the  quality  of  ab  measurements are initio  wavefunctions.  Collaborative experimental and theoretical studies, some of which are reported in this  thesis,  wavefunctions  have  been  instrumental  in quantum  in  the  chemistry. These new  construction of wavefunctions  more  accurate  [BB87, BM87,  BB87c, FB87a, FB87b] could be called 'universal' wavefunctions in the sense that they are sufficiently accurate to calculate all  electronic properties of the molecule  with good precision. Unlike other physical observables such as the total energy, dipole moment and  <r > 2  e  the XMP  probes regions of phase space which  contribute little to the total energy of the  system. These regions have only  8 recently  received  much  theoretical  calculating  wavefunctions  utilize  emphasizes)  regions of phase space  attention  the  because  variational  traditional  principle  which  methods  of  weights  (or  that contribute most to the total energy.  It  has been mentioned [FB87] that the error in the total energy is second order in the  error  XMPs)  in the  are  reasonable much  only  wavefunction whereas first  that X M P s  more  sensitive  order  in  the  the error  as well as other probe  error in one-electron properties in  the  wavefunction.  one-electron properties  of wavefunction  quality  than  (e.g.  Thus  it  is  should provide a  the  calculated  total  energy. As opposed to commonly quoted one-electron properties such as the dipole moment  and  quadrupole  FB87a,  CC87]  seems  moment,  to  increasing amount  indicate  that  XMPs  are  of evidence even  more  [BB87, sensitive  BM87, to  the  diffuse region of the electron distribution. In general, large r is not equivalent to small  p  because  there  is  no  one-to-one  correspondence  between  momentum. However there is still an inverse weighting of the  position  respective  and  spaces  due to the Fourier transform relationship.  In  summary,  the  renewed  interest  in momentum  space  chemistry that we  are  currently witnessing illustrates the importance of complementary views in scientific inquiry.  It  phenomena  can be said that traditional understanding from  the  position  space  perspective  corresponding understanding in momentum space.  is  of chemical and physical incomplete  without  the  9 1.4. SCOPE  OF  THESIS  The two physical observables  described in Sees.  1.2  (BES) and  the primary experimental goals of the E M S studies reported studies  can be  further  subdivided according to  1.3  (XMPs)  are  in this thesis.  The  the  secondary  goals  involved in  momentum  resolution  XMPs  each particular study namely, (A) E M S studies of small molecular This  involves  the  comparison  of  systems. high  calculated from a range of S C F wavefunctions beyond are  to include effects  extensively  of correlation  analyzed  and  and  compared  MDs  up to the Hartree-Fock limit and relaxation.  with  Comparisons with photoelectron spectroscopy  and  The  existing  inner  valence B E S  theoretical  predictions.  are also made in cases wherein P E S  data for the inner valence region is available. (B) Chemical applications of E M S to complex molecular systems. This  involves  the  study  and  identification  of chemical  trends  in  a  series  of  related molecules. Most of the studies are exploratory and have tended to involve observation  of the  XMPs  of those  orbitals  that  would contribute  most  to  the  and  are  chemistry of the respective molecules.  Type-(A)  E M S studies  were  reported  in chapters 4,  done  for  5, 6 and  H 0, 2  D 0,  NH  2  3  7, respectively. On the  and  H CO 2  other  hand,  type-(B)  E M S studies were done for the outer valence orbitals of para-dichlorobenzene,  the  outermost  and  the  and  10,  halogens  valence  orbitals  (Cl , Br 2  2  and  of N F I ) 2  respectively. A special chapter a  multichannel  3  and  and are  (Chapter  E M S spectrometer  and  the  substituted  reported  in  methyl amines chapters  8,  11) outlines developmental reports  preliminary  results  9  work done on obtained  for  10 helium  and  argon. Atomic  units  thesis unless otherwise stated.  (n = m  e  = e = l)  have  been  used  throughout  the  C H A P T E R 2. T H E O R E T I C A L METHOD  2.1.  THEORY  OF ELECTRON  MOMENTUM  SPECTROSCOPY  (EMS)  2.1.1. Introductory Remarks Electron momentum spectroscopy is based on the (e,2e) reaction, e  (Po , E ) + M  0  0  where  p^,  > e,(p,,E,)  E ^ (i = 0,l,2)  + e (p ,E ) 2  2  2  + M (p , E ) +  3  are the momentum and energy  3  [2.1]  of the incoming,  scattered and ejected electrons, respectively. The ion recoil momentum and energy are 2.1.  given by p  3  and E , respectively. The (e,2e) reaction is also shown in Fig. 3  Several early  [MW76,  WM78,  studies  [CG72,  GF81] have  WH73] and reviews  shown  that  under certain  of the (e,2e) reaction well-defined kinematic  conditions the (e,2e) cross section (reaction probability) t is largely a measure of the electron momentum distribution of the single-particle orbital from which the electron was ionized or knocked out. It is now established [MW76, WM78] that the necessary scattering conditions are, I. High impact energy ( E  0  > lOOOeV);  II. Symmetric energy sharing (E, = E  2  = E);  III. Maximal momentum transfer (| K* | = | p - p, | £ 5a ~ ); 1  0  IV. Non-coplanar ( 0 — 4 5 ° ,  0  <j> variable);  Under these conditions and within the, V. Independent-particle description of electronic motion;  t The (e,2e) cross section referred to is the probability of simultaneously detecting e ^ p ^ E , ) and e ( p , E ) . It is also referred to as the triple differential cross section, d a/dfi , dfi dE. 2  2  2  3  2  11  Fig. 2.1.  The (e,2e) reaction.  13 the (e,2e) cross section is given by,  a  e,2e  constant-(47r)~ /cin  |</>j(p)|  1  =  [2.2]  2  where p is the momentum of the electron in orbital, momentum distribution is spherically averaged (fdfi)  prior to ionization. The to account for the random  orientation of the gaseous targets.  The expression in Eqn. 2.2 follows from energy conservation E  = E, + E  0  *  E, + E  E  + 2  2  + E  3  + Ek [2.3]  b  and momentum conservation laws,  Po E  3  = Pi  + p  2  +  [2.4]  j?3  is the ion recoil energy and is negligible due to the large mass of the ion. is the binding energy (ionization potential) of the orbital electron. The ion  recoil  momentum, p  magnitude  but  3  opposite  within the in  sign  conditions' (I-IV) outlined to  the  orbital  electron  above  is  momentum  equal in prior  to  knock-out, i.e.  p = "P  3  [2.5]  Eqn.  2.5 is an approximation which assumes the target to be initially at rest  and  that the ion is a 'spectator' in the collision process. Further discussion of  this central assumption in EMS is made in the following section.  14 The fundamental result in Eqn. 2.2 has been and is the cornerstone its applications to chemical problems. The present chapter  will  of E M S and  give a  summary  of the conditions under which the '(e,2e) cross section and momentum distribution proportionality' is valid. The approximations leading to this proportionality relation will also be outlined and discussed.  2.1.2. Plane Wave Impulse Approximation (PWIA) The (e,2e) reaction probability amplitude can be written as [WM78], T __ f  = < X"(p,)  Q  X"(p )  *  2  N  and the (e,2e) cross section is therefore  a  X~  e,2e  are  the  (27r)Mp p /p ) 2  =  1  distorted  2  final  [SZ74]  ionic  (N-l-electron)  which 'models'  the  a n  and  a  sum  final  X (p )  N  >  +  0  0  [2.6]  |T ^_  v  f <  Q  (p ,p, ,p 0  and  2  ) |  [2.7]  2  outgoing(-)  spherical  ^ describe the exact initial (N-electron)  interaction  over  | T | *  with incoming( +)  systems.  Q  1  given by,  d  N-electron system. The notation £ degeneracies  a  electron waves  wave boundary conditions. ' P o ^ and  0  _  f  V  T  represents the  between refers  the  complicated  incoming electron  T-operator and  the  to an average over all initial state  state degeneracies.!  It  is  quickly  apparent  that the complications in Eqn. 2.6 are three-fold, namely: (1)  Accurate  quantum  mechanical  description  of  the  incoming  and  outgoing  electron waves.  t Since the present energy resolution in E M S experiments is not capable of resolving vibrational and rotational states, vibrational and rotational closure is applied [MW76, M73, M75].  15 N (2) Accurate quantum mechanical description of the initial C P (* " ). (3) Accurate representation N  0  )  a n  d  f" l  states  m a  1  f  of the many-body transition operator, T.  A l l aspects (1, 2 and 3) are approximate wherein  However from physical arguments  approaches to Eqn. 2.6 certain  reasonable  McCarthy  non-trivial.  and  understanding  Weigold  (PWIA) to the  [MW76]  can  be  applied  the  kinematic regimes attained.  It  plane  (e,2e) reaction. Starting from the  is  wave  quantum  in  and  can be identified this  impulse  spirit  that  approximation  mechanical  three-body  description [SZ74] of the (e,2e) reaction, the ionized electron can be considered to be initially bound to a quasi-particle ( N - l electron system). For this reason Eqn. 2.1 has been termed a quasi-three-body  The quasi three-body  T-operator  reaction.  can be expanded  [F61, A G 6 7 , F83] in terms of  two-body t-operators,t T = t  +  0  t,  + t G t 2  where  t  0  0  (G  0  =  2  represents the  represent the system).  + t GSt,  0  G  + toGSt,  direct electron-electron  interaction of electrons 0  is  the  + t,GSt  Green's  1 and function  0  +  -  Coulomb interaction,  2 with for  the  the  ion core  [2.8] t,  and  t  2  ( N - l electron  non-interacting  system  [E-Ho+ie]' ). 1  t Two-body collisions (i.e. elastic and inelastic scattering) are relatively well-understood [B83] and therefore have been used as the basis for approximate solutions to the quasi-three-body problem.  16 Within  the  impulse  representing [N66]  approximation  multiple  is generally  binding energy  scattering  valid  of the  [N66]  can  wherein  the  be  the  neglected.  impact  target electron.  higher  In  order The  energy  terms  in  impulse  the  2.8  approximation  is much higher  such cases  Eqn.  assumption  than  the  of single  collisions between the incoming electron and the target electron is generally valid and the that  the  ion can be treated as contribution from  t,  a 'spectator'.  is negligible (i.e. core  reactions involving high impact energy ^5a " )1  0  described  In as  cases  therefore  simply a  binary  Studies  [MW76] have excitations  also shown  are  small) for  ( E ) and large momentum transfer  (| R |  0  wherein  the  (e,2e)  encounter between  orbital electron that is to be ionized, the  the  T-operator  reaction  can  be  incoming electron  accurately and  the  is given approximately by  the two-body direct Coulomb t-operator, T This small  [2.9]  approximation is valid momentum  transfer  only in the  limit of high momentum  ( | r v | < 1 a ~ ) as 1  0  in the  transfer.  Ehrhardt-type  For  [EJ86] (e,2e)  reaction kinematics (asymmetric coplanar) the interaction of the incoming electron with  the  ion core  (t,)  and  double collisions (multiple scattering  terms in Eqn.  2.8) make significant contributions [EJ86] even if the impact energy is high.  In the limit of high impact energies  and high momentum  transfer,  the distorted  incoming and outgoing waves can be approximated as plane waves. For example, the incoming wave will be given as [F83],  = (1 + G J t , ) | p > 0  |p > = e x p ( i p V r ) 0  [2.10]  17 The plane wave approximation for the incoming and outgoing particles as well as the  impulsive (binary encounter)  approximation to the  T-operator  then  yields an  expression for the (e,2e) cross section,  °e,2e  =  47T (p p /p )a  fdflfdv|<p¥  3  1  2  0  M o t t  j  V  vibrational integral  |*  1  N 0  >|  in the  [2.11]  2  J  structure  (/dv),  ~  <  collision term The  N f  term  case of molecular  targets,  is  reasonably  approximated by calculating the neutral and ionic wavefunctions at the equilibrium geometry  [DM75, BB87]. The half-off-shellt Mott scattering cross section, °y[ tt 0  is given by [MW76],  a  Mott  where  °  K*' =  l*H  fc  =  (j?i"~P2)/2  2  [2  and k* =  (p -p)/2.  -  12]  The expression given by Eqn.  0  2.11 is known as the Plane Wave Impulse Approximation (PWIA).  The assumptions the  exact  Oftentimes necessary  scattering  more  approximation  Experimentally, [MW76,  the  are  very difficult  (even for  parameterization accurate  (DWIA) targets  PWIA  potentials  empirical in  multi-center  of the  calculations  [MW76, where  PWIA  WM78].  the can  of  be  to  atoms) the  are  assess theoretically because difficult  distorting  such  as  the  This  is  even  form  of  tested  and  the  to obtain  (optical) distorted more  potentials  much effort  [MW76].  potentials wave  impulse  complicated are  has  not  focused  is  for  known. on  this  CG80]. Weigold and co-workers [FM78, D M 7 8 , LW81] have shown that  t The Mott scattering amplitude is half-off-the-energy of the finite binding energy.  shell (i.e. 1c ^ k*") because  18 for the conditions employed in EMS increasing the impact energy from 400eV to 1200eV produced negligible changes in the shapes of the measured experimental momentum profiles  and that E > lOOOeV is required to obtain correct relative 0  cross sections. These results indicate that the (e,2e) cross section is not sensitive to the scattering dynamics under the present EMS conditions (I-IV) and therefore first-order theories  (e.g. PWIA) are sufficient.  It is now generally accepted that  at E >1000eV the PWIA as applied to EMS studies of atoms and molecules is 0  a very reasonable approximation giving quantitative results for all orbitals. Such considerations have been used in the present work.  Recent  studies  however  show  that  in  certain  cases  [MW85,  CM86,  CB87a],  especially in the prediction of absolute magnitude, the PWIA may be insufficient. Defining the regions of applicability of the PWIA and improvements to the PWIA are still the subject of current further investigations the comparison of PWIA the  low  momentum  intensity  in  agreement would  the  and DWIA calculations show agreement for shape in  region  high  (0-1.5a ~ )• 1  o  momentum  with experimental results.  expect  most  of  [AC86, AC87a]. In general,  the  The DWIA  region  (>l.5a ~ ) 1  0  however for  predicts  atomic  higher  targets  in  The high momentum region is where one  distortion  effects  because  high  ion-recoil  collisions  sample the electronic wavefunction much closer to the nucleus. It is for the same reasons high  that plane wave  momentum)  and  treatments  low  energy  of both high energy PES (which PES (which  results  in low  samples  energy  ejected  electrons) are grossly inadequate.  The  requirements  of  high  impact  energy  (condition  I),  symmetric  scattering  19 (condition II) and maximal momentum transfer consistent  with  the  PWIA.  It  is clear  (condition III) outlined earlier  (in the  semi-classical sense) that  are  under  these conditions the collisions will be mostly single (binary) collisions between  the  incoming and ionized electrons and the ion is largely a spectator in the collision.  In Eqn. 2.11 we are left with two terms, the scattering dynamics and a structure  namely a collision term sensitive to  term sensitive to the electronic  structure  of the target system. The resulting factorization of Eqn. 2.11 is a general result of impulse approximations to scattering problems and has  also been exploited by  many workers as for example in Compton scattering [C85]. The usefulness of the PWIA  in E M S however  comes from  the  fact  that under  well-chosen conditions  the collision term in Eqn. 2.11 is not only separable but is also nearly constant.  McCarthy and Weigold [MW76] discussed several possible scattering for  EMS  namely  the  symmetric  coplanar,  constant-angle  energy-varying arrangements.!  constancy  the  of  non-coplanar  collision  term  form of scattering  [MW76, WM78] that for the the  experimental  impact energy  momentum  [C81,  non-coplanar, profile (as  a  non-coplanar  is  best  achieved  (condition IV). It  has  symmetric arrangement  the magnitude  through been  the the  shown  the shape of  function of p) is independent  and is given essentially by the electronic structure  non-coplanar arrangement  and  Of these possible arrangements  LB83]  arrangement  symmetric  arrangements  of  the  term. In  the  of the orbital electron momentum, p is  given by,  t The asymmetric coplanar arrangement (Ehrhardt-type) [EJ86, LW84] is sensitive to the scattering dynamics and involves very low momentum transfer and is therefore not suited for electronic structure determination.  20 p = {(2p,cos0 - p  0  )  2  + [2p sin0sin(0/2)] } 2  [2.13]  1 / / 2  1  where the scattering angles d and 0 are defined in Fig. 2.1.  Another arrangement  which has been demonstrated to be equally suitable for  electronic structure determination is the high energy asymmetric coplanar (HEAC) arrangement of Lahmann-Benani et al. [LD86]. As pointed out in this article [LD86] the variations of the Mott scattering term in the HEAC (3.5%)  are only  slightly  larger  than  those  arrangement  in the non-coplanar  symmetric  arrangement (<2%) [LB83]. These variations are very small compared to that using  the coplanar symmetric  arrangement  (22%) [LD86]. Additionally, the  momentum transfer is also strictly constant for the HEAC and the non-coplanar symmetric arrangements.  From  the experimental end, the non-coplanar  advantageous  symmetric  arrangement is  because the angular correlation spectra on both sides of 0=0°  should be of identical shape and magnitude and thus provide a consistency check for the data. Furthermore, the collision volume 'seen' by the movable analyzer is constant as the angle 0 is sweeped. These non-coplanar symmetric  scattering arrangement  factors  all indicate  that the  is the most favored  for EMS  studies.  2.1.3. Target Hartree-Fock Approximation (THFA)  In  general much of our current understanding of electronic structure, ionization  phenomena and molecular properties is based  on the independent-particle (or  single-particle) picture of the  molecule. In this case the  N-electron wavefunction,  «P ^ is given by a single Slater determinant of one-electron orbital wavefunctions 0  Wj). *  N 0  ( 1 , . . .N)  =  |tf,,...* >  [2.14]  N  Note that antisymmetrization is implicit in the notation. In addition, the following discussion N-electron  is  limited  system  only cannot  to  closed-shell be  described  systems. by  The  an  extent  to  which  antisymmetrized  set  the of  independent-particle functions is called correlation effectsA  The best independent-particle picture is given by the restricted Hartree-Fock (HF) model [S082] of electronic motion. The H F model assumes  that the electrons in  a molecule 'move about' according to an average field created by the other (N-l) electrons. This assumption transforms the N-electron Schrodinger equation, H*( 1 , . . . N ) = E * ( 1 , . . . N ) into a  set  of N coupled one-electron  [2.15] (integro-differential)  equations  each  of the  form,  F  i ^ i  =  e  i ^ i  i =1 , . . .N  [2.16]  where Fj is the effective one-electron operator also known as the Fock yp ^  are  the  HF  canonical  orbital wavefunctions  and  e^  are  the  operator, H F orbital  energies.  t Correlation effects account for the fact that electrons interact with each other instantaneous^ rather than with an average field created by the other electrons. These effects are discussed in another section.  22 The canonical H F equations (SCF)  method  (Eqn. 2.16)  and for this reason  are  solved using the  wavefunctions  of the  self-consistent-field  single-determinant  form  are also called S C F wavefunctions. Another descriptive term which has been used is the  'molecular orbital (MO) approximation', however this terminology should be  discouraged  because,  as  unitary transformation the  total energy  H F equations  In  is well  of the  stationary  is invariant under  not unique  [S082].t  orbitals that  relates  orbital  words,  make  The canonical form of the  have  the  energies.  found  greater  applicability due  experimentally-derived The  derivation  of  ionization  Koopmans'  to  Koopmans'  potentials  theorem  to  the  involves  the  assumption that the ionization process leaves the N - l orbitals undisturbed other  a  (Eqn. 2.16) is more commonly used for various reasons.  which  canonical  total energy  orbitals. This means that the  are  E M S canonical orbitals  theorem  known, the  frozen.  effects  as  well  as  orbital  approximation  Within  this  correlation to  frozen-orbital  effects  E M S is  are  approximation  neglected.  straightforward  (FOA) relaxation  Application  and  the  or in  of the  ion-neutral  frozen overlap  amplitude given in Eqn. 2.11 reduces to,  <p*  N _ 1 f  |*  N 0  >  = <p|^ > c  = * (p)  [2.17]  c  where  \p (p) refers c  to the  momentum-space  representation  of the  characteristic  orbital that has been ionized.  A  more general  expression for the  ion-neutral overlap amplitude is obtained  t For example localized orbitals will give the molecular orbitals.  same total energy  by  as delocalized  23 assuming that the initial state is largely a single determinant H F wavefunction while the final ion state is represented (in general) by a linear combination of HF  configurations, |/3> with a hole in orbital  l * f  N  _  1  = j<3 j 0 j 0 ^ j l ^  >  Z  S  C  [ 2  -  1 8 ]  The resulting ion-neutral overlap amplitude is therefore given by, <p* " N  1  f  |* > = L j S N  0  j Q  C  uVj (p)  j 0  [2.19a]  where S J Q is the probability amplitude for finding the one-hole (\£j~) in the final ion state, * 1  N f  ~  [MW76]. C j ^ is a  1  coefficient that ensures that the configuration  configuration  Clebsch-Gordan  | B> belongs to the point group of  the system [WM78]. As can be seen from Eqn. 2.19a contributions from other orbitals especially of the same symmetry such as for example the 3 a,  4 a and  5a of carbon monoxide [DD77, FB87b] may contribute to the ion-neutral overlap amplitude. However in almost all cases only one term in Eqn. 2.19a dominates and the results obtained from the FOA and THFA are then the same for shape. In this case the (e,2e) cross section is given by, a  e,2e  =  4  * (PiP2/Po)tf  <  S rdfl 2  3  v  collision term  fdv|^ (p)|  c  M o t t  ^  2  c  v  [2.19b]  >  structure term  The result in Eqn. 2.19 is called the Target Hartree-Fock Approximation (THFA) as applied to EMS. A different derivation of the THFA using the method of second quantization is also given by McCarthy and Weigold [MW87].  One  particular difference between the FOA and the THFA is the fact that  final  24 stare correlation effects in  the  interpretation  valence  binding  correlation  coefficient  to  spectroscopic  correlation  measured  for in the  structure of  especially  most  intensity  corresponding  The  respective  spectra  state  the  of the  2.18.  of extra  energy  (satellite)  proportional  Eqn.  can be accounted  atoms  within  momentum  and  and  in  molecules.  is just  the  useful  the  The  symmetry  (xjj^ ^) in the  states are  profiles  This is very  observed  same  which  configuration  (satellite)  that  the  factor,  THFA.  inner  observed  manifold  weight  expansion  or  is the  given in  assigned  on the  basis  assuming  the  binding  energy  range that the  sum of  spectrum is obtained over a large enough binding energy  of their  the spectroscopic factors satisfy the spectroscopic sum rule [WM78].  2.1.4. Full Ion-Neutral Overlap  Amplitude  The full ion-neutral overlap amplitude given in Eqn. 2.11 could also be evaluated directly using correlated wavefunctions has  been  done  in  recent  amplitude  is  given  simply  molecular  orbital basis  as  studies if the  for both initial and final  [BB87,  the  final  initial  BM87]. ion  state  Computationally  state to  ion states. This  is  take  expanded  advantage  properties. In this case the ion-neutral overlap amplitude has  the in  the  overlap same  of orthogonality  the  same form  a molecular orbital expanded in the neutral basis. Details of the computation given  in  the  Transforming because  it  papers the  is  of  overlap  still  Martin, amplitude  expressed  in  Shirley to terms  and  Davidson  momentum of  basis  Gaussian-type)  which can be transformed  are also other  effective ways of obtaining accurate  space  [MS76, is  functions  using standard  methods  are  MD77].  straightforward (Slater-type [KS77].  ion-neutral overlap  such as the generalized overlap amplitude method of Williams et  as  or  There  amplitudes  al. [WM77] and  25 more  recently  separately methods  the  overlap method  optimized initial have  been  and  of Agren  Final  computational  and Jensen  states.  but  Presently  with  the  [AJ87] which the  assumes  limitations of these  rapid phase of development in  computing technology and software these limitations will be likely diminished.  Spherical averaging of the  absolute  square of the Fourier-transformed  ion-neutral  overlap amplitude yields the ion-neutral overlap distribution (OVD). OVDs give a more  accurate  representation  compared to the  of the  structure  single-particle momentum  factor  in the  (e,2e) cross  section  distributions. M D s in some sense  are  the zeroth order approximation to the more accurate OVDs. These calculations for H O 2  and  NH  3  in  the  present  work  were  done  in  collaboration  with  the  theoretical group of Prof. Ernest R. Davidson (Indiana University). The ion-neutral overlap  amplitudes  were  calculated by the  Indiana  group  and  the  OVDs  were  calculated from these quantities by the author using the H E M S package at U B C .  2.2. MOMENTUM The results EMS  of the  SPACE  CHEMISTRY  previous section (Eqns. 2.11  (symmetric non-coplanar  geometry)  is the  and  2.19)  appropriate  clearly indicate that experimental tool for  momentum space chemistry. Central to momentum space chemistry is the Fourier transform  relationship  representation  between  the  momentum  (Eqn. 1.2). Fourier transforms  representation  and  the  position  find applications in various  branches  of science [B65, C73] however in this section focus will be on its application to EMS.  The Fourier transform properties are mentioned in their most general form  to reflect the fact that they are used in other branches engineering.  These  are  then  related  to  specific  of physical science and  principles in momentum  space  26 chemistry [ET77] and E M S [CB82, L B 8 3 , LB83a].  Two  functions  are said to form a symmetrical Fourier transform  pair,  f(x) and  F(s) where, F(s)  = (27r)" / /  f (x)exp(-ixs)dx  [2.20a]  f(x)  = (27r)~ ' J  F(s)exp( + ixs)ds  [2.20b]  1  1  2  /2  In E M S the particular and  \p (p)  which  Fourier transform  are the position  respectively. The examples  space  pair  we are interested in are, *// (r)  and momentum  space  wavefunctions,  are shown for the case of one dimension to maintain  simplicity and clarity.  Several basic  theorems govern  the application of Fourier transforms  to physical  problems. These are listed below. (1) If  Similarity  Theorem  f(x) has the Fourier  | aj This  F(s), then  f(ax) has the Fourier  transform  F(s/a).  _ 1  theorem  is referred  means that an atomic will  transform  have  property  to in E M S as the inverse weighting principle.  function  a corresponding  momentum  is clearly illustrated  gases [LB83].  concentrated near the nucleus  However caution  i n position  space orbital which will  in the trend  of the observed  should be exercised  be diffuse.  XMPs  in applying this  polyatomic systems where other effects complicate the Fourier  This space This  of the noble principle to  transform.  27 (2)  Addition  If  f(x)  Theorem  and  g(x)  have  Fourier  transforms  F(s)  and  G(s),  respectively  then  f(x) + g(x) have the Fourier transform F(s) + G(s). This  theorem  illustrates  why the  atomic orbitals (LCAO-MO)  molecular orbital is preserved in momentum  addition, molecular symmetry represent the  angular  molecular symmetry of a linear combination  is preserved  part of the  space.  of In  because the spherical harmonics which  wavefunction  are  invariant under  the  Fourier  transform. In momentum space, inversion symmetry is also added (if not originally present). This ensures that the center of mass coordinate (3) If  electron has  no net  translational  motion in  the  system.  Shift Theorem f(x)  has  a  Fourier  transform  F(s),  then  f(x-a)  has  the  Fourier  transform  exp(-27rias)F(s). This theorem that  is the basis of the  momentum  distributions  bond oscillation principle  associated  with  chemical  in E M S which states bonds  will  exhibit  oscillations along the bonding direction with period = 2n7t7R (n = 0,l,...) where R is the  bond  length.  Antibonding orbitals  (n = 0,l,...)  which  Duncanson  [CD41]  expressed  in  are  out  showed  terms  of  a  of for  phase a  linear  exhibit oscillations with period = (2n+ l)7i7R with  the  diatomic  bonding  that  combination  if  of  the  atomic  orbitals.  Coulson  molecular orbitals  and  orbital  is  (LCAO-MO)  centered in the respective nuclei, the momentum density is given by,  IWP>I  2  =  ix  atom  (p>r  {i±co (p.5)}  [2.2i]  S  > „ — diffraction term  '  28 where the orbitals.  ( + ) sign refers  For  diffraction This  a  to bonding orbitals and (-) sign refers  bonding orbital it can  term  is maximal when p  observation  is referred  to as  also be  seen  from  is perpendicular  the  to antibonding  Eqn. 2.21  to the  that  the  bond direction, R.  bond directional principle  in E M S which  states that the momentum of an electron in a chemical bond is more likely to be  directed  perpendicular  to  the  bond  rather  than  phenomenon has been nicely illustrated in the case of H (4)  Definite  Integral  along  the  bond.  This  [CD41, LB83a].  2  Theorem  The definite integral of a function from -°° t o  +  00  is equal to the value of its  transform at the origin, i.e.  F(0)  = ff(x)dx  This  theorem  that  only  s-type  momentum relative  illustrates  density  ratio  of  basis at  [2.22] an  early observation  functions p = Oa "  1  0  symmetric  (1 = 0)  can  therefore  components  [LN75] in E M S which give  provides  in  the  intensity a  at  p = 0a " • 1  0  sensitive  molecular  mentioned  probe  of  The the  orbital, especially in  mixed s-p type orbitals.  (5)  Autocorrelation  Theorem  If  f(x)  Fourier  has  a  transform  F(s),  then  its  autocorrelation  function  * ff This  (u)f(u+x)du theorem  interpretation wavefunction  has  has the Fourier transform been  exploited by  of E M S data.  The  Coplan  j F ( S) | . 2  and  autocorrelation  in this case is referred  as  the  co-workers  function  [MT82]  of the  B(r) function.  It has  in  position been  their space argued  that the B(r) function which can also be derived from the momentum density is  29 more familiar to most people and therefore  2.3. BASIS  SETS  FOR AB  INITIO  easier to  understand.  WAVEFUNCTIONS  Currently there is no numerical solution to the  restricted  Hartree-Fock  equations  except  the  of diatomic  molecules  in the  case  of atoms  and  recently  in  case  [LS85].t As a more general solution a known set of spatial basis functions introduced  and  the  H F integro-differential  algebraic  equations  equations  are called the Roothaan equations [R51].  This known set  and  solved  by  equations  of spatial functions  standard  are  matrix  converted  techniques.  {bj} is called the  to  a  are  set of  The  resulting  basis set. The  molecular  orbitals can be expanded using this set,  \p = Z j c j b j i  i  j = 1 , . . .M  [2.23]  In the limit of a complete set (i.e. M — • °°) the expansion in Eqn. 2.23 is exact. This however is not feasible because of the hardware  and software  limitations of  an infinite basis set. What is generally done is to limit the basis set to a small finite number of well-chosen (i.e. optimal) basis functions.  Much  work  in  recent  years  has  focused  on  the  optimal  form  (in  terms  of  accuracy and computational ease) of these basis functions. One particular function is the atom-centered  Slater-type orbital (STO),  t Recent reports [F87] indicate that the H F equations can be solved for a general polyatomic molecule by combined use of basis functions and a numerical grid.  30 b-  = Nr "  S T 0  n  e"  1  Y (d,4>)  a r  [2.24]  lm  where a is the orbital exponent, N is a normalization constant and Y j ( 0 , 0 ) is m  a spherical harmonic. Another functional form, adopted first by Boys [B50], is the atom-centered cartesian Gaussian-type orbital (GTO), b j  G  T  = Nx y z  0  n  1  m  e  _  a  r  [2.25]  2  where k ( = n + l + m) is the angular momentum quantum number.  It is generally held [DF86, RF85] that =three GTOs are equivalent to one STO for the same level of accuracy. More GTOs are necessary because GTOs have the  wrong behaviour near the nucleus (r^O) and at large r [H85, DF86].  Although the relative merits of GTO and STO basis sets have been a matter of extensive debate, it should be noted that, in the limit of large basis sets, both types of basis sets should give the same result. Other types of basis sets exist in the literature and are discussed elsewhere [S77, H85]. we have concentrated  In. the present work  on the more popular functional forms namely the STO and  GTO basis sets.  One  particular advantage of GTOs is the ease of computation compared with  STOs. However GTOs are not optimum basis functions and therefore the basis functions are usually represented b.CGTO J  =  2  d  =  n l m =J x  y  z  as a linear combination of GTOs,  e  -ar  2 S  -1,...L  [2.26]  where L is the length of the contraction and d • is the contraction coefficient.  The functions, b-  Even-tempered  CGTO  basis  are called contracted Gaussian-type orbitals (CGTOs).  sets (GTO or  STO) were  Ruedenberg  [SR79]. The  {a 1 . . . a  } be generated according to a geometric  f  a  =  i  a/3  (i  1  even-tempered  first introduced by  = 1  restriction  requires  that  Schmidt the  and  exponents  progression, [2.27]  m)  This restriction then reduces the number of parameters to be optimized for each group of atomic functions B) instead way  belonging to the  of m. Even-tempered  basis  same symmetry  sets have  to just  therefore  of extending finite basis sets towards a complete basis  two  allowed a  (a  and  systematic  set.  Once a particular basis set is chosen, a choice often dictated by the researcher's goals,  the  S C F calculation  particular geometry atomic  molecular  performed.  geometry  which  center  and  the  Roothaan  as  the  GAUSSIAN  such  and  the  MUNICH  optimum coefficients  the  functions,  basis  the  that  orbital  involves  be  the  programs  are  then  of Pople  energies  the and  close  above  numerical  (depending  on  calculation the  choice  yields of  a  and  standard  co-workers  [BW76,  resulting solution  molecular orbitals in terms of the  total  wavefunctions basis  of  equilibrium  solved using  of Diercksen [D74]. The describe  assumption  are then placed on each  quantities other molecular properties are also derived and  The  the  experimental  Basis functions  H F equations  programs  yields the  This  could  or the S C F optimal geometry.  programs BW80]  is  set)  energy.  these  tabulated.  which can to  From  the  true  be  arbitrarily  Hartree-Fock  32 wavefunctions.  However much difficulty in interpretation  these approximate approximate [DF86].  results  wavefunctions  are  particular  problem  This  basis-set  numerical  truncation  error  has  occurred  unawaret  and  of is  of  basis  because most  the  set  considered  (or over interpretation)  limits  quality to  be  of  is a  of  users of these  their  also  applicability  known  serious  as  the  bottleneck  for  quantum chemistry [LS85, SP85].  2.4. ELECTRON The  CORRELATION  Hartree-Fock  model  in  model outlined in Sec. 2.1.3  chemistry  and  physics.  It  description of chemical reactivity and  has  has  also  been a successful  been  useful  in  predictive  the  theoretical  chemical bonding. However certain  definite  limitations exist within the H F model and experimental evidence indicate that in many  cases  there  is  a  need  to  go  beyond  the  inclusion of electron correlation which is neglected of 'electron correlation' is accounted  H F model. in the  This  means  the  H F model. Some form  for by the H F model since electrons  of the  same spin can not occupy the same space (Pauli exclusion principle).  A  concise  which  the  eigenvalue  discussion correlation of  the  of the  correlation  energy  Hamiltonian  is  defined and  its  problem as  is given by  "the  expectation  difference value  Lowdin between  in  the  [L59] in the  exact  Hartree-Fock  approximation" [L59], that is,  E  corr  = E  , - Ex™ exact HF  L  [2.28] J  t Much of the confusion may be due to the awe created for the uninitiated by the words ab initio (from first principles). Huzinaga [H85] refers to finite basis sets as the 'quasi-empirical element' in ab initio calculations.  33  In practice the correlation energy, E  c Q r r  the  which is  experimental energy  (E  and  e x a c t  ),  zero-point  vibrations,  the  mentioned  therefore refers to the  is calculated from the difference 'corrected'  for  energy.  The  Hartree-Fock  between  relativistic effects correlation  fact that in real systems electrons  and  usually  (mainly of  opposite spin [L59]) tend to 'avoid each other'.  Various methods exist for including electron correlation  and one of them  is  the  configuration interaction (CI) method. In the CI method the exact wavefunction is represented by a linear combination of Slater determinants which are built from the H F molecular orbitals [S082],  *exact  =  +  £  C  °*HF  +  Z  i j ij*i' c  ijkl ^ j S J  The first term is the  1  ]  +•••  Hartree-Fock wavefunction,  all possible singly-excited configurations, doubly-excited configurations, means the virtual  promotion  molecular  ^  (or  orbital  the  the  second term  third term is the  and so on. The nomenclature,  excitation) j  relative  of an to  electron  the  HF  is the  set of  set of all possible in "Pj^,  for example  in molecular configuration.  orbital i to The  a  numerical  calculation in Eqn. 2.29 quickly becomes intractable  as more terms are added in  the expansion and therefore in practice higher-order  excitations are neglected.  CI  wavefunction  is  usually  terminated  at  the  double-excitation  level  and  The the  resulting wavefunction is called a singles and doubles CI (SDCI) wavefunction. A useful  extension  other than all  single  the and  of the  SDCI method  H F wavefunction. double  is the  The final  excitations  inclusion of reference wavefunction  relative  to  many  wavefunctions  would therefore well-chosen  include reference  34 configurations  and  wavefunction.  Wavefunctions  percentage of the  this  is  referred of  to  this  as  type  the are  total correlation energy  multi-reference capable  and are  of  SDCI  (MRSDCI)  recovering  a  larger  considered very accurate  for  the prediction of molecular properties.  2.5. GREEN'S Green's  FUNCTION  functions  are  METHODS  particularly  AND  IONIZATION  attractive  because  of  SPECTRA the  properties  which  render them very useful to the interpretation of ionization spectra. In the Green's function approach the ionization potentials and the transition amplitudes, unlike in the CI approach, are given explicitly in the expression for the one-particle many body Green's function in energy space [CD77],  G(CJ) X X  n  IP  n  X /(u+IP-iT?)  = 2 {X *  = <*  n  f  N  n  1  n  |a |* n  0  N  (ionization potential) is the final  [2.30]  >  [2-31]  is the transition amplitude, a  and the exact  + E . A . term}  R  is the annihilation operator for orbital n and  energy difference between  the exact neutral state  ion state energies. The Green's function is very intuitive in  the sense that the ionization energies are given by its poles (i.e. energy where  G(CJ)-—*»).  The  electron  affinity  (E.A.)  term  in  Eqn.  2.30  regions is  not  relevant in the present application.  The  is  absolute square of the transition amplitude,  called the pole strength  or the  spectroscopic factor as  it is usually called in  35 CI  terminology.  Cederbaum  theoretical methods hence  the  for  ionization  and  finding  energies  co-workers  [CD77,  CD86]  have  outlined  the  the one-particle many body Green's function (and and  intensities).  This  is obtained  by  solving  the  Dyson equation, G(w) where  = G°(u>)  G° (u)  self-energy  is  + G°(w)I(o))G(u)  the  known Hartree-Fock  [VS84]. The poles of  whereas  the  energies  are  poles  [2.33]  of  extent  function  correspond to the  G°(CJ)  G(co) correspond  exact to the  Green's  to  the  that the  exact  £ ( G J ) is  and  the  H F orbital energies  ionization  self-energy term,  energies.  The  is calculated  Z(CJ),  exactly. Various solutions to Eqn. 2.31 involve varying degrees of approximation to  the  self-energy  algebraic order 3rd  term.  diagrammatic  of perturbation order  A systematic  construction theory  in perturbation  of approximations is  scheme  [VS84].  theory  set  and  [ADC(n)]  which  is  For example, ADC(3) thus  is equivalent  accurate  will to  given by  be  the  accurate  extended  particle-hole Tamm-Dancoff* approximation (ext. 2ph-TDA). ADC(4) would include  3hole-2particle  and  3particle-2hole  configurations  and  so  to  on.  the nth to two  therefore A  recent  review [VT86] covers the recent progress in the Green's function method.  One  special note that should be mentioned with regards to the CI approach and  the Green's function approach to calculating spectroscopic factors is their dependence basis  set  saturation  (poles strengths)  on basis set quality. It has been shown [CD77, CD86] that is critical to quantitative  ionization intensities. The basis set dependence  prediction of experimental relative of the  theoretical binding energy  spectrum can be understood following the arguments of Cederbaum et al. [CD86].  36 The  breakdown  from  the  near  configuration. states.  of the  single-particle  degeneracy of  This occurs  Furthermore,  localization of the  the  2h-lp  whenever  the  pole  picture  the  strength  of ionization arises configurations  neutral is  molecule  very  with  the  has  dependent  (theoretically) single-hole  low-lying  on  the  low-lying virtual orbital [CD86]. It is, therefore,  excited  degree evident  of that  basis set quality is critical in the sense that it should model accurately the few low-lying virtual molecular orbitals in addition to the occupied molecular orbitals.  2.6.  HEMS:  A  COMPUTER  PACKAGE  FOR  MOMENTUM-SPACE  CALCULATIONS During  the  course  of  the  present  study  the  existing computer  programs  for  calculating the spherically averaged momentum distributions and the generation of position-space and momentum-space package  called  HEMS  density maps  (H-compiler  optimized  were integrated  into one whole  programs  E M S ) . Various  subroutines were written into more efficient subprograms. extended HEMS  to include d-functions programs  are  shown  and in  f-functions  Fig. 2.2  and  in the  for  Subprograms  basis  outlined  in  set.  were also  The  Table  integrated  2.1. H E M S  operates using a link driver (HEMS*) which basically acts as a 'traffic director' for  the  package  and  desires of the user  routes  the  program  to  various  links  depending  on  the  (see Fig. 2.2 and Table 2.1). The links or subprograms  are  dynamically loaded and unloaded after use to save space and time. The resulting speed and economy is therefore a  single input  maps.  file  is necessary  substantial.  As can be seen from Fig. 2.2 only  for generating  all the  calculations  and  density  37  Table 2.1. Subprograms  (Links) in the H E M S  package.  HEMSL1*  Reads data input file and reformats input file to particular requirements in each link. For GTOs expands Cartesian GTOs in terms of complex spherical GTOs.  HEMSL2*  Calculates spherically averaged MDs  and prints out several  M D statistics useful for debugging. HEMSL3*  Calculates and plots the momentum-space density maps.  HEMSL4*  Calculates and plots the position-space density maps.  HEMSL5*  Plots calculated spherically averaged M D s . Has options to plot experimental data.  A l l programs are written in F O R T R A N . The Fourier transform of GTOs and STOs are analytic (see for example Kaijser and Smith [KS77]). The spherical averaging algorithm is similar to that reported by Levin et al. [LN75]. a  b  38  HEMS*  HEM5L2*  HEMSL4*  HEMSL5*  Spheri cal t y - o v « r a g a d  Mom e n t u m - s p a c e  Posi i i o n - s p a c e  Mom e n t u m DI i f r i but I o n  Densi t y  Densit y  Fig. 2.2.  Maps  Layout of the H E M S computer package.  Maps  39 A  sample input file for the H  molecule utilizing the double zeta basis set of  2  Snyder and Basch [SB72] is shown in Fig. 2.3. The format of the input file is very similar to the output of standard ab inito quantum chemical programs (e.g. GAUSSIAN76 quantum  [BW76] and M U N I C H [D74]). This allows easy interfacing between  chemical programs  and  HEMS.  The input structure  (see  F i g . 2.3)  is  very straightforward apart from the control cards (lines 1, 2 and the last line). Basically,  it  involves  definition and  finally  a  molecule  identifier,  geometry  specifications,  basis  the M O coefficients and angular momentum parameters.  set In  the example shown in Fig. 2.3 the control cards are set to 1, 2 and 5 which means the generation of spherically averaged M D s and their respective plots. The output generated  by the H E M S package for this particular example ( l ^ g orbital  of H ) is shown in Fig. 2.4. The integrated set of programs allows great ease 2  in the calculation and display of theoretical M D s . This particular run took about 1.117 sec C P U time at a cost of CC$0.21 (normal rate).  Replacing  the  control  cards  with  the  appropriate  link  numbers  (e.g.  3  for  momentum space and 4 for position space), the respective density maps can be generated.  The  difference  maps.  momentum  HEMS  difference [BC68].  position  special options  density difference  space  calculated  for  for  (PQJ H  2  generating ~  P^Qp)  (R=1.4a.u.).  6-configuration multiconfiguration S C F (MCSCF) [DW66] and the  of Cade  map The  and  is the  of Das and Wahl wavefunction  also has  Fig. 2.5 shows the  space  wavefunction  package  and  (Fig. 2.5a) quantitative  Wahl  density m  a  P  The  m  CI  wavefunction  S C F wavefunction is the Hartree-Fock limit  [CW74].  The  resulting position space  density  is identical to that obtained by Bader and Chandra agreement  demonstrate  the  accuracy  of  the  present  Options for L i n k D r i v e r H E M S * Options for I-ink #1  1  20 0 5 t 1 OOOOOOOOOOOOOOO H!2% 2 Geometry Specifications 0.0 1 0.0 0.0 2.64562 [Atomic number, x, y, z] 1 0.0 0.0 GAUSSIAN "DZ• SB Basts • •••»• demonstration run: ICPEAC 87 Basis type 2 3 0 1.0 Basis Set Definition 19.24060 0.032B2BO 2.69915 0.2312080 " [Exponents, Coefficients] 0.653410 0.81723BO 1 0 1.0 0.177580 1.0000000. Molecular Orbital (MO) label Msi'.g'A •+ — 4 No. of basis functions in M O 0 0 0 i~6~ 232363 — r 1 0 413260 2 0 0 0 E x p a n s i o n in terms of basis functions 1 2 0 232303 0 0 6 [Atom center, M O coefficient, basis function no.l 2 0 413260 0 0 0 2 & !u% [Angular dependence, n, 1, m] 4 1 - 199944 0 0 0 i 1 - 831113 2 0 0 0 1 2 0 199944 0 0 0 2 0 831113 2 0 0 0 END Options for L i n k #2 02000000000000 •*— 5 0 0 0 0 0 0 0 0 Options for L i n k #5 0 1 3 1 0 3 Plotting options O 1 31 0 3 1.5 o. 1 %  Fig. 2.3.  Input structure of H E M S package. Example is for the calculation and plots of the spherically averaged M D s of H  2  41  TOTAL 'E2E- INTENSITY 8.2006S60E-01 3 8595730E-01 4.8817169E-02 3.9240979E-03 6.8182475E-04 2 0263882E-04 5.8410529E-05  7.9533696E-01 7.2569108E-01 6.2337875E-01 2 7943492E-01 1922577SE-01 1.2632120E-01 2.9246323E-02 1.7360482E-02 1.0347713E-02 2.55O01O6E-03 1.7302211E-03 1.2249590E-03 5.2733975E-04 4 . 1316475E-043.25770S8E-O4 1 . S908635E-04 1.2438085E-04 9.6888078E-05 4.S348730E-05 3.5288176E-05 2 7571718E-05  S.0475305E-01 7.977157BE-02 6.2777996E-03 9 0038753E-04 2 . 57 17984E-04 7 5272706E-OS 2 1667816E-05  GAUSSIAN INSTRUMENTAL FUNCTION ASSUMED: DELP- 0.10O0A.U. INTENSITY  0.0 0 1CO00 0.20000 0 30000 0 40000 0.50000 0.60000 0 70000 0.80000 0.9OOO0  8.02546E-01 7 .794O6E-01 7 .14053E-01 6.17489E-01 5 04565E-01 3 90198E-01 2 86232E-01 1 99788E-01 1 33245E-01 8 53809E-02 5 .29402E-02 3 20488E-02 1 91487E-02 1 14337E-02 6 91395E-03 4 28835E-03 2 7S647E-03 1 B4740E-03 1 29222E-03 9 39779E-04 7 05578E-04 5 42245E-04 4 23029E-04 3 3267 1E-04 2 62307E-04 2 06635E-04 1 62295E-04 1 26993E-04 9 90172E-05 7 70432E-05 6 00650E-05 4 61652E-05 3 52679E-05 2 67720E-05 1 26638E-05  1 .ooooo  1 . 10OO0 1.20000 1 . 30OO0 1.40000 1 .5COOO 1.60000 1 . 70OO0 1.80000 1.90000 2 OOOOO 3.10000 2.20000 2.30000 2.400O0 2.50O00 2.60000 2 70000 2.80000 2.90000 3.00000 3.10000 3 . 2OOO0 3 3OO0O 3.40000  SPHERICALLY AVERAGED M O M E N T U M DISTRIBUTION  , f—  .UNI 10.0  MOMENTUM  -i  1  1  1  r  1  H  CD DC  1  2  _  < 9  ORB: 1 a ,  >  i— „  Z  UD  1 iJ r-  > Lu  DC  q b  ••• MOMENTUM DISTRIBUTION STATISTICS "••  .0  0.6 1.2 1.8 2.4 MOMENTUM (A.U.)  INTEGRATED INTENSITY: UNFITTED- 0.99407E*00 FITTED- 0.10357£»01 MAXIMUM INTENSITY(UNF): O.82007E•OO  AT P(A.U.)- 0.0  MAXIMUM INTENSITY(FIT): 0.80255E+00 FWHM(FIT): 0.49032£»00 S-TYP£(T»TRUE.F-FALSE): T  Fig. 2.4.  Sample output of H E M S package based on data input from Fig. 2.3.  3.0  42  Fig. 2.5.  Density difference (PQJ ~ P ^ C F ^ ^ momentum space and (b) position space for the H molecule calculated at the equilibrium geometry. The H-atoms are indicated in Fig. 2.5b by the solid dots. m  2  a  p  s  m  43 programs.  Currently, the package is installed in the U B C Computer System (Amdahl 5850) which runs on the Michigan Terminal System (MTS) operating system. The bitnet node address is (7serid@UBCMTSG.BITNET.  C H A P T E R 3. E X P E R I M E N T A L METHOD  3.1. ELECTRON MOMENTUM SPECTROMETER The  spectrometer  EMS  spectrometer  operational [L84].  details  used  in the  present  of the  symmetric,  of  spectrometer  the  studies  and  also  by  Cook  [C81].  a  non-coplanar have  It is a modification of an earlier  [HH77]  is  In  type.  been  The  described  instrument the  high momentum resolution  course  construction earlier  by Leung  developed by Hood of  the  and  present  et  work  al. the  instrument has been moved to a new laboratory where careful re-alignment about the collision interaction region has resulted in a significantly increased coincidence countrate. A brief outline of the design and operation of the spectrometer  in its  present form is given below.  3.1.1. Description of Spectrometer The  EMS  spectrometer  (width = 40cm, VHS-4,  height=40cm)  dia.)  arrangement base pressure fluid)  housed and  pump  pump (b) evacuates attached  to  the  in  pumped  1200L/sec). One diffusion  other diffusion (16cm  is  an by  O-ring two  sealed  aluminum  oil diffusion  (a) evacutes  the  gun  pumps  chamber (Varian  region while  the  the analyzer region through a U-shaped tube  top  of  the  aluminum  main  chamber.  This  allows for differential pumping of the gun region giving a differential ratio of 10:1 (analyzengun). Each oil diffusion  is backed by a Sargent  Welch two-stage  rotary  pump.  pump (Neovac S Y A similar  rotary  pump serves the gas inlet line. A n outlet system routes the exhaust gases from the rotary pumps to the fume hood. With the present system a base pressure of 5x10  torr is typically attained in both chambers.  44  45 To  reduce  (hydrogen  the  earth's  annealed)  magnetic  encloses  field  the  (500mG), a  cylindrical  whole spectrometer.  mu-metal  This reduces  the  shield  magnetic  field inside the spectrometer to =*5mG.  A basic schematic diagram of the E M S spectrometer is shown in Fig. 3.1. The spectrometer is composed mainly of an intense electron beam source (EBS), a gas cell (GC) and two identical energy-analyzer-detector  systems  (EADs). One of the  energy-analyzer-detector systems is stationary while the other is movable over the range $ = ± 3 0 ° . The movable E A D is placed on a turntable mounted on sapphire balls and driven by an external servo-motor.  The  components  of  the  electron  beam  source  are  shown  in  Fig. 3.1.  The  electrons are produced by thermionic emission using a commercial triode electron gun  body  A -shaped  (Cliftronics  of  a  (C),  a  (G), an accelerating anode (A) and an einzel lens (L). In  a  thoriated  Wehnelt-type grid  CE5AH). tungsten  Basically, filament  the  which  electron serves  gun as  is  the  composed cathode  typical operation the cathode is at a high negative potential (-1200eV + energy) electrons  and is heated are  by a D C current  extracted  by  the  anode  binding  (2.2-2.5A). The thermionically released (typically  at  + lOOeV  relative  to  the  cathode) and focused by the einzel lens.  The electron beam is further  collimated and transported  into the collision region  (GC). This is accomplished by adjusting the quadrupole deflector voltages (DI and D2)  and  minimizing the  current  collected on the  aperture  plates  (PI, P2  and  MAIN  DIFF PUMP 1200 L/sec  C M A  SERVO MOTOR  Fig.  3.1.  GUN  DIFF PUMP 1200 L/sec  r o  Schematic of Electron Momentum details.  5 cm.  Spectrometer. See text for  47  P3).t  The current  grid,  cathode,  on the Faraday cup (FC) is also maximized. In practice  anode and  lens voltages  are  optimized to  beam with minimal angular divergence. Typical current  give the  most  collected on the  cup is 50-60>A. In general this procedure is done after  the  intense Faraday  —2 hours to give time  for the gun and associated power supplies to stabilize.  Gas  is  introduced  via  a  Teflon  collision  point (see  Fig. 3.1).  tube  into  The sample  a  1mm  dia.  pressure in the  hole just  below  analyzer  chamber  the is  -5 <5xl0  torr. The pressure inside the gas cell (GC) is estimated to be a factor  of 10 higher than the analyzer chamber. Annular slots on both sides of the cell allow outgoing electrons to be transported The  electron-analyzer-detector  system  (see  gas  to the E A D s .  F i g . 3.1)  accurately  6 — 45°.  Since both E A D s are identical the following discussion will equally apply  (AIL) then  the  pass  energy  (lOOeV)  cylindrical  mirror  analyzer  apertures allow  immersion ( A l and  improved  collected at  and the movable E A D systems The asymmetric immersion  lens  asymmetric  are  and  via dowell pins  serves  outgoing electrons  aligned  mounted  to both the stationary  to ensure that the  is  to retard and  A2) are  outgoing electrons  focus  (CMA).  lens  momentum  the  The  system made  the  focal  have small  resolution  electrons  (2.0mm  [LB83].  into  properties  been  (typically at the and  discussed and  These  entrance  of  the  dimensions  of  the  earlier  1.0mm,  defining  [L84].  respectively) apertures  asymmetric immersion lens system, from geometric considerations,  t These apertures define a geometrical acceptance ± 1 . 3 ° [L84].  600eV) to  in  The to the  are capable of  angle for the electron beam of  48 accepting a cone of electrons defined by the half-angles, A0=A0=O.9°. A further consideration of these apertures and their effect on the momentum  resolution is  made in Sec. 3.2.  After  angular  selection,  deflectors  (D3).  necessary  to  that  since  introduce  A  extra  choice  are of  are  placed  uncertainty  introduced  X - and  optimal momentum  deflectors  any  electrons  particular  obtain  the  the  mean  the  Y-voltages  resolution. It  after the  in the  into  C M A via X - Y  was  found  should however  angular  selection  values of 9 and  this  to  be  be  noted  does  not  <j> which define  the orbital electron momentum that is being sampled.  The principle of the cylindrical mirror analyzer (CMA) is well-known [R72]. The C M A is an electrostatic deflection type of energy analyzer which uses two coaxial cylinders of widths a and b held at different  voltages, V  &  and V ^ , respectively.  It has the particular advantage of using the complete (27r) azimuthal angle, high order  focusing and high resolving power  instead  of the  more  usual  360°  (2ii)  [R72]. A C M A used  135° sector in  Auger  C M A was  used  spectroscopy  for  practical reasons as well as to be able to mount the C M A such that the launch angle in the C M A is at 4 2 . 3 ° . This allows for second-order  axis-to-axis focusing  of the C M A however with the present arrangement (with the deflector, D3) it is doubtful whether such second order focusing is achieved. To correct for edge field effects, and  logarithmically spaced  on both  secondary surface.  sides  of the  end  correctors  (EC) are  C M A . A l l surfaces  are  installed top and benzene-sooted  bottom  to minimize  electron emmision as well as to provide an even potential across The  dimensions  of the  present  C M A are  outlined  in  Fig. 3.2.  the The  135  Sector  u  CMA  P a r a m et e r s  a  = 24. 5m m  b = 67. 7m m d =(ri v  s  + d. ) \ '  d = 54. 2m m  z  o  =  1 56. 6 m m  co = 4 2 . 3 ° i9  = 45. 0 °  Design parameters of 135° sector cylindrical miror analyzer (CMA). In units of the inner cylinder radius (a), d = 2.21 and the calculated dispersion, D = 6.12.  50 theoretical energy resolution of a CMA [R72] is given by the base resolution (R),  R ~ {(W +W )/Dsin0} s  where W  g  + (W /a)  e  and W  2  s  + AZ (+A0,-A0)/D  [3.1]  T  are the widths of the source and exit slits, a is the inner  g  cylinder radius, D is the dispersion and AZ,p is the spread introduced by the uncertainty (angular aberrations) in the launch angle. With the parameters shown in Fig. 3.2 the 135° sector CMA has a calculated fwhm (derived from the base resolution)  of  =1.4%  (AE/E  pags  xl00).  The  energy  resolution  of  the EMS  spectrometer can therefore be calculated from,  AE « ( A E where  AECMA  + AE  2 C M A  *  s  t  n  e  l g a r  e r  2 G  u  n  } / 1  [3.2]  2  of the two CMA resolutions and A E Q  u n  is the  energy spread (=0.8eV) in the electron gun. Equation [3.2] results from the fact that the coincidence method (At<5ns fwhm) discriminates against most electrons, i.e. electrons with energy E ^ S are correlated only to electrons of energy E + 6. 2  Thus in the present case (E„  QCC  , = lOOeV) the elastic width is =1.6eV fwhm and  p«.ss  the EMS energy resolution is also =1.6eV fwhm.  Although the entrance to the CMA is defined by an aperture (1.0mm dia.) the exit is defined by an annular slit of dimensions (1.3mmx4.0mm).  The energy  dispersion is controlled by the (vertical) y-dimension (1.3mm) and the x-dimension (4mm)  is  relatively  wider  to  ensure  optimization  of  the  countrates.  The  momentum resolution is found to be critically dependent on the choice of the Y-deflector (D3) voltage. This is because the asymmetric immersion lens system has a wider acceptance angle than the electron analyzer slit system. Optimum  51 tuning of the (+A0)  Y-deflector voltage  (D3) ensures  selection of a  cone  of  electrons  8 of 4 5 ° . These conditions ensure a minimum value  about a mean angle  of the (e,2e) cross section at 0 = 0 ° .  Finally  the  multiplier  electrons  are  detected  ( C E M , Mullard  application. The capacitively  cascade  decoupled  and  B318AL)  of electrons  (0.0022pF)  amplified ( x l O )  by  8  whose  output  results  in a  [L84] and  fed  is  channel  closed  current  to  a  an  for  pulse  electron  the  present  which  external  is  then  charge-sensitive  preamplifier (500 input impedance) and the coincidence electronics system.  3.1.2. Coincidence Detection, Event Processing and Control The  electron  pulses  generated  at  the  channel  electron  multipliers (CEMs)t  are  detected and processed in a coincidence detection electronic system outlined in F i g . 3.3 and the particular components are detailed in Table 3.1. This system uses a single time delay on one of the and Weigold [MW76]. Briefly,  channels  and has  the pulses from the  which has a fixed gain (xlO) are further filter  amplifier  such  that  suitable  produced. With the appropriate 'ringing'  observed  in  the  negative  pulses from the  fast,  been described by McCarthy (charge-to-voltage)  preamplifier  amplified (typically x20) with a timing negative  pulses  (=5ns  fwhm,  choice of 50O cables and proper  C E M pulses  can  be  reduced  timing filter amplifiers are  then  to  -8V) are  termination,  < 15%.  The  introduced to  the fast,  constant  fraction discriminators (CFDs) which produce pulses whenever the signal is beyond the time  set  threshold,  'jitter'  usually -3.0V.  associated  with  Constant  conventional  fraction  discriminators  discriminators  [GM67]  are  free  from  and  are  very  t A cleaning procedure similar to one outlined recently [GR84] was found to be suitable in re-conditioning used C E M s with diminished gain.  L S I - I 1/03  CPU  SCA  r* l  xtnc  —  SCA  rand  INTERFACE  ADC  tru* (tart/stop inhibll/raaat DIGITAL  tcon E O  scan  0  Ditplay  Graphic  Lin*  CRT  Terminal  Printer  Floppy Disk Drive  Prog. P.S.  I/O  DAC  motor REAL-TIM  CLOCK  Fig. 3.3.  Timing coincidence electronics for E M S spectrometer.  Ol  to  53  Table 3.1. Components of E M S timing coincidence electronics. Model  #  /  9301  Type  1.  Pre-amplifier  Ortec  2.  Timing filter amplifier (TFA)  Or tec 454  3.  Constant fraction discriminator (CFD)  Ortec 463  4.  Ratemeter  NR-10  5.  Time to amplitude converter (TAC)  Ortec 467, has a built-in S C A  6.  Single channel analyzer (SCA)  Ortec 406A  7.  Programmable power supply  Fluke 412B (modified for voltage programming)  8.  Servo-motor and amplifier  from Leeds Northrup and Bristol chart recorder  9.  Computer  L S I 11/03 RT-11SJ Operating System  54 useful for applications wherein timing information is important. The fast,  negative  pulses (10ns fwhm, -800mV) from the constant fraction discriminators then as  the  pulse  stop  and  start pulses  is delayed by  a  fixed  of a time-to-amplitude amount  converter  (typically 30ns).  The  serve  (TAC). The  T A C which  stop  is  the  'heart' of the coincidence circuit generates voltages proportional to the time delay between the leading edges of the start and stop pulses. Typically the T A C is set at a full scale range of 200ns which is equivalent to an output of = 10V (2/xs width). The voltage  output  of the  T A C serves  analyzers (SCAs). One S C A (lower level = 0.4V, coincidence serves  window while the  as  the  random  other  window.  as  upper  S C A (lower This  input to two  level=1.4V) serves  level = 2.0V,  particular  single channel  choice  upper [LB83]  as  the  level=10V) of  windows  (random:coincidence = 8) gives a good compromise between optimal signal/noise ratio and  proper  1 = + 4V)  background  generated  The  NIM-logic  pulses  by the random and coincidence SCAs  home-built interface computer  subtraction.  [C81] before  finally  being stored  memory. The true coincidence rate  and  (logic  are  0 = 0V,  processed  in separate arrays  standard  logic  using a in the  deviation defined by  this method are therefore given by,  N  true  AN  where N  true  C Q  ^  n  = N =  coinc (N  and N  -  coinc  r a n £  j  N  rand  + N  /8  rand  are the  / 8  [3.3a]  2)1/2  number of counts registered  and random SCAs, respectively. The estimated = 100MS.  [3.3b]  in the coincidence  processing time for each event is  This places a practical limit to a maximum coincidence rate of  10 cps a  55  which is still very high compared to the present coincidence rate (^O.lcps).!  The  LSI  computer-interface  parameters the  user  and/or The  (e.g. E  computer  generated  by  allows  user  to  of running several sequential data  scan mode)  as  system  also  has  a  T A C to  be  digitized  the  the  control  experimental  and <t>) and collect E M S data. With the L S I 11/03 computer  0  is capable  angular  system  well  as  scans  (in energy  displaying and printing out the  12-bit  A D C that and  allows the  subsequently  mode results.  voltage levels  stored  in  computer  memory. The histogram of T A C voltages serves the useful function of monitoring the time spectrum in an E M S experiment and is thus a check for the of  spurious  which ' only The  rest  signals  or  external  noise.  The typical  time  resolution is  presence 4-5ns of  = Ins is due to the electronic components of the coincidence system. of the  time  spread  is mainly  due  to  the  different  flight  times of  electron trajectories in the C M A [VS83].  The E M S spectrometer  can be operated in three different modes namely, elastic  mode, binding energy mode and angular correlation mode. In the elastic mode the electron  beam  impact  energy  is  set  at  600eV  (corresponding  to  coincidence  operation at E = 1200eV), focussed and aligned. This permits the electron energy 0  analyzers to be optimized for scattered  electrons  of 600eV  as  observed in the  coincidence modes. Typically the grid voltage is varied to reduce the  countrates  and prevent the C E M s from overloading. The C M A voltages are then  separately  t In addition to the small lens acceptance angles the low coincidence countrate results from the fact that most of the ionizing collisions are due to small momentum transfer collisions ( K < l a " ) rather than large momentum transfer collisions [EJ86]. The E M S spectrometer selects the latter type of collision because of the symmetric energy sharing arrangement. 1  0  56 adjusted  so as  to pass 600eV electrons  (elastically scattered) into each channel  (movable and fixed EADs). The shapes of the elastic peaks are also monitored to check  the  performance  of the  C M A . Extra  caution is exercised in setting  movable C M A voltages because it is necessary  to ensure that the  the  elastic count  rate is also invariant with the angle <j>. This procedure is routinely done (every 2-3 days) to monitor any shift in surface potentials and beam characteristics.  The  binding energy  and  angular  (coincidence)  scans  are  done  by  setting  the  cathode back to 1200eV and re-focusing the beam. Either type of scan can then be initiated by the appropriate binding energy  software control in the L S I 11/03 computer. The  scan is done at fixed  angle  0 and variable impact energy  (E ) 0  supplied by a programmable high voltage power supply. The programmable power supply is ramped correlation  scan  variable angle  by  a  is done <j>.  12-bit digital at  an  The angle  to  analog converter  appropriately  selected  is varied through  a  (DAC).  fixed  The  angular  impact energy  and  12-bit D A C and a calibrated  servo-motor amplifier. More detailed description of the hardware and software  are  available [L84].  Under routine operating conditions in the coincidence modes the measured  energy  resolution  energy  of  the  E M S spectrometer  is  1.6-1.7eV  fwhm.  The  quoted  resolution includes the finite energy spread of the electron gun (^O.SeV) and the CMA  energy  resolution  (^l^eV).  The  momentum  and this is discussed further in the following section.  resolution  is  0.10-0.15a " 0  1  57  3.2. MODELLING THE EFFECTS OF FINITE MOMENTUM RESOLUTION Inherent in any general,  this  scattering experiment is the instrumental resolution function. In  instrumental  especially the  resolution  finite sizes of the  function  is due  to  the  collision  geometry,  apertures that define the collision process,  the  energy spread of the incoming and outgoing particles and the electron beam size. In  E M S , where  the  quantity  of critical interest is the  experimental  momentum  profile (XMP), the most important instrumental factor to be accounted for is the momentum measured  resolution  approximation,  the  ion recoil momentum is essentially equal (but opposite in sign) to  the  momentum and  is  of the  given  function.  particular  by  Eqn.  Within  the  binary  encounter  orbital electron prior to ionization [MW76,  2.13.  Mapping  the  orbital  electron  WM78]  momentum,  p  is  achieved by variation of the azimuthal angle 0 within the symmetric non-coplanar geometry  [MW76]  where  0 ,=0 2 = 4 5 °  (fixed)  and  E , = E =600eV 2  (fixed).  A  propagation of error type analysis [MC81, B84] shows that to a large degree the contributions  from  Ap ,  Ap j  0  and  A0  are  relatively minor.  Explicit  analysis  [MC81, B84] of the momentum resolution, Ap shows, Ap = p ~ { [ 2 p , s i n 0 ( p +  2  + (p  E  o  2  2  2  2  the  o  (2p, sin 0sin0/2cos0/2)A0  + [4p,(cos 0  For  4p, sin0cos0sin 0/2]A0  - 2p,cos0) +  1  0  + sin 0/2) 2  2p cos0]Ap o  1  [3.4]  - 2p,cos0)Ap } o  values  of  the  kinematic  parameters  used  = 1200eV, E = E = 6 0 0 e V , E = 1 5 . 7 e V , 0 = 45°, 1  2  b  in  the  present  work  A 0 = + 1° and A 0 = ± 1 °  i.e. Eqn.  3.4 can be approximated by, Ap = p ~ { [ 2 p p o 1  1  -  2p! (3 cos0)]A0 2  + 2p  2 1  sin0A0}  [3.5]  58 Clearly, Ap is strongly dependent on the value of <j> only at small values of momentum or 0.  The  analysis  of  EMS data  therefore  presents  an  unusual  challenge  to  the  experimentalist because the features  of the experimental momentum profile most  critical  theoretical  in  its  comparison  with  predictions  are  in  the  region  p < l . O a o " - Clearly an accurate knowledge of the momentum resolution function 1  is required for detailed evaluations calculations  of EMS experiments  and quantum chemical  reported in the present work. Although other workers in the field  have often neglected momentum resolution effects, a Ap as small as influences  both the  shape  and relative magnitudes  of the  O.l5a ~  1  0  observed momentum  profiles.  It could however be argued that an 'experimental' approach to the momentum resolution function problem is to further reduce the lens aperture sizes such that the  momentum  alternative  is  resolution  however  function  not  counts  that would result.  which  approach the  feasible  is  approximately  because  The present  Ap<0.15a  _1 0  the  delta  extremely  function. low  aperture sizes (half-angles,  practical sensitivity  spectrometers, result in a  of  a  limit  for  the  current  This  coincidence  A0=A0=1°), single  channel  -  Two main approaches in defining the momentum resolution function are presently used  by workers in the EMS field namely, the  analytic gaussian  function method. Both methods  relative advantages and disadvantages are discussed.  planar grid method and the are outlined below  and their  59 3.2.1. P l a n a r G r i d  Method  The planar grid method has been adopted by several groups [C81, F83]. The method basically assumes a planar grid in the collision region (see Fig. 3.4) defined by the half angles Ad  and A0 of both analyzers as well as the beam  size. For simplicity the grid is assumed to be rectangular with typical spacings of 0.2°. For a particular value of <p , 6  and E  detect contributions from  in the planar  0  p-values  0  defined  0  the detector is assumed to grid. The resulting  convoluted (resolution fitted) momentum distribution is therefore given by, <p(p)> =  W  i  j  *p(  P  i  where p(Py) is the theoretical MD  j  )}/ { 1 ^  W i j  }  [3.6]  at p(0^, 0 j ) and w- is a general weighting  factor. In most cases, w~ is assumed to be unity that is , a uniform is assumed and the convoluted MD  distribution  is just the uniformly averaged value of p(p^)  over the grid.  Three main assumptions are inherent in the planar grid method, (a) The interaction volume is assumed to be defined by a plane of dimensions (2A8) X (2A0). The  primary  electron  beam  diameter  is smaller  than  the  dimensions subtended by the acceptance angles in the collision region. The much smaller electron beam width ensures that the collision volume does not change when the azimuthal angle <j> is varied [C81]. (b) Trajectories originating  from  this planar  grid  are assumed  to be evenly  distributed (or uniformly weighted). This means that trajectories originating from (fi  0 )  0 ) are as likely as other trajectories (6^, O  #j) as shown in Fig. 3.4.  (c) The effective momentum representing the summed trajectories from the planar  Fig. 3.4.  Collision volume defined by spectrometer apertures ana beam size. Gridded area refers to the region of convolution using the planar grid method (see text).  61 grid  is  given by p  as  defined by  (6 ,  (p ). A s will  0  0  be  shown  in a  later  planar  grid  in Fig. 3.5.  The  subsection, this assumption may be incorrect.  A  typical  method  result  is  curves  Clementi  and  resolution  Comparison  of  values  momentum  values  [CR74].  results  the  3p  calculated  increasing  and  1  0  Ar  different  are  Roetti  (i.e.  p<0.3a ~  XMP.  the  given for  theoretical  optimum  for  Ap) in  curves  of A0  from  It  the  can  has  very  with  the  and  be  the  a  , A# = 1.0°  distribution using the (At9 = 1° =fixed) Hartree-Fock  seen effect  slight  of shift  measured  A0=1.O°  that  Ar  give an  limit  wavefunction  lowering  increasing towards 3p  the the  of  momentum intensity  higher  X M P shows  at  momentum. that  excellent fit to the  the  A r 3p  These values of AO and A0 also match closely the physical sizes of the  lens acceptance angles (A0=A0 = O.9°, see Sec. 3.1.1).  The choice of the A r 3p X M P as the  'calibrant' gas for E M S studies is based  on extensive S C F limit and CI studies  [MA84] which indicate that the  MD  is  only  very  slightly  (<1%)  Furthermore relativistic effects  affected  by  inclusion  of  A r 3p  correlation  effects.  are not expected to be important for the  valence  orbitals of a low Z atom such as argon.  Comparison of measured X M P s and convoluted M D s using the planar grid method have  been  routinely  used  in the  earlier  work  in the  present  elsewhere. The validity of the planar grid method can be further a careful evaluation of the assumptions optimum A6  laboratory  and  investigated by  (a-c) outined above instead of finding the  and A<j> values which give the best fit to the  A r 3p X M P . One  i  r  "i  q d  Ar P L A N A R GRID  1  r  3P  METHOD  ( 15. 7 eV) R e s o l ut i o n ( 1)  A *  =  0. 0*  (2)  A *  =  0. 4°  ( 3)  A *  =  0. 8°  (4)  A *  =  1. 0*  ( 5.) (6)  = 1.2° a* = 2. 0 °  (7)  A *  1.5 2.0 2.5 Momentum (a.u.) ^ Fig. 3.5.  =  Effects  3. 0 °  3.0  3.5  Sample results using the planar grid method. The unconvoluted curve is area normalized to the Ar 3p XMP and all other curves maintain the same relative normalization.  63 way of addressing this question is to devise alternative methods of convoluting the effective momentum spread and to observe the dependence of the convoluted MD  on the  form of the  instrumental function. An alternative  Ap convolution  method is provided by the analytic gaussian function method.  3.2.2. Analytic Gaussian Function Method This  method  involes  an  analytic  convolution  method  whereby  a  gaussian  instrumental function given by,  exp[-(q-q ) /a ] 2  [3.7]  2  0  is convoluted to the theoretical MD, p(p). The convoluted MD is therefore given by, <p(p)> =  * where  {fp(q)exp[-(q-q ) /a ]dq} 2  {;exp[-(q-q ) /a ]dq}"  a =Ap /(ln2) 2  2  2  0  2  2  [3.8]  1  0  and  Ap is  defined  as  integration of the numerator in Eqn. 3.8  the  1  1  resolution, t  is done numerically using a  integration routine and is found to be reasonably -3 . 5ao " ^p^ + 3 . 5ao "  momentum  The  4-point  converged within the range  in steps of O . l a ~ . The denominator in Eqn. 3.8 is 1  0  1/2 known and is equal to a(7r)  The analytic gaussian function method is attractive in several respects. First, the method does not assume a uniform distribution of trajectories  but weights the  trajectories preferentially at the nominal value of the momentum defined by t The momentum resolution half-width-at-half-maximum.  (Ap),  by  definition,  is  equal  to  6  0  the  64 and  <f> . Following the Central Limit Theorem [G85] the gaussian (or normal) 0  distribution  of electron trajectories  is the most probable  distribution function  especially if many small factors contribute to the final experimental uncertainty. In addition according to the Maximum Entropy Principle [J57] the most probable distribution, given the only requirement that the variance ( a ) be finite, is the 2  gaussian  (normal)  distribution. Secondly,  the gaussian  function method is also  computationally faster (by =50%) than the planar grid method.  The  analytic  gaussian  function  'unphysical' results at p = 0 a ~ 1  o  method  (for a  fixed  Ap) however  yields  This is due to the fact that p(p) exists only  for p > 0 a o - In practice, however, the fact that the 'effective' Ap increases as _1  p  decreases  towards p^O  (as predicted by the propagation of error analysis)  should be taken into account. This can be approximated  by keeping Ap constant  (i.e. independent of p) and folding the theoretical MD, p(p) about p = 0 a " o  1  and  doing the integration in Eqn. 3.8. This has the net effect of increasing the effective value of Ap at values of momentum, p<2Ap.  Sample results using the gaussian function method at different values of Ap are shown along with the Ar 3p XMP in Fig. 3.6. It can be seen that the results are very similar to that obtained using the planar grid method (compare Figs. 3.5 and 3.6). As Ap increases there is an increase in intensity at p<0.3a "  1  0  accompanied by a very slight shift in p shift of P  m a x  towards larger values of p. This  towards larger momentum can be seen by assuming a simple form  of p(p) such as, p(p)  = p exp(-$p ) 2  2  [3.9]  Ar GAUSSIAN FUNCTION  3P  METHOD  ( 15. 7 eV) R e s o l ut i o n (1) 0.0  au  (5)0.  15au  ( 2 ) 0. 0 5 a u  (6)0.  20au  ( 3 ) 0. 10au  (7)0.  30au  ( 4 ) 0. 12au  (8)0.  40au  1.5 2.0 2.5 Momentum (a.u.) Fig.  3.6.  Effects  3.5  Sample results using the analytic gaussian function method. The unconvoluted curve is area normalized to the A r 3p X M P and all other curves maintain the same relative normalization.  Ol  66  which  maximizes  at p. max =  (5) -1/2  It  can be shown  that  the resulting  substitution of Eqn. 3.9 into Eqn. 3.8 yields the convoluted MD, <p(p)> which maximizes at  - [0/5)+aV2] Clearly  it can be seen  1/2  [3.10]  that the first order  correction is very  small and  furthermore it is positive (i.e. towards larger momentum). This result is consistent with earlier observations regarding observed  discrepancies in the comparisons of  theory and experiment for the outermost valence orbitals of H 0 [BL85]. It was 2  originally  thought that the discrepancy  at lower  momentum  may be due to  uncertainties in the momentum resolution function that was used. The more exhaustive study of H 0 2  [BB87] incorporating correlated wavefunctions  for both  initial and final states, in fact, showed that the reported discrepancy [BL85] was not due to momentum resolution effects but due to the inadequate  inclusion of  electron correlation and relaxation.  3.2.3. Defining the O p t i m u m p-Value  A mental picture of the (e,2e) reaction can be envisioned as shown in Fig. 3.4. The finite dimensions of the lens apertures ( A  t  and A , A 2  s  and A ) define a 6  collision volume from which various combinations of electron trajectories are likely to register as (e,2e) events. The combination  of all these trajectories define the  signal strength ultimately registered at a particular value of 6  0  and #  0  and  therefore at a corresponding p. We shall define p as the optimum p-value that accurately represents (in a statistical sense) all the electron trajectories physically collected in the detection system for a given nominal setting of 6  0  (45°) and  67 <po (variable). The optimum p-value, p should therefore  be equal to p in  the  limit of very small A 0 and A 0 and very small beam size.  It is clear that increasing A 0 , A 0 or the beam size would have the effect of including electron trajectories different  values  regions  of the  of  p.  that correspond to different (6^ ,  This  would  orbital momentum  therefore  correspond  distribution but  to  all would  0 j ) and  therefore  sampling  different  be  recorded  at  a  given nominal value of p. In regions wherein the M D varies steeply the observed (e,2e) cross section would therefore be difficult to interpret. This has the effect of 'smearing' the  sharp  features in the M D if an otherwise  ideal E M S experiment  was performed. However not only do features in the measured X M P get smeared but  this  also  has  the  effect  of  introducing  an  uncertainty  in  the  optimum  p-value, p.  There are three possible ways of defining the optimum p-value, p, namely, (a) p = p (zeroth order); (b) p = (c) p  [L^ Wj_} (first order);  =  Statistical average  over the collision volume weighted by the lens and  analyzer transmission functions.  Option  (a)  has  been  used  option is inconsistent with planar  array  of electron  traditionally as the  planar  trajectories  grid that  outlined in Sec. method. are  If one  0  3.7.  The  series  of  p-value  histograms  at  6  0  different  however  assumes,  equally weighted  produce a p-value histogram for a given value of E , Fig.  3.2.1  then  and <j> values  0  of  this  in fact, one  a  could  as shown in 0  O  were  .68  PHI = 0 D E C  8  PHI =  DEC  2  §  1 ^ ?6 -0.05 0.00  0.05  OK  OO  0.20  O.IS  O.JO  PHI = 1 D E C  g  2  o  2  UJ  I -005 0.00  0 05  O.K)  0.15  0.20  0.25  O.JO  P H = 2 DEC  PHI = 3 0 D E C  i II III OB  O O  0.20  0 . »  0 JO  OJS  2J0  0.40  0J5  2.40  ill 230  2.«5  235  2.*0  Momentum (a.u.) Fig. 3.7.  P-value histogram at different values of 0 - E =1200eV, E =15.7eV, E , = E =600eV, e =45\ A0=A0 = 1°. The grid dimensions are X 2A0. Arrows refer to the zeroth order approximation to p and stars refer to first order approximation to p . O  b  2  0  0  69 obtained by  calculating all the  planar  (see  grid  F i g . 3.4)  values  of p^. as  and counting the  defined by  number  (6^,  <}>•) in  the  of p-values that occur  at  particular values of p ± 6 p  where  5p is the histogram halfwidth. The arrows in  each  histogram  define  the  refers  to the  (Fig. 3.7)  zeroth-order  p-value  approximation to p  given by  (6 ,  0 ),  0  that  O  is, it  (Option a). The stars (*) in each  histogram, on the other hand, refer to the first order approximation to p  which  is just the mean p-value of the distribution defined in each histogram. It can be seen  from  Fig. 3.7  that  at  values  between the zeroth-order and  of <f> =0°-3°  first-order  that in the most severe case  there  0  (<t>o=0°) the difference between the 1  is relatively small it may become correspondingly as  more  marked  important  accurate  differences  approximations to p. It should be noted  and first order approximation to p is only ^ O . O S a o " -  and  are  as  zeroth order  Although this correction  more precise E M S experiments  calculations are  done  in the  future.  At  larger values of <j> the distribution of p-values becomes nearly uniform and  the  0  two  approximations  preceding planar  analysis  grid  (arrows simply  method  is  and  stars)  states not  that  consistent  give essentially the assuming with  a  uniform  assuming  same  p-value. The  distribution in  p =p  (i.e.  zeroth  the order  approximation) at small values of <f> . 0  A  proper  definition  of p  could  be  attained  by  adopting the  (Option b) in the planar grid method or some other One should note  that the  present  weighted  p-value  realistic weighting scheme.  analysis is still limited to a planar grid. A  non-planar grid (Option c), for example a distorted cylinder, would give a better approximation  of the  optimum p-value. Clearly  this  would  involve  sophisticated  calculations and non-trivial analysis procedures (e.g. Monte Carlo methods [NT86]).  70 Another alternative which is consistent with given  by  assumed  the analytic to be  normally  gaussian  method.  distributed  present work the analytic gaussian although  variations of both  the first-order approximation  In  around  p  this (6  0r  case  electron  #o)-  m  a  u  to p is  trajectories are studies in the  method has therefore generally been employed  methods [D87] have been used for consistency  checks.  All conclusions reported in this thesis are however unaffected by this choice.  C H A P T E R 4. W A T E R : P A R T I  4.1. In  OVERVIEW this  chapter  momentum  an E M S study  resolution  accompanied by a momentum life,  related  has  also  chemistry [KK72]. Some of the  H 0  [HH77,  independently  been  a  earliest  DD77].  The  resolution  of H 0  determined  orbital density  benchmark  test  first  system  topography  binary  (e,2e)  [HH77] and  Dixon  (=0.4a ~ )  but  1  in  importance  on  [EK69,  H 0,  they  for done  2  [DD77] were  nevertheless  restricted  illustrated  the  sensitivity of the (e,2e) technique to details in the orbital wavefunction. Even low momentum between  the  resolution both studies  measured  water  and the  These  discrepancies  (p<0.6a ~ ) 1  0  momentum  distribution of the  and  were appear  than  studies  second  other  momentum  analogous phosphine  significant discrepancies  lb,  particularly  apparent  in  to  that  orbitals  suggest  predicted by the  and  3a ^ orbitals of  row  hydrides  distributions. In  third row atom [HH77a]  and  the  the  low in  momentum question  corresponding wavefunctions. such  ammonia [HH76] revealed similar apparently orbital  at  corresponding M D s calculated using ab initio S C F wavefunctions.  spatially extended on  strongly suggested  to  quantum  molecules were  experiments et al.  are  in both  theoretical  for  high  MDs  subject of several reviews  (e,2e) measurements  0  at  2  Quite apart from its fundamental  binary  The  shell orbitals  experimentally  molecule is the  by Hood et al.  low momentum  valence  discussion of the  The water  F72].  by  reported.  and position space.  water  2  is  of the  contrast,  as  hydrogen  fluoride  outermost  chloride  71  [BH80]  were  more  Subsequent [BH80]  found  sulfide to  and  outermost  orbital M D s of  containing hydrides, namely hydrogen hydrogen  are  anomalous behavior in the the  region  the  [CB80], be  quite  72 adequately  modelled  by  Although these results  wavefunctions  are  of  still not fully  consistent  with  the  well  known  properties  of second  and  third  and  double-zeta understood  unusual  row hydrides  (DZ)  or  better  quality.  they at least appear to be  relative such  as  chemical the  and  tendency  physical  for ligand  donor activity and hydrogen bonding.  A  very recent  with  (e,2e) study  of water  position sensitive detection  analyzers  has  complemented  additional structure  [CC84] using a binary (e,2e)  in the  the  energy  earlier  dispersive planes  studies  in the binding energy  [HH77,  spectra of water  spectrometer  of the  DD77]  by  electron  discussing  notably in the  inner  valence (2a ) region. The splitting of the inner valence ionization pole strength in 1  water  and  arising  other  from  Although the  molecules  the  has  breakdown  most recent  been  of the  study  attributed  to  independent  [CC84] of H 0 2  many-body  particle  effects  [SC78]  picture  for ionization.  emphasized the  binding energy  spectra and its comparison with many-body calculations, only a limited number of experimental  points  on  the  orbital  momentum  distributions  were  obtained.  Moreover the measured M D s were derived from peak areas in the binding energy spectra  obtained  uncertainty  in  at the  a  series  momentum  of  azimuthal  scale,  angles.  particularly  Despite  below  some  0.6a ~ , 1  o  appreciable these  MD  results t appear to reflect the same kinds of differences from Hartree-Fock theory observed other of  in the  earlier  various results H 0  have  2  binary  (e,2e) studies  [HH77,  the present direct measurement  been  undertaken  utilizing  the  ( ^ O . l a o " ) available in the present spectrometer 1  DD77].  In view  of these  of the valence orbital M D s high  momentum  resolution  [LB83]. In addition the binding  t It should be noted that Fig. 3 of reference [CC84] is incorrect as initially published. The correct figure has been published in a subsequent erratum [CC84].  73 energy spectrum in the inner valence region has been obtained and compared  directly  with  a  number  Furthermore our earlier discussions  of representative and  many-body  this is  calculations.  illustrations of the momentum space  chemistry of atoms [LB83], diatomics [LB83a] and linear triatomics [LB85a] are now extended to the bent triatomic system  H 0. 2  4.2. BINDING ENERGY SPECTRA OF WATER The binding energy spectra of water obtained at relative azimuthal angles of 0° and 8° and an impact energy of 1200eV plus the binding energy are shown in Fig. 4.1. The  spectrum at <p = 0°  thus tends to emphasize ionization  probes very low momentum components and of "s-type" orbitals  (ie.those which contain  totally symmetric components). The minimum momentum, P j m  n  accessible in the  experiment depends on the impact energy (E ) and the electron binding energy 0  (E^) and is given by, Pmin ~ E / ( 2 E - 2 E ) / 1  b  On  0  [4.1]  2  b  the other hand, the spectrum at <t> = 8°  (corresponding to p=  valence electrons) shows contributions from both "s-type" and  0.6a ~ 0  1  for  "p-type" orbitals  (Fig. 4.1). The two binding energy spectra were obtained by repeated sequential scans so that their relative intensities are automatically normalized. The energy scale was calibrated  by aligning  the spectrum with the vertical IPs established  by (UV and X-ray) photoelectron spectroscopy [BT68, SN69, MM77].  74  q 06  H 0 1200 eV  j  2  q id  cp = 0 d e g . _  q  q  CN  "c q d  D  10.0  15.0  20.0  25.0  30.0  Binding Energy  D  10.0  15.0  20.0  25.0  40.0  45.0  35.0  40.0  45.0  (eV)  30.0  Binding Energy Fig. 4.1  35.0  (eV)  Binding energy spectra of H O at azimuthal angles 0° and 8 ° . The impact energy is 1200eV + binding energy. The sitting binding energies where the M D s are measured are indicated by the arrows in the lower part of the <f>= 8° spectrum. 2  75 The  ground  state  electronic  configuration  of water  in the  independent  particle  approximation may be written as follows: 1  (1a,)  A 1•  (2a,)  2  db )  2  (3a,)  2  2  (1b,)  2  2  Comparing the two spectra in F i g . 4.1, relative increase of the low energy (10-20eV) takes place on going from 0 = 0° to 0 = 8°. "p-type"  character  of the  three  outermost  orbitals.  peaks  This is expected from the Gaussian curves  have  been  fitted to the spectra (Fig. 4.1) taking into account the known vertical IPs [BT68, SN69] and associated vibrational widths for the corresponding transitions together with  the  instrumental  outermost  three  character  of the  energy  valence inner  resolution.  orbitals, valence  the  In  contrast  peak  at  orbital (2a , )•.  to  32.2eV  The  the  behavior  reflects  of  the  the  "s-type"  Gaussian fitting to the  2a,  band reflects the many-body nature of the associated transition (see below).  The  present  results  binding energy  (Fig. 4.1)  spectra  are  consistent  [HH77, DD77] for H 0 2  Cambi et al. [CC84] given the differences measurements. high  energy  The 2 a , and  shows  with as  both  earlier  well as the  binary  (e,2e)  recent work of  in energy resolution in the  respective  band (Fig. 4.1) is quite broad with a tail extending to indication of partially  resolved  structure  on  the  high  energy side. Similar features in the inner valence band can be seen in the X P S work reported by Siegbahn et al. [SN69] in 1969 and also in the later work of Martenson  et al.  [MM 7 7].  A  re-examination  resolution (1.2eV fwhm) binary (e,2e) spectrum also strongly suggests that the  2a,  of  the  somewhat  higher  energy  reported by Hood et al. [HH77]  envelope consists  of two or more partially  resolved peaks. The present experimental binding energy spectrum  (</) = 0°)  in the  76 2a,  region is shown in greater detail in Fig. 4.2a  together with the  data of  Hood et al. [HH77] and Cambi et al. [CC84]. The solid line represents a fitted curve assuming three equal-width Gaussians (2.77eV fwhm) which give a good fit to  the  data.t  [HH77]  and  differences  There also  is good agreement  with  in the  that  sharpness  recently of the  with the  reported peak  earlier  by  onsets  differing energy resolutions used in the respective  data  of Hood  Cambi et are  also  al.  et  [CC84].  consistent  al. The  with  the  experiments.  In view of the interesting nature of ionization from the inner valence region and in particular  the  wide range of theoretical studies  [AS80, MO80, A M 8 2 ,  VC82] predicting a failure of the  single particle picture for H 0  ionization,  study  a  experimental recent  careful  comparison  spectrum  theoretical  in  the  studies  region  [AS80,  inner  2  has  been  23-45eV.  VC82,  made  The  NT82]  of  (Fig.  results the  of  NT82, valence  4.2)  with  three  of  many-body  the these  (satellite)  structure of the inner valence ionization of water are presented in Figs. 4.2b-4.2d and Table 4.1 for comparison. It should be noted that the recent Multireference double very  excitation  configuration  similar result  to  the  interaction  semi-internal  (MRD-CI)  at  the  appropriate  energies.  [CC84] gives  CI calculation by Agren and  [AS80]. The calculated pole strengths in each lines  calculation  In each  study  are  calculation the  a  Siegbahn  represented by vertical highest  pole  strength  has been arbitrarily normalized to unity to facilitate comparison with experiment. Gaussian curves with the experimentally determined halfwidth (fwhm = 2.77eV) are f If the instrumental halfwidth (1.6eV) is allowed for, a natural halfwidth of 2.26eV can be derived for each 2 a , pole. This (2a,)" state is entirely dissociative as has been shown by dipole (e,e+ion) coincidence studies [TB78]. Therefore the natural peak width would reflect the intersection of the Franck-Condon region with the repulsive potential energy curve for each 2 a , pole.  -l  1  1  1  1  1  1  1  1  r-  77  S e m i - I nt e r n o l  Cl  23.0  25.0  27.0  29.0  31.0 33.0 35.0 Binding Energy (eV)  37.0  39.0  41.0  43.0  23.0  25.0  27.0  29.0  31.0 33.0 35.0 Binding Energy (eV)  37.0  39.0  41.0  43.0  23.0  25.0  27.0  29.0  31.0 33.0 35.0 Binding Energy (eV)  37.0  39.0  41.0  43.0  1'  T  EXPERIMENT  6  rp =  1 i  1  r-  (l200eV)  0 deg.  £° 23.0  Fig.  4.2  25.0  27.0  29.0  31.0 33.0 35.0 Binding Energy (eV)  37.0  39.0  41.0  43.0  (a)Comparison of the inner valence (2a, ) binding energy spectrum (0 = 0°) of H 0 [solid circles] with previous experimental work (Ref.[HH77] [open square], Ref.[CC84] [open triangle]). The experimental points have been fitted with three Gaussians of 2.77eV fwhm (dashed lines) and the sum given by the solid line. Shown above is the comparison with several theoretical calculations (b)SAC-CI N V [NT82], (c)Ext. 2ph-TDA [VC82] and (d)Semi-Internal CI [AS80]. The calculated pole strengths are indicated by bars at appropriate energies and Gaussian curves (2.77eV fwhm) are convoluted, summed and scaled (x2.5) to yield the theoretical binding energy profile (solid curve) in each case. 2  Table  4.1.  Orbital  2a,  Ionization  E n e r g i e s (eV)  and  Peak NV  Intensities Ext.  of  Water  (e,2e)  b  XPS°  SAC-CI  32.2  (0.58)  32.2  32.39  (0.772)  30.48  (0.081)  27.24  (0.047)  (0.18)  34.89  (0.178)  33.41  (0.58)  31.48  (0.712)  (0.095)  40.70  (0.051)  33.99  (0.098)  32.29  (0.018)  36.56  (0.023)  35.37  (0.114)  38.09  (0.068)  35.97  (0.100)  d  2ph-TDA  S  Semi-Internal  (2.77] 35.0 12.77] 38.9 [2.77]  a b  All  Intensities  Binding fwhm  energy  Including  quoted results  In  parentheses  taken at  instrumental  .  »)=8 . 0  resolution  are  quoted  C  Ref.[SN69].  a  Ground s t a t e Ref.[vC82].  energy*-76.24749  a.u.  Ref.[NT82].  Ground s t a t e  energy»-76.04110  a.u.  Ref.[AS80].  S  f  Intensities  contributing  less  than  2% a r e  not  In  square  reported.  brackets!].  CI  f  79 folded  into each  of the  calculated  poles  in the  respective  calculations  and  the  resulting total envelope is given by the solid line in Figs. 4.2b, 4.2c and 4.2d. The  Green's  uses the As  function  extended  calculation due  two-particle-hole  to  von Niessen et  al.  [VC82]  Tamm-Dancoff approximation  opposed to an earlier version (2ph-TDA) [SC78], the  (Fig. 4.2c)  (Ext.  2ph-TDA).  Ext. 2ph-TDA is exact  to third order in the electron-electron interaction. The two other theoretical employ the method  studies  of configuration interaction (CI). Agren and Siegbahn [AS80]  used semi-internal CI to describe the electron correlation effects whereas Nakatsuji and Yonezawa [NT82] utilized the Though two calculations are  symmetry  reported  adapted cluster  in reference  (SAC) CI method.  [NT82], namely  (V) and a non-variational (NV) solution, only the  SAC-CI  a variational  N V calculation, which  should be more accurate since it involves fewer approximations, is included. It is not  the  purpose  methodologies comparative  of  of the  evaluation  the  present  theoretical of  the  study  to  studies  calculated  make  extensive  mentioned pole  but  strengths  comments rather  with  on  to  the  the  make  a  experimental  results for water. It is seen, in particular, that all calculations agree qualitatively with the experimental binding energy  spectra.  strengths (peak intensities)  positions  theory  gives the  and  peak  However, the envelope of the pole as  predicted by the  best overall agreement with experiment  in the  SAC-CI N V  case of water.  Below the main inner valence peak the present work suggests the presence of a very weak signal in the region =27eV consistent  with the findings of the recent  binary (e,2e) study by Cambi et al. [CC84] as well as with the  appearance of  the  data in an earlier X P S work [SN69]. It should be noted that the  NV  calculation does not predict corresponding  main  pole  although  the  semi-internal  CI  intensity  [AS80],  in the  MRD-CI  SAC-CI  region below the [CC84]  and  Ext.  80 2ph-TDA [VC82] calculations do indicate such intensity.  Other  theoretical  studies  by  Mishra  and  [AM82] also predict some structure both theoretical studies  Ohrn above  [MO80] and  Arneberg  et  and below 30eV. Whereas  attribute all the structure from 25-45eV to the  2a,  al.  most  molecular  orbital, the calculations of Arneberg et al. [AM82] predict appreciable contribution in this region from the 3a,  molecular orbital.  4.3. MOMENTUM DISTRIBUTIONS OF WATER The experimental  momentum  profiles  (XMPs)  and calculated spherically  averaged  MDs are presented together with r-space and p-space density contour maps and three dimensional surface and each  2a,  representations in Figs. 4.3-4.6 for the  molecular orbitals of water,  of the  integrated  obtained  M D s and  Fourier  transformed,  diagrams  various  respectively. The top left hand  presents  theoretical  l b , , 3a,,  a  comparison  M D s obtained  position space wavefunctions.  of the  from  It  experimentally  noted  experimental momentum resolution (0. l a o" ) has been folded into the 1  MDs  shown  in the  figure.  All XMPs  were  obtained  at  the  2  section in  spherically  should be  lb  averaged, that  the  theoretical  indicated  "sitting"  binding energies (see Fig. 4.1).  Two types of literature comparisons spherically  with averaged  wavefunctions,  experiment  [HH77,  M D s . These  representative DD77]  wavefunctions  of those used in the  have are  been  used  to  those of STO type  earlier  calculate reported  by Aung, Pitzer and Chan (APC) [AP68] and the G T O functions of Snyder and Basch (SB) [SB72]. A third wavefunction, namely that reported by Neumann and  81 Moskowitz  (NM) [NM68],  has  also been  used.  The theoretical  M D s have  been  height normalized in each case to the respective X M P s . Differences between  area  normalization and height normalization are insignificant (see discussion below).  To provide a  more  between  X M P and  each  plots  are  The  distribution  critical  presented  view  the  in the  of the  respective  lower left  difference  is  obtained  regions  of discrepancies  theoretical  hand by  or  agreement  M D s , distribution  section of each subtracting  difference  integrated  the  diagram.  M D (normalized  above) calculated from the best fitting wavefunction (APC) from the  as  X M P . The  distribution difference is on the same intensity scale as the respective M D shown immediately  above.  The  actual  value  of  the  distribution  difference  dependent upon how the theoretical M D s are normalized. The other  is  slightly  limit of the  hatched area represents the zero position if area normalization (in the  momentum  region  should be  0 to  l.5a " )  noted that the  1  0  was  used  shape of the  instead  of height  distribution difference  normalization. It  is largely independent  of the  normalization method.  For the respective valence orbitals the two dimensional (2D) density contour and the three dimensional (3D) boundary surfaces are shown in both momentum  maps  (at three values of the density)  and position space in the center  and right hand  sections respectively of Figs. 4.3-4.6. The double-zeta (DZ) quality wavefunction of Snyder and Basch [SB72] has been used to generate the and  surfaces  shown  in  Figs.  4.3-4.6. t  The  DZ  (more complex) maps  quality  wavefunctions  are  t The r-space coordinates (x,y,z) used are (0,0,0) and (0, ± 1.430456, 1.107118) for the oxygen atom and the hydrogen atoms, respectively. A l l numbers are in atomic units.  82 considered to be sufficiently accurate between  structural  triatomic  system,  somewhat  better  outermost  two  surfaces  features H 0. 2  in  While  agreement  to provide a qualitative working comparison  momentum the  sophisticated  for  the  bent  A P C wavefunction  gives  maps  and  would not be expected to differ drastically from those which have  been  below),  experimental  space  the  (see  the  position  for  orbitals  with  more  and  the  M D s , particularly  essential  features  of  the  obtained using the SB wavefunction. This is done as part of a continuing effort to  contribute  chemical  to  a  concepts  conventions  are  clearer in  understanding  atoms  similar  to  and  those  and  visualisation of momentum  molecules. used  in  The  earlier  mapping  work  space  procedures  [LB83a].  The  and  contour  values of all maps are 0.2, 0.4, 0.6, 0.8, 2, 4, 6, 8, 20, 40, 60 and 80% of the unit  maximum density value. The plane vectors  p-space.  spanning  the  of the  range -5.0 to  To emphasize the details of the  5.0  contour  map is defined by two  atomic units  2D density maps,  in both r-space  and  line projection plots  (along direction shown by dotted lines) parallel to the two axes are also included.  4.4.  COMPARISON  OF EXPERIMENTAL  AND  THEORETICAL  MOMENTUM  DISTRIBUTIONS The  characteristics  presented  in  Table  of 4.2.  the  literature  The  large  wavefunctions  number  used  in  this  study  of published wavefunctions  for  are the  water molecule precludes an exhaustive comparison of experiment and theory. The three wavefunctions used in the present work provide a reasonable  range in the  quality of wavefunctions for comparison with both the present results as well those of earlier experiments  as  [HH77, DD77]. Comparison of the X M P s with those  calculated from more sophisticated wavefunctions  for H 0 2  such as those reported  Table  4.2.  Comparison  of  Theoretical APC  a  SCF Wavefunctions  b  f o r Water  NM°  SB  EXPERIMENTAL  d  Bas i s  STO(IV) (3s3p1d/2s1p)  GTO [532/21]  GTO [42/2]  --  Energy(a.u.)  -76.00477  -76.044  -76.0035  -76.4376® (-76.067)  Dipole  moment(D)  The b a s i s n o t a t i o n b  ° d  2 .092  2.035 follows  Ref.[AP68]. See Ref.[DP72]  Ref.[NM68]. Contracted  basts sets are  1 .8546  2 .681 denoted  in  f  9  square b r a c k e t s [ ] .  for corrections.  Ref.[NM68] Ref.[SB72]. E s s e n t i a l l y of Total  experimental energy  f  Estimated Hartree-Fock SCF  9  Ref.[SC78].  DZ q u a l i t y . with r e l a t i v i s t i c and limit.  mass c o r r e c t i o n s .  Ref.[RS75].  Ref.[RS75].  oo co  84 by Rosenberg and Shavitt [RS75] and Davidson and Feller [DF84] is reported in Chapter 5.  In  general,  quality with  the  spherically  wavefunction  the  averaged  of Snyder  present experiment  and  for  M D s predicted Basch  the  two  from  [SB 7 2] are outer  findings  are  consistent  with  the  conclusions  not  valence  4.4) but a reasonable description is provided for the l b  the  essentially  in good  orbitals  reached  in  agreement  (Figs. 4.3  and 2 a ,  2  DZ  and  orbitals. These  earlier  work  at  low  momentum resolution [HH77, DD77]. This is not surprising since the variationally inferior SB wavefunction also gives poor agreement for molecular properties  such  as the dipole moment (see Table 4.2). In the case of the atomic M D s of A r , K r and  Xe it has  been  reported  [LB83]  that DZ quality wavefunctions  give M D s  very close to those calculated using Hartree-Fock quality wavefunctions. it should be noted  that this is evidently not the  However  case for Ne [LB83] which is  isoelectronic with H 0. 2  Consider now the  comparison with experiment  calculated from the wavefunctions different  (see  types  uses Gaussian APC  and  experimental  of basis  NM  spherically averaged M D s  variationally superior APC-Basis I V [AP68] and N M [NM68] Table 4.2).  type  of the  It  should be noted  functions  (APC uses  orbitals). For the  wavefunctions  predict  M D . However for the  Slater  case of the MDs  that these wavefunctions  in  two outermost  type lb  2  orbitals  whereas N M  orbital (Fig. 4.5),  close orbitals  agreement (lb,  use  and  with 3a,)  both the there  is a large discrepancy between  experiment and theory as can be seen clearly in  the  (Figs. 4.3 and  distribution difference  plots  4.4). In this connection it should  85 be  remembered  wavefunction  that  for  the  which  distribution  the  difference  discrepancies  are  shown  is  for  the  APC  smallest.  In  particular,  the  calculations seriously underestimate the low momentum region (<0.5a ~ ) for the 1  0  two outermost orbitals. Furthermore, the observed l b ,  and 3a,  asymmetric,  with  at  momentum,  than  their  respective  those  maxima  predicted  by  the  situated  XMPs are more  lower  respective  values  calculated  of  the  momentum  distributions. This contrasts with the close agreement observed between experiment and theory for the conclusions  of  the  lb  2  orbital  earlier  work  (Fig. 4.5). [HH77,  These results  DD77]  confirm the  carried  out  at  general  much lower  resolution.  Experimental  momentum profiles  in the  2a,  inner  valence  region  have  been  determined at the band maximum (32.2eV) and also on the shoulder at 35.6eV. These results are presented in Fig. 4.6. energy  The relative intensities  of the binding  spectra at the azimuthal angles of 0° and 8° (Fig. 4.1)  have already  suggested a dominant "s-type" character for the ionization strength above 28eV. This is confirmed, at least at 32.2 and 35.6eV, by the two MDs shown in Fig. 4.6.  It  can  be  seen  that  all  three  wavefunctions  reasonable agreement with experiment (at least below and 35.6eV.  However the  give  calculated  l.5a ~ )  shape of the APC MD has  than that predicted by both SB and N M wavefunctions.  1  0  MDs in  at both 32.2eV  a narrower half width This may reflect the  different types of basis functions used. In momentum space, Slater type orbitals tend to decay faster than Gaussian type orbitals. The reverse is true in r-space due to the Fourier Transform (FT) relation.  86 Earlier  measurements  theoretical features  work  above  measurements  of  [CC84,  binding NT82,  — 25eV are confirm this  energy  VC82,  spectra  AS80,  MO80]  principally due to assignment  at  [DD77,  2a,  CC84]  have  as  well  as  all suggested  that  ionization. The present E M S  35.6eV. This suggests  that  there are considerable correlation effects the main features of the 2 a ,  although  orbital are  preserved at 35.6eV as, is the case for the main pole at 32.2eV. This suggests that the dominant contribution to the many-body structure state  configuration  mentioned  earlier,  interaction the  associated  binding energy  with  spectra  the  results from final ionic  2a,  of water  hole.  As  in the  has  been  region 23-45eV  clearly indicate a breakdown of the simple M O picture for the ionization process.  The  present  spectrometer  results  obtained  with  binary  (e,2e)  confirm the existence of a significant discrepancy between the  XMPs  and those calculated using near for the these  lb,  two  and  3a,  orbitals  Considering  the  pronounced towards  high  momentum  show  more  intensity  F T relation this  in turn  at  indicates  (—  [AP68,  NM68]  In particular, the X M P s  low momentum that  than  is  for  predicted.  these orbitals  may  be  [HH77] than would be predicted by a near Hartree-Fock  The discrepancy is greatest for the shift  resolution  Hartree-Fock S C F wavefunctions  molecular orbitals of water.  more spatially extended wavefunction.  a  in  0.2a ~^) Q  low momentum  the  relative  to  position theory.  lb, of  The  the case  orbital in that there is a maximum for  the  of  the X M P  3a,  orbital is  similar although the shift ( = 0 . l a " ) in the maximum of the X M P is less than 1  0  that  in  the  momentum  case  of  region  is  experimental error.  the  lb,  orbital.  comparably  large  However, and  the  these  discrepancy effects  are  in far  the  low  outside  87 Several  possibilities exist for  experimental  momentum  understanding  profiles  and  the  the  significant discrepancies  theoretical  MDs  for  the  between outermost  orbitals of water. These differences can be atttributed to either limitations in the basic  (e,2e)  effects  as  agreement  theory  or  a  source  has  been  to of  an  inadequate  this  obtained  wavefunction.  discrepancy between  can  be  experiment  noble gas atoms [LB83] using the same  Anomalous  discounted  and  instrumental  since  Hartree-Fock  excellent  theory  for  spectrometer.  In terms of possible breakdowns of basic (e,2e) theory the various approximations to the  theoretical formulation of the  (e,2e) reaction model should be considered.  These are as follows: (1) the Plane Wave Impulse Approximation (PWIA); (2) the Target Hartree-Fock Approximation (THFA); (3) the neglect of vibrational and geometry  effects;  (4) the flexibility of the basis set and wavefunction quality.  The validity of the P W I A has been, the subject of intense investigations in early (e,2e)  work  [HM73,  HM74,  WH75].  A t high impact  energies  (>>400eV)  the  P W I A is found to be generally valid for atoms and molecules in that the shapes of  the  energy. orbitals,  measured In fact, such  as  M D s have the the  PWIA H 0 2  been  found  to  should be most l b , , since  they  be  independent  accurate have  for the  the  of electron  impact  outermost  valence  lowest binding  However in the case of water the worst agreement is obtained for the orbitals.  Recent  studies  on  the  distorted  which attempt to account for the effects  wave of the  impulse  energies. outermost  approximation  [GF80]  potential on the motion of the  88 scattered  electrons  have revealed some qualitative changes  but only in the high momentum region. For p < 1 . 5 a o considered to be valid. Perhaps reaction theory is the hydrogen  atom  the  in the predicted M D s the P W I A is generally  _ 1  best confirmation of the basic P W I A  fact that the measured  electron M D of the  [LW81] is in excellent agreement with the  (e,2e)  ground state  square of the  exact  solution of the Schro dinger equation.  As  has  (e,2e)  been  shown in Chapter  differential  interactions  are  cross  2, simple analytic results  section  neglected.  if  initial  Within  the  state  THFA  and the  are  final  obtained for  state  electronic  the  configuration  overlap  function  reduces to the momentum wavefunction of the characteristic orbital. In general, it has  been  systems. ground  assumed In  the  that ground  case  state correlations  of H 0  it has  2  state correlation effects  are  been  are  stated  involved. Final  negligible for closed shell [NT82] that  final  attempt  to  amplitude GOA by  include  such  effects  has  (GOA) method  applied  to  method  the  orbital  fact set.  state CI on the MDs of water  accounts that  the  been water  Interestingly,  this  are  study  are investigated in Chapter 5. A n made by  in  Williams  the  generalized  et al.  developed from the  [WM77a] showed  that  THFA lb,  using the  G O A method  Dunning wavefunction  orbital still persists.  as  against  overlap  [WM77a].  The  but is limited  H F ground a  slightly  agreement between calculation and experiment for the M D s of the 3 a , orbitals was obtained with the  thus  of including initial state  for some correlation and relaxation effects approximations  significant  state correlation effects  seem likely to be a more important factor. The effects CI and  no  state better  and l b  2  the  simple use  of the  [D71]. However the  discrepancy  in  the  89 The  discrepancy  orbitals effect  may of  between  therefore  electron  experiment indicate  correlation  and theory  the  can  need  be  for  quite  in the  case  of the  including correlation  significant.  For  outermost  effects.  example,  The  the  best  Hartree-Fock estimate of the dipole moment of water is 7.6% higher [RS75] than the  experimental  value  and  only  with  post-HF  methods  such  as  4th-order  Many-Body Perturbation Theory (1.2%error)  [DK83] and Multireference Single and  Double  can  Excitation  CI  (4.0%error)  [DF84]  more  reasonable  agreement  be  obtained.  Another factor that might be considered to affect the  question  of geometry  change  the  accompanying the  +  2  states.  The  112.5°  calculations  and  180°  show  for  that  the  2  B,  THFA  formation of the  state. S C F calculations [M71, SJ75] and experiments equilibrium geometries for the H 0  validity of the  final  ionic  [L76] have shown that the  ion are different for each of the electronic the  H O H angle  Ob," )  and  1  increases 2  from  ( 3 a  A,  104.5° 1  )  1  not reduce the  final  +  2  2  to the  to  states,  respectively. On the other hand, the H O H angle decreases to 58° for the ( 1 b ' ) state of the H O  is  2  B  2  ion. As a consequence the overlap amplitude does  characteristic  orbital since the  populated  molecular orbitals of  ionic state are not necessarily orthogonal to the  corresponding orbitals  in the neutral as is assumed in the T H F A model. However geometry effects not expected  to be significant because of the  very short collision times  are  (due to  high impact energy) relative to the nuclear motion.  For  meaningful  comparisons,  it  should  also  be  pointed  out  that  the  effect  vibrational motion on the theoretical M D should be considered. Vibrational  of  effects  90 on the  Compton profile of H  2  have been  estimated  to be small (1.0%)  [UB72]  and only at low momentum. Quite similar results have been obtained by Palalikit and Shavitt [PS85] on zero-point vibrational corrections to the Compton profile of water. However, the greater sensitivity of experimental momentum profiles to the details of the wavefunction compared to the total momentum density measured by Compton  scattering  [SW75]  are  well  known  [LB83a].  In  addition,  one  would  expect considerable vibrational effects on the M D of molecules of light atoms and weak  force  Theoretical revealed  constants  and  calculations that  also  [KE74]  Hartree-Fock  experimental force constants, potential energy water  surface.  involves various  vibrational  effects  if the on  force  potential  plus the  modes  detailed  greater  have  true  anharmonic  generally  of water  sometimes by up to 20% indicating a "flatter" factors  are  surface  [T72].  the  any  constants  energy  anharmonic  than  These  in  the  vibrational motion is  fact that vibrational motion of  suggest theoretical  a need  to  seriously consider  prediction  of  momentum  distributions, t  Lastly, with has  it is interesting  the  A P C wavefunction  a superior  S C F energy.  any simple manner made  by  properties  to note that better  Kern of  the  than  with  the  agreement (see N M wavefunction  Fig. 4.3) is found though  Karplus  water  latter  The theoretical M D does not seem to converge in  with improvement in energy. Similar observations  and  the  [KK72]  molecule.  This  in  their  again  survey emphasizes  of  certain the  fact  have  been  calculated that  the  t Preliminary calculations for H 0 by Leung and Langhoff [LL87] using a well known potential energy surface [H66] indicate that vibrational effects (for symmetric vibration) on M D s are very slight. In particular, the effects on the M D s of the non-bonding l b , (out-of-plane) and 3 a , (bonding) orbitals are found to be too small to explain the observed discrepancies. 2  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  °  0.0  0.5 1.0 1.5 2.0 Momentum (a.u.)  Fig. 4.3  MOMENTUM DENSITY  POSITION DENSITY  2.5  Spherically averaged momentum distribution (upper left) and distribution difference (lower left) for the l b , orbital of H O. The theoretical M D s are height normalized to the maximum of the X M P . The distribution difference is evaluated by taking the difference between the X M P and the theoretical M D calculated from the A P C wavefunction. The upper level of the hatched area indicates zero distribution difference in the case where area normalization is used. Two dimensional density contour maps and three dimensional density surface plots in momentum space (center) and position space (right) are evaluated using the SB wavefunction. 2  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  POSITION DENSITY  MOMENTUM DENSITY  H 0 3a | 2  o o  0.0  0.5 1.0 1.5 2.0 Momentum (a.u.)  2.5  0.0  0.5 1.0 1.5 2.0 Momentum (a.u.)  2.5  Fig. 4.4  -4.0  -2.0 0.0  2.0  4.0  0.5 1.0  -4.0  -2.0  0.0  *  • * * — .i . .a. , 2.0 4.0 0.S 10  Spherically averaged momentum distribution (upper left) and distribution difference (lower left) for the 3a, orbital of H O. The theoretical MDs are height normalized to the maximum of the XMP. The distribution difference is evaluated by taking the difference between the XMP and the theoretical MD calculated from the APC wavefunction. The upper level of the hatched area indicates zero distribution difference in the case where area normalization is used. Two dimensional density contour maps and three dimensional density surface plots in momentum space (center) and position space (right) are evaluated using the SB wavefunction. 2  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  MOMENTUM DENSITY  POSITION DENSITY H 0 2  1b  2  T  1  o o  (0.1.0)  0.0  0.5 1.0 1.5 Momentum (a.u.)  2.0  '2.5  0.0  0.5 1.0 1.5 2.0 Momentum (a.u.)  2.5  Fig.  4.5  -4.0  -2.0  0.0  2.0  4.0  0.5 1.0  -4.0 - 2 . 0  i—t-—i  0.0  2.0  Spherically averaged momentum distribution (upper left) and distribution difference (lower left) for the l b orbital of H O. The theoretical M D s are height normalized to the maximum of the X M P . The distribution difference is evaluated by taking the difference between the X M P and the theoretical M D calculated from the A P C wavefunction. The lower level of the hatched area indicates zero distribution difference in the case where area normalization is used. Two dimensional density contour maps and three dimensional density surface plots in momentum space (center) and position space (right) are evaluated using the SB wavefunction. 2  2  i  i  4.0  0.5  1.0  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION \  POSITION DENSITY  MOMENTUM DENSITY  xi  (32. 2eV)  $ x3  (35.6eV)  1  H 0 2  2a,  APC NM SB  T  0.0  0.5 1.0 d.5 "^.o' Momentum (a.u.) •  T  O c  0* (0,1.0)  \ a -4.0  {2.5  -2.0  0.0  2.0  4.0  0.5  1.0  -40  -2.0  —1—1—1—— *  00  2.0  1  Expt-Theor y  C V  Q  0.0  0.5  1.0  1.5  2.0  2.5  Momentum (a.u.)  Fig. 4.6  Spherically averaged momentum distribution (upper left) of the 2a, orbital of H 0 measured at 32.2eV (solid circle) and 35.6eV (open square). Distribution difference plot (lower left) is evaluated using the X M P at 32.2eV and the theoretical M D calculated from the A P C wavefunction. The lower level of the hatched area indicates zero distribution difference in the case where area normalization is used. Two dimensional density contour maps and three dimensional density surface plots in momentum space (center) and position space (right) are evaluated using the SB wavefunction. 2  4.0  0.5 10  95 variational principle is of itself an insufficient condition for providing a completely adequate  wavefunction.  This  is  particularly  because the discrepancy between experiment  noteworthy  in  the  present  and theory is found in the  work  outermost  orbitals which contribute least to the total energy.  In  summary,  theoretical  there  a  need  for  a  M D s calculated  from  a  more  wavefunctions Furthermore, considered.  is  ranging  from  geometry  This will  class of wavefunctions  and  comparison  Hartree-Fock  a meaningful  ORBITAL  The  analysis  and  momentum  DENSITY  MAPS  of orbital density space  to  and  post  alluded to  evaluation  (e,2e) results  comprehensive  with set  of  Hartree-Fock  quality.  earlier  also  of the  should  be  limitations of each  or the Hartree-Fock model itself in predicting experimental  momentum distributions. These questions  4.5.  sophisticated  vibrational effects  enable  of binary  is  AND  in the following chapter.  SURFACES  maps  facilitated  are addressed  by  (Figs. 4.3-4.6) in complementary the  now  position  well-known momentum  space  concepts [ET77]. These concepts can be summarized into: (a) symmetry preservation  with the addition of inversion symmetry in momentum  space; (b) inverse spatial reversal; (c) molecular density directional reversal; (d) molecular density oscillations which are the manifestation in momentum  of nuclear  geometry  space.  The r-space and p-space density maps and surfaces  of the  lb,  orbital (Fig. 4.3)  96 show  the  dominant  out-of-plane  oxygen  2p  character  x  of  (100) direction in r-space) therefore  this  orbital  (x-axis  is  it is normally referred  in  the  to as  a  non-bonding molecular orbital. The well defined nodal plane in the r-space map is retained of  in the corresponding p-space map (symmetry preservation  near-HF  wavefunctions  perturbations  which  have  in this orbital induced by the  study  of the  basis  set  water  shows  The difference  r-space density  between  the  present study  however  and the  addition of inversion symmetry (Figs.  introduce  Such a  al. [DP72] using a  maps generated from the  STO basis  orbitals  atoms.  would  theoretical  (541/31)  maps indicating the slight perturbative  (541/31)  all  functions  hydrogen  molecule by Dunning et  (essentially DZ) used in the  in  polarization  property). Use  is  very  4.3-4.6),  effects  SB [42/2] GTO basis  . set  and those maps calculated using the  minimal.  the  STO  Whereas  in momentum  concepts  of  symmetry  space are  inverse  preservation  quite  spatial  apparent  reversal  and  molecular density directional reversal cannot be readily applied in the case of the 3a,  and  general  lb  2  orbitals . The non-linear  Fourier  Transform  properties  situation for diatomics such as H CS  2  and  OCS  approximation  [LB85b]  [ES55]  2  which  bonding  in  geometry less  (Fig.  2  makes  straightforward  assignment  compared  to  of the  [LB83a] and linear polyatomics such as C 0 , 2  are  of  H 0 2  higher is  molecular orbitals. The bonding characteristics in r-space in terms of electron  of H 0  density  symmetry.  attributed of the  to  3a,  derealization across  4.4). In p-space bonding is manifested  the  In  the 3a,  orbital are the  simplest and  lb  2  manifested  hydrogen  atoms  by the expansion of the density in  the (001) direction. The r-space density map and surface of the  lb  2  orbital (Fig.  4.5) illustrates the familiar simple intuitive view of covalent bonding which is an accumulation of electron density along the bond direction. In contrast,  bonding is  97 not obvious in the corresponding p-space density map and surface.  In  a  more  exact  treatment, there is also some bonding attributed  to the  2a,  orbital as is clearly seen in the r-space density map in Fig. 4.6. Nevertheless it can be  seen  that the  2a,  orbital is predominantly an oxygen 2s orbital. The  contracted density in r-space an  expanded  (as seen in the  density in p-space  line projection plots) translates  in agreement  with the  to  inverse spatial reversal  property.  4.6.  WIDE RANGE  It is of interest coordinate  MOMENTUM  DENSITY  MAPS  to investigate the nature of momentum density maps  values  than  in  Figs.  4.3-4.6.  This  is  particularly  at larger  important  since  effects  due to molecular geometry will be manifested at large p as approach is  made  in  the  direction  of  the  virtual  p-space  boundary  [LB83a].  Wide  range  density maps up to 16 a.u. in momentum space are shown for the four valence orbitals  in F i g . 4.7. For comparison their  position space  counterparts  are  also  shown alongside. The nuclear geometry information of course appears directly in the  r-space  manifested These  in a  appear  [ET77] as the  density  lb,  at  maps  at  small  quite  different  form,  large  momentum  as  shown in Fig. 4.7 for the  r.  However  namely  as  in  the  p-space  modulations or oscillations in the 3a,,  lb  2  it  is  molecular density oscillations.  and  2a,  orbitals. In  density contrast,  orbital, which is essentially non-bonding and thus contains little or no  nuclear information, does not show such modulations (at least Detailed computational studies Rozendaal  maps  and  Baerends  of these modulations have been  [RB84].  However no  such  study  has  out to made  16  for F  a.u.). 2  by  previously been  MOMENTUM DENSITY  -12 B -f.4  Fig. 4.7  0.0  (.4  MOMENTUM DENSITY  OJt -». -J.  -12.8 -6 4  0.0 6.4  POSITION DENSITY  08  -6. -J.  -4.0 -2.0 0.0  POSITION DENSITY  2.0  4.0 OS 10  -4.0 -JO  00  1.0 4.0 OS 10  Wide range momentum density contour maps (left hand side) for the valence orbitals of H O calculated using the S B wavefunction. The 21 contour values are in seven decades ie. 0.00002, 0.00005, 0.00008, etc. up to 20, 50 and 80% of the maximum density. The projection plots on both primary axes are on a logarithmic scale. A l l dimensions are in atomic units. 2  CD 00  99 done on systems with lower symmetry such as water. The observed modulations are not as pronounced as those calculated for other diatomic molecules such H  [LB83a] and polyatomic systems  2  [LB85a]  and  CF  4  [LB84].  The  such as modest  C0  2  [LB85c], C S  modulations  for  2  H 0 2  as  [LB85a], COS may  be  a  consequence of the non-linear geometry as well as the low atomic number of the hydrogen atoms. The widely differing electron densities on the nuclear centers do not allow the effective interatomic coupling needed for strong bond oscillations. It has been shown that these modulations or oscillations occur with periodicities in multiples of 7T/R where R is the internuclear separation. However in the case of H 0 2  the  periodicities of the  straightforward  manner  modulations  do  not  seem  to  correspond  to integral multiples of n7R, where R is the  in  any  O-H bond  distance (1.80a.u.), in contrast to the situation observed for diatomics and linear polyatomic systems [LB85a, LB85b, LB85c].  C H A P T E R  5.1.  W A T E R :  5.  P A R T  II  OVERVIEW  In the previous chapter the following considerations were raised as being possible sources of the observed discrepancies between  the theoretical M D s and the  XMPs  of H 0 . 2  (1) Inadequacies for  in the plane wave impulse approximation (PWIA) and the  need  distorted wave treatments (DWIA);  (2) Inadequacies the  need  to  treatments(ie.  in the target Hartree-Fock approximation (THFA) consider  sufficiently  complete  target  and  molecule-final  therefore  ion  overlap  adequate treatment of relaxation and correlation);  (3) Neglect of nuclear motion, i.e.vibrational effects; (4)  Insufficient  flexibility  of  the  basis  set  and  resultant  deficiencies  in  the  theoretical wavefunction.  These and  matters concern both the the  accuracy  of  inaccuracies in the profiles  and  the  the  of the  wavefunctions.  assumed orbital  adequacy  theory  Explanations  proportionality between  momentum  distribution  the  of E M S cross 1-3 emphasize experimental  evaluated  at  the  sections possible  momentum equilibrium  geometry of the neutral molecule. Items 2 and 4 are of particular concern from the quantum in  the  present work. A further  standpoint effects  chemical theoretical standpoint  is the  however  area  and these are  investigated in detail  of possible concern from an  experimental  accuracy with which the momentum resolution is known. Such will  be  small at  higher  present experimental work) compared  momentum  to the  100  observed  resolution (as discrepancies  used  in  the  between  the  101 measured X M P s and the calculated M D s in the case of H 0 . 2  The validity of the P W I A for the present experimental conditions (impact energy E  0  = 1200eV, E = E = 6 0 0 e V , 1  previous chapter between  9  2  and has  E M S experiment  =  45°, p < 2 a " ) 1  0  been clearly demonstrated and  theory  for  atomic  has  been discussed in the  by the  complete  hydrogen  agreement  [LW81]  where  the  T H F A is not involved and the wavefunction is exact. There is thus good reason to use  the  PWIA  for other  atomic and molecular targets under  the  equivalent  conditions. The combined use of the P W I A for the kinematic factor together with use  of  the  THFA  for  the  electronic  structure  factor  in  the  description  interpretation of the binary (e,2e) reaction as studied by E M S is also by  the  excellent results  CM86] and metal atoms  obtained  for  the  valence orbitals  of noble  supported  gas  [FW82] and also for small molecules [LB83a,  and  [LB83, WM77].  In these cases where high quality Hartree-Fock limit wavefunctions were used it is clear that the shapes  of  the  combined use of these two approximations measured  XMPs  were  extremely  closely  is valid  since  the  reproduced  by  the  calculated orbital momentum distributions at least for electron momenta less than =2a "  for targets lighter than A r (Z=18).t  1  0  In  the  case of Ne  (Z=10),  which  is isoelectronic with  H 0, 2  the  PWIA  was  satisfactory [LB83] in describing the shapes of the momentum distributions out to p — 2a '^. 0  in the  In the region of higher electron momentum (i.e. larger than that used  present work for H 0 ) considerable distortion of the 2  electron waves  can  t Experiments [LB83, CM86] indicated that distortion becomes more important at lower momentum as the orbital energy and Z increase. For example even for targets as large as Xe (Z = 54) use of the PWIA was found [LB83, CM86] to be adequate for the valence orbitals out to l . 5 a " at 1200eV impact energy. 1  0  102 occur and the  distorted  wave impulse approximation  (DWIA) has  been  found  to  be necessary for an adequate interpretation of E M S measurements for example in A r [MW85], Xe [CM86], Ne and K r [LD86]. In the region of p below  =2a "  1  0  the shape of the X M P for the noble gases was equally well reproduced by either the  DWIA + T H F A  molecules have  or  PWIA + T H F A .  not been reported  Thus  far  due to the  distorted  wave  complex multicenter  treatments  for  nature of the  problem.  In  some  molecules  Hartree-Fock profiles  for  limit some  it  has  been  wavefunctions orbitals.  found  fail  This is  to most  that  THFA  reproduce likely  calculations  the  due  to  using  even  experimental  momentum  the  neglect of  inherent  electron correlation and relaxation in the  Hartree-Fock description. In such cases  calculation of the  overlap using CI wavefunctions  full  target ion-neutral  used to include correlation and relaxation effects. Overlap Amplitude (GOA) method) the  target  [BH80]  molecule-final  using  limited  ion  basis  be  A n earlier attempt (Generalized  by Williams et al. [WM77a] at evaluation of  state set  can  overlap  for  wavefunctions  H 0  [WM77a],  2  indicated  HF  small changes  and  HC1  (towards  lower momentum) from predictions based on the T H F A . However more  exhaustive  treatments  investigate  of overlap,  using  more completely the effects theoretical quantum account  ground  improved  wavefunctions  final  needed  to  of correlation and relaxation. From the standpoint of  chemistry it is important  and/or  are  state  electron  to assess the need for taking into correlation  effects  since  these  have  certainly been found to be crucial in reconciling theory and E M S experiments the  case of N O [BC82] as well  cases where the product ions of H  as 2  for  H  2  in  [WM77] and He [CM84] in those  and He are left in excited states.  103 As  has  been  discussed  shown  by  McCarthy  wavefunctions profiles  for  earlier  by  Weigold  [M85],  and  additional  co-workers quantitative  and the reaction model is possible if the the  scale t  with the  taken  into  valence full  orbitals  are  [DD77,  obtained  on  a  CM86]  assessment experimental  common  and  of  the  momentum  (relative)  intensity  Franck-Condon width of each final ion electronic state being  consideration  in  normalization of experiment  the  normalization  and theory  procedure.  With  a  at only one value of the  single  point  momentum  on  a given calculated orbital M D or O V D a stringent quantitative test of theory can be made at  all other  This procedure  experimental  data points  provides very much more  height or area normalization of theory which has previous  frequently  and experiment  basis set  optimization  Hartree-Fock  and the  accuracy of the  individual  (as for example  in the  energy  SCF  Hartree-Fock  are  given, is a necessary  properties,  functions  wavefunctions,  that all other as  is usually necessary  critically  on  are  important  even but  when  energies  near  the  usually insufficient condition  calculated properties  are  also close to  the  is well known [BC82, GC78]. The addition of more  such as the dipole and quadrupole depend  wavefunction  The usual variational treatment involving  of  alone for guaranteeing  which  the  for each separate orbital  been used in earlier E M S studies  concerns from the theoretical standpoint.  diffuse  specific information than  chapter).  Flexibility of the  energy  and calculations for all orbitals.  the  to permit  moments accurate  accurate calculation of  (and therefore modeling  of  the  properties  also likely the MDs) long  range  charge  t True absolute E M S measurements are very difficult to perform with high accuracy. Such measurements have for example been reported (±20%) for He and H [VK81]. 2  104 distribution. The total energy is, of course, not very sensitive to such details so that basis  sets designed  without unnecessary properties.  solely to yield a reasonable  computational expense  Further  considerations  value of the  total  may well give poor results  in addition to  the  total  energy  energy  for  are  other  therefore  necessary to ensure sufficient wavefunction accuracy for general applications where theoretical  quantum  chemistry  is  required  to  give  accurate  predictions  of  experimental observables. Such considerations are also of importance for a number of properties  including those dependent on the  long range distribution of charge  density.  After  convergence  energy  has  been  obtained  at  the  Hartree-Fock  level  and the predicted M D then the other possible sources  for  both  total  of any remaining  discrepancy between the calculated orbital M D s and the X M P s measured by E M S can  be investigated. As mentioned above the need to go beyond the T H F A  to incorporate correlation and relaxation effects) of  suitable  CI  wavefunctions  in  can be investigated with the  order  to  evaluate  mind  new  higher  the  ion-neutral  (i.e. use  overlap  amplitude with sufficient accuracy.  With  the  above  considerations  in  measurements of the momentum profile for the  lb,  statistical  precision E M S  orbital of H 0 2  as well  as  new momentum profile measurements for the three outer valence orbitals of D 0 2  were  made.  These  experimental data  new  measurements taken  [BL85] (see  have been placed on the on  a quantitative  basis  chapter  together  4) for the  same (relative) intensity  with T H F A  with  the  earlier  reported  four valence orbitals of H 0 2  scale and are  (MDs) and full  now compared  overlap calculations (OVDs)  105 using very accurate Hartree-Fock limit and CI wavefunctions, respectively. These include the already published 84-GTO wavefunction of Davidson and Feller [DF84] as  well  as  the  39-STO  wavefunction of Rosenberg and  Shavitt [RS75].  These  best existing literature wavefunctions [DF84, RS75] have been improved upon in the  present  work  wavefunctions  for  by the  use  of  THFA  previously  calculations.  unreported These  water, which give essentially converged resultst  99-GTO  new  and  109-GTO  S C F wavefunctions  for  for total energy, dipole moment  and M D are considered to be the most accurate to date.  Comparisons of theoretical M D s calculated from essentially converged Hartree-Fock quality  wavefunctions the  are  important  independence  of  calculated  minimisation.  Comparative studies  in order  M D s has of basis  to  been  determine achieved  sets for the  whether as  water  well  basis as  set  energy  molecule [DF84,  RS75] have shown that certain position space one-electron properties such as the dipole and quadrupole moments  are  much more sensitive to the basis set  than  the total energy.  Most theoretical efforts to date have been directed towards assessing the effect of "basis  set  truncation" on position space  much  less  attention  emphasize a  different  given  to  the  properties. There  conjugate  region of phase  space.  momentum Tanner  has  space  in general properties  and Epstein [TE74]  been which have  calculated the Compton profile and momentum expectation values for H 0 using 2  t These results are considered to be effectively converged since the total energy is within 0.5 millihartree of the estimated Hartree-Fock limit [RS75] and because the dipole moment, quadrupole moment and M D do not change significantly as even more basis functions are added. Other basis sets containing up to 140 GTOs were actually tested with no significant change in either the M D or O V D [FB87].  106 various S C F wavefunctions was  observed  wavefunctions difficult.  that  from minimal basis  distinctions  between  of DZ quality or better  However this  is not  to near Hartree-Fock quality. It  Compton  (e.g.  near  profiles  calculated  Hartree-Fock) were  so surprising since the  using  extremely  Compton profile refers  to  the total electron distribution rather than to an individual orbital.  Leung  and  Brion  [LB83a,  LB84a]  momentum distribution of H  have  (iCg)  2  investigated  assessment of the  "p-type" shape, t  set  effects  1 o"g orbital of H  calculated M D s than  affords much  2  M D s of  The maximum at non-zero momentum afforded by the  "p-type"  therefore  provides a sensitive probe of the basis  problem and the limitations of the Hartree-Fock model in momentum  The most accurate are  present  H 0 basis sets (ie. the  Hartree-Fock  as  99-GTO and  2  converged for both energy  work  the  do orbitals with  molecular orbitals in water  which  on  and found these to be small. However it  should be noted that the "s-type" shape of the less stringent  basis  the  foundations  (independent  particle)  and T H F A  for  more  picture.  109-GTO  space.  wavefunctions,  MDs) have been  accurate  To elucidate  treatments the  set  effect  used in the beyond  the  of correlation  and relaxation, a configuration interaction (CI) treatment (recovering a very large fraction  of the  calculation  of the  wavefunctions calculated  correlation energy)  are  of the  ion-neutral  overlap  generated  using  ion-neutral  overlap  initial  amplitude the  and final in the  highly  distributions  are  extended then  states is used in a  present basis  work. sets  compared  The and  with  CI the the  t A much greater specificity was observed by Leung and Brion [LB84a] for calculations using different basis sets compared to the measured cross section when the difference density ( p - 2 p „ ) was used instead of the density. H  107 experimental momentum profiles.  The  role  of vibrational motion on momentum  distributions  has  yet  to be  investigated but preliminary calculations for the symmetric stretch in H and  N H  for  [BM87]  3  have  indicated  that  the  effects  are  very  valence orbitals and essentially negligible for the non-bonding l b , respectively. Isotopic effects EMS  experiments  experimental investigate different  any  for  the  deuterated  effects  of  XMPs water  nuclear  vibrational frequencies  H  of (D 0)  D  therefore  2  vibration  and  2  on  [BT68] of D 0 2  mass (thus lower vibrational frequency) effects.  Consider  vibrational amplitudes. The and the  In  <(AR  summary  explanations H 0. orbital row  for  this  behavior.  (e.g.  2  the  the  momentum  hydrides  1  then  In turn  2  ) > ^ 2  H H  for  example  the  and H  2  1  the  and l e orbitals,  momentum 2  new  profiles.  D 0  due to its  2  to The the  higher  root  seeks  discrepancies  to  (ie. X M P s  address  between  H F ) which  and  the  various  experiment nature of the  theoretical  have  square  been  D 0 2  [C68].  2  study  mean  0 is 20% greater than  2  should provide insight into the  and  The  0 may reveal within  0 is 20% greater than D 0  2  present  distributions  [DM75].  2  molecular  for H  2  H  for H  3  all  would be expected to exhibit less of the  <(ARQ ) > ^  observed  N H  small for  provides opportunity  experimental sensitivity any vibrationally induced effects.  vibrational  0 [LL87]  2  have been investigated and found to be negligible in  comparing  data  fully  and  theory  outer  MDs) of other  found  to  possible in  valence second  exhibit similar  108 5.2. EXTENDED Several course  new, of  BASIS highly  the  Davidson  extended  present  (Indiana  SETS  FOR basis  work  University).  in  WATER sets for H 0  an  First,  have  2  interactive a  99  been  developed in the  collaboration  basis  function  with  set  Prof. E.R.  (referred  to  as  99-GTO) was constructed from an even-tempered (19s,10p,3d,lf/10s,3p,2d) primitive set  which  was  contracted  to  were  [10s,8p,3d,lf/6s,3p,ld]. The  deleted,  but  the  s-components  p-components  of  the  of  the  cartesian  d-functions  cartesian  f-functions  were retained. The (s,p) portion of the oxygen basis was created by  extending the (18s,9p) atom optimized exponents with one additional set of diffuse functions  of each  type.  The hydrogen s-exponents are  optimal for the isolated  atom.  In  an even-tempered basis [FB87] the i'th exponent is given as  with i = l .  The a  j3 = 2.29663 g  a  s  and /3 values for the  and  a = 0.04956, p  oxygen (18s,9p)  /3 =2.57217. p  The  set  are  hydrogen  af}  1  beginning  a = 0.07029, g  values  are  = 0.02891, / 3 = 2.58878. Exponents for the d- and f-type polarization functions  on  g  oxygen were  3.43,  1.18  partially optimized at  the  SD-CI  level.  The  d exponents  are  and 0.33. The f exponent is 1.20. For hydrogen the p- and d-type  polarization functions were taken from the work of Davidson and Feller [DF84].  The to set  109 basis function set (referred to as  109-GTO), a (23s,12p,3d,lf/10s,3p,2d)  [14s,10p,3d,lf/6s,3p,ld] contraction, was generated through  even-tempered  the  addition of  more  diffuse  restriction. The most diffuse  functions,  by extending the (19s,lOp) while  still  retaining  s function in this set possessed  exponent of 0.0649 compared to 0.1375 in the original set.  the an  Table  5.1  Properties of  Theoretical SCF and CI Wavefunctions  Basts Set  Energy(a.u.)  SB 14  [42/2] CGTO  -76.0035  RHF  APC (331/21) 27 STO  -76.00468  RHF  NM 36  [531/21] CGTO  -76.044  RHF  NM 58  (1062/42) CGTO  -76.059  RHF  RS  39-STO  -76.0642  RHF  DF 84  84-GTO CGTO  -76.06661  RHF  99-GTO 99 CGTO  Wavefunction  Dipole  Moment(0)  Hartree-Fock  limit  Reference [SB72]  5.462  [AP68]  5.307  [NM68]  5.371  [NM68]  1 .995  5.396  [RS75]  2.021  5.417  [DF84]  2 .092  -76.06689  109-GTO 109 CGTO 140-GTO 140 CGTO  2 -16 o < r >(10 cm"*)  5.430  -76.0673  this  work  this  work  [FB87]  -76.0675 a  1,980±0.01  5.432+0.OOI  CI  (84-GT0)CI  -76.3210  1 .929  5.490  this  work  CI  (109-GTO)CI  -76.3761  1 .895  5.500  this  work  CI  (140-GT01CI  -76.3963  1 .870  5.507  1.8546+0.0006*  5.1±07  Experimental * b  c  Ref. [RS75]. Ref. [D87]. Estimated non-vIbrating dipole  -76.437610.0004  a  moment Is 1.848D.  [FB87] 3  O CO  110 While these basis sets, incidentally, give the lowest S C F energy yet reported  for  water  the  at  diffuse  the  basis  (r-space)  experimental function  tail of the  109-GTO results  geometry,  limit  and  to  they  are  give  an  also  designed  improved  orbitals. The negligible difference  to  saturate  representation  between  for a wide variety of calculated properties  the  of  the  99-GTO  and  (Table 5.1)  indicates  that this goal has been essentially accomplished. The neutral molecule equilibrium geometry,  R Q J J = 0.9572A,  C9JJQJJ= 104.52°  was  used  for  Indiana  group  for  generating  all  wavefunctions.  CI calculations were  then  109-GTO  configurations  sets  using  done by the  the  built  from  Based on these small CI calculations, the and  used  in a  multireference  SD-CI  configurations. For example, for the reference  and  11011  wavefunction  X'A, and  set  17316  were  configurations  used. These are  in the  with  final  37  evaluate  the  molecule-ion  1  2  1  were  "overlap"  selection of  CI for the  2  reference  wavefunction  2  1  1  the  neutral molecule  in the  ion X B ,  <X B,|X A >,  selected  15 configurations in the  CI wavefunctions  2  and  SCF MOs.  still small CI calculations which recover only about  correlation energy in H 0 . These  2  energy  configurations  CI for the  estimated  <X B |X A >  final  set,  84-GTO  molecule  configurations  perturbation  in the  Similarly  neutral  important  109-GTO basis  configurations used.  the  the  were  were  83% of the then used to  <X A,|X A >, 2  set  1  1  and  which have the same form as an M O expanded in this basis.  5.3. EXPERIMENTAL  DETAILS:  Triply distilled H 0  NORMALIZATION  and D 0  2  OF  (MSDISOTOPES,  2  DATA  >99.8%  purity), each  degassed  by repeated freeze-thaw cycles, were used for these experiments. The vapour was admitted via a Granville Phillips leak valve to give an ambient pressure of 5 x 10  torr. Lengthy equilibration in the case of D 0 2  prior to data the  chamber  performed  acquisition allowed near  establish both  the  energy  = 1.6eV fwhm and =0.1 S a o ' ,  for the  quite  well  3p orbital of argon were  and momentum resolutions which  accordance with earlier work [LB83], that the  was  hours)  were  respectively. Under these conditions it was found,  1  in  24  complete hydrogen-deuterium exchange on  walls. Standard calibration runs  to  (not less than  described within  the  PWIA  A r 3p momentum distribution  and  THFA  using the Hartree-Fock  limit wavefunction of Clementi and Roetti [CR74].  The measured  momentum profiles (XMPs)  and  three  also  (relative)  the  absolute  outer  valence  intensity  scale  for the  orbitals  (i.e.  of  four valence orbitals of H 0 2  D 0 2  all relative  were  put  on  the  same  normalizations preserved)  by  normalization on the peak areas in angular selected binding energy spectra  (see  chapter  full  4)  [BL85,  BW87].  This  normalization involved  Franck-Condon width for production of each of the (2a,)" *  electronic  1.0:1.1:1.0:2.4 distribution  states  of  H O 2  yielding  a  (lb,)"''', (3a,)"''', relative  ratio  (lb )and  (at  2  0 = 8°)  . In this procedure it was necessary to take into account the  of satellite structure  [BL85]. With  +  integration over the  this procedure  (25-45eV) of the  a very stringent  (2a ,)*^  quantitative  state  (see  of full  F i g . 4.1)  comparison (to better  than 5%) with calculations is possible since experiment and theory are normalized to each other  at only a single point on one of the  four measured  momentum  112 profiles. A l l other experimental and all other calculated points are therefore  open  to quantitative scrutiny.  The  new measurements  of the  momentum  profile for the  lb,  orbital of H 0 2  have been added to the results obtained for this orbital in the earlier reported study [BL85] (see chapter  4). This results in a considerable improvement in the  quality of the data for this particular orbital which showed [BL85] the greatest discrepancy  between  theory  and  experiment  with  the  near  Hartree-Fock  level  wavefunctions used in the calculations in an earlier study [BL85]. This data for the  l b , orbital of H 0 and the existing data [BL85] for the 3a 2  1 ;  lb  2  and 2 a ,  X M P s for H 0 (open squares, Figs. 5.1-5.4) have each been placed on the same 2  relative XMPs  intensity scales using the  procedure  outined above.  Comparisons of the  with calculated M D s and OVDs were made in a quantitative manner  a single point normalization to the M D calculated for the 109-GTO  wavefunction  water  the  at  calculated data  which  has  Hartree-Fock level points  the (see  (including the  most Table  accurate 5.1).  lb  2  orbital using the  calculated  A l l other  109-G(CI) O V D results  by  properties  for  experimental  and  assuming unit pole  strength, see Sec. 5.6) for all four orbitals have their absolute values relative to this single point normalization.  The  measured X M P s for H 0 and D 0 are shown on the same intensity scale 2  2  in Figs. 5.1-5.4 in comparison with a wide range of already reported [BL85] as well as new calculations. Some of the presently reported calculations involve the new,  more  sophisticated  Hartree  Fock  wavefunctions  which  have  only  recently  been published [BB87] (see also Sec. 5.2 above). These wavefunctions are used in  113  T  1  1  1  i  1  1  1  1  1  r  Momentum (a.u.) Fig. 5.1  Detailed comparison of the experimental momentum profiles (XMPs) of the l b , orbital of D O (solid circles)and H O (open squares) with several spherically averaged momentum distributions calculated using the T H F A . Distributions calculated from the (1)SB [42/2], (2)NM [531/42], (3)NM (1062/42), (4)DF 84-GTO, (5)99-GTO, (6)109-GTO and the (6c)109-G(CI) full overlap calculation are placed on a common intensity scale by using a single point normalization of the 109-GTO calculation on the l b X M P (see text for details). The sitting binding energy for the measurements is 12.2eV. 2  2  2  114  T  1  1  1  1  1  1  1  1  1  r  Momentum (a.u.) Fig. 5.2  Detailed comparison of the experimental momentum profiles (XMPs) of the 3a, orbital of D O (solid circles)and H O (open squares) with several spherically averaged momentum distributions calculated using the T H F A . Distributions calculated from the (1)SB [42/2], (2)NM [531/42], (3)NM (1062/42), (4)DF 84-GTO, (5)99-GTO, (6) 109-GTO and the (6c)109-G(CI) full overlap calculation are placed on a common intensity scale by using a single point normalization of the 109-GTO calculation on the l b X M P (see text for details). The sitting binding energy for the measurements is 15.0eV. 2  2  2  115  T  1  1  1  1  1  1  1—~i  1  1  1  1  r  Momentum (a.u.)  Fig. 5.3  Detailed comparison of the experimental momentum profiles (XMPs) of the l b orbital of D O (solid circles)and H 0 (open squares) with several spherically averaged momentum distributions calculated using the T H F A . Distributions calculated from the (1)SB [42/2], (2)NM [531/42], (3)NM (1062/42), (4)DF 84-GTO, (5)99-GTO, (6)109-GTO and the (6c)109-G(CI) full overlap calculation are placed on a common intensity scale by using a single point normalization of the 109-GTO calculation on the l b X M P (see text for details). The sitting binding energy for the measurements is 18.6eV. 2  2  2  2  116  0.9 1.2 1.5 Momentum (a.u.) Fig.  5.4  Detailed comparison of the experimental momentum profiles (XMPs) of the 2a, orbital of H O (open squares) with several spherically averaged momentum distributions calculated using the T H F A . Distributions calculated from the (1)SB [42/2], (2)NM [531/42], (3)NM (1062/42), (4)DF 84-GTO, (6) 109-GTO and (6c)109-G(CI) are placed on a common intensity scale by using a single point normalization of the 109-GTO calculation on the lb X M P (see text for details). The sitting binding energy for the measurements is 32.2eV. 2  2  117 the target Hartree-Fock approximation (THFA) to investigate basis set  dependence  and the convergence of calculated properties in addition to the normal variational treatment minimising the  total energy.  Further  calculations model the  effects of  correlation (with up to 88% of the ground state correlation energy recovered) by evaluating the full overlap amplitude (see Eqn. 2.11) using CI wavefunctions for both the initial state target neutral molecule and the final ion state. The various results at the T H F A  level are discussed in the following sections with  reference  to Figs. 5.1-5.4.  5.4.  VIBRATIONAL  It can be difference  seen  EFFECTS from  between  Figs.  the  5.1-5.4  XMPs  that,  of H 0  within and  2  statistics,  D 0.  there  There  2  is no obvious  appear  to  possible slight differences in the high momentum region ( p > l . 5 a " ) 1  0  momentum H 0. 2  profile where  the  results  However these differences  for D 0 2  are  and D  2  2  some  of the  lb,  slightly lower than those for  are at best marginal considering the error bars  involved. In earlier E M S work [DM75] no differences X M P s of H  be  were observed between  which are the only other isotopically substituted  the  molecules  to be studied thus far by E M S . The present results for H 0 and D 0 indicate 2  2  that vibrational effects have no significant effect on orbital X M P s at least in the case  of water  at  the  current  level  of experimental  accuracy.  The  essentially  identical X M P s observed for H 0 and D 0 now extend the earlier conclusions of 2  Dey  et al.  the  more  2  [DM75] concerning the complex  polyatomic  isotopic diatomic molecules, H  water  system  which  antisymmetric modes of vibration. It had been thought between  measured  XMPs  and the  has  both  2  and D , to 2  symmetric  and  [DM75] that discrepancies  calculated M D s might result  either  from  the  118 failure  of  the  Born-Oppenheimer  approximation  or  from  failure  to  consider  accurately the vibrational integral in Eqn. 2.11.  Several  theoretical  molecular  studies  properties.  Compton  have  For  profile of H 0  investigated  example  a  study  [PS85] suggested  2  vibrational of  vibrational  that  the  might be expected to be most pronounced, though low  momentum  39-STO  wavefunction  averaged surface with  region.  over  the  The  Compton profile  [RS75] for SDQ  H 0 2  (singly,  doubly  (see  effects  calculated  corrections  vibrational effect  small (less than  was  calculated  later  discussion)  and  on  quadruply  to  on M D s  1%),  [PS85]  the  in the  using  the  and vibrationally  excited  CI)  potential  of Rosenberg et al. [RE76]. The present E M S results are also consistent a  series  of  studies  [KE74,  EK71]  on  vibrational  corrections  to  the  one-electron properties of H 0 . It was noted [KE74, EK71] that these corrections 2  are small and in the case of the dipole moment, the vibrational correction ranges from  0.3% in the ground state to 5% in the (0,3,0) excited vibrational state. A  recent  study  by Breitenstein et al. [BM86] of vibrational corrections to electron  impact differential cross sections also showed that such effects are small.  Intuitively it would seem that even if vibrational effects the  effect would be least on the  (non-bonding)  lb,  on M D s are significant,  orbital since it is effectively  a lone pair mainly lying perpendicular to the molecular plane and thus it would be largely unaffected differences  by the  between H 0 2  nuclear  and D 0 2  motion. The absence of momentum  even for the  bonding 3 a ,  and  lb  2  profile orbitals  suggests that the presently observed discrepancies [BL85] between calculations and experiments, particularly for the  lb,  orbital but also to a lesser extent  for  the  119 3a,  orbital, are not due predominantly to vibrational effects.  5.5. BASIS  SET  EFFECTS  The previous study of the M D s of H 0 in chapter  4 as well as several earlier  2  works, clearly indicated significant discrepancies Hartree  Fock wavefunctions  two outermost the  from  orbitals ( l b ,  the  between calculation using existing  literature  and  experimental  and 3a,). The greatest differences  (least tightly bound) essentially  non-bonding  lb,  at  least  wavefunctions  for used  the  relative  included  the  Snyder and Basch (SB [42/2]) contracted  gaussian)  shapes.  In  minimum  basis  [SB72] and  wavefunction  of  the  in the  wavefunction total  present  work  of Aung,  energy  of  are  shown  Pitzer and  -76.00468  a.u.  and  a  of  work  and  gaussian  Moskowitz  Table  The  (APC (331/21)) dipole  was  was very the  functions  of  (i.e.  extended  (NM  [531/21])  all other  5.1.  it  [HH77]  near Hartree-Fock  of these and  in  Chan  earlier  set  Neumann  [NM68]. The details as well as properties used  the  and 2 a , )  2  the  were observed for  orbital. In contrast  found that agreement for the third and fourth orbitals ( l b good  M D s for  wavefunctions  Slater  type  (STO)  [AP68] which gives  moment  of  2.035D  was  a  also  compared in the earlier work [BL85] and gave the best agreement of the three wavefunctions  These  results  energy  (i.e.  (SB, N M and A P C ) with the experimental momentum profiles.  of the best  earlier  from  a  study  [BL85]  variational  showed  standpoint)  that  for  H 0  wavefunction  2  the  lowest  (contracted  NM  [531/21]) does not give the best calculated M D . In fact the variationally inferior APC  wavefunction  inadequate  for  the  gave lb,  the and  better 3a,  overall fit orbitals)  (but to  the  one  that  measured  was  still  XMPs.  rather It  is  120 interesting in this regard to note that the A P C wavefunction also gave a dipole moment those  (2.035D)  given  slightly nearer  by  either  the  to  the  contracted  Hartree-Fock  N M [531/21]  limit  (1.98±0.01D)  (2.092D)  or  SB  than  (2.681D)  wavefunctions.  From this  rather limited comparison of theory  and experiment  it is clear  that  the calculated results for total energy, dipole moment and momentum distribution are far from converged (see Tables 5.1 and 5.2). It is also clear that the degree of  convergence  is  different  for  different  calculated  properties  for  each  of  the  wavefunctions. This is, of course, a straightforward manifestation of the fact that a wavefunction  is a model (in all cases for every molecule and for all neutral  atoms except for atomic hydrogen). Such a model will only be as reliable as approximations  used  those properties only the the  in its building and it will only be adequate for calculating  for  which these approximations  variational constraint  energy  the  of minimised energy)  minimisation stresses the  small r  surprising that use of variationally determined results for properties,  and  such as dipole moment  testing are  constraints  (usually  sufficiently valid. Since  region of wavefunctions wavefunctions  often  it is  not  leads to poor  and M D s , which depend sensitively  upon the longer range (r) charge distribution.  With  the  between  above measured  considerations XMPs  water, it is of interest including the  and  in mind, calculated  and  considering  M D s for the  to investigate the effects  the  outer  large  discrepancy  valence  orbitals of  of increased basis set  flexibility  addition of higher order polarization functions. In this it is of key  importance to ensure that convergence has been reached not only with regard  to  121 the H F limit of energy but also for those properties including the dipole moment and M D which are influenced by the large r (i.e. low momentum)  part of the  wavefunction. When this has been achieved, the remaining discrepancies between  theory  of  neglect  the  and experiment of  electron  can be investigated in terms of the  correlation  and  relaxation  effects  (if any)  significance  implicit  in  the  Hartree-Fock model.  A t the time of the first E M S experiments for H 0 the wavefunctions used were 2  in  the  (see  range of quality of the  also Table 5.1). The E M S results,  (reported in chapter to  SB, N M and A P C functions  facilitate  more  direct  sophisticated  referred  to  above  at much improved momentum resolution  4), were evaluated using these same wavefunctions in order comparison literature  with  the  original  wavefunctions  are  studies now  [HH77,  compared  DD77].  with  Other  experiment.  These include the 39-STO wavefunction of Rosenberg and Shavitt [RS75] and the 84-GTO wavefunction of Davidson and Feller single  determinant  S C F wavefunctions  wavefunctions  have  the  H F limit at the  estimated  moments  of 1.995D and  1.98±0.01D that  total energies  the  (experimental  variationally  currently  within  published for  0.003a.u.  value is  1.8546D).  84-GTO  2  These  two  and give calculated dipole  close to the It  H 0.  and O.OOla.u., respectively of  experimental geometry  2.02 I D which are  superior  [DF84] which are two of the best  estimated  is however  wavefunction  H F limit of  of interest  gives  a  less  to note  good  value  (2.02ID) for the dipole moment than the 39-STO (1.995ID). This may reflect the superiority of STOs at large distances. Consideration of these two relatively high quality  wavefunctions  energy  and  [DF84,  dipole moment  to  RS75] the  alone  indicates  H F limit  values  that has  convergence not  yet  been  of  both  reached.  122 Calculations [AP68]  in  fact  show  that  the  all  give  quite  wavefunctions  exhibiting in each the l b ,  and 3 a ,  case a similar  84-GTO  [DF84],  similar  39-STO  results  (considerable)  for  [RS75] and A P C  the  discrepancy  calculated  MDs  with experiment  for  orbitals of H 0 (compare results in previous chapter and Figs. 2  5.1 and 5.2)  Therefore  new  and  further  improved  109-GTO, have been generated University)  in the  adequately  the  course  wavefunctions,  namely  99-GTO  and  in collaboration with Prof. E.R. Davidson (Indiana  of the  present  work  in an  attempt  variationally insensitive but chemically important  to  model  large r  portion of the electron distribution. Details of these new wavefunctions  more  (low p)  are given  in Sec. 5.2 and pertinent properties are shown in Table 5.1. To give an idea of the basis extension, it should be noted  that the  109-GTO wavefunction uses a  [14sl0p3dlf/6s3pld] basis set whereas  a minimal basis set for water would use  only  be  seen  a.u.  of the H F limit) but that the dipole  a  [2slp/ls]  basis  set.  It  converged (at least to within moment  (2.006D) is still  variationally Even  inferior  larger  0.0005  farther  from  NM(1062/42)  gaussian  Hartree-Fock  can  basis  dipole moment  sets of  the  1.9803D  the  H F limit than  [NM68] than  that  energy  those given by  and  39-STO  [RS75]  those  reported  here  with  an  energy  (109-GTO) is  the  wavefunctions. have  of -76.0672  given  a  hartrees.  This value of the dipole moment is believed to be converged to ±0.0 ID.  Calculations spherical  of the  averaging  Ap = 0 . l 5 a " ) 1  o  momentum and  distributions for  incorporation of the  all 4 valence orbitals (including  experimental  momentum  resolution,  using these various existing and new wavefunctions are shown in  123 Figs.  5.1-5.4  in comparison with  The calculations were and  140-G(CI),  109-G(CI)  see  assuming  intensity  the  experimental results  carried out in the below) unit  THFA  pole  and  treatments.  strength  scale  established  by  single  to  experiment  on  the  calculations  PWIA  for  lb  (see  on the  same  scale  affords  a  very  to  the  all  four  orbitals  and  this  on  of  Fig.  109-G(CI)  (including  are  normalization  experimental points and calculations are for  2  (except for the  orbital)  orbital  2  and D 0 .  2  A l l calculations  each  point  for H 0  a  the  common 109-GTO  5.3).  Thus  relative absolute stringent  the  all  intensity  quantitative  comparison of theory and experiment.  Several  observations  can  be  made  in reference  earlier  comparisons  of  experiment and theory for the valence orbitals of H 0 (Chapter 4). It should be 2  noted that curves 1 (SB) and 2 (NM[531/21]) were shown with individual height normalizations for each orbital to the experiment. Comparison of curves 2 and 3 show the serious effect of the contraction of the same (NM) wavefunction on the calculated  M D s particularly  uncontracted  set,  in  NM(1062/42),  the  case  shows  of  (see  the  lb,  and  3a,  orbitals.  curve  3,  Figs.  5.1  and  The  5.2)  an  improved momentum distribution and a much better dipole moment than does the contracted set, NM[531/21] (see curve 2, Figs. 5.1 and 5.2) although the  energy  is only marginally affected by the contraction (see Table 5.1). This illustrates the extreme care  necessary  in choosing the  such as the M D or dipole ,moment in each case found [BL85].  to  are  the M D s calculated from  give rather  similar  results  to  proper  contraction  scheme  if  properties  required. Curve 4 (Figs. 5.1-5.4) the the  shows  84-GTO wavefunction and these A P C wavefunction  as  used  are  earlier  124 Considering first the  lb,  the  section and  maximum cross  orbital (Fig. 5.1) it can be seen decrease in p *max  as  that the increase in  well as  the increasingly b  improved modeling of the low momentum region in going from SB (curve NM(1062/42) (curve 3) to 84-GTO (curve 4) are carried even further the  99-GTO  set  also occur for the  the  further  dipole  (curve 5) wavefunction. Similar improvements  improvement  moment  84-GTO  3a,  for  the  wavefunction.  (Fig. 5.2) and 2 a ,  (see  Table  99-GTO The  5.1)  and  in both  of  calculated  the calculated M D s for the  lb  in going to  change  total  wavefunctions  the  theoretical  experimental momentum profile with expansion in basis In contrast,  in basis  energy  relative  MDs  the  towards  the  set is clearly illustrated. converge  at the NM(1062/42) level. However it is of importance  to note that no  change  orbitals  from  in the 99-GTO  calculated to  M D s for  109-GTO  (see  the  curves  two outermost 5 and  6 which  5.1-5.3). On the other hand in the case of the 2 a , already  converged  (114-GTO,  at  119-GTO  the  84 G T O level (see  and  140-GTO)  gave  no  and  to  orbital (curve 3, F i g . 5.3)  2  1) to  (Fig. 5.4) orbitals. This reflects  109-GTO  progression  with  J  are  occurs  further  in going  identical in  Figs.  orbital the calculated M D is  F i g . 5.4). Even noticeable  larger basis  change  so  sets  these M D  curves would seem to be at the Hartree-Fock limit.  These results indicate the importance of having s-p saturated  basis sets as well  as higher order polarization functions (i.e. d- and f-functions on the oxygen) when predicting moment.  properties  such  as  electron  momentum  The importance of very diffuse functions  distributions  and  the  dipole  (much more than expected) in  the basis set is shown by the corresponding improvement in the calculated M D s . As  more diffuse  functions  are  added, the  M D shifts  and also gives appreciably  125 more  intensity  at  lower  momentum.  The  balanced  addition  of  extra  diffuse  functions in H 0 evidently provides an improved description of the large r (low 2  p) part of the total wavefunction. In terms of the inverse weighting property of the Fourier transform this should contribute to a better description of the low momentum components of the calculated MD.  Summarising the above considerations it can be concluded that gaussian basis set saturation has been effectively  reached at the 99-GTO SCF level. However while  it is clear that considerable improvement over earlier calculations has been gained for the  l b , , 3a,  and 2a,  orbitals with this "best" HF level treatment it can  be seen that a considerable discrepancy with experiment still occurs, especially in the  low  momentum  region.  In particular, significant  additional low momentum  components are observed experimentally in the case of the the  observed  p  (0.60±0.02a " ) 1  m o  0  is  appreciably  lb,  lower  orbital and also than  even  that  i l l 3.X  (0.65a ~ ) predicted by the best (99-GTO and 109-GTO) H F level wavefunctions. 1  0  Smaller discrepancies exist between calculations and experiment for the 3a, and 2a,  orbitals at the HF level.  Properties of the various calculated MDs and XMPs, including P in Table 5.2.  m a x  ,  are given  The properties shown in this table summarize the convergence of  the calculated molecular orbitals of water in momentum space . To characterize the MDs two properties have been evaluated, namely the leading slope at half maximum (LSHM) and the momentum at which the MD maximizes (p should be pointed out that P  m a x  ). It  is not the most probable momentum which is  characterized by the p which maximizes p « ( M D ) . Note that the p 2  and the  Table  5.2 C h a r a c t e r i s t i c s  Basis  Set  of  Calculated Orbital  Momentum  lb  D i s t r i b u t i o n s and Experimental 3a  Momentum P r o f i l e s lb.  Pmax  LSHM  Pmax  Pmax  LSHM  SB  [42/2]  0.76  (0.0990)  0.75  (0.1104)  0.74  (0.1212)  NM  [531/21]  0.76  (0.0989)  0.76  (0.1078)  0.74  (0.1192)  NM  (1062/42)  0.72  (0.1121)  0.71  (0.1186)  0.72 (0.1216)  RS  39-STO  0.67  (0.1195)  0.69  (0.1240)  0.72 (0.1204)  DF  84-GTO  0.68  (0.12O4)  0.69  (0.1270)  0.72 (0.1210)  99-GTO  0.65  (0.1262)  0.68  (0.1280)  0.72 (0.1208)  109-GTO  0.65  (0.1264)  0.68  (O.1280)  0.72 (0.1209)  84-G(CI)  0.67  (0.1316)  0.68  (0.1365)  0.72 (0.1222)  109-G(CI)  0.63  (0.1398)  0.66  (0.1370)  0.72 (0.1216)  0.60  (0.115)  0.69  (0.110)  0.72  (O.IIO)  0.60  (0.115)  0.69  (0.110)  0.72  (0.110)  H0 2  Expt.  0 0  Expt.  2  a b  b b  Peak maxima and  LSHM (In p a r e n t h e s i s )  E s t i m a t e d experimental  uncertainty  are both quoted  In p  and  LSHM  In atomic  LSHM are i 0 . 0 2 a  Q  -1  units, and ± 0 . 0 0 5 ,  respectively.  CO 03  127 LSHM do not completely characterize the MD but they can be used as guides in comparing the different calculations with experiment. The P the  MDs  and XMPs  can be considered  m a x  as momentum  and the LSHM of space  analogs of  one-electron properties in position space in the sense that they can be used as "diagnostics" of wavefunction quality. The p , vv  ov  and L S H M values for the MDs  m 3.x  were obtained using a standard cubic spline fitting routine [UB84]. The statistical deviation (i.e. the individual error bars of the points in the distribution) were also considered in obtaining the best fits for the XMPs. It can be seen that the experimental  p-space  properties for H 0  and D 0  2  are the same within  2  experimental error. and L S H M at the SCF level. From Table 5.2 it  Consider first the trend of p is evident that the P  m a x  for the 3a, and l b orbitals are quite converged at 2  the 99-GTO level and NM(1062/42) level (58-GTO), respectively with the orbital converging  towards a p  (0.68a ~ ). In contrast, the p„,  higher  level  with  a  pmax  ov  of 0.65a ~ 0  1  0  of the lb  1  o  (0.72a ~ ) than 1  1  compared  lb  the 3a, orbital  orbital converges at the 99-GTO to the experimental  value of  0.60ao" ±0.02. Despite the estimated uncertainty it is clear from the trend of 1  the data points (Fig. 5.1) that the P  m a x  from experiment  is significantly lower  than that predicted by any calculation at the THFA level.  Since basis set saturation has been established the remaining discrepancy between theory and experiment  may be associated with one or more of the following  effects (1) Deficiencies in the PWIA treatment;  2  128 (2)  Further  uncertainties  Ap = 0 . I 5 a ~  in  the  experimental  momentum  resolution  beyond  already incorporated in the calculations;  1  0  (3) Failure of the  THFA  in H 0 2  and therefore  the need to consider the  fact  that correlation and relaxation effects may be significantly influencing the valence momentum  profiles.  This  would  amount  to  failure  of the  Hartree-Fock  model  description in these cases.  Deficiencies  in  the  PWIA  treatment  discussed in Sec.  5.1  other  molecules). Similarly  atoms  can be  and  discounted  since  a further of the  only significantly affect in Ap would  are  (i.e. good agreement  experimental geometry)  increase  (1)  the  unlikely  at  the  enlargement 1  0  reasons  H F level for  already  a number of  resolution effects  (unphysical with  respect  to  (2) the  already used would in any case  very low momentum the  the  unknown momentum  Ap ( O . l 5 a ~ )  not change  for  part of the  position of p  curve. A n y such  or the  majority of the  111 3.X  large  "mismatch" down the  distribution  (see  F i g . 5.1).  discrepancy between of  electron  leading edge (i.e. low p region) of the Therefore  H F theory  correlation  implicit  the  most  likely  source  momentum  of the  remaining  and experiment is item (3), namely the in the  Hartree-Fock  model used  in  the  neglect THFA  treatment. This possibility is investigated in detail in Sec. 5.6 following.  Before  proceeding to  observations  a consideration  of correlation effects  X M P s for H O are largest for the 2  somewhat smaller discrepancies are also found for the 3 a , other  following  further  are made. First, while the observed discrepancies between calculated  M D s and measured  the  the  hand  the  lb  2  orbital is apparently  well  lb,  orbital similar but  and 2 a ,  represented  orbitals. On  already  at  the  129 APC  and  NM(1062/42) levels. Second, it is instructive  using  the  best gaussian  basis  set  (i.e.  109-GTO or  to  compare  M D results  99-GTO) and  best  Slater  [RS75] basis sets (39-STO) available to date. This "best" GTO/STO comparison is shown together with the experimental orbitals  of  H 0 2  and  D 0. 2  The  measurements in Fig. 5.5 overall  good  agreement  for the  between  calculations confirms the generally held view that 2 to 3 GTOs are each STO. It can be seen that while the calculated use  of both STO and G T O (Fig. 5.5)  the  109-GTO calculation is marginally better  improvement  for the  that although the energy,  p  and  max  lb,  lb  the  the  two  needed for  orbital is identical for  2  and in good agreement with for  valence  3a,  experiment,  orbital and  orbital. It can also be seen from Tables  a  slight  5.1 and  5.2  109-GTO wavefunction gives superior values for calculated total <r >  the  2  39-STO wavefunction gives a better value for  the  dipole moment.  5.6. CORRELATION  AND  RELAXATION  EFFECTS  In view of the failure of even highly saturated basis sets to satisfactorily the observed momentum the  lb  2  distributions at the target Hartree-Fock level (except  orbital, at least on the basis of the present normalization) a  investigation  beyond  wavefunctions  developed in an interactive  the  Hartree-Fock  The H F limit of total energy -76.0675a.u.  while  the  -76.4376a.u.  [RS75].  for the  estimated  The  electron correlation neglected Correlation effects  predict  difference in the  model  has  been  made  for  theoretical using  CI  collaboration with Prof. E.R. Davidson.  ground  state of H 0  non-relativistic, (-0.370a.u.) Hartree-Fock  can be treated by configuration  2  is estimated  non-vibrating is  the  extra  total  to be  energy  energy  due  is to  single configuration S C F model. interaction  (CI) description for  Comparison of calculated valence orbital M D s of H O using the best gaussian (109-GTO, solid line) and best Slater (39-STO, broken line) basis sets. The T H F A calculations are placed on a common intensity scale by using a single point normalization of the 109-GTO calculation on the l b X M P (see text for details). 2  2  131 the  target molecule. For the  target  molecular  orbitals  relaxation effects. the  These  final  can be  ion species used  correlated  configuration interaction using  to describe  wavefunctions  both  electron  can then  be  full ion-neutral overlap (see Eqn. 2.11). A n y difference  overlap  and  the  corresponding  THFA  calculations  correlation  used  to  between  indicates  the  the and  evaluate  such a  importance  full of  relaxation and correlation effects.  Previous comparisons of X M P s and theoretical M D s for H 0 were all done using 2  the target Hartree-Fock approximation (THFA) except for the generalized overlap amplitude (GOA) calculation of Williams et al. [WM77a], These G O A calculations [WM77a], which utilized a rather limited basis set,  showed a small improvement  over the T H F A calculation but major discrepancies between experiment and theory remained.  The  G O A method  many-body Green's function  In  the  attempts calculation of the  overlap via  techniques.  present work calculations of the  wavefunctions  ion-neutral  ion-neutral  based on the two highly extended  overlap  amplitude  using CI  basis sets, namely the  [DF84] and 109-GTO basis sets were done. The CI wavefunctions  84-GTO  were developed  in an interactive collaboration with Prof. E.R. Davidson (Indiana University). For the CI treatment using the 84-GTO basis (labelled 84-G(CI)) the neutral molecule ( ' A ^ (SACs)  wavefunction calculated  ( E Q J =-76.3210  a.u.  is at  expanded the  which  into  experimental includes  69%  energy). For the CI treatment using the  5119  symmetry  neutral of  the  adapted  molecule total  configurations  equilibrium  ground  state  geometry correlation  109-GTO basis (labelled 109-G(CI)), the  neutral molecule is expanded into 11011 SACs  ( E Q J = -76.3761 a.u. which includes  132 83% of the  ground state correlation energy). Details of the  109-G(CI) final ion  state wavefunctions are to be found in Table 5.3.  The  spherically  averaged  amplitudes  using both  momentum  space  overlap  square  sets  are  compared  (ie.  84-G(CI)  corresponding T H F A  results  in Figs.  the  and D O X M P s  measured  H 0 2  point  for  experiment  the  have Table  (see  for the  The  are  5.1-5.4) and  2  case of Figs.  109-GTO  (THFA)  full  and  the  ion-neutral  compared  also Figs. lb  overlap  calculated  5.2.  l b , , 3a,,  in the  ion-neutral  been  109-G(CI))  5.6-5.8  same as and  in  and  2  orbitals. Normalization is the single  Fourier-transformed  of CI wavefunctions  properties  calculations  of the  to  the  along with  2a,  5.1-5.4  valence (i.e. at  calculation on the  a  lb  2  momentum profile).  Comparison  of the  1200eV) spectra 32.2eV  and  many-body  0°  [CC84,  35.6eV structure  and  8°  BL85] as  [BL85] found  have in the  binding energy well  as  the  confirmed  the  (at  a  total  momentum  energy  of  profiles measured  assignment  that  the  predicted  spreading  of  the  minor  poles  over  to  the  of minute poles  in this region due to small contributions from the ionization of the outer This  at  extensive  region 25-45eV is predominantly due  (2a,) ^ hole state. CI calculations however suggest the presence  orbitals.  about  the  wider  valence energy  spectrum (with much of the extra intensity at energies even higher than 45 eV) results in pole strengths for the main lines of each of the  three outer  orbitals which are slightly less than one (i.e. 0.87, 0.88 and 0.89 -  valence  see Table  5.4). These pole strengths for the outer valence orbitals are converged to ± 1 % .  Table  5.3. CI Calculations  (a.u. ) of the  Ground and Final  Ion States  of HD Using the 109-GTO Basis  Set  2  STATE  E(CI)  E(SCF)  AE(CI)  E(exptl)°  E(HF )  H 0 X 'A,  -76.37614  -76.0671  0.309  -76.4376  -76.0675  0.370  83  -75.92084  -75.5569  0.364  -75.9746  -75.5600  0.415  88  -75.83376  -75.4822  0.351  -75.8976  -75.4841  0.414  85  -75.68563  -75.3492  0. 336  -75.7576  -75.3515  0.406  83  2  H 0*  (1b,)"  H0  (3a ,)~  2  +  2  H,0  +  (1b )"  1 1 1  2  For the Ionic C  d  states the energy  f  E(corr) = E(exptl)  &E(CI) •  a  E(CI) - E(SCF)  - I.P.) a.u.  ionic states.  - E(HF) for neutral, E(corr) =  %E(corr) « AE(CI)/E(corr)  b  refers to a Koopmans' energy.  For the ion states, E(exptl)° -(76.4376 Includes relaxation for  8  3  x 100  E(exptl) -  E(Koopmans) for Ion.  E(corr)  e  y.E(corr)  d,f  134  i  1  1  i  1  1  1  1  1  1  r  Momentum (a.u.) Fig. 5.6  Correlation effects in the calculated momentum distributions of the l b , orbital of water. The more accurate CI overlap calculations (solid line), (4c)84-G(CI) and (6c)109-G(CI), and the T H F A calculations (broken line), (4)84-GTO and (6) 109-GTO are all placed on the same relative absolute scale by using a single point normalization of the 109-GTO calculation on the lb X M P (see text for details). Further comparisons are made with the measured H O (open squares) and D O (solid circles) experimental momentum profiles. 2  2  2  135  T  1  1  1  1  1  1  1  1  1  i  i  r  Momentum (a.u.) Fig. 5.7  Correlation effects in the calculated momentum distributions of the 3a, orbital of water. The more accurate CI overlap calculations (solid line), (4c)84-G(CI) and (6c)109-G(CI), and the THFA calculations (broken line), (4)84-GTO and (6) 109-GTO are all placed on the same relative absolute scale by using a single point normalization of the 109-GTO calculation on the lb X M P (see text for details). Further comparisons are made with the measured H O (open squares) and D O (solid circles) experimental momentum profiles. 2  2  2  136  Fig. 5.8  Correlation effects in the calculated momentum distributions of the l b orbital of water. The more accurate CI overlap calculations (solid line), (4c)84-G(CI) and (6c)109-G(CI), and the T H F A calculations (broken line), (4)84-GTO and (6) 109-GTO are all placed on the same relative absolute scale by using a single point normalization of the 109-GTO calculation on the lb X M P (see text for details). Further comparisons are made with the measured H O (open squares) and D O (solid circles) experimental momentum profiles. 2  2  2  2  Table  5.4. CI Calculations  (109-G(CI)) of the Pole Strengths and  Energy Spectrum of C (3a J  O.OOO  0.998  0.0357»  0.974  0.021  33. 1  0.4394»  0.992  0.077  33.6  0.2232*  0.999  0.000  34.2  0.0225*  0.659  0. 325  35.8  0.0218*  0.865  O. 133  37.8  0.0216*  0.863  O. 135  39.5  0.0730*  0.980  0.015  41 .0  0.0240  0.475  0.487  44 .0  0.OO39*  0.708  0.260  44.9  0.0002  0.336  0.525  45.0  0.0009  0.033  0.482  45.8  0.0051*  0.914  0.058  46.3  0.0043  0.393  0.504  47 . 1  0.0227*  0.827  O. 147  Energy  Pole Strength  -1 ( 1b,)  12.39  0.869  ( 3 a )  14 . 76  0.882  db )  18.79  0.888  -1 (2a,)  29.0  3  in the Binding  C (2a ,)  STATE  (  Energies (eV)  £(•)'0.869  2  H0 2  2  CO  138 Even  more  extensive  breakdown  ionization of the 2 a , and  experimentally  energies  for each  mixing  as  calculated  [CC84, final  BL85].  using  Table  the  are  main (2a,)"'''  5.9 in comparison with the spectrum  particle  5.4  shows  ion state together with the  strengths  (2.77eV) of the  single  picture  in the  case of  inner valence orbital is well-known both theoretically [VS84]  calculated pole  of the  109-G(CI)  the  pole  strengths  CI coefficients  correlated  for  2a, /3a,  wavefunctions.  These  convoluted  with the  peak  Figs. 4.1 and 4.2) and shown in F i g .  (see  relevant  estimated  and  experimental  part of the earlier reported  width  binding energy  [BL85] in the region 23-45 eV. The agreement between the calculation  and experiment is generally quite reasonable. at best semi-quantitative  However the calculated results  are  with respect to the distribution of intensity particularly  in the 31-36 eV region. Note that the calculated binding energy profile is slightly shifted at  (= leV) to the  higher energy  =35eV is still inadequately  side and that the  predicted  by the  position of the  shoulder  109-G(CI) calculation. It should  also be noted that the spectrum reported by Cambi et al. [CC84] extends out to 50eV  and  shows  weak  intensity  comparable  to that predicted  at  =47eV.  It  is  probable that, even at this level, the calculation is still basis set dependent with respect to the exact (2a,)"* pole strength strength  calculations reported are  shown  in  distribution (see also for example pole  by Cambi et al. [CC84] and Agren and  [AS80]  which  comparison  definite  conclusions would require  with  a convergence  experiment study  in  of the  Siegbahn  Fig. 4.2). energies  More  and pole  strengths of the many-body structure of the (2a,)" ^ hole state.  With  these  5.1-5.3,  considerations  5.6-5.8)  are  in  mind,  presented as  the  calculated  outer valence OVDs  normalized distributions  (Figs.  (i.e. renormalizing the  139 pole strength 2a,  of each  outer  valence  poles to unity). The  inner  O V D in F i g . 5.4 was obtained by calculating a pole-strength  of the greater  (slightly different) than  1%)  unity.  It  is  in  the  region  above  25eV and  then  to  note  that  this  summed  2a,  intensity  divided  in this region in order to renormalize the  of interest  valence  weighted sum  OVDs of all significant poles (i.e. those with  found  summed pole strength to  of the  by  the  resultant O V D  pole  strength  has  essentially the same value as that for the main pole of each of the three outer valence orbitals. In this normalization procedure any contributions from any minor outer  valence poles have been  neglected.  Calculations in any  case indicate  that  such pole strengths are less than 3% (in fact only one pole is 2: 1%, i.e. that at 41eV,  see Table 5.4).  On the basis of the above normalization procedure for the CI calculations, it can be seen (Fig. 5.6 and 5.7, Table 5.2) that use of the CI wavefunctions for the lb,  and 3 a ,  outer valence orbitals in the overlap treatment results in a further  significant progression of these "p-type" calculated OVDs towards low momentum and  higher  cross  section  and  thus  towards  Such a small shift towards low momentum  better  agreement  with  experiment.  with the inclusion of correlation has  also been observed earlier for A r (3p) by Mitroy et al. [MA84] but the  effect  was  good  very  small  in  that  case.  It  is  also  significant  agreement already attained  at the 84-GTO T H F A  5.8)  is unchanged  of the  can  be seen  by use  that, with the  and CI treatments give the range  for the  lb  2  84-G(CI) or  to  note  level for the  described  same absolute intensities over the case of the  lb  109-G(CI) overlap  normalization procedure  orbital. In the  that  2a,  inner  2  the  orbital (Fig.  treatments.  above, the entire  It  THFA  momentum  valence orbital (Fig.  140 5.4) inclusion of correlation (109-G(CI)) gives a slightly improved quantitative to the  data.  These  influences the  lb,  situation for the  findings  indicate the  and 3 a , lb  fact  that  correlation mainly  orbitals. This behaviour is in sharp contrast  orbital where  2  important  correlation and  fit  relaxation effects  to the  appear  to  have a negligible effect on the calculated overlap distribution.  The  interplay  of basis  set  effects  illustrated by the results for the  and  lb,  the  inclusion of  and 3 a ,  correlation  are  well  orbitals shown in Figs. 5.1, 5.2,  5.6 and 5.7. It can be seen that inclusion of correlation through the CI overlap treatment further  improves  agreement between  the  calculated  distributions (MDs  and OVDs) and experiment at both the 84-GTO level (compare curves 4 and 4c, Figs. 5.6 and 5.7) and the  109-GTO level (compare curves 6 and 6c, Figs.  and 5.7). In comparison with the 84-G(CI) wavefunction (p moment= 1.929D) shift of the increase  use  of the  109-G(CI) wavefunction  O V D to low momentum  =0.-67a  shows  ( p „ , „ = 0.63a " )  a further  in cross section and an improved value of the dipole moment  but significant discrepancy since  the  experimental  (pmax = 0.60ao ~ ±0.02) 1  the  3a,  orbital the  experiment. predicted higher  With  together  LSHM  example,  for  with a  momentum higher  109-G(CI) overlap  with more  lb,  (ie.  orbital  are  leading  profile  is  0.1316  interaction,  and  (1.895D).  treatments a small  at  at  lower  section. a  as  half  0.1398  momentum  In  result  smaller  O V D curves slopes  further  orbital even at the 109-G(CI)  calculation gives  asymmetric  steeper  lb,  maximum cross  inclusion of configuration  values the  still exists for the  > dipole  significant  together with a  1  0  Despite the dramatically improved agreement over the T H F A  level  _1 0  5.6  P  the quite  m  a  x  for  the  close  values  characterized maximum)  case of  by  to are the  which, for  84-G(CI)  and  141  —i—i—i—i—i—i—i—i—i—i—r  : H 0  (2a,)  2  —1  1  1  1  1  1  1  1  1  1  1  1  1  1  i  1  - 1  •v  109-G(CI )  Cal c n .  "  (b) ; 23.0  5  I—i—i 23.0  Fig. 5.9  25.0  i  25.0  27.0  i  i  27.0  29.0  i  i  29.0  i  i  i  i i i 31.0 33.0 35.0 37.0 39.0 Binding Energy («V)  i  31.0  i  i  i  i  I  I  33.0 35.0 37.0 39.0 Binding Energy («V)  i 41.0i i 43.0 i i i i i i.. ± 45.0 47.0 49.0  41.0  43.0  i  i  45.0  i  i  47.0  i  i  49.0  Binding energy spectrum of water in the (2a,) inner valence region. The experimental data (a) at 0 = 0 ° ( E = 1200eV + B.E.) is taken from Ref. [BL85]. The theoretical binding energy profile (b) was determined in the present work from pole strengths calculated using the 109-G(CI) wavefunctions (shown as solid vertical lines at the appropriate energies). The calculated poles have been convoluted with an experimental width of 2.77eV fwhm estimated from spectrum (a). Further details are given in Table 5.4. 0  142 109-G(CI) wavefunctions, respectively. The experimental L S H M  for H 0 2  ( l b , ) is  (114-G(CI), 119-G(CI) and  140-G(CI))  0.115±0.005a.u.  Calculations  using even  larger  basis  sets  which recover even more of the correlation energy of the neutral molecule and of the  cation gave  orbital.  These  84-G(CI),  OVDs indistinguishable from considerations  109-G(CI)  and  together  140-G(CI)  respectively of the estimated any  further  (very  with  the the  treatments  109-G(CI) results fact  (see  recover  correlation energy of H 0 2  difficult)  theoretical  investigations  Table 69%,  for 5.2)  83%  (-0.370a.u.), of  the  the  lb,  that  the  and  88%  suggest  effects  that  of  the  remaining correlation energy on the small residual discrepancies are not likely to produce (curves  much change.  In  this  regard  it  should be  4c and 6c, Fig. 5.6) which results  noted  in going from  that  the  large  84-G(CI) to 109-G(CI)  is mostly due to the improvement in the basis set. A smaller part of the can be  ascribed to improvement in the  consider the various calculations for the  shift  shift  CI description. It is also of interest lb,  orbital as shown on Figs.  5.1  to and  5.6 together  with the experimental results. In particular it is noticeable that all  calculations  are  0.9-1.230"'  region of the X M P for the  slightly  higher  than  the  (ie.  addition to the  at  the  CI  level)  reflects  electronic  the  'non-characteristic'  single-particle orbital M D that would have been  state.  The  overlap  sections  low momentum using the full  simple approximation of the more usual T H F A . ion  cross  in  the  l b , orbital.  The shift of the calculated OVDs towards treatment  measured  amplitude  can  contributions expected  Consider the case of the be  expressed  overlap  in terms  in  in the  (lb,) of  the  143 single-particle orbitals (of the ground Hartree-Fock configuration),  <p* where  N _ 1 f  |* > N  0  the  labels  = C , uV  1 and first  + C ^  n  2  2 refer  respectively.  The  term  nomenclature  of T H F A ) whereas  + ...  2  to  lb,  corresponds the  [5.1]  and to  2b,  the  p-space  molecular  characteristic  second term corresponds  orbital  the  l b , —> 2 b , 2  electron  correlation)  2  double with  excitation of the  the  l b , —» 2b,  neutral  single  the  be considered  'non-characteristic' contribution. The correction comes mostly from the  of  (in  to the lowest-lying  virtual orbital of the same symmetry. The latter can therefore a  orbitals,  as  integral  (which represents radial  excitation  of  the  ion (which  describes orbital contraction). Because of the signs involved, this orbital contraction makes  the  molecule-ion overlap more  diffuse  than  the  characteristic  orbital. To  illustrate this point consider a two orbital model: |*  N 0  = A , | 1b, >  >  A | 2b, >  -  2  [5.2]  2  2  There will be no large term in (lb , 2 b , ) The  2b,  term  2  represents  electron  because of Brillouin's theorem  correlation.  independent of the relative phases of l b , |* The  f  N _ 1  second  B /B, 2  >  = B,|lb,>  term  2b,  are  chosen  lb,  has  no radial node  small r.  A  will  2  be  positive,  and 2 b , . For the ion, [5.3]  2  relaxation  (i.e.  orbital  relative phases of l b ,  to be positive at large r, then  Consequently, the  and  - B |2b,>  represents  will depend on the  A,  [S082].  and  2b,  has  second term  one, will  contraction). and  The  2 b , . If both  sign of lb,  and  B / B , will be positive. Since  they  2  will  subtract  have  from the  opposite first at  signs large  at r  144 and  add at  small r.  ^>  will  be contracted  consider the molecule-ion overlap <  <*  N f  " |* 1  N 0  >  ^|'Po^ -  relative  to  | l b ! > . Now  Explicit evaluation gives  >  + A B |2b,>  = A ^ J l b ^  2  2  = C , | 1b,> + C | 2 b ! >  [5.4]  2  Notice  that C  amplitude  is  has the opposite  2  more  >.  is  position  space  so that the molecule-ion overlap than  | 1 b , >,  opposite  to  the  _  in  contraction  in  2  N 1  i  contraction  diffuse  sign to - B  clearly  This  behaviour  illustrated  by  and the  converse  considering  the  momentum difference  space density  plot for the l b , orbital shown in F i g . 5.10. Likewise the overlap  ( P Q J - p r p p j p ^ )  density (the spherical average of which yields the OVD) can be written as, p  = C, (lb,) 2  Q  2  + C  2 2  (2b )  Since  |B |<|B |,|A |<|A |,  small  compared  2  dominant  1  2  2  correction  + 2C,C (1b )(2b ) 2  1  it can be seen that  1  to ( 2 C i C ) .  2  1  Consequently  the cross  in the molecule-ion overlap  [5.5]  1  |C | < |C, |  and C  2  term  density.  is  2 2  in ( l b ) ( 2 b )  is the  This momentum  space  1  1  contraction on going to the CI overlap treatment can also be clearly seen in the distributions shown in Figs. 5.1 and 5.6.  5.7. CALCULATED PROPERTIES NEAR THE HF AND CI LIMITS To  summarize  the various basis  above,'  the pm„a x  orbital  are plotted  r  and L S H M  set, relaxation  and correlation effects  of the theoretical  as a function  of the number  M D s and OVDs of contracted  discussed  of the l b i  GTOs  1  in their  respective basis sets in F i g . 5.11. Attention is focussed on the l b , orbital since it  is most  sensitive  to correlation  and basis  set effects.  A s has been  shown  MOMENTUM DENSITY DIFFERENCE  Fig. 5.10  POSITION DENSITY DIFFERENCE  Two-dimensional density difference (p^j - P-pjjp^) plots in momentum space and position space for the l b , orbital of water calculated using the 84-GTO basis. Contours are at ± 8 0 , ± 4 0 , ± 8 , ± 4 , ± 0 . 8 , ± 0 . 4 , ± 0 . 0 8 and ±0.04% of the maximum density difference. A l l dimensions are in atomic units.  146 0.801 1 ^  =1  3  0.75'  V  0.70  a.  0.65 0.60  *d 0.13 d  HARTREEFOCK 5  Sen _l  0. 10  E o  4c 5  6  ci 4c  6  \6c b  1 SB [42/2] 2 NMQ53I/2Q 3 NMU062/42) 4 OF 84-GTO  99-GTO I09-GT0 84"G(CI) I09"G(CD  EXPT.  °I  TOTAL ENERGY  -76.00 _  .02-  lb, LSHM 3  .04-  ^. I?  2  2.4  5 6 4c 6c  '  H0 DIPOLE MOMENT ^  2.6-  ">  EXPT.  (a)  ~0.I2  Q  lb, Pm a x  HARTREE \ -FOCK oo  X  I  2  0  6  HARTREE -FOCK  .2  v  \ 3 iHFUmiKi56  .08  ci  5-763-4  CI EXPT  2.2-  o o  S  2.0  4c 6c 1  0  40  80  I20  1  f  6c  Exptl. Energy  -76.5-  A - - A  8  H  c  "(d)" ~ "  1  0  40  80  I20  Number of Contracted GTOs Fig. 5.11  Convergence of momentum distribution properties, (a)p and (b)leading slope at half maximum (LSHM), (c)dipole moment and (d)the total energy as a function of basis set complexity. The ^max' L S H M and dipole moment are plotted in the left hand side while the total S C F Hartree-Fock energy is plotted in the right hand side. The estimated Hartree-Fock limit and total experimental energy are shown by the dashed lines in Fig. 5 . l i d . The broken vertical line in Figs. 5.11a-c separates the single configuration (Hartree-Fock) and many configuration (CI) results. A l l quantities are quoted in atomic units except for the dipole moment which is quoted in debyes (D).  147 above the total  3a,  energy  and  which  2a, has  orbitals are been  used  also quite sensitive to these effects. traditionally as  the  principal  diagnostic  wavefunction quality is shown as a function of the number of contracted on  Fig. 5 . l i d .  represented  The  by  estimated  dashed  lines.  Hartree-Fock limit It  is  of  great  and  importance  the  of basis set (Fig. 5.11a  total energy.  and  5.11b)  Whereas for most purposes  is somewhat  to m a x  note  for  GTOs  experimental energy  convergence of the calculated momentum space properties ( P function  The  that  are the  and L S H M ) as a  slower than  that  for  one can consider the total energy  to be converged essentially at the H F limit (see Table 5.2) at the 84-GTO level the same is not true for the momentum space properties which converge at 99-GTO level as The  shown for the  lb,  orbital of water  (Figs.  5.11a  and  the  5.11b).  convergence of the dipole moment (Fig. 5.11c) is less regular by comparison  to the P  m a x  >  L S H M and total energy. These differences in convergence are  not  unexpected in view of the fact that the error in a calculated molecular property (e.g. the dipole moment or the momentum distribution) is first order with  respect  to the error in the wavefunction whereas the error in the total energy is second order [FB87].  The  findings  of  the  present  work  stress  the  Variational Theorem, i.e. the constraint of energy  need  for  proper  minimization  use  must  of  the  always be  accompanied by correct prediction of an adequate range of molecular properties. Since the low momentum regions of the regions of r-space  which make  orbital M D s include contributions  little contribution to the  total energy,  from  it is not  unreasonable to expect that the X M P s may be inadequately described by methods which emphasize only the energetics of the system when choosing the basis  set.  148  While wavefunctions they  cannot  more extended  give more  than  than  0.0005  109-GTO have been constructed [FB87]  hartree  improvement  to  the  R H F energy  since the Hartree-Fock limit has already been reached within that accuracy Table  5.2  and F i g . 5 . l i d ) .  The present  109-GTO basis  (see  is also saturated  diffuse basis functions (at least at the H F level) so further  with  significant changes in  the M D are not obtained at the T H F A level.  Finally it should be pointed out that comparisons of X M P s and  OVDs  demonstrate  the  extreme  sensitivity of the  and calculated M D s  E M S technique  certain details in the electronic wavefunction to which the  energy  towards  is much less  sensitive. As has been shown by E M S , agreement of theoretical orbital MDs with X M P s can be obtained for more tightly bound orbitals such as the l b (and  probably  also  la,)  orbitals  already  at  the  simple  DZ  2  and 2 a ,  level.  These  distributions could then tend to dominate the total electron momentum distribution and  thus  no  experiment  serious  for the  discrepancy  total momentum  might  be  detected  between  theory  distribution as observed for instance  and  in the  Compton profile [TE74, SW75]. The present work clearly demonstrates the orbital specificity as  of E M S which provides more detailed information than  Compton  scattering  [TE74,  SW75]  which  measure  the  methods  total  such  momentum  distribution.  Improving  the  accuracy  of  calculated  orbital  MDs  and  OVDs  would  importance not only from the computational point of view but also with to  the  theoretical  interpretation efforts  of  towards  current  EMS  the  solution  experiments. of  the  In  this  Hartree-Fock  regard,  be  of  respect recent  equations  in  149 momentum space [NT81], alternative theoretical approaches 'minimal energy would  be  of  [GB85] other than  criterion' and also new ways to model molecular  interest.  optimized small basis  On  the  other  hand  the  design  of  wavefunctions  momentum  sets, although computationally attractive,  the  density  will lose physical  meaning if the energy is non-optimal.  5.8.  SUMMARY  Considerable  improvement  has  been  obtained  in the  degree  of agreement  with  experiment of T H F A calculations of MDs using wavefunctions at the Hartree-Fock limit in the case of the two outermost orbitals ( l b , THFA case  level there still remains of the  lb,  data  and final for  all 4  appreciable  orbital. Incorporation  calculation of the initial  an  and 3 a , ) . However at  discrepancy most  of correlation  and  notably  relaxation  in  the  effects  by  ion-neutral overlap distribution using CI wavefunctions states results valence  especially for the  lb,  the  for  the  in generally very good agreement with the E M S  orbitals.  However quite  small discrepancies  still  exist,  orbital. With both the highly extended basis sets as well  as with the successively improved calculations of the ion-neutral overlap amplitude using  CI  wavefunctions,  the  shift  consistently been towards lower P  m  better agreement with experiment. description  of  the  experimental  accompanying improvement  x  the  theoretical  MDs  and  OVDs  has  and higher cross section, and thus toward  It is noteworthy momentum  in the  dipole moment as well as other  a  of  that an improved theoretical  profiles  prediction of both  occurs the  when  there  total energy  is  an  and  the  properties.  The present work shows that it is important to perform the overlap calculations  150 with  "near-complete"  difficult.  basis  sets  of  whether  Investigation  -  a  method  the  which  remaining  is  computationally  small  but  finite  between the measured X M P and calculated M D s and OVDs for the is due to the of  the  some  still unaccounted  correlation energy other  factor  for part of the  is accounted  for  differrence  lb,  of H 0 2  electron correlation energy  in the  such as a breakdown of the  very  (88%  140-G(CI) calculation), or  PWIA  due  to  to distortion of the  incoming and outgoing electron waves by the polar target molecule will have to await  further  reaction  developments  theory.  A t the  orbitals of water,  with  in  quantum  experimental further  mechanical  level  computation  E M S measurements  improved statistical  accuracy  and/or of the  will  be  (e,2e) valence  needed  as  even finer details of the target molecule-ion overlap and the (e,2e) reaction theory are investigated.  The  present  correlation  and  relaxation and the fact that good agreement is only obtained between theory  and  experiment  work  if  the  computed with obtained  clearly demonstrates  ion-neutral  overlap  the  effects  (i.e.  the  of electron  electronic  structure  sufficient accuracy. The good overall quantitative  between  experiment  and  theory  for  the  valence  factor)  is  agreement now  orbitals of H 0 2  also  indicates the general suitability of the P W I A for the study of small molecules by EMS at  at  impact energies  5 = 45°.  experiment targets  and  However  of 1200eV using the  small  discrepancies  symmetric non-coplanar  between  the  PWIA  treatment  suggest that a careful investigation of distortion effects in  present studies  particular  highly  polar  molecules  also demonstrate clearly the  momentum portion of molecular wavefunctions  need  would  be  geometry and  by molecular  informative.  The  to consider carefully the low  and the importance in many cases  151  of electron correlation in the These  effects  to  used  be  will have for  highly  valence orbitals of small molecules such as H 0 . 2  to be taken accurate  into account if molecular wavefunctions  investigation  of problems  calculation of charge, spin and momentum distributions.  of bonding  and  are the  C H A P T E R 6 . AMMONIA  6.1.  OVERVIEW  As  part  of  a  continuing series  of E M S studies  detailed experimental  and  molecule is reported  in this chapter.  [HH76],  theoretical  study  of the  of the  hydride  valence  orbitals of the N H  Previous studies  Camilloni et al. [CS76] and Tossell et al.  interesting  features.  momentum  resolution, significant  differences  momentum  profiles  appropriately  distributions  Although  (XMPs)  (MDs) were  these  and  observed  for  earlier  3  studies  were the  convoluted  outermost  a  by Hood et  [TL84] have  between  the  of N H  molecules  3  al.  shown several  limited  measured  by  poor  experimental  theoretical  'non-bonding'  momentum 3a,  orbital  [HH76, TL84] when even near-Hartree-Fock wavefunctions were used. Particularly interesting NH CH 2  has  been  comparison  of  the  'lone  pair't  XMPs  of N H  3  and  in the E M S study by Tossell et al. [TL84] in which they showed that  3  derealization of the more  the  detailed  'lone pair' occurs upon methyl substitution on N H . Recent 3  experiments  and calculations  (see  chapter  this study to compare the outermost orbitals of N H NH(CH ) 3  2  and N ( C H ) 3  3  9)  have  and N H C H 2  now 3  extended  with those of  which show even more extensive derealization [BB87a,  3  BB87b].  Reports appeared method  of  other  recently for  experimental [OM83,  obtaining  OM84,  the  probes  of  electronic  OI86]. Ohno et al.  information  about  the  density  distributions  [OM83] have quality  of  have  proposed LCAO  a  MO  t Extra caution must be exercised in equating lone pairs (a simple valence bond concept) with the outermost 3 a , molecular orbital. Even in a minimal basis set there is some H i s character attributed to the 3 a , orbital in N H [SB72]. 3  152  153 wavefunction  'tails' by calculating the exterior  densities.  They  assumed  molecule  is  boundary  surface  largely  (in  a  dependent  classical on  the  (EED) and interior (IED) electron  sense)  that  spatial  which is given by the  chemical  electron  envelope  reactivity  distribution  of  a  outside  a  of spheres obtained  from  the  van der Waals radii of the individual atoms that comprise the molecule. Thus by integrating the  the  M O wavefunction  exterior electron density  Although  the  EEDs  for  outside  the  van  der  Waals boundary  (EED) can be calculated for each each  orbital  cannot  be  surface,  molecular orbital.  measured  directly  in  an  experiment, they have compared the ratio of E E D s for separate orbitals with the ionization branching ratios  for these orbitals as  measured  by Penning ionization  *  electron spectra (PIES) [0M83, 0 M 8 4 ] obtained using He ( 2 S ) metastable atoms. 3  A  comparison of E M S measurements and the  measured  PIES ratios in N H  3  corresponding calculated E E D and  is given in the present work.  The ammonia sample used in the present study was supplied by Matheson Ltd. with a purity of 99.99%.  6.2. BASIS The  SETS  experimental  averaged  FOR results  momentum  LITERATURE in  distributions  S C F L C A O - M O wavefunctions. basis diffuse  the  up  to  effectively  WAVEFUNCTIONS  present  work  calculated  for  are a  compared  variety  with  of selected  spherically ab  initio  The selection was done in such a way, that the  sets cover a wide range from sets  SCF  double  Hartree-Fock  zeta limit  (DZ) quality quality.  to  These  together with selected calculated properties are shown in Table 6.1.  extended  and  wavefunctions  Table  6 . 1 . P r o p e r t i e s of  WavefunctIon  DZ  6-311G  6-311+G  H 0 D 2 G  119-GTO  126-GTO  Theoretical  SCF and  Ni t r o g e n  Hydrogen  Basis  Basis  Set  CI  Wavefunctions f o r  Energy(a.u)  Dipole Moment(D)  Set  (10s5p)/  (4s)/  [4s2p]  (2s)  (11s5p)/  (5s)/  [4s3p]  [3S]  (11s5p1d)/  (5s1p)/  [4s3p1d]  [3s1p]  (12s6p)/  (5s)/  [5s4pJ  13s]  (11s7p)/  (5s)/  C5s4p]  [3s]  (13s8p2d)/  (8s2p)/  [Bs5p2d]  [4s1p]  NH3  -56.1777  (19s10p3d1f)/  (10s3p1d)/  [10s8p3d1f]  [6s3p1d]  (23s12p3d1f)/  (10s3p1d)/  [14s10p3d1f]  [6s3p1d]  Hartree-Fock  P  P  EED  0.69  0.68  1.98  max 3a,  max 1e  -56.1813  2 . 2 2 9 2  0.56  0.67  2.58  -56.1790  2.2564  0.54  0.67  2.50  -56.22191  1.6598  0.58  0.66  2.41  -56.2245  1.6440  0.57  -56.2246  1.6417  0.54  -56.226  1.63°  b  0.63  I imt 119-G(CI)  -56.5155  1.5952  0.52  126-G(CI)  -56.5160  1.5891  0.52  1.47149*  0.5210.02  d  ExptI. Ratio refer*  to  Estimated  [FB87].  In  Oa,/1e)  Estimated uncertainty Is  10.00015D  2.60  Is  10.001  10.02.  Experimentally-derived non-vibrating, Uncertainty  0.6210.02  [0186].  Uncertainty is  0.63  In  n o n - r e l a t i v i s t 1c.  the g r o u n d  vibrational  I n f i n i t e n u c l e a r mass  state  [MD81].  total  energy  (PB75,  FB87).  Ratio  155 Some important features of the various basis sets are discussed below, (1)  Double zeta Two  sets  atomic  of contracted  orbital. No  Gaussian  additional  type  functions  polarization functions  are  used  are  for  each  employed.  this basis set, proposed by Snyder and Basch [SB72], the least s- and p-functions  of the  nitrogen  atom  are  In  tight  represented by a single  primitive Gaussian function. (2)  6-311G This  "split-valence" basis  three  contracted  Gaussian nitrogen  Gaussian  functions atom.  of Krishnan et al. functions  to represent the  A restriction  of the  and  [KB80]  two  valence  uses  uncontracted  part  2s  and  split-valence method  a  set  of  primitive 2p  of  the  is given by  the fact that the atomic s- and p-valence orbitals are described by a common set of exponents and differ only in the contraction coefficients. The  nitrogen  ls-shell  set  represented  functions.  The  contracted  to  three  Gaussian-type  are  used  for  . functions  basis  is  for  each  hydrogen  atomic  six  consists  orbitals.  regarded as a triple zeta class basis (3)  by  contracted of  five  Since three  valence  orbital,  Gaussian s-functions,  sets of  basis  6-311G can  be  set.  6-311G** An  augmentation  of 6-311G by including polarization functions  in the  nitrogen (3d) as well as in the hydrogen (2p) basis sets [KB80]. The total energy is improved from -56.1777a.u. to -56.2102a.u.. (4)  6-311 + G A different  approach  to augment 6-311G. In contrast  to 6-311G** no  156 polarization functions p-type  primitive  included have  in the  been  were  added  Gaussian nitrogen  chosen  but  function basis  following  a  one  diffuse  s- and  with  a  The  additional valence  set.  proposal  common  one  of Clark  diffuse  exponent  et al.  were  functions  [CC83]  for  anion calculations. The hydrogen basis set is the same as for 6-311G. (5)  HDD2G A  (9s5p)  set  of primitive Gaussian functions  for  nitrogen  given by  Huzinaga [H65] was contracted to [3s2p] by Dunning and Hay [DH77] and  augmented  with  two  diffuse  2s  and  two  diffuse  2p  functions  appropriate for describing atomic Rydberg orbitals. The hydrogen  atoms  are represented by the (5s)/[3s] set of 6-311G. (6)  56-GTO This extended basis  set of Rauk et al. [RA70] includes 91 primitive  Gaussian  functions,  contracted  to  56  contains  polarization  functions  in  form  Gaussian-type of  nitrogen  orbitals.  It , also  3d-orbitals  and  hydrogen 2p-orbitals. (7)  119-GTO This M O wavefunction of Feller et al. [FB87] with its 119 contracted Gaussian-type orbitals represents the most extended basis set currently available in the literature. It includes d- and f-orbitals for nitrogen as well as p- and d-orbitals for hydrogen. With a predicted total energy of -56.22456a.u.  119-GTO is one of the best available basis set  with  regard to the total Hartree-Fock energy. (8)  126-GTO In  addition  to  these  above  basis  sets,  already  available  in  the  157 literature,  a  new  126-GTO  wavefunction  has  been  developed  in  the  present work. This wavefunction is very close (within = 1 millihartree) to the estimated Hartree-Fock limit and is discussed in the Sec. 6.2.1. Further  significant improvement in the  wavefunction requires  inclusion  of electron correlation as discussed later in Sec. 6.2.2.  All  calculations (except  experimentally  for the  estimated  (r =1.012A,  0HNH  NH  56-GTO basis  equilibrium  = 1 O 6  - °) 7  [  B P 5 7  set)  geometry  1-  T  h  e  have  been  of  the  dependence  performed NH  of the  in  molecule  3  total energy  the chosen geometry was studied by Rauk et al. [RA70] for the  the  on  56-GTO basis  set. For the experimental equilibrium geometry they calculated an S C F energy of -56.22150a.u.. B y variation of bond length and bond angle the S C F energy could be  optimized  %NH  =  1 0 7  basis  set  later  section)  to - °  was  2  -56.22191a.u. (  S C F  performed  a  geometry  equilibrium geometry). in this  of Ohno et al.  equilibrium geometry.  in  of r ^ j j  =  The calculation for  S C F optimum geometry  [0186]  for  1.89033 bohr  56-GTO  are  since  also  the  the  sets  comparison  as  (see  given for the S C F  The E E D s for all other basis sets however are calculated  Ohno et al.  between  56-GTO  EEDs  for the experimental equilibrium geometry. The same geometry for the basis  and  the  [0186]  concepts  was  chosen  of exterior  averaged momentum distributions (see Sec. 6.7).  in order electron  to  allow  densities  respective  a conclusive  and  spherically  158 6.2.1. A 126-GTO Extended B a s i s Set for N H The  extended  basis  set  (23s,12p,3d,lf/10s,3p,2d) Gaussian Prof.  type  for  NH  primitive  (GTO) basis  E.R. Davidson  set  was  (Indiana  which  3  3  consists  contracted  developed  to  in an  University).  The  of  an  even  tempered  [14s,10p,3d,lf/6s,3p,ld]  interactive  collaboration with  s-components  of  the  cartesian  d-functions and the p-components of the f-functions were removed to avoid linear dependence, forming the  final  126-GTO basis. The s- and p-symmetry portion of  the basis for nitrogen was created from energy optimized exponents [SR79, FR79] using  an  even-tempered  restriction on the  exponents.  The hydrogen  s-exponents  are as previously employed by Feller et al. [FB87].  Even tempered exponents form geometric sequences, r\  i  =  a/3  1  = 1 , . . .N)  (i  where different (a,/3) pairs are used for each type of function (s,p,d...) and each atomic  number.  If  N  is  the  symmetry, then the dependence  number  of  of a^^XN)  Gaussian  primitives  of  a  certain  and P^p^N) on N can be described  by simpie functional forms so that it is possible to generate near-optimal sets of arbitrary  size.  a = 0.05012, s  The  a  /3  and  /3 = 2.12175  and  G  values  for  a = 0.03971 p  the and  nitrogen  (22s, l i p )  0 = 2.35056. P  set  are  Experience  has  shown that the energy-optimized even-tempered sets are considerably improved for many This  properties i f they extension  was  primitives,  which  simply  a.  as  are  made  would  The d-type  extended by one more  for  have  the had  current  basis  diffuse  sets.  optimum exponents  and f-type  s and p primitive.  Thus, of  a|3,  the  most  diffuse  are  now  given  polarization function exponents  were  taken  159 from  earlier  work  designed from the  by  Feller  et al.  [FB87].  The  basis  set  for  ammonia  was  109-GTO basis employed for water by Bawagan et al. [BB87],  which was essentially converged! for total energy, dipole moment and momentum distribution. function  The  basis  set  has  been  designed  to  saturate  limit so as to give improved representation  orbitals. It  should be noted  that due  to the  of the  large  the  diffuse  (r-space)  number  basis  tail of the  of functions  which  have been used the wavefunction is expected to be fairly insensitive to the  exact  choice of exponents.  As  in  previous  FB87a],  the  studies  of  momentum  assessment of the  distributions  quality of the  in  this  laboratory  wavefunction is based  [BB87,  upon  the  values of the one electron properties calculated from the wavefunction at several levels of sophistication (see Table 6.1) using the experimental geometry 1.012A and  =  (RHF),  singly  SD-CI)  and  and  106.7°. The methods doubly  multireference  excited singly  =  applied were restricted Hartree-Fock  Hartree-Fock and  r^jj  doubly  configuration excited  interaction  configuration  (HF  interaction  (MRSD CI). The all electron CI convergence has been shown to be improved (ie. more  correlation  Hartree-Fock and hence energy  energy  was  recovered  virtual orbitals are  with  transformed  fewer  to K-orbitals [FD81,  K-orbitals were used. The configurations  selected  based  on second  order  configurations)  used in the  Rayleigh-Schro dinger  when  CP82, MRSD-CI  perturbation  the  FB85] were theory  t These results are considered effectively converged since the total energy is within 0.5 millihartree of the estimated Hartree-Fock limit and because the dipole moment, quadrupole moment and the momentum distributions do not change significantly as even more basis funcitons are added. Other basis sets containing up to 140 GTOs were actually tested for water with no significant change in either the momentum distribution, total energy and dipole moment at the S C F or CI level.  160 and  coefficient  contribution  in  the  HF  SD-CI,  with  configurations being kept. This selection was necessary of  configurations  associated  with  the  extended  basis  all  singly  excited  due to the large  number  set  the  exceeding  the  current  variational capacity of the Indiana group.  Table 6.1 lists S C F and CI energies and one-electron properties for the  126-GTO  basis. The CI was done by the Indiana group in two steps. First, a small CI was  done. For this  configurations  were  15,868  spin-adapted  selected.  The  Hartree-Fock singly and doubly excited  selection  singly excited configurations  and  doubly excited configurations  such that the  procedure  using second  order  the  space  largest  was  chosen  coefficients.  A  from the  CI energy.  (2a  in the  threshold  systematically maintained for the ion states (3a for calculation of the  perturbation  configurations  coefficient  retaining theory  neglected configurations  contribution of less than one millihartree to the reference  involved  satellite  of  at  have  Then the  all  the  on  the  a total MRSD-CI  H F SD-CI's having least  0.030  was  and ( l e ) " \ but a higher one  was  used  states as  discussed  later  Sec.  6.3). The dimension of the neutral MRSD-CI was 31,845 out of the total of  3,465,270 possible singly and doubly excited configurations. This MRSD-CI  (see  energy  is -56.5160a.u..  Based on the energy  estimated  of -56.563a.u.  non-vibrating, non-relativistic, infinite nuclear  [FB87], the  correlation energy  limit is -0.337a.u. . Using this estimate,  based  on the  mass  total  Hartree-Fock  the calculations recovered 86.5% of the  total correlation energy (see Tables 6.1 and 6.2).  161 6.2.2.  Inclusion  of  Correlation:  Calculation  of  the  Ion-Neutral  Overlap  Distribution The  highly extended  126-GTO  basis  set  is  used  as  the  foundation  for  more  accurate calculation of the E M S cross section by incorporating correlation in both the  neutral  OVD  and the  (see  performed  Eqn.  2.11).  using the  The overlap then  final  the  same  calculation of the  The  calculations  on  full  ion-neutral overlap distribution,  the  ion  same molecular orbital basis  amplitudes  have  ion state, i.e. the  computed form  as  from an  the  MO  ion states involved  states  as  for  of  the  CI wavefunctions expanded  a similar  in  the  process  ammonia neutral  for  the  neutral  were done for the (3a i ) "  1  and (le)"  ionization processes.  1  molecule. ion states  basis.  to that of the  with the exception of no R H F calculations being done. H F SD-CI  were  and  The  neutral MRSD-CI  The coefficients for the  MRSD-CI were chosen based upon the coefficient contribution using a threshold of 0.030 for ( 3 a , ) " in  and ( l e ) '  1  1  . The energies for these two cation states are listed  Table 6.2. The calculated vertical L P . values  good  agreement  AC78].  with  the  experimental  values  (10.94eV and (10.85eV  16.50eV) are in  and  16.5eV)  [BS75,  The corresponding calculated pole strengths (=0.87) shown in Table  indicate  that  contrast  the  many-body  these (2a,)  effects.  states 1  process The  correspond leads  to  calculated  pole  to  essentially  a  manifold of  strengths  single  and  final  particle ion  energies  states.  states for  the  due  6.3 In to  (2a,)"  1  ionization process are also shown in Table 6.3.  In  order to get the pole distribution of the  performed.  The  calculation  used  a  (2a,)"  symmetrically  t A l l calculations were done in C symmetry point group operators were maintained. g  1  a different closedt  set  type of CI was of  configurations  and closure with respect to C g  y  \  Table  6.2. CI Calculations (a.u.) ofthe  Ground and Final Ion States  of NH  . *c,d E(corr)  -56.563  -56.226  0.337  86.5  0.326  -56.1642  -55.796  0.368  88.6  0.312  -55.9566  -55.597  0.360  86.6  E(CI)  E(SCF)  AE(CI)  E(exptl)  NH  -56.5160  -56.2246  0.289  (3a,)"  -56.1212  -55.7950  (1e)"  -55.9072  -55.5957  X 'A,  NH  +  3  NH* 3  * c  1  1  AE(CI) • E(CI)  Includes relaxation for  b  f  r  _. ,_, * c, e %E(corr)  - I.P.) a.u.  ionic states.  - E(HF) for neutral, E(corr) « E(exptl) -  %E(corr) • AE(CI)/E(corr) For the ionic  3  - E(SCF)  For the ion states, E(exptl)* -(56.563  E(corr) » E(exptl) 8  Using the 126-GTO Basis Set E(HF)  STATE  3  3  E(Koopraans) for Ion.  x 100  states the energy  refers to a Koopmans' energy.  Oi  to  Table  6 . 3 . CI C a l c u l a t i o n s  (126-G(CI))  of the  Pole Strengths  and  Energies CI  (eV)  in the Binding  Energy  Coefficients  Spectrum of  NH3  Predicted Intens1ty  STATE  Energy  (3a,)" (le)"  1  1  (2a,)"  1  3  Pole  Strength  10.94  d  0.8744  16.50  d  0.8781  b  C(2a,)  C(1e)  C  C(3a,)  C  -0.0027  C  2  (  1 .000  26.32  0.0137  0.9854  -0.1312  0.0133*  28.49  0.4818*  0.9938  0.1097  0.4758*  30.22  0.1358*  0.9877  -0.1439  0.1325*  31 .03  0.0750*  0.9897  0.0947  0.0735*  31 .05  0.0017*  0.9505  0.1320  0.0016*  32.86  0.0524*  0.9909  0.1128  0.0515*  34.61  0.0434*  0.7180  -0.6862  0.0223*  35.51  0.0004  0.5447  0.0493  0.0001  35.86  0.0003  -0.3094  0.5978  0.0000  36.08  0.0022  0.3064  0.8975  0.0002  36.59  0.0039*  0.9859  0.1226  0.0038*  reference  Spectroscopic  2  0.9996  r(*)=0.794 Neutral  S C (2a )  energy  factor  1s - 5 6 . 5 1 6 0 a . u .  (S )  as g i v e n  2  Coefficients  of primary  Experimental  vertical  I(*)=0.775  1n  hole o r b i t a l s  IPs  Eqn. 2.32. 1n  f o r the ( 3 a . ) 1  the generalized and (1e)  'overlap'  processes  orbital  a r e 10.85  (see  Eqn. 2.11).  and 16.5 eV,  respectively  [TB70].  1—  1  O) 00  164  containing  the largest  coefficient  contributions for 1 5  preliminary calculation using all single excitations from  2  A , roots  from  a  all hole states. These  reference configurations were then used in a calculation involving up to triple excitations from all hole states with perturbation selection.  The  accurate CI wavefunctions, as determined  properties  ,  were  then  <;qi^N 1|q ^N ^ ^ l  Q  >  c ^  used £j  g  w a v e  to f  u n c  from the study  evaluate  ti ns  the  of one-electron  molecule-ion  "overlap"  have the same form as an MO  0  expanded in the 126-GTO basis.  The overlap orbitals were then normalized using S w h e r e ,  S o  S  = | |<*  2  N_1 f  |* > I | N  [6.1]  2  0  takes values near 0.9 for primary hole states without strong satellites. This 2  procedure removes the spectroscopic factor (or S ) information from the calculated 2  OVDs (i.e. the OVDs are each normalized to unity) however the values of S are shown in Table 6 . 3 . The "overlap" orbitals can be expressed in terms of the molecular orbitals used in the CI calculation as f  I  =  L  i  C  i l ^ i  [6  Sum rules can be derived to show that the sum of the coefficients  2]  squared  times the spectral factor squared equals 1 / 2 P - when summed over all M 11  -  =  +  s  1/2 ion states. P J J is the occupation of orbital \[/^ in the neutral wavefunction. The  values of C- show clearly which primary hole state \p'^ is associated with T  165 each CI state. The spectral constants S , C ^ and S C 2  for  the  ion states of the  accounted for the 2 a , "  The  spherically  1  averaged  2  molecule. A t least  2  are listed in Table 6.3  78% of the  sum  ZCj Sj 2  2  T  was  process (Table 6.3).  momentum  distribution was  then  generated  for  each  S C F occupied orbital as well as the normalized overlap distribution (OVD) using the  ion-neutral overlap. In addition each  neutral  molecule ground  state  orbital resulting from the CI of the ground state molecule alone was To  account  for  the  finite  experimental  momentum  resolution  ,  a  natural  evaluated. momentum  resolution function (Ap = 0 . 1 5 a " ) was folded into the calculations. 1  o  6.3. BINDING  ENERGY  SPECTRA  Ammonia, in its ground neutral  1  A , state, has C g  y  symmetry and the electronic  configuration can be written as:  (1a,)  (2a,)  2  The binding energies from  high  2  (1e)«  (3a,)  2  of the two outer valence orbitals of N H  resolution  U V photoelectron  spectroscopy  using  3  are well known  H e l sources  [TB70,  RK73]. In addition, low resolution binding energy spectra including the 2 a , valence orbital are  available from X P S [BS75, AC78],  [VB72] and E M S [HH76]. F i g . 6.1  shows  the  dipole (e,2e)  binding energy  inner  spectroscopy  spectra  of N H  3  obtained in the present work at an impact energy of 1200eV + binding energy and at relative azimuthal angles of 0° and 8 ° . The relative intensities in the 0°  166 spectrum  (p^O.lao" ) 1  and  the  8°  spectrum  (p = 0 . 6 a ~ )  reflect  1  o  the  different  symmetries of the valence orbitals of N H . In particular the outermost  3a,  3  le  orbitals  are  clearly of dominant  valence  orbital is  aligning  the  's-type'. t  spectra  with  The the  'p-type'  energy  known  character  scale  in  whereas  Fig. 6.1  the  has  2a,  been  vertical ionization potential  of  and inner  set the  by 3 a,  orbital (10.85eV) as measured by photoelectron spectroscopy [RK73]. On this basis the respective maxima of the peaks observed for the le and 2 a , values  consistent  [BS75, AC78].  with  earlier  measurements  The binding energy  low momentum  U P S [TB70,  positions are  resolution E M S study  This earlier spectrum corresponds  in  RK73]  also consistent  (E =400  bands occur at  with  eV) [HH76] done  o  to p = 0.5 a ~  and has  1  0  and X P S an  at  earlier  <fi=  10°.  an energy resolution  of 1.44eV fwhm.  In  the  ionization of inner  valence  orbitals  many-body  structures  (satellites)  are  frequently observed [CD 8 6]. A similar case is seen in the inner valence region of NH  3  side  as shown by the broad peak and the structured tailing on the high energy (30-45eV) of the  (2a,)"  earlier E M S study of N H photoelectron study Mg  a  by  additional structure  [BS75,  Banna  [BS75] and the  hole.  Similar  broad  tailing  can  be  seen  in  an  [HH76] as well as in dipole (e,2e) [VB72] and X-ray  spectroscopies  reported K  3  1  Zr  et al.  AC78]. [BS87]  A  recent  also  shows  M ^ [AC78] X P S spectra  located on the  high energy  synchroton similar  radiation  PES  structure.  The  both show indications of  side of the  main ( 2 a , ) "  1  peak  although these authors did not comment specifically on this at the time. In these t The 's-type' and 'p-type' nomenclature is used to refer to whether the momentum profile (XMP) has a minima at p - 0. This follows from the fact that theoretical momentum distributions have a node at p = 0 for atomic orbitals with 1*0 [LN75].  1  1  1  1  I 3a^  1  1  1  1  1  1  1  1  1e  1  NH  IT  —i—i—[••  1  3  i  4>=8° i i i  1 5.0  ' 10.0  '  1 15.0  1  1 1 20.0  S W A *  1 25.0  I  30.0  BINDING ENERGY  T  I  1  5.0  1  1  i  i  10.0  1  1  i  i  15.0  1  1  i  I  20.0  1  1  i  i  25.0  BINDING ENERGY  • 6.1  35.0  l  40.0  i  (b) l  45.0  (EV)  1  1  i  i  30.0  1  1  i  i  35.0  1  1  1  1  40.0  r  1  45.0  1  (EV)  Binding energy spectra of N H measured by electron momentum spectroscopy at an impact energy of 1200eV binding energy (a) 0 = 0° and (b) 0 = 8 ° . 3  +  168 earlier X P S studies the choice of background (sloping) is somewhat  subjective and  this introduces some uncertainty in the determination of intensities and may also preclude  observation  of the  low intensity  structures  at  higher  binding energies.  This complication does not occur in the E M S spectra since the coincidence method ensures a level baseline.  To obtain improved statistics in the many-body region corresponding to the inner valence  ionization spectrum,  been  summed and the  well  fitted  by  a  0 = 0°  the  and  <j> = 8°  spectra  above  resulting profile is shown in Fig. 6.2a.  convolution of five  gaussian  peaks  21 eV  have  The spectrum is  of equal  width  (3.6  eV  fwhm). The fitted peak positions and intensities are reported in Table 6.4. This only  partially  resolved  compared - with present  full  four  Bieri  different  et al. pole  appropriate  energies  and  (x2.5)  scaled  and  strength  (see  Cacelli  distributions  and gaussian to  in  the  Tables  6.3  inner  valence  spectrum  6.2b-6.2e.  These  and  and  6.4)  is  involve the a  small CI  126-GTO basis as well as the Green's function caculations  [BA82]  theoretical  structure  calculations in Figs.  126-G(CI) calculation  calculation using the of  many-body  yield  the  et al.  [CM82]  (see  are  indicated  by  curves  theoretical  (3.6eV fwhm)  also  Table  vertical are  binding energy  6.4).  bars  at  The the  convoluted, summed profile  (solid line) in  each case.  The 126-G(CI) many-body calculation which is reported in the present work used the extended  126-GTO basis set (see earlier discussion). The actual S  pole strengths are strength  2  factors or  shown in Tables 6.3 and 6.4 and Fig. 6.2b. The total pole  'recovered' in the calculation in the energy range 20-37eV is 0.775. A  169  NH  S <  V i  Many-Body I nner  20.0  24.0  1  1  28.0  1  r  MJO  32.0  1  1  1  j  NH  40 X)  !  1  4AJ0  j  1  Structure  Va! e n c e  48.0  j  -?  1  1  1  1  1  1  -l  T~  1  Bierl  126-G(CI)  r-  (2a,)  NH,  (2a,)  3  Regi on  et  Greens  al .  Funct i on  ia:  Ul  i  (d)  2 _ J 20.0  1 24.0  '  •  28.0  L  H I  32JD  ' 1 1 1 1 1 1 J6-0 40 X) *4J0 4».0  2QJ0  •• • • 2*J>  12.0  2BX  - i — i — i  NH  i  (2a,)  3  Srrol I  XJO  *OS>  i — i — i — i — i — i  NH  3  (2a,)  Cocel I i  126-C(CI)  Greens  MR-lp2h-CI  **X> 4«J)  et  (e)  (c) 20.0  24J0  MJO  12J0  MA  40.0  **Jt  BINDING E N E R G Y ( E V )  Fig. 6.2  al .  F u n c t i on  20.0  24.0  2tU>  -I  I  I  I  32X1 XD  L.  *0.0  44J0  BINDING ENERGY ( E V )  Comparison of the inner valence binding energy spectrum with theoretical predictions. The experimental inner valence spectrum (a) corresponds to the <j>= 0 ° + 8° summed spectrum. The calculated spectra refer to (b) the present full 126-G(CI) calculation , (c) the MR-lp2h-CI calculation and the Green's function calculations of (d) Bieri et al. [BA82] and (e) Cacelli et al. [CM82].  i  Table 6.4. Experimental orbital of NH .  and  c a l c u l a t e d energies (eV)  and r e l a t i v e pole  strengths f o r the Ionization of  the 2a,  3  Experimental  '  Theoret ical  R e l a t i v e Pole Strengths(this  work)  Ref.[BA82] Greens Function  This Work(126-G(CI))  27.6  100.  100.  100.  26.32  Rei. Pole Strength 2.7  30. 3  13.  18.  16.  28.49  100.  27.48  100.  33.2  13.  13.  13.  30.22  27.8  29.50  31.0  35.5  12.  14.  13.  31 .03  15.4  30.28  18.0  41.8  8.  12.  10.  31 .05  0.3  32.86  10.8  34.61  4.7  36.59  0.8  Energy  EMS(0' )  EMS(8" )  EMS(0'+8* )  Energy  Energy  Energy  26.37  Rei. Pole d Strength 4.3  27.45  Rei. Pole Strength 36.0  29.04  100.  0  Pole strengths are r e l a t i v e I n t e n s i t i e s with the major pole Estimated experimental pole strength uncertainty 1s ±4%. S C -- see text. Table 6.3 and F1g. 6.2b. S , spectroscopic f a c t o r . 2  2  normalized  Ref.[CM82] Greens Function  to  100%.  171 smaller  CI similar  calculation,  in spirit  [D87] to  which is referred  to  as  the  one-particle  MR-lp2h-CI,  two-hole  was done  Tamm  using the  Dancoff 126-GTO  basis to obtain a many-body spectrum extending to a larger energy range and is shown in F i g . 6.2c. This calculation yielded 48  2  A , roots in the energy  range  20-47eV. Although the M R - l p 2 h - C I t is of lower accuracy than the full 126-G(CI) calculation  (Fig.  6.2b),  the  MR-lp2h-CI  calculation  relative energy spacings are reasonably accurate. was  'recovered'  with  this  method  in  the  gives  pole  energies  A total pole strength  energy  range  20-4 l e V ,  whose  of 0.857 which  is  comparable to that of the experimental binding energy spectrum. The -3.0eV shift of the calculated binding energy spectrum (see Fig. 6.2c) relative to experiment is due initial  to  the  particular  state neutral  [CM82] involved  choice of the  reference  energy  (E =-56.28lla.u.)  for  the  molecule. The Green's function calculation of Cacelli et  a renormalized optical potential and employed a  44-GTO  al.  basis  set. The calculated energy from the 44-GTO wavefunction is -56.4086a.u. and the calculated dipole moment is 1.845D [CM82]. This calculation however yields only two  poles  in the  inner  valence  region  and  a  experimental results at higher binding energies  profile quite  different  from  the  (compare Figs. 6.2a and 2e). The  calculations of Bieri et al. [BA82] employed a similar Green's function approach. The ionization energies lines  were  Approximation  of the  main  (2a,)transition  calculated  [BA82]  using  the  (2ph-TDA).  The calculation employed an  as  well  two-particle-hole extended  as  the  satellite  Tamm-Dancoff gaussian  basis  set of ( l l s , 7 p , l d / 6 s , l p ) contracted to [5s,4p,ld/3s,lp]. This calculation also fails to predict the higher energy poles (compare Figs. 6.2a and 6.2d).  t The MR-lp2h-CI was generated using 12 dominant configurations for the neutral state and all one-particle two-hole configurations for the ion. This gave a total of 8094 configurations.  172 As can be seen in F i g . 6.2 the agreement with the experimental binding energy profile full  in terms  126-G(CI)  of energy positions and intensities is somewhat better for calculation (Fig. 6.2b)  than  for  either  of the  Green's  the  function  calculations in the region below 35eV which represents the upper energy limit for these  three calculations. A few  higher  energy  poles are  predicted by  the  full  126-G(CI) calculation. These results likely reflect the basis set dependence of the calculated pole strength distribution. Increased flexibility in the basis set has been found  to provide a better description of the  pole strength over a larger energy  range [FG87]. The MR-lp2h-CI calculation (Fig. 6.2c) gives a fairly good estimate of  the  overall  pole  strength  distribution and  relative  energies  over the  entire  experimental energy range (20-45ev) although the absolute energy scale is shifted by  -3.0eV.  It  is expected that  the  extension of the  full  126-G(CI) calculation  beyond =36eV would provide a similar pole strength distribution.  The 0°  similar spectral shape obtained throughout the and  region  0= are  8°  (see F i g . 6.1  24-45eV region at both 0 =  and Table 6.4) indicates that the  predominantly s-type and therefore  belong to the  poles in this  (2a,)"''' hole state.  However it should be noted that although some p-type poles due to (3a,)"''' predicted  in this  region  they  are  of negligible intensity  (see  Table 6.3).  are The  (2a,)"''' assignment of the structure to the high energy side of the main (2a,)"^ peak 32.2  In  is further  confirmed by  the  experimental  momentum  profile  measured  at  eV (see later discussion).  the  comparison  photoelectron  of  spectroscopy  binding a  few  energy points  spectra  as  obtained  from  should  be  remembered.  E M S and The  relative  173 intensities  corresponding  measured  by  the  differences  in the  to  two  ionization  techniques  from  are  the  different  expected  intensity ratios between  to  be  molecular  different  orbitals  [M85].  E M S and X P S indicate the  The  different  regions of the electron distribution being probed. X-ray photoelectron  spectroscopy  probes  p=lOa " ;  Y  the  high  momentum  M ^ = 132.3eV, p = 3 a  _ 1 0  components  )  whereas  (e.g.  Mg K =1253.6eV, a  low momentum  are probed in E M S . Notwithstanding the differences where  comparison  between  relative  intensities  components  different  energy  (0.1-2.0a " ) 1  o  in momentum the only case  measured  by  likely to be valid is the comparison of relative intensities same symmetry manifold  1  0  both  techniques  for states within  is the  [M85]. In such cases close agreement (when effects of  resolutions  are  taken  into  account)  has  been  obtained  between  E M S and intermediate photon energy (80-120eV) photoelectron spectroscopy as for example  in  the  Ar  3s"  1  [BB87d]  many-body  structures  in  the  inner  valence  spectra. It should be noted that relative intensities of the different inner valence poles will be matrix  element  strongly influenced by the in  photoelectron  (photoelectron  spectroscopy  (or  energy  dipole e,2e)  dependent) dipole at  lower  photon  energies where the individual partial photoionization cross-sections are still rapidly changing. As such, attempts [A86, AC87] to compare intensities in lower energy PES  and in E M S (i.e. binary e,2e)  It should also be remembered  are  not meaningful on a quantitative  that any such comparisons are  the often large differences in energy resolution.  basis.  also influenced by  174 6.4.  COMPARISON  OF  EXPERIMENTAL  THEORETICAL  PREDICTIONS  The experimental  momentum  2a,)  of N H  profiles  MOMENTUM  (XMPs)  of the  PROFILES  valence  orbitals  WITH  (3a,,  le,  shown in Figs. 6.3a-6.3f have been placed on the same intensity-  3  scale (i.e. absolute to within a single factor) as described below.  <j>= 8°  The  binding  energy  spectrum  (Fig.  normalization of the experimental momentum outer  valence  binding energy  spectrum  have  6.1b)  serves  as  the  profiles. The peaks each  been  fitted  basis  in the with  a  for  <p= 8° gaussian  peak taking into account the energy resolution as well as the known vertical IPs and Franck-Condon widths. Likewise the inner valence region has been fitted (see Sec. 6.3) using a template comprising five peaks each with 3.6eV fwhm. A l l the strength below).  in the  region 22-45eV is assigned to the  Considering the  relative  intensity (area) ratio of 1.00  peak  : 1.65  (for each orbital) corresponding to similar  to  that  [BB87] permits  employed  in  a very stringent  calculated M D s and OVDs to  one  an  calculation  on  #  areas : 1.61  orbital name  energy at which the 6.3c  and  6.3f that  a  3 a , :16:2a,  at the respective momentum  earlier  E M S study  quantitative  of H 0 2  comparison of the  orbital.  All  the  other  experimental  (see  values  which  chapter  XMPs  is 5)  with all  and  theoretical  value in each diagram (Fig. 6.3) below  experimental  'sitting binding energy'  particular X M P was measured. the  yields  with only a single point normalization of experiment  one  indicates  method  = 8 ° . This normalization method  normalizations are preserved. The energy the  this  (2a ,)"* state (see discussion  X M P corresponding  to  the  ,  i.e.  the  It can be seen from Figs. satellite  at  32.2eV on  the  175 higher energy side of the main (2a,)" (2a,)" 2a, 3a,"  1  1  peak (see Fig. 6.1) clearly belongs to the  manifold. The satellite intensity has been height normalized to the 'main'  XMP to facilitate shape comparison in Fig. 6.3f. 1  and le"  1  Any contribution from the  hole states in the 24-45eV region is expected to be small since  = 90% of the 3a,"  and le*  1  pole strengths  1  are predicted to be in the main  lines (see Table 6.3).  The  experimental  results  (Figs.  6.3a-6.3f)  momentum resolution EMS experiments  are  consistent with  the  earlier low  of Hood et al. [HH76] taken at lower  impact energy (400eV). It is of particular interest to note that the P 3a,  m a x  of the  XMP (0.52a ~ ) measured in the present study is similar to those reported 1  0  (O.Sao" ) by Hood et al. [HH76] and by Camilloni et al. [CS76]. However the 1  3a,  XMP measured in a recent study of N H  an XMP with a P  m a x  =0.4la ~  3  by Tossell et al. [TL84] shows  and which is  1  0  also relatively broader. In  contrast the present study which has permitted the XMPs to be placed on a common intensity  scale also allows  a much more stringent quantitative  test of  the theoretical calculations. The much improved momentum resolution ( 0 . l 5 a ~ ) 1  0  in the present work compared with that used (0.4a ~ ) by Hood et al. [HH76] 1  0  also permits calculations discrepancies  a much more stringent particularly were  available in the  in  suggested  present  the to  comparison of the  low occur.  momentum At  the  work this comparison is  region higher  measured XMPs with where  the  momentum  largest resolution  no longer dominated by the  momentum resolution instrumental effect. In addition the correct relative intensity scale obtained in the present study provides an additional quantitative test of the different wavefunctions  which was not possible in the earlier studies where each  176  NT E R M E D I A T E  MOMENTUM (A.U.) Fig. 6.3a-c  Comparison of valence X M P s of N H with M D s calculated from intermediate quality wavefunctions (a-c). Momentum distributions calculated from the (1)DZ, (2)6-311G, (3)6-31 I G * * , (4)6-3ll + G, (5)HDD2G, (6)56-GTO, (7J119-GTO and (8) 126-GTO wavefunctions are placed on a common intensity scale by using a single point normalization of the 126-GTO (and 126-G(CI) calculation) calculation on the le X M P (see text for details). 3  NEAR  HARTREE-FOCK  177  MOMENTUM (A.U.) Fig.  6.3d-f  Comparison of valence X M P s of N H with M D s calculated from near Hartree-Fock wavefunctions (d-f). Momentum distributions calculated from the (1)DZ, (2)6-311G, (3)6-311G**, (4)6-311 + G, (5)HDD2G, (6)56-GTO, (7)119-GTO and (8)126-GTO wavefunctions are placed on a common intensity scale by using a single point normalization of the 126-GTO (and 126-G(CI) calculation) calculation on the le X M P (see text for details). 3  178 orbital X M P was separately height normalized to each theoretical M D .  Theoretical M D s were  calculated  (see  Sec. 6.2)  from  the  various  shown in Table 6.1. A l l calculations were spherically averaged molecules  are  randomly  oriented)  and  have  been  folded  wavefunctions  (since the  with the  target  experimental  momentum resolution.  The S C F wavefunctions can be placed in two main groups, namely (i) Intermediate quality without polarization (DZ, 6-311G) and with polarization or diffuse functions (6-311G**, 6-311 + G, HDD2G); (ii) Near Hartree-Fock quality (56-GTO, 119-GTO, 126-GTO). The calculations using these two groups of single determinant are  presented in Figs.  6.3a-6.3c  and  6.3d-6.3f,  S C F wavefunctions  respectively. The two  categories  are based largely on the quality of the total energy calculated using these basis sets. The details of these basis sets together with the respective energies, dipole moments and p  calculated total  are given in Table 6.1. In order to compare i l l 3.X  the  measured  calculations, essentially  the  XMPs  of  'best  identical  to  the  valence  quality' the  orbitals  wavefunction  126-G(CI)  of  (i.e.  NH  the  O V D , see  3  with  126-GTO  Fig.  6.4  the  theoretical  M D which  below)  is  normalized to the l e X M P . This normalization procedure was chosen on the of (see  the  relative  Figs.  6.3b,  insensitivity of the 6.3e  and  6.4b).  le  M D to basis  set  and  correlation  A l l calculations for all three orbitals  their correct values relative to the single point normalization between and the 126-G(CI) O V D on the  le orbital.  is  height basis effects  maintain experiment  T  1  1  1  1  1  1  1  1  1  1  1  1  r  MOMENTUM (A.U.) Fig. 6.4  Comparison of the valence X M P s of N H with ion-neutral CI overlap distributions (OVDs) calculated from correlated wavefunctions. Distributions calculated from the (7)119-GTO, (7c)119-G(CI), (8) 126-GTO, and (8c)126-G(CI) are placed on a common intensity scale by using a single point normalization of the 126-GTO (identical to the 126-G(CI) calculation) calculation on the l e X M P (see text for details). 3  180 Starting  with  wavefunctions magnitude  the  le  from  X M P (Figs.  D Z to  which are  in  near quite  calculated M D s maximize at le  X M P . A l l calculations  (0.1-0.5a ©" ) 1  somewhat  situation for the  3a,  6.3b  Hartree-Fock close  also  give  than  able  to predict  3a,  it  is  apparent  with  in  is observed  experiment.  the  low  all  P  In  to  m  a  the  of the  x  region  contrast,  6.3d) is quite different  quality wavefunctions  and  However  momentum  experimentally.  X M P with regard  that  M D s of similar shape  slightly higher than the  X M P (Figs. 6.3a and  the  give  intensity  M D s calculated from the intermediate not  6.3e)  agreement  a momentum  less  and  the  in that  the  (DZ and 6-311G) are  either  shape or  magnitude.  However inclusion of diffuse functions as employed in the H D D 2 G and 6-311 + G (Fig.  6.3a)  wavefunctions  produces  calculated  M D s in closer but  still  not  good  agreement with experiment. This H D D 2 G basis due to Dunning and Hay [DH77] is a variation of the functions basis  set  on the  6-311G basis  nitrogen  (-56.1790a.u.)  6-311G (-56.1777a.u.)  atom.  is very  Although the similar to  wavefunctions,  is quantitatively different  set with the  addition of diffuse 2s and  calculated total energy  that  of the  2p  with this  DZ (-56.1714a.u.)  and  the calculated H D D 2 G momentum distribution  and in particular provides a much improved description  of the low momentum  region. It is of interest however that the calculated dipole  moments  (Table  do  situation  is  6.1)  similar  not  in going  wavefunction  which includes  Since  diffuse  these  differ from diffuse  functions,  very the s-  unlike  much 6-311G  and  for  these  wavefunction  p-functions  polarization  wavefunctions.  on  functions,  appreciably to the bonding, there is very little improvement  to  the do  the  6-311+ G  nitrogen not  in energy.  This  atom.  contribute However  the improvement in the calculated M D s is dramatic as can be seen in F i g . 6.3a.  181 In  contrast the  nitrogen  6-311G** wavefunction  and hydrogen  which has  atoms yields a M D (see  polarization  Fig. 6.3a)  functions  only slightly  from that calculated using the unpolarized 6-311G wavefunction energy is significantly improved from -56.1777a.u.  on  the  different  although the  to -56.2102a.u.  and the  total dipole  moment is also correspondingly improved (see Table 6.1). This result is consistent with  the  well  known  fact  that  condition, is of itself not sufficient  energy  minimization,  although  to guarantee a wavefunction  a  necessary  good enough  to  calculate  all properties with high accuracy especially. Likewise inclusion of diffuse  functions  to  good  'universal'  6-311+ G  The 2 a , see  improve  the  calculated  wavefunction  as  M D is of itself insufficient can  be  seen  above  for  to  the  guarantee HDD2G  a  and  wavefunctions.  X M P (accounting for the intensity over the whole inner valence  above) shown in Figs. 6.3c and 6.3f is overestimated  S C F calculations is reasonably  region,  slightly (—5%) by all  (DZ, 6-311G, 6-311G**, 6-311+ G, HDD2G) although  the  shape  predicted.  Theoretical M D s calculated from the extended gaussian  basis sets of Rauk et  al.  [RA70] (56-GTO), Feller et al. [FB87] (119-GTO) and a new 126-GTO basis  set  are  also  compared  with the  valence  orbital X M P s  panels of Fig. 6.3. A t the S C F level the is optimal with regards to the next best wavefunction the  is the  56-GTO wavefunction  calculated  of N H  3  on the  126-GTO wavefunction total  energy and  119-GTO wavefunction  right  hand  (E = -56.2246a.u.)  dipole moment.  The  (E = -56.2245a.u.) followed by  (E =-56.2219a.u.). The near Hartree-Fock  (126-GTO, 119-GTO and 56-GTO) predict valence X M P s of N H  3  wavefunctions  quite similar to  182 those  using intermediate  quality wavefunctions  except  in the  case of the  3a,  M D . The simpler and variationally inferior 6-311+ G wavefunction yields an M D for the 3 a ,  orbital very close to that calculated, using the 119-GTO and 56-GTO  near Hartree-Fock wavefunctions. The l e and 2 a , Hartree-Fock wavefunctions  M D s calculated from the near  (Figs. 6.3e and 6.3f) show similar behaviour to those  calculated using intermediate  quality wavefunctions  there is some improvement  for the  (Figs. 6.3b and 6.3c) although  le orbital using the  126-GTO wavefunction.  The M D s calculated using the 126-GTO, 119-GTO and 56-GTO wavefunctions still slightly overestimate the magnitude of the 2 a , X M P .  The  reasonably  calculated large  close agreement for  from  the  dipole moment  greater  than  moments  the  6-311+ G  case  wavefunction  (2.2292D)  experimental  predicted value  of the  is  with  of  (1.6417D,  agreement  with  quadrupole  moment  1.6442D  experimental  quadrupole  results.  (-2.1960a.u.)  126-GTO wavefunction are moment  and  and  X M P with  surprising  wavefunction  1.47149 +0.0015D.  1.660D, Other <  3a,  perhaps this  predicted by the variationally superior  wavefunctions  the  the  r  2  >  126-GTO,  (-2.42±0.04a.u.)  [DD82] and  <r  >  as  the  using  the  values  for  obtained  g  dipole 56-GTO  such  experimental 2  the  in much better  properties  in good agreement with the  the  is significantly  119-GTO and  (26.7134a.u.)  e  because  However  respectively) are  calculated  the M D  (25.501a.u.)  [H67].  It can be seen that from an overall standpoint the  126-GTO wavefunction gives  the  investigated  best  work.  description  Since the  millihartree) basis  set  to  among  calculated the  saturation  the total  estimated has  S C F wavefunctions energy  with  Hartree-Fock  effectively been  this  limit  basis of  established.  set  in the  is  quite  present close  (1  -56.226 + 0.001a.u.  [FB87],  This can be  by  seen  the  183 fact that there is only a very slight improvement of the to the  119-GTO M D for each  previous E M S work on H 0  orbital as  126-GTO M D relative  shown in F i g . 6.3. Experience in the  [BB87] has shown that basis set saturation is only  2  achieved very close to the Hartree-Fock limit (— within 0.5 millihartree).  The  possible  and  the  reasons for  the  M D s calculated  remaining differences  with near Hartree-Fock  between  experimental  S C F wavefunctions  XMPs  have  been  discussed earlier [BB87]. Assuming the adequacy of the P W I A description [WM78] as  seems  reasonable  from  [BB87], these differences possibly  the  influence  the  are of  consideration  of a  most likely due  vibrational  wide range  to electron  motion.  The  of E M S studies  correlation effects  former  are  discussed  separate section below. Vibrational motion (symmetric modes) effects of  H 0 2  and in  a  on the M D s  have been found to be negligible by Leung and Langhoff [LL87] and it  is also of interest to consider whether this phenomenon is of importance in N H . 3  Vibrational effects acurate 1  in N H  prediction  A —t?A, 1  (3a,)"  of  Hel  and  1  have been extensively studied, most specifically in the  3  1  photoelectron  A —» E, 2  1  (le)"  vibronic transitions  1  are particularly interesting because the N H of  planar  geometry  vibrational effects  (i.e. D ^  symmetry)  + 3  intensities  accompanying  [AR82, CD76]. These  the  effects  ( A ,) ionic state is believed to be 2  [AR82]. In an  in N H , Feller [F87] has 3  investigated  effort the  to  effects  estimate  the  of bending  distortions on the calculated M D s . The symmetric bending distortions did not yield a  significantly different  equilibrium  geometry.  nowhere more than  vibrationally averaged  The  change was  1% [F87].  mostly  M D from at  the  that  peak  calculated  at  maximum and  the was  184 6.5. ION-NEUTRAL  OVERLAP  DISTRIBUTIONS  The recent E M S study of H 0 2  prediction  of  XMPs  requires  especially for the outermost with  Prof.  E.R.  (see chapter inclusion  (OVDS)  5) [BB87] has shown that accurate  of  correlation  and  relaxation  effects  ( l b , ) valence orbital. This was done (in collaboration  Davidson)  by  performing  separate  multi-reference  SD-CI  calculations for the neutral and respective ionic states having first achieved basis set  saturation  recovered were  then  study NH set  at  with least  used  extended  basis  86% of the  for  sets  at  the  S C F level.  total correlation energy.  calculation of the  ion-neutral  These  These  overlap  CI  procedures  wavefunctions  (OVD).  A similar CI  (see preceeding section) has been carried out in the present work for molecule and the respective N H ^ " " ions. Using the 1  3  reported  in  the  present  performed for the neutral N H ionic states at the Tables using  6.2 the  and  work  separate  basis  calculations  been  MRSD-CI  molecule and for the (3a,)" ,  experimental neutral  geometry.  6.3. The calculated total energy  126-G(CI) and  126-GTO extended  1  3  (le)"  The particulars for the  119-G(CI) wavefunctions  the  neutral  recovered  1  have and  (2a,)"  1  are shown in state of N H  86.5%  and  3  86.3%  respectively of the estimated total correlation energy (-0.337a.u.) [PB75, FB87]. A consideration of the  OVDs  gives the  spectroscopic  factors  in Table 6.3. It can be seen that the values for the are both approximately single  particle  2a,"  1  3a,"  1  and  le"  processes  1  0.87 indicating minimal splitting of the ionization (i.e. a  description is reasonably  indicate that the  (pole strengths) shown  process  adequate).  leads to many  final  However  the  CI  calculations  ion states of which a large  fraction (0.77) is recovered below a binding energy of 37eV which was the of  the  calculation.  MR-lp2h-CI  An  even  larger  fraction  calculation in the  energy  range  (0.885)  is  20-47eV. In  recovered Fig. 6.4  using the  limit the  valence  185 XMPs  are  overlap  compared  calculations  maximum  using  with with  the  the  OVDs  the  same  126-G(CI)  resulting  from  normalization calculation.  Hartree-Fock limit S C F calculations of the  the  as  before  The  CI  on  119-GTO  M D s reported  are also shown for comparison. For this purpose  above  ion-neutral  the  le X M P  and  126-GTO  in the previous section  the O V D calculations were each  normalized to unity. The 119-G(CI) O V D was also calculated for the 3 a ,  orbital  and is shown for comparison in F i g . 6.4a.  It  can  be  seen  quantitative  from  Fig.  to CI for the  le  low  orbital. In the  momentum  slightly smaller p of  the  been  incorporation  3a,  max  X M P (see  observed in the  of  CI  gives  O V D distributions for the  case of the  3a,  H 0 2  represented  in the  dramatic  3a,  orbital.  and ion states results  region.  This  results  in  orbital inclusion of electron  in an O V D with more  density  a  and  higher  in better agreement with experiment F i g . 6.4a case of the  and  Table  outermost  6.1). lb,  Quite  cross-section  THFA  momentum  similar behaviour  orbital of H 0 2  distributions.  [BB87], the ion-neutral overlap amplitude for N H  Similar 3  a  on the leading edge has  (see F i g . 5.6)  [BB87]. This has been understood in terms of the 'non-characteristic' not  a  that there is no detectable change in going from S C F  correlation in both neutral the  that  improvement in the predicted  It is however noteworthy  in  6.4  contributions  to the  case  can be expanded  for  in the  form, <p*  N f  ~  1  |*  N 0  >  = C,^  where the 'non-characteristic'  3  a  1  + C ^ 2  4  a  1  +  ...  [6.3]  orbital contribution is due to the lowest lying 4 a ,  virtual orbital (compare with F i g . 5.6 of chapter  5). These orbitals are  generally  186 very diffuse region of the  in position space and, conversely, contribute calculated O V D . In addition, it was  5 [J3B87] that the are  due to the  increased  low momentum  combined effects  to the  low  momentum  also shown earlier in chapter  components  in the  of initial state correlation and  calculated O V D a contraction in  the final ionic state.  In the case of the  le X M P (Fig. 6.4b) the calculated O V D using the 126-G(CI)  wavefunction yields quantitatively the same profile as the M D calculated from 126-GTO wavefunction. not  very  observed  important  in  This result predicting  in the case of the  some sense is like the  lb  2  indicates the  le  that correlation and/or X M P . Analogous  X M P of H 0 3  relaxation  results  have  is  been  (see chapter 5) [BB87] which in  2  le orbital in N H  the  (i.e. largely a bonding orbital).  The effect of correlation and relaxation in the calculated inner valence 2 a , O V D is shown  in Fig. 6.4c.  The  126-G(CI) O V D was  weighted average of the (very slightly different) 2a,"  1  final ion states found in the energy  obtained  126-GTO  wavefunction.  Both  calculations  region 26-36eV (see  1  pole-strength  are  Tables  6.3  and  M D calculated using  slightly above  This slight difference may be due to one or more of the following (a) additional (2a,)"  a  OVDs calculated for each of the  6.4). The resulting O V D (Fig. 6.4c) is very similar to the the  from  the  2a, X M P .  effects:  strength beyond the upper limit of the experimental binding  energy spectrum (Fig. 6.2a); (b) limitations in the present theoretical treatment or (c) distortion effects, i.e. failure of the plane wave impulse approximation. A  similar level of agreement between  calculation and experiment  was  found  for  187 the  2a,  X M P in H 0  [BB87] which is also split into several final ion states  2  spread over a wide binding energy range. These results for the M D s and OVDs as  well as  those for the  binding energy  overall  suitability  of  neutral  and  wavefunctions  correlation process  ion  for  as  the  126-GTO  describing  well  as  the  the  spectra  clearly show  basis  used  for  development  the  need  for  adequate  and subtle  features  X M P for  in F i g . 6.2  the  of  3a,  the  outer  the  of 126-G(CI)  incorporation  inner  of  valence ionization  valence  orbital  of N H . 3  Furthermore the good level of quantitative agreement achieved for both H 0 and 2  NH  indicates that the description provided by the P W I A is quite reasonable  3  for  E M S studies of these systems at an impact energy of 1200eV + binding energy. Thus neglect of electron correlation rather than distortion effects main reason  for discrepancies between  for the valence orbitals of N H  The effect of  the  and theory  and  2a,  In  126-G(CI) wavefunction  the  S C F level  2  orbitals but  properties.  at  and H 0 .  of electron correlation is not only manifested  3a,  molecular  3  experiment  seems to be the  particular,  in the calculated OVDs  also in improved calculated the  predicted  dipole  values  moment  (1.5891D) is in closer agreement with  the  of using  other the  experimental  dipole moment (1.47149 + 0.00015D) [MM81]. The only available calculations which yield  better  et al.  theoretical  [FB87] which  the  complete  and  Sadlej  4th  utilized  order  (1.4991D)  dipole moments an  theoretical estimates of the  the  even larger  many-body [DS86].  are  It  dipole moment  calculation of  130-GTO basis  perturbation should  MRSD-CI  theory  however do not  The dipole moment vibrational correction in N H  3  be  set  Feller  (1.5881D) and  calculation of Diercksen noted  that  in  general,  include vibrational averaging.  can be significant (as mentioned  188 in  the  earlier  [WM76,  section)  and  FB87] which  has  brings  been  the  estimated  theoretical  to be of the  order  of -0.025D  predictions in slightly closer accord  with experiment (see Tables 6.1 and 6.2). Likewise good agreement was obtained betweeen the theoretical and experimental quadrupole moments and < r >  values  2  using  extended  properties  basis  sets  (dipole moment,  sensitive  to  the  diffuse  (l'30-G(CI),  quadrupole region  119-G(CI))  moment,  of the  [FB87].  <r > )  are  2  e  spatial  electron  inverse weighting property of the Fourier transform,  These  e  molecular  known to be  distribution.  the experimental  quite  From  the  momentum  profiles are found to be similarly sensitive to the low momentum regions of the electron  distribution  agreement  (in  between  the  p-representation).  experiment  and  The  calculation  consistent not  pattern  only  for  of  good  momentum  distributions but also for a wide range of properties is indicative of the accuracy and general  suitability of highly extended  these studies for N H  In an effort (i.e.  for  3  3  correlated  wavefunctions  used in  and H 0 [BB87]. 2  to explore the possibility of using simpler treatments of correlation  the  neutral  distributions, target-natural NH  and  using the  target  molecule  only)  for  prediction  of  momentum  orbital (TNO) M D s for the valence orbitals of neutral  126-G(CI) wavefunctions  have  been  calculated. Earlier  nitric oxide showed excellent agreement (for shape) between  the  XMPs  work on and  the  TNOs [CC82] and similar results have been found recently for H S [FB87a] on 2  a quantitative  basis. The T N O results  for N H  3  are  shown in comparison with  the 126-G(CI) in the lower portion of Fig. 6.5 while the respective upper  sections  show  natural  the  density  differences  (TNO-SCF)  and  (OVD-SCF).  The  target  189  orbitalst OVDs.  were obtained from the Natural  orbitals  result  CI calculations used to produce the  from  diagonalization of  the  matrix [L59]. It has been known that optimal convergence can  be obtained  126-G(TNO) 126-GTO  with  treatment  natural  orbitals. It can be seen  does not  provide  (SCF) M D s . The biggest  any  difference  126-G(CI)  one-electron of the  density  CI expansion  from Fig. 6.5  that  the  improvement  over  the  orbital where  the  systematic is for the  3a,  126-G(TNO) is even lower than the 126-GTO result.  6.6. POSITION-SPACE  AND  MOMENTUM-SPACE  DENSITY  MAPS  Earlier reported density maps for other molecules have generally been carried out with low quality wavefunctions  to minimize computation costs. These calculations  have usually been adequate to represent the general features of the charge momentum  distributions  and  to  represent  chemistry. The present work on N H  3  essential  (like that of H O z  indicated the importance of extended and saturated diffuse for  of momentum [BB87]) has  momentum space have been calculated using the the experimental geometry 126-GTO  basis sets which include many maps  molecule in both position and  3  126-GTO S C F wavefunction  at  (Fig. 6.6). The corresponding X M P s and the calculated  spherically averaged  a cross  space  however  and polarization functions. Therefore two-dimensional density contour  the three valence orbitals of an oriented N H  show  ideas  and  section through  M D s are the  also shown for  comparison. The  electron distribution in the  maps  x-z plane in the  case of the position density map and through the p -p X  momentum  density  map.  The position density map  t The occupation numbers for the 1.963 and 1.980, respectively.  3a,,  le and 2 a ,  plane in the case of the z is in a plane that includes  natural orbitals are  1.966,  2  RELATIVE MTENSmr (ARB. UMTS) 2.0 4.0 6.0 8.0  0.0  <Jq"  1  -r=n  10.0  DTFERENCE DENSITY (ARB. UNfTS) -1.0 0.0 1.0 2.0  20.0  DIFFERENCE 0ENSITY (ARB. UNfTS) 2.0 0.0 2.0 4.0  r-  3 p  c q  3 3  t\3 O 3  6  P  3  Ir £ 2 . 3a  o 3  o 3  CT  2. 5<  c  (0  5'  CO  q  HQ  p  P  •a  2.  1 1  S  s  o  a 3" <  rt.'  sr 2  I"  -  P.  sr  -  5  -  i -  1 1 1  7  "f  K>  fV* f/_.  & Q.  1 0.0  — . * at at  1  1  1  1  — —  _i  RELATIVE NTENSTY (ARB. UNITS) 10.0 20.0 30.0 40.0  •  _»  a)  <  z 5  L _ _ l  2 X CH  •  M  crt o» ii A o -«  £ p  rt  1  cn  sr> o" 3 S. 3  X g.  061  o P ~  3"  <> t  CO  RELATIVE WTENSITY (ARB. UNITS) 4.0 8.0 12.0 16.0  0.0  3  a- I  o  cn P to  ^ p  EL »  J —  •  50.0  DIFFERENCE DENSITY (ARB. UMTS) -5.0 0.0 5.0 10.0  191 the  positions of the nitrogen atom and one of the three hydrogen atoms. The  employed molecular (x,y,z) orientation in atomic units is N (0.0, 0.0, 0.127872), HI  0.0,  (1.77164,  -0.592238),  H2  (-0.88582,  1.53428,  -0.592238)  and H 3  (-0.88582, -1.53428, -0.592238).  The  position density  contour  map for the outermost  3a,  valence  orbital (Fig.  6.6a) shows a nodal 'surface' displaced from the nitrogen center as can be seen from the one-dimensional position density distribution projection plots along the xand  z-axes,  respectively. Besides the dominating nitrogen  2p  function there is  also a considerable contribution from the nitrogen 2s and hydrogen Is orbitals. A minimal basis set decomposition of the 3 a , molecular orbital shows,  *3a,  -  X ( N  2s  )  Since the momentum  +  0 2p>  " ^ " i s *  ( N  density  C 6  at p = 0 is determined  solely  by the s-type  -  4 ]  basis  functions then, P  3  a  i  ( 0 ) « X - 3e  The fact that are  individually  [6.5]  £33 ^0) is very small does not necessarily imply that small.  The corresponding  shows some small s-character 2p orbital. The r-space  momentum  space  contour  X or e  map thus  although the dominating feature is still the nitrogen  map clearly shows the influence of the hydrogen  atoms  indicating that some bonding can also be attributed to the 3 a , orbital. In recent EMS  work  it has been  demonstrated  so-called 'lone pair' orbital in N H His  character)  3  that  this  noticeable  derealization  of the  (i.e. the orbital contains small but significant  is greatly enhanced by methyl substitution [BB87a, BB87b]. These  192 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  OJ U) UJ 13 MOMENTUM (A.U.) Fig. 6.6  MOMENTUM DENSITY  -4.0  -2.0  00  2.0  POSITION DENSITY  4.0  05 1.0  -4.0  -2.0  0.0  2.0  4.0  0.5 1.0  Position space and momentum space density contour maps for an oriented N H molecule. Maps were generated using the 126-GTO wavefunction at the neutral experimental geometry. Contours are at 80, 50, 20, 8, 5, 2, 0.8, 0.5, 0.2, 0.08, 0.05 and 0.02% of the maximum intensity. Positions of the atoms in a given plane are indicated. A l l dimensions are in atomic units. 3  193 trends are  also clearly predicted  have been found for H 0  and ( C H ) 0  2  clearly reflected of  NH  related  and  3  to  in molecular orbital calculations. Similar 3  [CB87]. These delocalization effects  2  (Fig. 6.6) in the experimental measurements for the emphasize  the  the  canonical  fact  that  molecular  effects  E M S measures  orbital  but  a quantity  significantly  3a,  are  orbital  very closely  different  from  the  simple valence bond idea of an atomic-like lone pair localised on the N atom.  In  the  case of the  l e orbital the  nitrogen  some delocalization of electron density  2p  x  function is dominant  distribution in the  but with  region of the  hydrogen  atoms (Fig. 6.6b). The nodal plane in the r-space map (yz plane) is preserved in the  p-space map  (p P y  z  plane).  The  2a,  r-space and p-space maps  (Fig. 6.6c)  clearly display the dominant contribution of the nitrogen 2s function. The r-space map also shows significant perturbations degree of bonding character  6.7. EXTERIOR  ELECTRON  Recent studies by Ohno et exterior  in the 2 a ,  electron  across the hydrogen atoms due to some orbital.  DISTRIBUTION  (EED)  RATIOS  AND  XMPS  al. [OM83, OM84, 0186] have investigated the use of  distribution  ratios  obtained  from  Penning  ionization  electron  spectroscopy (PIES) in probing the quality of the long range portion of theoretical wavefunctions. of the  The experimental  wavefunctions  used  ratio  in the  and  also the  ratios calculated using some  present E M S study  are  shown in Table 6.1.  Ohno and Ishida [OI86] have shown that the valence orbital E E D ratios for N H  3  (3a,/le)  (see Table 6.1) which have been calculated using the 6-311+ G, H D D 2 G  and  56-GTO  2.60  [0186].  wavefunctions In  contrast,  are the  quite 6-311G  close to and  the  experimental  PIES  6-311G** wavefunctions  value of  give  much  194 poorer E E D ratios (1.98 and 1.76, respectively).  These  E E D results  because  similar  are  of some  trends  calculated momentum  in  interest  behaviour  in relation to the  are  observed  distributions. This is not  for  present E M S work  the  unexpected  experimental  and  since both E M S and  Penning ionization are sensitive probes of the longer range charge distribution (i.e. the  wavefunction  tails). The 6-311 + G  agreement with the  3a,  M D which gives the  X M P amongst  the  best (but not good)  simple wavefunctions  shown in Fig.  6.3  also gives an E E D ratio (2.58) in best agreement (within the limited range  of  wavefunctions  wavefunctions with the 2.41,  compared)  which were  measured  respectively  Penning  ionization  found  X M P than in  with  experiment.  to the  give  branching  give ratios  slightly  ratio.  The  HDD2G  M D s in slightly less  6-311 + G,  correspondingly  The  less  6-311G  which gave M D s in poor agreement with the  3a,  56-GTO  good agreement  (3a,/le)  good and  and  of 2.50  agreement  6-311G**  with  and the  wavefunctions  X M P also gave E E D ratios in  poor agreement with the Penning ionization branching ratio.  Due to the  fact that the  only the outermost  3a,  He and  2 S 3  metastable atom has  le orbitals of N H  are  3  an energy of 19.82eV  accessible by PIES. It is  not possible therefore  to compare  E E D ratio. A further  limitation in the comparison of E E D ratios with experiment  is based  on the  branching  ratio  reacting section  and is  the inner valence 2 a ,  severe assumptions proportionality.  non-reacting  determined  by  the  (2a,/le)  made in deriving the E E D ratio and  Strictly speaking  regions  X M P with the  of the  overlap  there  electron  integral  is  no  boundary  distribution.  (over  all space)  The  PIES  between  PIES  cross  of the involved  195  molecular  orbital  wavefunction  with  the  vacant  inner  shell  orbital  of  metastable  atom. Furthermore correlation effects are also ignored in the derivation  of the proportionality of Penning ionization cross sections to E E D values 0186]. This is not a limitation in the in  fact  it has  accurately  the  been 3a,  shown  that  orbital in N H  interpretation  electron (see  3  the  [OM84,  of E M S cross sections  correlation is necessary  and  in describing  earlier discussion) as well as the  lb,  orbital of H 0 [BB87]. However even with the limitations in the PIES and E E D 2  comparison, molecules  the  results  [OM83,  of  0M84]  Ohno show  and  co-workers  interesting  for  parallels  NH with  3  [0186] the  and  results  other of  the  corresponding E M S studies and calculations in the present work and it is evident that  both  techniques  are  sensitive  to  the  outer  spatial  regions  of molecular  orbitals.  6.8.  SUMMARY  Earlier discrepancies between experimental momentum profiles and theoretical M D s especially for the outermost use  of highly extended  (3a,)  basis  orbital of N H  sets (saturated  3  are now largely resolved. The  in the  diffuse  function limit)  and  sufficiently correlated wavefunctions for both the initial neutral state and the final ionic states have led to better agreement with the X M P s momentum accurate wide  resolution E M S . These  momentum  range  quadrupole  moment  newly developed wavefunctions  distributions but  of molecular and  properties  also give the such  by high  yield not only  best calculated values for a  as  the  total  ionization potentials  (for  the  valence states) and therefore  as measured  energy, outer  dipole  valence  moment, and  inner  may be called 'universal' wavefunctions for N H . 3  C H A P T E R 7. F O R M A L D E H Y D E  7.1.  OVERVIEW  As  part  critical  of the  continuing series  evaluation  momentum  of  resolution  preliminary  study  (binary(e,2e))  at  molecular  wavefunctions,  study  the  of  the  of  binding  energy was  1  o  than  the  chapter  orbitals  of  spectra reported  at lower impact energy  (Ap = 0 . 4 a " )  of small molecules  this  valence  two azimuthal angles  1976. Although done resolution  of E M S studies  present  reports  H CO.  of  and  the  a  high  2  An  earlier  H CO  using  EMS  2  by Hood et al.  [HB76] in  (400eV) and at poorer  momentum  work,  this  earlier  study  [HB76]  illustrated the symmetry diagnostic capabilities of E M S in resolving a controversy regarding TB70].  the  energy  This  was  ordering of the  achieved  through  5a, a  and  lb  two-angle  orbitals  2  (0 = 0°  of H C O 2  [CD75,  0 = 20°)  binding  and  energy scan and observing the relative heights of the relevant peaks. No detailed experimental angular time  although  correlation study  of the  simple calculations of the  valence orbitals  valence  was  done  orbital momentum  at  the  distributions  were reported.  Other previous related studies on H C O  include theoretical investigations of basis  2  set is  [RE73] and electron correlation effects a measure of the  sensitive  to  resolution reported section  total electron  wavefunction  momentum  quality than  Penning ionization electron by  for  Ohno et al. the  5a,  [OT86].  orbital  was  [ST75] on the  In  distribution and therefore  E M S which spectrum this  noted  196  is  2  the  is less  orbital-specific. The high  (PIES) of H C O  study and  Compton profile which  large  compared  with  has  relative  also PIES  calculated  been cross  exterior  197 electron density (EED) values.  In the  present work the  experimental momentum  profiles (XMPs) of H G O  are  2  reported for the first time and compared with theoretical momentum distributions (MDs)  calculated  from  a  range  of ab  initio  wavefunctions.  The  inner  valence  binding energy region is also analyzed and compared with a many-body  Green's  function calculation [VB80].  Formaldehyde was was  prepared  allowed to pass through  by heating  paraformaldehyde  at  60°C.  The  vapour  a heated Granville-Philips leak valve. The binding  energy spectra indicated there were no significant impurities.  7.2. BASIS  SETS  FOR  SCF  WAVEFUNCTIONS  The measured X M P s in the present work are compared with spherically averaged momentum  distributions  LCAO-MO  wavefunctions.  calculated  for  a  variety  of  The experimental momentum  selected  ab  initio  SCF  resolution (Ap = 0 . 1 5 a " ) 1  0  was also folded into the caculations. The wavefunctions cover a wide range from the  simple  4-3 I G  Hartree-Fock  limit.  basis The  set  to  details  a of  more  extended  these  basis  wavefunctions  set  essentially  together  with  at  the  selected  calculated properties are shown in Table 7.1.  Important features of the various basis sets are discussed below, (1)  4-31G This  "split-valence" basis  comprised  of 0[3s2p],  set  C[3s2p]  [KB80] involves a and  H[2s]. Four  contracted  G T O set  primitive GTOs  are  Table  7.1. C h a r a c t e r i s t i c s of  SCF Wavefunctions f o r  Wavefunct i on  Carbon and Oxygen B a s i s Set  Hydrogen Basis Set  4-31G  (8s4p)/  (4s)/  [3s2p]  [2s]  4-31G+G  (9s4p)/ [4s2p]  (4s1p)/ [2s1p]  DZ  (9s5p)/ [4s2p]  (4s)/ [2s]  (19s10p2d1f)/ [10s5p2d1f]  (10s2p1d)/ [4s2p1d]  134-GTO  HjCO  Hartree-Fock  Energy(a.u.)  Dipole  Moment(D)  1 13.6911  3 .005  t h i s work  1 13.6962  3.002  t h i s work  -113.8209  3. 1 10  [SB72]  •113.9202  2.857  [DF86]  113.925  1 imlt Exptl.  114.562  Estimated Hartree-Fock l i m i t Includes r e l a t 1 v 1 s t 1 c e f f e c t s P o s i t i v e d i p o l e moment  2.3310.02  [GS74]. [NM69].  means negative end  on oxygen atom  Reference  (i.e. C+0 )  [K060, HL68]  199 contracted to form the  Is core and the respective valence orbitals are  'split' into three GTOs contracted GTO  left uncontracted.  are kept the  to one CGTO  and the  least  Exponents for both valence s- and  tight  p-functions  same.  4-31G + G This  basis  set  involves the  original  4-3 I G  basis  (1) plus  a  set of  diffuse functions. The diffuse functions have been used by Chong and co-workers  [ZS79,  LB83b].  The  specific  diffuse  functions  used  are:  C ( a = 0.02789), O ( a = 0.04216) and H ( a = 0.1160). It has been shown g  g  p  in earlier E M S studies  [BB87, BM87] that diffuse functions are very  important in the accurate prediction of orbital X M P s . Double zeta (DZ) Two  sets  atomic  of contracted  Gaussian  type  functions  are  orbital. No additional polarization functions  used  are  for  each  employed.  In  this basis set, proposed by Snyder and Basch [SB72], the least tight s- and p-functions  of the nitrogen atom are  represented  by a  single  and  Feller  primitive Gaussian function. 134-GTO This  extended  basis  set  reported  recently  by  Davidson  [DF86] involves a (19sl0p2dlf) primitive set contracted on  the  oxygen  and  (10s2pld)—>[4s2pld]  carbon  atoms.  contraction  The  scheme.  hydrogen The  to  basis  [10s5p2dlf] involved  s-components  of  a the  cartesian d functions were deleted. The calculated S C F total energy is -113.9202a.u. date.  which  is the  best S C F energy  for H C O 2  reported  to  200 All  wavefunctions  (r =1.2078A,  generated  r =1.116lA,  CQ  4-31G  were  using  + G wavefunctions were generated  ENERGY  Formaldehyde,  in  its  experimental  Z H C H = 116.52°)  C H  7.3. BINDING  the  equilibrium  [T063].  The  geometry  4-31G  and  using the G A U S S I A N 7 6 package  [BW76].  A,  and  SPECTRA  ground  neutral  1  state,  has  symmetry  the  electronic configuration can be written as:  i  (1a,)  (2a,)  2  v  (3a,)  2  *  (4a,)  2  v  (1b )  2  of  (2b )  2  2  2  —  valence  assignment  subject  (1b,)  2  .  v  core  The  (5a,)  2  2  (i.e. ordering)  debate  [BB68,  of the  TB70,  valence  CD75,  orbitals  NM69].  On  of H C O  has  the  of vibrational  2  basis  analysis and isotopic studies in a photoelectron spectroscopy  been  (PES) study,  et al. [TB70] assigned the third and fourth ionization bands to the  lb  Turner  and 5 a ,  2  orbitals, respectively. This was clearly the reverse of the order suggested calculations at that time [NM69] which were in agreement  the  by M O  with the configuration  shown above. Although it was acknowledged that the vibrational analysis may be uncertain  [BR72]  a  photoionization  that the original assignment Green's  function  agreement  with  study  Hood  by  orbitals are above) function  calculation the  [NM69]  and  spectrometric  study  [GC75]  suggested  of Turner et al. [TB70] was correct. A many-body [CD75]  however  M O calculations [NM69].  et al.  ordered  mass  [HB76]  showed  predicted  orbital  A t this juncture,  unequivocally that  the  assignments a  timely E M S  5a,  and  according to that predicted by earlier M O calculations Green's function results  [CD75].  Further  many-body  [VB80] and CI [K81] calculations also confirmed the earlier  in  lb  2  (see  Green's  assignments  201 based  on  the  simpler  M O calculations  [NM69].  The  preceding  results  clearly  illustrate the caution necessary in interpreting P E S data especially in the case of overlapping  states.  E M S with  its  distributions allows unambiguous  capability  assignment  of  determining  orbital  momentum  of ionic states sufficiently  separated  in energy.  Fig.  7.1  shows  the  binding energy  spectra  work at an impact energy of 1200eV + B . E . 0 = 0°  and  spectra  0 = 6°.  with the  The energy  vertical  for  peak  the  remaining  valence  derived from curve-fitting the  positions  and  widths  as  spectroscopy  orbitals  and  set  present  2b  by aligning the orbital (10.9eV)  2  [TB70]. The their  binding energy  given by  in the  and at relative azimuthal angles of  known vertical ionization potential of the  by high resolution photoelectron  intensities,  obtained  2  scale in F i g . 7.1 has been  as measured IPs  of H C O  relative  spectra  P E S [TB70] and  estimated ionization  (Fig. 7.1)  with  convoluted with  the  instrumental energy resolution, are shown in Table 7.2. The relative intensities in the  0 = 0°  the  different  well-resolved 'p-type' between  spectrum  (p = 0 . 1 a  symmetries peaks  and 's-type'  _ 1 o  of the  attributed  to  )  and the valence the  2b  0 = 6° spectrum orbitals  2  and  (p^O.Sao ~ ) 1  of H C O .  Quite  2  4a,  orbitals  have  reflect  clearly  the  dominantly  symmetries, respectively. The broad, partially resolved band  12-18eV (Fig. 7.1)  orbitals. The middle peak  is due  to  ionization from the  in this band (hatched)  lb,,  5a,  and  lb  2  and the righthand peak (solid)  shown in Fig. 7.1 clearly illustrate that the middle peak is of 's-type' symmetry whereas  the righthand peak is of 'p-type' symmetry. From this comparison it is  already clear that the lb  2  5a,  orbital (s-type) has  a lower binding energy than  the  orbital (p-type) in agreement with the earlier findings of Hood et al. [HH76]  202  20.0  10.0 i  1  1  1  1  25.0  1  1  1  1  1  1  1  1  1  1  r  o CO  H C0 2  o 1b, o  2  b  I  2  5 a ,  \  / 1—n  1 b  $=6°  2  4  0  3a  1  1—  q  o d -I  5.0 7.1  I  10.0  I  J  L  15.0  20.0  _l  L  25.0 30.0 35.0 BINDING ENERGY (EV)  I  '  40.0  '  45.0  Binding energy spectra of H CO measured at an impact energy of_1200eV + binding energy for (a) 0 = 0 ° and (b) 0 = 6 ° . The two spectra are on the same relative intensity scale. The 5a, (hatched) and l b (solid) peak areas are illustrated for clarity. 2  2  Table  7.2.  Binding energies  and I n t e n s i t i e s  In  the I o n i z a t i o n s p e c t r a  of H CO 2  Relative Orbital  Energy  (eV)  Fwhm (eV)  Intensity  b  EMS ( O ' ) (p*0.1a." )  EMS(6") (p»0.5ao- )  1  1  10.9  1.70  7.6  18. 1  1  14.5  1 .84  20. 1  32.2  5a j  16. 1  2 OO  100.0  18.8  1b  17.0  1 .92  20.7  40.6  21.4  2.40  77.2  53.3  27.5  3.0  7.4  31.5  3.0  7.7  34.25  3.0  30.3  37.0  3.0  6.8  41.5  3.0  5.9  2b  1b  4 a  2  2  1  Includes experimental width Normalized r e l a t i v e  to  of  1.7eV fwhm.  5 a , peak I n t e n s i t y  at *=0* . U n c e r t a i n t y  is ±5%.  CO  o  00  204 and contrary to the assignments  proposed by Turner et al. [TB70]. Hood et  [HB76] also observed from the binding energy centered at 34.2eV and assigned this feature although clearly suggested  scan at  0 = 0° a broad  to the 3 a ,  al.  structure  orbital. This assignment  by M O calculations was not confirmed in the  earlier  study [HB76] since the binding energy scan in the inner valence region was only obtained at one angle (i.e.0 = 0°). A more extensive analysis of this inner valence region has been carried out in the present work.  Over  the  past decade  molecules  have  revealed,  (single-particle) Such  theoretical  picture  behaviour  is  as  a  and experimental general  phenomenon,  of ionization in  also  observed  studies  the  in the  a  breakdown  inner-valence  inner  valence  of many atoms  region  and  of the M O  [CD86,  region of H C O 2  exhibits intensity (Fig. 7.2 and Table 7.2) due to many-body effects  VS84]. which  spread over  a wide range of the binding energy spectrum rather than a single peak expected from  a  single-particle (Koopmans-type)  approximation. The  0 = 0°  (scaled  x0.55)  and 6° inner valence spectra have been overlaid in F i g . 7.2a to emphasize common  symmetry  of  the  main  structure  in  the  24-44eV region. The  the inner  valence region was fitted with a four-peak  template each of equal widths (3.0eV  fwhm)  at  intensities  the and  energies the  given  similar  in  Table  7.2.  spectral  shape  indicates  mainly s-type and this is consistent process. significant  The  present  intensity  et al. [HH76].  results at  for  =28eV  with the  the  inner  which  was  A  consideration  not  the  actual  that the. 24-44eV region is  strength valence  of  being due to the  region  specifically  show  a  identified  small by  3a,"  1  but Hood  q  Green's  CN  Ul  205  Cal cul at i on  q o z  Function -1  ..(lb ) 2  o  5. . . (4a^)  to  6. . . (3a^)  ui  2°  K)  -1 -1  -  o d J  15.0 T  z  J  L  19.0 1  23.0 1  1  27.0 1  1  31.0  w  J  L  39.0  43.0  L  47.0  r  o  H C0  CD*  rr <  J  L  35.0  2  o  I  MB-St at es  4>=6°  o «|)=0 (x0. 55) o  o Ul  t/l  o O Ul  ui  t s  q o 15.0  19.0  23.0  27.0  31.0  35.0  BINDING ENERGY  Fig.  7.2  43.0  39.0  47.0  (EV)  Inner valence binding energy spectra of H CO. The experimental binding energy spectra at (a) 0=O°(xO.55) and 0 = 6° are overlaid. Four peaks of equal widths (3.0eV) are placed (see Table 7.2) and convoluted to yield the best fitting curve (solid line). The theoretical binding energy spectrum (b) is from the Green's function calculation of von Neissen et al. [VB80]. The calculated poles were convoluted with the experimentally derived widths, summed and scaled (x2.5) to produce the theoretical binding energy profile (solid line). 2  206 A  more  detailed experimental  provided by the symmetry by  measured  characteristic  von  Niessen  assignment  and  X M P (see  of the  et al. give  confirmation, at  a  3a,  [VB80] good  7.2).  This  many-body  in the  region of 34eV,  F i g . 7.9) which shows clearly the orbital. Recent  shown  Green's  in F i g . 7.2b  representation  = 36eV. Note that intensity at  least  of  the  function  also  's-type'  calculations  confirm the  observed  is  present  spectrum  up  to  — 29eV is also predicted by the calculations (Fig.  Green's  function  calculation  (Fig.  7.2b)  involved  a  ( l l s 7 p l d / 6 s l p ) G T O set which is a reasonably extended basis set.  Although  the  = 23eV ( l b level  and  region.  significant  3a," (Fig.  associated  cannot  limited  ionization strength  statistical  It  results  function  are  to the  is  and the  Table limit  at  agreement between  spectra best  poles  7.2)  in  also  of the  Green's  is due the  this show  to  the  experimental  however it should be noted only  at  very low signal  experiment  is believed that this strength  (Fig. 7.2b)  calculations  small p-type  precision of the  (Fig. 7.2a  Overall, there is reasonable  and theoretical  some  be confirmed due  beyond 36eV which  calculation (Fig. 7.2b).  7.2a)  Green's  calculation predicts  present experimental  process.  1  function  ^) their presence  the  The  function  2  Green's  semi-quantitative.  that  Increased  flexibility of the basis set has been shown [FG87] to be very important in the improved prediction of the inner-valence binding energy spectrum, especially in the higher energy region.  207  7.4. COMPARISON OF EXPERIMENTAL MOMENTUM PROFILES WITH THEORETICAL PREDICTIONS The  experimental  momentum  profiles  (XMPs)  of the  valence  orbitals  of  H CO 2  were obtained in two ways due to the multitude of states and the closeness of the  lb,,  Figs.  5a,  7.4,  7.8,  azimuthal the  and  lb  7.9)  angle  at  binding energies.  2  were  obtained  appropriate  (only partially resolved)  in  selected  lb,,  5a,  series of narrow range binding energy of  The the  and  fixed  widths  manner  5a,  and  each  of the  lb  orbitals  were  were  lb  orbitals  2  measured  plotted  The  as  a  plus  variation the  resulting  of  XMPs  of  from  a (0)  shown in Fig. relative  energy  experimental  spectroscopy  function  (see  at azimuthal angles  3 0 ° . These spectra,  width  XMPs  derived  deconvoluted using fixed  respectively.  then  (i.e.  and  (Franck-Condon  orbitals,  2  3a,  whereas  resolution) derived from high resolution photoelectron lb,,  and  binding energies)  0 ° , 2 ° , 4 ° , 6 ° , 8 ° , 10°, 1 4 ° , 20° and  positions  4a,  2  usual  spectra  7.3 on a common intensity scale, were  2b ,  energy  [TB70] for  deconvoluted  areas  of momentum  (see  the for Figs.  7.5, 7.6, 7.7).  The X M P s  have been  placed on a common intensity scale by normalizing using 0 = 0° and  the relative intensities in the Table with  7.2). a  The peaks  gaussian  peak  in the  0 = 0°  taking  into  0 = 6° binding energy  and  0 = 6°  consideration  spectra the  have  energy  by  summing  assignment)  relative  to  the  wide range binding energy  the  counts  whole spectra  in  the  binding energy  region,  also  each been  3a,  24-44eV,  spectrum  (see  resolution  known vertical IPs and Franck-Condon widths. The inner-valence normalized  spectra  fitted  and  the  X M P was (see  above  (9-44eV). The  (Fig. 7.1) thus provide two extra data  two  points  208  H CO 2  E  o  =  T200eV  4>=10  i i i ii $=14'  $=20  i  i  i i i i ii 4>=30'  ' ,iT*imii iTi i ,iT t  l  f>t  1  3O0  X>X> BJO  Fig.  7.3  1»J  »J3  BJO  BA  MJD  BA  l  | U  MX  210  BINDING ENERGY (EV) Binding energy scans in the region 14-22eV as a function of azimuthal angle, <p. Relative normalization is maintained since scans were sequential. The Fitted gaussian peaks are illustrated by dashed curves and the resulting envelope by a solid line.  209 and  therefore  a  consistency  check  on  the  respective  valence  XMPs  of  H CO 2  (open triangles in Figs. 7.4-7.9).  The  normalized  theoretical  valence  calculations.  XMPs The  of  H CO 2  theoretical  are  MDs  also were  compared calculated  with  several  from  various  wavefunctions shown in Table 7.1 and then spherically averaged and folded with the  experimental  momentum  resolution  (Ap = 0 . 1 5 a " )-  can  be compared on a common intensity scale using a single point normalization  1  0  Theory  and  of the best S C F calculation (134-GTO) to a single point on the Fig.  7.5). A l l calculations and all other  can  then  be  normalization.  compared This  affords  2  X M P (see  experimental data points for all orbitals  quantitatively  procedure  lb  experiment  a  on  the  critical  basis  of  this  quantitative  single  point  assessment  of all  measured  XMPs  experimental and theoretical data.  To  provide  a  comprehensive  framework  for  interpreting  the  two-dimensional density contour maps, in both position and momentum space, also presented for each reported  are  X M P in Figs. 7.4-7.9. The contour maps were  calculated using the DZ wavefunction and assumed the H C O 2  the xz plane with the z-axis serving as the  C  2  molecule to be in  axis. The contours shown  are  80, 50, 20, 8, 5, 2, 0.8, 0.5 and 0.2% of the maximum density of each orbital. Along the side of each contour map are projections of the density along the axes indicated by the dotted lines.  The set energies.  of theoretical M D s can be classified according to their total calculated The  4-3 IG,  4-31G + G  and  DZ  wavefunctions  could  therefore  be  210 considered  to be  wavefunction  of intermediate  quality  (see  Table  7.1)  whereas  wavefunction  (-113.9202a.u.)  energy (-113.925a.u.)  can  be  seen  calculation  and  (intermediate  is  from  2b  Figs.  7.4-7.9 over  outermost  2  and  very  near  that  quite  Hartree-Fock)  orbitals  differences  for  three outermost  lb,  the  estimated  of the  134-GTO  Hartree-Fock  (lb , 2  and 5 a ,  good  agreement  except  in  the  orbital. In each case both classes  valence  particular the  to  all six orbitals  innermost  the  close  essentially  limit  [GS74].  experiment  region of the  other  134-GTO  of Davidson and Feller [DF86] can be considered to be  of Hartree-Fock limit quality. The calculated S C F total energy  It  the  predict  4a,  very  and  (2b ,  similar  3a,)  lb,  momentum  of wavefunctions  M D s for  whereas  there  three  are  small  orbitals are more sensitive to the basis set than  the  XMPs  orbitals  the  In  orbitals. Experience in comparing measured  5a,)  low  between  of H C O .  2  and  exists  2  and calculated M D s for  other small molecules [BB87, BM87] has shown the need for quite extended  basis  sets  such  with  functions  addition were  of  found  to  diffuse  and  polarization  be  necessary  in  the  momentum region of the X M P s of the outermost and N H  particular  modelling of  the  valence orbitals of H 0  is in keeping with earlier observations  2  as to the  seen  limit  the  description of the  near 2b  lower (and therefore  2  Hartree-Fock  effect  134-GTO  X M P (Fig. 7.4) in the  4a,  for a wide variety of molecules.  for the less tightly held orbitals ( 2 b ,  do not really show any clear trend  low  [BB87]  2  2  The results  that  accurate  In  [BM87]. The insensitivity of the more tightly bound orbitals ( l b ,  3  and 3a,)  functions.  lb,  and 5 a , ,  Figs.  7.4-7.6)  of basis set quality. It is calculation  gives  the  sense that the predicted p  best is  in closer agreement with experiment). There is also a slight  improvement in the description of the MDs  calculated  from  intermediate  low momentum  quality  superior quality (see Table 7.1) of the  wavefunctions.  in shape and magnitude  The  lower  X M P peaks  2  region 0 . 1 - 0 . 6 a to the  - 1 0  -  at  The  P  m  a  and  x  has  from the  set  the  intensity  in  the  (at the Hartree-Fock level)  would lead to significantly improved agreement with  discrepancy  probably due to the  despite  measured X M P .  appreciable  It is unlikely that improvements  134-GTO basis  experiment.  a  However,  the  134-GTO wavefunction, the predicted M D  is still significantly different 2b  region compared with  observed  neglect of the  for  effects  the  outermost  2b  orbital  2  is  of electron correlation and/or  most  electron  relaxation. A higher level of theoretical treatment beyond the Hartree-Fock picture is thus desirable in order to include electron correlation in the neutral state and both electron correlation and relaxation in the  final  ion state in the  form of a  full ion-neutral overlap distribution (OVD) calculation. Such an O V D treatment as has  been  H 0 2  extremely  [BB87],  evidently  NH  necessary  3  effective  in comparing theory  [BM87]  and H S 2  particularly in the  studies are currently in progress  From  the  orbital  is  evidenced  r-space only by  maps  E M S measurements for  A similar  case of the  2b  overlap  relative  treatment is  X M P of H C O .  2  Such  2  [BB87c].  (right side of Fig. 7.4) it can be  approximately  the  [FB87a].  with  a  non-bonding  number  of  orbital  contours.  on  As  seen  the  noted  that the  oxygen by  atom  Neumann  Moskowitz [NM69] there is also some participation from the hydrogen atoms. in fact,  electron correlation is the  calculation and the 2b may  be  2  dominant reason  for the  discrepancy  X M P then this picture (Fig. 7.4, rhs) of the 2b  grossly inadequate.  A similar  situation exists  in the  2b  2  as and If,  between 2  orbital  case of the l7r  Fig. 7.4  Comparison of 2b experimental momentum profile of H CO with calculated M D s . Open triangles represent data points derived from the long range binding energy scans (Fig. 7.1). All calculations are normalized to experiment by a single point normalization of the 134-GTO M D on the l b X M P . On the center and right hand panels are the momentum and position space density contour maps, respectively calculated using the D Z wavefunction. A l l dimensions are quoted in atomic units. 2  2  2  RELATIVE INTENSITY (ARB. UNITS) 3.0 6.0 9.0 12.0  0.0  J  -4.0  1  -2.0  1  i  0.0  i  i  2.0  •  •  4.0  6.0  •  0.5 1.0  214 2it orbital of N O [BC82] where  orbital in CO [FB87b] and the that electron correlation  is necessary  for describing the  it is apparent  dipole moment  reversal  and the X M P . It is also possible that the neglect of electronic relaxation in the fmal et  ion state  (B)  will  2  2  have  critical  effect  in the  predicted  M D s . Ozkan  al. [OC75] have calculated large electronic relaxation (or reorganization)  in H C O , in particular, the 2 b 2  In  a  the  case  of the  lb,  2  and l b , molecular orbitals.  X M P it  can  be  seen  in Fig. 7.5  that  agreement is obtained between experiment and theory using the wavefunctions  whereas  wavefunctions  yield  the  MDs  variationally  of  effects  lower  inferior  intensity  with  m  a  and  4-31G+G  shifted  x  good  134-GTO and DZ  4-3 I G P  quite  to  higher  momentum. It can be seen from the r-space and p-space contour maps (Fig. 7.5, rhs) that the The 5 a ,  lb,  orbital can be considered  as  TQQ  largely a  type of orbital.  X M P (Fig. 7.6) on the other hand shows clearly a mixed s-p type of  profile similar to the  5 a orbital of CO [FB87a]. Such an analogy is  by  p-space  the  r-space  and  considered as a OQQ predicted by the  4-31G + G  likely that this difference 2  Z  the  show  and  expected  that  the  5a,  134-GTO calculations athough are  = 10% higher than  could be attributed  A , manifold and would affect the 4 a ,  of the 2  which  type of orbital. The shape of the 5 a ,  of the predicted cross sections  the  maps  mixing of ionization strength  the  orbital  supported can  be  X M P is reasonably the  absolute  measured  value  X M P . It is  to mixing of ionization strength and 3 a , in the  in  X M P s similar to the case 5 a, 4 a and 3 a poles (all  symmetry) of CO [FB87b, DD77]. Another point that should be mentioned is much  lower  cross  section  compared to 4-31G + G and  predicted  by  the  134-GTO wavefunctions.  4-3 I G  and  DZ  wavefunctions  This result reflects  the  need  Fig. 7.6  Same as Fig. 7.4 except for 5a, orbital of H CO 2  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  MOMENTUM DENSITY  MOMENTUM (A.U.)  Fig. 7.7  Same as Fig. 7.4 except for l b  2  orbital of H C O . 2  POSITION DENSITY  217 for diffuse functions in the basis set. It is also consistent with the conclusions of Ohno et al. [OT86] regarding their comparison of E E D values (calculated using a modified 4-3 I G basis) with the experimental PIES cross sections.  In  F i g . 7.7 the  well  with the  lb  X M P , which is largely a TT orbital, is shown to  2  theoretical  orbitals, the calculated  calculations.  lb  2  A s opposed  to  the  2b ,  lb,  2  compare and  5a,  M D s are less sensitive to basis set quality. This is  also one of the reasons why it was chosen for the  single point normalization of  theory to experiment to establish a quantitative basis for comparison.  Comparison of the theoretical  4a,  calculations  (Fig. 7.8) and 3 a , shows  (Fig. 7.9) X M P s with the  similar behaviour  to  that  observed  respective  for  X M P (see F i g . 7.6). A l l calculated MDs in both cases (i.e. for the 4 a , orbitals)  overshoot  magnitude of  the  of the  4a,  the  respective  X M P and  that  a larger  The broad  profile is expected  can  be  width at  considered  predicted half  as  a  presence  Besides  the of H  4a, Is  theory.  The any  the  type  2s and  of  2p  orbital  contributions  to the  orbital of H C O  and the  character  4a,  in the  3a, in  shape  momentum  calculated M D s .  7.8,  from the  4a, rhs).  orbital It  carbon  is and  4 a orbital of CO [FB87b] which  profile. The reason 2  (Fig.  and  the  experimental of the  5a,  difference  between  anti-bonding nature of the  This is again analogous  in shape between the  by  the  a*  however shows a 'p-like' momentum  is  = 10%.  maximum than  from  anti-bonding with respect to the oxygen atoms.  by  cross section there is also some difference  profile has  which  XMPs  the  for this large  difference  4 a orbital of CO [FB87b]  orbital of H C O 2  which adds  symmetrically (in-phase) to the C 2s. It is known that s-type atomic orbitals (in  SPHERICALLY AVERAGED  MOMENTUM DISTRIBUTION  MOMENTUM DENSITY  POSITION DENSITY  MOMENTUM (A.U.)  Fig.  7.8  Same as Fig. 7.4 except for 4a, orbital of H C O . 2  tsD OO  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION •2, 3, 4  H CO 3 a 2  POSITION  MOMENTUM DENSITY IO  1  ci  1 - - 4-31G 2 4-31G+G 3 DZ 4 134-GTO  q  s q o o  Tf(o,.t.o? 0.0  Fig.  0.5  7.9  1.0 1.5 MOMENTUM (A.U.)  2.0  12.5  -4.0  -2.0  0.0  2.0  4.0  Same as Fig. 7.4 except for 3a, orbital of H C O . 2  0.5 1.0  -4.0  -2.0  220 LCAO-MO p=0a "  picture)  3a,  the  only  contribution  X M P (Fig. 7.9),  (note spike in the  which  NH  nature 3  the  r-space  largely corresponds  momentum  density  have  been  to  the  oxygen  at  2s orbital  projection plot), is also slightly overestimated by  theoretical calculations although the this  to  [BB87b].  1  o  The  are  observed  shape is adequately  in the  inner  the  predicted. Differences of  valence X M P s  of H 0 2  [BB87],  [BM87] and Ne [DM78] and these differences were found to persist even  with inclusion of electron correlation in both initial neutral and final ion states. It has been postulated that this phenomenon could be due to: (a)  unaccounted  ionization  stength  beyond  the  limits  of  the  binding  energy  spectrum (i.e. E j ^ 4 5 e V ) . 3  (b)  limitations in  the  present  theoretical  treatment  (i.e.  basis  set  effects  and  insufficient electron correlation and/or relaxation) (c)  distortion  Distorted  effects  wave  (i.e.  failure  calculations are  of  the  presently  plane not  wave  feasible  impulse  but  could  approximation). be  investigated  experimentally by increasing the impact energy beyond 1200eV. This, however, is not possible with the present instrumentation.  7.5.  SUMMARY  The present E M S study of H C O shows good quantitative agreement between all 2  measured  valence  XMPs  and  M D s calculated  (near Hartree-Fock) except for the outermost 2b studies of H 0 2  observed  in  the  [BB87] and N H outermost  2b  2  3  from 2  the  are  wavefunction  orbital. Based on previous E M S  [BM87], it is suggested orbital  134-GTO  due  to  the  that the discrepancies neglect  of  electron  221 correlation also  and/or  indicate  that  electronic electronic  Extensive many-body energy  spectrum  structures are the  relaxation relaxation  structures  which  is  are  effects. may  Other be  important  also observed  assigned  largely  clearly predicted by Green's  theoretical  to  in the the  studies  in the inner  (3a,)"  1  2b  2  [OC75] orbital.  valence binding process.  These  function calculations [VB80] although  agreement with experiment is only semi-quantitative, especially in the higher  energy region.  C H A P T E R 8. PARA-DICHLOROBENZENE  8.1.  OVERVIEW  Chemical reactivity can be studied by measuring compounds compound changes  that differ and  in  substituents  from  each  substituents  the  are  properties  on the  basic  the  reactants  by  a  appropriately  of the parent  perturbation of the substituent which  other  properties  in  series of  If the  it is possible to  terms  of the  compound. Under favourable  perturb  of a  single substituent. chosen,  compounds  can serve  mutually  the  parent interpret  effects  of  the  circumstances  the  as a model for a chemical reaction in each  other's  basic  electronic  structure,  resulting ultimately in the breaking and making of chemical bonds. Examples of the  systematic  study  of substituent  effects  in a homologous series  of molecules  can be found in most forms of spectroscopy. With these considerations in mind, electron  momentum  spectrocopy  has  recently been  applied to the  study  nitrogen 'lone-pair' orbital in ammonia, methylamine, and triazine [TL84, These ideas now  are  presented  now further for  para-dichlorobenzene  the  two  f  chlorine  the  e  removes  present  outermost  (p-DCB), together  doubly degenerate l i g atoms  explored in the  with  (ir , 3  of  HC85].  work and E M S data 7 r ) valence 2  the  orbitals  are of  a comparison with the corresponding  orbital of benzene. In p-DCB the addition of the two degeneracy  and  it is of interest  to investigate  the  nature of this effect on the momentum distributions of the energetically separate p-DCB benzene  orbitals,  7T and  ir system,  3  while  7T .t 2  The interaction between  of intrinsic interest  t These orbital assignments are symmetry and p-DCB assuming ^2h. the C-Cl bond and the z-axis normal and angles are standard [EC73] with  s  itself, has  the  CI atoms  and  the  wider implications for  based on benzene assuming D g ^ planar y y - The x-axis is chosen to be along to the plane of the ring. The borid lengths the C - C l bond length set at 1.77A. m m e t r  222  223 chemistry. For example, organometallic chemistry and much of organic reactivity is dominated by  n interactions. Furthermore,  the  element on a tr system is of general interest sensitivity  of E M S to  Fine details  of the  influence of an  electronegative  and important to understand.  dynamics  of electron  The  motion should  provide new insights into 7T electron behaviour and reactivity.  The electronic structure of the benzene molecule has been extensively studied both theoretically Because  and  of  approaching  experimentally  the  large  size  the  self-consistent  of  incorporating many-body effects within  the  last  ten  or  since  the  the  beginnings  benzene  field  (SCF)  of  molecule, limit  quantum  mechanics.  detailed  calculations  [EK73]  and  calculations  [VC76, CD78, PB70, HS74] have only been done  fifteen  years.  The  outer-valence  binding  energies  of  benzene have been measured by photoelectron spectroscopy using excitation sources from  H e l and H e l l  [TB70, A L 7 0 , K M 7 6 , J L 6 9 ,  [GB74]. The outermost of  7T symmetry.  SP74, L F 7 2 ,  LA80] to X-rays  orbital of benzene has been identified as the  This doubly degenerate  e  orbital has  labelled ir ,  of 9.3  2u  ^al  (I =12.4 eV) is non-degenerate and is usually labelled T T ,  o r D  2  3  more  tightly bound  v  To date there has  been only a single E M S study  momentum  distribution  unfilled  orbitals  7T  7T ; the  vertical ionization  (I )  l a  is often  a  energy  v  eV and  l ig  l e ^ orbital  of  the  valence  orbitals  of benzene,  namely  the  of the  of benzene  itn*,  7r * 5  and  binding energies [FM81].  is  consideration  planar,  the  a  along  with  a  and  7T systems  detailed  knowledge  are of  effectively the  6  three have  [JM76]. Since  separated.  electronic  benzene makes it particularly favourable for studies of substituent  The  and  orbitals,  7T *  been extensively investigated with electron transmission spectroscopy benzene  [KK81].  structure  effects.  This of  224 In this initial E M S study of substituent substitution  at  the  degenerate  IT  level  momentum  1  is  (4-3 IG) that (see  are  are  4  ring  positions  examined.  distributions  para-dichlorobenzene  group  and  of  In  in  on  particular,  (=1.6  Because  Calculations  using  have  order  been  to  energy  target  binding 7T (lb^g) 2  the  Filled  doubly  energies  and  orbitals  of  simple  wavefunctions  carried out by  the Maryland  provide  further  insight  into  the  also played a role in the choice  resolution currently  available in E M S  to choose a molecule in  7T levels were sufficiently separated from each other as well as from  next-lowest  that  limited  the  eV fwhm in the present work) it was necessary  which filled the  of the  highest-lying  and  experimental results. Experimental considerations of p-DCB.  the  3  available for p-DCB  Acknowledgements)  in benzene the effect of chlorine  7T (2b2g)  the  measured.  effects  levels. The compound  densities  could be  also had  maintained  at  to have the  10  1  2  a vapor  pressure  - 1 0  cm"  1 3  such  level  3  at  room temperature. In addition the compound had to be sufficiently inert to resist decomposition  in  the  para-dichlorobenzene  spectrometer.  have  respectively, as measured The While  next-lowest these  nevertheless judicious  level  spacings  The  two  outer  vertical ionization energies by H e l photoelectron  (primarily CI 3p) do not  (I ) y  (7T3,7r2)  of 8.94  spectroscopy  is at  allow complete  filled  levels  of  9.84  eV  and  [KK81, SC73, PL80]  11.37  eV [KK81,  separation  of the  SC73,  bands,  PL80].  they  are  sufficient to provide specific information on the separate orbitals with  selection  measurements.  of  the  actual  sitting  binding  energies  used  for  the  225 8.2. EXPERIMENTAL  MOMENTUM  PROFILES  The addition of two chlorine atoms at the para positions in benzene removes the degeneracy it  2  of the benzene  l e ^ g orbitals (TT ,7T ) at I = 9.3 eV. In p-DCB 3  is shifted up in energy  shift  in energies  effects  associated  has  been  with  the  (I =9.8  2  eV) and the  v  7T down (I =8.9eV). This v  3  explained in terms  of the  substitution  two chlorine atoms.t  of the  the  v  inductive and  resonance The  next  orbital in p-DCB is mainly chlorine 3p in character.  The outer eV  and  valence binding energy  0 = 6°,  spectrum  is shown in F i g 8.1.  of p-DCB obtained at  The  error  bars  represent  E ^ = E g = 600 one  standard  deviation. The indicated curve fitting analysis and the absolute energy scale were established  with  measurements  reference  have  [PL80] and Kimura  been  to  high-resolution  reported  et al. [KK81].  by  Hel  Streets  and  PES Caesar  measurements. [SC73],  The vertical ionization energies  Such  Potts  et  of the  al. first  six bands were assessed from P E S to be at I = 8.94, 9.84, 11.49, 13.00, 14.74 y  and  15.90 eV respectively. The corresponding E M S binding energy  were likewise estimated to be 1.6,  1.6,  1.9,  1.8 and  peak  widths  1.7 eV fwhm, taking into  account the E M S resolution (1.6 eV) and also the natural Franck-Condon widths [KK81].  Only  the  first  two bands  (7r ,7r ) 3  2  are  single ionic states according to  the high-resolution P E S data [KK81]. The third band involves ionization from  the  orbitals with a predominantly C l 3p character. A gaussian fitting program using parameters  based on the experimental widths, peak positions and spacings  from  t A general discussion of resonance versus inductive effects is given in Ref. [EB73]. A discussion of resonance versus inductive effects in the context of the photoelectron spectrum of para-dichlorobenzene is given in Ref. [BM68]. A more detailed consideration of the influence of inductive and resonance effects on ionization potentials in substituted benzenes has been given in Ref. [FK81].  226 PES  and the present instrumental energy  resolution gave the  generally excellent  fit indicated in Fig. 8.1. The dashed lines are the individual gaussian peaks with the  solid line being their sum. The two vertical arrows  selected  experimental sitting binding energies  (8.7 and  be noted, are not exactly equal to the respective I  represent the  carefully  10.1 eV, which, it should  values) at which momentum  distributions were sampled. In this way contributions from neighbouring states are minimized and the  principal contribution will be from the  7T and  ir  3  orbitals,  2  repectively.  Experimental momentum profiles (XMPs) representative  of the  first two bands of  p-DCB, obtained as described above, are shown in Figs. 8.2a and 8.2b. The solid line drawn through the experimental points for each distribution provides a visual guide.  Since the  widths  relative cross sections as  momentum  profiles  3  are  clearly  very  (7r ) band shows a maximum ( P  the  (0.6au)  3  are  equal,  the  observed  for  the  m a x  higher  )  surprisingly,  the  at =0.8 au, which is higher than ( 7 r ) band.  I  of the  2  Thus,  an  increase  in  V  energy  probable  (Fig. 8.1)  different. Somewhat  I I I3.X  binding  bands  2  final states. The shapes and magnitudes  lower I p  7r  and  shown are in essentially the correct ratio, taking into  account summation over all the two  7T  of the  does  momentum.  experiments,  for  not  necessarily  This  fact  example  in  correspond  has  been  to  an  increase  demonstrated  the  case  of  C0  measurements of p-DCB [KK81] the  peaks  due  to  in  [LB85a].  2  7T and 3  in  earlier In  2  7 T ) contains  a small overlapping contribution from the  (essentially  7 T ) . Since P  m  3  2  a  v  for  the  first  band  is greater  than  EMS PES  of similar  X M P at  (essentially  most  Hel  7T are  intensity. Examination of F i g . 8.1 indicates that the measured  the  8.7 eV  second that  for  band the  227  to  4  6  8  10  Binding Energy  Fig. 8.1.  12  14  16  18  (eV)  Binding energy spectrum of para-dichlorobenzene at <p = &°. The dashed curves are from a gaussian fitting program based on parameters from high-resolution P E S [KK81]. The solid line is the sum of the individual gaussian peaks. Vertical arrows show the sitting binding energies at which the momentum profiles of Fig. 8.2 were sampled.  228 second band,  the  overlapping contribution to the  lower its observed P  m  a  (Cl 3p) from the  T T and ir 3  experimental  orbital.  3  of the third and  3  that the  cross  momentum  section near  This  is  not  resolution  p = 0 is larger  an  experimental  (Ap=*0. 1 a ~ )  of  1  0  m  a  2  values it is  x  for the artifact  the  ir  2  orbital  because  the  spectrometer  is  sufficient to clearly resolve any node in the momentum distribution at p = 0, has been demonstrated  in the case of the ir  regard the behaviour of the measured  orbital momentum distribution appears  2  the poor statistics on the benzene measurements assessment of the P with  m  the  a  x  value for the  present  as  3p orbitals of argon [LB83]. In this  somewhat closer to that measured for the benzene  comparison  ir .  clearly show that there is a significant difference in  7 T 3 orbital.  the  to  orbital X M P s of p-DCB, not withstanding the small difference in  2  also noteworthy for  ir  ir  that between  their energies. In addition to the different cross sections and P  that  only serve  since the energy separation  2  second is much larger than  The present measurements the  can  relative to that of a completely separated  x  The second band is mostly due to ir band  first band  l ig e  higher-precision  l e ^ ^ orbital [FM81]. However,  [FM81] preclude any unequivocal  orbital, and therefore p-DCB  measurements  any detailed is  at  best  speculative.  8.3. CALCULATED  MOMENTUM  In order to further  understand  7T 3 with  and a  ir  the experimental measurements,  orbital momentum  2  basis  set  of  4-3 I G  DISTRIBUTIONS  distributions  type  for  [DH71] using  p-DCB the  have  calculations of the been  GAUSSIAN  carried 80  out  program  [BW80]. The results are shown in Fig. 8.3. Using even such a simple basis set, the  results  are  quite instructive in that the  general characteristic  features  and  1  r  (a)  O d q 06  T  1  r  ~i  1  1  r  229  ci-Q-a Iw= 8.9eV -  (8.7eV)  I= v  9.8eV  (IQIeV)  Momentum (a.u.) Fig. 8.2.  Measured momentum profiles for the IT (a) and i r (b) orbitals of para-dichlorobenzene. The vertical ionization energies ( I ) are shown. The values in brackets are the sitting binding energies at which the measurements were taken. 3  v  2  230 trends of the experimental results predicted p whereas 7T . 3  w  the  The  for  max  7T  (dotted  3  calculated  P  and  m  a  x  positions  neither  cross-section contribution at  basis are  are  p = 0.t  that for j r  ir  somewhat  higher  momentum  further  than  than  of theory  7T and 3  ir  that  those  for  observed  shows  any  accurate  basis  and  experiment  using the  2  the  (solid curve),  distribution  comparisons  calculated for  2  is greater  2  Calculations using much more  The orbital energies  9.35 and  section for  calculated  evidently required before  can be made.  are clearly reproduced. Firstly,  curve) is larger than  predicted maximum cross  experimentally,  sets are  (Fig. 8.2)  10.02 eV compared with the experimental [KK81] I  4-3 I G values  of 8.9 and 9.8 eV.  Further  understanding  different P  m  of p-DCB, p-DCB  a  of the  observed  and  calculated  values of the momentum distributions for the 7 r  x  can be obtained by considering the  and  features,  benzene  degenerate (7r  3  shown  in  Fig.  8.4.  particularly  the  and ir  orbitals  simple orbital diagrams  for both  A  simple  3  orbital  2  diagram  of  the  and 7 T ) l e ^ g orbital of benzene is shown in Fig. 8.4a and the 2  effect of chlorine substitution in p-DCB is shown in F i g . 8.4b. For benzene,  one  ^ ig  an  representation  of  antisymmetric (ir ) 2  the  the  degenerate  e  P "" a  configuration with respect  n  a  s  a  symmetric  first  have  the  effect  of  lowering  3  to the perpendicular plane  1 and 4 positions [KK81]. Substitution of chlorines at the  would  ( 7 r ) and  the  energy  of  through  1 and 4 positions  the  two  levels  by  withdrawing charge from the ir system onto the electrophilic chlorine atoms. This inductive effect is offset for the 7T level of p-DCB by the antibonding resonance 3  t The present calculations, which were done without the incorporation of the instrumental momentum resolution, are not likely to give good quantitative agreement with the measured X M P s even if the momentum resolution is folded in.  231  CALCULATION 4-31G BASIS  0.1  C I - ^ - C I  CO  c  CD  p-DCB T T  Q  p-DCB T T  Iv(eV) 8.9  3  9.8  2  E o o o  1.0  2.0  3.0  Momentum (a.u.) Fig. 8.3.  Calculated momentum density distribution for the ff and 7T levels of para-dichlorobenzene and the doubly degenerate n , 7T levels of benzene. The calculations are based on 4-3 I G wavefunctions and the momentum densities have been spherically averaged. The benzene and 7T para-dichlorobenzene densities are indistinguishable and are shown by the solid line. The 7T density is shown by the dotted line. 2  3  3  2  3  2  232  (a)  BENZENE  (b)  I = 9.3eV p-DICHLOROBENZENE v  2b I =8.9eV  1b I = 9.8eV  2 g  lg  v  v  Schematic representation of wavefunction amplitudes for (a) the two degenerate levels (ir and it ) of the highest occupied 7T level of benzene and (b) the ir and it levels of para-dichlorobenzene. The areas of the circles are proportional to the magnitude of the wavefunction amplitudes. Solid circles represent positive amplitude and open circles negative amplitude. Nodes (regions of zero wavefunction amplitude) are indicated by dashed lines. 2  3  3  2  233 interaction of the C l 3p 4.  This  causes  benzene  ir  the  l ^ g - For  orbital  3  ir  the  e  with the  z  1,4  of the  chlorine  resonance  energy  to  exceed  eliminating any  ring. Though the substitution  of the carbon atoms  z  orbital in p-DCB,  2  amplitude to be zero, thereby system  2p  has  been  I  symmetry  ir ,ir 3  justified  decreases) requires  that  the  Cl 3p  levels of benzene  2  in  of z  interaction with the ir  resonance  splitting of the  long  (i.e.  at positions 1 and  terms  of  upon  inductive  and  effects, additional direct experimental evidence for the correctness of the  explanations has been lacking.  The measured  relative (e,2e) cross sections for the momentum profiles as shown  in  and  Figs.  The  ir  3  8.2a  8.2b  provide additional data  orbital of p-DCB has  a momentum  higher momentum  value (=0.8  au) than  profile of benzene,  which is at  =0.7  7T to  2  p-DCB momentum profile (=0.6 that  7 T . It  for  3  has  also  It is reasonable  profile with a P  that for the  above discussion. m  a  2  shifted  x  ^ ig  3  the P  m  a  au) is closer to that for benzene noted  above  that  the  finite  for the  x  than it is cross-section  is more like that observed for benzene  2  to a  momentum  e  (7r ,7r )  au [FM81]. In contrast,  been  behaviour near p = 0 for p-DCB ir  supporting the  le^g-  to expect the inductive effect of chlorine substitution to have,  at  most, a small influence on the momentum distributions of the electrons. However, ir  it can be seen from Fig. 8.4b that the antibonding interaction for the of  p-DCB  momentum increase  introduces is  in the  two  related  additional nodes  to  the  amplitude  momentum  distribution  amplitude.  In  is  particular,  derivative  of the  large  expected the  larger  for  into of  the  the  wavefunction.  position-space  momentum  the  number  of  level  Since electron  wavefunction,  components  wavefunctions  3  with nodes  an  of the  electron  sharply  varying  in  spatial  a  234 wavefunction, the larger the momentum  density, as has  ir* orbital of C 0 7T  3  2  amplitude of the high-momentum components  been observed for the momentum distribution of the  [LB85a]. This behaviour is precisely what is observed for the  orbital of p-DCB where P  where there is no resonance electrons,  the  m  a  ir  is quite high. For the  x  orbital of p-DCB,  2  interaction of the chlorine 3p orbitals with the ring  nodal structure  of the  benzene  level is unchanged,  and  the momentum distribution is expected to be more like that of the benzene  of the  therefore  unsubstituted  le-^g orbital, at least at the level of the 4-3 I G calculation. The observed  situation (Fig. 8.2) is generally in accord with the  foregoing arguments,  as  are  the trends of the 4-3 I G calculations (Fig. 8.3)  8.4.  SUMMARY  Distinct differences have been observed in the experimental momentum profiles for the separate 7 T and 3  of  the  chemical  ir  molecular orbitals of para-dichlorobenzene.  2  reactivity on  of  such  aromatic  inductive  and  resonance  arguments  based  correctness  of these ideas  has  mainly been  upon substitution. With  energy  orbital  electron  momentum  have  effects.  limited  levels and their changes selected  systems  Explanations  long  relied  Confirmation  of  E M S measurements of binding  distribution  it  is  now  details of electron densities.  The present E M S measurements of the  profiles  3  comparison provided  with  clear  outermost  7 T and  molecular evidence  for  orbital the  7 T orbitals of para-dichlorobenzene 2  calculations detailed  the  to measurements of energy  possible  provide more detailed insight into such matters by direct probing of the  of the  on  for  nature  p-DCB  and  of  ir  the  to  finer  momentum and  benzene, electron  their have charge  235 distributions resonance  normally  effects.  rationalized  by  arguments based  on  separate  inductive  and  C H A P T E R 9. M E T H Y L A T E D A M I N E S A N D N F  9.1.  3  OVERVIEW  Chemical properties  are best interpreted  sufficiently  understanding  level  of  detailed  understanding  could  at the fundamental  of the  in  molecular electronic  principle  chemistry from sufficiently accurate  level in terms of a  be  obtained  by  structure. theoretical  approximations which must be made  molecules  limit  the  theory  oftentimes  not  fully  understanding  as  attached in  realisable.  A  electronic structure  reactivity is the such  of  use  inductive,  and  simpler  use  resonance  in both  and  the  detailed  understanding  and  popular  intuitive  derived from  quantum  teaching  for  most  is  thus  approach  to  and predicting stability and  theories  polarization  to a particular 'molecular center'.  common  a  in complex systems  of concepts  a  solutions of the Schro dinger equation for the  system in question. However the accuracy  Such  of electron  effects  of  displacement  substituent  groups  Such concepts and rationalizations are  and  practise  of Organic  Chemistry. It is not surprising that such empirical approaches diverse and conflicting opinions about electronic effects.  and  Inorganic  have often led to  For instance a well-known  controversy exists as to whether methyl groups are intrinsically electron donating or electron accepting relative to hydrogen [HP70, SU83, MB83].  An  effective  questions EMS  charge  experimental  been  afforded  by  measurements  together  with  (Chapters  has  alternative  5 and  6) have  quantum  electron  mechanical  momentum  sophisticated  distribution in individual  molecular orbitals  236  spectroscopy.  quantum  shown that very accurate  approach  to  In  mechanical  particular calculations  mapping of the  is now feasible  these  at  electronic least  for  237 small molecules. The experimental by  EMS  provides  approximations for  direct  discussed  direct experimental  complex  molecules.  orbital  imaging  in chapter  in  momentum  particular  space  produced  within  2. As such, E M S should also prove  probing of orbital electron  In  detailed understanding  mapping of orbital electron densities  such  densities  E M S studies  are  the useful  in larger  and  likely  provide  to  more a  of structure and reactivity at the electronic level based on  experimental observations. Examples of such applications of E M S are to be found in  recent  publications  comparing  NH  NH CH  and  3  2  investigation of the 7T electrons in para-dichlorobenzene  In  this chapter  each results  2  3  NH (CH  ) ,  3  N (CH  2  published  [BB87a].  Tossell et al. provides  NF . 3  are  3  an  The present  [TL84] on N H experimental  and  distribution in the In particular  the  3  3  2  theoretical  outermost  validity  and NF  3  considerably  NH CH .  and  an  and  these  mechanical calculations of momentum  on N (CH )  study  also  valence orbitals for  reported  as well as density maps in both momentum  communication of preliminary studies  charge  3  are compared with several quantum  distributions  NF  and  )  3  and  (Chapter 8) [BB86].  measurements of the X M P s of the outermost  NH CH ,  of  [TL84]  3  3  basis  valence  The for  and position space. A has  3  extends the  recently earlier  present more discussion  orbitals of the  of commonly held concepts  been  work of  detailed  of the  electronic  methyl amines is evaluated  work  and  in  the  obtained  in  light of the present findings.  The  NH CH , 2  3  cylinders from  NH(CH ) , 3  2  N(CH )  commercial suppliers  3  3  and  NF  and  used  without  3  samples further  were  purification. Mass  spectral analysis showed the samples to be free of any significant impurities.  238 9.2. BASIS SETS FOR SCF  WAVEFUNCTIONS  Theoretical spherically averaged momentum distributions (MDs) and density contour maps  in  momentum  wavefunctions developed  of  and  position  varying quality  in this  laboratory  space by  have  means  [BB87].  The  been  of  the  calculated HEMS  existing H E M S  using  computer  programs  several package  have  been  modified for the present work in order to accomodate the large number of atoms and basis functions required for the larger molecules used in the present study. The experimental momentum resolution (Ap = 0.15a " ) has also been folded into 1  o  each calculated M D .  The wavefunctions investigated in the present study include: (1)  STO-3G This  is  a  minimal  basis  set  designed  originally  by  Pople  and  co-workers [HS69]. It involves a N[2slp], C[2slp], F[2slp] and H[ls] basis  set  in which  each  s-  and p-function is a contraction of three  primitive GTOs. (2)  4-3 I G This  "split-valence"  basis  involves  a  N[3s2p],  C[3s2p], F[3s2p]  and  H[2s] contraction scheme. Four primitive GTOs are contracted to form the  Is  core  contracted  and ' the  into  one  valence  CGTO  and  orbitals the  are other  'split'  into  GTO left  three  GTOs  uncontracted.  Exponents for both valence s- and p-functions have the same value. (3)  STO-3G + G This  basis  set  involves the  original  STO-3G(l)  basis  plus  a  set of  239  diffuse  functions. These  diffuse  and co-workers [ZC79, molecular [BB87,  LB83b,  polarizabilities. BM87]  important  in  that  the  functionst  It  MM83]  has  accurate  in the  been  incorporation  have been  used by  Chong  improved prediction of  shown in earlier E M S studies  of  diffuse  prediction of  functions  orbital  XMPs.  are  very  However  it  should be noted that small basis sets optimized for the calculation of momentum  distributions alone  may  yield  unreliable results  for  other  molecular properties (e.g. dipole moment and total energy). (4)  4-3 I G * This basis set involves the addition of a standard  set of polarization  functions (d-type) on the heavy atoms to the 4-3 I G basis described in (2). The standard exponent is (1^ = 0.80 for C, N , and F .  A l l wavefunctions were generated at the respective experimental geometries [TK71, WL67,  WL69,  SG50,  BP57]  using  the  Gaussian76  package  [BW76].  The  characteristics and properties of these wavefunctions are outlined in Table 9.1.  9.3. MEASURED  AND  CALCULATED  MOMENTUM  High momentum resolution E M S measurements the NF . 3  outermost  valence orbital  These  are  X M P of N H  results 3  have been made of the X M P s for  of each of NH CH ,  shown in F i g . 9.1  DISTRIBUTIONS  2  3  NH (CH 3)2;  together  with  the  N (CH ) 3  3  outermost  and (3a,)  reported earlier (see chapter 6) [BM87]. Below each molecule name  the 'sitting binding energy' at which the particular X M P was measured is quoted  t The specific diffuse functions used are: C(a =0.02789), F ( a =0.04993) and H ( a =0.1160) [ZC79, LB83b, MM83].  N ( a =0.03882,  Table  9.1.  Experimental  and  calculated properties  for  N H 3 , N F 3 and  the m e t h y l a t e d  amines.  Energy(a.u.)  • I p o l e Moment(0)  ST0-3G  -55.4540  1.786  9.59  ST0-3G+G  -55.5301  1 .736  11 .43  4-31G  -56.1025  2.299  11 .26  4-31G*  -56.1297  1 .922  11 .42  126-GTO  -56.2246  1 .642  126-G(CI)  -56.5160  1 .589  Exptl.  -56.563™  1.47  ST0-3G  -94.0268  1 .625  8.90  ST0-3G+G  -94.0656  1 .974  9.67  4-31G 4-31G»  -95.0625  2.071  10.48  -95.1128  1 .730  Exptl.  -?-  1 .23  ST0-3G  - 132.6089  1 .262  8. 13  ST0-3G+G  -132.6583  1.598  8.99  4-31G 4-31G*  -134.0352  1.519  9.72  -134.1O50  1 .222  Exptl.  -?-  1.01*  ST0-3G  -171.1886  1 .010  7.76  ST0-3G+G  -171.2517  1 .233  8.81  4-31G  -173.0080  1 .052  Exptl.  -?-  0.612  -347.7538  0.394  -347.8475  0 . 197  -352.0756  0.436  -352.2079  0.315  -7-  0.234°  Mo 1 ecu1e Basis NH  Second  Pmax(ao"')  I.P.(eV)  Set  3  NHjCH  10:94 0.5210.05  b  10.85°  3  10.64 0.7±0.1  d  NH(CH  N(CH ) 3  9.84 8.94°  0.8±0.1  3  3 ST0-3G ST0-3G+G 4-31Q 4-31G*  9.35 8.51°  1.1±0.1  f  N F  Exptl.  Experimentally-derived non-vibrating, Ref.[MM81J Ref.[K81]  .  Ref.[TK71l  "  Ref.[WL68]  Ref.[WL69J "»ef.lK54j Ref.[BL70]  non-relat1v1st1c  10.64 15.34 15.38 14.88 13.73  1.3±0.1 total  energy. Estimated  Hartree-Fock  limit  h  1s  -56.226a.u.  to O  241  OUTERMOST VALENCE MOMENTUM DI STRI BUT I ONS I EXPT.(1200eV) (1) (2) (3)  OJO  THEORY STO-3G (4) 4-31G (5) STO-3G+G (5c)  0.4  16  0£  4-31G* 126-GTO 126-C(CI)  2.0  2.4  2.8  MOMENTUM (A.U.) Fig. 9.1  Comparison of the experimental momentum profiles of the outermost valence molecular orbitals of (a) N H , Oo) NH CH , (c) N H ( C H ) , (d) N ( C H ) and (e) N F with (spherically averaged) momentum distributions calculated using various basis sets. The theoretical M D s are normalized to the respective X M P s at p = 1 . 0 a " using the best (in energetic terms) wavefunction in each case. The binding energy at which the X M P was measured is shown in brackets. 3  3  2  3  1  0  3  3  2  3  242 in brackets. These values correspond closely to the vertical IPs of the  outermost  'lone pair' orbitals of the respective molecules.  The  measured  several  outermost  theoretical  wavefunctions and  some  valence  momentum  XMPs  distributions  of varying quality. The relevant  wavefunctions  are  calculated  of modest 3  each  compared  calculated  from  characteristics  properties  quality  126-G(CI) treatments for N H  are  (see  are  with  chapter  p^l.Oao" -  Each  1  scale  with  procedure case  of  respect affords  NH  Hartree-Fock ion-neutral purposes.  to  the  other  wavefunctions  in  Table  of the  experimental  maintains  calculations  limit  overlap  published  (126-GTO)  and  (126-G(CI), see  Such high level  [BM87] CI Eqn.  state-of-the-art  used  9.1.  The  126-GTO  and  compared  'best' calculation (in terms of the  for  momentum  the  correct  the  particular  high  level  treatment 2.11)  are  at  intensity  molecule.  This  studied. In the  S C F calculations of  at  the  correlation  using  the  shown  comparison  also  calculations  profile  relative  a more critical assessment of the wavefunctions recently  3  therefore  with  initio L C A O - M O  6) [BM87]. The M D s are  see Table 9.1) to the  calculation  of the  exception  to the corresponding X M P s by normalizing the calculated total energy,  ab  shown  the  in F i g . 9.1  are  at  for  present impractical  for larger molecules such as the methyl amines and NF . 3  The general trends of the experimental results rather simple wavefunctions Experimental preliminary  results  and  communication  commonly used  'textbook'  employed in the STO-3G for  are clearly predicted even by present calculations (see  calculations  NH , N(CH ) 3  arguments based  3  only 3  and  have NF  been 3  on molecular geometry,  F i g . 9.1).  reported  [BB87a].  the  in  a  Following  steric  effects  243 and  simple valence bond hybridization concepts,  on N in an N X ( Z X N X = 90°, trigonal  show  decreasing  NH ,  i.e.  3  geometry).  predicted  that  Thus  the  %s-character  Predicted %s-character:  NH  on  to  as  100% p-character  this  measured  simple  XMPs  NH  3  the  are  2  3  3  microwave experiments  ( Z H N H = 106.7°)  2  3  [BP57].  In  3  3  [TK71,  (<£XNX=120°, basis  substituted  for NH CH , NH (CH ) 2  WL67,  100% s-character  outermost  >NH CH >NH(CH ) >N(CH )  3  show bond angles in the range 109°-111° for  intuitive  for  methyl groups  The bond lengths and bond angles well-known from  'lone pair' orbital  molecule would be predicted to range from  i.e. pyramidal geometry)  planar  generally  3  the outermost  it  would  the  be  orbitals would for hydrogen in  3  and N ( C H )  2  3  WL69].  These  3  measured  XMPs  are  molecules  (see F i g . 9.2) which are larger  contrast,  i.e.  than  show  a  trend, Observed %s-character:  N H <NH CH <NH(CH ) <N(CH ) 3  which is clearly exactly opposite  2  3  3  2  3  to that predicted on the  3  basis of the intuitive  arguments outlined above.  For N F , intuitive arguments based on geometry 3  ( Z F N F = 102.15°) [SG50] would  predict increased s-character relative to N H . In this case the prediction happens 3  to be consistent with the experimental results (Fig. 9.1e). It is obvious that such intuitive arguments are unreliable as a predictive tool.  In an effort  to provide further  understanding  of these seemingly complex results,  244  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  MOMENTUM DENSITY  POSITION DENSITY  Momentum (a.u.)  Fig. 9.2a-c  Two-dimensional contour plots (oriented molecules) of the position-space (xz plane) and momentum-space ( P P plane) densities of the outermost valence orbitals of N H , N H C H and N H ( C H ) calculated using the respective 4-3IG wavefunctions. N H is assumed to be of C g symmetry with the z-axis being the C axis whereas N H C H and N H ( C H ) molecules assume C symmetry. The positions of the atoms in the respective xz planes are indicated on the position-space maps. Experimental bond angles and bond distances were used. A l l dimensions are quoted in atomic units. X  Z  3  3  2  3  2  3  y  3  g  2  3  3  2  245  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  MOMENTUM DENSITY  POSITION DENSITY  Momentum (a.u.)  Fig. 9.2d-e  Two-dimensional contour plots (oriented molecules) of the position-space (xz plane) and momentum-space ( p P plane) densities of the outermost valence orbitals of N (CH ) and NF calculated using the respective 4-3 I G wavefunctions. Both molecules are assumed to be of C g symmetry with the z-axis being the C axis. The positions of the atoms in the respective xz planes are indicated on the position-space maps. Experimental bond angles and bond distances were used. A l l dimensions are quoted in atomic units. x  z  3  3  y  3  3  246 both  experimentally  density maps  and  for the  theoretically,  respective  the  position-space  molecules have  0.08,  0.05  wavefunction  and  was  0.02%  chosen  of the  for  the  respective  momentum-space  been calculated and the  are shown in F i g . 9.2. The contour values are 0.2,  and  80, 50, 20, 8, 5, 2, 0.8, maximum intensities.  calculation  of  density  contour  in  an  present study. The number  S C F calculation increases  as  =N  where  4  N  is  maps  functions [DF86]. For example, an S C F calculation for N H  3  as  a  moderately  of integrals the  0.5,  The 4-3IG  suitable compromise between computing economy and accuracy for the large molecules involved in the  results  involved  number  of  and N ( C H ) 3  basis using  3  the 4-3 IG basis set would involve 15 and 54 basis functions, respectively.  It can be seen that the observed increase in s-character (compare  Figs.  nitrogen  Tone  9.1a-9.1d) pair'  is related to the large H Is contribution trans to  (see  Figs.  more trans-H Is character a  maximum is reached  9.2a-9.2d).  A s more  Tost'  substitution.  A  comparison CH OH 3  The  of  as  in the  density  similar the  [MB81] and  case  understanding  is  of N ( C H ) 3  XMPs  (CH ) 0 3  2  present  of  the  [CB87] with experimental  of alkyl inductive effects  a following section.  effect  3  [BB87a].  valence (3a,)  increasingly  delocalization  measured  implications of the  methyl  groups  are  the  added,  is contributed (in phase) to the molecular orbital until  pair', atomic-like characteristic of the outermost increasingly  upon methyl substitution  delocalized  has outer the and  also  been  a, -type 3a,  Clearly,  the  Tone  orbital in N H with  each  observed valence  calculated results  2  for  is  methyl in  orbitals  orbital of H 0  3  the of  [BB87a]. current  in organic chemistry will be discussed in  247 The increased  %s-character  in N F  (Fig. 9.1e)  3  occurs  for  an  entirely  different  reason to that for the methyl amines. Close examination of the M O coefficients as well as inspection of the r-space the increased s-character situation in N H result,  which  and p-space maps  of the outermost valence orbital of NF  is intuitively predicted  on the  in terms of simple valence  noted  in the  absence  of steric  electronegativity of the substituent as  the  electronegativity  increasing amounts are  relative to the  3  is due to the larger N 2s contribution (relative to N 2p). This  3  rationalized that  (Fig. 9.2e) indicates that  qualitatively  of the  supported  by  of geometry,  has  bond hybridization concepts.  effects  the  increases.  substituent  of s-character  basis  bond  angle  often  been  Bent [B60]  decreases  as  the  It was also pointed out [B60] that increased,  the  central  atom  diverted  to the lone pair orbital. These ideas for NF the  present  E M S results  as  well  as  by  3  the  calculated density maps (Fig. 9.2e).  Another  way  (relative  to  outermost ( a ) p  viewing  N H ) is by valence  N(CH )  3  "  P  the  derealization  taking the  3  3  in  of  orbitals NH  ( F l g s 3  as  -  9  3 a  density  shown  - " - ) 9  3 b  of  a  in n  d  both position-space and momentum-space  (W  charge  in  difference Fig. P  N  F  "  3  maps  9.3. 3  N (CH )  of the  These P  N  H  3  NF  S s  3  respective  correspond  ( F i 3  and  to  9-3c-9.3d)  calculated using the respective 4-3 I G  wavefunctions. The contour values chosen are  ±80, ±40, ± 8 , ±4, ±0.8, ±0.4,  ±0.08  maximum  and  position-space  ±0.04% density  of  difference  nitrogen  center  in the  whereas  in the case of NF  contribution in N H  3  the  respective maps  the  case of N ( C H ) 3  3  3  'transfer  intensities.  of charge'  is clearly illustrated  away (see  From from  the the  Fig. 9.3a)  the higher N 2s contribution relative to the N  2s  is also very obvious (see N 2s spikes in projection plots in  248 r N(CH ) 9  3  P  n  H  3  3  POSITION DENSITY DIFFERENCE  1  1  1  1  1  1  1  1  MOMENTUM JDENSITY  DIFFERENCE  a  1  b I  1 1  I  I  I  j  • • t 1 -4.0 - 2 . 0  ^NF  3  I  I  1-  T  I  0.0  I  4.0  2.0  /°NH  I  iI  •<w.o)  1_  '  - 0 . 5 0.0  -4.0  •  I  1  0.0  I  I  2.0  I  4.0  1  1  1—  - 0 . 5 0.0  3  POSITION DENSITY DIFFERENCE  Fig. 9.3  '  -2.0  MOMENTUM DENSITY DIFFERENCE  Two-dimensional density difference contour maps in both position-space (xz plane) and momentum-space ( P P X  difference densities correspond to Pjg(CH 3b) and p  N  F  - p  N  H  )  Z  - Pjqpj  plane). The (3a and  (3c and 3d). Positive difference is  shown by solid contour lines while negative difference is shown as dashed lines. The positions of the atoms in the respective xz planes are indicated on the position-space maps. Experimental bond angles and bond distances were used. A l l dimensions  are quoted in atomic units.  249 Fig. 9.3c). The complementary momentum density difference maps (Figs. 9.3b and 9.3d)  illustrate  distribution  the  for  origin  N (CH ) 3  of  the  and  3  increased  NF  as  3  s-components also  of  shown  by  the  momentum  the  calculated  spherically-averaged MDs and revealed by the EMS measurements of momentum profiles (see Fig. 9.1). The derealization of electron density towards the trans-H (in the case of N (CH ) ) and the increase of the relative N 2s contribution (in 3  the  case  p = Oa o" . 1  of  NF ) is  projection plots p=Oa " 0  1  translates  3  This  is  3  in momentum  illustrated by the  in Figs. solely  9.3b  due  to  space  hatched  to  areas  increased in the  intensities  density  at  difference  and 9.3d.  The calculated increased intensity  at  s-orbitals.  This  by  aspect  can  be  understood  considering the Fourier transform relationship,  iMp) = ( 2 7 T ) "  1 / 2  T e "  1  [9.1]  ^ * ( r ) dr  At p = Oa o ~ , the 'area' of the position-space orbital wavefunction integrated over 1  all configuration space yields the momentum space wavefunction,  <MP>lp=0  =  (2TT)"  i / 2  [9.2]  J>(r) dr  Therefore only s-type orbitals can contribute at p = Oa " 0  d-type)  orbitals give  equal lobes of opposite  1  since pure p-type (or  sign [LN75] with a node at the  origin.  Consider now  a detailed comparison of the  measured XMPs  calculated MDs. Measurements of the outermost XMP of N H resolution (Fig. 9.1a)  has been reported earlier (see  3  with the various at high momentum  chapter 6) [BM87] as well  as the distributions calculated from the highly extended  Hartree-Fock limit SCF  250 (126-GTO)  and  CI  involves a full final  ion  (126-G(CI))  overlap between  wavefunctions.  The  CI wavefunctions  126-G(CI)  for the  calculation which  neutral target and  states was used for normalizing the theoretical M D s to the  the  3a, X M P  of N H . It can be seen that with increasing sophistication in the basis set (i.e. 3  from  STO-3G  region  to  4-3 I G to  126-GTO) better  of the- X M P is obtained.  correlation  and  relaxation  Further  effects  in  description of the  low  momentum  improved agreement is obtained  NH  by  the  126-G(CI) calculation (curve 5c, Fig. 9.1a). Additional details of experimental  and  3  are  considered  calculated results for all the valence orbitals of N H few  shown  are given in chapter  3  6. A  points however should be noticed. Firstly, the STO-3G + G M D (curve 3, F i g .  9.1a)  which involves the  p-functions  on the  addition of a diffuse  hydrogen  atoms,  s-function on nitrogen  (even slightly better than  the  126-G(CI)) although the  total  S C F energy  the  STO-3G  9.1).  In  compared  contrast  the  polarization functions give any improvement These results,  to  4-3 I G *  (£1^ = 0.80)  calculation,  improvement  value  is only marginal  diffuse  which  utilizes  the  in the calculated M D compared  to the  Table  standard  bond  XMPs  does not  4-3 I G calculation.  insufficiency of the variational  good description of the  in the  (see  and has better calculated total energy,  once again, illustrate the  guaranteeing  and  gives a surprisingly good description of the  XMP  in  as  when  especially when  procedure  intermediate  (or  poor) quality basis sets are employed. It should also be noted that caution should be exercised in interpreting the STO-3G + G results as will be discussed later.  Substitution of a methyl group in N H results  are  clearly  observed  3  yields the N H C H 2  shown in Fig. 9.1b. The increase both  experimentally  and  in  3  molecule for which  in s-character relative to N H the  calculations.  3  is  A n earlier E M S  251 study  of  the  outermost  orbital  of  NH CH 2  at  3  somewhat  lower  momentum  resolution [TL84] has indicated the same trend. The present study with improved momentum  resolution  (Ap = 0.15a " )  allows  1  0  the  low  momentum  region  (0.1-0.4a " ) to be more directly observed in that a dip in the XMP (Fig. 9.1b) 1  0  is seen at p = 0 . 2 a ~  in agreement with the general predictions of the various  1  o  calculated MDs. Similar to the situation for N H , the MD description provided 3  by  the  basis.  4-3 IG  basis is  However as  much better  than that given by the  clearly illustrated by  simpler STO-3G  a comparison of the  STO-3G+G and  4-3IG* MDs, it is again evident that diffuse functions are much more effective in describing the XMPs than standard bond polarization functions. It is also clear that even with the semi-quantitative.  'best fitting' calculation (STO-3G+G) the agreement is only  More extended  basis sets such as  those used  in N H  3  (Fig.  9.1a and chapter 6) are expected to give further improved quantitative agreement with experiment.  However use of more complex wavefunctions  for molecules  as  large as the methyl amines is too time consuming and prohibitively expensive at the present juncture.  Results for the di-methylated species, N H ( C H ) , 3  the same pattern of agreement with N H  3  and N H C H 2  3  2  are shown in Fig. 9.1c. Again  between experiment  and theory is obtained as  with very good agreement in shape of the MD (but not  necessarily for other properties, see Table 9.1) for the STO-3G + G calculation. In the case of the tri-substituted methyl amine ( N ( C H ) ) for which results are 3  shown in Fig. 9. Id  the apparently less effective  3  4-3 IG*  calculation (Figs.  9.1b  and  9.1c) was not performed because of program limitations. The other calculated  MDs  (STO-3G, 4-3 IG, STO-3G + G) however show the same pattern as for N H , 3  252 NH CH 2  and  3  NH(CH ) 3  with  2  agreement for shape with the  the  ST0-3G + G  experiment.  to the momentum profile at p = l . 0 a "  1  o  from  the  best wavefunction  and NH (CH ) 3  MDs.  the  best  The theoretical M D s were normalized  using the 4-3 I G M D which is calculated  (in energetic  X M P of N F  terms).  A s with the  case of  NH CH 2  3  is shown in F i g . 9.1e compared with the calculated  3  In this case the STO-3G + G calculation, which was quite successful for the  methyl  amines,  4-3 I G *  calculation gives a quite reasonable  Fig.  giving  the agreement is again only semi-quantitative.  2  The outermost  calculation  9.1e  grossly  overestimates the  s-character  of  this  orbital  but  the  description of the X M P . Note that in  the STO-3G + G M D has been scaled by a factor  (x0.70) to bring the  curve into better perspective with respect to the experimental measurements. This result  indicates  the  importance  of exercising caution  in the  functions to fairly simple basis sets when predicting X M P s [BB87].  These  designed  functions  as  field-induced  mainly  calculations energies  diffuse  of  molecular  from the  the case of N H  3  (see  Sec.  9.2)  polarization  polarizabilities  for  STO-3G + G wavefunction  are  diffuse  as pointed out  before  C, N , O, F  functions  [ZC79].  addition of  and  used  Furthermore,  and  H  for  improved  the  calculated  of poor quality, for example, in  the calculated total energy is =20eV higher than the  estimated  Hartree-Fock limit (see Table 9.1). Likewise the predicted dipole moments using  the  showed moments  STO-3G + G  greater  wavefunction  discrepancies  with  were  (for  all molecules  the  experiment  studied  compared  except with  obtained for N F ) 3  the  dipole  predicted by the corresponding STO-3G wavefunction. This is instructive  because the particularly  dipole moment sensitive  to the  is often longer  thought  of as  a molecular property  range part of the  electronic charge  that is density.  253 However as has been demonstrated by the EMS measurements and calculations for H 0  [BB87] and N H  2  3  [BM87] dipole moments and MDs of good precision  are only obtained when diffuse functions are included in sufficiently extended basis sets that demonstrate basis set saturation.  A closer examination of the experimental results (Fig. 9.1) shows an interesting trend in the 'secondary' p at p = l a max  _ 1 0  (i.e. the p corresponding to the max  p-type component in the MD) which occurs in the range  0.4a ~ ^p^2a ~ . 1  With increasing methyl substitution the observed secondary P 0.52a "  1  0  in N H  to = l . l a "  3  1  0  and =1.3a -' 0  1  o  o  increases from  m a x  for N ( C H ) 3  3  and NF , 3  respectively (see Table 9.1). The experimental results indicate increasing high momentum components as more methyl groups (or F atoms) are substituted into N H . The shift of the p momentum has been associated with 'max towards higher ° 3J  m  the presence of additional nodal surfaces as for instance in the outer 7T orbitals of p-dichlorobenzene (see chapter 8) [BB86] and C 0  2  [LB85a]. Tossell et al.  [TL84] suggested that the higher momentum components observed in N H C H 2  compared to N H  3  3  are due to an extra nodal surface. This node is attributed to  the trans-H Is which is antibonding with respect to the N 2p.t The steeper and more prominent nodal surfaces with increasing substitution can be seen in the corresponding position-space density maps in Fig. 9.2.  It should be noted that there is no direct relation between the increased high momentum  components  (i.e. shift  in -Pmax  to higher  momentum)  in the  t A recent EMS study by Rosi et al. [RC87] also suggested the presence of a 'region of minimum charge density between the nitrogen and carbons' to explain the secondary P tri-ethylamine (TEA). m  m a x  254 experimental as  the  momentum  'secondary'  profile and the  p  moves  orbital ionization potential  to higher  momentum  from  NH  (IP). In  fact,  to N ( C H 3 ) 3 ,  3  max the IPs decrease (see Table 9.1). Martin and Shirley [MS74] have suggested the decrease in IP that accompanies relatively  large  stabilization  polarizable  methyl substituents.  correlation between suggest that  the  in  methyl substitution on N H  the  final  In support  A(IP) of the outermost flow  of charge  yields the stabilization energy  from  state  ion  is due to  3  afforded  by  of this argument they  that  the  easily  show a linear  orbital and the relaxation energy,  the  alkyl  group  to  the  the  nitrogen  and  center  in the molecular ion [MS74, P47]. It is important  to note that the conclusions of the present E M S work concerning derealization of 'lone  pair' charge  density  towards  the  trans-H in C H  groups  3  (see  below)  are  not incompatible with the observations of Martin and Shirley [MS74]. This is due to the fact that the experimental momentum profile of the amines is largely an initial  state property  final state effect  9.4. THE The  [MB83].  chemical  reactivity, 'have  moderately prevalent results N (CH ) 3  no  In  their  Swain et inductive  by resonance'. view  shown 3  INDUCTIVE  inductive effect  chemistry  groups  trend  in the  vertical IPs  is largely a  [AB80].  METHYL  intrinsic  [WM78] while the  especially in  however  the  EFFECT  of methyl  groups  recent  extensive  al.  [SU83] have  (or  field)  is still  analysis  influence  a controversial issue in of substituent  suggested but  that  tend  to  CH  3  effects and  donate  C H 2  donate electrons  when  of  previous  clearly provide  section  for  the  stability  NH ,  experimental  3  NH CH ,  evidence  2  3  carbo-cations. NH (CH )  supporting  3  the  5  electrons  The notion that methyl groups considering  on  2  is a The and  general  255 conclusions  of early theoretical ab  HP70] that conclusions  CH  groups  3  of Hehre and  are  initio MO electron  calculations by  withdrawing. In  Pople [HP70] regarding  the  Pople et  al.  [P70,  particular the  early  methyl inductive effect,  which were based on calculations using the STO-3G basis set, have now  been  confirmed in the present work where larger basis sets (e.g. 4-31G*) have been used. The  present work places the early work of Pople [P70, HP70], which  criticized at the time, in proper perspective. Pople and  was  co-workers [HP70,  P70]  noted that the increase in base strength of alkyl amines with increasing alkyl substitution is not  associated  with  increasing electron density  on  nitrogen.  In  particular they noted [P70, HP70] the decreased electron density on the nitrogen center and increased electron density on the hydrogens trans to the lone pair as is also indicated by theoretical opinions,  studies methyl  the density map [HP70, P70]  groups  are  diagrams shown in Fig. 9.2.  indicated  that,  intrinsically  contrary  electron  to  The  commonly  withdrawing  early held  substituents.  However prior to the present direct experimental probing of the electron density of the outermost orbitals (using EMS) were also supported by  the  experimental studies. For  example, NMR  these theoretical conclusions  findings of a  [HP70,  number of evidences from  chemical shift data [JK70] has  P70] other also  indicated that methyl groups are intrinsically electron withdrawing. In particular it has been demonstrated that decreased electron density around a particular nucleus corresponds to deshielding (i.e. decrease in  13  C  chemical shifts with increasing  alkylation in substituted alcohols [JK70]). Furthermore the dipole moments of the alkyl amines are observed to decrease with increasing addition of methyl groups (see Table 9.1) quite contrary to what would be expected if methyl groups were to be intrinsically electron releasing or to have no significant electronic effect at  256 all.  Ingold [169] in his  alkylated amines  discussion of the  suggested  unusual  dipole moment  that N - H and O - H bonds are  trend  of the  associated with larger  moments than N - C and O-C bonds. These rationalizations are consistent with the present results.  The  fact  that  nitrogen  atom  substituents reconciled and gas [BB71]  base but  is  strength rather  now  is  with  recognized  not the  have  argued  that  the  increased  [MM75,  previously misunderstood phase basicity of the  associated  amines  electron  density  polarizability afforded  UM76,  phenomena  with  AB80].  These  regarding  the  on  by  methyl  views have gas  contradictory  now  phase acidity  [M65, B B 7 1 , BR71]. Brauman and  seemingly  the  increased  acidity  Blair and  increased basicity of the amines with increasing alkylation is due in fact to the larger polarizability of alkyl groups (relative to hydrogen) rather than to inductive effects.  Likewise,  carbo-cations  the  [MB83]  N M R chemical are  also  shift  recognized  as  data  [JK70]  and  due  to two  distinct  effects  (i.e. initial and final state effects, respectively).  9.5.  SUMMARY  The  present  E M S results  maps,  although  series,  are  and  accompanying  calculations  currently limited to the outermost  clearly consistent with the  the  stability of and  separate  of M D s and  density  valence orbitals of the  view that methyl groups  are  amine  intrinsically  electron withdrawing when bonded to nitrogen as earlier predicted in calculations by  Hehre  and  Pople  [HP70].  The  present  experimental  results  using  E M S to  probe directly the outermost orbital electron density clearly confirm the qualitative  257 aspects of the theoretical calculations based on S C F L C A O - M O wavefunctions. On the basis of the work for N H even  better  quality  (highly  3  (see chapter  extended)  6) [BB87] it can be expected  wavefunctions  would provide  even  that better  description of the experimental results for the methyl amines but would not alter the conclusions of the present work.  CHAPTER  10.1.  10. T H E H A L O G E N S : A T H E O R E T I C A L  STUDY  OVERVIEW  The  halogens  symmetric  (except  for  non-coplanar  F )  were  2  electron  studied  momentum  using  the  spectrometer  Flinders  with  University  position  sensitive  detectors in both exit channels. The polar angles are 45° and the impact energy is  lOOOeV plus the binding energy. Momentum resolution is = 0. l a "  and  1  0  energy resolution is 1.5eV fwhm. Details of the data  analysis  Flinders  procedures  spectrometer  has  are  discussed  spectrometer,  elsewhere  multichannel plates  [CM84,  in the  the  its operation CM86].  and  Since  energy dispersive  the  plane  of  each analyzer, the experimental momentum profiles are generated by deconvoluting the  binding energy  energy  scans  normalized  scan obtained  (done  XMPs  binding energy  at  sequentially at  for respective  0 angles.  different  different  angles)  This series  therefore  automatically gives  orbitals corresponding to the  ion states  in  the  spectrum.  A l l experimental measurements reported  in this section were made  by colleagues  at  U B C and Flinders University (see Acknowledgements) and further  to  be  found  of binding  in the  published results  for  Cl  2  [FG87],  Br  2  details  [FG87a]  and  are I  2  [GP87].  As  part  personnel,  of a the  collaborative E M S project spherically averaged  on  the  M D s and the  and position-space maps were generated  halogens  corresponding  by the author  developed at U B C .  258  involving  the  above  momentum-space  using the H E M S  package  259 The ground state electron configuration of the halogens  ( ' E g * ) [CF71] are given  below:  CI  Br  2  4a  core  :  4a  2 g  2  5<V  u  g  outer valence  inner valence  outer valence  core  2 :  1ua  I0a  2 g  11o  2 u  inner valence  10.2. MOLECULAR Measured  XMPs  2iTg, 27T u  27T «  inner valence  core  4o~  2 V  and 5 a  U  and  4c  g  g  (Cl ) 2  calculated M D s [FG87] corresponding  peak  inner  areas  valence  orbitals  of  the  binding  providing consistency checks at respective spectra  gaussian [FG87].  to ionization of  the  outer valence orbitals, as well as corresponding data for the  experimental data for the three outer fitted  g  u  outer valence  CHLORINE and  6 * « 6^ «  2 g  fits  to  Sections  10. la-10.5a.  The  valence orbitals have been evaluated  from  outer  and  c  shown  energy  # = 0° and  the b  are  spectra  Figs.  [FG87].  Additional  points  0 = 7° have been generated from  valence of  in  Figs.  region of the  wide energy  10.1-10.5  the  show  the  range  respective  two-dimensional momentum density and position density maps for an oriented C l molecule calculated using the M L * polarized wavefunction. Contour values are 0.02,  0.05,  0.08,  0.2,  0.5,  0.8,  2,  5,  8,  20,  50, and  2  at  80% of the maximum  density. The side panels in sections b and c show density profiles (on a relative scale) along the dashed lines on the density maps.  260 The  calculated spherically averaged M D s shown on Figs.  obtained using a range wavefunctions  of wavefunctions of varying  were generated  using the G A U S S I A N  10.1a-10.5a have been  quality. These 80 package  S C F (RHF)  [BW80] at  the  equilibrium geometry (3.7568a.u.) [CF71]. A whole range of wavefunctions ranging from the simplest STO-3G basis set to the extended M L * *  basis set (see Table  1) were constructed.  The  STO-3G and LP-41G* are internally stored basis sets of the G A U S S I A N  package  [BW80]  wavefunctions  and  were  were  generated  used  without  using  the  modifications.  (13sl0p)  basis  The set  ML of  series  McLean  80 of and  Chandler [MC80] contracted to [6s5p]. This contracted basis set is referred to as the  M L wavefunction.  To investigate  the  effect  of polarization functions,  polarized basis sets were developed namely the M L * and M L * *  two  basis sets. The  M L * basis was built from the M L basis augmented with one d-type polarization function having an exponent (0.56) suggested by Sakari et al. [ST81]. In similar manner the M L * * two  d-type  Huzinaga  polarization functions [H84].  wavefunctions favorably  basis was built from the M L basis and then augmented  The  are  with  with  calculated  SCF  shown in Table  the  best reported  exponents  (0.22,  0.797)  (RHF)  energies  for  SCF  equilibrium geometry (3.7619a.u.) [SM74].  The  momentum  distributions and  suggested the  maps  have  (-918.99012a.u.)  been  by  respective  10.1. The calculated total energies  total energy  with  compare  calculated at  calculated using the  the  HEMS  package developed at U B C . A l l calculated M D s shown in the various figures are spherically averaged and have been convoluted with the  experimental momentum  Table  10.1. Wavefunctions f o r  Molecular C h l o r i n e  Wavefunction  Type  B a s i s Set  Total  energy(a.u.)  Reference  ST0-3G  GTO  (9s5p)/ [3s2p]  -909.11163  [BW80]  LP-41G*  GTO  (5s5p1d)/  --a--  [BW80]  (13s10p)/ [6s5p]  -918.92629  [MC80]  (13s10p1d)/  -918.97376  --b--  -918.97803  --b--  [2s2p1d] ML  GTO  ML*  GTO  [6s5p1d] ML**  No t o t a l  GTO  energy  (13s10p2d)/ [6s5p2d]  a v a i l a b l e since frozen  P o l a r i z a t i o n f u n c t i o n s [ST81,  H84]  cores used.  have been  added to the  ML b a s i s  set [MC80].  OS  262 resolution (Ap=0. 1 a " )• It should be noted that the experimental results shown 1  0  for each outer valence orbital in Figs.  10. la-10.5a have the correct relative  normalization  were determined  to each other  since they  repeated sequential binding energy peak  areas  by accumulation of  scans. The data points are from integrated  covering the unresolved  rotational-vibrational  width  of each ion  electronic state. In order to place all the calculations and the XMPs of the three outer valence orbitals in Figs. 10.1a-10.3a on the same (relative) absolute scale a single point height normalization has been used between the (best fitting) ML* calculation (see Table  10.1) and the measured 2it  u  XMP  (Fig. 10.2a) at its  maximum. All other data points, both experimental and theoretical, maintain their correct normalization relative to this single point.  Consider first Figs. 10.1-10.3. It can be seen that a reasonably good quantitative fit to the  27Tg  (Fig. 10.1a) and the  27T  U  (Fig. 10.2a) XMPs is given by the  ML* [MC80] and LP-41G* [BW80] wavefunctions. These two wavefunctions, each with  a single d-polarization  function, give similar  results. However the  27T  U  experimental cross section maximizes at a lower momentum than predicted by the calculation.  The  minimum basis set STO-3G  wavefunction  [BW80] gives a  significantly poorer fit to experiment.  The  5tfg XMP (Fig. 10.3a) shows an s-p mixed symmetry as expected from its  atomic orbital (AO) composition (see also density maps in Figs. 10.3b and 10.3c). The  STO-3G wavefunction clearly grossly misrepresents the orbital shape, with  much too large a relative contribution from the p-type orbitals. The ML* and LP-41G* both represent the shape and relative contributions of the s- and p-type  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  MOMENTUM DENSITY  POSITION DENSITY  Momentum (a.u.)  Fig. 10.1  (a) Experimental and calculated spherically averaged momentum distribution for the 2ir orbital of C l . The solid circles are experimental values extracted from the appropriate peak areas in the binding energy spectra at given values of 0; the open triangles are data extracted from the respective peak areas in the 0 = 0° and 7° spectra. In (b) and (c) the orbital density are shown in momentum and position space respectively. A l l dimensions are in atomic units. 2  to W  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  MOMENTUM DENSITY  POSITION DENSITY  Momentum (o.u.)  Fig.  10.2  (a) Experimental and calculated spherically averaged momentum distribution for the 2ff orbital of C l . See Fig. 10.1 for details. u  2  tsD  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  MOMENTUM DENSITY  POSITION  DENSITY  Momentum (a.u.)  Fig. 10.3  (a) Experimental and calculated spherically averaged momentum distribution for the 5 o orbital of C l . See Fig. 10.1 for details. g  2  to Ol  Fig.  10.4  (a) Experimental and calculated distribution for the  4a  spherically averaged  orbital of C l . See Fig. 10.1 2  momentum for  details.  Ci OS  SPHERICALLY AVERAGED M O M E N T U M DISTRIBUTION  MOMENTUM  DENSITY  POSITION  DENSITY  Momentum (a.u.)  Fig. 10.5  (a) Experimental and calculated spherically averaged momentum distribution for the 4 a orbital of C l . See Fig. 10.1 for details. g  2  to  -J  268 orbitals  quite  well,  with  the  best  s-to-p  ratio  given  by  the  M L * calculation.  However the overall cross section is in both cases (at least on the basis of the single  point  explanation than  normalization  the  for this could be the  unity  effects  on  in  the  causing  measured  the  proposition comes  orbital  u  discussed  possibility that the  energy  splitting  from the  2TT  of  range  the  about  pole  5 ag  too  due  Some  to  from  that the  10. la-10.3a  sensitive test of the composition of the  Consider now the shown  in Figs.  measured 10.4  and  higher  space)  relative  binding  10.5, respectively.  the  The  regards to shape for the 4 a  The effect which orbitals  It is noticeable  the  together  d-functions  valence  agreement and 4 a  XMPs  (as  with the  without (ML*) and  in  polarization with  a  most  (4a  u  and  4ag)  that these  orbitals  space (and conversely diffuse in orbital  between  despite  experiment  wavefunction  Figs.  calculations  the  their and  significantly theory  with  orbitals is quite good.  of polarization built into the  shows  wavefunction  outer  predicts  It is obvious  X M P provides  X M P s of the inner valence orbitals  to  energies.  5ag  this  wavefunction.  are predicted to be quite compact in momentum position  for  calculation [FG87] which  of polarized wavefunction.  of Figs.  many-body  support  just such splitting using a similar type a comparison  high. A n  pole strength is less  16.23eV  strength.  ADC(3) many-body  above)  10.1-10.3) using  (ML), double  the  with set  of  is illustrated in Fig. 10.6  for  all  McLean one  three and  set  polarizing  of  outer  valence  Chandler  [MC80]  added  d-functions  polarizing (ML**),  respectively. The normalization is identical to that used in Figs. 10. la-10.5a.  1  1 1 1 1—i  1  r  i  ,  1—  g I =11. 63«V .  1  1  1i — i — i — i  \  : z  i;  • I  q  z  $. 0.5 .  Q o 0.0 tcj  . .  1.0  \ i 15  2.0  25  i — i —  27T,,  2 l r  v  i  u r =14. 41 «v .  • m  y  -  I , A  V 0.0  •  i  0.5  i  2  ——  ^\  if  ci_  1.0  1.5  2.0  ML ML * ML * • Expt.  2.5  UI  i  to o  2 si u  I  "g I =16. 18«V .  \ 5 3  UJ u . ot  i — i  i  i  5  ,\ •  i  \  I  v  \  •  . =23. 6eV  expt  \  \ » ft  ^\\  — i  i  i  0.0  0.5  Fig.  10.6  i  1.0  i_  1.5  2.0  2.5  0.5 1.0 1.5 2.0 MOMENTUM (A.U.)  2.5  The effect of basis set polarization on calculated spherically averaged momentum distributions for the valence orbitals of C l . A l l calculations are on the same (relative) intensity scale with the experiment normalized to the M L * calculation at a single point on the 2 7 T X M P . 2  U  to to  270 The calculated M D for the the  5a  g  orbital (see Fig. 10.3) is extremely sensitive to  degree of polarization selected. The best fit would be with  (ML) wavefunction, if it is assumed that the 5 a  the unpolarized  ionization is effectively confined  g  to a single pole at 16.2eV. However as discussed above there is some theoretical evidence above  for  a  , the  splitting of the  5a "*  pole  g  M L * wavefunction best describes  compared with the observed X M P , for the of interest  strength. the  5o  to note that polarization functions  As has  also  been  ratio of s to p  noted  components  orbital. In this connection, it is  g  in the  wavefunction  also  increase  the distribution of satellite intensity above the main line [FG87] and this is what is also observed in the experimental binding energy spectra [FG87].  10.3. MOLECULAR The  measured  from  the  MDs  for  BROMINE  XMPs  47T ,  47T  the  7o"  g  U  u  and and and  (Br ) 2  calculated 8o  outer  g  7o  M D s [FG87a]  valence orbitals as well  inner  g  corresponding  valence  orbitals  are  as  to ionization the  shown  calculated in  Figs.  10.7a-10.11a, respectively. The relative peak areas at 23.1eV and 25.6eV derived from the binding energy spectra plots of the 7 a  The  u  calculated  and 7 a y  MDs  contracted  G T O basis  contracted  basis  functions  in the  set  were set  [FG87c] at <p = 0° and 0 = 6° are shown on the  orbitals (Figs. 10.10a and  obtained (denoted  (denoted  as  using  as  an  V N ) and  V N * ) . The  V N * wavefunction had  unpolarized  (14sllp5d)/[9s6p2d]  a polarized  (14sllp7d)/[9s6p4d]  two  exponents  10.11a)  valence of 0.2  and  d-type 0.6.  polarization The  correct  relative normalization between the three outer valence X M P s has been maintained and a  single point normalization was  used  to compare  experiment  with  theory.  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  q d -  i  i  i  i  i  i  i — i  i  MOMENTUM  DENSITY  POSITION  i  Air Br 9 2 n  m  0  d  l =io. 74«V  oc  p si  v  t »)OOO«V • B . C .  o d  0  U  p W I  .o  p d CO  I  0.5 10 J.5 MOMENTUM (A.U.)  Fig. 10.7  2.0  2.5  -4.0  -2.0  0.0  2.0  4.0  0.5 1.0  (0.1.0)  — i — - I  -8.0  (a) Experimental and calculated spherically averaged momentum distribution for the Ait orbital of B r . The solid circles are experimental values extracted from the appropriate peak areas in the binding energy spectra at given values of <p; the open triangles are data extracted from the respective peak areas in the 0 = 0 ° and 6° spectra. In (b) and (c) the orbital density are shown in momentum and position space respectively. A l l dimensions are in atomic units. 2  i  -4.0  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  MOMENTUM DENSITY  POSITION  MOMENTUM (A.U.)  Fig.  10.8  (a) Experimental and calculated spherically averaged momentum distribution for the 47T orbital of B r . See Fig. 10.7 for details. U  2  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION •  o  i  i  i  i  •n  8G*  01  m <  o  MOMENTUM DENSITY  POSITION DENSITY  i  Br  Br 2  Br 2 8o\.  0  I =14. 62eV  6  ¥  CM  o u  o  2  o  1/1  u _l  $2 o  q o d  0.0  J0.5  -4.0  -2.0  0.0  2.0  4.0  0.5 1.0  (0,1.0) 1 I-4.0 1 -8.0  j |  1__0.0 J 1 4.0 1 1 8.0 1  0.5 tO  MOMENTUM  Fig.  10.9  (a) Experimental and calculated spherically averaged momentum distribution for the 80g orbital of B r . See Fig. 10.7 for details. 2  to CO  Fig. 10.10  (a) Experimental and calculated spherically averaged momentum distribution for the 7o" orbital of B r . The triangles are relative values obtained from the peak areas at 23.1eV in the energy spectra. See Fig. 10.7 for details. u  2  tso  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  MOMENTUM DENSITY  POSITION DENSITY  MOMENTUM (A.U.)  Fig. 10.11  (a) Experimental and calculated spherically averaged momentum distribution for the la orbital of B r . The triangles are relative values obtainea from the peak areas at 25.6eV in the energy spectra. See Fig. 10.7 for details. 2  to in  276 The best Fitting (VN*) calculated M D was height normalized to the  47T X M P . y  The momentum space  and position space density maps calculated using the V N *  wavefunction  shown in Figs.  are  also  0.05, 0.08, 0.2, 0.5, 0.8, The side panels  10.7-10.11. Contour values  are  at  0.02,  2, 5, 8, 20, 50, and 80% of the maximum density.  of sections b and c show density profiles (on a relative scale)  along the dashed lines on the density maps. A l l calculations are for an oriented Br  2  molecule at the experimental geometry (r = 4.2106a.u.)  The two wavefunctions  [CF71].  give very similar M D s and both give good  fits, in both shape and magnitude , to the measured  47r_,  4ir  y  quantitative  and 8o u  XMPs. y  The small discrepancy between calculation and experiment at low p for the  47T  orbital could likely be reduced by incorporating diffuse functions in the basis as  has  been  found  necessary  in obtaining  with E M S studies of H O [BB87], N H z  adequate basis  sets  for  U  set  comparison  [BM87] and H S [FB87a]. The (j> = 0°  3  2  and 0 = 6° relative peak areas for the 23.1eV and 25.6eV peaks are in excellent agreement with (Figs.  the  10.10 and  shapes of the  10.4. MOLECULAR  the  67Tg,  IODINE  and  U  and 70g  and  7o"g M D s , respectively  23.1eV and 25.6eV peaks  are  orbitals, respectively.  (I ) 2  H^g  respectively.  (16sl3pl0d)/[7s6p5d] calculated  u  u  X M P s and calculated MDs [GP87] corresponding to ionization from  67T  10.12a-10.14a,  7o"  10.11). This confirms that the  due to ionization from the 7 o  The measured  calculated  energy  Two  contracted of  outer  valence  different  G T O set  -13826.208a.u.  orbitals  are  wavefunctions (denoted  and  an  as  compared are  used,  VN1) which  unpolarized  in  Figs.  namely has  a  a  total  (16sl3p7d)/[6s5p2d]  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  0.0  0.5  1.0  1.5  2.0  2.5  MOMENTUM DENSITY  -4.0  -2.0  0.0  2.0  POSITION DENSITY  4.0  0.5 1.0  -8.0  -4.0  0.0  4.0  8.0  0.5 1.0  MOMENTUM (A.U.)  Fig.  10.12  (a) Experimental and calculated spherically averaged momentum distribution for the 6ff orbital of I • The solid circles are experimental values extracted from the appropriate peak areas in the binding energy spectra at given values of <p; the solid squares are data extracted from the respective peak areas in the <f> = l° and 7° spectra. In (b) and (c) the orbital density are shown in momentum and position space respectively. A l l dimensions are in atomic units. 2  to  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  MOMENTUM DENSITY  POSITION DENSITY  MOMENTUM (A.U.)  Fig.  10.13  (a) Experimental and calculated spherically averaged momentum distribution for the 6ff orbital of I . See Fig. 10.12 for details. u  2  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION I  vt  I  1  l  I  ni  l  l  l  , 1  "g  MOMENTUM DENSITY  POSITION DENSITY  I : m :  I =13. OOeV v  VN1 VN5  0.0  0.5  i  \  1.0  -  11.5  j  •  2.0  •  •  2.5  .oo i 1 1 (0.1,0)  -4.0  -2.0  0.0  2.0  4.0  0.5 1.0  -8.0  l  -4.0  0.0  l  1i  4.0  i  1  8.0  '2  0.5 1.0  MOMENTUM (A.U.)  Fig. 10.14  (a) Experimental and calculated spherically averaged momentum distribution for the 11a orbital of I . See Fig. 10.12 for details. 2  to  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  MOMENTUM DENSITY  POSITION DENSITY  1  -T-  '2 10<7  U  1  .oo 0.0  0.5  1.0  1.5  2.0  2.5  -4.0  -2.0  0.0  2.0  MOMENTUM (A.U.)  Fig. 10.15  4.0  0.5 1.0  "(0.1.0) -8.0 -4.0  11 1  I  0.0  1 1  1  4.0  1  1  8.0  i 1  0.5 1.0  1  (a) Experimental and calculated spherically averaged momentum distribution for the 10a orbital of I . See Fig. 10.12 for details. u  2  to  OO O  SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION  MOMENTUM DENSITY  POSITION DENSITY  MOMENTUM (A.U.)  Fig. 10.16  (a) Experimental and calculated spherically averaged momentum distribution for the 10a orbital of I . See Fig. 10.12 for details. 2  to 00  282 contracted G T O set (denoted as VN5) which has a total energy of -13826.031a.u. The  two  different  S C F wavefunctions  give  very  similar  distributions. Correct relative normalization of the 67Tg, 6TT maintained  as  explained  earlier.  In  order  to  calculated  momentum  and H ^ g X M P s  u  compare  experiment  are  with  theoretical calculation a single point normalization is made on the 6n  the  X M P . The y  momentum  space  and  position space  polarized wavefunction are  density  maps  also shown in Figs.  calculated  using  the  VN1  10.12-10.14. Contour values  are  at 0.02, 0.05, 0.08, 0.2, 0.5, 0.8, 2, 5, 8, 20, 50, and 80% of the maximum density.  The  side  dashed  lines  on  panels the  show density  density  maps.  profiles  (on a  A l l calculations  relative are  for  molecule at the experimental geometry (r = 5.039a.u.)  [CF71].  The  are  general  calculated  features  M D s . The  of  the  experimental  relative  magnitudes  data of  the  scale)  well  calculated  an  cross  U  data,  higher  as  well  momentum  as  the  fact  compared  to  that  the  the  calculations significantly overestimate  P  6TT  m  a  of the  x  67Tg  X M P . For  u  the  oriented  described  ionization of the 67Tg, 67T and l l a ^ orbitals are in reasonable the  along  I  2  by  the  sections  for  agreement with  X M P occurs at  the  XMP  the degree of symmetric components  a the  in the  wavefunction, that is the momentum density at p = 0a " . 1  0  The calculated MDs for the inner valence orbitals ( 1 0 a in  Figs.  10.15a  and  10.16a. Similarly  momentum-space  calculated from the V N 5 polarized wavefunction are reported range  XMPs binding  in Figs. energy  10.12-10.16.  scans  at  The  0=1°  data  and  and  lOcTg) are shown  and position-space  also presented  points  o> = 7°  u  obtained [GP87]  are  maps  alongside from  the  found  the long  to  be  283 consistent with that predicted by theory except for the 100" orbital. It has been u  postulated  [GP87] that the  different  shape  observed  is  due to the mixing of  states of different symmetry, most likely poles due to the 1 0 a  process. Such  g  mixing would account for the broad momentum profile at 16-20.4eV.  10.5. GENERAL  TRENDS  Several interesting trends can be observed in comparing the measured XMPs of the valence orbitals of C l , B r 2  2  and I  2  molecules.  One interesting feature of the measured XMPs of the halogens is the outermost valence 7f P  m a x  in P  orbital. It is seen in Figs. 10.2,  u  10.8 and 10.13 that the measured  in the respective TT XMPs decrease from C l as  m a x  MDs.  It is  the  2  to B r  2  to I . This decrease 2  atoms get heavier is also clearly predicted by the calculated  well known that within a 'family' (for example,  the  noble gases  [LB83]) the measured XMPs for the outer valence p-type orbitals maximize at lower values of momentum as the atom gets heavier. This effect is related to the inverse weighting property of the Fourier transform. As one proceeds from the  lightest  become  member of the  more diffuse  group towards  the  heaviest  (in position space) and consequently  member, become  the orbitals contracted in  momentum space.  Another trend in the halogens is observed in the measured XMPs of the outer valence a component  g  orbitals which are of mixed s-p character. The symmetric (i.e. s-type) in the  measured  XMPs  relative  to  the  anti-symmetric  (i.e.  p-type)  component is found to decrease within the group as the diatomic becomes heavier.  284 That is, the observed  percentage  s-character (intensity at p = Oa o ~  that at p = l a ~ ) of the outer valence a  1  relative to  orbital is found to be:  1  0  y  %s-character:  Cl >Br >I 2  2  2  This trend is predicted by the calculations especially for C l case of the l l o "  a  orbital of I  and B r . In the  2  2  the failure of the calculated MD  2  to predict the  y  measured  XMP  is probably  wavefunctions for the I  2  to  the  difficulty  in obtaining accurate  molecule which is a 106-electron system. In fact for all  these molecules ( C l , B r 2  related  2  and  I ) the experimental total energy is currently 2  not available and neither are accurate estimates of the Hatree-Fock limit total energies. Without these values it is not easy to evaluate the quality of the respective wavefunctions as was done for smaller molecules. In a simple LCAO-MO picture the  a  outer valence orbital in the  halogen  y  diatomic molecules can be considered as a sigma (s-p hybridized) bonding orbital. The  varying s and  p components in this orbital therefore reflect the optimum  balance necessary for bonding. Levin et theoretical orbital MDs for the first row contribution from  al.  [LN75] found in an investigation of  that for the first row diatomics ( L i , 2  hydrides (BH, CH  and  FH)  the 2p-component of the • a_  there was  N an  2  and  F ) 2  and  increase in the  hybrid orbitals with increasing  y  atomic number. This is manifested by the decreasing s-to-p ratio in the predicted MDs  as the atomic number of the heavy atom increased [LN75]. Similar results  were also predicted by Cade and Huo the respective outer valence o the first row  [CH67] in an AO  composition analysis of  molecular orbitals (calculated in position space) of y  hydrides. Bader et  al.  [BK67] correlated this increase in  2pa  285 component in the outer valence 30g  molecular orbital from B H to F H (first row  hydrides) to an increase in bonding character.  The decreasing s-to-p ratio in the  experimental momentum profiles of the outer valence atomic number  is now seen  as  a more general  o_ orbitals with increasing y  phenomenon  the present case proceeding down a chemical group ( C l , Br / 2  2  as is observed in 12)-  C H A P T E R 11. M U L T I C H A N N E L EMS: PROSPECTS A N D D E V E L O P M E N T S  11.1.  MULTICHANNEL  The  idea  of a  ELECTRON  MOMENTUM  multichannel E M S spectrometer  SPECTROMETER  (MEMS)  is of paramount  interest  since  coincidence rates in high resolution single channel E M S experiments are very low (<0.1cps).  Two approaches  have been attempted,  namely a multichannel detection  scheme in the angular (0) plane described by Moore et al. [MC78] and later by Cook  [C81] and  a multichannel detection scheme  in the  energy  dispersive plane  described by Weigold and co-workers [CM84, MW85]. The former scheme  [MC78]  used  scheme  correlated  pairs  of  channel  electron  multipliers  while  the  latter  [CM84, MW85] involved the straightforward adaptation of commercial channelplate assemblies existing results  with  pair  position sensitive detectors  of  180°  been  [C81].  analyzer  and  [CM84]. Some of experimental results  outlined in chapter has  spherical  (Surface  adapted  Science Laboratories) to  has  produced  quite  an  impressive  obtained using this instrument  are  10. The former scheme (multichannel detection in the 0-plane) in this laboratory and initial construction was  Preliminary testing  in  the  single  channel  mode  of  done by Cook  operation  (i.e.  one  stationary C E M and one movable C E M ) was done [L84a, M85a] as a necessary testing plate  phase  prior  to  (MCP) detector  coincidence problems  signals in  the  the  planned  system.  were original  These  extremely design  installation of a tests low and  [L84a, and  multichannel microchannel  M85a],  that  this  construction.  In  however, was  showed  possibly  addition  the  due  that to  angular  distribution of the (singles) elastic signal was found to be anisotropic.  The  present  chapter  outlines  significant changes  286  that  have  been  made  to  the  287 original design of the M E M S  as well as initial (single channel mode) results on  helium and argon.  11.1.1. Accurate Several  reasons  Alignment were  advanced  to explain the  difficulties  encountered  with  the  'prototype' spectrometer. These were: (1) Spurious magnetic  fields;  (2) Faulty design parameters  in the C M A and conical retarding lens system;  (3). Inaccurate alignment; (4) Background secondary electrons.  The present  multichannel E M S spectrometer  is shown in Fig. 11.1. One  major  change from the original version [C81] has been the redesign and construction of a spherical retardation stage [M85a]. Previous work [M85a] also concentrated on correcting  magnetic  field  inhomogeneity  in  the  spectrometer  magnetic components in and around the spectrometer the  scattering  region. This,  by  replacing all  especially in the area  however, did not improve the  near  very low coincidence  rate and therefore suggested items (2)-(4) should be examined.  From Fig. 11.1 it can be seen that the basic design parameters  of the original  [C81] 360° C M A have been retained. A review of the original design  parameters  and a comparison with the theoretical values recommended by Risley  [R72] are  shown in Table  11.1. As can be seen  the  with the theoretical values. Furthermore at E  present  design parameters  conform  =100eV the theoretical deflection pass voltage (93eV) is very close to the observed deflection voltage (90eV). This then  288  OUTER CYLINDER  TMP 150IVsec  Fig. 11.1.  Schematic of Multichannel E M S Spectrometer. The components are: height adjustment screws (HAS), end correctors (EC), retarding grid (RG), spherical retarder (SR), conical lens-deflector (LENS), channel electron multiplier (CEM), multichannel plate (MCP) mount, inner and outer cylinders of the C M A , gas cell (GC), Faraday cup (FC), spray plates (SP1, SP2, SP3), quadrupole deflectors ( Q l , Q2, Q3), focus (F), anode (A), grid (G), cathode (C) and turbomolecular pumps (TMP).  289 Table 11.1. Design parameters for 360° C M A . Actual*  Theoretical  Inner radius (a)  1.0  1.0  Outer radius(b)  2.0  2.0  0.5  0.5  Source to image distance (z )  2.5  2.5  V (eV)  90  93  a,b  0  c  d  In units of inner cylinder radius, a (r = 63mm). Values recommended by Risley [R72] for first order focusing at 0 = 4 5 ° . H  d  s i" E/V Deflection voltage for pass energy is lOOeV derived from (b/a) ==2.1. =  d  +  d  290 suggested for  that item (2), at least in regard to the C M A , is not the likely reason  the coincidence problem. The deflection voltage also increased linearly with an  increase in the C M A pass energy as expected.  A  simple consideration of the design tolerances  showed that accurate The M E M S therefore outer  and  alignment maybe more critical than was originally  spectrometer  could  easily  inner  in the construction of the C M A assumed.  is mounted on Teflon sheets, instead of ruby balls, and be  clinders,  mis-aligned. Assuming a it  can  be  difference in the electron trajectories  shown  that  slight this  non-coaxiality of  will  result  in  a  the large  that are eventually focussed on the exit slits  at opposite ( ± 1 8 0 ° ) sides of the C M A . For the present set-up (b/a = 2.0, z = 2.5) 0  and a pass energy of lOOeV, a 0.4mm shift of the inner cylinder axis relative to the outer cylinder co-axis results in a Following the same arguments  the  l.OeV difference  in deflection voltages!  alignment of the electron gun relative to the  C M A co-axis will also be crucial.  Instead of attempting a complete re-design and reconstruction of the simple accurate  but  significant  changes  have  been  made  to  observe  spectrometer,  whether,  alignment was critical to the coincidence problem. The following  in  fact,  changes  were made: (a) Installation of height adjustment  screws (HAS);  (b) Re-design of gas cell. Teflon-tipped allowed  very  height  adjustment  small  (=0.1mm)  inner and outer  screws and  at  accurate  the  bottom  changes  of  in the  the  outer  alignment  cylinder of  the  cylinder co-axes. The gas cell was re-configured by constructing  291 a brass tube that was attached to the lens system (see F i g . 11.1). This enabled the aperture plates  (SP2 and SP3) to be placed on the same tube. This allowed  the incoming electron beam to be well-defined with respect to the C M A co-axis.  To complement was  the  empirically  following physical changes,  devised.  This  involved  re-assembling the electron gun and the aligning  them  with  a  high tolerance  the  cylinder  co-axis, the  height  accurate  slow  and  alignment tedious  C M A mounting block and (12.7mm  specially designed jig. The inner and outer the C M A mounting block. To further  an  dia.)  stainless  cylinders are  process  of  simultaneously  steel  then  procedure  rod  and  a  placed on top of  align the outer cylinder axis with the inner  adjustment  screws  are  used  in conjunction  with  a  high precision level.  With this accurately aligned set-up the elastic signal ( E = 6 0 0 e V ) was monitored 0  with a single C E M mounted on top of the the  360°  could be  monitored.  Tests  ± 5 % , with  without  the  the  showed  detection. This was  argon,  application  detectable  has  the  elastic  of  deflection  signal was  set to  voltages  homogeneous  lOOeV. These in  the  lens  to  tests  were  system.  The  C E M (fixed) allowed the first tests for coincidence  done for helium. The tests showed that only a very weak  spectrum  which  that  C M A pass energy  subsequent addition of another  coincidence  C E M around  annular slit the homogeneity of the elastic signal as a function of <j>  better than done  C M A . B y moving the  could a  coincidence  be  smaller  obtained cross  spectrum.  for  helium. In  section  These  than  results  addition,  substitution  helium,  did  clearly  suggested  background signals were 'swamping' the coincidence signals.  not  give  of any  spurious  292 11.1.2. Suppression of Background Secondary Electrons Several are  procedures  and  methods  for  suppressing  background  available [FI75]. The best method discovered in the  simple, non-magnetic, stainless was  installed in front  of the  steel grid exit slit  (50 mesh, of the  secondary  electrons  present study  SM-53,  C M A (see  Ethicon  was  a  Inc.). This  Fig. 11.1)  and  an  external retarding voltage was applied. B y varying the external retarding voltage on  the  grid the  elastic signal could be optimized both in terms  of shape and  signal to background ratio. A n example of the improved performance  that could  be obtained with the grid is shown in Fig. 11.2. As can be seen the background signal is dramatically reduced. Later tests showed that secondary electrons could also  be  suppressed  by  applying a  negative  voltage  to  the  front  end  of  the  CEMs.  A series of tests were carried out on helium and argon using the retarding grid. These single channel mode tests were successful suggesting that the main reasons for  the difficulty in detecting the E M S coincidence signals were items (3) and (4),  namely inaccurate alignment and background secondary electrons. The results  will  be discussed later in Sec. 11.2.  11.1.3. C A M A C Interface A  CAMACt  Systems standard  interface  and some associated  Corp. Details are modular  interface  software  were obtained from  shown in Fig. 11.3 and Table (hardware  and  software)  for  Kinetic  11.2. C A M A C exchanging  data  is a and  control information between a computer and a particular instrument or groups of t C A M A C stands for Computer A N S I / I E E E Std. 583-1982.  Aided  Measurements  and  Control  adopted  by  293  ELASTIC SIGNAL-without GRID 120  -i  • • •  100-  m  OF corn  t—  80-  • •  60-  •  d  z  40-  •  • •  20-  397  398  399  • • 400  401  402  403  ENERGY ev  ELASTIC SIGNAL-with GRID 120-  • • •  100-  P  80-  NO. OF COl  z  • • —• - ^ • leV fwhm • •  60-  40-  • •  20-  • •  0 -  397  398  399  400  401  402  403  ENERGY «v  Fig.  11.2.  Comparison of elastic signal (a) without grid and (b) with grid. E =-400eV, E =-100eV, E . . = -395eV. 0  n  CAMAC  CEM  294  PREAMP CFD  TAC CEM  PREAMP  ADC AMP  CFD  •top  H V iP.S. | HV  Vary E " Q  Vary O -  DAC DAC  »,2  MOTOR  CRATE CONTROLLER  >  LSI-BUS  o CPU  o  o  o  TTO  RL02  [a] T I M E M O D E  CAMAC  CEM  PREAMP  1> 1  CFD TAC  CEM  PREAMP  SCA AMP  COUNTER  0  COUNTER  CFD  1  a  COUNTER  Vary E ~  0  DAC  Vary (J)"*  1.2  DAC  LAM  CRATE CONTROLLER  <  LSI-BUS  A A Oil PRINTER  Fig. 11.3.  3  >  [l>] C O U N T M O D E  M E M S modes of operation, (a) Time mode and (b) Count mode.  295  Table 11.2. Configuration of C A M A C  a  system.  Unit  Model #  C A M A C Crate  KSC  1502  25 stations  Crate Controller  KSC  3920-Z1B  Interfaced with P D P 11/23+ system using a K S C 2920 computer interface card  Analog to Digital converter (ADC)  KSC  3553-Z1B  12-bit  Digital to Analog converter (DAC)  KSC  3112-M1A  12-bit, 8-channel  Presettable counter (scaler)  KSC  3640  16-bit, 4-channel home-built, uses M N 3 4 6 chips  Sample and Hold unit Software  a  Comments  KSC  6410  Computer-Aided Measurements and Control.  R S X - 1 1 M compatible  296 instruments.  Due to the  modules the C A M A C  strict  standardization  and  high availability of  CAMAC  system is easy to use and is very cost-effective in the long  run.  For these reasons C A M A C is widely used in research and industry all over  the  world.  therefore  Extensive  literature  exists  for  the  CAMAC  system  [HL76]  and  the present description will focus more on its application to the present  study.  11.1.4. Modes of Due  Operation  to the particular needs of the E M S experiment two modes of operation were  designed, namely, a time mode (Fig. 11.3a) and a count mode (Fig. 11.3b). The different  modes  of  operation  involved  slightly  different  hardware  set-ups  and  likewise different computer programs.  The  time mode utilized the A M A program [B87] and involves the generation of a  time spectrum at each angle <p (or at each energy). The coincidence spectrum is then obtained by applying a 'software' channel analyzer to each time spectrum.  The  time  mode  (Fig.  11.3a)  operates  by  digitizing  each  signal  from  the  time-to-amplitude converter (TAC, Ortec 567) using a combination of a home-built sample-and-hold  unit  and  a  slow  analog-to-digital  converter  signal (voltage proportional to time difference between  (ADC).  start and  Each T A C  stop pulses)  is  stored in computer memory. The computer real time clock serves as the program timer.  The  timer  is  regularly  polled before  stepping  to  the  next  parameter  (energy or angle). A t the end of each complete scan, data collection is inhibited and a 'software'  channel analyzer is applied to each time spectrum. Each time  297 spectrum is scanned and the number of signals in the coincident window relative to the random window is obtained to determine the true coincidence count.  As can be seen the time mode is slow and ties up the computer unnecessarily because  the  present  A D C cannot  However, it has the advantage  store  more  than  one  word  at  a  time.t  of storing all available raw data and is therefore  useful for debugging the spectrometer as well as for routine checks.  The  count  mode  (see  Fig. 11.3b)  involves the  use  of the  operates using an interrupt service routine and therefore  E M S program.  It  allows the computer to  function more efficiently.  Briefly,  the  count  channel  analyzers  mode (Ortec  channels. Each S C A then or  scaler  whenever  the  operates 550)  by  corresponding  the  T A C output  to  the  sends a slow NIM-logic  C E M serves  as  random  to and  two  single  coincidence  pulse to an up/down counter  T A C signal occurs in the  scaler which monitors the countrate movable or fixed  sending  prescribed window. A third  (via the ratemeter, Ortec 541) of either a program timer. This therefore  the  ensures that  for each parameter  (angle or energy) scanned the observed coincidence counts  are  normalized  countrate  and  on  the  thus  allowing  beam intensity. The end of each parameter signal  generated  interrupt monitoring  service the  on the  third scaler  routine contents  (ISR). of  the  for  variations  interrupt  random  t A possible solution would be to use buffered, 13-bit LeCroy-3512 A D C .  pressure  scan is noted by a L A M (look-at-me)  (ie. countrate  The  in gas  and  a fast  monitor) which service  an  responds  by  clearing  the  A D C with memory such as  the  coincidence  routine  activates  scalers,  298 scalers  and  initiating  energy parameter spectrum  is  the  next  set  of parameters.  Once the  final  angular  or  is reached, data collection is inhibited and the true coincidence  calculated  and  displayed  on  the  video  screen.  A n accompanying  printout is also produced.  The  count  mode  has  several  advantages  over  the  time  mode.  Due  to  the  particular configuration the count mode does not tie up the computer. This the  computer to other  hardware  users and other jobs. Furthermore, the count mode uses  channel analyzers  and  does  not  use  the  hold units. The deadtime per event is therefore time mode which has earlier  the  count  frees  an estimated  mode  normalizes  deadtime the  slow  A D C and  sample  and  smaller (=5»us) compared to the  of =150jus.  Finally,  coincidence spectrum  on  as  mentioned  the  countrate  (instead of the dwell time) thus allowing for small experimental variations during the scan.  The  count  incapability  mode, of  however,  obtaining  a  suffers time  from  one  spectrum.  disadvantage  Therefore  in  and practice  that the  is  EMS  experiment is performed by doing initial runs using the time mode and then routine runs are done using the count mode. In the future  its  the  these two modes of  operation can be integrated into a more efficient and flexible" package.  299 11.2. PRELIMINARY  RESULTS  11.2.1. Helium With the  the  multichannel electron momentum spectrometer  alignment procedure  (Is ^)  was  energies. Ep  modified  a g s  measured  outlined above,  at  A sample run  = 200eV  is  different  the  impact  binding energy  energies  (single channel mode)  shown in Fig. 11.4. Also  and  done  at  at  (Fig. 11.1) and  spectrum  of helium  different  <j> = 0°,  shown in Fig. 11.4  C M A pass  E =800eV 0  (inset)  is  and  the  variation of the observed fwhm of the He Is binding energy peak as a function of C M A pass energy. Due to the very small natural line width of the He Is"''" ionization process it can be used as a measure It  is found  energy  that  at  E = 8 0 0 e V and 0  resolution ( 2 . 0 ± 0 . 5 e V  Ep  a g g  of the M E M S energy resolution.  = 200eV the  fwhm) is close to the  experimentally observed  predicted energy resolution  (2.4eV).t The slightly better energy resolution (%AE/E) of the multichannel E M S spectrometer  as  opposed  spectrometer  outlined  multichannel  spectrometer.  in  to the  high momentum  chapter  3  is  due  resolution  of  to  resolution single channel E M S the  smaller  exit  slit  in  the  11.2.2. Argon To  determine  spectrometer,  the  angular  the  present  multichannel  EMS  test runs in single channel mode were carried out using argon. Fig.  11.5 shows the angular correlation spectrum of the argon 3p orbital measured E =815.7eV, E 0  pass  at  =200eV. It can be seen that the angular correlation spectrum  t This is based on the present slit dimensions (exit=entrance = 1.0mm), a dispersion of 151mm and 0 = 45° which gives an analyzer energy resolution of — 1%. The contribution of the gun energy spread is estimated to be =0.8eV.  300  FWHM vs. CMA PASS ENERGY  O EXPTL BEST FIT 1  1  20  1  22  1  24  1  26  1  26  1  30  ENERGY(ev)  F i  g-  11-4.  He Is binding energy spectrum. E =800eV, E  =200 eV, pass 0 - 0 ° . The inset shows variation of He Is fwhm as a function of C M A pass energy. The dashed lines indicate error limits of the projected C M A energy resolution. 0  301 is  reasonably  symmetric  about  0 = 0°  indicating  that  the  introduced by any slight misalignment or retardation effects present operating 0 = 0°)  to  the  resolution  is  conditions. Furthermore maximum of the  estimated  to  be  from  angular  the  inhomogeneities  are small under  ratios  of the  correlation spectrum,  0. l - 0 . 2 a " .  This  1  0  result  is  minimum the  quite  momentum resolution of the single channel E M S spectrometer  the (at  momentum  close  to  the  outlined in chapter  3.  During  the  latter  energy  (lOOOeV)  part  of the  and with  study,  tests  much larger  were  retardation  performed  at  higher  impact  (E =100eV). However it pass  was observed that the coincidence count rate decreased dramatically. It is believed that much of the which  distorts  difficulty  the  is due to the  electron  trajectories  lens effects appreciably.  in the  retardation  Several  stage  suggestions  are  proposed to remedy this problem. One of them is to devise, design and test a 'softer'  retardation  stage  in  the  lens  system.  Possibly  a  two-stage  retardation  with focusing in between will improve the present situation.  The present studies, however, have clearly identified the critical aspects necessary in the design of an optimum multichannel E M S spectrometer. of  the  CMA  and  the  suppression  of  secondary  Accurate alignment  electrons  measurements (in single channel mode) of the binding energy as  well  as  preliminary and  the  angular  measurements  momentum  correlation spectra at  resolution are  E =800eV o  estimated  of the  A r 3p  (E .„„„ = 200eV) I  to  be  test  spectra of helium  orbital. From  the  2.0eV fwhm  allowed  energy and  these  resolution  O.l-0.2a ~ ,  respectively. These observed instrumental resolutions clearly indicate the  1  0  potential  302  A r 3 p M o m e n t u m Profile 600  500  V I ° V 1°  400  300 H  UJ CO  to to o a: u  \  I  200  \  /  \  Io  100  -100  -|—  -20  -30  -40  0  -10  V  10  20  30  ANGLE (deg)  Fig. 11.5.  Ar E P  ass  3p =  -  experimental  2 0 0 e V  '  E  grid " =  momentum 3 9 5 e V  -  profile.  E =-815.7eV, 0  303 of the present instrument spectrometer  in the  sensitive detector.  to function  very near future  as a  high performance  multichannel E M S  by installing a microchannel plate position  CONCLUDING  REMARKS  "In the present status of experimental techniques, no one has been able to observe the orbital pattern experimentally. However, to date, no one can assert that this is totally impossible.... If we could only experimentally obtain any knowledge of HOMO and LUMOt  patterns,  orbital pattern  chemistry  would  be profoundly  affected. In  that event the  concept, which is at present of a somewhat unreal nature, will be  provided with a certain empirical nature." Kenichi Fukui, 1977; Nobel Laureate in Chemistry (1981)  Although Prof. Fukui was unaware at that time of the yet developing technique of electron momentum spectroscopy (EMS), he was definitely prophetic in realizing its  vital  importance.  chemistry  was  The  shared  concerns  by  the  regarding  principal  the  'orbital  developers  of  spectroscopy [CG72, WH73] and by those who specifically to molecular quantum chemistry have  chemistry [HH77,  MC78].  been considerably extended  pattern  concept'  electron  in  momentum  sought to apply E M S  These applications to quantum  in the  series  of studies  reported in  this thesis.  The  experimental  several provided  momentum  molecules  as  much  needed  importantly  have  laid  profiles  measured  in  empirical a  solid  (XMPs)  the  present  basis  basis  and  for  to  the  binding  E M S studies, 'orbital  assessing  the  wavefunctions in quantum chemistry. Such an assessment the  fact  that  under  the  energy  spectra  of  not  only  have  concept',  but  most  quality  of theoretical  was made feasible by  kinematic conditions employed in E M S (E >1000eV, o  t H O M O refers to the highest ocupied molecular orbital whereas L U M O refers to the lowest unoccupied molecular orbital. 304  305 symmetric, non-coplanar),  the (e,2e) cross  section is proportional to the absolute  square of the ion-neutral overlap amplitude--& particular  molecular  ion-neutral  overlap  electronic  structure  amplitude, though  quantity 2).  (chapter  feasible  dependent  Exact  solely on the  calculations  of the  in principle, are simply intractable  for most chemical systems. A conceptual and computational advantage is provided by 'models' of electronic structure, considers  the electrons  such  case,  a  the  one of which is the Hartree-Fock model which  as independent  total  particles  wavefunction  is  moving in an average field. In  represented  by  an  anti-symmetrized  product of one-electron functions or so-called orbitals.  Within  the Hartree-Fock  E M S can be interpreted the  orbital  that  model, the experimental  momentum  as the electron momentum  has been  ionized.  Thus  profile obtained in  distribution corresponding to  it has often  been  said  that E M S  provides an experimental tool for orbital imaging and wavefunction mapping.  The limits of the Hartree-Fock model have been illustrated and discussed in this thesis.  The detailed  (chapter  E M S studies  6) and H C O 2  (chapter  of H  2  0 (chapter  7) clearly  4), D  illustrated  2  0 (chapter  the inadequacies  Hartree-Fock model in predicting the fine details of the experimental •profile,  especially in describing the X M P of the least  Collaborative theoretical discrepancies distribution state  by  performing  utilizing  and final  effort,  bound  NH  of the  momentum  molecular orbital.  integral to the present work, has resolved these accurate  highly correlated  ion states.  5),  Other  calculations  of  the  ion-neutral  overlap  wavefunctions  for both  the initial  neutral  considerations  such  vibrational  effects,  as  experimental inaccuracies, etc. have been found to be of minor consequence. The  3  306 need for going beyond the Hartree-Fock model and including explicitly the details of the  initial  neutral and final  ionic states is therefore  stressed.  In  retrospect,  the following results seem natural because quantum theory, which is the basis of our  understanding  processes and  of  the  physico-chemical world,  not of properties,  a physics  is  ultimately  of interactions and  a  'physics of  not of attributes'  [B35].  The  EMS  studies  of  small  molecular  systems  (chapters  4-7)  provided  a  benchmark for assessing the limitations not only of the theoretical framework for interpreting the  XMPs  but  also of the  theoretical interpretation of the  binding  energy spectra, most especially the inner valence binding energy region. The E M S studies  of the  showed  extended  interaction  inner valence binding structures  (CI) and  which  Green's  energy  were  function  region of H 0 , N H 2  qualitatively predicted  methods.  The  and  3  wavefunctions  (with extended  basis  sets  2  by configuration  improved prediction of  ionization intensities, however, was shown to be attained only when accurate  H CO  and the  the  sufficiently  inclusion of sufficient  correlation) were used.  The  extension  of these  E M S studies  to  much  bigger  systems  was  a  logical  offshoot following the successful studies of small molecular systems. E M S studies of  the  outermost  (chapter  9)  trends. trend  and  valence  para-dichlorobenzene  In particular, the clearly  however  predicted  suggested  orbitals of the  measured by  extensive  methylated  (chapter  XMPs  molecular electron  8)  of the  orbital  amines  showed  (chapter  9), N F  interesting chemical  methylated amines  calculations.  delocalization  away  3  These from  showed a calculations  the  nitrogen  307 center-— a picture contrary of  methyl  groups.  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