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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Evaluation of wavefunctions by electron momentum spectroscopy Bawagan, Alexis Delano Ortiz 1987
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Title | Evaluation of wavefunctions by electron momentum spectroscopy |
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Bawagan, Alexis Delano Ortiz |
Publisher | University of British Columbia |
Date Issued | 1987 |
Description | Electron momentum spectroscopy (EMS) provides experimental atomic and molecular electronic structure information in terms of the binding energy spectrum and the experimental momentum profile (XMP), which is a direct probe of the electron momentum distribution in specific molecular orbitals. The measured XMPs permit a detailed quantitative evaluation of theoretical ab initio wavefunctions in quantum chemistry and also provide a means to investigate traditional concepts in chemical reactivity at the fundamental electronic level. This thesis reports high momentum resolution EMS measurements of the valence orbitals of H₂0, D₂0, NH₃ and H₂CO obtained using an EMS spectrometer of the symmetric, non-coplanar type operated at an impact energy of 1200eV. The measured experimental momentum profiles for the valence orbitals of each molecule have been placed on a common intensity scale, which has allowed a stringent quantitative comparison between experiment and theory. These studies now confirm earlier preliminary investigations that suggested serious discrepancies between experimental and theoretical momentum distributions. Exhaustive consideration of possible rationalizations of these discrepancies indicate that double zeta quality and even near Hartree-Fock quality wavefunctions are insufficient in describing the outermost valence orbitals of H₂0 and NH₃. Preliminary results for H₂CO also indicate that near Hartree-Fock wavefunctions are incapable of describing the outermost 2b₂ orbital. Interactive and collaborative theoretical efforts have therefore led to the development of new Hartree-Fock limit and also highly correlated (CI) wavefunctions for H₂0, NH₃ and H₂CO. It is found that highly extended basis sets including diffuse functions and the adequate inclusion of correlation and relaxation effects are necessary in the accurate prediction of experimental momentum profiles as measured by electron momentum spectroscopy. New EMS measurements are also reported for the outermost valence orbitals of NF₃, NH₂CH₃, NH (CH₃)₂, N (CH₃)₃ and para-dichlorobenzene. These exploratory studies have illustrated useful chemical applications of EMS. In particular, EMS measurements of the outermost orbitals of the methylated amines have revealed chemical trends which are consistent with molecular orbital calculations. These calculations suggest extensive electron density derealization of the so-called nitrogen 'lone pair' in the methylated amines in comparison to the 'lone pair' in NH₃. EMS measurements of the non-degenerate π₃ and π₂ orbitals of para-dichlorobenzene show different experimental momentum profiles consistent with arguments based on inductive and resonance effects. These experimental trends, both in the case of the amines and para-dichlorobenzene, were qualitatively predicted by molecular orbital calculations using double zeta quality wavefunctions. However more accurate prediction of the 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 integrated computer package (HEMS) for momentum space calculations has also been developed based on improvements to existing programs. Development studies testing a new prototype multichannel (in the ɸ plane) EMS spectrometer are described. |
Subject |
Wave functions Electron spectroscopy |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-07-27 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0060438 |
URI | http://hdl.handle.net/2429/26958 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemistry |
Affiliation |
Science, Faculty of Chemistry, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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