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Orbital electron density from electron momentum spectroscopy : comparison with AB initio calculations Rolke, James M. 1997-12-31

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ORBITAL E L E C T R O N DENSITY F R O M E L E C T R O N M O M E N T U M SPECTROSCOPY: W I T H A B INITIO  COMPARISON  CALCULATIONS  By James M . Rolke B. Sc. (Hons.), University of British Columbia, 1992  A THESIS SUBMITTED IN P A R T I A L F U L F I L L M E N T OF THE REQUIREMENTS FOR THE D E G R E E OF DOCTOR OF PHILOSOPHY  in T H E . F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T OF CHEMISTRY  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH C O L U M B I A May 1997 ©James M . Rolke, 1997  In  presenting this  thesis  in partial  fulfilment  of the requirements  for an advanced  degree at the University of British Columbia, T agree that the Library shall: make it freely available for reference and study. I further copying  agree that permission for extensive  of this thesis for scholarly purposes may be granted  department  or  by  his or  her  representatives.  It  is  by the head of my  understood  that  copying or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department of The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  2.l<¥r7  11  Abstract  Orbital electron momentum profiles measured by electron momentum spectroscopy (EMS) at a total energy of 1200 eV are presented in this work for the HOMOs of open-shell molecules (NO and N 0 ) , phosphorus compounds ( P H , P F and P ( C H ) ) and transition 2  3  metal hexacarbonyls (Cr(CO) , Mo(CO) and W(CO) ). 6  6  3  3  3  In addition, the complete valence  6  shell binding energy spectra and all valence momentum profiles have been obtained for molecular oxygen ( 0 ) and methanol (CH OH) in the 9-59 eV and 6-47 eV binding energy 2  3  ranges, respectively. A highly sensitive energy-dispersive multichannel E M S spectrometer has been used to obtain most of the measurements in the present work. momentum profiles for N O , 0  2  The experimental  and C H O H are a significant improvement over previously 3  published single-channel EMS results. The measured momentum profiles are compared with calculated cross-sections using a range of Hartree-Fock (HF) wavefunctions from minimal basis set to near-Hartree-Fock limit in quality. The effects of correlation and relaxation on the calculated momentum profiles of NO, N 0 , 0 , P H and C H O H are investigated using multi-reference singles and doubles 2  2  3  3  configuration interaction (MRSD-CI) calculations of the full ion-neutral overlap distributions. The experimental measurements have also been compared with calculated density functional theory (DFT) momentum profiles using the target Kohn-Sham approximation. The measured valence experimental shell binding energy spectra for 0  2  and C H O H have been compared 3  with spectra calculated using MRSD-CI and many-body Green's function methods. Generally good agreement has been obtained between either high-level HF, MRSD-CI or DFT calculated momentum profiles and the experimental momentum profiles for NO, N 0 , 2  0 , P H , P F , P ( C H ) and C H O H . 2  3  3  3  3  3  However, there are still some small but significant  discrepancies at low momentum for the outermost orbitals of N O and 0  2  as well as the 2a"  (HOMO), 7a' and 5a' orbitals of C H O H . Post-Hartree-Fock calculations have little effect on 3  the shapes of the calculated momentum profiles for NO, N 0 and P H . In contrast, the 7a', 2  3  Ill  ( 6 a ' + l a " ) a n d 5a'  orbital experimental m o m e n t u m profiles for methanol display shapes and  i n t e n s i t i e s t h a t c a n o n l y b e p r e d i c t e d b y t h e M R S D - C I a n d / o r D F T c a l c u l a t i o n s a n d it i s c l e a r that  in  these cases  electron  correlation  is important.  calculated p o l e strengths and energies for 0  2  The  reported  MRSD-CI  a n d the m e a s u r e d m o m e n t u m p r o f i l e s h a v e b e e n  u s e d to a s s i g n the s i g n i f i c a n t satellite splitting o b s e r v e d i n the Experimental determinations  presently  2a  u  and  2rj  g  o f the p o l e strengths f o r the i n n e r v a l e n c e  i o n i z a t i o n strength.  (4a')"  1  and  (3a')  _1  satellite i o n i z a t i o n processes o f C H O H h a v e also been a c h i e v e d i n the present w o r k . 3  T h e o b s e r v e d i n c r e a s e i n s-type character f o r the o u t e r m o s t orbitals o f P F r e l a t i v e to the H O M O o f P H  3  is e x p l a i n e d b y m o l e c u l a r orbital calculations.  3  and P ( C H ) 3  3  This increased  s-type character has b e e n interpreted i n terms o f a d e r e a l i z a t i o n o f the charge density i n the H O M O  orbitals w h e n H is replaced b y F or C H  H O M O  o f C r ( C O ) g v e r i f y earlier m e a s u r e m e n t s i n that they d i s p l a y l a r g e i n t e n s i t y  3  .  T h e present experimental results for at  the low  m o m e n t u m w h i c h is not p r e d i c t e d b y either s y m m e t r y c o n s i d e r a t i o n s o r c a l c u l a t i o n s w i t h i n the p l a n e w a v e i m p u l s e a p p r o x i m a t i o n ( P W I A ) . H i g h i n t e n s i t y at l o w m o m e n t u m i s a l s o o b s e r v e d for the H O M O s o f M o ( C O ) g and W ( C O )  6  a n d it i s s u g g e s t e d t h a t t h i s i s d u e e i t h e r t o g r o u n d  state v i b r a t i o n a l effects o r d i s t o r t i o n o f the i n c o m i n g a n d o u t g o i n g e l e c t r o n s w a v e s . T h e latter suggestion is supported b y distorted w a v e i m p u l s e a p p r o x i m a t i o n C r 3d a n d M o 4d a t o m i c o r b i t a l s .  (DWIA)  calculations  for  iv  T a b l e o f Contents  Abstract  ii  Table of Contents  iv  List of Tables  vii  List of Figures  viii  List of Abbreviations  xi  Units  xiii  Acknowledgements  xiv  1 Introduction  1  1.1 Electron Momentum Spectroscopy  2  1.1.1 The EMS Experiment  2  1.1.2 Binding Energy Spectrum  6  1.1.3 Momentum Profile  9  1.2 Overview of the Thesis  2 Theory 2.1 The Calculation of Wavefunctions  12  15 15  2.1.1 Hartree-Fock Theory for Closed-Shell Molecules  18  2.1.2 Hartree-Fock Theory for Open-Shell Molecules  25  2.1.3 Configuration Interaction (CI)  28  2.1.4 Density Functional Theory  31  2.2 The Interpretation and Calculation of the EMS Cross-Section  37  2.2.1 The Binary Encounter and Impulse Approximations  37  2.2.2 The E M S Scattering Cross-Section  39  2.2.3 The Distorted Wave Impulse Approximation  40  2.2.4 The Plane Wave Impulse Approximation  42  2.2.5 The Target Hartree-Fock Approximation  45  2.2.6 The Target Kohn-Sham Approximation  46  2.2.7 Fourier Transform  49  2.2.8 Resolution Folding Effects  50  V  3 Experimental Method  51  3.1 The Multichannel Energy-Dispersive Spectrometer  51  3.2 Background (Random Coincidence) Correction  59  4 The HOMOs of NO and N 0  61  2  4.1 Introduction  61  4.2 Gas Samples and Sample Handling for N O  63  4.3 Calculations  67  4.3.1 Hartree-Fock Calculations  67  4.3.2 MRSD-CI, A C P F and QDVPT Calculations  72  4.3.3 DFT Calculations  73  4.4 Results and Discussion  74  4.4.1 Comparison of Experimental and Theoretical Momentum Profiles  74  4.4.2 Density Maps in Momentum and Position Space  83  5 Molecular Oxygen  86  5.1 Introduction  86  5.2 Hartree-Fock, MRSD-CI, A C P F and DFT Calculations  91  5.3 Results and Discussion  94  5.3.1 B inding Energy Spectra  94  5.3.2 Comparison of Experimental and Theoretical Momentum Profiles 5.3.2.1 The 5.3.2.2 The  l7t l7t  105  g  Orbital ( X ^ Final Ion State)  113  u  Orbital (a  114  2  4  n , AU  and n u (3) Final Ion States) 2  2  u  U  5.3.2.3 The 3 c Orbital (b IT and fl Z~ Final Ion States)  118  5.3.2.4 The 2 a Orbital (c S~ Final Ion State, Peak P and Peak Q)  120  5.3.2.5 The 2 o Orbital ( E~ and IT Final Ion States)  122  4  g  2  g  4  u  4  2  g  g  5.3.3 Density Maps in Momentum and Position Space  6 The HOMOs of PH ,PF and P(CH ) 3  3  3  3  123  128  6.1 Introduction  128  6.2 Calculations  131  6.2.1 Hartree-Fock Calculations  131  6.2.2 MRSD-CI Calculation  134  6.2.3 DFT Calculations  134  6.3 Results and Discussion  135  vi 6.3.1 Comparison of Experimental and Theoretical Momentum Profiles  135  6.3.2 Density Maps in Momentum and Position Space  146  6.3.3 The Methyl Inductive Effect  149  7 The HOMOs of Cr(CO) , Mo(CO) and W(CO) 6  6  6  154  7.1 Introduction  154  7.2 Calculations  157  7.2.1 Hartree-Fock Calculations  157  7.2.2 DFT Calculations  160  7.3 Results and Discussion  161  7.3.1 Comparison of Experimental and Theoretical Momentum Profiles  161  7.3.2 Density Maps in Momentum and Position Space  170  8 Methanol  172  8.1 Introduction  172  8.2 Calculations  174  8.2.1 Hartree-Fock Calculations  174  8.2.2 MRSD-CI Calculation  177  8.2.3 DFT Calculations  178  8.3 Results and Discussion  179  8.3.1 Binding Energy Spectrum  179  8.3.2 Comparison of Experimental and Theoretical Momentum Profiles  185  8.3.3 Density Maps in Momentum and Position Space  194  9 Concluding Remarks  201  Bibliography  206  vii  L i s t of Tables  4.1  Calculated and experimental properties for N O X Tl  68  4.2  Calculated and experimental properties for N 0 X Ai  69  5.1  Calculated and experimental properties for 0 X Z  92  5.2  Experimental ionization energies and peak widths for 0  5.3  MRSD-CI states and intensities (pole strengths) for 0  6.1  Calculated and experimental properties for P H , P F and P ( C H )  6.2  Experimentally determined properties for amines and phosphines  152  7.1  Calculated and experimental properties for Cr(CO) , Mo(CO) and W(CO)6  159  8.1  Calculated and experimental properties for C H O H  175  8.2  Ionization energies and intensities (pole strengths) for C H O H  2  2  2  2  g  3  96  2  103  2  3  3  6  6  3  3  +  3  132  180  viii  L i s t of F i g u r e s 1.1  A schematic of the symmetric non-coplanar (e, 2e) scattering geometry  2.1  Electron configuration diagram for  2.2  Electron configuration diagram for (y/f  3.1  Schematic of the energy-dispersive multichannel E M S spectrometer  3.2  Position sensitive detectors and signal processing electronics for the E M S  (yAjCU)( y/$)( yA 2 a)(  WiP)( W?> ) a  a)( y/f/f)( y/£oc)( yf/^X yf a)  3 26 26 52  spectrometer  54  3.3  Time spectrum output by the T A C  55  3.4  Non-binning and binning mode Ar 3p binding energy spectra  56  3.5  Non-binning and binning mode angle measurements and momentum profiles for A r 3p  4.1  58  Previous and presently reported experimental momentum profiles for the N O 2K orbital  4.2  64  Measured and calculated spherically averaged Hartree-Fock and postHartree-Fock momentum profiles for the 2K orbital of N O  4.3  Measured and calculated spherically averaged Hartree-Fock and postHartree-Fock momentum profiles for the 6aj orbital of N 0  4.4  75  2  Measured and calculated spherically averaged DFT momentum profiles for the 2K orbital of NO  4.5  76  81  Measured and calculated spherically averaged DFT momentum profiles for the 6aj orbital of N 0  82  2  5.1  E M S binding energy spectra of 0 from 9 to 59 eV  98  5.2  Measured and calculated binding energy spectra of 0 from 9 to 59 eV  2  2  102  ix 5.3  Summed experimental and calculated spherically averaged momentum profiles for the valence orbitals of 0  5.4  Measured and calculated spherically averaged momentum profiles for the l7C (X2n ) and l7t (a Tl , A Yl ) 4  g  5.5  g  2  u  u  u  orbitals of 0  109  2  Measured and calculated spherically averaged momentum profiles for the 2 2 3 a ( V B i r g ) and l7t ( n u (3)) orbitals of 0  110  Measured and calculated spherically averaged momentum profiles for the 2 a ( c S ; , P, Q) orbital of 0  111  Measured and calculated spherically averaged momentum profiles corresponding to sums of states and for the 2 a ( Z~, E~) orbital of 0  112  g  5.6  107  2  2  u  4  u  5.7  2  4  2  g  5.8  Momentum and position density contour maps for the l7C , l7t and 3 a g  orbitals of 0 5.9  Measured  and  calculated  spherically  averaged  calculated  spherically  averaged  t  Measured  Hartree-Fock  and  calculated  spherically 2  averaged  136  Hartree-Fock  137  3  Hartree-Fock 3  and  3  138  143  3  144  3  Measured and calculated spherically averaged DFT momentum profiles for the 8a orbital of P ( C H ) t  6.7  and  Measured and calculated spherically averaged DFT momentum profiles for the 8aj orbital of P F  6.6  and  Measured and calculated spherically averaged DFT momentum profiles for the 5a! orbital of P H  6.5  g  3  M R S D - C I momentum profiles for the 8a orbital of P ( C H ) 6.4  and 2 a  125  MRSD-CI momentum profiles for the 8a orbital of P F 6.3  u  2  and  Measured  g  124  MRSD-CI momentum profiles for the 5^ orbital of P H 6.2  u  2  Momentum and position density contour maps for the 2 a orbitals of 0  6.1  2  3  3  Position (charge) density difference maps for substituted phosphines  145 150  X  7.1  E M S binding energy spectra of Cr(CO) from 5 to 23 eV  7.2  Experimental and theoretical momentum profiles for the 2 t Cr(CO)  7.3  7.5  orbital of 162  Experimental and theoretical momentum profiles for the 2 t  2g  orbital of 163  6  Experimental and theoretical momentum profiles for the 2 t W(CO)  2g  6  Mo(CO) 7.4  156  6  2g  orbital of 164  6  Plane wave  impulse approximation and distorted  wave impulse  approximation calculations of the theoretical momentum profiles for the Cr 3d and M o 4d atomic orbitals  169  8.1  E M S binding energy spectra of C H O H from 6 to 47 eV  181  8.2  Measured and calculated binding energy spectra of C H O H from 6 to  3  3  47 eV 8.3  Measured and calculated spherically averaged momentum profiles for the 2a" and 7a' orbitals of methanol  8.4  195  Momentum and position density contour maps for the l a " and 5a' orbitals of methanol  8.8  188  Momentum and position density contour maps for the 2a", 7a' and 6a' orbitals of methanol  8.7  187  Measured and calculated spherically averaged momentum profiles for the 5a', 4a' and 3a' orbitals of methanol  8.6  186  Measured and calculated spherically averaged momentum profiles for the 6a', l a " and sum of the 6a' and l a " orbitals of methanol  8.5  184  196  Momentum and position density contour maps for the 4a' and 3a' orbitals of methanol  197  List of Abbreviations au  atomic units  ACPF  average coupled-pair functional  ADC  analog to digital converter  BES  binding energy spectrum  BW  background window  CFD  constant fraction discriminator  CGTO  contracted Gaussian type orbital  CI  configuration interaction  CW  coincidence window  D  Debye  DAC  digital to analog converter  DFT  density functional theory  DWIA  distorted wave impulse approximation  DZ  double zeta  ECP  effective core potential  EMS  electron momentum spectroscopy  eV  electron volt  FWHM  full width at half-maximum  G(CI)  a configuration interaction calculation based on a Gaussian type orbital basis set  GF  Green's function (calculation)  GTO  Gaussian type orbital  GW-PG  Gaussian weighted planar grid  HF  Hartree-Fock  HOMO  highest occupied molecular orbital  IP  ionization potential  LCAO  linear combination of atomic orbitals  LDA  local density approximation  KS  Kohn-Sham  MBPT  many-body perturbation theory  MBS  minimal basis set  MC  multichannel  MCP  microchannel plate  MD  momentum distribution  MO  molecular orbital  MP  M0ller-Plesset  MR-CI  multireference configuration interaction  MRSD-CI  multireference singly and doubly excited configuration interaction  NHOMO  next highest occupied molecular orbital  OVD  overlap distribution  PES  photoelectron spectroscopy  PSDA  position sensitive detector analyzer  PWIA  plane wave impulse approximation  QDVPT  quasi-degenerate variational perturbation  RA  resistive anode encoder  RHF  restricted Hartree-Fock  ROHF  restricted open-shell Hartree-Fock  SC  single-channel  SCA  single channel analyzer  SCF  self consistent field  STO  Slater type orbital  TAC  time to amplitude converter  THFA  target Hartree-Fock approximation  TKSA  target Kohn-Sham approximation  TMP  theoretical momentum profile  VWN  Vosko, Wilk and Nusair  UHF  unrestricted Hartree-Fock  XMP  experimental momentum profile  XPS  X-ray photoelectron spectroscopy  Xlll  Units  Atomic units are a system of reduced units that are useful for discussing quantities on the atomic scale. In this system, the Bohr radius a Planck's constant h, the charge on an 0  electron e and the permittivity of free space 47T£ are all considered to be dimensionless 0  quantities of magnitude 1.  Quantity  Atomic units (1 au)  Distance  a  0.52918 A 5.2918X10" m  0  11  Bohr radius h  Momentum  1 au converted to other units  1.9929xl0- kgms-! 24  a  0  e  2  Energy  18  1 hartree = 2 Rydberg Charge  4.2398X10" J 27.2114 eV  1.6022xlO-!9C  e  Elementary charge Dipole Moment  8.4785X10" C m 2.5418 Debye (D) 11  ae Q  chargexdistance Quadrupole Moment  chargexdistance  4.4867X10" C m 1.3451X10" esu-cm 21  ale 2  26  2  1 Debye is defined to be 10~ esu cm which is approximately 3.3xl0~ C m or 0.393427 au. 18  1 torr is equivalent to 1 mm Hg or 0.1333 kPa  30  xiv  Acknowledgements  First of all, I would like to thank my research supervisor, Prof. C.E. Brion. It has been a great privilege to work for Chris and to learn from our discussions. His academic and personal support as well as his enthusiasm for scientific research have enabled me to complete this work. I also owe special gratitude to the many past and present members of the research group over the last few years: Jennifer A u , Gordon Burton, Natalie Cann, Glyn Cooper, Monica Dyck, Bruce Hollebone, Noah Lermer, John Neville, Terry Olney, and Yenyou Zheng. I especially thank Bruce, John and Yenyou for showing me how to run the spectrometers and for assistance with my experiments and the various technical difficulties along the way. In addition to all of their help with my research, I must also thank all of the members of the group for the many enjoyable discussions and cups of coffee together. I would also like to acknowledge the staff members of the Chemistry Department for their technical support.  In particular, I would like to thank Ed Gomm, Ron Marwick,  Bill Henderson and Brian Snapkauskas of the Mechanical Shop as well as Brian Greene of the Electronics Shop for their assistance in keeping the spectrometers running. I would also like to thank Prof. Ernest R. Davidson, Yan Wang and S.J. Chakravorty of Indiana University for their invaluable contributions to the theory involved in this work. In addition, I thank Prof. I.E. McCarthy for his theoretical contributions to the work on the metal carbonyls. I gratefully acknowledge the financial assistance of the Natural Sciences and Engineering Research Council of Canada (NSERC) for a postgraduate scholarship. I thank my family for their support throughout this degree in addition to all their contributions throughout my education. The support and encouragement of Dad, Richie and Jodi has helped me reach this point. In addition, the contributions of my step-mother Barb to my life during this work cannot be forgotten.  XV  Although a complete list of all the friends who have influenced my life during the course of this work would be impossible, some of them deserve special mention for their contributions. In particular, I thank Kyle Marling, Anne Spademan and Raphael Chan for their invaluable support and advice. I would also like to thank Beth Gatehouse, Kaley Walker and Almira Blazek for their help and for adding some fun to the program. Other friends who deserve credit for their interest, support and advice during this work are Kelle Pegg, Lisa Wall, Dave Ham, Maura Bushell, Jennifer Jamieson, Chris Nichols, Mike Wong, Deanna Tan, Laurie Garret and K i m Yee.  1  Chapter 1. Introduction  Chapter 1 Introduction  It has long been known that chemical phenomena are largely determined by the behaviour of electrons in the outermost valence orbitals with the remaining electrons having relatively little effect.  In particular, the frontier molecular orbital theory of Fukui [1],  Woodward and Hoffman [2] has emphasized the importance of the electron density distribution in the highest-occupied molecular orbital (HOMO) in determining chemical reactivity. Thus, experimental techniques which provide measurements of the electron density in individual outer valence orbitals are highly desirable so that a wide range of chemical and physical phenomena can be better understood. In addition, ab initio quantum chemistry procedures can be used to provide theoretical electronic wavefunctions and charge distributions, but the quality of the results is highly dependent upon the choice of basis set and the various approximations that are required in the chosen quantum mechanical procedure. Experimental electron density distributions for individual orbitals therefore play an important role because, in addition to providing fundamental information regarding the potential for chemical reactivity and behaviour, they can also be used in the direct evaluation of the quality of theoretical methods. It should be noted that measurements of electron density distributions have long been provided by X-ray and 7-ray Compton scattering profile studies [3-6], by high energy electron impact ionization [7,8] and by positron annihilation [9,10] techniques, but these various methods all measure the total electron density due to all electrons in the target and do not provide specific electron density information for the individual atomic and molecular orbitals.  A unique  experimental technique that can investigate individual orbital electron densities (momentum  Chapter 1. Introduction  2  distributions) is electron momentum spectroscopy (EMS).  The E M S experiment and the  information provided by the technique are discussed in Section 1.1 below.  1.1 Electron Momentum Spectroscopy 1.1.1 The EMS Experiment The E M S experiment involves the complete kinematic determination of a high impact energy, high momentum transfer electron impact ionization (e, 2e) reaction: +e7 + e  er] + M^M  +  (1.1)  2  A schematic of the E M S (or binary (e, 2e)) collision process in the commonly used symmetric non-coplanar geometry  [11,12]  is shown in Fig.  1.1.  The reaction involves the ionization of a  target (M) by an energetic electron followed by the coincidence detection of the scattered and ejected electrons. The energies and directions (and thus the momenta) of the incoming and outgoing electrons are determined experimentally by the geometric arrangement  and  electrostatic potentials of the electron optics and energy analyzers. The incident electron e has 0  an impact energy of E with corresponding momentum p while the two outgoing electrons e 0  0  x  and e have energies h\ and E and momenta p and p , respectively. The collision results in 2  2  x  2  an ion M in a final state / with recoil energy and momentum E i  and q, respectively. In  +  KCoi  addition, t9j and 6 are the polar angles between the incident electron and the directions of the 2  two outgoing electrons. As can be seen from Fig. 1.1, the angle (j) is the relative azimuthal angle between the two planes modes  [11,12].  (PQ,P\)  and  (PQ,P ). 2  The E M S experiment operates in two  Variation of the impact energy at fixed Ey, E ,  binding energy spectrum (BES).  2  t9  1 ?  6  2  and ty results in a  Alternatively, as will be shown in Chapter 2, an orbital  electron momentum distribution for a binding (i.e. ionization) energy selected electron can be obtained by variation of the azimuthal angle ty. A n electron momentum distribution measured in E M S is often referred to as an experimental momentum profile (XMP).  Chapter 1. Introduction  e  o( o'P ) E  0  Fig. 1.1: A schematic of the symmetric non-coplanar (e, 2e) scattering geometry.  Chapter 1. Introduction  4  The possibility of using the (e, 2e) reaction in Eq. (1.1) as an experimental technique to investigate individual orbital electron densities was first suggested in 1960 by Baker, McCarthy and Porter [13] in the concluding paragraph of a paper dealing with the theory of the (p, 2p) reaction as a probe of proton momenta in nuclei. Because of these origins involving the (p, 2p) reaction in nuclear physics, E M S was known as binary (e, 2e) spectroscopy for much of its early history. The possibility of such (e, 2e) experiments was investigated theoretically in the late 1960s by Neudatchin and co-workers [14,15] and by Glassgold and Ialongo [16]. The first instrument developed to study (e, 2e) scattering was reported in 1969 by Ehrhardt et al. [17].  However, this instrument was specifically designed to observe the  scattering of slow (~ 100 eV) electrons from a gas in a coplanar geometry to provide a test for theories of low energy impact ionization rather than for studies of electron momentum distributions. In the same year, Amaldi et al. [18] developed a symmetric coplanar (e, 2e) instrument with high impact energy (14.6 keV) and measured the binding energy spectrum with an energy resolution sufficient to separate the carbon K and L shell electrons in a transmission experiment through a thin film graphite target.  This study [18] indicated that important  information concerning the electronic structure of the target could be obtained using the (e, 2e) technique. In 1972, Camilloni et al. [19] reported the first experimental orbital momentum distributions in a high energy transmission (e, 2e) study of the carbon K (Is orbital) shell for a thin graphite film target. However, due to the very low energy resolution of the instrument, the momentum distributions of the individual C 2s and 2p valence orbitals could not be obtained [19]. In 1973 the first gas phase binary (e, 2e) experiment was carried out by Weigold and co-workers who reported valence shell orbital momentum distributions for the individual 3p and 3s atomic orbitals of argon [20,21]. These atomic measurements were soon followed by the first molecular results which were for the valence electrons of methane [22]. In 1974 molecular E M S studies were started at the University of British Columbia and results were soon reported for the individual valence orbitals of N H [23] and H 0 [24]. In parallel with 3  2  Chapter 1. Introduction  5  these early experimental advances, the theoretical interpretation of the binary (e, 2e) scattering cross-section was further developed [25-27].  E M S studies were subsequently reported by  groups at the University of Maryland [28] and the University of Waterloo [29]. More recently, E M S experiments have also been carried out at Carleton University [30], Tsinghua University (Beijing, China) [31] and Tohoku University (Sendai, Japan) [32]. The E M S technique has been further developed over the past two decades and applied to a growing range of studies of chemical interest. These developments are summarized in a number of E M S review articles [11,12,33-40] and bibliographies [38,41].  The advent of  multichannel energy-dispersive [42-46] and momentum-dispersive [47] E M S spectrometers which provide greatly improved count rates (one or two orders of magnitude) compared to earlier single-channel spectrometers has greatly increased the range of studies possible. For example, Zheng et al. [48] used an energy-dispersive multichannel spectrometer to provide the first E M S results for an excited and oriented target (sodium atoms in the 3 P 2  3 / 2  , m = +1 state). t  Lermer et al. [49-51] have reported improved measurements of the low cross-section X M P s corresponding to the excited n = 2 and n = 3 ion states of helium as well as the 2 p a and 2 p a u  g  orbitals of H and D , obtained with a momentum dispersive E M S spectrometer. Recently, 2  2  Storer et al. [52] have reported an EMS spectrometer that is multichannel in both energy and momentum which has been used to study thin solid films and thus have extended E M S studies to condensed matter targets. In particular, this spectrometer [52] has been used to obtain E M S measurements for the valence shell of graphite as well as amorphous forms of carbon. Multichannel E M S spectrometers have also made feasible the study of larger molecules such as SF  6  [42] and acetone [45].  Results obtained very recently at the University of British  Columbia for glycine [53-55] have extended the EMS technique to prototype biomolecules. In parallel with these experimental developments, the theoretical treatments involved in the interpretation of E M S studies (and quantum chemistry in general) have advanced considerably [56,57]. As will be discussed in more detail in Chapter 2, the X M P s obtained from E M S are directly comparable to theoretical momentum profiles (TMPs) calculated from  6  Chapter 1. Introduction  quantum chemistry wavefunctions. Much larger basis sets are now employed in the calculation of TMPs from Hartree-Fock (HF) wavefunctions than in earlier E M S studies.  Substantial  advancement has also been made in the theory of post-Hartree-Fock methods such as configuration interaction (CI) as well as the capability to compute large CI wavefunctions [56,57] for calculation of TMPs. In other work, the incorporation of instrumental momentum resolution effects in TMPs has been found to be an extremely important consideration in the interpretation of the E M S experiment [58,59]. Lastly, the development of the target KohnSham approximation (TKSA) for the calculation of theoretical momentum profiles using density functional theory (DFT) [60] has permitted high-level calculations which include some accounting of electron correlation effects through the exchange-correlation potential. The DFT calculations are particularly important for larger molecules where large CI calculations are not feasible. In the last few years, the application of Kohn-Sham orbital momentum profiles has been extended to other larger molecules and, while the method is still relatively new, it has thus far provided very good descriptions of the momentum profiles of several molecules if sufficiently large basis sets are used [45,53-55,60-62].  1.1.2 Binding Energy Spectrum The binding energy mode is principally used to select the various orbitals for the study of momentum profiles. The energy resolution in existing EMS studies is typically limited to ~ 1.3 eV F W H M due to the Maxwell-Boltzmann distribution of electron energies from the heated electron gun cathode (~ 0.7 eV FWHM) and the contributions from the energy resolutions of the two electron energy analyzers.  Even with present multichannel  instrumentation, monochromation of the incident electron beam is not a realistic possibility in the E M S experiment because of the very low (e, 2e) coincidence cross-section and the resulting low count rates.  Further improved multichannel E M S spectrometers are currently being  developed with much increased count rates and thus incident electron beam monochromation may soon be employed.  Chapter 1.  Introduction  1  Despite the present limitations in energy resolution for the E M S experiment, BES have some distinct advantages compared with high resolution photoelectron spectroscopy (PES) studies which can provide very accurate ionization potentials.  Before the advent of  monochromated synchrotron radiation, discrete atomic emission resonance lamp sources were routinely used for PES due to their high intensity and very narrow natural line widths. The most commonly used resonance lines are He I and He JJ with energies of 21.2 eV and 40.8 eV, respectively. However, higher binding energy studies above ~ 20 eV are not possible with He I PES sources. In addition, studies with He II sources are normally contaminated above ~ 20 eV by a He I "shadow" and by self-ionization of He at 24.6 eV. Thus, the inner valence region of molecules (usually in the 20-40 eV binding energy range) has not been studied with these low energy photon line sources [63,64]. In contrast, the EMS experiment does not suffer from such spectral contamination effects and there are no restrictions on the incident energy since higher binding energies are readily accessible by simply increasing the electron beam impact energy. While X-ray line light sources involving the decay of atomic inner-shell hole states (e.g. the A l and M g Ka lines at 1486.6 eV and 1253.6 eV, respectively) can provide higher energy PES studies, the resolution is significantly poorer (0.7-1.0 eV F W H M ) than for the He I and He II sources.  A wide range of tunable photon energies is now available for PES studies using  monochromated synchrotron radiation sources, but such facilities are remotely located and beam time must be obtained.  In addition, the energy resolutions of monochromated  synchrotron radiation sources are typically wider than that of emission line sources such as He I and II resonance lamps and are often only of the same order of magnitude as the best overall E M S energy resolution, particularly at higher photon energies. It should also be noted that PES spectra usually have large sloping non-spectral backgrounds (particularly at lower kinetic energies) while the E M S experiment results in a constant background (due to random coincidences - see Chapter 3) which is easily and unambiguously removed during the collection of the data.  Chapter 1. Introduction  8  Within the Hartree-Fock independent particle model, one peak occurs in the binding energy spectrum corresponding to each occupied orbital (neglecting effects such as spin-orbit interaction and nuclear motion). This behaviour is normally observed for ionization from the outer valence orbitals.  However, electron correlation (many body effects) and electronic  relaxation can distort this simple picture and ionization from a more tightly bound valence orbital may occur at several ionization energies and produce a manifold of final ion states for a given orbital ionization [63-65]. This results in "satellite peaks" in the binding energy spectra. In terms of the CI model, these peaks can be modelled by linear combinations of ionizations and simultaneous excitations of other electrons which occur in addition to the primary (singlehole) ionization process. Alternatively, many body Green's function methods can be used to predict the spectra [65]. The presence of satellite peaks is most common in the higher energy inner valence region with binding energies typically in the range 20-60 eV [65]. Although these satellite peaks are generally of lower intensity than that due to the main ionization peak, they are frequently very numerous and can account for a significant portion of the total orbital ionization intensity (ionization strength).  The sum of the intensities of all of the ionization  peaks having the same parentage must also be equal to the number of electrons removed (the spectroscopic sum rule - see Chapter 2).  The capability of E M S to detect such electron  correlation effects has been known since the early measurements on argon [20,21] in which (3s) satellite processes were observed at energies above that of the main (3s)" peak. _1  1  It is difficult to assign the final ion state and overall symmetry of parent and satellite ionization processes with PES [66] and, although some information on the character and parentage of ionization processes can be obtained using angular resolved PES measurements [63,64] and/or the dependence of intensity on photon energy, the results are not always conclusive.  In contrast, the E M S cross-section provides more direct information  concerning the origin of satellite peaks in binding energy spectra.  Since the E M S binding  energy spectra can be obtained as a function of electron momentum (i.e. angle 0) consideration of the relative intensities in BES at two different angles can provide some symmetry  Chapter 1. Introduction  information.  9  Furthermore, the shapes of the electron momentum profiles often allow  assignment of the orbital parentage of the given ionization process by comparison with ab initio calculations. Different ionization processes sharing the same initial orbital parentage would be expected to have the same shape momentum profile if initial state correlation is negligible.  For example, the disputed energy ordering of the (5ai)  _1  and C ^ ) " orbital 1  ionizations of formaldehyde (H CO) has been clarified by E M S measurements and momentum 2  profile calculations [67,68].  1.1.3 Momentum Profile The unique feature of EMS is the measured electron momentum profiles obtained using the technique.  The spherically averaged square of the momentum space molecular orbital,  y/(p), generally provides good descriptions of the measured X M P (see Section 2.2.5) and thus E M S permits the "imaging" of electron density in momentum space for individual orbitals. This imaging ability is useful not only for the fundamental electronic structure information it provides but also because the experimental momentum profiles provide a stringent quantitative test of the basis sets and quantum chemistry procedures (e.g. HF, CI and DFT) used to obtain theoretical wavefunctions. The X M P is particularly sensitive to the low momentum region of a wavefunction.  It should be noted that position (r) and momentum (p) space are inversely  weighted according to the Fourier transform (see Section 2.2.7) so that large r generally corresponds to low p and vice versa. Therefore, EMS is particularly sensitive to the outermost (large r) spatial regions of electron density. Although a good description of the large r regions of electron density is vitally important in understanding chemical reactivity, this region is often poorly represented by theoretical wavefunctions commonly calculated with the variational method because of the insensitivity of the total energy to the larger r regions. The energy minimization constraints of the variational method tend to emphasize the opposite regions (i.e. small r, large p) of the wavefunction [69]. The momentum profiles from E M S therefore provide an additional criterion to variationally determined total energies and other properties  10  Chapter 1. Introduction  emphasizing medium r and p such as dipole and quadrupole moments.  Consideration of a  range of properties is necessary for the calculation of "universal" wavefunctions that provide reliable modelling in all regions of phase space. For example, even near-Hartree-Fock limit TMPs for H 0 and N H were initially found to underestimate the low momentum regions of 2  3  the H O M O experimental momentum profiles for these molecules [39,69,70]. As a result, the E M S studies on these molecules led to the development of more highly saturated basis sets with polarization and extremely diffuse functions that resulted in a significant improvement in the agreement between theory and experiment in the low momentum regions of these orbitals and also very accurate prediction of a wide range of properties [56,69]. However, because of the almost negligible effects of such diffuse and polarization functions on the small r region heavily emphasized by the variational method, the calculated total energies differed only very slightly when these diffuse and polarization functions were added to the basis set. The development of wavefunctions that provide good calculated results for a range of properties emphasizing different regions of phase space has been of particular interest in many E M S studies. High-level multi-reference single and double excitation configuration interaction (MRSD-CI) wavefunctions have been developed in ongoing collaborative work between this laboratory and co-workers at Indiana University and the corresponding theoretical momentum profiles have been compared with the experimental momentum profiles [37,39,71].  For  example, one factor contributing to the initial disagreement between the theoretical and experimental momentum profiles for the HOMOs of H 0 and N H mentioned above was 2  3  found to be the absence of a treatment of electron correlation effects in the theoretical calculations [39,69,70]. When electron correlation effects were accounted for by using large basis set MRSD-CI calculations (and also including the G W - P G resolution folding method [59] to account for the instrumental angular resolution) excellent agreement between theory and experiment was obtained for these hydrides [39,69,70]. The inclusion of electron correlation effects was also necessary to obtain reasonable agreement between theory and the experimental momentum profile for the H O M O of hydrogen fluoride [72].  However, the  Chapter 1. Introduction  11  momentum profiles of the third row hydrides S i H [73], P H [74], H S [75] and HC1 [76] as 4  3  2  well as C H [77] showed good agreement with near Hartree-Fock limit TMPs. The effects of 4  correlation and relaxation on the momentum profiles were found to be negligible for these molecules [73-77]. Further investigations of the effects of electron correlation have been provided by the recent comparison of theoretical momentum profiles obtained using D F T with measured X M P s [45,53-55,60-62]. The "orbital imaging" provided by the measured momentum profile facilitates the study of bonding and chemical behaviour of a series of compounds by measurement of the momentum profiles of similar orbitals. The sensitivity of the E M S cross-section to the low p (large r) region is particularly useful for the study of such effects. For example, Leung and Brion [78,79] have used this technique to study the changes in electron density distribution occurring on forming the chemical bond in molecular hydrogen. In other work, the methyl inductive effect has been studied by comparison of experimental and theoretical momentum profiles for the HOMOs of ammonia and the various methylamines together with a consideration of calculated momentum and position space density difference maps [70,80-82]. Methyl substitution effects have also been investigated in the calculated and experimental orbital momentum profiles for the HOMOs of the series formaldehyde, acetaldehyde and acetone [83] and for the series H 0 [69], C H O H [84] and C H O C H [85,86]. 2  3  3  3  Electronic structure information is more commonly studied and understood in terms of the position space representation.  The experimental results from E M S studies therefore  provide a complementary picture of chemical phenomena in the momentum space description. Although the Schrodinger equation is most commonly solved in the position space representation for the total r-space wavefunction, it has long been known that there is an analogous p-space equation that may be solved for the momentum space wavefunction. The p-space representation depends upon the momenta rather than the positions of the electrons. The two wavefunctions are related by the Fourier Transform (see Section 2.2.7).  This  alternative momentum space representation was investigated by Coulson and Duncanson in a  Chapter 1. Introduction  12  series of theoretical papers on the electron density for H and simple hydrocarbons [87-89]. It 2  was shown [87-89] that chemical bonding concepts could be discussed in terms of the p-space electron density and that the momentum space representation provided complementary information that was not readily obtained from the position space viewpoint.  Although  momentum space theoretical studies are not widely found in the literature, the effectiveness of such approaches is well illustrated by the recent theoretical work of Cooper, Allan and co-workers [90-92].  These calculations [90-92] have demonstrated the usefulness of the  momentum space representation and have shown that molecular similarity and dissimilarity indices calculated using momentum space electron densities are capable of rationalizing chemical properties and reactivity.  In particular, the momentum space approach gave  satisfactory predictions of the biomedical activity of molecules such as anti-HIV phospholipids [90-92] where r-space calculations were inconclusive. The experimental and theoretical results from E M S studies over the last quarter century have been used to illustrate chemical behaviour such as bonding and reactivity in the momentum space representation and to provide a pedagogical approach to this complementary viewpoint. The studies of the bonding in H  2  [78,79] and methyl substitution [70,81-83,85,86] mentioned above as well as studies of many other molecules have been aided by calculated momentum and position density contour maps. Brion [11], McCarthy and Weigold [35] and Leung [93] have discussed momentum and position space properties in various review articles. Zheng, Neville and Brion [53,54] have studied the distribution of "lone pair" density on the nitrogen and oxygen atoms of glycine and the role of correlation using E M S measurements and position density maps for the HOMOs generated with the best fitting DFT calculations for the three lowest energy conformers.  1.2  Overview of the thesis In the present work, the technique of EMS has been extended to the investigation of  several types of molecules of particular experimental and theoretical interest. The remaining chapters of the thesis are organized as outlined below.  13  Chapter 1. Introduction  The theoretical aspects of quantum chemistry related to E M S studies are discussed in Chapter 2. In addition, the methods for the interpretation and calculation of the E M S crosssection are also introduced. Chapter 3 gives details of the construction and operation of the multichannel electron momentum spectrometer used in the present work. The data processing procedures profiles are also explained in this chapter. Experimental and theoretical investigations of the electron momentum distributions for the outermost (HOMO) orbitals of the open-shell molecules N O and N 0 are described in 2  Chapter 4. Systematic comparisons of the experimental results with Unrestricted Hartree-Fock (UHF), Restricted Open-shell Hartree-Fock (ROHF), MRSD-CI and D F T calculations. A n averaged coupled-pair functional (ACPF) calculation is also reported for N O while a quasidegenerate variational perturbation theory (QDVPT) calculation is presented for N 0 . This 2  work has been published as J. Rolke, N . Cann, Y . Zheng, B.P. Hollebone, C.E. Brion, Y . A . Wang and E.R. Davidson, in Chem. Phys. 201 (1995) 1 [94]. E M S measurements of the complete valence shell of molecular oxygen are presented in Chapter 5.  The experimental E M S binding energy spectra are compared to theoretical  predicted binding energies and pole strengths from a new MRSD-CI calculation. In addition, the E M S valence momentum profiles are compared with calculations using both the U H F and ROHF methods, with MRSD-CI momentum profiles and also with density functional methods employing a range of exchange-correlation functionals. In the case of the outermost  l7t  g  orbital  ( TI final ion state) an A C P F calculation is also reported. A n assignment of binding energy 2  P o  peaks of disputed origin is provided by a comparison of the measured E M S binding energy spectra and momentum profiles with MRSD-CI calculations. E M S studies of phosphorus compounds and the investigation of inductive effects is discussed in Chapter 6, where results for the HOMOs of P H , P F , and P ( C H ) are presented. 3  3  3  3  The experimental momentum profiles for the HOMOs of P F and P ( C H ) are the first to be 3  3  3  reported and are used to investigate substituent effects of the methyl group relative to hydrogen  Chapter 1. Introduction  and fluorine.  14  The present experimental results for P F and P ( C H ) 3  3  are compared with  3  previous E M S results for P H [74]. In addition, the experimental momentum profiles for 3  P H [74], P F and P ( C H ) are compared to HF, MRSD-CI and D F T theoretical momentum 3  3  3  3  profiles. This work has appeared in the literature as J. Rolke and C.E. Brion, in Chem. Phys. 207 (1996) 173 [95]. The application of EMS to larger inorganic molecules is investigated in Chapter 7, where results for the HOMOs of the transition metal hexacarbonyls Cr(CO)g, Mo(CO)g and W(CO)6 are presented.  E M S measurements for the t  2g  HOMOs of these molecules are of  particular interest because these orbitals are primarily metal valence d in character.  The  experimental momentum profiles are compared with Hartree-Fock and DFT theoretical momentum profiles within the plane wave impulse approximation (PWIA) of the E M S cross-section. Distorted wave impulse approximation (DWIA) calculations for atomic Cr 3d and M o 4d orbitals have also been carried out to investigate the role of distorted wave effects in the E M S cross-section in systems involving these atoms (see Chapter 2 for descriptions of the PWIA and D W I A methods for calculation of the E M S cross-section). This work has been published as J. Rolke, Y . Zheng, C.E. Brion, S.J. Chakravorty, E.R. Davidson and I.E. McCarthy, in Chem. Phys. 215 (1997) 191 [96]. E M S measurements of the full valence shell of methanol (CH OH), including the 3  satellite structure up to 47 eV, are presented in Chapter 8. Calculated binding energy spectra obtained by combining large basis set Hartree-Fock calculations of the momentum profiles with Green's function pole strengths are compared to the experimental BES. In addition, a range of Hartree-Fock, MRSD-CI and DFT calculations are investigated and compared to experimental momentum profiles. Chapter 9 summarizes the general conclusions arising from the present work. The major results are summarized and some suggestions for future experimental and theoretical studies are presented.  Chapter 2. Theory  15  Chapter 2 Theory  2.1 The Calculation of Wavefunctions The focus of this section is the description of the theoretical methods which are commonly used to calculate the EMS cross-section for comparison to experiment. Over the past seventy years, theoretical models used to aid in the interpretation of experiment have advanced greatly. This chapter will be concerned with the following methods: (a) Hartree-Fock (for both closed- and open-shells) (b) Configuration interaction (CI) (c) Density functional theory (DFT) Since these methods are well-known and can be found in detail in standard quantum chemistry texts [57,97,98] only brief descriptions will be given here.  Both Hartree-Fock and  configuration interaction methods have been used in the interpretation of E M S studies for many years. However, the application of DFT to EMS is much more recent and it provides a useful alternative to the more established methods.  Before the individual methods are  described in Sections 2.1-2.4, it will be useful to review some basic concepts of quantum chemistry such as the Hamiltonian  Schrddinger  and the Born-Oppenheimer  equation,  the  variational  principle,  the  molecular  approximation.  According to quantum mechanics, the energy and properties of a stationary state of a molecule can be obtained by solution of the Schrddinger equation: H ¥=E¥ X  (2.1)  Chapter 2. Theory  16  In Eq. (2.1) H is the Hamiltonian,  an operator representing the total energy E for the stationary  state of the system described by the total wavefunction *F. The Schrodinger equation for a molecule will have many solutions, corresponding to different stationary states. The state with the lowest energy is the ground state. The most common techniques for the calculation of wavefunctions (e.g. Hartree-Fock and CI) are based on the variational  principle  [98].  If 4Q is the exact ground state  wavefunction for a system, then it satisfies the Schrodinger equation (Eq. (2.1)) such that: H* = E % 0  (2.2)  0  where H is again the Hamiltonian and E is the exact ground state energy. The expectation 0  value of the energy (E) of any normalized wavefunction <I> is given by: E=j$>*H&dr > E  (2.3)  Q  where integration is over the coordinates of all electrons. Thus, E is greater than or equal to E  0  (equality results only if the trial wavefunction O is identical to the ground state wavefunction %).  E is therefore an upper bound to the exact ground state energy and, in the variational  method, the lowest energy wavefunction is assumed to be the best approximation to the exact wavefunction.  However, the degree to which such a wavefunction can provide accurate  electronic properties (e.g. dipole moment, quadrupole moment, momentum profile) depends on the computational method (e.g. Hartree-Fock, CI, etc.) and also on the basis set chosen in the calculation. If we assume the nuclei and electrons to be point masses and neglect spin-orbit and other relativistic effects, then the molecular Hamiltonian is (2.4) A  A  e  Am  i  A B>A  AB  A  i  iA  r  j  i>]  V  r  where A and B refer to nuclei and i and j refer to electrons. The first term in Eq. (2.4) is the operator for the kinetic energy of the nuclei and the second term is the operator for the kinetic  Chapter 2. Theory  17  energy of the electrons. The third term is the potential energy of the repulsions between nuclei with atomic numbers Z and Z which are separated by a distance of r . A  B  AB  The fourth term is  the potential energy of the attractions between the electrons and the nuclei, with a distance of r  iA  between electron i and nucleus A. The last term is the potential energy of the repulsions  between the electrons, with r^ being the distance between electrons / and j. The full molecular Hamiltonian given by Eq. (2.4) is formidable and thus the BornOppenheimer approximation is used as a simplification. Since the electrons are much lighter than the nuclei, they move much faster than the nuclei. Thus, to a good approximation, we can regard the nuclei as fixed while the electrons carry out their motions (i.e. in a classical sense, the change in nuclear positions is negligible during time that the electrons carry out a "cycle" of their motions). Therefore, the terms involving the kinetic energy of the nuclei in Eq. (2.4) can be omitted to give a new molecular Hamiltonian: H=H +V el  where the purely electronic Hamiltonian  (2.5)  NN  H contains terms involving only the electrons: ei  4,=-^SvMI^l lEf +  e  Zm  and the nuclear repulsion term V  NN  i  A  r  r  is the only remaining term involving the nuclei:  ^ =1 1 — A B>A  The V  NN  AB  the electronic wavefunction of the system.  (2-7)  AB  r  term is constant for a given choice of the r  distances and therefore does not affect  The variables in Eq. (2.5) are the electronic  coordinates and the electronic Hamiltonian H  d  Schrddinger  (2.6)  j i> j ij  iA  i  can be used to solve the purely  electronic  equation: ^el^el^el^el  ( -8) 2  Chapter 2. Theory  18  and the resulting electronic energy E  d  can then be added to the nuclear repulsion to give the  total energy E: E = E VNN e]+  (2.9)  }  While the Born-Oppenheimer approximation simplifies the procedure, the determination of the electronic wavefunctions for a many-electron system is still a challenging problem mainly due to the inter-electron llr^ terms of the Hamiltonian. Hartree-Fock theory (see Section 2.1.1 below) provides the simplest and most common method for handling these problems and calculating the electronic wavefunctions.  2.1.1 Hartree-Fock Theory for Closed-Shell Molecules The Hartree-Fock method was developed in the late 1920s and early 1930s from intuitive, physical arguments as a model of electronic structure. The electrons in a molecule are assumed to be independent of one another in Hartree-Fock theory and thus they may be described by a series of one-electron functions called orbitals. In particular, the total n-electron Hartree-Fock wavefunction ^  H F  is expressed as a single antisymmetrized Slater determinant  composed of one-electron functions called spin orbitals.  This antisymmetric determinant  restriction is chosen so that the wavefunctions obey the Pauli principle and thus are physically acceptable. Each spin orbital is composed of the product of a spatial orbital \\f and a spin t  function (a or (3).  The one-electron spatial part of a molecular spin-orbital is called a  molecular orbital (MO). Using the variational method (Eq. (2.3)), the Hartree-Fock total molecular electronic energy £  H F  is given by: (2.10)  where H i and V e  NN  have the forms given in Eqs. (2.6) and (2.7). Substituting in the forms for  the operators, converting to atomic units and expanding *P yri, it can be shown that:  HF  in terms of its constituent orbitals  Chapter 2. Theory  19  nil  nllnll  £ F = 2]>>r + XS(2/y-K ) + V  (2.11)  e  H  i j  m  1=1 y=l  1=1  where  ^r =(vi(i)|^Tk-a)> B  / = (  Jy  I 1 j VA - (i) | £  V ZA I  V  (2.12)  - - vf - X—I V^i (D>  (2) | "^1 Wi (D V ; (2))  0)  (2.13)  ^=(^(l)^(2)|-^-|^(l)^(2)) The one-electron core Hamiltonian  H^  K  (2.14)  omits the interactions of electron i with the other  electrons. The sums involving / and j are over the nil occupied spatial orbitals y/j of the nelectron molecule (hence the factor of 2 in some terms). The quantities Jy and K j are known (  as the Coulomb and Exchange integrals, respectively and the integration is over the spatial coordinates of electrons 1 and 2. The next step in the Hartree-Fock method is to find the orbitals y/,- that minimize the variational integral in Eq. (2.10) such that the Hartree-Fock energy E  HF  to the energy E  HF  is as close as possible  from the exact wavefunction. As a computational simplification, the orbitals  are taken to be orthonormal. Minimization of E  HF  electron Hartree-Fock  with respect to I/A- results in nil one-  equations [57] each of which satisfy: F(l)y/^e  (2.15)  i¥i  F ( l ) is the one-electron Fock operator and e is the energy of orbital y/,-. The Fock operator is {  given (in atomic units) by: nil  F(l) =  + X[27y (1) - Kj(1)]  i  z  ^  A \A  j=i  = " o V f - X — + Z [ 2 i y (1) 1  r  ( 2  Kj(l)]  -  1 6 )  The Coulomb operator J j and the exchange operator Kj for interactions of electron (1) with  Chapter 2. Theory  20  electron (2) are defined by their operation on any arbitrary function f(\): Jj(\)f(\)=f{V)\Wj(2) —dr  (2.17)  2  2 r  12  r * Kj(\)f(l)  V0(2)/(2)  = xifj(l) —  d%  2  '12  (2.18)  It should be noted that these last two operators depend on the coordinates of one electron only. The Coulomb operator 7 (1) describes the potential energy of the interaction between ;  2  electron (1) and a smeared-out electron density y ( 2 ) ;  for electron (2). Thus, instantaneous  interactions between electrons are neglected in favor of an "average field" of electrons and this form for the Coulomb operator is required for the total electronic wavefunction to be separated into products of one-electron wavefunctions.  The exchange operator arises from the  antisymmetry requirement on the total wavefunction and has no obvious physical meaning. From the forms of the Coulomb and exchange operators, it can be seen that the Fock operator F depends on its own eigenfunctions y/;. However, the y/ are not known initially and thus the t  n Hartree-Fock equations (Eq. (2.15)) must be solved by an iterative process by which the calculations are repeated until the results are self-consistent (i.e. until the no change results with further iteration). Because the (average) electric field of the molecular orbitals is used to iteratively determine the orbitals themselves, the Hartree-Fock method is considered to be a self-consistent field (SCF) method. Until the 1950s, the Hartree-Fock MOs were evaluated numerically and were generally limited to atoms and small linear molecules because of computational difficulties. However, great advances were made in 1951 when Roothaan proposed [99] that the spatial orbitals y/i could be expanded as linear combinations of a set of N well behaved functions basis functions % '• s  N  Vi =  X «& c  s=\  (2.19)  Chapter 2. Theory  21  Substituting Eq. (2.19) into the Hartree-Fock equation (2.15) gives: (2.20)  ^ si P% s ^i ^ si X s s s c  c  Multiplication by % * and integration gives a set of N equations (the  Hartree-Fock-Roothaan  equations) with unknowns c : si  E « ( r - £ « 5 „ ) = 0, c  F  J  (2.21)  r = \,...,N  (2.22)  The non-trivial solution for the set of coefficients c requires that si  detF„-e,5„ =0 This is known as the Hartree-Fock  secular equation.  (2.23) Thus, Roothaan's procedure is very  amenable to computer-based calculation because of it can be reduced to a common matrix algebra determinant problem. The iterative procedure is started by an initial guess of the individual one-electron orbitals y/; and then the Fock operator is constructed according to Eqs. (2.16)-(2.18). The matrix elements, F and S , are then computed and used to find the set of orbital energies e rs  rs  t  from the secular equation (Eq. (2.23)). The values of  are used in turn to find the coefficients  c from Eq. (2.21) which gives an improved approximation of the orbitals y/,-. The procedure si  is then repeated using the improved orbitals until the improvement in total energy from one cycle to the next is smaller than a preset value, typically less than 10" hartrees. 5  The quality of the final Hartree-Fock result is very sensitive to the choice of basis functions % . For molecules, the basis set commonly consists of functions resembling atomic s  orbitals centered on each of the nuclei of the molecule. combination of atomic orbitals or LCAO  approximation.  This is known as the linear The two main types of basis  Chapter 2. Theory  22  functions used are Slater-type orbitals (STOs) and Gaussian-type orbitals (GTOs).  In  spherical polar coordinates, STOs are defined as X ~r - e< Y (Q,$) n  x  (2.24)  r  s  lm  where n is a positive integer, / and m are angular momentum quantum numbers, Yi (8,(j)) are m  the spherical harmonics, and £ is the orbital exponent (C, determines the diffuseness of the basis function). The main advantage of using Slater-type orbitals is that they have the proper short range (i.e. cusp at the nucleus) and long range (tail) behavior for calculation on atoms. This advantage is offset, however, by the fact that integrals involving STOs are often difficult to evaluate (particularly for the two-electron r^s  terms of the matrix elements F  rs  and S ) and rs  require more computing time than integrals involving GTOs. Cartesian Gaussian-type orbitals are very popular for molecular calculations because of the ease of computing the integrals required. The Cartesian GTOs are defined by X ~x y z e-y k  m  n  s  (2.25)  where £ is as above, and k,m,n are non-negative integers. Gaussian-type orbitals do not model wavefunction tails and cusps as well as STOs and thus, to make up for this deficiency, more Gaussian primitive functions are needed than STOs to model a wavefunction to the same level of accuracy. Like STOs, the exponents C, for GTOs are chosen to minimize the Hartree-Fock energy of the atom. L C A O s of these basis functions are used to describe the molecular orbital wavefunctions with the coefficients optimized variationally. Because of the relatively large number of basis functions in a Gaussian-type calculation, it is usual to constrain groups of functions to have a fixed ratio of coefficients. The individual functions are called primitives, and the resultant linear combination is called a contracted Gaussian. A n overview of basis sets and techniques has been given by Davidson and Feller [100]. It should also be noted that if the basis set were complete (i.e. N-°° in Eq. (2.19)), then it would be possible to express exactly any well-behaved function as an expansion of the basis  Chapter 2. Theory  23  set. The resulting wavefunction and total energy from such a complete basis set would then be the best possible from the Hartree-Fock method and thus these are referred to as the HartreeFock limit wavefunction and total energy. In practical calculations, however, the number of basis functions used must be finite and the resulting Hartree-Fock energy is greater than the Hartree-Fock limit energy. Therefore, successive calculations with increasing basis set size and flexibility give energies that come increasingly closer to the Hartree-Fock limit energy. If the number of basis functions is very large, the basis set offers great flexibility and there is only a small change in the Hartree-Fock total energy upon the addition of further basis functions to the set. The resulting wavefunction and total energy (and other calculated properties) from such a large and flexible basis set are referred to as being near-Hartree-Fock  limit quantities  because they closely approach the Hartree-Fock limit quantities. A brief description of the varying classes and features of basis sets is now considered. A minimal basis set (MBS) is one in which one basis function is included for each atomic orbital. If the calculation is Slater-type, each basis function consists of one Slater-type orbital. Thus, minimal basis sets are often described as single zeta basis sets because they are equivalent to using a single Slater function for each atomic orbital (i.e. a single £ exponent see Eq. (2.24)). For Gaussian-type calculations, however, the basis function may be a single primitive Gaussian or a linear combination (i.e. contraction) of primitives. For example, the STO-3G basis set [101] used in this thesis is a minimal basis set in which each basis function is a linear combination of three Gaussian functions.  Because it is the smallest calculation  possible, the minimal basis set is very limited in its flexibility and typically leads to an inaccurate quantitative description of molecular structure and properties. The flexibility of the basis set can be increased by including more than one function per atomic orbital. A double zeta basis has two Slater functions with different exponents. A n equivalent Gaussian basis would have to contain two contracted Gaussians per atomic orbital. A split valence basis set has different handling of the core and valence atomic orbitals. For example, the 4-3IG basis [102] used in this thesis has one basis function or contracted  Chapter 2. Theory  24  Gaussian representing each core atomic orbital and two for each valence orbital (the 4-31G basis set is thus said to be single zeta with respect to the core orbitals and double zeta with respect to the valence orbitals). Core orbitals contribute to the total energy but not very much to bonding, and thus split valence basis sets give results comparable to handling both core and valence in the same way. However, the main advantage of split-valence basis sets is decreased computation time. It should also be noted that the 4-31G basis set notation indicates that the single core contracted Gaussian function consists of four Gaussian primitive functions while the two valence contracted Gaussians consist of three and one primitives. In this thesis, the split-valence 6-311G basis set [103] of Pople and co-workers is used and it consists of a single zeta for the core (consisting of six primitives) and triple zeta modelling of the valence (three contracted Gaussians consisting of three, one and one primitives). Polarization  functions are often added to basis sets to provide a good description of  polar charge distributions [98]. For example, a spherical charge distribution is given for the H atom by minimal basis and double zeta basis sets (i.e. only s-orbitals in these basis sets), but such a description is not realistic when hydrogen is bonded to an electronegative atom, such as in HF. The charge distribution can be polarized to one side of the H atom by the addition of a p-type function to the H basis set. Augmentation of the basis set with d and f functions can also be done to allow for polarization of charge. Basis sets augmented by the minimal set of polarization functions on all non-hydrogen (heavy) atoms are commonly identified by an asterisk (*) following the " G " in the basis set name. If p-type polarization functions are also added to the hydrogen atom basis, two asterisks are used (**). For example, the 6-311G* and 6-311G** basis sets [104] are formed by augmentation of the 6-311G basis set [103] with spherical d-type polarization functions on the heavy atoms (6-311G*) and a p-type polarization function on all H atoms (6-311G**). In addition to polarization functions, diffuse functions (i.e. having small £ exponents) are often added to basis sets [98]. The inclusion of such functions often allows a better modelling of the large-r (long range) charge density that is generally not described well by  Chapter 2. Theory  GTOs.  25  If the energy minimization criterion is used for convergence, these diffuse orbitals  make very little contribution to the total energy. However, diffuse functions have been shown to be important in the calculation of properties such as electron affinities, proton affinities and valence orbital momentum profiles that are highly dependent upon the outermost regions of electron density and well as medium r and p properties such as dipole and quadrupole moments.  The addition of diffuse functions to all heavy (i.e. non-hydrogen) atoms is  designated by a single "+" preceding the " G " in the basis set name. The addition of diffuse functions to all atoms, including H , is designated by a double plus "++".  For example, the  commonly used 6-311+G and 6-311++G basis sets [105] include augmentation of the 6-311G basis set [103] with diffuse s- and p-type functions on the non-hydrogen atoms (6-311+G) and addition of a diffuse sp-shell on all hydrogen atoms (6-311++G). Polarization and diffuse functions are often combined when added to the basis set. For example, the 6-311+G* and 6-311++G** basis sets are commonly used in calculations to include descriptions of both the diffuse and polarized charge density.  2.1.2 Hartree-Fock Theory for Open-Shell Molecules For open-shell molecules, free radicals and triplet states there are unpaired electrons and the Hartree-Fock-Roothaan equations need modification. For doublet states, there will be one extra a electron, for triplet states, two extra a electrons, and so forth. Simple molecular orbital theory can be extended to open-shell systems in two ways. The first is described as restricted open-shell Hartree-Fock  (ROHF) theory [57,97,98,107]. In  this approach, a single set of molecular orbitals is used, some being doubly occupied and some being singly occupied with an electron of a spin. This is the case illustrated for a five-electron doublet state in Fig. 2.1. (xi/^aXy/^PX^ccX^PXy/^oc).  The spin orbitals used in the single determinant are then The coefficients c  si  are still defined by the expansion (2.19)  and their optimum values are still obtained from the secular equation (2.23).  However,  details are more complicated since different conditions apply to singly- and doubly-  Chapter 2. Theory  A  ^4  increasing Energy  Tl Fig. 2.1: Electron configuration diagram for (y/ia)(y/if3)(y/2(X)(y/2P)(yf3(x)>  A  4  Increasing ^ Energy ^«  Fig. 2.2: Electron configuration diagram for  4- V'2|8  {y^a)(\^P)(\^a)(\^fi)(\^a).  Chapter 2. Theory  27  occupied orbitals [57,97,98,106,107]. The second type of molecular orbital theory in common use for open-shell systems is spin-unrestricted Hartree-Fock (UHF) theory [57,97,98,108].  In this approach, different  spatial orbitals are assigned to a and P electrons. Thus, there are two distinct sets of molecular orbitals y/f and y/f (i = 1, be written as  AO. The electron configuration for a five-electron doublet may  (y/"a)(y/f ft )(y/f a)(y// P X y f « )  a  n  d  i s  illustrated in Fig. 2.2. It is important  to note that the previously doubly-occupied orbital y/j is now replaced by two distinct orbitals, y/f and y/f. Since the ROHF function is a special case of the U H F function, it follows from the variational principle that the optimized U H F energy must be below the optimized ROHF value.  On the other hand, U H F functions have the disadvantage that they are not true  eigenfunctions of the total spin operator S  2  [57,97,98,108], unlike exact wavefunctions which  are necessarily spin-pure. Thus, U H F wavefunctions which are designed for doublet states (as in the example shown in Fig. 2.2) are contaminated by functions corresponding to states of higher spin multiplicity, such as quartets. In U H F theory, the two sets of molecular orbitals are defined by two sets of coefficients: N s=l  N  V?=L4z,  (2-27)  s=\  These coefficients are varied independently, leading to the following U H F generalizations of the Hartree-Fock-Roothaan equations [98,108]:  £c2(F ?-£X) = 0, r  r=l,...,N  (2.28)  s  £cg(^--e?S„)=0,  r=l,...,N  (2.29)  Chapter 2. Theory  28  For non-trivial solutions of the two sets of coefficients c", and cf, the following secular equations for the a and (3 orbitals must be independently solved: det rs r  i °rs = 0  (2.30)  =0  (2.31)  c  det rs 1  c  i "rs  2.1.3 Configuration Interaction (CI) A Hartree-Fock wavefunction takes into account the interactions between electrons only in an average way.  For a more accurate model of electronic structure the instantaneous  electron interactions must be included. Since electrons repel each other, they tend to keep out of each other's way. For example, if one electron in helium is close to the nucleus at one instant, then it is energetically favorable for the other be far away because of the repulsive electron-electron force. The motions of electrons are therefore correlated with each other. To fully describe this electron correlation, methods other than the Hartree-Fock treatment must be used. It should also be noted that the correlation energy is the difference between the exact non-relativistic energy E  emct  and the Hartree-Fock limit energy is p: H  ^corr  =  ^exact ~ ^ H F  (2.32)  Although the Hartree-Fock limit energy differs from the exact energy usually by only a few percent [98], neglect of correlation often precludes the detailed and accurate description of many molecular properties.  Therefore, the role of electron correlation is an important  consideration since it may be a determining factor in predicting chemical phenomena [71]. One method that is used extensively to incorporate instantaneous electron correlation into the wavefunction is the configuration interaction (CI) approach. The CI calculation for an atom or molecule first involves choosing a basis set of one electron functions % . The Hartrees  Fock orbitals y/,- are written as LCAOs of the basis set functions and the Hartree-Fock equations (2.23) are then solved for the linear combination coefficients c . si  The number of  Chapter 2. Theory  29  resulting atomic or molecular orbitals equals N, the number of basis functions used. If the molecule has n electrons, nil of these orbitals will be occupied orbitals while the remaining N-nll orbitals will be unoccupied virtual orbitals. The Hartree-Fock wavefunction will be a determinant composed of spin orbitals formed from the occupied MOs. This Hartree-Fock ground state electron configuration is called the reference function or reference state. Using the set of occupied and virtual spin-orbitals, other electron configurations can be created that have different orbital occupancies.  Some of the occupied orbitals are replaced with virtual  orbitals corresponding to the excitation of one or more electrons to virtual orbitals. Each configuration has an associated many-electron function  *FP  onfl  8  (a Slater determinant) called a  configuration function or configuration state function. The configuration functions in a CI calculation are classified as singly excited, doubly excited, triply excited, ... , according to  whether 1,2,3, ... , electrons are excited from occupied to unoccupied (virtual) orbitals. The total electronic CI wavefunction *F  CI  is then written as a linear combination of the electron  configurations:  ^ci=I^-°  nfiS  (2-33)  i=0  where ^ g ^ '  8=  ^HF *  s  m  e  reference state function corresponding to the ground state H F  configuration. In a very similar manner to that outlined for the Hartree-Fock method above, the matrix elements Hy and S j are constructed from the n-electron Hamiltonian for the t  configurations i and j.  The CI expansion coefficients a are then solved variationally to t  minimize the total energy using the following secular equation: det ^ - £ ^ = 0  (2.34)  The coefficient of the ground state configuration is usually much larger than any of the excited configurations. It should be noted that the type and size of the basis set affects the final CI results greatly since the "quality" of the reference state HF wavefunction is determined by the basis  Chapter 2. Theory  30  set. Therefore, it is useful to perform the CI calculations with a basis set which gives initial near-Hartree-Fock limit results for several properties (if the CI calculation for such as a large basis set is computationally viable). The configuration coefficients a need not be the only variables in Eq. (2.33) and the t  molecular orbital coefficients c  si  Hartree-Fock reference  can be varied simultaneously with the a, after the initial  state is constructed.  Such a procedure  is obviously more  computationally demanding. A CI calculation which includes all the possible electron configurations possible for a given basis set in the CI expansion (Eq. (2.33)) is referred to as a full CI calculation. However, full CI calculations are only possible for small molecules and small basis sets because of the large number of configurations involved. Even with a modest-size basis set, the number of possible configurations for most molecules is extremely large and therefore  it is not  computationally feasible to include them all. Thus, a limited CI procedure is usually chosen in which only selected configurations from the total possible configurations are used. In addition, it is not generally necessary to include all of the possible configurations to obtain "good" total energies and other properties. Selecting the important configurations (i.e. those which have the largest coefficients and contributions to the total energy) is a difficult procedure that requires considerable experience. Because of these considerations, sufficiently accurate CI calculations even for systems with only more than a few electrons are extremely challenging and are not feasible for larger molecules.  It should also be noted that in the CI expansion given by  Eq. (2.33) one includes only configuration functions that have the same symmetry properties as the state being approximated (usually the molecular ground state). For example, the ground state of H is a 2  state, and thus a CI calculation of the H ground state would include only 2  configurations with this symmetry. One common way to decrease the size of CI calculations is to use the frozen-core approximation  in which only excitations out of the valence orbitals are used. Excitations out  of the inner-shell core MOs are not included in Eq. (2.33). The contributions to the total  Chapter 2. Theory  31  correlation energy due to inner-shell electrons are not particularly small, but appear to be relatively constant from one molecular environment to another. The procedure outlined above is referred to as single reference CI, since only the Hartree-Fock ground reference state is used to derive the configurations. However, if several configurations contribute strongly to the total wavefunction then the single reference technique is not very efficient and a multireference  configuration  interaction  (MR-CI) calculation  provides better results. A n M R - C I calculation is usually performed in two (or more) stages. First, a CI calculation is performed as described above. The most important configurations are then each used as reference states (i.e. multiple reference states are used) to generate further configurations and the coefficients computed again. The choice of which configurations to include in the second stage (or any further stages) is generally made depending on the coefficient of the configuration in the expansion and its contribution to the total energy. The CI calculations presented in this thesis (performed by Prof. E.R. Davidson and co-workers at Indiana University) involve multiple reference states as described above.  In particular, an  initial singles and doubles excitation CI calculation was done to produce  reference  configurations. A second CI calculation was then performed in which all singles and doubles excitations from the multiple reference configurations were included. This type of M R - C I calculation is therefore known as multireference  singly and doubly excited  configuration  interaction (MRSD-CI) because of its use of one- and two-electron excitations. Because of the two steps in the calculation, the final MRSD-CI wavefunction will include some triple and quadruple excitations.  2.1.4 Density Functional Theory The DFT method, its underlying assumptions and its computational implementation are introduced in this section.  In addition, it will be shown that implementation of density  functional theory is very similar to that of the conventional Hartree-Fock method.  Chapter!.  32  Theory  In 1964, Hohenberg and Kohn [109] showed that the total energy (and other properties) of a system of n interacting electrons can be expressed as a functional of the electron density p(r). While the Hohenberg-Kohn theorem proved that it was possible in principle to calculate the ground state molecular properties from p(r), it did not provide explain how such calculations could be done.  Kohn and Sham [110] gave the first steps to making DFT  computationally viable when they considered a fictitious system of n non-interacting electrons having a total density equal to the true electron density of the system of interest. Kohn and Sham showed that the total ground state electronic energy of such a system with ground state electron probability density p(r) is given by: (2.35) where T [p] is the kinetic energy, 7[p] is the classical Coulomb repulsion energy and &  J p(r)v(r)d(r) is the potential energy from any external fields v(r) present (usually just the field of the nuclei in the electron-nuclear attraction). The first three terms in Eq. (2.35) have similar terms in Eq. (2.11) for the Hartree-Fock total energy. The term E [p] xc  is known as the  exchange-correlation functional and it includes the exchange energy and all contributions to E[p] due to correlated electronic motion.  The principal difference between the density  functional energy and the Hartree-Fock energy arises from this term. Eq. (2.35) would simply be an alternative expression for the Hartree-Fock energy if E [p] xc  is replaced by the Hartree-  Fock exchange energy Ky in Eq. (2.14). Kohn and Sham also imposed the constraint that the density must integrate to the number of electrons in the system: (2.36) i=i  The Kohn-Sham orbitals y/f  s  are found from the following Kohn-Sham equations, which are  derived by minimization of the energy E[p] in Eq. (2.35) subject to the constraint in Eq. (2.36): KS  4s(D^ =e,W KS  (2.37)  Chapter 2. Theory  33  It should be noted that Eq. (2.37) is very similar to the Hartree-Fock equations in Eq. (2.15). The Kohn-Sham operator F  is also similar to the Fock operator in Eq. (2.16) and is given by:  K S  * k s ( D = - \ V? - X—+  X2//D+V (l) j=\  (2.38)  x c  A lA  2  r  The Coulomb repulsion operator J -(1) has exactly the same form as the corresponding Hartree-Fock operator in Eq. (2.17), except that 1/^(2) is the Kohn-Sham orbital y/f (2).  The  s  only difference between the K S equations and the H F equations is that the H F exchange operator Kj(l) is replaced by the exchange-correlation  potential V^ (l). The name of V ( l ) is c  xc  therefore derived from the fact that it handles the effects of both exchange (antisymmetry) and electron correlation similar to the H F exchange operator.  V ( l ) is formally defined as the xc  functional derivative of the exchange-correlation energy: <5E [p] xc  V  x  c  =  ~  (  2  -  3  9  )  How a functional derivative is computed is described in [111] but such detail it is not necessary in this thesis. The solution of the K S equations is also similar to that for the H F equations. The K S operator is first constructed and the equations solved for the y/f , then a new K S s  operator is constructed and the process repeated (an iterative solution is necessary because the KS operator depends on its solutions) until the results are self-consistent. In a manner similar to Eqs. (2.19)-(2.23), the Kohn-Sham orbitals may be expressed as a linear combination of a basis set and the coefficients solved for variationally. The important fact that should be noted from (2.39) is that V is known exactly only if x c  the form of E  xc  is known exactly. If V  xc  is known, then the Kohn-Sham equations can be  solved for the K S orbitals y/f and therefore for the density p and energy E[p]. The major s  complication of Kohn-Sham DFT therefore arises from the fact that the exact functional E  xc  is  not known for molecules. In practical calculations, approximations must be made and a functional form for E  xc  (and hence V ) must be chosen before any calculations may be xc  Chapter 2. Theory  performed.  34  Many approximate functionals have been devised in DFT. These are usually  constructed by choosing an analytical form for £  x c  and then comparing the properties from  DFT calculations with this approximate functional to experimental values. Unfortunately, such a procedure is not systematic and a suitable form for £  x c  is not always clear. Some of the  common choices for the exchange-correlation potential are described below. generally be divided into two classes: local and non-local  These may  functionals.  Local functionals employ the local density approximation  (LDA) in which the  exchange-correlation energy is approximated as [109,110]: (2.40) where e [ p(r)] is the exchange-correlation energy per electron in a homogeneous electron gas xc  with electron density p . The L D A therefore assumes that p varies slowly with position and is often very accurate for metallic and strongly delocalized systems with fairly constant electron density. However, the electron density in a molecule varies rapidly in a molecule (because the positive charge is located only at the nuclei instead of being uniformly distributed) and therefore this approximation can be expected to be less accurate for such a system. Despite these shortcomings for molecules, LDAs are commonly used in the DFT literature because of their ease of computation. For example, in the L D A a single point in space is used to calculate the exchange-correlation energy and potential whereas non-local density approximations (described below) are more complex because they use a region of space instead of just a single point in the calculation. In addition, for computational purposes it is often convenient to represent the results for e  xc  and V in an analytic form. The L D A used in this thesis is that of xc  Vosko, Wilk and Nusair (VWN) [112] which is commonly found in the D F T literature. The V W N functional [112] was obtained by combining the homogeneous electron gas correlation energy from Ceperly and Alder [113] with the exchange energy for such an electron gas from Dirac [114]. Vosko et al. [112] chose their analytical form for the correlation energy functional by performing a Pade fit on the results of Ceperly and Alder [113]. This form was chosen  Chapter 2. Theory  35  because the derivative is easily obtained and thus the correlation potential is easily determined (similar to Eq. (2.39), the correlation potential is the functional derivative of the correlation energy). This method is of separating the exchange-correlation energy into two separate terms is commonly used and the resulting total potential is therefore also the sum of two parts: V =V +V K  X  (2.41)  C  The V W N functional includes some accounting for correlation and thus is an improvement over older DFT methods that neglected correlation (e.g. the X a method [110] uses the same as the V W N method, but V is neglected). However, the correlation energies in such L D A c  approximations are still often overestimated by as much as 100% [115] while the exchange energy is sometimes underestimated by 10% [116]. Because the molecular geometries are unaffected by these errors, many molecular properties are still well described by L D A functionals.  However, these errors in the correlation and exchange energies produce  dissociation energies that are almost always too high (often by a few eV). This overbinding of molecules is due to the incorrect asymptotic form of the V W N exchange potential.  The  exchange potential V falls off too rapidly at large r so that an electron in the large r region x  feels the full Coulomb potential. The outermost regions of the electron density distribution are therefore too diffuse. Non-local functionals have been developed as a solution to the problems of overbinding and incorrect asymptotic exchange from L D A functionals.  The usual results involve the  addition of gradient corrections to the potential (i.e. derivatives of the charge density are added to the potential similar to a Taylor series expansion). The potential is referred to as non-local because it reflects the charge density about a region of space (due to the derivative terms) instead of just the charge density at one point in space. The non-local exchange potential used in this thesis is the gradient-corrected exchange functional of Becke [116]. Becke ensured that the exchange energy E showed the correct asymptotic behaviour at large r by adding gradient x  corrections. However, it should be noted that the exchange potential which arises from the  Chapter 2. Theory  36  functional derivative of Becke's exchange energy does not necessarily have such correct behaviour at large r. The two non-local correlation potentials used in this thesis are those of Perdew and Wang [115,117,118] and Lee, Yang and Parr [119]. Similar to the work of Becke, Perdew's correlation potential [115,117,118] uses coefficients for the gradient expression that are derived from least square fits. Hybrid non-local functionals have recently been proposed which include a mixture of Hartree-Fock exchange with DFT exchange-correlation.  One hybrid method (B3LYP)  employed in this thesis is Becke's three parameter functional [120] using the correlation functional of Lee, Yang and Parr [119]. This is generally the most commonly used hybrid functional in the DFT literature. Another hybrid functional used in this thesis (B3PW91) also consists of Becke's three parameter functional [120], but the correlation functional of Lee, Yang and Parr [119] is replaced by that of Perdew and Wang [118]. Even after a choice is made for the exchange and correlation potentials, there are still some approximations that often arise in the computational evaluation of the potentials.  In  particular, the evaluation of the functional forms of the exchange and correlation potentials involves integrals with fractional powers of the electron density [121]. One density functional program used to obtain the results presented in this thesis (the deMon [122,123] program) uses numerical approximations to facilitate the evaluation of such integrals.  First, the deMon  program evaluates the exchange and correlation potential at a series of points about the molecule called a grid. A secondary basis set called an auxiliary basis set is then used to fit the calculated potential via a least squares fit. The deMon program auxiliary basis sets used consist of Gaussian primitives and are described by the (j,k;m,n) notation, where j and k are the number of s-type and sets of s-, p-, d-type GTOs for the charge density fit, and m and n are the sizes of similar basis sets for the fit of the exchange-correlation potential. Each set of s-, p-, and d-type GTOs in the auxiliary deMon basis set has a common exponent for all / values. The fit to the potential converts the exchange and correlation potential to a simple sum of Gaussian functions that can be integrated more easily. Because the calculation of the Coulomb potential  Chapter 2. Theory  37  can also be difficult, the charge density is also fitted with the set of auxiliary functions to hasten integral evaluation. However, it should be stressed that the fit quality and precision of calculated properties (e.g. total energy) are highly dependent upon the choices of the grid and auxiliary basis sets. As a general guide, larger auxiliary or orbital basis sets and finer grids will provide more accurate results [121]. It should be noted that for the other programs employed to obtain density functional results in this thesis (the Gaussian 92 [124] and Gaussian 94 [125] programs) the least squares fit to the calculated potential is simply replaced by calculating a much finer grid.  2 . 2 The Interpretation and Calculation of the EMS Cross-Section 2.2.1 The Binary Encounter and Impulse Approximations The (e, 2e) reaction has been described in Section 1.1.1 and a schematic of the symmetric non-coplanar (e, 2e) geometry most commonly used in E M S experiments is shown in Fig. 1.1. If conservation of energy is applied to the reaction, then the following equation results: E =E +E + £ 0  l  2  (2.42)  where e is the energy transferred to the target molecule. It should be noted that the ion recoil energy £  r e c o i l  (the kinetic energy transfer to the molecule) is extremely small and therefore it  has been neglected in Eq. (2.42).  The quantity 8 is the binding (ionization) energy  corresponding to formation of the final ion state / . However, the ion recoil momentum q cannot be neglected and it is related (by conservation of momentum) to the momenta of the electrons by: Q=Po-Pi~P2  ( 2- 43)  The binary encounter [12,126] and impulse approximations [12,127] are used to  simplify the interpretation of the (e, 2e) reaction. Under these conditions, the ionization event is well described by a two-body electron collision (i.e. the ion resulting from the collision can  Chapter 2. Theory  38  be considered to be a spectator to the (e, 2e) collision) and the incident electron can be said to knock out the target electron in an essentially clean manner. In order to provide the necessary conditions for the binary encounter approximation and impulse approximations to be valid, the momentum transfer K=p -p\  from the incident electron to the target M is maximized in  0  E M S by the choice of high impact energy E and the symmetric non-coplanar scattering 0  geometry [12,33] (Fig. 1.1). The value of E is chosen to be very large in comparison with the 0  ionization energy of the orbital being studied (E is typically 1200 eV + e and e is usually in 0  the range 5-60 eV). In addition, the energies of the outgoing electrons and their corresponding polar angles are selected to be equal (i.e. E = E = 600 eV and 0\ = 6 = 6 = 45° so that the x  2  2  kinematics are "symmetric"). The kinematics are referred to as non-coplanar because the outof-plane azimuthal angle <> / is allowed to vary (in most E M S experiments, ty is typically varied over the range of ±30°). It should also be noted that the symmetric non-coplanar kinematics is chosen because experimental results obtained with this geometry is essentially independent of kinematic factors (see Section 2.2.3). Within the binary encounter and impulse approximations, the ion recoil momentum is equal in magnitude and opposite in sign to the momentum p of the struck electron immediately prior to the collision (i.e. its momentum while still bound to the target M): p = -q -PI+P2-P0  In this type of collision (binary encounter, high momentum transfer, symmetric non-coplanar), the momentum p of the ejected electron prior to knock-out is related to the out-of-plane azimuthal angletyas follows [12,33]: p = [(2p cos0-p f + (2p s i n 0 s i n ( 0 / 2 ) ) ] 2  l  Q  x  1/2  (2.45)  where pj = yJ2E\ and p = «J2E in atomic units. It can be seen from Eq. (2.45) that two 0  0  general types of information can be measured for electrons by E M S . First, scanning the impact energy of the incoming electron (and hence p in Eq. (2.45)) allows the binding energy 0  Chapter 2. Theory  39  spectrum (BES) of the target molecule to be obtained. Second, scanning the azimuthal angle (j> over a range while keeping the impact energy at a constant value corresponding to the binding energy of interest allows a measurement of the E M S cross-section as a function of momentum ($ is easily transformed to p by use of Eq. (2.45)). The measured distribution of cross-section as a function of momentum is known as the experimental momentum profile (XMP).  2 . 2 . 2 The EMS Scattering Cross-Section The (e, 2e) cross-section,  <7  E M S  ,  is given in atomic units by [12]: |2  AP\PI  ;  E M S - •-(27U)  (2.46)  Po  where Tj- is called the scattering amplitude (i.e. the reaction amplitude for ionization to final ion state/) and is given by: T = (x -\Pi)X ~\P2)Vf~ \nE)\% X \po)) i  l  i  N  (2.47)  (+  f  where X^~\P\)^  X^~\Pi)  & X^ \Po)  an  +  represent the wavefunctions for the outgoing (-) and  incoming (+) electrons, T(E) is the transition operator for the (e, 2e) process, and ^J^  -1  4^  and  are the total wavefunctions for the final ion state and the initial target ground state, 2  respectively. The quantity Tj- in Eq. (2.46) is therefore the probability of a transition from the initial state prior to collision to the final ion state. Within the binary encounter and impulse approximations (see Section 2.2.1 above) the operator Tj depends only on the coordinates of the incident and target electrons so that Eq. (2.46) becomes [12]: cr  The  X  E M S  ave  =  (2K f  Jjfa -\ )x -\ ) \T(E)fN-l {  Po  {  Px  Pl  12  Vi )x \Po) N  {+  (2.48)  ave  in Eq. (2.48) represents a sum over final ion state degeneracies and an  average over initial state degeneracies (i.e. a sum over vibrational, rotational and electronic states). However, rotational and vibrational effects are too small to be observed experimentally  Chapter 2. Theory  40  because of the relatively broad instrumental energy resolution (see Chapter 3). Using the BornOppenheimer approximation, the ion and target wavefunctions  and ^  can be expressed  as the product of electronic, vibrational and rotational wavefunctions and the sums over final vibrational and rotational states can be eliminated from Eq. (2.46) via a closure relation [12]. In addition, the E M S reaction is typically done in the gas phase so that a spherical average given by (47i) \dQ. is included in the E M S cross-section to account for the random orientation _1  of the gaseous target molecules. Vibrational effects are commonly approximated by evaluating the cross-section at the molecular equilibrium geometry of the target molecule [12] (a study by Leung et al. [128] has shown this to be a reasonable approximation for calculating the valence shell momentum profiles for the water molecule). The average over initial electronic state effects is generally approximated by evaluation of the cross-section in the initial ground electronic state.  The sum over degenerate final electronic states taken into account by  assuming that all of the degenerate final ion states give the same  CJ MS E  ar  >d therefore the sum  can be replaced by a factor gj- equal to the degeneracy of the final ion state. Taking all of these considerations into account, the EMS-cross-section is given by:  OEMS =  where y / " and ^ 1  4  *  3  8f  ^  J  \x (P0X H  \T(E)\(v?- \vf) ]  (PI)  H  (^ ))| d£l  (+) X  (2.49)  2  0  are now the total ion and neutral electronic wavefunctions , respectively.  2.2.3 The Distorted Wave Impulse Approximation In the distorted wave impulse approximation (DWIA), the wavefunctions X (Pi)> (-)  % (p ) and % (Po) of the incident and outgoing electrons are represented by distorted waves (-)  (+)  2  and the E M S cross-section in Eq. (2.49) becomes [12]:  <ws \ l( x -  L  (  f  Po  }  (JH  )x ~ (  }  to)^rh )z< N  +)  dQ (i*)f  (2.50)  Chapter 2. Theory  where < T  Mott  41  is the Mott scattering cross-section for electron-electron collisions. A n explicit  form for the Mott scattering cross-section in the symmetric non-coplanar geometry is given in Ref. [12]. This cross-section has been shown to be effectively constant [129] for the symmetric non-coplanar geometry with the electron angles and valence binding energies commonly used in E M S . For example, at 6 = 45° and EQ = 1215.6 eV in the symmetric non-coplanar geometry, the variation of  <7  Mott  over the range of azimuthal angle <> j = ±30° is small (< 2%) and  effectively negligible [129]. In addition, although greater differences (-13%) result in f J  M o t t  if  the impact energy is varied by 50 eV [49], the impact energy dependence of the Mott scattering factor is usually neglected (such a choice has been made in this thesis). Therefore, the E M S cross-section can be considered to be proportional to the electronic structure factor given by the integral in Eq. (2.50). The main advantage of the DWIA cross-section is that the distorted waves in Eq. (2.50) allow for the perturbation (distortion) of the incoming and outgoing electron waves by the target and the ion. In E M S studies of the p and s type valence orbitals of atoms at EQ ~ 1200 eV, distortion effects have been found to be significant at higher momenta (p > 1.5 au) and the D W I A has been found to give an excellent description of the E M S cross-section for atoms up to p ~ 3 au [12,33]. In addition, in agreement with the physical argument that the electron waves should be influenced less by the target and ion as their energy increases, such distortion effects have been shown to decrease as the impact energy and outgoing electron energies are increased [12]. Although the DWIA model is physically reasonable, the calculation of the transition amplitude using this equation is often prohibitively difficult . Only spherical potentials are feasible in calculations and thus DWIA calculations have only been done for atomic targets of spherical symmetry such as the noble gases [36,130,131], the Cd atom [132] and the Cr, M o and T i atoms studied in this thesis (see Chapter 7). DWIA calculations have thus far been found to be intractable for molecules.  Chapter 2. Theory  42  2.2.4 T h e P l a n e W a v e I m p u l s e A p p r o x i m a t i o n Due to the limitations in calculating the distorted wave cross-section for molecules, the plane wave impulse approximation (PWIA) must be used to provide a description of the E M S cross-section.  In the PWIA, the incident and outgoing electrons are assumed to have  sufficiently high energies such that they may be represented by plane waves {.%=e ). ipr  Therefore, it is assumed that the electron waves are not distorted by the molecular potential. With the use of plane waves and the binary encounter approximation (and hence the impulse approximation), the E M S cross-section is [12]:  ' E M S -  4  7  : ^M„,,J|(p*rhi R  3  „  2  dQ.  (2.51)  Po  where p represents the plane-wave target electron with momentum given by Eq. (2.45). Similar to Eq. (2.50) for the DWIA, the Mott scattering cross-section in Eq. (2.51) is effectively constant for the electron angles and energies used in the present work. Thus, the cross-section observed in an E M S experiment is essentially directly proportional to |2  dQ,, the spherically averaged square of the overlap of the final ion and initial  target electronic wavefunctions.  It should also be noted that the ion-neutral overlap is in  momentum (p) space. In the PWIA model, the collision is viewed effectively as a binary encounter between two hard spheres (i.e. like the collision of two billiard balls). It is assumed that the ejected electron does not feel the potential due to the target or the effect of the incoming electron until it is knocked out. Similarly, the incoming electron does not feel the target potential or the effects of the other electron until the collision takes place. Collision events are also reasonably treated as instantaneous so that geometry changes within the neutral target molecule and the resulting ion are assumed to be negligible during the collision time. Although it may initially appear that these approximations are so restrictive and that they might limit the quality of the results from the PWIA model, extensive EMS studies over more than 20 years have shown [12]  Chapter 2. Theory  43  the PWIA model to be generally adequate for describing the shapes of orbital momentum profiles at values of p < 1.5 au. If the energies of the incoming and outgoing electrons are kept sufficiently high (> 1000 eV) so that the momentum transfer is large (this is the case for the symmetric non-coplanar geometry with EQ = 1200 eV + binding energy and E\= E = 600 eV 2  that is typically used in E M S studies and in this thesis), the PWIA has been shown to be an effective and reliable model for the interpretation of E M S cross-section [11,12,37-40]. However, it should be noted that the PWIA begins to break down for target electron momenta (ion recoil momenta) above approximately 1.5 au. Beyond this value of momenta, distortion effects tend to result in greater experimental cross-section than that predicted with the PWIA and therefore  momentum profiles calculated with the DWIA provide generally better  agreement with experiment in this region (although DWIA calculations are only feasible for atomic systems at this time). The interpretation of the E M S cross-section given by Eq. (2.51) relates to calculated wavefunctions from theory in the following manner.  In theoretical quantum chemistry, the  overlap of final ion and initial target electronic wavefunctions is often referred to as a Dyson orbital  [60,62].  Dyson orbitals may be obtained directly as solutions of Dyson's  quasiparticle equation, or indirectly by calculating the  and (N-l) electron wavefunctions  (e.g. by CI or many-body perturbation theory (MBPT)) and then computing the overlap. The Dyson orbitals are also normalized such that: (2.52)  where Sj- is the normalization factor [60,62]. Therefore, the E M S cross-section simplifies to: (2.53) The Sj factor in Eq. (2.53) is known as the spectroscopic factor or pole strength of the ionization and is given by:  Chapter 2. Theory  44  5/= £ 5 /  (2.54)  and the quantity sj is the square of the overlap between ^Py, the ion wavefunction, and the target wavefunction with a hole in orbital y/j: (2.55) Therefore, SJ is the probability of finding the (y ) y  1  one-hole configuration in the final ion  state / . For the case of a many-body wavefunction, removal of one electron from the system always results in more than one "hole" (i.e. a range of hole states) and thus the total probability SJ is the sum of all the individual sJ (Eq. (2.54)). Each hole left by an electron that has been removed is described as having been "split up" into poles, each of which has a different binding energy. It should also be noted that the SJ for ionization of the j  th  electron obey the  spectroscopic sum rule: 5>/  =l  (2.56)  /  where the sum is over all final states/. Therefore, for each electron ionized, the removal of one electron always results in the creation of one complete hole, even if the hole is fragmented. The Dyson orbital and eigenvalue with the largest spectroscopic factor for any hole associated with the removal of a certain electron is generally referred to as the main pole, while all others are usually referred to as "satellites" because of their lower spectroscopic factor (lower probability) and because they are often found close in energy to the parent peak. A theoretical momentum profile evaluated using the quantity j|Oy(/>) d£2 in Eq. (2.53) is often referred to as the ion-neutral overlap distribution (OVD).  As discussed  above, computation of the O V D with many-body calculations (such as the configuration interaction treatments described in Section 2.1.3) for the neutral and ion wavefunctions can be used to assess the effects of electron correlation and electronic relaxation effects in the measured cross-section [11,12,37-40].  However, it should be noted that instrumental  Chapter 2. Theory  45  momentum (angle) resolution effects (discussed below) must first be incorporated into the O V D before it can be compared to the experimental momentum profile.  2.2.5 The Target Hartree-Fock Approximation Calculating the full O V D in Eq. (2.53) is often highly computationally intensive. A simpler alternative to calculating the full O V D is to modify Eq. (2.53) using the target HartreeFock approximation (THFA).  The T H F A [12] assumes that electron correlation effects are  negligible in the initial state (i.e. neutral ground state molecule) and thus the many-body wavefunction ^  can be replaced by the single determinant Hartree-Fock wavefunction  of canonical H F molecular orbitals. Within this model, the many-body ion wavefunction ^FJ^  -1  is also represented as a linear combination of independent particle determinants of ground state target H F orbitals. In addition, it is assumed that, for a given ion state manifold/, only one of the Sj is appreciably different from zero. If the one significant SJ is the one given by j = j(f), then the expression for the E M S cross-section is given by: OEtAs^gfSfJlvjipfdQ  (2.57)  where y/j(p) is the momentum space canonical Hartree-Fock orbital j from which the electron is ionized (the Fourier transform of the more familiar one-electron position space orbital wavefunction  y/j(r)).  It should also be noted that the spectroscopic factor (pole strength of  ionization) has been reduced to SJ in Eq. (2.57), which is the probability of finding the (y/j)~  l  one-hole configuration in the final ion state/. The integral in Eq. (2.57) is known as the spherically averaged one-electron momentum distribution (MD) and thus, within the PWIA and T H F A approximations, electron momentum spectroscopy provides imaging of the canonical Hartree-Fock orbital electron density in momentum space.  Therefore, within the T H F A , the E M S cross-section is an initial state  property and is independent of the final state of the ion. Theoretical calculations of the E M S cross-section are thus simplified because only one calculation (of the electronic structure of the  Chapter 2. Theory  46  neutral molecule) need be done for comparison to experiment. In should also be noted that, within the T H F A as described by Eq. (2.57), all momentum profiles within the ionization manifold from the same target orbital j will have the same shape, regardless of the final state/. Therefore, the assignment and interpretation of satellite structure in the E M S binding energy is greatly simplified. The relative intensities of the ionization peaks within a manifold are given by the spectroscopic factors Sj. In addition, the spectroscopic sum rule in Eq. (2.56) implies that by summing all the observed intensity of each final ion state manifold, all the orbitals in a molecule can be normalized to a common relative intensity scale, with appropriate allowances for degeneracies. A further approximation is that, if final state correlation is negligible, then SJ will be unity for primary hole ion states and zero for all other states (this is equivalent to the frozen orbital or Koopmans approximation). This is often a reasonable model for outer valence orbital ionizations which have been found in previous EMS studies [12] to have pole strengths near unity (i.e. only a single ionization peak typically arises from each outer valence orbital). However, ionization from inner valence orbitals has been observed to often produce a multitude (or manifold) of "satellite" ionization peaks due to the substantial electron correlation effects in these processes. Therefore, inner valence X M P s from the same ionization manifold are more likely to differ in their individual shapes than those from outer valence orbitals. In addition, theoretical momentum profiles calculated within the T H F A may have more noticeable shape discrepancies with inner valence X M P s because of these final ion state correlation effects.  2.2.6 The Target Kohn-Sham Approximation Dyson orbitals obtained from configuration interaction calculations according to Eq. (2.53) above have been shown to provide highly accurate predictions of the E M S crosssection [11,12,37-40]. In particular, in those cases where distortion effects are negligible and correlation effects are necessary to model the experimental cross-section, the Dyson orbital within the PWIA has often provided the most accurate theoretical interpretation of the  Chapter 2. Theory  47  experimental momentum profile [11,12,37-40].  However, such CI calculations are often  demanding and scale at least as N , where N is the basis set size. Therefore, CI calculations for 5  E M S studies (via Eq. (2.53)) are often limited to small molecules. The target Hartree-Fock approximation (Section 2.2.5) is an alternative calculation that scales as ~ i V , although the 4  quality of the resulting momentum profile is generally lower than that obtained from CI. While the T H F A allows calculations for somewhat larger molecules than are possible with the CI overlap, T H F A calculations still rapidly become computationally difficult for many large molecules that are of interest in E M S . The desired results for larger molecules are instead much more feasible with Kohn-Sham DFT because such calculations scale approximately as - A . In addition, despite the lowered computational complexity, the properties predicted by 7 3  Kohn-Sham DFT are generally comparable to or better than Hartree-Fock [60,111,133,134] since some accounting of electron correlation effects in the target ground state is included in DFT via the exchange correlation potential (see Section 2.1.4). However, it should be noted that the extension of Kohn-Sham DFT to EMS studies was not practical until recently because the Kohn Sham orbitals were originally introduced [110] as purely mathematical constructs to simplify calculations instead of as physically meaningful quantities. The necessary connection between Kohn-Sham orbitals and the E M S cross-section was provided by Casida [135] and Duffy et al. [60], who reinterpreted Eq. (2.53) in the context of Kohn-Sham density functional theory and the target Kohn-Sham approximation (TKSA). It was proposed [60,135] that the true Kohn-Sham orbitals (i.e. those that would arise from the exact exchange-correlation functional) could be interpreted as approximations to Dyson orbitals. Therefore, the T K S A gives a result similar to Eq. (2.57) in which the canonical Hartree-Fock orbital is replaced by a momentum space Kohn-Sham orbital  y/f (p): S  (2.58)  In the justification for using Kohn-Sham orbitals as approximations to Dyson orbitals, Casida [135] showed that the Kohn-Sham equation  is the variationally best local  Chapter 2. Theory  approximation to Dyson's quasiparticle equation.  48 However, while the direct solution of  Dyson's equation can in principle yield an infinite number of solutions (Dyson orbitals), only N occupied Kohn-Sham orbitals result from solution of the KS equation for an N electron molecule. Therefore, there is a spectroscopic factor included in the PWIA model of the E M S cross-section (Eq. (2.53)), but no corresponding factor in the T K S A cross-section (Eq. (2.58)). Although the E M S cross-sections within the T H F A and T K S A have similar forms (Eqs. (2.57) and (2.58)), it should be stressed that they differ when electron correlation is considered.  The T H F A neglects initial target electron correlation effects while the T K S A  includes such effects within the chosen form of the exchange-correlation functional. Therefore, the quality of the comparisons of the T K S A momentum profiles to experiment is often highly dependent on the choice of exchange-correlation functional (i.e. local, non-local or hybrid) and the computational methods (i.e. auxiliary basis sets, grid size, fitting procedures) used to evaluate it. It should be noted that even if the "exact" exchange-correlation functional could somehow be known, the resulting exact Kohn-Sham orbitals would still be, within the T K S A , approximations  to Dyson orbitals. Thus, there are really two approximations included in  practical DFT calculations of momentum profiles. The exact KS orbitals are approximations to the full Dyson overlap, but DFT calculations also involve approximate functionals (and limited basis set size) so that one can never actually compute the exact KS orbitals. Although these considerations may appear to be extremely restricting, it has been shown in several E M S studies [45,53-55,60-62] that the use of approximate Kohn-Sham orbitals as approximations to Dyson orbitals provides reasonable (and often excellent) comparisons with the X M P s .  The  accuracy of such approximations has been evaluated by comparison of the T K S A momentum profiles with experiment and theoretical momentum profiles computed within the PWIA (Eq. (2.53)) and T H F A (Eq. (2.57)) models [45,53-55,60-62]. The T K S A momentum profiles have been shown to generally provide results comparable to or better than those from the T H F A and gives similar results to those from CI in cases where electron correlation effects are  Chapter 2. Theory  49  important (e.g. H 0 , N H and H F [60]). Such comparisons of the T K S A , PWIA and T H F A 2  3  momentum profiles will also be continued in this thesis. Despite the fact that Kohn-Sham orbitals are frequently good approximations to Dyson orbitals, it has often been noted that the Kohn-Sham orbital and total energies differ from those obtained from Dyson's equation [60,111,136].  Therefore, orbital and total energies from  Kohn-Sham DFT are not readily comparable to those from Hartree-Fock and post-Hartree-Fock (e.g. CI) methods. The shift in Kohn-Sham orbital energies relative to the values for the Dyson orbitals has been discussed in detail by Duffy et al. [60].  2.2.7 Fourier Transform The usual wavefunctions produced by the variational method are position space wavefunctions \|/(r). However, as given by Eqs. (2.57) and (2.58), the E M S experimental results are more directly related to momentum space wavefunctions \|/(p).  Although the  Schrodinger equation can in principle be solved directly in momentum space, the integrals involved are difficult to evaluate and therefore the usual approach is to solve the Schrodinger equation in position space and Fourier transform the resulting wavefunctions to momentum space according to the following equation: y/(p) = (2Kr je- y/(r)dr 3/2  ipr  (2.59)  For this purpose, a F O R T R A N program (HEMS) has been developed in this laboratory to perform the necessary transformation and evaluation of the spherically averaged momentum profiles. This program is based on equations reported by Kaijser and Smith Jr. [137]. The chosen basis functions are transformed into momentum space according to Eq. (2.59), the value 2 of y/j is computed (where either HF or KS orbitals are used) and then spherically averaged so that, according to Eqs. (2.57) and (2.58), it is related to the E M S experimental cross-section. However, it should be noted that experimental angular (i.e. momentum) resolution effects (described in the next section) must be incorporated before the calculations can be compared to  Chapter 2. Theory  50  the X M P . Computation of the O V D (Eq. (2.53)) also involves a Fourier transform to p space, followed by the incorporation of momentum resolution effects.  2.2.8 Resolution Folding Effects Like all spectrometers, the E M S spectrometer does not have perfect resolution, but rather has some finite resolutions for instrumental polar (6) and azimuthal ((j)) angles. This angular resolution is reflected in the shape of the XMPs.  Therefore, for meaningful  comparison of theory and experiment, the calculated DWIA cross-sections (Eq. (2.50)), PWIA overlap distributions (Eq. (2.53)) or HF and T K S A momentum distributions (Eqs. (2.57) and (2.58)) must be momentum resolution folded by taking into account the finite instrumental acceptance angles [58,59]. Because the Gaussian-weighted planar grid (GW-PG) method of Duffy et al. [59] has been shown to incorporate these angular resolution effects and to provide excellent agreement between high-level theory and experiment, it has been used for momentum resolution folding in the present work.  For the multichannel energy-dispersive E M S  spectrometer used to collect the data presented in this thesis, finite instrumental half angles of AG = ±0.6° and A 0 = ±1.2° have been used in the G W - P G calculations (thus, the F W H M momentum resolution is approximately 0.1 atomic units). After momentum resolution folding, the O V D (Eq. (2.53)) or the M D (Eqs. (2.57) and (2.58)) is referred to as a theoretical momentum profile (TMP) and is then suitable for comparison with the experimental momentum profile (XMP). The resolution folded DWIA cross-section in Eq. (2.50) may also be compared with experimental measurements in the case of atoms but it should be noted that this is a cross-section and not a momentum profile.  Chapter 3. Experimental  Method  51  Chapter 3  Experimental Method  3.1 The Multichannel Energy-Dispersive Spectrometer The spectrometer  construction used  and  in the  operation  present  of the  work has  multichannel energy-dispersive E M S been  described  in detail  in earlier  publications [45,46]. A schematic of the spectrometer is shown in Fig. 3.1. Briefly, a high energy (1200 eV + binding energy) electron beam (25-30 | i A ) is produced from a directly heated thoriated tungsten filament (F) and an oscilloscope electron gun containing a grid (G) and accelerating anode (A). The beam is aligned and focussed into the collision region by three sets of electrostatic quadrupole deflectors (D1-D3) and two Einzel lenses ( F l and F2). The electron beam alignment and collimation are monitored by the currents on four spray plates (P1-P4) and a Faraday cup (FC). In the present work, the current of a well-focussed incident beam (~ 1 mm in diameter) into the Faraday cup is about 25-30 \iA. The electron beam undergoes ionizing collisions with the target gas and produces two outgoing electrons which are sampled at the polar angles 9 (45°) by electron lenses and decelerated into the hemispherical electrostatic electron analyzers. One analyzer is fixed while the other is movable under computer control through a -35° to +35° range of azimuthal angles (> | (see Fig. 1.1) to vary the measured electron momentum (see Eq. (2.45)). The analyzer pass energy is 50 eV. A n 8 eV range of outgoing electron energies (i.e. 600 ± 4 eV) is sampled by a double chevron configuration microchannel plate (MCP) detector [138-140] located in the exit plane of each analyzer. Coincidence events are recorded from the many electrons striking  Chapter 3. Experimental  52  Method  ELECTRON GUN Fig. 3.1: Schematic of the energy-dispersive multichannel EMS spectrometer used in the present work. The two analyzers are kept at fixed polar angles 0 = 45° and the out-of-plane azimuthal angle < > | of the movable analyzer is variable. This figure is from Refs. [45,46]. F Filament F1.F2 electron lenses D1-D3 quadrupole deflectors G Grid P1-P4 spray plate apertures RA resistive anode A Anode . FC Faraday cup MCP microchannel plates  Chapter 3. Experimental  the detectors.  Method  53  The position (energy) information is determined using a Gear-type [141]  resistive anode (RA) encoder at the back of each M C P array and charge division circuitry. The timing information is determined using a time-to-amplitude converter (TAC).  It should be  noted that the details of the position and timing signal processing as well as the computer control have been described previously in great detail [45,46]. A n overall schematic of the signal processing electronics is shown in Fig. 3.2. Fig. 3.3 shows a typical time spectrum obtained from the T A C output with time resolution of 16 ns. The aluminum vacuum housing is evacuated by two Seiko-Seiki T M P 450 Maglev turbomolecular pumps (base pressure ~ 10" torr) backed by a rotary pump. The electron gun is 7  differentially pumped by a Seiko-Seiki T M P 300 Maglev turbomolecular pump. Ambient magnetic fields are reduced below 1 mGauss by hydrogen-annealed |l-metal shields placed inside the vacuum chamber. Gas samples and volatile liquid samples are introduced into the spectrometer using a gas line that connects to the interaction region.  The pressure within the spectrometer is  controlled by use of a variable leak valve (sample pressures of approximately 6 x l O typically employed for gaseous targets).  - 6  torr are  Solids of low vapour pressure can be directly  introduced through a vacuum lock using a solid sample probe [55]. Use of this sample probe was necessary to obtain sufficient gas density at room temperature for the measurements of W ( C O ) in this work. 6  Data collection is carried out in either the non-binning or binning modes [45]. Typical results are shown in Fig. 3.4 for A r 3p ionization. The non-binning coincidence binding energy spectrum is obtained at a fixed impact energy with triangular response functions in both the true coincidence window (CW, open circles) and background random coincidence window (BW, filled circles). In the non-binning mode the detection efficiency is different at each collected energy (i.e. higher in the top region of the triangle and lower near the two ends). Due to this non-uniform detection efficiency, the non-binning mode provides relatively rapid data  Chapter 3. Experimental  MCP  RA  54  Method  A  u  A  M+BM  SUM AMP ELECTRON GUN POWER SUPPLY SUM AMP  A MCP  F  +  B  DELAY AMP  STEPPINO MOTOR CONTROLLER  F  RA SUM AMP  PSDA  PRESET  LABMASTER CARD  PC  T IMING FILTER AMP RATEMETER  COINC SCA  TRIPLE COINC  GATE/ DELAY  BACK SCA  TRIPLE COINC  GATE/ DELAY  LINEAR GATE ROUTER MULTIPLEXER  1— LINEAR TIMING FILTER  GATE  CFD  AMP RATEMETER  F i g . 3.2: Schematic diagram of the position sensitive detectors and signal processing electronics for the EMS spectrometer. Each position sensitive detector consists of a pair of microchannel plates (MCP) and a resistive anode (RA). Also shown are: position sensitive detector analyzer (PSDA), constant fraction discriminator (CFD); time-to-amplitude converter (TAC), single-channel analyzer (SCA), analogue-to-digital converter computer card (ADC), and personal computer (PC). This figure is from Refs. [45,46].  Chapter 3. Experimental  Method  55  channel number 1400  0  0  64  I  — r —  I  0  1  50  128  192  1  1  II-  L _  100  150  256 1  1.  200  time (ns) F i g . 3.3: Time spectrum output by the TAC and collected by the ADC computer card. The time ranges of the true coincidence window (CW) and background window (BW) as set by the single channel analyzers (SCA) are indicated. It should be noted that the C W is set by the S C A to collect true coincidence + random coincidence events while the B W is set by the SCA to collect random coincidence events. The data in the figure were collected in 45 minutes from non-binning mode measurements of the A r 3p ionization process. This figure is from Ref. [46].  Chapter 3. Experimental  10  56  Method  15  20  25  10  15  20  25  Binding Energy (eV) F i g . 3.4: Non-binning mode (squares) and binning mode (circles) Ar 3p binding energy spectra. The nonbinning data shown are the sum of BES collected in a period of 17 hours at 27 out-of-plane azimuthal angles ranging from -30° to +30°. The binning data are the sum of 21 BES collected in 21 hours. The binning and non-binning mode raw data (CW designates the coincidence window while BW designates the background window) are shown in the upper-left and lower-left panels, respectively. The BW data are of greater intensity than the CW data because of the greater width of the background window (as shown in Fig. 3.3). This is accounted for in the background correction procedure (see Section 3.2). The binning mode and non-binning mode background corrected BES are shown in the right panel, normalized for height to allow direct shape comparison of the two spectra. The result of a least-squares fit of a pair of Gaussian functions to the ionization peak is indicated by the solid line. This figure is from Ref. [46].  Chapter 3. Experimental  Method  57  collection for situations where the ionization energy for a single orbital is well-separated in energy from those for the remaining orbitals (e.g. the low-lying HOMOs of NO, N 0 , 0 2  2  and  the metal hexacarbonyls studied in the present work - see Chapters 4, 5 and 7). However, the non-binning data collection mode is not well suited for the simultaneous study of several ionization processes at different binding energies because the relative intensities of each process are not easily determined with high accuracy from such a non-linear background. In contrast, the binning data collection mode (see Fig. 3.4 and Ref. [45]) is used to collect binding energy spectra with a flat overall response function (see the open and filled squares in the C W and B W windows in Fig. 3.4). In the binning mode, the total energy range of the detector and the binding energy range to be measured are divided into a number of narrow energy ranges called "energy bins" that are measured for equal times (the width of each energy bin is typically ~ 0.3 eV) as the impact energy is scanned. In this manner, all BES ionization peaks obtained at all (> angles have a common intensity scale and several ionization processes may be studied in the same experiment. Therefore, the binning mode has been used to obtain the measurements for the full valence shell regions of molecular oxygen and methanol presented in Chapters 5 and 8 of this work, respectively. The overall instrumental energy resolution is monitored by measuring the BES for a calibration gas in the binning mode. The He (Is)  -1  ionization peak at 24.6 eV or the A r 3p  ionization peak at 15.8 eV are typically chosen for this purpose. As demonstrated by the data in Fig. 3.4 for the A r 3p ionization process, an energy resolution of ~ 1.5 eV F W H M is typically obtained. It should also be noted that the value of the spectrometer energy resolution is required for further analysis of the experimental data (see Chapters 5 and 8). The Ar 3p ionization process is also used to monitor the angular resolution of the spectrometer. As can be seen from Fig. 3.5, the momentum profile for this ionization process is "p-type" (i.e. the maximum of the profile is at a value of p > 0).  If a spectrometer with perfect angular  resolution existed, measurement of the Ar 3p ionization process with such a spectrometer would give a p-type momentum profile with zero intensity at p = 0 au due to orbital symmetry  Chapter 3. Experimental  Method  58  F i g . 3.5: Non-binning mode (open circles) and binning mode (open squares) angle measurements (left) and momentum profiles (right) for Ar 3p ionization. The data sets were collected in 17 and 21 hours, respectively. The binning mode data has been scaled by a factor of 1.9 to normalize the two data sets at the maximum of the momentum profile. The solid line in the right panel is the calculated Ar 3p theoretical momentum profile from the CI ion-neutral overlap calculation of Davidson and co-workers [142]. The experimental angular resolution (A0 = 0.6°, A<> | = 1.2°) has been incorporated into the TMP using the GW-PG method [59] and the TMP has been normalized to the experimental data at the maximum of the momentum profile. This figure is from Ref. [46].  Chapter 3. Experimental  Method  59  considerations. However, as can be seen in Fig. 3.5, some experimental intensity is always observed at p = 0 au due to finite instrumental angular resolution effects (i.e. effects due to the finite acceptance angles about the nominal values of 0 and (j) for each analyzer). The angular resolution parameters for the spectrometer have been determined to be A0 = 0.6° and A<j) = 1.2° by comparison of the A r 3p experimental momentum profile with a high quality CI calculated momentum profile [142] that incorporates the G W - P G resolution folding method [59]. As can be seen in Fig. 3.5, excellent agreement is obtained between theory and experiment using these parameters.  To ensure that the spectrometer is operating properly, results similar to those  shown in Fig. 3.5 should be obtained each time the instrumental angular resolution is monitored. It should be noted that the above tests of the energy and angular resolution of the spectrometer are generally performed before and after each experiment or following any adjustments to the spectrometer.  3.2 Background (Random Coincidence) Correction A coincidence event results in two electrons arriving at the detectors at the same time (i.e. the two electrons are "correlated" with respect to time). However, it is possible that two electrons that do not come from the same ionization event can strike the detectors at the same time. These "uncorrelated" electrons give a random coincidence event that must be taken into account. The random coincidence background (see Fig. 3.3) has a constant amplitude because all time differences for uncorrelated electron pairs have equal probability. Therefore, the total coincidence peak in Fig. 3.3 has two components: those from actual (e, 2e) coincidence events ("true" coincidences) in the peak and the constant underlying background resulting from random uncorrelated electron pairs ("random" coincidences). The random coincidences must be removed from the total coincidences in the peak to determine the number of true coincidence events. The amplitude of the background is determined by use of coincidence and background windows (CW and B W in Fig. 3.3) that encompass the coincidence peak and a large background region, respectively. The background window is typically a factor r larger in  Chapter 3. Experimental  Method  60  width than the coincidence window in order to obtain the random coincidence rate with greater precision. If the number of total coincidence counts in the C W window is A^ and the number of non-coincident counts in the B W window is N^, then the number of true coincidence counts N is obtained as follows: t  N = N -N /r t  c  b  (3.1)  The error in the number of true coincidence counts is: AN =^N +^2  t  (3.2)  Chapter 4. The HOMOs of NO and  N0  61  2  Chapter 4  The HOMOs of NO and N 0  2  4.1 Introduction The technique of E M S has been extensively applied to closed-shell molecules [11,12,37-40].  In contrast,  open-shell molecules have received much less  attention  experimentally because of their reactive nature and theoretically because of the more challenging nature of quantum mechanical calculations for such species.  In view of the  challenging experimental and theoretical problems associated with open-shell molecules, a detailed experimental and theoretical investigation of the electron momentum profiles for the outermost (HOMO) orbitals of NO and N 0 has been carried out in this laboratory. These are 2  benchmark molecules for quantum chemistry and are among the most common stable openshell species. These molecules are also of importance in processes such as nitrogen fixation [143], neurotransmission [144], smog and acid rain [143]. Previously reported E M S studies of the HOMOs of N O [145-147] show limited statistics and considerable scatter due to the low cross-sections involved for ionization from these partially filled orbitals. The Hartree-Fock wavefunctions employed in the earlier work on N O by Brion et al. [145] involved only minimal and double-zeta basis sets and more advanced calculations were not considered. Furthermore, the instrumental momentum resolution was not always taken into account in previous work when comparing the experimental and theoretical momentum profiles. The adequate incorporation of finite angular acceptances (i.e. momentum resolution effects) has been found to be essential for the meaningful comparison of experiment  Chapter 4. The HOMOs of NO and  N0  62  2  and theory, particularly at lower momentum [39,58,59].  The experimental and theoretical  study of N O by Fantoni et al. [146] included the use of the same Kouba and Ohm unrestricted Hartree-Fock (UHF) wavefunction [148] presented in the work of Brion et al. [145].  In  contrast, the theoretical treatment presented in the work of Tossell et al. [147] was limited in that calculations were presented only with small basis sets at the restricted open-shell HartreeFock (ROHF) level which may not provide an adequate computational model for open-shell molecules (see Section 2.1.2 for descriptions of the ROHF and U H F methods). In order to obtain improved E M S measurements for the H O M O of N O , the energydispersive multichannel EMS spectrometer [45] described in Chapter 3 was used in the present work.  A l l multichannel measurements in the present work were obtained using the non-  binning mode [45] (see Section 3.1). The differential detection efficiency of the non-binning mode (higher in the middle of the triangle and lower near the two ends) was advantageous for measurements on the single, well-separated binding energy peak of the H O M O of NO. The measurements for N 0 were carried out with an earlier single channel instrument at a binding 2  energy of 11.2 eV and the present results are the first to be reported for this molecule. The N O and N 0  2  samples were obtained from Matheson gas products.  A l l gas samples were of  > 99.0% purity. Some additional considerations regarding the handling of the N O gas sample are discussed in the next section. Much larger basis sets and more complex theoretical methods have been used in the present work for N O to calculate theoretical momentum profiles which also incorporate the improved resolution folding procedures [59]. Theoretical momentum profiles at the target Hartree-Fock approximation (THFA) level from both the U H F and ROHF methods are shown for both molecules. reference  Momentum profiles from post Hartree-Fock methods such as multi-  singles and doubles configuration interaction (MRSD-CI) methods,  averaged  coupled-pair functional (ACPF) [149] and quasi-degenerate variational perturbation theory (QDVPT) [150,151] are also presented in order to assess the role of electron correlation and relaxation effects.  In addition, no published work to date has compared E M S momentum  Chapter 4. The HOMOs of NO and  N0  63  2  profiles for open-shell molecules with momentum profiles from DFT. Thus, T K S A - D F T calculations using the local functional of Vosko, Wilk and Nusair [112], the non-local correlation functional of Perdew and Wang [115,117] and a non-local combination of the Perdew correlation and Becke exchange functional [116] are shown for the HOMOs of both molecules in the present work.  4.2  G a s Samples and Sample Handling for N O Experience in the present work has shown that great care is needed in handling and  introducing N O gas samples into the spectrometer.  In the initial phases of the present  experimental study of the H O M O X M P of N O using a single channel E M S spectrometer, surprising and unexpected results were obtained as shown by the open triangles in Fig. 4.1a. Previous experimental measurements [145-147] of the momentum profile of the 2n H O M O of NO display considerable scatter and uncertainties due to the very low E M S cross-section arising from the single occupancy of the 2rc orbital and the broad nature of the momentum profile. This can be seen in Fig. 4.1b where the previous results from Brion et al. [145] (filled squares), Fantoni et al. [146] (open inverted triangles), and Tossell et al. [147] (filled diamonds) are shown. The data from Fantoni et al. [146] is that collected at 800 eV, although some data were also presented at 2600 eV. Within the limited statistics the three experimental results appear to be in reasonable agreement with each other and with the theoretical momentum profile (TMP) from the Kouba and Ohrn (K&O-U) calculation [148].  In the  original publications, the K & O - U T M P was thought by Brion et al. [145] and also by Fantoni et al. [146] to agree well for shape with the H O M O momentum profiles as shown in Fig. 4.1b. It should be noted, however, that while it was reported [145] that the K & O - U wavefunction was a CI calculation, it has since been found that the momentum profile reported in Ref. [145] and presumably also that in Ref. [146] was calculated from the leading configuration of the CI wavefunction  only and thus was in fact  a Hartree-Fock  and not  a CI treatment  Chapter 4. The HOMOs of NO and  N0  64  2  NO  (a)  2TT  9 . 5 4 eV A SC ( N 0 / N 0 ~ i r n purity) o MC ( N O / N 0 impurity) MC (NO) - K & O - U (x88%) - 168-ACPF NO 2 0 7 - Q D V P T 2  2  N0  2  impurity  if) C CD  CD >  Brion et a l . Fantoni et al 8 0 0 eV Tossell et a l .  CD  K&O-U  0  0  1  2  M o m e n t u m (au) Fig. 4.1: (a) Experimental momentum profiles for the 2TC orbital of NO from the present work. Shown are the XMPs from a single channel EMS spectrometer (open triangles), a multichannel momentum dispersive multichannel EMS spectrometer (open circles) and a multichannel energy-dispersive EMS spectrometer (filled circles). The single channel and momentum dispersive results contain some degree of contamination from N 0 (see Section 4.2 for details). The estimated contribution due to N 0 impurity is represented by the N 0 207-QDVPT momentum profile (long dashed line) multiplied by 0.04. Also shown are the calculated theoretical spherically averaged K&O-U momentum profile [145] (multiplied by a factor of 0.88) and the 168-ACPF momentum profile for NO. (b) Previous experimental momentum profiles for the 2TC orbital of NO. Shown are the XMPs from Brion et al. [145] (filled squares), Fantoni et al. [146] (open inverted triangles) and Tossell et al. [147] (filled diamonds). The calculated spherically averaged K&O-U momentum profile is also shown. See text for further details. 2  2  2  Chapter 4. The HOMOs of NO and  N0  65  2  The K & O - U momentum profile shown in Fig. 4.1a is also from the leading configuration only. It should also be noted that the previous measurements [145,146] show very little data in the low momentum region below p ~ 0.2 au and that nearly all data points below 0.5 au are higher than predicted by the K & O - U theory. The measurements of Tossell et al. [147] which are reasonably consistent with the other experiments [145,146] were in poor agreement with calculations based on the Hartree-Fock wavefunction of Cade and Wahl [152]. A consideration of Figs. 4.1a and 4.1b shows that in the region above ~ 0.3 au the present single channel measurements (open triangles) are generally consistent with the three earlier published experiments [145-147].  However, below ~ 0.3 au the data points  unexpectedly rise as zero momentum is approached.  This is in marked contrast to all  calculations (see Fig. 4.1a and also Fig. 4.2 below) which, as expected from symmetry considerations, show a decreasing cross-section as zero momentum is approached. The high data points at low momentum persisted in repeated measurements, not withstanding the poor statistics inherent in the case of the single channel spectrometer. A further measurement in our laboratory (open circles in Fig. 4.1a) using the high precision and improved statistics of a recently built momentum dispersive multichannel E M S spectrometer [47] produced similar spuriously high results at low momentum.  It is noteworthy that above ~ 0.2 au these  measurements (like the earlier data [145-147] in Fig. 4.1b) are in close conformity with the less accurate K & O - U calculation but not with the very accurate and highly correlated 168-ACPF ion-neutral overlap calculation (dotted line in Fig. 4.1a) described in Section 4.3.2 below. The anomalous behaviour in the experimental results which is manifested particularly at lower momenta has been found to be due to N 0 impurities in the sample gas introduced to the 2  collision chamber. Consideration of the respective Franck-Condon widths and the fact that the H O M O ionization potential of N 0 is close to that of NO indicate that any small N 0 impurity 2  2  would cause the observed spurious effects particularly at the lowest momenta because of the very intense s-type N 0 H O M O momentum profile (see Fig. 4.3 below). Careful consideration 2  of the purity of the commercial gas cylinders and the results of other spectroscopic experiments  Chapter 4. The HOMOs of NO and  N0  66  2  [153] indicated that the N 0 impurity observed in the E M S measurements was not present in 2  the N O cylinders. Clearly, the N 0 impurity was being generated in the stainless steel sample 2  handling and introduction system of the EMS spectrometers. Since, as is well-known, N O is readily oxidized to N 0 by atmospheric oxygen an obvious possibility would be a leak in the 2  sample system.  However, careful leak testing eliminated this as the cause.  A further  possibility would be reaction of NO with oxygen containing species (e.g. H 0 , 0 , previously 2  2  studied molecules, etc.) adsorbed on the walls of the inlet manifold, valves and leak valve. Initial measurements for N O on the energy-dispersive multichannel E M S instrument [45] also showed a similar "turning-up" at low momentum. However, after repeated flushing to remove suspected N 0  2  impurities, overnight stagnation (to allow reaction) and evacuation, very  different results, as indicated by the solid circles in Fig. 4.1a, were eventually obtained. These procedures were repeated daily during the measurements.  It can be seen from this high  precision data that the low momentum "turning-up" has essentially gone and also that the X M P is shifted over to higher momentum below 1.0 au Furthermore these measurements are in generally very good agreement with the shape of the high-level 168-ACPF overlap calculation for N O (see Section 4.4.1) and in poor agreement with the shape of the K & O - U treatment. Note that the two calculations are separately height normalized to the experimental data sets in Fig. 4.1a. It can be concluded that the gas sample for the solid circle data points in Fig. 4.1a is dominantly NO. However, some minimal pollution from residual N 0 cannot be discounted 2  and this may be the reason for the remaining small discrepancy between the 168-ACPF theory and experiment at the lowest momenta. The estimated contribution from the (s-type - see Fig. 4.3) N 0 impurity in the earlier open circle and open triangle data sets is shown by the 2  shaded area under the long-dashed line in Fig. 4.1a. This long dashed line is the 207-QDVPT calculation for the H O M O of N 0 (see Section 4.3.2 for a description of this calculation) 2  scaled appropriately. Based on these observations and a consideration of the presently reported very high level calculations, we suggest that the earlier published measurements for N O [145-147] were also complicated by varying degrees of N 0 impurity. It should be noted 2  Chapter 4. The HOMOs of NO and  N0  67  2  (Fig. 4.1b) that these earlier measurements [145-147] all show higher cross-sections at lower momenta than even the K & O - U calculation. The present work indicates that the apparently good agreement of earlier measurements with the K & O - U calculation is fortuitous due to the probable presence of N 0 impurities. 2  4.3 Calculations 4.3.1 Hartree-Fock Calculations Spherically averaged theoretical momentum profiles have been calculated for the HOMOs of N O and N 0 using several basis sets of varying quality within the plane wave 2  impulse and the target Hartree-Fock approximations (via Eq. (2.57)).  Both the U H F and  ROHF methods have been used for all basis sets. The instrumental angular (momentum) resolution was included in the calculations using the Gaussian-weighted planar grid method [59] (see Section 2.2.8). Various other calculated and experimental electronic properties are listed in Tables 4.1 and 4.2. The experimentally derived geometries for N O [154] and N 0  2  [155] were used in all calculations. The dipole moment of N O is of special interest in quantum mechanical calculations because the predicted dipole moment reversal [156-158] gives a N~0+ polarity which is counter to that from intuitive electronegativity arguments. A n experimental measurement of the dipole moment [159,160] gave a value of 0.15872 Debye with an N " 0 polarity as reported by Gray et +  al. [161].  While the sign of the dipole moment for N O is often incorrectly predicted by  Hartree-Fock calculations, correlated treatments generally give the correct polarity [157]. Thus, a positive dipole moment for N O in Table 4.1 implies a polarity of N ~ 0 . +  The  experimental dipole moment for N O is from Ref. [160]. When considering the properties in Tables 4.1 and 4.2, it should be noted (see Section 2.1.2) that the U H F methods and ROHF methods give different values for the total energy and spin contamination. The increased variability of the U H F method causes the U H F  Chapter 4. The HOMOs of NO and N0  68  2  Table 4.1: Calculated and experimental properties for NO X H 2  Basis Set and Calculation Method  0  Total Energy (hartree)  (?)  (Debye)^  Pmax (au) c  lu lr 2u 2r 3u 3r 4u 4r 6u 6r  Hartree-Fock Calculations STO-3G-U STO-3G-R 6-311+G*-U 6-311+G*-R AUG5-U AUG5-R 168-U 168-R 210-U 210-R  -127.530 -127.526 -129.284 -129.277 -129.304 -129.296 -129.309 -129.299 -129.307 -129.297  0.9660 0.7500 0.8104 0.7500 0.8024 0.7500 0.7984 0.7500 0.7981 0.7500  +0.2492 +0.1559 -0.3019 -0.3617 -0.2391 -0.2873 -0.2440 -0.2753 -0.2397 -0.2689  1.27 1.20 0.94 0.89 0.91 0.85 0.91 0.86 0.91 0.89  4c 4a 6c 6a  Post-Hartree-Fock Calculations 168-CI 168-ACPF 210-CI 210-ACPF  -129.713 -129.730 -129.712 -129.728  0.7500 0.7500 0.7500 0.7500  +0.1388 +0.1779 +0.1432 +0.1805  0.86 0.85 0.88 0.86  3/ 3p 3b  DFT Calculations^ AUG5-L AUG5-P AUG5-BP  -128.981 -130.088 -129.967  +0.2536 +0.2322 +0.2316  0.77 0.76 0.79  Experimental  -129.900  +0.1587/  0.85S  — — — e  Calculations performed at the experimental equilibrium bond length of 1.151 A (2.175 au), Ref. [154]. A positive dipole moment u. implies a polarity of N"0 . Calculated dipole moments are for a non-relativistic, non-vibrating, non-rotating molecule. The p corresponds to the value of momentum where the intensity of the momentum profile is at a maximum (see Fig. 4.2). ^ The total energy from Kohn-Sham DFT has been suggested to be in error because of shifted orbital energies relative to ionization potentials from Dyson's equation [60] - see also Section 2.2.6. The "experimental" total energy is the estimated non-relativistic, non-vibrating, infinite nuclear mass total energy. Total energy obtained by adding the atomic energies from Ref. [162] to the D values from Ref. [159] plus the zero point energy from the frequencies in Ref. [159]. A numerical Hartree-Fock calculation with the ROHF method was also reported to give a total energy of -129.2991 hartree for NO [163]. / Refs. [159-161]. S Present work, + 0.05. a  b  +  c  m a x  e  0  Chapter 4. The HOMOs of NO and N0  69  2  Table 4.2: Calculated and experimental properties for N O 2 X A i 2  Basis Set and Calculation Method  Total Energy (hartree)  (?)  lu lr 2u 2r 3u 3r 5u 5r  Hartree-Fock Calculations STO-3G-U STO-3G-R 6-311+G*-U 6-311+G*-R AUG5-U AUG5-R 207-U 207-R  -201.273 -201.268 -204.087 -204.078 -204.121 -204.112 -204.129 -204.120  0.7619 0.7500 0.7702 0.7500 0.7712 0.7500 0.7713 0.7500  +0.2435 +0.2758 +0.7187 +0.8073 +0.6610 +0.7446 +0.6584 +0.7629  5c 5a 5q  Post-Hartree-Fock Calculations 207-CI 207-ACPF 207-QDVPT  -204.752 -204.795 -204.806  0.7500 0.7500 0.7500  +0.4373 +0.4740 +0.4857  3/ 3p 3b  DFT Calculations'^ AUG5-L AUG5-P AUG5-BP  -203.710 -205.400 -205.217  — — —  +0.2689 +0.2668 +0.2463  Experimental  -205.085^  0  (Debye)^  +0.289  e  a  Calculations performed at the experimental equilibrium bond length of 1.19455 A (2.25737 au) and bond angle of 133.851°, Ref. [155].  b  A positive dipole moment implies a polarity of N 0 . Calculated dipole moments are for a non-relativistic, non+  2  vibrating, non-rotating molecule. The total energy from Kohn-Sham DFT has been suggested to be in error because of shifted orbital energies relative to ionization potentials from Dyson's equation [60]. See also Section 2.2.6. ^ The "experimental" total energy is the estimated non-relativistic, non-vibrating, infinite nuclear mass total energy. Total energy obtained by adding the atomic energies from Ref. [162] to the D values from Ref. [164] plus the zero point energy from the frequencies in Ref. [164], Ref. [165]. c  0  e  Chapter 4. The HOMOs of NO and  N0  70  2  total energy for a given molecule and basis set to be generally lower than the R O H F total energy.  However, spin contamination can be problematic for U H F wavefunctions  [57,97,98,108] and thus U H F wavefunctions are not true eigenfunctions of the total spin operator S . 2  A measure of the spin contamination is found by calculation of the expectation  value of S .  The value of (s ^ is 0.75 for N O and N 0 when no spin contamination is present  2  2  2  and it can be seen that for all the U H F calculations in Tables 4.1 and 4.2 this value is exceeded due to interference from states of higher multiplicity. Spin contamination does not arise in the ROHF method and thus all ROHF calculations give ( § ) values of exactly 0.75 for N O and 2.0 2  for N 0 . A large amount of spin-contamination in any wavefunction may cause inaccuracy in 2  computed properties. There is a high degree of spin contamination for the STO-3G-U and 6-311+G*-U calculations for N O (Table 4.1).  However, the spin contamination for U H F  calculations on N 0 is reasonably small (Table 4.2). 2  Further details of the Hartree-Fock calculation methods and basis sets are described below. The total number of contracted Gaussian-type orbital functions (CGTO) used is also given. A l l calculations with the STO-3G, 6-311+G* and A U G 5 basis sets (described below) were done at the University of British Columbia with the Gaussian 92 program [124] while the calculations with the 168-GTO, 210-GTO and 207-GTO basis sets were done by Prof. E.R. Davidson at Indiana University with the M E L D program. Those calculations employing the U H F method have the " - U " extension added to the basis set symbol while the "-R" extension indicates the R O H F method. The ROHF calculations done at Indiana University involve both symmetry and equivalence restriction (i.e. all molecular orbitals are pure a and TZ orbitals). This symmetry and equivalence restriction affects the results for N O , making the 7i and n x  y  orbitals equivalent partner functions in the n irreducible representation of the C^y point group. Symmetry or equivalence restrictions were not done with the U H F calculations from the University of British Columbia or those from Indiana University.  Chapter 4. The HOMOs of NO and N0  71  2  (lu) STO-3G-U and (lr) STO-3G-R: These calculations employed a minimal basis set (effectively single zeta). Each function is a contraction of three primitive Gaussian functions. Both the nitrogen and oxygen atoms have a (6s,3p)/[2s,lp] contraction and thus 10 C G T O are used for N O while 15 CGTO are employed for N 0 . This basis was designed by Pople and co2  workers [101]. (2u) 6-311+G*-U and (2r) 6-311+G*-R: These calculations used an augmented version of the 6-311G basis of Pople and co-workers [103]. The 6-311+G* basis is formed by augmenting the 6-311G with diffuse s- and p-functions [105] and spherical d-type polarization functions [104] on both the nitrogen and oxygen atoms to produce a (12s,6p,ld)/[5s,4p,ld] contraction per atom. A total of 44 CGTO are employed for N O while 66 CGTO are used for N 0 . 2  (3u) AUG5-U and (3r) AUG5-R: The basis set for these calculations was taken from the work of Dunning et al. [166-169].  The AUG5 basis set used in the present work is actually a  truncated form of Dunning's aug-cc-pV5Z basis set in which all f-, g- and h-functions have been removed.  This truncation was adopted to provide compatibility with the density  functional calculations since the DFT program (deMon) cannot handle the higher / functions in its present form. Thus, the AUG5 consists of a (33s,13p,5d)/[7s,6p,5d] contraction per atom. In addition, the d functions have been changed from spherical to Cartesian so that the HartreeFock results from this basis set can be directly compared with the DFT results from this basis. Thus, a total of 110 CGTO are used for N O while 165 CGTO are used for N 0 . 2  (4u) 168-U and (4r) 168-R: The 168 CGTO basis set for N O has a contraction of (19s,14p,3d,2f,lg)/[7s,8p,3d,2f,lg] per atom. The primary (19s, 14p) functions are taken from Partridge [170], while the (3d,2f,lg) polarization functions are from Dunning [166]. The first fourteen s functions were contracted into two s functions using the first fourteen Is and 2s atomic orbital coefficients.  Similarly, the first seven p functions are contracted into one p  function using the first seven 2p atomic orbital coefficients. This contracted basis set is further augmented by additional diffuse s and p functions from Partridge's supplementary functions for  Chapter 4. The HOMOs of NO and  CT( P) and N~( P) [170]. 2  3  N0  72  2  A l l components of the d, f and g functions are kept in the  calculations. (5u) 207-U and (5r) 207-R: The 207 CGTO basis set for N 0 is obtained by removing all the 2  g functions from the 168-CGTO basis set, thus giving a  (19s,14p,3d,2f)/[7s,8p,3d,2f]  contraction per atom. (6u) 210-U and (6r) 210-R: This 210 CGTO basis set for N O is the augmented correlationconsistent,  polarized  valence  and co-workers [166-169].  quadruple-zeta  basis  set  The basis set consists of a  (aug-cc-pVQZ) of Dunning (13s,7p,4d,3f,2g)/[6s,5p,4d,3f,2g]  contraction per atom.  4.3.2 M R S D - C I , A C P F a n d Q D V P T C a l c u l a t i o n s (4c) 168-CI and (4a) 168-ACPF: The near Hartree-Fock limit 168-R calculation (described above in Section 4.3.1) was chosen as the initial reference calculation. The frozen-core, multireference singles and doubles excitations configuration interaction (MRSD-CI) and average coupled-pair functional (ACPF) [149] calculations were then performed (by Prof. E.R. Davidson at Indiana University with the M E L D program) on both the neutral molecules and cation radicals with the neutral symmetry restricted ROHF K-orbitals [171-173]. The full ionneutral overlap distributions were then calculated in the plane wave impulse approximation of the E M S cross-section via Eq. (2.53) to investigate the role of electron correlation and relaxation effects on the TMPs.  The instrumental angular (momentum) resolution was  included in the TMPs using the Gaussian-weighted planar grid method [59]. Various other electronic properties from these calculations are listed in Tables 4.1 and 4.2.  The  experimentally derived geometry for N O [154] was used in all calculations. (5c) 207-CI, (5a) 207-ACPF and (5q) 207-QDVPT: The Hartree-Fock 207-R calculation (described above in Section 4.3.1) was chosen as the initial reference calculation. M R S D - C I and A C P F calculations were performed (by Prof. E.R. Davidson at Indiana University) with this basis set for N 0 at the experimental geometry [155] according to the same procedure 2  Chapter 4. The HOMOs of NO and  N0  2  73  described above for calculations 4c and 4a for NO. In addition, frozen-core, quasi-degenerate variational perturbation theory (QDVPT) calculations were also performed for the N 0  2  molecule with this basis set. (6c) 210-CI and (6a) 210-ACPF: The Hartree-Fock 210-R calculation (described above in Section 4.3.1) was chosen as the initial reference calculation.  M R S D - C I and A C P F  calculations were performed (by Prof. E.R. Davidson at Indiana University) with this basis set for N O at the experimental geometry [154] according to the same procedure described above for calculations 4c and 4a.  4.3.3 D F T C a l c u l a t i o n s (30 AUG5-L, (3p) AUG5-P and (3b) AUG5-BP: Three density functional calculations were carried out at U B C using the deMon program [122,123] and the large A U G 5 orbital basis set (described above in Section 4.3.1 for calculations 3u and 3r) at the experimental geometries for N O [154] and N 0  2  [155]. A l l calculations used a random extra-fine grid and the energy  convergence was set at 10 hartree. The auxiliary basis set for fitting the charge density and -7  exchange-correlation potential was the N/O (5,4;5,4) from the deMon program [122,123]. The A U G 5 - L calculation employs the local density approximation of Vosko et al. [112] while the AUG5-P calculation uses the non-local correlation functional of Perdew and Wang [115,117]. The AUG5-BP calculation uses a non-local combination of the correlation functional of Perdew and Wang [115,117] and the exchange functional of Becke [116]. The Kohn-Sham orbitals have been extracted from the result and the momentum profiles have been calculated using the T K S A via Eq. (2.58). The instrumental angular resolution effects [59] were also incorporated into the TMPs. Various other electronic properties from the DFT calculations are listed in Tables 4.1 and 4.2. However, as noted in Section 2.2.6, DFT total energies differ from the exact result [60] and thus the DFT total energies in Tables 4.1 and 4.2 are not readily comparable to those from Hartree-Fock methods, CI, A C P F and QDVPT. In addition, while the DFT calculations reported in the present work result in separate a and P molecular orbitals  Chapter 4. The HOMOs of NO and N0  74  2  and thus contain some degree of spin contamination, the (s ) 2  values are not tabulated in  Tables 4.1 and 4.2 because the deMon program does not compute this expectation value (see Section 4.3.1 above for a discussion on spin contamination).  4.4 Results and Discussion 4.4.1  Comparison of Experimental and Theoretical Momentum Profiles High momentum resolution measurements have been made of the outermost valence  electron experimental momentum profiles (XMPs) for N O (2K, Fig. 4.2) and N 0 Fig. 4.3).  2  (6a  The X M P for N O was collected on the energy-dispersive multichannel E M S  spectrometer (see Chapter 3) while the X M P for N 0 was collected with a single channel 2  instrument.  On each figure the mean binding energy at which each particular X M P was  measured is noted (9.54 eV for N O and 11.23 eV for N 0 ) . These values correspond to the 2  vertical ionization potentials of the lowest lying peaks observed in the photoelectron spectra [174]. The outermost X M P for each molecule is well separated in energy from the rest of the valence ionization manifold, which ensures that all the observed intensity of the momentum profile is due to ionization to the lowest lying ion state only and contains no mixing with other ionization processes. While N 0 is known to exist in equilibrium with its dimer N 0 4 in the 2  2  gas phase, photoelectron spectroscopy [175] has shown that the dimer concentration is negligible at low pressures (0.01 torr) and room temperature. Furthermore, the first PES peak from the dimer has been recorded at low temperatures and does not overlap extensively with the band from the H O M O of the monomer [175]. Since the present E M S experiments were done at even lower pressures (~ 10" torr) and the dimer peak at 12.39 eV was not observed in 4  the E M S binding energy spectra, the present results contain no detectable contributions from N 0 . 2  4  Much better statistics have been obtained in the present work on N O compared to previous results [145-147].  In addition, the comparison of theory and experiment was  1?  Chapter 4. The HOMOs of NO and N0  75  2  4  r-  00 c  CD  2 h CD >  _o CD  o  o  omentum (au) MOMENTUM DENSITY  POSITION DENSITY NO  1 -8  0  (au) Fig. 4.2: Measured and calculated spherically averaged Hartree-Fock and post-Hartree-Fock momentum profiles for the 2n orbital of NO (upper panel). The lower panels show the momentum and position density contour maps for an oriented NO molecule calculated at the Hartree-Fock level using the 168-R basis set. The contours represent 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density.  Chapter 4. The HOMOs of NO and  76  N0  2  N0 6a 1 1 .23 eV  5a,5q  2  t± if)  o  2  c CD _j  Expt  j  5q 5a 5c 5r 5u 3r 3u 2u 1 u  CD  >  1  1  IE!  2 0 7 - Q D V P T 2 0 7 - A C P F 2 0 7 - C I 2 0 7 - R 2 0 7 - U A U G 5 - R A U G 5 - U 6 - 3 1 1 + G * -U S T 0 - 3 G - U  CD  i 0 0  1  omentum MOMENTUM DENSITY  2  (au) POSITION DENSITY  F i g . 4.3: Measured and calculated spherically averaged Hartree-Fock and post-Hartree-Fock momentum profiles for the 6aj orbital of N 0 (upper panel). The lower panels show the momentum and position density contour maps for an oriented N 0 molecule calculated at the Hartree-Fock level using the 207-R basis set. The contours represent 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density. 2  2  Chapter 4. The HOMOs of NO and  N0  11  2  much more limited in previous work [145-147] because of the small basis sets used and also because no post-Hartree-Fock calculations were done. In the present work, the experimental momentum profiles are compared on Figs. 4.2 and 4.3 with the theoretical momentum profiles (TMPs) calculated at the U H F and ROHF Hartree-Fock level and also with MRSD-CI, A C P F and Q D V P T ion-neutral overlaps as described in Sections 4.3.1 and 4.3.2 above.  Selected  properties for each calculation and corresponding experimental values are shown in Tables 4.1 and 4.2. The Hartree-Fock basis sets used range from a very modest minimal STO-3G basis to the very much larger 168-GTO, 207-GTO and 210-GTO basis sets developed in the course of the present work. The effects of many-body correlation and electronic relaxation are also seen on Figs. 4.2 and 4.3 from the MRSD-CI, A C P F and QDVPT calculations of the TMPs. The TMPs from the 210-CI and 210-ACPF treatments are very similar to their 168-CI and 168-ACPF counterparts and therefore these calculations are not shown on Fig. 4.2 but the properties are listed in Table 4.1. It should be noted that the experimental instrumental angular resolution has been accounted for in all theoretical momentum profiles in Figs. 4.2 and 4.3 using the G W - P G method [59]. Experiment and theory are placed on a common intensity scale for each molecule by normalizing the experiment to the 168-ACPF momentum profile for N O (Fig. 4.2) and to the 207-QDVPT momentum profile for N 0 (Fig. 4.3), assuming unit pole strength. 2  Since the  wavefunctions are normalized all of the calculations are on a common relative intensity scale for each molecule.  However, for profiles such as the N 0  2  H O M O where s character is  dominant (Fig. 4.3), it is often more difficult to distinguish the quality of TMPs than for p-type profiles such as the 27t orbital of NO. A change in relative intensity at low p is often the only major distinguishing characteristic between two TMPs for dominantly s-type momentum profiles such as the N 0 H O M O . As such, other normalizations to the N 0 X M P could be 2  2  selected. The U H F calculations of the momentum profiles in Figs. 4.2 and 4.3 show a trend of improving agreement with experiment as the basis set quality improves from the STO-3G  Chapter 4. The HOMOs of NO and  N0  78  2  (curve lu) which fits the observed XMPs very poorly, to the intermediate quality results (6-311+G*-U) to the large basis sets (curves 3u, 4u, 5u and 6u). There is a " U H F basis set limit" at which these Hartree-Fock momentum profiles tend to converge and little or no further improvement in shape fit with experiment is seen upon an increase in basis set size. A n improvement in shape and/or intensity fit is brought about only by use of the R O H F method (curves 3r, 4r, 5r and 6r). ROHF calculations were also obtained for both molecules with the STO-3G and 6-311+G* basis sets (calculations l r and 2r, respectively) and the results are given in Tables 4.1 and 4.2 (these TMPs are not shown in Figs. 4.2 and 4.3).  The STO-3G-R  momentum profile for N 0 is identical to the STO-3G-U momentum profile. However, in all 2  other cases in the present work the ROHF momentum profiles displayed a higher relative intensity at low momentum than their U H F counterparts. This effect is thought to be due to the influence of the energy minimization constraint in constructing Hartree-Fock wavefunctions. For an optimized U H F wavefunction, the additional flexibility introduced by different a and P spin manifolds leads to a lower total energy and a more contracted r-space H O M O density than an ROHF wavefunction with the same basis set. Thus, the ROHF method leads to a more diffuse r-space density distribution because of its additional orbital constraint. For the s-type H O M O of N 0 (Fig. 4.3), the increased intensity is concentrated near p = 0 and no significant 2  shape difference results between ROHF and U H F TMPs although R O H F leads to higher intensity at low p than UHF. As noted in Section 4.3.1 above, other calculated properties such as spin contamination differ considerably in the U H F and ROHF methods. Turning to a more detailed consideration of the various U H F and R O H F calculations, it can be seen that the minimal basis set STO-3G-U results are in very poor agreement with both the observed experimental momentum profiles (curves l u on Figs. 4.2 and 4.3) and electronic properties (Tables 4.1 and 4.2). For N 0 in particular the STO-3G-U momentum profile 2  exhibits a very different s to p ratio from the X M P and the other TMPs. The poor total energies for both molecules reflect the low degree of basis set saturation in the STO-3G-U calculations. While the dipole moments for N O and N 0 are closer to experiment than those from several 2  Chapter 4. The HOMOs of NO and  N0  79  2  large basis sets, it is well known that the STO-3G often produces accurate dipole moments because of a fortuitous error cancellation [57]. The 6-311+G*-U intermediate size basis set calculation (curve 2u on Figs. 4.2 and 4.3), including diffuse and polarization functions, provides a better description than the STO-3G-U wavefunction for each molecule. However, the low momentum regions of both X M P s are still underestimated. For N 0 the 6-311+G*-U momentum profile is comparable to the results of 2  the larger basis set Hartree-Fock (UHF and ROHF) calculations. With the exception of the dipole moment, general improvements are also obtained for other calculated properties for both molecules (Tables 4.1 and 4.2) relative to the STO-3G-U calculation. The A U G 5 - U T M P (curves 3u on Figs. 4.2 and 4.3) provides a further small improvement in shape compared to the 6-311+G*-U momentum profile for NO. The A U G 5 - U momentum profile for N 0  2  differs only in intensity from the 6-311+G*-U T M P .  Improvements are also seen in most other properties in Tables 4.1 and 4.2. As mentioned above, the U H F and ROHF momentum profiles from larger basis sets (curves 3u, 4u, 5u and 6u on Figs. 4.2 and 4.3) tend to converge. However, there is still some fluctuation in other properties with basis set size (Tables 4.1 and 4.2). In particular, there is still variation in total energy at the millihartree level. The dipole moments of N O and N 0 also 2  vary significantly with change in basis set size. Of the U H F calculations, the 168-U provides generally the best description of the range of properties considered here for N O while the 207-U provides the best description of the properties of N 0 . 2  The 168-R and 210-R  calculations provide good ROHF descriptions of the properties of N O while the 207-R gives the best ROHF description of the properties of N 0 (due to its lowest total energy). 2  Post-Hartree-Fock methods such as CI calculations [11,12,37-40] have often provided a further significant improvement in the shape fit with the X M P over Hartree-Fock level calculations, particularly where second row atoms are involved. However, from Fig. 4.2 it can be seen that a comparably good description of the observed momentum profile of N O is achieved by the larger basis set restricted Hartree-Fock (AUG5-R (3r), 168-R (4r) and  Chapter 4. The HOMOs of NO and  N0  80  2  210-R (6r)) and post-Hartree-Fock (168-CI (4c) and 168-ACPF (4a)) calculations. Although not shown on Fig. 4.2, the 210-CI and 210-ACPF TMPs give very similar results (see Table 4.1). There is still a small discrepancy between the N O X M P and all theory at low momentum below p ~ 0.4 au. This could be due to a very small amount of impurity N 0  2  (see Section 4.2) but this is thought to be unlikely after an analysis of the experimental and theoretical momentum profiles. Although the 207-CI (5c), 207-ACPF (5a) and 207-QDVPT (5q) post-Hartree-Fock calculations for N 0 are reasonably similar in shape to the ROHF 2  momentum profiles (3r and 5r), they do show higher intensity at lower p and fit the X M P quite well within the statistical precision of the data. Despite the only small differences in the shapes of the TMPs relative to the near-Hartree-Fock limit TMPs, it should be noted that the postHartree-Fock calculations provide significant improvement in the total energies and dipole moments of both molecules (Tables 4.1 and 4.2). Since the post-Hartree-Fock calculations are based on initial ROHF Hartree-Fock results, they are free of spin contamination. It should also be noted that the post-Hartree-Fock calculations for N O using the 168-GTO basis set generally yield better values of the properties than those using the 210-GTO basis set for a given type of calculation. The experimental momentum profiles are also compared on Figs. 4.4 and 4.5 with the DFT TMPs (see Section 4.3.3 above for a description of the calculation methods).  For  purposes of comparison, MRSD-CI, A C P F and QDVPT momentum profiles are also shown on Figs. 4.4 and 4.5.  The X M P s have been normalized to the AUG5-BP D F T theoretical  momentum profiles for the present comparisons in Figs. 4.4 and 4.5 (note that different normalizations were selected for Figs. 4.2 and 4.3). It can be seen that the D F T momentum profiles are more intense than the MRSD-CI, A C P F and QDVPT results.  This can occur  because overly spatially diffuse orbitals may result from some choices for the exchangecorrelation functional in the Kohn-Sham calculations [60] and thus the momentum profile may be more intense at low p (by the reciprocity of r- and p-space).  Chapter 4. The HOMOs of NO and N0  81  2  0  1  0  1  1  1  1  L  2  Momentum (au) Fig. 4.4: Measured and calculated spherically averaged DFT momentum profiles for the 2n orbital of NO. The solid circles represent the experimental multichannel or single channel measurements also shown on Fig. 4.2. Selected MRSD-CI and ACPF calculated momentum profiles are also shown for comparison with the DFT momentum profiles. Note that the normalization of experiment to theory differs from that used in Fig. 4.2. See text for details.  Chapter 4. The HOMOs  of NO and N0  2  82  Fig. 4.5: Measured and calculated spherically averaged DFT momentum profiles for the 6aj orbital of N0 . The solid circles represent the experimental multichannel or single channel measurements also shown on Fig. 4.3. Selected MRSD-CI, ACPF and QDVPT calculated momentum profiles are also shown for comparison with the DFT momentum profiles. Note that the normalization of experiment to theory differs from that used in Fig. 4.3. See text for details. 2  Chapter 4. The HOMOs of NO and  N0  83  2  It can be seen from Figs. 4.4 and 4.5 that there is quite good overall agreement between the D F T target Kohn-Sham momentum profiles and the experimental momentum profiles of both molecules. In terms of intensity of the DFT momentum profiles, the ordering is AUG5-P (curve 3p), A U G 5 - L (curve 3/) and AUG5-BP (curve 3b) for both molecules. For NO, there are some noticeable shape differences between the three DFT TMPs, with the Becke-Perdew non-local calculation (curve 3b) fitting the best. However, for N 0 the shape differences 2  between the D F T TMPs from the local density approximation and those from the non-local potentials are minimal. Thus, the local density approximation results appear to be at least as good a model of the momentum profiles for the H O M O of N 0 as the more sophisticated non2  local potential treatments. Somewhat better agreement with the shape of the observed X M P of N O is given by the DFT curve 3b (Fig. 4.4) in the low momentum region ( < 0.5 au) than for the high-level postHartree-Fock calculations (see Fig. 4.4, and also refer to curves 4c and 4a on Fig. 4.2). For N 0 , though, the intensities are more different and the shape fit with the X M P is roughly 2  comparable to those from the Hartree-Fock and post-Hartree-Fock TMPs (compare Figs. 4.3 and 4.5).  On comparing other properties, dipole moments from the D F T calculations  (Tables 4.1 and 4.2) are within 0.1 Debye of experiment. While the D F T dipole moments for N0  2  are closer to experiment than those from MRSD-CI or ACPF, the opposite is true for the  dipole moment of NO. It should be noted that some degree of spin contamination is present in the DFT calculations for open-shell molecules that is not present for the CI results but, as noted in Section 4.3.3, ^ S ^ from DFT was not computed. However, despite the spin contamination 2  and the well-known problem of obtaining absolute energies from DFT, it can be said that the DFT treatments provide quite good values of the other properties in the present work.  4.4.2 Density Maps in Momentum and Position Space The momentum and position space density contour maps corresponding to oriented N O and N 0 molecules presented in Figs. 4.2 and 4.3 (bottom panels) are of pedagogical interest 2  Chapter 4. The HOMOs of NO and NO  84  2  and also provide some insight into the (spherically averaged) experimental and theoretical 2  momentum profiles. These maps are slices of the orbital electron density y/  through the  molecular plane of each oriented molecule in position space and with momentum perpendicular and parallel with respect to the molecular plane in momentum space. The calculations are based on the near Hartree-Fock limit results for both molecules (168-R for N O and 207-R for N 0 ) which have been found to provide good descriptions of the X M P s of these molecules 2  at the Hartree-Fock level. The origin for the position space maps is the molecular center of mass and all dimensions are in atomic units. The side panels on each map show slices of the momentum or position space density along the vertical (right panel) and horizontal (top panel) dotted lines. The H O M O of N O is predicted (from simple Hartree-Fock theory) to be a completely antibonding it* orbital with no s-type contributions. Accordingly, the r-map (Fig. 4.2) for this molecule displays an orbital consisting mainly of atomic p-type functions on each atom. The p-type orbital components for N O are not of equal intensity on N and O. In addition, there is a "filling-in" along the perpendicular "nodal" plane in p-space.  Since the Kouba and Ohrn  treatment [148] was erroneously considered (see Section 4.2) to give the best fit to the momentum profiles in earlier work [145] the fact that the corresponding H O M O r-map of N O became more non-bonding (and hence less antibonding) as the amount of nodal character in the orbital decreased was thought to be significant. However, it has been found in the present work that the K & O - U wavefunction does not provide an acceptable fit to the N O X M P once the N 0 impurity has been eliminated (see Section 4.2 and Fig. 4.1a).  2  Thus any deductions [145]  arising from the non-bonding or antibonding character of this orbital predicted using the Kouba and Ohrn wavefunction are inappropriate. It should also be noted that while there is some filling in along the perpendicular plane, the intensity at the origin of the p-map is exactly zero (see top panel). Thus, spherical averaging of this momentum density should give a momentum profile with zero intensity at p = 0.  In this regard it should be noted that the calculated  spherically averaged momentum profiles in Fig. 4.2 have non-zero intensity at p = 0 due to the  Chapter 4. The HOMOs of NO and  N0  85  2  fact that they have been folded with the instrumental resolution function to compare with experiment. The p-map for the 6aj orbital of the triatomic species N 0 (Fig. 4.3) is more complex 2  than those for the diatomic N O molecule.  The complexity of the p-map indicates the  difficulties that often arise in interpreting momentum space representations  for larger  molecules. The multicenter nature of this molecule is reflected in the complex oscillatory behaviour in momentum space. It should be noted that the momentum space wavefunction contains all the symmetry characteristics of the position space wavefunction with the addition of an inversion center and thus the p-map contains a center of symmetry (even though such a symmetry element is not present in the  point group). Finally, on the p-map there is a large  momentum density at p = 0 (top and right panels in the p-map) and thus the spherically averaged N 0 6a momentum profile (Fig. 4.3) is strongly s-type in contrast to the p-type 2  x  momentum profiles of N O (Fig. 4.2). In the r-map, the H O M O displays the aj symmetry characteristic of a totally symmetric orbital in the C  2 v  molecular point group.  Chapter 5. Molecular  Oxygen  86  Chapter 5  Molecular Oxygen  5.1 Introduction The electronic structure  of the  oxygen molecule has  been  the  subject  of  many theoretical and experimental studies. This interest is due in part to the great abundance of the molecule and to the important role it plays in living systems and in atmospheric processes [143]. The open-shell nature of 0  2  adds a challenging aspect to quantum mechanical  calculations that is not present for closed-shell molecules. Experimentally, He I and He II experiments [174,176-179] using high-resolution photoelectron spectroscopy (PES) for the valence shell ionization processes of 0  2  have been reported up to ~ 28 eV binding energy.  Gardner and Samson [180] extended the valence shell He II PES studies to ~ 40 eV binding energy and high resolution threshold PES studies using synchrotron radiation have also been reported  [181].  In addition, x-ray photoelectron spectroscopy (XPS) studies up to  ~ 32 eV [182] and ~ 60 eV [183] have been reported. The full valence shell binding energy spectrum has also been measured at low resolution (1 eV F W H M ) by dipole (e, 2e) spectroscopy [184] and by electron momentum spectroscopy (2 eV F W H M ) [185] out to ~ 60 eV. The first reported study of 0 by E M S was the early single channel experimental work 2  of Suzuki et al. [185] in which valence electron momentum distributions corresponding to the major features in the binding energy spectrum up to 60 eV were obtained.  However, no  calculated binding energy spectra or theoretical momentum profiles were reported [185].  Chapter 5. Molecular  Oxygen  87  While no theoretical momentum profiles were presented in the work of Suzuki et al. [185], experimental momentum profiles were obtained for binding energies up to 60 eV. In some cases, the experimental momentum profiles corresponding to the production of the individual quartet and doublet ion states arising from ionization of the same orbital were obtained. The  fl n 4  and A TI ion states were difficult to deconvolute because of the limited statistics even 2  u  U  with the higher energy resolution (1.2 eV FWHM) available at impact energies of 400 eV. In addition, the l7t ionization satellite peak at 23.7 eV could not be resolved from the nearby u  c E~ state. On the basis of the shapes of the observed momentum profiles, the peaks at 39 and 4  47 eV were assigned to £ ~ and £~ ion states from 2 a ionization although no accounting for 4  2  g  the total 2cj pole strength and the high binding energy region out to 60 eV was made. A g  further limitation was the limited statistical precision of the data due to low instrumental sensitivities [185].  In particular, the momentum profiles corresponding to the C E ~ and 2  33.0 eV final ion states showed large scatter [185]. Subsequent E M S work on 0  2  by Tossell et al. [147] presented both experimental and  theoretical momentum profiles for the five valence orbitals of 0 but no binding energy spectra 2  were reported.  However, although the shapes of the observed momentum profiles agreed  qualitatively with theory, the calculations used very limited basis set size [147]. Very limited statistics were also apparent and the momentum profiles corresponding to individual quartet and doublet states were not resolved [147]. The binding energy spectra of oxygen are more complex than for closed-shell cases because of the presence of both quartet and doublet final ion states which make the deconvolution and interpretation of the spectra more difficult. The ability of theory to predict the relative intensity of the quartet and doublet ion states in the EMS spectrum of oxygen (or other open-shell molecules) and their corresponding momentum profiles has not been investigated. The initial orbital origin of the peak in the 33 eV region of the 0  2  binding energy  spectrum has been the subject of considerable dispute. Gardner and Samson [180] did not  Chapter 5. Molecular  Oxygen  88  include this energy region in their experiment while the X P S study of Siegbahn et al. [183] attributed the 33 eV peak to shake-up or more likely characteristic energy losses. The fact that the peak was observed at a comparable intensity in dipole (e, 2e) work [184], which was done at much lower pressure than the XPS study [183], suggested that energy losses undergone by ejected photoelectrons could be ruled out and that the peak at 33 eV was more likely due to final ion states arising from shake-up mechanisms.  The propagator techniques used by  Cederbaum and Domcke [65] calculated all of the n  intensity to be at energies < 27.3 eV  2  while Dixon and Hull [186] predicted all the n 2  u  u  intensity to be at < 22.7 eV. Other theory by  Honjou et al. [187,188] using configuration interaction wavefunctions placed all of the I1 2  U  intensity at energies < 24.3 eV and predicted that the peak at 33.0 eV was mainly due to £~ 2  and (to a lesser extent) Z ~ poles. The multiconfigurational electron propagator calculations of 2  Yeager and co-workers [189,190] predicted that almost all the intensity due to ionization of l7t  electron is below 24.56 eV and also indicated that a transition at ~ 34.7 eV is due to a E~ 2  u  state. The electron propagator techniques by Purvis and Ohrn [191] did not predict any poles in the 33 eV region. Gerwer et al. [192] suggested that the 33 eV peak be assigned as  2  n  u  since this led to reasonable agreement between their calculated total electronic ion state crosssections for formation of (ln^)" ions and those obtained by measurements using dipole (e, 2e) 1  spectroscopy [184] as well as He I PES experiments using a many line light source [193]. However, many corrections were required [192] for unresolved experimental final ion state cross-sections and the agreement between theory and experiment was only moderately good. On the basis of the observed p-type momentum profile and peak area arguments, Suzuki et al. [185] suggested that the peak at 33 eV was due to a  2  n  satellite associated with the  u  l7t  u  ionization process. However, this assignment [185] was not definitive since poles from 2 a  u  ionization with c E~ or C E ~ parent final ion states would also have p-type momentum 4  2  distributions similar to those from poles corresponding to n 2  u  ionic states. No momentum  profiles were reported for the 33 eV region in the EMS work of Tossell et al. [147].  Chapter 5. Molecular  Oxygen  89  The above history indicates that improved experimental and more detailed theoretical investigations of 0 0  2  2  are desirable. In the intervening years since the previous E M S studies on  [147,185] large improvements in sensitivity have been achieved with the development of  multichannel E M S spectrometers [42,44,45,47].  In addition, a range of much more  sophisticated calculation methods using large basis sets are now possible with current computing devices. The present work now reports multichannel energy-dispersive electron momentum spectroscopy measurements of all valence orbital momentum profiles and complete valence shell binding energy spectra (up to 59 eV) for 0  2  with improved statistical precision  compared with previous work. The comparison of theoretical intensities to the experimental E M S binding energy spectrum and the subsequent theoretical modelling of the measured momentum profile intensities has also been carried out using a range of computational methods and basis sets. The E M S binding energy spectra including the more complex inner valence region are compared with the results of a new multi-reference singles and doubles configuration interaction (MRSD-CI) calculation of the ionic states using very large basis sets. The measured orbital momentum profiles are compared with a variety of Hartree-Fock treatments ranging from a minimal STO-3G basis to near Hartree-Fock limit quality basis sets including a 168 contracted Gaussian type orbital (168-GTO) calculation developed in the course of the present collaborative work.  Both the Unrestricted Hartree-Fock (UHF) and  Restricted Open-shell Hartree-Fock (ROHF) methods have been used to calculate momentum profiles in the present work. In addition, the effects of correlation and electronic relaxation on all of the valence orbital momentum profiles are also investigated with new M R S D - C I ionneutral overlap calculations (168-G(CI)). In the case of the outermost  l7t  orbital ( TI final 2  g  g  ion state) an averaged coupled-pair functional (ACPF) [149] calculation using the 168-GTO basis is also reported.  Many-electron effects on all 0  2  momentum profiles are further  investigated with density functional methods, using the target Kohn-Sham approximation of the E M S cross-section [60] with local and non-local (gradient corrected) functionals.  Chapter 5. Molecular  Oxygen  90  In the present work, careful attention has also been given to obtaining individual experimental momentum profile measurements corresponding to the separate quartet and doublet final ion state exit channels arising from a common initial orbital origin. Experimental  X n , a Il , 2  momentum profiles are provided for all of the orbital ionizations leading to the  An, 2  u  fc Z , B YT , 4  2  g  2  4  g  u  n (3), c Z ; and C S " final ion states as well as for the peak at 33 eV. 4  2  u  e  The inner valence 2 a  ionization ( Z~ and E ~ final ion states) pole strengths have been 4  g  2  studied by an analysis of the binding energy and momentum profiles for the high energy 36-59 eV region.  The comparison of the calculated momentum profiles with individual  experimental momentum profiles has been used to determine pole strengths for the l7U  u  ionization peaks corresponding to the A I I 2  (23.7  eV) and these pole strengths  parent state and the IT (3) satellite state 2  U  U  have been compared with previously published  experimental [177,185] and theoretical [65,186-192] results. In a similar way, pole strengths for production of the c £ ~ and C E ~ states from 2 a ionization have also been determined. 4  2  u  An assignment of the peak at 33 eV in the binding energy spectra is provided by an analysis of the measured EMS and calculated MRSD-CI binding energy spectra and the pole strengths in the (l7C ) and ( 2 a ) _1  u  u  _1  manifolds.  Multichannel measurements in the present work were obtained using the binning mode [45] of the EMS spectrometer (see Section 3.1). In addition, earlier non-binning mode (see Section 3.1) measurements were obtained for the l7Cg (HOMO) orbital of 0  2  that is well-  separated from the rest of the valence ionization spectrum (these non-binning measurements have been published in Ref. [94] with the results for NO and N 0 shown in Chapter 4). The 2  0  2  sample was obtained from Medigas gas products and was of > 99.0% purity. No impurities  were observed in any of the spectra.  Chapter 5. Molecular  Oxygen  91  5.2 Hartree-Fock, MRSD-CI, ACPF and DFT Calculations The basis sets for the O atom, calculation methods and calculation notation for 0  2  are  the same as described for N O in Sections 4.3.1-4.3.3 (with the exception that the calculations involving the 210-GTO basis set were for NO only).  The number of C G T O for a given  calculation method is the same for both NO and 0 . For example, the A U G 5 - U calculation for 2  0  2  is designated as 3u, uses the same basis set for each O atom as described for the 3u  calculation for NO in Section 4.3.1 and therefore employs a total of 110 C G T O . It should also be noted that calculations lu, lr, 2u, 2r, 3u, 3r, 3/, 3p and 3b for 0  2  were performed at the  University of British Columbia while calculations 4u, 4r, 4c and 4a were performed by Prof. E.R. Davidson at Indiana University with the M E L D program. The experimentally derived geometry for 0  2  [194] was used in all calculations. Various other calculated and experimental  electronic properties for 0  2  are listed in Table 5.1. The same considerations regarding total  energy and spin contamination from the UHF, ROHF and DFT methods discussed in Sections 4.3.1 and 4.3.3 also apply to oxygen. It can be seen from Table 5.1 that all ROHF calculations (no spin contamination) give an (s J value of exactly 2.0 for 0 2S  in the U H F calculations.  2  while this value is exceeded  A large amount of spin-contamination' in any wavefunction may  cause inaccuracy in computed properties, but the spin contamination of the U H F wavefunctions in Table 5.1 is reasonably small (< 3%). Because of the two unpaired a electrons in the X ~Z~ ground state of neutral oxygen, the 3  calculation of the TMPs (and therefore also the theoretical binding energy  spectra)  corresponding to a given orbital electron ionization and final ion state is more detailed than for a closed-shell molecule.  If an orbital other than the  l7U  g  (HOMO) orbital is ionized and a  quartet final ion state is produced, then the ion state spin eigenfunction must result in a total spin of 3/2. One possible spin eigenfunction corresponds to the a(l)a(2)oc(3) configuration for the three unpaired electrons in the ion. In this case it can be said that a P electron ionization process from the orbital of interest contributes to the T M P corresponding to the quartet ion  Chapter 5. Molecular Oxygen  92  Table 5.1: Calculated and experimental properties for O 2 ^ 2 „ 3  Basis Set and Calculation Method  Total Energy (hartree)  0  (?)  (au)  b  (4 (au)  Pmax (au)«  c  lu lr 2u 2r 3u 3r 4u 4r  Hartree-Fock Calculations STO-3G-U STO-3G-R 6-311+G*-U 6-311+G*-R AUG5-U AUG5-R 168-U 168-R  -147.634 -147.632 -149.660 -149.638 -149.686 -149.663 -149.691 -149.667  2.0034 2.0000 2.0488 2.0000 2.0488 2.0000 2.0484 2.0000  -0.9307 -0.9258 -0.4304 -0.4708 -0.1574 -0.2304 -0.2178 -0.2768  40.652 40.649 43.744 43.673 43.446 43.385 43.446 43.386  1.36 1.36 1.10 1.07 1.07 1.05 1.07 1.05  4c 4a  Post-Hartree-Fock Calculations 168-CI 168-ACPF  -150.133 -150.146  2.0000 2.0000  -0.2766 -0.2716  43.337 43.410  1.03 1.03  3/ 3p 3b  DFT Calculations AUG5-L AUG5-P AUG5-BP  -149.338 -150.547 -150.418  -0.3259 -0.3390 -0.3224  43.878 43.924 43.708  0.96 0.95 0.98  -0.29^  44.4±0.2  Experimental  6  — — —  -150.326/"  /j  1.0*'  a  Calculations performed at the experimental equilibrium bond length of 1.20748 A (2.28181 au), Ref. [194].  b  The quadrupole moment is defined as 0 ^ = (1 / 2 ) ^ ^ q (3zf — r^fj, summing over all nuclei and electrons Calculated quadrupole moments are for a non-relativistic, non-vibrating, non-rotating molecule. t  c  The electronic spatial extent is defined as ^r ^ = 2  r^, summing over all electrons.  ^ The p corresponds to the value of momentum where the intensity of the momentum profile is at a maximum (see Figs. 5.4a and 5.4b). The total energy from Kohn-Sham DFT has been suggested to be in error because of shifted orbital energies relative to ionization potentials from Dyson's equation [60] - See also Section 2.2.6. / The "experimental" total energy is the estimated non-relativistic, non-vibrating, infinite nuclear mass total energy. Total energy obtained by adding the atomic energies from Ref. [162] to the DQ values from Ref. [159] plus the zero point energy from the frequencies in Ref. [159], S Recommended value from Ref. [195]. Ref. [196]. Present work. m a x  e  h  1  Chapter 5. Molecular  Oxygen  93  state [197]. However, a spin eigenfunction with total spin of 3/2 can also be constructed from a particular linear combination of the oc(l)a(2)p(3), a(l)(3(2)a(3) and p(l)oc(2)a(3) electron configurations for the ion. Thus, a electron ionization from the orbital of interest also makes a contribution to the T M P corresponding to the quartet final ion state. The relative contributions of these p and a electron ionizations to the T M P corresponding to a quartet ion state are 1 and 1/3, respectively [197].  Similarly, for ionization from an orbital (other than the HOMO)  producing a doublet final ion state, the ion state spin eigenfunctions must result in a total spin of 1/2. The ion state spin eigenfunction corresponds to a linear combination of cc(l)a(2)P(3), a(l)P(2)a(3) and P(l)a(2)a(3) electron configurations only and therefore it can be said that only a electron ionization from the orbital of interest contributes to the T M P corresponding to a doublet final ion state. Lastly, the case for the H O M O l7r. orbital is much simpler since only g  a electrons may be ionized and used for calculation of the TMP. TMPs for closed-shell molecules are simply multiplied by the respective orbital occupancies (number of electrons) to put the TMPs for all orbitals on the same intensity scale. However, the situation for an open-shell molecule is more complicated.  The rules for  predicting relative intensities of final ion states for open-shell molecules have been well-noted in the PES literature [63,64,198]. The relative intensity for a given ion state with total spin S is proportional to the degeneracy of the final ion state gf given by the product of the spin degeneracy factor P  s p m  equal to (25 +1) and orbital factor P bit equal to the orbital degeneracy or  of the ion state [63,64,198]: S/^spin'^rbit  ( -!) 5  It should also be noted that where several states arise from ionization of an orbital of an openshell molecule (i.e. doublets and quartets for 0 ) , the sum of the gf factors associated with the 2  various final ion states must equal the number of electrons in the orbital of origin [64]. This allows the intensities of the TMPs arising from ionization from the various orbitals of 0 to be 2  placed on a common scale. Thus in constructing the TMPs corresponding to the TI , 2  4  n, u  Chapter 5. Molecular Oxygen  2  n , Z~, £", % 4  u  2  and %  94  final ion states of 0  the factors g of 2, 8/3, 4/3, 4/3, 2/3, 4/3 and  2  f  2/3, respectively, are used to put them on a common relative intensity scale. The various TMPs of the valence orbitals were obtained using Eq. (2.53) for the M R S D - C I and A C P F calculations (4c and 4a, respectively), Eq. (2.57) for the Hartree-Fock calculations (lu, lr, 2u, 2r, 3u, 3r, 4u and 4r) and Eq. (2.58) for the DFT calculations (31, 3p and 3b) according to the above considerations of the a and P electron ionization contributions to the quartet and doublet final ion states and the factors gyfor the various final ion states. The finite experimental momentum resolution (Ap ~ 0.1 au) was folded into all TMPs using the G W - P G method [59].  5.3  Results a n d Discussion  5.3.1 B i n d i n g E n e r g y S p e c t r a Oxygen belongs to the D  T O n  point group and its (open-shell) ground state valence shell  electronic configuration in the Hartree-Fock approximation is 2cr 2cT 2  g  2 u  inner valence  3OI\KI\KI  (5.2)  X Z" 3  outer valence  Removal of an electron from the valence orbitals gives the following experimentally reported 0 + ionic states [174,176-183]: 2  configuration  ionic state  2oj2oS3aJl<l7cJ  x ng  2aj2<3ajl<l^  a n ,A n , n (3)  2  4  2  2  u  u  u  2oj2o;i3cJl<l7c;  2o;2oj3o;i<i7c; 2oj2aS3oJl<l7tJ  c % , C X 4  E-, Z-, Z-(2), Z-(2) 2  4  2  (5.3)  Chapter 5. Molecular  Oxygen  95  In addition, a satellite peak of disputed origin has been observed at 33.0 eV (labelled Q in the present work). The X Ti 2  g  ground ionic state from l7t ionization has a vertical ionization potential of g  12.30 eV and is well resolved in the measured photoelectron spectra [174,176-180]. The valence shell ionization potentials reported in the He(I) and He(H) studies of Edqvist et al. [177] and Baltzer et al. [179] and the XPS results of Siegbahn et al. [183] are presented in Table 5.2. Results from the earlier E M S study of Suzuki et al. [185] are also shown in Table 5.2. Ionization from orbitals other than the l7i H O M O results in an ion with three singly g  occupied orbitals and thus both quartet and doublet ion states are observed. Ionization of the l7r,  u  orbital gives the a Tl  state at 16.7 eV, the parent A Tl  4  2  u  u  state at 17.6 eV and a higher  energy ri (3) pole at 23.9 eV [179]. The presence of a n (2) pole of very low intensity is 2  2  u  u  predicted to be at ~ 20 eV by theory [65,186,189,190]. This weak I1 (2) pole has not been 2  U  identified experimentally [174,176-180]. It should be noted that other notation systems have been used for the predicted n (2) pole at ~ 20 eV [65,186,189,190] and the experimentally 2  u  observed I1 (3) pole at ~ 23.7 eV. Baltzer et al. [179] employed a similar notation of 2 n 2  2  U  and 3  2  ri  u  u  for these states, respectively, while Suzuki et al. [185] chose to ignore the predicted  pole at ~ 20 eV and instead applied the notation n (2) to their experimentally observed pole 2  u  at 23.7 eV. Suzuki et al. [185] also assigned the peak at ~ 33 eV in the EMS binding energy spectrum to ln ionization and labelled the final ion state of this peak as lT (3). In the present 2  u  u  work, we label the peak at 33 eV simply as Q and discuss its possible orbital origin on the basis of the present experimental results and calculations in Section 5.3.2.4. Ionization of a 3 a electron likewise leads to the separate quartet & E~ and doublet 4  g  J5 Z~ states which are observed in the photoelectron spectra at binding energies of 18.17 and 2  20.43 eV, respectively [174,177]. Similarly, removal of an electron from the inner closed shell 2a  orbital produces the O t c E~ and C E~ 4  u  2  states. Baltzer et al. [179] have assigned these  Chapter 5. Molecular  Oxygen  Table 5.2: Experimental ionization energies and peak widths for O2 0  Vertical ionization energies orbital origin^  l7C  final ion stated  xn 2  g  PES [177]  PES X P S [179] [183]  EMS [185]  EMS this work  c  Peak width EMS this work ' 6  g  12.30  12.31  13.1  12.5  12.30  0.80  17.0  16.8  16.7  1.00  16.8  17.5  0.90  1*11  an  u  16.70  16.70  l7t  A n  u  17.73  17.64  18.17  18.17  18.8  18.1  18.2  0.52  20.43  20.30  21.1  20.3  20.3  0.70  24.0  23.9  23.7  23.7  1.50  24.58  24.56  25.3  24.5  24.5  0.40  27.3  27.9  27.5  27.4  2.40  Q  33.6  32.5  33.0  1.40  4  2  u  *>% 3o  g  l7C  2  2a  u  2a  u  n (3) u  u  2a  g  %  39.6  39  38.9  2.25  2a  g  %  41.6  39  40.9  2.00  2a  g  X (2)  46  47  45.2  3.00  2a  g  X(2)  48  47  48.4  3.00  4  g  All ionization energies and peak widths are in eV. Assignments from Refs. [174,177,179,183,185] with the exception of the peak at 33.0 eV which has been labelled Q in the present work. Energy position of the maximum of the Gaussian function width for this final ion state used in the deconvolution procedure. °" Natural half-width of the Gaussian function for this final ion state used in the deconvolution procedure.  a  b  c  Chapter 5. Molecular  Oxygen  97  processes to ionization potentials of 24.56 eV and 27.3 eV, respectively. However, the present M R S D - C I results (Table 5.3) and other calculations [187-190] predict the presence of strong 4  X~ poles that overlap the peak traditionally associated with the C E ~ ionic state. 2  This  predicted dual character is supported by the present experiments and therefore we label the peak at 27.3 eV as peak P in the present work and discuss its composition in Section 5.3.2.4. The n (3) and c E~ states are difficult to separate because of their closeness in energy. The 2  4  u  observed ionization potentials for the peak corresponding to the c £~ final ion state reported by 4  high resolution experiments [177,179,180] are ~ 0.9 eV less than that reported in the XPS work of Siegbahn et al. [183]. Ionization of the inner-valence 2rj orbital produces the X ~ and E~ states. Due to the 4  2  g  more limited energy resolution of XPS, the vertical ionization potentials of these states are not as well-characterized as the lower energy states. Siegbahn et al. [183] reported Z ~ and £ ~ 4  2  vertical ionization potentials of 39.6 and 41.6 eV, respectively, while Gardner and Samson [180] reported values of 39.7 and 40.3 eV. The threshold PES experiments of Ellis et al. [181] gave similar values of 39.5 and 40.8 eV. In addition, the pole strength of these states is thought to be further split into several satellite processes both from considerations of experiment [181,183,184,185] and theory [187-191]. The earlier studies [181,183,184,185] all showed that the inner valence region of the binding energy spectrum of 0 is split into satellite peaks at 2  ~ 46 and 48 eV. In Table 5.2 these states are labelled [185] as I~(2) and £~(2). However, 4  2  additional satellite processes must occur since the intensity continues out to the limit of the data at 59 eV. Fig. 5.1 shows the binding energy spectra of 0  2  from 9 to 59 eV for measurements at  relative azimuthal angles of (a) § = 0.5° and (b) § = 8.5° (impact energy of 1200 eV + binding energy) on a common intensity scale and at an energy resolution of 1.5 eV F W H M . Fig. 5.1c shows the summed binding energy spectra of 0  2  from 9 to 59 eV for azimuthal (> | angles of  0.5°, 1.5°, 3.5°, 5.5°, 8.5°, 9.5°, 12.5°, 13.5°, 16.5°, 17.5°, 20.5°, 24.5° and 30.5°. The energy scale was calibrated with respect to the ln„ vertical ionization potential as measured by high  Chapter 5. Molecular Oxygen  10  98  20  30  Binding  40  50  60  Energy (eV)  F i g . 5.1: EMS binding energy spectra of 0 from 9 to 59 eV at (a) < > | = 0.5°, (b) < > | = 8.5° and (c) over all thirteen § angles, obtained at an impact energy of (1200 eV + binding energy). The dashed lines represent Gaussianfitsto the peaks and the solid curve is the summed fit. See text for details. 2  Chapter 5. Molecular  Oxygen  99  resolution photoelectron spectroscopy [174,177]. Gaussian peak shapes have been fitted to the main peaks throughout the spectra in Fig. 5.1a-c using vertical ionization potentials and Franck-Condon widths (folded with the E M S instrumental energy width of 1.5 eV F W H M ) estimated from photoelectron spectroscopy measurements [174,177,179,183].  The relative  energy spacings of the Gaussian peaks were estimated from the vertical ionization potentials, with small adjustments to compensate for the asymmetries in the shapes of the Franck-Condon envelopes. For the states above 30 eV, energies and widths were more difficult to determine because of the lower energy resolution of the XPS studies [183]. Measured and calculated ionization potentials from the literature and those used in the present work are shown in Table 5.2. The peak widths used in the present work to fit the experimental E M S binding energy spectra in Fig. 5.1 are also given in Table 5.2. In the outer valence region of the <j) = 0.5° and § = 8.5° experimental binding energy spectra, several features can be seen in each spectrum. A peak due to the process (the X U  l7C  g  ionization  state) is observed at a binding energy of 12.30 eV, which is consistent with  2  g  the results of photoelectron spectroscopy [174,177,179]. This peak displays characteristic "p-type" behaviour, having a greater intensity at <J) = 8.5° than at § = 0.5°. The band from 15 to 21 eV contains four ionization processes. The quartet and doublet states arising from  A n 2  ionization (the a TI and 4  u  u  l7U  u  states) are observed at 16.7 and 17.5 eV, respectively, in  agreement with the observed PES spectrum [174,177,179]. These two states are clearly p-type and are difficult to deconvolute because of their close spacing [177,179] and because of the low intensity of the A I I state. The next two quartet and doublet states (& £~ and 5 £ ~ ) arise 2  4  2  U  from ionization of the 3 o orbital. The energies of these peaks are also consistent with the g  literature values of 18.17 eV and 20.43 eV [174,177,179]. The Z> £~ state is very intense and 4  "s-type" so that it has greater intensity at § = 0.5° than at <b = 8.5°. Thus, although the £> E~ 4  state is very close to the A Yl state, the separation of the two states is aided in the E M S 2  u  binding energy spectra by their different symmetries (i.e. s-type and p-type, respectively). Since the B E ~ state also arises from the 3a„ orbital, it is likewise s-type. 2  Chapter 5. Molecular  Oxygen  100  The presence of the II (3) satellite peak arising from l7U ionization has been reported 2  U  u  to be at 23.90 eV [179]. The EMS binding energy spectra and associated curve fit in Fig. 5.1 show a p-type momentum profile at an energy of 23.7 eV which can be identified with the 2  n (3) satellite. This process is close in energy to the nearby c £~ state at 24.56 eV [179] and 4  u  it is clear from Fig. 5.1b that two ionization processes are present in this energy region. The c E~ state arises from 2 a ionization and is also p-type. The closeness of these two states 4  u  (and the fact that both are p-type) complicates the deconvolution of the spectrum in this region. A further ionization process, traditionally labelled [174,176-183] as the C Z~ state from 2  2 o ionization has been observed at 27.3 eV using high resolution photoelectron spectroscopy u  [179]. However, the present work (see Section 5.3.2.4 below) indicates that an additional ion state ( £~) contributes to the ionization in this energy region. In view of this situation the 4  peak observed (Fig. 5.1) at 27.4 eV is referred to as peak P. In addition, the satellite peak labelled Q in the present work is observed at 33.0 eV in Fig. 5.1. The photoelectron study of Siegbahn et al. [183] gives a vertical ionization potential of 33.6 eV for this peak while the value of 32.5 eV from Suzuki et al. [185] is in better agreement with the present work. The observed binding energy spectra (Fig. 5.1) in the high energy region from 36-59 eV are complex. A n XPS study by Siegbahn et al. [183] assigned the broad peak at ~ 39 eV to contain the two £ ~ and Z ~ states from 2rj ionization at vertical ionization energies of 39.6 4  2  g  and 41.6 eV, respectively. However, it should be noted that energy positions are more difficult to determine with the lower energy resolution of XPS. Deconvolution of the E M S binding energy spectra in the present work gives slightly different results (Table 5.2) of 38.9 and 40.9 eV.  Both states are clearly s-type, which is consistent with earlier E M S  measurements [185]. Siegbahn et al. [183] also observed states in the high energy region at 46 and 48 eV which they assigned to be £~(2) and £g(2) satellite states from 2 a ionization. 4  2  g  Suzuki et al. [185] supported this conclusion by measurement of the E M S momentum profiles. The E M S binding energy spectra in the present work also show intensity in this region and Gaussian peaks have been fitted to these dominantly s-type peaks at 45.2 and 48.4 eV  Chapter 5. Molecular  Oxygen  101  (Table 2). In addition, the spectra in Fig. 5.1 show a high energy tail (mainly s-type) out to the limit of the data at 59 eV. Calculated binding energy spectra are compared with the measured binding energy spectra in Fig. 5.2. Theory and experiment have been area normalized below 36 eV binding energy in the ty = 8.5° spectrum The calculated spectra are obtained by using the pole energies and pole strength SJ values given by the presently reported 168-CI M R S D - C I calculation (see Table 5.3) as well as the angular dependence from the presently reported 168-CI theoretical momentum profiles for each pole at each angle. The measured E M S instrumental energy resolution function, as well as the widths of the transitions as observed in the experimental E M S binding energy spectra (see Fig. 5.1 and Table 5.2) have been folded into the calculated spectra. Good agreement is obtained between experiment and theory for the  l7U  orbital (X H 2  g  g  final ion state). The correct shapes and positions are also reasonably well predicted at both angles for the band from 15 to 21 eV which contains four closely spaced ion states with two different symmetries. The intensity of the peak at ~ 20.5 eV corresponding to the B lT state 2  g  (Table 5.2) is underestimated by theory atty= 8.5° with the present normalization. In addition, the widths and positions in the 23-25 eV energy region are not well reproduced by theory (the closely-spaced I1 (3) and c E~ states are observed in this region - see Fig. 5.1 and Table 5.2). 2  4  U  The experimental and theoretical intensities for peaks P and Q observed experimentally at ~ 27.5 eV and ~ 33 eV are in agreement, but the energy positions predicted by theory are about 1 eV too high. Large discrepancies in energy positions and intensities also occur at both angles for the E ~ and Z ~ states at ~ 39 eV. The difference in intensity between experiment and 4  2  theory in this region can likely be attributed to the presence of higher energy satellite processes clearly visible in the experimental spectra out to the 59 eV limit not taken into account by the MRSD-CI calculation. The MRSD-CI calculation in the present work only takes into account states up to ~ 42 eV (see Table 5.3). In this regard, it should be noted that the MRSD-CI calculations are not expected to be accurate above the double ionization potential of O2  Chapter 5. Molecular Oxygen  Binding  102  Energy (eV)  Fig. 5.2: Measured and calculated binding energy spectra of 0 from 9 to 59 eV at (a) § = 0.5° and (b) (]) = 8.5°. The solid curves indicate the synthesized theoretical binding energy spectra with pole energies and pole strengths (Table 5.3) given by the 168-CI MRSD-CI calculation which was also used for the angular dependence. The same energy peak widths as used in Fig. 5.1 have been folded into the synthesized spectra. The theoretical and experimental binding energy spectra are area normalized on the 9-36 eV region at <> f = 8.5°. See text for details. 2  Chapter 5. Molecular Oxygen  103  Table 5.3: MRSD-CI states and intensities (pole strengths) for O j State  %  Energy (au)  Energy (eV)  -149.3924  20.53  0.807  0.993  -148.9729  31.94  0.039  0.048  -148.9360  32.94  0.020  0.025  (7t )-i(2a )-i(7t )i  0.80  -148.9023  33.86  0.053  0.065  (3a )-i(7t )-2(7t )2  0.63  (7t )-i(2a )-i(7t )i  0.53  (3a )-i(7t )- (7t )2  0.49  1.23S - b 2  Excitation (3a )-i  40.86  0.182  0.224  0.91  g  0.82 u  u  g  g  u  u  -148.6451  Coefficient  g  u  g  2  u  g  g  (2a )-i(2a )i(7t )- (7t ) 1  g  u  u  g  g  -149.2367  24.76  0.059  0.039  (3a )-i(l7r )-i(l7t )i  0.91  -149.1290  27.69  0.270  0.332  (3a )-i(l7t )-i(l7t )i  0.74  (2a )-i  0.55  (2a )-i  0.61  g  u  g  g  u  g  u  -148.9011  33.89  0.375  0.461  u  (3a )-i(l7r. )-i(l7t )i  0.52  g  X  X  0.52 0.45  (2a )-i  %  1  u  g  -148.7611  37.70  0.029  0.036  (l7l )-2(3rj )l  0.89  -149.2387  24.71  0.652  0.802  (2a )"l  0.84  -149.1007  28.46  0.102  0.125  (3a )-i(l7t )-i(l7t )i  0.77  -148.6945  39.52  0.025  0.031  (3a )-2(3a )i  0.71  -148.5913  42.32  0.006  0.007  (l7C )-2(2a )-i(l7t )2  0.82  -149.4768  18.23  0.853  1.049  (3a )-i  0.92  -149.0655  29.42  0.019  0.023  (7t )-i(7t )-i(3a )i  0.86  -148.7941  36.81  0.009  0.011  (7C )- (2o )-l(7C )l  0.66  (30 )-l(7C )-2(7C )2  0.48  u  u  u  g  u  g  g  u  u  u  g  g  u  g  u  1  u  u  g  u  g  g  -148.6705  40.17  0.773  0.951  (2o )-i  0.85  -149.6947  12.30  0.815  1.002  (IV  0.91  -149.2016  25.72  0.021  0.026  (l7U )-2(l7tg)l  0.91  -148.8933  34.11  0.011  0.014  (lTtJ-^lTtJl  0.80  g  1  u  Chapter 5. Molecular Oxygen  104  T a b l e 5.3: C o n t i n u e d State 2  n  u  Energy (au)  Energy (eV)  -149.4860  17.98  0.303  0.373  (l^u)"  1  -149.4366  19.16  0.004  0.005  (l^u)"  1  -149.2561  24.23  0.477  0.587  -148.8406  35.54  0.023  0.028  1.235)  b  Excitation  n  u  -149.5291  16.81  0.838  1.031  0.93 0.73  (3o )-i(l7t )- ^ C T u ) g  g  1  0.59  (3G )-2(l7t )-  0.43  (iKu)"  0.94  u  g  4  Coefficient  1  In the present work renormalized pole strengths are used when comparing with experiment - see discussion in Section 5.3.2. The renormalized pole strength is equal to 1.23 xS . f  The factor of 1.23 is the scaling factor used to normalize  the MRSD-CI calculation in Fig. 5.3a (see Section 5.3.2).  Chapter 5. Molecular Oxygen  105  (~ 36.3 eV) because of the presence of highly excited Rydberg states of O* converging on the double ionization limit.  5.3.2 Comparison of Experimental and Theoretical Momentum Profiles Experimental and theoretical spherically averaged momentum profiles have been obtained for each of the valence orbitals of 0 . 2  The experimental momentum profiles are  extracted from the multichannel binning mode binding energy spectra. The relative intensities for the different transitions are maintained by the sequential, angular-correlated data collection process.  The Gaussian fitting procedure, described above in Section 5.3.1 for the binding  energy spectra, is used to determine the relative intensities of the various transitions at each azimuthal angle ty. The experimental momentum profile corresponding to a particular final ion state is obtained by plotting the area under the corresponding fitted peak for each electronic state of the ion as a function of p (i.e. ty angle).  With this procedure all experimental  momentum profiles are automatically placed on a common relative intensity scale.  The  theoretical momentum profiles are obtained as described in Section 5.2. The experimental and theoretical momentum profiles (shown on an angle (ty) scale because of the wide range of binding energy included - see Eq. (2.45)) have been placed on a common intensity scale by normalizing the experimental data, summed from 9-36 eV, to the AUG5-BP theory summed over the outer valence orbitals (lrc , g  l7t , u  3a ) plus the 2 a orbital g  u  as shown in Fig. 5.3a. The 2o orbital has been left out of the summation on Fig. 5.3a because: g  (1) as can be clearly seen on Fig. 5.1, there is apparently 2o intensity beyond the limit g  of the experimental data which extends out to 59 eV (2) little or no 2 a intensity is predicted [187-191] above 36 eV u  (3) the momentum profile analysis of the 2 a ionization processes giving £ ~ and £ ~ 4  2  u  final ion states as shown in Figs. 5.6 and 5.7 and discussed in Section 5.3.2.4 below accounts for essentially all the intensity below 36 eV.  Chapter 5. Molecular  Oxygen  (4) this range is below the double ionization potential of 0  106  2  - see discussion in Section  5.3.1 above. It should be noted that all the TMPs in Fig. 5.3a are shown at unit pole strength except for the 168-CI momentum profile (curve 4c) which was constructed using the M R S D - C I poles in Table 5.3 each with their respective pole strength (SJ) value. It is clear from Fig. 5.3a that only the density functional theory calculations predict the correct shape of the observed X M P from 9-36 eV, especially in the low momentum region (below (J) = 10°). A shape very different from experiment is predicted at lower momenta (angles) by the Hartree-Fock and CI calculations. The A U G 5 - B P density functional theory calculation which includes both correlation and exchange has been chosen for normalization of the experimental data.  However, the local  (AUG5-L) and correlation-only (AUG5-P) density functional theory calculations give very similar results. The normalization factor obtained in the above procedure for Fig. 5.3a has been used for each individual orbital for all experimental and all theoretical comparisons (see Figs. 5.4-5.7) in the present work. The CI calculation (curve 4c) has much lower intensity than experiment because the chosen normalization to the AUG5-BP curve assumes that virtually all of the ionization intensity for the orbitals lies within the energy range 9-36 eV. The CI calculation however predicts typically only ~ 80% of each ionization manifold in the binding energy range below 36 eV (see Table 5.3) with the "missing" pole strength unaccounted for and presumably located at higher binding energies. In order to treat the CI calculation in a manner consistent with the above normalization scheme, a second curve (4C, equal to 1.23x4c) has been constructed on Fig. 5.3a to provide the best intensity and shape fit between CI and experiment at least in the region above 0 = 10°. At lower <j) angles the CI calculation provides a poor shape fit to experiment. It should be noted that this factor corresponds to the same normalization of CI to experiment as was used for the binding energy spectra in Fig. 5.2. Similarly, the summed experimental and theoretical momentum profiles over the entire measured binding energy range from 9-59 eV (Fig. 5.3b) for the valence orbitals  (lTC , g  l7t , u  Chapter 5. Molecular Oxygen  0  2  107 2r 2u 1r 1u  •  6 - 3 1 1 + G * - R 6-31 1 + G * - U S T O - 3 G - R S T O - 3 G - U MC expt.  4 c 1 6 8 - CI 4r 1 6 8 -R 4u 1 6 8 -U 3r AUG5- R 3u AUG5- U  3 b A U G 5 - BP 3p A U G 5 -P 31 A U G 5 - L 4C = 4c  x1 . 2 3  60 \,3u,4u -2u.3r.4r  (a) O u t e r Valence  I  O  9 - 3 6 eV  3 0 \-  X  c CD  0 100  CD >  CD 5 0 h  cr  0  o  10  20  30  Angle  F i g . 5.3: Summed experimental and calculated spherically averaged momentum profiles for the (a) outer valence + 2a orbital and (b) outer valence + 2a + 2(7g orbitals of O2. The experimental data points are obtained by summing the measured binding energy spectra from 9 to 36 eV (Fig. 5.3a) and 9 to 59 eV (Fig. 5.3b) at each azimuthal angle (momentum). The solid curves are the sum of the respective momentum profiles calculated by several quantum mechanical methods and a wide variety of basis sets (see Section 5.2 for descriptions of the calculations). It should be noted that the summed theoretical curves take into account the gjfactors for quartet and doublet states (see text for details). The momentum resolution has been folded into all calculations using the GW-PG method [59] before summation. The experimental data has been normalized to the AUG5-BP calculation in Fig. 5.3a and this same normalization factor has been applied to the data in Fig. 5.3b and all individual orbital momentum profiles (see Figs. 5.4-5.7). Note that the 168-CI calculation (curve 4c) is also shown as 1.23 x 4c (curve 4C). See text for details. u  u  Chapter 5. Molecular Oxygen  108  3 a , 2 G and 2a ) have also been constructed using the same normalization factor for g  u  g  experiment and theory from Fig. 5.3a. The 168-CI momentum profile (curve 4c in Fig. 5.3b) was constructed using the MRSD-CI poles in Table 5.3 each with their respective SJ value. Good agreement for shape is obtained between experiment and all calculations over the entire angular range of the data in Fig. 5.3b. Although the better Hartree-Fock calculations are close to the summed experimental data, the DFT calculations predict slightly greater intensity than experiment, particularly at the lower momenta. In this regard, additional extra 2 o intensity is g  expected above the upper limit of the experimental binding energy (59 eV) as discussed above. The discrepancy between theory and experiment above 0 = 20° is likely due to distorted wave effects [12] at higher momenta. The 168-CI momentum profile predicts a lower intensity than experiment and this may be due to the remaining pole strength (SJ) which is likely located in the high energy range above 36 eV. In a manner similar to Fig. 5.3a, a second CI curve (4C) has been constructed by multiplying curve 4c by a factor of 1.23. This curve matches the experiment better than curve 4c, although it still does not predict all the experimental intensity. The renormalized MRSD-CI pole strengths (\.23xSJ) are also shown in Table 5.3. The experimental and theoretical momentum profiles corresponding to each of the various ion states associated with the five valence orbitals of 0  2  (see Section 5.3.1) are  presented in Figs. 5.4-5.7 using the above normalization. The energies shown on Figs. 5.4-5.7 correspond to the binding energy peak.maxima of the ion states observed in the present work (see Fig. 5.1 and also Table 5.2). The Hartree-Fock and DFT TMPs are presented at unit pole strength on Figs. 5.4a-d, 5.5a-d and 5.7b-d. In the other panels of Figs. 5.4-5.7, the HartreeFock and DFT TMPs are presented at a pole strength scaled by a factor to shape fit the X M P and to determine an experimental pole strength (the factor required in each case is indicated in each panel).  The A C P F profile has only been calculated for the H O M O (see Figs. 5.4a  and 5.4b) and is presented at unit pole strength. The MRSD-CI theoretical momentum profiles (curves 4C) in Figs. 5.4-5.7 are presented at their renormalized pole strengths (1.23x5^) from Table 5.3.  Chapter 5. Molecular Oxygen  0  2  2r 2u 1r 1u  •  o  109  6 6 S S  -31 1+G*-R -31 1+G*-U T 0 - 3 G - R T 0 - 3 G - U  MC MC  expt expt  binning non-bin  4a 168-ACPF 4C 168-CI x1.23 4r 1 6 8 - R 4u 1 6 8 - U 3r AUG5-R 3u  3b 3p 31  A U G 5 -• B P AUG5-P AUG5- L  AUG5-U  Momentum (au) Fig. 5.4: Measured and calculated spherically averaged momentum profiles for the liig (X^H^) and lrc (a^n , A^riy) orbitals of O2. The solid circles represent the experimental. energy-dispersive multichannel measurements obtained in the present work with the binning mode of the EMS spectrometer. The open circles represent the experimental energy-dispersive multichannel measurements obtained earlier in this laboratory using the non-binning mode of the same EMS spectrometer. All calculations have been spherically averaged and folded with the experimental momentum resolution. See text for details. u  u  Chapter 5. Molecular Oxygen  0  2  110  2r 2u 1r 1u  •  6-311+G*-R 6 - 3 1 1 + G * - U S T 0 - 3 G - R STO-3G-U MC expt  4C 4r 4u 3r 3u  168-CI 168-R 168-U AUG5-R AUG5-U  xl.23  .(a) 2u,3u,4u v2r,3r,4r,4C  3a  b I  18.2  A U G 5 - BP AUG5-P AUG5-L  30g  4  g  3b 3p 31  g  eV  b I4  18. 2  eV  20  0 B  20.3  2  z  g  eV 8  0 4  0 0  Momentum (au Fig. 5.5: Measured and calculated spherically averaged momentum profiles for the 3o~g ( & £ , B £ ) and l7t 4  2  g  g  (2n (3)) orbitals of O2. The solid circles represent the experimental energy-dispersive multichannel measurements obtained in the present work with the binning mode of the EMS spectrometer. All calculations have been spherically averaged and folded with the experimental momentum resolution. See text for details. u  u  Chapter 5. Molecular Oxygen  0  2  111  2r 2u 1r 1u  6-3 6-3 STO STO  1 1 - 3 - 3  MC  expt  •  1+G+-R 1+G*-U G - R G - U  4C 4r 4u 3r 3u  168168168AUG5 AUG5  CI x 1 . 2 3 R U - R - U  3b 3p  31  AUG5--BP AUG5--P A U G 5 - -L  10 2u,3u,4u 2r,3r,4r  (a)  2a c I 2 4 . 5 eV  2cr c Z 2 4 . 5 eV  4  u  HF:  4  u  0.7 i: 4  u  u  DFT:  0.7 i: 4  0 (  C  )  2r,3r,4r, 2u,3u,4u  peak P 2 7 . 4 eV  4  peak P 27.4.eV  3p,3b  DFT: 0.2 IJ"+0.3 I~ 4  o.2 i;r+o.3 iu 4  / \ {e)  2r,3r,4r 2u,3u,4u 1 u, 1 r  z  0  (f)  peak Q 3 3 . 0 eV HF:  _3Z  2  peak Q 3 3 . 0 eV DFT:  0.57 IJ2  0.57 I2  0 0  0  omentum (au Fig. 5.6: Measured and calculated spherically averaged momentum profiles for the 2o~ ( c S , P, Q) orbital of 4  u  u  O2. The solid circles represent the experimental energy-dispersive multichannel measurements obtained in the present work with the binning mode of the EMS spectrometer. All calculations have been spherically averaged and folded with the experimental momentum resolution. See text for details.  Chapter 5. Molecular Oxygen  112  Angle Fig. 5.7: Measured and calculated spherically averaged momentum profiles corresponding to sums of states as 4 2 indicated and for the 2o"g ( Z g , Z g ) orbital of 02- The solid circles represent the experimental energydispersive multichannel measurements obtained in the present work with the binning mode of the EMS spectrometer. All calculations have been spherically averaged and folded with the experimental momentum resolution. See text for details.  Chapter 5. Molecular Oxygen  113  5.3.2.1 The \n Orbital (X Yl Final Ion State) 2  g  g  The experimental momentum profile for the ln orbital of 0 g  (X U 2  2  g  final ion state) is  shown in Figs. 5.4a and 5.4b. The outermost X M P is well separated in energy from the rest of the valence ionization manifold, which ensures that all the observed intensity of the momentum profile is due to ionization to the lowest lying ion state only and contains no mixing with other ionization processes.  The solid circles in Figs. 5.4a and 5.4b represent the experimental  energy-dispersive multichannel measurements obtained for the H O M O in the present work with the binning mode of the EMS spectrometer (see Section 3.1). The open circles in Figs. 5.4a and 5.4b represent experimental energy-dispersive multichannel measurements for the H O M O obtained earlier in this laboratory using the same E M S spectrometer in the non-binning mode (see Section 3.1). In Figs. 5.4a and 5.4b the non-binning data have been renormalized to the presently obtained measurements (solid circles) in the region of the maximum. The l7Cg experimental momentum profile is compared on Fig. 5.4a with the various theoretical momentum profiles (TMPs) calculated at the U H F and ROHF Hartree-Fock level and also with the MRSD-CI and A C P F ion-neutral overlaps as described in Section 5.2 above. Selected properties for each calculation and corresponding experimental values are shown in Table 5.1. The Hartree-Fock basis sets used range from a minimal STO-3G basis to the very much larger 168-GTO basis set.  The effects of many-body correlation and electronic  relaxation are also seen on Fig. 5.4a from the MRSD-CI and A C P F calculations of the TMPs. The 168-CI momentum profile in Fig. 5.4a corresponds to the lT MRSD-CI pole at 12.30 eV 2  g  (Table 5.3).  The slightly higher cross-section than is expected from theory or symmetry  considerations at very low p was persistently observed. The reason for this is not at present understood. Calibration using the A r 3p orbital showed the expected nodal behaviour at the lowest momenta. As might be expected, there is a large disagreement between the STO-3G minimal basis set calculations (curve l u and lr) and the observed shape and intensity of the X M P .  The  intermediate size basis set 6-311+G*-U calculation (curve 2u) gives better agreement with  Chapter 5. Molecular Oxygen  114  experiment than the ST0-3G calculations but it still underestimates the p < 1 au region of the X M P . The larger size basis set U H F calculations (curves 3u and 4u) are in better agreement with experiment with regard to shape and the similarity of curves 3u and 4u suggests that the U H F momentum profiles are essentially converged at this level of basis set size. Similarly, the higher level ROHF momentum profiles generally agree better with experiment as the basis set size is increased. The ROHF momentum profiles with intermediate and large size basis sets for this orbital generally display a higher relative intensity at low momentum than U H F calculations (of the same basis set size). The converged ROHF momentum profiles (curves 3r and 4r) match the shape and intensity of the X M P well but it should be noted that the U H F method predicts some other electronic properties (such as total energy, see Table 5.1) better than the R O H F method. It should be noted that, on the basis of the present normalization, the CI (curve 4C) and A C P F (curve 4a which is shown at unit pole strength) calculations fit the shape of the X M P comparably well and only slightly overestimate the intensity. The density functional momentum profiles for the H O M O  l7C  orbital (X Tl 2  g  g  final ion  state) are shown on Fig. 5.4b. It can be seen that all three DFT calculations give quite similar results and display slightly more intensity at low momentum than is observed experimentally. The M R S D - C I and A C P F calculations (curves 4C and 4a) most closely fit the experimental data for shape and intensity. However, all DFT calculations for the  l7t  g  orbital also provide a  good description of the shape of the X M P .  5.3.2.2  T h e l7C O r b i t a l (a Il , A Yl 4  u  2  u  u  and  2  n (3) u  F i n a l I o n States)  Experimental and theoretical momentum profiles for the  l7C  u  orbital of 0  (a n final 4  2  u  ion state) are shown in Figs. 5.4c and 5.4d. The best results for shape and intensity compared to the X M P are clearly the intermediate and large basis set U H F calculations of A U G 5 - U (curve 3u) and 168-U (curve 4u). The ROHF calculations from these basis sets are adequate for shape comparison, but the intensity is much too low. This is in contrast to the H O M O (Fig. 5.4a) and the results for the HOMOs of N O and N 0  2  in Chapter 4 where ROHF  Chapter 5. Molecular  Oxygen  115  calculations gave better shape comparisons with the X M P than U H F calculations. As noted above, the 168-CI momentum profile in Figs. 5.4c and 5.4d corresponds to the I1 MRSD-CI 4  U  pole predicted at 16.81 eV (Table 5.3). This 168-CI calculation predicts results (curve 4C) that are closer in intensity and shape to the intermediate and large basis sets R O H F results than U H F calculations. The DFT momentum profiles (Fig. 5.4d) are more intense than the 168-CI calculation (curve 4C), but are still unable to account for the experimental intensity at p < 1.2 au. The D F T momentum profiles predict the shape of the observed X M P well but not the intensity. If the density functional TMPs are multiplied by a factor of ~ 1.2, an excellent shape fit with experiment is obtained. The choice of functional has little effect on the overall shape of the TMPs. Experimental and theoretical momentum profiles for the corresponding to the  A n 2  u  l7t  u  orbital of  final ion state are shown in Figs. 5.4e and 5.4f.  0  2  The 168-CI  momentum profile in Figs. 5.4e and 5.4f corresponds to the II MRSD-CI pole at 17.98 eV 2  U  (Table 5.3). The statistics for this final ion state are very limited due to the low intensity of this state and the difficulty in deconvoluting it from the energetically close and intense a Tl 4  u  and  b 2,~ states. The Hartree-Fock and DFT theoretical momentum profiles in Figs. 5.4e and 5.4f 4  have been scaled by a factor of 0.4 to provide reasonable agreement with the intensity of the X M P and because of the additional  l7t  u  satellite intensity at higher energy discussed below  (i.e. the n (3) peak at 23.7 eV). Within the rather limited precision of the experimental data, 2  u  this experimentally obtained pole strength of 0.4 derived from a comparison of the intensities of the X M P and TMPs is consistent with the theoretical estimates of 0.44 by Cederbaum and Domcke [65], 0.34 by Dixon and Hull [186] and 0.35 by Honjou et al. [187,188]. The MRSD-CI calculation of the present work (see Table 5.3) predicts a normalized pole strength of 0.373 in excellent agreement with the experimental value. It should also be noted that the ROHF momentum profiles in Fig. 5.4e generally predict more intense TMPs than their U H F counterparts. The 168-CI momentum profile (curve 4C) predicts a shape and intensity more  Chapter 5. Molecular  Oxygen  116  similar to the ROHF limit curves 3r and 4r than the U H F limit curves 3u and 4u. The DFT momentum profiles (Fig. 5.4f) are very similar in shape and intensity to the 168-CI calculation. The experimental and theoretical momentum profiles for the  l7C  u  orbital corresponding  to the n (3) satellite state are shown in Figs. 5e and 5f. As expected, the X M P and TMPs are 2  u  similar to the momentum profiles corresponding to the parent  A n 2  u  state at 17.5 eV (see  Figs. 5.4e and 5.4f). It should also be noted that the 168-CI momentum profile in Figs. 5.5e and 5.5f corresponds to the n 2  u  MRSD-CI pole at 24.23 eV (Table 5.3). The peak is of low  E M S intensity and is close in energy to the peak corresponding to the c E~ state and thus 4  deconvolution leads to some scatter in the data. The satellite splitting has been accounted for by scaling the Hartree-Fock and DFT TMPs by a factor of 0.55 to give the large basis set ROHF calculations (curves 3r and 4r) reasonable agreement with the experimental intensity. Other choices could be made for the normalization of experiment to theory, but curves 3r and 4r were chosen since ROHF momentum profiles agreed well for the intensities of other doublet states (see Figs. 5.4a and 5.5c). It should be noted that this experimentally derived pole strength of 0.55 is identical to the calculated pole strength reported by Cederbaum and Domcke [65]. The experimental pole strength from Figs. 5.5e and 5.5f also agrees quite well with the M R S D - C I calculation of the present work which predicts a normalized  2  ri  u  pole  strength of 0.589 at 24.23 eV (see Table 5.3). Dixon and Hull [186] predicted a slightly higher pole strength of 0.64.  On comparing the relative fits of the various TMPs in Figs. 5.5e  and 5.5f, it is obvious that the minimal basis set STO-3G-U and STO-3G-R profiles are in poor agreement with the measured intensity and shape of the X M P .  The large basis set U H F  calculations (curves 3u and 4u) and the 168-CI calculation (curve 4C) predict reasonable shapes for the X M P , but have low intensity with the present normalization factor of 0.55. The DFT momentum profiles are all very similar and all fit the experiment well within the limited experimental statistics for this ionization process. The DFT momentum profiles adequately predict the experimental intensity in a manner similar to the large basis set ROHF calculations (curves 3r and 4r).  Chapter 5. Molecular  Oxygen  117  Previous work [185,192] assigned peak Q as I1 2  U  in character.  However, a  consideration of the theoretical and experimentally derived I1 pole strengths rules out the 2  U  possibility that the 33 eV peak consists of l7C processes. The MRSD-CI theory of the present u  work (Fig. 5.2 and Table 5.3) predicts three n  poles below ~ 24.3 eV. The M R S D - C I poles  2  u  at 17.98 eV (normalized pole strength 0.373) and 24.23 eV (normalized pole strength 0.589) can be identified with the  An  and n (3) states observed in the PES [174,177,179]. As  2  2  u  u  mentioned previously, the n (2) state has not been observed experimentally but has been 2  u  predicted to be formed with a very low cross-section at ~ 20 eV [65,186,189,190]. The weak 2  n  M R S D - C I pole at 19.16 eV in Table 5.3 (normalized pole strength of 0.005) can be  u  identified with this predicted n (2) state. The three MRSD-CI n 2  2  u  u  energies and normalized  pole strengths (0.373, 0.005 and 0.589) below 24.3 eV in the present work are in reasonable agreement with previous predictions [65,186-188]. Cederbaum and Domcke [65] gave  A n , n (2) 2  2  u  u  The propagator technique work of  and IT (3) energies of 17.47, 19.75 and 2  U  27.32 eV with relative intensities of 0.44, 0.01 and 0.55. The semi-empirical CI calculations of Dixon and Hull [186] predicted energies of 17.56, 19.97 and 22.72 eV with relative intensities of 0.34, 0.001 and 0.64. Honjou et al. [187,188] predicted energies of 17.84, 20.7 and 24.44 eV and relative intensities of 0.35, 0.000, and 0.47 for these states, respectively. In addition, Yeager and co-workers [189,190] obtained calculated ionization energies of 17.77, 20.82 and 24.56 eV for the  A n , n (2) 2  2  u  u  and ri (3) ionic states, respectively. It should be 2  u  noted that all previous calculations [65,186-190] and the present M R S D - C I results place virtually all the I1 intensity below ~ 27.3 eV and, in most cases, below 24.5 eV. However, 2  U  the II assignments of the previously published theory [65,186-190] and the presently reported 2  U  calculation differ from those of Suzuki et al. [185] who designated the observed 23.7 eV peak as n (2) and then used this assignment to characterize the peak at ~ 33 eV as n (3) on the 2  2  u  u  basis of the comparison of experimental intensity ratios with those from theory [65,186-188] and the theoretical study of Gerwer et al. [192]. It is suggested in the present work that the theoretical I1 intensity ratios were not compared with the correct experimental intensities in 2  U  Chapter 5. Molecular  Oxygen  118  the previous E M S work [185] due to the predicted I1 (2) intensity at ~ 20 eV being 2  U  incorrectly compared with the 23.7 eV experimental peak (and thus the theoretical 2  n (3) intensity being incorrectly compared with the 33 eV experimental peak intensity). u  The ionization energies and pole strengths listed above from previously published theory [65,186-190], the MRSD-CI calculation of the present work and the photoelectron study by Baltzer et al. [179] all indicate that the peak at 23.7 eV in the E M S binding energy spectra should be assigned as II (3) and that no significant I i 2  2  U  u  intensity is present at 33 eV. The  only predicted MRSD-CI TI pole above 24.3 eV is of too low intensity (0.028 normalized 2  U  pole strength) and too high energy (35.54 eV) to account for peak Q. In addition, while Gerwer et al. [192] included the 33 eV peak in the total  l7t  u  photoionization cross-section for  comparison of their theory to experiment, the agreement was only moderately good and several correction procedures were required for the summation of unresolved peaks in the dipole (e, 2e) and line-source data. In the present work, the experimentally derived pole strengths for the peaks corresponding to the A I I 2  and II (3) states are found to be 0.4 and 0.55, 2  U  U  respectively (Figs. 5.4e, 5.4f, 5.5e and 5.5f) and even with the limited statistics for these profiles these values correspond well with theory [65,186-188] and the presently reported MRSD-CI calculation. Thus, at least 0.95 of the experimental II 2  U  pole strength has been  accounted for at energies much lower than 33 eV and this represents essentially all the I1 2  U  pole strength, given the experimental statistics.  5.3.2.3 The 3a Orbital (b lr and B Ir Final Ion States) A  g  2  g  g  The experimental and theoretical momentum profiles for the 3 a orbital of 0 g  2  (b YT A  %  final ion state) are shown in Figs. 5.5a and 5.5b. A l l profiles are similar in shape except that the STO-3G calculations (curves l u and lr) are in serious disagreement, as might be expected. The intermediate and large ROHF calculations (curves 2r, 3r and 4r) show much better agreement with the shape and intensity of the observed X M P than the STO-3G-R calculation. It should be noted that the 168-CI momentum profile in Figs. 5.5a and 5.5b corresponds to the  Chapter 5. Molecular  4  Oxygen  119  S " M R S D - C I pole at 18.23 eV (Table 5.3). The MRSD-CI calculation (curve 4C) is formed  from an initial ROHF reference state and thus the intensity of the 168-CI momentum profile is similar to curves 2r, 3r and 4r. Similar to the X M P corresponding to the quartet (a II ) ionic 4  u  state in Figs. 5.4c and 5.4d, the best fit to the shape and intensity of the X M P corresponding the quartet & £~ ion state in Fig. 5.5a is given by the higher level U H F calculations (curves 2u, 3u 4  and 4u). However, these U H F TMPs still underestimate the experimental intensity slightly. The high momentum region (p > 1 au) is modelled reasonably well by all calculations. Excellent agreement for both shape and intensity is obtained for the D F T and experimental momentum profiles (Fig. 5.5b). A l l the DFT profiles are very similar in shape and intensity and provide the best match to the experimental intensity. The experimental and theoretical momentum profiles for the 3 a  g  orbital of 0  2  corresponding to the B TT final ion state are shown in Fig. 5.5c and 5d. Both the STO-3G-U 2  g  and STO-3G-R momentum profiles are poor predictions of the X M P shape and intensity. The higher level U H F calculations (curves 2u, 3u and 4u) show reasonable agreement with the shape of the observed X M P but slightly underestimate its intensity. In contrast, the ROHF calculations (curves 2r, 3r and 4r) show excellent agreement with the experimental shape and intensity, particularly in the low momentum region.  It appears that large basis set U H F  calculations are best for predicting the shape and intensity of X M P s associated with quartet final ion states (Figs. 5.4c, 5.4d, 5.5a and 5.5b) while large basis set ROHF calculations are a better model for the X M P s corresponding to doublet final ion states (Figs. 5.4a, 5.4b, 5.5c and 5.5d). The 168-CI momentum profile (curve 4C) in Figs. 5.5c and 5.5d corresponds to the 2  E~ MRSD-CI pole at 20.53 eV (Table 5.3).  This MRSD-CI momentum profile provides  excellent agreement with the experimental shape and intensity of the X M P .  The DFT  momentum profiles are too intense (Fig. 5.5d) although the shape of the X M P is adequately predicted.  Chapter 5. Molecular Oxygen  120  5.3.2.4 The 2c Orbital ( c Z ~ Final Ion State, Peak P and Peak Q ) 4  u  The experimental and theoretical momentum profiles for the 2 a orbital of 0 (c £~ 4  u  final ion state) are shown in Figs. 5.6a and 5.6b.  2  The E ~ MRSD-CI pole at 24.71 eV 4  (Table 5.3) has been chosen for the calculation of the 168-CI momentum profile (curve 4C). It is clear from Figs. 5.6a and 5.6b that the experimental and theoretical intensities are very different since the Hartree-Fock and DFT TMPs as shown have been multiplied by a factor of 0.7 to match the experimental intensity. This experimentally estimated pole strength of 0.7 is in good agreement with the (renormalized) pole strength of 0.802 at 24.71 eV calculated by the M R S D - C I method (Table 5.3). It is suggested that the ~ 30% difference in intensity between the TMPs and the X M P in Figs. 5.6a and 5.6b is due to splitting of the 2 c ionization process u  into additional £ ~ satellite states or "poles" due to electron correlation effects. 4  Considering next the broad peak Q at 33 eV, it should be noted that no E ~ states are 4  predicted in this energy region by the present MRSD-CI calculations (see Table 5.3) or by any of the other earlier published theoretical studies [187-190]. It should also be noted that the region of the binding energy spectrum (Fig. 5.1) above 36 eV is dominantly due to s-type (2fJ )" processes as discussed in Section 5.3.1 and therefore any contributions from 1  g  p-type processes (i.e. E ~ and £~) in this region are very small or negligible. Peak Q must 4  2  therefore contain essentially only E ~ states and a 0.57 E~ fraction of the Hartree-Fock and 2  2  DFT TMPs is found to fit the X M P for this peak (Figs. 5.6e and 5.6f). This fraction is in quite good agreement with the (renormalized) MRSD-CI £~ pole strength of 0.461 predicted at 2  33.89 eV (Table 5.3). Honjou et al. [187,188] also predicted strong Z ~ poles at 34.9 and 2  36.6 eV while Yeager and co-workers [189,190] identified the experimentally observed peak in the region of 33 eV with a E~ ionization process calculated at 34.70 eV. 2  The above analysis leaves experimental pole strengths of 0.3 £~ and 0.43 £~ 4  2  unaccounted for and this intensity must therefore be mostly in peak P at 27.4 eV since all other regions of the binding energy spectrum have been assigned (see Section 5.3.2.5 and Fig. 5.7 below for a discussion of the (2a,,)" contributions which are located above 36 eV). Peak P is 1  Chapter 5. Molecular  Oxygen  121  well-fitted by the T M P linear combination of (0.2 E and 0.3 Z ) as shown in Figs. 5.6c and 4  2  U  U  5.6d. The above analysis of the c E~ ionic state and peaks P and Q therefore account for 4  ~ 90% (0.7+0.2 for %  and 0.57+0.3 for % ) of the pole strength of the %  and  %  manifolds. The above assignment of peak P is further supported by the present M R S D - C I and earlier calculations [187-190] which all predict both £ ~ and X~ ionization strength in the 4  2  region of ~ 27 eV with relatively minor additional strength located in the higher energy region above 36 eV. Finally, it should be noted that 0.72 of the £ ~ pole strength would be needed for 4  a good fit if peak P was due to the quartet state and this is clearly not the case since ~ 0.7 of the 4  E~ pole strength is already contained in the c £~ state at 24.5 eV. 4  Figs. 5.7a and 5.7b give further evidence that almost all of the X ~ and E ~ pole 4  2  strength is accounted for if there are significant X~ processes in peak P and if peak Q has 4  mainly Z~ character. Fig. 5.7a shows the experimental data for the peak corresponding to the 2  c E~ ionic state (24.5 eV) and peak P (27.4 eV) added together and compared to several high4  level calculated 2 a  u  momentum profiles.  The individual Hartree-Fock and D F T TMPs  corresponding to £ ~ final ion states are included in the sum with a unit pole strength while a 4  pole strength of 0.3 has been used for the TMPs corresponding to Z ~ final ion states. The CI 2  calculation (curve 4C) in Fig. 5.7a consists of the summation of the poles at 24.71, 27.69 and 28.46 eV, each with their respective renormalized SJ values.  Good shape and intensity  agreement is obtained for the TMPs and summed X M P in Fig. 5.7a, especially for the CI calculation. In Fig. 5.7b, the experimental data for the peak at 24.5 eV corresponding to the c E~ ionic state, peak P (27.4 eV) and peak Q (33 eV) have been summed and compared to 4  several high-level Hartree-Fock and DFT 2 a momentum profiles corresponding to S~ and 4  u  2  E~ final ion states each with unit pole strength. The CI calculation (curve 4C) in Fig. 5.7b  consists of the summation of the poles at 24.71, 27.69, 28.46 and 33.89 eV, each with their respective renormalized SJ values.  Within the statistics of the data it can be seen that  experiment and theory are in good agreement for total intensity, indicating that essentially all of  Chapter 5. Molecular Oxygen  122  the £ ~ and E~ pole strength lies within the energy region containing the three peaks (c Z~, P 4  2  4  and Q).  5.3.2.5 The 2a Orbital ( E " and Z ~ Final Ion States) 4  2  g  The experimental and theoretical momentum profiles for the £ ~ and E ~ parent and 4  2  satellite states from 2rj orbital ionization are shown in Figs. 5.7c and 5.7d. Siegbahn et al. g  [183] gave vertical ionization potentials of 39.6 and 41.6 eV for the E ~ and E ~ states, but the 4  2  present E M S work gives slightly lower ionization potentials (Table 5.2). Gardner and Samson obtained ionization potentials of 39.7 and 40.4 for these states [180]. Due to the limited knowledge of the positions and widths of these states, an energy slice from 36-43 eV (open circles in Figs. 5.7c and 5.7d) encompassing the main peak centered at ~ 39 eV has been taken. The same normalization factor determined for all other orbitals (see Fig. 5.3) has been used in Figs. 5.7c and 5.7d. The MRSD-CI E ^ and Z " poles at 40.17 and 40.86 eV (Table 5.3) have 4  2  been chosen for the calculation of the 168-CI momentum profile (curve 4C) since their energies and pole strengths correspond most closely with the experimentally observed parent ion states. While the X M P is clearly s-type, it does not match the theoretical intensity and thus additional 2rj ionization intensity must occur in higher energy poles. A l l calculations (Hartree-Fock, g  MRSD-CI and DFT) model the experimental shape and intensity in a similar manner.  A  second energy slice from 43-59 eV (filled triangles in Figs. 5.7c and 5.7d) shows the same s-type shape and this is consistent with previous observations [183,185] indicating that the higher energy region is predominantly composed of (2ag)  _1  poles.  The peak centered at  ~ 47 eV accounts for the majority of the intensity in the 43-59 eV region (see Fig. 5.1) and the states assigned as £~(2) and E~(2) are found within this peak (see Refs. [183,185] and also 4  2  Table 5.2). Results from the MRSD-CI calculation in the present work only extend to ~ 42 eV and do not take into account the very high energy region because of complications involving the modelling of double ionization. While the shape of the 36-59 eV binding energy slice  Chapter 5. Molecular  Oxygen  123  (filled circles in Figs. 5.7c and 5.7d) agrees well with that of the summed theory for 2 o  g  ionization, some experimental intensity is apparently missing. However, as noted in Section 5.3.2, there is still missing intensity (Fig. 5.3) at a binding energy of 59 eV and therefore additional 2 o intensity remains at higher binding energies. g  5.3.3 Density Maps in Momentum and Position Space The momentum and position space density contour maps corresponding to each of the five valence orbitals of an oriented 0  2  molecule are presented in Figs. 5.8 and 5.9. These  density maps provide some insight into the corresponding (spherically averaged) experimental and theoretical momentum profiles. The position space maps are slices of the orbital electron 2  density y/  through a plane containing both nuclei.  The momentum space maps show  momenta perpendicular and parallel to the molecular plane. The calculations are based on the 168-R near-Hartree-Fock limit results which have been found to provide generally good descriptions of the X M P s of 0  2  at the Hartree-Fock level. The origin for the position space  maps is the molecular center of mass and all dimensions are in atomic units. The side panels on each map show slices of the momentum or position space density along the vertical (right panel) and horizontal (top panel) dotted lines. The momentum and position density maps for the  l7i  g  orbital of 0  2  are shown in  Figs. 5.8a and 5.8b. Simple molecular orbital Hartree-Fock models describe this M O as an antibonding n* orbital involving out-of-phase (destructive interference) overlap of atomic O 2p orbitals. The expected shape of a pure n* orbital can be seen from the r-map in Fig. 5.8b (i.e. four lobes of density and nodal planes both along the internuclear axis and perpendicular to the internuclear axis). The highly nodal character of this orbital in r-space is also reflected in p-space (r-space symmetry characteristics are conserved upon transformation to p-space and thus there are also two nodal planes in the p-map).  In view of this high degree of nodal  124  Chapter 5. Molecular Oxygen  MOMENTUM DENSITY  POSITION DENSITY  MOMENTUM DENSITY  POSITION DENSITY  0  (au)  4  8  -  8  -  4  0  (au)  Fig. 5.8: Momentum and position density contour maps for the lftg, l7t and 3cjg orbitals of an oriented molecule calculated at the Hartree-Fock level using the 168-R basis set. The contour values represent 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density. The side panels (top and right side) show the density along the dashed lines (dashed vertical and horizontal lines) for each density map. u  Chapter 5. Molecular  Oxygen  MOMENTUM  125  DENSITY  POSITION  DENSITY  o  '(b)  2  2cr  u  h  1P -4  0  4  1  (au)  MOMENTUM  DENSITY  POSITION  DENSITY  o  '(d)  —I  |-"~~  2  •  „  i.  0  •  0  (au)  Fig. 5.9: Momentum and position density contour maps for the 2a and 2o"g orbitals of an oriented 0 molecule calculated at the Hartree-Fock level using the 168-R basis set. The contour values represent 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density. The side panels (top and right side) show the density along the dashed lines (dashed vertical and horizontal lines) for each density map. u  2  Chapter 5. Molecular  Oxygen  126  behaviour it is not surprising that the experimental and theoretical momentum profiles for this orbital display such a high p  m a x  value of 1.0 au in view of the relation  p=ihd(y/)/dx  [145,199]. The interference of wavefunctions on the atomic centers is also reflected in the momentum density map as additional lobes called "bond oscillations" [78,200,201]. The momentum and position density maps for the Figs. 5.8c and 5.8d. The  l7t  u  l7C  u  orbital of 0  2  are shown in  orbital is a bonding orbital resulting from in-phase (constructive  interference) side-on overlap of atomic O 2p orbitals. The expected two-lobed shape of the orbital can be seen from the r-map (Fig. 5.8d). Because the overlap is off the internuclear axis and because the lobes are of opposite sign, a nodal plane exists along this axis. Similarly, one nodal plane is observed in the p-map in the p  para  n i direction (Fig. 5.8c). The momentum e  density at p = 0 is identically zero so that all (non-momentum resolution folded) profiles for this orbital have zero intensity at p = 0. The small intensity near p = 0 in the TMPs shown in Figs. 5.4a and 5.4b is due to momentum resolution folding effects. The momentum and position density maps for the 3 a orbital of 0 g  2  are shown in  Figs. 5.8e and 5.8f. Constructive overlap of atomic orbital components is clearly seen in the region between the O atoms to produce a a bond. The multiple lobes in the p-map are further evidence of this bonding behaviour. The large amount of momentum density at p = 0 (see top and side slices in the p-map) indicates the strongly s-type character of the experimental and theoretical momentum profiles for this orbital (see Figs. 5.5a-d).  In addition, the small  secondary maxima in the momentum density (see side slice in the p-map) are reflected in the spherically averaged momentum profiles as small p-type behaviour with maxima of p ~ 1.4 au. The 2 a orbital of 0 u  2  is predicted from molecular orbital theory to be antibonding and  this is reflected by the momentum and position density maps for this orbital (Figs. 5.9a and 5.9b). The r-map displays a nodal plane perpendicular to the internuclear axis. Strong multiple lobes are observed in the p-map and no momentum density is observed at p = 0.  Chapter 5. Molecular  Oxygen  127  Maxima in the momentum density observed at p ~ 0.8 au in the side slice of the p-map are reflected in the p-type shapes of the observed X M P and TMPs for this orbital. The weakly bonding character of the 2 o orbital of 0 g  2  is indicated by the a density  between the oxygen atoms in the r-map (Figs. 5.9c and 5.9d). The inner valence 2 a orbital is g  mostly composed of atomic-like O 2s character. On comparing with the density maps for the 3 a orbital (Figs. 5.8e and 5.8f) it is also clear that the 2 a is more spatially contracted (and g  g  hence more diffuse in momentum space) than the 3 a orbital. This is reflected by the broader g  momentum profiles for 2 a ionization in Figs. 5.7c and 5.7d than for the momentum profiles g  for 3 a ionization in Figs. 5.5a-d. The observed strongly s-type momentum profiles from 2 a  g  ionization are also reflected by the presence of a large amount of momentum density in the 2 a  g  g  p-map at p = 0.  Chapter 6. The HOMOs ofPH ,  PF and  3  3  P(CH )  128  3 3  Chapter 6  T h e H O M O s of P H , P F a n d P ( C H ) 3  3  3  3  6.1 Introduction Inductive effects in N H , N F and the methyl-substituted amines N ( C H ) H _ where 3  3  3  n  3  n  n = 0-3, have been studied previously by E M S and quantum mechanical calculations [39,70,80-82,202].  According to intuitive hybridization concepts based on simple Valence  Bond Theory, the "lone pair" on the N atom would be expected to become more "p-type" as the ligand-N-ligand bond angle increases. Since the bond angles of N ( C H ) , N H and N F are 3  3  3  3  111°, 107° and 102°, respectively, these intuitive concepts predict that the lone pair of N ( C H ) should show a higher degree of p-type character than the lone pair in N H while the 3  3  3  lone pair in N F should show increased s character relative to N H . The experimental 3  3  momentum profile (XMP) for N F was in qualitative agreement with these intuitive 3  expectations, but the results for N ( C H ) were completely contrary to those predicted [81,82]. 3  3  However, the correct qualitative trends were given by even the simplest M O (delocalized) treatments such as STO-3G minimal basis set calculations [81]. Improved basis sets gave better quantitative agreement [82]. The increase in s-type character for N F compared to N H 3  3  was attributed to an increase in the N 2s contribution to the molecular orbital while the observed increase in s-character upon methyl substitution was related to the large H Is contribution trans to the nitrogen "lone pair" [81,82]. Upon further methylation, more trans-H Is character contributes to the molecular orbital and thus, the 3aj orbital of N H loses its 3  localized "lone pair" character and becomes increasingly delocalized with each methyl substitution [81,82].  It was also suggested [81,82] that this derealization of the H O M O  Chapter 6. The HOMOs ofPH , 3  PF  andP(CH )  3  3  129  3  charge density caused by methyl substitution implies that the methyl group is an electronwithdrawing substituent (relative to hydrogen) when bonded to N . Other E M S studies on methyl substitution in the series H C O , C H C O H and ( C H ) C O [83] and H 0 [69], C H O H 2  3  3  2  2  3  [84] and CH3OCH3 [85,86] have also shown this derealization effect of the methyl group on the frontier orbitals. These results are in contrast to the well-known and often invoked methylinductive effect in organic chemistry, where concepts such as basicity and the stability of carbocations have traditionally been rationalized by considering the methyl group to be electron-donating relative to hydrogen [203]. However, consideration of other properties such as the decrease in the dipole moment of substituted amines upon increasing methylation give further evidence of the electron-withdrawing ability of the methyl group [82]. The latter view is also strongly supported by the  1 4  N and  1 5  N N M R chemical shifts which are moved  downfield by increasing methylation, thus indicating a deshielding of the nitrogen nucleus [204,205]. Much earlier low level M O calculations by Hehre and Pople [206] also suggested that methyl groups are electron attracting relative to hydrogen when bonded to nitrogen or oxygen. To further investigate the inductive effects of the methyl group relative to hydrogen and fluorine, the experimental momentum profiles for the HOMOs of P F and P ( C H ) are studied 3  3  3  in the present work and compared with experimental results for P H previously reported from 3  this laboratory [74]. A knowledge of whether methyl groups are intrinsically electron-donating or electron-attracting in these phosphorus compounds and the behaviour relative to the corresponding nitrogen compounds is of importance in predicting the general properties of alkyl groups in many compounds and chemical reactions. Furthermore, intuitive arguments based on geometry and hybridization concepts would predict the so-called "lone-pair' of a P X  3  molecule to range from 100% s-type character (for an X P X bond angle of 90°) to 100% p-type character (for an X P X bond angle of 120°, i.e. when the molecule is planar). The experimental ligand-P-ligand bond angles are 98.6°, 96.9° and 93.3° for P ( C H ) , P F and P H , respectively 3  3  3  3  [207-209]. Thus, intuitive arguments predict the HOMOs of P ( C H ) and P F to contain more 3  3  3  Chapter 6. The HOMOs ofPH , 3  PF  3  andP(CH ) 3  130  3  p-type character than the H O M O of P H . However, the bond angles in the phosphorus 3  compounds are smaller than in the corresponding amines and thus, according to intuitive concepts, the substituted phosphorus compounds should show greater s-type character in their H O M O electron momentum profiles than in their nitrogen analogs. The experimental momentum profiles for P H [74], P F and P ( C H ) are compared to 3  3  3  3  theoretical momentum profiles at the target Hartree-Fock approximation (THFA) level (see Section 2.2.5).  For P H , a post-Hartree-Fock momentum profile from the multi-reference 3  single and doubles configuration interaction (MRSD-CI) method taken from the work of Clark et al. [74] is also presented in order to assess the role of electron correlation and relaxation effects. Due to the increasing difficulty of applying the Hartree-Fock and CI methods to larger molecules, the ability of density functional theory (DFT) calculations to predict momentum profiles for P X molecules is also investigated in the present work. 3  As discussed in  Section 2.2.6, DFT allows considerable computational savings for calculations on larger molecules [60] such as those studied in the present work. Thus, T K S A - D F T calculations of the momentum profiles using (i) the local density approximation of Vosko et al. [112], (ii) the nonlocal correlation functional of Perdew and Wang [115,117] and (iii) a non-local combination of the Perdew correlation functional and Becke exchange functional [116] have been carried out for the HOMOs of all three molecules in the present work. E M S momentum profile measurements for the HOMOs of P F and P ( C H ) 3  3  3  were  obtained with a single channel E M S spectrometer [78] at binding energies of 12.2 eV and 8.6 eV, respectively. The P F sample was obtained from Ozark Mahoning gas products while 3  the P ( C H ) sample was obtained from Aldrich Chemicals. Both samples were of > 99.0% 3  purity.  3  Chapter 6. The HOMOs ofPH , 3  PF  3  andP(CH ) 3  131  3  6.2 Calculations 6.2.1 Hartree-Fock Calculations Spherically averaged theoretical momentum profiles have been calculated within the plane wave impulse and the target Hartree-Fock approximations (via Eq. (2.57)) for the H O M O of each of the three molecules using several basis sets of varying quality. In addition to  p , max  the maximum of the theoretical or experimental momentum profile, other calculated and experimental electronic properties for the three molecules are also listed in Table 6.1. The experimentally derived geometries for P H [207], P F [208] and P ( C H ) [209] were used in 3  all calculations and C  3 v  3  3  3  molecular symmetry was assumed for each molecule.  Details of the calculation methods are described below. The total number of contracted Gaussian-type orbital functions (CGTO) used is also given. A l l calculations were performed at the University of British Columbia using the Gaussian 92 program [124] with the exception of the 136-GTO calculation which is taken from the work of Clark et al. [74].  (1) S T O - 3 G : A calculation employing a minimal basis set (effectively single zeta).  Each  function is a contraction of three Gaussian functions and thus it consists of P(9s,6p)/[3s,2p], C/F(6s,3p)/[2s,lp] and H(3s)/[ls] contractions. Therefore, a total of 12 C G T O are employed for P H , 24 C G T O for P F and 33 CGTO for P ( C H ) . This basis was designed by Pople and 3  3  3  3  co-workers [101]. (2) 4-31G: This is a split-valence basis developed by Ditchfield, Hehre and Pople [102]. It uses a minimal basis for each of the core orbitals on P, C and F and essentially a double zeta description of the valence orbitals. This basis set has limited flexibility, with only one extra s and p wavefunction added to account for polarization and diffuseness. P(12s,8p)/[4s,3p], C/F(8s,4p)/[3s,2p] and H(4s)/[2p] contraction.  It consists of a  The number of C G T O is  then 19 for P H , 40 for P F and 58 for P ( C H ) . 3  3  3  3  (3) 6-311G: The 6-31 I G basis of Pople and co-workers [103] is a split-valence basis  Chapter 6. The HOMOs ofPH , 3  PF  3  andP(CH )  132  3 3  Table 6.1: Calculated and experimental properties for P H , P F and P(CH ) 3  Molecule^  PH  Wavefunction  1 2 3 4 5 6 7 8 8c  3  61  6p 6b  PF  1 2 3 4 6 7  3  61  6p 6b  P(CH ) 3  1 2 3 4 5 6 7 6/ 6p 6b  3  3  3  Total Energy (hartree)  (Debye)*  STO-3G 4-31G 6-311G 6-311+G* 6-311++G** AUG3 AUG5 136-GTO 136-G(CI) AUG3-L AUG3-P AUG3-BP Experimental  -338.634 -342.022 -342.423 -342.468 -342.477 -342.486 -342.493 -342.493 -342.683 -341.854 -343.333 -343.174 -343.42  0.537 1.171 1.159 0.757 0.755 0.684 0.665 0.665 0.624 0.701 0.528 0.592 0.578/'  STO-3G 4-31G 6-311G 6-311+G* AUG3 AUG5 AUG3-L AUG3-P AUG3-BP Experimental  -631.019 -638.294 -639.059 -639.230 -639.287 -639.327 -637.981 -641.554 -641.144  1.334 2.723 2.680 1.678 1.246 1.228 0.907 0.897 0.866 1.13*  STO-3G 4-31G 6-311G 6-311+G* 6-311++G** AUG3 AUG5 AUG3-L AUG3-P AUG3-BP Experimental  -454.401 -458.983 -459.517 -459.609 -459.623 -459.642 -459.656 -458.744 -461.476 -461.202  0.458 1.192 1.606 1.429 1.438 1.461 1.464 1.405 1.301 1.384 1.192  d  Pmax (au) c  7  0.62 0.55 0.42 0.47 0.47 0.45 0.44 0A5 0A5  d  d  d  d  e  3  d  0.44 0.43 0.45 0.5  d  A  Calculations used the experimental equilibrium geometry for P H [207], P F [208] and P(CH ) [209]. b Calculated dipole moments are for a non-relativistic, non-vibrating, non-rotating molecule. The p corresponds to the value of momentum where the intensity of the momentum profile is at maximum (see Fig. 6.1). Ref. [74]. Ref. [210]. /Ref. [224]. S Ref. [208]. Ref. [223]. a  3  c  m a x  d  e  h  3  3  3  Chapter 6. The HOMOs ofPH , 3  PF  andP(CH )  3  3  133  3  comprised of an inner shell of six s-type Gaussians and an outer valence shell that has been split into three parts represented by three, one and one primitives, respectively.  Thus, it  consists of a P(13s,9p)/[6s,5p], CYF(1 ls,5p)/[4s,3p] and H(5s)/[3s] contraction. The number of C G T O is then 30 for P H , 60 for P F and 87 for P ( C H ) . 3  (4)  3  3  6-311+G*: This is an augmented  3  version of the 6-31 I G basis of Pople and  co-workers [103]. The 6-311+G* basis is formed by augmenting the 6-31 I G with diffuse s- and p-functions [105] and spherical d-type polarization functions [104] on all heavy (non-hydrogen) atoms.  A (14s,10p,ld)/[7s,6p,ld] contraction is then used for P, a  (12s,6p,ld)/[5s,4p,ld] contraction for C and F, and a (5s)/[3s] contraction for H to give a total of 39 C G T O for P H , 96 CGTO for P F and 123 CGTO for P ( C H ) . 3  3  3  3  (5) 6-311++G**: A split valence basis set in which the 6-311+G* basis set has been further augmented by the addition of a diffuse sp shell and a p-type polarization function to all hydrogen  atoms  [104,105].  The  contractions  are  thus  P(14s,10p,ld)/[7s,6p,ld],  C(12s,6p,ld)/[5s,4p,ld] and H(6s,lp)/[4s,lp] to provide a total of 51 CGTO for P H and 159 3  C G T O for P ( C H ) . 3  3  (6) A U G 3 : The A U G 3 basis set used in the present work is a truncated form of Dunning's aug-cc-pVTZ basis set [166-169] in which all f-, g- and h-functions have been removed. This truncation was adopted to provide compatibility with the density functional calculations since the present version of the DFT program (deMon) cannot handle these higher / functions. In addition, the d functions have been changed from spherical to Cartesian so that the HartreeFock results are comparable to DFT results with this basis.  Thus, it consists of a  P(48s,21p,3d)/[6s,5p,3d], C/F(23s,8p,3d)/[5s,4p,3d] and H(5s,3p)/[4s,3p] contraction. A total of 78 C G T O are employed for P H , 144 CGTO for P F and 261 CGTO for P ( C H ) . 3  (7) A U G 5 :  3  3  3  The basis set for these calculations was taken from the work of Dunning and  co-workers [166-169]. The AUG5 basis set used in the present work is actually a truncated form of Dunning's aug-cc-pV5Z basis set in which all f-, g- and h-functions have been removed. Thus, the AUG5 consists of the atomic contractions of (65s,24p,5d)/[8s,7p,5d] for P,  Chapter 6. The HOMOs ofPH , 3  PF and  P(CH )  3  134  3 3  (33s,13p,5d)/[7s,6p,5d] for C/F and (13s,5p)/[6s,5p] for H . The d functions are spherical. Thus, a total of 177 CGTO are used for P H , 204 CGTO for P F and 393 C G T O for P ( C H ) . 3  3  3  3  (8) 136-GTO: The 136 CGTO extended basis set for P H designed by Davidson and Boyle 3  and reported in the work of Clark et al. [74] consists of P(21s,14p,4d,2f)/[12s,10p,4d,2f] and H(10s,3p,2d)/[6s,3p,ld] contractions. This basis set was designed to saturate the diffuse basis function limit and to give improved representation of the r-space tail of the orbitals [74].  6.2.2 MRSD-CI Calculation (8c) 136-G(CI): The 136-G(CI) calculation for P H is from the earlier work of Clark et al. [74] 3  and a much more detailed discussion of the calculation can be found in that paper. Briefly, the Hartree-Fock singly and doubly excited configuration interaction (HF SDCI) and MRSD-CI wavefunctions for the neutral and ion species were constructed from the 136-GTO basis set (described above in Section 6.2.1). The Hartree-Fock virtual orbitals were transformed into K orbitals to improve the CI convergence. The configuration space for the neutral M R S D - C I calculations was chosen from the largest coefficients of an initial H F SDCI calculation. In addition, the molecular orbitals for the neutral molecule were also used for the ion state calculations.  6.2.3 DFT Calculations (61) A U G 3 - L , (6p) A U G 3 - P and (6b) A U G 3 - B P : These three density functional calculations were carried out at the University of British Columbia using the deMon program [122,123] and the large A U G 3 orbital basis set (described above in Section 6.2.1 for calculation 6) at the experimental geometries for P H [207], P F [208] and P ( C H ) 3  3  3  3  [209] and C  3 v  molecular  symmetry. A l l calculations used a random extra-fine grid and the energy convergence was set at 10" hartree. The auxiliary basis sets for fitting the charge density and exchange-correlation 7  potential were the P(6,4;6,4), F/C(5,4;5,4), and H(4,l;4,l) from the deMon program [122,123]. The A U G 3 - L calculation employs the local density approximation of Vosko et al. [112] while  Chapter 6. The HOMOs ofPH , 3  PF  3  andP(CH ) 3  135  3  the AUG3-P calculation uses the gradient corrected correlation functional of Perdew and Wang [115,117]. The AUG3-BP calculation uses a combination of the correlation functional of Perdew and Wang [115,117] and the exchange functional Becke [116].  The Kohn-Sham  orbitals have been extracted from the result and the momentum profiles have been calculated using the T K S A via Eq. (2.58). The instrumental angular resolution effects [59] were also incorporated into the TMPs. Various other electronic properties from the DFT calculations are listed in Table 6.1. However, as noted in Section 2.2.6, the use of inexact functionals in practical DFT calculations causes the total energies to differ from the exact result [60] and thus the D F T total energies in Table 6.1 are not readily comparable to those from Hartree-Fock methods and MRSD-CI.  6.3 Results and Discussion 6.3.1  Comparison of Experimental and Theoretical Momentum Profiles The experimental momentum profile (XMP) for the outermost orbital (5a^ of P H , 3  obtained earlier by Clark et al. [74] using a single-channel E M S spectrometer [78] is compared in Fig. 6.1 with both earlier and presently performed calculations. In addition, the outermost X M P s for P F (8a Fig. 6.2) and P ( C H ) (8a Fig. 6.3) have been measured in the present 3  1;  3  3  ls  work with the same single-channel spectrometer [78].  On each figure, the mean binding  energy at which each particular X M P was measured is noted (10.6 eV for P H , 12.29 eV for 3  P F and 8.65 eV for P(CH ) ). The values correspond to the vertical ionization potentials of 3  3  3  the lowest lying peaks observed in the photoelectron spectra [211-213] for these molecules. Even at the modest EMS energy resolution, the outermost ionization process for each molecule is well separated in energy from the rest of the valence ionization manifold, and this ensures that all the observed intensity in each momentum profile is due solely to ionization to the lowest lying ion state with no contribution from higher ionization processes.  Chapter 6. The HOMOs ofPH , PF and 3  P(CH )  3  0  136  3 3  1  2  Momentum (au) MOMENTUM DENSITY  (au)  POSITION DENSITY  (au)  Fig. 6.1: Measured and calculated spherically averaged Hartree-Fock and MRSD-CI momentum profiles for the 5a^ orbital of PH3 (upper panel). The lower panels show the momentum and position density contour maps for an oriented PH3 molecule calculated at the Hartree-Fock level using the AUG3 basis set. The position density map shows the shape of the molecular orbital in one plane through the molecule containing PH and bisecting the angle between the other two ligands. The P atom is situated at the origin. The momentum map shows momenta perpendicular and parallel to the defined planes in position space. The contours represent 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density.  Chapter 6. The HOMOs ofPH , 3  PF  3  andP(CH )  137  3 3  12  PF  3  8  Q 1  12.29 eV  Cfl  • S C expt. 7 AUG5 6 AUG3 4 6-311+G* 3 6 - 3 1 1 G 2 4 - 3 1 G 1 STO-3G  6 h CD > H—'  _o CD  0 0  1  2  omentum (au) MOMENTUM DENSITY  POSITION DENSITY  1  PF 8a,  3  /\  1 1 1  PF 8a,  3  t11 I 1111t 1f [  -4  1 1 p  0 (au)  -8  0 (au)  Fig. 6.2: Measured and calculated spherically averaged momentum profiles for the 8aj orbital of PF3 (upper panel). The lower panels show the momentum and position density contour maps for an oriented PF3 molecule calculated at the Hartree-Fock level using the AUG3 basis set. The position density map shows the shape of the molecular orbital in one plane through the molecule containing PF and bisecting the angle between the other two ligands. The P atom is situated at the origin. The momentum map shows momenta perpendicular and parallel to the defined planes in position space. The contours represent 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density.  Chapter 6. The HOMOs ofPH  3>  PF  3  andP(CH )  138  3 3  P(CH )  12  >^  3  3  8a  fi.65 eV  C CD  SC  expt.  AUG5 AUG3 6-31 1+ + G** 6-311+G* 6-311G 4-31G ST0-3G  CD >  CD  0 0  omentum (au) POSITION DENSITY  MOMENTUM DENSITY •  P(CH ) 3  3  A  ! -8  0 4 8 (au) Fig. 6.3: Measured and calculated spherically averaged momentum profiles for the %&\ orbital of P(CH3)3 (upper panel). The lower panels show the momentum and position density contour maps for an oriented P(CH )3 molecule calculated at the Hartree-Fock level using the AUG3 basis set. The position density map shows the shape of the molecular orbital in one plane through the molecule containing PCH and bisecting the angle between the other two ligands. The P atom is situated at the origin. The momentum map shows momenta perpendicular and parallel to the defined planes in position space. The contours represent 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density. -4  3  Chapter 6. The HOMOs ofPH , 3  PF  3  andP(CH ) 3  139  3  The experimental momentum profiles for P H , P F and P ( C H ) 3  3  3  3  are compared on  Figs. 6.1-6.3 respectively with a range of theoretical momentum profiles (TMPs) calculated at the Hartree-Fock level.  Selected properties from each calculation and the corresponding  experimental values are shown in Table 6.1. The Hartree-Fock basis sets used range from a very modest minimal STO-3G basis to the very much larger A U G 5 and 136-GTO basis sets. The effects of many-body correlation and electronic relaxation are also seen on Fig. 6.1 from the M R S D - C I ion-neutral overlap reported earlier [74].  It should be noted that the  experimental instrumental angular resolution has been accounted for in all theoretical momentum profiles in Figs. 6.1-6.3 using the GW-PG method [59]. Experiment and theory have been placed on a common intensity scale for each molecule by normalizing the experiment to the 136-G(CI) momentum profile for P H , the 3  A U G 3 momentum profile for P F and the AUG5 momentum profile for P ( C H ) (assuming 3  unit pole strength).  3  3  This places the calculations for each molecule on a common relative  intensity scale. However, for dominantly s-type profiles such as the P F and P ( C H ) HOMOs 3  3  3  (Figs. 6.2 and 6.3), it is more difficult to assess the quality of different TMPs than in the case of p-type profiles such as the 5a! orbital of P H . This is because a change in relative intensity at 3  low p is often the only major distinguishing characteristic between two TMPs for dominantly s-type momentum profiles such as the P F  3  and P ( C H ) 3  3  HOMOs.  As such, other  normalizations to the X M P could be employed instead of the choices made in the present work. As in previous studies on the substituted amines, intuitive arguments fail to predict the correct amount of s- and p-type contributions to the H O M O momentum profiles of P F and 3  P ( C H ) . Specifically, the momentum profiles for P F and P ( C H ) display far more s-type 3  3  3  character than phosphine.  3  3  Intuitive arguments based on their larger bond angles (see  introduction) predict that these orbitals would have more p-type character than the H O M O of P H . While the presence of a small second maximum in the momentum profile at p ~ 0.8 au 3  for P ( C H ) indicates a small amount of p-type character in this orbital, it is obvious (Fig. 6.3) 3  3  that s-type character is dominant in this X M P . Thus, as might be expected, intuitive arguments  Chapter 6. The HOMOs ofPH , 3  PF  3  andP(CH ) 3  140  3  involving hybridization and localized bonding are unreliable. However, as can be seen from Figs. 6.1-6.3, delocalized molecular orbital concepts including even the simplest STO-3G molecular orbital description clearly provide a more realistic description of the HOMOs. The general shapes of the experimental momentum profiles are similar to those of their respective amine analogs [70,80-82]. The H O M O of N F displays relatively more p-type character than 3  the H O M O of P F as shown by the prominent maxima at p ~ 1.3 au that is clearly observed in 3  the N F X M P [81,82]. In contrast, the X M P of P F shows no strong second maxima in this 3  3  region. For N ( C H ) and P ( C H ) , the ratio of s to p character is similar in both momentum 3  3  3  3  profiles. A more detailed examination of the momentum profiles calculated using Hartree-Fock and M R S D - C I procedures will now be presented.  The minimal basis set STO-3G result  (curve 1 on Figs. 6.1-6.3) has generally the poorest agreement for shape and intensity with the experimental momentum profiles and the properties listed in Table 6.1. For example, the STO-3G calculation gives a seriously incorrect prediction of the s to p ratio for the P ( C H ) 3  3  X M P . The STO-3G total energy for all molecules is also very poor, as expected from the low degree of basis set saturation in this calculation. While the STO-3G dipole moment for P H is 3  better than those from several large basis sets, it is well known that the STO-3G calculation often produces fairly good dipole moments because of a fortuitous error cancellation [57]. In addition, the STO-3G dipole moment for P ( C H ) 3  3  is clearly in error.  The split-valence  description using the small 4-31G basis set calculation (curve 2 on Figs. 6.1-6.3) provides a moderately better description of the XMPs of all molecules than the STO-3G momentum profile. For other properties, the 4-31G total energies (Table 6.1) are surpassed by those from larger basis sets. However, the 4-31G dipole moments are interesting in that those for P H and 3  P F are clearly in error while the 4-31G dipole moment for P ( C H ) is in agreement with 3  3  3  experiment. The TMPs from the intermediate size 6-311G calculation (curve 3 on Figs. 6.1^6.3) provide slightly better energies and an improved fit to the X M P s of P H , P F and P ( C H ) 3  3  3  3  Chapter 6. The HOMOs ofPH , 3  PF  andP(CH )  3  3  141  3  compared with the low-level minimal basis set and 4-3IG calculations. For other properties, the 6-311G values are easily surpassed by those from larger basis sets (Table 6.1). The next level of calculation is the addition of diffuse and polarization character to the 6-31 I G basis set. While the 6-311+G* momentum profile (curve 4 on Figs. 6.1-6.3) gives a better shape fit than the 6-31 I G T M P for P H , little shape difference is seen for these two calculated TMPs for P F 3  3  and P ( C H ) . For other properties, though, the 6-311+G* gives lower total energies and much 3  3  improved dipole moments compared with the 6-31 I G value (Table 6.1). The 6-311++G** momentum profiles for P H and P ( C H ) (curve 5 on Figs. 6.1 and 6.3) show little or no shape 3  3  3  change from the 6-311+G* momentum profiles, thus indicating that added diffuse and polarization of the H atom basis sets has little effect on the calculated momentum profiles. Instead, the main improvements are in total energies and dipole moments for P H and 3  P(CH ) . 3  3  Despite the larger size of the AUG3 calculation, the 6-311++G** (curve 5) and A U G 3 (curve 6) momentum profiles for P ( C H ) are almost identical (see Fig. 6.3). Generally good 3  3  shape agreement with experiment is provided by the AUG3 and 6-311+G* TMPs for P F (Fig. 6.2). The A U G 3 calculation provides a slightly better momentum profile (see p ) max  3  for  P H than that from the 6-311++G** result. 3  The A U G 3 and A U G 5 momentum profiles (curves 6 and 7, respectively, on Figs. 6.16.3) are similar in shape for each molecule, although there are some differences in intensity, especially for P F . It can be noted that the AUG5 calculation for P H involves an even larger 3  3  basis set than the 136-GTO calculation thought to be near the Hartree-Fock limit in the work of Clark et al. [74]. The AUG5 and 136-GTO TMPs are in excellent agreement for shape and also give reasonably similar total energies and dipole moments. This again indicates that the Hartree-Fock limit is closely approached at this level. The AUG5 also gives the lowest total energies for P F and P ( C H ) . The AUG5 dipole moments for all molecules are also generally 3  3  3  good, especially for P H and P F . 3  3  Chapter 6. The HOMOs ofPH , 3  PF and 3  P(CH )  142  3 3  The inclusion of electron correlation causes virtually no change in the T M P for P H  3  (compare the 136-GTO and 136-G(CI) momentum profiles in Fig. 6.1). TMPs for the valence orbitals of other third row heavy atom containing molecules such as HC1 [76] and H S [75] 2  have also shown little change with inclusion of CI. This is in contrast to the equivalent second row heavy atom containing molecules such as N H , H F and H 0 [39,69,72]. However, in all 3  2  cases electron correlation and relaxation have a significant effect on the total energy and dipole moment (see Refs. [39,56,69] and calculations 8 and 8c in Table 6.1).  The 136-G(CI)  calculation gives the lowest total energy for P H and provides a dipole moment within 0.05 3  Debye of experiment. The X M P s for the outermost orbitals of P H , P F and P ( C H ) are also compared to the 3  3  3  3  DFT theoretical momentum profiles in Figs. 6.4-6.6. The X M P s have been normalized to the DFT TMPs in Figs. 6.4-6.6 (it should be noted that different normalizations were selected for Figs. 6.1-6.3).  For purposes of comparison, Hartree-Fock momentum profiles for each  molecule and the post-Hartree-Fock 136-G(CI) momentum profile for P H are also shown. It 3  can be seen that the DFT momentum profiles are slightly more intense than the Hartree-Fock and post-Hartree-Fock results in Fig. 6.4, with the exception of curve 6b for P H . 3  The D F T target Kohn-Sham momentum profiles on Figs. 6.4-6.6 provide very good overall agreement with the experimental momentum profiles of all three molecules.  The  correlation-only AUG3-P T M P (curve 6p) is the most intense for P F and P ( C H ) , followed 3  3  3  by the local density approximation A U G 3 - L T M P (curve 61). Curves 6p and 61 are identical for P H . For all molecules, the exchange and correlation AUG3-BP T M P (curve 6b) has the 3  lowest intensity of the DFT TMPs. It should also be noted that there are only very small shape differences between the three DFT TMPs for P H , with the Becke-Perdew non-local 3  calculation (curve 6b) fitting the best. However, for P F and P ( C H ) , the shape differences 3  3  3  between the DFT TMPs from the local density approximation and those from the non-local potentials are minimal.  Chapter 6. The HOMOs  ofPH , 3  PF and 3  P(CH )  3 3  143  Momentum (au) Fig. 6.4: Measured and calculated spherically averaged DFT momentum profiles for the 5ai orbital of PH3. The solid circles represent the single channel measurements also shown on Fig. 6.1. Selected Hartree-Fock and MRSD-CI calculated momentum profiles are also shown for comparison with the DFT momentum profiles. All calculations have been spherically averaged and folded with the experimental momentum resolution. See text and Table 6.1 for details of the wavefunctions and normalization procedures. Note that the normalization of experiment to theory differs from that used in Fig. 6.1.  Chapter 6. The HOMOs ofPH , 3  PF and  P(CH )  3  144  3 3  PF 8 a 12.29 eV  12 H  3  6b 6p 6  61 7  h  6  1  AUG3-BP AUG3-P AUG3-L AUG5 AUG3  0 0  1  2  omentum (au) F i g . 6.5: Measured and calculated spherically^averaged DFT momentum profiles for the 8a] orbital of PF3. The solid circles represent the single channel measurements also shown on Fig. 6.2. Selected Hartree-Fock calculated momentum profiles are also shown for comparison with the DFT momentum profiles. All calculations have been spherically averaged and folded with the experimental momentum resolution. See text and Table 6.1 for details of the wavefunctions and normalization procedures. Note that the normalization of experiment to theory differs from that used in Fig. 6.2.  Chapter 6. The HOMOs  0  ofPH , 3  PF and  P(CH )  3  145  3 3  1  2  Momentum (au) Fig. 6.6: Measured and calculated spherically averaged DFT momentum profiles for the 8aj orbital of P(CH3)3. The solid circles represent the single channel measurements also shown on Fig. 6.3. Selected Hartree-Fock calculated momentum profiles are also shown for comparison with the DFT momentum profiles. All calculations have been spherically averaged and folded with the experimental momentum resolution. See text and Table 6.1 for details of the wavefunctions and normalization procedures. Note that the normalization of experiment to theory differs from that used in Fig. 6.3.  Chapter 6. The HOMOs ofPH , 3  PF and 3  P(CH )  146  3 3  Of the DFT momentum profiles for P H , the AUG3-BP momentum profile (curve 6b 3  on Fig. 6.4) provides generally the best shape fit with the observed X M P . While this profile provides an acceptable fit to the X M P , it is difficult to see any improvement for shape compared to the high-level Hartree-Fock and post-Hartree-Fock momentum profiles for P H  3  (compare to curves 6-8c on Fig. 6.4). Similarly, all of the DFT momentum profiles for P F  3  presented in Fig. 6.5 fit the X M P well for shape, but these results are roughly comparable to those from the Hartree-Fock momentum profiles (curves 6 and 7 on Fig. 6.5, and also refer to Fig. 6.2). The DFT momentum profiles for P ( C H ) also provide good fits to the X M P (Fig. 6.6) 3  3  although there is some discrepancy in the region of the second maximum from p = 0.5 au to p = 1.3 au where the TMPs underestimate the X M P .  As mentioned above, much of the  problem in determining the "goodness" of a T M P for this orbital arises from the characteristics of an sp-type momentum profile. A slightly better shape fit is achieved by normalization to the Hartree-Fock momentum profile from the AUG3 calculation in Fig. 6.3. For other electronic properties, DFT dipole moments from the AUG3-P and AUG3-BP for P H and P ( C H ) are in 3  3  3  good agreement with experiment and are generally more accurate than those from Hartree-Fock and post-Hartree-Fock calculations (Table 6.1). DFT dipole moments for P F are also accurate 3  (within 0.27 Debye of experiment), although better dipole moments are obtained from the Hartree-Fock A U G 3 and A U G 5 calculations. In addition, as discussed in Section 2.1.4, there is the well-known problem of obtaining absolute energies from DFT [111,214]. However, it can be said that the DFT treatments provide quite good values of the other properties in the present work.  6.3.2 Density Maps in Momentum and Position Space Calculated momentum and position space density contour maps for the HOMOs of P H , P F and P ( C H ) are presented in Figs. 6.1-6.3 above. These maps are slices of the 3  3  3  3  Chapter 6. The HOMOs ofPH , 3  PF and P(CH ) 3  147  3 3  2  orbital electron density iff  in momentum and position space, respectively. It should be noted  that these maps are for oriented molecules, whereas the momentum profiles in Figs. 6.1-6.3 are for randomly oriented (i.e. spherically averaged) molecules. The maps were produced with the A U G 3 Hartree-Fock calculation since this provides a reasonably adequate description for all three molecules (see Section 6.3.1).  The position density maps show the shapes of the  molecular orbitals in one plane through the molecule containing PH, PF or P C H , respectively, and bisecting the angle between the other two ligands. The P atom is situated at the origin. The momentum maps show momenta perpendicular and parallel to the defined planes in position space. The side panels on each map show slices of the momentum or position space density along the vertical (right panel) and horizontal (top panel) axes. A l l dimensions for the p- and r-maps are in atomic units. The p- and r-maps in Figs. 6.1-6.3 are helpful in understanding the shapes of the experimental and theoretical momentum profiles. It is obvious from the p-maps of P F and 3  P ( C H ) that the intensity is largest at p = 0 and thus (after spherical averaging) there is 3  3  maximum intensity at p = 0 in the momentum profile (Figs. 6.2 and 6.3). However, for P H  3  there is only a small amount of intensity at p = 0 as indicated by the side panels on the momentum density map (Fig. 6.1). While some small degree of s-type character (due to the H atoms) is seen in the P H momentum profile, p-type character is dominant for this orbital. The 3  two maxima centered at p = ± 0.5 au in the right side panel of the P H p-map are also reflected 3  in the momentum profile as a p  m a x  (after spherical averaging) occurring at ~ 0.5 au. Similarly,  the two other large maxima in the P ( C H ) p-map at p = ±0.8 au (right panel) account for the 3  3  p-type behaviour and second maxima at p ~ 0.8 au in the momentum profile after spherical averaging. Similar to previous work on the methylamines [80-82], the observed increase in s-character upon methyl substitution in P H to give P ( C H ) can be related to an increased H 3  3  3  Is contribution to the M O from the H trans to the "lone pair". A consideration of the M O  Chapter 6. The HOMOs ofPH , 3  PF and  P(CH )  3  148  3 3  coefficients in both the AUG3 wavefunction and the simpler 4-3IG wavefunction indicates that although the H Is contribution in P H is relatively small the trans-H 3  H O M O in P ( C H ) 3  Is contribution to the  is quite large. In addition, the calculated increased intensity at p = 0 au  3  must be solely due to these increased s orbitals contributions. This is more readily understood if one considers the Fourier transformation between position and momentum space given by Eq. (2.59). Eq. (2.59) may be simplified to give the momentum space wavefunction at p = 0 au:  (6.1) Thus, only s-type orbitals can contribute at p = 0 au since pure p-type or d-type orbitals have equal lobes of opposite sign [27] with a node at the origin (i.e. they have zero integrated "area" in both position and momentum space). P(CH ) 3  3  The increased contribution of trans-H Is to the  H O M O relative to the P H H O M O thus corresponds to greater intensity at p = 0 3  according to Eq. (6.1) and thus the momentum profile is expected to be more s-type as is observed (Fig. 6.3). A similar effect (i.e. decreased %p character and increased %s character following methyl substitution) has also been observed in the comparison of the outermost orbitals of H C H O , C H C H O and ( C H ) C O [83]. The implications of these results for the 3  3  2  methyl inductive effect in organic chemistry will be discussed in Section 6.3.3. The increase in s-character in P F relative to P H occurs for a different reason than that 3  3  for P ( C H ) and P H . Analysis of the density maps and an examination of the M O coefficients 3  3  3  reveals a sharp increase in the P 3s atomic orbital contribution to the H O M O of P F compared 3  to the 3s contribution to the H O M O of P H  3  The higher P 3s contribution in P F relative to the 3  P 3s contribution in P H is clearly seen in the top and right panels of the r-maps for these 3  molecules as increased density at zero atomic units for P F . These results are analogous to the 3  sharp increase in N 2s contribution to the M O observed for N F relative to N H in previous 3  work [81].  3  Chapter 6. The HOMOs  ofPH , 3  PF  andP(CH )  3  3  149  3  Bawagan and Brion [82] commented on the "secondary" p  m a x  for the methyl-  substituted amines and noted that increasing methyl substitution led to an increase of p  m a x  from 0.52 au in N H to ~ 1.1 au in N ( C H ) . Although no experimental momentum profiles 3  3  3  are available for P H ( C H ) and P H ( C H ) , the momentum profiles for P H and P ( C H ) also 2  3  follow this trend O^max f °  3  ~ 0.8 au).  i  r  The shift of p  2  3  0-5 au while the secondary p  s  m a x  3  for P ( C H ) 3  3  3  occurs at  towards higher momentum has earlier been associated with  max  additional nodal surfaces [83,199,215].  Tossell et al. [80] and Bawagan and Brion [82]  suggested that the higher momentum components observed in the methyl-substituted amines compared to N H were due to these extra nodal surfaces. These nodes were attributed to the 3  trans-H Is which is antibonding with respect to the N 2p. present work that the shift of p  m a x  Similarly, it is suggested in the  of P ( C H ) towards higher momentum relative to p 3  3  max  of  P H is due to the increased trans-H Is character in the P ( C H ) H O M O relative the trans-H Is 3  3  3  character in the P H H O M O . The increase in number of nodal surfaces with increasing methyl 3  substitution can be seen by considering the corresponding position space density maps for P H  3  and P ( C H ) (Figs. 6.1 and 6.3, respectively). 3  3  6.3.3 The Methyl Inductive Effect To provide more insight into the relatively large amount of s-type character in the HOMOs of P F and P ( C H ) 3  3  3  and the inability of geometry and hybridization arguments to  rationalize the results, two-dimensional position (charge) density difference maps for the outermost orbitals of P F and P H , and also P ( C H ) and P H are shown in Fig. 6.7 calculated 3  3  3  3  3  with the 4-31G and AUG3 wavefunctions. These maps are slices of the orbital electron density differences p  -p  P F  3  and P ( C H )  P H  P  3  3 3  -  PPH  m  position space. The position density difference  3  maps show the shapes of the molecular orbitals in one plane through the molecule containing PH, PF or PCH, respectively, and bisecting the angle between the other two ligands. The P atom is situated at the origin. Slices of the position space density difference along the vertical  Chapter 6. The HOMOs  POSITION  ofPH ,  DENSITY  4 - 3 1 G  PF  3  andP(CH )  DIFFERENCE  p -p PF  3  PHt  3  150  3  POSITION  DENSITY  AUG3  DIFFERENCE  p -p PF  PHi  Fig. 6.7: Position (charge) density difference maps for substituted phosphines. The contour maps were generated with the 4-3IG and A U G 3 wavefunctions and all maps show the density for oriented molecules. The position density difference maps show the shapes of the molecular orbitals in one plane through the molecule containing P H , P F or P C H , respectively, and bisecting the angle between the other two ligands. The P atom is situated at the origin. The contour values represent ±0.01, ±0.03, ±0.1, ±0.3, ±1.0, ±3.0, ±10.0, ±30.0 and ±99.0% of the maximum intensities. Positive difference is shown by solid contour lines while negative difference is shown as dashed lines. The side panels (top and right side) show the density difference along the dashed lines (horizontal and vertical) in the density map.  Chapter 8. Methanol  181  10  20  30  40  Binding Energy (eV)  Fig. 8.1: EMS binding energy spectra of CH OH from 6 to 47 eV at (a) (j) = 0°, (b) < > | = 8° and (c) summed over sixteen § angles, obtained at an impact energy of (1200 eV + binding energy). The dashed lines represent Gaussian fits to the peaks and the solid curve is the summed fit. 3  Chapter 8. Methanol  182  by high resolution ultraviolet photoelectron spectroscopy [174,260] for the six outermost orbitals.  However, the (la")" and (6a')" bands are only partially resolved even at high  resolution.  1  1  A low resolution X-ray photoelectron spectroscopy experiment [183] gave an  ionization potential of 32.2 eV for the main inner valence peak corresponding to the (3a')"  1  state. The work of von Niessen et al. [260] has compiled the experimental ionization potential data up to 1979 together with the results of many-body Green's function calculations of the binding energy spectrum. Measured ionization potentials from the literature [174,183,260], the E M S ionization potentials from Minchinton et al. [84], and those used in the present work are shown in Table 8.2. Also given in Table 8.2 are calculated M R S D - C I ionization energies from the present work and Green's function ionization energies obtained by von Niessen et al. [260]. For the inner valence 3a' and 4a' orbitals, the corresponding spectroscopic strengths from the Green's function calculations [260] results are also shown. The removal of the least tightly bound (i.e. HOMO) 2a" electron leads to the ground state of the ion corresponding to the band at 11.0 eV in the E M S binding energy spectrum (Fig. 8.1). The nodal "p-type" character of this orbital which would be expected for such an oxygen "lone pair" of electrons is confirmed by the fact that the intensity is much larger at ty = 8° than atty= 0°. However, the second "lone pair" (i.e. 7a' orbital) results in the band at 12.7 eV which has only slightly larger intensity atty= 8° than atty= 0°, indicating a mixed s-p type character. The band at ~ 15.5 eV results from ionization from both the 6a' and l a " orbitals while the (5a')" ionization accounts for the partially resolved shoulder at 17.7 eV. The (4a')" 1  1  process produces the large bands in the inner valence region at 22.7 eV. Other than for the 2a" (HOMO) band, the intensities of each of the other bands are quite similar both atty= 0° and ty = 8° thus indicating that these all have appreciable totally symmetric ("s-type") components in the wavefunctions of the corresponding molecular orbitals. The region above 25 eV which has been studied earlier both by XPS [183] and E M S [84] shows a prominent peak at 32.2 eV due to the (3a')  _1  process. However, there is considerable additional ionization strength both  between 25-29 eV and also spread out over the binding energy spectrum above the 32.2 eV  Chapter 8. Methanol  183  peak to the limit of the data at 47 eV. The many-body Tamm-Dancoff Green's function inner valence calculations reported by von Niessen et al. [260] predict the (4a')" process to be 1  dominated by two poles at 23.01 and 23.38 eV each with a spectroscopic factor of 0.44 (Table 8.2). The calculations [260] also predict that the (3a')" process is severely split into at 1  least nine poles spread over a wide energy range with the main intensity in the ~ 33 eV region, in reasonable agreement with experiment. However, the remaining intensity observed above 36 eV is not predicted [260]. Calculated binding energy spectra are compared with the measured binding energy spectra in Fig. 8.2. Theory and experiment have been normalized at 15.2 eV in the (j) = 8° spectrum The calculated spectra are obtained by using the pole energies and pole strengths given by von Niessen et al. [260] in Table 8.2, and the Trun-pV5Z theoretical momentum profiles for each pole at each angle. For the first six main peaks below 25 eV, the measured E M S instrumental energy resolution function, as well as the widths of the transitions as observed in the experimental EMS binding energy spectra (see Fig. 8.1 and Table 8.2) have been folded into the calculated spectra. For the (3a') ionization parent and satellite processes _1  and their corresponding poles, both the EMS energy resolution and an width of 2.00 eV based on the observed XPS [183] have been folded into the calculated spectra. The region containing the 2a" and 7a' orbitals is overestimated at <> | = 8° and underestimated at (j) = 0°. However, the correct shapes and positions are also reasonably well predicted at both angles for the 14 to 17 eV energy region containing the (6a')" and (la")" 1  ionization processes.  1  The energy position of the (5a')" ionization process is also predicted 1  well, especially at <> ( = 8°, but the experiment has greater intensity at (i.e. more s-type character than predicted).  = 0° than predicted  The predicted energy of the (4a')" ionization 1  process is about 0.5 eV too high, thus causing the leading edge of the experimental peak at 22.7 eV to be underestimated.  In addition, the peak maxima of the (4a')  _1  peak is also  overestimated by the calculation at both angles while no poles model the experimental intensity in the 25-30 eV region, thus suggesting that the theory should contain lower pole strength at the  Chapter 8. Methanol  184  10  20  30  Binding Energy (eV)  40  Fig. 8.2: Measured and calculated binding energy spectra of C H 3 O H from 6 to 47 eV at (a) $ = 0° and (b) § = 8°. The solid curves indicate the synthesized theoretical binding energy spectra with pole energies and pole strengths (Table 8.2) given by the Green's function calculation [260] and including the angular dependence from the Trun-pV5Z Hartree-Fock calculation. The same energy peak widths as used in Fig. 8.1 have been folded into the synthesized spectra. See text for details.  Chapter 8. Methanol  185  stated energies of 23.01 and 23.38 eV (Table 8.2) and instead place more (4a')" poles in the 1  region 25-30 eV. The intensity of the theory in the region > 30 eV is in good agreement (at both angles) with the main peak corresponding to (3a')"  1  parent ionization at an  experimental ionization energy of 32.2 eV. However, the theoretical peak is ~ 0.8 eV too high in energy and this can be attributed to the energies of the major (3a')" poles predicted at 32.82 1  and 33.08 eV (Table 8.2). While the shape of the parent peak is well-predicted, the overall asymmetry and observed non-zero intensity of the experimental data out to 47 eV is not wellmodelled by theory. The highest energy predicted for any pole is 35.62 eV and it is clear from the experiment that more poles with considerable pole strength are needed at higher energies.  8.3.2 Comparison of Experimental and Theoretical Momentum Profiles High momentum resolution measurements have been made of the valence electron experimental momentum profiles (XMPs) for C H O H (Figs. 8.3-8.5) at each ty angle. The 3  fitted peak areas in the binding energy spectra (see Section 8.3.1) have been used for the relative intensities of the seven valence orbitals. On each figure the mean binding energy for each X M P is noted. These values correspond to the energy position of the main Gaussian function used to fit this peak in the deconvolution procedure. Although much improved statistics have been obtained in the present multichannel work for C H O H , the shapes and relative intensities obtained in the present work are consistent 3  with the previous measurements [84]. The most significant improvements are for the two outermost (2a" and 7a') orbitals and for the inner valence 4a' orbital. In addition, the level of theory was much more limited in the earlier work [84] in that only a single Hartree-Fock calculation with a small basis set was used and no post-Hartree-Fock calculations were performed.  In the present work, the experimental momentum profiles are compared on  Figs. 8.3-8.5 respectively with the theoretical momentum profiles (TMPs) calculated using a range of basis sets from the very modest minimal STO-3G basis set to the near-Hartree-Fock limit level (110-GTO and Trun-pV5Z) as described in Section 8.2.1. The effects of electron  Chapter 8. Methanol  0  186  1  2  0  Momentum (au)  1  2  Fig. 8.3: Measured and calculated spherically averaged momentum profiles for the 2a" and 7a' orbitals of methanol. The solid circles represent the experimental energy-dispersive multichannel measurements. All calculations have been spherically averaged and folded with the experimental momentum resolution. See text and Table 8.1 for details of the wavefunctions and normalization procedures.  Chapter 8. Methanol  187  Momentum (au) F i g . 8.4: Calculated spherically averaged momentum profiles for the 6a', la" and sum of the 6a' and la" orbitals of methanol. All calculations have been spherically averaged and folded with the experimental momentum resolution. See text and Table 8.1 for details of the wavefunctions and normalization procedures.  Chapter 8. Methanol  188  CH3OH O  MC e x p t  5 110-GTO 4 6-31 1 + + G** 3 6-31 1+G* 2 6-311G 1 ST0-3G  6c 6b 6a 6 5c  Trun-pV5Z Trun-pV5Z Trun-pV5Z Trun —pV5Z 11 0 — G ( C I )  x0.86 x0.75 x0.70  bp D F T - B 3 P W 9 1 bl D F T — B 3 L Y P Id D F T - L D A  5a 17.6 eV 1  • o  20  Deconvolution 2 8 - 4 7 eV  0 0  omentum (au) Fig. 8 . 5 : Measured and calculated spherically averaged momentum profiles for the 5a', 4a' and 3a' orbitals of methanol. The solid circles represent the experimental energy-dispersive multichannel measurements from the deconvolution procedure. The open circles for the 4a' orbital represent the experimental energy-dispersive multichannel measurements taken by summing over all angles within the energy "slice" of 20-28 eV while the open circles for the 3a' orbital represent the experimental energy-dispersive multichannel measurements taken by summing over all angles within the energy "slice" of 28-47 eV. See text and Table 8.1 for details of the wavefunctions and normalization procedures. Curves 6a, 6b and 6c are constructed by scaling curve 6 by factors of 0.70, 0.75 and 0.86, respectively.  Chapter 8. Methanol  189  correlation and electronic relaxation effects are also investigated in Figs. 8.3-8.5 by the llO-G(CI) MRSD-CI ion-neutral overlaps described in Section 8.2.2 above. The D F T TMPs calculated according to Section 8.2.3 are also shown on Figs. 8.3-8.5. A pole strength of unity has been used in the presentation of all TMPs including the MRSD-CI theoretical momentum profiles in Figs. 8.3-8.5. The experimental instrumental angular resolution has been accounted for in all theoretical momentum profiles using the GW-PG method [59]. Experiment and theory have been placed on a common intensity scale for each molecule by normalizing the experiment to the 110-GTO theoretical momentum profile for the 2a" (HOMO) orbital. It should be noted that alternative normalizations (e.g. on the HO-G(CD) could be chosen. Selected other properties for each calculation and corresponding experimental values are shown in Table 8.1. The observed experimental momentum profile for the 2a" orbital (Figs. 8.3a and 8.3b) displays some intensity at low momentum below p ~ 0.3 au that is not accounted for by even the more sophisticated theoretical treatments. The only intensity that the TMPs in Figs. 8.3a and 8.3b have at p = 0 is due to the very small effects of the resolution folding procedure [59]. Minchinton et al. [84] also observed similar low momentum discrepancies between their experimental data for the 2a" orbital and theory, although the differences were larger due to the use of the small double-zeta basis set which does not reproduce the low momentum region below 0.6 au. From the present work it is clear that the discrepancies at low momentum for the 2a" orbital are significantly decreased by the use of high-level wavefunctions. The use of electron correlation and relaxation in the 110-G(CI) calculation (curve 5c) only decreases the intensity of the T M P slightly compared to curve 5, but there are no large differences in shape between these curves. This is in sharp contrast to the X M P and TMPs for the outermost lbj orbital of H 0 [37,39,69] in which electron correlation and relaxation effects are necessary to 2  predict correctly the region of the X M P below the maximum ( p  m a x  = 0.5 au).  Although  slightly better agreement is obtained by the DFT momentum profiles in Fig. 8.3b,  Chapter 8. Methanol  190  some experimental intensity still remains unaccounted for in the very low momentum region (p < 0.3 au).  The experimental and theoretical momentum profiles for the 7a' orbital are shown in Figs. 8.3c and 8.3d. In contrast to the situation for the 2a" orbital, the calculation of the ionneutral overlap including electronic correlation and relaxation by the M R S D - C I method produces a very significant improvement over the near Hartree-Fock limit calculations in the level of agreement between the measured and calculated 7a' momentum profiles below 1 au (compare curves 5, 6 and 5c). The HO-G(CI) momentum profile is the only calculation in Fig. 8.3c that reasonably models both the shape and intensity of the 7a' X M P . However, even this high-level post-Hartree-Fock calculation is not able to account for all the experimental intensity at low momentum (p < 0.5 au). The DFT momentum profiles for the 7a' orbital (Fig. 8.3d) are slightly more intense than experiment in the region of the maximum about ~ 0.9 au. However, the DFT momentum profiles for the 7a' orbital predict a similar shape at low momentum as the HO-G(CI) momentum profile and provide an even further improved fit to the experimental data in this region. This effect can be attributed to the inclusion of electron correlation in the DFT exchange-correlation potential. The theoretical momentum profiles for the 6a' and l a " orbitals of CH3OH are shown in Figs. 8.4a-d.  The summed (6a'+la") experimental and theoretical momentum profiles are  shown in Figs. 8.4e and 8.4f. The shape and intensity of the (6a'+la") X M P is well-reproduced by the  HO-G(CI) calculation although the intensity at low momentum is slightly  underestimated. The Hartree-Fock momentum profiles do not predict the correct shape of the X M P in this region in that curves 2-6 tend to "flatten out" at low p while the X M P displays stronger p-type character. It is clear that the improved fit of the HO-G(CI) momentum profile for the (6a'+la") orbitals is due to the different shape it gives for the 6a' momentum profile in Figs. 8.4a and 8.4b, placing less intensity in the region p < 0.3 au and more in the region 0.3 < p < 1.0 au than the large basis set Hartree-Fock calculations. Except for the STO-3G momentum profile, all Hartree-Fock TMPs for the l a " orbital (Fig. 8.4c) are similar in shape.  Chapter 8. Methanol  191  Similar effects occur for the DFT and CI momentum profiles for the (6a'+la") orbital (Fig. 8.4f) and excellent shape and intensity fits with experiment are obtained. The experimental momentum profile for the 5a' orbital (Figs, 8.5a and 8.5b) is predominantly s-type.  A l l the calculated profiles for the 5a' molecular orbital  show considerably less s-type contributions at low p than is observed experimentally. Minchinton et al. [84] also observed this discrepancy between simple Hartree-Fock theory and experiment in the low p region of the 5a' orbital. In the present work the discrepancies between experiment and theory at low momentum are somewhat decreased by the use of larger basis set Hartree-Fock calculations and, in particular, by using the M R S D - C I method.  It  should be noted that in Fig. 8.5a the only T M P that provides reasonably adequate shape and intensity compared to the X M P is the high-level 110-G(CI) momentum profile. This suggests that electron correlation effects are important in the low momentum regions of the E M S crosssection for this orbital. A l l other calculations "flatten out" at low p or even predict a more p-type momentum profile in the case of very simple wavefunctions (STO-3G).  The DFT  momentum profiles for the 5a' orbital (Fig. 8.5b) do not agree with experiment well in that the low momentum region near p = 0 is seriously overestimated. It appears that an overly spatially diffuse orbital has resulted from the DFT calculations, especially for the D F T - L D A results (see curve Id on Fig. 8.5b). While the shapes of Kohn-Sham DFT momentum profiles are often in reasonable agreement with experiment, but it has been observed that the use of approximate functionals in D F T calculations may result in overestimations of the "diffuseness" of the electron density [60,265]. These momentum profile overestimations have also been observed more frequently from DFT calculations employing the local density approximation functionals [60,265] which are generally considered to be of lower quality relative to gradient corrected functionals. The experimental and theoretical momentum profiles for the 4a' inner valence orbital of C H O H are shown in Figs. 8.5c and 8.5d. The filled circles show the experimental energy3  dispersive multichannel E M S measurements for the peak at 22.7 eV obtained from the  Chapter 8. Methanol  192  deconvolution procedure. In addition, the open circles show the multichannel data taken from summing the experimental binning mode data over all angles within the given energy slice of 20-28 eV that contains the main peak centered at 22.7 eV in Fig. 8.1.. The theoretical momentum profiles in Fig. 8.5c are presented at 100% relative intensity with the exception of curves 6b and 6c. Curve 6b represents the Trun-pV5Z T M P for the 4a' orbital multiplied by a factor of 0.75 while curve 6c represents the Trun-pV5Z momentum profile multiplied by a factor of 0.86. It can be seen in Fig. 8.5c that all of curves 1-6 provide reasonable shape comparisons to the XMPs. The best shape fits for the 4a' X M P are given by the 6-311++G** and Trun-pV5Z momentum profiles. The 6-311G momentum profile (curve 2) for the 4a' orbital also provides a reasonable shape fit. However, the most noticeable difference between experiment and curves 1-6 is that all of these curves greatly overestimate the deconvoluted experimental data (filled circles) and slightly overestimate the energy slice data (open circles). If curve 6 is multiplied by 0.75 to produce curve 6b, excellent shape and intensity agreement with the deconvoluted data is obtained. The intensities of the experimental deconvoluted and theoretical momentum profiles thus suggest that the 4a' parent peak at 22.7 eV has a pole strength of 0.75 while the remaining 0.25 pole strength is split into higher energy "satellite" ionization processes up to 28 eV and beyond. A consideration of the data for energy region from 20-28 eV (open circles) accounts for most of the remaining pole strength. This data has the same shape as the filled circles, which indicates that the energy region from ~ 25-28 eV beyond the parent peak centered at 22.7 eV is composed of mainly (4a')  _1  satellite ionization  processes. If curve 6 is multiplied by 0.86 to produce curve 6c, excellent shape and intensity agreement is obtained with the energy slice data from 20-28 eV. Thus, 86% of the (4a')"  1  intensity can be accounted for over this energy region. This is in agreement with the discussion of Section 8.3.1, in which the spectroscopic strength for the inner valence 4a' orbital was found to be severely split. In the earlier study of methanol [84], a parent pole strength of 0.6 was indicated for the 4a' orbital when comparing the intensities of the X M P s with the lower quality double-zeta Snyder and Basch [266] calculation. The DFT momentum profiles for the 4a'  Chapter 8. Methanol  193  orbital are shown in Fig. 8.5d along with the experimental data for the energy region 20-28 eV. The density functional TMPs all agree exceptionally well for the shape of the 4a' X M P . In addition, DFT predicts intensities which are lower and more in agreement with experiment than the Hartree-Fock curve 6 and thus are consistent with essentially all of the (4a') pole strength _1  being in the 20-28 eV region. The experimental and theoretical momentum profiles for the 3a' inner valence orbital of C H O H are shown in Figs. 8.5e and 8.5f. 3  The filled circles represent the experimental  multichannel E M S measurements obtained from the deconvolution procedure by fitting a Gaussian peak to the main (3a')- parent ionization peak centered at 32.2 eV (see Fig. 8.1 and 1  also Table 8.2). The open circles show the data for the entire high energy region of 28-47 eV which contains the parent peak and the satellite processes present at energies > 35 eV in the E M S binding energy spectra (Fig. 8.1).  It can be seen from Figs. 8.5e and 8.5f that all  calculations provide reasonable shape comparisons to the two X M P s . However, curves 1-6 all overestimate the intensity of the experimental data from the deconvolution procedure.  If  curve 6 is be multiplied by a factor of 0.70 (i.e. curve 6a), excellent agreement is obtained for shape and intensity with the experimental data. On this basis the 3a' parent peak would have a pole strength of 0.70 with the remaining ~ 0.30 pole strength being split into higher energy satellite ionization processes. A similar parent pole strength of 0.6 was found in the earlier work [84]. This is consistent with the severe splitting of the inner valence 3a' orbital predicted by theory [260]. The fact that the 28-47 eV data (open circles) has the same shape as the deconvoluted data (solid circles) suggests that the EMS cross-section in the 35-47 eV region is dominantly due to (3a') satellite processes. The intensity of the 28-47 eV region is also only _1  slightly less than predicted by the higher-level TMPs (curves 4 and 6) and thus nearly all of the (3a') pole strength is accounted for within this energy region. The small amount of remaining _1  (3a') pole strength is presumably located above 47 eV. The DFT momentum profiles for this _1  orbital (Fig. 8.5f) generally provide excellent shape agreement with experiment and intensities similar to those obtained with the higher level Hartree-Fock calculations.  Chapter 8. Methanol  For  accurate computations  194 of chemical phenomena,  it is important to use a  wavefunction that gives good calculated values for a range of properties (e.g. total energy, dipole moment and momentum profile) that emphasize different regions of phase space [39,69].  On considering the other properties obtained from the various wavefunctions  (Table 1), it can be seen that the poor STO-3G total energy and low dipole moment value reflect the low degree of basis set saturation in this calculation. Much better total energies and dipole moments are obtained with larger basis sets that incorporate diffuse and polarization functions (the 6-311G, 6-311+G* and 6-311++G** calculations).  The 110-GTO and  Trun-pV5Z calculations provide the largest Hartree-Fock results in the present work.  It  appears that the Hartree-Fock limit is being closely approached by these calculations (5 and 6) since they provide momentum profiles (see Figs. 8.3-8.5) with much the same shape and intensity as well as similar energies, although the dipole moment is still converging (Table 8.1). In addition to the improved agreement it provides between theory and experiment for several XMPs, a significant improvement is made in the total energy with the 110-G(CI) calculation. While little can be said regarding the DFT total energies [60], it can be seen from Table 8.1 that the dipole moments predicted by the DFT calculations are generally excellent and within 0.1 Debye of experiment.  8.3.3 Density Maps in Momentum and Position Space The momentum and position space density contour maps corresponding to an oriented C H O H molecule presented in Figs. 8.6-8.8 provide some information regarding the 3  complementary nature of p- and r-space. Some knowledge of the shapes and intensities of the (spherically averaged) XMPs and TMPs can also be gained by an analysis of these maps. 2  These maps are slices of the orbital electron density y/  through the molecular plane of an  oriented molecule in position space and with momentum perpendicular and parallel with respect to the molecular plane in momentum space. The molecular planes chosen are indicated  Chapter 8. Methanol  195  MOMENTUM DENSITY  POSITION DENSITY  CH 0H 2a"  (a) 1 _1_ ^  CH3OH  '(b)  3  2a"  8  . /^v/->A _ 4 0 -4 ^  -8  -4  i**-—'  i 0 (au)  1  -8  4  MOMENTUM DENSITY 3  8  0 (au)  CHjOH 7a'  1  (d)  \ \ VI  -4  1  0 (au)  POSITION DENSITY  CH OH 7a'  '(c)  -4  (I(  0  \ V  0  -4  4  8  (au) POSITION DENSITY  MOMENTUM DENSITY  CH3OH  CH3OH  6a'  6a'  >  llllMTli ^^^^^^^^^^  1 i 0 (au)  1  0 (au) t  F i g . 8.6: Momentum and position density contour maps for the 2a", 7a' and 6a' orbitals of an oriented methanol molecule calculated at the Hartree-Fock level using the Trun-pV5Z basis set. The contour values represent 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density. The side panels (top and right side) show the density along the dashed lines (dashed vertical and horizontal lines) for each density map.  Chapter 8. Methanol  196  MOMENTUM DENSITY  POSITION DENSITY  CH OH  (a)  CH 0H  (b)  3  1a"  3  la"  g  i i i i  1  4  K 0.  0  ^C  —4  0  *  —A  1  (au) MOMENTUM DENSITY  1 0  1  (au) POSITION DENSITY |  (d)  CH3OH  5a'  1 t  i  (au) F i g . 8.7: Momentum and position density contour maps for the la" and 5a' orbitals of an oriented methanol molecule calculated at the Hartree-Fock level using the Trun-pV5Z basis set. The contour values represent 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density. The side panels (top and right side) show the density along the dashed lines (dashed vertical and horizontal lines) for each density map.  Chapter 8. Methanol  197  MOMENTUM DENSITY  POSITION DENSITY  CH 0H 4a'  K--J|3K5  ^^ttf-^l  CH3OH  '.(b)  3  4a'  > 1  0  0  1  (au)  (au)  MOMENTUM DENSITY  POSITION DENSITY  1  '(d)  CH OH 3a' 3  \ ,\  3 M -4  1  0  1  (au)  F i g . 8.8: Momentum and position density contour maps for the 4a' and 3a' orbitals of an oriented methanol molecule calculated at the Hartree-Fock level using the Trun-pV5Z basis set. The contour values represent 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, 30.0, and 99.0% of the maximum density. The side panels (top and right side) show the density along the dashed lines (dashed vertical and horizontal lines) for each density map.  Chapter 8. Methanol  198  in the right-hand side panels of Figs. 8.7-8.8. The calculations for each molecule are based on the near Hartree-Fock limit results (Trun-pV5Z) which has been found to provide good descriptions of the X M P s at the Hartree-Fock level. The origin for the position space maps is the molecular center of mass and all dimensions are in atomic units. The side panels on each map show slices of the momentum or position space density along the vertical (right panel) and horizontal (top panel) dotted lines.  Position space three dimensional molecular orbital  representations for the valence orbitals of methanol have been published by J0rgensen and Salem [267] and these are also shown in Ref. [84]. The 2a" H O M O of methanol is predicted in simple valence bond terminology to be an oxygen lone pair. Accordingly, the r-map (Fig. 8.6b) for this orbital mainly consists of atomic p-type functions concentrated on oxygen. However, there is a partial derealization of the 2a" electron density away from a pure O lone pair onto the C atom. While the general shape of this orbital is similar to that of a 7t* orbital, the p-type orbital components are not of equal intensity on C and O. In addition, there is only one nodal plane in p-space (Fig. 8.6a) due to the heteroatomic nature and asymmetry of C H O H [268]. The momentum density along the other 3  plane in p-space is very small and is identically zero at zero momentum.  Thus, spherical  averaging of this momentum density should give a momentum profile with zero intensity at p = 0. In this regard it should be noted that the calculated spherically averaged momentum profiles in Figs. 8.3a and 8.3b have a very small non-zero intensity at p = 0 which is entirely due to the fact that they have been folded with the instrumental resolution function to compare with experiment. However, as noted in Section 8.3.2 and shown in Figs. 8.3a and 8.3b, the experimental intensity is somewhat larger than theory below p - 0.3 au. The 7a' orbital is also considered (at least according to simple valence bond concepts) to be an oxygen lone pair. The more complex molecular orbital Trun-pV5Z wavefunction used in this work, however, predicts lobes of electron density on both C and O which interfere constructively or destructively (see r-map on Fig. 8.6d). Unlike the 2a" orbital, the p-map for the 7a' orbital contains no nodal planes and the momentum density at zero momentum is non-  Chapter 8. Methanol  199  zero. Thus, a significant s-type component to the momentum profile is predicted by all TMPs, in accordance with experiment.  However, p-type components in the wavefunction still  dominate. The r-map for the 6a' molecular orbital (Fig. 8.6f) shows that significant G exists while G  C H  c o  density  density due to the in-plane hydrogen of the methyl group is also present. In  addition, the methyl and hydroxyl hydrogen density adds in-phase to produce a spatially diffuse electron density and thus, according to the reciprocity of r- and p- space, appreciable electron density exists at low momentum in p-space (see p-map). This electron density at low p is reflected in the predominantly s-type shape of the momentum profiles for this orbital. The l a " orbital is considered to be basically an out-of-plane orbital with lobes like a it orbital involving the out-of-plane hydrogens of the methyl group, the C 2p and some lobes of lesser density on O. Thus, the M O coefficients show that C 2p, O 2p and H Is components dominate the electron density. The out-of-plane character is reflected by the nodal plane along the C - 0 axis (see r-map in Fig. 8.7b). The momentum density is zero at p = 0 and the side slice in the p-map also indicates appreciable electron density with a maximum at ±0.6 au. Thus, after spherical averaging, the momentum profiles for this orbital have zero intensity at zero momentum and a p  m a x  at ~ 0.6 au. It should be noted that, similar to situation for the 2a"  momentum profile, the calculated momentum profiles in Fig. 8.4c and 8.4d have non-zero intensity at low momentum due to folding with the experimental angular resolution function. The 5a' orbital shows two main regions of density (see r-map in Fig. 8.7d). Density between the out-of-plane hydrogens and the O H density adds to produce the region below the 0  C - 0 bond axis. The density in the top region involves C, O and the in-plane methyl group H . The p-map is complex, reflecting the multicenter nature of this orbital. In addition, the top and side slices of the p-map reflect large amounts of both s- and p-type components in the molecular orbital.  Thus, a large amount of s-type character appears in the calculated  momentum profiles for this orbital.  It should be noted, however, that the Hartree-Fock  Chapter 8. Methanol  200  Trun-pV5Z wavefunction used to construct these maps underestimates the s-type character in the X M P (Fig. 8.5a). Two boundary surfaces appear in the 4a' r-map (Fig. 8.8b) with maximum density on the C and O atoms. A n analysis of the molecular orbital coefficients reveals that O 2s and C 2s atomic components dominate this M O . It is not unexpected that such an inner valence orbital would display mostly atomic character. There is considerable density at p = 0 in the p-map. Some p-type components are also prevalent in the wavefunction as seen by the two maxima at ±0.4 au in the top slice of the p-map. The calculated momentum profiles for this orbital (Figs. 8.5c and 8.5d) are thus predominantly s-type with some levelling off of the profile below 0.4 au. The large electron density on O indicated in the top slice of the 3a' r-map (Fig. 8.8d) is due to large s-orbital components in the wavefunction. The M O coefficients reveal that this inner valence orbital is mostly composed of atomic-like O 2s character. The p-map contains very large momentum density at p = 0 and the TMPs for this orbital are very strongly s-type (Figs. 8.5e and 8.5f).  Chapter 9. Concluding Remarks  201  Chapter 9  Concluding Remarks  T h i s w o r k has presented E M S m e a s u r e m e n t s o f the b i n d i n g e n e r g y s p e c t r a a n d valence m o m e n t u m profiles for molecular oxygen and methanol.  In a d d i t i o n , the  all  momentum  p r o f i l e s f o r the h i g h e s t o c c u p i e d m o l e c u l a r orbitals h a v e b e e n s t u d i e d f o r N O a n d N02,.the series P H , P F 3  3  and P ( C H ) 3  3  a n d the series C r ( C O ) , M o ( C O ) 6  6  and W ( C O ) . 6  M o s t o f these  m e a s u r e m e n t s w e r e o b t a i n e d u s i n g the e n e r g y - d i s p e r s i v e m u l t i c h a n n e l s p e c t r o m e t e r d e s c r i b e d i n C h a p t e r 3. the  larger  F o r the studies o f the l o w i o n i z a t i o n c r o s s - s e c t i o n s f o r the o p e n - s h e l l m o l e c u l e s ,  molecules  and  some  orbitals  of  methanol,  the  increased  sensitivity  of  this  m u l t i c h a n n e l spectrometer m a d e p o s s i b l e the c o m p l e t i o n o f the m e a s u r e m e n t s i n a reasonable a m o u n t o f t i m e a n d to g o o d statistical p r e c i s i o n . T h e e x p e r i m e n t a l results h a v e b e e n c o m p a r e d w i t h a r a n g e o f H a r t r e e - F o c k c a l c u l a t i o n s p e r f o r m e d at t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . addition, comparisons have been made with high-level Hartree-Fock and  In  post-Hartree-Fock  (e.g. M R S D - C I ) calculations carried out i n c o l l a b o r a t i o n w o r k b y P r o f e s s o r E . R . D a v i d s o n a n d c o - w o r k e r s at I n d i a n a U n i v e r s i t y . o f the r e c e n t l y  T h e present w o r k has also p r o v i d e d quantitative assessments  d e v e l o p e d D F T target K o h n - S h a m a p p r o x i m a t i o n  calculation of momentum  (TKSA)  method  for  the  profiles.  T h e present E M S results f o r N O are an i m p r o v e m e n t o v e r p r e v i o u s m e a s u r e m e n t s a n d the d a t a s h o w c o n s i d e r a b l y less scatter.  T h e results f o r N O are s i g n i f i c a n t l y d i f f e r e n t f r o m  p r e v i o u s l y p u b l i s h e d m e a s u r e m e n t s i n the l o w m o m e n t u m r e g i o n a n d the d i f f e r e n c e s c a n b e attributed to c o n t a m i n a t i o n b y N 0  2  i n the earlier w o r k . In a d d i t i o n , the first m e a s u r e m e n t s a n d  Chapter 9. Concluding Remarks  202  calculations for the X M P for the 6a H O M O of N 0 are reported. A wide range of Hartreex  2  Fock and high level post-Hartree-Fock (MRSD-CI, A C P F and QDVPT) calculations have also been performed and generally good agreement was obtained between theoretical and experimental momentum profiles for both open-shell molecules. At a given Hartree-Fock level, ROHF calculations are found to give more intense momentum profiles than U H F methods. Post-Hartree-Fock calculations produce changes in the intensity of the momentum profiles of N O and N 0 but only small changes in the shape. 2  The small but significant  discrepancies remaining between theory and experiment for N O at lower momentum even at the post-Hartree-Fock level are reduced slightly by the density functional theory calculations. The presently reported experimental momentum profiles of 0 improved statistics compared with previously published E M S results.  2  display considerably The oxygen binding  energy spectra in the 9-59 eV range have been compared with MRSD-CI calculations. Strong splittings corresponding to E ~ and E ~ satellite states are observed experimentally and also 4  2  predicted by the MRSD-CI calculations for the inner valence region.  The measured  momentum profiles are compared with cross-sections calculated using both unrestricted (openshell) and restricted methods using basis sets ranging from minimal basis to near Hartree-Fock limit in quality.  In addition, the effects of correlation and relaxation on the calculated  momentum profiles are investigated using MRSD-CI calculations. Electron correlation effects in the ground state are further examined using several density functional approaches for the momentum profiles. The present EMS measurements and MRSD-CI calculations clearly show that the binding energy peak P at ~ 27.3 eV has significant contributions from both £ ~ and 4  2  £~ processes in contrast to earlier assignments which have attributed this peak to the C X~ 2  state alone. Similarly, the binding energy peak at 33 eV is shown to be due to £ ~ satellite 2  intensity rather than the earlier assignment of n 2  u  character.  Previous E M S measurements for P H [74] have been extended to include experimental 3  and theoretical studies of the orbital momentum profiles for the HOMOs of P F and P ( C H ) . 3  3  3  New calculations with larger basis sets have also been presented for the H O M O of P H . As in 3  Chapter 9. Concluding  Remarks  203  the case of N H , N F and N ( C H ) , intuitive arguments fail to predict the correct amounts of 3  3  3  3  s- and p-type character in the momentum distributions of P H , P F and P ( C H ) . 3  3  3  3  A  consideration of the E M S results and calculations together with an analysis of the respective H O M O density difference maps suggests that the methyl group is electron withdrawing relative to H when bonded to phosphorus. These results are analogous to the situation found previously [81,82] for the corresponding nitrogen compounds. E M S measurements for the metal hexacarbonyls M(CO)6, where M = Cr, M o or W , are compared with a range of Hartree-Fock and DFT calculations. The present measurements for Cr(CO)g are in good agreement with previously published work, particularly in the low momentum region where some unexpected intensity is observed in the experimental momentum profile contrary to symmetry expectations. The measurements of the X M P s of the 2t  2g  HOMOs of Mo(CO) and W(CO)g are the first to be reported. The large discrepancies 6  remaining between theory and experiment at low momentum for all molecules cannot be accounted for by any of the PWIA Hartree-Fock or DFT calculations. Vibrational effects distorting the symmetry of the ground state molecules are a possible explanation for the discrepancies at low momentum. Comparison of PWIA and DWIA calculations for atomic Cr 3d and M o 4d orbitals suggest that the discrepancies in the metal hexacarbonyl molecules may be due to distorted wave effects.  At present DWIA calculations are not possible for  molecular targets and clearly such theoretical developments are desirable. The methanol binding energy spectra in the 6-47 eV range have been compared with Green's function calculations [260]. In the inner valence region strong splitting of the 4a' and 3 a' orbitals due to final electron state correlation is observed experimentally and predicted by the calculations. The experimental momentum profiles of the complete valence orbitals have been used to evaluate various theoretical models including Hartree-Fock wavefunctions ranging from the minimal basis set to the near Hartree-Fock limit, many-body M R S D - C I treatments with large basis sets, and Kohn-Sham DFT calculations a large basis set and several exchange-correlation potentials. Inclusion of electron correlation using the M R S D - C I and DFT  Chapter 9. Concluding  Remarks  204  methods is found to be necessary for reasonable predictions of the shapes and intensities of the outer spatial regions of the valence electron momentum profiles of the 7a', (6a'+la") and 5a' orbitals. Some small discrepancy between theory and experiment remains at low momentum for the 2a", 7a' and 5a' orbitals.  For the 5a' orbital in particular, all theory severely  underestimates the s-type character in the X M P . The shapes of the inner valence 4a' and 3a' momentum profiles are well predicted by theory, but the theoretical momentum profiles must be multiplied by a factor of 0.75 for the 4a' and 0.70 for the 3a' orbital to match the experimental intensity. This is further evidence of the extensive pole strength splitting that occurs for ionization from these orbitals. The high sensitivity of the energy-dispersive multichannel E M S spectrometer (approximately a factor of ten times that of a single channel instrument) was crucial for the measurements reported in this thesis. However, further improvements in sensitivity and energy resolution are highly desirable so that even larger molecules as well as low density targets such as excited molecules, free radicals and ions may be studied. A new high performance E M S spectrometer is currently being constructed in this laboratory which is multichannel in both energy and angle [269]. This spectrometer is designed to detect all outgoing electrons around the entire 360° azimuth (i.e. electrons will be detected at all <() angles simultaneously) while also using the energy-dispersive capabilities of the present E M S spectrometer in the radial direction. This new instrument will complete the progression of symmetric non-coplanar E M S spectrometers developed over the past 25 years. 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