E V A L U A T I O N OF WAVEFUNCTIONS BY ELECTRON MOMENTUM SPECTROSCOPY by A L E X I S D E L A N O ORTIZ B A W A G A N B.Sc.(cum laude), University of the Philippines, 1979 M.Sc, University of Houston, 1982 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S FOR T H E D E G R E E O F DOCTOR O F P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E STUDIES Chemistry Department We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A August, 1987 c Alexis Delano Ortiz Bawagan, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia 1956 Main Mall Vancouver, Canada Department V6T 1Y3 DE-6(3/81) 'Sa aking mga Magulang' (To my Mother and Father) i ABSTRACT Electron momentum spectroscopy (EMS) provides experimental atomic and molecular electronic structure information in terms of the binding energy spectrum and the experimental momentum profile (XMP), which is a direct probe of the electron momentum distribution in specific molecular orbitals. The measured X M P s permit a detailed quantitative evaluation of theoretical ab initio wavefunctions in quantum chemistry and also provide a means to investigate traditional concepts in chemical reactivity at the fundamental electronic level. This thesis reports high momentum resolution E M S measurements of the valence orbitals of H 2 0 , D 2 0 , N H 3 and H 2 C O obtained using an E M S spectrometer of the symmetric, non-coplanar type operated at an impact energy of 1200eV. The measured experimental momentum profiles for the valence orbitals of each molecule have been placed on a common intensity scale, which has allowed a stringent quantitative comparison between experiment and theory. These studies now confirm earlier preliminary investigations that suggested serious discrepancies between experimental and theoretical momentum distributions. Exhaustive consideration of possible rationalizations of these discrepancies indicate that double zeta quality and even near Hartree-Fock quality wavefunctions are insufficient in describing the outermost valence orbitals of H 2 0 and N H 3 . Preliminary results for H 2 C O also indicate that near Hartree-Fock wavefunctions are incapable of describing the outermost 2b 2 orbital. Interactive and collaborative theoretical efforts have therefore led to the development of new Hartree-Fock limit and also highly correlated (CI) wavefunctions for H 2 0 , N H 3 and H 2 C O . It is found that highly extended basis sets including diffuse functions and the adequate inclusion ii of correlation and relaxation effects are necessary in the accurate prediction of experimental momentum profiles as measured by electron momentum spectroscopy. New E M S measurements are also reported for the outermost valence orbitals of NF 3 , NH 2 CH 3 , NH (CH 3 ) 2 , N (CH 3 ) 3 and para-dichlorobenzene. These exploratory studies have illustrated useful chemical applications of E M S . In particular, E M S measurements of the outermost orbitals of the methylated amines have revealed chemical trends which are consistent with molecular orbital calculations. These calculations suggest extensive electron density derealization of the so-called nitrogen 'lone pair' in the methylated amines in comparison to the Tone pair' in N H 3 . E M S measurements of the non-degenerate IT 3 and 7T 2 orbitals of para-dichlorobenzene show different experimental momentum profiles consistent with arguments based on inductive and resonance effects. These experimental trends, both in the case of the amines and para-dichlorobenzene, were qualitatively predicted by molecular orbital calculations using double zeta quality wavefunctions. However more accurate prediction of the experimental momentum profiles of these molecules will need more extended basis sets and the inclusion of correlation and relaxation effects as suggested by the studies based on the smaller molecules. A n integrated computer package (HEMS) for momentum space calculations has also been developed based on improvements to existing programs. Development studies testing a new prototype multichannel (in the 0 plane) E M S spectrometer are described. iii T A B L E O F C O N T E N T S A B S T R A C T " T A B L E OF C O N T E N T S * v LIST OF F I G U R E S v i i LIST OF T A B L E S x i LIST OF A B B R E V I A T I O N S xii A C K N O W L E D G E M E N T S xiv Chapter 1. INTRODUCTION 1 1.1. Historical Remarks 1 1.2. Binding Energy Spectrum (BES): A Physical Observable 3 1.3. Experimental Momentum Profile (XMP): A Physical Observable 5 1.4. Scope of Thesis 9 Chapter 2. T H E O R E T I C A L M E T H O D 11 2.1. Theory of Electron Momentum Spectroscopy (EMS) 11 2.1.1. Introductory Remarks 11 2.1.2. Plane Wave Impulse Approximation (PWIA) .. 14 2.1.3. Target Hartree-Fock Approximation (THFA) ... 20 2.1.4. Full Ion-Neutral Overlap Amplitude 24 2.2. Momentum Space Chemistry 1 25 2.3. Basis Sets for Ab initio Wavefunctions 29 2.4. Electron Correlation 32 2.5. Green's Function Methods and Ionization Spectra 34 2.6. H E M S : A Computer Package for Momentum-Space Calculations 36 Chapter 3. E X P E R I M E N T A L M E T H O D 44 3.1. Electron Momentum Spectrometer 44 3.1.1. Description of Spectrometer 44 3.1.2. Coincidence Detection, Event Processing and Control 51 3.2. Modelling the Effects of Finite Momentum Resolution 57 3.2.1. Planar Grid Method 59 3.2.2. Analytic Gaussian Function Method 63 3.2.3. Defining the Optimum p-Value 66 Chapter 4. W A T E R : P A R T I 71 4.1. Overview 71 4.2. Binding Energy Spectra of Water 73 iv 4.3. Momentum Distributions of Water 80 4.4. Comparison of Experimental and Theoretical Momentum Distributions 82 4.5. Orbital Density Maps and Surfaces 95 4.6. Wide Range Momentum Density Maps 97 Chapter 5. WATER: PART II 100 5.1. Overview 100 5.2. Extended Basis Sets for Water 108 5.3. Experimental Details: Normalization of Data I l l 5.4. Vibrational Effects 117 5.5. Basis Set Effects 119 5.6. Correlation and Relaxation Effects 129 5.7. Calculated Properties Near the HF and CI Limits 144 5.8. Summary 149 Chapter 6. AMMONIA 152 6.1. Overview 152 6.2. Basis Sets for Literature SCF Wavefunctions 153 6.2.1. A 126-GTO Extended Basis Set for N H 3 ... 158 6.2.2. Inclusion of Correlation: Calculation of the Ion-Neutral Overlap Distribution 161 6.3. Binding Energy Spectra 165 6.4. Comparison of Experimental Momentum Profiles with Theoretical Predictions 174 6.5. Ion-Neutral Overlap Distributions (OVDs) 184 6.6. Position-Space and Momentum-Space Density Maps 189 6.7. Exterior Electron Distribution (EED) Ratios and XMPs 193 6.8. Summary 195 Chapter 7. FORMALDEHYDE 196 7.1. Overview 196 7.2. Basis Sets for SCF Wavefunctions 197 7.3. Binding Energy Spectra 200 7.4. Comparison of Experimental Momentum Profiles with Theoretical Predictions 207 7.5. Summary 220 Chapter 8. PARA-DICHLOROBENZENE 222 8.1. Overview 222 8.2. Experimental Momentum Profiles 225 8.3. Calculated Momentum Distributions 228 8.4. Summary 234 Chapter 9. METHYLATED AMINES AND N F 3 236 9.1. Overview 236 9.2. Basis Sets for SCF Wavefunctions 238 9.3. Measured and Calculated Momentum distributions 239 9.4. The Methyl Inductive Effect 254 v 9.5. Summary 256 Chapter 10. T H E H A L O G E N S : A T H E O R E T I C A L S T U D Y 258 10.1. Overview 258 10.2. Molecular Chlorine ( C l 2 ) 259 10.3. Molecular Bromine ( B r 2 ) 270 10.4. Molecular Iodine ( l 2 ) 276 10.5. General Trends 283 Chapter 11. M U L T I C H A N N E L E M S : PROSPECTS A N D D E V E L O P M E N T S .. 286 11.1. Multichannel Electron Momentum Spectrometer (MEMS) 286 11.1.1. Accurate Alignment 287 11.1.2. Suppression of Background Secondary Electrons 292 11.1.3. C A M A C Interface 292 11.1.4. Modes of Operation 296 11.2. Preliminary Results 299 11.2.1. Helium 299 11.2.2. Argon 299 C O N C L U D I N G R E M A R K S 304 R E F E R E N C E S 308 vi List of Figures Fig. 2.1 The (e,2e) reaction 12 Fig . 2.2 Layout of the H E M S computer package 38 Fig. 2.3 Input structure of H E M S package 40 Fig. 2.4 Sample output of H E M S package 41 Fig. 2.5 Density difference (CI - SCF) maps for H 2 42 Fig. 3.1 Schematic of Electron Momentum Spectrometer 46 Fig. 3.2 Design parameters of 135° sector C M A 49 Fig. 3.3 Timing coincidence electronics of E M S spectrometer 52 Fig. 3.4 Collision volume defined by spectrometer apertures and beam size 60 Fig. 3.5 Sample results using planar grid method 62 Fig. 3.6 Sample results using analytic gaussian function method 65 Fig. 3.7 P-value histogram at different values of <p0 68 Fig. 4.1 Binding energy spectra of H 2 O at 0 = 0 ° and 8° 74 Fig. 4.2 Comparison of inner valence (2a, ) binding energy spectrum with previous experimental and theoretical work 77 Fig. 4.3 Spherically averaged momentum distribution and distribution difference for the l b , orbital of H 2 0 91 Fig. 4.4 Spherically averaged momentum distribution and distribution difference for the 3a, orbital of H 2 0 92 Fig. 4.5 Spherically averaged momentum distribution and distribution difference for the l b 2 orbital of H 2 0 93 Fig. 4.6 Spherically averaged momentum distribution and distribution difference for the 2a, orbital of H 2 0 94 Fig. 4.7 Wide range momentum density contour maps for the valence orbitals of H 2 0 98 Fig. 5.1 Detailed comparison of the X M P s of the l b , orbital of H 2 O and D 2 O with theoretical calculations 113 vii Fig. 5.2 Detailed comparison of the X M P s of the 3a, orbital of H 2 O and D 2 O with theoretical calculations 114 Fig. 5.3 Detailed comparison of the X M P s of the l b 2 orbital of H 2 O and D 2 O with theoretical calculations 115 Fig. 5.4 Detailed comparison of the X M P s of the 2a, orbital of H 2 O and D 2 O with theoretical calculations 116 Fig. 5.5 Comparison of the calculated valence MDs using the best Slater and Gaussian basis sets 130 Fig. 5.6 Correlation effects in the calculated MDs of the l b , orbital of water 134 Fig. 5.7 Correlation effects in the calculated MDs of the 3a, orbital of water 135 Fig. 5.8 Correlation effects in the calculated MDs of the l b 2 orbital of water 136 Fig. 5.9 Binding energy spectrum of water in the inner valence region 141 Fig. 5.10 Two-dimensional density difference (CI-THFA) plots in momentum space and position space 145 Fig. 5.11 Convergence of calculated properties 146 Fig. 6.1 Binding energy spectra of N H 3 at 0 = 0 ° and </» = 8° 167 Fig. 6.2 Comparison of the inner valence binding energy spectrum with theoretical predictions 169 Fig. 6.3 Comparison of valence X M P s of N H 3 with MDs calculated from various wavefunctions 176 Fig. 6.4 Comparison of valence X M P s of N H 3 with ion-neutral overlaps (OVDs) calculated from correlated wavefunctions 179 Fig. 6.5 Comparison of valence X M P s of N H 3 with MDs calculated from target natural orbitals (TNOs) 190 Fig. 6.6 Position space and momentum space density contour maps for N H 3 192 Fig. 7.1 Binding energy spectra of H 2 CO measured at 0 = 0° and 0 = 6° 202 Fig. 7.2 Inner valence binding energy spectra of H 2 CO 205 viii Fig. 7.3 Binding energy scans in the region 14-22eV as a function of azimuthal angles, </> 208 Fig. 7.4 Comparison of the 2b 2 experimental momentum profile of H 2 C O with calculated MDs 212 Fig. 7.5 Comparison of the l b , experimental momentum profile of H 2 C O with calculated MDs 213 Fig. 7.6 Comparison of the 5a, experimental momentum profile of H 2 C O with calculated MDs 215 Fig. 7.7 Comparison of the l b 2 experimental momentum profile o f , H 2 C O with calculated MDs 216 Fig. 7.8 Comparison of the 4a, experimental momentum profile of H 2 C O with calculated MDs 218 Fig. 7.9 Comparison of the 3a, experimental momentum profile of H 2 C O with calculated MDs 219 Fig. 8.1 Binding energy spectrum of para-dichlorobenzene 227 Fig. 8.2 Measured momentum profiles for the ir3 and ir2 orbitals of para-dichlorobenzene 229 Fig. 8.3 Calculated momentum distributions for para-diclorobenzene and benzene , 231 Fig. 8.4 Schematic representation of the wavefunction amplitudes for the IT orbitals of para-dichlorobenzene and benzene 232 Fig. 9.1 Comparison of the experimental momentum profiles of the outermost valence orbitals of NH 3 , NH 2 CH 3 , NH (CH 3 ) 2 , N ( C H 3 ) 3 and N F 3 241 Fig. 9.2 Two-dimensional contour plots of the position space and momentum space densities of the outermost valence orbitals of N H 3 , N H 2 C H 3 , N H ( C H 3 ) 2 , N ( C H 3 ) 3 and N F 3 244 Fig. 9.3 Two-dimensional density difference contour maps in both position space and momentum space 248 Fig. 10.1 Experimental and theoretical MDs for 2ir^ orbital of C l 2 263 Fig. 10.2 Experimental and theoretical MDs for 2iru orbital of C l 2 264 Fig. 10.3 Experimental and theoretical MDs for 5o^ orbital of C l 2 265 ix Fig. 10.4 Experimental and theoretical MDs for 4o"u orbital of C l 2 266 Fig. 10.5 Experimental and theoretical MDs for 4ag orbital of C l 2 267 Fig. 10.6 Effect of basis set polarization on calculated MDs of Cl 2 269 Fig. 10.7 Experimental and theoretical MDs for 4ir^ orbital of Br 2 271 Fig. 10.8 Experimental and theoretical MDs for 4TT u orbital of Br 2 272 Fig. 10.9 Experimental and theoretical MDs for 8a^ orbital of Br 2 273 Fig. 10.10 Experimental and theoretical MDs for 7a u orbital of Br 2 274 ^ Fig. 10.11 Experimental and theoretical MDs for la orbital of Br 2 275 Fig. 10.12 Experimental and theoretical MDs for 67Tg orbital of I 2 277 Fig. 10.13 Experimental and theoretical MDs for 67TU orbital of I 2 278 Fig. 10.14 Experimental and theoretical MDs for H ^ g orbital of I 2 279 Fig. 10.15 Experimental and theoretical MDs for 10a u orbital of I 2 280 Fig. 10.16 Experimental and theoretical MDs for 10<7g orbital of I 2 281 Fig. 11.1 Schematic of Multichannel EMS Spectrometer 288 Fig. 11.2 Comparison of observed elastic signal 293 Fig. 11.3 MEMS modes of operation 294 Fig. 11.4 He Is binding energy spectrum 300 Fig. 11.5 Ar 3p experimental momentum profile 302 x List of Tables Table 1.1 Highlights in Electron Momentum Spectroscopy 2 Table 2.1 Subprograms in the H E M S package 37 Table 3.1 Components of E M S timing coincidence electronics 53 Table 4.1 Orbital ionization energies and peak intensities of water 78 Table 4.2 Comparison of theoretical SCF wavefunctions 83 Table 5.1 Properties of theoretical SCF and CI wavefunctions 109 Table 5.2 Characteristics of calculated orbital MDs and experimental momentum profiles 126 Table 5.3 CI calculations of the ground and final ion states of H 2 0 133 Table 5.4 Binding energy spectrum of water in the inner valence region 137 Table 6.1 Properties of Theoretical SCF and CI wavefunctions for N H 3 154 Table 6.2 CI calculations of the ground and final ion states of N H 3 162 Table 6.3 CI calculations of the pole strengths and energies in the binding energy spectrum of N H 3 163 Table 6.4 Experimental and calculated energies and relative pole strengths for the ionization of the 2a, orbital of N H 3 170 Table 7.1 Properties of theoretical wavefunctions for H 2 CO 198 Table 7.2 Binding energies and relative ionization intensities for H 2 CO 203 Table 9.1 Charateristics of theoretical SCF wavefunctions 240 Table 10.1 Wavefunctions for C l 2 261 Table 11.1 Design parameters of 360° C M A 289 Table 11.2 Configuration of C A M A C system 295 xi L I S T O F A B B R E V I A T I O N S A D C Analog-to-Digital Converter ADC(n) Algebraic Diagrammatic Construction to nth order au atomic units B E S Binding Energy Spectrum CI Configuration Interaction CGTO Contracted Gaussian Type Orbital C M A Cylindrical Mirror Analyser D Debye D A C Digital-to-Analog Converter DZ Double Zeta E M S Electron Momentum Spectroscopy eV electron Volt F O A Frozen Orbital Approximation fwhm full width at half maximum GTO Gaussian Type Orbital H E M S H-compiler optimized programs for E M S H F Hartree-Fock IP Ionization Potential ISR Interrupt Service Routine L C A O Linear Combination of Atomic Orbitals M B P T Many Body Perturbation Theory M B S Minimal Basis Set xii M C S C F Multiconfiguration Self Consistent Field M D theoretical Momentum Distribution M O Molecular Orbital MRSDCI Multireference Singles and Doubles Configuration Interaction O V D Ion-Neutral Overlap Distribution PES Photoelectron Spectroscopy PWIA Plane Wave Impulse Approximation R base Resolution SAC-CI Symmetry Adapted Cluster Configuration Interaction SCF Self Consistent Field SDCI Singles and Doubles Configuration Interaction STO Slater Type Orbital T A C Time-to-Amplitude Converter T H F A Target Hartree-Fock Approximation X M P Experimental Momentum Profile X P S X-ray Photoelectron Spectroscopy 2ph-TDA 2 particle-hole Tamm Dancoff Approximation xiii A C K N O W L E D G E M E N T S First and foremost, I would like to thank my research supervisor, Prof. Chris E . Brion, for his unwavering support, both academically and personally, throughout the course of this work. It has been a great privilege to share with him many long discussions from which I have definitely learned much and, most of all, to have been a witness to his infectious enthusiasm for scientific research. I would also like to thank Dr. Tong Leung for his invaluable assistance during the early phase of this work. Special thanks are also extended to the many colleagues with whom I have interacted and had discussions in the course of this work, in particular (1) Prof. Ernest R. Davidson, Dr. David Feller and Caroline Boyle - for the extended basis set (SCF and CI) calculations for H 2 0 , N H 3 and H 2 C O ; (2) Prof. Mike Coplan - for his collaboration in the work on para-dichlorobenzene; (3) Dr. L . Frost, Dr. A . M . Grisogono, Prof. C. Brion, Prof. W. von Niessen, Prof. A . Sgamellotti, Prof. E . Weigold, M r . R. Pascual and Dr. P .K . Mukherjee - for the data obtained in the work on the halogens; (4) Prof. Erich Weigold - for assistance with the multichannel E M S spectrometer; (5) Prof. Larry Weiler, Prof. Del Chong and Dr. Mark Casida -for discussions pertaining to work on the amines; Special acknowledgements are also due to the many support staff at U B C who 'stretched that extra arm' to enable the continued optimal operation of the E M S spectrometer. In particular, I wish to thank Ed Gomm, Bill Henderson, Brunius Snapkauskas, Brian Greene and Philip Carpendale. xiv I also gratefully acknowledge the University of British Columbia for receipt of a University Graduate Fellowship (UGF) during the period 1983-87. Finally, I would like to thank Adelaida for typing the bibliography and, most specially, for her sincere and loving support. xv CHAPTER 1. INTRODUCTION 1.1. HISTORICAL REMARKS Electron momentum spectroscopy (EMS)t has emerged as a unique and powerful spectroscopic technique following the pioneering experimental studies of Amaldi et al. [AE69], Camilloni et al. [CG72] and Weigold et al. [WH73] and the insightful theoretical investigations of Neudatchin et al. [NN69] and Glassgold and Ialongo [GI68]. The novel idea of obtaining electronic structure information using the (e,2e) coincidence experiments came from the analogous nuclear quasi-elastic knockout (p,2p) reaction [BM60, SN66]. The theoretical framework, originally developed for (p,2p) reactions, was quickly applied to the analysis of (e,2e) experiments. Initial speculations at that time [GI68, NN69, L72] indicated the potential of (e,2e) reactions for probing individual orbital electron momentum distributions (MDs) as well as the investigation of correlation effects in atoms, molecules and thin films. These speculations were soon to be verified in several experiments around the world [CG72, WH73, HH76, MC78, N L 8 1 , RD84]. Since then several reviews [MW76, WM78, GF81, N L 8 1 , MT82, B86, MW87] have been published. The important highlights in E M S for the past decade and a half are listed in Table 1.1. The list illustrates the various capabilities and applications of E M S . For a more extensive survey of the literature the bibliography published by Leung and Brion [LB85] should be consulted. t Electron momentum spectroscopy is also known as binary (e,2e) spectroscopy. 1 2 Table 1.1. Highlights in Experimental Electron Momentum Spectroscopy. Amaldi et al. [AE69] Demonstrated feasibility of (e,2e) reactions for obtaining the binding energy spectra of K-shell electrons in carbon film. E 0 = 14.6keV, AE=150eV. Camilloni et al. [CG72] Obtained the first K-shell and L-shell momentum distributions of carbon film using the coplanar symmetric (e,2e) reaction. E 0 = 9 k e V , A E = 45eV. Weigold et al. [WH73] Obtained the first resolved 3p and 3s valence shell momentum distributions of argon using the non-coplanar symmetric (e,2e) reaction. Evidence for inner valence (3s""b correlation states were also reported. E 0 = 4 0 0 e V , A E = 5eV. Dixon et al. [DM76] Observed population of n = 2 states in He illustrating potential of E M S for observing ground state correlations. Hood et [HH76] al. Serious disagreement between the experimental and theoretical momentum distributions of water were observed in low momentum resolution E M S experiments. Inadequacies in the ab initio wavefunctions for water were postulated. Moore et al. [MC78] Introduced multichannel E M S spectrometer in the 0-plane. Lohmann and Weigold [LW81] Reported the first experimental verification of the exact solution of the Schro dinger equation for the H-atom. E M S measurements are reported for E 0 = 400eV, 800eV, 1200eV. Leung and Brion [LB83a, LB84a] Obtained an experimental estimation of the (spherically averaged) chemical bond in H 2 . Cook et al. [CM84] Introduced the multichannel microchannel plate E M S spectrometer in the energy dispersive plane. 3 1.2. BINDING ENERGY SPECTRUM (BES): A PHYSICAL OBSERVABLE The intimate relation between experiment and theory is well illustrated by the development of quantum theory. Heisenberg [H30] points out the classic experiments of 1900-1930 which marked the genesis of the fundamental concepts in quantum theory. One example is the series of collision experiments of Franck and Hertz [FH13, H23] which demonstrated that atoms can only assume discrete energy values. These findings gave valuable support to Bohr's ideas concerning the 'stationary states of an atom'. It became apparent that the energy (E) of an atom in its ground, excited or ionized state is a well-defined physical observable defined by, <* |H|*> = E [1.1] H is the linear Hamiltonian operator for the system and 4" represents the particular state (ground, excited or ionized) wavefunction. Quantum theory [D57] provides a prescription for calculating the energy observable, E (see Eqn. 1.1) which we then identify with the state wavefunction, «P. The study of energies of the different possible states (energy spectra) of atoms or molecules has been the focus of various types of experiments. Electron energy loss spectroscopy (EELS) [TR70, WB72] and photoabsorption [WM77b] experiments probe the energies of, the excited states of the system. Photoelectron spectroscopy (PES) [SN69, TB70, KK81], dipole (e,2e) spectroscopy [BH81, B82] and Penning ionization electron spectroscopy (PIES) [OM84] have been useful in the study of 4 the energies of the different ionic states, t Electron momentum spectroscopy also has the capability of obtaining the energy spectra of the different ionic states of an atom or molecule. Since it is not energy-limited by resonance light sources (e.g. H e l limit is 21.2eV), E M S is capable of observing spectra over the complete binding energy range including both the outer and the inner valence regions. E M S and PES studies in the inner valence region have shown extra structure to what one would expect from the single-particle picture of ionization and these structures are now attributed to correlation states. The understanding of these correlation (many-body) states as well as the accurate prediction of their transition probabilities is fairly recent [MS76, CD77, CD86]. Nevertheless, the generally semi-quantitative agreement obtained between the experimental binding energy spectra obtained in E M S and the calculated spectra is quite impressive. It is clear from the studies conducted so far that the breakdown of the single-particle picture of ionization, specially in the inner valence region, is a general phenomenon [CD86]. Although E M S and PES both give some information regarding the correlation state (satellite) intensities, the two techniques are of an entirely different nature. It should be pointed out that E M S probes the low momentum components (p = 0 - 3 a 0 ~ 1 ) whereas higher photon energy PES probes the high momentum components (e.g. A l K a , p = l O a 0 " 1 ) . of the target wavefunction. Thus it is only in the case of states within the same symmetry manifold that E M S and PES (synchroton and X-ray sources) might be expected to give similar satellite t Strictly speaking the observed excitation energies or ionization potentials refer to the energy difference between the initial neutral state and the final neutral or ionic state. 5 relative intensity distributions [M85, BB87d, K K 8 7 , SH87]. Accurate experimental and theoretical data on correlation state intensities are few and future studies in this area will clearly feature even higher energy resolution E M S studies. 1.3. EXPERIMENTAL MOMENTUM PROFILE (XMP): A PHYSICAL OBSERVABLE The correspondence of the momentum-space representation and the position-space representation in quantum theory [D57] is illustrated by the Fourier transform relationship, *<p) = ( 2 T T ) " 3 / 2 J e " 1 ^ <Mr) dr [1.2] Although the position-space representation is more intuitive and easy to visualize, new insights to chemical and physical problems are available in the momentum-space representation. Coulson and Duncanson [CD41] first initiated the study of the properties of momentum-space wavefunctions and densities in different atoms and molecules. Since then significant theoretical work [SW65, ET77, NT81, R83, DS84, RB84, KK86] has developed into what could be called the field of momentum-space chemistry. Theoretical quantum chemists have exploited the advantages of working in momentum space. Recent reports [NT81, DS84] have showed that the Hartree-Fock equations for molecular systems may be solved easier and more directly in momentum space because the singularities in the potential (due to the multi-center nature of molecules) are reduced to one singularity at p = 0. Others [HL80, DJ84] have investigated the use of optimal basis sets obtained from extensive analysis of the momentum space Hartree-Fock wavefunctions. 6 Epstein and Tanner [ET77] outline the major principles in momentum-space chemistry and how it applies to the interpretation of molecular momentum-space density distributions. These principles or properties can be summarized as [ET77, CB82], (i) Inverse spatial weighting; (ii) Preservation of molecular symmetry; (iii) Addition of inversion symmetry; (iv) Bond oscillation; (v) Bond directional reversal; These Fourier transform properties or principles have been invaluable in the interpretation of momentum-space experimental quantities and the corresponding momentum-space and position-space density maps. Further discussion of these properties will be made in the following chapter. Experimental momentum-space chemistry involves a wide variety of experimental techniques. Electron momentum spectroscopy [MW76, WM78, LB85, B86, MW87], Compton scattering [W77, L77, C85], high energy electron scattering [BF74, BW77] and positron annihilation [W73, B77] are some of the techniques used to measure the experimental momentum distribution. In E M S the angular correlation spectra obtained for a particular energy-selected state are called experimental momentum profiles (XMPs). The term X M P is the preferred name as opposed to previous designations such as experimental momentum distributions. This avoids confusion between the purely experimental quantity (XMP) and the purely theoretical quantity (MD). 7 Like the total energy (Sec. 1.2) the XMP is also a well-defined physical observable within the limits defined by the theory of EMS and is proportional to the spherically averaged orbital momentum distribution, p(p) = fdQ|tf(p) | 2 [1.3] \^ (p) is the single-particle (momentum-space) orbital from which the electron was ionized. Energy-selected XMPs therefore provide a very detailed probe of the electron distribution in atoms, molecules and solids. In fact, EMS has been referred to in the literature as an orbital imaging [BB87a] or as a wavefunction mapping [MW87] technique. Since EMS is orbital selective it has a distinct advantage compared to Compton scattering and high energy electron scattering which measure a quantity (the Compton profile) equal to an integral over the total electron momentum distribution. Similarly the two-photon coincidence rate in positron annihilation gives information on the total momentum distribution [B77]. With the current improvements in momentum resolution, EMS measurements are now providing a sensitive test of the quality of ab initio wavefunctions. Collaborative experimental and theoretical studies, some of which are reported in this thesis, have been instrumental in the construction of more accurate wavefunctions in quantum chemistry. These new wavefunctions [BB87, BM87, BB87c, FB87a, FB87b] could be called 'universal' wavefunctions in the sense that they are sufficiently accurate to calculate all electronic properties of the molecule with good precision. Unlike other physical observables such as the total energy, dipole moment and < r 2 > e the XMP probes regions of phase space which contribute little to the total energy of the system. These regions have only 8 recently received much theoretical attention because traditional methods of calculating wavefunctions utilize the variational principle which weights (or emphasizes) regions of phase space that contribute most to the total energy. It has been mentioned [FB87] that the error in the total energy is second order in the error in the wavefunction whereas the error in one-electron properties (e.g. XMPs) are only first order in the error in the wavefunction. Thus it is reasonable that X M P s as well as other one-electron properties should provide a much more sensitive probe of wavefunction quality than the calculated total energy. As opposed to commonly quoted one-electron properties such as the dipole moment and quadrupole moment, increasing amount of evidence [BB87, BM87, FB87a, CC87] seems to indicate that X M P s are even more sensitive to the diffuse region of the electron distribution. In general, large r is not equivalent to small p because there is no one-to-one correspondence between position and momentum. However there is still an inverse weighting of the respective spaces due to the Fourier transform relationship. In summary, the renewed interest in momentum space chemistry that we are currently witnessing illustrates the importance of complementary views in scientific inquiry. It can be said that traditional understanding of chemical and physical phenomena from the position space perspective is incomplete without the corresponding understanding in momentum space. 9 1.4. SCOPE OF THESIS The two physical observables described in Sees. 1.2 (BES) and 1.3 (XMPs) are the primary experimental goals of the E M S studies reported in this thesis. The studies can be further subdivided according to the secondary goals involved in each particular study namely, (A) E M S studies of small molecular systems. This involves the comparison of high momentum resolution X M P s and MDs calculated from a range of SCF wavefunctions up to the Hartree-Fock limit and beyond to include effects of correlation and relaxation. The inner valence B E S are extensively analyzed and compared with existing theoretical predictions. Comparisons with photoelectron spectroscopy are also made in cases wherein P E S data for the inner valence region is available. (B) Chemical applications of E M S to complex molecular systems. This involves the study and identification of chemical trends in a series of related molecules. Most of the studies are exploratory and have tended to involve observation of the X M P s of those orbitals that would contribute most to the chemistry of the respective molecules. Type-(A) E M S studies were done for H 2 0 , D 2 0 , N H 3 and H 2 C O and are reported in chapters 4, 5, 6 and 7, respectively. On the other hand, type-(B) E M S studies were done for the outer valence orbitals of para-dichlorobenzene, the outermost valence orbitals of N F 3 and the substituted methyl amines and the halogens ( C l 2 , B r 2 and I 2 ) and are reported in chapters 8, 9 and 10, respectively. A special chapter (Chapter 11) outlines developmental work done on a multichannel E M S spectrometer and reports preliminary results obtained for 10 helium and argon. Atomic units (n = m e = e = l) have been used throughout the thesis unless otherwise stated. CHAPTER 2. THEORETICAL METHOD 2.1. THEORY OF ELECTRON MOMENTUM SPECTROSCOPY (EMS) 2.1.1. Introductory Remarks Electron momentum spectroscopy is based on the (e,2e) reaction, e 0 (Po , E 0 ) + M > e , ( p , , E , ) + e 2 ( p 2 , E 2 ) + M + ( p 3 , E 3 ) [2.1] where p^, E^ (i = 0,l,2) are the momentum and energy of the incoming, scattered and ejected electrons, respectively. The ion recoil momentum and energy are given by p 3 and E 3 , respectively. The (e,2e) reaction is also shown in Fig. 2.1. Several early studies [CG72, WH73] and reviews of the (e,2e) reaction [MW76, WM78, GF81] have shown that under certain well-defined kinematic conditions the (e,2e) cross section (reaction probability) t is largely a measure of the electron momentum distribution of the single-particle orbital from which the electron was ionized or knocked out. It is now established [MW76, WM78] that the necessary scattering conditions are, I. High impact energy (E 0 > lOOOeV); II. Symmetric energy sharing (E, = E 2 = E); III. Maximal momentum transfer ( | K* | = | p 0 - p, | £ 5a 0 ~ 1); IV. Non-coplanar ( 0 — 4 5 ° , <j> variable); Under these conditions and within the, V. Independent-particle description of electronic motion; t The (e,2e) cross section referred to is the probability of simultaneously detecting e ^ p ^ E , ) and e 2 ( p 2 , E 2 ) . It is also referred to as the triple differential cross section, d 3 a/dfi , dfi 2 dE. 11 Fig. 2.1. The (e,2e) reaction. 13 the (e,2e) cross section is given by, a e , 2 e = constant- ( 4 7 r ) ~ 1 / c i n |</>j(p)|2 [2.2] where p is the momentum of the electron in orbital, prior to ionization. The momentum distribution is spherically averaged (fdfi) to account for the random orientation of the gaseous targets. The expression in Eqn. 2.2 follows from energy conservation E 0 = E , + E 2 + E 3 + Ek * E , + E 2 + E b [2.3] and momentum conservation laws, Po = P i + p 2 + j ? 3 [2.4] E 3 is the ion recoil energy and is negligible due to the large mass of the ion. is the binding energy (ionization potential) of the orbital electron. The ion recoil momentum, p 3 within the conditions' (I-IV) outlined above is equal in magnitude but opposite in sign to the orbital electron momentum prior to knock-out, i.e. p = " P 3 [2.5] Eqn. 2.5 is an approximation which assumes the target to be initially at rest and that the ion is a 'spectator' in the collision process. Further discussion of this central assumption in EMS is made in the following section. 14 The fundamental result in Eqn. 2.2 has been and is the cornerstone of E M S and its applications to chemical problems. The present chapter will give a summary of the conditions under which the '(e,2e) cross section and momentum distribution proportionality' is valid. The approximations leading to this proportionality relation will also be outlined and discussed. 2.1.2. Plane Wave Impulse Approximation (PWIA) The (e,2e) reaction probability amplitude can be written as [WM78], T f _ _ Q = < X " ( p , ) X " ( p 2 ) * f N _ 1 | T | * 0 N X + ( p 0 ) > [2.6] and the (e,2e) cross section is therefore given by, a e , 2 e = ( 2 7 r ) M p 1 p 2 / p 0 ) 2 a v | T f < ^ _ Q ( p 0 , p , , p 2 ) | 2 [2.7] X ~ are the distorted electron waves with incoming( +) and outgoing(-) spherical wave boundary conditions. 'Po^ a n d ^ describe the exact initial (N-electron) and final ionic (N-l-electron) systems. T represents the complicated T-operator [SZ74] which 'models' the interaction between the incoming electron and the N-electron system. The notation £ Q V refers to an average over all initial state degeneracies and a sum over final state degeneracies.! It is quickly apparent that the complications in Eqn. 2.6 are three-fold, namely: (1) Accurate quantum mechanical description of the incoming and outgoing electron waves. t Since the present energy resolution in E M S experiments is not capable of resolving vibrational and rotational states, vibrational and rotational closure is applied [MW76, M73, M75]. 15 N (2) Accurate quantum mechanical description of the initial CP 0 ) a n d f " m a l states ( * f N " 1 ) . (3) Accurate representation of the many-body transition operator, T. A l l aspects (1, 2 and 3) are non-trivial. However from physical arguments and approximate approaches to Eqn. 2.6 certain kinematic regimes can be identified wherein reasonable understanding can be attained. It is in this spirit that McCarthy and Weigold [MW76] applied the plane wave impulse approximation (PWIA) to the (e,2e) reaction. Starting from the quantum mechanical three-body description [SZ74] of the (e,2e) reaction, the ionized electron can be considered to be initially bound to a quasi-particle ( N - l electron system). For this reason Eqn. 2.1 has been termed a quasi-three-body reaction. The quasi three-body T-operator can be expanded [F61, AG67, F83] in terms of two-body t-operators,t T = t 0 + t , + t 2 G 0 t 0 + t 2 G S t , + t o G S t , + t , G S t 0 + - [2.8] where t 0 represents the direct electron-electron Coulomb interaction, t , and t 2 represent the interaction of electrons 1 and 2 with the ion core (N- l electron system). G 0 is the Green's function for the non-interacting system ( G 0 = [E-Ho+ie] ' 1 ) . t Two-body collisions (i.e. elastic and inelastic scattering) are relatively well-understood [B83] and therefore have been used as the basis for approximate solutions to the quasi-three-body problem. 16 Within the impulse approximation [N66] the higher order terms in Eqn. 2.8 representing multiple scattering can be neglected. The impulse approximation [N66] is generally valid wherein the impact energy is much higher than the binding energy of the target electron. In such cases the assumption of single collisions between the incoming electron and the target electron is generally valid and the ion can be treated as a 'spectator'. Studies [MW76] have also shown that the contribution from t, is negligible (i.e. core excitations are small) for reactions involving high impact energy ( E 0 ) and large momentum transfer (| R | ^5a 0" 1)- In cases therefore wherein the (e,2e) reaction can be accurately described as simply a binary encounter between the incoming electron and the orbital electron that is to be ionized, the T-operator is given approximately by the two-body direct Coulomb t-operator, This approximation is valid only in the limit of high momentum transfer. For small momentum transfer ( | r v|<1a 0~ 1) as in the Ehrhardt-type [EJ86] (e,2e) reaction kinematics (asymmetric coplanar) the interaction of the incoming electron with the ion core (t,) and double collisions (multiple scattering terms in Eqn. 2.8) make significant contributions [EJ86] even if the impact energy is high. In the limit of high impact energies and high momentum transfer, the distorted incoming and outgoing waves can be approximated as plane waves. For example, the incoming wave will be given as [F83], T [2.9] = (1 + GJt,)|p 0> |p0> = exp(ipVr) [2.10] 17 The plane wave approximation for the incoming and outgoing particles as well as the impulsive (binary encounter) approximation to the T-operator then yields an expression for the (e,2e) cross section, °e,2e = 4 7 T 3 ( p 1 p 2 / p 0 ) a M o t t f d f l f d v | < p ¥ f N ~ 1 | * 0 N > | 2 [2.11] j < J V collision term structure term The vibrational integral (/dv), in the case of molecular targets, is reasonably approximated by calculating the neutral and ionic wavefunctions at the equilibrium geometry [DM75, BB87]. The half-off-shellt Mott scattering cross section, °y[0tt is given by [MW76], a M o t t = fc° l * H 2 [ 2 - 1 2 ] where K* ' = (j?i"~P2)/2 and k* = ( p 0 - p ) / 2 . The expression given by Eqn. 2.11 is known as the Plane Wave Impulse Approximation (PWIA). The assumptions of the PWIA are very difficult to assess theoretically because the exact scattering potentials (even for atoms) are difficult to obtain [MW76]. Oftentimes empirical parameterization of the distorting (optical) potentials is necessary in more accurate calculations such as the distorted wave impulse approximation (DWIA) [MW76, WM78]. This is even more complicated for multi-center targets where the form of the potentials are not known. Experimentally, the PWIA can be tested and much effort has focused on this [MW76, CG80]. Weigold and co-workers [FM78, DM78, LW81] have shown that t The Mott scattering amplitude is half-off-the-energy shell (i.e. 1c ^ k*") because of the finite binding energy. 18 for the conditions employed in EMS increasing the impact energy from 400eV to 1200eV produced negligible changes in the shapes of the measured experimental momentum profiles and that E 0 > lOOOeV is required to obtain correct relative cross sections. These results indicate that the (e,2e) cross section is not sensitive to the scattering dynamics under the present EMS conditions (I-IV) and therefore first-order theories (e.g. PWIA) are sufficient. It is now generally accepted that at E0>1000eV the PWIA as applied to EMS studies of atoms and molecules is a very reasonable approximation giving quantitative results for all orbitals. Such considerations have been used in the present work. Recent studies however show that in certain cases [MW85, CM86, CB87a], especially in the prediction of absolute magnitude, the PWIA may be insufficient. Defining the regions of applicability of the PWIA and improvements to the PWIA are still the subject of current further investigations [AC86, AC87a]. In general, the comparison of PWIA and DWIA calculations show agreement for shape in the low momentum region (0-1.5ao ~ 1 )• The DWIA however predicts higher intensity in the high momentum region (>l .5a 0 ~ 1 ) for atomic targets in agreement with experimental results. The high momentum region is where one would expect most of the distortion effects because high ion-recoil collisions sample the electronic wavefunction much closer to the nucleus. It is for the same reasons that plane wave treatments of both high energy PES (which samples high momentum) and low energy PES (which results in low energy ejected electrons) are grossly inadequate. The requirements of high impact energy (condition I), symmetric scattering 19 (condition II) and maximal momentum transfer (condition III) outlined earlier are consistent with the PWIA. It is clear (in the semi-classical sense) that under these conditions the collisions will be mostly single (binary) collisions between the incoming and ionized electrons and the ion is largely a spectator in the collision. In Eqn. 2.11 we are left with two terms, namely a collision term sensitive to the scattering dynamics and a structure term sensitive to the electronic structure of the target system. The resulting factorization of Eqn. 2.11 is a general result of impulse approximations to scattering problems and has also been exploited by many workers as for example in Compton scattering [C85]. The usefulness of the PWIA in E M S however comes from the fact that under well-chosen conditions the collision term in Eqn. 2.11 is not only separable but is also nearly constant. McCarthy and Weigold [MW76] discussed several possible scattering arrangements for E M S namely the symmetric coplanar, symmetric non-coplanar and constant-angle energy-varying arrangements.! Of these possible arrangements the constancy of the collision term [C81, LB83] is best achieved through the non-coplanar form of scattering arrangement (condition IV). It has been shown [MW76, WM78] that for the non-coplanar, symmetric arrangement the shape of the experimental momentum profile (as a function of p) is independent of the impact energy and is given essentially by the electronic structure term. In the non-coplanar arrangement the magnitude of the orbital electron momentum, p is given by, t The asymmetric coplanar arrangement (Ehrhardt-type) [EJ86, LW84] is sensitive to the scattering dynamics and involves very low momentum transfer and is therefore not suited for electronic structure determination. 20 p = {(2p,cos0 - p 0 ) 2 + [ 2 p 1 s i n 0 s i n ( 0 / 2 ) ] 2 } 1 / / 2 [2.13] where the scattering angles d and 0 are defined in Fig. 2.1. Another arrangement which has been demonstrated to be equally suitable for electronic structure determination is the high energy asymmetric coplanar (HEAC) arrangement of Lahmann-Benani et al. [LD86]. As pointed out in this article [LD86] the variations of the Mott scattering term in the HEAC arrangement (3.5%) are only slightly larger than those in the non-coplanar symmetric arrangement (<2%) [LB83]. These variations are very small compared to that using the coplanar symmetric arrangement (22%) [LD86]. Additionally, the momentum transfer is also strictly constant for the HEAC and the non-coplanar symmetric arrangements. From the experimental end, the non-coplanar symmetric arrangement is advantageous because the angular correlation spectra on both sides of 0=0° should be of identical shape and magnitude and thus provide a consistency check for the data. Furthermore, the collision volume 'seen' by the movable analyzer is constant as the angle 0 is sweeped. These factors all indicate that the non-coplanar symmetric scattering arrangement is the most favored for EMS studies. 2.1.3. Target Hartree-Fock Approximation (THFA) In general much of our current understanding of electronic structure, ionization phenomena and molecular properties is based on the independent-particle (or single-particle) picture of the molecule. In this case the N-electron wavefunction, «P0^ is given by a single Slater determinant of one-electron orbital wavefunctions Wj). * 0 N ( 1 , . . . N ) = | t f , , . . . * N > [2.14] Note that antisymmetrization is implicit in the notation. In addition, the following discussion is limited only to closed-shell systems. The extent to which the N-electron system cannot be described by an antisymmetrized set of independent-particle functions is called correlation effectsA The best independent-particle picture is given by the restricted Hartree-Fock (HF) model [S082] of electronic motion. The H F model assumes that the electrons in a molecule 'move about' according to an average field created by the other (N-l) electrons. This assumption transforms the N-electron Schrodinger equation, H*( 1 , . . . N ) = E * ( 1 , . . . N ) [2.15] into a set of N coupled one-electron (integro-differential) equations each of the form, F i ^ i = e i ^ i i = 1 , . . . N [2.16] where Fj is the effective one-electron operator also known as the Fock operator, yp ^ are the H F canonical orbital wavefunctions and e ^ are the H F orbital energies. t Correlation effects account for the fact that electrons interact with each other instantaneous^ rather than with an average field created by the other electrons. These effects are discussed in another section. 22 The canonical H F equations (Eqn. 2.16) are solved using the self-consistent-field (SCF) method and for this reason wavefunctions of the single-determinant form are also called SCF wavefunctions. Another descriptive term which has been used is the 'molecular orbital (MO) approximation', however this terminology should be discouraged because, as is well known, the total energy is invariant under a unitary transformation of the orbitals. This means that the orbitals that make the total energy stationary are not unique [S082].t The canonical form of the H F equations (Eqn. 2.16) is more commonly used for various reasons. In E M S canonical orbitals have found greater applicability due to Koopmans' theorem which relates the experimentally-derived ionization potentials to the canonical orbital energies. The derivation of Koopmans' theorem involves the assumption that the ionization process leaves the N - l orbitals undisturbed or in other words, frozen. Within this frozen-orbital approximation (FOA) relaxation effects as well as correlation effects are neglected. Application of the frozen orbital approximation to E M S is straightforward and the ion-neutral overlap amplitude given in Eqn. 2.11 reduces to, < p * f N _ 1 | * 0 N > = <p|^c> = * c ( p ) [2.17] where \pc (p) refers to the momentum-space representation of the characteristic orbital that has been ionized. A more general expression for the ion-neutral overlap amplitude is obtained by t For example localized orbitals will give the same total energy as delocalized molecular orbitals. 23 assuming that the initial state is largely a single determinant HF wavefunction while the final ion state is represented (in general) by a linear combination of HF configurations, |/3> with a hole in orbital l * f N _ 1 > = Zj<3 S j 0 C j 0 ^ j l ^ [ 2- 1 8 ] The resulting ion-neutral overlap amplitude is therefore given by, <p* f N" 1 |* 0 N> = L j S j Q C j 0 uVj (p) [2.19a] where SJQ is the probability amplitude for finding the one-hole configuration (\£j~1) in the final ion state, * f N ~ 1 [MW76]. C j ^ is a Clebsch-Gordan coefficient that ensures that the configuration | B> belongs to the point group of the system [WM78]. As can be seen from Eqn. 2.19a contributions from other orbitals especially of the same symmetry such as for example the 3 a, 4 a and 5a of carbon monoxide [DD77, FB87b] may contribute to the ion-neutral overlap amplitude. However in almost all cases only one term in Eqn. 2.19a dominates and the results obtained from the FOA and THFA are then the same for shape. In this case the (e,2e) cross section is given by, a e , 2 e = 4 * 3 ( P i P 2 / P o ) t f M o t t S c 2 r d f l f d v | ^ c ( p ) | 2 [2.19b] < v ^ v > collision term structure term The result in Eqn. 2.19 is called the Target Hartree-Fock Approximation (THFA) as applied to EMS. A different derivation of the THFA using the method of second quantization is also given by McCarthy and Weigold [MW87]. One particular difference between the FOA and the THFA is the fact that final 24 stare correlation effects can be accounted for in the T H F A . This is very useful in the interpretation of extra structure especially that observed in the inner valence binding energy spectra of most atoms and molecules. The observed correlation (satellite) state intensity within the same symmetry manifold is proportional to the spectroscopic factor, which is just the weight or the coefficient of the corresponding configuration (xjj^ ^) in the expansion given in Eqn. 2.18. The correlation (satellite) states are assigned on the basis of their respective measured momentum profiles and assuming the binding energy spectrum is obtained over a large enough binding energy range that the sum of the spectroscopic factors satisfy the spectroscopic sum rule [WM78]. 2.1.4. Full Ion-Neutral Overlap Amplitude The full ion-neutral overlap amplitude given in Eqn. 2.11 could also be evaluated directly using correlated wavefunctions for both initial and final ion states. This has been done in recent studies [BB87, BM87]. Computationally the overlap amplitude is given simply if the final ion state is expanded in the same molecular orbital basis as the initial state to take advantage of orthogonality properties. In this case the ion-neutral overlap amplitude has the same form as a molecular orbital expanded in the neutral basis. Details of the computation are given in the papers of Martin, Shirley and Davidson [MS76, MD77]. Transforming the overlap amplitude to momentum space is straightforward because it is still expressed in terms of basis functions (Slater-type or Gaussian-type) which can be transformed using standard methods [KS77]. There are also other effective ways of obtaining accurate ion-neutral overlap amplitudes such as the generalized overlap amplitude method of Williams et al. [WM77] and 25 more recently the overlap method of Agren and Jensen [AJ87] which assumes separately optimized initial and Final states. Presently the limitations of these methods have been computational but with the rapid phase of development in computing technology and software these limitations will be likely diminished. Spherical averaging of the absolute square of the Fourier-transformed ion-neutral overlap amplitude yields the ion-neutral overlap distribution (OVD). OVDs give a more accurate representation of the structure factor in the (e,2e) cross section compared to the single-particle momentum distributions. MDs in some sense are the zeroth order approximation to the more accurate OVDs. These calculations for H 2 O and N H 3 in the present work were done in collaboration with the theoretical group of Prof. Ernest R. Davidson (Indiana University). The ion-neutral overlap amplitudes were calculated by the Indiana group and the OVDs were calculated from these quantities by the author using the H E M S package at U B C . 2.2. MOMENTUM SPACE CHEMISTRY The results of the previous section (Eqns. 2.11 and 2.19) clearly indicate that E M S (symmetric non-coplanar geometry) is the appropriate experimental tool for momentum space chemistry. Central to momentum space chemistry is the Fourier transform relationship between the momentum representation and the position representation (Eqn. 1.2). Fourier transforms find applications in various branches of science [B65, C73] however in this section focus will be on its application to E M S . The Fourier transform properties are mentioned in their most general form to reflect the fact that they are used in other branches of physical science and engineering. These are then related to specific principles in momentum space chemistry [ET77] and E M S [CB82, LB83 , LB83a]. 26 Two functions are said to form a symmetrical Fourier transform pair, f(x) and F(s) where, F(s) = (27r)" 1 / 2 / f (x)exp(-ixs)dx [2.20a] f(x) = (27r)~ 1 ' / 2J F(s)exp( + ixs)ds [2.20b] In E M S the particular Fourier transform pair we are interested in are, *// (r) and \p (p) which are the position space and momentum space wavefunctions, respectively. The examples are shown for the case of one dimension to maintain simplicity and clarity. Several basic theorems govern the application of Fourier transforms to physical problems. These are listed below. (1) Similarity Theorem If f(x) has the Fourier transform F(s), then f(ax) has the Fourier transform | a j _ 1 F(s/a). This theorem is referred to in E M S as the inverse weighting principle. This means that an atomic function concentrated near the nucleus in position space will have a corresponding momentum space orbital which will be diffuse. This property is clearly illustrated in the trend of the observed X M P s of the noble gases [LB83]. However caution should be exercised in applying this principle to polyatomic systems where other effects complicate the Fourier transform. 27 (2) Addition Theorem If f(x) and g(x) have Fourier transforms F(s) and G(s), respectively then f(x) + g(x) have the Fourier transform F(s) + G(s). This theorem illustrates why the molecular symmetry of a linear combination of atomic orbitals (LCAO-MO) molecular orbital is preserved in momentum space. In addition, molecular symmetry is preserved because the spherical harmonics which represent the angular part of the wavefunction are invariant under the Fourier transform. In momentum space, inversion symmetry is also added (if not originally present). This ensures that the electron has no net translational motion in the center of mass coordinate system. (3) Shift Theorem If f(x) has a Fourier transform F(s), then f(x-a) has the Fourier transform exp(-27rias)F(s). This theorem is the basis of the bond oscillation principle in E M S which states that momentum distributions associated with chemical bonds will exhibit oscillations along the bonding direction with period = 2n7t7R (n = 0,l,...) where R is the bond length. Antibonding orbitals exhibit oscillations with period = (2n+ l)7i7R (n = 0,l,...) which are out of phase with the bonding orbitals. Coulson and Duncanson [CD41] showed for a diatomic that if the molecular orbital is expressed in terms of a linear combination of atomic orbitals (LCAO-MO) centered in the respective nuclei, the momentum density is given by, I W P > I 2 = ixatom(p>r {i±co S(p.5)} [2.2i] > „ — ' diffraction term 28 where the ( + ) sign refers to bonding orbitals and (-) sign refers to antibonding orbitals. For a bonding orbital it can also be seen from Eqn. 2.21 that the diffraction term is maximal when p is perpendicular to the bond direction, R. This observation is referred to as the bond directional principle in E M S which states that the momentum of an electron in a chemical bond is more likely to be directed perpendicular to the bond rather than along the bond. This phenomenon has been nicely illustrated in the case of H 2 [CD41, LB83a]. (4) Definite Integral Theorem The definite integral of a function from -°° t o + 0 0 is equal to the value of its transform at the origin, i.e. F ( 0 ) = ff(x)dx [2.22] This theorem illustrates an early observation [LN75] in E M S which mentioned that only s-type basis functions (1 = 0) can give intensity at p = 0a 0 " 1 • The momentum density at p = Oa 0 " 1 therefore provides a sensitive probe of the relative ratio of symmetric components in the molecular orbital, especially in mixed s-p type orbitals. (5) Autocorrelation Theorem If f(x) has a Fourier transform F(s), then its autocorrelation function * ff ( u ) f ( u + x ) d u has the Fourier transform j F ( S ) | 2. This theorem has been exploited by Coplan and co-workers [MT82] in their interpretation of E M S data. The autocorrelation function of the position space wavefunction in this case is referred as the B(r) function. It has been argued that the B(r) function which can also be derived from the momentum density is 29 more familiar to most people and therefore easier to understand. 2.3. BASIS SETS FOR AB INITIO WAVEFUNCTIONS Currently there is no numerical solution to the restricted Hartree-Fock equations except in the case of atoms and recently in the case of diatomic molecules [LS85].t As a more general solution a known set of spatial basis functions are introduced and the H F integro-differential equations are converted to a set of algebraic equations and solved by standard matrix techniques. The resulting equations are called the Roothaan equations [R51]. This known set of spatial functions {bj} is called the basis set. The molecular orbitals can be expanded using this set, \pi = Zj ci j b j j = 1 , . . .M [2.23] In the limit of a complete set (i.e. M — • °°) the expansion in Eqn. 2.23 is exact. This however is not feasible because of the hardware and software limitations of an infinite basis set. What is generally done is to limit the basis set to a small finite number of well-chosen (i.e. optimal) basis functions. Much work in recent years has focused on the optimal form (in terms of accuracy and computational ease) of these basis functions. One particular function is the atom-centered Slater-type orbital (STO), t Recent reports [F87] indicate that the H F equations can be solved for a general polyatomic molecule by combined use of basis functions and a numerical grid. 30 b - S T 0 = N r n " 1 e " a r Ylm(d,4>) [2.24] where a is the orbital exponent, N is a normalization constant and Yj m(0,0) is a spherical harmonic. Another functional form, adopted first by Boys [B50], is the atom-centered cartesian Gaussian-type orbital (GTO), b j G T 0 = N x n y 1 z m e _ a r 2 [2.25] where k ( = n + l + m) is the angular momentum quantum number. It is generally held [DF86, RF85] that =three GTOs are equivalent to one STO for the same level of accuracy. More GTOs are necessary because GTOs have the wrong behaviour near the nucleus (r^O) and at large r [H85, DF86]. Although the relative merits of GTO and STO basis sets have been a matter of extensive debate, it should be noted that, in the limit of large basis sets, both types of basis sets should give the same result. Other types of basis sets exist in the literature and are discussed elsewhere [S77, H85]. In. the present work we have concentrated on the more popular functional forms namely the STO and GTO basis sets. One particular advantage of GTOs is the ease of computation compared with STOs. However GTOs are not optimum basis functions and therefore the basis functions are usually represented as a linear combination of GTOs, b.CGTO = 2 d x n y l z m e - a r 2 S-1,...L [2.26] J = = J where L is the length of the contraction and d • is the contraction coefficient. The functions, b-CGTO are called contracted Gaussian-type orbitals (CGTOs). Even-tempered basis sets (GTO or STO) were first introduced by Schmidt and Ruedenberg [SR79]. The even-tempered restriction requires that the exponents {a 1 f . . . a } be generated according to a geometric progression, This restriction then reduces the number of parameters to be optimized for each group of atomic functions belonging to the same symmetry to just two (a and B) instead of m. Even-tempered basis sets have therefore allowed a systematic way of extending finite basis sets towards a complete basis set. Once a particular basis set is chosen, a choice often dictated by the researcher's goals, the SCF calculation is performed. This involves the assumption of a particular molecular geometry which could be the experimental equilibrium geometry or the SCF optimal geometry. Basis functions are then placed on each atomic center and the Roothaan H F equations are then solved using standard programs such as the G A U S S I A N programs of Pople and co-workers [BW76, BW80] and the M U N I C H programs of Diercksen [D74]. The resulting solution yields the optimum coefficients that describe the molecular orbitals in terms of the basis functions, the orbital energies and the total energy. From these quantities other molecular properties are also derived and tabulated. a i = a/3 1 ( i = 1 m) [2.27] The above numerical calculation yields wavefunctions which can be arbitrarily close (depending on the choice of basis set) to the true Hartree-Fock 32 wavefunctions. However much difficulty in interpretation (or over interpretation) of these approximate numerical results has occurred because most users of these approximate wavefunctions are unawaret of the limits of their applicability [DF86]. This particular problem of basis set quality is also known as the basis-set truncation error and is considered to be a serious bottleneck for quantum chemistry [LS85, SP85]. 2.4. ELECTRON CORRELATION The Hartree-Fock model outlined in Sec. 2.1.3 has been a successful predictive model in chemistry and physics. It has also been useful in the theoretical description of chemical reactivity and chemical bonding. However certain definite limitations exist within the H F model and experimental evidence indicate that in many cases there is a need to go beyond the H F model. This means the inclusion of electron correlation which is neglected in the H F model. Some form of 'electron correlation' is accounted for by the H F model since electrons of the same spin can not occupy the same space (Pauli exclusion principle). A concise discussion of the correlation problem is given by Lowdin [L59] in which the correlation energy is defined as "the difference between the exact eigenvalue of the Hamiltonian and its expectation value in the Hartree-Fock approximation" [L59], that is, E = E , - Ex™ [2.28] corr exact H F L J t Much of the confusion may be due to the awe created for the uninitiated by the words ab initio (from first principles). Huzinaga [H85] refers to finite basis sets as the 'quasi-empirical element' in ab initio calculations. 33 In practice the correlation energy, E c Q r r is calculated from the difference between the experimental energy ( E e x a c t ) , which is 'corrected' for relativistic effects and zero-point vibrations, and the Hartree-Fock energy. The correlation usually mentioned therefore refers to the fact that in real systems electrons (mainly of opposite spin [L59]) tend to 'avoid each other'. Various methods exist for including electron correlation and one of them is the configuration interaction (CI) method. In the CI method the exact wavefunction is represented by a linear combination of Slater determinants which are built from the H F molecular orbitals [S082], *exact = C ° * H F + Z i j c i j * i ' ] + £ i jk l ^ j S J 1 +••• ^ The first term is the Hartree-Fock wavefunction, the second term is the set of all possible singly-excited configurations, the third term is the set of all possible doubly-excited configurations, and so on. The nomenclature, for example in " P j ^ , means the promotion (or excitation) of an electron in molecular orbital i to a virtual molecular orbital j relative to the H F configuration. The numerical calculation in Eqn. 2.29 quickly becomes intractable as more terms are added in the expansion and therefore in practice higher-order excitations are neglected. The CI wavefunction is usually terminated at the double-excitation level and the resulting wavefunction is called a singles and doubles CI (SDCI) wavefunction. A useful extension of the SDCI method is the inclusion of reference wavefunctions other than the H F wavefunction. The final wavefunction would therefore include all single and double excitations relative to many well-chosen reference 34 configurations and this is referred to as the multi-reference SDCI (MRSDCI) wavefunction. Wavefunctions of this type are capable of recovering a larger percentage of the total correlation energy and are considered very accurate for the prediction of molecular properties. 2.5. GREEN'S FUNCTION METHODS AND IONIZATION SPECTRA Green's functions are particularly attractive because of the properties which render them very useful to the interpretation of ionization spectra. In the Green's function approach the ionization potentials and the transition amplitudes, unlike in the CI approach, are given explicitly in the expression for the one-particle many body Green's function in energy space [CD77], G ( C J ) = 2 n { X n * X n / ( u + I P - i T ? ) + E . A . t e r m } [2.30] X n = < * f N 1 | a n | * 0 N > [2-31] X n is the transition amplitude, a R is the annihilation operator for orbital n and IP (ionization potential) is the energy difference between the exact neutral state and the exact final ion state energies. The Green's function is very intuitive in the sense that the ionization energies are given by its poles (i.e. energy regions where G ( C J ) - — * » ) . The electron affinity (E.A.) term in Eqn. 2.30 is not relevant in the present application. The absolute square of the transition amplitude, is called the pole strength or the spectroscopic factor as it is usually called in 35 CI terminology. Cederbaum and co-workers [CD77, CD86] have outlined the theoretical methods for finding the one-particle many body Green's function (and hence the ionization energies and intensities). This is obtained by solving the Dyson equation, G(w) = G°(u>) + G ° ( w ) I ( o ) ) G ( u ) [2.33] where G° (u ) is the known Hartree-Fock Green's function and £ ( G J ) is the self-energy [VS84]. The poles of G ° ( C J ) correspond to the H F orbital energies whereas the poles of G(co) correspond to the exact ionization energies. The energies are exact to the extent that the self-energy term, Z ( C J ) , is calculated exactly. Various solutions to Eqn. 2.31 involve varying degrees of approximation to the self-energy term. A systematic set of approximations is given by the algebraic diagrammatic construction scheme [ADC(n)] which is accurate to nth order of perturbation theory [VS84]. For example, ADC(3) will be accurate to 3rd order in perturbation theory and thus is equivalent to the extended two particle-hole Tamm-Dancoff* approximation (ext. 2ph-TDA). ADC(4) would therefore include 3hole-2particle and 3particle-2hole configurations and so on. A recent review [VT86] covers the recent progress in the Green's function method. One special note that should be mentioned with regards to the CI approach and the Green's function approach to calculating spectroscopic factors (poles strengths) is their dependence on basis set quality. It has been shown [CD77, CD86] that basis set saturation is critical to quantitative prediction of experimental relative ionization intensities. The basis set dependence of the theoretical binding energy spectrum can be understood following the arguments of Cederbaum et al. [CD86]. 36 The breakdown of the single-particle picture of ionization arises (theoretically) from the near degeneracy of the 2h-lp configurations with the single-hole configuration. This occurs whenever the neutral molecule has low-lying excited states. Furthermore, the pole strength is very dependent on the degree of localization of the low-lying virtual orbital [CD86]. It is, therefore, evident that basis set quality is critical in the sense that it should model accurately the few low-lying virtual molecular orbitals in addition to the occupied molecular orbitals. 2.6. HEMS: A COMPUTER PACKAGE FOR MOMENTUM-SPACE CALCULATIONS During the course of the present study the existing computer programs for calculating the spherically averaged momentum distributions and the generation of position-space and momentum-space density maps were integrated into one whole package called H E M S (H-compiler optimized programs for EMS). Various subroutines were written into more efficient subprograms. Subprograms were also extended to include d-functions and f-functions in the basis set. The integrated H E M S programs are shown in Fig. 2.2 and outlined in Table 2.1. H E M S operates using a link driver (HEMS*) which basically acts as a 'traffic director' for the package and routes the program to various links depending on the desires of the user (see Fig. 2.2 and Table 2.1). The links or subprograms are dynamically loaded and unloaded after use to save space and time. The resulting speed and economy is therefore substantial. As can be seen from Fig. 2.2 only a single input file is necessary for generating all the calculations and density maps. 37 Table 2.1. Subprograms (Links) in the H E M S package. H E M S L 1 * Reads data input file and reformats input file to particular requirements in each link. For GTOs expands Cartesian GTOs in terms of complex spherical GTOs. H E M S L 2 * Calculates spherically averaged MDs and prints out several M D statistics useful for debugging. H E M S L 3 * Calculates and plots the momentum-space density maps. H E M S L 4 * Calculates and plots the position-space density maps. H E M S L 5 * Plots calculated spherically averaged MDs. Has options to plot experimental data. a A l l programs are written in F O R T R A N . b The Fourier transform of GTOs and STOs are analytic (see for example Kaijser and Smith [KS77]). The spherical averaging algorithm is similar to that reported by Levin et al. [LN75]. 38 H E M S * H E M 5 L 2 * H E M S L 5 * S p h e r i c a l t y - o v « r a g a d Mom ent u m DI i f r i but I on Mom ent u m - s p a c e D e n s i t y Maps H E M S L 4 * P o s i i i o n - s p a c e D e n s i t y Maps Fig. 2.2. Layout of the H E M S computer package. 39 A sample input file for the H 2 molecule utilizing the double zeta basis set of Snyder and Basch [SB72] is shown in Fig. 2.3. The format of the input file is very similar to the output of standard ab inito quantum chemical programs (e.g. GAUSSIAN76 [BW76] and M U N I C H [D74]). This allows easy interfacing between quantum chemical programs and H E M S . The input structure (see Fig. 2.3) is very straightforward apart from the control cards (lines 1, 2 and the last line). Basically, it involves a molecule identifier, geometry specifications, basis set definition and finally the M O coefficients and angular momentum parameters. In the example shown in Fig. 2.3 the control cards are set to 1, 2 and 5 which means the generation of spherically averaged MDs and their respective plots. The output generated by the H E M S package for this particular example ( l ^ g orbital of H 2 ) is shown in Fig. 2.4. The integrated set of programs allows great ease in the calculation and display of theoretical MDs. This particular run took about 1.117 sec C P U time at a cost of CC$0.21 (normal rate). Replacing the control cards with the appropriate link numbers (e.g. 3 for momentum space and 4 for position space), the respective density maps can be generated. The H E M S package also has special options for generating density difference maps. Fig. 2.5 shows the density difference (PQJ ~ P^Qp) m a P m momentum space and position space calculated for H 2 (R=1.4a.u.). The CI wavefunction is the 6-configuration multiconfiguration SCF (MCSCF) wavefunction of Das and Wahl [DW66] and the SCF wavefunction is the Hartree-Fock limit wavefunction of Cade and Wahl [CW74]. The resulting position space density difference map (Fig. 2.5a) is identical to that obtained by Bader and Chandra [BC68]. The quantitative agreement demonstrate the accuracy of the present Options for Link Driver H E M S * Options for I-ink #1 1 2 0 0 5 t 1 OOOOOOOOOOOOOOO H!2% 2 1 0.0 1 0.0 GAUSSIAN "DZ• SB Basts • •••»• demonstration run: 2 3 0 1.0 19.24060 0.032B2BO 2.69915 0.2312080 0.653410 0.81723BO 1 0 1.0 0.177580 1.0000000. Msi'.g'A •+ — 4 0 . 0 0 . 0 0.0 2.64562 ICPEAC 87 Geometry Specifications [Atomic number, x, y, z] Basis type Basis Set Definition " [Exponents, Coefficients] i~6~ 232363 —r 0 0 0 1 0 413260 2 % 0 0 0 2 0 232303 1 0 0 6 2 0 413260 2 0 0 0 & !u% 4 1 - 199944 i 0 0 0 1 - 831113 2 0 0 0 2 0 199944 1 0 0 0 2 0 831113 2 0 0 0 END 02000000000000 •*— 5 0 0 0 0 0 0 0 0 0 1 3 1 0 3 O 1 31 0 3 Molecular Orbital (MO) label No. of basis functions in M O Expansion in terms of basis functions [Atom center, M O coefficient, basis function no.l [Angular dependence, n, 1, m] Options for Link #2 Options for Link #5 o. 1 1.5 Plotting options Fig. 2.3. Input structure of H E M S package. Example is for the calculation and plots of the spherically averaged MDs of H 2 41 TOTAL 'E2E-8.2006S60E-01 3 8595730E-01 4.8817169E-02 3.9240979E-03 6.8182475E-04 2 0263882E-04 5.8410529E-05 INTENSITY 7.9533696E-01 2 7943492E-01 2.9246323E-02 2.55O01O6E-03 5.2733975E-04 1 . S908635E-04 4.S348730E-05 7.2569108E-01 1922577SE-01 1.7360482E-02 1.7302211E-03 4 . 1316475E-04 1.2438085E-04 3.5288176E-05 6.2337875E-01 1.2632120E-01 1.0347713E-02 1.2249590E-03 3.25770S8E-O4 9.6888078E-05 2 7571718E-05 S.0475305E-01 7.977157BE-02 6.2777996E-03 9 0038753E-04 2 . 57 17984E-04 7 5272706E-OS 2 1667816E-05 GAUSSIAN INSTRUMENTAL FUNCTION ASSUMED: DELP- 0.10O0A.U. MOMENTUM INTENSITY 0.0 8 .02546E-01 0 1CO00 7 .794O6E-01 0.20000 7 .14053E-01 0 30000 6 .17489E-01 0 40000 5 04565E-01 , 0.50000 3 90198E-01 0.60000 2 86232E-01 f— 0 70000 1 99788E-01 .UNI 10.0 0.80000 1 33245E-01 .UNI 10.0 0.9OOO0 8 53809E-02 .UNI 10.0 1 .ooooo 5 .29402E-02 CD 1 . 10OO0 3 20488E-02 DC _ 1.20000 1 91487E-02 < 9 1 . 30OO0 1 14337E-02 1.40000 6 91395E-03 1 .5COOO 4 28835E-03 > 1.60000 2 7S647E-03 i— „ 1 . 70OO0 1 B4740E-03 Z UD 1 i J 1.80000 1 29222E-03 1.90000 9 39779E-04 2 OOOOO 7 05578E-04 r -3.10000 5 42245E-04 2.20000 4 23029E-04 2.30000 3 3267 1E-04 2.400O0 2 62307E-04 2.50O00 2 06635E-04 > 2.60000 1 62295E-04 2 70000 1 26993E-04 2.80000 9 90172E-05 2.90000 7 70432E-05 3.00000 6 00650E-05 Lu 3.10000 4 61652E-05 DC 3 . 2OOO0 3 52679E-05 q 3 3OO0O 2 67720E-05 3.40000 1 26638E-05 b S P H E R I C A L L Y A V E R A G E D M O M E N T U M DISTRIBUTION -i 1 1 1 1 1 r H 2 ORB: 1a, ••• MOMENTUM DISTRIBUTION STATISTICS "•• INTEGRATED INTENSITY: UNFITTED- 0.99407E*00 FITTED- 0.10357£»01 MAXIMUM INTENSITY(UNF): O.82007E•OO AT P(A.U.)- 0.0 0.80255E+00 S-TYP£(T»TRUE.F-FALSE): T .0 0.6 1.2 1.8 2.4 3.0 MOMENTUM (A.U.) MAXIMUM INTENSITY(FIT): FWHM(FIT): 0.49032£»00 Fig. 2.4. Sample output of H E M S package based on data input from Fig. 2.3. 42 Fig. 2.5. Density difference (PQJ ~ P ^ C F ^ m a p s m ^ momentum space and (b) position space for the H 2 molecule calculated at the equilibrium geometry. The H-atoms are indicated in Fig. 2.5b by the solid dots. 43 programs. Currently, the package is installed in the U B C Computer System (Amdahl 5850) which runs on the Michigan Terminal System (MTS) operating system. The bitnet node address is (7serid@UBCMTSG.BITNET. CHAPTER 3. EXPERIMENTAL METHOD 3.1. ELECTRON MOMENTUM SPECTROMETER The spectrometer used in the present studies is a high momentum resolution E M S spectrometer of the symmetric, non-coplanar type. The construction and operational details of the spectrometer have been described earlier by Leung [L84]. It is a modification of an earlier instrument developed by Hood et al. [HH77] and also by Cook [C81]. In the course of the present work the instrument has been moved to a new laboratory where careful re-alignment about the collision interaction region has resulted in a significantly increased coincidence countrate. A brief outline of the design and operation of the spectrometer in its present form is given below. 3.1.1. Description of Spectrometer The E M S spectrometer is housed in an O-ring sealed aluminum chamber (width = 40cm, height=40cm) and pumped by two oil diffusion pumps (Varian VHS-4, 1200L/sec). One diffusion pump (a) evacutes the gun region while the other diffusion pump (b) evacuates the analyzer region through a U-shaped tube (16cm dia.) attached to the top of the aluminum main chamber. This arrangement allows for differential pumping of the gun region giving a differential base pressure ratio of 10:1 (analyzengun). Each oil diffusion pump (Neovac S Y fluid) is backed by a Sargent Welch two-stage rotary pump. A similar rotary pump serves the gas inlet line. An outlet system routes the exhaust gases from the rotary pumps to the fume hood. With the present system a base pressure of 5x10 torr is typically attained in both chambers. 44 45 To reduce the earth's magnetic field (500mG), a cylindrical mu-metal shield (hydrogen annealed) encloses the whole spectrometer. This reduces the magnetic field inside the spectrometer to =*5mG. A basic schematic diagram of the E M S spectrometer is shown in Fig. 3.1. The spectrometer is composed mainly of an intense electron beam source (EBS), a gas cell (GC) and two identical energy-analyzer-detector systems (EADs). One of the energy-analyzer-detector systems is stationary while the other is movable over the range $ = ± 3 0 ° . The movable E A D is placed on a turntable mounted on sapphire balls and driven by an external servo-motor. The components of the electron beam source are shown in Fig. 3.1. The electrons are produced by thermionic emission using a commercial triode electron gun body (Cliftronics CE5AH). Basically, the electron gun is composed of a A -shaped thoriated tungsten filament which serves as the cathode (C), a Wehnelt-type grid (G), an accelerating anode (A) and an einzel lens (L). In a typical operation the cathode is at a high negative potential (-1200eV + binding energy) and is heated by a DC current (2.2-2.5A). The thermionically released electrons are extracted by the anode (typically at + lOOeV relative to the cathode) and focused by the einzel lens. The electron beam is further collimated and transported into the collision region (GC). This is accomplished by adjusting the quadrupole deflector voltages (DI and D2) and minimizing the current collected on the aperture plates (PI, P2 and C M A M A I N DIFF PUMP 1200 L/sec S E R V O M O T O R G U N DIFF PUMP 1200 L/sec r o 5 cm. Fig. 3 .1 . Schematic of Electron Momentum Spectrometer. See text for details. 47 P3).t The current on the Faraday cup (FC) is also maximized. In practice the grid, cathode, anode and lens voltages are optimized to give the most intense beam with minimal angular divergence. Typical current collected on the Faraday cup is 50-60>A. In general this procedure is done after —2 hours to give time for the gun and associated power supplies to stabilize. Gas is introduced via a Teflon tube into a 1mm dia. hole just below the collision point (see Fig. 3.1). The sample pressure in the analyzer chamber is -5 < 5 x l 0 torr. The pressure inside the gas cell (GC) is estimated to be a factor of 10 higher than the analyzer chamber. Annular slots on both sides of the gas cell allow outgoing electrons to be transported to the EADs . The electron-analyzer-detector system (see Fig. 3.1) is accurately aligned and mounted via dowell pins to ensure that the outgoing electrons are collected at 6 — 45°. Since both EADs are identical the following discussion will equally apply to both the stationary and the movable E A D systems The asymmetric immersion lens (AIL) then serves to retard the outgoing electrons (typically at 600eV) to the pass energy (lOOeV) and focus the electrons into the entrance of the cylindrical mirror analyzer (CMA). The focal properties and dimensions of the asymmetric immersion lens system have been discussed earlier [L84]. The apertures ( A l and A2) are made small (2.0mm and 1.0mm, respectively) to allow improved momentum resolution [LB83]. These defining apertures in the asymmetric immersion lens system, from geometric considerations, are capable of t These apertures define a geometrical acceptance angle for the electron beam of ± 1 . 3 ° [L84]. 48 accepting a cone of electrons defined by the half-angles, A0=A0=O.9°. A further consideration of these apertures and their effect on the momentum resolution is made in Sec. 3.2. After angular selection, the electrons are introduced into the C M A via X - Y deflectors (D3). A particular choice of X - and Y-voltages was found to be necessary to obtain optimal momentum resolution. It should however be noted that since the deflectors are placed after the angular selection this does not introduce any extra uncertainty in the mean values of 9 and <j> which define the orbital electron momentum that is being sampled. The principle of the cylindrical mirror analyzer (CMA) is well-known [R72]. The C M A is an electrostatic deflection type of energy analyzer which uses two coaxial cylinders of widths a and b held at different voltages, V & and V ^ , respectively. It has the particular advantage of using the complete (27r) azimuthal angle, high order focusing and high resolving power [R72]. A 135° sector C M A was used instead of the more usual 360° (2ii) C M A used in Auger spectroscopy for practical reasons as well as to be able to mount the C M A such that the launch angle in the C M A is at 42 .3° . This allows for second-order axis-to-axis focusing of the C M A however with the present arrangement (with the deflector, D3) it is doubtful whether such second order focusing is achieved. To correct for edge field effects, logarithmically spaced end correctors (EC) are installed top and bottom and on both sides of the C M A . A l l surfaces are benzene-sooted to minimize secondary electron emmision as well as to provide an even potential across the surface. The dimensions of the present C M A are outlined in Fig. 3.2. The 1 3 5 u S e c t o r CMA P a r a m et e r s a = 2 4 . 5 m m b = 67 . 7 m m d =(ri + d. ) v s \ ' d = 54 . 2 m m zo = 1 5 6 . 6 m m co = 4 2 . 3 ° i 9 = 4 5 . 0 ° Design parameters of 135° sector cylindrical miror analyzer (CMA). In units of the inner cylinder radius (a), d = 2.21 and the calculated dispersion, D = 6.12. 50 theoretical energy resolution of a CMA [R72] is given by the base resolution (R), R ~ {(W s +W e ) /Dsin0} + ( W s / a ) 2 + A Z T ( + A 0 , - A 0 ) / D [3.1] where W g and W g are the widths of the source and exit slits, a is the inner cylinder radius, D is the dispersion and AZ,p is the spread introduced by the uncertainty (angular aberrations) in the launch angle. With the parameters shown in Fig. 3.2 the 135° sector CMA has a calculated fwhm (derived from the base resolution) of =1.4% (AE/E p a g s xl00). The energy resolution of the EMS spectrometer can therefore be calculated from, AE « ( A E C M A 2 + A E G u n 2 } 1 / 2 [3.2] where A E C M A * s t n e l a r g e r of the two CMA resolutions and A E Q u n is the energy spread (=0.8eV) in the electron gun. Equation [3.2] results from the fact that the coincidence method (At<5ns fwhm) discriminates against most electrons, i.e. electrons with energy E ^ S are correlated only to electrons of energy E 2 + 6. Thus in the present case (E„ Q C C , = lOOeV) the elastic width is =1.6eV fwhm and p«.ss the EMS energy resolution is also =1.6eV fwhm. Although the entrance to the CMA is defined by an aperture (1.0mm dia.) the exit is defined by an annular slit of dimensions (1.3mmx4.0mm). The energy dispersion is controlled by the (vertical) y-dimension (1.3mm) and the x-dimension (4mm) is relatively wider to ensure optimization of the countrates. The momentum resolution is found to be critically dependent on the choice of the Y-deflector (D3) voltage. This is because the asymmetric immersion lens system has a wider acceptance angle than the electron analyzer slit system. Optimum 51 tuning of the Y-deflector voltage (D3) ensures selection of a cone of electrons ( + A 0 ) about a mean angle 8 of 45° . These conditions ensure a minimum value of the (e,2e) cross section at 0 = 0° . Finally the electrons are detected and amplified (x lO 8 ) by a channel electron multiplier (CEM, Mullard B318AL) whose output is closed for the present application. The cascade of electrons results in a current pulse which is then capacitively decoupled (0.0022pF) [L84] and fed to an external charge-sensitive preamplifier (500 input impedance) and the coincidence electronics system. 3.1.2. Coincidence Detection, Event Processing and Control The electron pulses generated at the channel electron multipliers (CEMs)t are detected and processed in a coincidence detection electronic system outlined in Fig. 3.3 and the particular components are detailed in Table 3.1. This system uses a single time delay on one of the channels and has been described by McCarthy and Weigold [MW76]. Briefly, the pulses from the (charge-to-voltage) preamplifier which has a fixed gain (xlO) are further amplified (typically x20) with a timing filter amplifier such that suitable fast, negative pulses (=5ns fwhm, -8V) are produced. With the appropriate choice of 50O cables and proper termination, the 'ringing' observed in the C E M pulses can be reduced to < 15%. The fast, negative pulses from the timing filter amplifiers are then introduced to constant fraction discriminators (CFDs) which produce pulses whenever the signal is beyond the set threshold, usually -3.0V. Constant fraction discriminators are free from time 'jitter' associated with conventional discriminators [GM67] and are very t A cleaning procedure similar to one outlined recently [GR84] was found to be suitable in re-conditioning used CEMs with diminished gain. t con E O scan 0 Prog. P.S. motor r* SCA l— SCA tru* (tart/stop inhibll/raaat xtnc INTER-FACE rand Ditplay Graphic Lin* Floppy Disk Drive CRT Terminal Printer L S I - I 1/03 CPU ADC DIGITAL I /O DAC REAL-TIM CLOCK Fig. 3.3. Timing coincidence electronics for E M S spectrometer. Ol to 53 Table 3.1. Components of E M S timing coincidence electronics. Model # / Type 1. Pre-amplifier Ortec 9301 2. Timing filter amplifier (TFA) Or tec 454 3. Constant fraction discriminator (CFD) Ortec 463 4. Ratemeter NR-10 5. Time to amplitude converter (TAC) Ortec 467, has a built-in SCA 6. Single channel analyzer (SCA) Ortec 406A 7. Programmable power supply Fluke 412B (modified for voltage programming) 8. Servo-motor and amplifier from Leeds Northrup and Bristol chart recorder 9. Computer LSI 11/03 RT-11SJ Operating System 54 useful for applications wherein timing information is important. The fast, negative pulses (10ns fwhm, -800mV) from the constant fraction discriminators then serve as the stop and start pulses of a time-to-amplitude converter (TAC). The stop pulse is delayed by a fixed amount (typically 30ns). The T A C which is the 'heart' of the coincidence circuit generates voltages proportional to the time delay between the leading edges of the start and stop pulses. Typically the T A C is set at a full scale range of 200ns which is equivalent to an output of = 10V (2/xs width). The voltage output of the T A C serves as input to two single channel analyzers (SCAs). One SCA (lower level = 0.4V, upper level=1.4V) serves as the coincidence window while the other S C A (lower level = 2.0V, upper level=10V) serves as the random window. This particular choice [LB83] of windows (random:coincidence = 8) gives a good compromise between optimal signal/noise ratio and proper background subtraction. The NIM-logic pulses (logic 0 = 0V, logic 1 = + 4V) generated by the random and coincidence SCAs are processed using a home-built interface [C81] before finally being stored in separate arrays in the computer memory. The true coincidence rate and standard deviation defined by this method are therefore given by, N true = N c o i n c - N rand /8 [3.3a] AN t r u e = (N c o i n c + N rand / 82 ) 1 / 2 [3.3b] where N C Q ^ n and N r a n £ j are the number of counts registered in the coincidence and random SCAs, respectively. The estimated processing time for each event is = 100MS . This places a practical limit to a maximum coincidence rate of 10 acps 55 which is still very high compared to the present coincidence rate (^O.lcps).! The LSI computer-interface system allows the user to control experimental parameters (e.g. E 0 and <t>) and collect E M S data. With the LSI 11/03 computer the user is capable of running several sequential data scans (in energy mode and/or angular scan mode) as well as displaying and printing out the results. The computer system also has a 12-bit A D C that allows the voltage levels generated by the T A C to be digitized and subsequently stored in computer memory. The histogram of T A C voltages serves the useful function of monitoring the time spectrum in an E M S experiment and is thus a check for the presence of spurious signals or external noise. The typical time resolution is 4-5ns of which ' only = Ins is due to the electronic components of the coincidence system. The rest of the time spread is mainly due to the different flight times of electron trajectories in the C M A [VS83]. The E M S spectrometer can be operated in three different modes namely, elastic mode, binding energy mode and angular correlation mode. In the elastic mode the electron beam impact energy is set at 600eV (corresponding to coincidence operation at E 0 = 1200eV), focussed and aligned. This permits the electron energy analyzers to be optimized for scattered electrons of 600eV as observed in the coincidence modes. Typically the grid voltage is varied to reduce the countrates and prevent the CEMs from overloading. The C M A voltages are then separately t In addition to the small lens acceptance angles the low coincidence countrate results from the fact that most of the ionizing collisions are due to small momentum transfer collisions (K< l a 0 " 1 ) rather than large momentum transfer collisions [EJ86]. The E M S spectrometer selects the latter type of collision because of the symmetric energy sharing arrangement. 56 adjusted so as to pass 600eV electrons (elastically scattered) into each channel (movable and fixed EADs). The shapes of the elastic peaks are also monitored to check the performance of the C M A . Extra caution is exercised in setting the movable C M A voltages because it is necessary to ensure that the elastic count rate is also invariant with the angle <j>. This procedure is routinely done (every 2-3 days) to monitor any shift in surface potentials and beam characteristics. The binding energy and angular (coincidence) scans are done by setting the cathode back to 1200eV and re-focusing the beam. Either type of scan can then be initiated by the appropriate software control in the LSI 11/03 computer. The binding energy scan is done at fixed angle 0 and variable impact energy ( E 0 ) supplied by a programmable high voltage power supply. The programmable power supply is ramped by a 12-bit digital to analog converter (DAC). The angular correlation scan is done at an appropriately selected fixed impact energy and variable angle <j>. The angle is varied through a 12-bit D A C and a calibrated servo-motor amplifier. More detailed description of the hardware and software are available [L84]. Under routine operating conditions in the coincidence modes the measured energy resolution of the E M S spectrometer is 1.6-1.7eV fwhm. The quoted energy resolution includes the finite energy spread of the electron gun (^O.SeV) and the C M A energy resolution ( ^ l ^ e V ) . The momentum resolution is 0.10-0.15a 0 " 1 and this is discussed further in the following section. 57 3.2. MODELLING THE EFFECTS OF FINITE MOMENTUM RESOLUTION Inherent in any scattering experiment is the instrumental resolution function. In general, this instrumental resolution function is due to the collision geometry, especially the finite sizes of the apertures that define the collision process, the energy spread of the incoming and outgoing particles and the electron beam size. In E M S , where the quantity of critical interest is the experimental momentum profile (XMP), the most important instrumental factor to be accounted for is the momentum resolution function. Within the binary encounter approximation, the measured ion recoil momentum is essentially equal (but opposite in sign) to the momentum of the particular orbital electron prior to ionization [MW76, WM78] and is given by Eqn. 2.13. Mapping the orbital electron momentum, p is achieved by variation of the azimuthal angle 0 within the symmetric non-coplanar geometry [MW76] where 0 ,=0 2 = 4 5 ° (fixed) and E , = E 2 =600eV (fixed). A propagation of error type analysis [MC81, B84] shows that to a large degree the contributions from A p 0 , A p j and A0 are relatively minor. Explicit analysis [MC81, B84] of the momentum resolution, Ap shows, A p = p ~ 1 { [ 2 p , s i n 0 ( p o - 2p,cos0) + 4 p , 2 s i n 0 c o s 0 s i n 2 0 / 2 ] A 0 + ( 2 p , 2 s i n 2 0 s i n 0 / 2 c o s 0 / 2 ) A 0 + [ 4 p , ( c o s 2 0 + s i n 2 0 / 2 ) - 2 p o c o s 0 ] A p 1 + ( p 0 - 2p,cos0)Ap o} [3.4] For the values of the kinematic parameters used in the present work i.e. E o = 1200eV, E 1 = E 2 = 6 0 0 e V , E b =15.7eV, 0 = 45° , A 0 = + 1° and A 0 = ± 1 ° Eqn. 3.4 can be approximated by, A p = p ~ 1 { [ 2 p 1 p o - 2 p ! 2 ( 3 c o s 0 ) ] A 0 + 2 p 1 2 s i n 0 A 0 } [3.5] 58 Clearly, Ap is strongly dependent on the value of <j> only at small values of momentum or 0. The analysis of EMS data therefore presents an unusual challenge to the experimentalist because the features of the experimental momentum profile most critical in its comparison with theoretical predictions are in the region p<l.Oao" 1 - Clearly an accurate knowledge of the momentum resolution function is required for detailed evaluations of EMS experiments and quantum chemical calculations reported in the present work. Although other workers in the field have often neglected momentum resolution effects, a Ap as small as O . l 5 a 0 ~ 1 influences both the shape and relative magnitudes of the observed momentum profiles. It could however be argued that an 'experimental' approach to the momentum resolution function problem is to further reduce the lens aperture sizes such that the momentum resolution function is approximately a delta function. This alternative is however not feasible because of the extremely low coincidence counts that would result. The present aperture sizes (half-angles, A0=A0=1°), which approach the practical sensitivity limit for the current single channel spectrometers, result in a Ap<0.15a 0 _ 1 -Two main approaches in defining the momentum resolution function are presently used by workers in the EMS field namely, the planar grid method and the analytic gaussian function method. Both methods are outlined below and their relative advantages and disadvantages are discussed. 59 3.2.1. Planar Grid Method The planar grid method has been adopted by several groups [C81, F83]. The method basically assumes a planar grid in the collision region (see Fig. 3.4) defined by the half angles Ad and A0 of both analyzers as well as the beam size. For simplicity the grid is assumed to be rectangular with typical spacings of 0.2°. For a particular value of <p0 , 60 and E 0 the detector is assumed to detect contributions from p-values defined in the planar grid. The resulting convoluted (resolution fitted) momentum distribution is therefore given by, <p(p)> = W i j * p ( P i j ) } / { 1 ^ W i j } [3.6] where p(Py) is the theoretical MD at p(0^, 0j) and w- is a general weighting factor. In most cases, w~ is assumed to be unity that is , a uniform distribution is assumed and the convoluted MD is just the uniformly averaged value of p(p^) over the grid. Three main assumptions are inherent in the planar grid method, (a) The interaction volume is assumed to be defined by a plane of dimensions (2A8) X (2A0). The primary electron beam diameter is smaller than the dimensions subtended by the acceptance angles in the collision region. The much smaller electron beam width ensures that the collision volume does not change when the azimuthal angle <j> is varied [C81]. (b) Trajectories originating from this planar grid are assumed to be evenly distributed (or uniformly weighted). This means that trajectories originating from ( f i 0 ) 0 O) are as likely as other trajectories (6^, #j) as shown in Fig. 3.4. (c) The effective momentum representing the summed trajectories from the planar Fig. 3.4. Collision volume defined by spectrometer apertures ana beam size. Gridded area refers to the region of convolution using the planar grid method (see text). 61 grid is given by p as defined by (60, (p0). As will be shown in a later subsection, this assumption may be incorrect. A typical result for the A r 3p momentum distribution using the planar grid method is given for different values of A0 (At9 = 1° =fixed) in Fig. 3.5. The theoretical curves are calculated from the Hartree-Fock limit wavefunction of Clementi and Roetti [CR74]. It can be seen that lowering the momentum resolution (i.e. increasing Ap) has the effect of increasing the intensity at p < 0 . 3 a 0 ~ 1 and results in a very slight shift towards higher momentum. Comparison of the curves with the measured A r 3p X M P shows that the optimum values , A# = 1.0° and A0=1.O° give an excellent fit to the A r 3p X M P . These values of AO and A0 also match closely the physical sizes of the lens acceptance angles (A0=A0 = O.9°, see Sec. 3.1.1). The choice of the A r 3p X M P as the 'calibrant' gas for E M S studies is based on extensive SCF limit and CI studies [MA84] which indicate that the A r 3p M D is only very slightly (<1%) affected by inclusion of correlation effects. Furthermore relativistic effects are not expected to be important for the valence orbitals of a low Z atom such as argon. Comparison of measured X M P s and convoluted MDs using the planar grid method have been routinely used in the earlier work in the present laboratory and elsewhere. The validity of the planar grid method can be further investigated by a careful evaluation of the assumptions (a-c) outined above instead of finding the optimum A6 and A<j> values which give the best fit to the Ar 3p X M P . One i r "i 1 r q d P L A N A R G R I D M E T H O D A r 3P ( 1 5 . 7 e V ) R e s o l ut i o n E f f e c t s ( 1) A * = 0. 0* (2 ) A * = 0. 4° ( 3) A * = 0. 8° (4 ) A * = 1. 0* ( 5.) = 1 . 2 ° (6 ) a* = 2. 0° (7 ) A * = 3. 0° 1.5 2.0 2.5 Momentum (a.u.) ^ 3.0 3.5 Fig. 3.5. Sample results using the planar grid method. The unconvoluted curve is area normalized to the Ar 3p XMP and all other curves maintain the same relative normalization. 63 way of addressing this question is to devise alternative methods of convoluting the effective momentum spread and to observe the dependence of the convoluted MD on the form of the instrumental function. An alternative Ap convolution method is provided by the analytic gaussian function method. 3.2.2. Analytic Gaussian Function Method This method involes an analytic convolution method whereby a gaussian instrumental function given by, e x p [ - ( q - q 0 ) 2 / a 2 ] [3.7] is convoluted to the theoretical MD, p(p). The convoluted MD is therefore given by, <p(p)> = { f p ( q ) e x p [ - ( q - q 0 ) 2 / a 2 ] d q } * { ; e x p [ - ( q - q 0 ) 2 / a 2 ] d q } " 1 [3.8] where a2=Ap2/(ln2) and Ap is defined as the momentum resolution, t The integration of the numerator in Eqn. 3.8 is done numerically using a 4-point integration routine and is found to be reasonably converged within the range -3 . 5ao " 1 ^p^ + 3 . 5ao " 1 in steps of O.la 0~ 1. The denominator in Eqn. 3.8 is 1/2 known and is equal to a(7r) The analytic gaussian function method is attractive in several respects. First, the method does not assume a uniform distribution of trajectories but weights the trajectories preferentially at the nominal value of the momentum defined by 60 t The momentum resolution (Ap), by definition, is equal to the half-width-at-half-maximum. 64 and <f>0. Following the Central Limit Theorem [G85] the gaussian (or normal) distribution of electron trajectories is the most probable distribution function especially if many small factors contribute to the final experimental uncertainty. In addition according to the Maximum Entropy Principle [J57] the most probable distribution, given the only requirement that the variance (a 2) be finite, is the gaussian (normal) distribution. Secondly, the gaussian function method is also computationally faster (by =50%) than the planar grid method. The analytic gaussian function method (for a fixed Ap) however yields 'unphysical' results at p = 0 a o ~ 1 - This is due to the fact that p(p) exists only for p>0ao _ 1- In practice, however, the fact that the 'effective' Ap increases as p decreases towards p^O (as predicted by the propagation of error analysis) should be taken into account. This can be approximated by keeping Ap constant (i.e. independent of p) and folding the theoretical MD, p(p) about p = 0 a o " 1 and doing the integration in Eqn. 3.8. This has the net effect of increasing the effective value of Ap at values of momentum, p<2Ap. Sample results using the gaussian function method at different values of Ap are shown along with the Ar 3p XMP in Fig. 3.6. It can be seen that the results are very similar to that obtained using the planar grid method (compare Figs. 3.5 and 3.6). As Ap increases there is an increase in intensity at p<0.3a 0" 1 accompanied by a very slight shift in p towards larger values of p. This shift of P m a x towards larger momentum can be seen by assuming a simple form of p(p) such as, p(p) = p 2 e x p ( - $ p 2 ) [3.9] G A U S S I A N F U N C T I O N M E T H O D Ar 3P ( 1 5 . 7 e V ) R e s o l ut i o n E f f e c t s (1 ) 0 . 0 au ( 5 ) 0 . 15au (2 ) 0. 05au ( 6 ) 0 . 20au (3 ) 0. 10au ( 7 ) 0 . 30au (4 ) 0. 12au ( 8 ) 0 . 40au 1.5 2.0 2.5 Momentum (a.u.) 3.5 Fig. 3.6. Sample results using the analytic gaussian function method. The unconvoluted curve is area normalized to the A r 3p X M P and all other curves maintain the same relative normalization. Ol 66 which maximizes at p. = (5) -1/2 It can be shown that the resulting max substitution of Eqn. 3.9 into Eqn. 3.8 yields the convoluted MD, <p(p)> which maximizes at Clearly it can be seen that the first order correction is very small and furthermore it is positive (i.e. towards larger momentum). This result is consistent with earlier observations regarding observed discrepancies in the comparisons of theory and experiment for the outermost valence orbitals of H 2 0 [BL85]. It was originally thought that the discrepancy at lower momentum may be due to uncertainties in the momentum resolution function that was used. The more exhaustive study of H 2 0 [BB87] incorporating correlated wavefunctions for both initial and final states, in fact, showed that the reported discrepancy [BL85] was not due to momentum resolution effects but due to the inadequate inclusion of electron correlation and relaxation. 3.2.3. Defining the Opt imum p-Value A mental picture of the (e,2e) reaction can be envisioned as shown in Fig. 3.4. The finite dimensions of the lens apertures (A t and A 2, A s and A 6) define a collision volume from which various combinations of electron trajectories are likely to register as (e,2e) events. The combination of all these trajectories define the signal strength ultimately registered at a particular value of 60 and #0 and therefore at a corresponding p. We shall define p as the optimum p-value that accurately represents (in a statistical sense) all the electron trajectories physically collected in the detection system for a given nominal setting of 60 (45°) and - [0/5)+aV2] 1/2 [3.10] 67 <po (variable). The optimum p-value, p should therefore be equal to p in the limit of very small A0 and A0 and very small beam size. It is clear that increasing A0, A0 or the beam size would have the effect of including electron trajectories that correspond to different (6^ , 0 j ) and therefore different values of p. This would therefore correspond to sampling different regions of the orbital momentum distribution but all would be recorded at a given nominal value of p. In regions wherein the M D varies steeply the observed (e,2e) cross section would therefore be difficult to interpret. This has the effect of 'smearing' the sharp features in the M D if an otherwise ideal E M S experiment was performed. However not only do features in the measured X M P get smeared but this also has the effect of introducing an uncertainty in the optimum p-value, p. There are three possible ways of defining the optimum p-value, p, namely, (a) p = p (zeroth order); (b) p = [L^ Wj_} (first order); (c) p = Statistical average over the collision volume weighted by the lens and analyzer transmission functions. Option (a) has been used traditionally as outlined in Sec. 3.2.1 however this option is inconsistent with the planar grid method. If one assumes, in fact, a planar array of electron trajectories that are equally weighted then one could produce a p-value histogram for a given value of E 0 , 60 and <j>0 as shown in Fig. 3.7. The series of p-value histograms at different values of 0 O were .68 2 1^ ?6 § PHI = 0 DEC -0.05 0.00 0.05 O K OO 0.20 O.IS O.JO PHI = 1 DEC g o UJ I 2 2 -005 0.00 0 05 O.K) 0.15 0.20 0.25 O.JO PH = 2 DEC O B O O 0.20 0 . » 0 JO O J S 0.40 PHI = 8 DEC PHI = 30 DEC i II III i l l 2J0 0J5 2.40 2.«5 230 235 2.*0 Momentum (a.u.) Fig. 3.7. P-value histogram at different values of 0 O- E 0 =1200eV, Eb=15.7eV, E , = E 2 =600eV, e0=45\ A0=A0 = 1°. The grid dimensions are X 2A0. Arrows refer to the zeroth order approximation to p and stars refer to first order approximation to p. 69 obtained by calculating all the values of p .^ as defined by (6^, <}>•) in the planar grid (see Fig. 3.4) and counting the number of p-values that occur at particular values of p ± 6 p where 5p is the histogram halfwidth. The arrows in each histogram (Fig. 3.7) define the p-value given by ( 6 0 , 0 O ) , that is, it refers to the zeroth-order approximation to p (Option a). The stars (*) in each histogram, on the other hand, refer to the first order approximation to p which is just the mean p-value of the distribution defined in each histogram. It can be seen from Fig. 3.7 that at values of <f>0=0°-3° there are marked differences between the zeroth-order and first-order approximations to p. It should be noted that in the most severe case (<t>o=0°) the difference between the zeroth order and first order approximation to p is only ^ O . O S a o " 1 - Although this correction is relatively small it may become important as more precise E M S experiments and correspondingly as more accurate calculations are done in the future. At larger values of <j>0 the distribution of p-values becomes nearly uniform and the two approximations (arrows and stars) give essentially the same p-value. The preceding analysis simply states that assuming a uniform distribution in the planar grid method is not consistent with assuming p = p (i.e. zeroth order approximation) at small values of <f>0. A proper definition of p could be attained by adopting the weighted p-value (Option b) in the planar grid method or some other realistic weighting scheme. One should note that the present analysis is still limited to a planar grid. A non-planar grid (Option c), for example a distorted cylinder, would give a better approximation of the optimum p-value. Clearly this would involve sophisticated calculations and non-trivial analysis procedures (e.g. Monte Carlo methods [NT86]). 70 Another alternative which is consistent with the first-order approximation to p is given by the analytic gaussian method. In this case electron trajectories are assumed to be normally distributed around p (60r #o)- m a u studies in the present work the analytic gaussian method has therefore generally been employed although variations of both methods [D87] have been used for consistency checks. All conclusions reported in this thesis are however unaffected by this choice. C H A P T E R 4. W A T E R : P A R T I 4.1. OVERVIEW In this chapter an E M S study of the valence shell orbitals of H 2 0 at high momentum resolution is reported. The experimentally determined MDs are accompanied by a related discussion of the orbital density topography in both momentum and position space. Quite apart from its fundamental importance to life, water has also been a benchmark test system in theoretical quantum chemistry [KK72]. The water molecule is the subject of several reviews [EK69, F72]. Some of the earliest binary (e,2e) measurements for molecules were for H 2 0 [HH77, DD77]. The first binary (e,2e) experiments on H 2 0 , done independently by Hood et al. [HH77] and Dixon et al. [DD77] were restricted by low momentum resolution ( = 0 . 4 a 0 ~ 1 ) but nevertheless they illustrated the sensitivity of the (e,2e) technique to details in the orbital wavefunction. Even at low momentum resolution both studies strongly suggested significant discrepancies between the measured momentum distribution of the l b , and 3a ^ orbitals of water and the corresponding MDs calculated using ab initio S C F wavefunctions. These discrepancies were particularly apparent in the low momentum region ( p < 0 . 6 a 0 ~ 1 ) and appear to suggest that the orbitals in question are more spatially extended than predicted by the corresponding wavefunctions. Subsequent studies on other second row hydrides such as hydrogen fluoride [BH80] and ammonia [HH76] revealed similar apparently anomalous behavior in the outermost orbital momentum distributions. In contrast, the outermost orbital MDs of the analogous third row atom containing hydrides, namely hydrogen sulfide [CB80], phosphine [HH77a] and hydrogen chloride [BH80] were found to be quite 71 72 adequately modelled by wavefunctions of double-zeta (DZ) or better quality. Although these results are still not fully understood they at least appear to be consistent with the well known and unusual relative chemical and physical properties of second and third row hydrides such as the tendency for ligand donor activity and hydrogen bonding. A very recent (e,2e) study of water [CC84] using a binary (e,2e) spectrometer with position sensitive detection in the energy dispersive planes of the electron analyzers has complemented the earlier studies [HH77, DD77] by discussing additional structure in the binding energy spectra of water notably in the inner valence (2a 1) region. The splitting of the inner valence ionization pole strength in water and other molecules has been attributed to many-body effects [SC78] arising from the breakdown of the independent particle picture for ionization. Although the most recent study [CC84] of H 2 0 emphasized the binding energy spectra and its comparison with many-body calculations, only a limited number of experimental points on the orbital momentum distributions were obtained. Moreover the measured MDs were derived from peak areas in the binding energy spectra obtained at a series of azimuthal angles. Despite some appreciable uncertainty in the momentum scale, particularly below 0 . 6 a o ~ 1 , these M D results t appear to reflect the same kinds of differences from Hartree-Fock theory observed in the earlier binary (e,2e) studies [HH77, DD77]. In view of these other various results the present direct measurement of the valence orbital MDs of H 2 0 have been undertaken utilizing the high momentum resolution ( ^ O . l a o " 1 ) available in the present spectrometer [LB83]. In addition the binding t It should be noted that Fig. 3 of reference [CC84] is incorrect as initially published. The correct figure has been published in a subsequent erratum [CC84]. 73 energy spectrum in the inner valence region has been obtained and this is compared directly with a number of representative many-body calculations. Furthermore our earlier discussions and illustrations of the momentum space chemistry of atoms [LB83], diatomics [LB83a] and linear triatomics [LB85a] are now extended to the bent triatomic system H 20. 4.2. BINDING ENERGY SPECTRA OF WATER The binding energy spectra of water obtained at relative azimuthal angles of 0° and 8° and an impact energy of 1200eV plus the binding energy are shown in Fig. 4.1. The spectrum at <p = 0° probes very low momentum components and thus tends to emphasize ionization of "s-type" orbitals (ie.those which contain totally symmetric components). The minimum momentum, P m j n accessible in the experiment depends on the impact energy (E 0) and the electron binding energy (E^) and is given by, Pmin ~ E b / ( 2 E 0 - 2 E b ) 1 / 2 [4.1] On the other hand, the spectrum at <t> = 8° (corresponding to p= 0.6a 0~ 1 for valence electrons) shows contributions from both "s-type" and "p-type" orbitals (Fig. 4.1). The two binding energy spectra were obtained by repeated sequential scans so that their relative intensities are automatically normalized. The energy scale was calibrated by aligning the spectrum with the vertical IPs established by (UV and X-ray) photoelectron spectroscopy [BT68, SN69, MM77]. 74 q 06 q id q q CN H 2 0 1200 eV j cp = 0 d e g . _ "c D D q d 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 Binding Energy (eV) 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 Binding Energy (eV) Fig. 4.1 Binding energy spectra of H 2 O at azimuthal angles 0° and 8° . The impact energy is 1200eV + binding energy. The sitting binding energies where the MDs are measured are indicated by the arrows in the lower part of the <f>= 8° spectrum. 75 The ground state electronic configuration of water in the independent particle approximation may be written as follows: Comparing the two spectra in Fig. 4.1, relative increase of the low energy peaks (10-20eV) takes place on going from 0 = 0° to 0 = 8° . This is expected from the "p-type" character of the three outermost orbitals. Gaussian curves have been fitted to the spectra (Fig. 4.1) taking into account the known vertical IPs [BT68, SN69] and associated vibrational widths for the corresponding transitions together with the instrumental energy resolution. In contrast to the behavior of the outermost three valence orbitals, the peak at 32.2eV reflects the "s-type" character of the inner valence orbital (2a , )•. The Gaussian fitting to the 2a , band reflects the many-body nature of the associated transition (see below). The present results (Fig. 4.1) are consistent with both earlier binary (e,2e) binding energy spectra [HH77, DD77] for H 2 0 as well as the recent work of Cambi et al. [CC84] given the differences in energy resolution in the respective measurements. The 2a, band (Fig. 4.1) is quite broad with a tail extending to high energy and shows indication of partially resolved structure on the high energy side. Similar features in the inner valence band can be seen in the X P S work reported by Siegbahn et al. [SN69] in 1969 and also in the later work of Martenson et al. [MM 7 7]. A re-examination of the somewhat higher energy resolution (1.2eV fwhm) binary (e,2e) spectrum reported by Hood et al. [HH77] also strongly suggests that the 2a , envelope consists of two or more partially resolved peaks. The present experimental binding energy spectrum (</) = 0°) in the 1 A 1 • ( 1 a , ) 2 ( 2 a , ) 2 d b 2 ) 2 ( 3 a , ) 2 ( 1 b , ) 2 76 2a , region is shown in greater detail in Fig. 4.2a together with the data of Hood et al. [HH77] and Cambi et al. [CC84]. The solid line represents a fitted curve assuming three equal-width Gaussians (2.77eV fwhm) which give a good fit to the data.t There is good agreement with the earlier data of Hood et al. [HH77] and also with that recently reported by Cambi et al. [CC84]. The differences in the sharpness of the peak onsets are also consistent with the differing energy resolutions used in the respective experiments. In view of the interesting nature of ionization from the inner valence region and in particular the wide range of theoretical studies [AS80, MO80, AM82 , NT82, VC82] predicting a failure of the single particle picture for H 2 0 inner valence ionization, a careful comparison study has been made (Fig. 4.2) with the experimental spectrum in the region 23-45eV. The results of three of these recent theoretical studies [AS80, VC82, NT82] of the many-body (satellite) structure of the inner valence ionization of water are presented in Figs. 4.2b-4.2d and Table 4.1 for comparison. It should be noted that the recent Multireference double excitation configuration interaction (MRD-CI) calculation [CC84] gives a very similar result to the semi-internal CI calculation by Agren and Siegbahn [AS80]. The calculated pole strengths in each study are represented by vertical lines at the appropriate energies. In each calculation the highest pole strength has been arbitrarily normalized to unity to facilitate comparison with experiment. Gaussian curves with the experimentally determined halfwidth (fwhm = 2.77eV) are f If the instrumental halfwidth (1.6eV) is allowed for, a natural halfwidth of 2.26eV can be derived for each 2a , pole. This (2a,)" state is entirely dissociative as has been shown by dipole (e,e+ion) coincidence studies [TB78]. Therefore the natural peak width would reflect the intersection of the Franck-Condon region with the repulsive potential energy curve for each 2a , pole. Fig. 4.2 - l 1 1 1 1 1 1 1 1 r-S e m i - I nt e r no l Cl 77 23.0 25.0 27.0 29.0 31.0 33.0 35.0 37.0 39.0 41.0 43.0 Binding Energy (eV) 23.0 25.0 27.0 29.0 31.0 33.0 35.0 37.0 39.0 41.0 43.0 Binding Energy (eV) 23.0 25.0 27.0 29.0 31.0 33.0 35.0 37.0 39.0 41.0 43.0 Binding Energy (eV) 1 ' 6 £ ° T 1 i 1 r-EXPERIMENT (l200eV) rp = 0 d e g . 23.0 25.0 27.0 29.0 31.0 33.0 35.0 37.0 39.0 41.0 43.0 Binding Energy (eV) (a)Comparison of the inner valence (2a, ) binding energy spectrum (0 = 0°) of H 2 0 [solid circles] with previous experimental work (Ref.[HH77] [open square], Ref.[CC84] [open triangle]). The experimental points have been fitted with three Gaussians of 2.77eV fwhm (dashed lines) and the sum given by the solid line. Shown above is the comparison with several theoretical calculations (b)SAC-CI N V [NT82], (c)Ext. 2ph-TDA [VC82] and (d)Semi-Internal CI [AS80]. The calculated pole strengths are indicated by bars at appropriate energies and Gaussian curves (2.77eV fwhm) are convoluted, summed and scaled (x2.5) to yield the theoretical binding energy profile (solid curve) in each case. T a b l e 4 . 1 . O r b i t a l I o n i z a t i o n E n e r g i e s ( e V ) a n d P e a k I n t e n s i t i e s o f W a t e r ( e , 2 e ) b XPS° S A C - C I N V d E x t . 2 p h - T D A S S e m i - I n t e r n a l C I f 2 a , 3 2 . 2 ( 0 . 5 8 ) 3 2 . 2 3 2 . 3 9 ( 0 . 7 7 2 ) 3 0 . 4 8 ( 0 . 0 8 1 ) 2 7 . 2 4 ( 0 . 0 4 7 ) ( 2 . 7 7 ] 3 5 . 0 ( 0 . 1 8 ) 3 4 . 8 9 ( 0 . 1 7 8 ) 3 3 . 4 1 ( 0 . 5 8 ) 3 1 . 4 8 ( 0 . 7 1 2 ) 1 2 . 7 7 ] 3 8 . 9 ( 0 . 0 9 5 ) 4 0 . 7 0 ( 0 . 0 5 1 ) 3 3 . 9 9 ( 0 . 0 9 8 ) 3 2 . 2 9 ( 0 . 0 1 8 ) [ 2 . 7 7 ] 3 6 . 5 6 ( 0 . 0 2 3 ) 3 5 . 3 7 ( 0 . 1 1 4 ) 3 8 . 0 9 ( 0 . 0 6 8 ) 3 5 . 9 7 ( 0 . 1 0 0 ) a A l l I n t e n s i t i e s q u o t e d I n p a r e n t h e s e s . b B i n d i n g e n e r g y r e s u l t s t a k e n a t » ) = 8 0 . fwhm I n c l u d i n g i n s t r u m e n t a l r e s o l u t i o n a r e q u o t e d I n s q u a r e b r a c k e t s ! ] . C R e f . [ S N 6 9 ] . a G r o u n d s t a t e e n e r g y * - 7 6 . 2 4 7 4 9 a . u . R e f . [ N T 8 2 ] . S R e f . [ v C 8 2 ] . f G r o u n d s t a t e e n e r g y » - 7 6 . 0 4 1 1 0 a . u . R e f . [ A S 8 0 ] . I n t e n s i t i e s c o n t r i b u t i n g l e s s t h a n 2% a r e n o t r e p o r t e d . 79 folded into each of the calculated poles in the respective calculations and the resulting total envelope is given by the solid line in Figs. 4.2b, 4.2c and 4.2d. The Green's function calculation due to von Niessen et al. [VC82] (Fig. 4.2c) uses the extended two-particle-hole Tamm-Dancoff approximation (Ext. 2ph-TDA). As opposed to an earlier version (2ph-TDA) [SC78], the Ext. 2ph-TDA is exact to third order in the electron-electron interaction. The two other theoretical studies employ the method of configuration interaction (CI). Agren and Siegbahn [AS80] used semi-internal CI to describe the electron correlation effects whereas Nakatsuji and Yonezawa [NT82] utilized the symmetry adapted cluster (SAC) CI method. Though two calculations are reported in reference [NT82], namely a variational (V) and a non-variational (NV) solution, only the SAC-CI N V calculation, which should be more accurate since it involves fewer approximations, is included. It is not the purpose of the present study to make extensive comments on the methodologies of the theoretical studies mentioned but rather to make a comparative evaluation of the calculated pole strengths with the experimental results for water. It is seen, in particular, that all calculations agree qualitatively with the experimental binding energy spectra. However, the envelope of the pole strengths (peak intensities) and peak positions as predicted by the SAC-CI N V theory gives the best overall agreement with experiment in the case of water. Below the main inner valence peak the present work suggests the presence of a very weak signal in the region =27eV consistent with the findings of the recent binary (e,2e) study by Cambi et al. [CC84] as well as with the appearance of the data in an earlier X P S work [SN69]. It should be noted that the SAC-CI N V calculation does not predict corresponding intensity in the region below the main pole although the semi-internal CI [AS80], MRD-CI [CC84] and Ext. 80 2ph-TDA [VC82] calculations do indicate such intensity. Other theoretical studies by Mishra and Ohrn [MO80] and Arneberg et al. [AM82] also predict some structure both above and below 30eV. Whereas most theoretical studies attribute all the structure from 25-45eV to the 2a , molecular orbital, the calculations of Arneberg et al. [AM82] predict appreciable contribution in this region from the 3a, molecular orbital. 4.3. MOMENTUM DISTRIBUTIONS OF WATER The experimental momentum profiles (XMPs) and calculated spherically averaged MDs are presented together with r-space and p-space density contour maps and three dimensional surface representations in Figs. 4.3-4.6 for the l b , , 3 a , , l b 2 and 2a , molecular orbitals of water, respectively. The top left hand section in each of the integrated diagrams presents a comparison of the experimentally obtained MDs and various theoretical MDs obtained from spherically averaged, Fourier transformed, position space wavefunctions. It should be noted that the experimental momentum resolution (0. l a o" 1 ) has been folded into the theoretical MDs shown in the figure. A l l X M P s were obtained at the indicated "sitting" binding energies (see Fig. 4.1). Two types of literature wavefunctions, representative of those used in the earlier comparisons with experiment [HH77, DD77] have been used to calculate spherically averaged MDs. These wavefunctions are those of STO type reported by Aung, Pitzer and Chan (APC) [AP68] and the GTO functions of Snyder and Basch (SB) [SB72]. A third wavefunction, namely that reported by Neumann and 81 Moskowitz (NM) [NM68], has also been used. The theoretical MDs have been height normalized in each case to the respective X M P s . Differences between area normalization and height normalization are insignificant (see discussion below). To provide a more critical view of the regions of discrepancies or agreement between each X M P and the respective theoretical MDs, distribution difference plots are presented in the lower left hand section of each integrated diagram. The distribution difference is obtained by subtracting the M D (normalized as above) calculated from the best fitting wavefunction (APC) from the X M P . The distribution difference is on the same intensity scale as the respective M D shown immediately above. The actual value of the distribution difference is slightly dependent upon how the theoretical MDs are normalized. The other limit of the hatched area represents the zero position if area normalization (in the momentum region 0 to l . 5 a 0 " 1 ) was used instead of height normalization. It should be noted that the shape of the distribution difference is largely independent of the normalization method. For the respective valence orbitals the two dimensional (2D) density contour maps and the three dimensional (3D) boundary surfaces (at three values of the density) are shown in both momentum and position space in the center and right hand sections respectively of Figs. 4.3-4.6. The double-zeta (DZ) quality wavefunction of Snyder and Basch [SB72] has been used to generate the (more complex) maps and surfaces shown in Figs. 4.3-4.6. t The DZ quality wavefunctions are t The r-space coordinates (x,y,z) used are (0,0,0) and (0, ± 1.430456, 1.107118) for the oxygen atom and the hydrogen atoms, respectively. A l l numbers are in atomic units. 82 considered to be sufficiently accurate to provide a qualitative working comparison between structural features in momentum and position space for the bent triatomic system, H 2 0. While the more sophisticated A P C wavefunction gives somewhat better agreement with the experimental MDs, particularly for the outermost two orbitals (see below), the essential features of the maps and surfaces would not be expected to differ drastically from those which have been obtained using the SB wavefunction. This is done as part of a continuing effort to contribute to a clearer understanding and visualisation of momentum space chemical concepts in atoms and molecules. The mapping procedures and conventions are similar to those used in earlier work [LB83a]. The contour values of all maps are 0.2, 0.4, 0.6, 0.8, 2, 4, 6, 8, 20, 40, 60 and 80% of the maximum density value. The plane of the contour map is defined by two unit vectors spanning the range -5.0 to 5.0 atomic units in both r-space and p-space. To emphasize the details of the 2D density maps, line projection plots (along direction shown by dotted lines) parallel to the two axes are also included. 4.4. COMPARISON OF EXPERIMENTAL AND THEORETICAL MOMENTUM DISTRIBUTIONS The characteristics of the literature wavefunctions used in this study are presented in Table 4.2. The large number of published wavefunctions for the water molecule precludes an exhaustive comparison of experiment and theory. The three wavefunctions used in the present work provide a reasonable range in the quality of wavefunctions for comparison with both the present results as well as those of earlier experiments [HH77, DD77]. Comparison of the X M P s with those calculated from more sophisticated wavefunctions for H 2 0 such as those reported Table 4.2. Comparison of Theoretical SCF Wavefunctions for Water APC b NM° SB d EXPERIMENTAL a Bas i s STO(IV) GTO GTO --(3s3p1d/2s1p) [532/21] [42/2] -76.4376® (-76.067)f Energy(a.u.) -76.00477 -76.044 -76.0035 Dipole moment(D) 2.035 2 .092 2 .681 1 .85469 The basis notation follows Ref.[NM68]. Contracted basts sets are denoted in square brackets[]. b Ref.[AP68]. See Ref.[DP72] for corrections. ° Ref.[NM68] d Ref.[SB72]. E s s e n t i a l l y of DZ quality. Total experimental energy with r e l a t i v i s t i c and mass corrections. Ref.[RS75]. f Estimated Hartree-Fock SCF l i m i t . Ref.[RS75]. 9 Ref.[SC78]. oo co 84 by Rosenberg and Shavitt [RS75] and Davidson and Feller [DF84] is reported in Chapter 5. In general, the spherically averaged MDs predicted from the essentially DZ quality wavefunction of Snyder and Basch [SB 7 2] are not in good agreement with the present experiment for the two outer valence orbitals (Figs. 4.3 and 4.4) but a reasonable description is provided for the l b 2 and 2a , orbitals. These findings are consistent with the conclusions reached in earlier work at low momentum resolution [HH77, DD77]. This is not surprising since the variationally inferior SB wavefunction also gives poor agreement for molecular properties such as the dipole moment (see Table 4.2). In the case of the atomic MDs of Ar , K r and Xe it has been reported [LB83] that DZ quality wavefunctions give MDs very close to those calculated using Hartree-Fock quality wavefunctions. However it should be noted that this is evidently not the case for Ne [LB83] which is isoelectronic with H 2 0. Consider now the comparison with experiment of the spherically averaged MDs calculated from the variationally superior APC-Basis IV [AP68] and N M [NM68] wavefunctions (see Table 4.2). It should be noted that these wavefunctions use different types of basis functions (APC uses Slater type orbitals whereas N M uses Gaussian type orbitals). For the case of the l b 2 orbital (Fig. 4.5), both A P C and N M wavefunctions predict MDs in close agreement with the experimental M D . However for the two outermost orbitals ( l b , and 3a,) there is a large discrepancy between experiment and theory as can be seen clearly in the distribution difference plots (Figs. 4.3 and 4.4). In this connection it should 85 be remembered that the distribution difference shown is for the APC wavefunction for which the discrepancies are smallest. In particular, the calculations seriously underestimate the low momentum region (<0.5a 0~ 1) for the two outermost orbitals. Furthermore, the observed lb, and 3a, XMPs are more asymmetric, with their respective maxima situated at lower values of the momentum, than those predicted by the respective calculated momentum distributions. This contrasts with the close agreement observed between experiment and theory for the l b 2 orbital (Fig. 4.5). These results confirm the general conclusions of the earlier work [HH77, DD77] carried out at much lower resolution. Experimental momentum profiles in the 2a, inner valence region have been determined at the band maximum (32.2eV) and also on the shoulder at 35.6eV. These results are presented in Fig. 4.6. The relative intensities of the binding energy spectra at the azimuthal angles of 0° and 8° (Fig. 4.1) have already suggested a dominant "s-type" character for the ionization strength above 28eV. This is confirmed, at least at 32.2 and 35.6eV, by the two MDs shown in Fig. 4.6. It can be seen that all three wavefunctions give calculated MDs in reasonable agreement with experiment (at least below l .5a 0 ~ 1 ) at both 32.2eV and 35.6eV. However the shape of the APC MD has a narrower half width than that predicted by both SB and NM wavefunctions. This may reflect the different types of basis functions used. In momentum space, Slater type orbitals tend to decay faster than Gaussian type orbitals. The reverse is true in r-space due to the Fourier Transform (FT) relation. 86 Earlier measurements of binding energy spectra [DD77, CC84] as well as theoretical work [CC84, NT82, VC82, AS80, MO80] have all suggested that features above — 25eV are principally due to 2a , ionization. The present E M S measurements confirm this assignment at 35.6eV. This suggests that although there are considerable correlation effects the main features of the 2a, orbital are preserved at 35.6eV as, is the case for the main pole at 32.2eV. This suggests that the dominant contribution to the many-body structure results from final ionic state configuration interaction associated with the 2a , hole. As has been mentioned earlier, the binding energy spectra of water in the region 23-45eV clearly indicate a breakdown of the simple M O picture for the ionization process. The present results obtained with a high momentum resolution binary (e,2e) spectrometer confirm the existence of a significant discrepancy between the X M P s and those calculated using near Hartree-Fock SCF wavefunctions [AP68, NM68] for the l b , and 3a , molecular orbitals of water. In particular, the X M P s for these two orbitals show more intensity at low momentum than is predicted. Considering the F T relation this in turn indicates that these orbitals may be more spatially extended [HH77] than would be predicted by a near Hartree-Fock wavefunction. The discrepancy is greatest for the l b , orbital in that there is a pronounced shift ( — 0.2aQ~^) in the position of the maximum of the X M P towards low momentum relative to theory. The case for the 3 a , orbital is similar although the shift ( = 0 . l a 0 " 1 ) in the maximum of the X M P is less than that in the case of the l b , orbital. However, the discrepancy in the low momentum region is comparably large and these effects are far outside experimental error. 87 Several possibilities exist for understanding the significant discrepancies between experimental momentum profiles and the theoretical MDs for the outermost orbitals of water. These differences can be atttributed to either limitations in the basic (e,2e) theory or to an inadequate wavefunction. Anomalous instrumental effects as a source of this discrepancy can be discounted since excellent agreement has been obtained between experiment and Hartree-Fock theory for noble gas atoms [LB83] using the same spectrometer. In terms of possible breakdowns of basic (e,2e) theory the various approximations to the theoretical formulation of the (e,2e) reaction model should be considered. These are as follows: (1) the Plane Wave Impulse Approximation (PWIA); (2) the Target Hartree-Fock Approximation (THFA); (3) the neglect of vibrational and geometry effects; (4) the flexibility of the basis set and wavefunction quality. The validity of the PWIA has been, the subject of intense investigations in early (e,2e) work [HM73, HM74, WH75]. A t high impact energies (>>400eV) the PWIA is found to be generally valid for atoms and molecules in that the shapes of the measured MDs have been found to be independent of electron impact energy. In fact, the PWIA should be most accurate for the outermost valence orbitals, such as the H 2 0 l b , , since they have the lowest binding energies. However in the case of water the worst agreement is obtained for the outermost orbitals. Recent studies on the distorted wave impulse approximation [GF80] which attempt to account for the effects of the potential on the motion of the 88 scattered electrons have revealed some qualitative changes in the predicted MDs but only in the high momentum region. For p < 1 . 5 a o _ 1 the PWIA is generally considered to be valid. Perhaps the best confirmation of the basic PWIA (e,2e) reaction theory is the fact that the measured electron M D of the ground state hydrogen atom [LW81] is in excellent agreement with the square of the exact solution of the Schro dinger equation. As has been shown in Chapter 2, simple analytic results are obtained for the (e,2e) differential cross section if initial state and final state configuration interactions are neglected. Within the T H F A the electronic overlap function reduces to the momentum wavefunction of the characteristic orbital. In general, it has been assumed that ground state correlations are negligible for closed shell systems. In the case of H 2 0 it has been stated [NT82] that no significant ground state correlation effects are involved. Final state correlation effects thus seem likely to be a more important factor. The effects of including initial state CI and final state CI on the MDs of water are investigated in Chapter 5. A n attempt to include such effects has been made in the generalized overlap amplitude (GOA) method applied to water by Williams et al. [WM77a]. The G O A method accounts for some correlation and relaxation effects but is limited by the fact that the approximations are developed from the H F ground state orbital set. Interestingly, this study [WM77a] showed that a slightly better agreement between calculation and experiment for the MDs of the 3a , and lb 2 orbitals was obtained with the GOA method as against the simple use of the T H F A using the Dunning wavefunction [D71]. However the discrepancy in the l b , orbital still persists. 89 The discrepancy between experiment and theory in the case of the outermost orbitals may therefore indicate the need for including correlation effects. The effect of electron correlation can be quite significant. For example, the best Hartree-Fock estimate of the dipole moment of water is 7.6% higher [RS75] than the experimental value and only with post-HF methods such as 4th-order Many-Body Perturbation Theory (1.2%error) [DK83] and Multireference Single and Double Excitation CI (4.0%error) [DF84] can more reasonable agreement be obtained. Another factor that might be considered to affect the validity of the T H F A is the question of geometry change accompanying the formation of the final ionic state. SCF calculations [M71, SJ75] and experiments [L76] have shown that the equilibrium geometries for the H 2 0 + ion are different for each of the electronic states. The calculations show that the H O H angle increases from 104.5° to 112.5° and 180° for the 2 B , O b , " 1 ) and 2 A , ( 3 a 1 ) states, respectively. On the other hand, the H O H angle decreases to 58° for the 2 B 2 ( 1 b 2 ' 1 ) state of the H 2 O + ion. As a consequence the overlap amplitude does not reduce to the characteristic orbital since the populated molecular orbitals of the final ionic state are not necessarily orthogonal to the corresponding orbitals in the neutral as is assumed in the T H F A model. However geometry effects are not expected to be significant because of the very short collision times (due to high impact energy) relative to the nuclear motion. For meaningful comparisons, it should also be pointed out that the effect of vibrational motion on the theoretical M D should be considered. Vibrational effects 90 on the Compton profile of H 2 have been estimated to be small (1.0%) [UB72] and only at low momentum. Quite similar results have been obtained by Palalikit and Shavitt [PS85] on zero-point vibrational corrections to the Compton profile of water. However, the greater sensitivity of experimental momentum profiles to the details of the wavefunction compared to the total momentum density measured by Compton scattering [SW75] are well known [LB83a]. In addition, one would expect considerable vibrational effects on the M D of molecules of light atoms and weak force constants and also if the vibrational motion is anharmonic [T72]. Theoretical calculations [KE74] on the potential energy surface of water have revealed that Hartree-Fock force constants are generally greater than the experimental force constants, sometimes by up to 20% indicating a "flatter" true potential energy surface. These factors plus the fact that vibrational motion of water involves various anharmonic modes suggest a need to seriously consider vibrational effects in any detailed theoretical prediction of momentum distributions, t Lastly, it is interesting to note that better agreement (see Fig. 4.3) is found with the A P C wavefunction than with the N M wavefunction though the latter has a superior SCF energy. The theoretical M D does not seem to converge in any simple manner with improvement in energy. Similar observations have been made by Kern and Karplus [KK72] in their survey of certain calculated properties of the water molecule. This again emphasizes the fact that the t Preliminary calculations for H 2 0 by Leung and Langhoff [LL87] using a well known potential energy surface [H66] indicate that vibrational effects (for symmetric vibration) on MDs are very slight. In particular, the effects on the MDs of the non-bonding l b , (out-of-plane) and 3a , (bonding) orbitals are found to be too small to explain the observed discrepancies. SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY ° 0.0 0.5 1.0 1.5 2.0 2.5 Momentum (a.u.) Fig. 4.3 Spherically averaged momentum distribution (upper left) and distribution difference (lower left) for the l b , orbital of H 2 O. The theoretical MDs are height normalized to the maximum of the X M P . The distribution difference is evaluated by taking the difference between the X M P and the theoretical M D calculated from the A P C wavefunction. The upper level of the hatched area indicates zero distribution difference in the case where area normalization is used. Two dimensional density contour maps and three dimensional density surface plots in momentum space (center) and position space (right) are evaluated using the SB wavefunction. SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY H 20 3a | o o * • * *— .i . .a. , 0.0 0.5 1.0 1.5 2.0 2.5 Momentum (a.u.) -4.0 -2.0 0.0 2.0 4.0 0.5 1.0 -4.0 -2.0 0.0 2.0 4.0 0.S 10 0.0 0.5 1.0 1.5 2.0 2.5 Momentum (a.u.) Fig. 4.4 Spherically averaged momentum distribution (upper left) and distribution difference (lower left) for the 3a, orbital of H 2 O. The theoretical MDs are height normalized to the maximum of the XMP. The distribution difference is evaluated by taking the difference between the XMP and the theoretical MD calculated from the APC wavefunction. The upper level of the hatched area indicates zero distribution difference in the case where area normalization is used. Two dimensional density contour maps and three dimensional density surface plots in momentum space (center) and position space (right) are evaluated using the SB wavefunction. SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY 0.0 0.5 1.0 1.5 2.0 '2.5 Momentum (a.u.) -4.0 - 2 . 0 0.0 2.0 4.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0 2.5 Momentum (a.u.) POSITION DENSITY H 2 0 1 b 2 o o (0.1.0) i—t-—i i i T 1 -4.0 - 2 . 0 0.0 2.0 4.0 0.5 1.0 Fig. 4.5 Spherically averaged momentum distribution (upper left) and distribution difference (lower left) for the l b 2 orbital of H 2 O. The theoretical MDs are height normalized to the maximum of the X M P . The distribution difference is evaluated by taking the difference between the X M P and the theoretical M D calculated from the A P C wavefunction. The lower level of the hatched area indicates zero distribution difference in the case where area normalization is used. Two dimensional density contour maps and three dimensional density surface plots in momentum space (center) and position space (right) are evaluated using the SB wavefunction. SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION \ x i (32. 2eV) $ x3 (35.6eV) APC NM SB T \ a 0.0 0.5 1.0 d.5 " ^ .o ' {2.5 Momentum (a.u.) • 1 O c C V Q T Expt-Theor y 0.0 0.5 1.0 1.5 2.0 Momentum (a.u.) 2.5 MOMENTUM DENSITY POSITION DENSITY 1 H 2 0 2 a , 0* (0,1.0) —1—1—1—*— -4.0 -2 .0 0.0 2.0 4.0 0.5 1.0 -40 -2.0 00 2.0 4.0 0.5 10 Fig. 4.6 Spherically averaged momentum distribution (upper left) of the 2a, orbital of H 2 0 measured at 32.2eV (solid circle) and 35.6eV (open square). Distribution difference plot (lower left) is evaluated using the X M P at 32.2eV and the theoretical M D calculated from the A P C wavefunction. The lower level of the hatched area indicates zero distribution difference in the case where area normalization is used. Two dimensional density contour maps and three dimensional density surface plots in momentum space (center) and position space (right) are evaluated using the SB wavefunction. 95 variational principle is of itself an insufficient condition for providing a completely adequate wavefunction. This is particularly noteworthy in the present work because the discrepancy between experiment and theory is found in the outermost orbitals which contribute least to the total energy. In summary, there is a need for a comparison of binary (e,2e) results with theoretical MDs calculated from a more sophisticated and comprehensive set of wavefunctions ranging from Hartree-Fock to post Hartree-Fock quality. Furthermore, geometry and vibrational effects alluded to earlier should also be considered. This will enable a meaningful evaluation of the limitations of each class of wavefunctions or the Hartree-Fock model itself in predicting experimental momentum distributions. These questions are addressed in the following chapter. 4.5. ORBITAL DENSITY MAPS AND SURFACES The analysis of orbital density maps (Figs. 4.3-4.6) in complementary position and momentum space is facilitated by the now well-known momentum space concepts [ET77]. These concepts can be summarized into: (a) symmetry preservation with the addition of inversion symmetry in momentum space; (b) inverse spatial reversal; (c) molecular density directional reversal; (d) molecular density oscillations which are the manifestation of nuclear geometry in momentum space. The r-space and p-space density maps and surfaces of the l b , orbital (Fig. 4.3) 96 show the dominant oxygen 2 p x character of this orbital (x-axis is in the out-of-plane (100) direction in r-space) therefore it is normally referred to as a non-bonding molecular orbital. The well defined nodal plane in the r-space map is retained in the corresponding p-space map (symmetry preservation property). Use of near-HF wavefunctions which have polarization functions would introduce perturbations in this orbital induced by the hydrogen atoms. Such a theoretical study of the water molecule by Dunning et al. [DP72] using a (541/31) STO basis set shows r-space density maps indicating the slight perturbative effects . The difference between the maps generated from the SB [42/2] GTO basis set (essentially DZ) used in the present study and those maps calculated using the (541/31) STO basis however is very minimal. Whereas symmetry preservation and the addition of inversion symmetry in momentum space are quite apparent in all orbitals (Figs. 4.3-4.6), the concepts of inverse spatial reversal and molecular density directional reversal cannot be readily applied in the case of the 3a, and l b 2 orbitals . The non-linear geometry of H 2 0 makes assignment of general Fourier Transform properties less straightforward compared to the situation for diatomics such as H 2 [LB83a] and linear polyatomics such as C 0 2 , C S 2 and OCS [LB85b] which are of higher symmetry. In the simplest approximation [ES55] bonding in H 2 0 is attributed to the 3a, and l b 2 molecular orbitals. The bonding characteristics of the 3a , orbital are manifested in r-space in terms of electron density derealization across the hydrogen atoms (Fig. 4.4). In p-space bonding is manifested by the expansion of the density in the (001) direction. The r-space density map and surface of the l b 2 orbital (Fig. 4.5) illustrates the familiar simple intuitive view of covalent bonding which is an accumulation of electron density along the bond direction. In contrast, bonding is 97 not obvious in the corresponding p-space density map and surface. In a more exact treatment, there is also some bonding attributed to the 2a , orbital as is clearly seen in the r-space density map in Fig. 4.6. Nevertheless it can be seen that the 2a, orbital is predominantly an oxygen 2s orbital. The contracted density in r-space (as seen in the line projection plots) translates to an expanded density in p-space in agreement with the inverse spatial reversal property. 4.6. WIDE RANGE MOMENTUM DENSITY MAPS It is of interest to investigate the nature of momentum density maps at larger coordinate values than in Figs. 4.3-4.6. This is particularly important since effects due to molecular geometry will be manifested at large p as approach is made in the direction of the virtual p-space boundary [LB83a]. Wide range density maps up to 16 a.u. in momentum space are shown for the four valence orbitals in Fig. 4.7. For comparison their position space counterparts are also shown alongside. The nuclear geometry information of course appears directly in the r-space density maps at small r. However in the p-space maps it is manifested in a quite different form, namely as molecular density oscillations. These appear at large momentum as modulations or oscillations in the density [ET77] as shown in Fig. 4.7 for the 3 a , , l b 2 and 2a, orbitals. In contrast, the l b , orbital, which is essentially non-bonding and thus contains little or no nuclear information, does not show such modulations (at least out to 16 a.u.). Detailed computational studies of these modulations have been made for F 2 by Rozendaal and Baerends [RB84]. However no such study has previously been MOMENTUM DENSITY MOMENTUM DENSITY POSITION DENSITY POSITION DENSITY -12 B -f.4 0.0 (.4 OJt -». -J. -12.8 -6 4 0.0 6.4 08 -6. -J. -4.0 -2.0 0.0 2.0 4.0 OS 10 -4.0 -JO 00 1.0 4.0 OS 10 Fig. 4.7 Wide range momentum density contour maps (left hand side) for the valence orbitals of H 2 O calculated using the SB wavefunction. The 21 contour values are in seven decades ie. 0.00002, 0.00005, 0.00008, etc. up to 20, 50 and 80% of the maximum density. The projection plots on both primary axes are on a logarithmic scale. A l l dimensions are in atomic units. CD 00 99 done on systems with lower symmetry such as water. The observed modulations are not as pronounced as those calculated for other diatomic molecules such as H 2 [LB83a] and polyatomic systems such as C 0 2 [LB85c], C S 2 [LB85a], COS [LB85a] and C F 4 [LB84]. The modest modulations for H 2 0 may be a consequence of the non-linear geometry as well as the low atomic number of the hydrogen atoms. The widely differing electron densities on the nuclear centers do not allow the effective interatomic coupling needed for strong bond oscillations. It has been shown that these modulations or oscillations occur with periodicities in multiples of 7T/R where R is the internuclear separation. However in the case of H 2 0 the periodicities of the modulations do not seem to correspond in any straightforward manner to integral multiples of n7R, where R is the O-H bond distance (1.80a.u.), in contrast to the situation observed for diatomics and linear polyatomic systems [LB85a, LB85b, LB85c]. C H A P T E R 5. W A T E R : P A R T I I 5.1. OVERVIEW In the previous chapter the following considerations were raised as being possible sources of the observed discrepancies between the theoretical MDs and the X M P s of H 2 0 . (1) Inadequacies in the plane wave impulse approximation (PWIA) and the need for distorted wave treatments (DWIA); (2) Inadequacies in the target Hartree-Fock approximation (THFA) and therefore the need to consider sufficiently complete target molecule-final ion overlap treatments(ie. adequate treatment of relaxation and correlation); (3) Neglect of nuclear motion, i.e.vibrational effects; (4) Insufficient flexibility of the basis set and resultant deficiencies in the theoretical wavefunction. These matters concern both the adequacy of the theory of E M S cross sections and the accuracy of the wavefunctions. Explanations 1-3 emphasize possible inaccuracies in the assumed proportionality between the experimental momentum profiles and the orbital momentum distribution evaluated at the equilibrium geometry of the neutral molecule. Items 2 and 4 are of particular concern from the quantum chemical theoretical standpoint and these are investigated in detail in the present work. A further area of possible concern from an experimental standpoint is the accuracy with which the momentum resolution is known. Such effects however will be small at higher momentum resolution (as used in the present experimental work) compared to the observed discrepancies between the 100 measured X M P s and the calculated MDs in the case of H 2 0 . 101 The validity of the PWIA for the present experimental conditions (impact energy E 0 = 1200eV, E 1 = E 2 = 6 0 0 e V , 9 = 45° , p < 2 a 0 " 1 ) has been discussed in the previous chapter and has been clearly demonstrated by the complete agreement between E M S experiment and theory for atomic hydrogen [LW81] where the T H F A is not involved and the wavefunction is exact. There is thus good reason to use the PWIA for other atomic and molecular targets under the equivalent conditions. The combined use of the PWIA for the kinematic factor together with use of the T H F A for the electronic structure factor in the description and interpretation of the binary (e,2e) reaction as studied by E M S is also supported by the excellent results obtained for the valence orbitals of noble gas [LB83, CM86] and metal atoms [FW82] and also for small molecules [LB83a, WM77]. In these cases where high quality Hartree-Fock limit wavefunctions were used it is clear that the combined use of these two approximations is valid since the shapes of the measured X M P s were extremely closely reproduced by the calculated orbital momentum distributions at least for electron momenta less than = 2 a 0 " 1 for targets lighter than A r (Z=18).t In the case of Ne (Z=10), which is isoelectronic with H 2 0 , the PWIA was satisfactory [LB83] in describing the shapes of the momentum distributions out to p — 2a0'^. In the region of higher electron momentum (i.e. larger than that used in the present work for H 2 0 ) considerable distortion of the electron waves can t Experiments [LB83, CM86] indicated that distortion becomes more important at lower momentum as the orbital energy and Z increase. For example even for targets as large as Xe (Z = 54) use of the PWIA was found [LB83, CM86] to be adequate for the valence orbitals out to l . 5 a 0 " 1 at 1200eV impact energy. 102 occur and the distorted wave impulse approximation (DWIA) has been found to be necessary for an adequate interpretation of E M S measurements for example in A r [MW85], Xe [CM86], Ne and K r [LD86]. In the region of p below = 2 a 0 " 1 the shape of the X M P for the noble gases was equally well reproduced by either the DWIA + T H F A or PWIA + T H F A . Thus far distorted wave treatments for molecules have not been reported due to the complex multicenter nature of the problem. In some molecules it has been found that T H F A calculations using even Hartree-Fock limit wavefunctions fail to reproduce the experimental momentum profiles for some orbitals. This is most likely due to the inherent neglect of electron correlation and relaxation in the Hartree-Fock description. In such cases calculation of the full target ion-neutral overlap using CI wavefunctions can be used to include correlation and relaxation effects. A n earlier attempt (Generalized Overlap Amplitude (GOA) method) by Williams et al. [WM77a] at evaluation of the target molecule-final ion state overlap for H 2 0 [WM77a], H F and HC1 [BH80] using limited basis set wavefunctions indicated small changes (towards lower momentum) from predictions based on the T H F A . However more exhaustive treatments of overlap, using improved wavefunctions are needed to investigate more completely the effects of correlation and relaxation. From the standpoint of theoretical quantum chemistry it is important to assess the need for taking into account ground and/or final state electron correlation effects since these have certainly been found to be crucial in reconciling theory and E M S experiments in the case of N O [BC82] as well as for H 2 [WM77] and He [CM84] in those cases where the product ions of H 2 and He are left in excited states. 103 As has been shown earlier by Weigold and co-workers [DD77, CM86] and discussed by McCarthy [M85], additional quantitative assessment of the wavefunctions and the reaction model is possible if the experimental momentum profiles for the valence orbitals are obtained on a common (relative) intensity scale t with the full Franck-Condon width of each final ion electronic state being taken into consideration in the normalization procedure. With a single point normalization of experiment and theory at only one value of the momentum on a given calculated orbital M D or OVD a stringent quantitative test of theory can be made at all other experimental data points and calculations for all orbitals. This procedure provides very much more specific information than the individual height or area normalization of theory and experiment for each separate orbital which has frequently been used in earlier E M S studies (as for example in the previous chapter). Flexibility of the basis set and the accuracy of the wavefunction are important concerns from the theoretical standpoint. The usual variational treatment involving energy optimization of SCF wavefunctions, even when energies near the Hartree-Fock energy are given, is a necessary but usually insufficient condition alone for guaranteeing that all other calculated properties are also close to the Hartree-Fock properties, as is well known [BC82, GC78]. The addition of more diffuse functions is usually necessary to permit accurate calculation of properties such as the dipole and quadrupole moments (and therefore also likely the MDs) which depend critically on the accurate modeling of the long range charge t True absolute E M S measurements are very difficult to perform with high accuracy. Such measurements have for example been reported (±20%) for He and H 2 [VK81]. 104 distribution. The total energy is, of course, not very sensitive to such details so that basis sets designed solely to yield a reasonable value of the total energy without unnecessary computational expense may well give poor results for other properties. Further considerations in addition to the total energy are therefore necessary to ensure sufficient wavefunction accuracy for general applications where theoretical quantum chemistry is required to give accurate predictions of experimental observables. Such considerations are also of importance for a number of properties including those dependent on the long range distribution of charge density. After convergence has been obtained at the Hartree-Fock level for both total energy and the predicted M D then the other possible sources of any remaining discrepancy between the calculated orbital MDs and the X M P s measured by E M S can be investigated. As mentioned above the need to go beyond the T H F A (i.e. to incorporate correlation and relaxation effects) can be investigated with the use of suitable CI wavefunctions in order to evaluate the ion-neutral overlap amplitude with sufficient accuracy. With the above considerations in mind new higher statistical precision E M S measurements of the momentum profile for the l b , orbital of H 2 0 as well as new momentum profile measurements for the three outer valence orbitals of D 2 0 were made. These new measurements taken together with the earlier reported experimental data [BL85] (see chapter 4) for the four valence orbitals of H 2 0 have been placed on the same (relative) intensity scale and are now compared on a quantitative basis with T H F A (MDs) and full overlap calculations (OVDs) 105 using very accurate Hartree-Fock limit and CI wavefunctions, respectively. These include the already published 84-GTO wavefunction of Davidson and Feller [DF84] as well as the 39-STO wavefunction of Rosenberg and Shavitt [RS75]. These best existing literature wavefunctions [DF84, RS75] have been improved upon in the present work by use of previously unreported 99-GTO and 109-GTO wavefunctions for the T H F A calculations. These new S C F wavefunctions for water, which give essentially converged resultst for total energy, dipole moment and M D are considered to be the most accurate to date. Comparisons of theoretical MDs calculated from essentially converged Hartree-Fock quality wavefunctions are important in order to determine whether basis set independence of the calculated MDs has been achieved as well as energy minimisation. Comparative studies of basis sets for the water molecule [DF84, RS75] have shown that certain position space one-electron properties such as the dipole and quadrupole moments are much more sensitive to the basis set than the total energy. Most theoretical efforts to date have been directed towards assessing the effect of "basis set truncation" on position space properties. There has in general been much less attention given to the conjugate momentum space properties which emphasize a different region of phase space. Tanner and Epstein [TE74] have calculated the Compton profile and momentum expectation values for H 2 0 using t These results are considered to be effectively converged since the total energy is within 0.5 millihartree of the estimated Hartree-Fock limit [RS75] and because the dipole moment, quadrupole moment and M D do not change significantly as even more basis functions are added. Other basis sets containing up to 140 GTOs were actually tested with no significant change in either the M D or O V D [FB87]. 106 various SCF wavefunctions from minimal basis to near Hartree-Fock quality. It was observed that distinctions between Compton profiles calculated using wavefunctions of DZ quality or better (e.g. near Hartree-Fock) were extremely difficult. However this is not so surprising since the Compton profile refers to the total electron distribution rather than to an individual orbital. Leung and Brion [LB83a, LB84a] have investigated basis set effects on the momentum distribution of H 2 ( i C g ) and found these to be small. However it should be noted that the "s-type" shape of the 1 o"g orbital of H 2 affords much less stringent assessment of the calculated MDs than do orbitals with MDs of "p-type" shape, t The maximum at non-zero momentum afforded by the "p-type" molecular orbitals in water therefore provides a sensitive probe of the basis set problem and the limitations of the Hartree-Fock model in momentum space. The most accurate H 2 0 basis sets (ie. the 99-GTO and 109-GTO wavefunctions, which are converged for both energy and T H F A MDs) have been used in the present work as the foundations for more accurate treatments beyond the Hartree-Fock (independent particle) picture. To elucidate the effect of correlation and relaxation, a configuration interaction (CI) treatment (recovering a very large fraction of the correlation energy) of the initial and final states is used in a calculation of the ion-neutral overlap amplitude in the present work. The CI wavefunctions are generated using the highly extended basis sets and the calculated ion-neutral overlap distributions are then compared with the t A much greater specificity was observed by Leung and Brion [LB84a] for calculations using different basis sets compared to the measured cross section when the difference density ( p H - 2 p „ ) was used instead of the density. 107 experimental momentum profiles. The role of vibrational motion on momentum distributions has yet to be fully investigated but preliminary calculations for the symmetric stretch in H 2 0 [LL87] and for N H 3 [BM87] have indicated that the effects are very small for all valence orbitals and essentially negligible for the non-bonding l b , and le orbitals, respectively. Isotopic effects have been investigated and found to be negligible in E M S experiments comparing the X M P s of H 2 and D 2 [DM75]. The new experimental data for deuterated water ( D 2 0 ) therefore provides opportunity to investigate any effects of nuclear vibration on the momentum profiles. The different vibrational frequencies [BT68] of D 2 0 and H 2 0 may reveal within the experimental sensitivity any vibrationally induced effects. D 2 0 due to its higher mass (thus lower vibrational frequency) would be expected to exhibit less of the vibrational effects. Consider for example the molecular root mean square vibrational amplitudes. The < ( A R Q H ) 2 > 1 ^ 2 for H 2 0 is 20% greater than D 2 0 and the < ( A R H H ) 2 > 1 ^ 2 for H 2 0 is 20% greater than D 2 0 [C68]. In summary then the present study seeks to address the various possible explanations for the observed discrepancies between experiment and theory in H 2 0 . In turn this should provide insight into the nature of the outer valence orbital momentum distributions (ie. X M P s and theoretical MDs) of other second row hydrides (e.g. N H 3 and H F ) which have been found to exhibit similar behavior. 108 5.2. EXTENDED BASIS SETS FOR WATER Several new, highly extended basis sets for H 2 0 have been developed in the course of the present work in an interactive collaboration with Prof. E.R. Davidson (Indiana University). First, a 99 basis function set (referred to as 99-GTO) was constructed from an even-tempered (19s,10p,3d,lf/10s,3p,2d) primitive set which was contracted to [10s,8p,3d,lf/6s,3p,ld]. The s-components of the cartesian d-functions were deleted, but the p-components of the cartesian f-functions were retained. The (s,p) portion of the oxygen basis was created by extending the (18s,9p) atom optimized exponents with one additional set of diffuse functions of each type. The hydrogen s-exponents are optimal for the isolated atom. In an even-tempered basis [FB87] the i'th exponent is given as af}1 beginning with i = l . The a and /3 values for the oxygen (18s,9p) set are a g = 0.07029, j3 g = 2.29663 and a p = 0.04956, /3 p=2.57217. The hydrogen values are a s = 0.02891, /3 g = 2.58878. Exponents for the d- and f-type polarization functions on oxygen were partially optimized at the SD-CI level. The d exponents are 3.43, 1.18 and 0.33. The f exponent is 1.20. For hydrogen the p- and d-type polarization functions were taken from the work of Davidson and Feller [DF84]. The 109 basis function set (referred to as 109-GTO), a (23s,12p,3d,lf/10s,3p,2d) to [14s,10p,3d,lf/6s,3p,ld] contraction, was generated by extending the (19s,lOp) set through the addition of more diffuse functions, while still retaining the even-tempered restriction. The most diffuse s function in this set possessed an exponent of 0.0649 compared to 0.1375 in the original set. Table 5.1 Properties of Theoretical SCF and CI Wavefunctions Wavefunction Basts Set RHF RHF RHF RHF RHF RHF Hartree-Fock limit CI CI CI Experimental SB [42/2] 14 CGTO APC (331/21) 27 STO NM [531/21] 36 CGTO NM (1062/42) 58 CGTO RS 39-STO DF 84-GTO 84 CGTO 99-GTO 99 CGTO 109-GTO 109 CGTO 140-GTO 140 CGTO (84-GT0)CI (109-GTO)CI (140-GT01CI Energy(a.u.) -76.0035 -76.00468 -76.044 -76.059 -76.0642 -76.06661 -76.06689 Dipole Moment(0) a -76.0673 -76.0675 -76.3210 -76.3761 -76.3963 -76.437610.0004a 2 .092 1 .995 2.021 1,980±0.01 1 .929 1 .895 1 .870 1.8546+0.0006* * Ref. [RS75]. b Ref. [D87]. c Estimated non-vIbrating dipole moment Is 1.848D. 2 -16 o < r >(10 cm"*) 5.462 5.307 5.371 5.396 5.417 Reference 5.430 5.432+0.OOI 5.490 5.500 5.507 5.1±07 3 [SB72] [AP68] [NM68] [NM68] [RS75] [DF84] this work this work [FB87] this work this work [FB87] O CO 110 While these basis sets, incidentally, give the lowest SCF energy yet reported for water at the experimental geometry, they are also designed to saturate the diffuse basis function limit and to give an improved representation of the (r-space) tail of the orbitals. The negligible difference between the 99-GTO and 109-GTO results for a wide variety of calculated properties (Table 5.1) indicates that this goal has been essentially accomplished. The neutral molecule equilibrium geometry, RQJJ = 0.9572A, C9JJQJJ= 104.52° was used for generating all wavefunctions. CI calculations were then done by the the Indiana group for the 84-GTO and 109-GTO sets using configurations built from the neutral molecule SCF MOs. Based on these small CI calculations, the important configurations were selected and used in a multireference SD-CI with perturbation energy selection of the configurations. For example, for the 109-GTO basis set, 15 configurations in the reference set and 11011 configurations in the final CI for the neutral molecule X'A, wavefunction were used. Similarly 37 configurations in the reference set and 17316 configurations in the final CI for the ion X 2 B , wavefunction were used. These are still small CI calculations which recover only about 83% of the estimated correlation energy in H 2 0 . These CI wavefunctions were then used to evaluate the molecule-ion "overlap" < X 2 B , | X 1 A 1 > , <X 2A,|X 1A 1>, and < X 2 B 2 | X 1 A 1 > which have the same form as an M O expanded in this basis. 5.3. EXPERIMENTAL DETAILS: NORMALIZATION OF DATA Triply distilled H 2 0 and D 2 0 (MSDISOTOPES, >99.8% purity), each degassed by repeated freeze-thaw cycles, were used for these experiments. The vapour was admitted via a Granville Phillips leak valve to give an ambient pressure of 5 x 10 torr. Lengthy equilibration in the case of D 2 0 (not less than 24 hours) prior to data acquisition allowed near complete hydrogen-deuterium exchange on the chamber walls. Standard calibration runs for the 3p orbital of argon were performed to establish both the energy and momentum resolutions which were = 1.6eV fwhm and =0.1 S a o ' 1 , respectively. Under these conditions it was found, in accordance with earlier work [LB83], that the A r 3p momentum distribution was quite well described within the PWIA and T H F A using the Hartree-Fock limit wavefunction of Clementi and Roetti [CR74]. The measured momentum profiles (XMPs) for the four valence orbitals of H 2 0 and also the three outer valence orbitals of D 2 0 were put on the same (relative) absolute intensity scale (i.e. all relative normalizations preserved) by normalization on the peak areas in angular selected binding energy spectra (see chapter 4) [BL85, BW87]. This normalization involved integration over the full Franck-Condon width for production of each of the (lb,)"' ' ' , (3a,)"''', ( l b 2 ) a n d (2a,)" * electronic states of H 2 O + yielding a relative ratio (at 0 = 8°) of 1.0:1.1:1.0:2.4 . In this procedure it was necessary to take into account the full distribution of satellite structure (25-45eV) of the (2a ,)*^ state (see Fig. 4.1) [BL85]. With this procedure a very stringent quantitative comparison (to better than 5%) with calculations is possible since experiment and theory are normalized to each other at only a single point on one of the four measured momentum 112 profiles. A l l other experimental and all other calculated points are therefore open to quantitative scrutiny. The new measurements of the momentum profile for the l b , orbital of H 2 0 have been added to the results obtained for this orbital in the earlier reported study [BL85] (see chapter 4). This results in a considerable improvement in the quality of the data for this particular orbital which showed [BL85] the greatest discrepancy between theory and experiment with the near Hartree-Fock level wavefunctions used in the calculations in an earlier study [BL85]. This data for the l b , orbital of H 2 0 and the existing data [BL85] for the 3a 1 ; l b 2 and 2a , X M P s for H 2 0 (open squares, Figs. 5.1-5.4) have each been placed on the same relative intensity scales using the procedure outined above. Comparisons of the X M P s with calculated MDs and OVDs were made in a quantitative manner by a single point normalization to the M D calculated for the l b 2 orbital using the 109-GTO wavefunction which has the most accurate calculated properties for water at the Hartree-Fock level (see Table 5.1). A l l other experimental and calculated data points (including the 109-G(CI) OVD results assuming unit pole strength, see Sec. 5.6) for all four orbitals have their absolute values relative to this single point normalization. The measured X M P s for H 2 0 and D 2 0 are shown on the same intensity scale in Figs. 5.1-5.4 in comparison with a wide range of already reported [BL85] as well as new calculations. Some of the presently reported calculations involve the new, more sophisticated Hartree Fock wavefunctions which have only recently been published [BB87] (see also Sec. 5.2 above). These wavefunctions are used in 113 T 1 1 1 i 1 1 1 1 1 r Momentum (a.u.) Fig. 5.1 Detailed comparison of the experimental momentum profiles (XMPs) of the l b , orbital of D 2 O (solid circles)and H 2 O (open squares) with several spherically averaged momentum distributions calculated using the T H F A . Distributions calculated from the (1)SB [42/2], (2)NM [531/42], (3)NM (1062/42), (4)DF 84-GTO, (5)99-GTO, (6)109-GTO and the (6c)109-G(CI) full overlap calculation are placed on a common intensity scale by using a single point normalization of the 109-GTO calculation on the l b 2 X M P (see text for details). The sitting binding energy for the measurements is 12.2eV. 114 T 1 1 1 1 1 1 1 1 1 r Momentum (a.u.) Fig. 5.2 Detailed comparison of the experimental momentum profiles (XMPs) of the 3a, orbital of D 2 O (solid circles)and H 2 O (open squares) with several spherically averaged momentum distributions calculated using the T H F A . Distributions calculated from the (1)SB [42/2], (2)NM [531/42], (3)NM (1062/42), (4)DF 84-GTO, (5)99-GTO, (6) 109-GTO and the (6c)109-G(CI) full overlap calculation are placed on a common intensity scale by using a single point normalization of the 109-GTO calculation on the l b 2 X M P (see text for details). The sitting binding energy for the measurements is 15.0eV. 115 T 1 1 1 1 1 1 1 — ~ i 1 1 1 1 r Momentum (a.u.) Fig. 5.3 Detailed comparison of the experimental momentum profiles (XMPs) of the l b 2 orbital of D 2 O (solid circles)and H 2 0 (open squares) with several spherically averaged momentum distributions calculated using the T H F A . Distributions calculated from the (1)SB [42/2], (2)NM [531/42], (3)NM (1062/42), (4)DF 84-GTO, (5)99-GTO, (6)109-GTO and the (6c)109-G(CI) full overlap calculation are placed on a common intensity scale by using a single point normalization of the 109-GTO calculation on the l b 2 X M P (see text for details). The sitting binding energy for the measurements is 18.6eV. 116 0.9 1.2 1.5 Momentum (a.u.) Fig. 5.4 Detailed comparison of the experimental momentum profiles (XMPs) of the 2a, orbital of H 2 O (open squares) with several spherically averaged momentum distributions calculated using the T H F A . Distributions calculated from the (1)SB [42/2], (2)NM [531/42], (3)NM (1062/42), (4)DF 84-GTO, (6) 109-GTO and (6c)109-G(CI) are placed on a common intensity scale by using a single point normalization of the 109-GTO calculation on the l b 2 X M P (see text for details). The sitting binding energy for the measurements is 32.2eV. 117 the target Hartree-Fock approximation (THFA) to investigate basis set dependence and the convergence of calculated properties in addition to the normal variational treatment minimising the total energy. Further calculations model the effects of correlation (with up to 88% of the ground state correlation energy recovered) by evaluating the full overlap amplitude (see Eqn. 2.11) using CI wavefunctions for both the initial state target neutral molecule and the final ion state. The various results at the T H F A level are discussed in the following sections with reference to Figs. 5.1-5.4. 5.4. VIBRATIONAL EFFECTS It can be seen from Figs. 5.1-5.4 that, within statistics, there is no obvious difference between the X M P s of H 2 0 and D 2 0 . There appear to be some possible slight differences in the high momentum region ( p > l . 5 a 0 " 1 ) of the l b , momentum profile where the results for D 2 0 are slightly lower than those for H 2 0 . However these differences are at best marginal considering the error bars involved. In earlier E M S work [DM75] no differences were observed between the X M P s of H 2 and D 2 which are the only other isotopically substituted molecules to be studied thus far by E M S . The present results for H 2 0 and D 2 0 indicate that vibrational effects have no significant effect on orbital X M P s at least in the case of water at the current level of experimental accuracy. The essentially identical X M P s observed for H 2 0 and D 2 0 now extend the earlier conclusions of Dey et al. [DM75] concerning the isotopic diatomic molecules, H 2 and D 2 , to the more complex polyatomic water system which has both symmetric and antisymmetric modes of vibration. It had been thought [DM75] that discrepancies between measured X M P s and the calculated MDs might result either from the 118 failure of the Born-Oppenheimer approximation or from failure to consider accurately the vibrational integral in Eqn. 2.11. Several theoretical studies have investigated vibrational effects on calculated molecular properties. For example a study of vibrational corrections to the Compton profile of H 2 0 [PS85] suggested that the vibrational effect on MDs might be expected to be most pronounced, though small (less than 1%), in the low momentum region. The Compton profile was calculated [PS85] using the 39-STO wavefunction [RS75] for H 2 0 (see later discussion) and vibrationally averaged over the SDQ (singly, doubly and quadruply excited CI) potential surface of Rosenberg et al. [RE76]. The present E M S results are also consistent with a series of studies [KE74, EK71] on vibrational corrections to the one-electron properties of H 2 0 . It was noted [KE74, EK71] that these corrections are small and in the case of the dipole moment, the vibrational correction ranges from 0.3% in the ground state to 5% in the (0,3,0) excited vibrational state. A recent study by Breitenstein et al. [BM86] of vibrational corrections to electron impact differential cross sections also showed that such effects are small. Intuitively it would seem that even if vibrational effects on MDs are significant, the effect would be least on the (non-bonding) l b , orbital since it is effectively a lone pair mainly lying perpendicular to the molecular plane and thus it would be largely unaffected by the nuclear motion. The absence of momentum profile differences between H 2 0 and D 2 0 even for the bonding 3a, and l b 2 orbitals suggests that the presently observed discrepancies [BL85] between calculations and experiments, particularly for the l b , orbital but also to a lesser extent for the 119 3a, orbital, are not due predominantly to vibrational effects. 5.5. BASIS SET EFFECTS The previous study of the MDs of H 2 0 in chapter 4 as well as several earlier works, clearly indicated significant discrepancies between calculation using existing Hartree Fock wavefunctions from the literature and experimental MDs for the two outermost orbitals ( l b , and 3a,) . The greatest differences were observed for the (least tightly bound) essentially non-bonding l b , orbital. In contrast it was found that agreement for the third and fourth orbitals ( l b 2 and 2a,) was very good at least for the relative shapes. In the earlier work [HH77] the wavefunctions used included the minimum basis set of gaussian functions of Snyder and Basch (SB [42/2]) [SB72] and the near Hartree-Fock (i.e. extended contracted gaussian) wavefunction of Neumann and Moskowitz (NM [531/21]) [NM68]. The details as well as properties of these and all other wavefunctions used in the present work are shown in Table 5.1. The Slater type (STO) wavefunction of Aung, Pitzer and Chan (APC (331/21)) [AP68] which gives a total energy of -76.00468 a.u. and a dipole moment of 2.035D was also compared in the earlier work [BL85] and gave the best agreement of the three wavefunctions (SB, N M and APC) with the experimental momentum profiles. These results of the earlier study [BL85] showed that for H 2 0 the lowest energy (i.e. best from a variational standpoint) wavefunction (contracted N M [531/21]) does not give the best calculated M D . In fact the variationally inferior A P C wavefunction gave the better overall fit (but one that was still rather inadequate for the l b , and 3a, orbitals) to the measured X M P s . It is 120 interesting in this regard to note that the A P C wavefunction also gave a dipole moment (2.035D) slightly nearer to the Hartree-Fock limit (1.98±0.01D) than those given by either the contracted N M [531/21] (2.092D) or SB (2.681D) wavefunctions. From this rather limited comparison of theory and experiment it is clear that the calculated results for total energy, dipole moment and momentum distribution are far from converged (see Tables 5.1 and 5.2). It is also clear that the degree of convergence is different for different calculated properties for each of the wavefunctions. This is, of course, a straightforward manifestation of the fact that a wavefunction is a model (in all cases for every molecule and for all neutral atoms except for atomic hydrogen). Such a model will only be as reliable as the approximations used in its building and it will only be adequate for calculating those properties for which these approximations and testing constraints (usually only the variational constraint of minimised energy) are sufficiently valid. Since the energy minimisation stresses the small r region of wavefunctions it is not surprising that use of variationally determined wavefunctions often leads to poor results for properties, such as dipole moment and MDs, which depend sensitively upon the longer range (r) charge distribution. With the above considerations in mind, and considering the large discrepancy between measured X M P s and calculated MDs for the outer valence orbitals of water, it is of interest to investigate the effects of increased basis set flexibility including the addition of higher order polarization functions. In this it is of key importance to ensure that convergence has been reached not only with regard to 121 the H F limit of energy but also for those properties including the dipole moment and M D which are influenced by the large r (i.e. low momentum) part of the wavefunction. When this has been achieved, the remaining discrepancies (if any) between theory and experiment can be investigated in terms of the significance of the neglect of electron correlation and relaxation effects implicit in the Hartree-Fock model. A t the time of the first E M S experiments for H 2 0 the wavefunctions used were in the range of quality of the SB, N M and A P C functions referred to above (see also Table 5.1). The E M S results, at much improved momentum resolution (reported in chapter 4), were evaluated using these same wavefunctions in order to facilitate direct comparison with the original studies [HH77, DD77]. Other more sophisticated literature wavefunctions are now compared with experiment. These include the 39-STO wavefunction of Rosenberg and Shavitt [RS75] and the 84-GTO wavefunction of Davidson and Feller [DF84] which are two of the best single determinant SCF wavefunctions currently published for H 2 0 . These two wavefunctions have total energies within 0.003a.u. and O.OOla.u., respectively of the estimated H F limit at the experimental geometry and give calculated dipole moments of 1.995D and 2.02 ID which are close to the estimated H F limit of 1.98±0.01D (experimental value is 1.8546D). It is however of interest to note that the variationally superior 84-GTO wavefunction gives a less good value (2.02ID) for the dipole moment than the 39-STO (1.995ID). This may reflect the superiority of STOs at large distances. Consideration of these two relatively high quality wavefunctions [DF84, RS75] alone indicates that convergence of both energy and dipole moment to the H F limit values has not yet been reached. 122 Calculations in fact show that the 84-GTO [DF84], 39-STO [RS75] and A P C [AP68] wavefunctions all give quite similar results for the calculated MDs exhibiting in each case a similar (considerable) discrepancy with experiment for the l b , and 3a , orbitals of H 2 0 (compare results in previous chapter and Figs. 5.1 and 5.2) Therefore new and further improved wavefunctions, namely 99-GTO and 109-GTO, have been generated in collaboration with Prof. E.R. Davidson (Indiana University) in the course of the present work in an attempt to model more adequately the variationally insensitive but chemically important large r (low p) portion of the electron distribution. Details of these new wavefunctions are given in Sec. 5.2 and pertinent properties are shown in Table 5.1. To give an idea of the basis extension, it should be noted that the 109-GTO wavefunction uses a [14sl0p3dlf/6s3pld] basis set whereas a minimal basis set for water would use only a [2slp/ls] basis set. It can be seen that the energy (109-GTO) is converged (at least to within 0.0005 a.u. of the H F limit) but that the dipole moment (2.006D) is still farther from the H F limit than those given by the variationally inferior NM(1062/42) [NM68] and 39-STO [RS75] wavefunctions. Even larger gaussian basis sets than those reported here have given a Hartree-Fock dipole moment of 1.9803D with an energy of -76.0672 hartrees. This value of the dipole moment is believed to be converged to ±0.0 ID. Calculations of the momentum distributions for all 4 valence orbitals (including spherical averaging and incorporation of the experimental momentum resolution, Ap = 0 . l 5 a o " 1 ) using these various existing and new wavefunctions are shown in 123 Figs. 5.1-5.4 in comparison with the experimental results for H 2 0 and D 2 0 . The calculations were carried out in the PWIA and (except for the 109-G(CI) and 140-G(CI), see below) T H F A treatments. A l l calculations (including the 109-G(CI) assuming unit pole strength for each orbital) are on a common intensity scale established by single point normalization of the 109-GTO calculations to experiment on the l b 2 orbital (see Fig. 5.3). Thus all experimental points and calculations are on the same relative absolute intensity scale for all four orbitals and this affords a very stringent quantitative comparison of theory and experiment. Several observations can be made in reference to the earlier comparisons of experiment and theory for the valence orbitals of H 2 0 (Chapter 4). It should be noted that curves 1 (SB) and 2 (NM[531/21]) were shown with individual height normalizations for each orbital to the experiment. Comparison of curves 2 and 3 show the serious effect of the contraction of the same (NM) wavefunction on the calculated MDs particularly in the case of the l b , and 3a , orbitals. The uncontracted set, NM(1062/42), shows (see curve 3, Figs. 5.1 and 5.2) an improved momentum distribution and a much better dipole moment than does the contracted set, NM[531/21] (see curve 2, Figs. 5.1 and 5.2) although the energy is only marginally affected by the contraction (see Table 5.1). This illustrates the extreme care necessary in choosing the proper contraction scheme if properties such as the M D or dipole ,moment are required. Curve 4 (Figs. 5.1-5.4) shows in each case the MDs calculated from the 84-GTO wavefunction and these are found to give rather similar results to the A P C wavefunction as used earlier [BL85]. 124 Considering first the l b , orbital (Fig. 5.1) it can be seen that the increase in the maximum cross section and decrease in p as well as the increasingly *max b J improved modeling of the low momentum region in going from SB (curve 1) to NM(1062/42) (curve 3) to 84-GTO (curve 4) are carried even further in going to the 99-GTO (curve 5) wavefunction. Similar improvements with change in basis set also occur for the 3a, (Fig. 5.2) and 2a , (Fig. 5.4) orbitals. This reflects the further improvement (see Table 5.1) in both calculated total energy and dipole moment for the 99-GTO and 109-GTO wavefunctions relative to the 84-GTO wavefunction. The progression of the theoretical MDs towards the experimental momentum profile with expansion in basis set is clearly illustrated. In contrast, the calculated MDs for the l b 2 orbital (curve 3, Fig. 5.3) converge at the NM(1062/42) level. However it is of importance to note that no further change in the calculated MDs for the two outermost orbitals occurs in going from 99-GTO to 109-GTO (see curves 5 and 6 which are identical in Figs. 5.1-5.3). On the other hand in the case of the 2a , orbital the calculated M D is already converged at the 84 GTO level (see Fig. 5.4). Even larger basis sets (114-GTO, 119-GTO and 140-GTO) gave no noticeable change so these M D curves would seem to be at the Hartree-Fock limit. These results indicate the importance of having s-p saturated basis sets as well as higher order polarization functions (i.e. d- and f-functions on the oxygen) when predicting properties such as electron momentum distributions and the dipole moment. The importance of very diffuse functions (much more than expected) in the basis set is shown by the corresponding improvement in the calculated MDs. As more diffuse functions are added, the M D shifts and also gives appreciably 125 more intensity at lower momentum. The balanced addition of extra diffuse functions in H 2 0 evidently provides an improved description of the large r (low p) part of the total wavefunction. In terms of the inverse weighting property of the Fourier transform this should contribute to a better description of the low momentum components of the calculated MD. Summarising the above considerations it can be concluded that gaussian basis set saturation has been effectively reached at the 99-GTO SCF level. However while it is clear that considerable improvement over earlier calculations has been gained for the l b , , 3a, and 2a, orbitals with this "best" HF level treatment it can be seen that a considerable discrepancy with experiment still occurs, especially in the low momentum region. In particular, significant additional low momentum components are observed experimentally in the case of the lb, orbital and also the observed p m o ( 0 . 6 0 ± 0 . 0 2 a 0 " 1) is appreciably lower than even that i l l 3.X (0.65a 0~ 1) predicted by the best (99-GTO and 109-GTO) HF level wavefunctions. Smaller discrepancies exist between calculations and experiment for the 3a, and 2a, orbitals at the HF level. Properties of the various calculated MDs and XMPs, including P m a x , are given in Table 5.2. The properties shown in this table summarize the convergence of the calculated molecular orbitals of water in momentum space . To characterize the MDs two properties have been evaluated, namely the leading slope at half maximum (LSHM) and the momentum at which the MD maximizes (p ). It should be pointed out that P m a x is not the most probable momentum which is characterized by the p which maximizes p 2 «(MD) . Note that the p and the Table 5.2 C h a r a c t e r i s t i c s of Calculated Orbi ta l Momentum D i s t r i b u t i o n s and Experimental Momentum P r o f i l e s Basis Set lb 3a lb . SB [42/2] NM [531/21] NM (1062/42) RS 39-STO DF 84-GTO 99-GTO 109-GTO 84-G(CI) 109-G(CI) b H 20 Expt. b 0 2 0 Expt. Pmax LSHM 0.76 (0.0990) 0.76 (0.0989) 0.72 (0.1121) 0.67 (0.1195) 0.68 (0.12O4) 0.65 (0.1262) 0.65 (0.1264) 0.67 (0.1316) 0.63 (0.1398) 0.60 (0.115) 0.60 (0.115) Pmax LSHM 0.75 (0.1104) 0.76 (0.1078) 0.71 (0.1186) 0.69 (0.1240) 0.69 (0.1270) 0.68 (0.1280) 0.68 (O.1280) 0.68 (0.1365) 0.66 (0.1370) 0.69 (0.110) 0.69 (0.110) Pmax LSHM 0.74 (0.1212) 0.74 (0.1192) 0.72 (0.1216) 0.72 (0.1204) 0.72 (0.1210) 0.72 (0.1208) 0.72 (0.1209) 0.72 (0.1222) 0.72 (0.1216) 0.72 (O.IIO) 0.72 (0.110) a Peak maxima and LSHM (In parenthesis) are both quoted In atomic u n i t s , b -1 Estimated experimental uncertainty In p and LSHM are i 0 . 0 2 a Q and ± 0 . 0 0 5 , r e s p e c t i v e l y . C O 0 3 127 LSHM do not completely characterize the MD but they can be used as guides in comparing the different calculations with experiment. The P m a x and the LSHM of the MDs and XMPs can be considered as momentum space analogs of one-electron properties in position space in the sense that they can be used as "diagnostics" of wavefunction quality. The p v v, o v and LSHM values for the MDs m 3.x were obtained using a standard cubic spline fitting routine [UB84]. The statistical deviation (i.e. the individual error bars of the points in the distribution) were also considered in obtaining the best fits for the XMPs. It can be seen that the experimental p-space properties for H 2 0 and D 20 are the same within experimental error. Consider first the trend of p and LSHM at the SCF level. From Table 5.2 it is evident that the P m a x for the 3a, and l b 2 orbitals are quite converged at the 99-GTO level and NM(1062/42) level (58-GTO), respectively with the l b 2 orbital converging towards a p higher (0.72a 0~ 1) than the 3a, orbital (0.68a o~ 1). In contrast, the p„, o v of the lb 1 orbital converges at the 99-GTO level with a p of 0.65a 0 ~ 1 compared to the experimental value of max 0.60ao" 1 ±0.02. Despite the estimated uncertainty it is clear from the trend of the data points (Fig. 5.1) that the P m a x from experiment is significantly lower than that predicted by any calculation at the THFA level. Since basis set saturation has been established the remaining discrepancy between theory and experiment may be associated with one or more of the following effects (1) Deficiencies in the PWIA treatment; 128 (2) Further uncertainties in the experimental momentum resolution beyond Ap = 0 . I 5 a 0 ~ 1 already incorporated in the calculations; (3) Failure of the T H F A in H 2 0 and therefore the need to consider the fact that correlation and relaxation effects may be significantly influencing the valence momentum profiles. This would amount to failure of the Hartree-Fock model description in these cases. Deficiencies in the PWIA treatment (1) are unlikely for the reasons already discussed in Sec. 5.1 (i.e. good agreement at the H F level for a number of other atoms and molecules). Similarly unknown momentum resolution effects (2) can be discounted since a further enlargement (unphysical with respect to the experimental geometry) of the Ap ( O . l 5 a 0 ~ 1 ) already used would in any case only significantly affect the very low momentum part of the curve. Any such increase in Ap would not change the position of p or the majority of the 111 3.X large "mismatch" down the leading edge (i.e. low p region) of the momentum distribution (see Fig. 5.1). Therefore the most likely source of the remaining discrepancy between H F theory and experiment is item (3), namely the neglect of electron correlation implicit in the Hartree-Fock model used in the T H F A treatment. This possibility is investigated in detail in Sec. 5.6 following. Before proceeding to a consideration of correlation effects the following further observations are made. First, while the observed discrepancies between calculated MDs and measured X M P s for H 2 O are largest for the l b , orbital similar but somewhat smaller discrepancies are also found for the 3a, and 2a , orbitals. On the other hand the l b 2 orbital is apparently well represented already at the 129 A P C and NM(1062/42) levels. Second, it is instructive to compare M D results using the best gaussian basis set (i.e. 109-GTO or 99-GTO) and best Slater [RS75] basis sets (39-STO) available to date. This "best" GTO/STO comparison is shown together with the experimental measurements in Fig. 5.5 for the valence orbitals of H 2 0 and D 2 0 . The overall good agreement between the two calculations confirms the generally held view that 2 to 3 GTOs are needed for each STO. It can be seen that while the calculated l b 2 orbital is identical for use of both STO and GTO (Fig. 5.5) and in good agreement with experiment, the 109-GTO calculation is marginally better for the 3a , orbital and a slight improvement for the l b , orbital. It can also be seen from Tables 5.1 and 5.2 that although the 109-GTO wavefunction gives superior values for calculated total energy, p and < r 2 > the 39-STO wavefunction gives a better value for the max dipole moment. 5.6. CORRELATION AND RELAXATION EFFECTS In view of the failure of even highly saturated basis sets to satisfactorily predict the observed momentum distributions at the target Hartree-Fock level (except for the lb 2 orbital, at least on the basis of the present normalization) a theoretical investigation beyond the Hartree-Fock model has been made using CI wavefunctions developed in an interactive collaboration with Prof. E.R. Davidson. The H F limit of total energy for the ground state of H 2 0 is estimated to be -76.0675a.u. while the estimated non-relativistic, non-vibrating total energy is -76.4376a.u. [RS75]. The difference (-0.370a.u.) is the extra energy due to electron correlation neglected in the Hartree-Fock single configuration S C F model. Correlation effects can be treated by configuration interaction (CI) description for Comparison of calculated valence orbital MDs of H 2 O using the best gaussian (109-GTO, solid line) and best Slater (39-STO, broken line) basis sets. The T H F A calculations are placed on a common intensity scale by using a single point normalization of the 109-GTO calculation on the l b 2 X M P (see text for details). 131 the target molecule. For the final ion species configuration interaction using the target molecular orbitals can be used to describe both electron correlation and relaxation effects. These correlated wavefunctions can then be used to evaluate the full ion-neutral overlap (see Eqn. 2.11). Any difference between such a full overlap and the corresponding T H F A calculations indicates the importance of relaxation and correlation effects. Previous comparisons of X M P s and theoretical MDs for H 2 0 were all done using the target Hartree-Fock approximation (THFA) except for the generalized overlap amplitude (GOA) calculation of Williams et al. [WM77a], These G O A calculations [WM77a], which utilized a rather limited basis set, showed a small improvement over the T H F A calculation but major discrepancies between experiment and theory remained. The GOA method attempts calculation of the ion-neutral overlap via many-body Green's function techniques. In the present work calculations of the ion-neutral overlap amplitude using CI wavefunctions based on the two highly extended basis sets, namely the 84-GTO [DF84] and 109-GTO basis sets were done. The CI wavefunctions were developed in an interactive collaboration with Prof. E.R. Davidson (Indiana University). For the CI treatment using the 84-GTO basis (labelled 84-G(CI)) the neutral molecule ( ' A ^ wavefunction is expanded into 5119 symmetry adapted configurations (SACs) calculated at the experimental neutral molecule equilibrium geometry ( E Q J =-76.3210 a.u. which includes 69% of the total ground state correlation energy). For the CI treatment using the 109-GTO basis (labelled 109-G(CI)), the neutral molecule is expanded into 11011 SACs ( E Q J = -76.3761 a.u. which includes 132 83% of the ground state correlation energy). Details of the 109-G(CI) final ion state wavefunctions are to be found in Table 5.3. The spherically averaged square of the Fourier-transformed ion-neutral overlap amplitudes using both sets of CI wavefunctions have been calculated and the momentum space properties are compared in Table 5.2. The full ion-neutral overlap calculations (ie. 84-G(CI) and 109-G(CI)) are compared to the corresponding T H F A results in Figs. 5.6-5.8 (see also Figs. 5.1-5.4) along with the measured H 2 0 and D 2 O X M P s for the l b , , 3 a , , lb 2 and 2a , valence orbitals. Normalization is the same as in the case of Figs. 5.1-5.4 (i.e. at a single point for experiment and the 109-GTO (THFA) calculation on the l b 2 momentum profile). Comparison of the 0° and 8° binding energy (at a total energy of about 1200eV) spectra [CC84, BL85] as well as the momentum profiles measured at 32.2eV and 35.6eV [BL85] have confirmed the assignment that the extensive many-body structure found in the region 25-45eV is predominantly due to the (2a,) ^ hole state. CI calculations however suggest the presence of minute poles in this region due to small contributions from the ionization of the outer valence orbitals. This predicted spreading of the minor poles over the wider energy spectrum (with much of the extra intensity at energies even higher than 45 eV) results in pole strengths for the main lines of each of the three outer valence orbitals which are slightly less than one (i.e. 0.87, 0.88 and 0.89 - see Table 5.4). These pole strengths for the outer valence orbitals are converged to ± 1 % . Table 5.3. CI Calculations (a.u. ) of the Ground and Final Ion States of H2D Using the 109-GTO Basis Set STATE E(CI) E(SCF)3 AE(CI)b E(exptl)° E(HF ) a E(corr) e y.E(corr)d,f H20 X 'A, -76.37614 -76.0671 0.309 -76.4376 -76.0675 0.370 83 H20* (1b,)" 1 -75.92084 -75.5569 0.364 -75.9746 -75.5600 0.415 88 H20+ (3a ,)~ 1 -75.83376 -75.4822 0.351 -75.8976 -75.4841 0.414 85 H,0+ (1b2)" 1 -75.68563 -75.3492 0. 336 -75.7576 -75.3515 0.406 83 For the Ionic states the energy refers to a Koopmans' energy. &E(CI) • E(CI) - E(SCF) C For the ion states, E(exptl)° -(76.4376 - I.P.) a.u. d Includes relaxation for ionic states. 8 E(corr) = E(exptl) - E(HF) for neutral, E(corr) = E(exptl) - E(Koopmans) for Ion. f %E(corr) « AE(CI)/E(corr) x 100 134 i 1 1 i 1 1 1 1 1 1 r Momentum (a.u.) Fig. 5.6 Correlation effects in the calculated momentum distributions of the l b , orbital of water. The more accurate CI overlap calculations (solid line), (4c)84-G(CI) and (6c)109-G(CI), and the T H F A calculations (broken line), (4)84-GTO and (6) 109-GTO are all placed on the same relative absolute scale by using a single point normalization of the 109-GTO calculation on the l b 2 X M P (see text for details). Further comparisons are made with the measured H 2 O (open squares) and D 2 O (solid circles) experimental momentum profiles. 135 T 1 1 1 1 1 1 1 1 1 i i r Momentum (a.u.) Fig. 5.7 Correlation effects in the calculated momentum distributions of the 3a, orbital of water. The more accurate CI overlap calculations (solid line), (4c)84-G(CI) and (6c)109-G(CI), and the THFA calculations (broken line), (4)84-GTO and (6) 109-GTO are all placed on the same relative absolute scale by using a single point normalization of the 109-GTO calculation on the lb 2 XMP (see text for details). Further comparisons are made with the measured H 2 O (open squares) and D 2 O (solid circles) experimental momentum profiles. 136 Fig. 5.8 Correlation effects in the calculated momentum distributions of the l b 2 orbital of water. The more accurate CI overlap calculations (solid line), (4c)84-G(CI) and (6c)109-G(CI), and the T H F A calculations (broken line), (4)84-GTO and (6) 109-GTO are all placed on the same relative absolute scale by using a single point normalization of the 109-GTO calculation on the l b 2 X M P (see text for details). Further comparisons are made with the measured H 2 O (open squares) and D 2 O (solid circles) experimental momentum profiles. Table 5.4. CI Calculations (109-G(CI)) of the Pole Strengths and Energies (eV) in the Binding Energy Spectrum of H20 STATE Energy Pole Strength C2(2a ,) C2(3a J ( 1b,) -1 ( 3 a ( ) d b 3 ) (2a,) -1 12.39 14 . 76 18.79 29.0 33. 1 33.6 34.2 35.8 37.8 39.5 41 .0 44 .0 44.9 45.0 45.8 46.3 47 . 1 0.869 0.882 0.888 0.0357» 0.4394» 0.2232* 0.0225* 0.0218* 0.0216* 0.0730* 0.0240 0.OO39* 0.0002 0.0009 0.0051* 0.0043 0.0227* O.OOO 0.974 0.992 0.999 0.659 0.865 0.863 0.980 0.475 0.708 0.336 0.033 0.914 0.393 0.827 0.998 0.021 0.077 0.000 0. 325 O. 133 O. 135 0.015 0.487 0.260 0.525 0.482 0.058 0.504 O. 147 £(•)'0.869 CO 138 Even more extensive breakdown of the single particle picture in the case of ionization of the 2a , inner valence orbital is well-known both theoretically [VS84] and experimentally [CC84, BL85]. Table 5.4 shows the pole strengths and energies for each final ion state together with the CI coefficients for 2a , /3a, mixing as calculated using the 109-G(CI) correlated wavefunctions. These calculated pole strengths are convoluted with the estimated experimental width (2.77eV) of the main (2a,)"''' peak (see Figs. 4.1 and 4.2) and shown in Fig. 5.9 in comparison with the relevant part of the earlier reported binding energy spectrum [BL85] in the region 23-45 eV. The agreement between the calculation and experiment is generally quite reasonable. However the calculated results are at best semi-quantitative with respect to the distribution of intensity particularly in the 31-36 eV region. Note that the calculated binding energy profile is slightly shifted (= leV) to the higher energy side and that the position of the shoulder at =35eV is still inadequately predicted by the 109-G(CI) calculation. It should also be noted that the spectrum reported by Cambi et al. [CC84] extends out to 50eV and shows weak intensity comparable to that predicted at =47eV. It is probable that, even at this level, the calculation is still basis set dependent with respect to the exact (2a,)"* pole strength distribution (see also for example pole strength calculations reported by Cambi et al. [CC84] and Agren and Siegbahn [AS80] which are shown in comparison with experiment in Fig. 4.2). More definite conclusions would require a convergence study of the energies and pole strengths of the many-body structure of the (2a,)" ^ hole state. With these considerations in mind, the calculated outer valence OVDs (Figs. 5.1-5.3, 5.6-5.8) are presented as normalized distributions (i.e. renormalizing the 139 pole strength of each of the outer valence poles to unity). The inner valence 2a , O V D in Fig. 5.4 was obtained by calculating a pole-strength weighted sum of the (slightly different) OVDs of all significant poles (i.e. those with intensity greater than 1%) found in the region above 25eV and then divided by the summed pole strength in this region in order to renormalize the resultant OVD to unity. It is of interest to note that this summed 2a , pole strength has essentially the same value as that for the main pole of each of the three outer valence orbitals. In this normalization procedure any contributions from any minor outer valence poles have been neglected. Calculations in any case indicate that such pole strengths are less than 3% (in fact only one pole is 2: 1%, i.e. that at 41eV, see Table 5.4). On the basis of the above normalization procedure for the CI calculations, it can be seen (Fig. 5.6 and 5.7, Table 5.2) that use of the CI wavefunctions for the l b , and 3a, outer valence orbitals in the overlap treatment results in a further significant progression of these "p-type" calculated OVDs towards low momentum and higher cross section and thus towards better agreement with experiment. Such a small shift towards low momentum with the inclusion of correlation has also been observed earlier for A r (3p) by Mitroy et al. [MA84] but the effect was very small in that case. It is also significant to note that the good agreement already attained at the 84-GTO T H F A level for the l b 2 orbital (Fig. 5.8) is unchanged by use of the 84-G(CI) or 109-G(CI) overlap treatments. It can be seen that, with the normalization procedure described above, the T H F A and CI treatments give the same absolute intensities over the entire momentum range for the l b 2 orbital. In the case of the 2a , inner valence orbital (Fig. 140 5.4) inclusion of correlation (109-G(CI)) gives a slightly improved quantitative fit to the data. These findings indicate the important fact that correlation mainly influences the l b , and 3a , orbitals. This behaviour is in sharp contrast to the situation for the l b 2 orbital where correlation and relaxation effects appear to have a negligible effect on the calculated overlap distribution. The interplay of basis set effects and the inclusion of correlation are well illustrated by the results for the l b , and 3a , orbitals shown in Figs. 5.1, 5.2, 5.6 and 5.7. It can be seen that inclusion of correlation through the CI overlap treatment further improves agreement between the calculated distributions (MDs and OVDs) and experiment at both the 84-GTO level (compare curves 4 and 4c, Figs. 5.6 and 5.7) and the 109-GTO level (compare curves 6 and 6c, Figs. 5.6 and 5.7). In comparison with the 84-G(CI) wavefunction (p =0.-67a 0 _ 1 > dipole moment= 1.929D) use of the 109-G(CI) wavefunction shows a further significant shift of the O V D to low momentum ( p „ , „ = 0.63a 0 " 1 ) together with a further increase in cross section and an improved value of the dipole moment (1.895D). Despite the dramatically improved agreement over the T H F A treatments a small but significant discrepancy still exists for the l b , orbital even at the 109-G(CI) level since the experimental momentum profile is at lower momentum (p = 0.60ao ~ 1 ±0.02) with a higher maximum cross section. In the case of max the 3a, orbital the 109-G(CI) overlap calculation gives a result quite close to experiment. With inclusion of configuration interaction, smaller P m a x values are predicted together with more asymmetric O V D curves as characterized by the higher L S H M values (ie. steeper leading slopes at half maximum) which, for example, for the l b , orbital are 0.1316 and 0.1398 for the 84-G(CI) and 141 —i—i—i—i—i—i—i—i—i—i—r : H 2 0 ( 2 a , ) - 1 — 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i •v 109-G(CI ) Cal c n . " ( b ) ; i i i i i i i i i i i i i i.. ± 23.0 25.0 27.0 29.0 31.0 33.0 35.0 37.0 39.0 41.0 43.0 45.0 47.0 49.0 Binding Energy («V) 5 I—i—i i i i i i i i i i i i I I i i i i i i 23.0 25.0 27.0 29.0 31.0 33.0 35.0 37.0 39.0 41.0 43.0 45.0 47.0 49.0 Binding Energy («V) Fig. 5.9 Binding energy spectrum of water in the (2a,) inner valence region. The experimental data (a) at 0 = 0 ° ( E 0 = 1200eV + B.E.) is taken from Ref. [BL85]. The theoretical binding energy profile (b) was determined in the present work from pole strengths calculated using the 109-G(CI) wavefunctions (shown as solid vertical lines at the appropriate energies). The calculated poles have been convoluted with an experimental width of 2.77eV fwhm estimated from spectrum (a). Further details are given in Table 5.4. 142 109-G(CI) wavefunctions, respectively. The experimental L S H M for H 2 0 ( l b , ) is 0.115±0.005a.u. Calculations using even larger basis sets (114-G(CI), 119-G(CI) and 140-G(CI)) which recover even more of the correlation energy of the neutral molecule and of the cation gave OVDs indistinguishable from the 109-G(CI) results for the l b , orbital. These considerations together with the fact (see Table 5.2) that the 84-G(CI), 109-G(CI) and 140-G(CI) treatments recover 69%, 83% and 88% respectively of the estimated correlation energy of H 2 0 (-0.370a.u.), suggest that any further (very difficult) theoretical investigations of the effects of the remaining correlation energy on the small residual discrepancies are not likely to produce much change. In this regard it should be noted that the large shift (curves 4c and 6c, Fig. 5.6) which results in going from 84-G(CI) to 109-G(CI) is mostly due to the improvement in the basis set. A smaller part of the shift can be ascribed to improvement in the CI description. It is also of interest to consider the various calculations for the l b , orbital as shown on Figs. 5.1 and 5.6 together with the experimental results. In particular it is noticeable that all calculations are slightly higher than the measured cross sections in the 0.9-1.230"' region of the X M P for the l b , orbital. The shift of the calculated OVDs towards low momentum using the full overlap treatment (ie. at the CI level) reflects the 'non-characteristic' contributions in addition to the single-particle orbital M D that would have been expected in the simple approximation of the more usual T H F A . Consider the case of the ( l b , ) ion electronic state. The overlap amplitude can be expressed in terms of the 143 single-particle orbitals (of the ground Hartree-Fock configuration), < p * f N _ 1 |* 0 N> = C , uV n + C 2 ^ 2 + . . . [5.1] where the labels 1 and 2 refer to l b , and 2b, p-space molecular orbitals, respectively. The first term corresponds to the characteristic orbital (in the nomenclature of THFA) whereas the second term corresponds to the lowest-lying virtual orbital of the same symmetry. The latter can therefore be considered as a 'non-characteristic' contribution. The correction comes mostly from the integral of the l b , 2 —> 2b, 2 double excitation of the neutral (which represents radial electron correlation) with the l b , —» 2b, single excitation of the ion (which describes orbital contraction). Because of the signs involved, this orbital contraction makes the molecule-ion overlap more diffuse than the characteristic orbital. To illustrate this point consider a two orbital model: | * 0 N > = A , | 1 b , 2 > - A 2 | 2 b , 2 > [5.2] There will be no large term in (lb ,2b , ) because of Brillouin's theorem [S082]. The 2 b , 2 term represents electron correlation. A , and A 2 will be positive, independent of the relative phases of l b , and 2b , . For the ion, | * f N _ 1 > = B , | l b , > - B 2 | 2 b , > [5.3] The second term represents relaxation (i.e. orbital contraction). The sign of B 2 / B , will depend on the relative phases of l b , and 2b , . If both l b , and 2b, are chosen to be positive at large r, then B 2 / B , will be positive. Since l b , has no radial node and 2b, has one, they will have opposite signs at small r. Consequently, the second term will subtract from the first at large r 144 and add at small r. ^> will be contracted relative to | l b ! > . Now consider the molecule-ion overlap < ^ | ' P o ^ > - Explicit evaluation gives < * f N " 1 | * 0 N > = A ^ J l b ^ + A 2 B 2 | 2 b , > = C , | 1b,> + C 2 | 2 b ! > [5.4] Notice that C 2 has the opposite sign to - B 2 so that the molecule-ion overlap amplitude is more diffuse in position space than | 1 b , >, opposite to the i N _ 1 contraction in > . This behaviour and the converse momentum space contraction is clearly illustrated by considering the difference density ( P Q J - p r p p j p ^ ) plot for the l b , orbital shown in Fig. 5.10. Likewise the overlap density (the spherical average of which yields the OVD) can be written as, p Q = C , 2 ( l b , ) 2 + C 2 2 ( 2 b 1 ) 2 + 2 C , C 2 ( 1 b 1 ) ( 2 b 1 ) [5.5] Since | B 2 | < | B 1 | , | A 2 | < | A 1 | , it can be seen that | C 2 | < | C , | and C 2 2 is small compared to (2CiC 2 ) . Consequently the cross term in ( lb 1 ) (2b 1 ) is the dominant correction in the molecule-ion overlap density. This momentum space contraction on going to the CI overlap treatment can also be clearly seen in the distributions shown in Figs. 5.1 and 5.6. 5.7. CALCULATED PROPERTIES NEAR THE HF AND CI LIMITS To summarize the various basis set, relaxation and correlation effects discussed above, the p „ and L S H M of the theoretical MDs and OVDs of the l b i ' r m a x 1 orbital are plotted as a function of the number of contracted GTOs in their respective basis sets in Fig. 5.11. Attention is focussed on the l b , orbital since it is most sensitive to correlation and basis set effects. As has been shown MOMENTUM DENSITY DIFFERENCE POSITION DENSITY DIFFERENCE Fig. 5.10 Two-dimensional density difference (p^j - P-pjjp^) plots in momentum space and position space for the l b , orbital of water calculated using the 84-GTO basis. Contours are at ± 8 0 , ±40 , ± 8 , ± 4 , ±0 .8 , ±0 .4 , ±0 .08 and ±0.04% of the maximum density difference. A l l dimensions are in atomic units. 146 0.801 ^ 0.75' =1 3 0.70 X I 0.65 a. 0.60 *d 0.13 d ~ 0 . I 2 Sen _l 0. 10 Q 2.6-"> 2.4 E o 2.2-o o S 2.0 8 1 2 V HARTREE \ 5 6 -FOCK o o (a) HARTREE-FOCK 5 6 lb, P m a x EXPT. 4c \ 6 c b ci 4c ' EXPT. ° I lb, LSHM H 20 DIPOLE 1 SB [42/2] 5 99-GTO 2 NMQ53I/2Q 6 I09-GT0 3 NMU062/42) 4c 84"G(CI) 4 OF 84-GTO 6c I09"G(CD -76.00 -_ .02-3 .04-.08 TOTAL ENERGY v.2 ^ . \ 3 I? 0 6 i H F U m i K i 5 6 HARTREE -FOCK MOMENT ^ 5-763-4 CI EXPT H 4c 6c A - - A -76.5-1 1 1 Exptl. Energy "(d)" ~ " c i f c 6c 0 40 80 I20 0 40 80 I20 Number of Contracted GTOs Fig. 5.11 Convergence of momentum distribution properties, (a)p and (b)leading slope at half maximum (LSHM), (c)dipole moment and (d)the total energy as a function of basis set complexity. The ^max' L S H M and dipole moment are plotted in the left hand side while the total S C F Hartree-Fock energy is plotted in the right hand side. The estimated Hartree-Fock limit and total experimental energy are shown by the dashed lines in Fig. 5 . l i d . The broken vertical line in Figs. 5.11a-c separates the single configuration (Hartree-Fock) and many configuration (CI) results. A l l quantities are quoted in atomic units except for the dipole moment which is quoted in debyes (D). 147 above the 3a , and 2a, orbitals are also quite sensitive to these effects. The total energy which has been used traditionally as the principal diagnostic for wavefunction quality is shown as a function of the number of contracted GTOs on Fig. 5 . l i d . The estimated Hartree-Fock limit and experimental energy are represented by dashed lines. It is of great importance to note that the convergence of the calculated momentum space properties ( P m a x and L S H M ) as a function of basis set (Fig. 5.11a and 5.11b) is somewhat slower than that for the total energy. Whereas for most purposes one can consider the total energy to be converged essentially at the H F limit (see Table 5.2) at the 84-GTO level the same is not true for the momentum space properties which converge at the 99-GTO level as shown for the l b , orbital of water (Figs. 5.11a and 5.11b). The convergence of the dipole moment (Fig. 5.11c) is less regular by comparison to the P m a x > L S H M and total energy. These differences in convergence are not unexpected in view of the fact that the error in a calculated molecular property (e.g. the dipole moment or the momentum distribution) is first order with respect to the error in the wavefunction whereas the error in the total energy is second order [FB87]. The findings of the present work stress the need for proper use of the Variational Theorem, i.e. the constraint of energy minimization must always be accompanied by correct prediction of an adequate range of molecular properties. Since the low momentum regions of the orbital MDs include contributions from regions of r-space which make little contribution to the total energy, it is not unreasonable to expect that the X M P s may be inadequately described by methods which emphasize only the energetics of the system when choosing the basis set. 148 While wavefunctions more extended than 109-GTO have been constructed [FB87] they cannot give more than 0.0005 hartree improvement to the R H F energy since the Hartree-Fock limit has already been reached within that accuracy (see Table 5.2 and Fig. 5 . l id) . The present 109-GTO basis is also saturated with diffuse basis functions (at least at the H F level) so further significant changes in the M D are not obtained at the T H F A level. Finally it should be pointed out that comparisons of X M P s and calculated MDs and OVDs demonstrate the extreme sensitivity of the E M S technique towards certain details in the electronic wavefunction to which the energy is much less sensitive. As has been shown by E M S , agreement of theoretical orbital MDs with X M P s can be obtained for more tightly bound orbitals such as the l b 2 and 2a , (and probably also l a , ) orbitals already at the simple DZ level. These distributions could then tend to dominate the total electron momentum distribution and thus no serious discrepancy might be detected between theory and experiment for the total momentum distribution as observed for instance in the Compton profile [TE74, SW75]. The present work clearly demonstrates the orbital specificity of E M S which provides more detailed information than methods such as Compton scattering [TE74, SW75] which measure the total momentum distribution. Improving the accuracy of calculated orbital MDs and OVDs would be of importance not only from the computational point of view but also with respect to the interpretation of current E M S experiments. In this regard, recent theoretical efforts towards the solution of the Hartree-Fock equations in 149 momentum space [NT81], alternative theoretical approaches [GB85] other than the 'minimal energy criterion' and also new ways to model molecular wavefunctions would be of interest. On the other hand the design of momentum density optimized small basis sets, although computationally attractive, will lose physical meaning if the energy is non-optimal. 5.8. SUMMARY Considerable improvement has been obtained in the degree of agreement with experiment of T H F A calculations of MDs using wavefunctions at the Hartree-Fock limit in the case of the two outermost orbitals ( l b , and 3a,) . However at the T H F A level there still remains an appreciable discrepancy most notably in the case of the l b , orbital. Incorporation of correlation and relaxation effects by calculation of the ion-neutral overlap distribution using CI wavefunctions for the initial and final states results in generally very good agreement with the E M S data for all 4 valence orbitals. However quite small discrepancies still exist, especially for the l b , orbital. With both the highly extended basis sets as well as with the successively improved calculations of the ion-neutral overlap amplitude using CI wavefunctions, the shift of the theoretical MDs and OVDs has consistently been towards lower P m a x and higher cross section, and thus toward better agreement with experiment. It is noteworthy that an improved theoretical description of the experimental momentum profiles occurs when there is an accompanying improvement in the prediction of both the total energy and the dipole moment as well as other properties. The present work shows that it is important to perform the overlap calculations 150 with "near-complete" basis sets - a method which is computationally very difficult. Investigation of whether the remaining small but finite differrence between the measured X M P and calculated MDs and OVDs for the l b , of H 2 0 is due to the still unaccounted for part of the electron correlation energy (88% of the correlation energy is accounted for in the 140-G(CI) calculation), or to some other factor such as a breakdown of the PWIA due to distortion of the incoming and outgoing electron waves by the polar target molecule will have to await further developments in quantum mechanical computation and/or (e,2e) reaction theory. A t the experimental level E M S measurements of the valence orbitals of water, with further improved statistical accuracy will be needed as even finer details of the target molecule-ion overlap and the (e,2e) reaction theory are investigated. The present work clearly demonstrates the effects of electron correlation and relaxation and the fact that good agreement is only obtained between theory and experiment if the ion-neutral overlap (i.e. the electronic structure factor) is computed with sufficient accuracy. The good overall quantitative agreement now obtained between experiment and theory for the valence orbitals of H 2 0 also indicates the general suitability of the PWIA for the study of small molecules by E M S at impact energies of 1200eV using the symmetric non-coplanar geometry at 5 = 45° . However small discrepancies between the PWIA treatment and experiment suggest that a careful investigation of distortion effects by molecular targets and in particular highly polar molecules would be informative. The present studies also demonstrate clearly the need to consider carefully the low momentum portion of molecular wavefunctions and the importance in many cases 151 of electron correlation in the valence orbitals of small molecules such as H 2 0 . These effects will have to be taken into account if molecular wavefunctions are to be used for highly accurate investigation of problems of bonding and the calculation of charge, spin and momentum distributions. CHAPTER 6. AMMONIA 6.1. OVERVIEW As part of a continuing series of E M S studies of the hydride molecules a detailed experimental and theoretical study of the valence orbitals of the N H 3 molecule is reported in this chapter. Previous studies of N H 3 by Hood et al. [HH76], Camilloni et al. [CS76] and Tossell et al. [TL84] have shown several interesting features. Although these earlier studies were limited by poor momentum resolution, significant differences between the measured experimental momentum profiles (XMPs) and appropriately convoluted theoretical momentum distributions (MDs) were observed for the outermost 'non-bonding' 3a , orbital [HH76, TL84] when even near-Hartree-Fock wavefunctions were used. Particularly interesting has been the comparison of the 'lone pair't X M P s of N H 3 and N H 2 C H 3 in the E M S study by Tossell et al. [TL84] in which they showed that derealization of the 'lone pair' occurs upon methyl substitution on N H 3 . Recent more detailed experiments and calculations (see chapter 9) have now extended this study to compare the outermost orbitals of N H 3 and N H 2 C H 3 with those of N H ( C H 3 ) 2 and N ( C H 3 ) 3 which show even more extensive derealization [BB87a, BB87b]. Reports of other experimental probes of electronic density distributions have appeared recently [OM83, OM84, OI86]. Ohno et al. [OM83] have proposed a method for obtaining the information about the quality of L C A O M O t Extra caution must be exercised in equating lone pairs (a simple valence bond concept) with the outermost 3a, molecular orbital. Even in a minimal basis set there is some H i s character attributed to the 3a, orbital in N H 3 [SB72]. 152 153 wavefunction 'tails' by calculating the exterior (EED) and interior (IED) electron densities. They assumed (in a classical sense) that chemical reactivity of a molecule is largely dependent on the spatial electron distribution outside a boundary surface which is given by the envelope of spheres obtained from the van der Waals radii of the individual atoms that comprise the molecule. Thus by integrating the M O wavefunction outside the van der Waals boundary surface, the exterior electron density (EED) can be calculated for each molecular orbital. Although the EEDs for each orbital cannot be measured directly in an experiment, they have compared the ratio of EEDs for separate orbitals with the ionization branching ratios for these orbitals as measured by Penning ionization * electron spectra (PIES) [0M83, 0M84] obtained using He (2 3 S) metastable atoms. A comparison of E M S measurements and the corresponding calculated E E D and measured PIES ratios in N H 3 is given in the present work. The ammonia sample used in the present study was supplied by Matheson Ltd. with a purity of 99.99%. 6.2. BASIS SETS FOR LITERATURE SCF WAVEFUNCTIONS The experimental results in the present work are compared with spherically averaged momentum distributions calculated for a variety of selected ab initio SCF L C A O - M O wavefunctions. The selection was done in such a way, that the basis sets cover a wide range from double zeta (DZ) quality to extended and diffuse sets up to effectively Hartree-Fock limit quality. These wavefunctions together with selected calculated properties are shown in Table 6.1. Tab le 6 . 1 . P r o p e r t i e s of T h e o r e t i c a l SCF and CI Wavefunctions f o r NH3 WavefunctIon DZ 6-311G 6-311+G H 0 D 2 G 119-GTO 126-GTO H a r t r e e - F o c k I i m t 119-G(CI) 126-G(CI) E x p t I . Ni t rogen B a s i s Set (10s5p) / [4s2p] (11s5p) / [4s3p] (11s5p1d) / [4s3p1d] (12s6p) / [5s4pJ (11s7p) / C5s4p] (13s8p2d) / [Bs5p2d] Hydrogen B a s i s Set ( 4 s ) / (2s) ( 5 s ) / [3S] ( 5s1p ) / [3s1p] ( 5 s ) / 13s] ( 5 s ) / [3s] ( 8s2p ) / [4s1p] (19s10p3d1f) / (10s3p1d) / [10s8p3d1f] [6s3p1d] (23s12p3d1f) / (10s3p1d) / [14s10p3d1f] [6s3p1d] E n e r g y ( a . u ) D i p o l e Moment(D) -56.1777 -56.1813 -56.1790 -56.22191 -56.2246 - 5 6 . 2 2 6 b -56.5155 -56.5160 d 2 . 2 2 9 2 2.2564 1.6598 -56.2245 1.6440 1.6417 1 . 6 3 ° 1.5952 1.5891 1.47149* P max 3a, 0 .69 0.56 0.54 0.58 0.57 0.54 P max 1e EED R a t i o 0.68 0.67 0.67 0.66 0.63 1.98 2.58 2 .50 2.41 0.52 0.52 0.63 0.5210.02 0.6210.02 2 .60 R a t i o r e f e r * to O a , / 1 e ) [0186]. Est imated In [FB87] . U n c e r t a i n t y Is 10.001 Est imated u n c e r t a i n t y i s 10.02. E x p e r i m e n t a l l y - d e r i v e d n o n - v i b r a t i n g , n o n - r e l a t i v i s t 1c. I n f i n i t e n u c l e a r mass t o t a l energy (PB75, FB87). U n c e r t a i n t y Is 10.00015D In the ground v i b r a t i o n a l s t a t e [MD81]. 155 Some important features of the various basis sets are discussed below, (1) Double zeta Two sets of contracted Gaussian type functions are used for each atomic orbital. No additional polarization functions are employed. In this basis set, proposed by Snyder and Basch [SB72], the least tight s- and p-functions of the nitrogen atom are represented by a single primitive Gaussian function. (2) 6-311G This "split-valence" basis of Krishnan et al. [KB80] uses a set of three contracted Gaussian functions and two uncontracted primitive Gaussian functions to represent the valence part 2s and 2p of the nitrogen atom. A restriction of the split-valence method is given by the fact that the atomic s- and p-valence orbitals are described by a common set of exponents and differ only in the contraction coefficients. The nitrogen ls-shell is represented by six contracted Gaussian functions. The basis set for hydrogen consists of five s-functions, contracted to three Gaussian-type orbitals. Since three sets of basis . functions are used for each atomic valence orbital, 6-311G can be regarded as a triple zeta class basis set. (3) 6-311G** An augmentation of 6-311G by including polarization functions in the nitrogen (3d) as well as in the hydrogen (2p) basis sets [KB80]. The total energy is improved from -56.1777a.u. to -56.2102a.u.. (4) 6-311 + G A different approach to augment 6-311G. In contrast to 6-311G** no 156 polarization functions were added but one diffuse s- and one diffuse p-type primitive Gaussian function with a common exponent were included in the nitrogen basis set. The additional valence functions have been chosen following a proposal of Clark et al. [CC83] for anion calculations. The hydrogen basis set is the same as for 6-311G. (5) H D D 2 G A (9s5p) set of primitive Gaussian functions for nitrogen given by Huzinaga [H65] was contracted to [3s2p] by Dunning and Hay [DH77] and augmented with two diffuse 2s and two diffuse 2p functions appropriate for describing atomic Rydberg orbitals. The hydrogen atoms are represented by the (5s)/[3s] set of 6-311G. (6) 56-GTO This extended basis set of Rauk et al. [RA70] includes 91 primitive Gaussian functions, contracted to 56 Gaussian-type orbitals. It , also contains polarization functions in form of nitrogen 3d-orbitals and hydrogen 2p-orbitals. (7) 119-GTO This M O wavefunction of Feller et al. [FB87] with its 119 contracted Gaussian-type orbitals represents the most extended basis set currently available in the literature. It includes d- and f-orbitals for nitrogen as well as p- and d-orbitals for hydrogen. With a predicted total energy of -56.22456a.u. 119-GTO is one of the best available basis set with regard to the total Hartree-Fock energy. (8) 126-GTO In addition to these above basis sets, already available in the 157 literature, a new 126-GTO wavefunction has been developed in the present work. This wavefunction is very close (within = 1 millihartree) to the estimated Hartree-Fock limit and is discussed in the Sec. 6.2.1. Further significant improvement in the wavefunction requires inclusion of electron correlation as discussed later in Sec. 6.2.2. A l l calculations (except for the 56-GTO basis set) have been performed in the experimentally estimated equilibrium geometry of the N H 3 molecule ( r N H =1 .012A, 0 H N H = 1 O 6 - 7 ° ) [ B P 5 7 1 - T h e dependence of the total energy on the chosen geometry was studied by Rauk et al. [RA70] for the 56-GTO basis set. For the experimental equilibrium geometry they calculated an SCF energy of -56.22150a.u.. By variation of bond length and bond angle the SCF energy could be optimized to -56.22191a.u. in a geometry of r ^ j j = 1.89033 bohr and % N H = 1 0 7 - 2 ° ( S C F equilibrium geometry). The calculation for the 56-GTO basis set was performed in this SCF optimum geometry since the EEDs (see later section) of Ohno et al. [0186] for 56-GTO are also given for the SCF equilibrium geometry. The EEDs for all other basis sets however are calculated for the experimental equilibrium geometry. The same geometry for the respective basis sets as Ohno et al. [0186] was chosen in order to allow a conclusive comparison between the concepts of exterior electron densities and spherically averaged momentum distributions (see Sec. 6.7). 158 6.2.1. A 126-GTO Extended Basis Set for N H 3 The extended basis set for N H 3 which consists of an even tempered (23s,12p,3d,lf/10s,3p,2d) primitive set contracted to [14s,10p,3d,lf/6s,3p,ld] Gaussian type (GTO) basis was developed in an interactive collaboration with Prof. E.R. Davidson (Indiana University). The s-components of the cartesian d-functions and the p-components of the f-functions were removed to avoid linear dependence, forming the final 126-GTO basis. The s- and p-symmetry portion of the basis for nitrogen was created from energy optimized exponents [SR79, FR79] using an even-tempered restriction on the exponents. The hydrogen s-exponents are as previously employed by Feller et al. [FB87]. Even tempered exponents form geometric sequences, r\i = a/31 ( i = 1 , . . .N) where different (a,/3) pairs are used for each type of function (s,p,d...) and each atomic number. If N is the number of Gaussian primitives of a certain symmetry, then the dependence of a^^XN) and P^p^N) on N can be described by simpie functional forms so that it is possible to generate near-optimal sets of arbitrary size. The a and /3 values for the nitrogen (22s, l i p ) set are a s = 0.05012, /3G = 2.12175 and a p = 0.03971 and 0P = 2.35056. Experience has shown that the energy-optimized even-tempered sets are considerably improved for many properties i f they are extended by one more diffuse s and p primitive. This extension was made for the current basis sets. Thus, the most diffuse primitives, which would have had optimum exponents of a|3, are now given simply as a. The d-type and f-type polarization function exponents were taken 159 from earlier work by Feller et al. [FB87]. The basis set for ammonia was designed from the 109-GTO basis employed for water by Bawagan et al. [BB87], which was essentially converged! for total energy, dipole moment and momentum distribution. The basis set has been designed to saturate the diffuse basis function limit so as to give improved representation of the (r-space) tail of the orbitals. It should be noted that due to the large number of functions which have been used the wavefunction is expected to be fairly insensitive to the exact choice of exponents. As in previous studies of momentum distributions in this laboratory [BB87, FB87a], the assessment of the quality of the wavefunction is based upon the values of the one electron properties calculated from the wavefunction at several levels of sophistication (see Table 6.1) using the experimental geometry r ^ j j = 1.012A and = 106.7°. The methods applied were restricted Hartree-Fock (RHF), singly and doubly excited Hartree-Fock configuration interaction (HF SD-CI) and multireference singly and doubly excited configuration interaction (MRSD CI). The all electron CI convergence has been shown to be improved (ie. more correlation energy was recovered with fewer configurations) when the Hartree-Fock virtual orbitals are transformed to K-orbitals [FD81, CP82, FB85] and hence K-orbitals were used. The configurations used in the MRSD-CI were energy selected based on second order Rayleigh-Schro dinger perturbation theory t These results are considered effectively converged since the total energy is within 0.5 millihartree of the estimated Hartree-Fock limit and because the dipole moment, quadrupole moment and the momentum distributions do not change significantly as even more basis funcitons are added. Other basis sets containing up to 140 GTOs were actually tested for water with no significant change in either the momentum distribution, total energy and dipole moment at the S C F or CI level. 160 and coefficient contribution in the H F SD-CI, with all the singly excited configurations being kept. This selection was necessary due to the large number of configurations associated with the extended basis set exceeding the current variational capacity of the Indiana group. Table 6.1 lists SCF and CI energies and one-electron properties for the 126-GTO basis. The CI was done by the Indiana group in two steps. First, a small CI was done. For this 15,868 spin-adapted Hartree-Fock singly and doubly excited configurations were selected. The selection procedure involved retaining all the singly excited configurations and using second order perturbation theory on the doubly excited configurations such that the neglected configurations have a total contribution of less than one millihartree to the CI energy. Then the MRSD-CI reference space was chosen from the configurations in the H F SD-CI's having the largest coefficients. A coefficient threshold of at least 0.030 was systematically maintained for the ion states (3a and ( le )" \ but a higher one was used for calculation of the (2a satellite states as discussed later (see Sec. 6.3). The dimension of the neutral MRSD-CI was 31,845 out of the total of 3,465,270 possible singly and doubly excited configurations. This MRSD-CI energy is -56.5160a.u.. Based on the estimated non-vibrating, non-relativistic, infinite nuclear mass total energy of -56.563a.u. [FB87], the correlation energy based on the Hartree-Fock limit is -0.337a.u. . Using this estimate, the calculations recovered 86.5% of the total correlation energy (see Tables 6.1 and 6.2). 161 6.2.2. Inclusion of Correlation: Calculation of the Ion-Neutral Overlap Distribution The highly extended 126-GTO basis set is used as the foundation for more accurate calculation of the E M S cross section by incorporating correlation in both the neutral and the final ion state, i.e. the full ion-neutral overlap distribution, O V D (see Eqn. 2.11). The calculations on the ion states of ammonia were performed using the same molecular orbital basis as for the neutral molecule. The overlap amplitudes computed from the CI wavefunctions for the ion states then have the same form as an M O expanded in the neutral basis. The calculation of the ion states involved a similar process to that of the neutral with the exception of no R H F calculations being done. H F SD-CI and MRSD-CI were done for the (3a i ) " 1 and (le)" 1 ionization processes. The coefficients for the MRSD-CI were chosen based upon the coefficient contribution using a threshold of 0.030 for (3a,)" 1 and ( l e ) ' 1 . The energies for these two cation states are listed in Table 6.2. The calculated vertical L P . values (10.94eV and 16.50eV) are in good agreement with the experimental values (10.85eV and 16.5eV) [BS75, AC78]. The corresponding calculated pole strengths (=0.87) shown in Table 6.3 indicate that these states correspond to essentially single particle states. In contrast the (2a,) 1 process leads to a manifold of final ion states due to many-body effects. The calculated pole strengths and energies for the (2a,)" 1 ionization process are also shown in Table 6.3. In order to get the pole distribution of the (2a,)" 1 a different type of CI was performed. The calculation used a symmetrically closedt set of configurations t A l l calculations were done in C g symmetry and closure with respect to C g y point group operators were maintained. \ Table 6.2. CI Calculations (a.u.) of the Ground and Final Ion States of NH 3 Using the 126-GTO Basis Set STATE E(CI) E(SCF) AE(CI)3 E(exptl)b E(HF)f r . *c,d _. ,_, * c, e E(corr) %E(corr) NH 3 X 'A, -56.5160 -56.2246 0.289 -56.563 -56.226 0.337 86.5 NH3+ (3a,)"1 -56.1212 -55.7950 0.326 -56.1642 -55.796 0.368 88.6 NH3* (1e)"1 -55.9072 -55.5957 0.312 -55.9566 -55.597 0.360 86.6 * AE(CI) • E(CI) - E(SCF) For the ion states, E(exptl)* -(56.563 - I.P.) a.u. c Includes relaxation for ionic states. E(corr) » E(exptl) - E(HF) for neutral, E(corr) « E(exptl) - E(Koopraans) for Ion. 8 %E(corr) • AE(CI)/E(corr) x 100 For the ionic states the energy refers to a Koopmans' energy. Oi to T a b l e 6 . 3 . CI C a l c u l a t i o n s ( 126 -G(C I ) ) of the P o l e S t r e n g t h s and E n e r g i e s (eV) i n the B i n d i n g Energy Spect rum o f N H3 CI C o e f f i c i e n t s P r e d i c t e d I n t e n s 1 t y STATE E n e r g y 3 b P o l e S t r e n g t h C ( 2 a , )C C ( 1 e ) C C ( 3 a , ) C S 2 C 2 ( 2 a ( ) ( 3 a , ) " 1 1 0 . 9 4 d 0.8744 -0 .0027 0 .9996 ( l e ) " 1 1 6 . 5 0 d 0.8781 1 .000 ( 2 a , ) " 1 26.32 0.0137 0.9854 -0 .1312 0 . 0 1 3 3 * 28 .49 0 .4818* 0 .9938 0 .1097 0 . 4 7 5 8 * 30.22 0 .1358* 0 .9877 - 0 . 1 4 3 9 0 . 1 3 2 5 * 31 .03 0 .0750* 0.9897 0 .0947 0 . 0 7 3 5 * 31 .05 0 .0017* 0 .9505 0 .1320 0 . 0 0 1 6 * 32 .86 0 .0524* 0 .9909 0 .1128 0 . 0 5 1 5 * 34.61 0 .0434* 0 .7180 - 0 . 6 8 6 2 0 . 0 2 2 3 * 35.51 0.0004 0 .5447 0 .0493 0.0001 35.86 0 .0003 -0 .3094 0 .5978 0 .0000 36.08 0.0022 0.3064 0 .8975 0 .0002 36 .59 0 .0039* 0 .9859 0 .1226 0 . 0 0 3 8 * r ( * )=0.794 I ( * ) = 0 . 7 7 5 N e u t r a l r e f e r e n c e energy 1s - 5 6 . 5 1 6 0 a . u . S p e c t r o s c o p i c f a c t o r ( S 2 ) as g i v e n 1n Eqn . 2 . 3 2 . C o e f f i c i e n t s o f p r i m a r y h o l e o r b i t a l s 1n the g e n e r a l i z e d ' o v e r l a p ' o r b i t a l ( s e e E q n . 2 . 1 1 ) . E x p e r i m e n t a l v e r t i c a l IPs f o r the ( 3 a . ) and (1e) p r o c e s s e s a r e 10.85 and 16.5 eV, r e s p e c t i v e l y [ T B 7 0 ] . 1— 1 1 O) 00 1 6 4 containing the largest coefficient contributions for 1 5 2 A , roots from a preliminary calculation using all single excitations from all hole states. These reference configurations were then used in a calculation involving up to triple excitations from all hole states with perturbation selection. The accurate CI wavefunctions, as determined from the study of one-electron properties , were then used to evaluate the molecule-ion "overlap" <;qi^ N 1|ql^N>^ ^ Qc ^ g £j w a v e f u n c t i 0 n s have the same form as an MO expanded in the 126-GTO basis. The overlap orbitals were then normalized using S w h e r e , S 2 = | |< * f N _ 1 |*0N> I | 2 [ 6 . 1 ] o S takes values near 0.9 for primary hole states without strong satellites. This 2 procedure removes the spectroscopic factor (or S ) information from the calculated 2 OVDs (i.e. the OVDs are each normalized to unity) however the values of S are shown in Table 6 . 3 . The "overlap" orbitals can be expressed in terms of the molecular orbitals used in the CI calculation as f I = L i C i l ^ i [ 6- 2 ] Sum rules can be derived to show that the sum of the coefficients squared times the spectral factor squared equals 1 / 2 P - when summed over all M = + 11 s 1/2 ion states. P J J is the occupation of orbital \[/^ in the neutral wavefunction. The values of C-T show clearly which primary hole state \p'^ is associated with 165 each CI state. The spectral constants S 2 , C ^ and S 2 C 2 are listed in Table 6.3 for the ion states of the molecule. A t least 78% of the sum Z C j T 2 S j 2 was accounted for the 2 a , " 1 process (Table 6.3). The spherically averaged momentum distribution was then generated for each SCF occupied orbital as well as the normalized overlap distribution (OVD) using the ion-neutral overlap. In addition each neutral molecule ground state natural orbital resulting from the CI of the ground state molecule alone was evaluated. To account for the finite experimental momentum resolution , a momentum resolution function (Ap = 0 .15a o " 1 ) was folded into the calculations. 6.3. BINDING ENERGY SPECTRA Ammonia, in its ground neutral 1 A , state, has C g y symmetry and the electronic configuration can be written as: The binding energies of the two outer valence orbitals of N H 3 are well known from high resolution U V photoelectron spectroscopy using He l sources [TB70, RK73]. In addition, low resolution binding energy spectra including the 2a , inner valence orbital are available from X P S [BS75, AC78], dipole (e,2e) spectroscopy [VB72] and E M S [HH76]. Fig. 6.1 shows the binding energy spectra of N H 3 obtained in the present work at an impact energy of 1200eV + binding energy and at relative azimuthal angles of 0° and 8 ° . The relative intensities in the 0° ( 1a , ) 2 (2a , ) 2 (1e)« (3a , ) 2 166 spectrum ( p ^ O . l a o " 1 ) and the 8° spectrum (p = 0 . 6 a o ~ 1 ) reflect the different symmetries of the valence orbitals of N H 3 . In particular the outermost 3a , and le orbitals are clearly of dominant 'p-type' character whereas the 2a , inner valence orbital is 's-type'. t The energy scale in Fig. 6.1 has been set by aligning the spectra with the known vertical ionization potential of the 3 a , orbital (10.85eV) as measured by photoelectron spectroscopy [RK73]. On this basis the respective maxima of the peaks observed for the le and 2a , bands occur at values consistent with earlier measurements in UPS [TB70, RK73] and X P S [BS75, AC78]. The binding energy positions are also consistent with an earlier low momentum resolution E M S study ( E o = 4 0 0 eV) [HH76] done at <fi= 10°. This earlier spectrum corresponds to p = 0.5 a 0 ~ 1 and has an energy resolution of 1.44eV fwhm. In the ionization of inner valence orbitals many-body structures (satellites) are frequently observed [CD 8 6]. A similar case is seen in the inner valence region of N H 3 as shown by the broad peak and the structured tailing on the high energy side (30-45eV) of the (2a,)" 1 hole. Similar broad tailing can be seen in an earlier E M S study of N H 3 [HH76] as well as in dipole (e,2e) [VB72] and X-ray photoelectron spectroscopies [BS75, AC78]. A recent synchroton radiation PES study reported by Banna et al. [BS87] also shows similar structure. The M g K a [BS75] and the Zr M ^ [AC78] X P S spectra both show indications of additional structure located on the high energy side of the main (2a,)" 1 peak although these authors did not comment specifically on this at the time. In these t The 's-type' and 'p-type' nomenclature is used to refer to whether the momentum profile (XMP) has a minima at p - 0. This follows from the fact that theoretical momentum distributions have a node at p = 0 for atomic orbitals with 1*0 [LN75]. 1 1 1 1 1 1 1 1 1 1 1 1 I 1e 3a^ IT 1 1 NH 3 4>=8° (b) S W A * —i—i—[•• i i i i i 1 ' ' 1 1 1 1 1 I l l 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 BINDING ENERGY (EV) T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r I 1 i i i i i I i i i i i i 1 1 1 1 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 BINDING ENERGY (EV) • 6.1 Binding energy spectra of N H 3 measured by electron momentum spectroscopy at an impact energy of 1200eV + binding energy (a) 0 = 0° and (b) 0 = 8 ° . 168 earlier X P S studies the choice of background (sloping) is somewhat subjective and this introduces some uncertainty in the determination of intensities and may also preclude observation of the low intensity structures at higher binding energies. This complication does not occur in the E M S spectra since the coincidence method ensures a level baseline. To obtain improved statistics in the many-body region corresponding to the inner valence ionization spectrum, the 0 = 0° and <j> = 8° spectra above 21 eV have been summed and the resulting profile is shown in Fig. 6.2a. The spectrum is well fitted by a convolution of five gaussian peaks of equal width (3.6 eV fwhm). The fitted peak positions and intensities are reported in Table 6.4. This only partially resolved many-body structure in the inner valence spectrum is compared - with four different calculations in Figs. 6.2b-6.2e. These involve the present full 126-G(CI) calculation (see Tables 6.3 and 6.4) and a small CI calculation using the 126-GTO basis as well as the Green's function caculations of Bieri et al. [BA82] and Cacelli et al. [CM82] (see also Table 6.4). The theoretical pole strength distributions are indicated by vertical bars at the appropriate energies and gaussian curves (3.6eV fwhm) are convoluted, summed and scaled (x2.5) to yield the theoretical binding energy profile (solid line) in each case. The 126-G(CI) many-body calculation which is reported in the present work used the extended 126-GTO basis set (see earlier discussion). The actual S 2 factors or pole strengths are shown in Tables 6.3 and 6.4 and Fig. 6.2b. The total pole strength 'recovered' in the calculation in the energy range 20-37eV is 0.775. A 169 S < V i NH Many-Body S t r u c t u r e I nner Va! ence Regi on 20.0 24.0 28.0 32.0 MJO 40 X) 4AJ0 48.0 i Ul a: i 2 1 1 1 r 1 1 1 j ! 1 j 1 j 1 N H 3 (2a,) 126-G(CI) _ J 1 ' • H I L ' 1 1 1 1 1 1 -? 1 1 1 1 1 T~ NH, - l 1 r-(2a,) B i e r l et al . Greens Funct i on (d) • • • • 20.0 24.0 28.0 32JD J6-0 40 X) *4J0 4».0 2QJ0 2*J> 2BX 12.0 XJO *OS> **X> 4«J) N H 3 (2a,) Srrol I 126-C(CI) M R - l p 2 h - C I ( c ) 20.0 24J0 MJO 12J0 MA 40.0 **Jt BINDING ENERGY ( E V ) - i — i — i i i — i — i — i — i — i i N H 3 (2a,) Cocel I i et al . Greens Funct i on ( e ) - I I I I L . 20.0 24.0 2tU> 32X1 XD *0.0 44J0 BINDING ENERGY ( E V ) Fig. 6.2 Comparison of the inner valence binding energy spectrum with theoretical predictions. The experimental inner valence spectrum (a) corresponds to the <j>= 0°+ 8° summed spectrum. The calculated spectra refer to (b) the present full 126-G(CI) calculation , (c) the MR-lp2h-CI calculation and the Green's function calculations of (d) Bieri et al. [BA82] and (e) Cacelli et al. [CM82]. Table 6.4. Experimental and calculated energies (eV) and relative pole strengths for the Ionization of the 2a, orbital of NH3. E x p e r i m e n t a l ' T h e o r e t i c a l Relative Pole Strengths(this work) This Work(126-G(CI)) Ref.[BA82] Greens Function Ref.[CM82] Greens Function Energy EMS(0' ) EMS(8" ) EMS(0'+8* ) Energy Rei. Pole Energy Rei. Pole Energy Rei. Pole Strength 0 d Strength Strength 27.6 100. 100. 100. 26.32 2.7 26.37 4.3 27.45 36.0 30. 3 13. 18. 16. 28.49 100. 27.48 100. 29.04 100. 33.2 13. 13. 13. 30.22 27.8 29.50 31.0 35.5 12. 14. 13. 31 .03 15.4 30.28 18.0 41.8 8 . 12. 10. 31 .05 0.3 32.86 10.8 34.61 4 . 7 -36.59 0.8 Pole strengths are relative Intensities with the major pole normalized to 100%. Estimated experimental pole strength uncertainty 1s ±4%. S 2C 2-- see text. Table 6.3 and F1g. 6.2b. S , spectroscopic factor. 171 smaller CI similar in spirit [D87] to the one-particle two-hole Tamm Dancoff calculation, which is referred to as MR-lp2h-CI, was done using the 126-GTO basis to obtain a many-body spectrum extending to a larger energy range and is shown in Fig. 6.2c. This calculation yielded 48 2 A , roots in the energy range 20-47eV. Although the MR-lp2h-CIt is of lower accuracy than the full 126-G(CI) calculation (Fig. 6.2b), the MR-lp2h-CI calculation gives pole energies whose relative energy spacings are reasonably accurate. A total pole strength of 0.857 was 'recovered' with this method in the energy range 20-4 l eV, which is comparable to that of the experimental binding energy spectrum. The -3.0eV shift of the calculated binding energy spectrum (see Fig. 6.2c) relative to experiment is due to the particular choice of the reference energy (E =-56.28lla.u.) for the initial state neutral molecule. The Green's function calculation of Cacelli et al. [CM82] involved a renormalized optical potential and employed a 44-GTO basis set. The calculated energy from the 44-GTO wavefunction is -56.4086a.u. and the calculated dipole moment is 1.845D [CM82]. This calculation however yields only two poles in the inner valence region and a profile quite different from the experimental results at higher binding energies (compare Figs. 6.2a and 2e). The calculations of Bieri et al. [BA82] employed a similar Green's function approach. The ionization energies of the main ( 2 a , ) t r a n s i t i o n as well as the satellite lines were calculated [BA82] using the two-particle-hole Tamm-Dancoff Approximation (2ph-TDA). The calculation employed an extended gaussian basis set of ( l ls ,7p, ld/6s, lp) contracted to [5s,4p,ld/3s,lp]. This calculation also fails to predict the higher energy poles (compare Figs. 6.2a and 6.2d). t The MR-lp2h-CI was generated using 12 dominant configurations for the neutral state and all one-particle two-hole configurations for the ion. This gave a total of 8094 configurations. 172 As can be seen in Fig. 6.2 the agreement with the experimental binding energy profile in terms of energy positions and intensities is somewhat better for the full 126-G(CI) calculation (Fig. 6.2b) than for either of the Green's function calculations in the region below 35eV which represents the upper energy limit for these three calculations. A few higher energy poles are predicted by the full 126-G(CI) calculation. These results likely reflect the basis set dependence of the calculated pole strength distribution. Increased flexibility in the basis set has been found to provide a better description of the pole strength over a larger energy range [FG87]. The MR-lp2h-CI calculation (Fig. 6.2c) gives a fairly good estimate of the overall pole strength distribution and relative energies over the entire experimental energy range (20-45ev) although the absolute energy scale is shifted by -3.0eV. It is expected that the extension of the full 126-G(CI) calculation beyond =36eV would provide a similar pole strength distribution. The similar spectral shape obtained throughout the 24-45eV region at both 0 = 0° and 0= 8° (see Fig. 6.1 and Table 6.4) indicates that the poles in this region are predominantly s-type and therefore belong to the (2a,)"''' hole state. However it should be noted that although some p-type poles due to (3a,)"''' are predicted in this region they are of negligible intensity (see Table 6.3). The (2a,)"''' assignment of the structure to the high energy side of the main (2a,)"^ peak is further confirmed by the experimental momentum profile measured at 32.2 eV (see later discussion). In the comparison of binding energy spectra as obtained from E M S and photoelectron spectroscopy a few points should be remembered. The relative 173 intensities corresponding to ionization from the different molecular orbitals measured by the two techniques are expected to be different [M85]. The differences in the intensity ratios between E M S and X P S indicate the different regions of the electron distribution being probed. X-ray photoelectron spectroscopy probes the high momentum components (e.g. Mg K a =1253 .6eV, p = l O a 0 " 1 ; Y M ^ = 132.3eV, p = 3 a 0 _ 1 ) whereas low momentum components (0.1-2.0a o " 1 ) are probed in E M S . Notwithstanding the differences in momentum the only case where comparison between relative intensities measured by both techniques is likely to be valid is the comparison of relative intensities for states within the same symmetry manifold [M85]. In such cases close agreement (when effects of different energy resolutions are taken into account) has been obtained between E M S and intermediate photon energy (80-120eV) photoelectron spectroscopy as for example in the Ar 3s" 1 [BB87d] many-body structures in the inner valence spectra. It should be noted that relative intensities of the different inner valence poles will be strongly influenced by the (photoelectron energy dependent) dipole matrix element in photoelectron spectroscopy (or dipole e,2e) at lower photon energies where the individual partial photoionization cross-sections are still rapidly changing. As such, attempts [A86, AC87] to compare intensities in lower energy PES and in E M S (i.e. binary e,2e) are not meaningful on a quantitative basis. It should also be remembered that any such comparisons are also influenced by the often large differences in energy resolution. 174 6.4. COMPARISON OF EXPERIMENTAL MOMENTUM PROFILES WITH THEORETICAL PREDICTIONS The experimental momentum profiles (XMPs) of the valence orbitals (3a, , le , 2a,) of N H 3 shown in Figs. 6.3a-6.3f have been placed on the same intensity-scale (i.e. absolute to within a single factor) as described below. The <j>= 8° binding energy spectrum (Fig. 6.1b) serves as the basis for normalization of the experimental momentum profiles. The peaks in the <p= 8° outer valence binding energy spectrum have each been fitted with a gaussian peak taking into account the energy resolution as well as the known vertical IPs and Franck-Condon widths. Likewise the inner valence region has been fitted (see Sec. 6.3) using a template comprising five peaks each with 3.6eV fwhm. A l l the strength in the region 22-45eV is assigned to the (2a ,)"* state (see discussion below). Considering the relative peak areas this method yields a 3a , :16:2a, intensity (area) ratio of 1.00 : 1.65 : 1.61 at the respective momentum values (for each orbital) corresponding to # = 8 ° . This normalization method which is similar to that employed in an earlier E M S study of H 2 0 (see chapter 5) [BB87] permits a very stringent quantitative comparison of the X M P s with all calculated MDs and OVDs with only a single point normalization of experiment to one calculation on one orbital. A l l other experimental and theoretical normalizations are preserved. The energy value in each diagram (Fig. 6.3) below the orbital name indicates the experimental 'sitting binding energy' , i.e. the energy at which the particular X M P was measured. It can be seen from Figs. 6.3c and 6.3f that the X M P corresponding to the satellite at 32.2eV on the 175 higher energy side of the main (2a,)"1 peak (see Fig. 6.1) clearly belongs to the (2a,)"1 manifold. The satellite intensity has been height normalized to the 'main' 2a, XMP to facilitate shape comparison in Fig. 6.3f. Any contribution from the 3a," 1 and le"1 hole states in the 24-45eV region is expected to be small since = 90% of the 3a," 1 and le*1 pole strengths are predicted to be in the main lines (see Table 6.3). The experimental results (Figs. 6.3a-6.3f) are consistent with the earlier low momentum resolution EMS experiments of Hood et al. [HH76] taken at lower impact energy (400eV). It is of particular interest to note that the P m a x of the 3a, XMP (0.52a 0~ 1) measured in the present study is similar to those reported (O.Sao"1) by Hood et al. [HH76] and by Camilloni et al. [CS76]. However the 3a, XMP measured in a recent study of N H 3 by Tossell et al. [TL84] shows an XMP with a P m a x =0 .4 la 0 ~ 1 and which is also relatively broader. In contrast the present study which has permitted the XMPs to be placed on a common intensity scale also allows a much more stringent quantitative test of the theoretical calculations. The much improved momentum resolution (0.l5a 0 ~ 1 ) in the present work compared with that used (0.4a 0~ 1) by Hood et al. [HH76] also permits a much more stringent comparison of the measured XMPs with calculations particularly in the low momentum region where the largest discrepancies were suggested to occur. At the higher momentum resolution available in the present work this comparison is no longer dominated by the momentum resolution instrumental effect. In addition the correct relative intensity scale obtained in the present study provides an additional quantitative test of the different wavefunctions which was not possible in the earlier studies where each NT E R M E D I AT E 176 MOMENTUM (A.U.) Fig . 6.3a-c Comparison of valence X M P s of N H 3 with MDs calculated from intermediate quality wavefunctions (a-c). Momentum distributions calculated from the (1)DZ, (2)6-311G, (3)6-31 I G * * , (4)6-3ll + G, (5)HDD2G, (6)56-GTO, (7J119-GTO and (8) 126-GTO wavefunctions are placed on a common intensity scale by using a single point normalization of the 126-GTO (and 126-G(CI) calculation) calculation on the le X M P (see text for details). N E A R H A R T R E E - F O C K 177 MOMENTUM (A.U.) Fig. 6.3d-f Comparison of valence X M P s of N H 3 with MDs calculated from near Hartree-Fock wavefunctions (d-f). Momentum distributions calculated from the (1)DZ, (2)6-311G, (3)6-311G**, (4)6-311 + G, (5)HDD2G, (6)56-GTO, (7)119-GTO and (8)126-GTO wavefunctions are placed on a common intensity scale by using a single point normalization of the 126-GTO (and 126-G(CI) calculation) calculation on the le X M P (see text for details). orbital X M P was separately height normalized to each theoretical M D . 178 Theoretical MDs were calculated (see Sec. 6.2) from the various wavefunctions shown in Table 6.1. A l l calculations were spherically averaged (since the target molecules are randomly oriented) and have been folded with the experimental momentum resolution. The SCF wavefunctions can be placed in two main groups, namely (i) Intermediate quality without polarization (DZ, 6-311G) and with polarization or diffuse functions (6-311G**, 6-311 + G, HDD2G); (ii) Near Hartree-Fock quality (56-GTO, 119-GTO, 126-GTO). The calculations using these two groups of single determinant SCF wavefunctions are presented in Figs. 6.3a-6.3c and 6.3d-6.3f, respectively. The two categories are based largely on the quality of the total energy calculated using these basis sets. The details of these basis sets together with the respective calculated total energies, dipole moments and p are given in Table 6.1. In order to compare i l l 3.X the measured X M P s of the valence orbitals of N H 3 with the theoretical calculations, the 'best quality' wavefunction (i.e. the 126-GTO M D which is essentially identical to the 126-G(CI) O V D , see Fig. 6.4 below) is height normalized to the le X M P . This normalization procedure was chosen on the basis of the relative insensitivity of the le M D to basis set and correlation effects (see Figs. 6.3b, 6.3e and 6.4b). A l l calculations for all three orbitals maintain their correct values relative to the single point normalization between experiment and the 126-G(CI) OVD on the le orbital. T 1 1 1 1 1 1 1 1 1 1 1 1 r MOMENTUM (A.U.) Fig. 6.4 Comparison of the valence X M P s of N H 3 with ion-neutral CI overlap distributions (OVDs) calculated from correlated wavefunctions. Distributions calculated from the (7)119-GTO, (7c)119-G(CI), (8) 126-GTO, and (8c)126-G(CI) are placed on a common intensity scale by using a single point normalization of the 126-GTO (identical to the 126-G(CI) calculation) calculation on the le X M P (see text for details). 180 Starting with the le X M P (Figs. 6.3b and 6.3e) it is apparent that all wavefunctions from DZ to near Hartree-Fock give MDs of similar shape and magnitude which are in quite close agreement with experiment. However the calculated MDs maximize at a momentum slightly higher than the P m a x of the le X M P . A l l calculations also give intensity in the low momentum region (0.1-0.5a ©" 1 ) somewhat less than is observed experimentally. In contrast, the situation for the 3a, X M P (Figs. 6.3a and 6.3d) is quite different in that the MDs calculated from the intermediate quality wavefunctions (DZ and 6-311G) are not able to predict the 3a , X M P with regard to either shape or magnitude. However inclusion of diffuse functions as employed in the H D D 2 G and 6-311 + G (Fig. 6.3a) wavefunctions produces calculated MDs in closer but still not good agreement with experiment. This HDD2G basis due to Dunning and Hay [DH77] is a variation of the 6-311G basis set with the addition of diffuse 2s and 2p functions on the nitrogen atom. Although the calculated total energy with this basis set (-56.1790a.u.) is very similar to that of the DZ (-56.1714a.u.) and 6-311G (-56.1777a.u.) wavefunctions, the calculated HDD2G momentum distribution is quantitatively different and in particular provides a much improved description of the low momentum region. It is of interest however that the calculated dipole moments (Table 6.1) do not differ very much for these wavefunctions. This situation is similar in going from the 6-311G wavefunction to the 6-311+ G wavefunction which includes diffuse s- and p-functions on the nitrogen atom. Since these diffuse functions, unlike polarization functions, do not contribute appreciably to the bonding, there is very little improvement in energy. However the improvement in the calculated MDs is dramatic as can be seen in Fig. 6.3a. 181 In contrast the 6-311G** wavefunction which has polarization functions on the nitrogen and hydrogen atoms yields a M D (see Fig. 6.3a) only slightly different from that calculated using the unpolarized 6-311G wavefunction although the total energy is significantly improved from -56.1777a.u. to -56.2102a.u. and the dipole moment is also correspondingly improved (see Table 6.1). This result is consistent with the well known fact that energy minimization, although a necessary condition, is of itself not sufficient to guarantee a wavefunction good enough to calculate all properties with high accuracy especially. Likewise inclusion of diffuse functions to improve the calculated MD is of itself insufficient to guarantee a good 'universal' wavefunction as can be seen above for the HDD2G and 6-311+ G wavefunctions. The 2a , X M P (accounting for the intensity over the whole inner valence region, see above) shown in Figs. 6.3c and 6.3f is overestimated slightly (—5%) by all S C F calculations (DZ, 6-311G, 6-311G**, 6-311+ G, HDD2G) although the shape is reasonably predicted. Theoretical MDs calculated from the extended gaussian basis sets of Rauk et al. [RA70] (56-GTO), Feller et al. [FB87] (119-GTO) and a new 126-GTO basis set are also compared with the valence orbital X M P s of N H 3 on the right hand panels of Fig. 6.3. A t the SCF level the 126-GTO wavefunction (E = -56.2246a.u.) is optimal with regards to the calculated total energy and dipole moment. The next best wavefunction is the 119-GTO wavefunction (E = -56.2245a.u.) followed by the 56-GTO wavefunction (E =-56.2219a.u.). The near Hartree-Fock wavefunctions (126-GTO, 119-GTO and 56-GTO) predict valence X M P s of N H 3 quite similar to 182 those using intermediate quality wavefunctions except in the case of the 3a , M D . The simpler and variationally inferior 6-311+ G wavefunction yields an M D for the 3a , orbital very close to that calculated, using the 119-GTO and 56-GTO near Hartree-Fock wavefunctions. The le and 2a , MDs calculated from the near Hartree-Fock wavefunctions (Figs. 6.3e and 6.3f) show similar behaviour to those calculated using intermediate quality wavefunctions (Figs. 6.3b and 6.3c) although there is some improvement for the le orbital using the 126-GTO wavefunction. The MDs calculated using the 126-GTO, 119-GTO and 56-GTO wavefunctions still slightly overestimate the magnitude of the 2a , X M P . The reasonably close agreement for the case of the 3a, X M P with the M D calculated from the 6-311+ G wavefunction is perhaps surprising because the large dipole moment (2.2292D) predicted with this wavefunction is significantly greater than the experimental value of 1.47149 +0.0015D. However the dipole moments predicted by the variationally superior 126-GTO, 119-GTO and 56-GTO wavefunctions (1.6417D, 1.6442D and 1.660D, respectively) are in much better agreement with experimental results. Other calculated properties such as the quadrupole moment (-2.1960a.u.) and < r 2 > e (26.7134a.u.) obtained using the 126-GTO wavefunction are in good agreement with the experimental values for the quadrupole moment (-2.42±0.04a.u.) [DD82] and < r 2 > g (25.501a.u.) [H67]. It can be seen that from an overall standpoint the 126-GTO wavefunction gives the best description among the SCF wavefunctions investigated in the present work. Since the calculated total energy with this basis set is quite close (1 millihartree) to the estimated Hartree-Fock limit of -56.226 + 0.001a.u. [FB87], basis set saturation has effectively been established. This can be seen by the 183 fact that there is only a very slight improvement of the 126-GTO M D relative to the 119-GTO M D for each orbital as shown in Fig. 6.3. Experience in the previous E M S work on H 2 0 [BB87] has shown that basis set saturation is only achieved very close to the Hartree-Fock limit (— within 0.5 millihartree). The possible reasons for the remaining differences between experimental X M P s and the MDs calculated with near Hartree-Fock SCF wavefunctions have been discussed earlier [BB87]. Assuming the adequacy of the PWIA description [WM78] as seems reasonable from the consideration of a wide range of E M S studies [BB87], these differences are most likely due to electron correlation effects and possibly the influence of vibrational motion. The former are discussed in a separate section below. Vibrational motion (symmetric modes) effects on the MDs of H 2 0 have been found to be negligible by Leung and Langhoff [LL87] and it is also of interest to consider whether this phenomenon is of importance in N H 3 . Vibrational effects in N H 3 have been extensively studied, most specifically in the acurate prediction of H e l photoelectron vibronic intensities accompanying the 1 A 1 — t ? A , (3a,)" 1 and 1 A 1 — » 2 E , (le)" 1 transitions [AR82, CD76]. These effects are particularly interesting because the N H 3 + ( 2 A ,) ionic state is believed to be of planar geometry (i.e. D ^ symmetry) [AR82]. In an effort to estimate the vibrational effects in N H 3 , Feller [F87] has investigated the effects of bending distortions on the calculated MDs. The symmetric bending distortions did not yield a significantly different vibrationally averaged M D from that calculated at the equilibrium geometry. The change was mostly at the peak maximum and was nowhere more than 1% [F87]. 184 6.5. ION-NEUTRAL OVERLAP DISTRIBUTIONS (OVDS) The recent E M S study of H 2 0 (see chapter 5) [BB87] has shown that accurate prediction of X M P s requires inclusion of correlation and relaxation effects especially for the outermost ( l b , ) valence orbital. This was done (in collaboration with Prof. E.R. Davidson) by performing separate multi-reference SD-CI calculations for the neutral and respective ionic states having first achieved basis set saturation with extended basis sets at the SCF level. These procedures recovered at least 86% of the total correlation energy. These CI wavefunctions were then used for calculation of the ion-neutral overlap (OVD). A similar CI study (see preceeding section) has been carried out in the present work for the N H 3 molecule and the respective N H ^ " 1 " ions. Using the 126-GTO extended basis set reported in the present work separate MRSD-CI calculations have been performed for the neutral N H 3 molecule and for the (3a,)" 1 , ( le)" 1 and (2a,)" 1 ionic states at the experimental neutral geometry. The particulars are shown in Tables 6.2 and 6.3. The calculated total energy for the neutral state of N H 3 using the 126-G(CI) and 119-G(CI) wavefunctions recovered 86.5% and 86.3% respectively of the estimated total correlation energy (-0.337a.u.) [PB75, FB87]. A consideration of the OVDs gives the spectroscopic factors (pole strengths) shown in Table 6.3. It can be seen that the values for the 3 a , " 1 and l e " 1 processes are both approximately 0.87 indicating minimal splitting of the ionization (i.e. a single particle description is reasonably adequate). However the CI calculations indicate that the 2 a , " 1 process leads to many final ion states of which a large fraction (0.77) is recovered below a binding energy of 37eV which was the limit of the calculation. An even larger fraction (0.885) is recovered using the MR-lp2h-CI calculation in the energy range 20-47eV. In Fig. 6.4 the valence 185 X M P s are compared with the OVDs resulting from the above CI ion-neutral overlap calculations with the same normalization as before on the le X M P maximum using the 126-G(CI) calculation. The 119-GTO and 126-GTO Hartree-Fock limit S C F calculations of the MDs reported in the previous section are also shown for comparison. For this purpose the OVD calculations were each normalized to unity. The 119-G(CI) OVD was also calculated for the 3a, orbital and is shown for comparison in Fig. 6.4a. It can be seen from Fig. 6.4 that incorporation of CI gives a dramatic quantitative improvement in the predicted O V D distributions for the 3a, orbital. It is however noteworthy that there is no detectable change in going from SCF to CI for the le orbital. In the case of the 3 a , orbital inclusion of electron correlation in both neutral and ion states results in an O V D with more density in the low momentum region. This results in a higher cross-section and a slightly smaller p in better agreement with experiment on the leading edge max of the 3a, X M P (see Fig. 6.4a and Table 6.1). Quite similar behaviour has been observed in the case of the outermost l b , orbital of H 2 0 (see Fig. 5.6) [BB87]. This has been understood in terms of the 'non-characteristic' contributions not represented in the T H F A momentum distributions. Similar to the case for H 2 0 [BB87], the ion-neutral overlap amplitude for N H 3 can be expanded in the form, < p * f N ~ 1 | * 0 N > = C , ^ 3 a 1 + C 2 ^ 4 a 1 + . . . [6.3] where the 'non-characteristic' orbital contribution is due to the lowest lying 4a , virtual orbital (compare with Fig. 5.6 of chapter 5). These orbitals are generally 186 very diffuse in position space and, conversely, contribute to the low momentum region of the calculated O V D . In addition, it was also shown earlier in chapter 5 [J3B87] that the increased low momentum components in the calculated O V D are due to the combined effects of initial state correlation and a contraction in the final ionic state. In the case of the le X M P (Fig. 6.4b) the calculated O V D using the 126-G(CI) wavefunction yields quantitatively the same profile as the M D calculated from the 126-GTO wavefunction. This result indicates that correlation and/or relaxation is not very important in predicting the le X M P . Analogous results have been observed in the case of the l b 2 X M P of H 2 0 (see chapter 5) [BB87] which in some sense is like the le orbital in N H 3 (i.e. largely a bonding orbital). The effect of correlation and relaxation in the calculated inner valence 2a , O V D is shown in Fig. 6.4c. The 126-G(CI) OVD was obtained from a pole-strength weighted average of the (very slightly different) OVDs calculated for each of the 2 a , " 1 final ion states found in the energy region 26-36eV (see Tables 6.3 and 6.4). The resulting OVD (Fig. 6.4c) is very similar to the M D calculated using the 126-GTO wavefunction. Both calculations are slightly above the 2a , X M P . This slight difference may be due to one or more of the following effects: (a) additional (2a,)" 1 strength beyond the upper limit of the experimental binding energy spectrum (Fig. 6.2a); (b) limitations in the present theoretical treatment or (c) distortion effects, i.e. failure of the plane wave impulse approximation. A similar level of agreement between calculation and experiment was found for 187 the 2a , X M P in H 2 0 [BB87] which is also split into several final ion states spread over a wide binding energy range. These results for the MDs and OVDs as well as those for the binding energy spectra in Fig. 6.2 clearly show the overall suitability of the 126-GTO basis used for development of 126-G(CI) neutral and ion wavefunctions and the need for adequate incorporation of correlation for describing the subtle features of the inner valence ionization process as well as the X M P for the 3a , outer valence orbital of N H 3 . Furthermore the good level of quantitative agreement achieved for both H 2 0 and N H 3 indicates that the description provided by the PWIA is quite reasonable for E M S studies of these systems at an impact energy of 1200eV + binding energy. Thus neglect of electron correlation rather than distortion effects seems to be the main reason for discrepancies between experiment and theory at the SCF level for the valence orbitals of N H 3 and H 2 0 . The effect of electron correlation is not only manifested in the calculated OVDs of the 3a , and 2a, orbitals but also in improved calculated values of other molecular properties. In particular, the predicted dipole moment using the 126-G(CI) wavefunction (1.5891D) is in closer agreement with the experimental dipole moment (1.47149 + 0.00015D) [MM81]. The only available calculations which yield better theoretical dipole moments are the MRSD-CI calculation of Feller et al. [FB87] which utilized an even larger 130-GTO basis set (1.5881D) and the complete 4th order many-body perturbation theory calculation of Diercksen and Sadlej (1.4991D) [DS86]. It should however be noted that in general, theoretical estimates of the dipole moment do not include vibrational averaging. The dipole moment vibrational correction in N H 3 can be significant (as mentioned 188 in the earlier section) and has been estimated to be of the order of -0.025D [WM76, FB87] which brings the theoretical predictions in slightly closer accord with experiment (see Tables 6.1 and 6.2). Likewise good agreement was obtained betweeen the theoretical and experimental quadrupole moments and < r 2 > e values using extended basis sets (l'30-G(CI), 119-G(CI)) [FB87]. These molecular properties (dipole moment, quadrupole moment, < r 2 > e ) are known to be quite sensitive to the diffuse region of the spatial electron distribution. From the inverse weighting property of the Fourier transform, the experimental momentum profiles are found to be similarly sensitive to the low momentum regions of the electron distribution (in the p-representation). The consistent pattern of good agreement between experiment and calculation not only for momentum distributions but also for a wide range of properties is indicative of the accuracy and general suitability of highly extended and correlated wavefunctions used in these studies for N H 3 and H 2 0 [BB87]. In an effort to explore the possibility of using simpler treatments of correlation (i.e. for the neutral target molecule only) for prediction of momentum distributions, target-natural orbital (TNO) MDs for the valence orbitals of neutral N H 3 using the 126-G(CI) wavefunctions have been calculated. Earlier work on nitric oxide showed excellent agreement (for shape) between the X M P s and the TNOs [CC82] and similar results have been found recently for H 2 S [FB87a] on a quantitative basis. The TNO results for N H 3 are shown in comparison with the 126-G(CI) in the lower portion of Fig. 6.5 while the respective upper sections show the density differences (TNO-SCF) and (OVD-SCF). The target natural 189 orbitalst were obtained from the CI calculations used to produce the 126-G(CI) OVDs. Natural orbitals result from diagonalization of the one-electron density matrix [L59]. It has been known that optimal convergence of the CI expansion can be obtained with natural orbitals. It can be seen from Fig. 6.5 that the 126-G(TNO) treatment does not provide any systematic improvement over the 126-GTO (SCF) MDs. The biggest difference is for the 3a , orbital where the 126-G(TNO) is even lower than the 126-GTO result. 6.6. POSITION-SPACE AND MOMENTUM-SPACE DENSITY MAPS Earlier reported density maps for other molecules have generally been carried out with low quality wavefunctions to minimize computation costs. These calculations have usually been adequate to represent the general features of the charge and momentum distributions and to represent essential ideas of momentum space chemistry. The present work on N H 3 (like that of H z O [BB87]) has however indicated the importance of extended and saturated basis sets which include many diffuse and polarization functions. Therefore two-dimensional density contour maps for the three valence orbitals of an oriented N H 3 molecule in both position and momentum space have been calculated using the 126-GTO SCF wavefunction at the experimental geometry (Fig. 6.6). The corresponding X M P s and the calculated 126-GTO spherically averaged MDs are also shown for comparison. The maps show a cross section through the electron distribution in the x-z plane in the case of the position density map and through the p -p plane in the case of the X z momentum density map. The position density map is in a plane that includes t The occupation numbers for the 3 a , , le and 2a, natural orbitals are 1.966, 1.963 and 1.980, respectively. 2 <Jq" RELATIVE MTENSmr (ARB. UMTS) 0.0 2.0 4.0 6.0 8.0 10.0 -r=n 1 r -DTFERENCE DENSITY (ARB. UNfTS) -1.0 0.0 1.0 2.0 t\3 6 o 3 3 P O 3 3 a-o 3 o 3 3" £ 2 . 3 cn 2. P < (0 <t> P X g. to C T CO q HQ p •a P . 2. 3 p c ^ q p EL » 3 I S Ir s cn a o 5- a c 3" 5' CO < r t . ' o sr P ~ 2 I" p sr> o" 3 S. 3 & rt Q . sr RELATIVE WTENSITY (ARB. UNITS) 0.0 4.0 8.0 12.0 16.0 20.0 DIFFERENCE 0ENSITY (ARB. UNfTS) 2.0 0.0 2.0 4.0 5 i 1 1 1 - --1 1 .—* 2 - 7 1 at X • at CH " f K> M fV* crt o» • < _» f ii A a) /_. o -« £ — z — 5 1 1 1 1 1 L _ _ l _ i J — • RELATIVE NTENSTY (ARB. UNITS) 0.0 10.0 20.0 30.0 40.0 50.0 DIFFERENCE DENSITY (ARB. UMTS) -5.0 0.0 5.0 10.0 061 191 the positions of the nitrogen atom and one of the three hydrogen atoms. The employed molecular (x,y,z) orientation in atomic units is N (0.0, 0.0, 0.127872), H I (1.77164, 0.0, -0.592238), H2 (-0.88582, 1.53428, -0.592238) and H3 (-0.88582, -1.53428, -0.592238). The position density contour map for the outermost 3a , valence orbital (Fig. 6.6a) shows a nodal 'surface' displaced from the nitrogen center as can be seen from the one-dimensional position density distribution projection plots along the x-and z-axes, respectively. Besides the dominating nitrogen 2p function there is also a considerable contribution from the nitrogen 2s and hydrogen Is orbitals. A minimal basis set decomposition of the 3a, molecular orbital shows, *3a, - X ( N 2 s ) + 0 ( N 2p> " ^ " i s * C 6 - 4 ] Since the momentum density at p = 0 is determined solely by the s-type basis functions then, P 3 a i ( 0 ) « X - 3e [6.5] The fact that £33 ^0) is very small does not necessarily imply that X or e are individually small. The corresponding momentum space contour map thus shows some small s-character although the dominating feature is still the nitrogen 2p orbital. The r-space map clearly shows the influence of the hydrogen atoms indicating that some bonding can also be attributed to the 3a, orbital. In recent E M S work it has been demonstrated that this noticeable derealization of the so-called 'lone pair' orbital in N H 3 (i.e. the orbital contains small but significant H i s character) is greatly enhanced by methyl substitution [BB87a, BB87b]. These 192 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY OJ U) UJ 13 -4.0 -2.0 00 2.0 4.0 05 1.0 -4.0 -2.0 0.0 2.0 4.0 0.5 1.0 MOMENTUM (A.U.) Fig. 6.6 Position space and momentum space density contour maps for an oriented N H 3 molecule. Maps were generated using the 126-GTO wavefunction at the neutral experimental geometry. Contours are at 80, 50, 20, 8, 5, 2, 0.8, 0.5, 0.2, 0.08, 0.05 and 0.02% of the maximum intensity. Positions of the atoms in a given plane are indicated. A l l dimensions are in atomic units. 193 trends are also clearly predicted in molecular orbital calculations. Similar effects have been found for H 2 0 and ( C H 3 ) 2 0 [CB87]. These delocalization effects are clearly reflected (Fig. 6.6) in the experimental measurements for the 3a, orbital of N H 3 and emphasize the fact that E M S measures a quantity very closely related to the canonical molecular orbital but significantly different from the simple valence bond idea of an atomic-like lone pair localised on the N atom. In the case of the le orbital the nitrogen 2 p x function is dominant but with some delocalization of electron density distribution in the region of the hydrogen atoms (Fig. 6.6b). The nodal plane in the r-space map (yz plane) is preserved in the p-space map ( p y P z plane). The 2a , r-space and p-space maps (Fig. 6.6c) clearly display the dominant contribution of the nitrogen 2s function. The r-space map also shows significant perturbations across the hydrogen atoms due to some degree of bonding character in the 2a , orbital. 6.7. EXTERIOR ELECTRON DISTRIBUTION (EED) RATIOS AND XMPS Recent studies by Ohno et al. [OM83, OM84, 0186] have investigated the use of exterior electron distribution ratios obtained from Penning ionization electron spectroscopy (PIES) in probing the quality of the long range portion of theoretical wavefunctions. The experimental ratio and also the ratios calculated using some of the wavefunctions used in the present E M S study are shown in Table 6.1. Ohno and Ishida [OI86] have shown that the valence orbital E E D ratios (3a,/le) for N H 3 (see Table 6.1) which have been calculated using the 6-311+ G, HDD2G and 56-GTO wavefunctions are quite close to the experimental PIES value of 2.60 [0186]. In contrast, the 6-311G and 6-311G** wavefunctions give much 194 poorer E E D ratios (1.98 and 1.76, respectively). These E E D results are of some interest in relation to the present E M S work because similar trends in behaviour are observed for the experimental and calculated momentum distributions. This is not unexpected since both E M S and Penning ionization are sensitive probes of the longer range charge distribution (i.e. the wavefunction tails). The 6-311 + G M D which gives the best (but not good) agreement with the 3a , X M P amongst the simple wavefunctions shown in Fig. 6.3 also gives an E E D ratio (2.58) in best agreement (within the limited range of wavefunctions compared) with experiment. The HDD2G and 56-GTO wavefunctions which were found to give MDs in slightly less good agreement with the measured X M P than the 6-311 + G, give ratios (3a,/le) of 2.50 and 2.41, respectively in correspondingly slightly less good agreement with the Penning ionization branching ratio. The 6-311G and 6-311G** wavefunctions which gave MDs in poor agreement with the 3a , X M P also gave E E D ratios in poor agreement with the Penning ionization branching ratio. Due to the fact that the He 2 3 S metastable atom has an energy of 19.82eV only the outermost 3a , and le orbitals of N H 3 are accessible by PIES. It is not possible therefore to compare the inner valence 2a , X M P with the (2a,/le) E E D ratio. A further limitation in the comparison of E E D ratios with experiment is based on the severe assumptions made in deriving the E E D ratio and PIES branching ratio proportionality. Strictly speaking there is no boundary between reacting and non-reacting regions of the electron distribution. The PIES cross section is determined by the overlap integral (over all space) of the involved 195 molecular orbital wavefunction with the vacant inner shell orbital of the metastable atom. Furthermore correlation effects are also ignored in the derivation of the proportionality of Penning ionization cross sections to E E D values [OM84, 0186]. This is not a limitation in the interpretation of E M S cross sections and in fact it has been shown that electron correlation is necessary in describing accurately the 3a, orbital in N H 3 (see earlier discussion) as well as the l b , orbital of H 2 0 [BB87]. However even with the limitations in the PIES and E E D comparison, the results of Ohno and co-workers for N H 3 [0186] and other molecules [OM83, 0M84] show interesting parallels with the results of the corresponding E M S studies and calculations in the present work and it is evident that both techniques are sensitive to the outer spatial regions of molecular orbitals. 6.8. SUMMARY Earlier discrepancies between experimental momentum profiles and theoretical MDs especially for the outermost (3a,) orbital of N H 3 are now largely resolved. The use of highly extended basis sets (saturated in the diffuse function limit) and sufficiently correlated wavefunctions for both the initial neutral state and the final ionic states have led to better agreement with the X M P s as measured by high momentum resolution E M S . These newly developed wavefunctions yield not only accurate momentum distributions but also give the best calculated values for a wide range of molecular properties such as the total energy, dipole moment, quadrupole moment and ionization potentials (for the outer valence and inner valence states) and therefore may be called 'universal' wavefunctions for N H 3 . C H A P T E R 7. F O R M A L D E H Y D E 7.1. OVERVIEW As part of the continuing series of E M S studies of small molecules and the critical evaluation of molecular wavefunctions, this chapter reports a high momentum resolution study of the valence orbitals of H 2 C O . A n earlier preliminary study of the binding energy spectra of H 2 C O using E M S (binary(e,2e)) at two azimuthal angles was reported by Hood et al. [HB76] in 1976. Although done at lower impact energy (400eV) and at poorer momentum resolution (Ap = 0 . 4 a o " 1 ) than the present work, this earlier study [HB76] illustrated the symmetry diagnostic capabilities of EMS in resolving a controversy regarding the energy ordering of the 5a , and l b 2 orbitals of H 2 C O [CD75, TB70]. This was achieved through a two-angle (0 = 0° and 0 = 20°) binding energy scan and observing the relative heights of the relevant peaks. No detailed experimental angular correlation study of the valence orbitals was done at the time although simple calculations of the valence orbital momentum distributions were reported. Other previous related studies on H 2 C O include theoretical investigations of basis set [RE73] and electron correlation effects [ST75] on the Compton profile which is a measure of the total electron momentum distribution and therefore is less sensitive to wavefunction quality than E M S which is orbital-specific. The high resolution Penning ionization electron spectrum (PIES) of H 2 C O has also been reported by Ohno et al. [OT86]. In this study the large relative PIES cross section for the 5a , orbital was noted and compared with calculated exterior 196 197 electron density (EED) values. In the present work the experimental momentum profiles (XMPs) of H 2 G O are reported for the first time and compared with theoretical momentum distributions (MDs) calculated from a range of ab initio wavefunctions. The inner valence binding energy region is also analyzed and compared with a many-body Green's function calculation [VB80]. Formaldehyde was prepared by heating paraformaldehyde at 60°C. The vapour was allowed to pass through a heated Granville-Philips leak valve. The binding energy spectra indicated there were no significant impurities. 7.2. BASIS SETS FOR SCF WAVEFUNCTIONS The measured X M P s in the present work are compared with spherically averaged momentum distributions calculated for a variety of selected ab initio SCF L C A O - M O wavefunctions. The experimental momentum resolution (Ap = 0 . 1 5 a 0 " 1 ) was also folded into the caculations. The wavefunctions cover a wide range from the simple 4-3 IG basis set to a more extended basis set essentially at the Hartree-Fock limit. The details of these wavefunctions together with selected calculated properties are shown in Table 7.1. Important features of the various basis sets are discussed below, (1) 4-31G This "split-valence" basis set [KB80] involves a contracted GTO set comprised of 0[3s2p], C[3s2p] and H[2s]. Four primitive GTOs are Table 7.1. Characteristics of SCF Wavefunctions for HjCO Wavefunct i on 4-31G Carbon and Oxygen Basis Set (8s4p)/ [3s2p] Hydrogen Basis Set (4s)/ [2s] Energy(a.u.) 1 13.6911 Dipole Moment(D) Reference 3 .005 this work 4-31G+G (9s4p)/ [4s2p] (4s1p)/ [2s1p] 1 13.6962 3.002 this work DZ (9s5p)/ [4s2p] (4s)/ [2s] -113.8209 3. 1 10 [SB72] 134-GTO (19s10p2d1f)/ [10s5p2d1f] (10s2p1d)/ [4s2p1d] •113.9202 2.857 [DF86] Hartree-Fock 1 imlt 113.925 Exptl. 114.562 2.3310.02 Estimated Hartree-Fock li m i t [GS74]. Includes relat1v1st1c e f f e c t s [NM69]. Pos i t i v e dipole moment means negative end on oxygen atom ( i . e . C + 0 ) [K060, HL68] 199 contracted to form the Is core and the respective valence orbitals are 'split' into three GTOs contracted to one CGTO and the least tight GTO left uncontracted. Exponents for both valence s- and p-functions are kept the same. 4-31G + G This basis set involves the original 4-3 IG basis (1) plus a set of diffuse functions. The diffuse functions have been used by Chong and co-workers [ZS79, LB83b]. The specific diffuse functions used are: C(a g = 0.02789), O(a g = 0.04216) and H ( a p = 0.1160). It has been shown in earlier E M S studies [BB87, BM87] that diffuse functions are very important in the accurate prediction of orbital X M P s . Double zeta (DZ) Two sets of contracted Gaussian type functions are used for each atomic orbital. No additional polarization functions are employed. In this basis set, proposed by Snyder and Basch [SB72], the least tight s- and p-functions of the nitrogen atom are represented by a single primitive Gaussian function. 134-GTO This extended basis set reported recently by Davidson and Feller [DF86] involves a (19sl0p2dlf) primitive set contracted to [10s5p2dlf] on the oxygen and carbon atoms. The hydrogen basis involved a (10s2pld)—>[4s2pld] contraction scheme. The s-components of the cartesian d functions were deleted. The calculated SCF total energy is -113.9202a.u. which is the best SCF energy for H 2 C O reported to date. 200 A l l wavefunctions were generated using the experimental equilibrium geometry ( r C Q =1 .2078A, r C H = 1.116lA, Z H C H = 116.52°) [T063]. The 4-31G and 4-31G + G wavefunctions were generated using the GAUSSIAN76 package [BW76]. 7.3. BINDING ENERGY SPECTRA Formaldehyde, in its ground neutral 1 A , state, has symmetry and the electronic configuration can be written as: ( 1 a , ) 2 ( 2 a , ) 2 ( 3 a , ) 2 ( 4 a , ) 2 ( 1 b 2 ) 2 ( 5 a , ) 2 ( 1 b , ) 2 ( 2 b 2 ) 2 i v * v v . — core valence The assignment (i.e. ordering) of the valence orbitals of H 2 C O has been the subject of debate [BB68, TB70, CD75, NM69]. On the basis of vibrational analysis and isotopic studies in a photoelectron spectroscopy (PES) study, Turner et al. [TB70] assigned the third and fourth ionization bands to the l b 2 and 5a , orbitals, respectively. This was clearly the reverse of the order suggested by M O calculations at that time [NM69] which were in agreement with the configuration shown above. Although it was acknowledged that the vibrational analysis may be uncertain [BR72] a photoionization mass spectrometric study [GC75] suggested that the original assignment of Turner et al. [TB70] was correct. A many-body Green's function calculation [CD75] however predicted orbital assignments in agreement with the M O calculations [NM69]. A t this juncture, a timely E M S study by Hood et al. [HB76] showed unequivocally that the 5a, and l b 2 orbitals are ordered according to that predicted by earlier M O calculations (see above) [NM69] and Green's function results [CD75]. Further many-body Green's function [VB80] and CI [K81] calculations also confirmed the earlier assignments 201 based on the simpler M O calculations [NM69]. The preceding results clearly illustrate the caution necessary in interpreting PES data especially in the case of overlapping states. E M S with its capability of determining orbital momentum distributions allows unambiguous assignment of ionic states sufficiently separated in energy. Fig. 7.1 shows the binding energy spectra of H 2 C O obtained in the present work at an impact energy of 1200eV + B . E . and at relative azimuthal angles of 0 = 0° and 0 = 6° . The energy scale in Fig. 7.1 has been set by aligning the spectra with the known vertical ionization potential of the 2b 2 orbital (10.9eV) as measured by high resolution photoelectron spectroscopy [TB70]. The estimated vertical IPs for the remaining valence orbitals and their relative ionization intensities, derived from curve-fitting the binding energy spectra (Fig. 7.1) with peak positions and widths as given by P E S [TB70] and convoluted with the instrumental energy resolution, are shown in Table 7.2. The relative intensities in the 0 = 0° spectrum (p = 0 . 1 a o _ 1 ) and the 0 = 6° spectrum (p^O.Sao ~ 1 ) reflect the different symmetries of the valence orbitals of H 2 C O . Quite clearly the well-resolved peaks attributed to the 2b 2 and 4a, orbitals have dominantly 'p-type' and 's-type' symmetries, respectively. The broad, partially resolved band between 12-18eV (Fig. 7.1) is due to ionization from the l b , , 5a , and l b 2 orbitals. The middle peak in this band (hatched) and the righthand peak (solid) shown in Fig. 7.1 clearly illustrate that the middle peak is of 's-type' symmetry whereas the righthand peak is of 'p-type' symmetry. From this comparison it is already clear that the 5a, orbital (s-type) has a lower binding energy than the l b 2 orbital (p-type) in agreement with the earlier findings of Hood et al. [HH76] 202 10.0 20.0 25.0 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r o CO o o q o d 1 b , 5 a , 1 b 2 2 b2 \ / 4 0 1 I 1 — n 1— H 2C0 $ = 6 ° 3a -I I I L J L _l I ' ' 5.0 7.1 10.0 15.0 20.0 25.0 30.0 35.0 BINDING ENERGY (EV) 40.0 45.0 Binding energy spectra of H 2 CO measured at an impact energy of_1200eV + binding energy for (a) 0 = 0 ° and (b) 0 = 6 ° . The two spectra are on the same relative intensity scale. The 5a, (hatched) and l b 2 (solid) peak areas are illustrated for clarity. Table 7.2. Binding energies and In tens i t ies In the Ionizat ion spectra of H2CO b Rela t ive Intensi ty Orb i ta l Energy (eV) Fwhm (eV) EMS (O') (p *0 .1a . " 1 ) EMS(6") ( p » 0 . 5 a o - 1 ) 2b 2 10.9 1.70 7.6 18. 1 1 b 1 14.5 1 .84 20. 1 32.2 5a j 16. 1 2 OO 100.0 18.8 1b 2 17.0 1 .92 20.7 40.6 4 a 1 21.4 2.40 77.2 53.3 27.5 3.0 7.4 31.5 3.0 7.7 34.25 3.0 30.3 37.0 3.0 6.8 41.5 3.0 5.9 Includes experimental width of 1.7eV fwhm. Normalized r e l a t i v e to 5a, peak Intensity at *=0* . Uncertainty is ±5%. CO o 00 204 and contrary to the assignments proposed by Turner et al. [TB70]. Hood et al. [HB76] also observed from the binding energy scan at 0 = 0° a broad structure centered at 34.2eV and assigned this feature to the 3a, orbital. This assignment although clearly suggested by M O calculations was not confirmed in the earlier study [HB76] since the binding energy scan in the inner valence region was only obtained at one angle (i.e.0 = 0°). A more extensive analysis of this inner valence region has been carried out in the present work. Over the past decade theoretical and experimental studies of many atoms and molecules have revealed, as a general phenomenon, a breakdown of the M O (single-particle) picture of ionization in the inner-valence region [CD86, VS84]. Such behaviour is also observed in the inner valence region of H 2 C O which exhibits intensity (Fig. 7.2 and Table 7.2) due to many-body effects spread over a wide range of the binding energy spectrum rather than a single peak expected from a single-particle (Koopmans-type) approximation. The 0 = 0° (scaled x0.55) and 6° inner valence spectra have been overlaid in Fig. 7.2a to emphasize the common symmetry of the main structure in the 24-44eV region. The inner valence region was fitted with a four-peak template each of equal widths (3.0eV fwhm) at the energies given in Table 7.2. A consideration of the actual intensities and the similar spectral shape indicates that the. 24-44eV region is mainly s-type and this is consistent with the strength being due to the 3 a , " 1 process. The present results for the inner valence region show a small but significant intensity at =28eV which was not specifically identified by Hood et al. [HH76]. q CN q o z U l o to ui 2 ° K ) -o d z o CD* rr < w o o U l t/l o O U l ui t q s o Green's F u n c t i o n Cal cul at i on . . ( l b 2 ) 5. . . (4a^) 6. . . (3a^) -1 -1 -1 J L J L J L J L 205 15.0 19.0 23.0 27.0 31.0 35.0 39.0 43.0 47.0 T 1 1 1 1 1 r H 2 C 0 MB-St at es I 4>=6° o « | )=0 o (x0 . 55) 15.0 Fig. 7.2 19.0 23.0 43.0 47.0 27.0 31.0 35.0 39.0 BINDING ENERGY (EV) Inner valence binding energy spectra of H 2 CO. The experimental binding energy spectra at (a) 0=O°(xO.55) and 0 = 6° are overlaid. Four peaks of equal widths (3.0eV) are placed (see Table 7.2) and convoluted to yield the best fitting curve (solid line). The theoretical binding energy spectrum (b) is from the Green's function calculation of von Neissen et al. [VB80]. The calculated poles were convoluted with the experimentally derived widths, summed and scaled (x2.5) to produce the theoretical binding energy profile (solid line). 206 A more detailed experimental confirmation, at least in the region of 34eV, is provided by the measured X M P (see Fig. 7.9) which shows clearly the 's-type' symmetry characteristic of the 3a , orbital. Recent Green's function calculations by von Niessen et al. [VB80] shown in Fig. 7.2b also confirm the present assignment and give a good representation of the observed spectrum up to = 36eV. Note that intensity at — 29eV is also predicted by the calculations (Fig. 7.2). This many-body Green's function calculation (Fig. 7.2b) involved a ( l l s7pld/6s lp) GTO set which is a reasonably extended basis set. Although the Green's function calculation predicts some small p-type poles at = 23eV ( l b 2 )^ their presence cannot be confirmed due to the very low signal level and the associated limited statistical precision of the experiment in this region. The present experimental results (Fig. 7.2a and Table 7.2) also show significant ionization strength beyond 36eV which is the limit of the Green's function calculation (Fig. 7.2b). It is believed that this strength is due to the 3 a , " 1 process. Overall, there is reasonable agreement between the experimental (Fig. 7.2a) and theoretical (Fig. 7.2b) spectra however it should be noted that Green's function calculations are at best only semi-quantitative. Increased flexibility of the basis set has been shown [FG87] to be very important in the improved prediction of the inner-valence binding energy spectrum, especially in the higher energy region. 207 7.4. COMPARISON OF EXPERIMENTAL MOMENTUM PROFILES WITH THEORETICAL PREDICTIONS The experimental momentum profiles (XMPs) of the valence orbitals of H 2 C O were obtained in two ways due to the multitude of states and the closeness of the l b , , 5a , and l b 2 binding energies. The 2 b 2 , 4a , and 3a , X M P s (see Figs. 7.4, 7.8, 7.9) were obtained in the usual manner (i.e. variation of azimuthal angle at appropriate selected binding energies) whereas the X M P s of the (only partially resolved) l b , , 5a , and lb 2 orbitals were derived from a series of narrow range binding energy spectra measured at azimuthal angles (0) of 0° , 2° , 4 ° , 6 ° , 8° , 10°, 14°, 20° and 30° . These spectra, shown in Fig. 7.3 on a common intensity scale, were deconvoluted using fixed relative energy positions and fixed widths (Franck-Condon width plus experimental energy resolution) derived from high resolution photoelectron spectroscopy [TB70] for the l b , , 5a, and l b 2 orbitals, respectively. The resulting deconvoluted areas for each of the orbitals were then plotted as a function of momentum (see Figs. 7.5, 7.6, 7.7). The X M P s have been placed on a common intensity scale by normalizing using the relative intensities in the 0 = 0° and 0 = 6° binding energy spectra (see also Table 7.2). The peaks in the 0 = 0° and 0 = 6° spectra have each been fitted with a gaussian peak taking into consideration the energy resolution and the known vertical IPs and Franck-Condon widths. The inner-valence 3a , X M P was normalized by summing the counts in the region, 24-44eV, (see above assignment) relative to the whole binding energy spectrum (9-44eV). The two wide range binding energy spectra (Fig. 7.1) thus provide two extra data points Fig. 7.3 H2CO E = T2 00eV o 4>=10 i i i i i $=14 ' $=20 i i i i i i i i 4>=30' ' t , i T * i m i i l i T i f > t i 1 , i T l X>X> BJO 1»J » J 3 BJO B A MJD B A 3O0 | U M X 210 208 BINDING ENERGY (EV) Binding energy scans in the region 14-22eV as a function of azimuthal angle, <p. Relative normalization is maintained since scans were sequential. The Fitted gaussian peaks are illustrated by dashed curves and the resulting envelope by a solid line. 209 and therefore a consistency check on the respective valence X M P s of H 2 C O (open triangles in Figs. 7.4-7.9). The normalized valence X M P s of H 2 C O are also compared with several theoretical calculations. The theoretical MDs were calculated from various wavefunctions shown in Table 7.1 and then spherically averaged and folded with the experimental momentum resolution (Ap = 0 . 1 5 a 0 " 1)- Theory and experiment can be compared on a common intensity scale using a single point normalization of the best SCF calculation (134-GTO) to a single point on the l b 2 X M P (see Fig. 7.5). A l l calculations and all other experimental data points for all orbitals can then be compared quantitatively on the basis of this single point normalization. This procedure affords a critical quantitative assessment of all experimental and theoretical data. To provide a comprehensive framework for interpreting the measured X M P s two-dimensional density contour maps, in both position and momentum space, are also presented for each reported X M P in Figs. 7.4-7.9. The contour maps were calculated using the DZ wavefunction and assumed the H 2 C O molecule to be in the xz plane with the z-axis serving as the C 2 axis. The contours shown are 80, 50, 20, 8, 5, 2, 0.8, 0.5 and 0.2% of the maximum density of each orbital. Along the side of each contour map are projections of the density along the axes indicated by the dotted lines. The set of theoretical MDs can be classified according to their total calculated energies. The 4-3 IG, 4-31G + G and DZ wavefunctions could therefore be 210 considered to be of intermediate quality (see Table 7.1) whereas the 134-GTO wavefunction of Davidson and Feller [DF86] can be considered to be essentially of Hartree-Fock limit quality. The calculated S C F total energy of the 134-GTO wavefunction (-113.9202a.u.) is very close to the estimated Hartree-Fock limit energy (-113.925a.u.) [GS74]. It can be seen from Figs. 7.4-7.9 that quite good agreement exists between calculation and experiment over all six orbitals except in the low momentum region of the 2b 2 outermost orbital. In each case both classes of wavefunctions (intermediate and near Hartree-Fock) predict very similar MDs for the three innermost valence orbitals ( l b 2 , 4a , and 3a,) whereas there are small differences for the three outermost (2b 2 , l b , and 5a,) orbitals of H 2 C O . In particular the l b , and 5a , orbitals are more sensitive to the basis set than the other orbitals. Experience in comparing measured X M P s and calculated MDs for other small molecules [BB87, BM87] has shown the need for quite extended basis sets with addition of diffuse and polarization functions. In particular such functions were found to be necessary in the accurate modelling of the low momentum region of the X M P s of the outermost valence orbitals of H 2 0 [BB87] and N H 3 [BM87]. The insensitivity of the more tightly bound orbitals ( l b 2 , 4a , and 3a,) is in keeping with earlier observations for a wide variety of molecules. The results for the less tightly held orbitals (2b 2 , l b , and 5 a , , Figs. 7.4-7.6) do not really show any clear trend as to the effect of basis set quality. It is seen that the near Hartree-Fock limit 134-GTO calculation gives the best description of the 2b 2 X M P (Fig. 7.4) in the sense that the predicted p is lower (and therefore in closer agreement with experiment). There is also a slight improvement in the description of the low momentum region compared with the MDs calculated from intermediate quality wavefunctions. However, despite the superior quality (see Table 7.1) of the 134-GTO wavefunction, the predicted M D is still significantly different in shape and magnitude from the measured X M P . The 2b 2 X M P peaks at a lower P m a x and has appreciable intensity in the region 0 . 1 - 0 . 6 a 0 - 1 - It is unlikely that improvements (at the Hartree-Fock level) to the 134-GTO basis set would lead to significantly improved agreement with experiment. The discrepancy observed for the outermost 2b 2 orbital is most probably due to the neglect of the effects of electron correlation and/or electron relaxation. A higher level of theoretical treatment beyond the Hartree-Fock picture is thus desirable in order to include electron correlation in the neutral state and both electron correlation and relaxation in the final ion state in the form of a full ion-neutral overlap distribution (OVD) calculation. Such an O V D treatment as has been extremely effective in comparing theory with E M S measurements for H 2 0 [BB87], N H 3 [BM87] and H 2 S [FB87a]. A similar overlap treatment is evidently necessary particularly in the case of the 2b 2 X M P of H 2 C O . Such studies are currently in progress [BB87c]. From the r-space maps (right side of Fig. 7.4) it can be seen that the 2b 2 orbital is only approximately a non-bonding orbital on the oxygen atom as evidenced by the relative number of contours. As noted by Neumann and Moskowitz [NM69] there is also some participation from the hydrogen atoms. If, in fact, electron correlation is the dominant reason for the discrepancy between calculation and the 2b 2 X M P then this picture (Fig. 7.4, rhs) of the 2b 2 orbital may be grossly inadequate. A similar situation exists in the case of the l7r Fig. 7.4 Comparison of 2b 2 experimental momentum profile of H 2 CO with calculated MDs. Open triangles represent data points derived from the long range binding energy scans (Fig. 7.1). Al l calculations are normalized to experiment by a single point normalization of the 134-GTO M D on the l b 2 X M P . On the center and right hand panels are the momentum and position space density contour maps, respectively calculated using the DZ wavefunction. A l l dimensions are quoted in atomic units. RELATIVE INTENSITY (ARB. UNITS) 0.0 3.0 6.0 9.0 12.0 6 . 0 J 1 1 i i i • • • -4.0 -2.0 0.0 2.0 4.0 0.5 1.0 214 orbital in CO [FB87b] and the 2it orbital of N O [BC82] where it is apparent that electron correlation is necessary for describing the dipole moment reversal and the X M P . It is also possible that the neglect of electronic relaxation in the fmal ion state ( 2B 2) will have a critical effect in the predicted MDs. Ozkan et al. [OC75] have calculated large electronic relaxation (or reorganization) effects in H 2 C O , in particular, the 2b 2 and l b , molecular orbitals. In the case of the l b , X M P it can be seen in Fig. 7.5 that quite good agreement is obtained between experiment and theory using the 134-GTO and DZ wavefunctions whereas the variationally inferior 4-3 IG and 4-31G+G wavefunctions yield MDs of lower intensity with P m a x shifted to higher momentum. It can be seen from the r-space and p-space contour maps (Fig. 7.5, rhs) that the l b , orbital can be considered as largely a T Q Q type of orbital. The 5a , X M P (Fig. 7.6) on the other hand shows clearly a mixed s-p type of profile similar to the 5 a orbital of CO [FB87a]. Such an analogy is supported by the r-space and p-space maps which show that the 5a , orbital can be considered as a OQQ type of orbital. The shape of the 5a , X M P is reasonably predicted by the 4-31G + G and 134-GTO calculations athough the absolute value of the predicted cross sections are = 10% higher than the measured X M P . It is likely that this difference could be attributed to mixing of ionization strength in the 2 A , manifold and would affect the 4a , and 3a, X M P s similar to the case of the expected mixing of ionization strength in the 5 a, 4 a and 3 a poles (all 2 Z symmetry) of CO [FB87b, DD77]. Another point that should be mentioned is the much lower cross section predicted by the 4-3 IG and DZ wavefunctions compared to 4-31G + G and 134-GTO wavefunctions. This result reflects the need Fig. 7.6 Same as Fig. 7.4 except for 5a, orbital of H 2 CO SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY MOMENTUM (A.U.) Fig. 7.7 Same as Fig. 7.4 except for l b 2 orbital of H 2 C O . 217 for diffuse functions in the basis set. It is also consistent with the conclusions of Ohno et al. [OT86] regarding their comparison of E E D values (calculated using a modified 4-3 I G basis) with the experimental PIES cross sections. In Fig. 7.7 the l b 2 X M P , which is largely a TT orbital, is shown to compare well with the theoretical calculations. As opposed to the 2 b 2 , l b , and 5a , orbitals, the calculated l b 2 MDs are less sensitive to basis set quality. This is also one of the reasons why it was chosen for the single point normalization of theory to experiment to establish a quantitative basis for comparison. Comparison of the 4a, (Fig. 7.8) and 3a , (Fig. 7.9) X M P s with the respective theoretical calculations shows similar behaviour to that observed for the 5a , X M P (see Fig. 7.6). A l l calculated MDs in both cases (i.e. for the 4a , and 3a , orbitals) overshoot the respective X M P s by = 10%. Besides the difference in magnitude of the cross section there is also some difference between the shape of the 4a , X M P and that predicted by theory. The experimental momentum profile has a larger width at half maximum than any of the calculated MDs. The broad profile is expected from the anti-bonding nature of the 4a , orbital which can be considered as a a* type of orbital (Fig. 7.8, rhs). It is anti-bonding with respect to the 2s and 2p contributions from the carbon and oxygen atoms. This is again analogous to the 4 a orbital of CO [FB87b] which however shows a 'p-like' momentum profile. The reason for this large difference in shape between the 4a , orbital of H 2 C O and the 4a orbital of CO [FB87b] is the presence of H Is character in the 4a , orbital of H 2 C O which adds symmetrically (in-phase) to the C 2s. It is known that s-type atomic orbitals (in SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY MOMENTUM (A.U.) Fig. 7.8 Same as Fig. 7.4 except for 4a, orbital of H 2 CO. tsD OO SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION •2, 3, 4 H 2CO 3a 1 1 - - 4-31G 2 4-31G+G 3 DZ 4 134-GTO 0.0 0.5 1.0 1.5 2.0 12.5 MOMENTUM (A.U.) MOMENTUM DENSITY -4.0 -2.0 0.0 2.0 4.0 0.5 1.0 POSITION I O ci q s q o o Tf(o,.t.o? -4.0 -2.0 Fig. 7.9 Same as Fig. 7.4 except for 3a, orbital of H 2 CO. 220 L C A O - M O picture) are the only contribution to the momentum density at p = 0 a o " 1 [BB87b]. The 3a , X M P (Fig. 7.9), which largely corresponds to the oxygen 2s orbital (note spike in the r-space projection plot), is also slightly overestimated by the theoretical calculations although the shape is adequately predicted. Differences of this nature have been observed in the inner valence X M P s of H 2 0 [BB87], N H 3 [BM87] and Ne [DM78] and these differences were found to persist even with inclusion of electron correlation in both initial neutral and final ion states. It has been postulated that this phenomenon could be due to: (a) unaccounted ionization stength beyond the limits of the binding energy spectrum (i.e. Ej 3 ^45eV). (b) limitations in the present theoretical treatment (i.e. basis set effects and insufficient electron correlation and/or relaxation) (c) distortion effects (i.e. failure of the plane wave impulse approximation). Distorted wave calculations are presently not feasible but could be investigated experimentally by increasing the impact energy beyond 1200eV. This, however, is not possible with the present instrumentation. 7.5. SUMMARY The present E M S study of H 2 C O shows good quantitative agreement between all measured valence X M P s and MDs calculated from the 134-GTO wavefunction (near Hartree-Fock) except for the outermost 2b 2 orbital. Based on previous E M S studies of H 2 0 [BB87] and N H 3 [BM87], it is suggested that the discrepancies observed in the outermost 2b 2 orbital are due to the neglect of electron 221 correlation and/or electronic relaxation effects. Other theoretical studies [OC75] also indicate that electronic relaxation may be important in the 2b 2 orbital. Extensive many-body structures are also observed in the inner valence binding energy spectrum which is assigned largely to the (3a,)" 1 process. These structures are clearly predicted by Green's function calculations [VB80] although the agreement with experiment is only semi-quantitative, especially in the higher energy region. CHAPTER 8. PARA-DICHLOROBENZENE 8.1. OVERVIEW Chemical reactivity can be studied by measuring the properties of a series of compounds that differ from each other by a single substituent. If the parent compound and substituents are appropriately chosen, it is possible to interpret changes in the properties of the compounds in terms of the effects of the substituents on the basic parent compound. Under favourable circumstances the perturbation of the substituent can serve as a model for a chemical reaction in which the reactants mutually perturb each other's basic electronic structure, resulting ultimately in the breaking and making of chemical bonds. Examples of the systematic study of substituent effects in a homologous series of molecules can be found in most forms of spectroscopy. With these considerations in mind, electron momentum spectrocopy has recently been applied to the study of the nitrogen 'lone-pair' orbital in ammonia, methylamine, and triazine [TL84, HC85]. These ideas are now further explored in the present work and E M S data are now presented for the two outermost (ir 3 , 7r 2) valence orbitals of para-dichlorobenzene (p-DCB), together with a comparison with the corresponding doubly degenerate l e i g f orbital of benzene. In p-DCB the addition of the two chlorine atoms removes the degeneracy and it is of interest to investigate the nature of this effect on the momentum distributions of the energetically separate p-DCB orbitals, 7T 3 and 7 T 2 . t The interaction between the CI atoms and the benzene ir system, while of intrinsic interest itself, has wider implications for t These orbital assignments are based on benzene assuming Dg^ planar symmetry and p-DCB assuming ^2h. s y m m e t r y - The x-axis is chosen to be along the C-Cl bond and the z-axis normal to the plane of the ring. The borid lengths and angles are standard [EC73] with the C-Cl bond length set at 1.77A. 222 223 chemistry. For example, organometallic chemistry and much of organic reactivity is dominated by n interactions. Furthermore, the influence of an electronegative element on a tr system is of general interest and important to understand. The sensitivity of E M S to Fine details of the dynamics of electron motion should provide new insights into 7T electron behaviour and reactivity. The electronic structure of the benzene molecule has been extensively studied both theoretically and experimentally since the beginnings of quantum mechanics. Because of the large size of the benzene molecule, detailed calculations approaching the self-consistent field (SCF) limit [EK73] and calculations incorporating many-body effects [VC76, CD78, PB70, HS74] have only been done within the last ten or fifteen years. The outer-valence binding energies of benzene have been measured by photoelectron spectroscopy using excitation sources from Hel and H e l l [TB70, A L 7 0 , K M 7 6 , JL69 , SP74, LF72 , LA80] to X-rays [GB74]. The outermost orbital of benzene has been identified as the l e ^ orbital of 7T symmetry. This doubly degenerate l e i g orbital has a vertical ionization energy (I v) of 9.3 eV and is often labelled ir2, 7T 3; the more tightly bound l a 2 u o r D ^ a l (I v=12.4 eV) is non-degenerate and is usually labelled T T , [KK81]. To date there has been only a single E M S study of the binding energies and momentum distribution of the valence orbitals of benzene [FM81]. The three unfilled 7T orbitals of benzene, namely the itn*, 7 r 5 * and 7 T 6 * orbitals, have been extensively investigated with electron transmission spectroscopy [JM76]. Since benzene is planar, the a and 7T systems are effectively separated. This consideration along with a detailed knowledge of the electronic structure of benzene makes it particularly favourable for studies of substituent effects. 224 In this initial E M S study of substituent effects in benzene the effect of chlorine substitution at the 1 and 4 ring positions on the highest-lying Filled doubly degenerate IT level is examined. In particular, the binding energies and momentum distributions of the 7T3(2b2g) and 7 T 2 ( lb^g) orbitals of para-dichlorobenzene are measured. Calculations using the simple wavefunctions (4-3 IG) that are available for p-DCB have been carried out by the Maryland group (see Acknowledgements) in order to provide further insight into the experimental results. Experimental considerations also played a role in the choice of p-DCB. Because of the limited energy resolution currently available in E M S (=1.6 eV fwhm in the present work) it was necessary to choose a molecule in which filled 7T levels were sufficiently separated from each other as well as from the next-lowest levels. The compound also had to have a vapor pressure such that target densities could be maintained at the 1 0 1 2 - 1 0 1 3 cm" 3 level at room temperature. In addition the compound had to be sufficiently inert to resist decomposition in the spectrometer. The two outer filled ( 7 T 3 , 7 r 2 ) levels of para-dichlorobenzene have vertical ionization energies (Iy) of 8.94 and 9.84 eV respectively, as measured by He l photoelectron spectroscopy [KK81, SC73, PL80] The next-lowest level (primarily CI 3p) is at 11.37 eV [KK81, SC73, PL80]. While these spacings do not allow complete separation of the bands, they are nevertheless sufficient to provide specific information on the separate orbitals with judicious selection of the actual sitting binding energies used for the measurements. 225 8.2. EXPERIMENTAL MOMENTUM PROFILES The addition of two chlorine atoms at the para positions in benzene removes the degeneracy of the benzene le^g orbitals (TT 3,7T 2) at I v = 9.3 eV. In p-DCB the it 2 is shifted up in energy (I v =9.8 eV) and the 7T 3 down (I v=8.9eV). This shift in energies has been explained in terms of the inductive and resonance effects associated with the substitution of the two chlorine atoms.t The next orbital in p-DCB is mainly chlorine 3p in character. The outer valence binding energy spectrum of p-DCB obtained at E ^ = E g = 600 eV and 0 = 6° , is shown in Fig 8.1. The error bars represent one standard deviation. The indicated curve fitting analysis and the absolute energy scale were established with reference to high-resolution He l PES measurements. Such measurements have been reported by Streets and Caesar [SC73], Potts et al. [PL80] and Kimura et al. [KK81]. The vertical ionization energies of the first six bands were assessed from PES to be at I y = 8.94, 9.84, 11.49, 13.00, 14.74 and 15.90 eV respectively. The corresponding E M S binding energy peak widths were likewise estimated to be 1.6, 1.6, 1.9, 1.8 and 1.7 eV fwhm, taking into account the E M S resolution (1.6 eV) and also the natural Franck-Condon widths [KK81]. Only the first two bands (7r 3,7r 2) are single ionic states according to the high-resolution PES data [KK81]. The third band involves ionization from the orbitals with a predominantly C l 3p character. A gaussian fitting program using parameters based on the experimental widths, peak positions and spacings from t A general discussion of resonance versus inductive effects is given in Ref. [EB73]. A discussion of resonance versus inductive effects in the context of the photoelectron spectrum of para-dichlorobenzene is given in Ref. [BM68]. A more detailed consideration of the influence of inductive and resonance effects on ionization potentials in substituted benzenes has been given in Ref. [FK81]. 226 PES and the present instrumental energy resolution gave the generally excellent fit indicated in Fig. 8.1. The dashed lines are the individual gaussian peaks with the solid line being their sum. The two vertical arrows represent the carefully selected experimental sitting binding energies (8.7 and 10.1 eV, which, it should be noted, are not exactly equal to the respective I values) at which momentum distributions were sampled. In this way contributions from neighbouring states are minimized and the principal contribution will be from the 7T 3 and ir2 orbitals, repectively. Experimental momentum profiles (XMPs) representative of the first two bands of p-DCB, obtained as described above, are shown in Figs. 8.2a and 8.2b. The solid line drawn through the experimental points for each distribution provides a visual guide. Since the widths of the 7T 3 and 7r 2 bands (Fig. 8.1) are equal, the relative cross sections as shown are in essentially the correct ratio, taking into account summation over all the final states. The shapes and magnitudes of the two momentum profiles are clearly very different. Somewhat surprisingly, the lower I (7r 3) band shows a maximum ( P m a x ) at =0.8 au, which is higher than the p (0.6au) observed for the higher I (7r 2 ) band. Thus, an increase in I I I 3.X V binding energy does not necessarily correspond to an increase in the most probable momentum. This fact has been demonstrated in earlier E M S experiments, for example in the case of C 0 2 [LB85a]. In He l PES measurements of p-DCB [KK81] the peaks due to 7T 3 and 7T 2 are of similar intensity. Examination of Fig. 8.1 indicates that the measured X M P at 8.7 eV (essentially 7T 3) contains a small overlapping contribution from the second band (essentially 7T 2 ) . Since P m a v for the first band is greater than that for the 227 to 4 6 8 10 12 14 16 18 Binding Energy (eV) Fig. 8.1. Binding energy spectrum of para-dichlorobenzene at <p = &°. The dashed curves are from a gaussian fitting program based on parameters from high-resolution PES [KK81]. The solid line is the sum of the individual gaussian peaks. Vertical arrows show the sitting binding energies at which the momentum profiles of Fig. 8.2 were sampled. 228 second band, the overlapping contribution to the first band can only serve to lower its observed P m a x relative to that of a completely separated ir3 orbital. The second band is mostly due to ir2 since the energy separation of the third band (Cl 3p) from the second is much larger than that between ir 3 and ir 2. The present measurements clearly show that there is a significant difference in the TT 3 and ir2 orbital X M P s of p-DCB, not withstanding the small difference in their energies. In addition to the different cross sections and P m a x values it is also noteworthy that the cross section near p = 0 is larger for the ir2 orbital that for the 7 T 3 orbital. This is not an experimental artifact because the experimental momentum resolution (Ap=*0. 1 a 0 ~ 1 ) of the spectrometer is sufficient to clearly resolve any node in the momentum distribution at p = 0, as has been demonstrated in the case of the 3p orbitals of argon [LB83]. In this regard the behaviour of the measured ir 2 orbital momentum distribution appears somewhat closer to that measured for the benzene le^^ orbital [FM81]. However, the poor statistics on the benzene measurements [FM81] preclude any unequivocal assessment of the P m a x value for the l e i g orbital, and therefore any detailed comparison with the present higher-precision p-DCB measurements is at best speculative. 8.3. CALCULATED MOMENTUM DISTRIBUTIONS In order to further understand the experimental measurements, calculations of the 7 T 3 and ir 2 orbital momentum distributions for p-DCB have been carried out with a basis set of 4-3 IG type [DH71] using the G A U S S I A N 80 program [BW80]. The results are shown in Fig. 8.3. Using even such a simple basis set, the results are quite instructive in that the general characteristic features and Fig . 8.2. O d q 06 1 r (a) T 1 r ~ i 1 1 r c i - Q - a Iw= 8.9eV -(8.7eV) Iv= 9.8eV (IQIeV) 229 Momentum (a.u.) Measured momentum profiles for the IT 3 (a) and i r 2 (b) orbitals of para-dichlorobenzene. The vertical ionization energies (I v) are shown. The values in brackets are the sitting binding energies at which the measurements were taken. 230 trends of the experimental results (Fig. 8.2) are clearly reproduced. Firstly, the predicted p w for 7T3 (dotted curve) is larger than that for j r 2 (solid curve), max whereas the predicted maximum cross section for ir2 is greater than that for 7 T 3 . The calculated P m a x positions are somewhat higher than those observed experimentally, and neither calculated momentum distribution shows any cross-section contribution at p = 0.t Calculations using much more accurate basis sets are evidently required before further comparisons of theory and experiment can be made. The orbital energies calculated for 7T3 and ir2 using the 4-3 IG basis are 9.35 and 10.02 eV compared with the experimental [KK81] I values of 8.9 and 9.8 eV. Further understanding of the observed and calculated features, particularly the different P m a x values of the momentum distributions for the 7r 3 and ir2 orbitals of p-DCB, can be obtained by considering the simple orbital diagrams for both p-DCB and benzene shown in Fig. 8.4. A simple orbital diagram of the degenerate (7r 3 and 7T 2 ) le^g orbital of benzene is shown in Fig. 8.4a and the effect of chlorine substitution in p-DCB is shown in Fig. 8.4b. For benzene, one representation of the degenerate ^eig P a "" n a s a symmetric (7r 3 ) and an antisymmetric (ir2) configuration with respect to the perpendicular plane through the 1 and 4 positions [KK81]. Substitution of chlorines at the 1 and 4 positions would first have the effect of lowering the energy of the two levels by withdrawing charge from the ir system onto the electrophilic chlorine atoms. This inductive effect is offset for the 7T 3 level of p-DCB by the antibonding resonance t The present calculations, which were done without the incorporation of the instrumental momentum resolution, are not likely to give good quantitative agreement with the measured X M P s even if the momentum resolution is folded in. 2 3 1 0.1 CALCULATION 4-31G BASIS CO c CD Q E o C I - ^ - C I o o Iv(eV) p-DCB T T 3 8.9 p-DCB T T 2 9.8 1.0 2.0 Momentum (a.u.) 3.0 Fig . 8.3. Calculated momentum density distribution for the ff3 and 7T 2 levels of para-dichlorobenzene and the doubly degenerate n 3 , 7T 2 levels of benzene. The calculations are based on 4-3 IG wavefunctions and the momentum densities have been spherically averaged. The benzene and 7T 2 para-dichlorobenzene densities are indistinguishable and are shown by the solid line. The 7T 3 density is shown by the dotted line. 232 (a) BENZENE Iv = 9.3eV (b) p-DICHLOROBENZENE 2 b 2 g 1b l g Iv=8.9eV Iv = 9.8eV Schematic representation of wavefunction amplitudes for (a) the two degenerate levels (ir 3 and it 2 ) of the highest occupied 7T level of benzene and (b) the ir3 and it2 levels of para-dichlorobenzene. The areas of the circles are proportional to the magnitude of the wavefunction amplitudes. Solid circles represent positive amplitude and open circles negative amplitude. Nodes (regions of zero wavefunction amplitude) are indicated by dashed lines. 233 interaction of the Cl 3p with the 2p of the carbon atoms at positions 1 and z z 4. This causes the ir 3 orbital energy to exceed (i.e. I decreases) that of benzene l e ^ g - For the ir2 orbital in p-DCB, symmetry requires the C l 3p z amplitude to be zero, thereby eliminating any resonance interaction with the ir system of the ring. Though the splitting of the ir3,ir2 levels of benzene upon 1,4 chlorine substitution has long been justified in terms of inductive and resonance effects, additional direct experimental evidence for the correctness of the explanations has been lacking. The measured relative (e,2e) cross sections for the momentum profiles as shown in Figs. 8.2a and 8.2b provide additional data supporting the above discussion. The ir 3 orbital of p-DCB has a momentum profile with a P m a x shifted to a higher momentum value (=0.8 au) than that for the ( 7 r 2 , 7 r 3 ) ^eig momentum profile of benzene, which is at =0.7 au [FM81]. In contrast, the P m a x for the 7T 2 p-DCB momentum profile (=0.6 au) is closer to that for benzene than it is to that for 7T 3 . It has also been noted above that the finite cross-section behaviour near p = 0 for p-DCB ir2 is more like that observed for benzene le^g-It is reasonable to expect the inductive effect of chlorine substitution to have, at most, a small influence on the momentum distributions of the electrons. However, it can be seen from Fig. 8.4b that the antibonding interaction for the ir 3 level of p-DCB introduces two additional nodes into the wavefunction. Since electron momentum is related to the derivative of the position-space wavefunction, an increase in the amplitude of the large momentum components of the electron momentum distribution is expected for wavefunctions with sharply varying amplitude. In particular, the larger the number of nodes in a spatial 234 wavefunction, the larger the amplitude of the high-momentum components of the momentum density, as has been observed for the momentum distribution of the ir* orbital of C 0 2 [LB85a]. This behaviour is precisely what is observed for the 7T 3 orbital of p-DCB where P m a x is quite high. For the ir2 orbital of p-DCB, where there is no resonance interaction of the chlorine 3p orbitals with the ring electrons, the nodal structure of the benzene level is unchanged, and therefore the momentum distribution is expected to be more like that of the unsubstituted benzene le-^g orbital, at least at the level of the 4-3 IG calculation. The observed situation (Fig. 8.2) is generally in accord with the foregoing arguments, as are the trends of the 4-3 IG calculations (Fig. 8.3) 8.4. SUMMARY Distinct differences have been observed in the experimental momentum profiles for the separate 7T 3 and ir2 molecular orbitals of para-dichlorobenzene. Explanations of the chemical reactivity of such aromatic systems have long relied on arguments based on inductive and resonance effects. Confirmation of the correctness of these ideas has mainly been limited to measurements of energy levels and their changes upon substitution. With E M S measurements of binding energy selected electron orbital momentum distribution it is now possible to provide more detailed insight into such matters by direct probing of the finer details of electron densities. The present E M S measurements of the momentum profiles of the outermost 7T 3 and 7T 2 orbitals of para-dichlorobenzene and their comparison with molecular orbital calculations for p-DCB and benzene, have provided clear evidence for the detailed nature of the ir electron charge 235 distributions normally rationalized by arguments based on separate inductive and resonance effects. C H A P T E R 9. M E T H Y L A T E D A M I N E S A N D N F 3 9.1. OVERVIEW Chemical properties are best interpreted at the fundamental level in terms of a sufficiently detailed understanding of the molecular electronic structure. Such a level of understanding could in principle be obtained by theoretical quantum chemistry from sufficiently accurate solutions of the Schro dinger equation for the system in question. However the approximations which must be made for most molecules limit the accuracy of theory and a detailed understanding is thus oftentimes not fully realisable. A simpler and popular intuitive approach to understanding electronic structure in complex systems and predicting stability and reactivity is the use of concepts derived from theories of electron displacement such as inductive, resonance and polarization effects of substituent groups attached to a particular 'molecular center'. Such concepts and rationalizations are in common use in both the teaching and practise of Organic and Inorganic Chemistry. It is not surprising that such empirical approaches have often led to diverse and conflicting opinions about electronic effects. For instance a well-known controversy exists as to whether methyl groups are intrinsically electron donating or electron accepting relative to hydrogen [HP70, SU83, MB83]. A n effective alternative experimental quantum mechanical approach to these questions has been afforded by electron momentum spectroscopy. In particular E M S measurements together with sophisticated quantum mechanical calculations (Chapters 5 and 6) have shown that very accurate mapping of the electronic charge distribution in individual molecular orbitals is now feasible at least for 236 237 small molecules. The experimental mapping of orbital electron densities produced by E M S provides direct orbital imaging in momentum space within the approximations discussed in chapter 2. As such, E M S should also prove useful for direct experimental probing of orbital electron densities in larger and more complex molecules. In particular such E M S studies are likely to provide a detailed understanding of structure and reactivity at the electronic level based on experimental observations. Examples of such applications of E M S are to be found in recent publications comparing N H 3 and N H 2 C H 3 [TL84] and also an investigation of the 7T electrons in para-dichlorobenzene (Chapter 8) [BB86]. In this chapter measurements of the X M P s of the outermost valence orbitals for each of NH 2 CH 3 , NH (CH 3 ) 2 , N (CH 3 ) 3 and NF 3 are reported and these results are compared with several quantum mechanical calculations of momentum distributions as well as density maps in both momentum and position space. A communication of preliminary studies on N (CH 3 ) 3 and NF 3 has recently been published [BB87a]. The present study considerably extends the earlier work of Tossell et al. [TL84] on N H 3 and N H 2 C H 3 . The present more detailed work provides an experimental and theoretical basis for discussion of the electronic charge distribution in the outermost valence orbitals of the methyl amines and NF 3 . In particular the validity of commonly held concepts is evaluated in the light of the present findings. The N H 2 C H 3 , N H ( C H 3 ) 2 , N ( C H 3 ) 3 and NF 3 samples were obtained in cylinders from commercial suppliers and used without further purification. Mass spectral analysis showed the samples to be free of any significant impurities. 238 9.2. BASIS SETS FOR SCF WAVEFUNCTIONS Theoretical spherically averaged momentum distributions (MDs) and density contour maps in momentum and position space have been calculated using several wavefunctions of varying quality by means of the H E M S computer package developed in this laboratory [BB87]. The existing H E M S programs have been modified for the present work in order to accomodate the large number of atoms and basis functions required for the larger molecules used in the present study. The experimental momentum resolution (Ap = 0.15a o" 1 ) has also been folded into each calculated M D . The wavefunctions investigated in the present study include: (1) STO-3G This is a minimal basis set designed originally by Pople and co-workers [HS69]. It involves a N[2slp], C[2slp], F[2slp] and H[ls] basis set in which each s- and p-function is a contraction of three primitive GTOs. (2) 4-3 I G This "split-valence" basis involves a N[3s2p], C[3s2p], F[3s2p] and H[2s] contraction scheme. Four primitive GTOs are contracted to form the Is core and ' the valence orbitals are 'split' into three GTOs contracted into one CGTO and the other GTO left uncontracted. Exponents for both valence s- and p-functions have the same value. (3) STO-3G + G This basis set involves the original STO-3G(l) basis plus a set of 239 diffuse functions. These diffuse functionst have been used by Chong and co-workers [ZC79, LB83b, MM83] in the improved prediction of molecular polarizabilities. It has been shown in earlier E M S studies [BB87, BM87] that incorporation of diffuse functions are very important in the accurate prediction of orbital X M P s . However it should be noted that small basis sets optimized for the calculation of momentum distributions alone may yield unreliable results for other molecular properties (e.g. dipole moment and total energy). (4) 4-3 I G * This basis set involves the addition of a standard set of polarization functions (d-type) on the heavy atoms to the 4-3 I G basis described in (2). The standard exponent is (1^ = 0.80 for C, N , and F. A l l wavefunctions were generated at the respective experimental geometries [TK71, WL67, WL69, SG50, BP57] using the Gaussian76 package [BW76]. The characteristics and properties of these wavefunctions are outlined in Table 9.1. 9.3. MEASURED AND CALCULATED MOMENTUM DISTRIBUTIONS High momentum resolution E M S measurements have been made of the X M P s for the outermost valence orbital of each of NH 2 CH 3 , NH (CH 3)2; N (CH 3 ) 3 and NF 3 . These results are shown in Fig. 9.1 together with the outermost (3a,) X M P of N H 3 reported earlier (see chapter 6) [BM87]. Below each molecule name the 'sitting binding energy' at which the particular X M P was measured is quoted t The specific diffuse functions used are: C(a =0.02789), N(a =0.03882, F(a =0.04993) and H(a =0.1160) [ZC79, LB83b, MM83]. T a b l e 9 . 1 . Experimental and c a l c u l a t e d p r o p e r t i e s f o r N H 3 , N F 3 and the methylated amines. Mo 1 ecu1e B a s i s Set N H 3 ST0-3G ST0-3G+G 4-31G 4-31G* 126-GTO 126-G(CI) E x p t l . E n e r g y ( a . u . ) -55.4540 -55.5301 -56.1025 -56.1297 -56.2246 -56.5160 - 5 6 . 5 6 3 ™ • I p o l e Moment(0) 1.786 1 .736 2.299 1 .922 1 .642 1 .589 1 .47 b Second Pmax(ao"') I .P . (eV ) 0.5210.05 9.59 11 .43 11 .26 11 .42 10:94 1 0 . 8 5 ° NHjCH 3 ST0-3G ST0-3G+G 4-31G 4 -31G» E x p t l . -94.0268 -94.0656 -95.0625 -95.1128 - ? -1 .625 1 .974 2.071 1 .730 1 . 2 3 d 0 . 7 ± 0 . 1 8.90 9.67 10.48 10.64 NH(CH ST0-3G ST0-3G+G 4-31G 4-31G* E x p t l . - 132.6089 -132.6583 -134.0352 -134.1O50 - ? -1 .262 1.598 1.519 1 .222 1.01* 0 . 8 ± 0 . 1 8. 13 8.99 9.72 9.84 8 . 9 4 ° N ( C H 3 ) 3 ST0-3G ST0-3G+G 4-31G E x p t l . -171.1886 -171.2517 -173.0080 - ? -1 .010 1 .233 1 .052 0 . 6 1 2 f 1 . 1 ± 0 . 1 7.76 8.81 9.35 8 . 5 1 ° N F 3 ST0-3G ST0-3G+G 4-31Q 4-31G* E x p t l . -347.7538 -347.8475 -352.0756 -352.2079 - 7 -0.394 0. 197 0.436 0.315 0 . 2 3 4 ° 1 . 3 ± 0 . 1 10.64 15.34 15.38 14.88 13.73h E x p e r i m e n t a l l y - d e r i v e d n o n - v i b r a t i n g , non - re la t1v1s t1c t o t a l energy. Ref.[MM81J . Ref . [K81] Ref.[WL69J Re f . [TK71 l " " » e f . l K 5 4 j Ref.[WL68] Ref . [BL70] Est imated H a r t r e e - F o c k l i m i t 1s - 5 6 . 2 2 6 a . u . to O 241 OUTERMOST VALENCE MOMENTUM DI STRI BUT I ONS I EXPT.(1200eV) THEORY (1) STO-3G (4) 4-31G* (2) 4-31G (5) 126-GTO (3) STO-3G+G (5c) 126-C(CI) OJO 0.4 0£ 16 2.0 2.4 2.8 MOMENTUM (A.U.) Fig. 9.1 Comparison of the experimental momentum profiles of the outermost valence molecular orbitals of (a) N H 3 , Oo) NH 2 CH 3 , (c) N H ( C H 3 ) 2 , (d) N ( C H 3 ) 3 and (e) N F 3 with (spherically averaged) momentum distributions calculated using various basis sets. The theoretical MDs are normalized to the respective X M P s at p=1 .0a 0 " 1 using the best (in energetic terms) wavefunction in each case. The binding energy at which the X M P was measured is shown in brackets. 242 in brackets. These values correspond closely to the vertical IPs of the outermost 'lone pair' orbitals of the respective molecules. The measured outermost valence X M P s are each compared in Fig. 9.1 with several theoretical momentum distributions calculated from ab initio L C A O - M O wavefunctions of varying quality. The characteristics of the wavefunctions used and some relevant calculated properties are shown in Table 9.1. The wavefunctions are of modest quality with the exception of the 126-GTO and 126-G(CI) treatments for N H 3 (see chapter 6) [BM87]. The MDs are compared to the corresponding X M P s by normalizing the 'best' calculation (in terms of the calculated total energy, see Table 9.1) to the experimental momentum profile at p ^ l . O a o " 1 - Each calculation therefore maintains the correct relative intensity scale with respect to the other calculations for the particular molecule. This procedure affords a more critical assessment of the wavefunctions studied. In the case of N H 3 recently published [BM87] high level SCF calculations at the Hartree-Fock limit (126-GTO) and CI treatment of correlation using the ion-neutral overlap (126-G(CI), see Eqn. 2.11) are also shown for comparison purposes. Such high level state-of-the-art calculations are at present impractical for larger molecules such as the methyl amines and NF 3 . The general trends of the experimental results are clearly predicted even by the rather simple wavefunctions employed in the present calculations (see Fig. 9.1). Experimental results and STO-3G calculations only have been reported in a preliminary communication for N H 3 , N ( C H 3 ) 3 and NF 3 [BB87a]. Following commonly used 'textbook' arguments based on molecular geometry, steric effects 243 and simple valence bond hybridization concepts, the outermost 'lone pair' orbital on N in an N X 3 molecule would be predicted to range from 100% s-character ( Z X N X = 90° , i.e. pyramidal geometry) to 100% p-character (<£XNX=120°, i.e. trigonal planar geometry). Thus on this simple intuitive basis it would be generally predicted that the measured X M P s for the outermost orbitals would show decreasing %s-character as methyl groups are substituted for hydrogen in N H 3 , i.e. Predicted %s-character: N H 3 > N H 2 C H 3 > N H ( C H 3 ) 2 > N ( C H 3 ) 3 The bond lengths and bond angles for NH 2 CH 3 , NH (CH 3 ) 2 and N (CH 3 ) 3 are well-known from microwave experiments [TK71, WL67, WL69]. These molecules show bond angles in the range 109°-111° (see Fig. 9.2) which are larger than for N H 3 ( Z H N H = 106.7°) [BP57]. In contrast, the measured X M P s show a trend, Observed %s-character: N H 3 < N H 2 C H 3 < N H ( C H 3 ) 2 < N ( C H 3 ) 3 which is clearly exactly opposite to that predicted on the basis of the intuitive arguments outlined above. For N F 3 , intuitive arguments based on geometry ( Z F N F = 102.15°) [SG50] would predict increased s-character relative to N H 3 . In this case the prediction happens to be consistent with the experimental results (Fig. 9.1e). It is obvious that such intuitive arguments are unreliable as a predictive tool. In an effort to provide further understanding of these seemingly complex results, 2 4 4 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY Momentum (a.u.) Fig. 9.2a-c Two-dimensional contour plots (oriented molecules) of the position-space (xz plane) and momentum-space (P X P Z plane) densities of the outermost valence orbitals of N H 3 , N H 2 C H 3 and N H ( C H 3 ) 2 calculated using the respective 4-3IG wavefunctions. N H 3 is assumed to be of C g y symmetry with the z-axis being the C 3 axis whereas N H 2 C H 3 and N H ( C H 3 ) 2 molecules assume C g symmetry. The positions of the atoms in the respective xz planes are indicated on the position-space maps. Experimental bond angles and bond distances were used. A l l dimensions are quoted in atomic units. 245 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY Momentum (a.u.) Fig. 9.2d-e Two-dimensional contour plots (oriented molecules) of the position-space (xz plane) and momentum-space (p x P z plane) densities of the outermost valence orbitals of N (CH 3 ) 3 and N F 3 calculated using the respective 4-3 I G wavefunctions. Both molecules are assumed to be of C g y symmetry with the z-axis being the C 3 axis. The positions of the atoms in the respective xz planes are indicated on the position-space maps. Experimental bond angles and bond distances were used. A l l dimensions are quoted in atomic units. 246 both experimentally and theoretically, the position-space and momentum-space density maps for the respective molecules have been calculated and the results are shown in Fig. 9.2. The contour values are 80, 50, 20, 8, 5, 2, 0.8, 0.5, 0.2, 0.08, 0.05 and 0.02% of the respective maximum intensities. The 4-3IG wavefunction was chosen for the calculation of density contour maps as a suitable compromise between computing economy and accuracy for the moderately large molecules involved in the present study. The number of integrals involved in an SCF calculation increases as = N 4 where N is the number of basis functions [DF86]. For example, an S C F calculation for N H 3 and N ( C H 3 ) 3 using the 4-3 IG basis set would involve 15 and 54 basis functions, respectively. It can be seen that the observed increase in s-character upon methyl substitution (compare Figs. 9.1a-9.1d) is related to the large H Is contribution trans to the nitrogen Tone pair' (see Figs. 9.2a-9.2d). As more methyl groups are added, more trans-H Is character is contributed (in phase) to the molecular orbital until a maximum is reached in the case of N ( C H 3 ) 3 [BB87a]. Clearly, the Tone pair', atomic-like characteristic of the outermost valence (3a,) orbital in N H 3 is increasingly Tost' as density is increasingly delocalized with each methyl substitution. A similar delocalization effect has also been observed in the comparison of the measured X M P s of the outer a, -type valence orbitals of C H 3 O H [MB81] and ( C H 3 ) 2 0 [CB87] with the 3a , orbital of H 2 0 [BB87a]. The implications of the present experimental and calculated results for current understanding of alkyl inductive effects in organic chemistry will be discussed in a following section. 247 The increased %s-character in N F 3 (Fig. 9.1e) occurs for an entirely different reason to that for the methyl amines. Close examination of the M O coefficients as well as inspection of the r-space and p-space maps (Fig. 9.2e) indicates that the increased s-character of the outermost valence orbital of NF 3 relative to the situation in N H 3 is due to the larger N 2s contribution (relative to N 2p). This result, which is intuitively predicted on the basis of geometry, has often been rationalized in terms of simple valence bond hybridization concepts. Bent [B60] noted that in the absence of steric effects the bond angle decreases as the electronegativity of the substituent increases. It was also pointed out [B60] that as the electronegativity of the substituent increased, the central atom diverted increasing amounts of s-character to the lone pair orbital. These ideas for NF 3 are qualitatively supported by the present E M S results as well as by the calculated density maps (Fig. 9.2e). Another way of viewing the derealization of charge in N (CH 3 ) 3 and NF 3 (relative to N H 3 ) is by taking the density difference maps of the respective outermost valence orbitals as shown in Fig. 9.3. These correspond to ( a ) p N ( C H 3 ) 3 " P N H 3 ( F l g s - 9 - 3 a " 9 - 3 b ) a n d (W P N F 3 " P N H 3 ( F i S s - 9-3c-9.3d) in both position-space and momentum-space calculated using the respective 4-3 I G wavefunctions. The contour values chosen are ± 8 0 , ± 4 0 , ± 8 , ± 4 , ± 0 . 8 , ± 0 . 4 , ± 0 . 0 8 and ±0 .04% of the respective maximum intensities. From the position-space density difference maps the 'transfer of charge' away from the nitrogen center in the case of N ( C H 3 ) 3 is clearly illustrated (see Fig. 9.3a) whereas in the case of NF 3 the higher N 2s contribution relative to the N 2s contribution in N H 3 is also very obvious (see N 2s spikes in projection plots in r 9 N ( C H 3 ) 3 P n H 3 POSITION DENSITY DIFFERENCE a 1 1 1 1 1 1 1 1 1 1 1 j i • • t 1 I I I T 1- I I I 1 _ -4.0 - 2 . 0 0 .0 2 .0 4.0 - 0 . 5 0.0 248 MOMENTUM JDENSITY DIFFERENCE b IIII •<w.o) ' ' • 1 I I I I 1 1 1— -4.0 - 2 . 0 0.0 2 .0 4.0 - 0 . 5 0.0 ^ N F 3 / ° N H 3 POSITION DENSITY DIFFERENCE MOMENTUM DENSITY DIFFERENCE Fig. 9.3 Two-dimensional density difference contour maps in both position-space (xz plane) and momentum-space (P XP Z plane). The difference densities correspond to Pjg(CH ) - Pjqpj (3a and 3b) and p N F - p N H (3c and 3d). Positive difference is shown by solid contour lines while negative difference is shown as dashed lines. The positions of the atoms in the respective xz planes are indicated on the position-space maps. Experimental bond angles and bond distances were used. All dimensions are quoted in atomic units. 249 Fig. 9.3c). The complementary momentum density difference maps (Figs. 9.3b and 9.3d) illustrate the origin of the increased s-components of the momentum distribution for N (CH 3 ) 3 and NF 3 as also shown by the calculated spherically-averaged MDs and revealed by the EMS measurements of momentum profiles (see Fig. 9.1). The derealization of electron density towards the trans-H (in the case of N (CH 3 ) 3) and the increase of the relative N 2s contribution (in the case of NF 3 ) translates in momentum space to increased intensities at p = Oa o" 1 . This is illustrated by the hatched areas in the density difference projection plots in Figs. 9.3b and 9.3d. The calculated increased intensity at p = O a 0 " 1 is solely due to s-orbitals. This aspect can be understood by considering the Fourier transform relationship, iMp) = ( 2 7 T ) " 1 / 2 T e " 1 ^ * ( r ) dr [9.1] At p = Oa o ~ 1 , the 'area' of the position-space orbital wavefunction integrated over all configuration space yields the momentum space wavefunction, <MP>lp=0 = ( 2 T T ) " i / 2 J>(r) dr [9.2] Therefore only s-type orbitals can contribute at p = Oa 0 " 1 since pure p-type (or d-type) orbitals give equal lobes of opposite sign [LN75] with a node at the origin. Consider now a detailed comparison of the measured XMPs with the various calculated MDs. Measurements of the outermost XMP of N H 3 at high momentum resolution (Fig. 9.1a) has been reported earlier (see chapter 6) [BM87] as well as the distributions calculated from the highly extended Hartree-Fock limit SCF 250 (126-GTO) and CI (126-G(CI)) wavefunctions. The 126-G(CI) calculation which involves a full overlap between CI wavefunctions for the neutral target and the final ion states was used for normalizing the theoretical MDs to the 3a , X M P of N H 3 . It can be seen that with increasing sophistication in the basis set (i.e. from STO-3G to 4-3 IG to 126-GTO) better description of the low momentum region of the- X M P is obtained. Further improved agreement is obtained when correlation and relaxation effects in N H 3 are considered as shown by the 126-G(CI) calculation (curve 5c, Fig. 9.1a). Additional details of experimental and calculated results for all the valence orbitals of N H 3 are given in chapter 6. A few points however should be noticed. Firstly, the STO-3G + G M D (curve 3, Fig. 9.1a) which involves the addition of a diffuse s-function on nitrogen and diffuse p-functions on the hydrogen atoms, gives a surprisingly good description of the X M P (even slightly better than the 126-G(CI)) although the improvement in the total SCF energy compared to the STO-3G value is only marginal (see Table 9.1). In contrast the 4-3 IG* calculation, which utilizes the standard bond polarization functions (£1^ = 0.80) and has better calculated total energy, does not give any improvement in the calculated M D compared to the 4-3 IG calculation. These results, once again, illustrate the insufficiency of the variational procedure in guaranteeing good description of the X M P s especially when intermediate (or poor) quality basis sets are employed. It should also be noted that caution should be exercised in interpreting the STO-3G + G results as will be discussed later. Substitution of a methyl group in N H 3 yields the N H 2 C H 3 molecule for which results are shown in Fig. 9.1b. The increase in s-character relative to N H 3 is clearly observed both experimentally and in the calculations. A n earlier E M S 251 study of the outermost orbital of N H 2 C H 3 at somewhat lower momentum resolution [TL84] has indicated the same trend. The present study with improved momentum resolution (Ap = 0.15a0" 1) allows the low momentum region (0.1-0.4a0" 1) to be more directly observed in that a dip in the XMP (Fig. 9.1b) is seen at p = 0.2a o ~ 1 in agreement with the general predictions of the various calculated MDs. Similar to the situation for N H 3 , the MD description provided by the 4-3 IG basis is much better than that given by the simpler STO-3G basis. However as clearly illustrated by a comparison of the STO-3G+G and 4-3IG* MDs, it is again evident that diffuse functions are much more effective in describing the XMPs than standard bond polarization functions. It is also clear that even with the 'best fitting' calculation (STO-3G+G) the agreement is only semi-quantitative. More extended basis sets such as those used in N H 3 (Fig. 9.1a and chapter 6) are expected to give further improved quantitative agreement with experiment. However use of more complex wavefunctions for molecules as large as the methyl amines is too time consuming and prohibitively expensive at the present juncture. Results for the di-methylated species, N H ( C H 3 ) 2 , are shown in Fig. 9.1c. Again the same pattern of agreement between experiment and theory is obtained as with N H 3 and N H 2 C H 3 with very good agreement in shape of the MD (but not necessarily for other properties, see Table 9.1) for the STO-3G + G calculation. In the case of the tri-substituted methyl amine (N(CH 3 ) 3 ) for which results are shown in Fig. 9. Id the apparently less effective 4-3 IG* calculation (Figs. 9.1b and 9.1c) was not performed because of program limitations. The other calculated MDs (STO-3G, 4-3 IG, STO-3G + G) however show the same pattern as for N H 3 , 252 N H 2 C H 3 and N H ( C H 3 ) 2 with the ST0-3G + G calculation giving the best agreement for shape with the experiment. The theoretical MDs were normalized to the momentum profile at p = l . 0 a o " 1 using the 4-3 IG M D which is calculated from the best wavefunction (in energetic terms). As with the case of N H 2 C H 3 and NH (CH 3 ) 2 the agreement is again only semi-quantitative. The outermost X M P of N F 3 is shown in Fig. 9.1e compared with the calculated MDs. In this case the STO-3G + G calculation, which was quite successful for the methyl amines, grossly overestimates the s-character of this orbital but the 4-3 I G * calculation gives a quite reasonable description of the X M P . Note that in Fig. 9.1e the STO-3G + G M D has been scaled by a factor (x0.70) to bring the curve into better perspective with respect to the experimental measurements. This result indicates the importance of exercising caution in the addition of diffuse functions to fairly simple basis sets when predicting X M P s as pointed out before [BB87]. These diffuse functions (see Sec. 9.2) for C, N , O, F and H were designed mainly as field-induced polarization functions and used for improved calculations of molecular polarizabilities [ZC79]. Furthermore, the calculated energies from the STO-3G + G wavefunction are of poor quality, for example, in the case of N H 3 the calculated total energy is =20eV higher than the estimated Hartree-Fock limit (see Table 9.1). Likewise the predicted dipole moments obtained using the STO-3G + G wavefunction (for all molecules studied except for N F 3 ) showed greater discrepancies with the experiment compared with the dipole moments predicted by the corresponding STO-3G wavefunction. This is instructive because the dipole moment is often thought of as a molecular property that is particularly sensitive to the longer range part of the electronic charge density. 253 However as has been demonstrated by the EMS measurements and calculations for H 2 0 [BB87] and NH 3 [BM87] dipole moments and MDs of good precision are only obtained when diffuse functions are included in sufficiently extended basis sets that demonstrate basis set saturation. A closer examination of the experimental results (Fig. 9.1) shows an interesting trend in the 'secondary' p at p = l a 0 _ 1 (i.e. the p corresponding to the max max p-type component in the MD) which occurs in the range 0 . 4 a o ~ 1 ^ p ^ 2 a o ~ 1 . With increasing methyl substitution the observed secondary P m a x increases from 0.52a 0" 1 in NH 3 to = l . l a 0 " 1 and =1.3a 0-' for N ( C H 3 ) 3 and NF 3, respectively (see Table 9.1). The experimental results indicate increasing high momentum components as more methyl groups (or F atoms) are substituted into NH 3. The shift of the p m towards higher momentum has been associated with J 'max ° the presence of additional nodal surfaces as for instance in the outer 7T orbitals of p-dichlorobenzene (see chapter 8) [BB86] and C 0 2 [LB85a]. Tossell et al. [TL84] suggested that the higher momentum components observed in NH 2CH 3 compared to NH 3 are due to an extra nodal surface. This node is attributed to the trans-H Is which is antibonding with respect to the N 2p.t The steeper and more prominent nodal surfaces with increasing substitution can be seen in the corresponding position-space density maps in Fig. 9.2. It should be noted that there is no direct relation between the increased high momentum components (i.e. shift in -Pmax to higher momentum) in the t A recent EMS study by Rosi et al. [RC87] also suggested the presence of a 'region of minimum charge density between the nitrogen and carbons' to explain the secondary P m a x m tri-ethylamine (TEA). 254 experimental momentum profile and the orbital ionization potential (IP). In fact, as the 'secondary' p moves to higher momentum from N H 3 to N ( C H 3 ) 3 , max the IPs decrease (see Table 9.1). Martin and Shirley [MS74] have suggested that the decrease in IP that accompanies methyl substitution on N H 3 is due to the relatively large stabilization in the final state ion afforded by the easily polarizable methyl substituents. In support of this argument they show a linear correlation between A(IP) of the outermost orbital and the relaxation energy, and suggest that the flow of charge from the alkyl group to the nitrogen center yields the stabilization energy in the molecular ion [MS74, P47]. It is important to note that the conclusions of the present E M S work concerning derealization of 'lone pair' charge density towards the trans-H in C H 3 groups (see below) are not incompatible with the observations of Martin and Shirley [MS74]. This is due to the fact that the experimental momentum profile of the amines is largely an initial state property [WM78] while the trend in the vertical IPs is largely a final state effect [AB80]. 9.4. THE METHYL INDUCTIVE EFFECT The intrinsic inductive effect of methyl groups is still a controversial issue in chemistry [MB83]. In their recent extensive analysis of substituent effects on chemical reactivity, Swain et al. [SU83] have suggested that C H 3 and C 2 H 5 groups 'have no inductive (or field) influence but tend to donate electrons moderately by resonance'. The notion that methyl groups donate electrons is a prevalent view especially when considering the stability of carbo-cations. The results shown in the previous section for N H 3 , NH 2 CH 3 , NH (CH 3 ) 2 and N (CH 3 ) 3 however clearly provide experimental evidence supporting the general 255 conclusions of early theoretical ab initio MO calculations by Pople et al. [P70, HP70] that CH 3 groups are electron withdrawing. In particular the early conclusions of Hehre and Pople [HP70] regarding the methyl inductive effect, which were based on calculations using the STO-3G basis set, have now been confirmed in the present work where larger basis sets (e.g. 4-31G*) have been used. The present work places the early work of Pople [P70, HP70], which was criticized at the time, in proper perspective. Pople and co-workers [HP70, P70] noted that the increase in base strength of alkyl amines with increasing alkyl substitution is not associated with increasing electron density on nitrogen. In particular they noted [P70, HP70] the decreased electron density on the nitrogen center and increased electron density on the hydrogens trans to the lone pair as is also indicated by the density map diagrams shown in Fig. 9.2. The early theoretical studies [HP70, P70] indicated that, contrary to commonly held opinions, methyl groups are intrinsically electron withdrawing substituents. However prior to the present direct experimental probing of the electron density of the outermost orbitals (using EMS) these theoretical conclusions [HP70, P70] were also supported by the findings of a number of evidences from other experimental studies. For example, NMR chemical shift data [JK70] has also indicated that methyl groups are intrinsically electron withdrawing. In particular it has been demonstrated that decreased electron density around a particular nucleus corresponds to deshielding (i.e. decrease in 1 3C chemical shifts with increasing alkylation in substituted alcohols [JK70]). Furthermore the dipole moments of the alkyl amines are observed to decrease with increasing addition of methyl groups (see Table 9.1) quite contrary to what would be expected if methyl groups were to be intrinsically electron releasing or to have no significant electronic effect at 256 all. Ingold [169] in his discussion of the unusual dipole moment trend of the alkylated amines suggested that N - H and O-H bonds are associated with larger moments than N-C and O-C bonds. These rationalizations are consistent with the present results. The fact that base strength is not associated with electron density on the nitrogen atom but rather with the increased polarizability afforded by methyl substituents is now recognized [MM75, UM76, AB80]. These views have now reconciled previously misunderstood phenomena regarding the gas phase acidity and gas phase basicity of the amines [M65, BB71, BR71]. Brauman and Blair [BB71] have argued that the seemingly contradictory increased acidity and increased basicity of the amines with increasing alkylation is due in fact to the larger polarizability of alkyl groups (relative to hydrogen) rather than to inductive effects. Likewise, the N M R chemical shift data [JK70] and the stability of carbo-cations [MB83] are also recognized as due to two distinct and separate effects (i.e. initial and final state effects, respectively). 9.5. SUMMARY The present E M S results and accompanying calculations of MDs and density maps, although currently limited to the outermost valence orbitals of the amine series, are clearly consistent with the view that methyl groups are intrinsically electron withdrawing when bonded to nitrogen as earlier predicted in calculations by Hehre and Pople [HP70]. The present experimental results using E M S to probe directly the outermost orbital electron density clearly confirm the qualitative 257 aspects of the theoretical calculations based on SCF L C A O - M O wavefunctions. On the basis of the work for N H 3 (see chapter 6) [BB87] it can be expected that even better quality (highly extended) wavefunctions would provide even better description of the experimental results for the methyl amines but would not alter the conclusions of the present work. C H A P T E R 10. T H E H A L O G E N S : A T H E O R E T I C A L S T U D Y 10.1. OVERVIEW The halogens (except for F 2 ) were studied using the Flinders University symmetric non-coplanar electron momentum spectrometer with position sensitive detectors in both exit channels. The polar angles are 45° and the impact energy is lOOOeV plus the binding energy. Momentum resolution is = 0. l a 0 " 1 and the energy resolution is 1.5eV fwhm. Details of the spectrometer, its operation and data analysis procedures are discussed elsewhere [CM84, CM86]. Since the Flinders spectrometer has multichannel plates in the energy dispersive plane of each analyzer, the experimental momentum profiles are generated by deconvoluting the binding energy scan obtained at different 0 angles. This series of binding energy scans (done sequentially at different angles) therefore automatically gives normalized X M P s for respective orbitals corresponding to the ion states in the binding energy spectrum. A l l experimental measurements reported in this section were made by colleagues at U B C and Flinders University (see Acknowledgements) and further details are to be found in the published results for C l 2 [FG87], B r 2 [FG87a] and I 2 [GP87]. As part of a collaborative E M S project on the halogens involving the above personnel, the spherically averaged MDs and the corresponding momentum-space and position-space maps were generated by the author using the H E M S package developed at U B C . 258 259 The ground state electron configuration of the halogens ( ' E g * ) [CF71] are given below: CI 2 : c o r e 4 a g 2 4 a u 2 5<V 2 V 2 7 T g « inner valence outer valence Br 2 : c o r e inner valence outer valence c o r e 1ua g 2 I 0a u 2 11o g 2 6 * u « 6^ g« inner valence outer valence 10.2. MOLECULAR CHLORINE ( C l 2 ) Measured X M P s and calculated MDs [FG87] corresponding to ionization of the 2iTg, 27TU and 5 a g outer valence orbitals, as well as corresponding data for the 4o~u and 4 c g inner valence orbitals are shown in Figs. 10. la-10.5a. The experimental data for the three outer valence orbitals have been evaluated from fitted peak areas of the binding energy spectra [FG87]. Additional points providing consistency checks at # = 0° and 0 = 7° have been generated from the respective gaussian fits to the outer valence region of the wide energy range spectra [FG87]. Sections b and c of Figs. 10.1-10.5 show the respective two-dimensional momentum density and position density maps for an oriented C l 2 molecule calculated using the M L * polarized wavefunction. Contour values are at 0.02, 0.05, 0.08, 0.2, 0.5, 0.8, 2, 5, 8, 20, 50, and 80% of the maximum density. The side panels in sections b and c show density profiles (on a relative scale) along the dashed lines on the density maps. 260 The calculated spherically averaged MDs shown on Figs. 10.1a-10.5a have been obtained using a range of wavefunctions of varying quality. These SCF (RHF) wavefunctions were generated using the G A U S S I A N 80 package [BW80] at the equilibrium geometry (3.7568a.u.) [CF71]. A whole range of wavefunctions ranging from the simplest STO-3G basis set to the extended M L * * basis set (see Table 1) were constructed. The STO-3G and LP-41G* are internally stored basis sets of the G A U S S I A N 80 package [BW80] and were used without modifications. The M L series of wavefunctions were generated using the (13sl0p) basis set of McLean and Chandler [MC80] contracted to [6s5p]. This contracted basis set is referred to as the M L wavefunction. To investigate the effect of polarization functions, two polarized basis sets were developed namely the M L * and M L * * basis sets. The M L * basis was built from the M L basis augmented with one d-type polarization function having an exponent (0.56) suggested by Sakari et al. [ST81]. In similar manner the M L * * basis was built from the M L basis and then augmented with two d-type polarization functions with exponents (0.22, 0.797) suggested by Huzinaga [H84]. The calculated SCF (RHF) energies for the respective wavefunctions are shown in Table 10.1. The calculated total energies compare favorably with the best reported total energy (-918.99012a.u.) calculated at the S C F equilibrium geometry (3.7619a.u.) [SM74]. The momentum distributions and maps have been calculated using the H E M S package developed at U B C . A l l calculated MDs shown in the various figures are spherically averaged and have been convoluted with the experimental momentum Table 10.1. Wavefunctions for Molecular Chlorine Wavefunction Type Basis Set Total energy(a.u.) Reference ST0-3G GTO (9s5p)/ -909.11163 [BW80] [3s2p] LP-41G* GTO (5s5p1d)/ --a-- [BW80] [2s2p1d] ML GTO (13s10p)/ -918.92629 [MC80] [6s5p] ML* GTO (13s10p1d)/ -918.97376 --b--[6s5p1d] ML** GTO (13s10p2d)/ -918.97803 --b--[6s5p2d] No total energy available since frozen cores used. P o l a r i z a t i o n functions [ST81, H84] have been added to the ML basis set [MC80]. OS 262 resolution (Ap=0. 1 a 0 " 1 )• It should be noted that the experimental results shown for each outer valence orbital in Figs. 10. la-10.5a have the correct relative normalization to each other since they were determined by accumulation of repeated sequential binding energy scans. The data points are from integrated peak areas covering the unresolved rotational-vibrational width of each ion electronic state. In order to place all the calculations and the XMPs of the three outer valence orbitals in Figs. 10.1a-10.3a on the same (relative) absolute scale a single point height normalization has been used between the (best fitting) ML* calculation (see Table 10.1) and the measured 2itu XMP (Fig. 10.2a) at its maximum. All other data points, both experimental and theoretical, maintain their correct normalization relative to this single point. Consider first Figs. 10.1-10.3. It can be seen that a reasonably good quantitative fit to the 27Tg (Fig. 10.1a) and the 27T U (Fig. 10.2a) XMPs is given by the ML* [MC80] and LP-41G* [BW80] wavefunctions. These two wavefunctions, each with a single d-polarization function, give similar results. However the 27T U experimental cross section maximizes at a lower momentum than predicted by the calculation. The minimum basis set STO-3G wavefunction [BW80] gives a significantly poorer fit to experiment. The 5tfg XMP (Fig. 10.3a) shows an s-p mixed symmetry as expected from its atomic orbital (AO) composition (see also density maps in Figs. 10.3b and 10.3c). The STO-3G wavefunction clearly grossly misrepresents the orbital shape, with much too large a relative contribution from the p-type orbitals. The ML* and LP-41G* both represent the shape and relative contributions of the s- and p-type SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY Momentum (a.u.) Fig. 10.1 (a) Experimental and calculated spherically averaged momentum distribution for the 2ir orbital of C l 2 . The solid circles are experimental values extracted from the appropriate peak areas in the binding energy spectra at given values of 0; the open triangles are data extracted from the respective peak areas in the 0 = 0° and 7° spectra. In (b) and (c) the orbital density are shown in momentum and position space respectively. A l l dimensions are in atomic units. to W SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY Momentum (o.u.) Fig. 10.2 (a) Experimental and calculated spherically averaged momentum distribution for the 2ffu orbital of C l 2 . See Fig. 10.1 for details. tsD SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY Momentum (a.u.) Fig. 10.3 (a) Experimental and calculated spherically averaged momentum distribution for the 5o g orbital of C l 2 . See Fig. 10.1 for details. to Ol Fig. 10.4 (a) Experimental and calculated spherically averaged momentum distribution for the 4a orbital of C l 2 . See Fig. 10.1 for details. C i OS S P H E R I C A L L Y A V E R A G E D MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY Momentum (a.u.) Fig. 10.5 (a) Experimental and calculated spherically averaged momentum distribution for the 4a g orbital of C l 2 . See Fig. 10.1 for details. to - J 268 orbitals quite well, with the best s-to-p ratio given by the M L * calculation. However the overall cross section is in both cases (at least on the basis of the single point normalization on the 2TT u orbital discussed above) too high. A n explanation for this could be the possibility that the 5 ag pole strength is less than unity in the measured energy range about 16.23eV due to many-body effects causing the splitting of the pole strength. Some support for this proposition comes from the ADC(3) many-body calculation [FG87] which predicts just such splitting using a similar type of polarized wavefunction. It is obvious from a comparison of Figs. 10. la-10.3a that the 5ag X M P provides the most sensitive test of the composition of the wavefunction. Consider now the measured X M P s of the inner valence orbitals ( 4 a u and 4ag) shown in Figs. 10.4 and 10.5, respectively. It is noticeable that these orbitals are predicted to be quite compact in momentum space (and conversely diffuse in position space) relative to the outer valence orbital despite their significantly higher binding energies. The agreement between experiment and theory with regards to shape for the 4 a and 4 a orbitals is quite good. The effect of polarization built into the wavefunction is illustrated in Fig. 10.6 which shows the X M P s (as in Figs. 10.1-10.3) for all three outer valence orbitals together with the calculations using the McLean and Chandler [MC80] wavefunction without polarization (ML), with one set of added polarizing d-functions (ML*) and with a double set of polarizing d-functions (ML**), respectively. The normalization is identical to that used in Figs. 10. la-10.5a. z • I z q Q o t-cj U I to o 2 si u I 5 3 UJ u. ot 1 1 1 1 1 — i 1 r i 1— 2 l r g : \ Iv=11. 63«V . $. . . . \ i i ; 0.0 0.5 1.0 15 2.0 25 , \ 5 " g \ • \ Iv=16. 18«V . \ • \ \ » ft ^\ \ — i i i i i_ 0.0 0.5 1.0 Fig. 1 0 . 6 1.5 2.0 2.5 , 1 1 1 i — i — i — i i i — i — 27T, , u ry=14. 41 «v . • m -if ^ \ • i i V 0.0 0.5 1.0 1.5 2.0 2.5 i i i — i i i I . =23. 6eV expt 0.5 1.0 1.5 MOMENTUM (A.U.) 2.0 2.5 ci_2 — — ML ML * ML * • I , A Expt. The effect of basis set polarization on calculated spherically averaged momentum distributions for the valence orbitals of C l 2 . A l l calculations are on the same (relative) intensity scale with the experiment normalized to the M L * calculation at a single point on the 27T U X M P . to to 270 The calculated M D for the 5 a g orbital (see Fig. 10.3) is extremely sensitive to the degree of polarization selected. The best fit would be with the unpolarized (ML) wavefunction, if it is assumed that the 5 a g ionization is effectively confined to a single pole at 16.2eV. However as discussed above there is some theoretical evidence for a splitting of the 5 a g " * pole strength. As has also been noted above , the M L * wavefunction best describes the ratio of s to p components compared with the observed X M P , for the 5 o g orbital. In this connection, it is of interest to note that polarization functions in the wavefunction also increase the distribution of satellite intensity above the main line [FG87] and this is what is also observed in the experimental binding energy spectra [FG87]. 10.3. MOLECULAR BROMINE ( B r 2 ) The measured X M P s and calculated MDs [FG87a] corresponding to ionization from the 4 7 T g , 4 7 T U and 8 o g outer valence orbitals as well as the calculated MDs for the 7o"u and 7 o g inner valence orbitals are shown in Figs. 10.7a-10.11a, respectively. The relative peak areas at 23.1eV and 25.6eV derived from the binding energy spectra [FG87c] at <p = 0° and 0 = 6° are shown on the plots of the 7 a and 7 a orbitals (Figs. 10.10a and 10.11a) u y The calculated MDs were obtained using an unpolarized (14sllp5d)/[9s6p2d] contracted GTO basis set (denoted as VN) and a polarized (14sllp7d)/[9s6p4d] contracted basis set (denoted as VN*) . The two valence d-type polarization functions in the V N * wavefunction had exponents of 0.2 and 0.6. The correct relative normalization between the three outer valence XMPs has been maintained and a single point normalization was used to compare experiment with theory. q d -m oc U S P H E R I C A L L Y A V E R A G E D M O M E N T U M D I S T R I B U T I O N i i i i i i i — i i i Airn Br 0 9 2 l v=io. 74«V t 0 »)OOO«V • B . C . 0.5 10 J.5 MOMENTUM (A.U.) 2.0 2.5 M O M E N T U M D E N S I T Y -4.0 -2.0 0.0 2.0 4.0 0.5 1.0 P O S I T I O N d p si o d p W I p CO I . o d (0.1.0) — i — - I i -8.0 -4 .0 Fig. 10.7 (a) Experimental and calculated spherically averaged momentum distribution for the Ait orbital of B r 2 . The solid circles are experimental values extracted from the appropriate peak areas in the binding energy spectra at given values of <p; the open triangles are data extracted from the respective peak areas in the 0 = 0 ° and 6° spectra. In (b) and (c) the orbital density are shown in momentum and position space respectively. A l l dimensions are in atomic units. SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION MOMENTUM (A.U.) Fig. 10.8 (a) Experimental and calculated spherically averaged momentum distribution for the 47TU orbital of B r 2 . See Fig. 10.7 for details. m < o u 1/1 2 u _l o •n 0 1 o 6 CM o o $2 o SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION • i i i i i o d 0.0 8G* B r 0 I ¥=14. 62eV J0.5 MOMENTUM MOMENTUM DENSITY Br 2 8o\. -4.0 -2.0 0.0 2.0 4.0 0.5 1.0 POSITION DENSITY Br 2 q j (0,1.0) | 1 I 1 1__J 1 1 1 1 -8.0 -4.0 0.0 4.0 8.0 0.5 tO Fig. 10.9 (a) Experimental and calculated spherically averaged momentum distribution for the 80g orbital of B r 2 . See Fig. 10.7 for details. to CO Fig. 10.10 (a) Experimental and calculated spherically averaged momentum distribution for the 7o"u orbital of B r 2 . The triangles are relative values obtained from the peak areas at 23.1eV in the energy spectra. See Fig. 10.7 for details. tso SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY MOMENTUM (A.U.) Fig. 10.11 (a) Experimental and calculated spherically averaged momentum distribution for the la orbital of B r 2 . The triangles are relative values obtainea from the peak areas at 25.6eV in the energy spectra. See Fig. 10.7 for details. to i n 276 The best Fitting (VN*) calculated M D was height normalized to the 47T X M P . y The momentum space and position space density maps calculated using the V N * wavefunction are also shown in Figs. 10.7-10.11. Contour values are at 0.02, 0.05, 0.08, 0.2, 0.5, 0.8, 2, 5, 8, 20, 50, and 80% of the maximum density. The side panels of sections b and c show density profiles (on a relative scale) along the dashed lines on the density maps. A l l calculations are for an oriented B r 2 molecule at the experimental geometry (r = 4.2106a.u.) [CF71]. The two wavefunctions give very similar MDs and both give good quantitative fits, in both shape and magnitude , to the measured 47r_, 4ir and 8o X M P s . y u y The small discrepancy between calculation and experiment at low p for the 47T U orbital could likely be reduced by incorporating diffuse functions in the basis set as has been found necessary in obtaining adequate basis sets for comparison with E M S studies of H z O [BB87], N H 3 [BM87] and H 2 S [FB87a]. The (j> = 0° and 0 = 6° relative peak areas for the 23.1eV and 25.6eV peaks are in excellent agreement with the shapes of the calculated 7o"u and 7o"g MDs, respectively (Figs. 10.10 and 10.11). This confirms that the 23.1eV and 25.6eV peaks are due to ionization from the 7 o u and 70g orbitals, respectively. 10.4. MOLECULAR IODINE ( I 2 ) The measured X M P s and calculated MDs [GP87] corresponding to ionization from the 67Tg, 67T U and H ^ g outer valence orbitals are compared in Figs. 10.12a-10.14a, respectively. Two different wavefunctions are used, namely a (16sl3pl0d)/[7s6p5d] contracted GTO set (denoted as VN1) which has a total calculated energy of -13826.208a.u. and an unpolarized (16sl3p7d)/[6s5p2d] SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION 0.0 0.5 1.0 1.5 MOMENTUM (A.U.) 2.0 2.5 MOMENTUM DENSITY -4.0 -2.0 0.0 2.0 4.0 0.5 1.0 POSITION DENSITY -8.0 -4.0 0.0 4.0 8.0 0.5 1.0 Fig. 10.12 (a) Experimental and calculated spherically averaged momentum distribution for the 6ff orbital of I 2 • The solid circles are experimental values extracted from the appropriate peak areas in the binding energy spectra at given values of <p; the solid squares are data extracted from the respective peak areas in the <f> = l° and 7° spectra. In (b) and (c) the orbital density are shown in momentum and position space respectively. All dimensions are in atomic units. to SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY MOMENTUM (A.U.) Fig. 10.13 (a) Experimental and calculated spherically averaged momentum distribution for the 6ff u orbital of I2. See Fig. 10.12 for details. vt ni SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION I I 1 I l l l l , 1 " g I v=13. OOeV VN1 VN5 i \ - j • • • 0.0 0.5 1.0 11.5 2.0 2.5 MOMENTUM (A.U.) MOMENTUM DENSITY -4.0 -2.0 0.0 2.0 4.0 0.5 1.0 POSITION DENSITY I '2 : m : . o o (0.1,0) i l 1 1 1 l i i 1 -8.0 -4.0 0.0 4.0 8.0 0.5 1.0 Fig. 10.14 (a) Experimental and calculated spherically averaged momentum distribution for the 11a orbital of I 2 . See Fig. 10.12 for details. to SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION - T -0.0 0.5 1.0 1.5 MOMENTUM (A.U.) 2.0 2.5 MOMENTUM DENSITY -4.0 -2.0 0.0 2.0 4.0 0.5 1.0 POSITION DENSITY 1 '2 10<7U 1 i 1 . o o "(0.1.0) 1 1 1 1 1 I 1 1 1 1 -8.0 -4.0 0.0 4.0 8.0 0.5 1.0 Fig. 10.15 (a) Experimental and calculated spherically averaged momentum distribution for the 10au orbital of I 2 . See Fig. 10.12 for details. to OO O SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY MOMENTUM (A.U.) Fig. 10.16 (a) Experimental and calculated spherically averaged momentum distribution for the 10a orbital of I 2 . See Fig. 10.12 for details. to 00 282 contracted GTO set (denoted as VN5) which has a total energy of -13826.031a.u. The two different SCF wavefunctions give very similar calculated momentum distributions. Correct relative normalization of the 67Tg, 6TT u and H ^ g X M P s are maintained as explained earlier. In order to compare experiment with the theoretical calculation a single point normalization is made on the 6n X M P . The y momentum space and position space density maps calculated using the V N 1 polarized wavefunction are also shown in Figs. 10.12-10.14. Contour values are at 0.02, 0.05, 0.08, 0.2, 0.5, 0.8, 2, 5, 8, 20, 50, and 80% of the maximum density. The side panels show density profiles (on a relative scale) along the dashed lines on the density maps. A l l calculations are for an oriented I 2 molecule at the experimental geometry (r = 5.039a.u.) [CF71]. The general features of the experimental data are well described by the calculated MDs. The relative magnitudes of the calculated cross sections for ionization of the 67Tg, 67TU and l l a ^ orbitals are in reasonable agreement with the data, as well as the fact that the P m a x of the 67Tg X M P occurs at a higher momentum compared to the 6TT u X M P . For the X M P the calculations significantly overestimate the degree of symmetric components in the wavefunction, that is the momentum density at p = 0a 0 " 1 . The calculated MDs for the inner valence orbitals ( 1 0 a u and lOcTg) are shown in Figs. 10.15a and 10.16a. Similarly momentum-space and position-space maps calculated from the V N 5 polarized wavefunction are also presented alongside the reported X M P s in Figs. 10.12-10.16. The data points obtained from the long range binding energy scans at 0 = 1 ° and o> = 7° [GP87] are found to be 283 consistent with that predicted by theory except for the 100"u orbital. It has been postulated [GP87] that the different shape observed is due to the mixing of states of different symmetry, most likely poles due to the 10a g process. Such mixing would account for the broad momentum profile at 16-20.4eV. 10.5. GENERAL TRENDS Several interesting trends can be observed in comparing the measured XMPs of the valence orbitals of C l 2 , Br 2 and I 2 molecules. One interesting feature of the measured XMPs of the halogens is the outermost valence 7f u orbital. It is seen in Figs. 10.2, 10.8 and 10.13 that the measured P m a x in the respective TT XMPs decrease from C l 2 to B r 2 to I 2 . This decrease in P m a x as the atoms get heavier is also clearly predicted by the calculated MDs. It is well known that within a 'family' (for example, the noble gases [LB83]) the measured XMPs for the outer valence p-type orbitals maximize at lower values of momentum as the atom gets heavier. This effect is related to the inverse weighting property of the Fourier transform. As one proceeds from the lightest member of the group towards the heaviest member, the orbitals become more diffuse (in position space) and consequently become contracted in momentum space. Another trend in the halogens is observed in the measured XMPs of the outer valence a g orbitals which are of mixed s-p character. The symmetric (i.e. s-type) component in the measured XMPs relative to the anti-symmetric (i.e. p-type) component is found to decrease within the group as the diatomic becomes heavier. 284 That is, the observed percentage s-character (intensity at p = Oa o ~ 1 relative to that at p = l a 0 ~ 1 ) of the outer valence a orbital is found to be: y %s-character: C l 2 > B r 2 > I 2 This trend is predicted by the calculations especially for C l 2 and Br 2. In the case of the l l o " a orbital of I 2 the failure of the calculated MD to predict the y measured XMP is probably related to the difficulty in obtaining accurate wavefunctions for the I 2 molecule which is a 106-electron system. In fact for all these molecules ( C l 2 , B r 2 and I 2 ) the experimental total energy is currently not available and neither are accurate estimates of the Hatree-Fock limit total energies. Without these values it is not easy to evaluate the quality of the respective wavefunctions as was done for smaller molecules. In a simple LCAO-MO picture the a outer valence orbital in the halogen y diatomic molecules can be considered as a sigma (s-p hybridized) bonding orbital. The varying s and p components in this orbital therefore reflect the optimum balance necessary for bonding. Levin et al. [LN75] found in an investigation of theoretical orbital MDs that for the first row diatomics ( L i 2 , N 2 and F 2) and for the first row hydrides (BH, CH and FH) there was an increase in the contribution from the 2p-component of the • a_ hybrid orbitals with increasing y atomic number. This is manifested by the decreasing s-to-p ratio in the predicted MDs as the atomic number of the heavy atom increased [LN75]. Similar results were also predicted by Cade and Huo [CH67] in an AO composition analysis of the respective outer valence o molecular orbitals (calculated in position space) of y the first row hydrides. Bader et al. [BK67] correlated this increase in 2pa 285 component in the outer valence 30g molecular orbital from B H to F H (first row hydrides) to an increase in bonding character. The decreasing s-to-p ratio in the experimental momentum profiles of the outer valence o_ orbitals with increasing y atomic number is now seen as a more general phenomenon as is observed in the present case proceeding down a chemical group ( C l 2 , Br 2 / 12)-CHAPTER 11. MULTICHANNEL EMS: PROSPECTS AND DEVELOPMENTS 11.1. MULTICHANNEL ELECTRON MOMENTUM SPECTROMETER (MEMS) The idea of a multichannel E M S spectrometer is of paramount interest since coincidence rates in high resolution single channel E M S experiments are very low (<0.1cps). Two approaches have been attempted, namely a multichannel detection scheme in the angular (0) plane described by Moore et al. [MC78] and later by Cook [C81] and a multichannel detection scheme in the energy dispersive plane described by Weigold and co-workers [CM84, MW85]. The former scheme [MC78] used correlated pairs of channel electron multipliers while the latter scheme [CM84, MW85] involved the straightforward adaptation of commercial channelplate assemblies with position sensitive detectors (Surface Science Laboratories) to an existing pair of 180° spherical analyzer and has produced quite impressive results [CM84]. Some of experimental results obtained using this instrument are outlined in chapter 10. The former scheme (multichannel detection in the 0-plane) has been adapted in this laboratory and initial construction was done by Cook [C81]. Preliminary testing in the single channel mode of operation (i.e. one stationary C E M and one movable CEM) was done [L84a, M85a] as a necessary testing phase prior to the planned installation of a multichannel microchannel plate (MCP) detector system. These tests [L84a, M85a], however, showed that coincidence signals were extremely low and that this was possibly due to problems in the original design and construction. In addition the angular distribution of the (singles) elastic signal was found to be anisotropic. The present chapter outlines significant changes that have been made to the 286 287 original design of the M E M S as well as initial (single channel mode) results on helium and argon. 11.1.1. Accurate Alignment Several reasons were advanced to explain the difficulties encountered with the 'prototype' spectrometer. These were: (1) Spurious magnetic fields; (2) Faulty design parameters in the C M A and conical retarding lens system; (3). Inaccurate alignment; (4) Background secondary electrons. The present multichannel E M S spectrometer is shown in Fig. 11.1. One major change from the original version [C81] has been the redesign and construction of a spherical retardation stage [M85a]. Previous work [M85a] also concentrated on correcting magnetic field inhomogeneity in the spectrometer by replacing all magnetic components in and around the spectrometer especially in the area near the scattering region. This, however, did not improve the very low coincidence rate and therefore suggested items (2)-(4) should be examined. From Fig. 11.1 it can be seen that the basic design parameters of the original [C81] 360° C M A have been retained. A review of the original design parameters and a comparison with the theoretical values recommended by Risley [R72] are shown in Table 11.1. As can be seen the present design parameters conform with the theoretical values. Furthermore at E =100eV the theoretical deflection pass voltage (93eV) is very close to the observed deflection voltage (90eV). This then 288 OUTER CYLINDER Fig. 11.1. TMP 150IVsec Schematic of Multichannel E M S Spectrometer. The components are: height adjustment screws (HAS), end correctors (EC), retarding grid (RG), spherical retarder (SR), conical lens-deflector (LENS), channel electron multiplier (CEM), multichannel plate (MCP) mount, inner and outer cylinders of the C M A , gas cell (GC), Faraday cup (FC), spray plates (SP1, SP2, SP3), quadrupole deflectors ( Q l , Q2, Q3), focus (F), anode (A), grid (G), cathode (C) and turbomolecular pumps (TMP). 289 Table 11.1. Design parameters for 360° C M A . Inner radius (a) Outer radius(b) V d ( e V ) c Actual* 1.0 2.0 0.5 Source to image distance 2.5 (z 0 ) 90 Theoretical a,b 1.0 2.0 0.5 2.5 93 In units of inner cylinder radius, a (r = 63mm). Values recommended by Risley [R72] for first order focusing at 0 = 45° . H d = d s + d i " E / V Deflection voltage for pass energy is lOOeV derived from (b/a) ==2.1. 290 suggested that item (2), at least in regard to the C M A , is not the likely reason for the coincidence problem. The deflection voltage also increased linearly with an increase in the C M A pass energy as expected. A simple consideration of the design tolerances in the construction of the C M A showed that accurate alignment maybe more critical than was originally assumed. The M E M S spectrometer is mounted on Teflon sheets, instead of ruby balls, and therefore could easily be mis-aligned. Assuming a slight non-coaxiality of the outer and inner clinders, it can be shown that this will result in a large difference in the electron trajectories that are eventually focussed on the exit slits at opposite (±180° ) sides of the C M A . For the present set-up (b/a = 2.0, z 0 = 2.5) and a pass energy of lOOeV, a 0.4mm shift of the inner cylinder axis relative to the outer cylinder co-axis results in a l.OeV difference in deflection voltages! Following the same arguments the alignment of the electron gun relative to the C M A co-axis will also be crucial. Instead of attempting a complete re-design and reconstruction of the spectrometer, simple but significant changes have been made to observe whether, in fact, accurate alignment was critical to the coincidence problem. The following changes were made: (a) Installation of height adjustment screws (HAS); (b) Re-design of gas cell. Teflon-tipped height adjustment screws at the bottom of the outer cylinder allowed very small (=0.1mm) and accurate changes in the alignment of the inner and outer cylinder co-axes. The gas cell was re-configured by constructing 291 a brass tube that was attached to the lens system (see Fig. 11.1). This enabled the aperture plates (SP2 and SP3) to be placed on the same tube. This allowed the incoming electron beam to be well-defined with respect to the C M A co-axis. To complement the following physical changes, an accurate alignment procedure was empirically devised. This involved the slow and tedious process of re-assembling the electron gun and the C M A mounting block and simultaneously aligning them with a high tolerance (12.7mm dia.) stainless steel rod and a specially designed jig. The inner and outer cylinders are then placed on top of the C M A mounting block. To further align the outer cylinder axis with the inner cylinder co-axis, the height adjustment screws are used in conjunction with a high precision level. With this accurately aligned set-up the elastic signal (E 0 =600eV) was monitored with a single C E M mounted on top of the C M A . By moving the C E M around the 360° annular slit the homogeneity of the elastic signal as a function of <j> could be monitored. Tests showed that the elastic signal was homogeneous to better than ± 5 % , with the C M A pass energy set to lOOeV. These tests were done without the application of deflection voltages in the lens system. The subsequent addition of another C E M (fixed) allowed the first tests for coincidence detection. This was done for helium. The tests showed that only a very weak coincidence spectrum could be obtained for helium. In addition, substitution of argon, which has a smaller cross section than helium, did not give any detectable coincidence spectrum. These results clearly suggested spurious background signals were 'swamping' the coincidence signals. 292 11.1.2. Suppression of Background Secondary Electrons Several procedures and methods for suppressing background secondary electrons are available [FI75]. The best method discovered in the present study was a simple, non-magnetic, stainless steel grid (50 mesh, SM-53, Ethicon Inc.). This was installed in front of the exit slit of the C M A (see Fig. 11.1) and an external retarding voltage was applied. By varying the external retarding voltage on the grid the elastic signal could be optimized both in terms of shape and signal to background ratio. An example of the improved performance that could be obtained with the grid is shown in Fig. 11.2. As can be seen the background signal is dramatically reduced. Later tests showed that secondary electrons could also be suppressed by applying a negative voltage to the front end of the CEMs. A series of tests were carried out on helium and argon using the retarding grid. These single channel mode tests were successful suggesting that the main reasons for the difficulty in detecting the E M S coincidence signals were items (3) and (4), namely inaccurate alignment and background secondary electrons. The results will be discussed later in Sec. 11.2. 11.1.3. CAMAC Interface A C A M A C t interface and some associated software were obtained from Kinetic Systems Corp. Details are shown in Fig. 11.3 and Table 11.2. C A M A C is a standard modular interface (hardware and software) for exchanging data and control information between a computer and a particular instrument or groups of t C A M A C stands for Computer Aided Measurements and Control adopted by ANSI / IEEE Std. 583-1982. 293 ELASTIC SIGNAL-without GRID 120 -i 100- • • • m t— 80-OF corn 60-• • d z 40- • • 20-• • • • 397 398 399 400 401 402 403 ENERGY ev ELASTIC SIGNAL-with GRID 120-100- • • • P 8 0 -z NO. OF COl 6 0 -40-• • — • - ^ • leV fwhm • • 2 0 - • • • • 0 -397 398 399 400 401 402 403 ENERGY «v Fig. 11.2. Comparison of elastic signal (a) without grid and (b) with grid. E 0 =-400eV, E n =-100eV, E . . = -395eV. C A M A C 294 Fig. 11.3. C E M P R E A M P CFD C E M P R E A M P A M P CFD TAC •top Vary E Q " | H V i Vary O -H V P.S. MOTOR ADC DAC »,2 DAC C R A T E CONTROLLER L S I - B U S o o o o > CPU TTO RL02 [a] T I M E M O D E C A M A C C E M P R E A M P 1> 1 CFD C E M P R E A M P A M P CFD T A C SCA Vary E ~ Vary (J)"* 0 COUNTER 1 COUNTER a LAM COUNTER 0 DAC 1.2 DAC CRATE CONTROLLER < L S I - B U S A A Oil 3 > [l>] C O U N T M O D E PRINTER M E M S modes of operation, (a) Time mode and (b) Count mode. 295 Table 11.2. Configuration of C A M A C a system. Unit Model # Comments C A M A C Crate K S C 1502 25 stations Crate Controller K S C 3920-Z1B Interfaced with PDP 11/23+ system using a K S C 2920 computer interface card Analog to Digital converter (ADC) K S C 3553-Z1B 12-bit Digital to Analog converter (DAC) K S C 3112-M1A 12-bit, 8-channel Presettable counter (scaler) K S C 3640 16-bit, 4-channel Sample and Hold unit home-built, uses MN346 chips Software K S C 6410 RSX-11M compatible a Computer-Aided Measurements and Control. 296 instruments. Due to the strict standardization and high availability of C A M A C modules the C A M A C system is easy to use and is very cost-effective in the long run. For these reasons C A M A C is widely used in research and industry all over the world. Extensive literature exists for the C A M A C system [HL76] and therefore the present description will focus more on its application to the present study. 11.1.4. Modes of Operation Due to the particular needs of the E M S experiment two modes of operation were designed, namely, a time mode (Fig. 11.3a) and a count mode (Fig. 11.3b). The different modes of operation involved slightly different hardware set-ups and likewise different computer programs. The time mode utilized the A M A program [B87] and involves the generation of a time spectrum at each angle <p (or at each energy). The coincidence spectrum is then obtained by applying a 'software' channel analyzer to each time spectrum. The time mode (Fig. 11.3a) operates by digitizing each signal from the time-to-amplitude converter (TAC, Ortec 567) using a combination of a home-built sample-and-hold unit and a slow analog-to-digital converter (ADC). Each T A C signal (voltage proportional to time difference between start and stop pulses) is stored in computer memory. The computer real time clock serves as the program timer. The timer is regularly polled before stepping to the next parameter (energy or angle). A t the end of each complete scan, data collection is inhibited and a 'software' channel analyzer is applied to each time spectrum. Each time 297 spectrum is scanned and the number of signals in the coincident window relative to the random window is obtained to determine the true coincidence count. As can be seen the time mode is slow and ties up the computer unnecessarily because the present A D C cannot store more than one word at a time.t However, it has the advantage of storing all available raw data and is therefore useful for debugging the spectrometer as well as for routine checks. The count mode (see Fig. 11.3b) involves the use of the E M S program. It operates using an interrupt service routine and therefore allows the computer to function more efficiently. Briefly, the count mode operates by sending the T A C output to two single channel analyzers (Ortec 550) corresponding to the random and coincidence channels. Each SCA then sends a slow NIM-logic pulse to an up/down counter or scaler whenever the T A C signal occurs in the prescribed window. A third scaler which monitors the countrate (via the ratemeter, Ortec 541) of either the movable or fixed C E M serves as a program timer. This therefore ensures that for each parameter (angle or energy) scanned the observed coincidence counts are normalized on the countrate thus allowing for variations in gas pressure and beam intensity. The end of each parameter scan is noted by a L A M (look-at-me) signal generated on the third scaler (ie. countrate monitor) which activates an interrupt service routine (ISR). The interrupt service routine responds by monitoring the contents of the random and coincidence scalers, clearing the t A possible solution would be to use a fast A D C with memory such as the buffered, 13-bit LeCroy-3512 A D C . 298 scalers and initiating the next set of parameters. Once the final angular or energy parameter is reached, data collection is inhibited and the true coincidence spectrum is calculated and displayed on the video screen. An accompanying printout is also produced. The count mode has several advantages over the time mode. Due to the particular configuration the count mode does not tie up the computer. This frees the computer to other users and other jobs. Furthermore, the count mode uses hardware channel analyzers and does not use the slow A D C and sample and hold units. The deadtime per event is therefore smaller (=5»us) compared to the time mode which has an estimated deadtime of =150jus. Finally, as mentioned earlier the count mode normalizes the coincidence spectrum on the countrate (instead of the dwell time) thus allowing for small experimental variations during the scan. The count mode, however, suffers from one disadvantage and that is its incapability of obtaining a time spectrum. Therefore in practice the E M S experiment is performed by doing initial runs using the time mode and then the routine runs are done using the count mode. In the future these two modes of operation can be integrated into a more efficient and flexible" package. 299 11.2. PRELIMINARY RESULTS 11.2.1. Helium With the modified multichannel electron momentum spectrometer (Fig. 11.1) and the alignment procedure outlined above, the binding energy spectrum of helium (Is )^ was measured at different impact energies and at different C M A pass energies. A sample run (single channel mode) done at <j> = 0°, E 0 = 8 0 0 e V and E p a g s = 200eV is shown in Fig. 11.4. Also shown in Fig. 11.4 (inset) is the variation of the observed fwhm of the He Is binding energy peak as a function of C M A pass energy. Due to the very small natural line width of the He Is"''" ionization process it can be used as a measure of the M E M S energy resolution. It is found that at E 0 = 8 0 0 e V and E p a g g = 200eV the experimentally observed energy resolution (2.0±0.5eV fwhm) is close to the predicted energy resolution (2.4eV).t The slightly better energy resolution (%AE/E) of the multichannel E M S spectrometer as opposed to the high momentum resolution single channel E M S spectrometer outlined in chapter 3 is due to the smaller exit slit in the multichannel spectrometer. 11.2.2. Argon To determine the angular resolution of the present multichannel E M S spectrometer, test runs in single channel mode were carried out using argon. Fig. 11.5 shows the angular correlation spectrum of the argon 3p orbital measured at E 0 =815 .7eV, E =200eV. It can be seen that the angular correlation spectrum pass t This is based on the present slit dimensions (exit=entrance = 1.0mm), a dispersion of 151mm and 0 = 45° which gives an analyzer energy resolution of — 1%. The contribution of the gun energy spread is estimated to be =0.8eV. 300 FWHM vs. CMA PASS ENERGY O EXPTL BEST FIT 1 1 1 1 1 1 1 20 22 24 26 26 30 ENERGY(ev) F i g - 11-4. He Is binding energy spectrum. E 0 =800eV, E =200 eV, pass 0 - 0 ° . The inset shows variation of He Is fwhm as a function of C M A pass energy. The dashed lines indicate error limits of the projected C M A energy resolution. 301 is reasonably symmetric about 0 = 0° indicating that the inhomogeneities introduced by any slight misalignment or retardation effects are small under the present operating conditions. Furthermore from the ratios of the minimum (at 0 = 0°) to the maximum of the angular correlation spectrum, the momentum resolution is estimated to be 0. l - 0 . 2 a 0 " 1 . This result is quite close to the momentum resolution of the single channel E M S spectrometer outlined in chapter 3. During the latter part of the study, tests were performed at higher impact energy (lOOOeV) and with much larger retardation (E =100eV). However it pass was observed that the coincidence count rate decreased dramatically. It is believed that much of the difficulty is due to the lens effects in the retardation stage which distorts the electron trajectories appreciably. Several suggestions are proposed to remedy this problem. One of them is to devise, design and test a 'softer' retardation stage in the lens system. Possibly a two-stage retardation with focusing in between will improve the present situation. The present studies, however, have clearly identified the critical aspects necessary in the design of an optimum multichannel E M S spectrometer. Accurate alignment of the C M A and the suppression of secondary electrons allowed test measurements (in single channel mode) of the binding energy spectra of helium as well as the angular correlation spectra of the Ar 3p orbital. From these preliminary measurements at E o = 8 0 0 e V (E I.„„„ = 200eV) the energy resolution and momentum resolution are estimated to be 2.0eV fwhm and O . l - 0 . 2 a 0 ~ 1 , respectively. These observed instrumental resolutions clearly indicate the potential 3 0 2 600 500 400 300 H UJ 200 CO to to o a: u 100 -100 -40 Ar 3p Momentum Profile \ -30 - | — -20 I I / Io 1° V ° V \ \ V -10 0 ANGLE (deg) 10 20 30 Fig . 11.5. A r 3p experimental momentum profile. E 0 =-815 .7eV, EP a s s = - 2 0 0 e V ' E g r i d = " 3 9 5 e V -of the present instrument to function as spectrometer in the very near future by sensitive detector. 303 a high performance multichannel E M S installing a microchannel plate position C O N C L U D I N G R E M A R K S "In the present status of experimental techniques, no one has been able to observe the orbital pattern experimentally. However, to date, no one can assert that this is totally impossible.... If we could only experimentally obtain any knowledge of HOMO and LUMOt patterns, chemistry would be profoundly affected. In that event the orbital pattern concept, which is at present of a somewhat unreal nature, will be provided with a certain empirical nature." Kenichi Fukui, 1977; Nobel Laureate in Chemistry (1981) Although Prof. Fukui was unaware at that time of the yet developing technique of electron momentum spectroscopy (EMS), he was definitely prophetic in realizing its vital importance. The concerns regarding the 'orbital pattern concept' in chemistry was shared by the principal developers of electron momentum spectroscopy [CG72, WH73] and by those who specifically sought to apply E M S to molecular quantum chemistry [HH77, MC78]. These applications to quantum chemistry have been considerably extended in the series of studies reported in this thesis. The experimental momentum profiles (XMPs) and binding energy spectra of several molecules as measured in the present E M S studies, not only have provided much needed empirical basis to the 'orbital concept', but most importantly have laid a solid basis for assessing the quality of theoretical wavefunctions in quantum chemistry. Such an assessment was made feasible by the fact that under the kinematic conditions employed in E M S (E o >1000eV, t H O M O refers to the highest ocupied molecular orbital whereas L U M O refers to the lowest unoccupied molecular orbital. 304 305 symmetric, non-coplanar), the (e,2e) cross section is proportional to the absolute square of the ion-neutral overlap amplitude--& quantity dependent solely on the particular molecular electronic structure (chapter 2). Exact calculations of the ion-neutral overlap amplitude, though feasible in principle, are simply intractable for most chemical systems. A conceptual and computational advantage is provided by 'models' of electronic structure, one of which is the Hartree-Fock model which considers the electrons as independent particles moving in an average field. In such a case, the total wavefunction is represented by an anti-symmetrized product of one-electron functions or so-called orbitals. Within the Hartree-Fock model, the experimental momentum profile obtained in E M S can be interpreted as the electron momentum distribution corresponding to the orbital that has been ionized. Thus it has often been said that E M S provides an experimental tool for orbital imaging and wavefunction mapping. The limits of the Hartree-Fock model have been illustrated and discussed in this thesis. The detailed E M S studies of H 2 0 (chapter 4), D 2 0 (chapter 5), N H 3 (chapter 6) and H 2 C O (chapter 7) clearly illustrated the inadequacies of the Hartree-Fock model in predicting the fine details of the experimental momentum •profile, especially in describing the X M P of the least bound molecular orbital. Collaborative theoretical effort, integral to the present work, has resolved these discrepancies by performing accurate calculations of the ion-neutral overlap distribution utilizing highly correlated wavefunctions for both the initial neutral state and final ion states. Other considerations such as vibrational effects, experimental inaccuracies, etc. have been found to be of minor consequence. The 306 need for going beyond the Hartree-Fock model and including explicitly the details of the initial neutral and final ionic states is therefore stressed. In retrospect, the following results seem natural because quantum theory, which is the basis of our understanding of the physico-chemical world, is ultimately a 'physics of processes and not of properties, a physics of interactions and not of attributes' [B35]. The E M S studies of small molecular systems (chapters 4-7) provided a benchmark for assessing the limitations not only of the theoretical framework for interpreting the X M P s but also of the theoretical interpretation of the binding energy spectra, most especially the inner valence binding energy region. The E M S studies of the inner valence binding energy region of H 2 0 , N H 3 and H 2 C O showed extended structures which were qualitatively predicted by configuration interaction (CI) and Green's function methods. The improved prediction of the ionization intensities, however, was shown to be attained only when sufficiently accurate wavefunctions (with extended basis sets and the inclusion of sufficient correlation) were used. The extension of these E M S studies to much bigger systems was a logical offshoot following the successful studies of small molecular systems. E M S studies of the outermost valence orbitals of the methylated amines (chapter 9), N F 3 (chapter 9) and para-dichlorobenzene (chapter 8) showed interesting chemical trends. In particular, the measured X M P s of the methylated amines showed a trend clearly predicted by molecular orbital calculations. These calculations however suggested extensive electron delocalization away from the nitrogen 307 center-— a picture contrary to conventional views regarding the inductive effects of methyl groups. 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Evaluation of wavefunctions by electron momentum spectroscopy Bawagan, Alexis Delano Ortiz 1987
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Title | Evaluation of wavefunctions by electron momentum spectroscopy |
Creator |
Bawagan, Alexis Delano Ortiz |
Publisher | University of British Columbia |
Date Issued | 1987 |
Description | Electron momentum spectroscopy (EMS) provides experimental atomic and molecular electronic structure information in terms of the binding energy spectrum and the experimental momentum profile (XMP), which is a direct probe of the electron momentum distribution in specific molecular orbitals. The measured XMPs permit a detailed quantitative evaluation of theoretical ab initio wavefunctions in quantum chemistry and also provide a means to investigate traditional concepts in chemical reactivity at the fundamental electronic level. This thesis reports high momentum resolution EMS measurements of the valence orbitals of H₂0, D₂0, NH₃ and H₂CO obtained using an EMS spectrometer of the symmetric, non-coplanar type operated at an impact energy of 1200eV. The measured experimental momentum profiles for the valence orbitals of each molecule have been placed on a common intensity scale, which has allowed a stringent quantitative comparison between experiment and theory. These studies now confirm earlier preliminary investigations that suggested serious discrepancies between experimental and theoretical momentum distributions. Exhaustive consideration of possible rationalizations of these discrepancies indicate that double zeta quality and even near Hartree-Fock quality wavefunctions are insufficient in describing the outermost valence orbitals of H₂0 and NH₃. Preliminary results for H₂CO also indicate that near Hartree-Fock wavefunctions are incapable of describing the outermost 2b₂ orbital. Interactive and collaborative theoretical efforts have therefore led to the development of new Hartree-Fock limit and also highly correlated (CI) wavefunctions for H₂0, NH₃ and H₂CO. It is found that highly extended basis sets including diffuse functions and the adequate inclusion of correlation and relaxation effects are necessary in the accurate prediction of experimental momentum profiles as measured by electron momentum spectroscopy. New EMS measurements are also reported for the outermost valence orbitals of NF₃, NH₂CH₃, NH (CH₃)₂, N (CH₃)₃ and para-dichlorobenzene. These exploratory studies have illustrated useful chemical applications of EMS. In particular, EMS measurements of the outermost orbitals of the methylated amines have revealed chemical trends which are consistent with molecular orbital calculations. These calculations suggest extensive electron density derealization of the so-called nitrogen 'lone pair' in the methylated amines in comparison to the 'lone pair' in NH₃. EMS measurements of the non-degenerate π₃ and π₂ orbitals of para-dichlorobenzene show different experimental momentum profiles consistent with arguments based on inductive and resonance effects. These experimental trends, both in the case of the amines and para-dichlorobenzene, were qualitatively predicted by molecular orbital calculations using double zeta quality wavefunctions. However more accurate prediction of the experimental momentum profiles of these molecules will need more extended basis sets and the inclusion of correlation and relaxation effects as suggested by the studies based on the smaller molecules. An integrated computer package (HEMS) for momentum space calculations has also been developed based on improvements to existing programs. Development studies testing a new prototype multichannel (in the ɸ plane) EMS spectrometer are described. |
Subject |
Wave functions Electron spectroscopy |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-07-27 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0060438 |
URI | http://hdl.handle.net/2429/26958 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemistry |
Affiliation |
Science, Faculty of Chemistry, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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