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Studies of valence electron densities using electron momentum spectroscopy and computational quantum… Neville, John J. 1997

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Studies of Valence Electron Densities Using Electron Momentum Spectroscopy and Computational Quantum Chemistry by John J . Neville B . S c , The University of New Brunswick, 1991 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F C H E M I S T R Y W e accept th is thesis as c o n f o r m i n g to the r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A F e b r u a r y 1997 © J o h n J . N e v i l l e , 1997 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C HE M zi j~ Y-^y The University of British Columbia Vancouver, Canada Date / 7 DE-6 (2/88) Abstract The valence shell binding energy spectra and momentum profiles of acetone, dimethoxymeth-ane and glycine have been studied by electron momentum spectroscopy at a total energy of 1200 eV. These experimental measurements present several challenges, in both data collec-t ion and analysis, that are typically not encountered in electron momentum spectroscopy studies of smaller molecules. A newly constructed energy-dispersive multichannel electron momentum spectrometer yields improvements in both sensitivity and energy resolution as compared wi th single-channel spectrometers used in previous studies. These instrumental i m -provements address several of the challenges of performing electron momentum spectroscopy studies of larger molecules and make possible the experimental measurements presented in this thesis. The studies of acetone and dimethoxymethane each encompass the approximate binding energy range of 5-60 eV, while the study of glycine is l imited to the binding energy range of 6-27 eV. The current work provides the first experimental values of the inner-valence (6a)" 1 , ( 4 b ) - 1 and ( 5 a ) - 1 ionization potentials of dimethoxymethane, which are 20.4, 22.6 and 23.9 eV, respectively. For all three molecules, many-body effects are evident in the ionization spectra at binding energies greater than 16 or 17 eV. The experimental momentum profiles of all three molecules are compared w i t h theoretical profiles obtained from Hartree-Fock (HF) and density functional theory ( D F T ) calculations using the target Hartree-Fock and target K o h n - S h a m approximations, respectively. The i i convergence of the theoretical results with basis set is investigated by performing the cal-culations with a range of basis sets. In the case of the D F T calculations, the sensitivity of the theoretical profiles to the choice of exchange-correlation functional is also investi-gated. The outermost experimental momentum profiles of acetone (i.e., 5b 2 and 2bi) and dimethoxymethane (10b + 11a) are also compared with theoretical profiles calculated by E . R. Davidson from multi-reference single and double excitation configuration interaction ( M R S D - C I ) calculations of the corresponding neutral molecules and ions. The theoretical momentum profiles obtained from D F T calculations are generally in good agreement with the experimental profiles. The notable exceptions are the 5b 2 profile of ace-tone and the 9b profile of dimethoxymethane, for which none of the theoretical methods considered here reproduce the experimental profiles. Agreement is poorer between the H F profiles and the experimental data. This is particularly so for the outer-valence momentum profiles, where the H F calculations tend to underestimate the intensity at low momentum. There is little difference between the large-basis-set H F calculations and the M R S D - C I calcu-lations of the outermost momentum profiles of acetone and dimethoxymethane. Convergence of the theoretical momentum profiles with respect to basis set occurs with the 6-311-1—t-G** basis set for both the H F and D F T calculations; theoretical momentum profiles calculated using smaller basis sets tend to underestimate the intensity at low momentum, particularly for the outer-valence momentum profiles. The D F T profiles are relatively insensitive to changes in the exchange-correlation functional. m Table of Contents Abstract i i List of Tables v i i List of Figures v i i i List of Abbreviations x i Preface x i i i Acknowledgements xv 1 Introduction 1 1.1 Electron momentum spectroscopy 3 1.1.1 Development of E M S 4 1.1.2 Applications of E M S 6 1.1.3 Recent advances in E M S 10 1.2 E M S of larger molecules 12 1.3 Overview of the thesis 15 2 Theoretical Background and Computational Methods 19 2.1 The (e,2e) reaction 20 2.2 The E M S differential cross-section 23 2.2.1 The distorted-wave impulse approximation 24 2.2.2 The plane-wave impulse approximation 25 2.2.3 The target Hartree-Fock approximation 27 2.2.4 The target Kohn-Sham approximation 29 2.2.5 Angular resolution effects 32 2.3 Electronic structure calculations 32 2.3.1 Hartree-Fock theory 35 iv 2.3.2 Post-Hartree-Fock methods: electron correlation 38 2.3.3 Density functional theory 42 2.3.4 Basis sets 46 2.4 Molecular conformation 51 2.4.1 Geometry optimizations 52 2.4.2 Relative conformer energies 53 3 Experimental Methods 56 3.1 Mult ichannel energy-dispersive electron momentum spectrometer 56 3.2 Instrument electronics 61 3.2.1 Position electronics 61 3.2.2 T i m i n g electronics 63 3.3 Spectrometer calibration and operating conditions 66 3.4 D a t a collection and treatment 70 3.4.1 Non-binning data collection mode 72 3.4.2 B inning data collection mode 73 3.4.3 Background correction 74 3.4.4 Experimental angle and momentum profiles 75 4 Acetone 79 4.1 Introduction 79 4.2 Experimental details 81 4.3 Computat ional considerations 82 4.3.1 Molecular conformation and theoretical momentum profiles 82 4.3.2 Choice of basis set 87 4.4 Valence binding energy spectra 90 4.5 Valence momentum profiles 97 4.5.1 Outer-valence momentum profiles 102 4.5.2 Inner-valence momentum profiles 112 5 Dimethoxymethane 119 5.1 Introduction 119 5.2 Experimental details 122 5.3 Computat ional details 123 5.4 Valence binding energy spectra 124 5.5 Experimental and theoretical momentum profiles 129 5.5.1 Outer-valence momentum profiles 132 5.5.2 Inner-valence momentum profiles 143 5.6 Computat ional method and basis set effects 153 v 5.7 Consideration of other conformers 161 5.8 Optimized geometries 165 6 Glycine 173 6.1 Introduction 173 6.2 Experimental details 179 6.3 Computational Details 180 6.4 Valence binding energy spectra 187 6.5 Momentum profiles 190 6.5.1 General observations 209 7 Conclusions 211 Bibliography 217 vi List of Tables 2.1 Basis sets util ized for electronic structure calculations 48 4.1 Conformers of acetone 84 4.2 Hartree-Fock and L S D A density functional theory calculations of acetone . . 88 4.3 Measured and calculated valence ionization potentials of acetone 92 4.4 Calculations used to generate T M P s of acetone 100 5.1 Measured ionization potentials and calculated orbital energies of the valence shell of dimethoxymethane 127 5.2 Calculated properties of the gauche-gauche and gauche-anti conformers of dimethoxymethane 131 5.3 Basis set and computational method dependence of calculated properties of the gauche-gauche conformer of dimethoxymethane 158 5.4 Relative thermodynamic quantities of the conformers of dimethoxymethane . 162 5.5 Theoretical and experimental geometrical parameters of the gauche-gauche conformer of dimethoxymethane 167 5.6 Opt imized geometrical parameters of the gauche-anti conformer of dimeth-oxymethane 168 5.7 Calculated properties of the gauche-gauche conformer of dimethoxymethane at three levels of geometry optimization 171 6.1 Summary of calculations performed and calculated properties of glycine . . . 183 6.2 Relative thermodynamic quantities and abundances of the conformers of glycine. 184 6.3 Measured and calculated valence ionization potentials of glycine 190 v i i List of Figures 2.1 E M S scattering kinematics 21 3.1 Mult ichannel energy-dispersive electron momentum spectrometer 57 3.2 Direct probe heated sample system 59 3.3 Spectrometer electronics 62 3.4 Sample time spectrum 65 3.5 Electron analyser performance 68 3.6 Non-binning and binning mode argon ( 3 p ) _ 1 binding energy spectra 71 3.7 Non-binning and binning mode argon 3p angle and momentum profiles. . . . 76 4.1 Conformers of acetone 83 4.2 Effect of molecular conformation on acetone valence momentum profiles . . . 86 4.3 Dependence of the 5b 2 T M P of acetone on basis set 89 4.4 B i n d i n g energy spectra of acetone from 6-60 eV 91 4.5 Experimental and calculated valence shell binding energy spectra of acetone 96 4.6 Summed 2bi , 4b 2 and 8ai + l a 2 experimental and theoretical angle profiles of acetone 101 4.7 Experimental and theoretical 5b 2 momentum profiles of acetone 103 4.8 Experimental and theoretical 2bi momentum profiles of acetone 106 4.9 Experimental and theoretical 4b 2 momentum profiles of acetone 108 4.10 Experimental and theoretical 8ai + l a 2 momentum profiles of acetone . . . . 109 4.11 Experimental and theoretical 7ai + 3b 2 +. l b i momentum profiles of acetone 111 4.12 Experimental and theoretical 6ai momentum profiles of acetone 113 4.13 Experimental and theoretical 2b 2 momentum profiles of acetone 114 4.14 Experimental and theoretical 5ai momentum profiles of acetone 116 4.15 Experimental and theoretical 4ai angle profiles of acetone 117 5.1 Conformations of dimethoxymethane 121 5.2 B inding energy spectra of dimethoxymethane from 4.5-58.7 eV 126 v i i i 5.3 Experimental and theoretical 10b + 11a momentum profiles of dimethoxy-methane 133 5.4 Theoretical 10b and 11a momentum profiles of dimethoxymethane 134 5.5 Experimental and theoretical 9b momentum profiles of dimethoxymethane . 136 5.6 Experimental and theoretical 10a + 8b momentum profiles of dimethoxymethane 138 5.7 Experimental and theoretical 9a + 7b 4- 8a momentum profiles of dimethoxy-methane 140 5.8 Experimental and theoretical 6b + 7a + 5b momentum profiles of dimethoxy-methane 142 5.9 Experimental and theoretical 6a momentum profiles of dimethoxymethane . 144 5.10 Experimental and theoretical 4b momentum profiles of dimethoxymethane . 146 5.11 Experimental and theoretical 5a momentum profiles of dimethoxymethane . 147 5.12 Experimental angle profile of the 25-44 eV binding energy range of dimeth-oxymethane and individual and summed 3b and 4a theoretical profiles . . . . 149 5.13 Experimental angle profiles of a series of binding energy intervals in the 25-44 eV binding energy range of dimethoxymethane 152 5.14 Variat ion of the fitted 3b and 4a scale factors of Figure 5.13 wi th binding energyl53 5.15 Basis set dependence of the 10b and 11a Hartree-Fock T M P s of dimethoxy-methane 156 5.16 Basis set dependence of the 10b and 11a B 3 L Y P - D F T T M P s of dimethoxy-methane 157 5.17 Comparison of dimethoxymethane outer-valence momentum profiles of the gauche-gauche conformer and of a conformer sum 164 5.18 Variat ion wi th geometry of the 10b, 11a and 9a T M P s of dimethoxymethane 172 6.1 Conformers of glycine 177 6.2 Experimental and theoretical He l s momentum profiles 181 6.3 Effect of conformer abundances on the conformer sum theoretical momentum profiles of glycine 186 6.4 B i n d i n g energy spectra of glycine 189 6.5 Experimental and theoretical summed outer-valence momentum profiles of glycine 191 6.6 Experimental and theoretical momentum profiles for the H O M O of glycine . 194 6.7 Posit ion space electron density maps for the H O M O s of the three lowest energy conformers of glycine 195 6.8 Experimental and theoretical momentum profiles for M O 19 of glycine . . . . 197 6.9 Experimental and theoretical momentum profiles for M O 18 of glycine . . . . 198 6.10 Experimental and theoretical momentum profiles for the sum of M O s 19 and 18 of glycine 201 ix 6.11 Experimental and theoretical angle profiles for the sum of M O s 17-11 of glycine202 6.12 Experimental and theoretical momentum profiles for M O 11 of glycine . . . . 204 6.13 Experimental and theoretical momentum profiles for M O 10 of glycine . . . . 206 6.14 Experimental and theoretical momentum profiles for M O 9 of glycine . . . . 208 x List of Abbreviations A D C analogue-to-digital converter B E S binding energy spectrum B W background window C F D constant fraction discriminator C G F contracted Gaussian function CI configuration interaction C I S D configuration interaction, single and double excitations C W coincidence window D F T density functional theory D W I A distorted-wave impulse approximation E M S electron momentum spectroscopy fwhm ful l width at half maximum G G A generalized gradient approximation G W - P G Gaussian-weighted planar grid H F Hartree-Fock H O M O highest occupied molecular orbital IP ionization potential K S K o h n - S h a m L C A O linear combination of atomic orbitals x i L S D A local spin-density approximation M O molecular orbital M P M0ller-Plesset M R S D - C I multi-reference single and double excitation configuration interaction P E S photoelectron spectroscopy P S D A position sensitive detector analyser P W I A plane-wave impulse approximation S C A single-channel analyser S C F self-consistent field T A C time-to-amplitude converter T A P theoretical angle profile T H F A target Hartree-Fock approximation T K S A target K o h n - S h a m approximation T M P theoretical momentum profile X A P experimental angle profile X M P experimental momentum profile x u Preface Some of the work included in this thesis has been published previously. 1. Y . Zheng, J . J . Neville, C . E . Br ion, Y . Wang and E . R. Davidson, " A n electronic structure study of acetone by electron momentum spectroscopy: a comparison wi th S C F , M R S D - C I and density functional theory." Chem. Phys. 188 (1994) 109-129. 2. Y . Zheng, J . J . Neville and C . E . Br ion , "Imaging the electron density i n the highest occupied molecular orbital of glycine." Science 270 (1995) 786-788. 3. John J . Nevil le, Y . Zheng and C . E . Br ion , "Glycine valence orbital electron densities— comparison of E M S experiments with Hartree-Fock and density functional theories." J. Am. Chem. Soc. 118 (1996) 10533-10544. 4. J . J . Nevil le, Y . Zheng, B . P. Hollebone, N . M . Cann, C . E . Br ion , C . - K . K i m and S. Wolfe, " E M S studies of larger molecules of chemical and biochemical interest." Can. J. Phys. 74 (1996) 773-781. Publ icat ion 1 presents the results of an E M S study of acetone based upon the same ex-perimental data described in Chapter 4 of this thesis and also includes a description of the spectrometer constructed by research associate Dr . Yenyou Zheng and myself and de-scribed in Chapter 3. D r . Zheng was the principal author of this publication, w i t h the remaining authors, including myself, contributing to the analysis of the data and revision x i i i of the manuscript. The acetone study presented in Chapter 4 includes a reanalysis of the data presented in publication 1 and considerable additional computational work performed subsequent to the wri t ing of publication 1. Publ icat ion 2 is a short communication presenting preliminary E M S results of the H O M O of glycine. A l l three authors made significant contributions to the wri t ing of this publication, wi th D r . Zheng wri t ing the ini t ia l draft and myself and then Prof. Br ion performing exten-sive redrafts of the manuscript. A comprehensive report of the E M S study of the valence shell of glycine is given in publication 3, for which I was the principal author. A s such, I performed the electronic structure calculations and data analysis and wrote a l l drafts of the manuscript, incorporating the invaluable suggestions of my co-authors during the data analysis and manuscript editing and revising processes. The work reported in publication 3 is presented i n Chapter 6. Publ icat ion 4 is the text of an invited talk I prepared and presented at the International Symposium on (e, 2e), Double Photoionization and Related Topics, a satellite conference of the X I X I C P E A C . This paper describes the experimental and computational techniques used in extending electron momentum spectroscopy to the study of larger molecules and includes preliminary results from the studies of glycine (Chapter 6) and dimethoxymethane (Chapter 5). Elements of this publication have been incorporated into Chapters 1-3. A l l of the E M S data presented in the publications listed above and in this thesis was collected by myself wi th the assistance and guidance of D r . Zheng and Prof. B r i o n . x iv Acknowledgements W h i l e performing the research presented in this thesis, I have benefitted from the contri-butions of many individuals, whom I gratefully acknowledge. Foremost among them is my research supervisor, Prof. Chris Br ion , who introduced me to electron momentum spec-troscopy and whose wisdom and guidance has impacted (!) upon al l aspects of this work. A special thanks is owed to D r . Yenyou Zheng, with whom I worked closely during the construction, testing, debugging and operation of the energy-dispersive spectrometer. His experience and expertise were key to the success of the experimental aspects of this work. The successful design, construction and operation of the energy-dispersive spectrometer also depended greatly upon the contributions of the staff of the mechanical and electronic en-gineering services of the Department of Chemistry. In particular, the expertise and technical skills of B r i a n Snapkauskas, E d G o m m and Br ian Greene are gratefully acknowledged. I thank Prof. Saul Wolfe of Simon Fraser University for suggesting that dimethoxy-methane would be an interesting molecule to study and for posing many thoughtful and challenging questions concerning E M S . Our meetings not only increased my knowledge of organic chemistry but also improved my understanding of electron momentum spectroscopy. Prof. Ernest Davidson of Indiana University performed the M R S D - C I calculations of argon, acetone and dimethoxymethane and provided many insightful comments and sugges-tions concerning the interpretation of the results of these studies. I thank Prof. Dennis Salahub of the Universite de Montreal for generously providing a copy of Reference [85] prior to publication and for several helpful comments concerning the study of glycine. I gratefully acknowledge the assistance of Prof. Delano Chong, who provided many helpful comments and brought to my attention several relevant publications. M a n y of my early questions concerning E M S were answered by Bruce Hollebone, who also helped me learn to program in C and clarified many programming and computational issues. I thank D r . Natalie Cann for her development and refinement of the H E M S and resfold programs, which were used throughout this research, and for many helpful and interesting conversations. xv I gratefully acknowledge the many others with whom I have had the pleasure to work, talk and drink numerous cups of coffee during my studies at U . B . C . , in particular Jennifer A u , G o r d Burton , D r . G l y n n Cooper, Noah Lermer, Terry Olney, J i m Rolke and D r . Bruce Todd. They have helped to make my graduate studies an enjoyable experience and I have learned much from them. The financial support of the Natural Sciences and Engineering Research Counci l is grate-fully acknowledged. Special thanks go to 'the gang' at the Lutheran Campus Centre, for their friendship and support throughout my time at U . B . C . , and for helping me to develop an appreciation of fine beer. Thanks also to Vaughan and Janet Roxborough for their support and interest throughout my graduate studies. I gratefully acknowledge the contributions of my parents, not only during my graduate studies but throughout my education. I would not have reached this point without their support and encouragement. F inal ly , I thank Margie for her support, patience and understanding and Alexander for ensuring there was some excitement and play in every day. J O H N J . N E V I L L E The University of British Columbia February 1997 x v i my mother and father. x v i i C h a p t e r 1 I n t r o d u c t i o n Knowledge of the electron density distribution of a molecule is important for the under-standing and prediction of reactivity and chemical behaviour. It is straightforward to obtain theoretical electron density distributions, either from the square of the electronic wave-function obtained from, for example, Hartree-Fock (HF) or configuration interaction (CI) calculations or by calculating the density directly using density functional theory ( D F T ) . These theoretical densities, however, are of necessity approximations of the true electron density as a result of the simplifying assumptions associated wi th each theoretical method. Therefore, experimental measurements of electron distributions are invaluable not only for the information they provide directly concerning the electronic structure of the molecule but also as a means of assessing the quality of calculated electron densities and electronic wave-functions. Detailed experimental information of total charge distributions is available using x-ray and electron scattering techniques [1,2]. However, as the success of the frontier orbital theory of Fukui [3,4] has demonstrated, chemical reactivity is influenced primari ly by the electron density distributions of the outer-valence orbitals and in particular the highest oc-cupied molecular orbital ( H O M O ) . Experimental measurement of individual valence orbital electron distributions is possible using electron momentum spectroscopy ( E M S ) [5-9]. 1 Chapter 1. Introduction 2 The electron distributions obtained using E M S are in momentum space (i.e., they are electron momentum distributions) rather than in position space (electron position or charge distributions). In a series of theoretical papers entitled "Momentum distributions in molecu-lar systems," published in the 1940's, Coulson and Duncanson explored this alternative mo-mentum space representation of the electron density for molecular hydrogen and a number of simple hydrocarbons [10-12]. Their work illustrates the correspondence between momentum and position space and demonstrates how chemical concepts such as molecular bonding are reflected in the momentum space electron density. More recently, the usefulness of working in momentum space when considering chemical reactivity and physical properties has been demonstrated in theoretical work by A l l a n , Cooper and co-workers [13-15]. They have found that molecular similarity and dissimilarity indices calculated using momentum space elec-tron densities are capable of rationalizing physical and chemical properties and biological act ivity in cases where there is no obvious relationship between the bonding structure of the molecule and the property of interest. For example, such theoretical momentum space studies have been able to predict satisfactorily the relative effectiveness of ant i -HIV phos-pholipids observed in clinical testing [13-15]. In Reference [14], A l l a n and Cooper summarize the effectiveness of working with the momentum space density as follows: "In marked contrast to the position space density, the momentum density high-lights some of the most chemically interesting parts of the electron distr ibution, without over-emphasizing the bonding topology." The above quote could have been written just as readily concerning the effectiveness of E M S for studying molecular electron density, as demonstrated by the many experimental studies performed in the quarter-century since the first successful implementation of the E M S technique. 1 The majority of these E M S studies have been of atoms and simple polyatomic 1 Brief outlines of the history and applications of E M S are given in Sections 1.1.1 and 1.1.2, respectively. Chapter 1. Introduction 3 molecules such as the hydrides of the elements of the second row of the periodic table (i.e., C H 4 , N H 3 , H 2 0 and H F ) . The purpose of the present work is to extend E M S to the study of larger and more complicated molecules than have been studied previously, specifically acetone ( ( C H 3 ) 2 C O ) , dimethoxymethane ( ( C H 3 0 ) 2 C H 2 ) and glycine ( N H 2 C H 2 C O O H ) . To address some of the challenges posed by the study of these molecules (discussed in Section 1.2 below), an energy-dispersive multichannel electron momentum spectrometer has been constructed. This spectrometer, similar in design to one constructed previously at the Flinders University of South Austra l ia [16-18], has a sensitivity approximately one order of magnitude better than that of the single-channel spectrometers used previously in this group [19] and elsewhere to perform E M S measurements. In addition to the experimental measurements, the recently formulated application of K o h n - S h a m density functional theory to E M S [20,21] is evaluated for the molecules studied and compared with the results of Hartree-Fock and configuration interaction calculations. 1.1 Electron momentum spectroscopy Electron momentum spectroscopy involves the ionization of a (usually gas phase) atom or molecule by impact with a high-energy electron and the coincident detection of the two outgoing electrons (the scattered electron and the ionized electron). This ionization process is referred to as the (e, 2e) reaction, the name reflecting the origins of E M S in nuclear physics. The kinematics of the ionization are fully determined—i.e., the energies and trajectories of the incident electron and the two detected outgoing electrons are known. If the energies of the incident and outgoing electrons are sufficiently high and an appropriate experimental geometry is employed, electronic structure information of the neutral atom or molecule, independent of kinematic factors, can be obtained. Chapter 1. Introduction 4 From conservation of energy, the electron binding energies (also called ionization poten-tials or IPs) of the target atom or molecule can be determined. B y varying the energy of the incident electron beam and/or using a multichannel spectrometer capable of recording simultaneously data at a range of binding energies, a binding energy spectrum (BES) can be obtained. This contains information similar to that provided by photoelectron spectroscopy ( P E S , where ionization occurs as the result of the absorption of a photon rather than by elec-tron impact) , albeit wi th poorer energy resolution than is possible using P E S . W h a t makes E M S unique is the additional ability to determine, to a good approximation, the spherically-averaged distribution of electron momenta corresponding to the single-particle orbital from which the electron has been ionized, i.e., J \ipj{p)\2 dCl. This is done by recording the vari-ation in the (e, 2e) differential cross-section at a particular binding energy as a function of the trajectories of the two outgoing electrons. This momentum distr ibution—or experimen-tal momentum profile ( X M P ) , as it is more commonly termed i n the field of E M S — i s the momentum-space analogue to the position-space orbital electron density J |-0j(r)|2dQ. The momentum-space and position-space orbitals themselves (^(p) and ip(r)) are related by a Fourier transform (Equation (1.1) below). Note that the momentum distributions provided by other experimental techniques such as Compton scattering and positron annihilat ion are considerably more l imited than those from E M S as they are of the total molecular electron density (the sum of al l orbital densities) rather than of the electron density of individual orbitals [5]. 1.1.1 Development of EMS It was first suggested that the (e, 2e) reaction might be used to obtain the momentum distributions of electrons in binding energy-selected states by Baker, M c C a r t h y and Porter in 1960 by analogy wi th the (p, 2p) reaction used to obtain momentum distributions of protons Chapter 1. Introduction 5 in nuclei [22]. Further theoretical exploration of the possibilities of (e, 2e) experiments was performed by Neudatchin and co-workers [23,24] and Glassgold and Ialongo [25] in the late 1960's. The first experimental work involving the (e, 2e) reaction in which the scattering kine-matics were fully determined was reported by Ehrhardt et al . in 1969 and involved the scattering of low energy electrons 100 eV) off helium using an asymmetric coplanar ge-ometry [26]. The terms used to describe the (e, 2e) experimental kinematics indicate if the energies and polar angles of the detected electrons are equal (symmetric or asymmetric) and whether or not the trajectories of the two detected electrons are in the same plane (coplanar or non-coplanar). The reaction kinematics are discussed in more detail i n Chapter 2. The purpose of Ehrhardt 's work was to test theories of electron impact ionization at low impact energies rather than to obtain electronic structure information of the helium target. That same year, A m a l d i et al . reported the results of high energy (14.6 keV) coplanar symmetric electron impact experiments measuring the binding energy spectrum of the carbon K and L (i.e., core and valence) shells using a thin carbon film target [27]. Al though no momentum profile data were obtained, this very low energy resolution study helped to demonstrate the feasibility of obtaining electronic structure information using the (e, 2e) reaction. The first momentum profiles determined experimentally by E M S were reported i n 1972 by C a m i l l o n i et al . [28]. Momentum profiles of the K and L shells of carbon were obtained using a thin carbon foil target but the energy resolution was not sufficient to resolve the individual C 2s and 2p orbitals of the L (valence) shell. The following year, Weigold and co-workers reported the first atomic and molecular valence orbital momentum profiles, of gas phase argon [29] and methane [30], respectively. These measurements were obtained using symmetric non-coplanar kinematics, which, for reasons discussed in Chapter 2, have been found to be particularly well-suited to studies of electronic structure [6]. In the case of the Chapter 1. Introduction 6 argon measurements, clear evidence of the breakdown of the independent particle model of ionization was observed in the form of B E S peaks at binding energies greater than those for ionization from the single-particle 3p and 3s orbitals. The X M P s of these argon "satellite peaks" indicated that they correspond to ionization of a 3s electron of neutral argon and the simultaneous excitation of a second electron. These results confirmed the prediction of Levin made one year previously that (e, 2e) experiments would be capable of detecting such electron correlation effects [31]. Following the success of these ini t ia l gas-phase experiments, electron momentum spec-troscopy (originally known as binary (e, 2e) spectroscopy) has been used by several research groups [32-34] to study the electronic structure of a large number of gaseous atoms and molecules. M u c h of this research has been summarized in review articles 2 and bibliographies of al l studies reported up unti l 1990 have been published [36,38]. Several studies are dis-cussed i n the following section to demonstrate some of the applications of E M S . A d d i t i o n a l E M S studies that serve to validate the theoretical interpretation of the E M S differential cross-section are discussed in Chapter 2. 1.1.2 Applications of EMS A s mentioned above, E M S provides both binding energy spectra and binding energy-selected momentum profiles. W i t h currently available instrumentation, the E M S energy resolution is approximately two orders of magnitude worse than that of photoelectron spectroscopy. Consequently, E M S is not the best choice of experimental technique if accurate ionization potentials are desired. However, E M S does have one important advantage over P E S for the study of ionization—namely that the B E S are obtained as a function of electron momentum. This provides symmetry as well as energy information and aids greatly in the assignment 2 R e c e n t reviews of E M S include References [7-9,35-37]. Chapter 1. Introduction 7 of the ionization peaks. For example, E M S measurements have resolved the question of the energetic ordering of the ( 5 a i ) _ 1 and ( l b 2 ) _ 1 valence ionizations of formaldehyde ( H 2 C O ) [32,39]. This symmetry information is also very useful when studying inner-valence ionization (typically, binding energies in the range 20-60 eV) . In this energy region, the single-particle model of ionization tends to break down (see for example Reference [40]). Ionization from an inner-valence orbital ipj may occur at several ionization energies, each leading to a different final ion state. This results in "satellite peaks" in the B E S , as mentioned i n the previous section for argon. If initial-state correlation effects are small , as is typical ly the case for inner-valence ionization, the intensities of al l ionization peaks resulting from the removal of an electron from orbital ipj w i l l exhibit the same electron-momentum dependence, thus facil i tating the determination of the "parentage" of these satellite peaks, i.e., the orbital from which the electron has been ionized. In addition, in contrast to P E S , the intensities of the E M S binding energy peaks are directly proportional to the spectroscopic factors (discussed in Chapter 2) of the ionization processes [5, 41], which can be calculated using C I [39] or many-body Green's function theory [40]. The E M S studies of hydrogen sulphide [42], carbon monoxide [43] and molecular fluorine [44] include particularly detailed analyses of inner-valence ionization. The abil i ty of E M S to "image" the momentum space electron density of indiv idual molec-ular orbitals (within the approximations discussed in Chapter 2) has been the principal mo-tivation for most E M S studies. Comparisons of momentum profiles from E M S experiments wi th the corresponding theoretical profiles from ab initio calculations have resulted in new insights into what constitutes an "accurate" electronic wavefunction [9,45,46]. In particular, the accurate theoretical modeling of E M S momentum profiles has proven sensitive to basis set and electron correlation effects. Chapter 1. Introduction 8 M u c h of the ut i l i ty of E M S studies is a consequence of the experimental electron density profiles being in momentum space rather than position space, the perspective from which chemists typically view electronic structure. The momentum space molecular orbital ip(p), the spherically-averaged square of which generally provides a good description of the E M S X M P , is related to the position space molecular orbital ip(r) by the Fourier transform rela-tionship This relationship results in an approximately inverse weighting of position (r) and momentum [p) space. E M S is most sensitive to regions of low momentum (small p) and, consequently, strongly emphasizes the chemically sensitive outermost spatial regions of electron density. Al though the outermost (large r) region of electron density plays a significant role in chemical reactivity, this area is often represented poorly by variationally determined wavefunctions as a result of its small contribution to the total energy, a property dominated by the core (small r) electron density. This has been demonstrated clearly in E M S studies of H 2 0 and N H 3 , where theoretical momentum profiles ( T M P s ) generated f r o m Hartree-Fock wavefunctions considered at the time to be of high quality underestimated the intensity at low momentum of the outermost valence orbital X M P s [9,45,47]. Several factors were found to contribute to this disagreement, one being that the basis sets used for the calculations d id not describe adequately the large r regions of electron density. The addition of further extremely diffuse functions and polarization functions to the basis sets resulted in a significant improvement in the theoretical description of the low momentum regions of the momentum profiles while having very litt le impact on the calculated total energies. E M S measurements of many small molecules [9] have confirmed the importance of using sufficiently diffuse basis sets in order to calculate accurate momentum profiles. This has (1.1) Chapter 1. Introduction 9 been found to be of particular significance for outermost (highest energy) valence orbitals, which should be of the most significance to chemical reactivity. In addition, there has been a growing appreciation of the potential pitfalls of over-emphasising the total energy when assessing the "goodness" of a wavefunction, especially when the properties of interest are pr imari ly dependent upon the large r (or small p) regions of electron density. To ensure that a wavefunction accurately models al l regions of electron density, it is important to consider a range of properties that each emphasizes a different region of space. The total energy (small r ) , dipole or quadrupole moment (medium r) and E M S momentum profiles (large r) have proven to be effective choices to guide the development of "universal" wavefunctions [9,45,46]. In situations where a universal wavefunction is not necessary, care must be taken to choose a basis set well-suited to the regions of electron density emphasized by the particular properties of interest. In the case of the H 2 0 and N H 3 studies mentioned above, an additional factor contribut-ing to the disagreement between theory and experiment for the H O M O momentum profiles was found to be the absence of a treatment of electron correlation effects in the theoretical cal-culations. 3 W h e n electron correlation effects were accounted for (using multi-reference single and double excitation configuration interaction, or M R S D - C I ) and the G W - P G resolution-folding method was used to account for the instrumental angular resolution (see footnote), excellent agreement between theory and experiment was obtained [9]. In general, it has been found that electron correlation effects are important in the momentum-space outer-valence orbital electron densities of the polar hydrides of the elements of the second-row of the peri-odic table (i.e., N H 3 , H 2 0 and H F ) , but not of the corresponding hydrides of the third-row elements or small hydrocarbons [9]. 3 The final factor responsible for the disagreement was an inadequate treatment of the instrumental angular resolution [9]. This has been corrected with the development by Duffy et al. of the Gaussian-weighted planar grid method ( G W - P G ) [48], discussed in Chapter 2. Chapter 1. Introduction 10 In addition to the study of individual atoms or molecules, the study by E M S of series of chemically related molecules has provided some interesting insights into aspects of molec-ular electronic structure. A prime example is the series of increasingly methylated amines N H X ( C H 3 ) 3 _ X , x = 0 , . . . , 3 [49,50]. A n examination of the H O M O X M P s of these molecules indicated that the methyl group is electron withdrawing wi th respect to hydrogen in the H O M O s of the methyl amines. Similar findings have been obtained for the series water [45], methanol [51], dimethyl ether [52,53] and formaldehyde, acetaldehyde, acetone [54]. 1.1.3 Recent advances in EMS The coincidence nature of the E M S experiment and the necessary kinematics result i n a small experimental differential cross-section and long data collection times. This was so particu-larly for the first generation of E M S spectrometers [6,19]. These single-channel spectrometers were only capable of collecting data at a single binding energy and electron momentum at a time, thereby disregarding the majority of the (e, 2e) events. Consequently, several weeks were required to collect either one momentum profile or a valence shell B E S at a single mo-mentum value. Comprehensive studies of the entire valence shells of even small molecules routinely required several months of data collection. The advent of multichannel electron momentum spectrometers has significantly increased the sensitivity of the technique and made possible many interesting new experiments. M u l -tichannel spectrometers improve data collection efficiency by collecting data simultaneously over a range of binding energies [16,17], electron momenta [55] or a combination of the two [56]. Improvements in sensitivity of one to two orders of magnitude over that of single-channel spectrometers are obtained. The development of an energy-dispersive electron mo-mentum spectrometer has facilitated the first E M S study of an excited and oriented target, sodium atoms pumped by a polarized laser to the 32P3/2,?Tfy = +1 state [57]. In addit ion, a Chapter 1. Introduction 11 detailed study of the experimental momentum profiles arising from ionization to excited ion states of helium (n = 2 and n = 3 ion states) and H 2 and D 2 (2pa u , 2s<rg) has been performed recently using a momentum dispersive E M S spectrometer [58-60]. Measurements of these momentum profiles had been reported previously [18,61-65] but were l imited because of the very small cross-sections for ionization to the excited ion states of H e + and H^j-. F inal ly , Storer et al . have reported a spectrometer for the study of th in solid films that is multichan-nel in both energy and momentum [56]. Valence measurements of amorphous and graphitic carbon using this new spectrometer demonstrate the relationship between electronic and physical structure and the use of E M S for the characterization of condensed matter [56]. In addit ion to these experimental advances, there are recent theoretical developments of considerable significance to E M S . A s discussed above, E M S studies of many small molecules have demonstrated the importance of using relatively large basis sets containing diffuse and polarization functions and in some cases correlated methods to calculate accurate theoret-ical momentum profiles. Al though for small molecules it is generally possible to perform calculations of the necessary complexity, this becomes increasingly challenging as the size of the molecules being studied increases, to the point where only very approximate theoretical results are available for comparison with experiment. This is, of course, a general problem in computational chemistry rather than one specific to E M S . Consequently, the application of density functional theory ( D F T ) to chemical problems [66, 67] has been of great inter-est to chemists and this has been an area of considerable research activity in recent years. Reference [68] describes the current range of chemical applications of D F T . D F T is compu-tationally less demanding than Hartree-Fock theory and has been shown, in at least some applications, to provide results as good as or better than H F and some pos t -HF methods. In recent work by Duffy et al . [20,21], D F T has been applied to E M S and used to calcu-late theoretical momentum profiles ( T M P s ) . Comparisons of D F T T M P s of several small Chapter 1. Introduction 12 molecules wi th experimental data and T M P s obtained using H F and C I methods indicate that the D F T results are at least as good as those from the H F calculations and are in better agreement wi th experiment than the H F results in those cases where an accounting of electron correlation effects was found to be important, such as was described above for the H O M O of water [20,21]. 1.2 E M S o f l a r g e r m o l e c u l e s A s has been discussed in Section 1.1 above, over the past two decades electron momentum spectroscopy has been used extensively for the study of atoms and small molecules typi -cally having only a few heavy (i.e., non-hydrogen) atoms. Al though a few studies of larger systems have been performed, they have been l imited principally to molecules of high sym-metry, such as sulphur hexafluoride [16] and transition-metal hexacarbonyls [69]. Given the demonstrated success of E M S for the investigation of molecular electronic structure and also as an experimental tool to aid in the evaluation and design of electronic wavefunctions for molecules such as water [9,45] and ammonia [9], it is natural to consider applying E M S to the study of more complicated molecules. The extension of E M S measurements and the complementary quantum mechanical calcu-lations to larger molecules poses a number of challenging problems both experimentally and theoretically. The E M S signal-to-noise ratio decreases as the number of electrons increases, thereby necessitating greater data collection times to obtain data of reasonable precision. E x -perimental measurements are more complicated in many cases due to the low volati l i ty of the target molecule. This may necessitate the use of a heated sample system which must provide sufficient target density for gas phase studies over long measuring times and concurrently avoid thermal decomposition of the sample. One of the primary reasons that E M S studies Chapter 1. Introduction 13 have been l imited principally to simple systems is the rather poor energy resolution that has been typical of E M S experiments. The number of closely-spaced (in energy) valence orbitals tends to increase with increasing molecular size and complexity. Difficulties i n resolving ionization processes corresponding to the removal of electrons from orbitals of similar energy result i n increased challenges in data analysis and interpretation. Furthermore, i n studies of many larger molecules the target w i l l be present in more than one stable conformation at the experimental temperature, thus considerably complicating the data analysis. The increased coincidence count rates of recently developed multichannel electron mo-mentum spectrometers, discussed in Section 1.1.3 above, makes it possible to address and significantly overcome many of these challenges posed by the study of larger molecules. In-creased sensitivity allows measurements to be made using a lower target density than was previously practical , which is of benefit when studying compounds which have a low vapour pressure or which are prone to thermal decomposition. Addit ional ly , some of the increase in coincidence count rate can be sacrificed to obtain improved energy resolution and thus enable E M S measurements of a wider range of molecules. Even if further improvements in energy resolution (i.e., beyond those realized using the multichannel spectrometers described above) were possible however, they would not fully solve the problem of isolating peaks arising from ionizations from closely spaced orbitals in many molecules because of the natural (vibronic) widths of the ionization peaks. This is clear from an examination of the photoelectron spectra of, for example, many of the larger molecules reported in Reference [70]. Despite an energy resolution approximately two orders of magnitude better than that of E M S experiments, it is not possible to resolve fully the majority of the valence shell ionization peaks of many of these molecules. Consequently, an equally important result of the considerably increased coincidence count rates of mul t i -channel spectrometers is the increased feasibility of recording extensive B E S over a range Chapter 1. Introduction 14 of azimuthal angles. Using these B E S in conjunction with data from high resolution photo-electron spectroscopy (PES) , it is possible in many cases to obtain experimental momentum profiles for ionization processes closely spaced in energy. This can be done by fitting the B E S using Gaussian peaks located at the known P E S IPs and wi th widths based upon the P E S Franck-Condon widths and the experimental energy resolution of the E M S instrument. W h e n the energy spacing is such that this analysis is not possible, an experimental angle profile may st i l l be obtained by summing data over a range of binding energies corresponding to ionizations from several closely spaced orbitals, or by representing several closely spaced ionization processes by a single fitted peak. Al though this reduces the amount of available information, these studies may st i l l be useful for the evaluation of theoretical methods and the design and evaluation of wavefunctions. From a theoretical standpoint, computational complexity rapidly increases wi th number of electrons even at the Hartree-Fock level and high-level correlated treatments such as C I quickly become infeasible for larger molecules. The conformational mobil i ty of many larger molecules further increases the computational difficulties since separate theoretical momen-tum profile calculations are required for each stable conformer (conformational potential energy minimum) present at the experimental conditions. The individual conformer calcu-lations must then be Bol tzmann weighted at the experimental temperature and summed together prior to comparison wi th the E M S experimental data. For these reasons, the less computationally intensive approach to quantum mechanical calculations provided by D F T is of particular interest. A s previously mentioned, ini t ia l applications of D F T to E M S [20,21,71] have been encouraging. However, comparisons have thus far only been performed in a l i m -ited number of cases, all of them small molecules. Further evaluation of D F T as applied to E M S , involving comparisons wi th the results of H F and post -HF (e.g., CI) calculations and experimental data, is necessary to determine whether it w i l l provide a viable approach to Chapter 1. Introduction 15 the calculation of valence electron densities and other properties of larger molecules. 1.3 O v e r v i e w o f t h e t h e s i s The work presented in this thesis involves the extension of E M S to the study of molecules of greater complexity than have previously been investigated using this technique. Several experimental and theoretical considerations—many of them outlined above—arising from such studies are explored. This is done in the context of experimental E M S and theoretical studies of the valence shells of acetone, dimethoxymethane and glycine. The remainder of the thesis is organized as outlined below. Chapter 2 presents the theoretical background necessary for the interpretation of the experimental E M S differential cross-section and the calculation of theoretical momentum profiles for comparison wi th the experimental results. In addition, the computational meth-ods used i n performing the studies presented in Chapters 4-6 are described. The multichannel electron momentum spectrometer constructed in the course of this work is described i n Chapter 3, along with the calibration and operating procedures used in the collection of the experimental data presented in subsequent chapters. This energy-dispersive instrument is similar to one developed previously by Weigold et al . [7,16] and is capable of collecting data simultaneously over a binding energy range of ~ 16 eV. Procedures for sample handling and the processing of the experimental data are also discussed. E M S measurements of the valence shell of acetone over the binding energy range 6-60 eV are presented in Chapter 4. These measurements serve as a "bridging" study between previous E M S experiments and the studies presented in Chapters 5 and 6. Experimental results for the 5b 2 H O M O of acetone have been reported previously by Hollebone et al . [54] in a paper comparing the H O M O momentum profiles of acetone, acetaldehyde and Chapter 1. Introduction 16 formaldehyde. These single-channel E M S results for acetone had rather poor statistical precision, making the comparison with theoretical momentum profiles, including the results of large basis set H F and M R S D - C I calculations, somewhat inconclusive. Nevertheless, there appeared to be a discrepancy between the 5b 2 X M P and al l of the reported T M P s at electron momenta > 0.8 au. The previous single-channel study was l imited to the H O M O as a consequence of the low experimental count rates of the single-channel spectrometer and the experimental energy resolution of 1.7 eV full width at half maximum (fwhm). The improved energy resolution (1.4 eV fwhm) of the multichannel results presented in the current work has enabled a comprehensive study of the valence shell of acetone, including the collection of individual X M P s for six of the valence orbitals. The experimental results are compared wi th T M P s calculated using the H F and M R S D - C I wavefunctions reported i n Reference [54] as well as the results of additional H F calculations and D F T calculations using a range of basis sets and several exchange-correlation functionals. Furthermore, the effect of the orientation of the acetone methyl groups on the calculated momentum profiles is investigated. The E M S studies of dimethoxymethane and glycine, reported i n Chapters 5 and 6 re-spectively, represent the first E M S studies of these molecules. B o t h of these molecules are of biochemical interest, glycine as the simplest amino acid and dimethoxymethane both as a model compound for the glycosidic linkage in polysaccharides and for the study of the anomeric effect [72], a conformational effect observed in carbohydrates and related molecules. These molecules are both important for use as prototypes for the evaluation of theoretical methods that are being considered for use wi th much larger molecules—proteins i n the case of glycine and polysaccharides in the case of dimethoxymethane. To be feasible, theoretical studies of these much larger molecules w i l l require that the calculations be performed at as efficient a level as is possible and w i l l necessitate compromises between computational expense and accuracy of the results. Pr ior to performing such calculations, it is necessary to Chapter 1. Introduction 17 have an understanding of the potential impact on calculated properties of, for example, the neglect of electron correlation effects or the use of small basis sets. Benchmark calculations on molecules such as dimethoxymethane and glycine, for which a broad range of theoretical methods are st i l l practical and for which experimental data are available for comparison, are therefore of considerable importance. Experimental momentum profiles obtained v i a E M S should be particularly useful experimental "checks" of theoretical results for these molecules as a result of the sensitivity of E M S to the outer regions of electron density that may be ex-pected to play significant roles in conformation (through interactions such as intramolecular hydrogen bonding), reactivity and molecular recognition. The E M S study of dimethoxymethane reported in Chapter 5 covers the valence shell from 5 - 59 eV. X M P s are compared wi th T M P s obtained from H F and D F T calculations spanning a considerable range of basis set size and complexity. In addition, the results of M R S D - C I calculations for the two lowest energy ionizations (outermost orbitals), performed by Ernest R. Davidson of Indiana University, are compared with the experimental and other theoret-ical results. The sensitivity of the calculated momentum profiles to changes in molecular geometry and conformation are also investigated. The study of glycine is of the outer-valence orbitals, covering the binding energy range of 6 - 27 eV, and is reported in Chapter 6. The roles of basis set size and composition and also electron correlation effects in modeling the outer-valence orbitals of this molecule are studied by comparing the experimental measurements to H F and D F T calculations using a range of basis sets. A number of experimental [73-78] and theoretical [79-85] studies indicate that several stable conformers of glycine w i l l be present at the experimental conditions used for the E M S measurements. Consequently, a thorough study of the outer-valence T M P s of al l conformers likely to contribute to the observed experimental signal is performed. General observations and conclusions arising from this work are presented in Chapter 7, Chapter 1. Introduction 18 along w i t h a summary of the major findings and some suggestions for possible areas of future research. C h a p t e r 2 Theoretical Background and Computational Methods Two of the principal applications of E M S experiments are the evaluation of theoretical mod-els used to describe molecular electronic structure, in particular valence electron densities, and the investigation of the relationships between molecular geometry and composition, elec-tronic structure as described by E M S momentum profiles, and other molecular properties. For such applications of E M S to be possible, a thorough understanding of the E M S dif-ferential cross-section and how it may be related to or predicted from electronic structure calculations is necessary. Considerable work has been published describing the E M S differ-ential cross-section and deriving tractable forms for the calculation of theoretical momentum profiles ( T M P s ) for comparison with and to aid in the interpretation of their experimental counterparts ( X M P s ) . See for example the reviews of M c C a r t h y and Weigold [5-7] and work referenced therein. A brief overview of the theoretical representations of the E M S differential cross-section used in this work is given in the following sections, following a description of the (e, 2e) reaction that is the basis of E M S . The latter half of this chapter consists of an overview of the methods used to calculate theoretical representations of the experimental E M S data reported in subsequent chapters of this thesis. 19 Chapter 2. Theoretical Background and Computational Methods 20 2.1 The (e, 2e) reaction The (e, 2e) reaction, upon which E M S experiments are based, is shown in Equat ion (2.1) and is i l lustrated in Figure 2.1. e 0 + M ->• Mf + ei + e 2 (2.1) A s discussed briefly in Chapter 1, the reaction consists of the ionization of a target atom or molecule M, assumed to be at rest and in its ground electronic state, by an inelastic collision wi th an incident electron eo having a kinetic energy EQ and a momentum p 0 . The collision results in an ion M + in state / , wi th recoil energy and momentum -E^cou and q, respectively, and two outgoing electrons ei and e 2 , wi th respective kinetic energies and momenta of E\, p i , and E2,p2-The sum of the kinetic energies of the two outgoing electrons is the total energy (E) of the reaction (E = E\ 4- E2). The electron binding energy ey for ionization to the ion state / can be determined from conservation of energy: 6f = EQ — Ei — E2. (2-2) The ion recoil energy ^ r e C o i i is v e r y small and can be neglected. In contrast, the ion recoil momentum q is not negligible. It can be determined using conservation of momentum: q = p 0 - p i - p 2 . (2.3) If the kinetic energies of the incident electron and the two outgoing electrons are sufficiently high and the momentum transfer K of the reaction is high ( K = p 0 — P i ) , then the (e, 2e) reaction can be regarded as an electron-electron collision with the ion acting only as a spectator [7]. This simplification of the ionization reaction involves the binary encounter approximation and the impulse approximation, which are discussed below i n more detail . Chapter 2. Theoretical Background and Computational Methods 21 e0( E 0 . ft ) Figure 2.1: E M S scattering kinematics. The figure depicts the symmetric non-coplanar (e, 2e) kinematics used in the current work, where 0\ = 62 = 9 = 45°, Ei = E2 = 600 eV and 4> is variable over the range ± 3 0 ° . Chapter 2. Theoretical Background and Computational Methods 22 Under these conditions, the ion recoil momentum is equal in magnitude and opposite in sign to the momentum p of the ionized electron prior to ionization (i.e., while s t i l l part of the target molecule M ) : The binary encounter and impulse approximations are almost always invoked in the inter-pretation of E M S experiments and the work presented in this thesis is no exception. The scattering kinematics of the (e, 2e) reaction are illustrated in Figure 2.1. The ar-rangement shown represents symmetric non-coplanar kinematics. The polar angles 9\ and 6>2 between the two outgoing electron trajectories and the direction of the forward-scattered electron beam are equal and fixed (9i = 92 = 9), as are their kinetic energies. The out-of-plane azimuthal angle <f> is variable to allow access to a range of recoil momentum values q and hence electron momenta p. W i t h symmetric non-coplanar kinematics, the magnitude of p is If the electron energies are high in comparison to the binding energy and 9 = 45°, the first term of Equation (2.5) w i l l be near zero and p w i l l be approximately proportional to <f> for small values of 4>. Using symmetric non-coplanar kinematics, the experimental differential cross-section is essentially independent of kinematic factors (i.e, variation of EQ and and is thus a probe only of electronic structure. This kinematic arrangement also results i n a large momentum transfer K and high kinetic energies for both outgoing electrons, conditions necessary for the validity of the binary encounter and impulse approximations. Consequently, the majority of E M S experiments, including those reported in this thesis, have been performed using this kinematic arrangement. A second kinematic arrangement that is also a good probe of electronic structure is asymmetric coplanar kinematics near the Bethe ridge [7]. In this p = -q (2.4) p = [(2pi cos# - po) 2 + (2pi sin0 s i n ( 0 / 2 ) ) 2 ] 1 / 2 . (2-5) Chapter 2. Theoretical Background and Computational Methods 23 experimental arrangement, E1 » E2, (f) = 0°, 9X ~ 20° and 92 is varied. This arrangement is used in the recently reported spectrometer of Storer et al . [56] for the E M S study of thin solid films. 2.2 The EMS differential cross-section The differential cross-section, in atomic units, for the (e, 2e) reaction is [7] = ( 2 * ) 4 ? ^ S ? / P > ( « / ; P . . P . . P . ) ^ P-6) where 5^av represents a sum over final state degeneracies and an average over in i t ia l state degeneracies and the spherical averaging ( j dQ, ) accounts for the random orientations of the gas-phase target molecules. T/(e/; p0, P i , p2) is the reaction amplitude for ionization leading to the final ion state / . It can be written as Tf = < X ( - ) ( P i ) x ( " ) ( P 2 ) * / | r | ^ X ( + ) ( P o ) > • (2.7) The electrons have been represented by incoming (+) and outgoing (—) distorted waves (x*), \&f and are the wavefunctions of the final ion in state / and the in i t ia l neutral target in its ground electronic state and r is the operator describing the ionization reaction. The in i t ia l approximation upon which the interpretation of the E M S cross-section is based is the binary encounter approximation, which assumes that r depends upon the coordinates of the two electrons but not on those of the particles constituting the ion [7]. This allows Equat ion (2.7) to be factored into a probe part, dependent upon the reaction kinematics, and a structure part, dependent upon the electronic structure of the target and ion. Chapter 2. Theoretical Background and Computational Methods 24 2.2.1 The distorted-wave impulse approximation A further level of approximation is required to obtain a representation of the cross-section that is amenable to calculation. In the distorted-wave impulse approximation ( D W I A ) , the electrons are represented by distorted waves, as in Equation (2.7), and r is the free two-electron amplitude, i.e., the amplitude for inelastic collisions between two free electrons. In essence, the assumption is that the collision occurs so fast that the presence of the ion does not affect the collision process. The experimental cross-section then becomes [7] = 4 ^ / e e y j / !(X<->(PI)X<->(P2)*,| * . x « ( p „ ) > f * (2.8) where fee is the M o t t scattering cross-section for electron-electron collisions. A n explicit form of /ee in the symmetric non-coplanar geometry is given in Reference [7]. It depends only upon known kinematic factors and has been shown [19] in symmetric non-coplanar kinematics at E = 1200 eV to be essentially independent of the out-of-plane azimuthal angle <f> over the range of ± 3 0 ° typically used in E M S experiments. The M o t t scattering cross-section does have a small dependence on the energy of the incident electron. For the symmetric non-coplanar kinematics used in this work (E = 1200 eV, 9 = 45°) the variation in /ee is small over an electron binding energy range corresponding to the outer-valence orbitals of atoms or molecules. Consequently, the E M S cross-section can be considered to be proportional to the final, electronic structure, factor of Equation (2.8) for most purposes. However, over a binding energy range of 50 eV, fee has been shown to vary by approximately 13% [59,86]. Thus, when the precise relative intensities of ionization processes well separated in binding energy are of prime concern, the impact-energy dependence of the M o t t scattering cross-section should be taken into account. For the purposes of the present work, this impact-energy dependence has been neglected. The representation of the electrons using distorted waves accounts for perturbations of Chapter 2. Theoretical Background and Computational Methods 25 the electron trajectories by the target both prior to and following ionization. The validity of the D W I A and the underlying binary encounter approximation has been tested extensively for atoms [5-7]. The D W I A has been found to give an excellent description of the E M S experimental cross-section for electron momenta up to at least p ~ 3 au when symmetric non-coplanar kinematics are used wi th a total energy E > 1000 eV [5-7]. In the case of molecules, the multiple nuclei prevent the use of a centrally symmetric potential for calculation of the distorted waves and D W I A calculations are not, at present, feasible [6]. Consequently, a different approximation is necessary when considering the E M S cross-section of molecules. 2.2.2 The plane-wave impulse approximation If the energies of the electrons are sufficiently high, perturbations of their trajectories by the target and by each other w i l l be small and the electrons can be represented as plane waves. If this is done and the impulse approximation is also invoked (i.e., r is taken to be the free two-electron amplitude), the E M S differential cross-section can be written as [7] where p here represents a plane-wave electron having a momentum determined using Equa-tions (2.3) and (2.4). This is the plane-wave impulse approximation ( P W I A ) . The E M S experimental energy resolution is such that rotational and vibrat ional states of molecules are not resolved, thus necessitating the sum over unresolved final states and average over unresolved ini t ia l states indicated by ^ a v in Equation (2.9) and previous equa-tions. Using the Born-Oppenheimer approximation, the molecular and ion wavefunctions and \&/ can be expressed as the products of electronic, vibrational and rotational wave-functions. The sums over final vibrational and rotational states can be eliminated from Equat ion (2.9) by closure [7]. Rotat ion of the molecular target is accounted for by the spher-(2.9) Chapter 2. Theoretical Background and Computational Methods 26 ical averaging J dfl that is required because of the random orientations of the gas-phase target molecules. Initial-state vibrations are typically accounted for in E M S studies by cal-culating the electronic wavefunctions at the equilibrium molecular geometry. The validity of these approximations has been clearly demonstrated by E M S studies of H 2 and D 2 [87] and of H 2 0 and D 2 0 [45]. In both cases, isotopic substitution had no effect on the experimental momentum profiles. Taking into account the considerations outlined in the previous paragraph and the near-invariance under E M S conditions of the kinematic factors in Equation (2.9), the E M S dif-ferential cross-section for closed-shell molecules is essentially proportional to the spherically averaged square of the overlap of the neutral and ion electronic wavefunctions: where and \T// now represent the electronic wavefunctions of the target and ion. It is this representation of the E M S cross-section that w i l l be referred to as the P W I A in the remainder of this thesis. It is straightforward to calculate the cross-section as given by Equat ion (2.10) using the many-body wavefunctions of the neutral molecule in its ground electronic state and the ion in electronic state / . Such descriptions of the E M S cross-section are sometimes referred to as ion-neutral overlap distributions. Procedures for calculating electronic wavefunctions are described in Section 2.3. A n examination of the body of E M S studies that have been conducted i n the past twenty years [7] indicates that the plane-wave impulse approximation is generally adequate for describing the shapes of orbital momentum profiles for p < 1.5 au. For higher values of electron momentum, distortion effects tend to result in greater observed intensity than is predicted using the P W I A . The P W I A has been found to predict inaccurately the relative ionization cross-sections of atoms (e.g., it overestimates the ( 3 s ) - 1 : ( 3 p ) _ 1 ratio of argon) but typical ly gives good predictions of the relative ionization amplitudes of molecules [7]. (2-10) Chapter 2. Theoretical Background and Computational Methods 27 2.2.3 The target Hartree-Fock approximation The evaluation of Equation (2.10) for all valence ionization processes of a molecule can be a considerable task, since separate many-body electronic wavefunctions must be calculated for the neutral molecule and the ion in each accessed electronic state. (Each peak in the binding energy spectrum corresponds to ionization to a different electronic state of the ion.) Often, electron correlation effects in the neutral molecule are not significant when describing the (e, 2e) ionization process. If this is the case, the electronic wavefunction of the neutral molecule can be replaced in Equation (2.10) by the Hartree-Fock wavefunction \&0 ( H F theory is discussed in Section 2.3.1). This is the target Hartree-Fock approximation ( T H F A ) . If, in addit ion, the ion is represented as a linear combination of electron configurations (Slater determinants) obtained by various occupations of the target H F orbitals, the E M S differential cross-section then becomes [7,8] j j - g ^ s j / l ^ r i f a , (2.11) where ipj (p) is the momentum-space target orbital j from which the electron is ionized and Sj is the probabil i ty of the final ion state / having a contribution from the electron configuration formed by annihilating an electron in orbital j of the target configuration \&0- Sj is known as the spectroscopic factor or pole strength of the ionization. It can be calculated by taking the square of the overlap between the ion wavefunction \?; and the target wavefunction wi th a hole i n orbital j {^^): (2.12) Equat ion (2.11) indicates that, within the T H F A , the momentum dependence of the E M S differential cross-section (i.e., the experimental momentum profile) is an initial-state property, described by the spherically-averaged square of the target orbital from which the qJ Chapter 2. Theoretical Background and Computational Methods 28 electron is removed, and is independent of the final state of the i o n . 1 This greatly simplifies theoretical predictions of the E M S differential cross-section since only a single electronic structure calculation, of the neutral molecule, is necessary. W i t h i n the T H F A , a l l ionizations from the same target orbital j (i.e., ionization manifold j), regardless of the final ion state / , w i l l give rise to momentum profiles of the same shape, greatly facil i tating the assignment of "many-body" or "satellite" ionization peaks. The relative intensities of the ionization peaks wi th in a manifold are given by the spectroscopic factors Sj. The spectroscopic sum rule [5] states that the sum of all spectroscopic factors for a given ionization manifold is unity: Equat ion (2.13) indicates that if E M S measurements are performed for a binding energy range encompassing al l ionization peaks in a particular manifold j, the absolute spectro-Previous E M S studies (see for example References [44,71,88]) and many-body Green's func-t ion calculations [40] indicate that the spectroscopic factors for ionizations from outer-valence orbitals are usually near unity; i.e., each outer-valence orbital typically gives rise to a single ionization peak. In contrast, ionizations from inner-valence orbitals often give rise to mul -tiple ionization peaks corresponding to different final electronic states of the ion. These are the so-called "many-body" or "satellite" peaks. Their presence is a consequence of electron correlation effects (many-body effects) in the ion. Breakdown of the T H F A is evidenced by shape discrepancies between experimentally measured momentum profiles and the corre-sponding theoretical momentum profiles calculated using Equation (2.11) and by variations in the shapes of momentum profiles belonging to the same ionization manifold. 1 T h i s is also suggested by the description of the (e, 2e) reaction kinematics within the binary encounter and impulse approximations, in which the experimentally determined ion recoil momentum (q) is of equal magnitude to that of the struck electron (p) just prior to ionization (see Section 2.1 and Reference [7]). (2.13) scopic factors Sj for ionization from target orbital tjjj can be determined experimentally. Chapter 2. Theoretical Background and Computational Methods 29 2.2.4 The target Kohn-Sham approximation Use of the T H F A considerably reduces the computational effort necessary to obtain the-oretical momentum profiles. It has proven adequate to describe the E M S experiment for many atoms and small molecules. However, the neglect of target correlation and electronic relaxation of the ion implici t in the T H F A has proven significant i n some cases, particularly for the outermost valence orbitals of polar hydrides such as H F , H 2 0 and N H 3 [9]. In the case of these molecules, calculation v ia Equation (2.10) of the overlap of the ion and neutral C I wavefunctions (obtained using saturated basis sets) was found to be necessary in order to obtain quantitative agreement [9,48,89] between experiment and theory. A s increasingly larger molecules are studied by E M S , the feasibility of performing C I calculations (and, to a lesser extent, H F calculations) to obtain theoretical momentum profiles becomes increasingly l imited, particularly in light of the need to employ extended basis sets in order to obtain the-oretical results that agree with experiment. One computational alternative is K o h n - S h a m (KS) density functional theory. The comparatively low computational demands of D F T , par-t icularly in comparison with other methods that include electron correlation effects, make D F T an attractive method for use in theoretical studies of chemical problems. Recent work by Casida [90] and Duffy et al . [20,21] has provided the theoretical basis necessary to apply D F T to E M S . The key results of their work, as it pertains to this thesis, are presented below. A general overview of K o h n - S h a m D F T is given in Section 2.3.3. The target K o h n - S h a m approximation ( T K S A ) , introduced by Duffy et al . [20], approxi-mates the ion-neutral overlap of Equation (2.10) by a momentum-space K o h n - S h a m orbital Al though the form of the E M S differential cross-section in the T K S A is similar to that in Vf S(p): (2.14) Chapter 2. Theoretical Background and Computational Methods 30 the T H F A (Equation (2.11)), the basis for the approximation is different from that discussed above for the T H F A . The overlap of the ion and neutral wavefunctions in Equat ion (2.10) is a Dyson orbital . Instead of calculating the Dyson orbital by taking the overlap of the two wavefunctions, as discussed above, it can be determined [20] by solving Dyson's quasipar-ticle equation, which describes both ionization and electron attachment. Solving Dyson's quasiparticle equation or calculating the overlap of the ion and neutral wavefunctions yields identical results in the l imit of a ful l treatment of electron correlation. The T K S A is based upon the work of Casida [90] which showed that the K o h n - S h a m equation is the variationally best local approximation to Dyson's quasiparticle equation. Consequently, the K o h n - S h a m orbitals may be viewed as approximations to Dyson orbitals. 2 There are an infinite number of Dyson orbitals but, for an JV electron molecule, only N occupied K o h n - S h a m orbitals. The T K S A implies that the shape of each Dyson orbital w i l l correspond to that of a K o h n -Sham orbital . This is equivalent to the consequence of the T H F A that al l momentum profiles from the same ionization manifold have an identical shape which is determined by the tar-get orbital from which the electron has been removed. Despite the similarities between the target Hartree-Fock and target K o h n - S h a m approximations, they differ in the significant respect that the T K S A allows for the inclusion of target electron correlation effects and the T H F A does not. There are, in practice, two levels of approximation involved in the T K S A . First ly, K o h n -Sham orbitals are approximations of Dyson orbitals. This is true even i n the case of exact D F T , i.e., even if the exact exchange-correlation functional could be used in the D F T cal-culations, thereby yielding K o h n - S h a m orbitals whose densities summed to the true total density. Secondly, D F T calculations of molecules use approximate functionals and l imited basis sets and consequently do not produce "exact" K o h n - S h a m orbitals. The T K S A says 2 T h e T H F A may be regarded in a similar manner; the H F equation is the exchange-only case of Dyson's quasiparticle equation and H F orbitals are thus equivalent to Dyson orbitals in the exchange-only limit. Chapter 2. Theoretical Background and Computational Methods 31 nothing of the accuracy of these approximations. The approximation therefore must be eval-uated by comparing momentum profiles calculated using the T K S A (Equation (2.14)) wi th those obtained using the P W I A only (Equation (2.10)) and/or momentum profiles obtained experimentally. In their paper introducing the T K S A [20], Duffy et al . performed such com-parisons for the valence orbital momentum profiles of six atoms and small molecules for which the results of high-quality CI calculations and E M S experiments were available. For al l of the molecules considered, the T M P s derived from the CI calculations were in excellent agree-ment w i t h the experimental momentum profiles. Consequently, comparisons were made with the C I T M P s and not with the X M P s . They concluded that the T K S A performed about as well as the T H F A , but wi th the differences from the C I results (i.e., making only the P W I A ) tending to be in the opposite direction than was found for the T H F A . Whereas the T H F A T M P s tended to underestimate the intensity at low momentum, the reverse was most often the case for the T K S A T M P s . W i t h the exception of the valence T M P s of H 2 0 , al l of the D F T calculations reported by Duffy et al . [20] were performed using the local spin-density approximation ( L S D A ) . The calculations of H 2 0 performed using a gradient-corrected func-tional resulted in improved agreement between the T K S A and P W I A (i.e., CI) T M P s . This suggests that some of the differences observed for the other molecules may be attributable to the approximate functional used for the D F T calculations rather than l imitations inherent i n the T K S A . Subsequent E M S studies of ethylene [71], P H 3 and derivatives [91] and the open-shell molecules 0 2 , N O and N 0 2 [92] have included comparisons w i t h T M P s obtained using the T K S A . The results suggest that the T K S A yields T M P s in as good or better agreement wi th the experimental data than is the case for the T H F A . Further evaluation of the T K S A , as applied to larger molecules, w i l l be performed in subsequent chapters of this thesis by comparing T K S A T M P s with experimental data and wi th the results of H F and CI calculations for acetone, dimethoxymethane and glycine. Results obtained using several Chapter 2. Theoretical Background and Computational Methods 32 exchange-correlation f u n c t i o n a l and basis sets of varying complexity w i l l be compared in an attempt to separate to some extent inaccuracies resulting from the T K S A itself from those resulting from the choice of functional and basis set. 2.2.5 Angular resolution effects The theoretical descriptions of the E M S differential cross-section discussed above, whether they are obtained using Equation (2.10), (2.11) or (2.14), correspond to a hypothetical E M S experiment performed using a spectrometer having perfect resolution. In practice, of course, the instrumental energy and angular resolutions (in both the polar (9) and azimuthal (4>) angles) are finite and are reflected in the experimental binding energy spectra and momen-t u m profiles, respectively. In order to make a rigorous comparison between theoretical and experimental momentum profiles, it is necessary to account for the effects of the finite exper-imental angular resolution. The Gaussian-weighted planar grid ( G W - P G ) method of Duffy et al . [48] has proven to be an effective choice to account for these angular resolution effects, resulting in excellent agreement between experiment and high-quality theoretical calcula-tions for a wide range of atoms and molecules [9,48]. A l l theoretical momentum profiles presented in this thesis (unless otherwise noted) have been corrected for angular resolution effects using the G W - P G method prior to comparison with the experimental data. 2.3 Electronic structure calculations To make use of the expressions for the E M S differential cross-section discussed above re-quires knowledge of the molecular and ionic electronic wavefunctions in the case of the P W I A (Equation (2.10)) or either the H F or K S molecular orbitals (MOs) in the case of the T H F A or T K S A (Equations (2.11) and (2.14)). In the work discussed in this thesis, these quanti-Chapter 2. Theoretical Background and Computational Methods 33 ties have been obtained using standard theoretical models of molecular electronic structure, namely the Hartree-Fock and configuration interaction ab initio methods and K o h n - S h a m density functional theory. These methods are used widely throughout chemistry and many detailed references are available. For example, both the theoretical and computational as-pects of the ab initio methods are discussed in References [93] and [94]. Details of the application of density functional theory to molecules can be found in Reference [66]. A sum-mary of the basic concepts of D F T and a review of current chemical applications are given in Reference [68]. Consequently, only a general overview of these methods is presented here. In addition to those methods mentioned above, M0ller-Plesset ( M P ) perturbation theory, used in some of the studies presented in this thesis, is also discussed. The H F , C I and M P methods are al l based upon finding solutions of the time-independent Schrodinger equation, M = EV (2.15) where E is the energy of the stationary state of the system described by the wavefunction \I/ and H is the total-energy operator or Hamil tonian of the system. Equation (2.15) has an infinite number of solutions >^. The ground state of the system is described by the solution having the lowest energy E. The non-relativistic Hamil tonian for a molecule contains terms describing the kinetic energies of al l of the constituent particles (i.e., the electrons and nuclei) and the potential energy of the system resulting from the Coulombic forces between particles. The number of moving particles present in al l but the simplest atomic systems makes impossible the exact solution of the Schrodinger equation for most systems of chemical interest. The H F , M P and C I approaches yield approximate solutions of the Schrodinger equation by making simplifying assumptions of varying severity. Al though approximate, these three approaches are al l ab initio methods—they find solutions of (2.15) without the use of experimental data, Chapter 2. Theoretical Background and Computational Methods 34 other than the values of some fundamental physical constants. A n approximation common to al l of the methods discussed here is that the electrons, being much lighter than the nuclei, move so much faster than the nuclei that the electronic motion is dependent only upon the instantaneous positions of the nuclei and not upon their velocities. Consequently, the problems of electronic and nuclear motion can be separated and the electronic problem solved in the static field of the fixed nuclei. This is known as the Born-Oppenheimer approximation. It is the solutions to the electronic problem—i.e. , electronic wavefunctions—that are of interest in E M S . The non-relativistic electronic Hamil tonian for an n electron molecule w i t h m nuclei, in the absence of an external field and written in atomic units, is * = - £ v ? - £ £ ^ + £ £ 4 + £ £ ^ (,16) i=l i=l A=l 1 i=l j>i LJ A=l B>A A U where the indices i and j refer to electrons and A and B to nuclei. The atomic numbers of the nuclei are denoted by ZA and is the distance between particles i and A . The first term of Equat ion (2.16) represents the kinetic energies of the electrons and the remaining terms represent the potential energy. The final term, describing the Coulombic repulsion between nuclei, is constant for a given nuclear arrangement and does not affect the electronic wavefunction of the system. Despite the simplification resulting from the Born-Oppenheimer approximation, determination of the electronic wavefunction for a many-electron system remains a formidable problem because of the interdependence of the electrons arising from the 1/vij terms of the Hamil tonian. The most common means of addressing this is using Hartree-Fock theory. Chapter 2. Theoretical Background and Computational Methods 35 2.3.1 Hartree-Fock theory Hartree-Fock theory assumes that the electrons in a molecule are independent of one another so that their behaviour can be described by a series of one-electron functions or orbitals. The H F electronic wavefunction \I/HF has the form of a single determinant of one-electron spin orbitals, wi th each spin orbital being the product of a spatial orbital ipi (a function of the position of the electron) and a spin function a or B. The determinant is a sum of products of spin orbitals formed such that the wavefunction is antisymmetric w i t h respect to the exchange of any two electrons. The H F energy S H F (i-e, the expectation value of the energy corresponding to \&H F) is given by ^ H F = ( * H F | ^ | * H F > ( 2 . 1 7 ) Substituting for H using Equation ( 2 . 1 6 ) and expanding \I>HF in terms of the constituent orbitals, the Hartree-Fock energy of a closed-shell molecule is n/2 n/2 n/2 £ H F = 2 $ > + £ ^ { 2 J i j - Kij) + V N N i = l i = l j=l where hi = ( ^ ( 1 ) 2 1 ^rlA J , 13 ^ ( 1 ) ^ ( 2 ) 1 Ka = ^ ( 1 ) ^ ( 2 ) 1 ^(1)^(2) ^•(1M(2) ( 2 . 1 8 ) ( 2 . 1 9 ) ( 2 . 2 0 ) ( 2 . 2 1 ) V N N is the internuclear repulsion energy given by the final term of Equation ( 2 . 1 6 ) and and are called respectively the Coulomb and exchange integrals. ( 1 ) and ( 2 ) refer to different electrons. In a closed-shell molecule, the n occupied spin orbitals consist of electrons of a and B spin i n n/2 different spatial orbitals ipi7 hence the sums over n/2 occupied spatial Chapter 2. Theoretical Background and Computational Methods 36 orbitals and the factors of 2 in the above equations. Two identical spatial orbitals containing equal-energy electrons of opposite spin are often referred to collectively as a molecular orbital. The variational principle states that any normalized tr ia l wavefunction w i l l yield an expectation value for the energy which is greater than or equal to the energy of the true lowest-energy solution of the Schrodinger equation for the system. For a molecule, the lowest-energy solution of the electronic Schrodinger equation is the ground-state electronic wavefunction \&0 and the corresponding energy is the ground-state electronic energy EQ. Since any single determinant wavefunction w i l l always be an approximation of the true wavefunction, therefore J5HF > EQ. The H F wavefunction of a molecule is defined as the single determinant wavefunction yielding the lowest value of E-RF (and hence the energy closest to EQ). It is obtained by finding the set of orthonormal molecular orbitals tpi that minimizes £ - H F • These are the H F orbitals. Minimizat ion of . E H F wi th respect to tpi results i n a set of n/2 coupled one-electron differential equations [93] H = CiV* (2.22) where e; is the energy of orbital tpi and F is the Fock operator: F(l) = -1-Vl-Y/^ + T,[^-KJ] (2.23) The Coulomb and exchange operators Jj and Kj can be defined by their operation on an arbitrary function / ( l ) : 4 / ( 1 ) = / ( I ) / ^ ( 2 ) * ^ ( 2 ) ^ - ^ 2 (2.24) kjf{l) = ^ ( 1 ) / ^ - ( 2 ) 7 ( 2 ) — dv2 (2.25) J ri2 Jj and Kj are one-electron operators; they depend upon the coordinates of one electron only and describe the interaction between that electron and the average electric field of the Chapter 2. Theoretical Background and Computational Methods 37 remaining electrons. This approximation neglects the instantaneous interactions between electrons but is necessary to allow separation of the total electronic wavefunction into prod-ucts of one-electron wavefunctions. The one-electron operator F depends upon the solutions tpi to (2.22) through the Coulomb and exchange operators. Therefore, the equations must be solved iteratively unt i l self-consistency is achieved, i.e., unt i l no change in the solutions occurs upon further iteration. This is referred to as the self-consistent field or S C F method because the molecular orbitals are determined using their own effective electric field. To allow for the practical implementation of H F theory, the problem of finding the n/2 molecular orbitals ipi that provide the min imum H F energy is simplified by expressing the molecular orbitals as linear combinations of known functions fa, called basis functions. The set of a l l basis functions used for the expansion is the basis set. If a basis set consisting of iV functions is used for the expansion, each molecular orbital can then be expressed as N Rather than having to determine the functional form of the molecular orbitals, the problem has been reduced to determining the expansion coefficients c^. If the basis set constitutes a complete set, then by definition it w i l l be possible using a basis set expansion to express exactly any well-behaved function. The wavefunction result-ing from the S C F procedure w i l l be the best possible single-determinant wavefunction for the system, yielding the lowest possible H F energy. These are referred to as the HF-limit wavefunction and energy. To be complete, a basis set must generally consist of an infinite number of basis functions (i.e., N — oo). In practice, a basis set consisting of a l imited number of basis functions is used, resulting in an energy greater than the H F - l i m i t energy. B y performing calculations using successively more complete basis sets and monitoring the decrease in energy, it is possible to get a sense of the proximity of the calculated energy Chapter 2. Theoretical Background and Computational Methods 3 8 to the H F - l i m i t energy. Unfortunately, this does not necessarily indicate how similar the calculated wavefunction is to the H F - l i m i t wavefunction. For molecular calculations, the basis set consists most commonly of functions resembling atomic orbitals centred on each of the nuclei of the molecule. This is called the linear combination of atomic orbitals or L C A O approximation. If one basis function is used for each core and valence-shell atomic orbital of each atom, then the basis set is a min imal basis set. Use of a min imal basis set generally results in calculated molecular properties of poor accuracy as a result of the lack of flexibility inherent i n such a basis set. Further discussion of basis sets, including descriptions of those used for the calculations reported in this thesis, is given in Section 2 . 3 . 4 below. 2.3.2 Post-Hartree-Fock methods: electron correlation H F theory neglects the instantaneous interactions between electrons and consequently does not yield the exact wavefunction and energy of the molecule. The difference between the exact non-relativistic energy of a molecule and the corresponding H F - l i m i t energy is defined as the correlation energy: EC0TT = -Bexact — -EHF- ( 2 . 2 7 ) Although the H F energy generally differs from the exact energy by only ~ 1%, the accu-rate description of electron correlation is of key importance for calculating many chemical properties. In a recently published review article [95], Raghavachari and Anderson survey the techniques commonly used for the treatment of electron correlation in molecules and discuss the applicability and limitations of the various methods. M a n y of these techniques use the H F wavefunction as their starting point and thus are often called pos t -HF methods. Computat ional results of two post -HF methods, configuration interaction (CI) and M0ller Plesset ( M P ) perturbation theory, are discussed in the latter chapters of this thesis and are Chapter 2. Theoretical Background and Computational Methods used to aid in the interpretation of the E M S experimental data. 39 Configuration interaction A H F calculation of a closed-shell molecule using N basis functions w i l l result in N molecular orbitals. If the molecule has 2n electrons, n of these orbitals w i l l be occupied and N — n w i l l be vacant or virtual orbitals. The H F wavefunction w i l l be a determinant composed of the spin orbitals formed from the occupied M O s . This is referred to as the ground-state electron configuration. Other electron configurations can be created by forming determinants wi th some of the occupied orbitals replaced wi th vir tual orbitals. This corresponds to the exci-tation of one or more electrons to vir tual orbitals. A configuration interaction wavefunction \I/CI is formed by taking a linear combination of electron configurations. This can be written as where * 0 is the H F configuration and ^x>o are excited electron configurations. The C I expansion coefficients ax are determined variationally so as to minimize the total energy. Typical ly , the coefficient of the H F configuration is much larger than that of any of the excited configurations. If al l possible electron configurations for a given basis set are included i n the C I expan-sion (2.28), the calculation is described as full CI . F u l l C I calculations using basis sets large enough to provide chemically useful results are only practical for very small molecules. Con-sequently, some subset of al l possible electron configurations is normally chosen. The most common choice, known as configuration interaction, single and double excitations or C I S D , includes only those excited configurations formed by exciting one or two electrons. The con-figurations formed by double excitations account for the largest fraction of the correlation energy. Al though making a much smaller contribution to the total energy, inclusion of the (2.28) x>0 Chapter 2. Theoretical Background and Computational Methods 40 single excitation configurations has been found to be important for the accurate computation of some molecular properties [95]. A second common method of l imit ing the size of C I calculations is to make the frozen-core approximation. This approximation l imits excitations to those involving the valence electrons. In other words, configurations involving excitation of core electrons are neglected. This assumes that the contribution of inner-shell electrons to the total correlation energy is reasonably independent of molecular geometry and environment. The C I calculations compared with the experimental data in this thesis were performed by Ernest Davidson and co-workers at Indiana University and use a somewhat more compli-cated technique than that described above. Instead of using a single reference configuration (i.e., the H F configuration) to form the remaining electron configurations, multiple reference configurations are employed. These reference configurations are chosen by performing an ini t ia l C I S D calculation and identifying those configurations that have significant expansion coefficients. A second C I calculation is then performed in which al l single and double exci-tations from the reference configurations are included. A s a result, the calculation includes some configurations involving triple and quadruple excitations from the H F configuration. This is called a multi-reference single and double excitations CI ( M R S D - C I ) calculation. M0ller-Plesset perturbation theory A n alternative method of treating electron correlation is that introduced by M0ller and Plesset [96], in which electron correlation is treated as a perturbation of the Hartree-Fock problem. The electronic Hamil tonian for the system is expressed as the sum of two operators HX = H0 + XV (2.29) where H0 is the sum of the one-electron Fock operators given by Equation (2.23), V is the difference between the correct electronic Hamil tonian H (Equation (2.16)) and H0, and A is Chapter 2. Theoretical Background and Computational Methods 41 a dimensionless parameter indicating the magnitude of the perturbation. The wavefunction and energy of the system are then expressed as power series in A, that is oo * A = 5>l% (2.30) oo .Ex = ] T A * £ ( J ) . (2.31) i=0 Setting A equal to 1, the Equations (2.30) and (2.31) give the exact wavefunction and energy of the system (in the l imit of a complete basis set). This is equivalent to a ful l C I treatment. In M0ller-Plesset perturbation theory, the two series above are truncated and the level of the theory is referred to by the highest-order energy term retained in expansion (2.31), i.e., M P 2 , M P 3 , etc. ^(o) of Equation (2.30) is the H F wavefunction; E^ is the sum of the indiv idual H F orbital energies (e; of Equation (2.22)) and E^+E^ is the H F energy. First-order and higher terms of Equation (2.30) consist of electron configurations formed by excitations from the H F configuration, as also appeared in the CI expansion of the electronic wavefunction discussed in the previous section. The M P calculations performed for this thesis are second order (i.e., M P 2 ) , the lowest order that accounts for some electron correlation. A t the M P 2 level, only configurations involving double excitations from the H F configuration contribute to the M P expansion of the wavefunction. Configurations involving other excitations (e.g., singles and triples) do not appear unti l the M P 4 level. The M P 2 method is an efficient means of part ial ly accounting for electron correlation effects, requiring only a single step following the iterative determination of the H F wavefunction. However, it has the disadvantage of being non-variational, so that the resulting energy is not an upper bound on the true energy of the system. Also , perturbation theory tends to converge slowly and oscillate wi th order [95], so results calculated to different orders can differ significantly. Chapter 2. Theoretical Background and Computational Methods 42 2.3.3 Density functional theory A n alternative approach to problems of molecular electronic structure is provided by density functional theory. In contrast to the wavefunction-based electronic structure theories dis-cussed in the preceding sections, D F T is based upon the electron density distr ibution p(r). Although D F T has been used by physicists for several decades to describe the electronic structure of solids, the developments necessary for the routine application of D F T to chem-ical problems have only occurred in the past decade or so. Despite this, the computational advantages of D F T over the more traditional methods of quantum chemistry discussed above have resulted in a rapid growth in research concerning chemical applications of D F T . The foundation of D F T was laid by Hohenberg and K o h n [97], who showed that the to-tal energy of a system of n interacting electrons (e.g., the electronic energy of an ^-electron molecule) can be expressed (as can many other properties) as a functional of the electron density. Furthermore, analogous to the variational theorem discussed in Section 2.3.1, Ho-henberg and K o h n showed that for any arbitrary electron density chosen for a system, the corresponding total energy is an upper bound of the ground-state energy of the system. Subsequent work by K o h n and Sham [98] provided the framework necessary to make D F T a computationally practical technique. They considered a model system of n non-interacting electrons having a total density equal to the true electron density of the system (of interacting electrons) of interest. A s in H F theory, in K o h n - S h a m D F T the motions of the independent electrons are described by one-electron functions, in this case the K o h n - S h a m orbitals ipKS. Thus, the total density can be written as p ( r ) = E K S ( r ) f • ( 2 - 3 2 ) Integration of p(r) over al l space gives n , the total number of electrons in the system. A l -though K o h n and Sham introduced the orbitals ipKS purely as a computational convenience, Chapter 2. Theoretical Background and Computational Methods 43 recent work by Casida [90] and Duffy et al . [20] discussed i n Section 2.2.4 above indicates that they are of considerable significance to E M S . The total energy in K o h n - S h a m D F T is [68] E[p(r)} = TB[p(r)] + J v{v)p{v)dv + J[p(r)] + Exc[p(r)} (2.33) where the four terms contributing to the total energy of the non-interacting system wi th density p(r) are respectively the kinetic energy of the system, the potential energy resulting from the interactions between the external field v(r) and the density, the classical Coulomb repulsion and the exchange-correlation energy. For a free molecule in the absence of any external fields, v(r) is the field of the nuclei, given by the second term of Equat ion (2.16). The exchange-correlation functional Exc[p(r)} is defined to be the functional necessary for E of (2.33) to be the true ground-state energy of the system. The first three terms of Equation (2.33) have equivalent terms in the expression for the H F energy (Equation (2.18)) and if Exc[p(r)] is replaced by the H F exchange energy defined by Equations (2.18) and (2.21) then Equat ion (2.33) is simply a different formulation of the H F electronic energy expression. M i n i m i z a t i o n of E[p(r)} wi th respect to the K o h n - S h a m orbitals yields the K o h n - S h a m equations FKVfS = S (2.34) where the operator F K S is [68] i T K S = 1 y 2 + v { r ) + f P^]_dr, + U x c ( r ) ( 2 3 5 ) 2 J r - r' and the exchange-correlation potential vxc(r) is defined by The similarity of the K o h n - S h a m equations to the Hartree-Fock equations (2.22) and (2.23) is evident, the only difference being the presence of the exchange-correlation potential in the Chapter 2. Theoretical Background and Computational Methods 44 K S equations in place of the exchange operator in the H F equations. A s is the case in H F theory, the K o h n - S h a m equations are solved iteratively for ipfs and ^ unt i l self-consistency is achieved. The K o h n - S h a m orbitals are typically expressed as linear combinations of atomic orbitals in a manner analogous to that described in Section 2.3.1 for the H F orbitals, so determination of the K o h n - S h a m orbitals consists of choosing a basis set and determining the corresponding set of orbital coefficients. Despite the obvious similarities between the K o h n - S h a m and the Hartree-Fock equations, they differ in a significant manner: the K o h n - S h a m equations are in principle exact while the H F equations are not. In order to realize this "exactness", the exact exchange-correlation functional Exc[p(r)] must be used. The form of this functional is not known so approximations for Exc[p(r)] are used, thereby introducing errors into the theory. M u c h of the research activity concerning D F T has involved the derivation and assessment of exchange-correlation functionals. Several commonly used functionals, employed for the work presented in this thesis, are described below. Exchange-correlation functionals The simplest exchange-correlation functionals make use of the local spin-density approxi-mation ( L S D A ) , in which Exc[p(r)] is taken to be equal to the exchange-correlation energy of a homogeneous electron gas of density p. This may seem a drastic approximation for molecules, where the electron density can vary considerably, and indeed calculations using the L S D A seriously overestimate the binding of atoms in molecules. However, the L S D A does provide reasonably accurate structural properties and consequently has been applied to many chemical problems. The formulation of the L S D A exchange-correlation functional used i n this thesis is that recommended by Vosko, W i l k and Nusair [99].3 They use the an-3 T h i s is the functional selected using the keyword L O C A L in the deMon [ 1 0 0 , 1 0 1 ] density functional program and S V W N 5 in the G A U S S I A N 9 2 [102] and G A U S S I A N 9 4 [103] programs. Chapter 2. Theoretical Background and Computational Methods 45 alytic expression derived by Dirac [104] for the exchange energy of a homogeneous electron gas and a function obtained by a least-squares fit to the results of Monte Carlo calculations of homogeneous electron gases of various densities for the correlation energy. This separation of the exchange-correlation energy into separate exchange and correlation terms is common to al l of the functionals discussed here. Several exchange and correlation functionals have been developed that improve upon the L S D A by introducing a dependence on the gradient of the density in addit ion to that on the density itself. This is known as the generalized gradient approximation ( G G A ) [68]. A key feature addressed in the development of these functionals is the overestimation of atomic binding energies mentioned above. This overbinding is a consequence of the incorrect asymptotic form of the L S D A exchange potential. The potential vx (obtained from Ex v ia Equat ion (2.36)) falls off too rapidly with r , so that electrons at large r experience a greater Coulomb repulsion than they should, causing the outer-most regions of the electron density distr ibution to be too diffuse. The gradient-corrected exchange functional proposed by Becke in 1988 [105] has been constructed such that Ex has the correct asymptotic form and results in much-improved calculated thermochemical properties. It should be noted, however, that having the correct asymptotic form for the exchange energy Ex does not ensure that the same is true for the exchange potential vx. Also , this functional contains one semiempirical parameter, set by performing a fit to exact atomic H F data [105]. The Becke '88 functional is the gradient-corrected exchange functional that has been used in this thesis. Two gradient-corrected correlation functionals have been used in this thesis: Perdew's 1986 functional [106] and that of Lee, Yang and Parr [107]. In general, gradient corrections to the correlation energy have been found to be of lesser significance than those to the exchange energy [68]. A th i rd class of functional has recently been proposed which is a hybrid of the G G A functionals and the H F exchange energy and which is generally of greater accuracy than the Chapter 2. Theoretical Background and Computational Methods 46 G G A functionals alone. The most widely used hybrid functional is Becke's three-parameter functional [108], in which the relative contribution of the H F exchange energy to the total exchange energy was determined by a fit to experimental thermochemical data. For the work i n this thesis, a popular variant of Becke's three-parameter functional has been used i n which the gradient-corrected correlation functional of Perdew and Wang [109] has been replaced by that of Lee, Yang and Parr [107]. 2.3.4 Basis sets A l l of the electronic structure methods discussed in the preceding sections make use of a basis set expansion to express the molecular wavefunction (in the case of the H F and post-H F methods) or the electron density (in the case of D F T ) . The use of a finite basis set for this expansion, although necessary for practical reasons, introduces an addit ional level of approximation (and therefore source of error) into the calculations. The magnitude of this error can be minimized by choosing the basis functions carefully. A large number of basis sets of widely varying size and accuracy have been described in the literature. The choice of basis set for a particular calculation is influenced by the size of the system being studied, the available computational resources, the theoretical method employed, the properties to be calculated and the level of accuracy required. This choice becomes increasingly l imited as the size of the molecule increases, thereby necessitating trade-offs between accuracy and tractability. One of the goals of this thesis is to assess the sensitivity of calculated momentum-space electron density profiles to computational method and basis set. It is hoped that this may provide additional information to aid in the choice of basis set for the calculation of other chemical properties of similar molecules. Consequently, theoretical momentum profiles for comparison wi th the experimental data have been obtained from electronic structure calcu-Chapter 2. Theoretical Background and Computational Methods 47 lations using a range of basis sets. A l l of the basis sets considered here are composed of functions centred on each nucleus, w i t h the specific functions used dependent upon the respective atomic number (i.e., nuclear charge). The functions are chosen such that they have the same symmetry properties as atomic orbitals and are classified as s, p, d , . . . according to these properties. Basis sets typical ly consist of either Slater or Gaussian functions. Slater functions have an exponential dependence exp(—£r) on the radial distance r . The exponent £ determines the size (spatial extent) of the orbital . Gaussian functions have an exponential dependence on the square of the radial distance: exp(—(r 2 )- Slater functions are better choices to represent atomic orbitals because they have the correct form at the nucleus (a cusp). For this reason, they are commonly used for electronic structure calculations of atoms. However, Gaussian functions are commonly used for calculations of molecules because al l of the multi-centre integrals can be evaluated analytically; this is not possible using Slater functions. To minimize errors resulting from the incorrect shape of Gaussian functions, a linear combination of Gaussian functions wi th different exponents £ is often used to form one basis function. The individual Gaussian functions are then referred to as primitive Gaussian functions and the resulting basis function as a contracted Gaussian function ( C G F ) . A l l of the basis sets used in this thesis consist of Gaussian functions. Those common to al l of the studies performed are described below. The information is summarized in Table 2.1. Further discussion of Gaussian basis sets can be found in Reference [94]. S T O - 3 G The S T O - 3 G basis set designed by Pople and co-workers [110] is a m i n i m a l basis set i n which each basis function is a linear combination of three Gaussian functions. The coefficients and exponents of the primitive Gaussians are determined by a least-squares fit to a Slater function. M i n i m a l basis sets are often described as single-^ (single-zeta) basis Chapter 2. Theoretical Background and Computational Methods 48 Table 2.1: Basis sets util ized for electronic structure calculations. In addition to the basis sets listed, others formed by augmenting the 6-31G and 6-311G basis sets wi th diffuse (+,++) and polarization (*,**) functions have been used. Name Ref. Pr imit ive functions (C, N and O / H ) Contracted functions [C, N and O / H ] S T O - 3 G [110] (6s,3p/3s) [2s, lp/ls] 4-31G [111] (8s,4p/4s) [3s,2p/2s] 6-31G [111,112] (10s,4p/4s) [3s,2p/2s] 6-311G [113] ( l ls ,5p/5s) [4s,3p/3s] aug-cc -pVTZ [114,115] (lls,6p,3d,2f/6s,3p,2d) [5s,4p,3d,2f/4s,3p,2d] augmentations3. ** [116] ( l d / l p ) [ l d / l p ] ++ [117] ( l s , l p / l s ) [ l s , l p / l s ] a U s e d wi th the 6-31G and 6-311G basis sets to form, e.g., the 6-31G** and 6-311++G** basis sets. sets because they are equivalent to using a single Slater function for each atomic orbital . For example, a minimal basis set for the hydrogen atom consists of a single s-type basis function and for carbon, nitrogen or oxygen consists of two s-type functions and one set of three p-type functions. The lack of flexibility in minimal basis sets typically leads to inaccuracies in describing the electronic structure of molecules. 4 - 3 1 G This basis set [111] is an example of a split-valence basis set, i n which the valence atomic orbitals are each represented by multiple basis functions. The basis set label is derived from the number of primitive Gaussians used for each C G F , wi th the number before the dash referring to core A O s and the numbers following the dash referring to the valence A O s . In the case of the 4-31G basis set, one C G F consisting of three primit ive Gaussians and a second consisting of a single (uncontracted) Gaussian are used for each valence atomic orbital , e.g., the l s orbital in the case of hydrogen and the 2s and three 2p orbitals for carbon, nitrogen and oxygen. This is also referred to as valence double-zeta. The use of multiple basis functions with different exponents to describe the valence electrons helps to Chapter 2. Theoretical Background and Computational Methods 49 account for the perturbation of the electron density that may occur upon the formation of molecular bonds by allowing the electron density about a nucleus to expand or contract. This perturbation is much smaller in the case of the core electrons and consequently the core description has been kept at single-zeta quality in the 4-31G basis set. The use of four primit ive functions to describe the core rather than the three used i n the S T O - 3 G basis set allows for a more accurate representation of the shape of these orbitals near the nucleus, resulting in lower S C F energies. 6-31G The 6-31G basis set [111, 112] is very similar to the 4-31G basis set described above, differing however in the number of primitive functions used to form the core A O basis functions. A s a result, total energies obtained wi th the 6-31G basis set are lower than those obtained using the 4-31G basis set but properties that depend primari ly upon the valence electrons differ by a much smaller amount. 6-311G This valence triple-zeta basis set [113] features increased flexibility for the descrip-tion of the valence orbitals. It forms the basis of the 6-311G** basis set (see polarization functions, below) which, in contrast to the basis sets described above, has been optimized for use in correlated calculations (specifically M P 2 ) . Polarization functions Frequently, basis sets are augmented to include functions of higher angular quantum number than is necessary to describe the atoms in their ground electronic states. For example, p-type basis functions are added to hydrogen and d-type functions to carbon. These augmentations are called polarization functions because they allow for the polarization of electron density. This is particularly important for the accurate description of the electronic structure of highly polar molecules [94]. Augmentations of the above basis sets by the minimal set of polarization functions on al l non-hydrogen (heavy) Chapter 2. Theoretical Background and Computational Methods 50 atoms are commonly identified by an asterisk (*) following the " G " i n the basis set name. If (p-type) polarization functions are also added to the hydrogen atom basis, two asterisks are used (**). Alternatively, the functions added may be stated more explicit ly by their atomic symmetry labels with the heavy-atom polarization functions stated first by convention. For example, the 6-31G** basis set, consisting of the 6-31G basis set augmented by a set of d-type polarization functions on heavy atoms and p-type polarization functions on hydro-gens, may also be referred to as 6-31G(d,p). This notation is commonly employed when polarization functions beyond the minimum set are added. In this thesis, * and * * denote the polarization functions recommended by Pople and co-workers [116]. Diffuse functions A second common augmentation is the addition of highly diffuse func-tions (i.e., having small exponents) to the basis set. Such functions are necessary for the accurate description of the large-r (long-range) electron density and are important for cal-culations of anions and molecules involving long-range interactions (e.g., hydrogen bonds) as well as for the calculation of properties highly dependent upon the outermost regions of electron density such as electron affinities and outer-valence-orbital momentum profiles. The addition of a single s-type diffuse function and a set of three p-type diffuse functions [117] to the above-described basis sets for heavy atoms is denoted by a single "+" preceding the " G " in the basis set name. The further addition of diffuse s-type functions to the hydrogen atom basis is denoted by a double plus "++". aug-cc-pVTZ This basis set of Dunning and co-workers [114,115] is a polarized valence-triple-zeta basis set (cc-pVTZ) augmented with diffuse functions (aug-). It has been de-veloped to give accurate results for calculations including electron correlation effects and includes f-type functions on heavy atoms and d-type functions on hydrogen. Chapter 2. Theoretical Background and Computational Methods 2.4 Molecular conformation 51 The size and conformational mobil i ty of the molecules studied in this thesis introduces sev-eral complications that have generally not been present in the analysis of E M S experiments on smaller molecules. The common practice in E M S studies has been to make use of experi-mental molecular geometries when calculating theoretical momentum profiles for comparison wi th the corresponding experimental profiles. Such experimental geometries are readily avail-able in the literature for most small gas-phase molecules. However, this is often not the case for molecules of the size studied in this thesis (10+ atoms). The number of independent geometrical parameters in these larger molecules makes the complete experimental determi-nation of geometries considerably more challenging. In most cases, only part ial experimental geometries, in which some parameters have been assumed to equal "standard" values, are available. Furthermore, the conformational mobility of these molecules results in the pos-sibil i ty of multiple stable conformational isomers (conformers) contributing to the observed experimental signal. It then becomes necessary, when calculating T M P s for comparison wi th the experimental data, to consider al l likely conformers. This requires knowledge not only of the geometries of al l stable conformers but also of their relative energies, so that the contributions to the total observed signal from individual conformers can be weighted appro-priately by their relative abundances in the experimental sample. A s a consequence of the incomplete availability in the literature of experimental conformer geometries and relative energies for the molecules studied, theoretical predictions have been used. A brief overview of the standard procedures that have been used for calculating the theoretical geometries and free energies follows. Chapter 2. Theoretical Background and Computational Methods 52 2.4.1 Geometry optimizations The potential energy surface of a molecule describes the molecule's total energy as a function of the relative nuclear coordinates (i.e., the molecule's geometry). A n electronic structure calculation at a fixed geometry yields one point on this surface. The purpose of geometry optimizat ion calculations is to find those sets of relative nuclear coordinates that correspond to m i n i m a on the potential energy surface; these are equilibrium structures. Simple molecules usually have a single energy minimum; more complicated molecules can have several. The geometry optimization procedure is a multi-step process in which an input geometry is varied so as to minimize the total energy of the molecule. A n electronic structure calculation is performed at the in i t ia l geometry (using, for example, one of the methods discussed in Section 2.3). This is followed by calculation of the energy gradient, i.e., the first derivative of the energy wi th respect to displacements of the nuclei. If the gradient is less than a previously chosen threshold, then a stationary point has been located on the potential energy surface and the calculation is stopped. Otherwise, the geometry is modified using the forces on the nuclei determined from the gradient calculation and the process is repeated using the new geometry. The stationary-point geometry determined from a geometry optimization calculation is not necessarily an energy minimum; it may instead correspond to a saddle point on the potential energy surface. A saddle point is an energy min imum wi th respect to variation of some relative nuclear coordinates and an energy maximum with respect to variation of others. Stationary points can be characterized as energy m i n i m a or saddle points by calcu-lat ing normal-mode vibrational frequencies from the second derivative of the energy at the stationary-point geometry. If the geometry is a true min imum, the calculated vibrational frequencies w i l l al l be real. Transition structures (saddle points) are characterized by one or more imaginary vibrational frequencies. Chapter 2. Theoretical Background and Computational Methods 53 Performance of a geometry optimization calculation and characterization of the station-ary point to verify that an energy minimum has been found does not ensure that the global m i n i m u m on the potential energy surface has been located. For molecules wi th a large num-ber of geometrical parameters, it may be necessary to perform many geometry optimization calculations using different starting points (initial geometries) in order to maximize the likelihood that al l equil ibrium structures (conformers) are located. This can require a con-siderable amount of computational resources, especially if the molecule and basis set are large and the electronic structure method used is a costly one. Rather than sampling al l regions of conformational space, the number of ini t ia l geometries considered is often l imited to those thought likely to be near energy min ima on the potential energy surface. F inal ly , it is important to note that the potential energy surface and the equil ibrium structures derived from it depend not only on the molecule but also on the basis set and theoretical method used for the energy calculations. In some cases, a minimum-energy conformation obtained at one level of theory may be a transition structure at a different level of theory. 2.4.2 Relative conformer energies In an E M S experiment on a conformat ional^ mobile molecule, each stable conformation present i n the experimental sample w i l l contribute to the observed E M S signal in proportion to its relative abundance in the sample. Thus, the observed signal w i l l be the sum of the individual signals arising from each conformer, weighted by the respective fractional populations of the conformers. The fractional conformer populations pi at temperature T are described by a Bol tzmann distribution Chapter 2. Theoretical Background and Computational Methods 54 where GI;T is the Gibbs free energy of conformer i at temperature T , gi is the degeneracy of conformer i and ^ . is a sum over al l conformers. The energy obtained from an electronic structure calculation is not the free energy of a conformer but rather corresponds to the energy E0 of an isolated molecule wi th stationary nuclei at 0 kelvin. The contributions of the zero-point vibrational energy EZPV, the enthalpy change AH(T) from 0 K to T and the entropy SV at T must be accounted for to determine GT- Expl ic i t ly , GT = E0 + EZPV + AH(T) -T-ST- (2.38) EQ + EZPV is H0, the enthalpy at 0 K , which when summed wi th AH(T) gives HT, the enthalpy at temperature T. Statistical mechanics provides the techniques necessary to calculate these thermodynamic quantities knowing the normal-mode vibrational frequencies and equil ibrium geometry of the conformer 4 . The zero-point vibrational energy per mole is EZPV = ^NhJ2^ (2-39) i where z/j is the vibrational frequency of normal mode i, N is Avogadro's number, h is Planck's constant and the sum is over all normal modes of the conformer. The thermal enthalpy change AH(T) can be written as the sum of contributions from translational, rotational and vibrat ional degrees of freedom. The individual terms for non-linear molecules are AtftransCT) = \ ^ (2.40) A t f r o t ( T ) = ^RT (2.41) AHvih(T) = RQUi{eQ^'T - l ) " 1 (2.42) 4 T h e relationships used assume ideal-gas behaviour and the validity of the harmonic oscillator and rigid rotor approximations and are derived in many texts. See for example McQuarrie [118] or Mayer [119]. Chapter 2. Theoretical Background and Computational Methods 55 where R is the gas constant and QVI is the vibrational temperature for the normal-mode vibrat ional frequency given by (2.43) where k is Boltzmann's constant. The PV work term has been included in A / f t r a n s ( T ) . The entropy can also be partitioned into translational, rotational and vibrational terms: where M is the mass of the molecule, V/N is the average volume occupied per molecule (an inverse density), &A is a rotational temperature derived from the rotational constant A: QA = ~ (2.47) and o is a symmetry number determined by the symmetry of the conformer. There is an additional contribution to ST if the ground electronic state is degenerate. This is not the case for any of the molecules considered in this thesis. The form of Equation (2.37) is such that it is not necessary to calculate the absolute free energies GT of the conformers; the relative conformer free energies AGT are sufficient to determine the fractional conformer populations. Therefore, al l terms contributing to Equation (2.38) that are invariant between conformers can be neglected. These terms are A i T t r a n s ( T ) , AHROT(T) and 5 t r a n s , T - In the remainder of this thesis, all thermodynamic quan-tities are reported relative to those of the most stable conformer. Si (2.44) (2.45) (2.46) Chapter 3 Experimental Methods A l l of the experimental E M S data reported in this thesis were obtained using a multichannel energy-dispersive electron momentum spectrometer constructed as part of this project. The spectrometer design is based upon that of a similar instrument developed at the Flinders University of South Austra l ia [16-18]. The main features of the spectrometer as well as the procedures used for the collection and treatment of the experimental data are described below. 3.1 Multichannel energy-dispersive electron momentum spectrometer The multichannel energy-dispersive electron momentum spectrometer constructed in the course of the present work is shown schematically in Figure 3.1. The spectrometer is housed wi th in a 60 cm diameter cylindrical aluminum vacuum chamber. Two Seiko-Seiki T M P 450 Maglev turbomolecular pumps are used to maintain vacuum in the main chamber. A T M P 300 Maglev turbomolecur pump (located directly behind the electron gun shown in Figure 3.1) differentially pumps the electron gun, reducing the amount of sample that reaches 56 Chapter 3. Experimental Methods 57 7 ti PUMP O D2 P D 1 ' F l E L E C T R O N G U N Li PUMP 5 cn 10 Figure 3.1: Schematic diagram of the energy-dispersive multichannel electron momentum spectrometer constructed and used in the present work. The polar angles 6 of the two analysers are kept fixed at 45° and the out-of-plane azimuthal angle cp of the movable analyser is varied. The labels used are: F , filament; G , grid; A , anode; F l and F2 , electron lenses; D 1 - D 3 , parallel plate quadrupole deflectors; P 1 - P 4 , spray plate apertures; F C , Faraday cup; M C P , microchannel plates; R A , resistive anode. Chapter 3. Experimental Methods 58 the electron gun filament and thereby prolonging the lifetime of the filament. The output ports of the three turbomolecular pumps are connected to a common backing line that is maintained at a pressure of « 10~ 2 torr using a rotary pump. Base pressures of 1 0 - 7 torr i n both the main chamber and the electron gun region are achieved. The vacuum chamber is lined w i t h hydrogen annealed mumetal shielding to reduce the magnetic field i n the centre of the spectrometer (horizontal field « 30 mGauss after shielding) and hence minimize perturbation of the electron trajectories. Electron momentum spectroscopy experiments require a source of energetically and d i -rectionally well defined electrons. In the case of the spectrometer constructed and uti l ized i n the current work, the electron beam originates from an electron gun mounted underneath the base plate of the spectrometer and is directed upwards through the interaction region in the centre of the instrument. The electrons are produced by a dirrect-current-heated thoriated tungsten wire (diameter 0.05 mm) filament (F in Figure 3.1) and are accelerated and focussed using a commercial electron gun body 1 , consisting of a grid (G), anode (A) and electron lens (FI ) . The interaction region of the spectrometer is at ground potential and the electron gun filament is floated at the negative potential corresponding to the desired elec-tron energy, which under typical operating conditions is 1200 eV greater than the electron binding energy being monitored. A second electron lens (F2) located approximately midway between the electron gun and the interaction region is used to further focus the incident electron beam. Three sets of parallel plate quadrupole deflectors (D1-D3) are used to align the electron beam, which is monitored v ia four spray plate apertures (P1-P4) and a Faraday cup ( F C ) . A l l gaseous samples and most that are l iquid at room temperature are introduced into the spectrometer using a gas line connected to the side of the interaction region. The sample iCliftronics C E 5 A H Chapter 3. Experimental Methods 59 v a c u u n c h a n b e r wa I I s a n p l e t u b e p r o b e ^ 4 7K c o p p e r h e a t j n g b l o c k s A i n t e r a c t i o n r e g i o n g a s l i n e i n c i d e n t e l e c t r o n b e a n Figure 3.2: Direct probe heated sample system for use in performing E M S measurements on low vapour pressure liquids and solids. The line used for the introduction of gaseous samples to the spectrometer is also shown. (Not to scale.) vessels are connected to a sample vacuum manifold evacuated by a dedicated rotary pump. A variable leak valve is used to control the sample pressure in the spectrometer. To allow for the study of substances that exist as solids or low vapour pressure liquids at room temperature, a heated sample probe has also been constructed. A schematic diagram of the sample probe and gas line is shown in Figure 3.2. The view shown is perpendicular to that of Figure 3.1. The solid sample is placed in a Pyrex tube which is attached to the end of a metal probe and inserted through a vacuum lock in the wall of the spectrometer vacuum chamber. This allows sample replacement without removing the vacuum chamber and while maintaining a part ia l vacuum, thus considerably decreasing the delay between data collection periods. The Pyrex tube rests in the spectrometer in a non-inductively wound electrically heated copper block adjacent to the experimental interaction region. A second heating block surrounds the t ip of the sample tube, allowing separate temperature control of this area. The temperatures are monitored by thermocouples attached to the copper blocks. In this way, a solid sample Chapter 3. Experimental Methods 60 can be vapourized directly into the interaction region, providing a stable gas target density over a period of several days. In the case of some high vapour pressure solids, the proximity of the sample tube to the interaction region results in sufficient target density for E M S measurements without heating of the sample [69]. The two outgoing electrons produced by the electron impact ionization of a target atom or molecule (i.e., one scattered electron and one ejected electron) are each angle and energy selected and detected using hemispherical analysers [120] of mean radius 70 m m . The polar angles of the two analysers (9 in Figure 2.1) can be varied over the range 45 ± 5°. Variat ion of 9 is necessary for the study of the lower momentum regions (p < 1.0 au) of the momentum profiles of core (IP > 250 eV) electrons (see Equation (2.5)). For valence orbital studies, including a l l work reported in this thesis, the polar angles of the two analysers are fixed at 45°. The azimuthal position of one of the analysers is also fixed. The second analyser is mounted on a rotatable turntable driven by a computer-controlled stepper motor, allowing the out-of-plane azimuthal angle of the two analysers (cj) in Figure 2.1) to be varied over a range of ± 5 0 ° and thereby varying the electron momentum that is monitored. To be detected, electrons must first pass through one of the five-element electron lenses at the entrance of each analyser. These lenses act as two zoom lenses [121], the first defining the acceptance angle and thus the angular resolution of the spectrometer and the second decelerating the electrons. B y decelerating the electrons prior to energy analysis, a lower pass energy (typically 50 eV) can be used for the hemispherical analysers, resulting i n improved energy resolution. The hemispherical analysers (inner radius 50 m m , outer radius 89 mm) linearly disperse the electrons according to their input energies along the radial direction of the exit plane. One dimensional position sensitive detectors located at the exit plane of each analyser allow the determination of the radial position, and thus the energy, of each detected Chapter 3. Experimental Methods 61 electron. Each detector consists of a pair of 25 m m diameter microchannel plates 2 in the double chevron configuration [122] and a Gear type resistive anode [123]. The microchannel plates are operated in the pulse saturated counting mode, 3.2 Instrument electronics The electronics used for processing the signals from each detector and for controlling the spectrometer are shown schematically in Figure 3.3. The signal processing electronics per-form two tasks: screening of the data on the basis of the time correlation between signals received at the two detectors, and determination of the positions at which the electrons strike the detectors, and hence their kinetic energies. In Figure 3.3, the position electronics are shown i n the upper portion of the figure and the t iming electronics in the lower portion. The time correlation between the two signals is used to determine if the two electrons were de-tected i n coincidence, a necessary condition if they are the result of the same E M S scattering event (i.e., an ionization with both outgoing electrons having approximately equal energies). Accidental coincidences are also detected (see Section 3.2.2 below). 3.2.1 Position electronics A single electron reaching one of the detectors is amplified on the order of 10 6 times by the cascade effect of the microchannel plates, causing a charge pulse to strike the resistive anode. Output pulses are taken from both ends of each resistive anode (labelled in Figure 3.3 as A ^ j and Bjy[ and A p and B p , referring to pulses originating from the movable and fixed analyser detectors, respectively) and are amplified by preamplifiers ( P R E ) and amplifiers ( A M P ) . 3 The amplitudes of the A and B pulses from each detector are indicative of where on the resis-2 Electro Optical Sensors, VUW8946ZS 3 Ortec 1421H and 855, respectively. Chapter 3. Experimental Methods 62 MCP R A A M MCP R A SUM AMP SUM AMP TIMING FILTER AMP T1M1NG FILTER AMP CFO 1 AM+BU Af+ E SUM AMP ELECTRON GUN POWER SUPPLY DELAY AMP STEPPING MOTOR CONTROLLER CO INC SCA BACK SCA TRIPLE CO INC GATE/ DELAY LI NEAR GATE LABMASTER CARD P C TRIPLE CO INC L INEAR GATE GATE/ DELAY ROUTER MULTIPLEXER Figure 3.3: Schematic diagram of the position sensitive detectors and signal processing elec-tronics. The detectors each consist of a pair of microchannel plates ( M C P ) and a resistive anode ( R A ) . Other components illustrated are: P S D A , position sensitive detector analyser; C F D , constant fraction discriminator; T A C , time-to-amplitude converter; S C A , single chan-nel analyser; A D C , analogue-to-digital converter computer card; P C , personal computer. Chapter 3. Experimental Methods 63 tive anode the charge pulse struck, since this w i l l determine how much resistance (provided by the resistive anode) the pulse must pass through before reaching the position processing electronics. The A pulse and the sum of the two pulses ( A + B , obtained using a summing am-plifier ( S U M A M P ) 4 ) are input to a position sensitive detector analyser ( P S D A ) 5 , producing a voltage pulse (0-10 V ) with amplitude A + B ' proportional to the position on the detector where the electron struck. This pulse w i l l have maximum amplitude if an electron strikes the end of the detector from which the A pulse is taken (B=0, ^ + B = 1) and min imum amplitude if an electron strikes the opposite end of the detector (A=0, = 0). The A pulses are taken from the outer ends of the detectors (corresponding to greater electron en-ergy) and the B pulses are taken from the inner ends (lesser electron energy). Consequently, the amplitude of the position pulse (^+Fj) 1 S proportional to the energy of the detected electron (E) less the minimum detectable electron energy of the analyser/detector (Emin). That is: J^_-Q (x E — Em\n. The position pulses originating from each detector are then passed through an attenuator 6 and summed using a third summing amplifier, producing a pulse proportional in amplitude to the summed energy of the two detected electrons less the m i n i m u m detectable summed energy. 3.2.2 Timing electronics T i m i n g signals from each detector are taken directly from gold plated copper strip plates at the back of a ceramic plate behind each resistive anode and passed through a preamplifier and t iming filter amplifier, 7 followed by a constant fraction discriminator ( C F D ) 8 . The C F D 4 T w o sum and invert amplifiers (Ortec 533) are used in succession to obtain a summed but not inverted output pulse. 5 Ortec 464. Constructed by the U B C Chemistry Department Electronics Shop. 7 Ortec 474 8 Ortec 935. Chapter 3. Experimental Methods 64 only produces an output pulse when the amplitude of the input pulse is greater than a set threshold. This processing helps to eliminate spurious signals (i.e., "noise"). Ratemeters are used at this point to monitor the individual count rates of the two detectors (the "singles" rates). After delaying the t iming signal from one of the detectors by a fixed amount ( « 30 ns), the two t iming signals are passed to a time-to-amplitude converter ( T A C ) 9 to determine their time correlation. The T A C functions as a "stop-watch," producing an output pulse wi th amplitude proportional to the time between detection of the start and stop pulses. If two electrons arrive at the detectors in coincidence, the amplitude of the T A C output pulse w i l l correspond to the fixed delay of the t iming signal input to the T A C stop channel. A typical time spectrum, obtained from the output of the T A C , is shown i n Figure 3.4. A con-stant amplitude background is obtained in the time spectrum as a result of the detection of uncorrelated electron pairs. To determine the amplitude of this background and thus make possible the correction for random coincidences in the coincidence peak (i.e., determine the number of true coincident events) single channel analysers ( S C A ) 1 0 are used to separate coin-cident and non-coincident events by setting coincidence and background windows ( C W and B W , respectively, as illustrated in Figure 3.4). Further details of the background correction procedure are discussed in Section 3.4.3 below. The S C A output and pulses from each of the P S D A s are entered into a triple coincidence unit (constructed by the U B C Chemistry Department Electronics Shop) to further reduce spurious signals. If a triple coincidence is detected, a linear gate 1 1 is opened and the summed position pulse enters a router/mult iplexer 1 2 . Delay uni t s 1 3 are used to ensure the t iming and position pulses arrive at the linear gates concurrently. The router/multiplexer adds an offset 9 Ortec 467. 1 0 Ortec 550A and one S C A located on the T A C . n Tennelec T C 310. 1 2Tennelec T C 306. 1 3 Ortec 416A for the timing pulses and Ortec 427A for the position pulse. Chapter 3. Experimental Methods 65 channel number 0 64 128 192 256 1400 i 1 1 1 1 0 1 1 1 1 1 0 50 100 150 200 time (ns) Figure 3.4: Sample time spectrum, collected by the A D C computer card from the output of the T A C . The coincidence (true + random coincidence events) and background (random events) windows, set using single channel analysers, are indicated ( C W and B W , respec-tively). The data are from non-binning mode measurements (see Section 3.4) of the A r ( 3 p ) _ 1 ionization and were collected in 45 minutes. Chapter 3. Experimental Methods 66 to pulses coming through the "background" gate and produces an output pulse appropriately scaled for processing by a personal computer card analogue-to-digital converter ( A D C ) 1 4 . The A D C card divides the input voltage range into 512 equal-width channels, counts the number of pulses received in each channel and stores the data in memory on the card. A s a consequence of the offset added by the router/multiplexer to pulses coming through the background gate, the first 256 A D C channels contain the coincident window pulses and the second 256 the background window pulses. Each of the A D C channels corresponds directly to a narrow range of total energy of the detected electron pair. The subsequent computer processing of the experimental data is described in Section 3.4 below. A second computer c a r d 1 5 is used to control the energy of the incident electron beam (via the electron gun power supply) and the azimuthal (<f>) position of the movable analyser (via a stepping motor controller) and to count the number of processed position pulses originating from the movable analyser ( P R E S E T in Figure 3.3). The P R E S E T is used to determine the amount of time spent collecting data at each angle (f) and incident electron beam energy, so as to minimize the effects of small fluctuations in the electron beam intensity and the sample density. 3.3 Spectrometer calibration and operating conditions The spectrometer is operated wi th a nominal total energy of 1200 eV. This corresponds to an incident electron energy of (1200 eV + binding energy) and nominal analyser detection energies of 600 eV. The five-element lens systems prior to the entrance of each analyser retard the electrons by 550 eV before entering the hemispherical analysers. The analyser pass energy is 50 eV, meaning that 50 eV electrons (600 eV prior to retardation) take 1 4Series II Personal Computer Analyzer, Nucleus, Inc. 1 5 LabMaster D M A , Scientific Solutions, Inc. Chapter 3. Experimental Methods 67 the central trajectory through the analysers and strike the centre of the position sensitive detectors. The energy dispersive nature of the analysers allows electrons to be monitored over approximately an eight eV energy range, resulting in a detectable energy range of 600.0 ± 4.0 eV for each analyser. The total energy range monitored is thus 1200.0 ± 8.0 eV. Therefore, for an incident electron energy of 1200 eV + X , a binding energy range of X ± 8.0 eV is monitored simultaneously. A s a consequence of contact potentials, in each E M S experiment there is a constant offset between the energy set by the incident electron beam power supply and the monitored electron binding energy. Energy scale calibration is performed by either introducing a small amount of a calibration gas into the spectrometer during data collection or by assigning the energy of a sharp peak in the B E S to the IP determined using high-energy-resolution P E S . Note that the relative energy separations of the E M S ionization peaks are not affected by this energy offset. In the present work, an incident beam current of 25-30 fiA (as measured by the Faraday cup) is used. A sample pressure of ~ 10~ 5 torr (measured by an ion gauge on the main vacuum chamber) is used for samples that are gaseous at room temperature. W i t h this combination of beam current and sample pressure, single count rates of « 1500 Hz and a coincidence count rate of up to 40 true coincidences per minute are achieved. 1 6 In the case of the E M S measurements of glycine reported in this thesis, the solid sample was heated to a temperature (165°C) sufficient to produce single and coincident count rates comparable to those typical ly achieved for gaseous samples. Pr ior to performing any E M S studies, a number of testing and calibration procedures must be undertaken to verify the proper operation of the spectrometer. The performance of each analyser is separately calibrated and tested to ensure a uniform response and linear 1 6 T h e true coincidence count rate is dependent upon the target being ionized as well as the incident electron beam energy and out-of-plane azimuthal angle. The stated value was obtained from ionization of argon at a binding energy of 15.8 eV and an "out-of-plane azimuthal angle of 8°, i.e., the maximum of the A r 3p X M P . Chapter 3. Experimental Methods 68 0 30 60 2500 £ 2000 [ Channel number 90 120 0 30 60 90 120 Movable Analyser 2 4 - 4 - 2 Relative energy (eV) 0 2 4 Figure 3.5: A typical test of electron analyser performance using incident electrons elastically scattered off argon. For each analyser test, incident electron energies of 597, 599, 601 and 603 eV were used. In addition to the experimental data (•), a fitted curve (;—), obtained from a least-squares fit of Gaussian functions to the experimental data, is shown. relationship between detected position and electron energy across the entire detected energy range. Also , the relationship between detected position and electron energy must be the same for each detector. The uniformity of the detector response is tested by monitoring the P S D A output ( A ^ j j ) f ° r e a c h detector using typical operating conditions. A square response function should be obtained. The linear response and energy resolution of each analyser-detector assembly are tested using elastically scattered electrons produced wi th an incident electron beam energy of ~ 600 eV. The results of a typical test are shown in Figure 3.5. The linear relationship between electron energy and detected position is evident. The analyser energy resolution is not constant across the tested energy range, wi th the resolution better Chapter 3. Experimental Methods 69 for the higher energy electrons that take an outer trajectory through the analysers. This is thought to be the result of non-uniform electric field lines at the analyser exit. It w i l l not adversely affect spectrometer performance since coincident electron pairs w i l l consist of electrons having al l detectable energies with equal probability and the effect w i l l therefore be averaged out. The analyser power supplies are very stable so consequently these tests of individual analyser performance need only be performed infrequently. Before beginning each E M S experiment, calibration of the monitored total energy range is performed under normal operating conditions. This is done by collecting a B E S using the A D C computer card. The He ( I s ) - 1 ionization peak is most commonly used for this purpose. The incident electron beam energy is ini t ia l ly set to correspond wi th the binding energy of the He ( I s ) - 1 peak (24.6 eV) and sufficient data is collected to determine accurately the data channel of the A D C card corresponding to the centre of the peak. The incident electron beam energy is then adjusted higher and then lower by a few eV and the resulting shifts i n the ionization peak position (in terms of data channels of the A D C card) are noted. This information is then input to the data acquisition software to allow conversion from A D C data channel to electron binding energy. The overall instrumental energy resolution function is determined by recording the B E S of the He ( I s ) - 1 ionization peak at 24.6 eV or the A r ( 3 p ) _ 1 ionization peak at 15.8 eV. A n energy resolution of 1.4-1.6 eV is typically achieved (see Figure 3.6 in Section 3.4, below). The angular resolution of the spectrometer is determined using the A r 3p momentum profile. This profile is p-type, wi th the maximum occurring at 0.65 au. A s a consequence of the symmetry of the orbital from which the electron is ionized, the A r 3p X M P would have no intensity at zero momentum if recorded using a spectrometer having perfect angular resolution. In other words, the observation of any experimental intensity at p = 0 au is a consequence of the instrumental angular resolution. Instrumental angular resolution Chapter 3. Experimental Methods 70 parameters have been determined by a comparison of the A r 3p X M P wi th a high quality C I calculation of the A r 3p momentum profile [124] resolution-folded using the G W - P G method [48] and a range of values for the angular resolution parameters. The use in the resolution-folding procedure of values of At? = 0.6° and A0 = 1.2° results in excellent agreement between the A r 3p experimental and theoretical momentum profiles, as shown in the right panel of Figure 3.7 in Section 3.4.4. These angular resolution parameters have been used in the resolution-folding of T M P s performed in al l subsequent studies. B o t h the energy and angular resolutions of the spectrometer are monitored regularly to ensure that the spectrometer is operating properly and to determine the energy resolution for use i n the data analysis. Cal ibrat ion runs of A r and He are customarily recorded prior to and following each molecular study and also following any adjustments to the electron beam and analyser settings. The ratio of the observed intensity at the maximum of the A r 3p profile to that at p = 0 provides a convenient test of the angular resolution and is a sensitive indicator of a correctly tuned instrument. 3.4 Data collection and treatment D a t a collection is performed under computer control, allowing "hands-off" operation of the spectrometer—a practical necessity when continuous data collection times of several weeks are required to complete one study. Experimental data are collected using one of two data collection modes, termed "non-binning" and "binning" . The two data collection algorithms are described below. Argon ( 3 p ) _ 1 B E S obtained using both the non-binning and binning modes wi th comparable collection times are shown in Figure 3.6. Chapter 3. Experimental Methods 71 1200 10 15 20 25 10 15 20 Binding Energy (eV) Figure 3.6: Non-binning (squares) and binning (circles) mode argon ( 3 p ) _ 1 b inding energy spectra. The non-binning data shown are the sum of B E S collected in a period of 17 hours at 27 out-of-plane azimuthal angles ranging from —30° to + 3 0 ° . The binning data are the sum of 21 B E S collected in 21 hours. The binning and non-binning mode raw data ( C W : coincidence window; B W : background window) are shown in the upper-left and lower-left panels, respectively. The B W data are of greater intensity than the C W data because of the greater width of the background window (as shown in Figure 3.4). This is accounted for in the background correction procedure (Section 3.4.3). The binning mode and non-binning mode background corrected B E S are shown in the right panel, normalized for height to allow direct shape comparison of the two spectra. The result of a least-squares fit of a pair of Gaussian functions to the ionization peak is indicated by the solid line. Chapter 3. Experimental Methods 72 3.4.1 Non-binning data collection mode In the non-binning mode, the incident electron beam energy is fixed at an appropriate value to allow monitoring of a binding energy range containing the ionization processes of interest. B i n d i n g energy spectra are collected sequentially for each of a set of out-of-plane azimuthal angles. D a t a are collected at each angle by the A D C computer card unt i l a preset number of pulses are detected by the movable analyser detector. These "singles" pulses (as opposed to coincident pairs of pulses, one from each detector) are monitored using a counter on the LabMaster computer card. Pr ior to stepping to the next angle, the data for the current angle are saved to computer disk and al l channels on the A D C card are reset to zero. W h e n the data collection routine has cycled through a ful l set of angles (constituting one scan), the scan number is incremented and the process repeated. A computer display of the collected data is updated following each angle change and is used by the operator to monitor the data collection procedure and to determine when sufficient data has been collected to allow the experiment to be stopped. The collected data for each angle consist of the number of coincident and non-coincident events (from the coincidence and background S C A windows, respectively) recorded in each A D C channel. The A D C channel numbers are converted by the acquisition software to electron binding energy using calibration data obtained v ia the procedure described in Sec-tion 3.3. The square response functions of each analyser combine to produce a triangular response function for the coincidence spectrum (evident in the lower left panel of Figure 3.6). A s a result of this non-uniform detection efficiency across the monitored binding energy range, the non-binning mode is not well suited to situations where multiple ionization pro-cesses at different binding energies are of interest, particularly if the relative intensities of the ionization processes are of interest. However, for the study of a single ionization process that is energetically well separated from neighbouring processes, the non-binning data collection Chapter 3. Experimental Methods 73 mode is the most efficient choice. This mode is often used when collecting diagnostic spectra using argon or helium and has been used in the study of acetone, reported i n Chapter 4. 3.4.2 Binning data collection mode The binning data collection mode is designed to circumvent the problem of the triangular response function of the energy dispersive spectrometer and to allow data collection over a binding energy range greater than that simultaneously monitored by the spectrometer [17,18]. This is done by dividing the energy range monitored by the spectrometer into many narrow energy ranges, or "energy bins." In the experiments reported in this thesis, each energy bin is ~ 0.36 eV wide and consists of the summed data from four consecutive A D C channels. W h e n collecting the experimental data, once the preset number of singles counts from the movable analyser detector are received by the counter on the LabMaster card, the A D C data are transferred to the appropriate energy bin in computer memory and al l A D C channels are reset to zero. The computer display is then updated and the incident electron beam energy is incremented by an energy step equal to the width of the energy bins. B y systematically shifting the monitored binding energy range in this manner, data at each binding energy are collected for an equal period of time at al l positions of the spectrometer's response function, resulting in equal instrumental sensitivity for a l l binding energies. Consequently, al l experimental data from one study are on the same relative intensity scale and only multipl icat ion by a single normalization factor is necessary to allow direct comparison with theory. After data have been collected at one azimuthal angle at a l l incident electron beam energies necessary to cover the desired binding energy range, the B E S data for that angle are saved to computer disk, the movable analyser is stepped to the next angle, the incident electron beam energy is reset to its in i t ia l value and B E S data for that angle are collected. A s is the case in the non-binning mode, following a complete cycle Chapter 3. Experimental Methods 74 through a ful l set of azimuthal angles (i.e., one scan), the scan number is incremented and the process repeated unt i l the precision of the collected data reaches an acceptable level. The binning data collection mode is well suited to the study of several ionization pro-cesses spread throughout a broad binding energy range, such as ionizations from a molecule's valence orbitals or some subset thereof. The uniform detection efficiency over the entire en-ergy range results in al l processes being on a common intensity scale. This is important when investigating the breakdown of the single particle model of ionization and when per-forming detailed assessments of theoretical models of the E M S momentum profiles. These advantages of the binning mode are tempered by a loss in collection efficiency in comparison to the non-binning mode. Because data for al l binding energies are collected at a l l points in the monitored energy range of the spectrometer, the incident electron beam energy must be varied over a range beginning below and ending above the binding energy range of interest. A t these extreme energies, only a fraction of the data collected are wi th in the desired binding energy range. This amounts to only a small proportion of the total data collected when the energy range being studied is large, but is more significant for smaller binding energy ranges wi th widths comparable to the simultaneously monitored energy range of the spectrometer. 3.4.3 Background correction The first step in the data analysis process is the correction of the B E S for false coincidences. A s discussed above in Section 3.2.2, some fraction of the events detected by the spectrometer w i t h a time correlation corresponding to a coincidence (as set wi th the coincidence ( C O I N C ) S C A in Figure 3.3) are the result of a constant background (in the time spectrum) of uncor-related electron pairs. This background is constant because al l t ime intervals between the detection of two electrons constituting an uncorrelated electron pair are equally likely. The magnitude of this background is determined by the collection of the non-coincident data Chapter 3. Experimental Methods 75 gated through the background ( B A C K ) S C A indicated in Figure 3.3. The true coincident signal (Nt) is then obtained from the difference between the coincident (JVC) and background (Nb) data, taking into account the ratio of the widths of the background and coincident windows (r) set by the respective S C A s : The width of the background data window is typically set larger than that of the coincident window to minimize A i V t . 3.4.4 Experimental angle and momentum profiles D a t a collection, using either the binning or non-binning mode and background correction as described above, results in a series of binding energy spectra recorded at different azimuthal angles between the two outgoing electrons (i.e., different positions of the movable analyser). The variation in intensity of an ionization peak in the B E S as a function of the out-of-plane azimuthal angles at which the B E S are collected is referred to as an experimental angle profile ( X A P ) . Experimental momentum profiles ( X M P s ) are obtained from the X A P s by conversion of the out-of-plane azimuthal angles <f> to electron momentum p using Equation (2.5). This is illustrated in Figure 3.7 for the A r 3p profile. The experimental angle and momentum profiles shown are from the same data sets used to generate the B E S shown in Figure 3.6. X A P s are obtained from the B E S data in one of two ways. In cases where the ionization peak of interest is fully resolved from adjacent peaks, it is sufficient to sum the experimental Nt = Nc- Nb/r. (3-1) The error in the true coincident signal is (3-2) Chapter 3. Experimental Methods 76 1400 cn •4—> c Z3 o O O O c 0 ~D O C ' o O r o\ Ar 3p o non-binning • binning i •V i — Azimuthal Angle (degrees) Momentum (a.u.) Figure 3.7: Non-binning (o) and binning (•) mode argon 3p angle (left) and momentum (right) profiles. The data sets were collected in 17 and 21 hours, respectively. The binning mode data has been scaled by 1.9 (the factor necessary to normalize the two data sets at their momentum profile maxima) to facilitate comparison of the two data sets. The solid line in the right panel is the A r 3p theoretical momentum profile from the C I ion-neutral overlap calculation of Davidson [124], wi th the experimental angular resolution (At? = 0.6°, Acfi = 1.2°) accounted for using the G W - P G method [48]. The T M P has been normalized to the experimental data at the profile maxima. Chapter 3. Experimental Methods 77 B E S data over an energy range encompassing the peak in question, as has been done to obtain the profiles shown i n Figure 3.7. Such a B E S sum w i l l yield one data point of the X A P for each out-of-plane azimuthal angle at which a B E S has been collected. This method is also used to obtain angle profiles corresponding to multiple ionization processes by summing B E S data over a broader energy range. In cases where it is not possible to fully resolve al l of the ionization processes of interest, a fitting procedure can be used to obtain individual angle and momentum profiles. This method has been used in al l of the studies presented in the subsequent chapters of this thesis. Gaussian functions are used to fit the various ionization peaks i n the B E S . The widths of the fitted functions are fixed at values obtained by convoluting the spectrometer resolution function (determined from measurements of the He ( I s ) - 1 ionization) wi th the Franck-Condon widths of the ionization processes determined, where available, from high resolution photoelectron spectroscopy data. The fitted peak positions are also fixed at val-ues obtained from photoelectron spectroscopy. These positions normally correspond to the vertical IPs, although in cases where the Franck-Condon envelope is highly asymmetric slight adjustments to the peak positions are sometimes necessary. A least-squares fit to the B E S is then performed, wi th only the peak areas allowed to vary. The distribution of fitted peak areas for a particular ionization process as a function of the out-of-plane azimuthal angles at which the B E S were collected is the experimental angle profile for that ionization process. In some cases, ionization processes are not sufficiently separated in energy to allow the de-termination of individual profiles by a fitting procedure. In these instances, single functions can be used to fit a B E S peak representing multiple ionization processes. The resulting angle and momentum profiles w i l l correspond to the sum of the profiles for the individual ionization processes. Experimental angle profiles provide a useful check of correct spectrometer operation. Chapter 3. Experimental Methods 78 Positive and negative out-of-plane azimuthal angles </> correspond to the same electron mo-mentum p (Equation (2.5)), so B E S collected at the same absolute out-of-plane azimuthal angles should be the same and X A P s should be symmetric about 0 = 0°, as is the case for the A r 3p profiles shown in Figure 3.7. In particular, mis-alignment of the electron analysers and the incident electron beam w i l l result in asymmetric angle profiles. Chapter 4 Acetone 4.1 Introduction Acetone (dimethyl ketone, ( C H 3 ) 2 C O ) has been chosen as the first molecule for study using the multichannel electron momentum spectrometer described in Chapter 3. Several consider-ations motivated this choice. W i t h 10 atoms and 32 electrons, acetone presents several of the challenges inherent i n performing E M S studies of larger molecules, such as a comparatively large number of valence orbitals resulting in a congested binding energy spectrum, some de-gree of conformational flexibility and increased computational costs. However, the physical properties of acetone (i.e., a l iquid having a high vapour pressure at room temperature) make E M S measurements relatively straightforward. Furthermore, theoretical momentum profiles obtained from large-basis-set H F and M R S D - C I calculations [54] have been published for the H O M O of acetone, thus allowing for evaluation of the T K S A using a molecule larger than those considered by Duffy et al . [20, 21] in their ini t ia l investigation of the application of D F T to E M S . Such comparisons between the T K S A and more-established E M S theoretical models (i.e., the P W I A and T H F A ) are necessary to gain some indication of whether or not the T K S A - D F T method can be applied with any confidence to larger molecules for which 79 Chapter 4. Acetone 80 high-level H F and C I methods become increasingly impractical . Previously published single-channel E M S data of the 5b 2 H O M O of acetone by Holle-bone et al . [54] further motivated the current study. The single-channel measurements were performed as part of an investigation of the effects of methyl substitution on the H O M O s of carbonyl compounds, specifically formaldehyde ( H 2 C O ) , acetaldehyde ( C H 3 C H O ) and acetone. The low sensitivity of the single-channel spectrometer, in combination wi th the low cross-section of the acetone ( 5 b 2 ) _ 1 ionization, resulted in an experimental momentum profile of poor precision [54]. This l imited the evaluation that could be performed of the high-level H F and M R S D - C I theoretical calculations mentioned above and presented in Ref-erence [54]. Nevertheless, there appeared to be a discrepancy between the 5b 2 X M P and all of the reported T M P s at electron momenta > 0.8 au. A remeasurement of this momen-t u m profile wi th greater precision is desirable to verify this discrepancy between theory and experiment. If this discrepancy is confirmed, the application of D F T v ia the T K S A to the 5b 2 X M P of acetone is of particular interest in order to investigate whether this discrepancy between theory and experiment can be eliminated. The ( 5 b 2 ) _ 1 ionization is energetically well-separated from the remainder of the outer-valence ionizations of acetone. However, ionizations from the eight remaining outer-valence orbitals occupy a relatively narrow binding energy range, as indicated by a He (II) P E S study [125]. The low count rates and 1.7 eV fwhm energy resolution of the single-channel spectrometer thus l imited the E M S study of Hollebone et al . [54] to the H O M O X M P . The improved sensitivity and energy resolution of the multichannel spectrometer used in the present work facilitates E M S measurements of the complete valence shell of acetone. Chapter 4. Acetone 81 4.2 Experimental details Valence-shell E M S measurements of acetone have been performed at a total energy of 1200 eV using the energy-dispersive multichannel spectrometer described i n Chapter 3. Three data sets were collected. The first consists of 6-60 eV binding energy spectra collected using the binning data collection mode (Section 3.4.2) at out-of-plane azimuthal angles ((f)) of 0° and 5°. These data are referred to subsequently as the long-range B E S data ( L R -B E S ) . A set of six shorter energy range binning-mode B E S were also recorded ( S R - B E S ) , covering the binding energy range 6-40 eV at cp = 0°, 2.5°, 7.5°, 10.5°, 14.5° and 20°. Final ly , in order to allow a precise description of the 5b 2 ( H O M O ) momentum profile of acetone, non-binning-mode data (non-bin) were collected at 26 out-of-plane azimuthal angles ranging from 0-30° . The incident electron beam energy during the non-binning data collection was set so as to centre the 5b 2 ionization peak (IP=9.8 eV from P E S [125]) in the monitored binding energy range. The acetone sample was of spectroscopic grade (> 99.9% pure) from B D H chemicals. Repeated freeze-pump-thaw cycles were performed using l iquid nitrogen to remove any gaseous impurities dissolved in the sample. The sample pressure was maintained at 1.0 x 10~ 5 torr during the data collection. Measurements of the A r ( 3 p ) _ 1 ionization were performed both before and after the acetone data collection to ensure proper operation of the spectrometer and to determine the experimental energy resolution, which was found to be 1.4 eV fwhm. Chapter 4. Acetone 4 . 3 Computational considerations 82 4.3.1 Molecular conformation and theoretical momentum profiles In the single-channel E M S study of the H O M O of acetone [54] and in the publication describ-ing the data discussed in this chapter [126], two different geometries were used in performing electronic structure calculations to obtain T M P s . The majority of the calculations used a zero-point average structure determined from microwave spectroscopy and electron diffrac-t ion data [127]. The orientations of the two methyl groups were rather arbitrari ly chosen to be "half-staggered," meaning that one methyl group was staggered wi th respect to the carbonyl group (i.e., wi th one hydrogen in the plane defined by the carbon and oxygen atoms and anti to the carbonyl group, resulting in an O C C H dihedral angle of 180°) and the other methyl group was eclipsed (i.e., wi th an O C C H dihedral angle involving the in-plane hydro-gen of 0°) [54,126]. In contrast, the M R S D - C I calculations (and therefore the H F calculations used as a basis for the CI calculations) used an M P 2 / 6 - 3 1 G * * optimized geometry [54] in which both methyl groups were eclipsed with respect to the carbonyl group. This confor-mation, referred to as "eclipsed," was found by Hollebone et al . to be the global m i n i m u m structure on the M P 2 / 6 - 3 1 G * * potential energy surface. Two other stationary points were also located, one corresponding to the half-staggered conformation discussed above and a second, referred to as "staggered," in which both methyl groups were staggered wi th re-spect to the carbonyl group. These two other conformations were, respectively, 1.2 and 3.6 mhartree higher in energy than the eclipsed conformation [54]. These three conformations of acetone are illustrated in Figure 4.1. In discussing their choices of geometry, Hollebone et al . noted that an E M S study of dimethyl ether [53] indicated that [54], "methyl group orientation has only a small effect on the resulting T M P s at this level of calculation." A comparison of the 2 0 4 - C G F basis set Chapter 4. Acetone 83 o o o I I  H H || H C H H C .•• C i\ i\ i\ I I I HH HH Tlfl H H H eclipsed half-staggered staggered Figure 4.1: Conformers of acetone [54] H F T M P s calculated using the M P 2 / 6 - 3 1 G * * eclipsed geometry and H F / 6 - 3 1 1 + + G * * T M P s calculated using the experimental half-staggered geometry (presented in F i g . 8 of Reference [126]) generally supports this statement. However, the use of more than one geometry for the theoretical calculations makes it impossible to determine to what extent the small observed differences are the result of differences in the basis set employed rather than the geometries used. In view of the above considerations and the increasing significance of conformational flexibility that may generally be expected in E M S studies of larger molecules, a more thor-ough investigation of the relationship between acetone conformation and the corresponding theoretical momentum profiles has been performed. Fu l l geometry optimizations have been performed at the M P 2 ( f u l l ) / 6 - 3 1 G * * level and the three stationary points reported by Holle-bone et al . [54] (i.e., the eclipsed, half-staggered and staggered conformations) have been identified. The nature of the stationary points was subsequently determined by performing frequency calculations at each of the stationary points. The computational results (obtained using GAUSSIAN94 [103]) are reported in Table 4.1 along with the zero-point average struc-tural parameters [127] mentioned above and used in Reference [54] and [126]. Examinat ion of the optimized structural parameters reported i n Table 4.1 indicates that the bond lengths and angles are relatively invariant wi th rotation of the methyl groups. The Chapter 4. Acetone 84 Table 4.1: Conformers of acetone. M P 2 ( f u l l ) / 6 - 3 1 G * * optimized bond distances (r), angles (a) and dihedral angles (d), conformer energies and other properties. Hj refers to the hydro-gen atom of each methyl group in the plane defined by the heavy atoms; H 0 refers to the out-of-plane hydrogens. In the case of the half-staggered conformation, where two numbers are given the first involves the atoms of the eclipsed methyl group and the second involves those of the staggered methyl group. Distances are in angstroms and angles in degrees. eclipsed half-staggered staggered experiment 3 . r C O 1.226 1.225 1.226 1.210 ± 0 . 0 0 4 r C C 1.511 1.509 1.516 1.516 1.517 ± 0 . 0 0 3 r C H , 1.085 1.085 1.086 1.085 1.091 ± 0 . 0 0 3 r C H c 1.089 .1.090 1.088 1.088 a O C C 121.8 122.1 120.7 120.3 a C C C 116.4 117.2 119.4 116.00 ± 0 . 2 5 a C C H i 109.6 109.8 112.8 113.2 a C C H 0 110.2 110.0 108.9 108.7 a H j C H 0 109.8 110.0 109.3 109.4 108.5 ± 0 . 5 a H 0 C H 0 107.2 107.2 107.5 107.3 dOCCR0 ± 1 2 0 . 9 ± 1 2 1 . 1 ± 5 8 . 5 ± 5 8 . 3 £ ( M P 2 ) (au) -192.589477 -192.588260 -192.585861 AE (k J /mol ) 0.0 3.195 9.494 symmetry C2v C s C2v # imag. freq. 0 1 2 dipole m o m . (D) 3.345 3.297 3.207 2.90 E x p e r i m e n t a l s t ructura l parameters are zero-point average values reported i n Reference [127] and determined from microwave spectroscopy and electron diffraction data. T h e experimental value of the dipole moment is f rom Reference [128], a microwave spectroscopy study of acetone. notable exceptions are the C C C angle and the C C H angle to the in-plane hydrogens, which increase in going from the eclipsed to the staggered conformation. This behaviour may be attributed to steric repulsion between the in-plane hydrogens in the staggered conformation. The optimized parameters are consistent with the experimental parameters of I i j ima [127] that are reproduced in Table 4.1. It should be noted that some differences between the optimized and experimental parameters are expected since the former are equil ibrium values while the latter are zero-point average values. Valence theoretical momentum profiles for the three conformations of acetone are shown in Figure 4.2. The T M P s have been obtained using the T H F A (Equation (2.11)) from the Chapter 4. Acetone 85 H F / 6 - 3 1 G * * molecular orbitals of the M P 2 / 6 - 3 1 G * * optimized conformers. Since the T M P s are compared only with each other and not with experimental data, no angular resolution folding (see Section 2.2.5) has been performed. Each panel of Figure 4.2 indicates the M O number of the T M P s shown as well as the orbital symmetry labels, w i t h the C 2 v labels (corresponding to the eclipsed and staggered conformations) printed above the C s labels (corresponding to the half-staggered conformation). 1 For al l of the valence orbitals of acetone, the T M P s of the three conformations are qualitatively the same and, in many instances, essentially identical. There are, however, several cases in which differences amongst T M P s of different conformations are evident. Not surprisingly, in those cases where differences are evident, the half-staggered-conformation T M P s are intermediate between the fixed- and staggered-conformation T M P s . Examinat ion of the coefficients of the M O s used to generate the T M P s in Figure 4.2 indicates that those M O s contributing to C - H a-bonding (MOs 15, 14, 12, 11, 10 and 9) have T M P s that vary wi th acetone conformation. In contrast, the T M P s for the remainder of the orbitals show very l i tt le variation wi th methyl-group orientation. These results indicate that, at least in the case of acetone, methyl-group orientation does not significantly affect the T M P s of orbitals that make litt le or no contribution to C - H bonding. However, in the case of those orbitals that do contribute to C - H bonding, methyl-group orientation should not be disregarded. The frequency calculations indicate that only the eclipsed conformation is an energy m i n i m u m . The half-staggered and staggered conformations are saddle points wi th , respec-tively, one and two imaginary frequencies corresponding to rotations of the methyl groups. Consequently, the M P 2 / 6 - 3 1 G * * optimized eclipsed conformation of acetone has been used for a l l further theoretical calculations reported in this chapter. It should be noted that the lowest vibrat ional frequency of this conformation, corresponding to opposing rotations of the 1 Note that, in Reference [126], it was stated that acetone has C s symmetry but the M O s were subsequently referred to by C 2 V symmetry labels. Chapter 4. Acetone 86 Figure 4.2: Effect of molecular conformation on acetone valence momentum profiles. The T M P s were calculated using the T H F A from H F / 6 - 3 l G * * / / M P 2 ( f u l l ) 6 - 3 l G * * molecular orbitals for the eclipsed (solid lines), half-staggered (long-dashed lines) and staggered (short-dashed lines) conformations. The molecular orbital number and C2v (eclipsed and staggered conformations) and C s (half-staggered conformation) symmetry labels are indicated in each panel. No angular resolution folding of the T M P s has been performed. Chapter 4. Acetone 87 methyl groups, is quite low (58.17 c m - 1 according to the above calculations). Consequently, appreciable rotation of the methyl groups w i l l occur at room temperature and the use of a single equil ibrium geometry for the T M P calculations may result in discrepancies w i t h the experimental momentum profiles of those orbitals contributing significantly to C - H bonding. 4.3.2 Choice of basis set Theoretical momentum profiles of the valence orbitals of acetone have been calculated using the T H F A (Equation (2.11)) and the T K S A (Equation (2.14)) and the results of single-point H F and L S D A - D F T calculations, respectively, performed using G A U S S I A N 9 4 [103]. The exchange-correlation functional proposed by Vosko, W i l k and Nusair [99] has been used for the L S D A calculations. A range of basis sets has been used in order to investigate the sensitivity of the T M P s to basis set quality. A l l of the basis sets used are described i n Section 2.3.4 wi th the exception of the 204 -CGF basis set taken from the single-channel E M S study of the acetone H O M O [54].2 This basis consists of an (18sl3p2d)/[6s7p2d] contraction for carbon and oxygen and a (10slp)/[5slp] contraction for hydrogen 3 and is based upon the highly converged basis sets of Partridge [129,130] augmented wi th polarization functions on al l atoms. A summary of the electronic structure calculations is given in Table 4.2. The corresponding 5b 2 T M P s of the acetone H O M O are shown in Figure 4.3. B o t h the H F and L S D A - D F T T M P s show a similar variation wi th basis set, as is also the case for the other calculated properties shown in Table 4.2. Only the 5b 2 T M P s are shown here as they exhibit the greatest variation with choice of basis set. Similar but less dramatic variations also occur for the other outer-valence T M P s of acetone. In contrast, the inner-valence T M P s are relatively insensitive to the choice of basis set. In the case of the 5b 2 T M P s , 2 T h i s basis set was referred to incorrectly as 196-GTO in Reference [54]. 3 T h e hydrogen s function contraction was described incorrectly as (10s)/[7s] in Reference [54]. Chapter 4. Acetone 88 Table 4.2: Hartree-Fock and L S D A density functional theory calculations of acetone. A l l calculations were performed using the M P 2 / 6 - 3 1 G * * optimized eclipsed conformation de-scribed in Table 4.1. The dipole moment of acetone has been determined experimentally to be 2.90 D [128]. K e y Basis set Hartree -Fock L S D A - D F T Total energy Dipole Total energy Dipole (hartree) moment (D) (hartree) moment (D) St S T O - 3 G -189.534399 1.945 -189.006897 1.817 4 g 4-31G -191.677185 3.623 -191.244868 3.047 6g 6-311G -191.918651 3.657 -191.498425 3.158 6p 6-311++G** -192.014838 3.592 -191.560744 3.213 tz aug-cc -pVTZ -192.035127 3.509 -191.581259 3.141 204 2 0 4 - C G F , Ref. [54] -192.043241 3.511 -191.593916 3.143 all of the calculations predict a two-peaked momentum profile. However, the minimal-basis-set S T O - 3 G calculations (st on Figure 4.3) differ significantly from the others in predicting that the peak at higher momentum is of appreciably greater intensity than that at lower momentum. A s the size and flexibility of the basis set used is increased, the magnitude of the low-momentum peak in the corresponding T M P increases and the positions of both peak maxima shift to lower momentum. This behaviour is typical of that observed in previous E M S studies (see Section 1.1.2 and References [9,35]). Small and medium-sized basis sets, particularly those lacking diffuse functions, tend to underestimate the intensity of valence T M P s at low momentum. This is a consequence of the approximately inverse relationship between position and momentum space and a reflection of the poor representation of the outermost (large-r) regions of the electron density by these basis sets. Of greater interest for the purposes of the present work is that the acetone T M P s obtained using the three largest basis sets considered here (6-311-f-t-G**, aug-cc -pVTZ and 204-CGF) are essentially indistinguishable, indicating that this property has converged and that nothing is gained by calculating acetone H F or D F T T M P s using the 2 0 4 - C G F or aug-cc -pVTZ (322-C G F ) basis sets rather than the considerably smaller 130-CGF 6-311++G** basis set. It Chapter 4. Acetone • • 89 Momentum (au) Figure 4.3: Dependence of the 5b 2 T M P of acetone on basis set. H F T M P s obtained using the T H F A are shown in the left panel and L S D A - D F T T M P s obtained using the T K S A are shown in the right panel. The key to the T M P labels is given in Table 4.2. No angular resolution folding of the T M P s has been performed. should be noted, however, that variations of other calculated properties of acetone do occur amongst these three basis sets, as indicated in Table 4.2. Interestingly, the 2 0 4 - C G F basis set [54] results in lower H F and L S D A - D F T total energies and very similar dipole moments when compared wi th the larger aug-cc-pVTZ basis set, suggesting that the former is generally a better choice for H F and D F T calculations of acetone. In view of the agreement amongst the 6-311++G**, aug-cc-pVTZ and 204-CGF valence T M P s of acetone and the respective sizes of the basis sets, the 6-311++G** basis set has been used in calculating T M P s for comparison wi th the experimental momentum profiles of acetone discussed in Section 4.5. Chapter 4. Acetone 90 4.4 Valence binding energy spectra Acetone has 32 electrons occupying four core orbitals and 12 valence orbitals. The indepen-dent particle ground-state electron configuration of the C 2 V symmetry conformations is ( l a 1 ) 2 ( 2 a 1 ) 2 ( l b 2 ) 2 ( 3 a 1 ) 2 ( 4 a 1 ) 2 ( 5 a 1 ) 2 ( 2 b 2 ) 2 ( 6 a 1 ) 2 c o r e inner-valence ( 3 b 2 ) 2 ( 7 a 1 ) 2 ( l b 1 ) 2 ( l a 2 ) 2 ( 8 a 1 ) 2 ( 4 b 2 ) 2 ( 2 b 1 ) 2 ( 5 b 2 ) 2 v v ' outer-valence The distinction between inner-valence and outer-valence orbitals is somewhat arbitrary. Those M O s which consist primari ly of non-bonding carbon or oxygen 2s character have been classified as inner-valence. B y analogy wi th atomic orbitals, momentum profiles having a m ax imum at zero momentum are conveniently referred to as s-type and those having a m i n i m u m at zero momentum and a maximum at some other momentum value are referred to as p-type. In the case of acetone, al l ionizations from a 2 , b i and b 2 symmetry orbitals w i l l result in p-type momentum profiles because of the nodal properties of these orbitals. These momentum profiles should have no intensity at zero momentum except for the small contribution resulting from the finite instrumental momentum resolution [48]. In contrast, ionizations from a.i symmetry orbitals may give rise to either s-type or p-type momentum profiles, depending upon the nature of the orbital in question. B i n d i n g energy spectra of acetone over the energy range 6-60 eV are shown i n Figure 4.4. The spectrum recorded at <f> = 0° is shown in the upper panel and that recorded at cj) = 5° is shown in the lower panel. The close energy spacing of the acetone ionization peaks, in conjunction wi th their Franck-Condon widths and the E M S energy resolution of 1.4 eV fwhm, results in most of the ionization peaks being only partial ly resolved. The notable exception is the peak arising from ionization of the 5b 2 H O M O , a predominantly non-bonding oxygen lone-pair orbital . The p-type nature of the 5b 2 momentum profile is Chapter 4. Acetone 91 400 Binding energy (eV) Figure 4.4: B i n d i n g energy spectra of acetone from 6-60 eV. The B E S (•) at <fi = 0° and 5°, recorded at a total energy of 1200 eV, are shown in the upper and lower panels, respectively. The dashed lines represent the results of a least-squares fit of Gaussian functions to the ionization peaks and the solid curves are the summed fits. The positions of the fitted peaks are indicated by vertical lines. The peaks 1-9 correspond to ionizations from the 5b2, 2b i , 4b2, 8a x + la2, 7ai , 3b 2 + l b i , 6ai , 2b2 and 5ai orbitals, respectively. Chapter 4. Acetone 92 Table 4.3: Measured and calculated valence ionization potentials of acetone. Calculated pole strengths are indicated in parentheses. A l l energies are in eV. BES Orbital E M S b P E S C Green's function 0 ' d peak a origin 1 5b 2 9.8 9.8 9.85(0.91) 2 2bi 12.6 12.6 12.65(0.90) 3 4b 2 13.4 ~ 13.4 13.45(0.93) 4 I 8ai 1 14.3 14.1 14.05(0.92) I l a 2 J ~ 14.4 14.40(0.93) 5 7ai 15.7 1 15.7 15.66(0.91) 6 i 3b 2 l 16.1 / 15.93(0.89) I l b i J ~ 16.0 16.08(0.92) 7 6ai 18.0 18.0 18.21(0.90) 8 2b 2 22.8 23.0 23.78(0.45) 24.76(0.37) 26.04(0.021) 9 5ai 24.6 24.6 25.42(0.13) 25.56(0.043) 25.60(0.13) 25.89(0.12) 26.06(0.27) 26.30(0.096) 4ai 29.36(0.024) 33.32(0.032) 33.51(0.052) 33.59(0.032) 33.63(0.044) 33.73(0.022) 33.79(0.021) 34.94(0.022) 35.30(0.023) 36.25(0.052) 36.44(0.036) 36.60(0.027) 36.71(0.038) 36.76(0.060) 36.94(0.026) 41.38(0.027) aPeak numbering corresponds to that used in Figure 4.4. bIonization peak energies used to fit the E M S binding energy spectra. The E M S absolute energy scale was determined by setting the energy of the ( 5 b 2 ) - 1 ionization peak to 9.8 eV. c From Reference [125]. d T h e outer-valence Green's function (OVGF) method was used for the outer-valence ionizations and the extended two-particle-hole Tamm-Dancoff approximation was used for the inner-valence ionizations. Basis set [9s5p/4s]/(4s2p/2s). Chapter 4. Acetone 93 evident by the increased intensity of this peak (number 1 in Figure 4.4) in the 4> = 5° B E S in comparison to that in the </> = 0° spectrum. The absolute energy scale of the B E S has been fixed by setting the position of the ( 5 b 2 ) _ 1 ionization peak to the P E S vertical ionization potential [125] of 9.8 eV. Also evident in Figure 4.4 is the s-type nature of the ( 6 a i ) _ 1 and ( 5 a i ) - 1 ionization peaks (peaks 7 and 9, respectively), the only two peaks in the valence B E S that are of significantly greater intensity at 0 = 0° than at <f> = 5°. M O calculations indicate that the 6ai and 5ai orbitals are of primari ly non-bonding oxygen 2s and methyl carbon 2s (in-phase) character, respectively. In order to allow separation of individual ionization processes and generation of exper-imental momentum profiles, the B E S have been fitted with a series of Gaussian functions indicated by the dashed lines in Figure 4.4. The sums of the fitted peaks are indicated by solid lines. The relative positions of the fitted peaks (indicated by vertical lines) have been taken from the He (II) P E S study of Bier i et al . [125]. The peak positions used to fit the B E S data and the P E S IPs are shown in Table 4.3. The widths used for the fitted peaks are a convolution of the Franck-Condon widths estimated from the P E S data [125] and the E M S energy resolution function. Only the peak amplitudes were allowed to vary during the least-squares fitting procedure. A single peak at 14.3 eV (peak 4) has been used to fit the closely spaced ionizations from the 8ai and l a 2 M O s , which were assigned vertical IPs of 14.1 and 14.4 eV, respectively, by Bier i et al . The close energy spacing and Franck-Condon widths of these ionization peaks makes them nearly unresolvable in the P E S , which was ob-tained wi th an energy resolution nearly two orders of magnitude better than that of the E M S B E S data. A similar situation exists for the ionizations from the 7ai , 3b 2 and l b i orbitals, which appear in the P E S as a single broad asymmetric band located at approximately 16 eV. Despite the paucity of structure on this ionization band, B ier i et al . assigned IPs of 15.7 eV to the ( 7 a i ) _ 1 and ( 3 b 2 ) _ 1 processes and ~ 16.0 eV to the ( l b i ) - 1 ionization. In the Chapter 4. Acetone 94 current work, two peaks (numbers 5 and 6) located at 15.7 and 16.1 eV have been used to fit this region of the B E S . It should be noted that the P E S IP assignment of Reference [125], which was based primari ly on a Green's function calculation (shown in Table 4.3), places the ( l b i ) - 1 ionization at higher energy than the ( 7 a r ) _ 1 and ( 3 b 2 ) - 1 ionizations. This differs from the energetic ordering of the acetone H F M O s shown in Figure 4.2 and used to gener-ate the ground-state electron configuration given above and likely represents a breakdown of Koopmans ' theorem. The small adjustment of the ( 2 b 2 ) _ 1 peak (peak 8) to 22.8 eV in the current work from the vertical IP of 23.0 eV reported by Bier i et al . [125] was necessary to obtain a good fit to the band extending from 20 to 27 eV in the B E S and appears reasonable from an examination of the P E S . In both the <f> = 0° and 4> = 5° B E S of Figure 4.4, there is experimentally observed intensity in the 20-22 eV binding energy range that is not accounted for by the least-squares fit. This could simply be the result of a poor fit to the ( 6 a ! ) - 1 and ( 2 b 2 ) - 1 peaks that border this energy range. However, the energies and widths used to fit these peaks appear reasonable upon examination of the P E S data of Bier i et al . [125]. There does appear to be a very weak feature at « 20 eV in the P E S ionization spectrum [125]. However, no mention of this feature was made in the P E S study and it is not clear that it is anything other than noise in the P E S data. This "extra" intensity in the B E S is examined further in Section 4.5.2 as part of a discussion of the inner-valence momentum profiles. The binding energy region > 27 eV, containing ionizations from the 4ai M O and possibly satellite peaks resulting from ionizations from other valence orbitals of acetone, has not been fitted because of the comparatively structureless nature of the E M S data in this energy range and the lack of P E S data to serve as a guide in the fitt ing procedure. The considerable ionization intensity evident over this broad energy range (27-60 eV) indicates the presence of significant many-body effects in the inner-valence ionization of acetone. Al though most Chapter 4. Acetone 95 of this intensity is contained within a broad peak between approximately 30 and 38 eV, the ionization strength is further spread over a large range of binding energies up to the l imit of the current data at 60 eV with no apparent peak structure. Comparison of the two B E S i n Figure 4.4 indicates that the intensity in the 27-60 eV region is greater at <f> = 0° than at 4> = 5°, suggesting that the bulk of the intensity in this energy region may be assigned to the s-type 4a i , 5ai and/or 6ai ionization manifolds. Further discussion of the assignment of this inner-valence ionization intensity is given in Section 4.5.2 below. The Green's function calculation of Reference [125], the results of which are reproduced in Table 4.3, is consistent with the B E S in that considerable split t ing of the ionization inten-sity of the three inner-most valence orbitals is predicted. A more comprehensive comparison of the Green's function (GF) calculation and the current E M S data is shown in Figure 4.5. The B E S data of Figure 4.4 are compared with calculated B E S . The calculated spectra have been obtained by convoluting the pole energies and pole strengths from the Green's function calculation wi th the E M S experimental energy resolution function and the same Franck-Condon widths used for the B E S fits shown in Figure 4.4. The momentum (and hence <f>) dependence of each of the B E S has been accounted for by mult ip ly ing each of the Green's function pole strengths by the intensity of the corresponding H F / 2 0 4 - C G F theoret-ical momentum profile [54] at the appropriate <f> value. The widths of the 4ai poles in the synthesized spectra have been chosen to be the same as that of the 5ai peak since there are no P E S data available for this ionization process. The calculated spectra have been scaled by a single factor in order to place them on the same relative intensity scale as the experimental data. Agreement between experiment and theory is good in the outer-valence region of the spectra shown in Figure 4.5, particularly in terms of the positions of the ionization peaks. The experimentally measured intensity at the lower momentum ( 0 = 0°) is underestimated Chapter 4. Acetone 96 400 10 20 30 40 50 60 Binding energy (eV) Figure 4.5: Experimental and calculated valence shell binding energy spectra of acetone at 0 = 0° and 0 = 5°. The experimental data (•) are the L R - B E S data shown i n Figure 4.4. The calculated spectra (solid lines) are based upon the ionization energies and intensities (indicated by vertical lines) given by the Green's function (GF) calculation of Reference [125] and reproduced in Table 4.3; the angular dependence has been accounted for using T M P s obtained from a H F / 2 0 4 - C G F calculation [54] of acetone. The same peak widths used to fit the spectra of Figure 4.4 have been incorporated into the calculated spectra. Chapter 4. Acetone 97 slightly and this is consistent with the comparison of the experimental and theoretical mo-mentum profiles presented in Section 4.5.1 (i.e., it is attributable to the H F T M P s used to account for the <j> dependence of the spectra and not to the G F calculation). The relative intensities of the ( 2 b 2 ) _ 1 and ( 5 a i ) - 1 main peaks are well described by the calculations. However, the calculated ionization energies of the ( 2 b 2 ) _ 1 , ( 5 a i ) - 1 and ( 4 a i ) _ 1 main peaks are shifted by 1-2 eV in comparison to the experimental spectra. Furthermore, the ex-perimentally observed intensity in the 30-60 eV energy region is underestimated by the theoretical calculations. This discrepancy is most reasonably attributed to the inadequacy of the Green's function calculation [125] because the H F T M P s give a fair description of the shape of the 2b 2 and 5a x experimental momentum profiles of acetone, as discussed in Section 4.5.2. 4.5 Valence momentum profiles Experimental momentum profiles have been obtained for the valence shell of acetone from the results of least-squares fits of the three data sets described in Section 4.2 (i.e., L R - B E S , S R - B E S and non-bin). The fits to the L R - B E S data are shown in Figure 4.4 and discussed in the previous section. The same procedure was used to fit the other binning-mode data set, the S R - B E S data extending from 6-40 eV. The positions and widths were fixed at the values used to fit the L R - B E S and only the peak amplitudes were allowed to vary. A n X M P for each fitted peak was obtained using the procedure described in Section 3.4.4. The ratio of the intensity scales of the S R - B E S and L R - B E S data was determined by performing a least-squares fit between the two <f> — 0° B E S over their common energy range of 6-40 eV. The S R - B E S and L R - B E S X M P s were then placed on a common intensity scale by scaling the L R - B E S X M P s by the ratio determined from this fit. The ionization peaks of binning-Chapter 4. Acetone 98 mode binding energy spectra are collected on the same relative intensity scale as a result of the linear background of the binning data collection mode and the fact that the B E S data are collected sequentially for many data collection cycles. Consequently, following the above-mentioned scaling of the L R - B E S X M P s , al l of the acetone binning-mode X M P s are on the same relative intensity scale. In the subsequent figures in this chapter, the L R - B E S data are indicated by filled squares (•) and the S R - B E S data by filled circles (•). Obtaining X M P s from the non-binning mode data is a more complicated undertaking because of the non-linear detection efficiency in this data collection mode (refer to Section 3.4.1 for more details). In the case of acetone, the large energy spacing between the (5b 2 )~ 1 and ( 2 b i ) - 1 ionizations allows the H O M O (5b 2) X M P to be obtained by summing the non-binning B E S data over an appropriate binding energy range, as was done in the original publication [126] of the data discussed in this chapter. The disadvantage to this approach is that it neglects the considerable additional data of the other outer valence ionizations that is collected as a result of the energy-dispersive nature of the spectrometer. A n alternative approach has been employed in this thesis which does not suffer from this disadvantage. A s was done for the binning-mode data, X M P s have been obtained from the non-binning data by performing a least-squares fit to the 26 non-binning mode B E S using the same peak positions and widths, wi th the exception that the fit only included the outer-valence ionizations because of the shorter binding energy range of the non-binning data. To account for the distortion of the non-binning B E S caused by the triangular detection efficiency, prior to performing the least-squares fit the B E S were mult ipl ied by the inverse of the detection efficiency function, which was determined by a least-squares fit of two straight lines to the background data (Section 3.4.3). The resulting non-binning X M P s are indicated by inverted triangles (v) in the subsequent figures in this chapter. Each non-binning X M P has been independently scaled to the corresponding binning-mode X M P using the </> = 0° and <f> = 20° Chapter 4. Acetone 99 data points common to both the binning and non-binning data. Theoretical momentum profiles of the valence shell of acetone have been obtained for comparison w i t h the X M P s from single-point H F / 6 - 3 1 1 + + G * * and D F T / 6 - 3 1 1 + + G * * cal-culations using the M P 2 / 6 - 3 1 G * * optimized eclipsed conformation described in Section 4.3.1. B o t h the H F and D F T calculations were performed using GAUSSIAN94 [103]. In the case of the D F T calculations, a number of exchange-correlation functionals (described in Section 2.3.3) have been used to investigate the sensitivity of the T M P s to the functional used for the electronic structure calculations. The T M P s have been obtained using the T H F A and T K S A wi th the results of the H F and D F T calculations, respectively. In the case of the 5b 2 and 2bi momentum profiles, the experimental data are also compared w i t h the corre-sponding T M P s calculated by Y . Wang and E . R. Davidson using Equation (2.10) from the neutral and ion M R S D - C I wavefunctions of acetone. The 5b 2 T M P was reported originally in the single-channel E M S study of acetone [54]. The CI calculations are based upon the H F / 2 0 4 - C G F wavefunction [54] discussed in Section 4.3.2 above. The effects on the T M P s of the finite spectrometer acceptance angles (i.e., the momentum resolution) have been folded into the calculated profiles using the G W - P G method [48]. The calculations compared to the X M P s are summarized in Table 4.4. The experimental and theoretical momentum profiles have been placed on a common intensity scale by normalizing the sum of the 2b i , 4b 2 and 8ai + l a 2 experimental angle profiles (fitted peaks 2-4) to the corresponding B L Y P - D F T theoretical angle profile ( T A P ) , as shown in Figure 4.6. This comparison is done using an angle rather than momentum scale because of the binding-energy dependence of the conversion between <j> and p (Equation (2.5)). These outer-valence momentum profiles were chosen for the normalization because, as is generally the case for outer-valence ionizations, their pole strengths are predicted by the Green's function calculations [125] (see Table 4.3) to be high and approximately equal. Chapter 4. Acetone 100 Table 4.4: Calculations used to generate T M P s of acetone. A l l calculations were performed using the M P 2 / 6 - 3 1 G * * optimized eclipsed conformation described in Table 4.1 and the 6-311++G** basis set, except for the M R S D - C I calculation [54] which uses the 2 0 4 - C G F basis set described i n Section 4.3.2. Method Total energy Dipole moment (hartree) (D) H F -192.014838 3.592 D F T L S D A a -191.560744 3.213 B L Y P b -193.138579 3.114 B P 8 6 C -193.210978 3.110 B 3 L Y P d -193.217830 3.230 M R S D - C P -192.629 3.29 experiment 2.90 f a T h e Vosko, W i l k and Nusair [99] local exchange-correlation functional was used. b T h e Becke exchange [105] and Lee, Yang and Parr correlation [107] functionals were used. c T h e Becke exchange [105] and Perdew correlation [106] functionals were used. d The B 3 L Y P functional is a modification of the hybrid functional proposed by Becke [108] and incorporating the exact exchange energy, wi th the Lee, Yang and Parr [107] correlation potential replacing that of Perdew and Wang [109]. e Reference [54]. fReference [128]. In comparing the experimental and calculated dipole moments, it should be remembered that the calculations ignore the effects of molecular vibrations and rotations. Chapter 4. Acetone 101 4 0 3 0 c C D Isda, blyj , b 3 l y p - b p 8 6 2b1+4^+88^182 hf • S R - B E S p k . 2 - 4 • L R - B E S p k . 2 - 4 — D F T / 6 - 3 1 1 + + G ** H F / 6 - 3 1 1 + + G ** C D > CD DC 2 0 1 0 Q I I I I I I I 0 5 1 0 1 5 2 0 2 5 3 0 Azimuthal angle (degrees) Figure 4.6: Summed 2bi , 4b 2 and 8ai + l a 2 experimental and theoretical angle profiles of acetone. The experimental profiles are the the sum of the profiles of fitted peaks 2-4 (see Figure 4.4) from the S R - B E S (•) and L R - B E S (•) data sets and the theoretical profiles are the sums of the corresponding individual profiles. The experimental angular resolution has been accounted for i n al l theoretical profiles using the G W - P G method [48]. The experimental data have been normalized to the B L Y P theoretical profile. Chapter 4. Acetone 102 The 5b 2 profile was not included in this sum because of the disagreement between the theoretical and experimental profiles, as shown in Figure 4.7, below. In performing the normalization, least-squares fits were performed between the experimental profile and each of the theoretical profiles shown in Figure 4.6. The best agreement between experiment and theory was obtained for the fit to the D F T - B L Y P profile. Consequently, this normalization factor has been used on the experimental profiles prior to al l comparisons of experiment and theory in the remainder of this chapter. Note that the D F T profiles are a l l of similar shape and intensity; consequently, the normalization factor would differ by less than 2% if either the L S D A or B P 8 6 profiles (the most and least intense, respectively, at the profile maximum) were used instead of the B L Y P profile. In contrast, the H F profile, although of similar intensity to the D F T profiles, is quite different in shape from the experimental profile. In particular, the peak maximum of the H F profile is at a greater momentum than is observed experimentally. 4.5.1 Outer-valence momentum profiles The 5b 2 experimental and theoretical momentum profiles are shown in Figure 4.7. The 5b 2 M O is primari ly a non-bonding oxygen p orbital . However, there is also significant C - C cr-bonding character and some C - H c-bonding character, as can be seen from the electron density plot in the inset of Figure 4.7. A s discussed by Hollebone et al . [54], this addit ional orbital density on the carbon and hydrogen atoms and the accompanying nodal surfaces leads to the observed double-lobed form of the 5b 2 momentum profile. The current experimental data provides a considerable improvement in precision over that of the previous single-channel study [54] and in particular provides a much better characterization of the position and intensity of the high-momentum lobe of the X M P . A l l of the theoretical calculations shown in Figure 4.7 predict a similar two-lobed shape Chapter 4. Acetone 103 7 0 1 2 3 Momentum (au) Figure 4.7: Experimental and theoretical 5b 2 momentum profiles of acetone. The X M P s are of fitted peak 1 (IP=9.8 eV, see Figure 4.4) from the S R - B E S (•), L R - B E S (•) and non-bin (V) data sets. The T M P s have been determined from the theoretical calculations listed in Table 4.4. The M R S D - C I T M P is that of Wang and Davidson from Reference [54]. The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. A contour plot of the electron density of the H F / 6 - 3 1 1 + + G * * 5b 2 M O of acetone in the plane containing the heavy atoms is shown in the inset. The contour lines represent densities of 0.0001, 0.0003, 0.001, 0.003, 0.01, 0.03, 0.1 and 0.3 au. Chapter 4. Acetone 104 for the T M P but differ in their predictions of the relative intensity of the two lobes and the momentum position of the maximum of the high-momentum lobe. The T M P s are i n good agreement wi th the experimental data in terms of the position and shape of the low-momentum lobe, which peaks at approximately 0.3 au. However, for a l l calculations consid-ered here, there is considerable disagreement between theory and experiment at momenta greater than 1 au. The T M P s consistently overestimate the intensity of the second lobe of the 5b 2 profile. This incorrect prediction of the relative intensities of the two lobes is worst for the H F and CI T M P s and slightly improved for the D F T T M P s , part icularly the one obtained from the B L Y P - D F T calculation. However, it is important to note that the differ-ences between the various T M P s are relatively small when compared wi th the considerable discrepancy between theory and experiment. The similarity of the H F , M R S D - C I and D F T T M P s suggests that this discrepancy is not the result of inadequate accounting of electron correlation and relaxation effects, which were the cause of disagreements between theory and experiment i n several previous E M S studies of other molecules (see Section 1.1.2). Likewise, the examinations of the basis-set and conformational dependencies of the acetone T M P s , performed in Section 4.3.2 and 4.3.1 above, would appear to rule out both of these as plausible explanations for the observed disagreement between theory and experiment. It should also be noted that the discrep-ancies at higher momentum are in the contrary direction to that typical ly observed for distorted wave effects in the experimental data [5,7]. However, in this regard, recent E M S results [69,131] suggest that distortion of the incoming and outgoing electrons may impact significantly upon experimental momentum profiles resulting from ionization from M O s re-sembling atomic d orbitals. In contrast to the generally accepted view [7] that distortion effects are only significant at high momentum, where they typically result i n an increase in the observed experimental cross-section, a study of the momentum profiles of the H O M O s Chapter 4. Acetone 105 of several transition-metal carbonyls and the corresponding metal atoms [69] indicates ap-preciable distortion effects at low momentum. Determination of whether this explains the discrepancies observed in the present work for ionization from the 5b 2 M O of acetone (which bears a qualitative resemblance to an atomic d orbital , see inset of Figure 4.7) must await further investigation. Work is currently underway [131] to investigate the role of distor-t ion i n the T M P s of molecules. However, such studies are particularly challenging because (as mentioned in Section 2.2.1) there is at present no tractable computational method for performing distorted-wave calculations on molecules. The 2bi experimental and theoretical momentum profiles are shown in Figure 4.8. This orbital is responsible primari ly for C - 0 7r-bonding, but there is also significant a-bonding between the methyl carbons and the out-of-plane hydrogens. The decreased precision of this X M P in comparison to that of the H O M O (5b 2) reflects the relatively close proximity of the (2b ! ) " 1 ionization peak (IP=12.6 eV) to the ( 4 b 2 ) - 1 ionization peak (IP=13.4 eV) in contrast to the larger energetic spacing between the ( 5 b 2 ) - 1 and ( 2 b i ) _ 1 ionizations. A s expected from symmetry, this momentum profile is p-type. A l l of the calculations result in very similar 2bi momentum profiles, indicating that the electron correlation and relaxation effects included in the CI wavefunctions are not significant for this momentum profile. The T M P s are in good agreement wi th the X M P in terms of the position of the profile maximum (PMAX), which occurs at « 0.9 au, but underestimate the experimental intensity below « 1 au. This could be due, in part, to errors in choosing the scale factor used to place theory and experiment on the same intensity scale (Figure 4.6). However, even if the X M P was rescaled to agree wi th the T M P s at pMAX, the experimental data would st i l l be higher than theory near zero momentum, where the T M P s drop to essentially zero intensity. Al though the reason for this discrepancy is not clear, it is worth noting two things: the low momentum region, which corresponds approximately to the outermost spatial regions of the orbital , tends to Chapter 4. Acetone 106 10 Momentum (au) Figure 4.8: Experimental and theoretical 2bi momentum profiles of acetone. The X M P s are of fitted peak 2 (IP=12.6 eV, see Figure 4.4) from the S R - B E S (•), L R - B E S (•) and non-bin (v) data sets. The T M P s have been determined from the theoretical calculations listed in Table 4.4. The M R S D - C I T M P is that of Wang and Davidson from Reference [54]. The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. Chapter 4. Acetone 107 be the region most sensitive to inadequacies in the theoretical calculations (see Section 1.1.2 and References [9,35]), and the 2bi T M P is amongst those most strongly affected by the orientations of the methyl groups (Figure 4.2)—suggesting that the neglect of vibrat ional effects when calculating the T M P s may be contributing to the observed discrepancy. The 4b 2 X M P , wi th an IP of 13.4 eV, is compared with the corresponding T M P s in Figure 4.9. This orbital consists primari ly of C to in-plane H and C - C cr-bonding. A l l of the T M P s shown in Figure 4.9 predict a two-lobed momentum profile. The D F T profiles are in close agreement at high momentum (above ~ 0.8 au), wi th somewhat greater differ-ences at low momentum. The H F T M P differs from the D F T profiles i n predicting that the high-momentum lobe is of appreciably greater intensity than the low-momentum lobe. Unfortunately, the scatter in the 4b 2 X M P precludes comment on the relative quality of the T M P s . If just the binning mode data are considered (• and •), a significant disagree-ment wi th theory is evident for the three high-momentum data points. In contrast, the non-binning mode data (v) are in good agreement with the T M P s at high momentum (at least above 1.2 au) but disagree considerably at intermediate momenta. This is most likely because of difficulties in fitt ing the ( 4 b 2 ) _ 1 ionization peak as a result of the close proximity of the much more intense ( 8 a i ) _ 1 ionization (see Figure 4.10, below), which from P E S has an IP of 14.1 eV [125]. This explains the two particularly high data points in the non-bin 4b 2 X M P near 0.5 au, which correspond to low points in the 8a x + l a 2 X M P . The X M P of fitted peak 4 at a binding energy of 14.3 eV is shown in Figure 4.10 and compared wi th the sum of 8ai and l a 2 T M P s . M O calculations indicate that the 8ai orbital is responsible primari ly for C - 0 cr-bonding and the l a 2 M O is almost solely involved in bonding between the methyl carbons and out-of-plane hydrogens. The T M P s shown in Figure 4.10 are very similar; the various D F T T M P s differ only in the predicted intensity at P M A X and, i n comparison to the D F T T M P s , the H F T M P is shifted slightly to higher Chapter 4. Acetone 108 Momentum (au) Figure 4.9: Experimental and theoretical 4b 2 momentum profiles of acetone. The X M P s are of fitted peak 3 (IP=13.4 eV, see Figure 4.4) from the S R - B E S (•), L R - B E S (•) and non-bin (v) data sets. The T M P s have been determined from the theoretical calculations listed in Table 4.4. The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. Chapter 4. Acetone 109 * 0 1 2 3 Momentum (au) Figure 4.10: Experimental and theoretical 8a! + l a 2 momentum profiles of acetone. The X M P s are of fitted peak 4 (IP=14.3 eV, see Figure 4.4) from the S R - B E S (•), L R - B E S (•) and non-bin (v) data sets. The T M P s are the sum of the 8ai and l a 2 T M P s determined from the theoretical calculations listed in Table 4.4. The individual B L Y P - D F T 8ai and l a 2 T M P s are also shown. The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. Chapter 4. Acetone 110 momentum. W i t h the exception of two data points near 0.5 au, the agreement between experiment and theory is very good. A s discussed in the preceding paragraph, the two low points i n the non-bin 8ai + l a 2 X M P correspond to high points in the 4b 2 X M P , indicating a failure to fully deconvolute these closely spaced ionization peaks in the fitting procedure. The experimental and theoretical 7a x + 3b 2 + l b i momentum profiles of acetone are shown in Figure 4.11. A s is discussed in Section 4.4 above, two peaks (numbers 5 and 6) have been used to fit the binding energy region corresponding to ionization from the 7ai , 3b 2 and l b i orbitals. This was found to be necessary to obtain a good fit to the B E S data. The very close energy spacing (0.4 eV) of these two fitted peaks makes it unlikely that peaks 5 and 6 w i l l individual ly correspond to ionizations from different M O s . Consequently, only their sum is considered in Figure 4.11. The individual T M P s of these three orbitals are a l l p-type and consequently sum to give a p-type profile. The calculations al l predict essentially the same profile shape and position of p M A X wi th only small differences in intensity at £ W x -The X M P is of similar shape to the T M P s , but of considerably less intensity. This represents a breakdown of the single-particle model of ionization and indicates that the spectroscopic factors (see Section 2.2.3) of one or more of these ionization peaks are significantly less than unity (or more correctly, less than those of the outer-valence ionizations used to normalize theory and experiment in Figure 4.6). This differs from the results of the Green's func-tion calculation [125] shown in Table 4.3, which predict that the pole strengths of al l of the outer-valence ionizations are approximately equal. A scale factor of 0.67 is necessary to bring the B L Y P - D F T T M P into agreement wi th the X M P in Figure 4.11. It is not possible to comment further regarding the individual spectroscopic factors of the ( 7 a i ) _ 1 , ( 3 b 2 ) _ 1 and ( l b i ) - 1 ionization processes because of their similar ionization potentials and momentum profiles. Chapter 4. Acetone 111 35 Momentum (au) Figure 4.11: Experimental and theoretical 7a x + 3b 2 + l b i momentum profiles of acetone. The X M P s are the sums of fitted peaks 5 and 6 (IPs=15.7 and 16.1 eV, see Figure 4.4) from the S R - B E S (•), L R - B E S (•) and non-bin (v) data sets. The T M P s are the sum of the 7 a l 5 3b 2 and l b i T M P s determined from the theoretical calculations listed i n Table 4.4. The short-dashed line is the B L Y P - D F T T M P scaled by 0.67 to better match the experimental data. The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. Chapter 4. Acetone 112 4.5.2 Inner-valence momentum profiles The 6a i , 2b 2 , 5ai and 4ai inner-valence orbitals of acetone are essentially non-bonding or-bitals resembling atomic 2s orbitals of the oxygen and carbon atoms. The experimental and theoretical momentum profiles corresponding to ionization from the 6a x M O at 18.0 eV are shown i n Figure 4.12. A s was clear from the binding energy spectra in Figure 4.4, this is a strongly s-type momentum profile that drops off quickly in intensity wi th increasing mo-mentum. The largest contribution to the 6ai M O comes from 2s density on the carbonyl carbon. However, there are also significant contributions, of opposite phase to that from the carbonyl carbon, from 2s density on the methyl carbons and oxygen. The opposite phases of the carbonyl carbon contribution and those from the remaining heavy atoms results in a nodal surface surrounding the carbonyl carbon and produces the m i n i m u m i n the momen-t u m profile near 0.5 au. A l l of the 6ai T M P s shown in Figure 4.12 are in good agreement wi th one another. In contrast, the binning and non-binning X M P s are of somewhat different shape, particularly in the momentum range 0.5-1.0 au. The non-binning X M P has been obtained from the very edge of the binding energy range monitored during the non-binning mode data collection. The detection efficiency at the edge of the monitored binding energy range is very low and consequently the non-binning 6ai X M P is of questionable reliability. Therefore, the comparison between theory and experiment for the 6ai momentum profiles w i l l be performed using the binning-mode experimental data only. The 6a x X M P is of similar shape to the T M P s shown in Figure 4.12, but of lesser intensity. This is particularly evident near zero momentum. The most likely explanation for this is that the spectroscopic factor for this ionization is less than one. If the B L Y P - D F T T M P is scaled by 0.86 (determined by a fit of the B L Y P - D F T T M P shape to the X M P ) reasonable agreement in terms of both shape and intensity is obtained, although the two data points near 0.5 au are somewhat higher than the T M P . Chapter 4. Acetone 113 40 Momentum (au) Figure 4.12: Experimental and theoretical 6ai momentum profiles of acetone. The X M P s are of fitted peak 7 (IP=18.0 eV, see Figure 4.4) from the S R - B E S (•), L R - B E S (•) and non-bin (v) data sets. The T M P s have been determined from the theoretical calculations listed in Table 4.4. The short-dashed line is the B L Y P - D F T T M P scaled by 0.86 to better match the binning-mode (SR-BES and L R - B E S ) data. The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. Chapter 4. Acetone 114 30 Momentum (au) Figure 4.13: Experimental and theoretical 2b 2 momentum profiles of acetone. The X M P s are of fitted peak 8 (IP=22.8 eV, see Figure 4.4) from the S R - B E S (•) and L R - B E S (•) data sets. The T M P s have been determined from the theoretical calculations listed in Table 4.4. The short-dashed line is the B L Y P - D F T T M P scaled by 0.70 to better match the experimental data. The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. The inset shows the momentum profile of the B E S fit residual summed from 19.3-21.5 eV. Chapter 4. Acetone 115 The experimental and theoretical 2b 2 momentum profiles of acetone are shown in Figure 4.13. The 2s intensity of the 2b 2 orbital is centred on the methyl carbons and is of opposite phase, resulting in the p-type momentum profile seen in Figure 4.13. A s was the case for the 6ai profiles, the 2b 2 T M P s are al l essentially the same and of significantly greater intensity than the X M P . A fit of the B L Y P - D F T T M P to the X M P results in a scale factor of 0.70 and the rescaled T M P shown in Figure 4.13. The agreement between the rescaled T M P and the X M P is poor. This is most likely at least partly the result of l imitations of the fitting procedure resulting in contributions from the neighbouring and strongly s-type 6ai and/or 5ai momentum profiles. If this is the case, it would suggest that the spectroscopic factor for the 2b 2 ionization at 22.8 eV is even less than 0.7. There is additional intensity observed in the B E S between 20 and 22 e V that is not accounted for by the least-squares fit, as was mentioned in Section 4.4. The momentum profile of the fit residual in this binding energy region is shown in the inset of Figure 4.13. The magnitude of the intensity unaccounted for in the B E S fit is clearly small i n comparison to that of the adjacent ( 6 a i ) - 1 and ( 2 b 2 ) _ 1 peaks. The shape of the resulting X M P may be classified as p-type, but it differs from all of the valence T M P s of acetone. The considerable intensity at low momentum suggests that some of this intensity is from the s-type ( 6 a i ) _ 1 ionization. The remaining p-type intensity could be from the ( 2 b 2 ) _ 1 ionization at 22.8 eV. However, this seems unlikely considering the 1.4 eV fwhm experimental energy resolution. A more likely explanation is that there is a low-intensity p-type ionization pole near 20 eV. Such a pole could belong to the 7ax, 3b 2 , l b i or 2b 2 ionization manifolds, al l of which have the correct symmetry and are missing some intensity in their main ionization peaks. The final two inner-valence profiles of acetone are shown in Figures 4.14 and 4.15. A n an-gle rather than momentum scale has been used for the 4ai profiles because the experimental profiles have been obtained by summing data over a wide energy range and the conversion Chapter 4. Acetone 116 80 Momentum (au) Figure 4.14: Experimental and theoretical 5&\ momentum profiles of acetone. The 5ai X M P s are of fitted peak 9 (IP=24.6 eV, see Figure 4.4) from the S R - B E S (•) and L R - B E S (•) data sets. The T M P s have been determined from the theoretical calculations listed in Table 4.4. The short-dashed line is the B L Y P - D F T T M P scaled by 0.59 to better match the experimental data. The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. Chapter 4. Acetone 117 C D > C D DC 70 60 50 • 27-40 eV SR-BES • 27-40 eV LR-BES • 27-60 eV LR-BES —DFT/6-311++G' ** ** ---HF/6-311++G, -48^0.33(73^3^+1 +0.14*6a +0.30*2F +0.41 *5a. BLYP Azimuthal angle (degrees) Figure 4.15: Experimental and theoretical 4ax angle profiles of acetone. The 4a x X A P s have been obtained by summing the S R - B E S (•) and L R - B E S (•) binding energy data from 27-40 eV and by summing the L R - B E S data from 27-60 eV (•). The T A P s have been determined from the theoretical calculations listed in Table 4.4. The short-dashed line is the sum of the indicated B L Y P - D F T T A P s . The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. A binding energy of 34.0 eV has been use to resolution fold the 4ai T A P s . Chapter 4. Acetone 118 between azimuthal angle and momentum is binding energy dependent (see Equat ion (2.5)). The 5a! M O consists of in-phase 2s character on the three carbon atoms and opposite phase 2s character on the oxygen. In contrast, the 4ai M O consists almost entirely of oxygen 2s character. Ionization from both orbitals results in s-type momentum profiles, wi th l itt le variation evident between the T M P s from different calculations. In both cases, the experi-mental profiles differ significantly in intensity from the theoretical profiles. Mul t ip l i ca t ion of the B L Y P - D F T T M P i n Figure 4.14 by 0.59 results i n reasonably good agreement between experiment and theory for the 5ai momentum profiles. The 4ax experimental profiles in Figure 4.15 obtained by summing binding energy data from 27-40 eV are of greater intensity than the corresponding T M P s . This result is readily explained by the presence of ionization poles from other valence ionization manifolds i n this binding energy range. In view of the considerable intensity present at binding energies greater than 40 eV, it is most likely that some of the 4ai ionization intensity is located in poles occurring outside of the 27-40 eV binding energy range. The shape of the experimental profiles i n Figure 4.15 indicates that the majority of the intensity at binding energies greater than 27 eV is the result of ionization from the 5ai and 4ai orbitals. The short-dashed curve in Figure 4.15 represents the sum of the B L Y P - D F T 4ai T A P and the 7ai, 3b2, l b i , 6ai, 2b 2 and 5ai T A P s scaled by the intensity absent from their main ionization peaks. The discrepancy between the dashed line and the 27-60 eV sum X A P (•) suggests that additional valence ionization intensity of acetone is present above 60 eV, which is also suggested by the B E S shown in Figure 4.4. Chapter 5 Dimethoxymethane 5.1 Introduction A n understanding of the electronic structure and electron density distribution of dimethoxy-methane ( C H 2 ( O C H 3 ) 2 ) is of interest in the fields of polymer and carbohydrate chemistry, as it may be viewed as a model compound for both polyoxymethylene ( ( C H 2 0 ) x ) and the acetal linkage in polysaccharides. Dimethoxymethane is of particular interest as a model for studying the anomeric effect. The anomeric effect was originally observed in sugars as the preference of a methoxy substituent at the anomeric carbon to be in the axial rather than the equatorial position, contrary to what would be expected from steric arguments alone. It has since been recognized as a more general effect found in molecules of the type R - X - C - Y where Y is electronegative and X has at least one pair of non-bonding electrons. The prefer-ence of electronegative substituents for the axial position in sugars corresponds to a gauche orientation about the X - C bond in R - X - C - Y . The anomeric effect is also characterized by bond lengths that differ from typical values. Several explanations for the anomeric effect have been proposed. They may be generally grouped into two categories: 119 Chapter 5. Dimethoxymethane 120 1. Electrostatic interactions between the dipoles associated with X and Y destabilize the anti orientation and stabilize the gauche orientation. 2. The gauche orientation maximizes the stabilizing interaction between the non-bonding electron density of X and the vacant o* C - Y antibonding orbital . A detailed overview of the anomeric effect may be found in Reference [72]. Reference [132] is a collection of more recent research in this area, including attempts to identify the primary factor(s) responsible for the anomeric effect. In the case of dimethoxymethane, both R X and Y are methoxy groups, so in light of the above discussion one might expect gauche orientations about each of the C H 2 - 0 bonds to be favoured. This was found to be the case by K u b o [133] well before the anomeric effect had itself been recognized. From a consideration of the experimental dipole moment of dimeth-oxymethane, K u b o concluded that the two methyl groups were located on opposite sides of the O - C - 0 plane. A n electron diffraction study [134] confirmed that dimethoxymethane exists in the gas phase predominantly in the gauche-gauche (gg) conformation (Figure 5.1). The temperature dependence of the gas phase dipole moment [135] and 1 3 C - 1 H N M R vicinal coupling constants [136] as well as vibrational spectroscopy studies [137] of dimethoxymeth-ane indicate that the gauche-anti (ga) conformer is also present in small amounts. Estimates of the gas-phase energy difference between these two conformers, A 6 7 g a _ g g , based upon these experimental results range from 5.9 to 10.5 k J / m o l [135-138]. M u c h of the theoretical work involving dimethoxymethane has focused on the geometries and relative energies of the various conformations. B o t h ab initio and D F T calculations have been shown to reproduce successfully the anomeric effect [139-142]. In addition to the global energy m i n i m u m gg conformation and the next most stable ga conformation mentioned above, two other energy minima have been identified [142]: the g + g ~ conformation in which Chapter 5. Dimethoxymethane 121 H H 3 C n^A / C H 3 H gg H C H , H ga H iV \ / ° ^ C H 3 H C H 3 H g + g _ aa Figure 5.1: Conformations of dimethoxymethane the orientations about each C H 2 - 0 bond are gauche but, i n contrast to the gg conformer, both methyl groups are on the same side of the O - C - 0 plane, and the aa conformation in which al l carbon and oxygen atoms are in approximately the same plane. These four energy-m i n i m u m conformations are shown in Figure 5.1. The computational studies of W i b e r g and Murcko [140] and Smith et al . [141] indicate that increasing basis set size by the addition of diffuse and polarization functions lowers the energies of the less stable conformers relative to that of the gg conformer. The inclusion of electron correlation was found to have the opposite effect on the relative conformer energies. However, Smith et al . [141] concluded that electron correlation effects were not significant in obtaining optimized geometries. In contrast, Kneisler and All inger found that electron correlation did have a significant effect on the C - O - C - 0 dihedral angles and concluded that it should be included in geometry optimizations [142]. They also found that B 3 L Y P - D F T and M P 2 calculations resulted in optimized geometries in good agreement with each other and wi th experiment. Chapter 5. Dimethoxymethane 122 In the present work, a comprehensive investigation of the momentum space valence orbital electron densities of dimethoxymethane has been performed. Experimental data obtained using E M S have been compared with theoretical results obtained from H F , C I and D F T calculations in order to assess the importance of electron correlation effects and basis set size and composition to the accurate description of the valence orbital electron densities of d i -methoxymethane. The need to account for conformations other than the most stable gg when interpreting the experimental results and the relationship between molecular conformation and orbital electron density are also investigated. 5.2 Experimental details Three sets of binding energy spectra of dimethoxymethane have been recorded: long binding energy range spectra ( L R - B E S ) from 4.5-58.7 eV at out-of-plane azimuthal angles of 0° and 9°; outer-valence spectra ( O V - B E S ) from 5.3-23.6 eV at 16 azimuthal angles from 0° to 30°; and inner valence spectra ( IV-BES) from 17.9-44.0 eV at 14 azimuthal angles between 0° and 30°. The B E S were recorded in the binning data collection mode (Section 3.4.2) at a total energy of 1200 eV. Many scans were accumulated over an appreciable measuring time (~ 640 hours) in order to improve the signal to noise ratio. The experimental energy resolution function (1.5 eV fwhm) and momentum resolution •(« 0.1 au fwhm) of the spectrometer were determined from measurements of the helium ( I s ) - 1 ionization peak and the argon 3p momentum profile, respectively. The l iquid dimethoxymethane sample was obtained from A l d r i c h Chemical Company, Inc. and had a stated purity of 99%. Dissolved gaseous impurities were removed by repeated freeze-pump-thaw cycles. The sample gas pressure was maintained at 1 x 10~ 5 torr. Temperature related pressure fluctuations were minimized by immersing the glass sample tube in a constant temperature water bath. Chapter 5. Dimethoxymethane 123 5.3 Computational details Theoretical momentum profiles of the valence orbitals of dimethoxymethane have been ob-tained using either the target Hartree-Fock approximation (Equation (2.11)) or the target K o h n - S h a m approximation (Equation (2.14)) and the results of single-point H F or D F T calculations, respectively. A range of basis sets of increasing complexity have been used. They are described in Section 2.3.4, wi th the exception of the 2 2 9 - C G F basis set described below. The dependence of the K o h n - S h a m D F T results on the exchange-correlation potential energy functional has been investigated by performing D F T calculations using three different functionals. The Vosko, W i l k and Nusair functional [99] was used to perform local spin-density approximation ( L S D A ) calculations. In addition, two non-local functionals were used: the B P 8 6 functional including the Perdew correlation [106] and Becke exchange [105] gradient corrections and the B 3 L Y P functional, a modification of the hybrid functional proposed by Becke [108] and incorporating the exact exchange energy, wi th the Lee, Yang and Parr [107] gradient corrections to the correlation potential replacing those of Perdew and Wang [109]. The majority of the calculations have been performed using geometries optimized at the MP2( fu l l ) level wi th the 6-31+G* basis set. Addi t iona l geometry optimizations have been performed at the B 3 L Y P / 6 - 3 1 + G * and B 3 L Y F 7 6 - 3 1 1 + + G * * levels. The optimized geometries are discussed in Section 5.8. G A U S S I A N 9 2 [102] and G A U S S I A N 9 4 [103] were used to perform all of the above mentioned calculations. To investigate further the effects of electron correlation and relaxation, T M P s corre-sponding to the two lowest energy ionization processes ( (10b) _ 1 and (11a) - 1 ) of the gg conformer of dimethoxymethane have been calculated by E . R. Davidson wi th in the plane Chapter 5. Dimethoxymethane 124 wave impulse approximation using the results of M R S D - C I calculations of the in i t ia l neutral and respective final ion states of dimethoxymethane in the ion-neutral overlap expression of Equat ion (2.10). The configuration spaces for the multi-reference singles and doubles con-figuration interaction calculations of both the molecular target and ion wavefunctions were chosen from the results of respective single-reference singles and doubles configuration inter-action perturbation calculations. For the C I calculations of both the ion and the neutral, the molecular orbitals of the neutral H F wavefunction were used; the H F vir tua l orbitals were first converted to K-orbitals [143-145] in order to improve the energy convergence. The CI calculations used frozen core electrons. A 229-CGF basis set has been used for the CI calculations and the H F calculation upon which they are based. This basis set is based on the highly converged atomic basis sets of Partridge [146,147] wi th the addition of Dunning's "double d " polarization functions [114] for carbon and oxygen and a single p polarization function (exponent 1.30) for hydrogen [148]. A n (18sl3p2d)/[6s7pld] contraction has been used for carbon and oxygen and a (10slp)/[5slp] contraction for hydrogen. A l l six compo-nents of the Cartesian d functions were kept. This 229 -CGF basis set for dimethoxymethane is the same as the 204 -CGF acetone basis set described in Chapter 4 and Reference [54] l, w i t h the exception that the two d polarization functions for carbon and oxygen have been contracted to a single two-term d function. 5.4 Valence binding energy spectra Dimethoxymethane is a 42 electron molecule having, within an independent particle model, 16 valence orbitals and 5 core orbitals. The predominant gg conformer has C 2 symmetry, w i t h three core orbitals of a symmetry and two of b symmetry. According to H F calculations, 1 I n Reference [54], this basis set was referred to incorrectly as 196-GTO and the hydrogen s function contraction was incorrectly stated to be (10s)/[7s] rather than the correct (10s)/[5s]. Chapter 5. Dimethoxymethane 125 the valence orbitals alternate in symmetry between a and b, from the inner-valence 4a orbital to the 10b H O M O . From symmetry arguments, ionization from orbitals of b symmetry w i l l result in p-type momentum profiles, or conversely all s-type momentum profiles must be the result of ionization from orbitals having a symmetry. B i n d i n g energy spectra of dimethoxymethane from 4.5 to 58.7 eV at out-of-plane az-imuthal angles of cp = 0° and 9° are shown in Figure 5.2 on a common intensity scale. The close energy spacing of the ionization peaks, their Franck-Condon widths and the E M S i n -strumental energy resolution of 1.5 eV fwhm allow only part ial resolution of the ionization peaks. In order to obtain X M P s of the valence shell of dimethoxymethane, Gaussian func-tions representing the various valence ionization processes have been fitted to the B E S . The indiv idual fitted peaks are represented in Figure 5.2 by the dashed lines and the sums of the fitted peaks by solid lines. For the peaks occurring at binding energies less than 20 eV, the relative energy spacings and Franck-Condon widths were determined from the high resolution He (I) P E S measurements of J0rgensen [149], shown in the upper panel of Figure 5.2. Smal l adjustments to the energy spacings of the peaks were made to account for asym-metries in the Franck-Condon envelopes. The peak widths used to fit the E M S data are a convolution of the Franck-Condon widths estimated from the P E S data [149] and the E M S instrumental energy resolution function (1.5 eV fwhm). A s a consequence of the close energy spacing of many of the ionization peaks, it was necessary in some instances to fit a function at a single binding energy to represent multiple ionization processes. This has been done for ionizations from M O s 10b and 11a, M O s 9a, 7b and 8a, and M O s 6b, 7a and 5b. In the case of the (10b) _ 1 and (11a ) - 1 ionizations, only a slight split t ing was observed using P E S [149] and, in the latter two cases, individual ionization peaks were not resolvable (see Figure 5.2), even w i t h the considerably better P E S energy resolution of 0.03 eV fwhm [149,150]. The absolute energy scale of the E M S B E S was fixed by setting the position of the first peak, Chapter 5. Dimethoxymethane 126 T 1 1 1 r Binding energy (eV) Figure 5.2: B inding energy spectra of dimethoxymethane from 4.5-58.7 eV. The B E S (•) at </> = 0° and 9°, obtained at a total energy of 1200 eV, are shown i n the upper and lower panels, respectively. The dashed lines indicate the result of a least-squares fit of Gaussian functions to the ionization peaks and the solid curves are the summed fits. The peak positions and assignments are indicated by the vertical lines. The high resolution (0.03 eV fwhm) He (I) P E S of dimethoxymethane, reproduced from Reference [149], is shown i n the upper panel on the same energy scale as the E M S data. Chapter 5. Dimethoxymethane 127 Table 5.1: Measured ionization potentials and calculated orbital energies of the valence shell of dimethoxymethane. Orb i ta l energies are H F / a u g - c c - p V T Z values for the M P 2 / 6 - 3 1 + G * optimized gg and ga conformers. A l l energies are in eV. O r b i t a l P E S E M S b - O r b i t a l energy origin 3 . [149' [151 this work gg ga 10b 11a 10.29 10.53 } 10.42 10.41 12.055 12.059 11.674 12.011 9b 11.44 11.48 11.5 12.696 13.375 10a 12.98 12.92 13.0 14.268 13.692 8b 13.42 13.45 13.6 14.409 14.589 9a ) 15.532 15.833 7b } 14.8 14.60 15.0 16.281 15.993 8a J 16.598 16.774 6b ) 18.244 18.342 7a } 17.0 17.0 18.742 18.572 5b J 18.795 18.908 6a 20.4 22.843 23.128 4b 22.6 25.343 25.174 5a 23.9 26.598 26.575 3b 36.501 36.569 4a 38.391 38.439 a O r b i t a l symmetry labels are those of the C 2 symmetry gg conformer. b E s t i m a t e d uncertainty ± 0 . 1 eV. The E M S absolute energy scale was determined by setting the first ionization energy to 10.41 eV. resulting from ionization from the 10b and 11a orbitals, to 10.41 eV, the average of the cor-responding high resolution P E S ionization potentials [149]. Ionization processes requiring energies greater than ?a 20 eV are not observable by He (I) P E S . Consequently, the positions and widths of the fitted peaks representing the (6a ) _ 1 , ( 4 b ) - 1 and ( 5 a ) _ 1 ionization processes have been determined by a fit to the 17.9-44.0 eV inner-valence B E S ( IV-BES) collected at 14 out-of-plane azimuthal angles ranging from <fi = 0° to 30°. The ionization potentials used in the present work are compared in Table 5.1 with the published He (I) P E S values of J0rgensen [149] and Zverev et al . [151] and calculated H F / a u g - c c - p V T Z orbital energies of the M P 2 / 6 - 3 1 ± G * optimized gg and ga conformers. In the outer-valence region (below 20 eV) of the 0 = 0° and 9° experimental B E S , several Chapter 5. Dimethoxymethane 128 features are evident. Peaks resulting from ionizations from orbitals 10b and 11a at 10.41 eV, 10a at 13.0 eV, 9a, 7b and 8a at 15.0 eV and from orbitals 6b, 7a and 5b at 17.0 eV have a spacing which is consistent with the results of P E S [149,151], as indicated i n Table 5.1 and Figure 5.2. The largest discrepancy is for the ( 9 a ) - 1 + ( 7 b ) - 1 + (8a)" 1 peak located at 15.0 eV in the current work. In the case of the P E S work of Zverev et al . [151], only the leading edge of this peak is contained within the energy range reported, precluding an accurate determination of the position and not surprisingly resulting in a low value. In contrast, J0rgensen reports P E S data of dimethoxymethane up to approximately 19.5 eV [149] (Figure 5.2), a range which fully contains the ionization band in question. This broad band, resulting from ionization from three different orbitals, extends from 14.0 to 16.2 eV i n the P E S data and lacks sufficient structure to allow the determination of individual vertical IPs. The asymmetric nature of the P E S band results in a small shift to higher energy in the E M S data relative to the P E S data as a consequence of the poorer E M S energy resolution. The ( 9 a ) - 1 + ( 7 b ) - 1 + ( 8 a ) - 1 peak at 15.0 eV and the ( 6 b ) - 1 + ( 7 a ) - 1 + ( 5 b ) - 1 peak at 17.0 eV each exhibit characteristic p-type behaviour, wi th greater intensity at 4> — 9° than at (p = 0°. In contrast, the (10a ) - 1 peak at 13.0 eV is strongly s-type and the ( 1 0 b ) - 1 4- ( 1 1 a ) - 1 peak at 10.41 eV has similar intensity at the two azimuthal angles. The positions and intensities of the ( 9 b ) - 1 and ( 8 b ) - 1 ionizations do not allow resolution of these individual peaks in the B E S . However, their inclusion in the fitting procedure, using physically realistic positions and widths determined from the P E S data [149], is necessary to obtain a reasonable fit to the experimental E M S data. The origins of the 15.0 and 17.0 eV bands, not discussed i n the P E S studies [149,151], have been assigned as indicated in Figure 5.2 and Table 5.1 and discussed above based upon an examination of the corresponding experimental momentum profiles of these peaks (see Section 5.5.1, below) and the calculated H F orbital energies shown i n Table Chapter 5. Dimethoxymethane 129 Three bands are evident in the inner-valence region (above 20 eV) of the binding energy spectra in Figure 5.2. The strong s-type peak at 20.4 eV can be readily assigned to the ( 6 a ) - 1 ionization on the basis of the calculated orbital energies (Table 5.1). The band between 22 and 25 eV appears to consist of two ionizations of differing symmetry, as indicated by the differences in the shape of the peak at cb = 0° and <f> = 9°. The low energy side of the peak displays p-type character, evidenced by the increased intensity in the 0 = 9° spectrum, while the high energy side is of greater intensity at cb = 0°. This observation, in combination wi th the calculated orbital energies in Table 5.1, supports an assignment of this band to ( 4 b ) - 1 and ( 5 a ) - 1 ionizations at 22.6 and 23.9 eV, respectively. The assignments of these ionization peaks are confirmed by an examination of the respective X M P s in Section 5.5.2 below. The th i rd ionization band in the inner-valence region is a broad s-type feature between approximately 28 and 37 eV. This is the region in which the parent ( 3 b ) - 1 and ( 4 a ) - 1 ionization processes are expected. However, no attempt has been made in the present work to fit peaks representing ( 3 b ) - 1 and ( 4 a ) - 1 ionizations because of the lack of structure in this B E S band, the likelihood that there are many 'satellite' states in this energy range of the B E S , and the lack of many-body calculations and synchrotron radiation or X - r a y P E S studies of dimethoxymethane to serve as a guide for the fitting procedure. Al though there are no clear features i n the B E S above 37 eV, considerable intensity is s t i l l observed, particularly at cj) — 0°, indicating the presence of predominantly s-type many-body ionization processes (satellite states) in this region. 5.5 Experimental and theoretical momentum profiles Experimental momentum profiles of the main peaks in the outer- and inner-valence regions of the dimethoxymethane B E S have been obtained by performing least-squares fits of the L R -Chapter 5. Dimethoxymethane 130 B E S (4.5-58.7 eV) , O V - B E S (5.3-23.6 eV) and I V - B E S (17.9-44.0 eV) data sets described in Section 5.2. Gaussian functions were fixed at those binding energies and widths used to fit the L R - B E S (Figure 5.2) and only the peak amplitudes were allowed to vary. For each fitted peak, the distr ibution of peak areas as a function of momentum, converted from <f> using Equation (2.5), yields the corresponding experimental momentum profile, as discussed i n Section 3.4.4. The three sets of B E S data have been placed on the same intensity scale by normalizing the long range and inner-valence spectra to the outer-valence spectra using the respective data collected at (j) = 0° and 0 = 9° over the corresponding overlapping energy ranges of the data sets (i.e., 5.3-23.6 eV for the normalization of the L R - B E S to the O V - B E S and 17.9-23.6 eV for the normalization of the I V - B E S to the O V - B E S ) . Following these two normalizations, al l X M P s share a common relative intensity scale. The resulting momentum profiles are shown in Figures 5.3 and 5.5 to 5.11, with data obtained from the O V - B E S shown as filled circles (•), data from the I V - B E S shown as open circles (o) and data from the L R - B E S shown as open squares (•). The error bars are based upon the quality of the fits. Also shown in Figures 5.3-5.11 are theoretical momentum profiles of the corresponding orbitals or sums of orbitals, calculated using the M P 2 / 6 - 3 1 + G * optimized gg conformer of dimethoxymethane and a number of theoretical methods, as outlined in Section 5.3. The key to the T M P labels i n Figures 5.3-5.11 is given in Table 5.2, along with selected calculated properties. T M P s of the gg conformer alone have been used for this comparison wi th the experimental data because, considering the calculated energy differences between the gg and ga conformers listed i n Table 5.2 and the experimental temperature of 298 K , the sample is expected to consist overwhelmingly of dimethoxymethane in the gg conformation. The significance of contributions from other conformers is investigated in Section 5.7. The experimental and theoretical results have been placed on a common intensity scale by normalization of the 10b + 11a (10.41 eV) X M P to the corresponding B 3 L Y P / 6 - 3 1 1 + + G * * T M P (Figure 5.3 Chapter 5. Dimethoxymethane 131 T a b l e 5.2: C a l c u l a t e d proper t ies of the gauche-gauche (gg) a n d g a u c h e - a n t i (ga) conformers of d i m e t h o x y m e t h a n e a n d key to the t h e o r e t i c a l m o m e n t u m profi les s h o w n i n F i g u r e s 5 . 3 -5.11. T h e c a l c u l a t i o n s were p e r f o r m e d u s i n g the M P 2 / 6 - 3 1 + G * o p t i m i z e d geometr ies . K e y C a l c u l a t i o n T o t a l energy A £ 0 D i p o l e moment gg g a - g g gg ga (hartree) (k J /mol ) (debye) hf-st H F / S T O - 3 G -264.55318 10.23 0.135 1.787 hf-6g H F / 6 - 3 1 G -267.82738 16.64 0.342 2.855 H F / 6 - 3 1 + G * -267.95876 9.70 0.294 2.325 hf-6p H F / 6 - 3 1 1 + + G * * -268.03074 8.68 0.277 2.307 hf-229 H F / 2 2 9 - C G F -268.06916 0.256 M P 2 ( f u l l ) / 6 - 3 1 + G * -268.73856 14.17 ci-229 C I / 2 2 9 - C G F -268.71508 0.263 ld-6p D F T L S D A a / 6 - 3 1 1 + + G * * -267.43708 11.85 0.324 1.950 bp-6p D F T B P 8 6 b / 6 - 3 1 1 + + G * * -269.63211 10.62 0.312 1.882 b3-6p D F T B 3 L Y P c / 6 - 3 1 1 + + G * * -269.63910 10.10 0.301 2.028 a T h e Vosko, W i l k and Nusair [99] local exchange-correlation funct ional was used. b T h e Becke exchange [105] and Perdew correlation [106] functionals were used. c T h e B 3 L Y P funct ional is a modif icat ion of the h y b r i d funct ional proposed by Becke [108] and incorporat ing the exact exchange energy, w i t h the Lee, Y a n g and Parr [107] correlat ion potent ia l replacing that of Perdew and W a n g [109]. Chapter 5. Dimethoxymethane 132 below). This theoretical momentum profile was chosen for the normalization since it agrees most closely in shape wi th the X M P . 5.5.1 Outer-valence momentum profiles The 10b + 11a experimental and theoretical momentum profiles are shown in Figure 5.3. These two orbitals are, to a first approximation, non-bonding orbitals predominantly centred on the two oxygen atoms and represent antisymmetric and symmetric combinations of the oxygen n^ orbitals 2 , respectively. There is reasonable agreement between the three D F T calculations (ld-6p, bp-6p and b3-6p) and the experimental profile. In contrast, none of the H F calculations predict the rise in intensity observed experimentally below 0.5 au. In all cases, the theoretical calculations predict greater intensity at high momentum (> 1 au) than is observed experimentally. To investigate the possibility that this discrepancy between the H F and D F T results is a consequence of the neglect of electron correlation effects in the case of the H F calculations, M R S D - C I calculations of the neutral target and two final ion states have been performed by E . R. Davidson and a theoretical momentum profile has been calculated using the CI wavefunctions and the target-ion overlap expression (Equation (2.10)). The resulting T M P (ci-229, dashed line in Figure 5.3) is fairly similar to the H F results and certainly does not reproduce the low p behaviour of either the D F T calculations or the experimental profile. To understand better the shapes of the 10b + 11a momentum profiles and the differ-ences between the computational methods, it is useful to consider the indiv idual 10b and 11a T M P s , shown in Figure 5.4. Note that the H F / 6 - 3 1 G calculation (hf-6g) reverses the energetic ordering of the (very closely spaced in energy) 10b and 11a orbitals w i t h respect to 2nn refers to non-bonding oxygen orbitals having the majority of the electron density perpendicular to the C O C planes; n f f refers to those having the majority of the electron density in the C O C planes. Chapter 5. Dimethoxymethane 133 0 I . 1 . L 0 1 2 Momentum (au) Figure 5.3: Experimental and theoretical 10b + 11a momentum profiles of dimethoxymeth-ane. The X M P s are of the peak fitted at 10.4 eV to the O V - B E S (•) and L R - B E S (•) data sets. The T M P s have been calculated at the M P 2 / 6 - 3 1 + G * optimized gauche-gauche geom-etry. The key to the T M P labels is given in Table 5.2. The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. Chapter 5. Dimethoxymethane 134 0 1 2 3 0 1 2 3 Momentum (au) Figure 5.4: Theoretical 10b (left panel) and 11a (right panel) momentum profiles of d i -methoxymethane. The profiles have been calculated using the M P 2 / 6 - 3 1 + G * optimized gauche-gauche geometry. The key to the T M P labels is given in Table 5.2. The experimen-tal angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. al l of the other calculations considered here. It is clear from Figure 5.4 that the intensity in the 10b + 11a profile near p = 0 au is almost entirely due to orbital 11a, while the 10b + 11a profile maximum near 0.8 au contains significant contributions from both orbitals. The dif-ferences between the H F and D F T results come almost entirely from the 11a T M P s . A l l of the calculations of the 10b T M P s , wi th the exception of the small basis set H F calculations hf-st and to a lesser extent hf-6g, are in reasonable agreement. There is, however, a small but definite shift in the maximum of the 10b profile (pMAx) towards lower momentum in com-paring the D F T profiles to the H F profiles. In the case of the 11a T M P s , the calculations shown al l predict a momentum profile of the same qualitative shape, wi th max ima at 0 and 1.2 au and a min imum near 0.7 au. However, the D F T calculations predict substantially Chapter 5. Dimethoxymethane 135 greater intensity at 0 au than do the CI and H F calculations. The similarity between the respective 10b and 11a H F and CI T M P s seems to suggest that electron correlation effects for these momentum profiles are not sufficiently great to account for the differences between the H F profiles and the D F T and experimental profiles. In view of the disagreement between the 11a D F T and CI T M P s , the reason for the agreement of the 10b + 11a X M P wi th the corresponding D F T T M P s , but not with the H F and C I T M P s , is not clear. Previous E M S studies [20,71,152,153] of other molecules have noted a tendency for the D F T T M P s to have greater intensity at low momentum than is the case for H F and C I T M P s , particularly in the case of L S D A D F T calculations. This has been attributed [20] to the underbinding of the large-r electron density in the D F T calculations, resulting in position space orbitals that are too diffuse and, consequently, momentum space profiles wi th too much intensity at low momentum. This deficiency of the L S D A has been addressed as part of the development of gradient-corrected functionals (Section 2.3.3) and consequently should be less of a problem for D F T T M P s calculated using gradient-corrected functionals. Al though an excess of intensity at low momentum can explain the large differences between the H F and C I and the D F T 11a T M P s , it does not explain the good agreement obtained between the experimental data and the D F T profiles. A n alternative explanation is that, in order to make these very large C I calculations tractable, too few electron configurations were included, yielding a result that neglects a significant fraction of the electron correlation effects. A d d i t i o n a l C I calculations, including a greater number of configurations, are neces-sary to verify this hypothesis. The computational challenges in performing such calculations on a molecule of the size of dimethoxymethane highlight the appeal of D F T . Ionization from M O 9b at 11.5 eV results in the experimental momentum profile shown in Figure 5.5. M O 9b may be described as the antisymmetric combination of two 0 n a orbitals. However, the M O calculations performed in the current work also predict some Chapter 5. Dimethoxymethane 136 6 Momentum (au) Figure 5.5: Experimental and theoretical 9b momentum profiles of dimethoxymethane. The X M P s are of the peak fitted at 11.5 eV to the O V - B E S (•) and L R - B E S (•) data sets. The T M P s have been calculated using the M P 2 / 6 - 3 1 + G * optimized gauche-gauche geometry. The key to the T M P labels is given in Table 5.2. The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. Chapter 5. Dimethoxymethane 137 methylene C - H and C H 2 - 0 bonding character for this orbital . The relatively large error bars and the scatter of the experimental data reflect the difficulties in obtaining an accurate experimental momentum profile of an ionization process separated by only ?» 1 e V from the nearest neighboring ionization ( (11a) - 1 ) . It is clear that al l of the theoretical profiles are i n poor agreement with the X M P , particularly for p < 0.5 au. In contrast, there are only small differences between the T M P s shown in Figure 5.5, wi th the notable exception of the min imal basis set H F / S T O - 3 G T M P . The H F and D F T B 3 L Y P and B P 8 6 T M P s calculated using the 6-311++G** basis set are very similar to one another. W h e n the H F calculation is performed wi th the substantially less flexible 6-31G basis set, the resulting T M P has slightly less intensity at low momentum and slightly more at high momentum. This trend continues when the S T O - 3 G minimal basis set is used. The 9b T M P derived from the D F T L S D A calculation has somewhat greater intensity in the vicinity of pMAx than do the other calculations using the 6-311++G** basis set. Al though the discrepancies between the 9b X M P and T M P s could be the result of inadequacies in the theoretical calculations, it is more likely that these differences are at least partly the result of l imitations in the fitting procedure used to obtain the 9b X M P . The small energy separations from the ( 1 0 b ) - 1 + ( 1 1 a ) - 1 peak at 10.41 eV and in particular the very intense s-type ( 1 0 a ) - 1 peak at 13.0 eV likely result in some contamination of the 9b experimental momentum profile w i t h intensity from these adjacent ionization processes. Alternative theoretical explanations for this discrepancy between experiment and theory are investigated in Sections 5.6-5.8. Experimental and theoretical momentum profiles for orbitals 10a + 8b are shown in F i g -ure 5.6. Al though separate Gaussian functions at 13.0 and 13.6 eV have been used for these ionization processes when fitting the B E S (Section 5.4 and Figure 5.2), the resulting X M P s show considerable cross-contamination as a consequence of the small energy spacing between the ( 1 0 a ) - 1 and ( 8 b ) - 1 ionization processes. Therefore, these two X M P s have been summed Chapter 5. Dimethoxymethane 138 20 15 c CD C D > C D c r 10 -5 h 0 Id-6p,bp-6p,b3-6p 10a + 8b 13.0&13.6eV • OV-BES • LR-BES —10a+8b 10a 8b b3-6p yf / / ht-6p 0 1 2 Momentum (au) Figure 5.6: Experimental and theoretical 10a + 8b momentum profiles of dimethoxymethane. The X M P s are of the sum of the peaks fitted at 13.0 and 13.6 eV to the O V - B E S (•) and L R - B E S (•) data sets. The T M P s are sums of the 10a and 8b T M P s calculated using the M P 2 / 6 - 3 1 + G * optimized gauche-gauche geometry. The individual hf-6p and b3-6p 10a and 8b T M P s are also shown. The key to the T M P labels is given in Table 5.2. The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. Chapter 5. Dimethoxymethane 139 together prior to further analysis and comparison with the theoretical results. The consider-able intensity near zero momentum is almost entirely attributable to the s-type 10a profile. B o t h the p-type 8b profile and the 10a profile contribute to the intensity observed near p = 1 au. The 10a M O consists primari ly of the symmetric combination of 0 n^ orbitals plus methyl C - H bonding character. In contrast, M O 8b of the gg conformer of dimeth-oxymethane is highly delocalized throughout the molecule, wi th significant C - 0 and C - H cr-bonding character. The three D F T theoretical profiles are very similar and are i n excellent agreement wi th the experimental data. The H F profiles differ considerably from the X M P , particularly at low momentum, and also exhibit considerable basis set dependence. A s was the case for the 10b + 11a profiles, almost al l of the differences between the H F and D F T results arise from differences in the a symmetry profiles (10a in this case), wi th the 8b T M P s showing l i tt le variation with theoretical method. The X M P obtained from the peak fitted to the B E S at 15.0 eV is shown in Figure 5.7 and compared wi th sums of the 9a, 7b and 8a theoretical profiles. The general agreement in terms of both shape and intensity between the X M P and the three D F T T M P s supports the assignment in Section 5.4 of the 15.0 eV peak to the ( 9 a ) _ 1 + ( 7 b ) _ 1 + ( 8 a ) - 1 ionization processes. The B 3 L Y P / 6 - 3 1 1 + + G * * T M P (b3-6p) best matches the shape of the experi-mental profile. The remaining D F T T M P s have the correct qualitative shape, but predict too much intensity at pMAx and too narrow a peak about pMAx- None of the H F T M P s shown in Figure 5.7 are in good agreement with the experimental data. The H F / 6 - 3 1 G T M P has the shape most similar to the X M P , but is considerably lower in intensity at p < 1 au. The H F / 6 - 3 1 1 + + G * * T M P , obtained using the same basis set as was used for the D F T T M P s in Figure 5.7, has a very different shape, predicting a maximum near p = 0. A n examination of the individual T M P s of orbitals 9a, 7b and 8a indicates that the discrepancy between the H F / 6 - 3 1 1 + + G * * T M P and the D F T T M P s near zero momentum is pr imari ly a consequence Chapter 5. Dimethoxymethane 140 30 £ 20 C D C D > ® 10 CL 0 9a+7b+8a 15.0 eV OV-BES LR-BES 9a+7b+8a 0 Momentum (au) Figure 5.7: Experimental and theoretical 9a + 7b + 8a momentum profiles of dimethoxy 1 methane. The X M P s are of the peak fitted at 15.0 eV to the O V - B E S (•) and L R - B E S (•) data sets. The T M P s are sums of the 9a, 7b and 8a T M P s calculated using the M P 2 / 6 -3 1 + G * optimized gauche-gauche geometry. The individual hf-6p and b3-6p 9a, 7b and 8a T M P s are also shown. The key to the T M P labels is given in Table 5.2. The experimen-tal angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. Chapter 5. Dimethoxymethane 141 of differences between the H F and D F T 8a profiles. The D F T profiles of both orbitals 9a and 8a have greater intensity than the corresponding H F profiles between « 0.5 and 1.0 au, giving rise to the differences in this momentum region evident in Figure 5.7. The H F / 6 -311++G** T M P of orbital 7b and the corresponding D F T T M P s differ only slightly. The considerable shape difference between the H F / 6 - 3 1 1 + + G * * and H F / 6 - 3 1 G T M P s of orbitals 9a + 7b + 8a is predominantly a result of the considerably greater intensity near p = 0 of the 8a T M P obtained from the larger basis set calculation. Al though the agreement between the experimental data and the B 3 L Y P / 6 - 3 1 1 + + G * * T M P in Figure 5.7 is reasonably good, the experimental data do consistently fall below the theoretical profile for p between m 0.5 and 1.5 au. This could be the result of many-body effects in ionization from one or more of these orbitals, resulting in reduced spectroscopic factors (Section 2.2.3). If this is the case, the missing intensity w i l l be contained in one or more ionization poles at other energies. The other likely explanation for this intensity difference is small errors in the f i t t ing procedure used to obtain the X M P arising from the approximation of using a single peak to fit three separate (albeit closely spaced) ionization processes. The shape of the X M P obtained at 17.0 eV, shown in Figure 5.8, is qualitatively repro-duced by al l of the theoretical momentum profiles shown. The considerable difference in intensity between the experimental results and the T M P s can be attributed to a breakdown of the single particle model of ionization, an effect which tends to become more significant at higher ionization energies (see Section 2.2.3 and Reference [154]). The B 3 L Y P / 6 - 3 1 1 + + G * * T M P scaled by a factor of 0.82 provides good agreement wi th the X M P in terms of both shape and intensity, whereas the H F / 6 - 3 1 1 + + G * * T M P predicts a somewhat greater p M A x than is observed experimentally. The scaled T M P s do underestimate the observed intensity at low p. This is likely a result of the rather crude approximation of scaling the 6b, 7a and 5b T M P s equally, even though it is highly unlikely that the spectroscopic factors for Chapter 5. Dimethoxymethane 142 0 1 2 3 Momentum (au) Figure 5.8: Experimental and theoretical 6b + 7a + 5b momentum profiles of dimethoxy-methane. The X M P s are of the peak fitted at 17.0 eV to the O V - B E S (•), L R - B E S (•) and I V - B E S (o) data sets. The T M P s are sums of the 6b, 7a and 5b T M P s , calculated using the M P 2 / 6 - 3 1 + G * optimized gauche-gauche geometry. The key to the T M P labels is given in Table 5.2. Chapter 5. Dimethoxymethane 143 these three ionizations are identical. In comparing the H F results to the experimental data, agreement improves with increasing basis set size as the location of pMAX shifts to lower momentum. The location of pMAX shifts even further towards low momentum in the case of the D F T calculations, wi th the B 3 L Y P / 6 - 3 1 1 + + G * * result giving the best agreement wi th the experimentally observed P M A X -5.5.2 Inner-valence momentum profiles The five inner-valence molecular orbitals of dimethoxymethane (6a, 4b, 5a, 3b and 4a) can all be described as consisting primari ly of combinations of the carbon and oxygen 2s atomic orbitals. The momentum profiles of these orbitals are al l characterized by spectroscopic factors of significantly less than unity as a result of many-body effects w i t h ionization from these orbitals, as is common with inner-valence ionization [154]. The strongly s-type ( 6 a ) - 1 ionization peak at 20.4 eV in Figure 5.2 corresponds to the X M P shown in Figure 5.9. M O 6a consists of significant 2s electron density on al l carbon and oxygen atoms, alternating in phase from methyl C to O to methylene C . The largest fraction of the electron density for this M O is located on the central (methylene) carbon atom. A l l of the theoretical momentum profiles shown in Figure 5.9 for orbital 6a predict essentially the same shape for this profile and are in good agreement wi th the experimental results. The theoretical results do differ, however, in the predicted intensity at low momentum, with the D F T and the minimal basis set H F calculations predicting considerably more intensity near p = 0 than is the case for the larger basis set H F calculations. It is apparent from the differences in intensity between the experimental and theoretical results that the spectro-scopic factor for this ionization process is less than one. Wh en the B3LYP/6-311+-1-G** and HF/6-311+- f -G** T M P s are scaled by 0.80 and 0.85 respectively, a more accurate assessment of the shapes of the theoretical profiles can be made. B o t h of the scaled profiles match the Chapter 5. Dimethoxymethane 144 60. h '(f) B 40 c CD > WW DC 20 0 0 ld-6p bp-6p,hf-st b3-6p hf-6p,hf-6g 6a 20.4 eV OV-BES LR-BES o IV-BES •--0.80 b3-6p --0.85 hf-6p Momentum (au) Figure 5.9: Experimental and theoretical 6a momentum profiles of dimethoxymethane. The X M P s are of the peak fitted at 20.4 eV to the O V - B E S (•), L R - B E S (•) and I V - B E S (o) data sets. The T M P s have been calculated using the M P 2 / 6 - 3 1 + G * optimized gauche-gauche geometry. The key to the T M P labels is given in Table 5.2. The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. Chapter 5. Dimethoxymethane 145 X M P fairly well. The D F T T M P slightly overestimates the ratio between the intensity near zero momentum and the intensity at the 'plateau' in the profile between 0.5 and 1.0 au. The 4b experimental momentum profile, obtained at a binding energy of 22.6 eV, and the corresponding theoretical profiles, are shown in Figure 5.10. The majority of the electron density for this orbital is located on the two methyl carbon nuclei. The opposite phase of the M O on these two nuclei results in the intensity min imum near zero momentum. A l l of the theoretical calculations correctly predict the p-type nature of this momentum profile and are in good agreement with the shape of the X M P . In contrast to the outer-valence p-type momentum profiles discussed above, there is l itt le variation wi th calculation of pMAX of the 4b T M P s . A scale factor of m 0.9 is necessary to bring the B 3 L Y P and H F / 6 -311++G** T M P s into intensity agreement with the experimental data. Using the slightly different scale factors indicated in Figure 5.10, the high degree of consistency between the shapes of the H F and B 3 L Y P momentum profiles is evident. There is some indication of a slight disagreement between theory and experiment at very low momentum, wi th the experimental results somewhat higher than the theoretical profiles. This could be due to small contributions from the neighboring ( 6 a ) - 1 ionization. Al though the energy separation between the two ionization processes is greater than 2 eV, the 6a profile is so intense near p = 0 that some intensity from this ionization process may be ' leaking' into the 4b fit at low momentum. Another possibility is that there are one or more low intensity s-type poles in the same binding energy region as the ( 4 b ) - 1 ionization at 22.6 eV. The 5a experimental and theoretical momentum profiles are shown in Figure 5.11. This M O consists primari ly of C 2s character, wi th the greatest contribution coming from the central carbon atom. The theoretical profiles are very similar in shape but differ somewhat in intensity. A s was the case for the 6a and 4b T M P s , the 5a T M P s appear to be less sensitive to the basis set used for the calculation than was the case for the outer-valence Chapter 5. Dimethoxymethane 146 "co C D C C D > C D DC hf-6p,hf-6g -b3-6p bp-6p ld-6p 4b 22.6 eV OV-BES LR-BES o IV-BES 0.90 b3-6p 0.87 hf-6p Momentum (au) Figure 5.10: Experimental and theoretical 4b momentum profiles of dimethoxymethane. The X M P s are of the peak fitted at 22.6 eV to the O V - B E S (•), L R - B E S (•) and I V - B E S (o) data sets. The T M P s have been calculated using the M P 2 / 6 - 3 1 + G * optimized gauche-gauche geometry. The key to the T M P labels is given in Table 5.2. The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. Chapter 5. Dimethoxymethane 147 2 5 0 1 2 3 Momentum (au) Figure 5.11: Experimental and theoretical 5a momentum profiles of dimethoxymethane. The X M P s are of the peak fitted at 23.9 eV to the O V - B E S (•), L R - B E S (•) and I V - B E S (o) data sets. The T M P s have been calculated using the M P 2 / 6 - 3 1 + G * optimized gauche-gauche geometry. The key to the T M P labels is given in Table 5.2. The experimental angular resolution has been accounted for using the G W - P G method [48]. Chapter 5. Dimethoxymethane 148 T M P s of dimethoxymethane, as illustrated by the similarity between the H F / 6 - 3 1 G and H F / 6 - 3 1 1 + + G * * T M P s . It is not surprising that the use of diffuse and polarization func-tions becomes less important for the inner-valence orbitals, which tend to be more spatially localized and atomic-like than the outer-valence orbitals. The B 3 L Y P and H F 6 - 3 1 1 - H - G * * T M P s , when scaled by 0.55 and 0.53 respectively, can be seen to have nearly the same shape and are generally consistent with the X M P , with the exception of a few points near p = 0. A n examination of the B E S in Figure 5.2 reveals that there is significant intensity on the high energy side of the (5a ) _ 1 peak, indicating the likely presence of one or more satellite ionization processes in this energy region. Consequently, the 'extra' intensity observed in the 5a experimental momentum profile near p = 0 can be reasonably attributed to s-type satellite ionization processes in this binding energy region. H F orbital energies (Table 5.1) indicate that the parent ( 3 b ) _ 1 and (4a)""1 ionization peaks should be located within the broad peak evident in the B E S (Figure 5.2) between 28 and 37 eV. However, as discussed above (Section 5.4), the relatively structureless nature of the B E S in this region and the lack of many-body theoretical calculations or higher energy resolution experimental studies to serve as a guide preclude the determination of X M P s for the ( 3 b ) _ 1 and ( 4 a ) - 1 ionizations by fitting of the B E S . It is possible, however, to obtain an experimental profile for the entire region simply by summing the B E S data over the appropriate energy range. This has been done for the 25-44 eV binding energy range, wi th the results indicated by the circles (o) and filled squares (•) in Figure 5.12. The data are shown on an angle rather than momentum scale because of the binding energy dependence of the (f> to p conversion (Equation (2.5)). The experimental angle profile is clearly s-type, as expected from an examination of the 0° and 9° B E S (Figure 5.2). The experimental data are compared in Figure 5.12 with B 3 L Y P and H F 6-311++G** theoretical profiles obtained by summing the individual 3b and 4a profiles. A s was observed for the other inner-valence Chapter 5. Dimethoxymethane 149 Inner-valence region •f—• CO c 0 0 > 0 DC o 25.0-44.1 eVIV-BESsum • 25.3-58.7 eV LR-BES sum • 25.3-44.4 eV LR-BES sum b3-6p 3b&4a hf-6p 3b&4a b3-6p 3b+4a+0.18(6b+7a+5b) \ +0.20 6a+0.10 4b+0.45 5a \Q ^ hf-6p3b+4a+0.12(6b+7a+5b^ +0.15 6a+0.13 4b+0.45 5a 8 \ o CP 30 Azimuthal angle (degrees) Figure 5.12: Experimental angle profile of the 25-44 eV binding energy range of dimeth-oxymethane and the individual and summed 3b and 4a theoretical profiles. The X A P s have been obtained by summing the I V - B E S (o) and L R - B E S (•) binding energy data from 25-44 eV and by summing the L R - B E S data from 25-59 eV (•). The theoretical profiles have been calculated using the M P 2 / 6 - 3 1 + G * optimized gauche-gauche geometry. The key to the T M P labels is given in Table 5.2. The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. B i n d i n g energies of 31.1 and 33.1 eV have been used to resolution fold the 3b and 4a T A P s , respectively. Chapter 5. Dimethoxymethane 150 ionization processes, ionization intensity from these orbitals is expected to be spread over multiple poles. It is also likely that satellite peaks due to ionization from valence orbitals other than 3b and 4a are present in the 25-44 eV energy range. Consequently, the intensity agreement evident in Figure 5.12 between the experimental and 3b + 4a theoretical profiles at low azimuthal angles is likely coincidental. The disagreement between the experimental and theoretical profiles at higher azimuthal angles suggests the presence in this energy range of satellite poles resulting from ionizations from other valence orbitals. Alternatively, the greater experimental intensity at high momentum could be the result of distortion effects, which are more likely to occur wi th inner-valence ionization. The two open squares i n Figure 5.12, obtained by summing L R - B E S data from 25.3 to 58.7 eV, indicate the significant amount of additional intensity present above 44 eV. If al l of the 'missing intensity' from orbitals 6b, 7a, 5b, 6a, 4b and 5a is added to the the 3b and 4a theoretical profiles, the upper curves in Figure 5.12 are obtained. The H F / 6 - 3 1 1 + + G * * curve matches well wi th the two experimental data points, while the B 3 L Y P / 6 - 3 1 1 + - I - G * * curve is somewhat higher at 0 = 0°. The discrepancy between the experimental data and the D F T calculation is perhaps not surprising considering the 'extra' experimental intensity observed near zero momentum in the 9b, 6b + 7a + 5b, 4b and 5a X M P s (Figures 5.5, 5.8, 5.10 and 5,11, respectively). Alternatively, this discrepancy could be due to additional s-type valence orbital ionization intensity at binding energies greater than 58.7 eV, the l imit of the current study. This is supported by the B E S shown in Figure 5.2, which indicate the presence of ionization intensity up to the l imit of the monitored binding energy range. The good intensity agreement between the experimental data and the H F calculation could then be partly fortuitous, since had the normalization between theory and experiment been done using the H F / 6 - 3 1 1 + + G * * results rather than the B3LYP/6 -311 - r - t -G** (beginning of Section 5.5 above), the experimental data would be lower relative to the theoretical results. Chapter 5. Dimethoxymethane 151 Despite the featureless nature of the broad peak i n the B E S between 25 and 44 eV, an attempt has been made to further investigate the nature of the ionization processes in this energy range. This has been done by dividing the 25-44 eV binding energy data into 11 energy ranges and considering the experimental angle profiles of each energy range, as shown in Figure 5.13. A s can be seen, the experimental profiles for a l l 11 energy ranges are s-type, although the widths of the profiles differ. Each profile has been fitted using the 3b (p-type) and 4a (s-type) B 3 L Y P / 6 - 3 1 1 + + G * * theoretical profiles, wi th the resulting scale factors and the fitted function shown in each panel of Figure 5.13. The reasoning behind this analysis is that the bulk of the ionization intensity in this binding energy range is likely from the ( 3 b ) _ 1 and ( 4 a ) _ 1 ionization manifolds, and that even if this is not the case, some indication of the general distribution of s-type and p-type ionization poles in this region might be obtained. Al though al l profiles are s-type, none of them are narrow enough to be fitted by the s-type 4a profile alone. This suggests that al l of these energy ranges contain intensity from multiple many-body ionization poles. In Figure 5.14, the fitted scale factors have been plotted as a function of the central binding energy of each range. The binding energy error bars in Figure 5.14 indicate the widths of the energy ranges and the scale factor error bars are the calculated uncertainties from the least-squares fit. The data shown in Figure 5.14 suggest that p-type ionization poles are concentrated at lower binding energy than are s-type poles, but w i t h significant ionization intensity of both symmetries spread throughout this binding energy range. It is interesting to note that the H F / a u g - c c - p V T Z 3b and 4a orbital energies reported in Table 5.1 are in the same order and are separated by a similar energy as the maxima of the plots of the 3b and 4a scale factors shown in Figure 5.14. Chapter 5. Dimethoxymethane 152 15 10 5 0 15 iSio CD DC 5 0 15 25.0-26.5 eV 0.14:0.06 32.2-33.'6 eV 0.20:0.21 •e—© Inner valence o IV-BES sums b3-6p 3b&4a 26.8-28.3 eV 0.12:0.04 34.0-35.4 eV 0.14:0.15 15 10 28.6-30.1 eV 0.18:0.07 j z : 10 h H 5 — i — • — i — 1 — 30.4-31.8 eV 0.28:0.12-0 15 35.8-37.2 eV 0.12:0.08^ 10 - 5 h 10 h © 0 15 37.6-39.0 eV 0.10:0.05. 10 H 5 0 39.4-40.8 eV 0.09:0.04 -©—0—© - 1 1 1 1 41.2-42.6 eV 0.08:0.04 43.0-44.'l eV 0.07:0.03 0 10 20 30 0 10 20 30 0 10 Azimuthal angle (degrees) -©-20 30 Figure 5.13: Experimental angle profiles (o) of a series of binding energy intervals in the 25-44 eV binding energy range of dimethoxymethane. The results of least-squares fits of the 3b and 4a B 3 L Y P / 6 - 3 1 1 + + G * * theoretical profiles (—) to each of the experimental angle profiles are also shown. The binding energy range used for each profile and the fitted scale factors (3b:4a) are indicated in each panel. Chapter 5. Dimethoxymethane 153 25 30 35 40 45 Binding energy (eV) Figure 5.14: Variat ion of the fitted 3b and 4a scale factors of Figure 5.13 w i t h binding energy. The scale factors were obtained by a least-squares fit of the 3b and 4a B 3 L Y P / 6 -311+-l-G** theoretical angle profiles to experimental angle profiles obtained by summing data over selected energy ranges of the experimental B E S . Each data point is located at the average energy of the binding energy range used for that fit, wi th the binding energy error bars indicating the width of each binding energy range and the scale factor error bars indicating the calculated uncertainties of the fits. The solid lines are a cubic spline fit to the data. 5.6 Computational method and basis set effects The theoretical momentum profiles in Figures 5.3-5.11 and the other calculated properties reported i n Table 5.2 indicate a dependence upon the calculation by which they have been obtained. Effects resulting from both changes in the basis set and theoretical method used are evident. Comparing the H F and D F T ( L S D A , BP86 and B 3 L Y P ) T M P s calculated using the 6-311++G** basis set and shown in Figures 5.3-5.11, several general observations can be made. For the outer-valence orbital momentum profiles (10b-5b), the D F T calculations predict more intensity at low momentum than do the H F calculations. This results in much Chapter 5. Dimethoxymethane 154 higher max ima at p = 0 for the s-type momentum profiles and a shift in the position of the profile maxima towards low p combined with an increase in the profile max ima for the p-type T M P s . In contrast, this behaviour is reversed for the inner-valence 4b and 5a orbital momentum profiles and there are only very slight differences between the 6-311+-f-G** H F and B 3 L Y P T M P s of the most tightly bound 3b and 4a orbitals. The differences between the H F and D F T T M P s are greatest for the L S D A calculations and least for the D F T calculations using the B 3 L Y P functional. Keeping in mind the approximately inverse weighting of p and r space resulting from the Fourier transform relating the two (Equation (1.1)), two separate effects can be used to explain the observed behaviour: 1. A s discussed by Duffy et al . [20], in the local spin-density approximation ( L S D A ) , the incorrect asymptotic form of the exchange-correlational functional results in large-r electrons that are less bound than they should be, causing the orbitals to be more diffuse and consequently yielding momentum profiles that are too high at low momentum. The improved asymptotic form of Becke's gradient corrected exchange functional [105] used in the B P 8 6 calculations lessens the under-binding of the large-r electrons and therefore lowers the T M P s near p = 0. The inclusion of the H F exchange term i n Becke's three parameter exchange functional [108] ( B 3 L Y P ) further improves the description of the large-r electron density and hence further decreases the low p intensity of the calculated momentum profiles. This effect may be expected to become less significant for the inner-valence orbitals because of their more contracted (localized) electron density when compared with the outer-valence orbitals. 2. Ini t ia l state correlation, not accounted for in the H F calculations but included in the D F T calculations, may result in an r-space contraction of the core orbitals as a result of the abil ity of the electrons to avoid each other [20]. If this also occurs for the Chapter 5. Dimethoxymethane 155 4b and 5a inner-valence orbitals, this would explain the decreased intensity at low momentum of the 4b and 5a D F T T M P s in comparison to the H F T M P s . In addition, this contraction of the core and possibly inner-valence orbitals would result in more effective screening of the nuclear charge and lead to more diffuse outer-valence orbitals and consequently outer-valence T M P s with greater intensity at low momentum. In previous E M S studies of N H 3 , H 2 0 and H F [20,45,47,155], the effect of electron correlation on the outer-valence T M P s , accounted for either by extremely large M R S D -C I calculations or density functional methods, has been similar to that observed here for dimethoxymethane, resulting in increased intensity at low momentum. Overal l , the D F T calculations provide better agreement wi th the experimental momentum profiles than do the H F calculations, suggesting that target electron correlation is an i m -portant factor in the momentum profiles of dimethoxymethane. A s was discussed i n Section 5.5.1, the differences between the D F T and C I 11a T M P s , i l lustrated in Figure 5.4, do not contradict this conclusion if they are the result of an insufficiently large M R S D - C I calcula-t ion. The energies reported in Table 5.2 also display a sensitivity to theoretical method. A l -though it is not meaningful to directly compare the total energies calculated by the H F and pos t -HF methods with those from the D F T calculations, the relative electronic ener-gies of the gg and ga conformers ( A E ^ g a - g g ) are comparable. A s noted i n References [140] and [141], the inclusion of electron correlation effects results in an increase in the calculated energy difference between the gg and ga conformers. This may be seen in Table 5.2 by comparing A # 0 , g a - g g from the H F / 6 - 3 1 + G * and M P 2 / 6 - 3 1 + G * calculations and from the H F / 6 - 3 1 1 + + G * * and the various D F T / 6 - 3 1 1 + + G * * calculations. The comparatively large value of Ai?o ,ga-gg obtained by the M P 2 / 6 - 3 1 + G * calculations is likely too large as a result of the rather modest basis set used and the incomplete accounting of electron correlation. Chapter 5. Dimethoxymethane 156 c 0 Q) 4 -C D £ 2 0 5 C D DC 0 0 tz,229.6+g*,6++g HF TMPs 10b I — 1 — )++g** *6p,6+g*,229 4z 11a o 2 3 0 1 Momentum (au) Figure 5.15: Basis set dependence of the 10b (left panel) and 11a (right panel) Hartree-Fock theoretical momentum profiles of dimethoxymethane. The T M P s were calculated using the M P 2 / 6 - 3 1 + G * optimized gauche-gauche geometry. T M P s shown are the results of H F calculations using the following basis sets: S T O - 3 G (st), 6-31G (6g), 6-31+G (6+g), 6-31G* (6g*), 6-31+G* (6+g*), 6-31++G** (6++g**), 6-311++G** (6p), aug-cc -pVTZ (tz) and the 2 2 9 - C G F basis set (229) described in Section 5.3. The experimental angular resolution has been accounted for using the G W - P G method [48]. This is supported by the observation by Wiberg and Murcko [140] that increasing the size of the basis set results in a decrease in A£Jo,ga-gg (this trend is also evident from the H F results in Table 5.2) and the fact that the M P 2 / 6 - 3 1 + G * total energy for the gg conformer is 0.02 hartree lower than that obtained from the C I / 2 2 9 - C G F calculation, which may be expected to be more accurate. To further assess the dependence of the calculated results upon basis set, additional H F and D F T - B 3 L Y P calculations have been performed. 10b and 11a theoretical momentum profiles of the M P 2 / 6 - 3 1 + G * optimized gg conformer of dimethoxymethane, calculated using a range of basis sets, are shown in Figure 5.15 obtained using the H F method and in Figure Chapter 5. Dimethoxymethane 157 Momentum (au) Figure 5.16: Basis set dependence of the 10b (left panel) and 11a (right panel) B 3 L Y P - D F T theoretical momentum profiles of dimethoxymethane. The T M P s were calculated using the M P 2 / 6 - 3 1 + G * optimized gauche-gauche geometry. T M P s shown are the results of D F T calculations using the B 3 L Y P exchange-correlation functional and the following basis sets: S T O - 3 G (st), 6-31G (6g), 6-31+G* (6+g*), 6-311++G** (6p) and aug-cc -pVTZ (tz). The experimental angular resolution has been accounted for using the G W - P G method [48]. 5.16 obtained using D F T wi th the B 3 L Y P functional. Several variations of the 6-31G basis set involving augmentation wi th diffuse and polarization functions (Section 2.3.4) have been used. In addit ion, the T M P s obtained using the minimal S T O - 3 G basis set and the 6-311++G** , 2 2 9 - C G F and aug-cc-pVTZ basis sets are shown. The 10b and 11a T M P s have been selected for this comparison because the results in Figures 5.3 to 5.11 indicate that the outer-valence orbital T M P s are the most sensitive to variations in the computational method and because the outer-valence orbitals are of greatest significance to chemical behaviour. The corresponding total energies and dipole moments from these calculations are shown in Table 5.3, along wi th the momentum values corresponding to the maxima of the 10b T M P s (pMAx) and the maximum intensities of the 11a T M P s . Chapter 5. Dimethoxymethane 158 Table 5.3: Basis set and computational method dependence of calculated properties of the gauche-gauche (gg) conformer of dimethoxymethane. The calculations were performed using the M P 2 / 6 - 3 1 + G * optimized geometry. Calculat ion Total energy Dipole moment 10b p M A X 11a max (hartree) (debye) (au) (arb. units) H F / S T O - 3 G -264.55318 0.135 1.510 3.211 H F / 6 - 3 1 G -267.82738 0.342 0.809 5.163 H F / 6 - 3 1 + G -267.83452 0.314 0.793 6.051 H F / 6 - 3 1 G * -267.95148 0.301 0.816 4.934 H F / 6 - 3 1 + G * -267.95876 0.294 0.794 5.624 H F / 6 - 3 1 + + G * * -267.97126 0.289 0.791 5.683 H F / 6 - 3 1 1 + + G * * -268.03074 0.277 0.787 5.648 H F / a u g - c c - p V T Z -268.06017 0.241 0.780 5.526 H F / 2 2 9 - C G F -268.06916 0.256 0.780 5.631 C I / 2 2 9 - C G F -268.71508 0.263 0.779 5.548 L S D A / 6 - 3 1 1 + + G * * -267.43708 0.324 0.735 8.028 BP86/6 -311++G** -269.63211 0.312 0.736 7.664 B 3 L Y P / S T O - 3 G -266.03937 0.153 1.457 2.986 B 3 L Y P / 6 - 3 1 G -269.46239 0.361 0.789 6.176 B 3 L Y P / 6 - 3 1 + G * -269.56528 0.333 0.752 7.150 B 3 L Y P / 6 - 3 1 1 + + G * * -269.63910 0.301 0.743 7.155 B 3 L Y P / a u g - c c - p V T Z -269.66590 0.261 0.739 6.926 Chapter 5. Dimethoxymethane 159 Considering the H F calculations, it is clear that the S T O - 3 G basis set is inadequate for describing the valence orbital momentum densities of dimethoxymethane. Calculations using this basis set tend to seriously underestimate the intensity at low momentum, reflecting inadequacies in modeling the outer spatial regions of electron density, which approximately correspond to low momentum because of the Fourier-transform relationship between p and r space. In addition, the total energy and dipole moment from the H F / S T O - 3 G calculation (Table 5.3) differ considerably from those calculated using larger basis sets. The use of the 6-31G basis set results in a considerable improvement in the calculated energy for the gg conformer as well as a large increase in the predicted low p intensity for the 10b and 11a T M P s . This behaviour is typical of that observed for the other valence orbitals of dimethoxymethane shown in Figures 5.5-5.11. In general, the H F / 6 - 3 1 G calculations are in much better agreement wi th experiment than are the H F / S T O - 3 G T M P s . However, even for those momentum profiles where the H F calculations are in reasonable agreement with the experimental data (Figures 5.8-5.11), the H F / 6 - 3 1 G T M P s tend to underestimate the intensity at low momentum. The addition of diffuse functions to the heavy atoms (6-31+G) improves the description of the low momentum region, resulting in a shift in the location of the max i m um of the p-type 10b profile (PMAX) towards lower momentum and a considerable increase i n the maximum of the 11a T M P (Figure 5.15 and Table 5.3). The calculated dipole moment also changes significantly from the H F / 6 - 3 1 G value. Note, however, that only a small improvement in the total energy results. These effects are reasonable since the addition of diffuse functions w i l l improve the description of the outer spatial regions of the electron density and consequently the low p region of the momentum density but w i l l have litt le effect on the energy, to which the outer regions of electron density contribute l i tt le. In the case of the 11a T M P , the H F / 6 - 3 1 + G calculation predicts more intensity near zero momentum than do any of the other H F calculations considered here, including those using Chapter 5. Dimethoxymethane 160 substantially more complete basis sets, suggesting that in this case, the under-emphasis of the diffuse electron density has been over-corrected. In contrast, the addition of heavy-atom polarization functions to the 6-31G basis set, giving the 6-31G* basis set, results i n only a small change in the T M P s , particularly for the 10b orbital , but a considerable (> 0.1 hartree) improvement in the total energy and a somewhat greater change in the dipole moment than was obtained by adding the diffuse functions. The dependence of the dipole moment on the intermediate regions of electron density results in the observed significant changes in the calculated values for this property using both the 6-31+G and 6-31G* basis sets. The H F / 6 - 3 1 G * T M P s actually have slightly less intensity at low p than do the H F / 6 - 3 1 G T M P s , suggesting that the addition of polarization functions to the basis set improves the description of the small and medium r electron density at some expense to the description of the large r density. The inclusion of both heavy-atom diffuse and polarization functions has the expected effect, resulting (in comparison to the H F / 6 -3 1 G * results) in only a slight improvement in the total energy but markedly increasing the predicted low p intensity in the 10b and 11a T M P s . The further addition of diffuse and polarization functions to the hydrogen atoms (the 6-31++G** basis set) results in litt le change in the 10b and 11a T M P s . This is not unexpected, since these two outer-valence orbitals consist of electron density centred predominantly on the oxygen atoms, wi th some electron density on the carbons but little on the hydrogens. Interestingly, a comparison of the remaining H F / 6 - 3 1 + G * and H F / 6 - 3 1 + + G * * valence T M P s of dimethoxymethane (not shown) indicates that the addition of diffuse and polarization functions to the hydrogen atoms also has l i tt le effect on the T M P s of orbitals with significant C - H bonding character. Slight changes i n the total energy and dipole moment of the gg conformer of dimethoxymethane do result w i t h the inclusion of hydrogen diffuse and polarization functions. The use of even larger basis sets for the H F calculations (i.e., 6-311++G**, 2 2 9 - C G F and aug-cc-pVTZ) Chapter 5. Dimethoxymethane 161 results in l itt le further change in the T M P s , indicating that the valence momentum profiles of dimethoxymethane are reasonably converged at the H F level using the 6-31+G* basis set. In the case of the inner-valence 6a, 4b and 5a momentum profiles (Figures 5.9-5.11), the H F / 6 - 3 1 G and H F / 6 - 3 1 1 + + G * * T M P s are nearly the same, indicating that the inclusion of diffuse and polarization functions is less important for describing the inner-valence orbitals, which are more atomic-like than the outer-valence orbitals. In the case of the D F T calculations using the B 3 L Y P functional, a comparison similar to the one just discussed for the H F method (but using a smaller number of basis sets) indicates essentially the same basis set dependence as was observed for the H F calculations (Figure 5.16 and Table 5.3). These results indicate the importance of considering basis set effects and electron density emphasis when performing quantum mechanical calculations. If only a single property is being calculated, it is essential to choose a basis set that w i l l accurately model the appropriate region of the electron density. For example, a basis set that is well suited to calculating total energies may give very poor results for properties pr imari ly dependent upon the large r regions of electron density. W h e n several molecular properties are off interest, the basis set must be sufficiently flexible—containing both polarization and diffuse functions—to reasonably model al l regions of electron density [45,46]. 5.7 Consideration of other conformers In the analysis of the experimental momentum profiles of dimethoxymethane in Section 5.5, it was assumed that the experimental results are dominated by the gauche-gauche conformer; consequently, contributions from other conformers of dimethoxymethane were ignored. The M P 2 / 6 - 3 1 + G * electronic energy difference between the gg and ga conformers ( A . E 0 , g a - g g ) is 14.17 k J / m o l (Table 5.2), suggesting that at the experimental temperature of 25°C it is Chapter 5. Dimethoxymethane 162 Table 5.4: Relative thermodynamic quantities of the conformers of dimethoxymethane. The values are based upon B 3 L Y P / 6 - 3 1 + G * geometry optimization and frequency calculations. Reported values are relative to those of the gg conformer at the experimental temperature of 298 K . Conf. A £ 0 AEzpv A ( A i J ( 2 9 8 ) ) AS298 AG298 9 A b u n d . (kJ /mol) (kJ/mol) (kJ/mol) ( J /mol -K) (kJ/mol) (%) gg 0.0 0.0 0.0 0.0 0.0 2 89.87 ga 10.618 -0.894 0.142 7.106 7.748 4 7.90 g + g ~ 15.878 -0 .739 0.418 21.409 9.173 2 2.22 aa 24.535 -2 .179 0.594 6.321 21.066 1 0.01 reasonable to consider only the gg conformer. However, the values for A£o,ga-gg obtained by the other calculations shown in Table 5.2 are, wi th the exception of the H F / 6 - 3 1 G value, less than the M P 2 / 6 - 3 1 + G * value. In addition, published estimates of the energy difference between these conformers determined using various experimental techniques range from 5.9 to 10.5 k J / m o l [135-138]. In view of this considerable uncertainty in A£'o,ga-gg and the discrepancies between the X M P s and the gg T M P s for some momentum profiles (particu-larly 9b), an investigation of the effect of including contributions from other conformers of dimethoxymethane when analyzing the E M S data has been performed. The relative Gibbs free energies (AG 2 9s) of the gg, ga, g + g ~ and aa conformers of d i -methoxymethane have been determined using the procedures described in Section 2.4.2. The calculations are based upon optimized geometries and vibrational frequencies obtained using D F T w i t h the B 3 L Y P functional and the 6-31+G* basis set. This level of theory was used rather than the M P 2 / 6 - 3 1 + G * level because of the considerable gain in efficiency realized by the D F T calculations over M P 2 calculations using the same basis set. The results of these free energy calculations are shown in Table 5.4. The B 3 L Y P and M P 2 optimized ge-ometries are discussed i n Section 5.8 below. In the absence of a specific recommendation for a scale factor for use with B 3 L Y P / 6 - 3 1 + G * vibrational frequencies, al l vibrational fre-quencies were scaled by a factor of 0.963 (recommended by Rauhut and Pulay [156] for use Chapter 5. Dimethoxymethane 163 with F J3LYP/6 -31G* vibrational frequencies) prior to the calculation of the thermodynamic quantities shown in Table 5.4. The relative free energy of each conformer has been calculated using Equation (2.38) at the experimental temperature T = 298 K . A l l values are relative to those of the gg conformer. The relative populations of the conformers have been calculated using Equation (2.37). The consideration of vibrational and rotational contributions to the relative conformer free energies (Table 5.4) results in a considerable decrease in the energies of a l l other con-formers relative to that of the most stable gg conformer. The predicted population of the aa conformer remains negligible at the experimental temperature, so contributions from this conformer may reasonably be neglected when analyzing the experimental data. A popula-t ion of « 8% for the ga conformer, however, is sufficiently large that it could potentially have a discernible impact on the momentum profiles. Theoretical momentum profiles of the weighted conformer sum have been obtained by summing the individual gg, ga and g + g ~ conformer T M P s , weighted by the calculated conformer populations at 298 K , for each or-bital or sum of orbitals corresponding to the experimental momentum profiles. The results for orbitals 21+20, 19, 18+17 and 16+15+14 are indicated by the dashed lines in Figure 5.17 and are compared with the respective gg conformer T M P s (solid lines). M O numbers rather than symmetry labels have been used because of the different symmetries of the conform-ers. In obtaining these conformer sum T M P s , the variation of the ionization potentials wi th conformer has been neglected. This assumes that the ionizations from a particular orbital for a l l conformers w i l l be contained with in the corresponding peaks fitted to the binding energy spectra. A n examination of the H F / a u g - c c - p V T Z / / M P 2 / 6 - 3 1 + G * orbital energies for the gg and ga conformers (Table 5.1) suggests that this is a good assumption for the inner-valence orbitals but is less reasonable for the outermost valence orbitals. It is apparent from Figure 5.17 that contributions from conformers other than gg do Chapter 5. Dimethoxymethane 164 c/5 C D 10 MOs 21+20 5 ^ MO 19 8 4 6 \ 3 -I \ 4 \ 2 7 \ 2 \ 1 \ -0 I , I 0 i i i i •- 0 £ 20 10 0 0 0 MOs 18+17 ggoniy + gg+ga+g g 30 20 10 h 0 1 2 0 1 Momentum (au) Figure 5.17: Comparison of dimethoxymethane outer-valence theoretical momentum profiles of the gauche-gauche (gg) conformer (solid lines) and of a conformer sum (dashed lines) weighted using calculated conformer abundances at 298 K . The T M P s were obtained from B 3 L Y P / 6 - 3 1 + G * / / B 3 L Y P / 6 - 3 1 + G * calculations of each conformer. The experimental an-gular resolution has been accounted for using the G W - P G method [48]. Chapter 5. Dimethoxymethane 165 not dramatically affect the momentum profiles of dimethoxymethane at the experimental temperature of 298 K . Of the profiles shown in Figure 5.17, only that for M O 19 shows any significant change when the ga and g + g ~ conformers are included. In this case, the indiv id-ual ga and g + g ~ T M P s are both s-type, leading to the observed increase in intensity at low momentum. This results in slightly improved agreement between theory and experiment for M O 19 (9b in the case of the gg conformer, Figure 5.5). However, the bulk of the discrepancy between theory and experiment for the 9b momentum profile is not accounted for by the inclusion of contributions from other conformers of dimethoxymethane. This change in the shape of the M O 19 T M P upon inclusion of other conformers does suggest an alternative explanation for the observed disagreement between the 9b X M P and the T M P s (Figure 5.5)—namely vibrational effects. The low calculated vibrational frequency corresponding to torsions about the 0 - C H 2 bonds (the B 3 L Y P / 6 - 3 1 + G * value is 98.2 c m - 1 ) indicates that there is appreciable excitation of this vibrational mode at the experimental temperature. Consequently, the approximation of using the equilibrium geometry in performing the T M P calculations (Section 2.2.2) may be inaccurate. Testing of this hypothesis requires the cal-culation of vibrationally averaged T M P s of dimethoxymethane (as was done recently for the l b 3 u T M P of ethylene [71]), a considerable computational task for a molecule of the size of dimethoxymethane. In general, the profiles shown in Figure 5.17 suggest that the population of the gg conformer is sufficiently great at 298 K to allow the E M S data to be interpreted on the basis of the presence of the gg conformer alone, as was assumed in Section 5.5. 5.8 Optimized geometries The calculated theoretical momentum profiles discussed in Sections 5.5 and 5.6 above have been obtained using M P 2 / 6 - 3 1 + G * optimized geometries, while those in Section 5.7 were Chapter 5. Dimethoxymethane 166 calculated using B 3 L Y P / 6 - 3 1 + G * optimized geometries. In comparing the experimental and theoretical momentum profiles and interpreting any differences, the sensitivity of the con-former geometries to the theoretical method used for the geometry optimizations as well as the sensitivity of the T M P s to small changes in the conformer geometries should be consid-ered. To this end, additional geometry optimizations of the gg and ga conformers of dimeth-oxymethane have been performed using density functional theory wi th the B 3 L Y P functional and the 6-311+-t-G** basis set. Optimized parameters of the gg conformer from this work are compared in Table 5.5 with experimental values obtained by electron diffraction [134] and w i t h the results of M P 2 / 6 - 3 1 G * * , M P 2 / 6 - 3 1 1 + + G * * and B 3 L Y P / 6 - 3 1 G * * geometry optimizations recently reported by Kneisler and All inger [142]. A s noted by Kneisler and All inger , the calculated bond lengths show little variation between calculations. In the case of the C H 2 - 0 bonds, the calculated equilibrium values exceed the experimental average value by approximately 0.02 A. Kneisler and All inger [142] recommend a value of —0.0079 A for conversion from average bond lengths to equilibrium values. The calculated bond angles are smaller than the experimental values (with the exception of the B 3 L Y P / 6 - 3 1 G * * O C O an-gle), wi th the B 3 L Y P results in somewhat closer agreement wi th experiment than the M P 2 results. The dihedral angle d C O C O is the parameter most sensitive to the computational method. A l l of the calculated values of d C O C O are greater than the experimental value of 63.3°, by about 2-3° in the case of the M P 2 calculations and about 5° in the case of the density functional theory results. A comparison of the calculated parameters of the ga conformer (Table 5.6) results in similar observations. The M P 2 and B 3 L Y P C O bond lengths calculated using the three basis sets differ by no more than 0.010 A and generally by much less. A s was noted for the gg conformer, the D F T bond angles are greater than the corresponding M P 2 values. In the case of the dihedral angles, there is little disagreement between M P 2 and D F T for the one Chapter 5. Dimethoxymethane 167 cu a co - d QJ a >> X o - f l +J cu a S - l cu o o OT CD CU S H tuO CU T3 co CD co -3 fl co fl CD f-i CD , x fl R ° 2 ^ S> p CO CO r-J bO - f l cb cu -fl ' CJ fl OT co <D bo o -fl - 1 - 3 OT o + + i - H i - H CO I co PH CO PQ CO b H CO fl o r d co CU S H CO OT S H O) - 4 J CU B CO S H CO PH cj — •r! OT cu O CO CD ,—i bO CO I , S H CO -£i CD a S CD I f l .a -a OT fl O - Q co o •I o fl CD cO OH cu a C OT . . CD 0 3 'bb fl CO cu cu o O 0 fl co OT fl O - Q 0 1 -O •X-a co i co * + + i - H i - H CO I co CM PH CO o o o CD OT H fl cO LO " g LO co o H PH CO rH -|H> fl a> cj cu r f l s O I* 0 r - H 1 co I CO .O a CO I co CO r H , V <D r H CN CM 00 CO CO CO CO CO i - H r - H co r - H i - H CO I-H r - H LO LO o CM 00 CO t> oo Ol oq r - H r - H CO CO oo i - H T-H CO i - H i - H X H CM O CM O O J^H LO 1-H r - H CO i—1 i - H CO t - H r - H CO LO o CM CO CO a i i - H o I—1 i—1 CO 00 T—H t - H CO T-H T-H CM CM o CM CM LO CO o CO LO r - H i - H CM CO co i - H r - H CO 1—1 i—1 LO LO o CM CO 05 t~ LO LO i - H r H T-H CO I—H r - H CO r - H I-H oo 05 o CM LO CO i - H r - H CM CO L6 r H i - H CO i - H i—1 o u o O O o o o O o o o o U S-. <3 C3 co cu CJ fl CD in M H <D CO - u CO fl O O CO CM CD CJ fl CU S H . 0 3 CU fl o rH o .2 a V r f l 0 W H & Chapter 5. Dimethoxymethane 168 o £ a 3 C O co LO o T " H i—i T - H l>- o T - H C O CM O CO o T - H T - H T - H T " H T - H \ T - H T - H T—1 lO ~ -— 00 CM CD o C O ^ CM CM i—i T—i T - H CD T " H CM OS CM Oi T - H T - H CD o ,—1 T - H CM O L6 T—1 1—1 T 1 T - H T - H l>- T - H T - H T - H T - H LO oo 00 CM T - H CM co CD 00 i — i i—1 C O T - H CD T " H CD T - H o CM CN T - H T - H 00 T - H CD CM o CO o 1—1 T - H T - H T - H t -H T - H T - H T - H CD 0O " ^ • ^ 00 CM C O C O CM T - H T - H ^t* T - H CD T - H CM T - H T - H T - H T - H T - H O O oo l>- C O CD oi oo T - H T " H T " H • T - H o c- T - H T - H T - H T - H TO CM ~-00 CM -T r - H C O CM C O T—i T - H CM LO T - H CD T - H ^t 1 T - H OJ T - H T - H CM LO T - H LO ^ T - H oi L6 r - H T - H T - H o T - H ^ T - H T - H T - H r~ LO — oo CM CM C O T - H oo T - H T—i CM T - H CD T - H 00 CM o o l>-T - H CM -tf LO -tf T - H r - i oi CO T - H T - H T - H o t>- T - H \ T - H T - H T—1 Ol o — ~" 00 C O : — ' T - H C O 00 OJ T—i T—i CM -tf T - H CD T - H o o o o o o o s~ s- e O O U O O O O < SH o T ^ EH 18 Chapter 5. Dimethoxymethane 169 which is anti . In the case of the gauche dihedral angle, the D F T values are approximately 2° greater than the M P 2 results. The generalized anomeric effect is evident in comparing the two C H 2 - 0 ( r C O ) bond lengths of the ga conformer of dimethoxymethane. The length of the C H 2 - 0 bond about which the dihedral angle is approximately 65° is about 0.03 A less than that corresponding to the near 180° dihedral angle. Averaging these two bond lengths results in values very similar to (0.004 A less than) the calculated C H 2 - 0 bond lengths for the gg conformer. This is reasonable in light of the nn —» o* explanation of the anomeric effect; in the case of the ga conformer, the C H 2 - 0 bond on the gauche side of the molecule is shortened because of increased C - 0 bonding character and the one on the anti side is lengthened as a consequence of increased C - 0 antibonding character. For the gg conformer, each C H 2 - 0 bond w i l l be affected by both of these effects, resulting i n a bond length approximately the average of the two for the ga conformer. In the case of the 0 - C H 3 bond lengths for the ga conformer, the ones for the gauche methoxy group are essentially the same as the corresponding gg conformer values, while those for the anti methoxy group agree wi th the corresponding aa conformer values reported in Reference [142]. The good agreement between B 3 L Y P / 6 - 3 1 G * * and M P 2 / 6 - 3 1 1 + + G * * relative energies (Table 5.6), bond lengths and bond angles noted by Kneissler and Al l inger [142] does not appear to be fortuitous upon consideration of the additional B 3 L Y P calculations performed in the current work. The D F T results vary less with changes in basis set than do the M P 2 results. Consequently, the B 3 L Y P / 6 - 3 1 1 + + G * * results are in good agreement w i t h both the B 3 L Y P / 6 - 3 1 G * * and M P 2 / 6 - 3 1 1 + + G * * results. In summary, the optimized M P 2 and B 3 L Y P geometries are reasonably consistent with one another, wi th the most significant differences occuring for the calculated dihedral angles. Single point H F and B 3 L Y P D F T calculations using the 6-311++G** basis set have been performed using the three optimized gg conformer geometries calculated i n the present work. Chapter 5. Dimethoxymethane 170 Selected calculated properties are compared in Table 5.7. The same trends are evident wi th both the H F and D F T calculations. The total energy improves from the M P 2 / 6 - 3 1 + G * geometry to the B 3 L Y P / 6 - 3 1 + G * and B 3 L Y P / 6 - 3 1 1 + + G * * geometries. The calculated dipole moment and location of pMAX of the 10b T M P decrease in the same order as do the total energies and the intensity of the 11a T M P at zero momentum follows the reverse trend. Smaller differences are obtained between the calculations using the two B 3 L Y P geometries than between the M P 2 and B 3 L Y P geometries. The changes in these properties track the changes in the dihedral angle d C O C O reported in Table 5.5. The significance of the variations of the T M P s calculated using different optimized geometries has been assessed by comparing valence orbital B 3 L Y P / 6 - 3 1 1 + + G * * T M P s calculated using the M P 2 / 6 - 3 1 + G * , B 3 L Y P / 6 -3 1 + G * and B 3 L Y P / 6 - 3 1 1 + + G * * optimized geometries of the gg conformer. The results for the 10b, 11a and 9a T M P s are shown in Figure 5.18. For al l of the valence orbitals of the gg conformer, T M P s calculated using the B 3 L Y P / 6 - 3 1 + G * and B 3 L Y P / 6 - 3 1 1 + + G * * optimized geometries are essentially indistinguishable. For about half of the valence orbitals, T M P s calculated using the M P 2 and D F T geometries are also essentially identical. A l l but two of the remaining orbitals show very small variations of a magnitude similar to that seen for the 10b T M P s in Figure 5.18. The 11a and 9a T M P s are the only ones that display significant changes wi th geometry. In view of these observations, it may be concluded that small differences in the geometries used to calculate T M P s of dimethoxymethane do not, for the most part, result in significant changes in the resulting momentum profiles. However, the possibility of the choice of an inaccurate conformer geometry for the calculation of T M P s cannot be ruled out as a factor contributing to differences between theoretical and experimental results for a small number of the dimethoxymethane momentum profiles. Chapter 5. Dimethoxymethane >> S H CD a o CD fc>0 H-H o CO > CD CD CD S-H co CD S3 CO X3 O CD O SH CD a S H o o 'So CD X5 0 co hO 1 CD X5 o co CD +J <4H O ai CD CD P H o CD o "al • U § • r-H LO .N ^ a CO +s X 'fH co PI co s-< r-H r-H a, o d g f—\ S3 CD <—i O H £ o Q a bO s-< SH CD CD CD 73 £ E-i s3 cd Pi O O 00 o o -tf r - l l > C P CM CM LO co co r- c- -tf oo co co C— 1^— O O O C— I > LO N H CO CM CM r-H o d d -tf C O 00 r> C M to O r-H r-H C O C O C O O C P o co oo oo co co cp CM CM CM * o * + T " r-H r-H r-H CO CO I I C P C P r-H r-H co co PQ PQ # •*• * -x- *• O O O + + + + + + CO C O C O I I I co^  co^  co^  PH PH MH MH MH LO C P C P LO CO CO r-H CT) CT) N N N co co co ^ CM CM t -C P C P C P H 00 o O C M C O CM CM o d d O CT) C O T-H CO -tf CT) CJ5 CJ5 CO CO C O CO C D C D CT) CT) CT) C D C O C D CM CM CM -X-o * + + + -r T - H -x o , . , . + co co C O C D C D C O PH P  r-H r J C O C O PQ m CM PH -X- -X- * * * * o o o + + + + + + r-H I—I r-H r-H r-H r-H CO CO C O I I I CO^ CO^ C O PH PH PH P>H P>H r H r J r J CO CO C O CQ P3 pq Chapter 5. Dimethoxymethane 172 level of geometry optimization MP2/6-31+G* B3LYP/6-31+G* B3LYP/6-311++G, 1 1 1 - A i 1 9a --i i • 2 3 0 1 Momentum (au) Figure 5.18: Variat ion with geometry of the 10b (upper left panel), 11a (lower left panel) and 9a (lower right panel) theoretical momentum profiles of dimethoxymethane. The T M P s have been obtained from single-point B 3 L Y P / 6 - 3 1 1 + + G * * calculations using gauche-gauche conformer geometries optimized at the M P 2 / 6 - 3 1 + G * (solid lines), B 3 L Y P / 6 - 3 1 + G * (long dashed lines) and B 3 L Y P / 6 - 3 1 1 + + G * * (short dashed lines) levels. The experimental angular resolution has been accounted for in the theoretical profiles using the G W - P G method [48]. Chapter 6 Glycine 6.1 Introduction A n understanding of the electronic structure and electron density distribution of amino acids is of considerable importance because of their role as the basic structural units of proteins. Information concerning the electron density distributions of proteins is relevant to areas of protein biochemistry concerned wi th conformation, molecular recognition and reactivity. However, the large size and structural complexity of proteins considerably l imits the range of experimental and theoretical methods that can be applied to the study of these fundamental biochemical molecules. Consequently, studies of simpler systems are necessary both to gain experimental information that can possibly be applied to larger molecules and to evaluate theoretical methods prior to their application to larger systems. A s the simplest amino acid, glycine ( N H 2 C H 2 C O O H ) is an important model compound in biochemistry. Glycine is small enough that a broad range of theoretical methods can be employed and experimental data are attainable for comparison with theory, making this molecule an obvious test case for the evaluation of approximate theoretical methods being considered for use w i t h larger molecules. 173 Chapter 6. Glycine 174 Despite being the smallest amino acid, glycine has proven to be a challenging system for theoretical study. The neutral glycine molecule, the form found in the gas phase, has consid-erable conformational flexibility as a result of the three bond axes about which torsions can occur—namely the N - C , C - C and C-O(hydroxyl ) bonds. This conformational flexibility, in combination wi th several intramolecular hydrogen bonding possibilities, results i n numer-ous stable conformers and thus considerably increases the challenge of accurate theoretical modeling. M u c h of the theoretical work on glycine to date has focussed on determining the geometries and relative energies of these stable conformers. It is apparent from this quan-t u m mechanical work that the results for glycine are very sensitive to the theoretical method employed and to the nature of the basis set used. In particular, it has been found that the results from semiempirical and H F calculations vary considerably [79]. Conclusions based on H F calculations regarding the geometries and relative energies of the glycine conformers that are energy min ima on the glycine potential energy surface are strongly basis set de-pendent [79,80,82]. The use of flexible basis sets containing both diffuse and polarization functions appears necessary in order to perform meaningful geometry optimizat ion calcula-tions on glycine. This has been attributed to the presence of several non-bonding electron pairs and intramolecular hydrogen bonds [82]. Electron correlation effects have also been shown to play a significant role in glycine, wi th their inclusion, whether through M 0 l l e r -Plesset ( M P ) perturbation theory [81,82,157], CI [83] or D F T [84,85], having a particularly significant effect on the relative conformer energies. W i t h respect to determining geometries, there is some disagreement as to the necessity of employing theoretical methods that take account of electron correlation effects in glycine [157-159]. Regardless, recent high level calculations using large basis sets and M P 2 [81], CI [83] and D F T wi th hybrid function-als [84, 85] are in fairly close agreement not only for the conformer geometries but also for their relative energies. Chapter 6. Glycine 175 The need to perform relatively large calculations in order to obtain dependable results for the geometries and relative energies of the glycine conformers raises the question as to the sensitivity of other calculated properties of glycine to basis set size and composition and to electron correlation effects. Because of the importance of the outer valence orbitals—and in particular the H O M O electron density—to chemical reactivity [3,4], it is of particular interest to investigate the importance of the considerations mentioned above to the valence orbital electron density distributions of glycine. A s mentioned in Section 1.1.2, E M S mea-surements and associated calculations have clearly shown that electron correlation effects are a determining factor in the long range (low momentum) parts of the outer-valence or-bi ta l electron density distributions in N H 3 [9,47] and the methylamines [160]. Therefore, since for most conformers of glycine the H O M O electron density is predominantly located on the nitrogen atom, it is of interest to see if similar electron correlation effects are also a determining factor in the valence orbital electron density distributions of glycine. In this chapter, the first detailed and comprehensive E M S study of the outer-valence momentum profiles of glycine is presented. The improved sensitivity provided by the mul-tichannel electron momentum spectrometer described in Chapter 3 and the development of the heated sample probe shown in Figure 3.2 have been key to making these experimental measurements feasible. The roles of basis set size and composition and also electron corre-lation effects in modeling the outer-valence electron densities of glycine have been studied by comparing the experimental measurements to H F and D F T calculations using a range of basis sets. In view of the success of D F T , v ia the T K S A , in describing electron correlation effects in the valence momentum profiles of small molecules [20,21,71], this study of glycine is of particular interest to further evaluate the effectiveness of the T K S A for describing the valence electron densities of larger molecules. Several other previously published studies are pertinent to the current work. The outer-Chapter 6. Glycine 176 valence photoelectron spectrum of glycine has been reported by Debies and Rabalais [161] in 1974 and by Cannington and H a m [162] in 1983 using He(I) and He(II) resonance radiation respectively. B o t h studies found three peaks in the low binding energy region, at 10.0, 11.1 and 12.1 eV, assigned as ionizations from non-bonding nitrogen ( H O M O ) , carbonyl oxygen and hydroxyl oxygen orbitals, respectively. The two studies made the same assignments for the symmetry of the second and third orbitals (a' and a", respectively) but differed on the symmetry of the H O M O , with Debies and Rabalais predicting an out of plane (a") orbital and Cannington and H a m predicting an in plane (a') orbital . Recently, H u , Chong and Casida [163] have calculated the IPs of the two most stable conformers of glycine using a parametrized second-order Green function times screened interaction (pGW2) method. Their calculated IPs of the global-minimum conformer (Ip, see .below) are in very good agreement w i t h the experimental values reported by Cannington and H a m [162]. A s to the assignment of the photoelectron spectrum, the calculations of H u et al . [163] support the assignment by Cannington and H a m [162] for the H O M O (i.e., a'), but indicate that the previous assignments [162] of the ninth and tenth IPs (i.e., ionizations from M O s 12 and 11) were reversed and should be a' and a", respectively. A s stated above, there are several stable conformers of glycine present in the gas phase. In fact, theoretical studies have found as many as eight conformers [79-84], which are il lustrated in Figure 6.1. The conformer labels are those used by Csaszar [81], where p and n denote planar ( C s symmetry) and non-planar (Ci ) heavy-atom frameworks, respectively. Only a few of these conformers have been observed experimentally. B o t h conformers Ip and Hp have been identified by microwave spectroscopy [73-77]. In the recent study by Godfrey and Brown [77], the non-observance of conformers other than Ip and Hp, in combination wi th a consideration of their instrumental sensitivity and estimated absorption coefficients for conformers Hp and Hip, ini t ial ly lead to the prediction of an upper l imi t of 0.2 for Chapter 6. Glycine 177 H H O - H H H O H H O \) / / \1 / c - c c - c c - c / \ H 1 ^ H / / \ N O N O H H H H H H Ip Hp H i p 52.6% 9.0% 29.6% H ^ H H O- H H O H H x O c-c H ^ c - c 1 c - c / \ / \ / \ O H - N O - H N O \ \ h H H H H IVn V n VIp 6.8% 1.6% 0.3% H \ H H O H H 9 / -y c- -c c - c / \ / \ N ? o h / H H H H VIIp V l l l n 0.1% 0.0 % Figure 6.1: Conformers of glycine. Following the labeling scheme used i n Reference [81], p denotes a C s symmetry conformer and n a C i symmetry conformer. The calculated abundance of each conformer based on the relative free energies (see Section 6.3 and Table 6.2) at the experimental temperature of 165°C is also indicated. Chapter 6. Glycine 178 the abundance of Hip relative to that of Hp at the experimental temperature of 235°C. In contrast, the most elaborate theoretical studies [81,83,84] predict that conformer IIIp should be present in a proportion considerably larger than this. In this regard, Godfrey and Brown [77] point out that their l imited observations (only Ip and Hp) could be due to relaxation of the other conformers to Ip in the expanding gas jet. A recently published theoretical study [164] of the barriers to interconversion between glycine conformers indicates that this explanation is likely correct. This is further supported by recently reported evidence for a th i rd conformer, obtained by an infrared spectroscopy study of glycine trapped in inert gas matrices [78]. The results of the microwave studies [76,77] are in agreement wi th an electron diffraction study [165] and the various theoretical calculations that predict that the most stable con-former is Ip. In the case of conformer II, the microwave data indicates that this conformer has C s symmetry with a planar heavy-atom framework (i.e. Hp). A t the highest levels of theory employed thus far [81,83,84], the geometry-optimized structure is a C i symmetry conformer (Iln) resulting from a slight out-of-plane twist of the planar heavy-atom skeleton. These calculations also indicate that structure Hp is a saddle-point between two equivalent Iln structures [79,81-83]. However, the energy difference between the Hp and Iln forms is calculated to be quite small and is less than the calculated zero-point vibrat ional energy. Therefore, while the equilibrium geometry may correspond to Iln, the average geometry is expected [77,79-81,83] to be lip, in agreement with the microwave studies. In the case of the th i rd conformer, the calculated energy minimum is either the C s symmetry IIIp or the C i symmetry Hln form, depending upon the theoretical method and basis set used. However, the highest level post -HF and D F T calculations [81,83-85] al l predict IIIp to be the true energy min imum, although the energy difference from IHn is very small (~ 0.03 k J / m o l ) . Of the remaining predicted energy minima, three (IVn, Vn and VHIn) are of C i Chapter 6. Glycine 179 symmetry and two (VIp and VIIp) are of C s symmetry. These C i symmetry conformers (IVn, Vn and Vllln) differ considerably from their C s symmetry analogs both in terms of geometry and energy, in contrast to the situation for Hn and IHn described above. 6.2 Experimental details B i n d i n g energy spectra of gaseous glycine over the energy range of 6-27 eV were recorded sequentially at relative azimuthal angles of 0°, 1°, 2°, 3°, 4°, 6°, 8°, 11°, 14°, 17°, 20°, 25° and 30° using the binning data collection mode at a total energy of 1200 eV. M a n y scans were accumulated over an appreciable measuring time (~ 1100 hours) i n order to improve the signal to noise ratio of the data. The heated sample probe described in Chapter 3 and shown in Figure 3.2 was used to admit gaseous glycine into the experimental interaction region by sublimation of the solid sample (from M C B Chemicals) at 165°C. This temperature was chosen so that the singles and coincidence count rates were similar to those typical ly obtained for gaseous samples (see Section 3.3). Earlier experiments have shown that when glycine is sublimated in this temperature range using glass sample holders, sample decomposition does not occur [166,167]. The experimental energy resolution function (1.5 eV fwhm) and the momentum resolution ( « 0.1 au fwhm) of the spectrometer were determined from measurements of the helium ( I s ) - 1 b inding energy peak and the argon 3p momentum profile, respectively. Hel ium gas was admitted into the collision chamber during the data acquisition for glycine to serve as an internal standard for energy calibration and to aid in the monitoring of the experimental conditions. In particular, it was necessary to ensure that no charging effects occurred because of condensed glycine deposits which built up on the spectrometer during the experiment. N o change in the width or position of the sharp He ( I s ) - 1 peak was detected during the long Chapter 6. Glycine 180 data accumulation period. Furthermore, the He Is X M P , obtained by a least-squares fit of Gaussian functions to the E M S B E S (see Figure 6.4 below) is in excellent agreement wi th the T M P calculated from the highly-correlated He wavefunction of Davidson [168], as shown in Figure 6.2. 6.3 Computational Details In the present work, H F and D F T calculations were performed for each of the eight M P 2 / 6 -311++G** geometry optimized conformers shown in Figure 6.1 and predicted by Csaszar [81] to be energy-minima on the glycine conformational potential energy surface. Calculations were carried out for the C s symmetry conformers Ip, Hp, IIIp, VIp and VIIp (having a planar heavy-atom skeleton) and the C i symmetry conformers IVn, Vn and VHIn. The Hp and IIIp geometries were used for this study even though they are saddle points rather than energy m i n i m a on the M P 2 / 6 - 3 1 1 + + G * * potential energy surface because, as dis-cussed in Section 6.1, in both cases the effective ground state structures are expected to be the C s symmetry conformers. The H F calculations for each of these eight conformers were performed using a series of basis sets of increasing complexity, ranging from the S T O -3 G (st) min imal basis set wi th 30 contracted Gaussian functions ( C G F s ) to the 3 4 5 - C G F near-Hartree-Fock l imit aug-cc-pVTZ (tz) basis set of Dunning and co-workers [114,115]. Theoretical momentum profiles for the valence orbitals of each of the conformers were ob-tained using the T H F A (Equation (2.11)) with each of the H F wavefunctions. Momentum profiles were also calculated using the T K S A (Equation (2.14)) from the Kohn-Sham or-bitals obtained from D F T calculations. The K S - D F T calculations were performed wi th a variant of the aug-cc-pVTZ basis set, referred to in the present work as the t r u n - p V T Z (tt) basis set, from which the heavy-atom f polarization functions and hydrogen d polar-Chapter 6. Glycine 181 0 1 2 3 Momentum (au) Figure 6.2: Experimental and theoretical He Is momentum profiles. The experimental profile (•) is that of the He ( I s ) - 1 ionization peak (see Figure 6.4) and was monitored during the glycine data collection to ensure correct spectrometer operation. The theoretical profile (—) was obtained using Davidson's highly-correlated He wavefunction [168]. Chapter 6. Glycine 182 ization functions have been removed. The resulting 2 4 0 - C G F basis set s t i l l contains d and p polarization functions on the heavy atoms and hydrogens, respectively. This truncation was necessary because the version of the deMon program [100,101] used to perform some of the D F T calculations cannot process f functions. The D F T calculations were performed with in the local spin-density approximation using the Vosko, W i l k and Nusair [99] func-tional ( L S D A ) and also using two gradient-corrected functionals: the first ( B P 8 6 ) consisting of the Perdew 1986 correlation [106] and Becke 1988 exchange [105] functionals and the second ( B 3 L Y P ) being the hybrid functional proposed by Becke [108] and incorporating the exact exchange energy, but with the Lee, Yang and Parr [107] correlation functional replacing that of Perdew and Wang [109]. A summary of the calculations performed and selected calculated properties are shown in Table 6 .1. The basis sets (with the exception of the t r u n - p V T Z basis set described above) and exchange-correlation functionals used are discussed in greater detail in Sections 2.3.4 and 2.3.3, respectively. A l l H F calculations were performed using G A U S S I A N 9 2 [102]. D F T calculations wi th the L S D A and B P 8 6 functionals were performed using the deMon [100,101] density functional program w i t h an "extrafine nonrandom" grid and the B 3 L Y P D F T calculations were performed using G A U S S I A N 9 2 wi th the " In t=F ineGr id" option. The effects on the T M P s of the finite spectrometer acceptance angles (i.e., the momentum resolution) have been folded into the calculated profiles using the G W - P G method [48]. . For comparison with the experimental momentum profiles, the T M P s for the five lowest energy conformers (Ip, Hp, IIIp, IVn and Vn) were Bol tzmann weighted using Equat ion (2.37) according to the experimental temperature of 165°C and their calculated Gibbs free energies at that temperature relative to the free energy of the most stable conformer (Ip), and summed together. The remaining three conformers (VIp, VIIp and VHIn) were not included in this sum since the free energy calculations (see below) indicate that they w i l l Chapter 6. Glycine 183 Table 6.1: Summary of calculations performed and calculated properties of glycine. A l l calculations were performed using the M P 2 / 6 - 3 1 1 + + G * * optimized geometries reported in Reference [81]. K e y Method Basis Set Total energy Dipole moment 3 . (hartree) (debye) Ip Ip H p st H F S T O - 3 G -279.114538 1.1.85 4.638 4g H F 4-31G -282.403960 1.271 6.936 6g H F 6-311G -282.768248 1.258 6.914 6p H F 6-311++G** -282.921717 1.286 6.302 tt H F t r u n - p V T Z f -282.942077 1.275 6.098 tz H F aug-cc-pVTZ -282.951366 1.276 6.097 Id D F T L S D A b t r u n - p V T Z f -282.326642 1.210 5.715 bp D F T B P 8 6 C t r u n - p V T Z f -284.572595 1.182 5.591 b3 D F T B 3 L Y P d t r u n - p V T Z f -284.548905 1.199 5.689 D F T B P 8 6 e c c - p V Q Z 1.213 5.583 experiment 1.0-1.4S4.5-4.6h a Theoret ical values are for a non-rotating, non-vibrating molecule. b T h e Vosko, W i l k and Nusair [99] local exchange-correlation functional was used. c T h e Becke exchange [105] and Perdew correlation [106] functionals were used. d T h e B 3 L Y P functional is a modification of the hybrid functional proposed by Becke [108] and incorporating the exact exchange energy, wi th the Lee, Yang and Parr [107] correlation potential replacing that of Perdew and Wang [109]. C a l c u l a t e d by Chong [169] using the C C S D / D Z P geometries of H u et al . [83]. Also reported in Reference [169] and from the same calculations are: pa — 1.001 D and pb = 0.685 D for conformer Ip and pa — 5.514 D and u,b = 0.873 D for conformer Hp. f T h e t r u n - p V T Z basis set is the aug-cc-pVTZ basis set wi th the f functions on the heavy atoms and the d functions on the hydrogens removed. s Suenram and Lovas [76] determined pa of conformer Ip to be 1.0 ± 0 . 1 5 D and pb ^ 0, pa > h B r o w n et al . [74] found pa of conformer H p to be 4.5 D and pb < 1 D . Chapter 6. Glycine 184 Table 6.2: Relative thermodynamic quantities and abundances (Abund.) of the conformers of glycine. A l l values are relative to those of conformer Ip and were calculated at the experimental temperature of 438 K . Vibrat ional frequencies (scaled by 0.97) and rotational constants from Reference [81] were used to calculate the relative entropies (AS), zero-point vibrat ional energies ( A i ? z p v ) and thermal energies (A(AH)) of al l conformers except IIIp, for which the vibrational frequencies from Reference [83] were used. Conf. A £ 0 a AEZpy A ( A # ( 4 3 8 ) ) AS438 A G 4 3 8 A b u n d . (kJ /mol) (kJ/mol) (kJ/mol) ( J /mol -K) (kJ /mol) (%) Ip 0.0 0.0 0.0 0.0 0.0 52.61 Iln 2.058 1.613 -1 .305 -9 .321 6.449 8.96 IIIp 6.663 0.072 0.081 10.764 2.100 29.56 IVn 5.156 0.203 -0 .402 -5 .720 7.463 6.78 Vn 10.503 0.482 -0 .456 -4.852 12.655 1.63 VIp 19.750 -1.102 -0 .499 1.498 18.491 0.33 VIIp 24.081 -1.369 0.351 0.136 23.004 0.10 VHIn 25.265 -0.662 -0 .049 -4 .420 26.491 0.04 Best M P estimates from Reference [81]. account for only « 0.5% of the sample at the experimental temperature. The use of free energies for the T M P weighting is a more physically realistic treatment than the use of elec-tronic energies alone, as was done in a preliminary analysis of the glycine H O M O electron density [152]. The calculated thermodynamic quantities are summarized i n Table 6.2. The relative free energies at 165°C differ significantly from the relative electronic energies of the glycine conformers. Free energies were calculated with the harmonic osci l lator-r igid rotor approximation using the standard statistical-mechanical formulae discussed in Section 2.4.2. The final predictions of Csaszar [81] for the relative conformer energies were used for the rel-ative electronic energies AE0. The rotational constants and vibrational frequencies (scaled by 0.97 as recommend in Reference [81]) reported by Csaszar [81] for the M P 2 / 6 - 3 1 1 + + G * * geometry optimized conformers were then used to calculate the relative zero-point vibra-t ional energies AEZPV, thermal energies A(AH(T)) and entropies AST of the conformers. The vibrat ional frequencies of conformer IIIp were not reported by Csaszar, so the relative thermodynamic properties for this conformer were calculated using the vibrat ional frequen-Chapter 6. Glycine 185 cies reported by Schaefer et al . [83] and calculated at the H F / D Z P level. Relative Gibbs free energies at 438 K were then calculated using Equation (2.38) of Section 2.4.2. The energy of conformer Hp relative to that of conformer Ip has been determined to be 5.4-6.7 k J / m o l by infrared spectroscopy of glycine isolated i n an inert gas matr ix [78], and 5.9 ± 1.8 k J / m o l by microwave spectroscopy [76]. Both of these experimental results are in agreement wi th the calculated relative Gibbs free energy of Hp reported in Table 6.2. In contrast, the calculated relative energy of IIIp is less than the single reported experimental value of 3.8-6.3 k J / m o l , obtained in the same infrared spectroscopy study mentioned above [78]. The disagreement between the calculated and experimental values for IIIp could be a result of the use of H F [83] rather than M P 2 [81] vibrat ional frequencies to calculate the free energy of this conformer or uncertainties i n the experimental value, determined in a challenging experiment involving infrared spectroscopy of glycine trapped in an inert gas matr ix [78]. In this regard, in a recent density functional study of glycine [85], Hagler et al . found that the use of vibrational frequencies calculated using the M P 2 or B 3 L Y P - D F T methods, rather than the H F method, resulted in a significantly greater relative free energy for conformer IIIp. This difference was primari ly attributed to a very low value for the first vibrational frequency of this conformer when calculated at the H F level of theory. The impact this has on the Bol tzmann weighted conformer sum T M P s discussed in the following sections has been investigated by setting the relative free energy of conformer IIIp to 12.02 k J / m o l , the value reported by Hagler et al . [85] for the temperature of 473 kelvin and obtained using vibrational frequencies calculated at the M P 2 / D Z P level of theory. Using this new value for the free energy of IIIp, the relative populations of conformers Ip-VHIn become 72.68, 12.38, 2.68, 9.37, 2.25, 0.45, 0.13 and 0.05%, respectively. Comparing these values wi th those in Table 6.2 indicates that this change in the free energy of IIIp has a rather large effect on the estimated conformer abundances, particularly for conformers Ip Chapter 6. Glycine 186 b3 J V 1 MO 20 / / vt / / t z \ / / w PI ^ Jl I / / i i • , 8 0 MO 19 conf. sum 1 conf. sum 2 2 3 0 1 Momentum (au) Figure 6.3: Effect of conformer abundances on the conformer sum theoretical momentum profiles of glycine. Bol tzmann weighted conformer sum T M P s of M O s 20-18 obtained using the conformer abundances reported in Table 6.2 (conf. sum 1, solid lines) and the abundances obtained if the free energy of IIIp is set to 12.02 k J / m o l (conf. sum 2, dashed lines) are compared. Refer to the text for further details. The key to the T M P labels is given in Table 6.1. Chapter 6. Glycine 187 and IIIp. In Figure 6.3, conformer sum T M P s of M O s 20-18 of glycine obtained using these abundances (conf. sum 2, the dashed lines) are compared wi th the corresponding T M P s obtained using the abundances reported in Table 6.2 (conf. sum 1, the solid lines). Despite the considerable differences in conformer abundances, the changes in the conformer sum T M P s of glycine are minor and do not alter the discussion in the remainder of this chapter, which is based upon the abundances reported in Table 6.2. 6.4 Valence binding energy spectra Glycine is a 40 electron molecule, wi th 15 valence orbitals. The P E S study of Cannington and H a m [162], as well as the H F calculations performed in the present study, indicate that ionizations from the 12 highest energy valence orbitals should be observable in the energy range studied in the current work. In the case of the C s symmetry conformers, eight of these orbitals have a' symmetry and four have a" symmetry. The remaining three valence and five core orbitals outside of the energy range of this study are al l of a' symmetry. Ionizations from the a" symmetry orbitals of C s symmetry conformers w i l l result in p-type momentum profiles. A s a consequence of the nodal plane in these orbitals, the corresponding momentum profiles w i l l have no intensity at zero momentum except for the small contribution from instrumental momentum resolution effects. Ionizations from a' symmetry orbitals of the C s symmetry conformers and from the orbitals of the C i symmetry conformers may give rise to either s-type or p-type momentum profiles, depending upon the nature of the orbital in question. For example, molecular orbitals that strongly resemble atomic p orbitals w i l l have p-type momentum profiles, although because of the lack of a nodal plane in these molecular orbitals, greater intensity at zero momentum may be observed than in the case of the a" momentum profiles. In the following discussion of the E M S experimental results for glycine, Chapter 6. Glycine 188 the molecular orbitals are referred to by number rather than by symmetry label because of the expected presence in the experimental sample of multiple conformers of different symmetries. The glycine binding energy spectra obtained at relative azimuthal angles of 0° and 8° are shown in Figure 6.4, wi th the fitted Gaussian peaks used to obtain the experimental momen-t u m profiles (using the procedure described in Section 3.4.4) shown as dashed lines. The high resolution P E S spectrum, digitized from figure 1 of Reference [162], from which the vertical ionization peak positions and Franck-Condon widths used for the fitting procedure were determined, is shown in the bottom panel of Figure 6.4. The reported ionization potentials and fitted Gaussian functions are indicated by vertical bars and dashed lines, respectively. The absolute energy scale of the B E S was established using the He ( I s ) - 1 ionization peak at 24.58 eV. The peak positions used to fit the B E S data and the experimental P E S [162] and theoretical p G W 2 [163] valence IPs of glycine are shown in Table 6.3. A s a result of the close energy spacing of the glycine valence orbitals, it is not possible for the most part to identify individual ionization peaks in the E M S B E S . However, examination of the 0 = 0° and 8° B E S shown in Figure 6.4 reveals five distinct regions. The lowest energy region, between approximately 9 and 12 eV, contains peaks due to ionization from the three outermost orbitals of glycine (MOs 20, 19 and 18). The observed intensity in this energy region is greater at (f> = 8°, indicating that these orbitals are predominantly p-type. This is as expected since these orbitals, centred predominantly on the nitrogen and two oxygen atoms, are largely "lone-pair" in character. The second region, from approximately 12 to 16 eV, is of greater overall intensity than the first one, particularly in the case of the (j> = 0° spectrum. This is because of the presence of ionizations from four orbitals ( M O s 17-14, wi th IPs of 13.6, 14.4, 15.0 and 15.6 eV, from P E S [162]) in this energy range and also because the overall symmetry of these bands is s-type, as a consequence of the predominantly a-bonding Chapter 6. Glycine 189 10 15 20 25 Binding Energy (eV) Figure 6.4: B inding energy spectra of glycine. The top two panels show the E M S B E S from this work for the binding energy range of 6 to 27 eV at out-of-plane azimuthal angles of (j> = 0° and 8°, obtained at a total energy of 1200 eV. The approximate corresponding electron momenta, calculated using Equation (2.5), are also indicated. The dashed lines represent the results of a least-squares fit of Gaussian functions to the ionization peaks and the solid curves are the summed fits. The areas indicated by a-g, plotted as a function of momentum or angle, give the experimental profiles shown in Figures 6.5, 6.6 and 6.8-6.14. The lower panel shows the He(II) P E S spectrum reported by Cannington and H a m [162]. The vertical ionization energies are indicated by vertical lines. The fitted Gaussian functions used to estimate the Franck-Condon widths of the ionization processes are indicated by dashed lines. The sharp peak in the P E S at 12.7 eV has been attributed by Debies and Rabalais [161] to HC1. Chapter 6. Glycine 190 Table 6.3: Measured and calculated valence ionization potentials of glycine. A l l energies are in eV. M O E M S a P E S b p G W 2 c origin Ip Hp 20 10.1 10.0 9.94 9.81 19 11.2 11.1 10.64 11.10 18 12.1 12.2 11.87 11.22 17 13.6 13.6 13.65 13.28 16 14.4 14.4 14.62 13.86 15 15.0 15.0 14.70 15.01 14 15.6 15.6 15.34 15.30 13 16.6 16.6 16.84 16.39 12 16.9 16.9 16.88 17.81 11 17.6 17.6 . 17.58 18.34 io 20.0 20.2 9 23.2 23.2 a Ionizat ion peak energies used to fit the E M S binding energy spectra. b F r o m Reference [162]. c Parametr ized second-order Green function times screened interaction method from Refer-ence [163]. nature of these orbitals. The third region, of comparable intensity to the second, runs from 16 to « 18.5 eV and is due to ionization from M O s 13, 12 and 11 (IPs 16.6, 16.9 and 17.6 eV) [162]. The overall symmetry in this region is p-type. The fourth "region," located at 20 eV, arises from ionization from M O 10 and is s-type, although the intensity at 8° is comparable to that at 0°. The final region in the B E S shown in Figure 6.4 is dominated by the He ( I s ) - 1 calibration peak at 24.6 eV. However, a p-type lower energy shoulder is evident at ~ 23 eV due to ionization from M O 9 of glycine. 6.5 Momentum profiles The experimental momentum profiles ( X M P s ) of the valence orbitals of glycine are shown in Figures 6.5, 6.6 and 6.8 to 6.14 (see below). The Franck-Condon widths and close energy Chapter 6. Glycine 191 Momentum (au) Figure 6.5: Experimental and theoretical summed outer-valence momentum profiles of glycine. The experimental profile (•) is the sum of the profiles of fitted peaks a, b and c (IPs 10.1, 11.2 and 12.1 eV, see Figure 6.4) and the corresponding theoretical profiles (—) are the sum of the T M P s of M O s 20-18. The key to the T M P labels is given i n Table 6.1. The X M P has been normalized to the b3 T M P . Chapter 6. Glycine 192 spacing of the ionization peaks necessitated the use of a f itt ing procedure to obtain the individual X M P s . A s shown in Figure 6.4 for the c/> = 0° and </» = 8° B E S , the set of thirteen B E S were fitted using Gaussian peaks for each ionization process located at the P E S vertical ionization energies [162]. The widths of the fitted peaks were fixed at values obtained by convoluting the E M S instrumental energy resolution function (fwhm=1.5 eV) wi th the Franck-Condon widths of the ionization peaks estimated by fitting Gaussian functions to the high resolution P E S spectrum reported by Cannington and H a m [162] and shown in the lower panel of Figure 6.4. The distribution of fitted peak areas for an individual ionization process as a function of momentum is the desired X M P . Also shown on Figures 6.5, 6.6 and 6.8 to 6.14 are the calculated T M P s produced from the calculations discussed i n Section 6.3 and summarized in Table 6.1. The T M P s used for comparison wi th the experimental data are the sums of the T M P s of the five conformers (Ip, Hp, IIIp, IVn and Vn) predicted to be the most prevalent in the sample mixture (> 99.5%), Bol tzmann weighted according to their relative Gibbs free energies and the experimental temperature of 165°C, as explained in Section 6.3. Also shown on Figures 6.6, 6.8, 6.9 and 6.12 are the individual T M P s for these five most stable conformers. Al though the X M P s as measured are on a common relative intensity scale, the data are not absolute. Therefore, a single normalization between theory and experiment is necessary before the calculations and measurements can be compared. This has been done by comparing the X M P obtained by summing fitted peaks a, b and c (IPs 10.1, 11.2, 12.1 eV) wi th the corresponding summed T M P s of M O s 20-18, as shown in Figure 6.5. A least-squares fit of the X M P to the B 3 L Y P D F T T M P (b3) was then performed because this theoretical distribution provides the best overall agreement for shape wi th the experimental data. A l l experimental profiles for glycine were then scaled using the factor resulting from this fit, wi th al l H F and D F T theoretical profiles kept on a common intensity scale. Chapter 6. Glycine 193 The momentum profile for the sum of the three outermost valence orbitals of glycine (Figure 6.5) exhibits p-type symmetry, as expected from the examination of the B E S i n Sec-tion 6.4. The atomic p orbital-like nature of M O s 20, 19 and 18 results in the m i n i m u m at zero momentum. Al though al l of the theoretical profiles correctly predict the general shape of the experimental profile, only the D F T results (Id, bp and b3) are in good quantitative agreement wi th the experimental data. The H F calculations predict the profile max imum to occur at a higher momentum than is observed experimentally for these orbitals and also underestimate the intensity in the low momentum region. Not surprisingly, the discrepancy between the H F T M P s and the X M P is greatest for the 3 0 - C G F min imal basis set calculation (st) and decreases as the basis set size increases. However, there are no significant differences between the 145-CGF 6-311++G** (6p) T M P and the 3 4 5 - C G F aug-cc -pVTZ (tz) T M P , suggesting that no further improvement in the agreement between the H F T M P s and exper-iment would be obtained by using a yet larger basis set. Al though the calculated momentum profile, which emphasizes the small p (predominantly large r) region of electron density, appears converged at the H F level using the 6-311++G** basis set, further improvements in the total energy and dipole moments (see Table 6.1), which emphasize small and medium r , respectively, are obtained using the larger t r u n - p V T Z (tt) and aug-cc -pVTZ (tz) basis sets. This demonstrates the importance of considering a number of properties emphasizing different regions of electron density when evaluating the accuracy of a wavefunction. The highest occupied molecular orbital of glycine, M O 20, has a binding energy of 10.1 eV as determined by E M S and a p-type momentum profile (Figure 6.6). The theoretical calculations indicate that this orbital is predominantly centred on the nitrogen atom and, in the case of the C s symmetry conformers, is symmetric with respect to reflection through the plane of the heavy-atom framework (i.e., of a' symmetry). This agrees wi th the P E S assignment of Cannington and H a m [162] and the p G W 2 calculations of H u et al . [163], Chapter 6. Glycine 194 10 0 " > 0 DC 10 ld,bp,b3 MO 20 tz,tt,6P HOMO conformer sum • peak a 10.1 eV ld,bp,b3 tz,tt,6p ld,bp,b3 tz,tt,6p IVn 6.8% 20a(N) ld,bp,b3 tz,tt,6p ld,bp,b3 tz,tt,6p ld,bp,b3 tz,tt,6p IIIp 29.6% 16a'(N)" Vn 1.6% 20a(N) Q I , i , r - ^ , t i , i , r - ^ . l 0 1 2 0 1 2 3 Momentum (au) Figure 6.6: Experimental and theoretical momentum profiles for the H O M O ( M O 20, IP 10.1 eV) of glycine. The X M P (•) and the T M P s (—) for the conformer sum are shown in the top-left panel. The remaining panels contain profiles for each of the five most abundant conformers, as indicated. The corresponding M O symmetry labels are indicated i n each panel, along wi th the atom on which the greatest proportion of the electron density for that M O is centred. The key to the T M P labels is given in Table 6.1. Chapter 6. Glycine 195 - 9 - 6 - 3 0 3 6 9 - 6 - 3 0 3 6 9 - 6 - 3 0 3 6 9 X Position (au) Figure 6.7: Posit ion space electron density maps for the H O M O s of the three lowest energy conformers of glycine. The maps were calculated using the results of the B 3 L Y P D F T calculation. The electron density in the symmetry plane of the molecule (i.e., the plane containing the heavy-atom nuclei) is shown. The contour lines in the upper panels represent 0.01, 0.03, 0.1, 0.3, 1.0, 3.0, 10.0, 30.0 and 99.0% of the maximum density. but not the earlier assignment by Debies and Rabalais [161]. The exception is the Hp conformer, for which the H O M O electron density is predominantly located on the carbonyl oxygen (indicated by O c on Figure 6.6). This is i l lustrated by the position-space electron density maps for this orbital for conformers Ip, Hp and IIIp, shown in Figure 6.7. This difference in the nature of the H O M O is most likely a result of conformer Hp being the only one of the five most stable glycine conformers containing an intramolecular hydrogen bond to the nitrogen atom (see Figure 6.1). This w i l l stabilize the nitrogen "lone pair" electron density and consequently increase the binding energy of the electrons i n this orbital . Chapter 6. Glycine 196 Concurrently, the carbonyl oxygen non-bonding orbital of conformer H p is destabilized by the unfavorable anti orientation of the acid group. The min imum in the experimental profile at zero momentum results from the nodal surface between the two lobes of electron density in this largely atomic p-like orbital . The intensity that is observed at zero momentum results from the small contributions to this orbital from electron density at other parts of the molecule, such as the a-bonding character between the two carbon atoms evident from the electron density maps in Figure 6.7. The multiple nodal surfaces in this orbital result in two-lobed M O 20 momentum profiles for many of the conformers, as shown i n Figure 6.6. The three D F T T M P s (Id, bp and b3) provide the best agreement w i t h the experimental data. The higher level H F T M P s (tz,tt and 6p) are essentially indistinguishable from one another and in fair agreement wi th the X M P , correctly predicting the position of the profile maximum (pMAx), but underestimating the intensity at low momentum. The smaller basis set H F calculations, in contrast, differ markedly from the experimental data. The 6g and 4g calculations correctly predict the qualitative shape of the momentum profile but place PMAX at too great a momentum. The minimal basis set st profile fails to predict correctly even the qualitative features of the momentum profile. Because of this poor agreement, the 4g and st T M P s have not been included for comparison with the other X M P s of glycine in the present work. Note that the use in the present work of free energies for the Bol tzmann weighting of the T M P s together wi th the determination of the X M P by fitting the B E S has resulted in further improved agreement between the H O M O X M P and the D F T T M P s compared wi th the earlier preliminary analysis [152]. In this earlier analysis, the X M P was obtained by summing the experimental data over a range of binding energies and electronic energies (rather than free energies) were used for the Bol tzmann weighting of the T M P s . Molecular orbitals 19 and 18 (Figures 6.8 and 6.9) have binding energies of 11.2 and 12.1 eV, respectively. Whi le both may generally be described as p-type profiles, the M O Chapter 6. Glycine 197 10 0 10 "co c 0 C © 5 0 DC 10 0 Id ' b? MO 19 tz,tt,6p k conformer sum KfW 11.2eV" Id ' tz,tt,6p 52.6% "M i5a'(O r) / / 6 g \ ^ i i i i i N l d ' 1 1 1 * \ \tz,tt ,6p_ "P ~¥ }HF 9.0% - 15a'(N) -HFand b 3 \ from MO 18 i i i i i IIIp ld,bp,b3 29.6% tz,tt,6p 15a'(Oc) -'HF from \ MO 18 i i i i ~~ i id IVn $ 6.8% tz,tt,6p i i i i i S Vn b3 1 .6% A^tz,tt ,6p >/^ \ 19a(Or) 0 2 0 1 2 Momentum (au) Figure 6.8: Experimental and theoretical momentum profiles for M O 19 (IP 11.2 eV) of glycine. The X M P (•) and the T M P s (—) for the conformer sum are shown i n the top-left panel. The remaining panels contain profiles for each of the five most abundant conformers, as indicated. The corresponding M O symmetry labels are indicated in each panel, along wi th the atom on which the greatest proportion of the electron density for that M O is centred. The key to the T M P labels is given in Table 6.1. Chapter 6. Glycine 198 10 ^ 10 "co C D CD > CD DC 10 0 MO 18 r conformer sum tz,tt,6p TL • peakc T JtZx 12.1 eV" lo/bpjd 1 X l r 1 . • X "T. i i ip 52.6% tz,tt,6p A 4a"(°c'°H)" /bp,lA IIP 9.0% l_ipf tz,tt,6p " 1 J% 4a"(O c,0 H)-V^bp,ld\ HFandB3 / / \ f r omM019 i | i | i IIIp 29.6% tz,tt,6p A 4a"(O c ,0 H ) -/ p ' ' \ HFfrom / \ MO 19 IVn 6.8% tz,tt,6p " /A, 1 8 a ( 0 C ' 0 H ) " Ap,ld\ 1 1 1 . 1 1 Vn 1.6% tz,tt,6p Jl 18a(O c,0 H)" /bp,ld\ 0 2 0 1 2 Momentum (au) Figure 6.9: Experimental and theoretical momentum profiles for M O 18 (IP 12.1 eV) of glycine. The X M P (•) and the T M P s (—) for the conformer sum are shown in the top-left panel. The remaining panels contain profiles for each of the five most abundant conformers, as indicated. The corresponding M O symmetry labels are indicated in each panel, along with the atoms on which the greatest proportion of the electron density for that M O is centred. The key to the T M P labels is given in Table 6.1. Chapter 6. Glycine 1 9 9 1 9 profile has considerably more intensity at zero momentum than does the profile for M O 1 8 . There is some disagreement amongst the theoretical calculations as to the symmetries of these orbitals. In the case of conformers Ip, IVn and Vn, a l l calculations performed predict M O s 1 9 and 1 8 to be predominantly oxygen "lone-pair" orbitals centred on the carbonyl oxygen (Oc) , and on both the carbonyl and hydroxyl oxygens ( O C , O H ) , respectively. In the case of conformer IIIp, the D F T calculations predict M O 1 9 to be predominantly centred on the carbonyl oxygen and of a' symmetry and M O 1 8 to be centred on both oxygen atoms and of a" symmetry, wi th the situation reversed for the H F calculations. For conformer Hp, the ld and bp D F T results predict the electron density of M O 1 9 to be predominantly located on the nitrogen atom and of a' symmetry and that of M O 1 8 to be concentrated on the two oxygen atoms and of a" symmetry. The reverse is true for the H F and b3 D F T calculations. A comparison of the two X M P s and the various T M P s helps to clarify the situation. The X M P obtained at 1 1 . 2 eV ( M O 1 9 ) has significant intensity at zero momentum while the 1 2 . 1 eV X M P ( M O 1 8 ) essentially drops to zero near zero momentum. This drop to zero is consistent wi th the a" symmetry momentum profiles, which have no intensity at zero momentum because of the nodal plane in these orbitals. In contrast, the a' symmetry profiles for conformers Hp and IIIp have some intensity at zero momentum. Therefore, for conformers Hp and IIIp, M O 1 9 is assigned as an a' symmetry orbital and M O 1 8 as an a" symmetry orbital . This symmetry assignment is consistent wi th that of Cannington and H a m from a consideration of the P E S of glycine and related molecules [ 1 6 2 ] . In Figures 6 . 8 and 6 . 9 , the inverse ordering of orbitals 1 9 and 1 8 for conformers Hp and IIIp has been corrected as indicated to allow a more meaningful comparison of the theoretical and experimental momentum profiles. Reasonable overall shape agreement between the X M P and T M P s for each of M O s 1 9 and 1 8 is obtained. However, a comparison of the X M P s for these orbitals reveals a correspondence between some experimental data points which Chapter 6. Glycine 200 are lower than the theoretical profiles in the one case and higher in the other (consider in particular the points between 0.3 and 0.7 au). This is most likely a result of l imitations in the energy resolution (1.5 eV fwhm) and the resulting uncertainties i n the deconvolution procedure used to obtain experimental momentum profiles for these energetically closely spaced orbitals. In addition to the challenge of separating two momentum profiles differing in binding energy by only 0.9 eV, the situation for glycine is complicated by the fact that the energies for particular molecular orbitals vary to at least some degree w i t h changes in molecular conformation. The H F / a u g - c c - p V T Z calculations conducted in the present study indicate variations in orbital energy between conformers Ip through V n of approximately 0.5 eV for each of M O s 19 and 18 and, in the case of some conformers, an energy spacing between M O s 19 and 18 of considerably less than 0.9 eV. However, a sufficient gap is predicted between the highest energy M O 20 and the lowest energy M O 19 and likewise between M O s 18 and 17 to provide reasonable confidence that a sum of fitted peaks b and c w i l l account for al l intensity resulting from ionization from M O s 19 and 18 without introducing intensity resulting from ionization from M O 20 or 17. In Figure 6.10, the X M P obtained by summing these two experimental profiles is compared with the corresponding T M P s for the sum of M O s 19 and 18. The resulting X M P is in good quantitative shape agreement wi th the three D F T T M P s . In contrast, the H F T M P s give a poor representation of the experimental data in the low momentum region below p = 0.7 au. In the case of molecular orbitals 17 through 12 (binding energy r<ni^6 13-17 eV) , the energy separation between orbitals is quite small and prevents the determination of meaningful momentum profiles for individual orbitals. For this reason, a sum of fitted peaks has been considered in this region and is compared in Figure 6.11 w i t h the summed theoretical profiles of M O s 17-11. M O 11 has also been included i n the sum to avoid complications resulting from differing predictions of orbital ordering by the H F and D F T Chapter 6. Glycine 201 10 0 1 2 3 Momentum (au) Figure 6.10: Experimental (•) and theoretical (—) momentum profiles for the sum of M O s 19 and 18 of glycine. The key to the T M P labels is given in Table 6.1. Chapter 6. Glycine 202 50 0 0 MOs 17-11 peaks d+e b3-20%MO11 10 20 30 Azimuthal angle (degrees) Figure 6.11: Experimental (•) and theoretical (—) angle profiles for the sum of M O s 17-11 of glycine. The key to the T M P labels is given in Table 6.1. The dashed line represents the b3 D F T T M P sum for M O s 17-11 minus 20% of the b3 profile of M O l l (see text for details). Chapter 6. Glycine 203 calculations. This issue is addressed further below for M O 11. The data are plotted using an out-of-plane azimuthal angle (</>) rather than momentum scale because of the variation of momentum wi th electron binding energy for a given azimuthal angle (Equation (2.5)). These orbitals are responsible for the bulk of the molecular bonding i n glycine and the cr-bonding nature of several of them results in the considerable intensity observed i n the profiles at low values of </>. Only qualitative agreement is obtained between the shapes of the experimental profile and al l of the theoretical profiles. The D F T profiles are consistent with the experimental data at small and large values of cb, but are higher than the experimental results for intermediate angles. The H F calculations, in contrast, fal l below the experimental data at low (f> and agree at intermediate and high values. In view of the good agreement between the D F T T M P s and the experimental data for M O s 20-18, a likely explanation for the behaviour observed for M O s 17-11 is that one or more of the p-type ionization processes occurring in this region have spectroscopic factors of less than one (see Equat ion (2.11)). If 20% of the pole strength for M O 11 is removed from the b3 D F T T M P sum for M O s 17-11, the resulting dotted line (Figure 6.11) shows somewhat improved agreement wi th experiment (see the following paragraph and Figure 6.12 below, where the remaining 80% of the M O 11 b3 D F T T M P fits the X M P for M O 11 quite well). A s discussed i n Sections 1.1.2 and 2.2.3, inner-valence ionization processes frequently show a range of energy poles as a consequence of final ion state electron correlation effects [154]. Experimental and theoretical momentum profiles for the ionization process at 17.6 eV, arising from ionization from M O 11, are shown in Figure 6.12. There is disagreement between the H F and D F T calculations as to the symmetry of this orbital . However, a comparison of the shape of the X M P with those of the various T M P s clearly indicates that, i n the case of the C s symmetry conformers, this is an a" orbital , in agreement wi th the D F T predictions and the p G W 2 calculations of H u et al . [163]. A n examination of the M O calculations indicates Chapter 6. Glycine 204 CZ 0 0 > 15 10 W b 3 MO 11 conformer sum 80%b3 • peak e 17.6 eV 15 10 9.0% £ 5 0 DC 10 0 HFfrom M 0 1 2 [\ DFT "IP u 29.6% HF HF from M 0 1 2 0 2 0 1 2 Momentum (au) Figure 6.12: Experimental and theoretical momentum profiles for M O 11 (IP 17.6 eV) of glycine. The X M P (•) and the T M P s (—) for the conformer sum are shown in the top-left panel. The remaining panels contain profiles for each of the five most abundant conformers indicated. The corresponding M O symmetry labels are indicated in each panel. The key to the T M P labels is given in Table 6.1. The dashed line represents the b3 D F T T M P scaled by 80% (see text for details). Chapter 6. Glycine 205 that this orbital may be primari ly thought of as a pseudo-7r orbital responsible for N H 2 cr-bonding, but wi th significant contributions to C H 2 bonding as well. For comparison between theory and experiment, the T M P s for the H F orbitals having the appropriate a" symmetry have been shown i n Figure 6.12 (i.e., M O 13 for Ip and M O 12 for H p and IIIp). A l though there are also clear differences between the H F and D F T T M P s for the C i conformers I V n and V n , the lack of symmetry in these conformers makes it difficult to determine the correct M O s to use for comparison with the X M P . Consequently, the predicted M O 11 T M P s have been used. Because of the small relative populations of conformers I V n and V n , this w i l l not significantly affect the Boltzmann-weighted conformer sum T M P s that are compared with experiment in the upper-left panel of Figure 6.12. It is evident from Figure 6.12 that there are only slight differences between calculations for this orbital and that a l l T M P s shown are consistent wi th the shape of the X M P . There is, however, a considerable discrepancy in terms of intensity between theory and experiment. The b3 T M P must be scaled by 80% to bring it into agreement wi th the X M P . This represents a break-down of the single particle model of ionization due to final state correlation and relaxation effects. The remaining intensity would be expected to be observed as satellite peaks at other binding energies, possibly outside of the energy range examined in the current study. This "missing intensity" at least partly explains the disagreement between theory and experiment seen for the sum of M O s 17-11 in Figure 6.11 at intermediate out-of-plane azimuthal angles (see the discussion in the preceding paragraph). The shape of the M O 11 momentum profile would result in a significant drop in experimental intensity for the M O 17-11 sum at intermediate angles, but very l i tt le change at lower or higher angles if the spectroscopic factor for this ionization process at 17.6 eV is lower than unity. In the case of M O 10 (Figure 6.13, electron binding energy 20.0 eV) , al l theoretical momentum profiles are in reasonably close agreement, differing, however, i n the relative Chapter 6. Glycine 206 10 0 0 Id b3 bp,tz 6g,6p,tt MO 10 20.0 eV peak f b3 MO10+ 16%M011 o XMP-b3 —16% M011 (b3) 1 2 Momentum (au) Figure 6.13: Experimental (•) and theoretical (—) momentum profiles for M O 10 (IP 20.0 eV) of glycine. The key to the T M P labels is given in Table 6.1. The dashed line represents the b3 D F T T M P sum ( M O 10 + 16% M O 11). The inset shows the ( X M P - b3 T M P ) difference (o) compared with 16% of the b3 T M P for M O 11. Chapter 6. Glycine 207 magnitudes of the maximum at zero au and the minimum near 0.5 au. A l l H F and D F T calculations indicate that M O 10 consists primari ly of carbon 2s electron density. However, there is also a significant a-bonding component, particularly between the hydroxyl oxygen and hydrogen. The X M P is consistent with the general features of the theoretical profiles, although there is a significant difference in intensity for the region between approximately 0.2 and 1.0 au. One possible explanation for this additional experimentally observed intensity is the presence in this binding energy region of satellite peaks from the M O 11 ionization process (see discussion in previous paragraph and Figure 6.12). A n examination of the momentum profile obtained by subtracting the b3 T M P of M O 10 from the 20.0 eV X M P (see inset of Figure 6.13) supports this explanation. The resulting profile is consistent w i t h the b3 M O 11 T M P scaled by 16%. Including this additional intensity in the b3 T M P (dashed line, Figure 6.13) eliminates the discrepancy between theory and experiment. Molecular orbital calculations indicate that M O 9, like M O 10, is comprised, to a con-siderable degree, of carbon 2s electron density. However, the two orbitals differ in that M O 9 also contains significant nitrogen 2s density and no appreciable bonding character. U p o n examining the X M P for this inner-valence orbital (Figure 6.14), it is apparent that the spec-troscopic factor for this ionization process is significantly less than one. In addit ion to this intensity difference between theory and experiment, the X M P (solid circles) is higher than al l of the T M P s near zero momentum and appears to peak at a slightly lower momentum than do the T M P s . These last two observations could be a result of contamination of this X M P w i t h signal from the adjacent and considerably more intense s-type He ( I s ) - 1 ioniza-tion calibration peak (see Figure 6.4). This seems likely since, as shown in Figure 6.14, the X M P (•) minus the scaled He Is T M P [168] gives a momentum profile (o) which agrees well for shape w i t h the b3 D F T T M P . Chapter 6. Glycine 208 0 1 2 3 Momentum (au) Figure 6.14: Experimental (•) and theoretical (—) momentum profiles for M O 9 (IP 23.2 eV) of glycine. The key to the T M P labels is given in Table 6.1. The dot-dashed line corresponds to the shape of the He Is T M P obtained using the highly correlated He wavefunction reported by Davidson [168]. The short dashed line represents 56% of the b3 D F T T M P of M O 9. The long dashed line is the sum of the dot-dashed and short dashed lines. The open circles are the result of subtracting the He Is T M P (dot-dashed line) from the measured X M P . Chapter 6. Glycine 209 6.5.1 General observations In comparing the various glycine theoretical momentum profiles wi th the experimental data, it can be seen that the profiles for the chemically important outer-valence orbitals (MOs 20-18) are most sensitive to changes in the basis set or theoretical method used and those for M O s 11-9 are relatively insensitive to changes in the computational method. There is a steady improvement in the agreement between the H F T M P s and the X M P s wi th increasing basis set size. The minimal basis set S T O - 3 G H F calculations (st on Figures 6.5 and 6.6) produce T M P s that differ dramatically from the experimental profiles. This basis set clearly does not have the necessary flexibility to accurately describe the outer-valence molecular orbitals of glycine. Some improvement occurs in going to the 4-31G (4g) and 6-311G (6g) split-valence basis sets. However, the outer-valence T M P s calculated using these basis sets st i l l tend to agree poorly with the X M P s , particularly at low momentum. The addition of diffuse and polarization functions to the 6-311G basis set (the 6-311+-hG** (6p) calculations) results i n improved agreement between theory and experiment for many of the profiles. This is typified by an increase in the predicted intensity at low momentum and a shift of the maxima of the momentum profiles (p M A X ) towards zero momentum in the case of p-type profiles. This is a consequence of the improved description of the spatially diffuse regions of the electron density when using the 6-311++G** basis set. Because of the Fourier transform relationship between position and momentum space, these outer regions of electron density roughly correspond to the low momentum regions in momentum space. Further increases in basis set size from the 145-CGF 6-311++G** basis set to the 2 4 0 - C G F t r u n - p V T Z (tt) and 3 4 5 - C G F aug-cc-pVTZ (tz) basis sets result in no further appreciable change i n the momentum profiles. It would therefore appear unnecessary to employ such large basis sets for Hartree-Fock calculations of momentum profiles and perhaps other properties of glycine that are predominantly dependent on the large-r regions of the electron density. It is important Chapter 6. Glycine 210 to note, however, that further improvements in both total energy and dipole moment are obtained by using the two larger basis sets (see Table 6.1). The apparent near convergence of the H F results for momentum profile, total energy and dipole moment suggests that the H F l imit has been closely approached. However, a discrepancy st i l l remains between these theoretical profiles and the experimental data, particularly in the region of low momentum where the H F results tend to underestimate the intensity. This discrepancy is essentially removed when the K S - D F T profiles are considered. The D F T calculations predict a further shift towards low momentum in the maxima of the p-type momentum profiles from those predicted by the H F calculations. The orbital ordering predicted by the D F T calculations also appears to be more consistent wi th the experimental data than that predicted by the H F calculations. The improved agreement between theory and experiment, particularly at lower momentum (< 1 au), can be attributed to the inclusion of electron correlation effects v ia the exchange-correlation functional in D F T . Al though there are small differences in the momentum profiles obtained from the D F T calculations using different functionals, it is not evident that one calculation provides markedly better agreement with experiment. However, it is important to note that a recent theoretical study [84] of the geometries and relative energies of the conformers of glycine determined using D F T found that, amongst the functionals considered, only the hybrid B 3 L Y P functional reproduced the energetic ordering of the glycine conformers predicted in studies using post -HF methods. Chapter 7 Conclusions In this work, E M S measurements of the valence binding energy spectra and momentum profiles of acetone, dimethoxymethane and glycine have been presented. This is the first E M S study of dimethoxymethane and glycine and the first complete valence study of ace-tone. These E M S measurements, of what are fairly large molecules in comparison with most previous E M S studies, were possible because of the construction of the energy-dispersive multichannel spectrometer described in Chapter 3. The increased sensitivity of this mul t i -channel spectrometer was necessary for the collection of the considerable data required for complete valence shell studies with reasonable statistical precision and i n a practical period of time. The three molecules studied have also allowed the assessment of the target K o h n - S h a m approximation ( T K S A ) for larger molecules than have been considered previously [20,21]. The study of acetone (Chapter 4) confirms the disagreement between experiment and theory for the 5b 2 momentum profile suggested by the previous single-channel E M S study [54]. None of the theoretical methods considered in the present work (i.e., H F , M R S D - C I and D F T ) adequately describes the 5b 2 X M P of acetone. The similarity of the C I and D F T T M P s to the H F T M P s suggests that this discrepancy is not the result of electron correlation effects. A possible explanation is that distortion of the incoming and outgoing electrons is 211 Chapter 7. Conclusions 212 occurring in the (e, 2e) reaction, particularly for p > 1 au. Investigation of this possibility must await the development of D W I A methods for molecules (Section 2.2.1) and/or E M S spectrometers wi th much higher impact energies. For the remaining outer-valence momentum profiles of acetone, agreement between the X M P s and H F T M P s is fair. D F T T M P s calculated using the T K S A and the 6-311++G** basis set are in somewhat better agreement wi th the X M P s . There is l itt le difference be-tween the inner-valence H F and D F T T M P s , both of which reproduce the shapes of the corresponding experimental profiles. Of the three molecules studied, the greatest differences between the H F and D F T T M P s are observed for dimethoxymethane (Chapter 5). These differences are most significant for the low momentum regions of the outer-valence s-type momentum profiles (i.e., 11a, 10a and 8a). The D F T T M P s are generally in good agreement with the X M P s of dimethoxymeth-ane. This is not the case for the H F outer-valence T M P s , which tend to underestimate the intensity observed experimentally at low momentum. M R S D - C I calculations of the 10b and 11a T M P s of dimethoxymethane yielded momentum profiles similar to the ones obtained from large-basis set H F calculations and in disagreement with both the D F T T M P s and the experimental 10b + 11a momentum profile, perhaps indicating that an insufficient number of electron configurations were included in the M R S D - C I calculations. These results under-score the considerable challenge in performing accurate post -HF calculations of a molecule of the size of dimethoxymethane and illustrate the appeal of the considerably less costly density functional methods. The present study of dimethoxymethane also provided the first experimental determination of the inner-valence ( 6 a ) - 1 , ( 4 b ) - 1 and ( 5 a ) - 1 IPs. The study of glycine (Chapter 6) has demonstrated the feasibility of obtaining gas-phase experimental momentum profiles of low vapour pressure solids and shown that useful results may be obtained even when the molecule is present in a number of conformations. A joint Chapter 7. Conclusions 213 consideration of the experimental and theoretical results has allowed for the determination of the nature and ordering of many of the valence ionizations of glycine. A s was observed in the study of dimethoxymethane, the H F T M P s of glycine tend to underestimate the low momentum intensity observed in the outer-valence experimental profiles. Agreement between experiment and theory is improved significantly when the D F T momentum profiles are considered. In each of the three molecules studied, the single-particle model of ionization breaks down for binding energies of 16 or 17 eV and higher. This is evidenced by X M P s that are of sig-nificantly less intensity than the corresponding theoretical profiles, indicating spectroscopic factors (pole strengths) of less than unity. Furthermore, in the cases of acetone and dimeth-oxymethane, for which measurements were performed up to « 60 eV, ionization intensity is spread over the entire energy range from 25-60 eV, with l itt le structure evident i n this region of the binding energy spectra and wi th indications that the ionization intensity extends to higher binding energies. This is indicative of the presence of many low-intensity ionization poles throughout this energy range. The respective experimental angular profiles in this binding energy range indicate that most of this ionization intensity can be attributed to the ( 5 a i ) _ 1 and ( 4 a i ) _ 1 ionization manifolds in the case of acetone and the ( 3 b ) _ 1 and ( 4 a ) _ 1 ionization manifolds in the case of dimethoxymethane. However, for both molecules, the experimental results indicate that other ionization manifolds also contribute to the observed intensity at binding energies greater than 25 eV. Further many-body theoretical studies of inner-valence ionization of acetone and dimethoxymethane would allow a more detailed analysis of the present experimental results. Assessments of the effects of the basis set on the calculated T M P s results in similar conclusions for acetone, glycine and dimethoxymethane. In al l three molecules, use of a min imal basis set (e.g., S T O - 3 G ) or a split-valence basis set (e.g., 4-31G, 6-31G, 6-311G) Chapter 7. Conclusions 214 without diffuse and polarization functions results in T M P s in poor agreement wi th experi-ment. This was found to be the case for both H F and D F T calculations. Basis set effects are most significant for the outer-valence momentum profiles, where use of any of the small basis sets mentioned above tends to result in T M P s that underestimate the low momentum intensity. The addition of diffuse and polarization functions to the basis sets generally re-sults i n improved agreement between experiment and theory, typified by shifts of the p M A X of p-type T M P s towards lower momentum and increases in the intensity of the p = 0 max-i m u m of s-type profiles. Considering the Fourier transform relationship between position and momentum space (Equation (1.1)), this demonstrates the importance of ut i l iz ing dif-fuse and polarization functions when describing the large r (small p) outer-valence electron density of acetone, glycine and dimethoxymethane. Basis set effects have been found to be of considerably less importance for the inner-valence T M P s of al l three molecules. Further increasing basis set size beyond that of the 6-311-1—h-G** basis set does not result in appre-ciable changes to the T M P s of these molecules. Consequently, it is recommended that the 6-311+-f-G** basis set be used for the calculation of valence momentum profiles and other large-r properties of these and similar molecules. It should be noted, however, that the results for dimethoxymethane (Section 5.6) indicate that the valence T M P s of this molecule are essentially converged using the 6-31+G* basis set, suggesting that this basis set may be a more efficient choice for the calculation of T M P s of larger molecules. If other molecular properties (e.g., the total energy) that are more dependent upon the small-r electron density are also of interest, however, then use of a larger basis set should be considered. The present study involving larger molecules further supports the validity of the T K S A for the application of D F T to E M S , ini t ia l ly evaluated by Duffy et al . [20,21] using atoms and small molecules. Only small variations of the D F T T M P s are observed wi th changes in the exchange-correlation functional. In general, the L S D A T M P s tend to have the greatest Chapter 7. Conclusions 215 intensity at low momentum, most likely as a result of overestimation of the large-r electron density when using this functional. The use of gradient-corrected functionals tends to de-crease the theoretical intensity at low momentum. Unfortunately, the statistical precision of the present data is insufficient to comment on the relative accuracy of the various functionals for calculating theoretical momentum profiles. The discrepancies between even the large basis set H F calculations and the experimen-tal momentum profiles of acetone, dimethoxymethane and glycine indicate the inadequacy of non-correlated methods for calculating the outer spatial regions of the valence electron distributions of these molecules. This finding is consistent with the observation that some accounting of electron correlation is necessary for the accurate calculation of other properties (e.g., relative conformer energies and/or geometries) of glycine [157] and dimethoxymeth-ane [142]. The good agreement between the D F T momentum profiles and the experimental data for these molecules suggests that D F T is an effective choice for accounting for these elec-tron correlation effects and modeling the longer range electron density. This is particularly encouraging considering the relative efficiency of D F T as compared w i t h pos t -HF methods for the inclusion of electron correlation. W h i l e the present work would not have been possible without the construction of the energy-dispersive multichannel spectrometer, the experimental results also demonstrate the l imitations of this instrument. Determination of individual X M P s corresponding to energet-ically close ionization processes was, in many cases, very difficult or impossible. In order to obtain more accurate experimental results for acetone, glycine and dimethoxymethane or to perform E M S studies of yet larger molecules, further improvements in instrumentation are necessary. 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