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Studies of molecular orbitals by electron momentum spectroscopy Clark, S. A. 1990

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STUDIES OF MOLECULAR ORBITALS BY ELECTRON M O M E N T U M SPECTROSCOPY By S. A. Clark B. Sc. (Physics) McMaster University A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES CHEMISTRY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1990 © S. A. Clark, 1990 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. Chemistry The University of British Columbia 6224 Agricultural Road Vancouver, Canada V6T 1W5 Date: Abstract The binding energies and momentum distributions of all of the valence orbitals of (CH 3) 20, PH3, C H 4 and SiH4 have been measured by high momentum resolution electron momen-tum spectroscopy. The binding energy spectra have been compared to Green's function and configuration interaction calculations from the literature and with new calculations performed in collaboration with co-workers at Indiana University and Universitat Braun-schweig. For P H 3 , C H 4 and SiH4, near Hartree-Fock limit calculations of the momentum distributions and very accurate calculations of the ion-neutral overlap using MRSD-CI wavefunctions to describe the ion and target have been performed in collaboration with co-workers at Indiana University. Good agreement is obtained between the (CH 3) 20 measurements and the momentum distributions calculated from relatively simple wavefunctions, except in the case of the outermost orbital. The effects of diffuse and polarization functions in the basis sets, and also the influence of molecular geometry, have been investigated. Comparison of the momentum distributions of the outermost orbitals of H 2 0, CH 3 OH and (CHs)20 demonstrates a derealization of charge density with methyl substitution. The measured momentum distributions of PH 3 , C H 4 and SiH4 are compared with near Hartree-Fock limit calculations as well as ion-neutral overlap calculations in which the ion and neutral wavefunctions are described by multireference, singly and doubly excited, configuration interaction calculations. In each case, the experimental results are well modelled by the near Hartree-Fock limit calculations, and there is little difference between the Hartree-Fock limit and ion-neutral overlap calculations. A significant splitting of the 11 4ai (inner valence) pole strength is observed for PH3 , but the inner valence strength is largely contained in the main peak for both C H 4 and SiH4. Green's function calculations quantitatively reproduce these results. Ion-neutral overlap calculations using MRSD-CI wavefunctions to describe the ion and target have been performed for HF, HC1, Ne and Ar. These are compared with previously published EMS measurements of the momentum distributions. Very poor agreement between theory and experiment is obtained for HF and HC1. The theoretical and experimental results for all of the hydrides CH 4 -HF and SiH 4-HCl as well as Ne and Ar are reviewed. 111 Table of Contents Abstract ii List of Tables viii List of Figures x List of Abbreviations xii Units xiv Acknowledgement xv 1 Introduction 1 1.1 Electron Momentum Spectroscopy 3 1.1.1 Binding Energy Spectrum 3 1.1.2 Momentum Distribution 6 1.2 Context of this work 8 1.2.1 Hydrides 9 1.2.2 Methyl substitution 10 1.3 Overview of this work 11 1.3.1 Layout of the thesis 13 2 Theory 16 3 Experimental Method 21 iv 3.1 Description of the Spectrometer 21 3.2 Signal Processing 25 3.2.1 Details of TPHC operation 27 3.3 Operating Procedures ,29 3.4 Data Analysis 32 4 Dimethyl Ether 35 4.1 Binding Energy Spectra 36 4.2 Choice of a Basis Set to Describe a Large Molecule 40 4.3 Comparison of Experimental and Calculated Momentum Distributions 44 4.3.1 Overall Results for Valence Orbitals of (CH 3) 20 44 4.3.2 Results of the basis set investigation 51 The effects of diffuse functions on the carbon and oxygen . 51 The effect of adding more functions of the hydrogen . . . . 52 The effect of d functions on the oxygen and carbon . . . . 53 Summary : 53 4.3.3 The effect of molecular geometry on the calculations 54 4.3.4 The observed shape of the 2bi momentum distribution 57 4.4 Comparison of the Electron Density in the Outermost Orbitals of HOH, CH 3 OH, C H 3 O C H 3 60 5 Phosphine 72 5.1 Calculation of the Momentum Distributions of PH3 73 5.2 Binding Energy Spectra 76 5.3 Comparison of Experimental and Calculated Momentum Distributions . 83 v 6 Methane 92 6.1 Calculation of the Momentum Distributions of CH4 93 6.2 Binding Energy Spectra 95 6.3 Comparison of Experimental and Calculated Momentum Distributions 102 7 Silane 107 7.1 Calculation of Binding Energy Spectra 107 7.2 Calculations of Momentum Distributions of SiH4 109 7.3 Binding Energy Spectra I l l 7.4 Comparison of Experimental and Calculated Momentum Distributions . 118 7.5 Momentum Distributions of Satellite Structure 122 8 Review of the Hydrides 125 8.1 Calculations 126 8.1.1 HF 127 8.1.2 HC1 128 8.1.3 Ne 129 8.1.4 Ar 131 8.2 Results of Overlap Calculations 141 8.2.1 HF 142 8.2.2 HC1 143 8.2.3 Ar and Ne 150 8.3 Overview of the Hydrides 151 8.3.1 A simple model to explain EMS results 151 8.3.2 Binding Energy Spectra of the Hydrides 154 8.3.3 Momentum Distributions of the Hydrides: General Trends . . . . 157 vi 8.3.3.1 The ion-neutral overlap amplitude and the EMS cross section 158 8.3.3.2 When is correlation negligible? 158 8.3.3.3 Contraction in p-space due to correlation 160 8.3.3.4 Disagreement for inner valence MDs 161 8.3.4 The Hartree-Fock Limit 162 8.3.5 Location of pmax 163 8.3.6 Energy localization versus position localization 166 9 Conclusions 167 Bibliography 172 vii List of Tables 4.1 Measured and calculated binding energies for (CHa^O 39 4.2 Diffuse s and p exponents for oxygen and carbon 41 4.3 Total energy and dipole moment of dimethyl ether 42 4.4 Total energy and dipole moment of methanol 63 4.5 Total energy and dipole moment of water 63 5.6 Calculated PH3 properties 75 5.7 Measured and calculated binding energies for P H 3 82 6.8 Total energy of C H 4 95 6.9 Measured and calculated binding energies for C H 4 . 96 7.10 Total energy of SiH4 109 7.11 Measured and calculated binding energies for SiH4 117 8.12 HF properties 133 8.13 Measured and calculated binding energies for HF 133 8.14 HC1 properties 134 8.15 Measured and calculated binding energies for HC1 135 8.16 Ne properties 136 8.17 Measured and calculted binding energies for Ne 136 8.18 Ar properties 137 8.19 Measured and calculated binding energies for Ar 137 viii 8.20 Experimental and calculated valence IPs of the hydrides 138 8.21 References for experimental ionization potentials 139 8.22 Maximum height of the calculated MDs 140 ix List of Figures 2.1 Scattering kinematics 17 3.2 Schematic of the spectrometer 23 3.3 Schematic of the signal processing electronics 26 3.4 Time spectrum 28 4.5 Binding energy spectra of the valence shell of (CH 3) 20 37 4.6 Comparison of experimental and calculated binding energy spectra of (CH 3) 20 38 4.7 Momentum distributions for all of the valence orbitals of (CH 3) 20 . . . . 45 4.8 Effect of diffuse p functions on 0 and C 46 4.9 Effect of diffuse s functions on 0, C and H 47 4.10 Geometries at which calculations were performed 55 4.11 Effects of molecular geometry 56 4.12 A simple atomic orbital picture of the 2&i orbital 57 4.13 Partitioning of the 2fci momentum distribution 58 4.14 Momentum distributions of (CH 3) 20, CH 3 OH, and H 2 0 65 4.15 Density maps for the HOMO orbitals of (CH 3) 20, CH 3 0H, and H 2 0 . . 66 4.16 Density maps for the NHOMO orbitals of (CH 3) 20, CH 3 OH, and H 2 0 . 67 5.17 Binding energy spectra of the valence shell of PH 3 79 5.18 Comparison of experimental and calculated binding energy spectra of PH 3 80 x 5.19 Momentum distributions of the oai orbital of PH 3 85 5.20 Momentum distributions of the 2e orbital of P H 3 86 5.21 Momentum distributions of the 4aj orbital of P H 3 87 5.22 Comparison of PH 3 MDs to minimal basis set calculation 89 6.23 Binding energy spectra of the valence shell of C H 4 98 6.24 Binding energy spectra of the inner valence region of C H 4 99 6.25 Comparison of experimental and calculated binding energy spectra of C H 4 100 6.26 Momentum distributions of the lt 2 orbital of C H 4 104 6.27 Momentum distributions of the 2aj orbital of C H 4 105 7.28 Binding energy spectra of the valence shell of SiH4 113 7.29 Binding energy spectra of the inner valence region of SiH4 114 7.30 Comparison of experimental and calculated binding energy spectra of SiH4 115 7.31 Momentum distributions of the 2t2 orbital of SiH4 119 7.32 Momentum distributions of the 3ai orbital of SiH4 120 7.33 Momentum distributions of the inner valence satellites of SiH4 124 8.34 Momentum distributions of the valence orbitals of CH 4 and SiH4 144 8.35 Momentum distributions of the valence orbitals of NH 3 and PH 3 145 8.36 Momentum distributions of the valence orbitals of H2O and H2S 146 8.37 Momentum distributions of the valence orbitals of H2O and H2S 147 8.38 Momentum distributions of the valence orbitals of HF and HC1 148 8.39 Momentum distributions of the valence orbitals of Ne and Ar 149 8.40 Molecular orbitals of the hydrides 152 8.41 Spectroscopic factors 156 8.42 The maximum in the momentum distribution 165 xi List of Abbreviations 2ph-TDA two-particle-hole Tamm-Dancoff approximation ADC algebraic diagrammatic construction BES binding energy spectrum CFD constant fraction discriminator CI configuration interaction CMA cylindrical mirror analyser DWIA distorted wave impulse approximation—see chapter 2. EMS electron momentum spectroscopy GTO Gaussian type orbital HOMO highest occupied molecular orbital IP ionization potential MAGW momentum averaged gaussian weighted resolution folding MD momentum distribution calculated within the target Hartree-Fock ap-proximation, equation 2.5. See chapter 2. MRSD-CI multireference singly and doubly excited configuration interaction NHOMO next highest occupied molecular orbital OVD overlap distribution—the momentum distribution calculated from equation 2.4. See chapter 2. PES photoelectron spectroscopy PT perturbation theory xii PWIA plane wave impulse approximation—see chapter 2. RHF restricted Hartree-Fock SAC symmetry adapted configuration SCA single channel analyser SCF self consistent field—usually used to describe a restricted Hartree-Fock calculation (RHF) TDA Tamm-Dancoff approximation THFA target Hartree-Fock approximation—see chapter 2. TPHC time to pulse height converter XMP experimental momentum profile—the angular distribution from an EMS experiment plotted as a function of momentum. Also known as a momentum distribution. xin Units aQ equals 1 bohr and is denned to be 4^j sp- which is approximately 0.529177 A au 1 atomic unit of charge x distance is defined to be aae which is approx-imately 8.5 • IO" 3 0 C m or 2.54177 D au 1 atomic unit of energy is denned to be % 8.854-10"12 C 2 / J m ) which is approximately 4.4 • I O - 1 8 J or 27.212 eV au 1 atomic unit of momentum is defined to be ^ which is approximately 2.0 • IO" 2 4 kgm/s D 1 debye is denned to be 1 0 - 1 8 esucm which is approximately 3.3 • IO" 3 0 C m or 0.393427 au eV 1 eV is approximately 1.6 • 1 0 - 1 9 J or 0.036749 au torr 1 torr is equivalent to 1 mm of Hg or 0.1333 kPa xiv Acknowledgement I am deeply indebted to my research supervisor, C E . Brion, from whom I have learned a great deal. The strengths of this thesis draw much on his ideas and efforts, while its weaknesses are largely my own. It would have been impossible to complete the work contained herein without the help of my many collaborators. It would also have been a considerably less interesting endeavor. I would like to thank all those with whom I have shared the limelight in the scientific literature: L. Adamowicz, A.O. Bawagan, C M . Boyle, E.R. Davidson, P. Du, D. Feller, R.F. Frey, W. von Niessen, T.J . Reddish, J . Schirmer, and E. Weigold. For invaluable technical assistance, discussions of things I could not grasp on my own, for advice, for guidance, for a few good references, for lending a hand, an ear, or a sharp drill bit, I am very grateful to: E. Burnell, G. Burton, P. Carpendale, M . E . Casida, S.O. Chan, W.F. Chan, D.P. Chong, G. Cooper, M . Coschizza, P. Duffy, C L . French, E. Gomm, B. Greene, R. Hamilton, M . Hatton, W.J . Henderson, B. Hollebone, D. Jones, T. Koga, N . Lermer, R .K. Jones, R. Marwick, B. Snapkauskas, R.N.S. Sodhi, K . H . Sze, B. Todd, D. Tonkin, E. Zarate, W. Zhang, and all the staff of the Chemistry department. Finally, I thank the National Sciences and Engineering Research Council for funding this research and for providing a postgraduate scholarship. xv Chapter 1 Introduction One of the earliest demonstrations of quantization was the classic Frank-Hertz experi-ment [1] which detected the energy loss of an electron which had excited an atom. This experiment is not very different in principle than the one proposed more than fifty years later as a quantitative test of quantum chemistry [2,3]. The proposed "(e,2e) reaction" involved the ionization of a target molecule M by a high energy electron: e~ + M —> M+ + e~ + e~ (1.1) and the coincident detection of the two outgoing electrons. If the momenta of all of the electrons were measured, it was proposed, then the angular distribution of the scat-tered electrons would yield information about the momentum distributions of the bound electrons in the target. The first (e,2e) measurements of atomic systems were soon reported [4,5]. Subse-quent detailed development of the (e,2e) reaction theory took place in conjunction with experimental measurements that tested the proposed theories [6,7,8,9,10]. The first electron momentum distributions measured for individual orbitals were those for carbon films [11] and Ar [12]. The study of Ar demonstrated that the particular scat-tering geometry now usually employed in electron momentum spectroscopy (symmetric 1 Chapter 1. Introduction 2 non-coplanar), in which the two outgoing electrons have equal energies and polar an-gles, yields results which are, to the greatest possible extent, independent of kinematic factors [12,13]. Hence electron momentum spectroscopy is primarily a probe of the prop-erties of the target. While the experimental results from electron impact ionization coincidence experi-ments employing other scattering geometries continue to provide challenges to scattering theorists [14,15,16], electron momentum spectroscopy has been used for some years as a test of quantum chemical calculations of the target wavefunctions and as a probe of the valence electronic structure of atoms and molecules. It has provided insight into the nature of electron distributions that takes us well beyond the confirmation, provided by Frank and Hertz, of Bohr's energy shell model. Layout of this chapter In section 1.1, electron momentum spectroscopy is explained in greater detail. The method of measuring binding energy spectra and momentum distributions and the rea-sons these measurements are of interest is discussed. A more complete discussion of these topics can be found in several review articles [10,17,18,19,20]. In section 1.2 a review of relevant studies is given which defines the context of the work presented in this thesis. Section 1.3 briefly summarizes the work performed and section 1.3.1 describes the layout of the thesis. Chapter 1. Introduction 3 1.1 Electron Momentum Spectroscopy Electron Momentum Spectroscopy (EMS) is a technique capable of measuring the vertical ionization potentials and electron momentum distributions of atoms and molecules. In an EMS experiment, a beam of high energy electrons ionizes a target and the outgoing electrons are detected in coincidence (see figure 2.1 in chapter 2). The outgoing electrons have equal, fixed polar angles (0a = 62). The energy of the electron beam, known as the impact energy, is Eo + £ where E0 is very large compared to the binding energy of the orbital being studied. The scattering event results in an energy loss of £ to the target, hence e is the binding energy. The coincidence count rate may be measured as a function of the binding energy e, in which case it is referred to as a binding energy spectrum, or as a function of the azimuthal angle (f>, in which case it is referred to as a momentum distribution. 1.1.1 Binding Energy Spectrum A binding energy spectrum is measured by varying the impact energy at a constant 4>. Peaks in the coincidence count rate occur at energies for which ionization is most favourable, much as in photoelectron spectroscopy (PES). The small coincidence cross sections (and therefore low count rates) of EMS experiments usually preclude the use of monochromators in the primary electron beam, hence the energy resolution (typically 1-2 eV FWHM) of an EMS experiment is determined by the temperature of the electron source and the response function of the electron analysers. (A notable exception is the Ar measurements by Williams [21], who used a monochromator to obtain an energy resolution of 0.06 eV.) PES measurements often have higher energy resolution than EMS Chapter 1. Introduction 4 measurements. Nevertheless, electron momentum spectroscopy offers information not readily available from PES, and this will be discussed below. Prior to the advent of synchrotron radiation, photoelectron spectroscopy was routinely performed with discrete resonant lamp sources. The most commonly used lines, He I and He II, have energies of 21.2 eV and 40.8 eV respectively. The He II spectra are frequently contaminated above ~20 eV by a He I 'shadow' and by self ionization of helium at a binding energy of 24.6 eV. Hence although very high energy resolution can be obtained, such sources are inappropriate for investigation of the inner valence region of molecules (typically in the range 20-40 eV), which often shows interesting structure well above the usable range of these sources. In contrast, changing the impact energy in an EMS experiment is trivial and observation of the inner valence region is routine. Some of the earliest identifications of extensive satellite structure in the inner valence region were by EMS [22,23,24,25,26]. The experimental results presented in chapters 4 through 7 are for molecules for which PES data in the inner valence region was sparse or non-existent. Synchrotron sources can, in principle, provide high energy, high resolution, high inten-sity, tuneable light over a wide energy range. Synchrotrons are, however, large, expensive, and not readily available. Furthermore, techniques for monochromation of light in the soft X-ray region [hv = 50-150 eV) are not yet routine. Energy resolution may only be moderate, and, like other PES light sources, stray light and higher order radiation may lead to large backgrounds (often not constant, or even linear, with photon energy or energy loss) which confuse the interpretation of the spectra. As will be discussed in more detail in chapter 3, EMS spectra are obtained with zero background. As has been mentioned already, there is considerable interest in the observation and Chapter 1. Introduction 5 identification of satellite lines. Within the simple Hartree-Fock picture, one peak occurs in the binding energy spectrum corresponding to each occupied energy shell, or orbital. If one assumes that there are some small effects due to correlation and relaxation, one might predict the existence of weak satellite bands in the ionization spectrum which "borrow" intensity from the main peak. Within the language of configuration interaction, one would view these peaks as corresponding to different roots of the final ion state. They can also be viewed in a simple model as excitations which accompany the "normal" ionization process. Satellite peaks typically occur in the inner valence region, and for some molecules, the intensity distribution is such that it is inappropriate to refer to "main lines" and "satellite lines". Theoretical descriptions of the observed binding energy spectra are, of course, of great interest. In the case of photoelectron spectroscopy, it is possible to predict the energies of the observed structures, but accurate predictions of the intensites are extremely difficult and dependent on kinematic factors like the incident photon energy [27]. By contrast, an EMS experiment is designed to make the calculation of the cross section relatively simple. The cross section is essentially independent of kinematic factors, and depends only on the structure (both initial and final state) of the target. As will be shown in chapter 2, the cross section is proportional to the spectroscopic factor, which can in principle be accurately calculated by Green's function, configuration interaction, or related methods.1 Hence the intensity distribution in a binding energy spectrum obtained from an EMS experiment is readily interpreted, useful as a test of quantum chemical calculations, and directly comparable to intensities as measured in other EMS experiments, independent of 1 To be more precise, the momentum integrated cross section as a function of e is proportional to the spectroscopic factor. Measurement of the differential cross section at a given momentum is proportional to the spectroscopic factor for measurements made over a given symmetry manifold. Chapter 1. Introduction 6 the choice of impact energy. All of the binding energy spectra in this thesis are compared with the results of Green's function calculations. In the case where an ion state, corresponding to a peak in the binding energy spec-trum, is theoretically described by one dominant hole configuration, the peak is said to be assigned to that one hole state. Particularly for simple molecules in which the num-ber of occupied orbitals of a given symmetry is small, it is possible to assign all of the satellite structure to various parent ion states. Reliable assignment of satellite structure by experiment is therefore of great interest in order to confirm theoretical assignments. In photoelectron spectroscopy the cross section for a given photoionization process may be studied either as a function of photon energy or of electron ejection angle in order to aquire information leading to an assignment of the process. However, the results of such investigations of anisotropy parameters (8) are very difficult to interpret theoreti-cally and may lead to incorrect assignments [28,29,30,31,32]. In contrast, the variation in intensity with <b as measured in an EMS experiment is a simple probe of the symmetry of the satellite peak and is easily interpreted both qualitatively and quantitatively. In the absence of initial state correlation, the <f> distribution of all of the satellites in the same symmetry manifold should have the same shape, and therefore such a measurement provides an ideal test of satellite parentage. Assignment of the inner valence satellite structures is made for all of the molecules studied in this thesis. 1.1.2 Momentum Distribution A momentum distribution is measured by varying (b at constant impact energy. The momentum of the bound electron prior to ionization, p, is a simple function of the angle Chapter 1. Introduction 7 4>. Hence the intensity as a function of <j> can be transformed into intensity vs. p, and is therefore called a momentum distribution. In measuring a momentum distribution, we are selecting a particular final ion state for study, to be precise, we are chosing that final ion state whose energy differs from the ground state by e. The cross section as a function of momentum is independent of kine-matic factors and is dependent only on the initial and final states of the target. Within the plane wave impulse approximation used for interpreting the EMS cross section, the intensity is given by the spherically averaged amplitude of the ion-neutral overlap. Mak-ing a more drastic approximation, known as the target Hartree-Fock approximation, the ion-neutral overlap can be equated to the momentum space wavefunction of the canonical Hartree-Fock orbital from which the ionization has taken place. Experimental measure-ments of momentum distributions are of great interest for the following reasons: • A complete, quantitative theoretical description of all aspects of the cross section is available. This is rare in scattering experiments. Where the plane wave impulse approximation is correct we can expect that the experimental momentum distri-butions of small systems can be predicted with great accuracy, and for any larger system, it can be accurately calculated at least in principle. It has been said that "the high-energy low-momentum (e,2e) reaction [EMS] is understood better than any other reaction in atomic or nuclear physics in the sense that we can calculate right answers with more certainty" (I.E. McCarthy, ref. [33]). • As the cross section depends only on the initial and final state of the target, it provides a quantitative check of theoretical calculations of target wavefunctions. • As was mentioned in section 1.1.1, the ability to select a particular state for study Chapter 1. Introduction 8 allows one to identify and assign structure. • The simple relationship between the cross section and the amplitude of the canon-ical Hartree-Fock orbital, though only approximate, is intriguing in itself. Within this approximation, EMS allows one to "see" canonical Hartree-Fock orbitals, which are one of the most important theoretical constructs in chemistry. It is straightforward to calculate momentum distributions from wavefunctions when the basis function exponents and orbital coefficients are provided, as they often are in the literature. The calculation [34] involves performing a Fourier transform of the basis functions into momentum space and a spherical average to take account of the random orientation of the target species. These calculations are performed at U.B.C. using wave-functions from the literature, those provided by collaborators, or those calculated here using standard molecular orbital programs [35]. 1.2 Context of this work Prior to 1984, EMS measurements had been published for a number of atoms, several diatomics, a number of small hydrides, various halogen substituted methanes, various methyl substituted amines, and the following molecules: CO, C 2 H 2 , NO, H 2 CO, C 2 H 4 , C 2 H 6 , CH 3 OH, C 0 2 , N 2 0, CH 3 CN, C 3 H 6 , C 2 H 3 F , OCS, butadiene, C 2H 3C1, CS 2, ben-zene, SF 6 and Cr(C0)6 [36]. A number of these studies were performed with poor momentum resolution, and some did not provide quantitative comparison with theory by normalizing the momentum distributions to the peak areas in the binding energy spectrum (see chapters 2 and 3). Very few studies compared the experimental results to ion-neutral overlap calculations, but instead comparisons were made to the spherically Chapter 1. Introduction 9 averaged, canonical Hartree-Fock orbital. Only in the case of atoms were the SCF calcu-lations at the Hartree-Fock limit, since typical SCF calculations for molecules (and the few ion-neutral overlap calculations) employed very small basis sets. Tests of the plane wave impulse approximation were made by varying the impact energy; and energies on the order of or greater than 1000 eV were found to be generally sufficient to give impact energy-independent results (for p below ~ 1.5 au). Observed discrepancies between the-ory and results therefore placed suspicion on the quality of the structure calculation, as the reaction theory was experimentally confirmed. 1.2.1 Hydrides The hydrides of the first and second row atoms C-F and Si-Cl are particularly interesting for study by EMS. While not as simple as atomic targets, they are sufficiently small to lend themselves to very accurate quantum chemical calculations. The valence orbitals are in all cases easily resolved in energy. At the time this work began (1986), high momentum resolution EMS measurements existed for NH 3 [37], H 2 0 [38], H2S [28], HF and HC1 [39] and the noble gases with which these molecules are isoelectronic, Ne and Ar [40]. Momentum distribution calculations essentially at the Hartree-Fock hmit, as well as highly accurate calculations of the ion-neutral overlap using multireference singly and doubly excited configuration interaction wavefunctions to describe the initial and final target states, were performed in collab-oration with E.R. Davidson's group at Indiana Univeristy for the molecules NH 3 [37], H 2 0 [38] and H2S [28]. It was demonstrated that in the case of H 2 0, previously published SCF calculations Chapter 1. Introduction 10 were not at the Hartree-Fock limit. For both H2O and N H 3 , a correct description of the experimental intensity was not obtained within the target Hartree-Fock approximation, even at the Hartree-Fock limit. Good agreement was only obtained for ion-neutral overlap calculations of the cross section. The momentum distributions of H2S, on the other hand, showed good agreement within the target Hartree-Fock approximation, and the effect of correlation and relaxation on the momentum distributions was found to be neghgible. In order to complete the study of the hydrides, high momentum resolution EMS mea-surements of C H 4 , SiH4 and P H 3 were made. In addition, near Hartree-Fock limit and CI wavefunctions for CH 4 , SiH4, PH 3 , HF, H C 1 , Ar and Ne were calculated by E.R. David-son's group in Indiana, and these were used to calculate the theoretical momentum distributions at U . B . C 1.2.2 Methyl substitution Prior to this work, several studies had been done examining the effects of methyl sub-stitution on the electronic charge density by electron momentum spectroscopy [41,42,43] and more appeared in the literature while this work was in progress [44,45]. Tossell et al. [41] measured the momentum distribution of the outermost orbital in both C H 3 N H 2 and N H 3 and discussed the effect of methyl substitution on this orbital, which they expected to be predominantly N 2p. The experimentally observed trends in the momentum distributions were qualitatively reproduced by SCF calculations using small basis sets. The use of these wavefunctions to calculate the charge density auto-correlation function showed that significant contributions to the outermost orbital were made by the C 2p, and in particular, the H Is on the methyl hydrogen trans to the N 2p. Chapter 1. Introduction 11 They conluded that methylation results in a derealization of the charge density in the outermost orbital. Bawagan and Brion measured the momentum distribution of the outermost orbital in each of NH 3 , NH 2 CH 3 , NH(CH 3) 2 ) N(CH 3) 3 [37,42,43]. They observed an increase in the low momentum intensity with increasing methyl substitution which was qualitatively supported by SCF calculations using small basis sets. They argued that the increase in intensity at low momentum corresponded to an increasing contribution from H s orbitals, and like Tossell et al., they concluded that a derealization of charge density occured with increasing methylation [42,43]. Since measurements had already been published of the momentum distributions of CH 3 OH and H 2 0, it was logical to make a measurement of the momentum distribution of the outermost orbital of (CH 3) 20 so as to a continue the studies of methyl substitution. In fact, the complete valence binding energy spectrum and the momentum distributions of all of the orbitals which could be resolved in energy were measured. 1.3 Overview of this work This thesis presents EMS measurements of the binding energy spectra and momentum distributions of (CH 3) 20, PH 3 , CH 4 and SiH 4 made on a non-coplanar, symmetric spec-trometer at an energy of EQ = 1200 eV. The binding energy spectra are compared with Green's function calculations from the literature, and in the case of SiB.4, with new calculations performed in collaboration with Chapter 1. Introduction 12 co-workers at Universitat Braunschweig. Good agreement between theory and experi-ment is obtained for all molecules studied. In each case, the satellite structure observed in the inner valence region is found to be derived primarily from the innermost valence orbital. The momentum distributions for the valence orbitals of (CH 3) 20 are compared with the results of SCF calculations using GAUSSIAN76 [35]. An investigation is made of the effects of the choice of the basis set on the calculated momentum distribution. Satis-factory agreement is obtained between theory and experiment for all but the outermost orbital. The momentum distributions of the two outermost orbitals are compared to the corresponding orbitals in CH 3 OH and H 2 0. The measured momentum distributions of C H 4 , SiH 4 and P H 3 are compared with near Hartree-Fock limit calculations of the momentum distributions. The effects of correlation and relaxation on the calculated momentum distributions is investigated by performing calculations of the ion-neutral overlap using CI wavefunctions. These calculations were done in collaboration with co-workers at Indiana University. Near Hartree-Fock limit and CI calculations of the wavefunctions for the ground and neutral states of HF, HC1, Ne and Ar performed at Indiana University were used to calculate momentum distributions for the valence orbitals of these targets. The results are compared with previously measured experimental data [39,40]. The experimental and theoretical results for the momentum distributions of all of the hydrides CH 4 -HF and SiH 4-HCl are presented and reviewed. Chapter 1. Introduction 13 1.3.1 Layout of the thesis The organization of the thesis is as follows: Chapter 2 presents the scattering theory necessary for an understanding of the rela-tionship between the theoretical calculations and the EMS cross section. This has been presented in detail previously [10,27]. The basic methods for performing the theoreti-cal calculations of interest (restricted Hartree-Fock, Green's functions, and configuration interaction) are not reviewed here, as they are well known and reviews abound in the literature [46,47,48,49,50,51]. Details of the methodologies employed for each molecule are given in the appropriate chapter. Chapter 3 describes the spectrometer and the data collection procedures employed in this work. The spectrometer was constructed at UBC with the aid of the electronics and machine shops, and was first described by Hood, Hamnett and Brion [52]. Subsequent modifications were made by Cook [53] and Leung [54]. Chapter 4 presents the EMS results for dimethyl ether. The binding energy spec-trum is compared with a previously published, extended 2ph-TDA, Green's function calculation. The momentum distributions are calculated using restricted Hartree-Fock wavefunctions calculated from small basis sets and compared to the experimental results. A systematic investigation was made of the effects of diffuse and polarization functions in the basis sets. The shape of the experimental momentum distributions is compared to those previously measured for H 2 O and C H 3 O H , and discussed in terms of the methyl inductive effect. This work has appeared in the literature as: S.A.C. Clark, A.O. Bawagan and C E . Brion, Chem. Phys. 137 (1989) 407. Chapter 1. Introduction 14 Chapter 5 presents the EMS results for phosphine. The binding energy spectrum is compared with previously published PES and EMS spectra, as well as with Green's function calculations at the 2ph-TDA and simplified ADC(4) levels, and a multirefer-ence, singly and doubly excited, configuration interaction (MRSD-CI) calculation using a large basis set. A significant breakdown of the molecular orbital picture of ioniza-tion is observed experimentally and quantitatively predicted by the calculations. The satellite structure is assigned based on the angular variation of the intensity. The exper-imentally measured momentum distributions are compared with calculations based on near Hartree-Fock limit wavefunctions and MRSD-CI calculations using large basis sets. These calculations were performed by E.R. Davidson and C. Boyle of Indiana University. This work has appeared in the literature as: S.A.C. Clark, C.E. Brion, E.R. Davidson and C. Boyle, Chem. Phys. 136 (1989) 55. Chapter 6 presents the EMS results for methane. The satellite structure in the binding energy spectra is assigned and compared with a previously published, 2ph-TDA Green's function calculation, and MRSD-CI calculations performed by E.R. Davidson and R.F. Frey of Indiana University. These calculations are also used to compute momentum distributions for comparison with the experimental results. This work has appeared in the literature as: S.A.C. Clark, T.J . Reddish, C.E. Brion, E.R. Davidson and R.F. Frey, Chem. Phys. 143 (1990) 1. Chapter 7 presents the EMS results for silane. The binding energy spectra are compared to ADC(3) and simplified ADC(4) Green's function calculations performed by W. von Niessen and J . Schirmer of the Universitat Braunschweig, and to MRSD-CI Chapter 1. Introduction 15 calculations, performed by E.R. Davidson's group at Indiana University. The MRSD-CI calculations, along with restricted Hartree-Fock calculations near the Hartree-Fock limit, also performed at Indiana Univeristy, are compared to the measured momentum distributions. This work has appeared in the literature as: S.A.C. Clark, E. Weigold, CE. Brion, E.R. Davidson, R.F. Frey, CM. Boyle, W. von Niessen and J. Schirmer, Chem. Phys. 134 (1989) 229. Chapter 8 presents the results of near Hartree-Fock limit and MRSD-CI calcu-lations of HF, HC1, Ne and Ar. The momentum distributions calculated from these wavefunctions are compared to previously published [39,40] experimental results. The experimental and theoretical results for all of the hydrides are reviewed. This work is abstracted from: S.A.C. Clark, CE. Brion, E.R. Davidson, CM. Boyle and R.F. Frey, to be published. Chapter 9 contains concluding remarks. Every attempt has been made to make each chapter complete and clear in itself; however it is recommended that chapter 2 be read prior to any of chapters 3 to 8. Chapter 2 Theory An (e,2e) reaction involves the ionization of a target species M by an electron beam and the measurement of the momenta of all of the incoming and outgoing electrons (see figure 2.1 and equation 1.1). The energy of the incoming electron is E0 + e. The "scattered" electron has energy E\ and the "ejected" electron has energy E2 < E\. The incoming electron has momentum k0 and the outgoing electrons have momenta fca and k2. We define p = ki -f- k2 — fc0. Under the binary encounter approximation, p is the momentum of the bound electron just prior to ionization. For the purposes of this thesis, we understand EMS to be an (e,2e) experiment per-formed with <f> variable, Ex = E2 = E0/2 and 6i = 62. This is often referred to as a "symmetric, non-coplanar" geometry. All of the experiments performed in this thesis were carried out with E0 = 1200 eV, £ a = E2 = 600 eV, and 6l = 62 = 45°. These conditions ensure: (1) large momentum transfer to the struck electron, (2) large energy transfer to the struck electron, (3) a rapid collision, (4) good agreement [33] between the plane wave impulse approximation and the distorted wave impulse approximation, which are discussed later. 16 Chapter 2. Theory 17 Figure 2.1: The scattering kinematics in the (e,2e) reaction. For a symmetric, non-coplanar EMS experiment, E\ = E2 = E«/2, 6\ =62, and <j> is variable. A detailed discussion of (e,2e) reaction theory and its relation to structure calculations is given in refs. [10,27]. The results of interest to EMS are reproduced here. Other discussions of heuristic value can be found in references [2,20,33,55,56]. Within the binary encounter approximation [57] and assuming an impulsive colli-sion [58], the (e,2e) scattering amplitude can be shown to be °t oc \{x[-\[-)*ri\T\*?X{:))\2 (2-2) XJ ^ and x\ ^ a r e outgoing electron waves, xi+^ *s the incoming electron wave, is the TV electron target wavefunction, is the wavefunction of the ion in final state / , and T = V + VGT where the Green's function G and potential V are denned appropriately for the above two body (binary encounter) interaction. Under the conditions of no core excitation, high momentum transfer, and weak coupling between non-elastic and elastic channels, T is simply the Mott scattering T-matrix, T m . Equation 2.2, with T equated Chapter 2. Theory 18 to T m , is referred to as the distorted wave impulse approximation (DWIA). Various approaches can be used to calculate the distorted waves x [10,59,60]. As dis-torted wave calculations are at present not tractable for molecules, all of the calculations in this thesis were performed within the plane wave impulse approximation, described below. The distorted wave impulse approximation will factor if the Xi a r e plane waves. (In fact, it will factor under much less drastic assumptions, as discussed in ref. [10], sec-tion 8.3; however this is not of interest here.) One then obtains: Of cx (k\Tm\k')\2 K x S - ^ * ? - 1 1*^X^)1' (2-3) where k = \(ko — p) and k' = |(iti — k2). The half-off-the-energy-shell Mott scat-tering cross section, | T m | 2 , is constant under the scattering conditions employed in a non-coplanar, symmetric EMS experiment [40]. Writing the explicitely as plane waves with momenta ki, and dropping the constant term IT^ I2, we have: o~j oc |(e'>f S^- 1 | * f ) | 2 (2.4) The ion-neutral overlap, < elP'r I $ f >, can be evaluated directly; however the target Hartree-Fock approximation (THFA) is usually applied. Under the THFA, in which the ion-neutral overlap is assumed to be proportional to a canonical Hartree-Fock orbital, the cross-section reduces to °1 « \UP)\" (2-5) where <j>g(p) is the momentum space wavefunction for the canonical Hartree-Fock orbital q from which the electron was removed. The spectroscopic factor, Chapter 2. Theory 19 =|< | a g | tyF >|2, (where a, is an annihilation operator for orbital q), is the probability of finding the one-hole configuration in the expansion of the final ion state. Where final state electron correlation effects are negligible, this expansion consists of only one term. It can be shown [10,18,27] that the following spectroscopic sum rule applies under approximations less drastic than the target Hartree-Fock approximation: £ S { ' > = 1 (2.6) / where the summation is over all final configurations, though only states within the same symmetry manifold q will contribute. When this is applied to equation 2.5 we obtain E * / « l*.(p)la (2-7) Equation 2.7 states that measurements of different orbitals in the same molecule can be made to have the correct intensities relative to each other by summing the experimental results over all final ion state configurations for each orbital. This amounts to normalizing the momentum distributions to the observed peak areas in the binding energy spectrum, as discussed in section 3.4. This derivation neglects several points which we now mention: • We have ignored spin for simplicity. Spin-orbit coupling is negligible at the ener-gies at which the experiment is performed, and all wavefunctions could have been written as products of space and spin functions. • As the targets studied in this thesis are not oriented, all expressions for the cross section should include an integral over angles /d$l. • Rotations and vibrations are not resolved experimentally. All calculations are per-formed at the equihbrium geometry and assuming vertical ionization. See also Chapter 2. Theory 20 comments in ref. [6,8,10]. Notation Spherically averaged calculations based on equation 2.4 are referred to in the text as overlap distributions or OVDs. Spherically averaged calculations based on equation 2.5 are referred to as momentum distributions or MDs. Experimental measurements are referred to as X M P s . Chapter 3 Experimental Method An electron momentum spectroscopy experiment requires: 1) Production of a coUimated, high energy electron beam. 2) Collision of this beam with a gas sample to induce ionization. 3) Selection of the scattered and ejected electrons which satisfy the conditions E\ = E2 — E0/2 and 6l = 62 = 45°. 4) Detection of the scattered and ejected electrons in coincidence. 5) Measurement of the coincidence count rate as a function primarily of (a) energy loss and (b) azimuthal angle <f>; but also as a function of (c) time, which is important for diagnostic purposes. The following sections will discuss how these ends are achieved. 3.1 Description of the Spectrometer A schematic of the EMS spectrometer used in this work is shown in figure 3.2. This spectrometer has been described in detail previously [40,52,53,54]. The spectrometer is constructed of brass, with the exception of the apertures, which are made of molybdenum to decrease the amount of backscattering of electrons, and 21 Chapter 3. Experimental Method 22 the gun, which is made of stainless steel due to the high temperatures in the region of the filament. The vacuum housing is composed of a thick-walled aluminum cylinder with O-ring seals at each end, and is pumped by two Varian VHS-4 diffusion pumps (1200 1/s air). The gun region is sealed from the rest of the spectrometer (except for an aperture to permit passage of the electron beam) and is pumped from below. The analyzers and gas chamber are pumped from above. A large diameter (16 cm) copper tube with two 90° bends connects the top flange to a diffusion pump mounted on the side of the spectrometer. This differential pumping system is less than ideal since the pumping speed of the side pump is considerably reduced by the long pumping path. The electron beam is produced with a DC, directly heated, thoriated tungsten fila-ment. Typical heating currents are 1.8-2.2 A at constant voltage. The gun is a commer-cially available Cliftronics CE5AH normally used in oscilloscope tubes. Modifications to the gun as supplied are minor, and consist of mechanical changes to facilitate mounting in the spectrometer. The filament, anode, grid, and second element of the lens are floated at Eo = —1200 eV with respect to the grounded collision region. The grid is typically set to between 0 and -5 V (with respect to the filament) but occasionally, particularly for fil-aments that have moved away from the grid aperture, may be operated at up to +100 V. The anode is typically at +50 to +150 V. The first and third elements of the lens are at true ground potential, and the second element is adjusted in the range 0 to +200 V with respect to the filament. The collimation and direction of the electron beam can be monitored on apertures A l , A2 and A3 and the faraday cup, all of which are connected to ground through microam-meters. The beam can be steered with deflectors Dl and D2, each of which consist of two pairs of electrostatic plates. Chapter 3. Experimental Method 23 1 * S E R V O | M O T O R I P U M P Figure 3.2: Schematic diagram of the EMS spectrometer used in this work. Legend: A anode EC end correctors L lens A1-A7 apertures F filament SL supporting legs C channeltron FC faraday cup T turntable CMA analysers G grid ZL zoom lens D1-D4 deflectors GC gas cell Chapter 3. Experimental Method 24 The gas cell is open to the rest of the spectrometer only through apertures A2 and A3 and the exit apertures to the lenses. A 70° slot is cut in the collision region to allow A4 a clear view of the collision region as it rotates about <f>. A sliding blind is used to cover that part of the slot which is not being used at any given value of <p. The angle of the scattered beam is selected by apertures A4 and A6 (A5 and A7). The energies of the two scattered electrons are selected by the energy analyser (labelled CMA in figure 3.2). Each energy analyser is a 135° sector of a cylindrical mirror analyser [61]. End correctors (logarithmically spaced plates with voltages that vary hnearly between the inner cylinder and outer cylinder voltage) are placed at the top and bottom and on the ends of each CMA segment. Second order focussing is attained if the entrance angle of the beam is 42.3°. Each CMA is tilted by 2.7° with respect to the table on which the spectrometer is mounted to allow an entrance angle of 42.3" while selecting a scattering angle (with respect to the incident beam) of 45°. The energy resolution of the analysers, AE/E, is 1% FWHM, where E is the pass energy of the beam. Clearly it is desirable to keep the pass energy low in order to obtain sufficient energy resolution. This is achieved with a three element zoom lens (ZL) between the collision region and the analyser. The zoom lens decreases the scattered beam energy by 500 eV, thus allowing the analyser to be operated with a pass energy of 100 eV for an incident energy E0 = 1200 eV. The first element of the lens is at ground potential and the third element of the lens (V3) is at -500 V, which necessitates that the CMA be floated at -500 V. 1 The voltage on the centre element of the lens is typically -8 V with respect to V3. Deflectors D3 and D4, which consist of two pairs of electrostatic plates, are adjusted 1 The C M A inner cylinder is kept at 0 V with respect to V3, while the outer cylinder is set to the approximately -77 V with respect to V3. The value of -77 V is the appropriate potential difference to analyse 100 eV electrons. Chapter 3. Experimental Method 25 to optimize the count rate, energy resolution, momentum resolution, and uniformity of the scattered beam about (b, simultaneously. One entire zoom lens and analyser system is rotatable about the vertical axis under computer control (see ref. [53], section 3.2.7) on ruby balls, which ride in a track as indicated in figure 3.2. The other CMA is fixed in position. This allows the angle 4> to be scanned between -35° and 35°. The coincidence cross section is symmetric about d> = 0°. The ability to scan d> about positive and negative angles, and therefore to test for this symmetry, is a useful diagnostic. The electrons are detected with commercial channeltron electron multipliers (Mullard B318AL). The entrance of each channeltron is floated at V3. The signal is capacitatively decoupled from the exit potential, which is typically -f 3.0 to +3.5 kV with respect to V3. 3.2 Signal Processing The channeltron signal is decoupled inside the vacuum housing using a high voltage capacitor at the channeltron exit. A schematic of the signal processing circuitry is shown in figure 3.3. From the channeltron exit the signal then travels 30 cm to a vacuum feedthrough on the atmospheric side of which is a preamplifier (Ortec 9301).2 After the preamp, the signal has a pulse height on the order of 300 mV and can be sent to a series of NIM-standard modules that perform most of the signal processing. As can be 2 The original transistors of the 9301 preamplifier have been replaced with Motorola 2N5179 and NSC PN4917 (sometimes PN4916s have been used, the PN4917s are slightly better). These inexpensive replacements for the original items are found to provide satisfactory preformance at the 5 ns time resolution of the experiment. Chapter 3. Experimental Method 26 NIM BIN CFD NIM FAST NIM SLOW CFD \ DELAY RATEMETER START TPHC OUT STOP E SCAN«-SCAN*-COMPUTER TRUE START TRUE STOP TPHC OUT TRUE COUNTS" "RANDOM COUNTS" SCA1 SCA2 Figure 3.3: Schematic of the signal processing electronics. CFD—constant fraction discriminator, SCA—single channel analyser, TPHC—time to pulse height con-verter. Chapter 3. Experimental Method 27 seen in figure 3.3, the signals are amplified by variable gain amplifiers (Ortec 454), and low amplitude noise is removed using constant fraction discriminators (Ortec 463). The "singles" count rate can then be sent directly to the computer for diagnostic purposes. One CFD signal is used as the start pulse for a time to pulse height converter (Or-tec 467) and the other is delayed (approximately 20 ns) by using a loop of wire before being sent as the stop pulse. The TPHC output is sent directly to the computer for diagnostic purposes as well as to the two single channel analysers (one is an Ortec 406A SCA, the other SCA is built into the Ortec 467 TPHC). One SCA is set to pass only those amplitudes which correspond to "true coincidences" while the other is set to scan a region which corresponds only to "random coincidences". A typical time spectrum is shown in figure 3.4. The "random" or background count rate is subtracted automatically from the "true" signal so that spectra are coDected with zero background. The computer interface has been described in detail previously (see ref. [53], section 8.5.3). 3.2.1 Details of TPHC operation The TPHC gives an output of 0 to -f 10 V which is Hnearly related to the time between the start and stop pulses. The high end (+10 V) corresponds to the time selected using the Range and Multipier on the front panel. These are normally set for 200 ns. The software must be told this setting through the use of the run control file. SCA inhibit mode is not employed as the built-in SCA is used to detect the "trues" while the TPHC output is sent to another SCA to detect the "randoms". SCA1 is actually built into the TPHC. It is used in window mode with front panel settings for the LLD and ULD of 0.4 and 1.0. Because window mode is used, the width of Chapter 3. Experimental Method 28 2 0 16 1 2 -C/3 W o Q 8 U I S o o 0 1 1—1—1—1—1—1 \ 1 1 - i H l i i i i i i i I i -I i j i i i j i ( i | i i i j i i i | i i i ' t 1 i • \ i t \i i — , ; t J I i — : : I i i i f i — i i ; f i • 0 2 0 4 0 6 0 8 0 100 120 140 D E L A Y TIME (ns) 160 180 2 0 0 Figure 3.4: A time spectrum as measured while collecting data on the 2ax and lt2 MDs of C H 4 over a period of several days. The number of coincidences is plotted as a function of the time between the start and stop pulses. Total data collection time is 23.4 hours. The time spectrum is fit by a Gaussian of FWHM 4.87(8) ns centred at 22.44(4) ns. Chapter 3. Experimental Method 29 the window is given by the upper level discriminator setting. Note that the window width range is 0-10 V, not 0-1 V, as may be indicated by the Ortec technical data sheet. Hence for a ULD setting of 1.0, the width is 1.0 V. SCA1 passes all signals with amplitudes between 0.4 and 1.4 V, corresponding to 8 to 28 ns. The peak in the time spectrum typically occurs near 22 ns. SCA2 is run in normal mode with the lower level and upper level settings at 2.0 and 10.0. Hence it passes all signals with amplitudes between 2 and 10 V (40 to 200 ns). The ratio between the "true" and "random" windows is therefore 1:8. This ratio must be set in the software through the run control file. The software calculates the coincidence count rate to be (8 • true — random). Computer control The TPHC is used in COINC mode, hence the gate logic must be high to permit detection of a start signal. At the end of a scan, when the computer is adjusting <f>, Eo, writes to a disk, writes to the printer, updates the screen, or is otherwise busy processing data, acceptance of signals is inhibited by setting the gate low and the inhibit/reset high. (It is not actually necessary to do both.) At the beginning of a scan and after a true stop pulse, the inhibit/reset is flipped. In this case it is being used as a reset. Except in the cases mentioned above, the gate is normally high and the inhibit/reset is normally low. 3.3 Operating Procedures The interaction of a gas with a surface will change the work function of the surface. This fact, combined with the extreme conditions of temperature under which electron Chapter 3. Experimental Method 30 gun filaments are operated, causes some of the properties of an electron spectrometer to "drift" with time. Since typical measurements span days or weeks, careful attention to the stability of the instrument may be required. The following procedures are followed when taking a spectrum. The system is pumped down to 10~7 torr and the sample gas is admitted to a pressure of 1.5-2.5-10-5 torr on the top ion gauge. The actual pressure used takes into account the ionization efficiency of the gas. The gun and deflectors are adjusted to give a suitable beam at E0 = 1200 eV. The analysers must be calibrated to ensure that they are passing 100 eV electrons and, more importantly, to ensure that the pass energy in both analysers is identical. Small deviations in the pass energy from 100 eV have a minimal effect on the results provided the error in both analysers is identical. The drift in the pass energy is typically less than 0.1 V over a week, and the relative change is typically zero under normal operating conditions, including gas changes. The analysers are calibrated by setting E0 = 600 eV. Under these conditions, electrons elastically scattered at 45° will be passed by the analysers. Hence this is referred to as looking at the "elastic beam". For this procedure, the beam current should be reduced because of the large scattered signal which could damage the channeltrons and distort the measurement. Therefore the grid is set to a large negative value to decrease the beam current, while the anode is kept constant to minimize the change in the operating conditions. With the elasic peak now falling on the channeltrons, the CMA outer voltages are adjusted individually to maximize the count rate (that is, to find the peak) on each channeltron. The energy resolution of each analyser is measured, as is the uniformity of the elastic count rate with d> in the movable analyser. Chapter 3. Experimental Method 31 The signal processing electronics can be checked with E0 set to 1200 eV (normal operating conditions). The only aspect of the signal processing electronics that typically changes with time is the amplitude of the signal that enters the variable gain amplifiers (Ortec TFA 454). To some extent this is due to changes in the properties of the chan-neltrons, but the main cause is the frequent replacement of the transistors in the fast preamps, the gains of which may vary considerably. Care must be taken to ensure that the gain of the Ortec TFA 454s is set appropriately in accordance with the level on the CFDs to ensure that the signal is sufficiently large to pass the discriminator and the noise sufficiently small to be removed. The time spectrum provides a diagnostic of system performance. A typical time spectrum, with a width of about 5 ns, is shown in figure 3.4. Time spectra are measured automatically during data collection. The location and width of the time peak do not drift with time. With E0 set at 1200 eV +e, binding energy spectra (BES) are then measured by recording the coincidence rate as a function of e at a selected constant <p. Typically, the energy scale of the spectrometer will drift considerably (1-2 eV) for the first few hours after a new gas is introduced. It is usually desirable to measure the binding energy spectrum for at least two different values of <j>\ these measurements are made by repeating short, sequential scans to ensure that changes in the operating conditions are reflected equally in the relative intensities of all of the spectra. Measurement of the momentum distributions (MDs) is made by recording the coin-cidence rate as a function of <j> at a constant e corresponding to the characteristic energy of a peak in the binding energy spectrum. The binding energy spectrum is remeasured typically after every 12 to 24 hours of measurement of the MDs to check the stability of the energy scale. Chapter 3. Experimental Method 32 Each of the studies reported in this work required several months to complete, there-fore it was necessary to check all of the parameters already mentioned, including the properties of the elastic beam, periodically during data collection. The spectrometer performance was further checked by measuring the BES of Ar and the MD of the 3p orbital of Ar before and after each study.3 The energy and momentum resolution were found to be consistent throughout these studies at 1.7 eV FHWM and 0.15 au HWHM respectively. All the gases used were supplied by Matheson and used without further purification. No impurities were detected in the binding energy spectra. 3.4 Data Analysis Virtually no further mathematical processing of the data collected by the computer is required. No form of data manipulation not explicitely mentioned here or in the chapters devoted to each molecule has been performed; in particular, the data has not been smoothed or deglitched. While the relative energy scale from the experiment is taken to be exact, the absolute energy is calibrated using an appropriate vertical ionization energy from high resolution PES measurements of the molecule under study. The PES measurements from which the energy scale is calibrated are referenced in the chapters in which the spectra appear. The binding energy spectrum is fitted with Gaussians whose widths reflect both the energy resolution of the spectrometer and the natural line widths (mostly vibrational 3 Ar is inexpensive and readily available. The shape of the 3p momentum distribution provides a suitable test of the momentum resolution. Chapter 3. Experimental Method 33 broadening) which can be estimated from the band shapes in high resolution photoelec-tron spectra. The sum of the areas that correspond to ionization from a given symmetry manifold are then used to normalize the momentum distributions. For example, if the relative peak areas of the 2£2~1 and 3a]"1 transitions in the BES of SiH4 are found to be 3:2 at 0 = 6°, then the heights of the MDs are adjusted so that the amplitudes at the momenta corresponding to <b = 6° are in the ratio of 3:2. In this way, all of the momen-tum distributions for a given molecule can be placed on the same relative intensity scale. Making the measurements absolute for comparison with theoty therefore requires only one normalization factor for all of the MDs of a given molecule. This is usually done by a single point normalization to theory for one momentum distribution so that all other relative normaHzations are preserved in both theory and experiment. The magnitude of the momentum of the bound electron, p, as denned in chapter 2, is related to the azimuthal angle (f> by p= {(2fci cos e-p0)2 + [2fca sin^ sin(<£/2)]2}5 (3.8) where 8 — 45°. This equation was used to transform the data from q> space to p space. It assumes that the reaction can be treated as a binary encounter. Occasionally, spurious signals appear in the form of a huge number of "true" coinci-dences. These are easily distinguished by their large size (at least two orders of magnitude larger than the coincidence count rate). The method of data collection used allows for the location and removal of such signals. Rapid scans of between 15-30 s per point (e or <f>) are performed over many hours and the results of each individual scan are recorded on paper. This hard copy can later be checked for spurious signals, and the entire scan discarded if they are found. Fortunately such signals are very infrequent, with no more Chapter 3. Experimental Method than one occurring in a two to three month period of data collection. Chapter 4 Dimethyl Ether Several recent EMS studies have examined the outermost (HOMO) orbitals of alkyl- and fluoro- amines [41,42,43,44]. The work in this laboratory [42,43] has investigated methyl substitution, particularly with regard to the methyl inductive effect. The EMS measure-ments are consistent with a variety of other experimental and theoretical studies which suggest, contrary to commonly held views, that methyl groups are electron attracting relative to hydrogen when bonded to nitrogen. The EMS work on nitrogen compounds is extended in the present study to investigate the effects of methyl groups bonded to oxygen. The experimental momentum profiles (XMPs) of the two outermost orbitals and the three inner valence orbitals of dimethyl ether have been measured. Good agreement is obtained between the measurements and the momentum distributions calculated from relatively simple wavefunctions except in the case of the outermost 26j orbital. An examination has been made of the effects of the inclusion of diffuse and polarization functions on the basis sets, as well as the choice of the molecular geometry for the calculation. The XMPs of the outermost orbitals are compared to those for H 2 0 and C H 3 O H . The two outermost orbitals for each of these molecules have large contributions from the oxygen 2p atomic orbital, and can be thought of as the 'lone pairs' to a first 35 Chapter 4. Dimethyl Ether 36 approximation. The NHOMO XMPs show an increase in low momentum components with increasing methyl substitution. The observed features of the binding energy spectra, which measure the ionization potential of the Za, orbital for the first time, are quite well reproduced by earlier published Green's function calculations. 4.1 Binding Energy Spectra The dimethyl ether molecule has C 2 v symmetry [65]. In the ground state the twenty-six electrons are arranged in thirteen doubly-occupied orbitals within the independent particle description. The electronic configuration of the a A i ground state can be written as (la,) 2 (2a,)2 ( l i 2 ) 2 (3a!)2 (2b2)2(4a,)2 (lb,)2 (5a,)2 (3fc2)2 (la 2) 2 (4fc2)2 (6aa)2 ( 2 ^ > V - N V ' » „ ' core inner valence outer valence The assignment of the order of occupation from 3a! through 2b, is based on previous photoelectron spectroscopy studies [62,63,64]. This ordering is found to be consistent with the shapes of the XMPs measured in the present work. All of the calculations in this work predict the order of the first three orbitals to be (la,)2 (lfc2)2 (2ai)2, which is incorrect. Such errors for core orbitals are common in Hartree-Fock calculations, and this problem has been examined and explained by Koga et ai. [66]. The binding energy spectra of (CH 3) 20, shown in figure 4.5, were obtained at an impact energy of 1200 eV (plus the binding energy) and at relative azimuthal angles q> of 1° and 9°. As can be seen in equation 3.8, the momentum of the ionized electron before collision is a function of the azimuthal angle q> and the binding energy. As a result Chapter 4. Dimethyl Ether 37 B I N D I N G E N E R G Y (eV) Figure 4.5: Binding energy spectra of the valence shell of at (CH 3 ) 2 0 at azimuthal angles <j>—\" and 9". The solid line is a sum of Gaussian functions with widths and peak locations given bv PES studies [62,63,64] and convoluted with experimental resolution. Chapter 4. Dimethyl Ether 38 6 0 > 6 < W A K 4 - i — — i — i — i — i — — i — i — i — i — — i — i — i — i — — h 1 0 _ 20 3'0 4'0 E X P E R I M E N T 0 -i 1 i_ 1 0 2 0 3 0 4 0 B I N D I N G E N E R G Y (eV) Figure 4.6: Comparison of experimental and calculated binding energy spectra of ( C H 3 ) 2 0 . (a) many-body Green's function calculation, ref. [63]; (b) EMS spectra, sum of tf>=l° and 9°. Chapter 4. Dimethyl Ether 39 Table 4.1: Measured and calculated binding energies (eV) for the valence shell of (CH 3) 20 State Experimental vertical IP Calculation EMSQ PES RHF Green's function6 this work ref. [62,63,64] this work ref. [63] 26a 10.0 10.04 11.54 10.10 [0.92] 6aa 11.9 11.91 13.15 11.77 [0.92] 4fc2 13.4 13.43 14.43 13.60 [0.93] la 2 14.2 14.20 14.99 14.26 [0.93] 362 17.46 16.19 [0.92] 5ai 16.2 16.0-16.5 17.78 16.35 [0.92] I6i 17.97 16.38 [0.92] 4a! 21.2 21.2 23.81 21.72 [0.81] 22.53 [0.090] 262 23.4 23.4 26.39 24.04 [0.35] 24.30 [0.46] 25.55 [0.028] 3a a 32.3 — 37.43 32.53 [0.026] 32.55 [0.043] 33.15 [0.050] 33.34 [0.13] 33.44 [0.059] 33.57 [0.072] 33.74 [0.023] 36.39 [0.022] "Estimated uncertainty ±0.1 eV. ''Extended 2ph-TDA and OVGF. Pole strengths are given in parentheses. eBasis set 4-31G+spd+s. of this, "s type" components make the dominant contribution to the spectrum at <^ >=1° while the spectrum at <f>=9° contains both s and p type contributions.1 The two binding energy spectra shown in figure 4.5 are on the same intensity scale. The energy scale was calibrated with respect to vertical ionization potentials measured by 1hi electron momentum spectroscopy, the momentum distributions which decrease monotonically from a maximum at p = 0 au are referred to as 's type'. Momentum distributions which have zero intensity at p = 0 au are referred to as 'p type'. It should be noted that momentum distributions of orbitals which are not members of the totally symmetric irreducible representation of the molecule are necessarily p type [67]. Chapter 4. Dimethyl Ether 40 photoelectron spectroscopy [62,63,64]. Gaussian curves have been fitted to the spectra using known vertical IPs and their associated Franck-Condon widths folded with the energy resolution of the spectrometer. As the 30^ 1 ionization process has not previously been observed to the best of our knowledge, the peak location and width were chosen to give the best fit to the measured spectra. Some further intensity evidently exists at higher binding energies. The sum of the </>=l° and 9° spectra is compared to many body Green's function cal-culations of the ionization pole strengths for (CH 3) 20 by von Niessen and co-workers [63] in figure 4.6. The outer valence IPs have been calculated by the Outer Valence Green's Function method [48,68] while the inner valence IPs have been calculated by the ex-tended 2ph-TDA method [69]. The peaks in the calculated spectrum are assigned widths taking into account the experimental energy resolution and the Franck-Condon width as estimated from high resolution PES (outer valence) and the present EMS (inner valence) spectra. The spectral intensity above 26 eV is all ascribed to the 3ai orbital since the distribution of intensity is qualitatively similar to that of the numerous poles predicted by the calculation for the Sa^1 ionization process [63]. Reasonable agreement is observed be-tween the calculation and the experimental results. The calculated and observed binding energies are listed in table 4.1. 4.2 Choice of a Basis Set to Describe a Large Molecule Molecular orbital calculations can become very costly as the size of the molecule in-creases. For a program which iteratively solves the Hartree-Fock-Roothan equations, like GAUSSIAN76, CPU time rises geometrically with the number of atoms in the molecule Chapter 4. Dimethyl Ether 41 Table 4.2: Diffuse s and p exponents for oxygen and carbon" Atom and Basis set function type STO-3G 4-31G 0 a 0.14170 0.0946 0 p 0.14170 0.0946 C 8 0.0348 0.123 cP 0.09036 0.0663 "Taken from tables 4 and 6 of ref. [70]. and the number of primitives in the basis set. (For a better discussion see reference [51].) These constraints effectively limit the quality of wavefunction attainable at present for a molecule as large as dimethyl ether. In a recent article [70], Casida and Chong suggest that a small basis set supplemented with appropriately chosen diffuse p functions on the heavy atoms can yield something close to the Hartree-Fock limit momentum distribution in cases where the linear com-bination of atomic orbitals approximation is reasonably good. That is the momentum distribution calculated from such a basis will closely resemble that calculated from a large basis which is essentially at the Hartree-Fock limit. In their paper Casida and Chong [70] give appropriate diffuse functions for atoms B through Ne to supplement each of STO-3G, 4-31G and 6-21G basis sets. Their results on the valence orbitals of H2O, HF and NH3 all show that any of the above basis sets supplemented with suitable diffuse functions give similiar results for a given calculated momentum distribution and all are relatively close to the Hartree-Fock limit momentum distribution [70]. Similar conclusions were reached by French et. al. in a study of H2S [28]. If this method can be satisfactorily extended to bigger systems, then it represents an excellent method of calculating MDs of large molecules while maintaining computing costs at a reasonable level. Chapter 4. Dimethyl Ether 42 Table 4.3: Total energy and dipole moment of dimethyl ether Basis set Dipole moment Total energy (D) (au) STO-3G 1.28 -152.132 STO-3G+p 2.17 -152.253 STO-3G+pp 2.09 -152.258 4-31G 2.01 -153.836 4-3lG+p 2.07 -153.841 4-3lG+pp 2.06 -153.842 4-31G+p+sss 2.04 -153.843 4-3lG+sp 2.08 -153.843 4-3lG+sp+s 2.05 -153.844 4-3lG+sp+p 1.93 -153.847 Snyder and Basch 2.06 -153.885 4-3lG+spd+s 1.64 -153.915 Experiment 1.31(1)° "Ref . [65]. The various basis sets tested in the present work and the conventions used for their names are described below. All calculations were performed with GAUSSIAN76 [35]. Throughout this paper, the term 'function' is used to mean a Gaussian or a contracted lin-ear combination of Gaussians. In naming the basis sets, we use the convention: X-fY+Z, where X=STO-3G or 4-31G, Y=s, p or d or a combination thereof, and Z=s or p or a combination thereof. X is a standard basis in GAUSSIAN76. A basis named X-fY refers to standard basis X supplemented with functions Y on the oxygen and carbon atoms. X+Y+Z means X has been supplemented with functions Y on the oxygen and carbon atoms and functions Z on the hydrogen atoms. For example: 4-3lG-fsp-(-s means the 4-31G basis with s and p functions added to the O and C and s functions added to the H. The specific basis sets used are as follows: (a) STO-3G. This is a minimal basis set and so allows for two s functions and one p Chapter 4. Dimethyl Ether 43 function on each of 0 and C and one s function on H. Each function is a contraction of three Gaussians. This basis set was designed by Pople and coworkers [71]. (b) 4-31G. This is a split-valence basis in which four primitive gaussians are contracted to form the Is core and the valence orbitals are split into one uncontracted Gaussian and one contracted Gaussian composed of three primitives [72]. (c) Snyder and Basch. This basis of contracted Gaussians is taken from Snyder and Basch [73]. The O and C bases each consist of four contracted s type Gaussians (two to describe the Is atomic orbital, two to describe the 2s) and two contracted p type Gaussians. The hydrogen basis consists of two contracted s type Gaussians. The choice of basis functions used has been shown to yield a similar energy to a calculation using a Slater basis, with two Slaters per atomic orbital, in which the Slater exponents have been optimized for the atoms. Hence this is referred to as a double zeta basis. Each of these basis sets (except Snyder and Basch) has been supplemented in the present work with various combinations of diffuse and/or polarization functions. The diffuse s and p functions for O and C are taken from the work of Casida and Chong [70]. Since this method [70] could not calculate the exponent for the s functions on oxygen, we have chosen the oxygen s exponent to be the same as for the p functions. The exponents used are shown in table 4.2. The d function exponent was chosen to be 0.8 as suggested by Hariharan and Pople [74]. When more than one diffuse function of the same type is indicated (eg. 4-3lG+pp), the exponent of the second diffuse function is calculated in an even-tempered manner by dividing the exponent of the first diffuse function by three. Chapter 4. Dimethyl Ether 44 The diffuse s function exponent on hydrogen is 0.036 as recommended by Hehre et. al. [75] and the exponent of the p function on hydrogen is 0.1160 after the work of Zeiss et. al. [76]. The former hydrogen exponent is believed to have little application except in calculating H" and in situations where hydrogen is expected to have considerable negative charge [75]. The latter is calculated as a suitable function to describe how a hydrogen Is function is perturbed by a uniform, static electric field. 4.3 Comparison of Experimental and Calculated Momentum Distributions 4.3.1 Overall Results for Valence Orbitals of (CH 3) 20 The experimental momentum profiles (XMPs) and calculated momentum distributions (MDs) of the valence orbitals of dimethyl ether are shown in figure 4.7(a)-(h). All cal-culated MDs have been folded with the experimental momentum resolution. The exper-imental momentum profiles of the 2bi, 6a l 5 4a l 5 2b2 and 3a! orbitals were determined at the appropriate binding energies. The remaining orbitals cannot be resolved sufficiently in energy by the present spectrometer to permit direct measurement of binding energy selected XMPs. However, the deconvoluted peak areas (figure 4.5) provide (with a single normalization) two extra points for the XMPs of the 26i, 6a!, 4a!, 2b2 and 3aa orbitals and two points for the (unresolved) 4fc2, la 2 and (362+5ai+lfc2) orbitals (see below). All the XMPs have been placed on a common intensity scale by normalizing on the peak areas in the binding energy spectrum (fig. 4.5). In the case of the 3ai orbital, all inten-sity above 28 eV to the limit of the data at 43 eV (and not the fitted peak area alone) was used for the normalizaton procedure. Such an assignment is supported by Green's function calculations [63] and also by the good quantitative fit observed in figure 4.7(h). Chapter 4. Dimethyl Ether 45 Figure 4.7: Momentum distributions for all of the valence orbitals of (CH 3 ) 2 0. See text for details concerning the experimental points and theoretical calculations. Chapter 4. Dimethyl Ether Figure 4.8: Effect of diffuse p functions on 0 and C. Chapter 4. Dimethyl Ether Figure 4.9: Effect of diffuse s functions on 0, C and H. Chapter 4. Dimethyl Ether 48 Additional points (solid squares) on each XMP reflect the relative peak areas at (f>=l° and 9° in figure 4.5. The XMPs and MDs are placed on a common intensity scale with only a single point normalization of the (4-3lG+spd+s) calculation to experiment for the 6ai orbital (at p=0.25 au). In figure 4.7, three MDs are shown for each orbital as calculated from three differ-ent quality wavefunctions. The STO-3G is a minimum basis set while the Snyder and Basch basis [73] is a standard Gaussian basis equivalent to double zeta in quality. The 4-3lG+spd+s wavefunction has the lowest total energy of any calculation presented in this paper (see table 4.3) and gives the best overall description of the data. Except where otherwise noted, all calculations in this paper were done at the equihbrium geometry as measured by Blukis et. al. [65]. The XMPs were measured by fixing the impact energy at 1200eV plus the energy shown in the diagram. This energy is slightly different in some cases from the vertical IP in order to avoid or minimize overlap with adjacent bands. It can be seen (figure 4.7(f),(g) and (h)) that quite good quantitative agreement between all three calculations and experiment is obtained for the 4ai, 2b2 and 3a: orbitals for shape and intensity. This agreement is found to be independent of the choice of basis set (see figures 4.7-4.9). While the sp character of the 6a! MD (figure 4.7(b)) predicted by all calculations is clearly present in the experimental data, the poor fit in the region of about 1 au is mainly due to inadequacies of the experiment since, as can be seen from figure 4.5, an overlapping contribution from the 462 state cannot be avoided at the binding energy corresponding to the maximum of the 6ax state. Approximately 27% of the "6ai" state intensity as observed will be due to the 4b2 state at the momentum (0.74 au) corresponding to (b=9°. Chapter 4. Dimethyl Ether 49 A correction has been made by subtracting the appropriate fraction of the calculated MD (4-3lG+spd+s) of the 462 orbital from the nominal 6a, measurements and renormalizing the result. The corrected data is shown as open circles on figure 4.7(b). This gives improved agreement with the 4-3lG+spd+s calculation. The corrected results are in good agreement with recent EMS measurements obtained by Zheng et. al. [77]. The calculated MDs for the 6a! orbital are quite basis set dependent. The limited experimental results for the 462, la 2 and (3fc2 + 5ai + lb,) orbitals are consistent with the more detailed data of Zheng et. al. [77]. The shape of the XMP for the 2b, orbital shown in figure 4.7a is somewhat surprising. Two noteworthy features are the large cross-section near p=0 au and the apparent dou-ble bumped distribution. The 2b, orbital is expected from symmetry arguments to be p type and therefore should have zero intensity at p=0 au outside of contributions from momentum resolution effects. Similar anomalies at low momentum have been observed in the HOMO orbital of H 2 CO [78], and in a series of methyl- and fluoro- substituted acetylenes [45]. One possible explanation for the large non-zero intensity observed for the 2b, XMP in this region could be that the true (resolutionless) shape of the momentum distribution rises very sharply near p=0 so that the spectrometer sees, with its limited resolution, something that does not in practice go down to zero intensity at low mo-mentum. However, the calculated MD (which is resolution-folded) fails to predict this despite the use of very diffuse functions in the basis sets. A further possible explana-tion is incomplete separation from the neighboring (dominantly s type) 6ai XMP at the particular value of the binding energy used to measure the 2b, XMP. However a consider-ation of the binding energy spectrum and expected peak widths suggests that significant overlap does not occur. Furthermore, the observed shape for the 2b, XMP is consistent Chapter 4. Dimethyl Ether 50 with the independent measurements of Zheng et. al. [77]. Another possibility for the finite cross-section near p=0 may be that significant changes in geometry (and there-fore in symmetry) may be occurring within the time scale of ionization. Since the time scale of ionization is electronic, this is equivalent to saying that the Born-Oppenheimer approximation is breaking down. Theoretical and mass spectral studies have indicated significant structural rearrangements following ionization for ( C H 3 ) 2 0 [79,80]. However the predicted geometries and the long time-scale mass spectral studies are not directly relevant to the question of a breakdown in the Born-Oppenheimer approximation in the EMS experiment. The shape of the 2^ X M P is particularly intriguing as it is found to be extremely broad and possibly double peaked (see section 4.3.4 below). Two components are clearly present in the momentum distribution predicted by the STO-3G calculation (figures 4.7 and 4.8) but are not so apparent in the higher quality calculations. However all calcula-tions clearly show a very broad MD in agreement with the observed overall shape of the 2&i X M P . The predicted intensity of the 2fci orbital is significantly lower than observed. There exist several possible explanations for this difference between theory and experiment. One possibility is the incompleteness of the rather limited basis sets used in the present SCF calculations. Although these supplemented basis sets are small, earlier SCF calculations of similar quality for H 2 0 , N H 3 and HF yielded MDs very similar to the MDs predicted by HF limit wavefunctions [70]. If the presently calculated MDs for ( C H 3 ) 2 0 are in fact significantly different from the Hartree-Fock limit M D , this may reflect the increased complexity of the ( C H 3 ) 2 0 molecule. Another possibility for this intensity discrepancy is the failure of the target Hartree-Fock approximation to correctly predict the X M P s of the Chapter 4. Dimethyl Ether 51 2b, and 6ai orbitals. Such a failure has been observed for the outermost valence orbitals of H 2 0 [38], NH 3 [37] and HF. To test this hypothesis, some form of calculation which takes into account correlation and relaxation in the ion and molecule should be performed on dimethyl ether. The most significant contribution to the intensity difference, however, may be highlighted by the recent results of Zheng et. al. [77] which show a shape for the 2b, orbital consistent with the present results but with a lower cross-section, in better agreement with theory. This suggests that the difference in intensity between theory and experiment may be largely due to uncertainties in the determination of the peak area from the binding energy spectrum. In summary, basis 4-3lG+spd+s is considered to be the best basis of those considered in the present work for calculating the MDs of dimethyl ether. We also note that while a Snyder and Basch basis [73] does relatively well for total energy and dipole moment (table 4.3) the momentum distributions for the 2b, and 6a! orbitals are poorly predicted (fig. 4.7). 4.3.2 Results of the basis set investigation Investigation of the effects of the use of diffuse functions and polarization functions in small basis sets are discussed in the following sections. The effects of diffuse functions on the carbon and oxygen It is found that the MDs calculated using the ST0-3G and 4-31G basis sets do not match the data for the 2b, and 6ai outer valence orbitals (figure 4.8). The 4-31G basis set gives quite good agreement, however, for the three inner valence orbitals 4ai, 262 and Chapter 4. Dimethyl Ether 52 3ai (see figure 4.8). The addition of diffuse p functions on oxygen and carbon improves the agreement with experiment (see figure 4.8, basis sets ST0-3G-fp and 4-3lG+p). For the 2b, and 2b2 orbitals, which are predicted to be p type, the ST0-3G+p shows much improvement over the ST0-3G. The ST0-3G+p and 4-3lG-fp calculations are almost identical for these two orbitals. In the case of the 4a, and 3aj orbitals, which are observed to be mostly s type, the addition of diffuse functions makes less difference in the calculation and the larger basis, 4-31G, is seen to be superior to the ST0-3G as expected. The Qa, orbital is measured to have mixed s and p character, and when corrected for overlap with the adjacent 4fc2 orbital, shows good agreement with the 4-31G+p calculation. It was found that additional p functions (4-3lG+pp) with even lower exponents did not greatly improve the overall description of the momentum distributions. The addition of a diffuse s to the 4-31G basis (i.e. 4-3lG+sp) improves the descrip-tion of the 6aj MD but makes little change to the MDs of the other measured orbitals (figure 4.9). The effect of adding more functions to the hydrogen The hydrogen atoms are believed to play an important role in determining the shape of the 2b, momentum distribution of (CH 3) 20 (see section 4.3.4). Thus we may expect that the addition of diffuse s functions on the hydrogens to improve the calculations considerably. The contribution to the wavefunction was found to be significant (the coefficient of the diffuse s on the four out of plane hydrogens in the 4-3lG+sp+s and 4-3lG+spd+s calculations was 0.032) and some improvement was observed in the low momentum intensity of the 2b, and 6a, orbitals (figure 4.9). Unfortunately, this did not Chapter 4. Dimethyl Ether 53 improve further on addition of even more s functions (4-3lG+p-fsss) with even lower exponents. Putting p functions on the hydrogens to allow polarization had almost no effect on the calculated MDs. The effect of d functions on oxygen and carbon The addition of d functions on oxygen and carbon (4-3lG+spd+s) to describe polar-ization did not significantly improve the calculated MDs (compare figures 4.7 and 4.9) but did considerably improve the calculated dipole moment and energy (table 4.2). Summary We summarize the results of our basis set investigations by comparing our work on (CH 3) 20 to that of Casida and Chong's study of H 2 0, HF and NH 3 [70]. While we find that the results from the 4-31G basis are superior to those from the STO-3G basis in describing the inner orbitals, it is true that for the 2fca orbital, these basis sets are roughly equivalent once they have been supplemented by diffuse functions (figure 4.8). Such an equivalency between the STO-3G, 4-31G and 6-21G supplemented basis sets was also noted by Casida and Chong for the outer valence orbitals of the small hydrides. We also found that addition of more diffuse p functions on the heavy atoms in addition to the one recommended by Casida and Chong gives no further improvement. Unlike the findings of Casida and Chong for hydrides, we find that the basis sets supplemented with diffuse functions on the heavy atoms are still quite inadequate to reproduce the present experimental results for (CH 3) 20. Furthermore, some small improvement is obtained by putting additional functions on the hydrogens. It is surprising to find that polarization Chapter 4. Dimethyl Ether 54 functions (d on 0,C and p on H), which are thought to improve the description of bonding, did not result in any improvement of the calculated MDs. The 4-3lG+spd+s (figure 4.7) and 4-3lG+p+sss (not shown) basis sets are close to the maximum size basis that our version of GAUSSIAN76 can handle (for (CH 3) 20). Further investigations of basis set effects are therefore at present not posssible. 4.3.3 The effect of molecular geometry on the calculated momentum distri-butions Dimethyl ether is accepted as being of C2v symmetry [65]. This being the case, the 2bi orbital should be p type. If the observed intensity at low momentum (figure 4.7) is not due to effects of finite angular resolution or overlapping intensity from the 6a! orbital, but is in fact due to some s character, then it implies that dimethyl ether does not have C2v symmetry. Keeping in mind that the methyl groups are freely rotating at room temperature [81,82], we have investigated the effects of changes in geometry on the calculated MDs. Two calculations, STO-3G and a 4-3lG+spd+s, were performed at the three geome-tries indicated in figure 4.10. Technically, the orbital name should change with each geometry but we have used the C2v designation of "26i" throughout for simplicity. For the cases of C2v and C, symmetry, the 2bi orbital is required to be pure p type. For the case of Ci symmetry, the orbital may be a mixture of s and p type. The results of the calculations using 4-3lG-fspd-fs on all orbitals are shown in fig-ure 4.11. Calculations using the STO-3G basis show similar trends. As can be seen, the Chapter 4. Dimethyl Ether 55 C2v C s C 1 ECLIPSED STAGGERED Figure 4.10: Geometries at which calculations of the momentum distributions of (CHaJjO were performed. geometry effects are most prevalent in the 2b, and 6a, MDs , for which there is relatively more contribution from the hydrogens than for the three inner orbitals. However, even for the 26i and 6a!, the changes with geometry are small and they do not account for the shape at low momentum of the 26i momentum distribution. Wh i l e no geometry effects were observed to be of any signifigance in this investigation, it should be cautioned that other changes in the ground state geometry than those considered here may be of importance. In a recent study, Goruganthu et. ai. showed that a large intensity appeared in the low momentum region of the calculated M D for the outermost ir orb i ta l of 2-butyne when the calculation was performed with a distorted cis bent geometry [45]. Chapter 4. Dimethyl Ether 56 Figure 4.11: Spherically averaged momentum distributions and theoretical MDs calculated using 4-31G+spd+s basis with different hydrogen configurations, according to the geometries in figure 4.10. Calculations have been folded with the experimental momentum resolution and normalized on the 2b2 orbital. Chapter 4. Dimethyl Ether 57 Figure 4.12: A simple atomic orbital picture of the 2bi orbital. 4.3.4 Understanding the observed shape of the 1bx momentum distribution There is some indication that the measured XMP of the 2bx orbital exhibits a double peak with maxima at approximately 0.5 au and 1.1 au respectively, (figures 4.7 and 4.11), although the relatively large uncertainties in the data prevent an unambiguous interpreta-tion. What is clear is that the XMP is very broad compared to other p type distributions observed in most previous work. (Compare, for example, the 2b2 orbital of (CH 3) 20). It is the unusually broad nature of this XMP that causes the observed signal at any given p to be correspondingly smaller for the 26j orbital (see figure 4.5) and this is reflected in the poor statistical precision of the data. Our approach to interpreting the experimental data for the 2bx orbital is to try to develop an understanding of the result of the STO-3G calculation. This was done for several reasons. Firstly the STO-3G calculation is small enough that it can be understood in terms of the simplest notions of atomic orbital participation. It is also the calculation Figure 4.13: Partitioning of the 26x momentum distribution as described by eq. 4.11. (a) STO-3G calculation; (b) 4-31G+sp+s calculation. Chapter 4. Dimethyl Ether 59 which most clearly predicts the double bump (see figure 4.8). The STO-3G is a minimal basis set. By looking at the results of this calculation, we are led to think of the 2b, orbital as consisting of out-of-plane p functions centered on the oxygen and carbon atoms and s functions on the out-of-plane hydrogens with the signs on the s functions arranged so that a pair of hydrogens taken together looks like a p function (see figure 4.12). Note that the in-plane hydrogens make no contribution to the 2b, orbital. This picture of atomic orbitals centred on the nuclei is easy to visualize and understand, and will guide the discussion of the 2b, orbital. In the figure 4.12, the carbon p functions are out of phase with the oxygen p function. This fact exhibits itself in the position space density maps (figure 4.15) as a nodal surface between the carbon and oxygen nuclei. Can we understand the momentum distribution in terms of separate contributions from different atomic centres? Suppose $r(p) is the STO-3G wavefunction describing the 2b, orbital. Then 3>r(p) c a n be written as a sum of Gaussian functions centred on the oxygen, carbon and hydrogen atoms. Designating these terms §>o(p)> $c(p) and 3>H(P) respectively, we have $ r(p) = $o(p) + $c(p) + *H(P) (4.9) The momentum distribution for $T(P) is given by MDx{p) MDT(p) = J | * r(p) | 2 dfi (4.10) and we can clearly write MDT{p) = MD0(p) + MDc{p) + MDH{p) + Ioc{p) + W?) + ICH(P) (4.11) where MD0(p) = I \ *o(p) | 2 , and Ioc{p) = J{$o(p)$b(p) + $C(P)*3(P)) dtl, etc. Chapter 4. Dimethyl Ether 60 The six terms on the right hand side of equation 4.11 and their sum are plotted in figure 4.13(a). Two important conclusions can be reached by studying this diagram. The first is that in this orbital, participation from all of the atomic centres is important in determining the shape and intensity of the momentum distribution. The second is that the "interference terms", Ioc, IOH and ICH, are just as important as MDo, MDc and MDg. In this sense, it is impossible to understand the momentum distribution in terms of contributions from separate atomic centres - the molecule must be studied as a whole. Finally, we examine why the other calculations do not show the double bump as clearly or at all by plotting the various terms in equation 4.11 in figure 4.13(b). As can been seen by comparing this with figure 4.13(a), most of the terms have shifted to lower momentum, (as expected, see chapter 8), and MDo has shifted so far as to smooth out the double bump. 4.4 Comparison of the Electron Density in the Outermost Orbitals of HOH, C H 3 O H , C H 3 O C H 3 It is of interest to consider the effect of successive substitution of C H 3 groups for H in H 2 0 by considering the electron density distributions in (CH 3) 20, CH 3 OH and H 2 0 as probed by EMS and other experimental and theoretical methods. In earlier related work [43], the effect of methyl substitution relative to hydrogen when bonded to nitrogen has been investigated by comparing the measured XMPs for the highest occupied molec-ular orbital (HOMO) of each of NH 3 , NH 2 CH 3 , NH(CH 3) 2 and N(CH 3) 3 with calculated spherically averaged momentum distributions. In addition a comparison of the measured and calculated spherically averaged momentum distributions with the associated charge Chapter 4. Dimethyl Ether 61 (position) density and density difference maps for oriented molecules provided further insight into the changes in distribution of electronic charge in the HOMO orbital with increased methylation [43]. The results of earlier EMS studies of the HOMO XMPs in ammonia and the amines [41,43] clearly indicate that there is an increase in the low momentum component in the orbital, indicating a significantly increased derealization of electron density away from nitrogen in the outermost orbital as the hydrogens on NH 3 are successively replaced by methyl groups. The charge is principally displaced onto the trans hydrogens [41,43,83]. These experimental observations by EMS for the HOMO or-bitals are consistent with the following other experimental and theoretical data involving properties of the total electron density in ammonia, the methylamines, and other related molecules: (i) The systematic decrease in dipole moment with successive methylation in going from NH 3 to N(CH 3) 3 [84,85,86,87]. (ii) The N Is excitation spectra of NF 3 [88] and N(CH 3) 3 [89] as measured by inner shell electron energy loss spectroscopy (ISEELS) are very similar and are dominated by (innerwell) shape resonances trapped by a Coulombic barrier and accompanied by a dras-tically decreased intensity of the (outerwell) Rydberg states. In contrast NH 3 [90] has an N Is spectrum with prominent Rydberg excitations and an intense ionization contin-uum. The ISEELS spectra exhibit increased resonance structure with successive methyl substitution into NH 3 [89]. It is evident that C H 3 and F ligands act in a qualitatively similar manner producing a Coulombic barrier on the periphery of the molecule of the type observed in SF6 [91,92] and other related molecules [92]. Such behaviour indicates that C H 3 is relatively electron attracting compared to H when bonded to N. Chapter 4. Dimethyl Ether 62 (iii) NMR measurements of 1 4 N [93] and 1 3 C [94] chemical shifts for NH 3 , the methyl amines, ethyl amines and related molecules, indicate an increased deshielding of the nitrogen and carbon atoms upon successive alkylation. These results are consistent with the notion that the electronic charge distribution is delocalized towards the hydrogens on the methyl groups. (iv) Recent calculations of the electrostatic potentials of amine nitrogens [95] have also indicated that methyl groups are electron attracting relative to hydrogen in these molecules. These observations for the amines together with the EMS measurements point to the fact that the net effect of methyl substitution into NH 3 is a movement of charge away from the N. This suggests that the methyl groups are more electron attracting than hydrogen when bonded to nitrogen. This latter deduction is at variance with commonly used chemical rationalizations [96] in which the inductive effect of methyl groups is usually considered to be electron donating relative to hydrogen. These ideas have held sway despite the fact that much contrary evidence has been in the literature for a long time [97]. With the foregoing considerations in mind we now extend the earlier EMS work on the effects of methyl substitution in N containing molecules to the analogous situation for Hgands bonded to 0. This is approached by considering the present EMS results for (CH 3 ) 2 0 together with earlier reported EMS studies of CH 3 0H [98] and H 2 0 [38]. Momentum distributions of the outermost orbitals (HOMO and NHOMO) in each of H 2 0 (l&i and 3a,,), CH 3 0H (2a" and 7a'), and (CH 3) 20 (2fcj and 6ax) are shown in figure 4.14 together with calculations performed in the present work using two different Chapter 4. Dimethyl Ether 63 Table 4.4: Total energy and dipole moment of methanol °Ref. [101]. Basis set Dipole moment Total energy (D) (au) STO-3G 1.57 -113.541 4-31G+p 2.57 -114.874 Snyder and Basch 2.42 -114.879 4-3lG+spd+s 2.07 -114.925 Experiment 1.698(5)° Table 4.5: Total energy and dipole moment of water Basis set Dipole moment Total energy (D) (au) STO-3G 1.73 -74.963 4-31G+P 2.73 -75.916 Snyder and Basch 2.68 -75.876 4-3lG+spd+s 2.31 -75.942 109-GTO 2.01° -76.0671° Hartree-Fock Limit 1.98(1)° -76.0675(10)b 109-G(CI) 1.90° -76.3761° Experiment 1.8546(4)c -76.4376(24)°" "Ref. [38]. 6Refs. [102,103,104]. cRefs. [105,106]. dNon-relativistic, frozen nuclei, ref. [107]. basis sets (STO-3G and 4-3lG+spd+s). The water and methanol calculations were per-formed using the equilibrium geometry as measured by microwave experiments [99,100]. Also shown are two other calculations for H2O. One is a Hartree-Fock limit calculation, labelled 109-GTO in figure 4.14; the other, labelled 109-G(CI), is a calculation of the ion-neutral overlap (equation 2.4) using multireference singly and doubly excited config-uration interaction calculations of the wavefunction of the ion and neutral states. Both of these calculations are taken from reference [38]. Chapter 4. Dimethyl Ether 64 For all orbitals shown in figure 4.14, the use of an improved basis increases the calcu-lated intensity and moves the maximum in the calculated MD to lower momentum. Even where good quantitative agreement is not obtained, the observed trend is in a direction towards better correspondence with experiment. In general, the HOMO orbitals have relatively more intensity at low momentum than is predicted by the SCF calculations. Minchinton et. al. [98] noted this type of discrepancy for the 2a" orbital of methanol and suggested that it might be due to a lack of diffuse functions in the Snyder and Basch basis set [73] originally used. As already discussed above, the addition of functions even more diffuse than those in the 4-3lG+spd+s basis does not improve the calculated MD for the 2bi orbital of dimethyl ether. In the case of H 2 0, it was found that the calculated momentum distribution showed a considerable discrepancy with the experimental result even when the calculation was taken to the Hartree-Fock limit. The target Hartree-Fock approximation had broken down in this case. An ion-neutral overlap calculation was found to adequately model the results [38]. This suggests that the difference between ex-periment and the present level of theory for the HOMO and NHOMO orbitals of dimethyl ether and methanol may be mainly due to neglect of correlation and relaxation effects. Momentum- and position-space density contour maps for selected planes of the HOMO and NHOMO orbitals of H 2 0, CH 3 OH and (CH 3) 20 are shown in figures 4.15 and 4.16. These are based on calculations with the ST0-3G basis set. Only the 16: orbital of H 2 0 (figure 4.15) resembles a simple atomic-like orbital located on the oxygen atom. In con-trast, all other HOMO and NHOMO orbitals are significantly molecular in nature. As in the case for the HOMO of dimethyl ether (26a), the wavefunction around the carbon in the HOMO of methanol (2a") is out of phase with that around the oxygen, giving rise to a nodal surface between these atoms (see figure 4.15). Chapter 4. Dimethyl Ether 65 109-G(CI) 4-31G+spd + s 109-GTO - - - - ST0-3G HOMO 0 '.111, ( C H 3 ) 2 0 2 b i I / X . . . . . i . . . 1 II •1 NHOMO 0. (0 . . . . , . * H 2 0 3a , . . . i 2 3 0 1 MOMENTUM (au) Figure 4.14: Experimental momentum profiles and calculated spherically averaged momentum distri-butions for the HOMO and NHOMO orbitals of (CH 3 ) 2 0 (2bx and 6oi) C H 3 O H (2a" and 7a'), and H 2 0 (lti and 3ai). The theoretical MDs are resolution folded and height normalized to the lfc2 orbital in H 2 0 , the 6a'+la" in CH 3 OH and the 262 in (CH 3 ) 2 0. The 109-GTO and 109-G(CI) calculations for H 2 0 are from reference [38]. Chapter 4. Dimethyl Ether 66 Figure 4.15: M o m e n t u m and posit ion space density contour maps for the H O M O orbitals of H2O ( l fei ) , C H 3 O H (la"), and ( C H 3 ) 2 0 (26!) . The contour maps were generated wi th the S T O - 3 G calcula -t ion. The contour values represent 0.02,0.04,0.06,0.08,0.2, . . . ,20,40,60 and 8 0 % of the m a x i m u m density. The numbers on the axes are in atomic uni ts of m o m e n t u m or length, as appropriate. T h e first co lumn shows the m o m e n t u m density in the pT=0 plane, the second column shows the posit ion density in the x = 0 plane and the th i rd co lumn shows the posit ion density in the y=0.5 au plane. (There is no density in the y=0 plane.) In the posit ion density maps, the relative positions of the nuclei in the x = 0 or y—0 plane are indicated in the lower right corner. For s impl ic i ty , the hydrogens bonded to carbon are not shown. A l l in-plane hydrogens bonded to oxygen are shown. Chapter 4. Dimethyl Ether 67 Figure 4.16: Momentum and position space density contour maps for the NHOMO orbitals of H 2 0 (3ai), C H 3 O H (7a'), and (CHa^O (6ai). The contour maps were generated with the STO-3G calcula-tion. The contour values represent 0.02,0.04,0.06,0.08,0.2,... ,20,40,60 and 80% of the maximum density. The numbers on the axes are in atomic units of momentum or length, as appropriate. The first column shows the momentum density in the px—0 plane, the second column shows the position density in the x=0 plane and the third column shows the position density in the t/=0.5 au plane. In the position density maps, the relative positions of the nuclei in the x=0 or y=0 plane are indicated in the lower right corner. For simplicity, the hydrogens bonded to carbon are not shown. A l l in-plane hydrogens bonded to oxygen are shown. Chapter 4. Dimethyl Ether 68 A consideration of the measured XMPs and calculated spherically averaged MDs (figure 4.14) together with the momentum and position space density maps (figure 4.15 and 4.16) of the HOMO and NHOMO orbitals of the three molecules H 2 0, CH 3 OH and (CH 3) 20 leads to the following observations: (a) The XMPs and associated calculated MDs (figure 4.14) show that increasing methylation leads to increasing cross-section at low momentum in the NHOMO orbitals. This implies that in a given orbital the low momentum components are increasingly enhanced at the expense of high momentum components with increased methylation. This may be rationalized as a derealization of the position space density away from the 0 atom and onto the methyl groups with increased methylation. A similar result has been observed for the HOMO orbital in the case of NH 3 and its methylated derivatives as discussed. (b) The position density maps show that the localized nature of the I6j orbital on 0 in H 2 0 contrasts strongly with the increasing derealization away from 0 in going to the HOMO and NHOMO orbitals of CH 3 OH and (CH 3) 20. This indicates that the orbital electron density in the immediate locality of the 0 is progressively reduced in going from H 2 0 to CH 3 OH to (CH 3) 20. A further noticeable feature of the position density maps (figures 4.15 and 4.16) is the additional nodal structure between the C and 0 atoms in the HOMO and NHOMO orbitals of CH 3 OH and (CH 3) 20. This nodal structure arises in the calculation because the orbitals centred on the oxygen and carbon are out of phase. As noted earlier in the case of paradichlorobenzene [108] and amines [43], such increased nodal structure contributes to increased high momentum components of the momentum distributions and this can also be seen in figure 4.14 in comparing results for CH 3 OH and (CH 3 ) 2 0 with those for H 2 0. The combined effects of the delocalization by C H 3 Chapter 4. Dimethyl Ether 69 groups and attendant increase in nodal structure with methylation serve to redistribute the momentum density to low and high momentum respectively both in C H 3 O H and (CH 3) 20 in comparison to the corresponding orbitals in H 2 0 (see figure 4.14). (c) The momentum density maps (figures 4.15 and 4.16) provide further complemen-tary insight into the electronic behaviour in the HOMO and NHOMO orbitals of these molecules. Specifically the smooth nature of the momentum density contours for the lbx HOMO orbital of H 2 0 (each lobe is like a cut through an onion) are direct evidence of the essentially non-bonding, atomic-like, character of this orbital. In marked contrast the momentum density maps of the HOMO orbitals of C H 3 O H and (CHs)20 are "wrin-kled", i.e., each lobe is much like the view obtained by slicing through a cabbage. The wrinkles require the existence of more than one centre of high charge density, hence they demonstrate that the charge density is less localized in C H 3 O H and (CH 3) 20 than in H 2 0. The view that C H 3 groups are electron donating relative to H when bonded to 0 is not consistent with the present observations at least as far as the HOMO and NHOMO orbitals are concerned. Clearly the reverse situation pertains, i.e., CH3 groups act in these molecules in a way that could be described as electron attracting. Finally we compare the present conclusions from EMS measurements on the HOMO and NHOMO orbitals of (CH 3) 20, CH 3 OH, and H 2 0 with other experimental and the-oretical investigations of the total charge distribution as was done in the case of the alkylated nitrogen compounds. Specifically, a relative decrease in electron density away from 0 on successive methylation of H 2 0 is reflected as follows: (i) The systematic decrease in dipole moment in going from H 2 0 (1.85 D) to CH 3 OH Chapter 4. Dimethyl Ether 70 (1.70 D) to (CH 3 ) 2 0 (1.30 D) [65,101,105,106,109]. (ii) The 0 Is ISEELS excitation spectra show increased innerwell shape resonance phenomena with successive methylation [90]. This indicates a Coulombic charge barrier in the region of the C H 3 groups (see discussion for nitrogen containing compounds above). (iii) From 1 3 C NMR chemical shift measurements of a series of alcohols ROH (R=CH 3, C H 3 C H 2 , (CH 3) 2CH, (CH 3) 3C) Jackman and Kelly [110] have concluded that methyl groups have a negative inductive effect, that is, C H 3 is electron attracting relative to H when bonded to 0. This conclusion is also supported by 1 7 0 NMR studies of the same compounds [111]. Similar conclusions arise from a consideration of the 1 3 C and 1 7 0 chemical shifts for ethers ROR (R= CH 3 , C H 3 C H 2 , (CH 3) 2CH) [94,111]. Similarly the 1 7 0 NMR chemical shift measurements for the series HOH, C 2 H 5 OH, (C 2 H 5 ) 2 0 [111] are also consistent with a negative alkyl inductive effect. However the 1 7 0 chemical shifts of the series HOH, CH 3 OH, (CH 3) 20 are consistent with a positive methyl inductive effect. It is evident that caution is necessary in the interpretation of chemical shifts. Al-though a consistent conclusion can be drawn from the NMR studies of the amines, the inconsistencies that result in the case of the methylated oxygen compounds serve as a re-minder that chemical shift measurements are not only complicated by solvent effects [112] (particularly hydrogen bonding, which would have a large effect in the cases of H 2 0 and CH 3OH) but also by contributions from the paramagnetic term in the expression for the field at the nucleus [113] which becomes significant when measurements are made of nuclei other than *H. Where this latter effect is significant it is not possible to interpret the chemical shift solely in terms of the electron density in the vicinity of the measured nucleus, even in the absence of solvent effects. Chapter 4. Dimethyl Ether 71 In summary, many studies including the present work all suggest that methyl groups are electron attracting relative to hydrogen when bonded to oxygen, contrary to the commonly held view which is that methyl groups are electron donating. Chapter 5 Phosphine A low impact energy (400 eV), low momentum resolution study of P H 3 has already been reported [23]. These results showed strong spbtting of the 4a a inner valence ionization strength due to many body effects. This earlier study for P H 3 suggested that even quite modest SCF wavefunctions provided a fairly good description of the shape of the mea-sured X M P s for the three valence orbitals, which where individually height normalized to the calculations. The X M P s of P H 3 measured earlier also showed some evidence of distortion above 1 au due to the low impact energies used (400 eV). It is therefore of interest to remeasure the X M P s of the valence orbitals of P H 3 using the higher impact energy (1200 eV) and much improved momentum resolution now available on our spec-trometer and also to place the X M P s on the same relative intensity scale to provide a more stringent quantitative comparison with calculations than was possible with the individual height normalizations used for each orbital in the earlier study [23]. This chapter presents high resolution EMS measurements of the binding energies and electron momentum distributions for the three valence orbitals of P H 3 . The measured binding energy spectrum, which shows extensive structure in the inner valence region, is compared with CI and many body Green's function calculations. The theoretical calculations accurately predict the complete breakdown of the single particle picture of 72 Chapter 5. Phosphine 73 ionization for the inner valence region of this molecule. The momentum distributions are compared on a quantitative basis with momentum distributions calculated from a range of SCF wavefunctions up to near the Hartree-Fock limit in quality and also with full ion-neutral overlap calculations carried out using correlated wavefunctions by the method of configuration interaction. Inclusion of electron correlation is found to have minimal effect on the calculated momentum distributions. 5.1 Calculation of the Momentum Distributions of P H 3 Within the target Hartree-Fock approximation, the experimental momentum profiles can be compared to the spherical average of the square of the momentum space wavefunction (equation 2.5). In addition to a minimum basis set calculation, a near-Hartree-Fock-limit calculation was also performed using a 136 GTO basis set described below. Calculation of the ion-molecule overlap (equation 2.4) was also performed using CI wavefunctions for the ion and molecule with which it is bebeved that more than 80% of the valence shell correlation energy is recovered. This 136 G(CI) configuration interaction wavefunction is also described below. Details of the wavefunctions and basis sets are given below and in table 5.6. The computations for PH 3 were performed at the equilibrium geometry with rp#=1.420 au and the PHP angle=93.3° [114]. The CI calculations include only valence shell correlation and neglect core correlation and relativistic effects. MBS-j-3d Basis Set. This calculation of Boyd and Lipscomb [115] used a minimum ba-sis set of Slater orbitals supplemented with a 3d orbital on the phosphine. The exponents were: P Is 14.7, P 2s and 2p 5.425, P 3s and 3» 1.6, P 3d 1.4 and H Is 1.2. Chapter 5. Phosphine 74 136 GTO Basis Set. The extended basis set for PH 3 consists of an even-tempered (21s, Up,4d,2f/10s,Zp,2d) primitive set contracted to a [12s, 10p, Ad, 2//6s, 3p, ld) Gaussian type (GTO) basis. The s components of the cartesian d functions and the p components of the / functions were removed to avoid linear dependence, forming the final 136 GTO basis. The basis set was taken from a calculation on PH 3 using energy optimized exponents [116,117] and employing an even-tempered restriction on the exponents. The / exponents are taken from an analogous calculation on H2S by Feller et al. [118]. The basis set was designed to saturate the diffuse basis function limit and to give improved representation of the (r-space) tail of the orbitals. Due to the very large number of functions which have been used, the wavefunction is expected to be fairly insensitive to the exact choice of exponents. 136-G(CI) Ion-molecule Overlap. The CI wavefunctions for the neutral and ion species are calculated using the 136-GTO basis. The methods applied were Hartree-Fock singly and doubly excited configuration interaction (HFSD-CI) and multireference singly and doubly excited configuration interaction (MRSD-CI). The valence electron CI conver-gence has been shown to be improved (i.e. more correlation energy was recovered with fewer configurations) when the Hartree-Fock virtual orbitals are transformed into K or-bitals [119,120,121] and hence K orbitals were used. The singly- and doubly-excited con-figurations used in the MRSD-CI were energy selected based on second order Rayleigh-Schrodinger perturbation theory and the reference space was selected based upon the coefficient contribution in the HFSD-CI. This selection was necessary due to the large number of configurations associated with the extended basis set exceeding our current variational capacity. The calculations on PH 3 were done with neutral PH 3 ground state molecular orbitals. Chapter 5. Phosphine 75 Table 5.6: Calculated P H 3 properties Type of Wavefunction Total energy Dipole moment calculation label (au) P) RHF° MBS+3d -341.3094 0.86 RHF 136-GTO -342.4934 0.665 MRSD-CI6 136G(CI) -342.6833 0.624 expt. -343.42c 0.578d "This calculation taken from ref. [115]. 6 In comparing with experimental quantities it should be noted that the calculations are for a non-vibrating molecule at the equilibrium geometry and do not include relativistic effects. Only valence shell correlation is included. The core correlation and relativistic effects are expected to be slightly larger for P H 3 than H2S (see discussion in ref. [28]). cRef. [122]. dRef. [123]. The neutral molecule HFSD-CI calculation selected 10,366 spin-adapted Hartree-Fock singly and doubly excited configurations, keeping all the singly excited and using second order perturbation theory on the doubly excited, with the neglected configurations having a total contribution of less than one millihartree to the CI energy. The MRSD-CI space was chosen from the HFSD-CI's largest coefficients, as discussed above; a coefficient threshold of at least 0.030 was systematically maintained for the 5a]"1 and 2e_ 1 ion states, but a higher one was needed for the calculation of multiple roots of 2 A i symmetry (the 5a]"1 and 4a]"1 included) as discussed later. The dimension of the neutral MRSD-CI was 22,589 out of the total of 2,495,566 possible singly and doubly excited configurations. This MRSD-CI energy is -342.6833 au, with the core electrons of phosphorous uncorrelated. Some calculated properties of the neutral molecule are listed in table 5.6 along with experimental values. The calculations for the ion states of phosphine were performed using molecular or-bitals for the neutral molecule. The OVDs computed from the CI wavefunctions for the ion states then have the same form as an MO expanded in the neutral basis. The Chapter 5. Phosphine 76 calculation of the ion states involved a similar process to that of the neutral with the exception that no RHF calculations were done. HFSD-CI and MRSD-CI were done for the oa^1 and 2e - 1 states. The coefficients for the MRSD-CI were chosen based upon the coefficient contribution using a threshold of 0.030 for oa^1 and 2e_ 1. The ionization potentials for these two cation states are listed in table 5.7. The LP. values are seen to be in good agreement with the experimental values. The 2 Aj multi-root calculations employed a modified reference space selection. The calculation was done in C, symmetry but used a symmetrically closed set of configurations containing the largest coefficient contributions for the first 15 states from a calculation in which all the single excitations from all valence hole states were kept. The IP values and spectroscopic factors for the roots of the CI calculation are compared with Green's function results [124] in table 5.7 and figure 5.18. 5.2 Binding Energy Spectra The phosphine molecule has C$v symmetry [114]. The electronic configuration of P H 3 can be written (la a) 2 (2a!)2 (Ie)4 (3 f l l) 2 (4aa)2 (2e)4(5a1)2 > ^ < v ' core valence The EMS binding energy spectra of PH 3 from 8-38 eV, shown in figures 5.17(a) and (b), were obtained at an impact energy of 1200eV (plus the binding energy) and at relative azimuthal angles <j> of 0° and 6° respectively. The two binding energy spectra are Chapter 5. Phosphine 77 on a common intensity scale. The energy scale was calibrated with respect to the ba, vertical ionization potential measured by high resolution photoelectron spectroscopy [125, 126,127,128]. Measured and calculated binding energies are summarized in table 5.7. Peaks due to the ha,1 and 2e _ 1 ionization processes are found at vertical ioniza-tion potentials of 10.6 and 13.6 eV consistent with the results of photoelectron spec-troscopy [125,126,127,128]. The inner valence region shows a multipeaked spectrum consistent with previous EMS work [23]. The main features of the inner valence part of the experimental binding energy spectrum at both <f>=0° and 6° are seen (figure 5.17) to be three large peaks at approximately 20.0, 23.2 and 25.6 eV, with the heights of the peaks in the ratio of approximately 3:1:2. The earlier low momentum resolution measurements of the binding energy spectra and the XMPs of the three principal inner valence satellite peaks of P H 3 [23] lead to the conclusion that the intensity above 17 eV is predominantly due to the 4a^1 ionization process. This assignment was subsequently supported by Green's function calculations using several different basis sets [124]. Until the recent studies of Cauletti et ai. [129] using synchrotron radiation, no comprehen-sive PES measurements of the inner valence region of the PH 3 binding energy spectrum had been reported (see table 5.7). These PES spectra [129] are at higher resolution (FWHM=0.8 eV) and show more fine structure in the inner valence region than either the present or the earlier [23] EMS results. Cauletti et ai. [129] suggest, on the basis of a deconvolution analysis, that at least nine peaks exist in the 17-30 eV binding energy region. It is apparent both from the present work (figure 5.17) and the PES spectra [129] that there is probably some additional spectral intensity above 38 eV. The lower rela-tive intensity observed at higher binding energies in the PES spectrum (see figure 2 of ref. [129]) compared to the EMS spectrum is not surprising since at the low photon energy Chapter 5. Phosphine 78 used (hi/=50 eV) the partial cross sections of the higher energy satellites are expected to be relatively lower than those at lower binding energy since they are closer to threshold. PES spectra at significantly higher photon energies are necessary in order to make any meaningful comparison with the intensities observed in EMS [27]. The deconvolution of the EMS spectrum shown in figure 5.17 above 17 eV is based on the deconvolution analysis of the PES spectrum reported by Cauletti et ai. [129] which shows nine peaks between 17 and 30 eV. However due to the slightly lower resolution only six peaks have been fitted in the EMS spectrum between 17 and 30 eV. Two further peaks, not identified in the PES spectrum, are fitted to the EMS spectrum at 32 and 36 eV. Each of the eight peaks fitted in the range 17-39 eV is more intense at <b = 0° than at <f> = 6° thus confirming the dominant 4a\~l character of the inner valence spectrum above 17 eV. In particular, the peak at 20.6 eV is clearly "s-type", i.e., it is dominantly due to the Aa^ process. On the basis of a relatively small decrease in the 3 value compared to that for the 19.4 eV peak, Cauletti et ai. [129] have suggested that the 20.6 eV peak is a satellite associated with an outer valence ionization process. Such an interpretation is inconsistent with the present experimental findings. The ADC(4) many-body Green's function calculations also predict that the satellite structures in this region are dominantly 4.a\~l in character (see table 5.7). The sum of the binding energy spectra taken at <f>=0° and d>=6° is compared in figure 5.18(a) with the results of two Green's function calculations [124,130]. The Green's function calculations have been folded with the experimental energy resolution and the Franck-Condon widths obtained from the high resolution photoelectron spectra [125,126, 127,128,129] to obtain figures 5.18(c) and (d). Chapter 5. Phosphine 79 E-< E—1 > E—1 3 2 1 0 3 2 1 0 5 a , 2e ~i—r~i—i—i i 1--T \ 1 —i—i—i—r— I I I I I I I I I I I I I I I I I I I i—I—r—I—I—\—I—I 1 I I I J — i — i_ _j i I i i—i—i 1—i—i— 10 20 30 B I N D I N G E N E R G Y (eV) Figure 5.17: Binding energy spectra of the valence shell of P H 3 at azimuthal angles <^ =0" and 6". The solid line is a sum of Gaussian functions centred on peak locations using experimental widths and convoluted with the instrumental energy resolution. Chapter 5. Phosphine 80 10 20 30 10 20 30 B I N D I N G E N E R G Y (eV) Figure 5.18: Comparison of experimental and calculated binding energy spectra of P H 3 . (a) EMS spectra: sum of <j>=0" and 6", this work; (b) CI calculation, this work; (c) many-body Green's function calculation, ref. [124]; (d) many-body Green's function calculation, ref. [130]. Chapter 5. Phosphine 81 The Green's function calculation shown in figure 5.18(c) is based on the two-particle-hole Tamm-Dancoff approximation (2ph-TDA) [131,48] and is taken from the work of Domcke et ai. [124]. A rather small basis set, (12s, Qp, 2d/4s) contracted to [6s, Ap, 2d/2s], was used. The calculation shown in figure 5.18(d) [130] used the simplified ADC(4) approximation [132] and also used a rather small basis set: (12s, 9p, Id/As, lp) contracted to [6s, Ap, \d/2s, lp). Also shown (in figure 5.18(b) and table 5.7) are the roots and spectroscopic factors from the MRSD-CI calculation done in the present work. Again the theoretical results are folded with observed widths. All calculations predict significant splitting of the Aa\~l ionization strength. The calculated spectral envelope shows quite reasonable agreement in all cases with the EMS data. Although all calculations predict qualitatively similar re-sults, the third band appears at too high an energy (~ 27 eV) and with too little intensity. As has been observed for other molecules, the 2ph-TDA method is found to underesti-mate pole strengths at higher binding energies [28,37,39,133,134]. The CI calculation predicts almost exact binding energies for the 5ax 1 and 2e - 1 ionization processes, while giving similar results to the Green's function calculations in the inner valence region. Table 5.7: Measured and calculated binding energies (eV) for the valence shell of P H 3 State Experimental vertical i r Calculation R.HF Grcen'n function " M R S D - C I " E M S PES 136-GTO 2ph-TDA ADC(4) 136-G(CI) (hit work11 ref. [23] ref. [125] ref. [127] ref. [124] ref. [129] this work ref. [124] ref. [130] thin work 5a j 10.6 10.69 1(1.56(1) 10.60 — 10.58 9.95 (0.02) 10.25 (0.911) 10.44 (0.8626) 30.44 (0.02) 19.28 (0.004) 24.97 (0.006) 26.49 (0.004) 28.16 (0.015) 30.05 (0.007) 2r 13.6 13.44 13.50(5) 13.6 — 14.17 13.23 (0.93) 13.59 (0.904) 13.62 (0.8399) 21.30 (0.006) 26.88 (0.003) 28.52 (0.006) 30.11 (0.004) 31.88 (0.003) 31.95 (0.004) U , 19.4 19.46 19.0 19.6(2) 19.4 19.37 (0.53) 19.28 (0.342) 19.70 (0.4373) 20.6 20.5(2) 20.6 20.98 (0.100) 21.7 22.32 (0.058) 23.2 22.61 23.2 23.23 22.39 (0.13) 22.79 (0.231) 22.01 (0.1180) 24.1 23.20 (0.03) 24.97 (0.005) 22.77 (0.0444) 23.46 (0.03) 25.98 (0.003) 23.15 (11.0050) 20.22 (0.03) 24.67 (0.0468) 26.78 (0.02) 25.41 (0.0185) 27.73 (0.02) 26.20 (0.0093) 25.0 25.46 25.6 28.04 (0.13) 26.49 (0.031) 27.61 (0.1184) 27.2 26.5 27.8 20.53 (0.083) 28.14 (0.0398) 29.1 20.1 31.8 31.60 (0.02) 30.06 (0.009) 3G.ll 33.96 (0.01) 31.18 (0.008) 31.54 (0.004) 32.90 (0.007) aThe pole strength is shown in parentheses. fcThese represent observed peaks for the ha, and 2e states, but are the centres of the Gaussians in the deconvolution for the inner valence (4ai) region. The deconvolution was based largely on the PES work of Cauletti et ai. [129] (see text for details). Chapter 5. Phosphine 83 5.3 Comparison of Experimental and Calculated Momentum Distributions The experimental momentum profiles (XMPs), calculated spherically averaged momen-tum distributions (MDs), and ion-molecule overlaps (OVDs) are shown in figures 5.19-5.21 together with momentum and position space density maps calculated in selected planes for an oriented PH 3 molecule using the 136 GTO wavefunction. The momentum resolution of the XMPs is clearly improved over the earlier work [23]. The XMPs were placed on the same realtive intensity scale by normabzation on the binding energy spec-tra (figure 5.17). The XMPs, MDs and OVDs have been placed on a common intensity scale by a single point normalization of the OVD calculation to the 5aa orbital. All other relative normalizations for calculations and experiment have been preserved. All calcu-lations have been folded with the experimental momentum resolution (0.15 au HWHM). For the reasons discussed in section 5.2, all of the inner valence satellite intensity above 17 eV (figure 5.17) has been assigned to the 4a]"1 process. The 136-G(CI) ion-molecule overlap (OVD) calculations using correlated wavefunc-tions and the 136-GTO SCF calculation of the momentum distributions (MD) are seen (figures 5.19-5.21) to be very similar for a given ionization process and essentially identi-cal in the case of the 5a! orbital (figure 5.19). The calculations also give an excellent fit to the shape of the experimental data for the 5ai orbital. Agreement with the data for the 2e orbital, however, is somewhat less satisfactory in that the MD and OVD calcula-tions maximize at shghtly too high a momentum and do not completely account for the intensity near p=0 au. If this additional intensity at low momentum in the experimental data is due to an unresolved (4aa) symmetric pole under the 2e binding energy peak, Chapter 5. Phosphine 84 then one would expect the experimental momentum profile to contain more total inten-sity than the calculated momentum distribution. However the calculation shows more intensity than the data between 0.6 and 1.0 au. The small difference at low momentum could possibly be due to uncertainty in the momentum resolution fitting procedure. The 136-GTO MD (and 136-G(CI) OVD) for the 4a1_1 process predicts significantly greater (~25%) intensity than is observed. This discrepancy may, at least in part, re-flect the existence of ia^ satellite intensity lying above 38 eV, which is the limit of the presently reported binding energy spectra data. It is also possible that some absorption is occuring for this more deeply bound orbital even at the impact energy of 1200 eV. It can be seen from the 0.74x(136-G(CI)) curve that the shape of the 4a! XMP is re-produced exactly by the 136-G(CI) and 136-GTO calculations, except in the region of approximately 1.2-2 au, where the difference between theory and experiment may be due to a breakdown of the plane wave approximation which is not unexpected at higher values of momentum [135]. In the previous (lower momentum resolution) EMS study of PH 3 [23], the MBS+3d wavefunction [115] was found to model the data reasonably satisfactorily (for shape) when the XMP of each orbital was individually height normalized to the calculation. However in the present work, with experiment and theory now on a common intensity scale for all orbitals, the MBS+3d calculation is found to be inadequate. In order to facilitate comparison with the earlier work [23], the MBS-f3d calculation is shown (figure 5.22) separately height normalized to each of the presently determined high momentum resolu-tion XMPs so that only the shape of the theoretical and experimental curves is compared for each orbital. It can be seen that the shapes of the 5a! and 2e orbitals show about the same level of agreement with experiment as was found in the earlier work. It is clear that Chapter 5. Phosphine 85 MOMENTUM (au) MOMENTUM (au) POSITION (au) Figure 5.19: Measured and calculated spherically averaged momentum distributions for the 5a] orbital of P H 3 . Density contour maps in momentum and position space are also shown. The contour maps were generated with the 136-GTO calculation. The contour values represent 0.02,0.04,0.06,0.08,0.2,... ,20,40,60 and 80% of the maximum density. The side panels (top and right side) show the density along the dashed lines (horizontal and vertical) in the density map. Chapter 5. Phosphine 86 MOMENTUM (au) M O M E N T U M (au) POSITION (au) Figure 5.20: Measured and calculated spherically averaged momentum distributions for the 2e orbital of PH3. Density contour maps in momentum and position space are also shown. The contour maps were generated with the 13G-GTO calculation. The contour values represent 0.02,0.04,0.06,0.08,0.2,... ,20,40,60 and 80% of the maximum density. The side panels (top and right side) show the density along the dashed lines (horizontal and vertical) in the density map. Chapter 5. Phosphine 8 7 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY PH 1 136-C(CI) 2 138-CTO 3 MBS + 3d 1 2 MOMENTUM (au) P H 3 4 o , . . . . , . . . . -> . o J < 0. w> -8.0 -4.0 0.0 4.0 8.0 0.5 1.0 MOMENTUM (au) P H 3 4o, . . . . , . . . . ! p H 't \ ! H H (o.o. 0 •4 -8.0 -4.0 0.0 4.0 8X> OS) 1.0 POSITION (au) Figure 5.21: Measured and calculated spherically averaged momentum distributions for the 4ai (main peak) orbital of P H 3 . Density contour maps in momentum and position space are also shown. The contour maps were generated with the 136-GTO calculation. The contour values1 represent 0.02,0.04,0.06,0.08,0.2,...,20,40,60 and 80% of the maximum density. The side panels (top and right side) show the density along the dashed lines (horizontal and vertical) in the density map. Chapter 5. Phosphine 88 much poorer agreement is obtained between the MBS+3d calculation and experiment when correct relative intensities are established for all orbitals (see figures 5.19-5.21). This shows that independent height normalization could lead to erroneous conclusions as to the quality of the wavefunction. In the momentum and position space density maps, all coordinates are in atomic units. The phosphorous nucleus is at the origin of the position space maps, and the hydrogens located at approximately (0,2.25,-1.46) (in the plane of the picture) and (±1.95, —1.13, —1.46) (out of the plane of the picture). These pictures are consistent with the simplest notions of how we expect these orbitals to be built up from atomic orbitals. To a first approximation, the 4a, is composed of a phophorous 3s orbital bonding with hydrogen Is orbitals. The 2e orbital consists of a phosphorous Zp orbital in the "plane" of the molecule (although the molecule isn't quite planar) combined appropriately with hydrogen Is orbitals. It is degenerate since there are two orthogonal p functions in the plane. (For the orientation of PH3 shown in the density maps, this is the xy-plane.) The 5ax orbital consists of a p function perpendicular to this plane bonding with the hydrogen s functions. The 5a, orbital in PH 3 shows much more phosphorous 3s contribution (this is seen as the symmetric density near the origin) than the corresponding orbital in NH3 (3ai) shows nitrogen 2s intensity [37]. In phosphine, all three orbitals show more density near the origin of position space (despite strong contributions from the hydrogens) than the corresponding orbitals in NH 3 . Both from EMS and PES results it has been observed that there is much more satel-lite structure in the inner valence region of the binding energy spectrum of PH 3 (see figures 5.17,5.18 and also references [23,129] than of NH 3 [37,136,137,138]. Although the independent particle picture of ionization breaks down completely in the inner valence Chapter 5. Phosphine 89 0 (a) . f e . 5a , PH MBS+3d f *\ t h e o r y x l . 2 4 i \ 0 0 6 4 2 0 (b) 2e J »\ MBS+3d PH t i t h e o r y x l . 0 5 4 0 (c) 4a, PH MBS+3d t h e o r y x0.56 0 1 2 3 MOMENTUM (au) Figure 5.22: Measured and calculated (MBS+3D) momentum distributions for the valence orbitals of P H 3 . Each calculation shown is separately height normalized to experiment for each orbital to facilitate direct comparison with earlier measurements [23]. Chapter 5. Phosphine 90 region of P H 3 , CI calculations of the ion-molecule overlap (OVD) of the inner valence orbitals of P H 3 and N H 3 vary little, if at all, from calculations based on the target Hartree-Fock approximation [37]. In fact, this is true for all valence orbitals of PH 3 . Yet a significant difference is observed between the ion-molecule overlap (OVD) and the target Hartree-Fock approximation momentum distribution (MD) for the Zax orbital of NH 3 [37]. This may suggest that the main factor which causes the CI calculation of the OVD to differ from the MD is the incorporation of relaxation. The present results and findings for the valence orbitals of PH 3 follow closely those found for H2S [28], i.e. the XMPs of the hydrides with second row heavy atoms from groups V and VI are already quite well described by SCF wavefunctions near to the Hartree-Fock limit. Incorporation of correlation and relaxation in P H 3 , H2S [28] and HC1 (chapter 8) produces a negligible change in the description of the momentum distri-bution. This is in sharp contrast to the situation for the corresponding first row hydrides, NH 3 [37], H 2 0 [38], and HF [139] where incorporation of the correlation and relaxation has been found to be of crucial importance for the outermost valence orbitals since ma-jor discrepancies exist between the measured XMPs and calculated MDs even at the Hartree-Fock limit for the valence orbitals. In summary, the present EMS results show good agreement between the measured XMPs for the 5ax orbital and the corresponding 136-GTO MD and somewhat less good agreement for the 2e orbital. The inclusion of correlation and relaxation (136-G(CI)) is not required to predict the momentum distribution provided a sufficiently saturated and diffuse basis set is used for the SCF calculation. A significant discrepancy in intensity but not shape between calculation and experiment for the Aa\~1 ionization process. Some of this discrepancy is likely due to missing satellite intensity beyond the upper limit of Chapter 5. Phosphine 91 the present binding energy spectra. \ Chapter 6 Methane The first reported study of a molecule using EMS was the work of Hood et ai. [7] on the valence shell binding energy spectra and momentum distributions of methane. In addition to the main ( l^) - 1 and (2ai)_ 1 processes, this work [7] and later studies [52,140, 141] clearly showed the existence of satellite structure of dominantly "s-type" symmetry extending over an energy range of at least 30 eV above the main (2a!) -1 peak. Since the earlier EMS studies [7,52,140,141] had in general significantly poorer momentum and/or energy resolution than is now available, detailed study of the satellite region was precluded. In the present work, momentum distributions and binding energy spectra have been measured for methane over a greater binding energy range and with better momentum resolution than in previous studies. The binding energy spectra are compared with a previously published 2ph-TDA Green's function calculation [141] and found to compare favourably. The measured momentum profiles have been quantitatively compared with new and very accurate calculations using SCF wavefunctions up to near the Hartree-Fock limit in quality and also with ion-neutral overlap calculations using correlated wavefunc-tions obtained by the method of configuration interaction. Excellent agreement is found with theory at the Hartree-Fock level and inclusion of relaxation and correlation is found 92 Chapter 6. Methane 93 to have minimal effect on the calculated momentum distributions. 6.1 Calculation of the Momentum Distributions of C H 4 Experimental momentum profiles (XMPs) are compared to momentum distributions (MDs) calculated with a range of SCF wavefunctions of essentially double zeta and Hartree-Fock-limit quality. In order to assess the importance of electron correlation and relaxation effects, the results are also compared with ion-molecule overlap distribu-tions (OVDs) calculated from configuration interaction (CI) wavefunctions for the initial (molecule) and final (ion) states. The calculations were carried out at the C H 4 experi-mental geometry given in ref. [142], which gives Rc#=2.052 Bohr. Details of the various wavefunctions are shown in table 6.8 and discussed below. Snyder and Basch. This basis of contracted Gaussians is taken from Snyder and Basch [73]. The C basis consists of four contracted s-type Gaussians (two to describe the Is atomic orbital, two to describe the 2s) and two contracted p-type Gaussians. The hydrogen basis consists of two contracted s-type Gaussians. The choice of basis functions used has been shown to yield a similar total energy in many cases to a calculation using a Slater basis, with two Slaters per atomic orbital, in which the Slater exponents have been optimized for the atoms. Hence this basis is often referred to as being double zeta in quality. Note that we have not used the results of the calculation on C H 4 presented in refer-ence [73], but have redone the calculation using GAUSSIAN76 [35] at the geometry given in [142], the same geometry used for all the calculations presented in this paper. Chapter 6. Methane 94 146-GTO Basis Set. The 146-GTO basis set was constructed from an even-tempered (23s, 12p, 3d, l//10s, 3p, 2d) primitive gaussian basis contracted to [14s, lOp, 3d, If/6s, 3p, Id]. For the s and p type primitives of the carbon basis set, the exponents were obtained from reference [143] as the optimum 22s, lip set plus one additional diffuse s and p function. The exponents of the d type primitives were 2.74, 0.94 and 0.32 and the exponent for the / type primitive was 1.06. The hydrogen portion of the basis is identical to that used previously for calculations on H2O [38,144]. The self-consistent-iield (SCF) energy for the neutral molecule was found to be -40.2169 hartrees. This is shghtly lower than the SCF energy -40.2166 (calculated for RCjy=2.050 Bohr) in the best previously published calculation [145] and is close to the estimated Hartree-Fock limit of-40.219 [146]. 146-G(CI) Ion-Molecule Overlap. The correlation and relaxation of the electrons was treated using the method of configuration interaction. For the CI calculations, K orbitals [119] obtained from the SCF calculation of the neutral molecule were used as the molecular-orbital (MO) basis in the CI. Frozen-core multi-reference singles and doubles CI (MRSD-CI) calculations, using a selection procedure based on perturbation theory (PT), were performed with all reference configurations that had coefficient contributions of at least 0.03. For the CPs using PT selection, the zeroth-order wave function was defined by diagonalizing the electronic Hamiltonian over the reference space and the second-order Rayleigh-Schrodinger (RS) PT energy contributions of the configurations outside the reference space were calculated. The subset of all of the configurations having an RS energy contribution of at least 10 - 6 hartrees was treated variationally. MRSD-CI calculations on the ionic states of CH 4 were performed using the same molecular orbital basis and geometry as for the neutral molecule. Chapter 6. Methane 95 Table 6.8: Total energy of CH 4° Calculation Total energy (au) Snyder and Basch -40.1826 146-GTO -40.2169 Hartree-Fock limit -40.2175±0.0005b -40.219±0.001 c 146-G(CI) -40.4297 experimental -40.46d -40.512±0.003 e "Calculations performed at experimental equilibrium bond length 2.052 bohr [142]. fcBased on expected error in 146-GTO basis set, this work and ref. [38]. eSuggested Hartree-Fock limit as quote in ref. [146]. dEstimated frozen-core CI limit. e Estimated non-vibrating, non-relativistic energy as quote in ref. [146]. As shown in table 6.8, the MRSD-CI energy for the neutral molecule was calculated to be -40.42968 hartrees. This accounts for approximately 89% of the valence shell correlation energy. The vertical IPs of the lt2 and 2aa states as calculated by this method can be found in table 6.9, along with the results of the other calculations and a variety of experimental values, including those measured in this work. 6.2 Binding Energy Spectra The methane molecule has Tj symmetry [142]. The ground state configuration can be written as: (loaf (2a 1) 2(lt 2) 6 core valence Chapter 6. Methane 96 The binding energy spectra of C H 4 , shown in figures 6.23(a) and (b), were obtained at an incident energy of 1200 eV (plus the binding energy) and at relative azimuthal angles 4> of 0.5° and 7.5° respectively. These spectra, which are on a common intensity scale, are dominated by two peaks at 14 and 23 eV which can be considered to be ionization from the lt2 and 2a, orbitals respectively. The energy scale was calibrated with respect to the vertical ionization potentials for the main (2a,)~1 process as measured by photoelectron spectroscopy [147,148,149,150]. By centering the main 2a, peak at 23.05 eV, we find the centre of the lt2 peak to be at 14.13±0.15 eV, in good agreement with most previous PES and electron impact studies [128,140,141,147,148,149,150,151,152,153,154,155,156]. The widths are also consistent with previous work, when the instrumental energy resolution of 1.7 eV FWHM is considered. (Note that the lt2 and 2a, peaks have both been vibrationally resolved by high resolution PES: see especially references [154,157].) The lf2 peak is very broad due to Jahn-Teller distortion [154,156,158,159]. Table 6.9: Measured and calculated binding energies (eV) for the valence shell of C H 4 State Experimental vertical IP Calculation EMS PES RHF M R S D - C I ° Green's f. 2ph-TDA a IP intensity ref. [148] S+B" 146-GTO 146-G(CI) ref. [141] 1*2 14.13(15) 1.00 14.2(2) 14.79 14.54 14.3 [0.8721] 14.10 [0.94] 2a, 23.05 25.4(10) 28.56(45) 31.45(30) 35.28(55) 0.673(30) 0.046(10) 0.041(5) 0.105(10) 0.041(5) 23.05(2) 25.76 25.18 23.3 [0.7776] 23.03 [0.84] 31.92 [0.04] 35.83 [0.05] 55.77 [0.04], 58.65 [0.01] "The spectroscopic factor is shown in parentheses. *Snyder and Basch basis set. Measurements of ionization efficiency curves by the method of retarding potential Chapter 6. Methane 97 difference [151,160] as well as electron energy loss experiments [161,162] have reported an unexpected feature in the vicinity of 19.5 eV. Such a feature has been observed in only one photon impact experiment [163], in which a maximum was observed in the ultraviolet absorption spectrum. The fact that no evidence for such a process has been observed in this EMS study or in previous EMS studies [7,140,141] is consistent with the assignment [151,162] of this feature to an autoionisation process (involving an excited 2ax electron which autoionizes down to the lt 2 limit). The fact that no direct ionization is observed at 19.5 eV is also consistent with the assignment [164,165] of such a feature to ion-molecule interactions (of the form CH+-+CH4 -»CHf+CH 2 ) . Weak satellite structure is observed in the inner valence region of C H 4 out to at least 55 eV in agreement with earlier lower resolution EMS studies [7,140,141]. The intense inner valence peak at 23.05 eV can clearly be identified as the "main" or "parent" peak of the 2ax1 ionisation process, with additional low intensity structure lying between 25-60 eV. This is in qualitative agreement with the prediction of Cederbaum et ai. [166] that ionization of inner valence electrons of saturated hydrocarbons can be adequately described as the creation of quasiholes. It is also consistent with the trend that hydrides of first row elements (C-F) show less satellite structure than their analogues in the second row (Si-Cl) [28,29,167,38,37,39]. Approximately 67% of the (2a!)-1 intensity is found experimentally in the main peak at 23.05 eV (fig. 6.23). In order to assign the more intense region of the satellite structure, additional mea-surements of higher statistical precision were made in the inner valence region. The mea-surements were made at <j}—0.5° and d>=7.5° in the binding energy range of 22-39 eV. Chapter 6. Methane 98 10 20 30 40 50 60 10 20 30 40 50 60 B I N D I N G E N E R G Y ( eV) Figure 6.23: Binding energy spectra for the valence shell of C H 4 at azimuthal angles $=0.5" and 7.5". The solid line is a sum of Gaussian functions centred on peak locations using experimental widths and convoluted with the instrumental energy resolution. (See text for other details of peak widths and positions). Chapter 6. Methane 99 B I N D I N G E N E R G Y ( e V ) Figure 6.24: Binding energy spectra of the inner valence region of C H 4 . (a) sum of angles <£=0.5" and $=7.5"; (b) $=7.5°; (c) $=0.5". The solid line is a sum of Gaussian functions (dashed lines). Chapter 6. Methane 100 2 0 E—1 10 i n H E-i 0 2a, CH EMS <?=0.5°+7.5 10 2 0 3 0 4 0 50 6 0 > 20 E-< W 10 0 CH G R E E N ' S F U N C T I O N " C A L C U L A T I O N _ 2 p h - T D A i r ~ \ — r 10 2 0 3 0 4 0 50 6 0 B I N D I N G E N E R G Y (eV) Figure 6.25: Comparison of experimental and calculated binding energy spectra of C H 4 . (a) EMS spectra, sum of 0=0.5° and 7.5", this work; (b) many-body Green's function calculation, ref [141]. Chapter 6. Methane 101 These results and their sum are shown, on the same relative intensity scale, in figure 6.24. The satellite structure between 25 and 40 eV has been deconvoluted into four peaks. These peaks, which are common to both spectra (0.5° and 7.5°) are at 25.4, 28.6, 31.5 and 35.3 eV with FWHM widths of 3.0, 2.0, 3.0 and 3.0 eV respectively. While the limited statistical precision of the data prevents us from presenting this as a unique deconvolu-tion, it is consistent with an earlier EMS measurement of the inner valence region made at a single angle with a shghtly better energy resolution [52] and with a dipole (e,2e) measurement of the same region, also with better energy resolution [168]. The satellite intensity in figure 6.24 is clearly lower over all binding energies at <j)=7.5° than at ^=0.5°, suggesting that the intensity should all be assigned to the 2a, ionization process. This is consistent with the general assignments reported in previous EMS studies as well as the general conclusions of Green's function calculations [7,52,140,141]. The energies of the satellite peaks observed in figure 2 are shown in table 6.9, along with the results of a 2ph-TDA Green's function calculation [141]. The 2ph-TDA calcu-lation gives excellent values for the IPs of the lt2 and 2a, orbitals. (Other, higher-level, Green's function calculations of the orbital ionization potentials have been performed and they also give good results. See references [166,169,170].) A comparison of the summed 0.5 ° and 7.5° spectra with the Green's function calculation reported by Cambi et al. [141] (shown in table 6.9) is also shown in figure 6.25 folded with the experimentally observed widths. While the 2ph-TDA approach is found in many cases to underestimate the intensity of satellite structure [28,29,37,39,133,134] this does not seem to be the case here. The Green's function calculation agrees quite well with the intensity distribution of the measured spectra in both the outer and inner valence regions. However very little (if any) intensity was observed between 55-60 eV in contrast to the prediction of the Chapter 6. Methane 102 calculation. 6.3 Comparison of Experimental and Calculated Momentum Distributions Experimental momentum profiles (XMPs), calculated momentum distributions (MDs), and calculated ion-molecule overlaps (OVDs) are shown in figures 6.26(a) and 6.27(a). The XMPs, MDs and OVDs are on a common intensity scale. The OVDs have been normalized to unit area and in addition both the OVDs and the MDs have been multiplied by the orbital occupancy. Hence the OVDs and MDs integrate to give the respective orbital occupancies (six for lt2 and two for 2a,). The experimental data for the two orbitals has been placed on a common relative intensity scale using peak areas and satellite intensities from the binding energy spectra. As discussed above (section 6.2) all the inner valence intensity between 20 and 60 eV can be assigned to the 2a,1 process. A single point normalization of the lt2 orbital (14 eV) data to the OVD calculation was performed, retaining the relative normalization of the 2a, and lt2 orbitals which had been derived from the binding energy spectrum. All other relative normalizations for both experiment and calculation are preserved. The instrumental momentum resolution (Ap = 0.15 au HWHM) has been folded into the calculations. Momentum and position density contour maps for a given plane of the oriented methane molecule are also shown for each orbital in figures 6.26(b),(c) and 6.27(b),(c). These were calculated using the Snyder and Basch wavefunction. All numbers shown are in atomic units. In the position density maps, the C nucleus is located at the origin and the hydrogen nuclei are at approximately (0, ±1.68, 1.18) (in the picture) and ( ±1.68, Chapter 6. Methane 103 0, -1.18) (out of the picture). Looking at the position density map in figure 6.27(c) we see that, as for the case of the 3ax orbital of SiH 4, most of the density of the 2ax orbital of C H 4 is found near the nucleus of the heavy atom. (For the equivalent diagram for SiH 4, see figure 7.32(c).) The momentum density is observed in both cases to be spherical near the origin of momentum space, while the position density is found to be spherical far from the origin of position space. This is as expected considering the inverse spatial relationship between position and momentum. For both the lt 2 orbital of CH 4 and the 2t2 orbital of SiH 4, intuitive MO theory predicts each t2 orbital to be composed of the valence atomic p orbital on the heavy atom bonding with the Is orbitals on the hydrogens, where two of the hydrogen s functions are positive and two negative as appropriate to bond with the p on the heavy atom. In the case of the lt 2 orbital of C H 4 , the space between the carbon nucleus and the hydrogen nuclei is "filled up" with charge density, in contrast to what occurs in the 2t2 orbital of SiH4. This is clearly seen in the position density map, particularly in the top panel for figure 6.26(c) which shows the density along the line x—0,y—Q (compare with figure 7.31(c) for SiH4). The localised areas of high charge density in the 2t2 orbital of SiH4 "interfere" with each other when Fourier-transformed into momentum space. The result is a more complex momentum density map for the 2t2 orbital of SiH4 (figure 7.31(b)), while that for C H 4 is relatively simple (figure 6.26(b)). Turning now to a consideration of the measured and calculated momentum distribu-tions, it can be seen from figures 6.26(a) and 6.27(a) that while the Snyder and Basch calculation agrees with the data reasonably well, the 146-GTO (near Hartree-Fock limit) Chapter 6. Methane 104 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY POSITION DENSITY CH4 1t 2 —i—i—'—i— (o.w) Wiuffl— -8.0 - 4 . 0 4.0 8.0 0.5 1.0 MOMENTUM (au) MOMENTUM (au) CH4 11 o yyv ' ' ' • I • • • • . o" J o* , '<°-'-o> -8.0 -4 . 0 0.0 4.0 8.0 0.5 1.0 POSITION (au) Figure 6.26: Measured and calculated spherically averaged momentum distributions for the U 2 or-bital of CH.). The solid circles represent the direct measurements. Open square is obtained from the respective lt2 and 2ax intensities in the binding energy spectra shown in figure C.23. Calculations are folded with experimental momentum resolution. This amounts to convoluting the curves with a Gaus-sian of 0.15 au HWHM. Density contour maps in momentum and position space are also shown. The contour maps were generated with the Snyder and Basch calculation. The contour values represent 0.02,0.04,0.06,0.08,0.2,... ,20,40,60 and 80% of the maximum density. The side panels (top and right side) show the density along the dashed lines (horizontal and vertical) in the density map. Chapter 6. Methane 105 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION MOMENTUM DENSITY 8.0 0.5 1.0 MOMENTUM (au) MOMENTUM (au) POSITION DENSITY (c) (0.1.0) -8.0 -4 .0 0.0 4.0 8.0 0.5 1.0 POSITION (au) Figure 6.27: Measured and calculated spherically averaged momentum distributions for the 2a\ (main peak) orbital of CH4.. The solid circles represent the direct measurements. Open squares are obtained from the relative U2 and 2a] intensities in the binding energy spectra shown in figure 6.23. Calculations are folded with experimental momentum resolution. This amounts to convoluting the curves with a Gaussian of 0.15 au HWHM. Density contour maps in momentum and position space are also shown. The contour maps were generated with the Snyder and Basch calculation. The contour values represent 0.02,0.04,0.06,0.06,0.2,...,20,40,60 and 80% of the maximum density. The side panels (top and right side) show the density along the dashed lines (horizontal and vertical) in the density map. Chapter 6. Methane 106 calculation gives a slightly improved fit to the data for the \t2 orbital. The CI treatment is found to give no noticeable difference in the calculated momentum distributions, sug-gesting that relaxation and correlation are not important in evaluating the overlaps for ionization for the lt 2 and 2ai orbitals of methane. This also appears to be the case for SiH4 for the corresponding 1t2 and 3aj valence orbitals. Chapter 7 Silane This chapter presents the first measurement of the binding energies and momentum distributions of the valence orbitals of SiH4 by electron momentum spectroscopy. The measured binding energy spectrum is compared with ADC(3) and ADC(4) many body Green's function calculations. The momentum distributions are compared on a quantita-tive basis with SCF wavefunctions near the Hartree-Fock limit and also with ion-neutral overlap calculations carried out using correlated wavefunctions by the method of configu-ration interaction. Quite good agreement with theory is found at the Hartree-Fock limit and inclusion of electron correlation has minimal effect on the calculated momentum distributions. A study of the angular dependence of the satellite structure in the inner valence region shows that these processes arise predominantly from final state correlations associated with the production of the 3aJ"1 hole state. 7.1 Calculation of Binding Energy Spectra A Green's Function calculation was used to predict the poles and pole strengths ex-pected in the inner valence binding energy spectrum of SiH4. The general features of the calculation, which uses the 58-GTO basis set (see section 7.2), are described below. 107 Chapter 7. Silane 108 The ionization energies and their relative intensities or pole strengths appear as poles and residues of the one-particle Green's function [171] and can be calculated directly via solution of the Dyson equation. The Green's function method takes the electronic correlation and relaxation effects into account and has proved to be a very useful tool for calculating and interpreting ionization spectra. In the present work we use the Green's function method in approximations developed by Cederbaum and coworkers [48,50,69, 131]. The principal approximation used in this work is the ADC(n) method (n-th order algebraic diagrammatic construction [69]). It is a systematic procedure for deriving approximations to the Green's function which are accurate to order n of perturbation theory and which have the correct analytical structure so that they can be applied in the entire valence region. The cases for n=3 and n=4 have been worked put in detail [69]. The ADC(3) method, also referred to as the extended two-particle-hole Tamm-Dancoff approximation [131], involves all single-hole, single particle, two-hole-one-particle (2hlp) and 2plh configurations. The mixing of configurations of different particle number in the Dyson equation introduces ground state correlation effects in the Green's function method. The ADC(3) method gives the main ionization energies of those states which have strong contributions from 2hlp configurations accurate to first order only. Therefore for the interesting satellite lines it is only qualitatively or at most semi-quantitatively correct. To obtain higher accuracy for these states one has to use the ADC(4) method, which includes the 3h2p and 3p2h configurations in addition to the configurations given above. The main lines are calculated accurately to fourth order and the satellite lines accurately to second order. However, the computational problems are severe for the full ADC(4) method. We have therefore resorted to a somewhat simplified ADC(4) Chapter 7. Silane 109 method in the present work. The very large 3p2h space is completely omitted as well as some of the higher-order coupling elements. This simplified ADC(4) method is discussed in ref. [132,172] and has been used earlier for C l 2 [173], Br 2 [174] and I2 [175]. The simplifications should retain the dominant effects appearing in fourth order. They have little influence on the satellite fines but experience shows that the energies for the main fines are slightly too low due to the neglect of a part of the ground state correlation energy arising in fourth order. The ADC(4) results presented here are considered preliminary. Work is in progress on improved implementations. The eigenvalues and eigenvectors are extracted from the equations with the multi-root Davidson [176] procedure due to Tarantelli [177]. Table 7.10: Total energy of SiH4 Calculation Silicon basis set Hydrogen basis set Total energy (au) (1) STO-3G (9s,6p)/[3s,2p] (3*)/[l-] -287.910 (2) 58-GTO (12s,9p,2d)/[6s,4p, 2d] (4s, lp)/[2s, lp] -291.135 (3) 126-GTO (235,15p, 5d, l/)/[8s, 7p, 2d, If] (10s,3p,2d)/[65,3p,ld] -291.266 (4) 126-G(CI) (23s, 15p, 5d, l/ )/[8 5 ,7p,2d, l/ ] (10s,3p,2d)/[6s,3p,ld] -291.515 experimental0 -291.896 aNon-vibrating, non-relativistic, estimate based on experimental AHj, \hv, and atomic energies. 7.2 Calculations of Momentum Distributions of SiH 4 Experimental momentum profiles (XMPs) were compared to momentum distributions (MDs) calculated with a range of SCF wavefunctions from minimum basis set up to near-Hartree-Fock-limit as well as to ion-molecule overlap distributions (OVDs) calculated from configuration interaction (CI) wavefunctions for the initial (molecule) and final (ion) states. The calculations were carried out at the SiH4 experimental geometry given Chapter 7. Silane 110 in ref. [178]. Details of the various wavefunctions are shown in table 7.10 and discussed below. (1) STO-3G Basis Set. This is a minimal basis set and so allows for three s functions and two p functions on the Si and one s function on H. Each function is a contraction of three Gaussians. This basis set was designed by Pople and coworkers [71]. (2) 58-GTO Basis Set. The 58-GTO basis set is constructed from a (12s,9p,2d/4s,lp) primitive gaussian basis contracted to [6s,4p,2d/2s,lp], taken from ref. [179]. This basis set was also used for the Green's function calculation of the binding energy spectrum (see section 7.1). (3) 126-GTO Basis Set. The 126-GTO basis set was constructed from an even-tempered (22s,15p,5d,lf/10s,3p,2d) primitive gaussian basis contracted to [8s,7p,2d,lf/6s,3p,ld]. In an even-tempered basis, the exponents £ form a geometric sequence: £ = ct8l (i = 0,1,... , N — 1) where a and 8 are optimized for each atomic number and type of function (s or p). For silicon, a^O.02484, 8,=2.3, ap=0.02484, /3P=2.3, ad=0.1311, 8d=2.3. The / exponent was optimized to be 0.30153. The hydrogen portion of the basis is identical to that used previously for calculations on H2O [38,144]. (4) 126-G(CI) Ion-Molecule Overlap. In order to investigate the effects of electron correlation and relaxation ion-molecule overlap distributions (OVDs) were constructed using frozen-core configuration interaction wavefunctions for both the neutral molecule ( a Ai) and the 2 T 2 and 2 A X ion states. Reference configurations for a multireference SDCI calculation were selected from HF SDCI calculations on each state. At the multireference SDCI level the important configurations were energy selected according to second order Rayleigh-Schrodinger perturbation theory. Chapter 7. Silane 111 7.3 Binding Energy Spectra The silane molecule has T j symmetry [178]. The independent particle electronic config-uration can be written ( l a 1 ) 2 ( 2 a 1 ) 2 ( U 2 ) 6 (3a 1 ) 2 (2f 2 ) 6 core valence The binding energy spectra of S i H 4 , shown in figures 7.28(a) and (b), were obtained at an impact energy of 1200 eV (plus the binding energy) and at relative azimuthal angles (j> of 0° and 6° respectively. The two binding energy spectra are on the same (relative) intensity scale. The energy scale was calibrated with respect to vertical ionization poten-tials measured by He I photoelectron spectroscopy [128,180]. Gaussian curves have been fitted to the two main peaks using known vertical IPs and their associated Frank-Condon widths, folded with the energy resolution of the spectrometer. The inner valence satellite region is fitted with four peaks as discussed below. (See also figure 7.29(a).) Measured and calculated binding energies are Usted in table 7.11. Peaks due to the 24J1 and 3a,1 processes are found at vertical ionization potentials of 12.8 and 18.2 eV in agreement with the results of He I photoelectron spectroscopy [128, 180]. The 2t2 peak is broadened due to the fact that the T2 radical cation states are split by a T j —>D2d Jahn-Teller distortion [128]. The larger intensity of the 2i 2 " 1 peak at <^ >=6° is indicative of its expected "p type" character (that is, dominantly Si 3p). The peak due to the ionization of the totally symmetric 3a, orbital has a larger intensity at cf>=0° than at e>=6° as expected for an orbital of "s type" character. Since the satellite structure Chapter 7. Silane 112 appearing at 24.2 eV shows the same intensity variation with <j> as the 3ai peak it can also be assigned to transitions arising from the 3a]"1 hole state. The satellite intensity observed above 25 eV is more difficult to assign on the basis of the data shown in fig 7.28. Therefore additional measurements with higher statistical precision have been made in the inner valence satellite region. Figure 7.29(b)-(h) shows binding energy spectra measured at a range of <j> angles (0°-15°) for binding energies between 20-40 eV and fig 7.29(a) shows the sum of the spectra over all angles. Gaussian peaks were fitted to the spectra with peak energies at 24.2, 26.4, 30.5 and 35.1 eV. The width of the first two peaks is the same (2.04 eV) as that of the main 3ax peak at 18.2 eV while the other two peaks at 30.5 and 35.1 eV each have a width of 3.2 eV. As can be seen, the Gaussian peaks decrease in size with increasing angle at each binding energy, demonstrating that the satellite intensity is "s-type" and therefore associated with ionization from the 3a, (totally symmetric) orbital. (See also section 7.5). The sum of the binding energy spectra taken at <f>—0° and </>=6°, which highlights contributions from both the 2t2 and 3ai orbitals, is compared with the ADC(3) and ADC(4) Green's function calculations of the expected pole strengths in figure 7.30. The Green's function calculations have been folded with the experimentally determined widths (see above discussion). Quite good qualitative agreement is observed for peak positions and intensities. Although the ADC(4) shows more splitting of the pole strength, the spectral envelopes of the ADC(3) and ADC(4) poles are very similar. The prediction that the pole strength in the region of 25 eV is due dominantly to the 3a^ ionization process is consistent with our assignment based on the observed intensity variation with d> (figures 7.28 and 7.29). Both calculations underestimate the higher energy structure Chapter 7. Silane 113 1 0 2 0 3 0 4 0 5 0 6 0 B I N D I N G E N E R G Y (eV) Figure 7.28: Binding energy spectra for the valence shell of SiH.) at azimuthal angles 4>—Q" and 6". Solid line is a sum of Gaussian functions centred on peak locations using experimental widths and convoluted with the instrumental energy resolution. (See text for other details of peak widths and positions). Chapter 7. Silane 114 S1H INNER VALENCE SATELLITES 20-40 eV B I N D I N G E N E R G Y ( e V ) Figure 7.29: Binding energy spectra of the inner valence region of SiH 4 . (a) sum over angles $=0",2",4",6"; (b)-(h) spectra at $=0",2",4",6" 8",10" and 15" respectively. The solid line is a sum of Gaussian functions (dashed lines). Chapter 7. Silane 115 1 .2 1 MANY-BODY GREEN'S FUNCTION CALCULATION ADC (3 ) 1 2t. 2 3a," (c) 1 1 I . I 1 1 10 1 20 .... . -30 1 40 • • i 50 i — • — 60 - 1 2 MANY-BODY GREEN'S FUNCTION CALCULATION ADC(4) 1 2t. 2 3a," (d) 10 20 30 40 50 60 10 20 30 40 50 60 BINDING ENERGY (eV) Figure 7.30: Comparison of experimental and calculated binding energy spectra of SiH4. (a) EMS spectra, sum of <£=0" and 6°, this work; (b) PES spectrum at hi> = 100 eV from ref. [181]; (c) and (d) many-body Green's function calculations, this work. Chapter 7. Silane 116 in the binding energy spectrum as has been observed earlier in the case of the diatomic halogens [173,174,175]. Also shown in figure 7.30(b) is the recently reported photoelectron spectrum of SiH4 at a photon energy of hf=100 eV [181]. The general features of the EMS and PES spectra are seen to be in agreement, with intensity variations reflecting the difference in the two ionization mechanisms. In particular, the intensity distribution is similar in the EMS and PES spectra throughout the inner valence binding energy range (16 - 60 eV) consistent with the present assignment of this region to final states associated with the 3a]"1 process. Table 7.11: Measured and calculated binding energies (eV) for the valence shell of SiH4 State Experimental vertical IP Calculation EMS PES PES RHF MRSD-CI" Green's function" this work refs. [180,128] ref. [181] STO-3G 58-GTO 126-GTO 126-G(CI) ADC(3) ADC(4) 2to 3a i 12.8 18.2 24.2 26.4 30.5 35.1 12.82 18.17 12.8 18.2 11.46 13.23 13.24 12.87 (0.87) 18.72 19.79 19.86 18.63 (0.78) 12.83 (0.90) 12.49 (0.895) 25.74 (0.005) 21.01 (0.002) 30.28 (0.009) 22.74 (0.003) 23.23 (0.005) 24.53 (0.005) 25.98 (0.005) 26.52 (0.007) 18.21 (0.76) 17.78 (0.723) 25.43 (0.16) 24.18 (0.137) 24.72 (0.047) 26.57 (0.004) "The spectroscopic factor is shown in parentheses. Chapter 7. Silane 118 7.4 Comparison of Experimental and Calculated Momentum Distributions Experimental momentum profiles (XMPs), calculated momentum distributions (MDs), and calculated ion-molecule overlaps (OVDs) are shown in figures 7.31(a) and 7.32(a). The XMPs, MDs and OVDs are on a common intensity scale. A single point normal-ization of the OVD calculation to the 2t2 orbital has been used and all other relative normalizations have been preserved. The OVDs have been normalized to unit area and both the OVDs and the MDs have been multiplied by the orbital occupancies. Hence the OVDs and MDs all integrate to give the orbital occupancy. The experimental data for the two orbitals have been placed on a common intensity scale using peak areas from the binding energy spectra. As discussed above (section 7.3) all the inner valence satellite intensity has been assigned to the 3a^1 process. The instrumental momentum resolution (Ap = 0.15 au HWHM) has been folded into the calculations. Position (in the xz plane) and momentum (in the pxpz plane) density contour maps for the silane molecule are shown for each orbital in figures 7.31(b),(c) and 7.32(b),(c). These maps were calculated using the 58-GTO wavefunction and all dimensions are in atomic units. In the position density maps, the Si nucleus is located at the origin and the hydrogen nuclei are at approximately (±2.29, 0, 1.62) (in the picture) and ( 0, ±2.29, -1.62) (out of the picture). The simplest MO description of the 2t2 orbital of silane is that it is the bonding combination of a 3p orbital on Si with the Is orbitals on the hydrogen, where two of the hydrogen s functions are positive and two negative as appropriate to bond with the Si 3p. With this in mind, the electron density in the planes shown in the maps in figures 7.31(b) and 7.32(b) may be interpreted as follows. In figure 7.31(b), the various p contributions Chapter 7. Silane 119 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION E-t——i in E— 0 (a) 2t S i H ST0-3G 2 58-CTO 3 1Z6-CT0 4 126-G(CI) 0 1 2 3 MOMENTUM (au) MOMENTUM DENSITY : M : Si H4 2t 2 • (Mm IgC) • v^i . q* o (1.0.0) - 8 . 0 -4.0 0.0 4.0 8.0 0.5 1.0 MOMENTUM (au ) POSITION DENSITY 4~ Si H4 2» 2 1 1 .a" | o" , - 8 . 0 -4.0 0.0 4.0 8.0 0.5 1.0 POSITION ( au ) F i g u r e 7 .31 : Measured and calculated spherically averaged momentum distributions for the 2t; orbital of SiH 4 . The solid circles represent the direct measurements. Open squares are obtained from the re-spective 2t2 and 3ai intensities in the binding energy spectra shown in figure 7.28. Density contour maps in momentum and position space are also shown. The contour maps were generated with the STO-3G calculation. The contour values represent 0.02,0.04,0.0C,0.0S,0.2,... ,20,40,60 and 80% of the maximum density. The side panels (top and right side) show the density along the dashed lines (horizontal and vertical) in the density map. Chapter 7. Silane 120 SPHERICALLY AVERAGED MOMENTUM DISTRIBUTION SlH 1 STO-3G Z 58-GTO 3 126-CTO 4 136-G(CI) MOMENTUM DENSITY ' 0 1 2 3 MOMENTUM (au) -8.0 -4.0 0.0 4.0 8.0 0.5 1.0 MOMENTUM (au) POSITION DENSITY Si H4 q A 3a 1 A d . , . . | . , . . o J CO ] o o o p -V -°! ! o" , p co (1.0.0) Si H4 3a 1 -8.0 -4.0 0.0 4.0 8.0 0.5 1.0 POSITION (au) Figure 7.32: Measured and calculated spherically averaged momentum distributions for the 3aj orbital of SiH 4 . The solid circles represent the direct measurements. The open square is obtained from the re-spective Hi and 3aj intensities in the binding energy spectra shown in figure 7.28. Density contour maps in momentum and position space are also shown. The contour maps were generated with the ST0-3G calculation. The contour values represent 0.02,0.04,0.06,0.08,0.2,... ,20,40,G0 and 80% of the maximum density. The side panels (top and right side) show the density along the dashed lines (horizontal and vertical) in the density map. Chapter 7. Silane 121 on the silicon He primarily along x=y, so that one lobe of the silicon p is to the left coming out of the paper and one lobe to the right going into the paper. Because the silicon p passes through the plane of the paper, the silicon contribution to the orbital is somewhat de-emphasized by this diagram (except near sc=0, 2=0). The s functions on the hydrogens on the left and right side of the diagram can be seen to be of opposite sign by the nodal surface between them. The slight asymmetry observable in the diagram about x=0 is an artifact of the calculation. The 3aj orbital can be thought of in the simplest terms as the Si 3s orbital bonding with all of the H Is orbitals. The large amount of density near x=0, 2=0 in figure 7.32(b) indicates that the Si is the main contributor to this orbital. It is interesting to note that outside of about 3 au in position space, the 3ai orbital appears roughly spherical. This is not so in momentum space where it can be seen that the orbital is spherical at low p. This is a manifestation of the fact that in position space, the way the nuclear geometry information is stored in the wavefunction (that is, the place in which the nuclei most drastically influence the electron density) is concentrated near the nuclei which are near the origin of position space. This contrasts with momentum space where this information is Fourier-transformed out to all space. Returning to figures 7.31(a) and 7.32(a), we see both the 58-GTO and 126-GTO SCF calculations give identical MDs so it would appear that these calculations are at the Hartree-Fock limit for momentum distributions. Further, the ion-molecule overlap calculations (126-G(CI)) are indistinguishable from the THFA results for each orbital. There is a slight discrepancy between the calculations and the experimental results for the 2f 2 orbital at low momentum. The theoretical results predict too little intensity at Chapter 7. Silane 122 low momentum and slightly too much intensity at high momentum. If alternatively the calculation is height normalized on the 3ai experimental results, then excellent agreement for shape is found for the Zax XMP. Such a renormalization would change the comparison of the theory to experiment for the 2£2 momentum profile. With this normalisation, the theory would fit the data quite well above 0.7 au but have too little intensity below this point. The intensity normalization as shown in figures 7.31 and 7.32 is however preferred since loss of intensity is not entirely unexpected for an inner valence process such as the 3a]"1 due to distortion effects which will be more prominent for deeply bound orbitals [135]. Furthermore any 3a]"1 poles existing beyond the binding energy limit of 60 eV will result in a net loss of intensity for this process in the present study. In chapter 8 it is shown that the slight discrepancy for the shape of the 2t2 momentum distribution can be removed by redefining the momentum resolution function. 7.5 M o m e n t u m D i s t r i b u t i o n s o f S a t e l l i t e S t r u c t u r e Using the areas of the Gaussian peaks calculated to fit the satellite structure of the inner valence region at each angle d> (fig. 7.29), experimental momentum profiles were determined for the bands at 24.2, 26.4, 30.5 and 35.1 eV. These XMPs are shown in figure 7.33 along with shape of the calculated MD for the 3ai orbitals. Separate height normalization of the MD is chosen in each case. All of the satellite structure is clearly seen to be predominantly s type, suggesting that the observed intensity in the binding energy range 20 - 40 eV arises dominantly from the 3a]"1 ionization process. A small but significant shape mismatch occurs in the case of the most prominent satellite at 24.2 eV and possibly also in that at 30.5 eV (figures 7.33(a) and (c) respectively). The differences are reasonably consistent with the shape of the p type SiH4 2£2 outer valence orbital. Chapter 7. Silane 123 The ADC(3) and ADC(4) calculations (table 7.11) suggest that such small contributions from the 2t process are occurring in these energy regions. If this is the case then the normalization of theory and experiment as used in figures 4 and 5 would be changed in a direction that would further improve the quantitative agreement between the calculated MDs and experiment. Alternatively the differences in shape for the satellites could be due to unaccounted for initial state correlation effects. Chapter 7. Silane 124 SlH 4 INNER VALENCE XMPs 24.2 eV — 1 2 6 - G T O 26.4 eV — 1 2 6 - G T O J i _ (c) (d) 2 ° 0 30.5 eV — 1 2 6 - G T O ' — i -35.1 eV — 1 2 6 - G T O MOMENTUM (au) Figure 7.33: Measured and calculated momentum distributions for the inner valence satellites of SiH4 at the indicated binding energies. Experimental points (solid circles) are given by the respective peak areas at the indicated binding energies from the spectra in figure 7.29. The 126-GTO calculation (given by the solid line) is individually height normalized in (a)-(d). Chapter 8 Review of the Hydrides It ranks you far beneath him that you seek to estabhsh the excep-tions while he seeks to estabhsh the rule. — Friedrich Nietzsche All of the hydrides of the first row atoms C through F and the second row atoms Si through Cl as well as the noble gases Ne and Ar have been studied by high resolution electron momentum spectroscopy [28,29,37,38,39,40,167,182]. With the appearance of this work, calculations of the momentum distributions for all of the above systems using restricted Hartree-Fock calculations very close to the Hartree-Fock limit as well as cal-culations of the ion-neutral overlap employing multireference singly and doubly excited configuration interaction calculations of the ion and neutral wavefunctions have been completed. The completion of this systematic investigation provides an opportunity to review the results of a study of a series of molecules which has been carried out in an experimentally and theoretically consistent fashion. Recent work has suggested that the application of the MAGW resolution folding technique can eliminate some discrepancies between experimental and theoretical mo-mentum distributions [183,184]. The MAGW method assumes a Gaussian distribution for the error in 9 and <f>, instead of the usual assumption of a Gaussian distribution in p. 125 Chapter 8. Review of the Hydrides 126 The application of this model to a large number of systems would be of interest. Such calculations have been carried out in this work. The purpose of this chapter is to: (1) present new ion-neutral overlap calculations for the valence orbitals of HF, HC1, Ne and Ar; (2) present the hydride results folded with the MAGW procedure; and (3) review and discuss the theoretical and experimental results for the momentum distributions of the first and second row hydrides. 8.1 Calculations Restricted Hartree-Fock (RHF) and multireference singly and doubly excited configura-tion interaction (MRSD-CI) calculations have been performed using large basis sets of Gaussian type orbitals (GTOs) for all of the molecules discussed in this chapter [28,29,37, 38,118,167,182]. The calculations on HF, HC1, Ar and Ne in this chapter are being pre-sented for the first time. The methodology is similar to that for the other molecules [118]. General aspects of the methods are discussed below, followed by specific comments for each molecule. The basis sets consist of relatively large numbers of s- and p-symmetry Gaussian-type orbitals which are energy optimized for the ground state of the appropriate atom. The choice of d and / functions is discussed in the text devoted to each molecule. Two types of CI procedures were followed: Hartree-Fock singles and doubles (HFSD-CI) and multireference singles and doubles (MRSD-CI). Valence electron CI convergence has been shown to be improved when the Hartree-Fock virtual orbitals are transformed to K orbitals [119,120,121] hence K orbitals obtained from the RHF calculation of the Chapter 8. Review of the Hydrides 127 neutral molecule were used as the molecular orbital basis in the CI. In the CI calculations, the core orbitals were kept frozen. The large size of the basis sets used in this study caused the total number of singly and doubly excited configurations to exceed that number which could be handled variation-ally. All configurations singly excited relative to any function in the reference space were kept; but the doubly excited configurations were energy selected based on second order Rayleigh-Schrodinger perturbation theory (RSPT). The reference configurations for the MRSD-CIs were chosen on the basis of the size of the configuration's coefficient in an HFSD-CI, employing a coefficient threshold of 0.030 unless otherwise stated. The subset of all the configurations having an RSPT energy contribution of at least 10 - 6 hartrees was treated variationally. The CI calculations on the ionic states were performed using the same molecular orbital basis and geometry as for the neutral molecule. 8.1.1 H F The basis set used in the calculations on HF consists of a large number of s and p-symmetry Gaussian-type orbitals (GTOs) which are energy optimized for the ground state of the respective atoms using an even-tempered restriction on the exponents. Very diffuse functions are included in the basis set. The basis set contained 117 functions with the contraction scheme denoted as (23s, 12p, 6d, 2/) —> [14s, lOp, 6d, 2/] on the F, and (10s, 4p, 2d) —• [6s, 4p, 2d] on the H. On the fluorine, the d exponents were chosen in an even-tempered fashion and the / exponents were based on the work of Sekino and Bartlett [185] 1 . The hydrogen d exponents were taken from the work of Davidson and J We used 3.7 and 1.1, while Sekino and Bartlett used 3.7 and 0.925 Chapter 8. Review of the Hydrides 128 Feller [144]. The s component of the Cartesian d's and the p component of the Cartesian /'s were removed to avoid linear dependency problems. Three calculations were performed with this basis set. These were a restricted Hartree-Fock calculation and two multireference singles and doubles CI calculations. In the case of the calculation which was used to generate the overlap distribution (figure 8.38) the lo core was kept frozen. Configurations were energy selected based on Rayleigh-Schrodinger perturbation the-ory and reference configurations for the MRSD-CIs were chosen from an HFSD-CI. For the neutral molecule, more than 98% of the estimated singles and doubles correlation energy of this basis set was variationally recovered by the chosen configurations. This represents about 87% of the experimental correlation energy. All calculations were carried out at the experimental ground state bond length of 0.9170A, 1.7328a.u. [186,187]. The calculated properties of HF are listed in tables 8.12 and 8.13. 8.1.2 HC1 The basis set for HC1 consists of an energy optimized, even-tempered primitive set of Gaussians (21a, Up, Ad, 2//10s, 3p, 2d) contracted to [12s, lOp, Ad, 2f/6s, 3p, Id}. The d and / exponents for CI are taken from table III in ref. [118]. These exponents have been estimated based on energy optimized exponents for sulphur. The s-components of the cartesian d functions and the p-components of the / functions were removed to avoid linear dependence, forming the final 100 GTO basis. Chapter 8. Review of the Hydrides 129 The MRSD-CI space was chosen from an HFSD-CI calculation, as discussed in sec-tion 8.1. A coefficient threshold of 0.030 was systematically maintained for the ion states 5s - 1 and 2p _ 1, but a higher one was needed for calculation of multiple roots of 2E+ symmetry (5s - 1 and 4s - 1 included). The 2 £ + multi-root calculation was done in C2v symmetry, but employed a symmetrically closed set of configurations containing the largest coefficient contributions from a calculation in which all the single excitations for all valence hole states were included. The calculations on HC1 were performed at the experimental equilibrium bond length of 1.2746 A, 2.4086 bohr [188,189]. A previous calculation on HC1 [118] has been per-formed using the same methodologies presented here and a shghtly larger 107 GTO basis set. The energies and properties of the present calculations are compared with the previous results in table 8.14 and found to be almost identical. 8.1.3 Ne The basis set for Ne consists of an even-tempered (28s, 17p, 8d, 3/) primitive set con-tracted to [13s, Up, 5d, 3/] Gaussian type basis. The s-components of the cartesian d functions and the p-components of the / functions were removed to avoid Hnear depen-dence, forming the final 92 GTO basis. This basis is based upon the infinite basis for the low lying states of atomic hydrogen, as discussed in an earHer paper by Feller et. ai. [118]. The hydrogenic states were determined as a function of the orbital scaling parameter 8 of the even tempered sequence. The even tempered basis set, even at the Hmit of an infinite number of basis functions, does not give the exact energies for a fixed value of 8 different from unity. A minimum in the choice of the value of 8 was computationaUy Chapter 8. Review of the Hydrides 130 set at 2.0 due to the number of significant figures in the two electron integrals. The specific basis functions for neon were then chosen using a criteria on the exponents such that the truncation error was limited to twice the infinite basis error for B = 2.0. The criteria required an uncontracted basis of 28s, 17p using smallest exponent of 0.03125. This basis gives an energy of -128.547096 au for a restricted Hartree-Fock calculation, compared to the numerical Hartree-Fock value of -128.547098 au [190]. Contraction of the s and p functions raised the SDCI energy from -128.7345 au to -128.7330 au. Eight primitive d functions, with exponents (8.0,4.0,... 0.125,0.0625), and three / functions with exponents (4.0,2.0,1.0), were added. The d functions were contracted using SDCI natural orbitals with a 1,1,1,1,3 contraction scheme. The d contraction raised the SDCI energy from -128.8845 au to -128.8839 au. Considering the large number of functions used, the wavefunction should be insensitive to the specific choice of exponents. For the calculation of the neutral molecule, the MRSD-CI space was chosen as outlined in section 8.1. The MRSD-CI energy is -128.8936 au compared to the estimated CI limit of the basis set of -128.8976 au. We estimate the nonrelativistic, infinite mass atomic energy to be -128.9400 au [118]. This gives a correlation energy of -0.393 au, which is comparable to the estimate of Veillard and Clementi [191] of-0.390 au.2 The difference is due largely to the inclusion of a Lamb shift correction and a slightly different relativistic correction, as discussed in the appendix of reference [118]. Using this estimate, our calculations recovered 88% of the total correlation. It should be noted that previous studies have found some amount of the correlation energy is in the functions g and higher [192,193,194,195]. An HFSD-CI valence calculation gave an L-shell correlation 2The value -0.390 au is based on table IX of ref. [191]. This includes the Lamb correction. The quantities EC(2) given in table X of ref. [191] are apparently in error, due to an inadvertent reversal of the sign of the Lamb corrections. Chapter 8. Review of the Hydrides 131 energy of -0.2818 au which is close to the estimate of Sasaki and Yoshimine for the total correlation energy of the L-shell contained in the functions s through / , which is -0.2870 au [196]. The non-zero values for the quadrupole moment were induced due to the program limitation of only D2h symmetry, which does not maintain spherical symmetry for the atom during the configuration selection from the initially closed set of configurations used for the reference space. The calculation of the ion states involved a similar process to that of the neutral with the exception that no RHF calculations were done. The coefficients for the MRSD-CI were chosen based upon the coefficient contribution using a threshold of 0.03, requiring a symmetrically closed set of configurations containing the largest coefficient contributions for the 2p - 1 and 2s _ 1 hole states. 8.1.4 Ar Initial calculations on Ar employed an even-tempered (21s, 14p, Ad, 2/) —> [12s, lOp, Ad, 2/] basis which we will refer to as the 76 GTO basis. Close examination of the composition of the CI states with 2 5 symmetry revealed that the basis set functions were not sufficiently diffuse to describe the excited states of the ion; the excited states being composed mostly of atomic 3d, 4s, and Ad orbitals. Using a He + hydrogenic model, it was determined that an exponent of 2.5-10-4 was necessary to correctly characterize the Ad (hydrogenic) orbital. The need for very diffuse d functions on Ar and Ar + has recently been reported by Stark and Peyerimhoff [197]. Chapter 8. Review of the Hydrides 132 They also found that the d functions can be contracted without a loss of accuracy [197]. A larger basis set was developed consisting of (23s, 15p, 9d, 2/) functions contracted to [14s, lip, 6d, 2/]. In this 91 GTO basis, the nine d function exponents were 3.906, 1.563, ...0.00256 contracted with a 1,1,1,1,2,3 contraction scheme. The / function exponents were 1.4 and 0.5. The smallest s exponent was 0.015 and the smallest p exponent 0.020. In studying the intial RHF calculation, it was observed that the use of K orbitals destroyed the diffuseness of the outer orbitals. The desirable property associated with K orbitals is the faster convergence of these orbitals over canonical orbitals [119,120,121]. A three step procedure was devised to avoid the loss of the diffuseness of these orbitals, while retaining the favourable convergence properties. First a standard RHF calculation was performed. Secondly, improved virtual orbitals (IVO) [198] were determined, chosing the hole state to be the 2 5 state of interest in Ar. During the IVO calculation, the unoccupied 3s orbital was held frozen. Thirdly, the occupied and virtual orbitals of interest (3d, 4d, 4p and 4s) were held frozen and K orbitals were generated for the rest of the virtual space. This procedure maintained the diffuseness in the valence and virtual regions of interest, but also allowed for fast convergence for the remaining virtual space. Chapter 8. Review of the Hydrides 133 Table 8.12: Calculated properties of hydrogen fluoride" P R O P E R T Y SCF MRSD-CI Expt. (a.u.) 117-GTO Numerical 117-G(CI) 117-G(CI) Hartree-Fock (all electrons) (frozen core) this work ref. [190] this work this work energy -100.0706 -100.0708 -100.4251 -100.3661 -100.460" -100.530c dipole moment < qz > -0.7562 -0.7561 -0.7143 -0.7106 -0.71846d quadrupole moment BZI 1.7324 1.7321 . 1.7043 1.7017 1.75(2)' < qz/r3F > 0.0025 0.0003 0.0017 0.0 < qzlr% > -0.0201 0.0035 0.0053 0.0 el. field grad. qp 2.8054 2.7988 2.6678 2.6570 2.5' el. field grad. qn 0.5178 0.5119 0.5384 0.5405 0.527" "Calculations performed at experimental geometry, JJKF=0.9170A, 1.7328 bohr [186,187]. 'Experimentally derived, nonrelativistic, nonvibrating, infinite nuclear mass energy [118]. cRef. [199]. ''References [200,201]. 'Experimentally derived, non-vibrating value from ref. [202], 'Based on experimental and theoretical investigations using the 5/2+ state of 1 9 F ; the experimental measurements were made by the proton beam induced, nuclear radiative resonance technique of per-turbed angular correlation. The value for qF is derived from the experimental result (eqQ)p = 39 MHz in solid HF [203] and Qf — 6.76 • 10~26 cm2. This value for QF is derived from the experimen-tal value (eqQ)F = 77.9 MHz in solid FC1 [204] and a theoretical calculation on FC1 which gives qF = 4.908 au [205]. "Based on {eqQ)D - 0.354238 MHz, ref. [200] and QD - 2.860 • 10"27 cm2, ref. [206]. Table 8.13: Measured and calculated ionization potentials (eV) for the valence shell of HF Experimental Expt. SCF MRSD-CI a Dominant ion states Label ref. [148] 117-GTO 117-G(CI) in MRSD-CI l?r 16.12 17.7 16.1 (0.88) l T T " 1 3cr 19.89 20.9 20.0 (0.89) 3 0 - 1 2a 39.65 43.6 36.5 (0.11) I f f - 1 + Iff -> 3s Rydberg; 11% 2c-" 1 40.1 (0.67) 69% 2a'1 42.6 (0.03) 3<r - 1 + 3o- -» 3s Rydberg; 3% 2o--] "The spectroscopic factor is shown in parentheses. Core electrons were kept frozen in the calculation. Chapter 8. Review of the Hydrides 134 Table 8.14: HC1 properties P R O P E R T Y (a.u.) R H F 100-GTO this work R H F 107-GTO ref. [118] MRSD-CI 100-GTO this work MRSD-CI 107-GTO ref. [118] Expt. Energy -460.1112 -460.1123 -460.3415 -460.5578 -460.886c SACs kept 1 1 13969 39451 SACs generated 1 1 787887 2045002 dipole moment p 0.4767 0.4767 0.4479 0.4570 0.43615(12)*' quadrupole moment 6ZZ 2.8078 2.8081 2.7019 2.7339 2.78(9)e < r2 >e 34.1389 34.1387 34.0259 34.0287 < P4 >e 891222 889918 891264 890171 < &H >e 0.3866 0.3866 0.3941 0.3932 < Sd >e 3184.59 3186.17 3184.68 3186.20 < l/rH >e 8.0010 8.0010 8.0038 8.0081 < l/rci >e 64.8207 64.8208 64.8347 64.8208 < qz/r*B > 0.0054 0.0054 -0.0033 0.0003 0.0 < q*/rcl > 0.0013 0.0019 -0.0003 0.0019 0.0 0.2803 0.2801 0.2857 0.2829 qci 3.6002 3.5999 3.4515 3.5283 "Calculations performed at experimental equilibrium bond length 1.2746A, 2.4086 bohr [188,189]. 6The properties (in atomic units) are defined by: /x =< >; 6ZZ — 0.5 < Y>qdZz\ — rf) >, where the sum runs over nuclei and electrons; < r 2 > e=< 5>,2 > and < p 4 >c=< Sp 4 >, where the sums run over electrons only; < c5n >c=< L6(rjn) >, where the sum runs over all electrons for a given nucleus n; qn= the field gradient at nucleus n. cExperimentally derived, nonrelativistic, nonvibrating, infinite nuclear mass energy, ref. [118]. dRef. [202,207]' eExperimentally derived, nonvibrating, ref. [202] Chapter 8. Review of the Hydrides 135 Table 8.15: Measured and calculated binding energies (eV) for the valence shell of HC1° State Theory Experiment RHF MRSD-CI Green's Functionb EMS PES this work this work ref. [208] ref. [39] ref. [209] 2TT 12.98 12.55 (0.8533) 12.66 (0.92) 12.8 12.8 39.24 (0.011) 42.43 (0.011) 5cr 17.03 16.54 (0.8620) 16.67 (0.92) 16.5 16.6 45.98 (0.012) 24.36 (0.2219) 23.65 24.50 (0.0072) 30.39 26.85 (0.3437) 25.67 (0.13) 25.8 (0.51) 25.85 30.08 (0.0002) 28.32 (0.57) 28.5 29.8 31.28 (0.0463) 32.43 (0.22) 32.8 (0.37) 32 32.48 (0.0040) 33.2 33.94 (0.0157) 34.65 36.44 (0.0253) 36.90 (0.2372) 37.8 (0.12) 37.41 (0.0034) 58.18 (0.013) 61.33 (0.014) "Spectroscopic factors are given in parentheses. ^Ionization energies of outer valence orbitals calculated by OVGF-method [48,68] using [12s, 9p, bd, If/4s, lp] —* (7s, bp, bd, If/2s, lp) basis set. Ionization energies in the inner valence region are calculated by the 2ph-TDA [48,69,131] using basis set [12s,9p/4s] -+ (6s,4p/2s). Chapter 8. Review oi the Hydrides 136 Table 8.16: Ne properties PROPERTY (a.u.) RHF this work Numerical HF ref. [190] HFSD-CI this work MRSD-CI this work Expt. Energy -128.5471 -128.5471 -128.8836 -128.8936 -128.9400° SACs kept 1 5964 18681 SACs generated 1 13570 1271622 quadrupole moment 0ZZ 0.00000 -0.00065 -0.00014 0.00000 < r2 >e 9.37186 9.37184 9.53369 9.54426 9.61(ll)b <P4 >e 98314.2 98409.3 98408.5 < 1>Ne >e 618.198 618.141 618.133 31.113 31.113 31.111 31.109 °Ne 0.00000 0.00072 0.00001 0.00000 "Nonrelativistic, infinite mass atomic energy, ref. [118]. *Ref. [210]. Table 8.17: Measured and calculated binding energies (eV) for the valence shell of Ne State RHF Numerical HF MRSD-CIa Expt. this work ref. [190] this work ref. [211] 2p 23.14 23.14 21.44 (0.9104) 21.559 2s 52.52 52.53 48.45 (0.8715) 48.476b aThe spectroscopic factor is shown in parentheses. ''Obtained by adding the 2p6 1 S o energy of Nel to the 2s2p6 energy of Nell. Chapter 8. Review of the Hydrides 137 Table 8.18: Ar properties P R O P E R T Y (a.u.) RHF 76 GTO this work RHF 79 GTO ref. [212] Numerical HF ref. [213,214] MRSD-CI 76 GTO this work MRSD-CI 91 G T O ° this work Expt. Energy -526.8169 -526.8174 -526.8175 -527.0531 -527.0370 SACs kept 1 1 6486 9328 SACs generated 1 1 502434 345202 < r 2 > e 26.0351 26.0352 26.0344 25.9743 26.1753 24.39(19)* < SAr >e 3799.8 3839.8 3800.0 3799.9 < l/rAr >e 69.7253 69.7248 69.7256 69.7344 69.7310 "Basis set: (23s, 15p, bd, 2/) [14s, l i p , 6d, 2/]. 'From diamagnetic susceptibility measurements, ref. [215]. Note that this experimental value is in doubt, see references [210,216]. Table 8.19: Measured and calculated binding energies (eV) for the valence shell of Ar State RHF Numerical HF MRSD-CI " Dominant ion configuration Expt. 91 GTO 91 GTO in MRSD-CI this work ref. [213,214] this work ref. [217,218,219] 3 P 16.08 16.08 15.58 (0.8737) 3s 23p 5 15.82 3s 34.76 34.76 29.53 (0.5629) 29.30 34.40 34.88 35.63 37.19 (0.0038) 3s 23p 44s 36.48 37.23 39.29 (0.0014) 3s 23p 43d 38.06 38.62 39.63 39.71 (0.1113) 3s 23p 44ci 41.23 41.80 42.56 (0.0705) 3s 23p 45ci 42.69 43.32 (0.0167) 3s23p46<i 43.49 (0.0032) 3s 23p 47ci 43.84 (0.0608) 3s 23p 45s 43.47 45.26 (0.0011) 3s 23p 48d "The spectroscopic factor is shown in parentheses. Chapter 8. Review of the Hydrides 138 Table 8.20: Experimental and calculated valence ionization potentials (eV) Molecule Orbital Expt." K T " MRSD-CI C IP IP IP S d KTIP-C I IP e C H 4 1*2 14.2 14.54 14.3 0.8721 0.24 2ai 23.05 25.18 23.3 0.7776 1.88 N H 3 3a i 10.87 11.69 10.94 0.8744 0.75 le 16.53 17.11 16.50 0.8781 0.61 2d! 27.74 31.07 28.49 0.4818 2.58 H 2 0 12.62 13.89 12.39 0.869 1.50 3oi 14.74 15.92 14.76 0.882 1.16 16a 18.51 19.53 18.79 0.888 0.74 2 a i 32.2 36.82 33.1 0.4394 3.72 HF lTT 16.12 17.69 16.10 0.8802 1.59 3(7 19.89 20.90 20.01 0.8937 0.89 2a 39.65 43.55 40.09 0.6724 3.46 Ne 2p 21.56 23.14 21.44 0.9104 1.70 2s 48.48 52.52 48.45 0.8715 4.07 S iH 4 2t2 12.82 13.24 12.87 0.8662 0.37 3oj 18.17 19.86 18.63 0.7834 1.23 P H 3 5a! 10.58 10.58 10.44 0.8626 0.14 2e 13.6 14.17 13.62 0.8399 0.55 4ai 19.4 23.23 19.70 0.4373 3.53 H 2 S 2bx 10.5 10.48 10.26 0.843 0.22 5a i 13.45 13.68 13.57 0.855 0.11 2b2 15.5 16.13 15.58 0.834 0.55 4a i 22.0 26.77 23.00 0.2884 3.77 HC1 2TT 12.8 12.98 12.55 0.8533 0.43 ba 16.6 17.03 16.54 0.8620 0.49 4a 25.85 30.39 26.85 0.3437 3.54 Ar 3P 15.82 16.08 15.58 0.8737 0.50 3s 29.30 34.76 29.53 0.5629 5.23 "Vertical IPs are listed. References are given in table 8.21 6Koopman's theorem IP based on calculations using extended Gaussian basis sets. This work. cRelative energy of the neutral molecule to the main pole of the cation having main configuration q (where q is the orbital listed in column two), as calculated by MRSD-CI. This work. d The spectroscopic factor of the main pole, as calculated by MRSD-CI. This work. e The difference in the IP as calculated by Koopman's theorem and the IP based on MRSD-CI. Chapter 8. Review of the Hydrides Table 8.21: References for experimental ionization potentials Molecule Reference C H 4 [148] N H 3 [138,137,220] H 2 0 [221,222] HF [148] Ne [211] S iH 4 [180,128] P H 3 [29] and refs. therein H 2 S [223] HC1 [209] Ar [218] Chapter 8. Review of the Hydrides 140 Table 8.22: Maximum height of the calculated MDs. The third column shows the per-centage difference in the height of the MD calculated from the ion-neutral overlap as compared to the THFA result.  Molecule Orbital RHF rriaximum intensity MRSD-CI maximum intensity (CI-RHF)/RHF % C H 4 l t 2 1.2275E-01 1.2308E-01 0.27 2o x 6.9229E-01 7.0741E-01 2.18 N H 3 3aj 1.0495E-01 1.1212E-01 6.83 le 8.7043E-02 8.7087E-02 0.05 2oi 4.3804E-01 4.3701E-01 -0.23 H 2 0 16i 6.1128E-02 6.6429E-02 8.67 3ai 6.4382E-02 6.8767E-02 6.81 1&2 6.2876E-02 6.3068E-02 0.31 2oi 2.8507E-01 3.1373E-01 10.05 HF l7T 3.7139E-02 3.9298E-02 5.81 3cr 4.3731E-02 4.6501E-02 6.33 2cr 1.8724E-01 2.0770E-01 10.93 Ne 2p 2.2568E-02 2.3555E-02 4.37 2s 1.2267E-01 1.2973E-01 5.76 S iH 4 2t2 2.3350E-01 2.3420E-01 0.30 3oi 8.5032E-01 8.4971E-01 -0.07 P H 3 5oj 2.4794E-01 2.4787E-01 -0.03 2e 1.8722E-01 1.8768E-01 0.24 4oj 9.8319E-01 9.4215E-01 -4.17 H 2 S 2ox 1.8253E-01 1.8383E-01 0.71 1.6500E-01 1.6872E-01 2.26 2b2 1.4476E-01 1.4492E-01 0.11 40! 7.0358E-01 6.7830E-01 -3.59 HC1 2TT 1.2280E-01 1.2387E-01 0.87 5cr 1.2771E-01 1.2165E-01 -4.74 4cr 5.1546E-01 4.9962E-01 -3.07 Ar 3p 8.4637E-02 8.5004E-02 0.43 3s 3.7132E-01 3.7730E-01 1.61 Chapter 8. Review of the Hydrides 141 8.2 Results of Overlap Calculations Figures 8.34-8.39 show the RHF and MRSD-CI calculations of the momentum distribu-tions of all the hydrides along with the experimental measurements. In all cases except HF and HC1, the results have been folded using the MAGW resolution folding proce-dure [183,184]. This method of resolution folding was not applied to the HF and HC1 data as it was measured on a different spectrometer [39] than the one for which the MAGW method was developed. The theoretical curves for HF and HC1 have been folded with a Gaussian of half width 0.10 au. The experimental data is taken from the following references: CH 4 [182], NH 3 [37], H 2 0 [38], HF [39], Ne [40], SiH4 [167], PH 3 [29], H2S [28], HC1 [39], Ar [40]. Except where noted, the experiments were performed with an impact energy of .En = 1200 eV. The relative intensities of the experimental momentum distributions are taken from the experiment by normalization on the peak areas of the binding energy spectrum, and only one normalization factor is used to make measured values absolute for each molecule. The normalization factor is chosen to give the best fit to the theoretical calculations for that molecule. All of the calculated momentum distributions, whether they are based on the ion-neutral overlap or the target Hartree-Fock approximation, integrate to give the orbital occupancy of the corresponding canonical Hartree-Fock orbital. The units on the y axis in figures 8.34-8.39 are the absolute intensity xlO2. Hence one can compare the relative heights of the calculations for different orbitals within a molecule and between molecules.3 3 The values of the heights of the calculated MDs reported in table 8.20 may differ from those in the figures by a factor of 2 or 3 which reflects the orbital occupancy. Chapter 8. Review of the Hydrides 142 The agreement between the calculated and experimental results when compared using the MAGW resolution folding procedure is, in each case, either similar to or better than that obtained previously. 8.2.1 HF Figure 8.38 shows the experimental data for HF as taken from ref. [39] plotted with the RHF and MRSD-CI calculations of the momentum distributions discussed in sec-tion 8.1.1. The experimental data was measured at two different impact energies, Eo = 400 eV and 1200 eV. The two data sets are in good agreement with each other and this suggests that the PWIA model is obeyed. With the overlap calculation normalized to the height of the measured 3tr distribution, it can be seen that the fit to the shape of the 3CT MD is good. However, considerable discrepancies exist for the other two momentum distributions.4 The RHF calculation of the l7r MD peaks at too high a value of momentum and falls short of the measured intensity. While the CI calculation shows an improvement in both the value of p m a a : and the intensity, this improvement is not sufficient to bring the calculation into agreement with the data. Both the RHF and CI wavefunctions give a reasonably good description of the shape of the 2cr MD, except above ~ 1.5 au. In this high momentum region, it is expected Calculations show that the application of the MAGW resolution folding procedure [183] slightly reduces, but does not eliminate, these discrepancies. Application of the resolution function which appears in ref. [183] would not change any of the conclusions reached herein. In any event, the use of the MAGW resolution folding procedure on these results would be inappropriate as the experimental data was obtained on a spectrometer with a different response function to that modelled in reference [183]. Chapter 8. Review of the Hydrides 143 that the experimental points for such a deeply bound inner valence orbital will be too high due to a breakdown in the plane wave description of the scattering process [135]. The good agreement in the shape is contrasted with a large mismatch in the intensity of the calculated MD and OVD and the measured XMP. The experimental points seem to contain only about 60% of the expected intensity. It seems unlikely that this is due to unmeasured poles in the ionization spectrum, as the inner valence binding energy spectrum of HF appears to be very simple [39,149]. Further, the spectroscopic factors from the CI calculation (table 8.13) predict that at least 90% of the 2<r intensity5 is in three poles, all below 45 eV (well within the the range of the experimental measurement, which extends to 55 eV). Green's function calculations [224,208] make essentially the same prediction as the CI calculation. This discrepancy is discussed further in section 8.3.3.4. 8.2.2 HC1 Previous work on HC1 [39] compared the experimental data to calculations of the ion-neutral overlap and calculations based on the target Hartree-Fock approximation. These used rather small basis sets. With the theoretical curves normalized to the 2ir experi-mental MD, the calculations of the ho and ACT MD S each predicted more intensity than was observed experimentally (see page 88 of ref. [39]). Figure 8.38 shows the experimental data for HC1 as taken from ref. [39] plotted with RHF and MRSD-CI calculations of the momentum distributions discussed in sec-tion 8.1.2. The shape of both the 2cr and 5<r MDs are predicted incorrectly at the 5This value was obtained by adding the 2c contributions as listed in table 8.13 and dividing the sum by the spectral factor of the 3c state, to which the experiment was normalized. 40 \ M R S D - C I " \ i - — R H F 1 ^ T — — . SiH4 3a , — M R S D - C I --• R H F AVERAGE MOMENTUM (au) 0 1 2 3 AVERAGE MOMENTUM (au) Figure 8.34: Momentum distributions of the valence orbitals of CH 4 and SiH4. CH 4 experimental data, as well as RHF and MRSD-CI calculations, are taken from ref. [182]. SiH4 experimental data is taken from ref. [167]. RHF and MRSD-CI calculations for SiH4 are taken from ref. [167] where they were labelled 126-GTO and 126-G(CI) respectively. Momentum distributions have been treated with the MAGW resolution folding method. 20 10 1 1 NH 3 3a, M R S D - C I _ R H F 1 ~*—M •. 40 20 0. 20 i - r -PH 3 A 5a, - A L M R S D - C I " / \ R H F 1 \ V t \ c • 0 1 2 3 AVERAGE MOMENTUM (au) 10 1 i NH 3 K le - V M R S D - C I " R H F i — M R S D - C I - - • R H F i i * 1 2 3 AVERAGE MOMENTUM (au) o 1 2 3 AVERAGE MOMENTUM (au) F igure 8.35: Momentum distributions of the valence orbitals of N H 3 and P H 3 . N H 3 experimental data taken from ref. [37]. RHF and MRSD-CI calculations for N H 3 are taken from ref. [37] where they were labelled 126-GTO and 126-G(CI) respectively. P H 3 experimental data, as well as RHF and MRSD-CI calculations, are taken from ref. [29]. Momentum distributions have been treated with the M A G W resolution folding method. i ' ' ' i v H 2 S ' \ 2b, - i \ MRSD-CI " \ RHF 0 20 10 0 H2S - MRSD-CI -- RHF 4^ AVERAGE MOMENTUM (au) AVERAGE MOMENTUM (au) Figure 8.36: Momentum distributions of the lfci and 3aj orbitals of H2O and the 2ii and 5ai orbitals of H2S. H2O experimental data taken *rom ref. [38]. Circles correspond to measurements of H2O; squares are measurements of D2O. RHF and MRSD-CI calculations for H2O taken from ref. [38] where they were labelled 109-GTO and 109-G(CI) respectively. H2S experimental data taken from ref. [28]. RHF and MRSD-CI calculations for H2S taken from ref. [28] where they were labelled 122-GTO and 122-G(CI) respectively. Momentum distributions have been treated with the MAGW resolution folding method. H 2 0 l b , - M R S D - C I -• R H F AVERAGE MOMENTUM (au) 80 40 0, 1 V i H2S V \ M R S D - C I _ — - R H F i 0 1 2 3 AVERAGE MOMENTUM (au) Figure 8.37: Momentum distributions of the lb? and 2ai orbitals of H2O and the 2b2 and 4ox orbitals of H2S. H2O experimental data taken from ref. [38]. Circles correspond to measurements of H2O; squares are measurements of D 20. RHF and MRSD-CI calculations for H2O taken from ref. [38] where they were labelled 109-GTO and 109-G(CI) respectively. H2S experimental data taken from ref. [28]. RHF and MRSD-CI calculations for H2S taken from ref. [28] where they were labelled 122-GTO and 122-G(CI) respectively. Momentum distributions have been treated with the MAGW resolution folding method. MOMENTUM (au) MOMENTUM (au) MOMENTUM (au) Figure 8.38: Momentum distributions of the valence orbitals of HF and HC1. HF experimental data taken from ref. [39]. Circles correspond to measurements made with an impact energy of Eo = 1200 eV; triangles are measurements with EQ = 400 eV. RHF and MRSD-CI calculations for HF from this work, see section 8.1.1. HC1 experimental data taken from ref. [39]. Circles correspond to measurements made with an impact energy of Ef\ = 1200 eV; triangles are measurements with EQ — 400 eV. RHF and MRSD-CI calculations for HC1 from this work, see section 8.1.2. Theoretical calculations have been folded with a Gaussian of half width 0.10 au. oo -1 A r 3p 30 MRSD-CI RHF 20 i 10 n 0 A r 3s MRSD-CI - — RHF J i _ J _ i _ 1 2 3 0 1 2 3 AVERAGE MOMENTUM (au) AVERAGE MOMENTUM (au) Figure 8.39: Momentum distributions of the valence orbitals of Ne and Ar. Ne and Ar experimental data taken from ref. [40]. RHF and MRSD-CI calculations from this work, see sections 8.1.3 and 8.1.4. The Ar calculations shown used the 91 GTO basis set. Momentum distributions have been treated with the MAGW resolution folding method. Chapter 8. Review of the Hydrides 150 Hartree-Fock limit, and the overlap calculation gives little improvement. New measure-ments of these momentum distributions are in progress. This discrepancy is discussed further in section 8.3.3.1. The disagreement between theory and experiment above 1 au for the 4tr orbital can be confidently ascribed to a breakdown in the plane wave description of scattering. This sort of discrepancy has been observed by Lahmam-Bennani et aJ. for insufficiently high values of the incident energy [135]. 8.2.3 Ar and Ne The data for Ar and Ne shown in figure 8.39 is taken from ref. [40]. This data has been height normalized according to peak areas in the binding energy spectra in ref. [40]. In the original publication, it was shown that momentum distributions calculated from the wavefunctions of Clementi and Roetti [225], which are at the Hartree-Fock limit, agreed well for shape with the experimental data for both valence orbitals in both molecules. The RHF and MRSD-CI calculations discussed in sections 8.1.3 and 8.1.4 are shown in fig. 8.39. The momentum distributions calculated from the RHF wavefunctions presented here are identical to those calculated from Clementi and Roetti's wavefunctions. The overlap calculation is found to be identical to the RHF calculation in the case of Ar, but not so for Ne. While excellent agreement between theory and experiment is obtained for shape, the intensity of the inner valence orbital is overestimated by the theoretical calculation in Chapter 8. Review of the Hydrides 151 each case. This has been found to be due to the failure of the plane wave impulse ap-proximation, as a distorted wave treatment of the theoretical calculations gives excellent agreement with experiment [59,60]. This is further discussed in section 8.3.3.4. 8.3 Overview of the Hydrides 8.3.1 A simple model to explain EMS results In section 8.3.3 we will discuss the level of.theory required to accurately reproduce the EMS results. But it is also of interest to consider what could be invoked as the simplest level of theory that will adequately explain the most basic aspects of the observations. In this section we review a form of simple molecular orbital theory; the basic ideas are taught to all undergraduates under various guises [226]. Within this model, electrons are arranged in orbitals with well defined energy (this is true for any Hartree-Fock model). Spin is of no interest to us, and can be ignored. Given the symmetry, occupation numbers, and energies of the atomic orbitals, we wish to predict the effects of bonding between atoms. In this simple model, a bond between atoms will result in the electrons rearranging themselves; but they will remain in orbitals of well defined energy. Only three rules govern the rearrangement: (1) Atomic orbitals mix only if they have similar atomic energies. (2) Atomic orbitals mix only if they have the same molecular symmetry. (3) The molecular orbitals will form in such a way as to satisfy the symmetry restrictions imposed by the molecular geometry, that is, the molecular orbitals will all be members 14.2 53.0 291 C J L 2p» + C>i + >>a - *i - >>2) 1<2 ~ 2J>K + C>1 + h« - *>j) 2pi + C«i + hj - h , -h 4) J 0 l ~ 2J + (h t + h 2 + hj + h 4) 1<H ~ l i N H , 10. s U.6 38.6 408 3ai 1« 1-1 ~ 1* 2p* +C>1 + h2 + h,) 2p» +(2hi - h ? ->>s) 2py + (h j - h j ) 2» + C»l + h j + h j ) H,0 12.4 14.8 18.8 38.1 640 16! 3ai 2<>1 l a t 2px *Pz +C>1 +/>2) 2P« + (^l - *a ) 2j + (hi + hj) 1* 16.1 19.9 89.7 894 H F l IT 3<7 ~ 2<r ~ Iff ~ 2p» 2p» 2pV + h 7i + h It Ne 21.4 48.6 870 2p 2i 1« 12.8 S i l L H o S 2<2 ~ 18.7 3 d ! ~ 107 l l S ~ 144 2at ~ 1844 lai ~ 3p* + (hi + hj - h« - hj) Spy + (hi + h 4 — hj - hj) 3p« +(hi +h 2 - h , -h«) 3« + (h! + h 2 + h, + h 4) 2p 2i U 10.8 18.6 6a, 2« 19.6 4at 187 8at 187 le 191 2160 2ot 10! Sp* + (h t + h 2 + hj) Sp* + (2hi - hj - hj) Spy + (h2 - hj) ». + (h t + h 2 + h 8) 2p, 2p* Spy 2< l i 10.4 18.8 16.6 22.0 170 170 170 284 2478 26i 6a! 26j 4<H It! 8ai lb] 2«i l a i Sp* Sp* +(hi + h 2 l sPy + ( A l - hi) »i + (hl + hj) »P* 2p* 2py 2< U HC1 12.7 2ir ~ 16.8 6<r 26.8 4cr 207 lir 207 8<r 277 2ff 2829 Iff Sp* Spi Spy + h 3* + h 2p» 2p* 2py 2< 1< A r 16.6 29.3 249 328 3206 3p 3< 2P 2» 1» H i ! o A i l H, Hi H F igure 8.40: The molecular orbitals of all of the hydrides are listed along with their approximate experimental energies in eV. The molecular orbitals are shown as a sum of atomic orbitals. The first term in the sum is the atomic orbital on the heavy atom. The second term is the sum over the hydrogen Is orbitals. The orientation of the hydrogens with respect to the cartesian coordinate system that defines the heavy atom p functions is shown in row three. Valence binding energies from table 8.20. Core binding energies from refs. [227,228]. Chapter 8. Review of the Hydrides 153 of an irreducible representation of the point group to which the molecule belongs. To apply these rules, we need a priori knowledge of the atomic orbital energies and the molecular geometry. Despite the small number and simplicity of the rules, there is a great deal of predictive power in this approach. Let us apply this model to the hydrides. As we are only allowed to work with atomic orbitals, the hydrides are made par-ticularly easy to study by the fact that the hydrogen atom has only one orbital. In those hydrides which have more than one hydrogen atom, the H Is orbitals will mix very strongly because they are all at exactly the same energy. These orbitals will mix in such a way that they will become members of the various irreducible representations of the molecule. They will then combine with the atomic orbitals on the heavy atom according to symmetry and energy considerations. The expected molecular orbitals are shown in figure 8.40, along with their approximate experimental energies in eV. The molecular orbitals are shown as a sum of atomic orbitals. The first term in the sum is the atomic orbital on the heavy atom. The second term, which is written as Tihi, refers to the sum of the Is orbitals on each of the hydrogens, as labelled in figure 8.40. For example, the 3a! orbital of H 2 0 is expanded as 2pz + ( ^ i + ^ 2 ) ) implying that it consists of the atomic 2pz on O and the atomic Is orbitals on hydrogens 1 and 2. What experimental observations are rationalized by this simple approach? (1) Core orbitals are atomic like. (2) A binding energy spectrum should consist of sharp peaks corresponding to energies of these energy-localized orbitals. In fact, this is an excellent first approximation. Further, Chapter 8. Review of the Hydrides 154 the number of peaks in each case is correctly predicted. (3) The basic shape, s type or p type, of the momentum distribution can be correctly predicted for every orbital studied by EMS of which we are aware. In conclusion, we re-iterate that all of the basic observations of EMS are easily ex-plained through this very simple energy-localized MO picture. It can be said that elec-tron momentum spectroscopy has provided no surprises. If any results have appeared counter-intuitive, it is because of the incorrect application of other models. 8.3.2 Binding Energy Spectra of the Hydrides It is presently possible to predict the qualitative features of the ionization spectra of molecules in both the core and valence regions and in some cases to make accurate quan-titative predictions. Reviews of the many-body Green's function and CI methods used for such calculations are found in the literature [47,48,49,50,229,230]. A very important review of the results of such computations has appeared [231] which discusses the general aspects of ionization spectra and provides rationalizations for the observed spectra based on the theoretical calculations. Our discussion of the observed trends in the hydride valence spectra will be made within the context of the models outlined in ref. [231]. As pointed out in the previous section, the binding energy spectra can be understood to a first approximation in terms of simple molecular orbital theory. Peaks in the spectra correspond to ionization from canonical Hartree-Fock molecular orbitals. Closer exami-nation of the binding energy spectra of the hydrides yields the following observations: (1) The simple MO picture may completely break down. (2) When the MO picture breaks down, it does so in the inner valence region and not Chapter 8. Review of the Hydrides 155 the outer valence region. (3) The MO picture applies less well to the second row hydrides than to the first row hydrides (see figure 8.41). (4) Group IVA hydrides and the noble gases show the least breakdown of the MO picture (see figure 8.41). Both CI and Green's function calculations have been shown to quantitatively predict the spectra of all of the hydrides [182,37,38,232,167,29,28,39]. We are more interested, however, in a qualitative description of the above phenomenon. The rationahzation of these observations as provided by Cederbaum et al. [231] can be couched in a molecular orbital type of language as borrowed from CI. To first order, the breakdown of the MO picture should occur when there exist two hole one particle (2hlp) configurations which are nearly degenerate in energy with, and which couple strongly with, a single hole configuration. Low-lying 2hlp configurations will tend to occur when the neutral molecule possesses low-lying excited states. The interaction between the 2hlp configuration and the lh configuration will be strong if the virtual orbital corresponding to the lp in the 2hlp configuration is localized in space as are the occupied orbitals. How does this explain the observed trends? The energies of the 2hlp configurations are generally much higher than that of the single holes in the outer valence region, hence the MO model applies well there. The increased number of electrons in the second row hydrides compared to the first row can lead to a large number of 2hlp configurations, and therefore an increased probability of a near degeneracy with a lh configuration. The molecules lacking symmetry will tend to have non-bonding orbitals and hence low-lying virtuals. Molecules with high symmetry will have a greater number of degenerate orbitals and therefore a smaller density of 2hlp configurations, and the large number of Chapter 8. Review of the Hydrides 156 N u m b e r of H y d r o g e n s Figure 8.41: The spectroscopic factor for the main pole of the inner valence orbital of each of the hydrides as calculated by MRSD-CI, this work. The same trend is observed experimentally in EMS and PES if one considers the percentage of experimental intensity in the main peak compared to the total observed. Green's function calculations also give the same result. Chapter 8. Review of tie Hydrides 157 irreducible representations will make it less likely that a given virtual orbital will have the same symmetry with (and therefore overlap strongly with) any of the occupied orbitals. While this model of ionization is still relatively simple, it clearly explains all of the observations. 8.3.3 Momentum Distributions of the Hydrides: General Trends An examination of the momentum distributions for the hydrides reveals that the target Hartree-Fock approximation (equation 2.5) works reasonably well for all of the molecules presented provided the Hartree-Fock limit is reached by the calculation. This is an im-portant result as it considerably simphfies our ability to calculate approximate theoretical MDs by eliminating the need to perform a full overlap calculation (equation 2.4). Careful examination of the hydride results leads to the following four general conclu-sions: 1) There is good agreement in almost all cases when comparison of the experimental data is made to a good overlap calculation. 2) The CI calculation of the overlap differs from the RHF calculation of the momentum distribution to a much greater extent for a given hydride in the first row than for its analogue in the second row. 3) For those cases where they differ, the CI calculation tends to have a smaller pmox than the RHF MD, and a larger height than the RHF MD. 4) The inner valence orbital is often missing experimental intensity. We will now discuss each of these observations in more detail. Chapter 8. Review of the Hydrides 158 8.3.3.1 The ion-neutral overlap amplitude and the EMS cross section Equation 2.4 is validated for the experimental conditions used in this study by the good agreement between the experimental data and theoretical calculations based on the ion-neutral overlap using correlated wavefunctions. The only exceptions are HF and HC1. These molecules are rather outstanding chemically, being very reactive and highly polar. One might speculate as to whether this high polarity dictates the need for a more complete CI expansion, but the calculations presented have converged. It should also be noted that there is nothing remarkable or unique in the comparison of calculated and experimental properties of HF and HC1 other than the momentum distributions. Many one electron properties of these two molecules are calculated with a degree of accuracy comparable to that for the other hydrides studied (see tables 8.12, 8.13, 8.14 and 8.15). Another possibility is that the plane wave impulse approximation has broken down. Yet the measurements made for these two molecules at different impact energies are in good agreement, and it seems unlikely that the plane wave impulse approximation should fail for these molecules while applying well in a large number of other studies. A new measurement of the momentum distributions is suggested by these results, and such work is in progress. 8.3.3.2 When is correlation negligible? We have noted that there is little difference between the momentum distributions calcu-lated using the THFA and those using the ion-neutral overlap for many of the orbitals of the second row molecules SiH4 through Ar. For a given first row hydride, the change on going to a THFA to overlap description of the MD is always greater than its second Chapter 8. Review of the Hydrides 159 row analogue. In each case, the overlap calculation agrees better with the experimental results than does the Target Hartree-Fock calculation. That the CI calculation differs from the RHF more for the molecules of the first row is not surprising. It is well known that the effects of both relaxation and correlation on ionization potentials become smaller as one goes down the periodic table. For larger atomic numbers, the difference between parent correlation and cation correlation tends to diminish and Koopman's theorem gets better. So perhaps it is not unreasonable to suppose that analogous trends exist for electron densities. For a given molecule, the difference between the CI and RHF calculation is often most pronounced for the inner valence orbital. Again, this is not surprising as it mirrors the trend observed for IPs. However, as pointed out in section 8.3.3.3, the sign of the difference in the heights of the calculated momentum distributions is not necessarily the same as the sign of the difference in the calculated IPs. A quantitative measure of the difference between the MRSD-CI and the RHF cal-culations of the momentum distributions is given by the difference in their maximum heights. The height of the calculated MDs at p m a x , and their difference for each orbital, is tabulated in table 8.22. Examination of table 8.22 reveals that the difference in the height of the CI and RHF calculations is rarely more than 10% for the molecules studied, and is often much smaller. In 1977, V.H. Smith asked: "how large, in fact, are the corre-lation corrections to p and do we need to worry about their effect . . . in the chemically interesting regions of the molecule. The available evidence is rather scanty." [233]. This systematic study has gone a long way toward answering that question. Chapter 8. Review of the Hydrides 160 8.3.3.3 Contraction in p-space due to correlation With only a few exceptions, the MD calculated from the CI expansion of the ion and neutral maximizes at a smaller momentum (pm a x) and has a larger height than the RHF MD. This can be rationalized in the following manner. For a lone symmetry state [234], the (neutral) CI expansion is a sum of an occupied RHF orbital and some virtual orbitals. The virtual orbitals will tend to be diffuse in position space, or tight in momentum space. Hence, compared to the RHF orbital, the CI maximizes at a smaller value of momentum. Given that it has more intensity at lower momentum, it must have a large maximum in order to integrate to unity: / <j>(p)p2 dp = 1. This rationalization is born out by the fact that all of the counterexamples of this trend, (the cases where the CI MD has a lower maximum than the RHF MD), are for orbitals which aren't lone symmetry states, and almost all appear in second row molecules: 2a! of N H 3 , 3ax of SiH 4, 5a\ and 4ax of PH 3 , 4aa of H 2S, and 5cr and 4<r of HC1. Since the effect of relaxation on the charge distribution is to bring the electrons closer to the nucleus, one would expect an expansion in p-space due to relaxation. Obviously, this is not the dominant effect for most of the hydride valence orbitals. The contraction in p-space due to the ion-neutral overlap description cannot be ex-plained solely by energy considerations. It is well known that lowering the total energy of a system (by describing it with an improved wavefunction or causing it to undergo a chemical change) will tend to shift (total) momentum density from regions of lower momentum to regions of higher momentum. This follows from the virial theorem re-quirement that the electronic kinetic energy equal the negative of the total energy, and Chapter 8. Review of the Hydrides 161 that the kinetic energy is proportional to < p 2 >. However it should be strongly empha-sized that this does not apply on an orbital basis: there appears to be no justification for the notion that a decrease in the ionization potential for a given orbital will necessarily cause a shift to higher momentum. For all of the orbitals listed above as counterexam-ples, there is a shift in the MD toward higher momentum on going from an RHF to CI description, but the change in the IP is in the same direction as for the orbitals which show a shift in the MD toward lower momentum (see tables 8.20 and 8.22). 8.3.3.4 Disagreement for inner valence MDs For several molecules, there is a significant mismatch between theory and experiment for the intensity of the inner valence momentum distribution. This occurs for Ne, Ar, PH 3 and HF. In every case, the shape of the MD is correctly predicted by the overlap calculation, but the theory predicts more intensity than is experimentally observed. One possible explanation for this phenomenon is the breakdown of the plane wave impulse approximation, or distortion. Attempts can be made to account for distortion of electron waves by the scatter-ing centre, as well as for the mutual interaction of the two outgoing electrons, using a distorted wave impulse approximation [20,59]. It has been shown that momentum distri-butions calculated within the target Hartree-Fock DWIA give excellent agreement with the experimental results for Ne and Ar [59,60]. It would appear that for these two atoms, all of the intensity discrepancy for the respective inner valence orbitals can be accounted for by distortion effects. One may therefore speculate that the disagreement for intensity of the inner valence Chapter 8. Review of the Hydrides 162 MDs for PH3 and HF may also be caused by distortion effects. In the case of PH3, which has a rather complex ionization spectrum, unobserved pole strength may mean that the experimental data has too low an intensity [29]. This is unlikely for HF, for which the spectrum has been measured to a reasonably high binding energy and which is predicted by theory to have a relatively simple spectrum [139]. Hydrogen fluoride is a particularly disturbing case in that the discrepancy between theory and experiment for the 2<r orbital is considerably larger than the change in theoretical intensity that has been observed in going from a PWIA to a DWIA treatment [60,59,235,236]. However, DWIA calculations have only been performed on a small number of atomic systems, so an ability to predict the effects of such calculations, especially on molecular systems, is lacking. Calculations on HF and PH3 using distorted waves would be most interesting in light of the disagreement between theory and experiment; but another question that needs addressing is the reason why Ar and Ne require a distorted wave treatment while many molecules do not. 8.3.4 The Hartree-Fock Limit We have already discussed the relative simplicity with which momentum distributions of second row hydrides can be calculated due to the fact that correlation and relaxation effects are unimportant. Another aspect worth considering is the complexity of basis set required to reach the Hartree-Fock limit. While a wavefunction may still be far from the true Hartree-Fock limit, it is possible that it already gives a momentum distribution indistinguishable from an MD calculated with a wavefunction at the HF limit. One might predict that a momentum distribution would be one of the last properties to converge to its Hartree-Fock limit value as a basis set is increased in size. The reason for this is the Chapter 8. Review of the Hydrides 163 fact that the low momentum region studied by EMS makes little contribution to the total energy, which is the optimization criterion of the calculation. This slow convergence has been verified in practice [38]. As has been discussed alreadjr (section 8.1), the calculations reported here employed very large, even tempered, energy optimized sets of s andp Gaussian orbitals. Experience gained in this series of studies showed that such basis sets are considerably improved for many properties if they are extended by one more diffuse s and p primitive [118]. These diffuse (r-space) functions contribute density to the low momentum region probed by EMS; so it is not surprising (in hindsight) that they have been found to be absolutely essential for the correct description of momentum distributions [38]. For H2S (ref. [28], 4-31G+ basis set) and NH 3 (ref. [37], 6311+G basis set) it has been shown that relatively small basis sets containing diffuse functions will give momentum distributions very close to the Hartree-Fock limit for MDs (but not for other proper-ties). This is certainly not the case generally, however, as the investigation of H 2 0 has shown [38]. In this study [38], a large number of different calculations were investigated and it was found that most had not converged to the Hartree-Fock limit for the MD. It was shown that the amount of computational effort required to reach the Hartree-Fock limit was considerably more than had previously been realized. 8.3.5 Location of p m a x We denote the value of momentum at which the momentum distribution reaches its maximum as p m o x • The value of p m a x for the outermost orbital of each of the hydrides is plotted against the number of hydrogens in the molecule in figure 8.42(a). Perhaps Chapter 8. Review of the Hydrides 164 surprisingly, the simplest way to explain the results is in terms of the ionization potentials of the these orbitals. It has been shown that the asymptotic behaviour (in position space) of atomic orbital wavefunctions is characterized entirely by their angular momentum and ionization poten-tials, assuming certain simplifications and within the Hartree-Fock description [237,238]. It would be reasonable to guess that the small momentum region probed by EMS cor-responds to this large r region [70,237]. Further, Casida and Chong have argued that momentum distributions of some small hydrides may be accurately described by the asymptotic behaviour of the corresponding atomic orbital on the heavy atom [70]. It can easily be shown that a momentum distribution calculated from Lasettre's descrip-tion of the asymptotic behaviour of the electron density [237] will have a maximum at Pmox= y/2e/3 where e is the Koopman's theorem ionization potential in atomic units. Using the values of the Koopman's theorem IP as listed in table 8.20, we have calcu-lated pmax from the formula above and the result is shown in fig. 8.42(b). This gives an excellent reproduction of the observed experimental trend. Note that in this context, we have emphasized the connection between momentum and energy which we have dismissed as an adequate explanation of other aspects of the experimental results in section 8.3.3.3. It is clear from figures 8.34-8.39 and figure 8.42 that the MD of any orbital in any second row hydride is more contracted in p space than the MD of the corresponding orbital in the corresponding first row hydride. The virial theorem predicts this correctly from the fact that the IPs, in all cases, are smaller for the orbitals of the second row hydrides. Chapter 8. Review of the Hydrides 165 3 CO 1 . 0 0 . 8 \ 0 . 6 ' C U ' 0 . 4 i — i -X-0 . 2 co 5^ o.o Ne P H 3 H ? S H C 1 4 Number of Hydrogens Figure 8.42: The maximum in the momentum distribution, (a) The experimentally observed vali of Pmax\ (b) theoretical p m o l based on Lasettre's asymptotic formula for atoms. Chapter 8. Review of the Hydrides 166 8.3.6 Energy localization versus position localization While we have generally referred to canonical Hartree-Fock orbitals throughout this the-sis, we have done so only for convenience. One can describe momentum distributions using localized molecular orbitals, hybrid orbitals, valence bond theory, or any other model. This may not be immediately obvious and warrants comment. Localized molecular orbitals are the ones that most closely resemble the Lewis dot structures with which chemists are very familiar. Consider C H 4 as an example. Within a localized MO picture, C H 4 has four identical valence orbitals which are equated with the four C-H bonds. These are spatially-localized orbitals. If we set out to calculate the momentum distributions of the valence orbitals of C H 4 , we will, in the process of performing the calculation, force the orbitals to energy-localize themselves. They will do this by mixing together in just such a way so as to produce results identical to what we have labelled the lt 2 and 2a, momentum distributions. (A more formal discussion of this important point is given in ref. [239].) Hence this description is ultimately equivalent to the canonical Hartree-Fock description. Chapter 9 Conclusions This work has presented EMS measurements of the binding energy spectra and momen-tum distributions of (CH 3) 20, PH 3 , C H 4 and SiH 4. The experimental results have been compared to calculations from the literature and to new calculations performed here and in collaboration with co-authors elsewhere. The good agreement between the experimental and theoretical results using large basis sets provides further verification of the validity of the plane wave impulse approximation for the scattering conditions employed in this work. The binding energy spectrum of (CH 3) 20 shows reasonable agreement with an ex-tended 2ph-TDA Green's function calculation. Satellite structure is observed in the region above and below the main 3a! peak, and is experimentally assigned as having predominantly 3ai character, in agreement with the Green's function calculation. The results of the calculations on (CH 3) 20 have added to the small body of data that has been collected concerning the effects of rotations and vibrations on momentum distribu-tions [38,45,67,240]. The results presented here suggest that the orientation of the methyl groups does not affect the momentum distributions. The theoretical investigations have also shed light on the ability of small basis sets to accurately describe momentum dis-tributions. The calculations presented here seem to have converged in some sense, in as 167 Chapter 9. Conclusions 168 much as further additions of diffuse functions to the small basis sets do not significantly change the calculated momentum distributions. It is not clear, however, that the results have converged to near the Hartree-Fock limit, as no Hartree-Fock limit calculations are available and poor agreement with experiment is obtained for the outermost orbital. The experimental results, when compared with those for CH 3 OH and H 2 O , demonstrate the derealization of charge density with increasing methyl substitution. The measured binding energy spectrum of phosphine shows reasonable agreement with Green's function calculations at the level of 2ph-TDA and simplified ADC(4), as well as reasonable agreement with an MRSD-CI calculation. However, the agreement is not entirely satisfactory and more accurate calculations would be of great interest, particularly since P H 3 is observed to be one of the most outstanding examples of the breakdown of the molecular orbital picture of ionization. The extensive satellite struc-ture observed in the inner valence region is observed to be dominantly of 4aJ"a character, in agreement with the Green's function and CI calculations. The momentum distribu-tions show good agreement with the near Hartree-Fock limit and MRSD-CI calculations presented, except for the intensity of the Aa, momentum distribution. This may be due to unobserved intensity in the binding energy spectrum leading to an incorrect normal-ization, although the discrepancy seems too large to be accounted for entirely in this wajr. Correlation and relaxation have no effect on the calculated momentum distributions as the calculated MDs and OVDs are identical. The binding energy spectrum of CH 4 , measured to 60 eV, shows excellent agreement with a 2ph-TDA Green's function calculation which predicts a small splitting of the main 2a, pole strength. The weak satellites observed are experimentally assigned to the 2a\l ionization process, in agreement with the prediction of the Green's function Chapter 9. Conclusions 169 calculation. Near Hartree-Fock limit calculations show excellent quantitative agreement with the measured momentum distributions and inclusion of correlation and relaxation is found to have a minimal effect on the calculated momentum distributions. As in the case of CH 4 , the binding energy spectrum of SiH 4 shows little satellite structure. Almost all of the satellite structure is between 18 eV (the location of the main 3ax peak) and about 35 eV, in good agreement with the ADC(3) and simplified ADC(4) Green's functions calculations presented. The satellites are assigned to the Zax ionization process, in agreement with the Green's function calculations. The measured momentum distributions show good agreement with the near Hartree-Fock limit and MRSD-CI calculations presented. Correlation and relaxation is found to have no effect on the calculated momentum distributions. In the course of this work, momentum distributions have been calculated using near Hartree-Fock limit wavefunctions for C H 4 , HF, Ne, SiH 4, PH 3 , HC1 and Ar. MRSD-CI calculations of the ion and neutral wavefunctions for all of the above systems have been performed and used to calculate momentum distributions. This work was done in collaboration with E.R. Davidson's group at Indiana University. This completes a systematic theoretical study of the momentum distributions of the hydrides CH 4 through HF and SiH4 through HC1, as well as Ne and Ar. With the completion of the experimental measurements of PH3, C H 4 and SiH 4, all of the above systems have also been measured at high momentum resolution. Analysis of the results led to the following conclusions: • The large disagreement between the high-level calculations and the experimental measurements for HF and HC1, seen in light of results for all the other molecules, strongly suggests that the experimental data should be remeasured. This conclusion Chapter 9. Conclusions 170 was not reached in the initial study of these two molecules [39] due to the limited calculations originally performed and the absence of a large, consistent body of experimental and theoretical results on different molecules. • The generally good agreement between theory and experiment (except in the case of HF and HC1 as mentioned above) shows that the plane wave impulse approximation, and in particular equation 2.4, provides a good description of the experimental results. • The effects of correlation and relaxation on momentum distributions tend to be greater for first row hydrides than second row hydrides. • In the case where there is only one orbital of a given symmetry in a molecule, the effect of the inclusion of correlation and relaxation in the calculated momentum distribution for that orbital will tend to be a contraction in p-space. • The ionization potential of an orbital may, in the cases of the outermost valence orbitals of the hydrides, provide a qualitative guide to the shape of the momen-tum distribution. Similar conclusions have been reached in other studies [70,237]. However, it has also been shown that the relative ionization potentials from dif-ferent calculations is in no way a measure of the relative shapes of the calculated momentum distributions. This systematic theoretical and experimental investigation has laid the groundwork for future studies. The range of validity of both the theoretical and experimental ap-proaches is now well understood, and will lead to more confident assessments of theoret-ical and experimental findings. This will be of great importance as EMS measurements Chapter 9. Conclusions 171 are extended to larger systems, like dimethyl ether, for which accurate theoretical calcu-lations are difficult, and experimental measurements of some orbitals may be impossible due to an inability to resolve closely lying states. In such cases, the experimentalist and the theorist must rely on each other to fill in the gaps in the understanding of the electronic structure. 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