A Combined Experimental and Theoretical Study of The
A COMBINED EXPERIMENTAL AND THEORETICAL STUDY OF THE ELECTRONIC STRUCTURE OF MOLECULES BY ELECTRON MOMENTUM SPECTROSCOPY AND DENSITY FUNCTIONAL THEORY By Patrick Duffy B.Sc. (Hons.), U.B.C. , 1987
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
in
THE FACULTY OF GRADUATE STUDIES
DEPARTMENT OF CHEMISTRY
We accept this thesis as conforming to the required standard
THE UNIVERSITY OF BRITISH COLUMBIA March 1995 © Patrick Duffy, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.
Department of Chemistry The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1Z1
Date: WMCL if IK Abstract
New theoretical models for the instrumental angular resolution function and for the experimental cross-section for electron momentum spectroscopy (EMS) are proposed, tested, and evaluated. For the experimental resolution function, the existing Gaussian Ap method is discussed and found inadequate as it is not representative of the spectrometer itself. A new model for the experimental resolution function, first proposed by Bawagan and Brion (Chem. Phys. 144, 167 (1990)), is considered in detail. An analytic form for the resulting resolution function, which involves the use of Gaussians in 6 and (f>, is developed, tested, and evaluated, and found to be both more realistic physically and to cause high-level quantum-mechanical calculations of the EMS angular cross-section to come into better agreement with experiment than when the Ap method (commonly employed previously) is used. The new experimental resolution function is further tested, in conjunction with exist ing theoretical methods, in an EMS study of acetylene. Three common ah initio methods for the calculation of approximate EMS angular cross-sections (Roothaan-Hartree-Fock, configuration interaction, and Green function) are considered. It is found that, for this molecule, the comparatively simple Hartree-Fock method provides an adequate descrip tion of the outer-valence electron angular cross-sections, and that the Green Function and multi-reference singles and doubles configuration interaction calculations also dis cussed are therefore not required when calculating angular cross-sections for acetylene. However, because of the pronounced breakdown of the single-particle model of ionization in the inner-valence region, these latter methods are found to be necessary for a quantita tive prediction of the angular cross-sections in that region. Likewise, simple Koopmans'
ii ionization potentials predict the binding energy spectrum of acetylene relatively accu rately in the outer-valence region, while correlated treatments are found to be necessary to successfully predict the observed breakdown of the single-particle picture of ionization in the inner-valence region. Because of the general complexity of the calculations necessary for the calculation of EMS binding energies and cross-sections by conventional ab initio methods, and because electron momentum spectroscopy studies are evolving towards more complex systems, including biomolecules, a new method, less computationally intensive than conventional ab initio methods, is proposed for use: density functional theory (DFT). Binding energies and EMS cross-sections are calculated using DFT and compared with the appropriate ab initio (SCF and CI) results. DFT is found to provide very good predictions of binding energies for a wide range of molecules, and to provide EMS cross-sections of comparable quality to Hartree-Fock. The overall charge density provided by DFT within the local density approximation is further tested, in order to better determine the reliability of DFT for both EMS calculations and the prediction of other properties based on the electron density. It is found that while DFT (within the local density approximation) generally predicts outer-spatial properties based on the density with reasonable accuracy, it is unreliable for calculation of properties which depend on an accurate description of the electron density near the nucleus. Use of (spatially) non-local functionals is found to not improve calculated results significantly.
iii Table of Contents
Abstract ii
List of Tables ix
List of Figures xi
Acknowledgements xiv
Preface xvi
Quotation xvii
1 Models Used in Electronic Structure 1
1.1 Theoretical Models for Atomic and Molecular Structure 1 1.1.1 Early Concepts 2 1.2 Quantum Mechanics 3 1.2.1 The Born-Oppenheimer Approximation 4 1.3 The Hartree-Fock Approximation 5 1.3.1 The LCAO approximation 6 1.3.2 Post Hartree-Fock Methods 6 1.3.3 Limitations of Hartree-Fock and Post Hartree-Fock Methods ... 7 1.4 Density Functional Theory 8 1.4.1 Two Early Models 8 1.4.2 Hohenberg and Kohn 9
iv 1.4.3 Kohn and Sham 10 1.4.4 Better Approximations to the Exchange-Correlation Potential. . . 11 1.5 Electron Momentum Spectroscopy 11 1.5.1 A Brief History of EMS 12 1.6 Scope of This Thesis 14
2 Modeling the Experimental Cross-Section 15 2.1 A Brief Overview of the EMS Experiment 15 2.2 The Distorted Wave Impulse Approximation 17 2.2.1 Assumptions in the Model 17 2.2.2 Limitations of the Model 18 2.3 The Plane Wave Impulse Approximation 19 2.3.1 Assumptions in the Model 19 2.3.2 Limitations of the PWIA 21 2.4 Information Obtained Within the PWIA 21 2.4.1 Binding Energy Spectra 22 2.4.2 Experimental Momentum Profiles 22 2.5 Calculation of the PWIA Cross-Section: The Generalized Overlap and the Target Hartree-Fock Approximation 23 2.5.1 Configuration Interaction 24 2.5.2 Green Functions 25 2.5.3 The Spectroscopic Sum Rule 25 2.5.4 The Target Hartree-Fock Approximation 26 2.6 Considerations for Calculations 27 2.6.1 Basis Sets 27 2.6.2 Computational Requirements of the Methods 28
v 3 Experimental Details 30 3.1 The Vacuum System 30 3.2 Electron Optics 32 3.3 Detection Circuitry and Coincidence Electronics 34 3.4 Computer Control 37
4 Modeling Finite Resolution Effects 39 4.1 Introduction 39 4.2 Conceptual Framework 45 4.3 Computational Details 53 4.3.1 Monte Carlo Method 54 4.3.2 Analytic Method 56 4.4 Results 58 4.4.1 Evaluation of the Ap, UW-PG, and GW-PG Models 59 4.4.2 Comparison of the GW-PG and MAGW Methods 62 4.4.3 Non-Gaussian Angular Resolution Functions 64 4.5 Conclusions 65
5 Evaluation of Existing EMS Models: A Study of Acetylene 67 5.1 Introduction 67 5.2 Experimental Method 70 5.3 Computational Details 71 5.3.1 Preliminary Details 71 5.3.2 Post-Target Hartree-Fock Calculations and the Effects of Correla tion and Relaxation 73 5.4 Results and Discussion 76 5.4.1 Binding Energy Spectra 77
vi 5.4.2 Experimental and Theoretical Momentum Profiles for the Valence
Orbitals of C2H2 79 5.4.3 Detailed Studies of the Binding Energy Spectra and Momentum Distributions in the Inner Valence Region 88 5.5 Conclusions 95
6 Density Functional Theory 96 6.1 Hartree-Fock Theory 96 6.1.1 Computational Implementation 98 6.2 Density Functional Theory 99 6.3 Computational Implementation 101 6.3.1 Approximations to the Exchange-Correlation Potential 101
6.3.2 The Xa Potential 102 6.3.3 Local Potentials Including Correlation 103 6.3.4 Non-local Exchange and Correlation Potentials 103 6.3.5 Evaluation of the Exchange and Correlation Potential 105
7 EMS from DFT: The Target Kohn-Sham Approximation 107 7.1 Introduction 107 7.2 Theoretical Background 108 7.2.1 Dyson's equation and target approximations 109 7.2.2 Kohn-Sham density-functional theory 114 7.2.3 Approximate Dyson orbitals from DFT 116 7.3 Computational details 123 7.4 Results and discussion 130 7.4.1 Target Kohn-Sham Approximation in the LDA 132 7.4.2 Ad hoc DFT approximations for Dyson orbitals 144
vii 7.4.3 Effect of the Functional 148 7.5 Conclusions 152
8 Calculation of Ionization Potentials from Density Functional Theory 157 8.1 Introduction 157 8.2 Density Functional Computations 159 8.3 Results and Discussion 160
9 Testing the Accuracy of the DFT Electron Density 166 9.1 Introduction 166 9.2 Computational Details 167 9.2.1 Preliminary Considerations 171 9.3 Results and Discussion 174 9.3.1 Non-local Functionals 185
9.4 Conclusions 186
10 Conclusions 188
List of Abbreviations 190
Bibliography 193
A Quadrature for the Analytic Method 207
B Contributions to the Width of the Momentum Resolution Function 208
C Orbital Energy Shifts 211
viii List of Tables
5.1 Total energies of C2H2 calculated using several basis sets and methods. . 72 5.2 Vertical ionization potentials for the principal ionic states of C2H2". ... 75
7.1 Basis set dependence of the total energy, dipole moment, and principle components of the dipole polarizability of H20 127 7.2 Absolute peak heights and positions of the maxima of MDs for the valence
orbitals of CH4, NH3, H20, HF, Ne, and C2H2 137 7.3 Symmetry breaking in DFT calculations on excited states of C2H2 as in dicated by the nonzero dipole moment 146 7.4 Comparison of the negative of Kohn-Sham orbital energies at full (KS orbital energy) and half (transition state method) occupancy with exper
imental ionization potentials, for C2H2 and H20, using the LDAxc func tional and the ANO+ basis set 147 7.5 EiFect of exchange-correlation functional on the dipole moments and po-
larizabilities of H20, using the ANO+ basis set 151
8.1 Ionization potentials of C2H2 as calculated using different methods in den sity functional theory. 161 8.2 Ionization potentials of several molecules as calculated by various methods within density functional theory. 162 8.3 Comparison of vertical ionization potentials (VIPs) for sixteen 22-electron molecules 164
ix 9.1 Auxiliary basis functions used in conjunction with the ANO and Dunning basis sets for one-electron property calculations 169 9.2 One-electron properties of water as calculated using several different or bital basis sets 172 9.3 Electron densities at the nuclei as calculated by DFT and compared to Hartree-Fock and MR-SDCI 175 9.4 Field gradients of several small molecules as calculated by DFT within the LDA 177 9.5 Electric field and total Hellmann-Feynman forces at the nucleus calculated by DFT within the LDA and compared to Hartree-Fock and MR-SDCI . 180 9.6 Dipole moments of several small molecules as calculated by DFT within the LDA 181 9.7 (r2) for several small molecules as calculated by DFT within the LDA and compared to Hartree-Fock, MR-SDCI, and experiment 182 9.8 Quadrupole moments of several small molecules calculated by DFT within the LDA and compared to Hartree-Fock, MR-SDCI, and experiment. . . 183 9.9 One-electron properties of water as calculated using different model ex change and correlation functionals 185
B.l Contributions of the various dependent variables to the standard deviation of the momentum p used in the resolution function 209
x List of Figures
2.1 Schematic showing electron trajectories in an EMS experiment for the symmetric non-coplanar geometry. 16
3.1 Cutaway diagram of the EMS spectrometer 31 3.2 Coincidence and detection circuitry for the EMS spectrometer 35 3.3 Example time spectrum for the EMS spectrometer 36
4.1 Schematic summarizing terminology and showing the relationship between theoretical and experimental quantities in EMS 46 4.2 Flowchart of the Monte Carlo method 55 4.3 Momentum resolution functions for Eo = 1215.8 eV and (f>o = 0°, 1°, 2°, 3°, 5°, 10°, 15°, and 30° 57 4.4 Comparison of experimental data and GW-PG resolution-folded near SCF limit or CI theory. 60 4.5 Comparison of GW-PG resolution-folded calculations using three different independent variables 63
5.1 The long range binding energy spectrum of acetylene at (a)
(b)
5.2 Experimental and theoretical momentum profiles for the l7ru (HOMO) orbital of acetylene, measured at a binding energy of 11.4 eV 81
5.3 Experimental and theoretical momentum profiles for the 3(rg orbital of acetylene, measured at an energy of 16.2 eV 83
xi 5.4 Experimental and theoretical momentum profiles for the 2au orbital of acetylene, measured at an energy of 19.3 eV 85
5.5 Experimental and theoretical momentum profiles for the 2crg orbital of acetylene 86 5.6 Binding energy spectra in the inner valence region (25 - 34 eV) of acetylene. 90 5.7 Angular distributions in the inner valence region (25 - 34 eV) of acetylene. 93
7.1 Effect of basis set on the TKSA MD for the lbi orbital of H20 128 7.2 Comparison of the experimental XMP and resolution-folded CI/109CGTO
MD for the lbi orbital of H20 131 7.3 Comparison of several DFT approximations for MDs with target Hartree- Fock and CI MDs, for the valence orbitals of H20 133 7.4 Comparison of several DFT approximations for MDs with target Hartree- Fock and CI MDs for the valence orbitals of C2H2 134 7.5 Correlation plot of the MD peak heights obtained in the target Kohn- Sham approximation (KS) or in the target Hartree-Fock approximation (HF) with configuration interaction (CI) 136 7.6 Comparison of target Kohn-Sham approximation (KS), target Hartree- Fock approximation (HF), and CI TMPs with XMPs for the valence or
bitals of H20 139 7.7 Comparison of target Kohn-Sham approximation (KS), target Hartree- Fock approximation (HF), and CI TMPs with XMPs for the valence or
bitals of CH4 140 7.8 Comparison of target Kohn-Sham approximation (KS), target Hartree- Fock approximation (HF), and CI TMPs with XMPs for the valence or
bitals of NH3 141
xii 7.9 Comparison of target Kohn-Sham approximation (KS), target Hartree- Fock approximation (HF), and CI TMPs with XMPs for the valence or- bitals of HF 142 7.10 Comparison of target Kohn-Sham approximation (KS), target Hartree- Fock approximation (HF), and CI TMPs with XMPs for the valence or- bitals of Ne 143 7.11 Effect of the exchange-correlation functional on the target Kohn-Sham approximation MDs for the valence orbitals of water 149
B.l The standard deviation of the momentum resolution function as a function
of (j>0 for 61=02= 45°, EX = E2 = 600 eV, and E0 = 1215.8 eV 210
xiii Acknowledgements
Many people have helped me a great deal up to this point in my academic career, both with the academic side of things and the personal. I'd like to take this opportunity to thank a few of them. I hope I don't miss anybody important. First and foremost, I would like to thank Mark Casida for his unflagging support all throughout my academic career at U.B.C. The work presented in this thesis is just as much a product of his insight as mine and my supervising professors', and much of the strength of it is directly attributable to him. I have learned much from Mark, both in and out of academia. I would also like to thank my supervising professors, Dr. Chris Brion and Dr. Delano Chong, for insight, guidance, support, and for putting up with the times when I had to "do it my way", whether or not my way was actually the right way to do it. I would also like to acknowledge the outside co-authors who, over the years, have made invaluable contributions to my published work. Chris Maxwell and Ernest David son provided the high-level Hartree-Fock and configuration interaction calculations for acetylene. Alain St. Amant provided me with an early copy of deMon, helped me to understand the code, and made my week-long stay in Montreal a pleasant one. Michel Dupuis provided invaluable help during the debugging of the deMon-HONDO interface and took the time afterwards to run the one-electron properties calculations as well. Thanks go as well to the many members (both past and present) of the Brion and Chong research groups for helpful discussions and good coffee breaks: Jennifer Au, Gor don Burton, Natalie Cann, Jon Carter, Wing Fat Chan, Steve Clark, Mike Cohen, Glyn Cooper, Jingang Guan, Bruce Hollebone, Noah Lermer, John Neville, Terry Olney, Tim
xiv Reddish, Jim Rolke, Bruce Todd, Wenzhu Zhang, and Yenyou Zheng. All have helped in some way to make my time at U.B.C. enjoyable and productive. I am very grateful also to Dr. John Jantzi, my optometrist. Without the specialized tools which he provided so that I might see things more clearly, my academic pursuits would have ended at high school, if not before. Finally, thanks to my family and friends. Thanks to my parents for not asking how the thesis was going too often and for free food and laundry from time to time. Thanks to Nick and Jan for the occasional pizza and the cheap accommodation, to Kim and Don for helping to keep me sane during all this and, lastly, to Carol and Basil for some of the best dental care a guy could hope for.
xv Preface
Much of this thesis was derived from published work:
Chapter Reference P. Duffy, M.E. Casida, C.E. Brion, and D.P. Chong, Chem. Phys. 159, 347 (1992). P. Duffy, S.A.C. Clark, C.E. Brion, M.E. Casida, D.P. Chong, E.R. Davidson and C. Maxwell, Chem. Phys. 165, 183 (1992). P. Duffy, D.P. Chong, M.E. Casida, and D.R. Salahub, Phys. Rev. 50, A4707 (1994). 8 P. Duffy and D.P. Chong, Org. Mass. Spectrom. 28, 321 (1993). 9 P. Duffy, D.P. Chong, and M. Dupuis, J. Chem. Phys. 102, 3312 (1995).
xvi Quotation
There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable.
There is another theory which states that this has already happened.
- Douglas Adams, "The Restaurant at the End of the Universe"
xvii Chapter 1
Models Used in Electronic Structure
Over the past hundred years, experimental techniques and the theoretical models used to aid in their interpretation have advanced greatly. This thesis will focus on two of these areas, namely the development and advancement of both the theoretical models and experimental techniques used in Electron Momentum Spectroscopy (EMS). It is the purpose of this thesis to present the development and evaluation of two new models for use in EMS, namely an improved model for the spectrometer angular resolution function, and a new approximate method for calculation of the experimental cross-section using density functional theory (DFT). In addition, EMS experiments together with Hartree-Fock (HF), Configuration Interaction (CI), and second order Green function (GF2) calculations are presented to aid in the evaluation of existing models. This chapter will provide a brief historical background and perspective for the material to appear in the rest of this thesis. A brief history of some of the major developments in quantum chemistry will be given, up to and including modern DFT. A brief history of EMS will also be given.
1.1 Theoretical Models for Atomic and Molecular Structure
This section and the ensuing subsections will provide a brief outline of some of the important developments in the theory of atomic and molecular structure which have occurred during the course of the past hundred years [1].
1 Chapter 1. Models Used in Electronic Structure 2
1.1.1 Early Concepts
In the late nineteenth and early twentieth century, many models for the structure of the atom and molecule were proposed. Each model was developed to explain some experi mentally observed phenomenon. For example, the "plum pudding" model, proposed in 1907 by J.J. Thomson, was an attempt to account for electrical neutrality in matter when electrons (whose e/m ratio Thomson had measured some years earlier) were the only par ticles then known (see ref. [2], p. 165). Thomson suggested that atoms were clouds of positive charge (the "pudding") with electrons deposited throughout (the "plums"). While Thomson's model successfully accounted for the electrical neutrality in matter, it failed to explain the results of scattering experiments by Geiger and Marsden in 1909 in which a-particles fired at metal foils were, sometimes, found to rebound directly back at the a-particle source (see ref. [2], pages 165 - 166). The Rutherford model of the atom was thus proposed to replace the Thomson model [3]. The discovery of the neutron in 1932 by James Chadwick allowed the model to be further refined by replacing an electron/proton pair in the nucleus (neutrons were not known at the time Rutherford proposed his model) with one electrically neutral particle of approximately the same mass as a proton (see ref. [2], p. 165).) This model for the atom remains largely unchanged to the present day. Mathematical models for the electronic structure of the atom were also advancing in step with experimental observations. Perhaps the most well-known series of experiments involved passing an electric discharge or heat through a cloud of hydrogen atoms and observing the spectra obtained. What was found was not a continuum of light, as was expected, but a discrete spectrum consisting of only a few lines. Each series of lines, some in the visible and some in the infra-red, were named and are generally known for the physicists who discovered them: Lyman for the ultraviolet, Balmer for the visible Chapter 1. Models Used in Electronic Structure 3
(though it was Angstrom who performed the actual measurements), etc. (see ref. [4], pages 23 - 24, and 29). Balmer (in 1885) was the first person to establish a pattern in the observed spectra, finding that the series of spectral lines he measured fit a simple empirical formula with integer parameters. Other spectroscopists measuring the hydrogen spectrum in different regions of the UV and infra-red found their spectra fit the same formula, only with different integer parameters. A new model of the atom (specifically the hydrogen atom) was proposed in 1913 by Niels Bohr to explain the observed spectra of hydrogen. In this model, electrons moved only in certain discrete circular orbits about a central nucleus [5]. In addition, the angular momentum of the electron in these orbits was to be quantized in multiples of h/2ir, and any electron in one of these orbits would remain in that orbit until perturbed. Bohr accounted for the fact that his models contradicted those of classical physics by simply assuming that conventional physics did not apply here, and that new physics was needed for electrons and protons in atoms. The Bohr model, though quite successful for hydrogen, was not readily suited to other atoms, and was not adaptable at all to molecules. By the mid-1920s, it was apparent that a new model was needed.
1.2 Quantum Mechanics
Schrodinger [6-8] and Heisenberg [9,10] introduced quantum mechanics in 1926, arriving at the formalism independently (Heisenberg the matrix formulation and Schrodinger the differential equation formulation). Quantum mechanics introduced the idea of a wave- function for the system, and postulated that all that could be known about a quantum- mechanical system was contained in this wavefunction. Quantum mechanics was refined to include relativistic effects in 1928 by Dirac [11,12]. V«-
Chapter 1. Models Used in Electronic Structure 4
Quantum mechanics enjoyed much initial success; the answers it yielded for the hy drogen atom restricted themselves naturally to the forms required for agreement with the available experimental data. In addition, the formalism could be extended to larger atoms and molecules with relative ease, and calculated observable quantities obtained from computations based on sound approximations agreed well with available experimen tally determined quantities. The basic formalisms in quantum mechanics have remained largely unchanged. The formulation of approximate solutions to the quantum-mechanical problem has remained a primary focus in the study of quantum mechanics. It is interesting to note that some of the earliest approximations, developed very soon after the introduction of quantum mechanics itself, remain some of the most important and widely used approxi mations today.
1.2.1 The Born-Oppenheimer Approximation
The Schrodinger equation cannot be solved exactly for systems consisting of more par ticles than do hydrogen or the hydrogenic cations. The primary reason for this is that there are too many bodies in motion. This difficulty must be addressed if useful prob lems are to be attempted. One intuitively reasonable approach is to assume that electron motion about the nucleus is much faster than the nuclear motion itself, so limiting the heavy particles in a system to defining a static potential in which the electrons move, thus reducing the many-particle Schrodinger equation to one involving electrons only. The Born-Oppenheimer approximation was proposed in 1927 [13]. It simplifies the Schrodinger equation describing the system of interest by fixing the nuclei in space. This renders any potential and/or kinetic energy originating in the motion of the nuclei a constant factor which may be separated from the electronic problem. Use of this model dictates that isotope effects are to be disregarded (different isotopes of an element will Chapter 1. Models Used in Electronic Structure 5 present the same static potential to an electron). The Bom-Oppenheimer approximation has become a central part of nearly every quantum chemistry program currently in use, and is used implicitly any time a program shows the total energy of a system as being partitioned into an electronic and a nuclear component. Using this approximation has allowed Hj to be solved accurately using numerical methods, although no analytic solutions exist for larger systems than this.
1.3 The Hartree-Fock Approximation
In 1928, Hartree introduced two ideas [14] which led eventually to the Hartree-Fock equations, enabling the calculation of approximate solutions of relatively high quality for many-electron systems. He assumed first that electrons in an atom or molecule moved independently of one another in orbits described by a unique one-electron function. This caused the electronic part of the Schrodinger equation for the system to factor into a series of one-electron equations. Second, to simplify the electronic potential in the Schrodinger equation, he assumed that the electrons moved in an average field created by the other electrons in the system. These two assumptions result in a series of coupled differential equations which may be solved iteratively to self-consistency — that is, until the functions obtained after solution of the equations are the same as the functions input to the equations (the Hartree equations depend internally upon their solutions). The resultant total wavefunction for the system is a simple product of the one-electron functions. The Hartree equations resulted in a total wavefunction which did not change sign on exchange of any two of the electrons in it. (It was not antisymmetric with respect to the exchange of any two electrons.) In addition, the model used for the potential allowed an electron to feel the potential due to itself (called self-interaction). Both of Chapter 1. Models Used in Electronic Structure 6 these problems were addressed and corrected by Fock in 1930 [15]. The wavefunction was made antisymmetric by taking a single determinant of the one-electron functions instead of a straight product. This, combined with the introduction of the exchange operator, whose contribution to the one-electron energy exactly cancels the self-interaction energy, solved the problem of self-interaction. Fock's formulation had the additional advantage that all differential equations for all electrons were of identical form, making it easier to implement computationally. The ease of computational implementation was offset somewhat, however, by the multicentre nature of the integrals which were required (up to four centers for a molecule consisting of four or more atoms).
1.3.1 The LCAO approximation
Roothaan introduced one of the last major approximations in the Hartree-Fock system of equations [16]. He proposed that molecular orbitals (MOs) be treated as the sum of a set of fixed atomic orbitals, and that the coefficients which summed up the atomic orbitals to make the molecular orbital would be what varied in a Hartree-Fock calculation. Furthermore, since three-centre integrals were not possible using Slater-type orbitals, Gaussian-type orbitals (GTOs) (as proposed by Boys in the 1930s) were introduced to make up the atomic orbitals.
1.3.2 Post Hartree-Fock Methods
A conceptually relatively simple solution to the non-inclusion of correlation in Hartree- Fock calculations is to employ methods which incorporate electron correlation by using as a wavefunction a combination of several configurations constructed from Hartree-Fock orbitals. These methods are collectively referred to as "post Hartree-Fock", as they generally use a Hartree-Fock calculation as a starting point. Chapter 1. Models Used in Electronic Structure 7
Configuration interaction [17] is the most general of all the post Hartree-Fock meth ods, and derives its extra configurations by exciting electrons from ground-state orbitals to (previously) virtual ones, either one or more at a time. The final (total) wavefunc- tion is taken as a linear combination of all configurations, and the coefficients for these configurations are found variationally within the CI calculation. The name given to a CI calculation is generally related to the number and type of configurations included in the calculations. CISD, for example, indicates a calculation including all single and double excitations from the ground state wavefunction (usually Hartree-Fock).
Multiconfiguration SCF (MSCSF) calculations have a heritage in configuration in teraction calculations. Just as in configuration interaction calculations, several config urations are used to describe the system of interest. The difference, though, is that in an MCSCF calculation, both the molecular orbital coefficients and the configuration coefficients are varied to find the optimum ground-state wavefunction. The first such calculations were in fact reported by Hartree in 1939 for the atomic oxygen cation [18].
1.3.3 Limitations of Hartree-Fock and Post Hartree-Fock Methods
Hartree-Fock and post Hartree-Fock methods are computationally intensive; this was exacerbated by the lack of computers during the early years of the method's development, thereby limiting calculations to atoms or relatively simple molecular systems. Specifically, because of the coulomb and exchange integrals involved in the procedure (see chapter 6), the time required for a Hartree-Fock calculation scales formally as N4 (where N is the number of atomic orbital basis functions employed in the calculation - see chapter 2), so that (for instance) doubling the number of basis functions used in the calculation will cause the time required for the calculation to increase by a factor of 16. This limitation of the Hartree-Fock method becomes more pronounced if post Hartree- Fock methods are employed; any method which depends on a Hartree-Fock calculation Chapter 1. Models Used in Electronic Structure 8 as a starting point will have at least an N4 dependence.
1.4 Density Functional Theory
Because the limitations mentioned previously for Hartree-Fock and post Hartree-Fock methods were recognized early in their development, alternative methods were quickly proposed in an attempt to circumvent them. Density functional theory is one alternative method, and its development will be outlined in the sections to follow.
1.4.1 Two Early Models
Density Functional Theory postulates that the total energy of a molecular system may be expressed as a function al of the total electron density. Because of this definition, any theory which has built into itself some function of the total electron density may be called a density functional theory. The first such theory was proposed by Thomas in 1927 [19], (and Fermi one year later) who attempted to model the total energy of a quantum-mechanical system by examination of a homogeneous electron gas. Using the homogeneous electron gas as a starting point, they derived an expression for the total energy which depended only on the electron density in the system of interest. The model was simple, but the answers it provided were not of as high quality as those from other methods then being introduced. As a result, the method came to be viewed as too oversimplified, and of not much use for solid-state or molecular physics when accurate predictions were desired. The method had another rather more serious drawback: it could not predict binding of atoms in molecules [20]. During and after this time, the Hartree-Fock method had increased rapidly in popu larity, as it seemed to be the easiest method from which relatively accurate answers could be obtained for molecular properties (M0ller and Plesset demonstrated that this should Chapter 1. Models Used in Electronic Structure 9 be the case [21]), and many calculations were performed with encouraging results. How ever, there still remained the problem of the many-centre integrals and how to evaluate them for large systems. In 1951, Slater postulated an approximation to Hartree-Fock theory and, unknown to him at the time, the computational method employed by modern density functional theory [22]. Slater sought to simplify the Hartree-Fock problem, and with the most computationally difficult part of the calculation being the exchange integrals, started with these. The approximation chosen was the exchange potential for the homogeneous electron gas, as derived by Dirac some twenty years previously [23] for use in Thomas- Fermi theory. Calculations were carried out with some success, and any calculation referred to in the literature as "Hartree-Fock-Slater" is very likely of this type.
1.4.2 Hohenberg and Kohn
In 1964, Hohenberg and Kohn provided the foundation for modern DFT [24]. In their ground-breaking paper, they proposed a new way of examining the electronic structure problem. By assuming some unique (and non-degenerate) ground state density for an atom or molecule, they proved that the external potential felt by the electrons in the system of interest was in fact uniquely determined for the system of interest, up to an additive constant. Treating the electronic structure problem in this manner has some very real potential advantages. For one, it reduced the problem to a function of six (space and spin) coor dinates rather than 3n ("n" is the number of electrons in the system) for every system to which it could be applied. The one main disadvantage of the formalism as proposed, however, negated all its advantages: There was no way (except in the case of hydrogen) to find the density being sought. Chapter 1. Models Used in Electronic Structure 10
1.4.3 Kohn and Sham
In 1965, Kohn and Sham proposed the first approximations to make DFT computation ally viable [25]. They assumed that, in an analogous manner to Hartree-Fock, electrons moved as independent particles in orbitals described by one-electron functions. This made the kinetic energy part of the energy expression possible to evaluate, although the kinetic energy was no longer the true kinetic energy of the electrons in the system; any difference between the Kohn-Sham (KS) kinetic energy and the true kinetic energy would be absorbed into the exchange-correlation potential for the electrons in the systems, dis cussed below (see chapter 6 for further details). The second approximation made was for the exchange and correlation potential. What was needed was a form which was a functional of the electron density, so that once the density was known the exchange and correlation energy would be known as well. The form chosen, based on variational arguments for the exchange potential of the homogeneous electron gas, neglected the requirement that the chosen potential absorb any differences between the true and Kohn-Sham kinetic energy in lieu of calculational simplicity. The form chosen was simply the exchange potential used by Slater in the Hartree-Fock-Slater calculations, but multiplied by 2/3. This potential also neglects correlation effects. Kohn and Sham imposed two additional constraints on their calculations. The first was that the squares of the orbitals used should sum to the total electron density, thus making the total electron density easy to obtained from their calculations. In addition, they required that any approximations to the exchange and correlation potential be mathematically local for computational simplicity.1 Though strictly not a necessary part
1An operator O is said to be mathematically local if its use is equivalent to multiplication by some function /, i.e. Orj>(r) = f(r)if>(r). The exchange and correlation operator in Kohn-Sham DFT is math ematically local because vxerj>(r) = vxc(r)ip(r). Chapter 1. Models Used in Electronic Structure 11 of "true" DFT, this restriction on the functional has remained in wide use.
1.4.4 Better Approximations to the Exchange-Correlation Potential.
Atoms and molecules do not have electron densities which resemble those of homogeneous electron gases (at least not globally). In spite of this, the exchange potential used by Kohn and Sham (or a slight alteration of it) remained in wide use for some time. Generally these alternate forms were obtained by multiplying Slater's potential by some number other than 2/3, where the number used was usually chosen so that the total energy of the atom studied matched the Hartree-Fock energy for that same atom. Adjusting the multiplicative factor in this way came to be called the Xa method [26], where "X" indicated that the functional was for exchange only (and not correlation), and a was the adjustable parameter it contained. Other, more physically reasonable approximations to the exchange and correlation potentials have since been developed, and some of these will be discussed in chapters 7, 8, and 9.
1.5 Electron Momentum Spectroscopy
All the methods discussed so far are theoretical in nature. In order to check the validity of any theory or model, it is important to have or design an experiment which may be directly correlated with the theory being investigated. For instance, if vertical IPs have been determined theoretically, then the results of a photoelectron spectroscopy (PES) experiment for the same molecule would be desired. Various types of calculations may be tested in this manner. However, if what is desired is an examination of the wavefunction or electron density (for there are no wave- functions per se in density functional theory), then this becomes rather more challenging. There is, after all, no direct experimental probe of the electron density (though Compton Chapter 1. Models Used in Electronic Structure 12 scattering and positron annihilation do indirectly probe the total electron density), and certainly not of the wavefunction (in the case of conventional ab initio methods that use a wavefunction) from which it came. There are, however, methods which approach this ideal, and one, specifically Electron Momentum Spectroscopy, is a main concern of this thesis. Electron momentum spectroscopy is a unique and versatile technique for the detailed study of atomic and molecular electronic structure [27-30]. In common with photo- electron spectroscopy, electron momentum spectroscopy can obtain the binding energy spectra (BES) of atoms and molecules in both valence and core regions. However EMS has the powerful additional advantage of being able to obtain binding energy spectra differential with respect to electron momentum over the momentum range typically up to ~3 atomic units. Therefore in EMS an electron momentum distribution (normally re ferred to as an experimental momentum profile, or XMP) can be obtained at each binding energy (BE). Thus EMS effectively provides (to within close approximations) a means of imaging the orbital electron density in momentum space (i.e. | tp(p) |2 ). The initial orbital symmetry information present in the XMP affords a straightforward means of investigating orbital ordering and the parentage of both main [31,32] and satellite peaks [33] in binding energy spectra. These unique capabilities of EMS are now also providing a powerful experimental quantum chemical approach to molecular wavefunction evaluation and design [28,30] as well as for the investigation of the relation between orbital electron density distribution and chemical bonding, structure and reactivity [34,35].
1.5.1 A Brief History of EMS
EMS is an electron scattering experiment and so, like other scattering experiments, has a heritage in particle physics. The first EMS experiments were performed in Italy and reported in 1969 [36], using a coincidence apparatus with a symmetric coplanar geometry Chapter 1. Models Used in Electronic Structure 13 for the analyzers (see chapter 3 for further details). In this experiment a beam of incident electrons was fired at a carbon film, and ionization potentials (IPs) (at very low energy resolution, ~ 150 eV full-width at half-maximum) of the valence and K shells of carbon were obtained. During this time, it was speculated that, in addition to binding energy spectra, the (e,2e) technique could be used to obtain information about the wavefunction in a molecule [37]. The first such study confirming this postulation was carried out in 1972 (again in Italy) [38], In this study, the K- and L-shell XMPs of electrons in a thin carbon film were obtained. The energy resolution in these experiments was again very low, but they at least demonstrated that the technique would work. The first EMS experiment to resolve the valence orbital binding energy spectrum of a gaseous target molecule was carried out in 1973 in Australia [39]. The binding energy spectrum of argon was obtained, as were the XMPs of the (valence) 3p and 3s orbitals. In addition to this, evidence was also presented that there were ionization events occurring at energies above that of the main 3s peak which showed a similar shape to that of the 3s peak, supporting the idea that the independent particle model of electronic structure breaks down for inner valence ionization. Many other EMS studies of atoms and molecules followed this initial work. Of par ticular importance are the papers concerned with the valence structure of NH3 [40] and
H20 [41] by Hood et al. in 1976 and 1977, which were the first EMS papers to ascribe disagreement between theory and experiment to inadequate basis sets (see chapter 2 for a further discussion of basis sets). In addition, Bawagan et al. have published a landmark series of studies of small hydrides [28,32,42] comparing EMS measurements to quantum mechanical calculations, which show that careful consideration of all aspects of the the oretical calculation of the electronic structure part of the calculated EMS cross-section (among which are saturation of the basis set and whether or not configuration interaction Chapter 1. Models Used in Electronic Structure 14 is employed) are necessary for agreement of theory with experiment.
1.6 Scope of This Thesis
This thesis will propose and evaluate two new models which can be incorporated into the theoretical description of the EMS cross-section. The first will be a new model for angular (and thus momentum) resolution effects in the spectrometer. For this part of the work, a new angle-space resolution function is proposed and critically evaluated by application to a range of target atoms and molecules. The new model resolution function is then tested in the context of a typical EMS study, in this case of acetylene, in chapter 5, in which the existing theoretical models for the experimental cross-section will also be evaluated. This thesis will also propose and evaluate the use of the Kohn-Sham DFT orbital as an alternative method for predicting cross-sections for comparison with EMS measurements. The model itself will be proposed and tested in chapter 7, and subsequent chapters will evaluate the overall quality of DFT for EMS, both for ionization potentials and for overall accuracy of the calculated electron density, to ensure that any good agreement obtained by DFT with EMS cross-sections is not merely coincidence. Accuracy of the electron density calculated by DFT will be assessed by employing DFT in the calculation of a series of one-electron properties weighting different regions of the electron density. Chapter 2
Modeling the Experimental Cross-Section
If the experimental cross-section is to be related to theoretical calculations, a set of models and model approximations must be used. This chapter will outline the models used in the interpretation of EMS spectra in the context of a perfect angular resolution experiment. Angular (momentum) resolution effects will be discussed separately, in chapter 4. All experimental data presented in this thesis are interpreted within the Plane Wave Impulse Approximation (PWIA). The Distorted Wave Impulse Approximation (DWIA) will be presented (briefly) first, however, to establish a context for the PWIA, which will be discussed in more detail. Following the discussion of the PWIA, the information which may be obtained using this model, i.e. the binding energy spectrum and the angle-dependent cross-section, will be discussed. In particular, theoretical approximations to the PWIA cross-section will be considered in detail. Before any discussion of the theory of EMS begins, however, a brief description of the experiment will be given to establish a background for the theory to follow.
2.1 A Brief Overview of the EMS Experiment
Figure 2.1 shows a schematic of the electron trajectories employed in the symmetric non-coplanar spectrometer; a spectrometer of this type was used to obtain the EMS data presented in this thesis. Chapter 3 contains a description of the construction and operation of the spectrometer.
15 Chapter 2. Modeling the Experimental Cross-Section 16
(E0, Po)
Figure 2.1: Schematic showing electron trajectories in an EMS experiment for the sym metric non-coplanar geometry. The polar angles 0\ and 02 are equal to 45°, and the azimuthal angle
Briefly, in a typical EMS experiment, a collimated beam of electrons of definite energy Eo is fired at an atomic or molecular target (which is usually in the gas phase). Within this beam of electrons, a single electron may impact on a single target atom or molecule. When this occurs, the target atom or molecule is ionized and a second electron is ejected:
e" + M -*• M+ + e~ + e~ (2.1) Eo, PO -£reccul>q E\, Pi -E25P2
Ejected and scattered electrons with the appropriate energies (Ej = E2 = 600 eV in the experimental work reported in this thesis) and angles ($1 = $2 = 45°) are then detected for a range of detector angles (f>0 at fixed Eo, resulting in an angle-dependent spectrum, or at fixed
2.2 The Distorted Wave Impulse Approximation
The most general of the theoretical expressions for modeling the EMS angle-dependent cross-section is found in the Distorted Wave Impulse Approximation. The DWIA has been studied extensively [29,43,44], and so will not be discussed in detail here. Instead, a brief outline of the main features and assumptions of the model will be given.
2.2.1 Assumptions in the Model
The EMS angle-dependent cross-section is given, in the DWIA, by [43]:
3 , cPa 2ir m 2m ,„ „ .,/,^^ / 1.,,2 ,^ ,„ „x °EMS - JMhiEl = T5SpHF<**>*£/ m "• (2-2) Chapter 2. Modeling the Experimental Cross-Section 18 where
M = (x
In equation 2.3, XA > XB » and Xo represent the two outgoing electrons and one incoming electron respectively as distorted waves, and and \pW are the wave- functions for the product ion (in excited state I) and the ground state of the system before ionization, respectively. The integral over angles (0) accounts for the random orientation of target molecules in the gas phase, and the sum represents both a sum over vibrational and rotational states and an average over degenerate initial states. In addi tion, v is the scattering potential for processes which involve core excitations in addition to ionization. TM(p2) is called the "knockout" term, and concerns processes involving ionization only.
The main assumption in the DWIA becomes apparent upon its examination: the electrons in the system under study are to be treated independently of the target atom or molecule. That is, the assumption is made that the first Born approximation holds for this system, and that the scattering potential felt by the electrons does not couple them to the target being ionized.
2.2.2 Limitations of the Model
The model, though physically reasonable, has the practical limitation that the potentials v and TM(P2) may only be spherical (to facilitate calculations), and so the calculation may only be done for targets of spherical symmetry, i.e. atoms. As a consequence, DWIA calculations have only been performed for the noble gases.
In addition, the DWIA only holds for suitably fast electrons, to ensure that interaction with the system is minimal (this is the first Born approximation). Because electrons of relatively low energy will interact too heavily with the target system of interest, they Chapter 2. Modeling the Experimental Cross-Section 19 cannot be considered independent, and so do not factor out of the N+l - electron system as the DWIA requires. Further discussion of low energy (e,2e) spectroscopy is beyond the scope of this thesis.
2.3 The Plane Wave Impulse Approximation
All of the experimental data presented in this thesis have been interpreted within the Plane Wave Impulse Approximation [29,43]. The PWIA makes several approximations in addition to those in the DWIA discussed briefly above; these will be discussed below.
2.3.1 Assumptions in the Model
Firstly, in the PWIA the incoming and outgoing electrons are treated as plane waves. This renders the calculation of approximate EMS cross-sections possible for many-centered targets (i.e. molecules) in addition to atoms. A sufficiently high impact energy is required to fulfill both the plane wave and first Born descriptions. In the PWIA, the EMS cross- section is given by
2 ^M=|(i(p1-p2)|rA,^|p1-p2| )||(p0 + ?))| • (2.5) In equation 2.4 q is the ion recoil momentum, equal in magnitude and opposite in sign (within the PWIA) to the momentum of the ejected electron (p) before it was removed from the atom or molecule (see equation 2.1). This will be further discussed below. In addition, all the terms preceding <7jvf in equation 2.4 are constant for an EMS experiment for a given impact energy EQ- The Mott scattering cross-section, CTM? an explicit form Chapter 2. Modeling the Experimental Cross-Section 20 for which is given in ref. [45], has been shown to be effectively constant for the electron angles and energies collected by the UBC spectrometer used to obtain the data presented in this thesis ([45]). It is therefore convenient to write, for this case, that •»»- «^KS/|<<.*r'>i*"")r* M One further assumption is made during the course of the calculations, which is that the average over vibrational and rotational states represented by the summation above may be approximated simply by performing the necessary calculations at the equilibrium geometry. A study by Leung et al. [46] has shown that this is a reasonable approximation for the water molecule. Within this approximation, the PWIA cross-section simplifies still further: When reference is made to the PWIA cross-section in the rest of this thesis, it will be equation 2.7 which is the intended expression. The PWIA reduces the collision process to what is effectively a collision between two hard spheres, a sort of "billiard ball" or "binary encounter" model of collision physics. Within this model, the ejected electron does not feel the effect of the incoming electron until it is knocked out. The incoming electron does not feel the potential due to the target or to the electron until the collision occurs. Within the PWIA, collisions are treated as instantaneous, so that neither the daughter ion nor the parent molecule have sufficient time during the scattering event to adjust their geometry. In effect, the target does not "see" the electron in the same way that the electrons do not "see" the target. Chapter 2. Modeling the Experimental Cross-Section 21 2.3.2 Limitations of the PWIA The many approximations which comprise the PWIA would seem at first to not lend much credibility to the model. However, experimental studies carried out at Eo = 1200 eV + binding energy with Ei = E2 = 600 eV have shown it to be both an effective and reliable model for the interpretation of EMS cross sections (see for example ref's. [28,33,42,47-53]). The model is, however, not without its limitations. For one, the approximation begins to break down for target electron momenta (ion recoil momenta) above approximately 1.5 a.u. After this point, cross-section predictions are unreliable, and the DWIA must be used (at least for atoms) in order to obtain better agreement with experiment. In addition, the PWIA is best suited to experimental conditions where high momen tum transfer occurs between the scattered and ejected electrons. The present symmetric non-coplanar experimental conditions (outlined in figure 2.1) correspond to this situation, and have been used to obtain all the data presented in this thesis. 2.4 Information Obtained Within the PWIA Within the PWIA, two different types of information may be obtained from an EMS measurement: a "Momentum Distribution" (MD), or a "Binding Energy Spectrum". The type of information obtained depends upon which instrumental parameters are held fixed and which are variable during the experiment. This section will outline in brief the methods by which each is obtained using the experimental apparatus. For further details, see chapter 3). Chapter 2. Modeling the Experimental Cross-Section 22 2.4.1 Binding Energy Spectra If the relative azimuthal angle BE = EQ — E\ — E%. (2-8) During an EMS experiment (as discussed in chapter 3), Eo (1200 eV + binding energy) is varied while Ei and E2 are fixed at 600 eV each. In this fashion, a histogram of intensity versus (E0 - 1200) provides a BES. 2.4.2 Experimental Momentum Profiles Application of conservation of momentum to equation 2.1 gives q = Po - Pi - P2 (2.9) q in equation 2.9 is the ion recoil momentum. Within the PWIA, q is equal in magnitude and opposite in sign to the momentum p of the electron in the orbital prior to its ejection. The momentum of this electron prior to ejection may be calculated (see figure 2.1) according to the relation: 2 2 p = y/[2p! cosdi - po] + [2p2 sin62 sin ( Therefore, if E0 is fixed at one value and the relative azimuthal angle (f>0 (see figure 2.1) of the two electron analyzers is varied, the signal collected in an EMS experiment is a function of that relative azimuthal angle. Using equation 2.10, the angular variable may (in the limit of perfect resolution) be converted unambiguously to a momentum scale (see Chapter 2. Modeling the Experimental Cross-Section 23 chapter 4 for a further discussion of resolution effects). When this is done, the resulting signal is called a "Momentum Distribution", or MD. Calculation of momentum distribu tions is the primary focus of this thesis. The current common theoretical approximations to the MD will be discussed in the sections to follow. 2.5 Calculation of the PWIA Cross-Section: The Generalized Overlap and the Target Hartree-Fock Approximation According to equation 2.7, the calculated cross-section is proportional to the spherically averaged square of the Fourier transform of an ion-neutral overlap; this quantity will hereinafter also be denoted II (p). The overlap by itself is also referred to as a generalized overlap or Dyson orbital. Dyson orbitals can be obtained either directly, as solutions of Dyson's quasiparticle equation, or indirectly, by calculating the N- and (N — l)-electron wavefunctions (e.g. by CI or many-body perturbation theory (MBPT)) and then obtaining the Dyson orbital as the generalized overlap (equation 2.7). In either case, use of the Dyson orbital is referred to as the Generalized Overlap Approximation, or GOA, and is entirely equivalent to evaluation of equation 2.7. Both the Green function (GF) and CI approaches give the same result in the limit of a full treatment of correlation (even if the basis set is incomplete [54].) Thus, whichever approach is most convenient as a starting point for developing approximate treatments may be used. The one-electron picture used in the quasiparticle equation is often convenient for the development of approximations [54-56], and the quasiparticle equation approach is advantageous for making connections with density functional theory, as will be discussed in chapter 6. The Target Hartree-Fock Approximation is an approximation to the full GOA. It will yield answers of similar quality to the GOA when correlation and relaxation effects in Chapter 2. Modeling the Experimental Cross-Section 24 the parent and daughter ion are unimportant when describing OTEMS within the PWIA. It will be discussed separately below (in section 2.5.4), and both the Green function and Hartree-Fock methods will be discussed in greater detail in chapter 6. 2.5.1 Configuration Interaction Configuration interaction is perhaps the conceptually simpler of the two methods intro duced previously for calculation of the GOA. A configuration interaction calculation is, as its name implies, a set of (many-electron) configurations combined to make a total wavefunction for the system of interest. That is, if \& is the total wavefunction for the system of interest, and $,• is a many-electron configuration which forms a part of it, then * = £>$,-. (2.11) « The many-electron configurations $,• are usually derived from a Hartree-Fock calcu lation on the ground state of the molecule by exciting electrons from their (ground state) orbitals to (previously unoccupied) virtual ones (see below and chapter 6 for a further discussion of the Hartree-Fock method). The number of configurations is usually lim ited by computational constraints (one ground state Hartree-Fock calculation may have many virtual orbitals and hence provide many possible configurations) and so only the configurations which contribute most to the total wavefunction are chosen (usually by perturbation theory). Highly accurate Dyson orbitals have been calculated via CI by Davidson et oil. [28,33, 42,47-53,57]. While this type of calculation leads to theoretical momentum distributions in excellent agreement with experiment, the cost of this level of accuracy is prohibitive for all except very small molecules. Chapter 2. Modeling the Experimental Cross-Section 25 2.5.2 Green Functions Green Functions have the advantage that, unlike CI, the Dyson orbital may be obtained from them in a single calculation instead of two. Dyson orbitals are found as the eigen- solutions to Dyson's quasiparticle equation [58]: [F + £(-I)]i{> = -Iif> (2.12) In equation 2.12 above, F is the usual Fock operator (see chapter 6), and E(—/) is the self-energy operator. The Dyson orbital is ij>, and / is the ionization potential. The self-energy operator describes all the many-body effects on ionization not present in Hartree-Fock theory. These effects are in general too complicated to treat exactly, so approximations are required. MDs of Dyson orbitals obtained by direct solution of Dyson's quasiparticle equation have only been calculated using smaller basis sets and lower order treatments of correla tion [52,59-62] (see also ref. [63]). It is clear from these studies that both the direct and indirect approaches to calculation of MDs suffer from similar computational exigencies with regard to the accurate treatment of correlation, including the need to use extended basis sets. 2.5.3 The Spectroscopic Sum Rule Dyson orbitals obtained as the result of CI or Green function calculations give rise to more eigenvalues (ionization potentials) and eigenvectors (Dyson orbitals) than there are electrons in the system of interest. This arises from the fact that removal of one electron from the system always results in more than one "hole'' (i.e. a range of hole states). Because of this, the hole left by the electron once removed is described as having been "split up" into poles, each of which has a different binding energy. For each pole the associated momentum distribution has a similar shape. The probability of electron Chapter 2. Modeling the Experimental Cross-Section 26 removal causing a hole at a given energy is given by a "spectroscopic factor", which obeys the sum rule (for the jth electron) E^ = l (2-13) t for each electron removed, so that the removal of one electron always results in the creation of one complete hole, even if the hole is fragmented. The Dyson orbital and eigenvalue with the largest spectroscopic factor for any hole associated with the removal of a certain electron is generally referred to as the main pole, while all others are usually referred to as "satellites", both because of their lower spectroscopic factor (probability of occurrence) and because they are often found close in energy to the parent peak. 2.5.4 The Target Hartree-Fock Approximation It is not always possible to carry out a full GOA calculation, as they are generally cost and computationally intensive. It is therefore convenient, in situations where initial (ground) state correlation and (final) relaxation effects are negligible, to invoke the Target Hartree- Fock Approximation (THFA) [29]. In this case, final (ion) state correlation can occur and equation 2.7 reduces to j) 2 CTEMS oc Sj J |0,(p)| dOf (2-14) where is the so-called spectroscopic factor (as discussed above), although in this case it represents the probability of finding the (0.)-1 one-hole configuration in the final ion state j. The quantity / |^i(p)| dtop is the spherically averaged one-electron independent particle momentum distribution. Chapter 2. Modeling the Experimental Cross-Section 27 2.6 Considerations for Calculations Before carrying out a calculation, there are factors which must be taken into consideration that will affect the type and accuracy of the calculation performed. The two main considerations are the basis set to be employed and the method (Hartree-Fock, CI, or Green function) in which it will be used. Basis set considerations will be discussed first, followed by a brief discussion of the manner in which the computational intensity of the methods discussed above scale according to the basis set. 2.6.1 Basis Sets All but the simplest quantum mechanical problems require the use of approximations to render the problem computationally tractable. It is, for instance, common practice in Hartree-Fock and post Hartree-Fock calculations to model the molecular orbitals as linear combinations (sums and differences) of atomic orbitals: ^• = £ Xi = EckXry'zie-°"'r\ (2.16) k The values of r, s, and t are restricted to be integral, and must add to the lvalue of the atomic orbital desired (e.g. 2 for a d-orbital). Each orbital on an atom will be represented by a set of gaussians as in equation 2.16 above. In addition, virtual orbitals for the atom may also be represented, so that a carbon atom set may include a d-orbital. These collections of Gaussians for individual atoms are called basis sets. Basis sets range in Chapter 2. Modeling the Experimental Cross-Section 28 size from very small (where only the occupied ground-state orbitals are modeled) to very extended, in which many virtual orbitals for the atom may be included in the basis set as well. As an example, a minimal basis set for carbon will consist of two s orbitals and one set of p orbitals. Each of the atomic orbitals might, for a limited basis set of this type, be constructed from three Gaussian functions. Generally speaking, as the size of the basis set (both the number of Gaussians used in an individual atomic orbital and the number of atomic orbitals in the basis set) increases, the precision of the answer obtained from the method used (though not necessarily the accuracy) will improve. For example, a carbon basis set employing five s-type atomic orbitals, four p-type atomic orbitals, and three d-type atomic orbitals will give a more precise answer than a minimal basis set. Eventually, it will be found that further ex tension of the basis set will not result in any appreciable improvement in calculated properties. At this point, the calculation is said to be converged. Use of larger basis sets does not come without cost, as any quantum-mechanical problem will depend geometrically on the size of the basis set used. There is therefore always a practical limitation on the size of the basis set which may be chosen which is imposed by the type of calculation being performed. In general, a trade-off is made between computational expediency on the one hand (which comes at the expense of loss of precision due to the smaller basis sets), and precision (which in general will be more computationally intensive as more extended basis sets are employed). 2.6.2 Computational Requirements of the Methods There are two factors which affect the computational intensity of a quantum-mechanical calculation: the basis set and the formalism employed. These will be discussed briefly below. Chapter 2. Modeling the Experimental Cross-Section 29 Of all the Hartree-Fock and post-Hartree-Fock methods discussed above, the Hartree- Fock calculation increases in computational difficulty the least with basis set size, scaling formally as the fourth power of the number of basis functions used in the calculations. Use of special symmetry methods will reduce the order of this dependence somewhat. The post Hartree-Fock methods discussed above scale typically as N5 or greater, and introducing extra refinement into the calculation will increase computing time as well. For example, a full configuration interaction calculation (full CI) employing a relatively small basis set may take a greater amount of computer time to complete than a calculation limited to singly and doubly excited configurations only which employs a larger basis set. Chapter 3 Experimental Details The EMS spectrometer used in the experimental work reported in this thesis is of the symmetric, non-coplanar type [29]. That is, it collects scattered and ejected electrons at equal polar angles to the z-axis (#o = 45°), and over a range of relative azimuthal angles ( 3.1 The Vacuum System The spectrometer itself (i.e. the various electron optics and detectors) is located inside an o-ring sealed cylindrical aluminum chamber of diameter and height 40 cm. The system is evacuated by two diffusion pumps (Varian VHS-4, 1200L/s, Neovac SY fluid), one of which evacuates the differentially pumped gun region while the other evacuates the main chamber. Differentially pumping the spectrometer in this fashion allows for relatively high gas pressures in the analyzer region and relatively low pressures inside the gun chamber, thus extending filament life (a typical filament lasts approximately 30 days). Pressures in the main chamber during the course of an experiment are approximately 2 x 10~5 torr with the gun chamber at an order of magnitude less. 30 Chapter 3. Experimental Details 31 FIXED MOVABLE CMA v**?**a**aa*KKVcc*Kl ^ J.Serv o Motor FlIanent Grld Figure 3.1: Cutaway diagram of the EMS spectrometer. See text for a detailed description of all components. SP[1,2,3]: spray plates; CEM: single channel electron multiplier; CMA: cylindrical mirror analyzer section; FC: faraday cup; GP: ground plate; D[l,2]: deflectors. Chapter 3. Experimental Details 32 The diffusion pumps are each backed by a rotary pump (Sargent-Welch); one addi tional rotary pump services the gas sample inlet line. The vacuum chamber is surrounded by a high-permeability fimetal shield to reduce the ambient magnetic field to below 30 mG. 3.2 Electron Optics Production and detection of electrons are the primary functions of the spectrometer, and the components which perform these operations are described in this section. Electrons are produced by thermionic emission from a V-shaped thoriated tungsten filament in a commercial electron gun (Cliftronics CE5AH), floating at a high negative potential (1200eV + binding energy). The emitted electrons are first extracted with a grid and then accelerated with an anode (generally at 100 volts positive with respect to the filament). An Einzel lens provides finer focusing of the beam into the collision region. The electron beam is further collimated and aligned by passing it through two sets of quadrupole x-y deflectors; an aperture is placed after each set of deflectors to define the beam direction and to provide further collimation. After passing through the aper tures, the beam passes into the gas cell. Any part of the beam not scattered there (by impact with the target atoms or molecules) passes through a third aperture and into a faraday cup. All apertures and the faraday cup are metered. Typical total beam currents traversing the collision region are on the order of 50 microamperes. The incident beam is approximately 1 mm in diameter in the collision region. After collision, scattered and ejected electrons are selected according to their scat tering angles (polar angle 0O = 45°, relative azimuthal angle 3.3 Detection Circuitry and Coincidence Electronics A schematic diagram of the coincidence electronics is shown in figure 3.2. The pulses exiting each multiplier are capacitively coupled to a current-sensitive preamplifier (Ortec 9301), which amplifies the pulse (typically by 10), and converts it to an output voltage. The capacitive coupling isolates the amplifier from the high voltage necessary for oper ation of the channel multiplier (~3.7 keV). The pulse from each preamplifier is further amplified by a timing filter amplifier (Ortec 454) which takes the pulse and multiplies it by a factor of 20. The resulting pulses are typically about 5 ns (fwhm) wide, and with proper termination and 50 Cl cables, ringing in the pulses can be kept low. After amplifi cation, each pulse is fed into a constant fraction discriminator (CFD) (Ortec 463), which eliminates noise pulses whose peaks do not exceed the chosen -3.0 volt threshold. If the pulse received exceeds the CFD threshold, a fast negative pulse (10 ns fwhm, -0.8 V) Chapter 3. Experimental Details 35 *| START PR AMF CFD OUT TAC DELAY PR! AMF CFD -i STOP SCA1 SCA2 \ r RATEMETER .SINGLES TRUE START I ENERGY SCANl ..TRUE STOP COMPUTER LTACOUT I ANGLE SCAN TRUE + ACCIDENTA . .ACCIDENTAL Figure 3.2: Coincidence and detection circuitry for the EMS spectrometer. See text for details. PRE: pre-amplifier; AMP: amplifier; CFD: constant fraction discriminator; TAC: time to amplitude converter; SCA: single channel analyzer. The diagram is taken from ref. [53]. is emitted by the constant fraction discriminator and used as a start pulse on the time- to-amplitude converter (TAC) (Ortec 467). One of the pulses from the pair of CFDs is delayed (typically by 30 ns) so that the full distribution of true and random coincident events may be seen in the time spectrum. After amplification and discrimination, it must be determined whether or not in coming pulse pairs originate from the same ionization event. This is the task of the coincidence detection electronics in the experiment. The TAC is the heart of the coinci dence detection electronics; it is used to convert the time between start and stop pulses sent by the CFDs to a voltage. The TAC scale is linear, and designed so that time delays of 0 to 200 ns between CFD pulses generates output pulses in the range from 0 to 10 V. Chapter 3. Experimental Details 36 1200 CO 800- 'o I1 i FWHM 5 ns 'g 400-I —* i J k- trues and accidentals, 8 — 28 ns J * i * i k— accidentals only, 40 - 200 ns OH pmnffanHw ab_dka«a-__WMB,—_____ i 40 80 120 160 200 Time (ns) Figure 3.3: Example time spectrum for the EMS spectrometer. See the text for a de scription of the function of the coincidence electronics. The TAC output pulses are sent to two single channel analyzers (SCAs) (Ortec 406A, and an Ortec 467 built into the TAC), each of which is set to detect pulses having certain voltage ranges. The first SCA detects pulses with amplitudes between 0.4 and 1.4 V, corresponding to true plus accidental coincidences (JVj) — see figure 3.3 — in the window from 8 to 28 ns, and the other from 2.0 to 10 v, corresponding to random coincidences (N2) which are in the time window from 40 to 200 ns, for a ratio of 1:8 between the two time (voltage) windows. The SCAs generate output which is analyzed by a home-built interface [66] before being stored in the LSI-11/03 computer. Taking into account the different sizes of the windows used for the two SCAs and because of the differing nature of the signal observed in each window (the first window detects a combination of accidental and true coincidences, and the second window contains accidental coincidences only), the Chapter 3. Experimental Details 37 signal observed is calculated by the following relation: Nirue = Ni-N2/8, (3.1) and the error in the count rate, AN, is calculated using 2 1/2 AN = (JV, + iV2/8 ) . (3.2) The output of the TAC may also be sent directly to an analogue-to-digital converter (ADC), which will digitize each voltage received to a 12-bit word. This allows direct monitoring of the time spectrum, to ensure correct settings for the SCA windows. 3.4 Computer Control Computer control for the operation of the spectrometer and the associated electronics as well as data collection is provided by an LSI-11/03 system. The computer runs the spectrometer in one of three modes, each of which will be described here. All of these modes may be run with the appropriate software, stored in the computer. Elastic scattering is used to ensure that the CMAs are passing electrons of the correct energy with good energy resolution. In this mode, the incident electron beam is set to an energy of 600 eV and the electrons collected are those which are elastically scattered off the gaseous target molecules. The various CMA voltages may be adjusted at this point to ensure that symmetric distributions of electrons about the desired pass energy (100 eV for this experiment) are obtained, and to ensure that the singles signal intensity does not change with variation of the nominal relative azimuthal analyzer angle Modeling Finite Resolution Effects 4.1 Introduction The importance of considering angular (or momentum) resolution effects was already rec ognized in the first reported comparison of calculated and measured electron momentum distributions by Camilloni et al. in 1972 for Is electrons in carbon [38]. Following these early investigations, electron momentum spectroscopy has been developed into a powerful technique in experimental quantum chemistry [27,29,67,68]. Comparison of XMPs with resolution folded spherically averaged theoretical momentum distributions have revealed deficiencies in the basis sets used in theoretical calculations (see for example references [28,31,41]) and, going beyond the orbital (Target Hartree-Fock) approximation has per mitted a detailed evaluation of the effects of electron correlation and relaxation [28,69] on electron momentum distributions. The conclusions from such comparisons are dependent upon the consideration of a number of factors, including an adequate estimation of an gular (or momentum) resolution effects for incorporation into the calculated momentum distributions. EMS experiments and theoretical studies of small hydrides such as H2O revealed some serious deficiencies in even the best wavefunctions existing at the time [70] and led to the design of new highly accurate near Hartree-Fock (NHF) limit self-consistent field and Configuration Interaction wavefunctions for H20 [28,57] and NH3 [42,57]. Similar experimental and theoretical studies have also now been made for H2S [33], SiH4 [47], 39 Chapter 4. Modeling Finite Resolution Effects 40 PH3 [48], CH4 [49], HF [50,71], and HC1 [51], as well as for the noble gases Ne and Ar [53], Although in most cases quite good agreement was found between theory and experiment, in some situations, notably H2O [28], NH3 [42], and HF [50], small but significant discrepancies remained between theory and experiment, even when using the best available quantum-mechanical treatments at both the SCF and CI levels [28,42,50, 57]. The possible reasons for these remaining discrepancies have been discussed earlier in some detail [28,72] and are summarized below. First, since essentially converged results were obtained for the calculated MDs and OVDs1 (these quantities, which include spherical averaging and resolution folding, will be referred to collectively as Theoretical Momentum Profiles (TMPs) in the present work) as well as for other properties such as total energy and dipole moment [28,42,57], the wavefunctions and thus the electronic structure part of the predicted EMS cross-section are considered to be highly accurate and therefore not the cause of the observed discrepancies. Similarly, the kinematic factors involved in the cross-section are by now well-understood [29] and have been shown to be effectively constant [73] under the conditions used in the symmetric non-coplanar geometry. Likewise, the plane-wave impulse approximation has been shown to provide a good description of the collision process for valence electrons below momenta of about 1.5 a.u. at the incoming (1200 eV + binding energy) and outgoing (600 eV) electron energies used. Furthermore, recent experimental [28] and theoretical [46] investigations have also strongly suggested that the discrepancies are not due to vibrational averaging effects. Since existing experience in EMS over a wide range of studies would seem to eliminate any of the above experimental or theoretical factors as major contributing causes of the persisting discrepancies between theory and experiment in the case of H2O, NH3, and HF, attention has recently been focused on the adequacy of existing procedures used for xIn much of the previous EMS literature, the term MD referred specifically to a theoretical momen tum profile calculated from an independent particle wavefunction, while the term OVD referred to a theoretical momentum profile obtained from an ion-neutral overlap, usually via configuration interaction. Chapter 4. Modeling Finite Resolution Effects 41 incorporating the effects of finite instrumental angular (or momentum) resolution into the calculated TMP [72]. For an accurate quantitative comparison of theory and experiment, it is essential to include the effects of finite angular resolution (arising from a finite collision region as well as instrumental acceptance angles and response functions) into the calculations in order to permit assessment of wavefunction quality. However, the accurate determina tion of the instrumental resolution function is an extremely difficult task. As a result, in earlier work, a variety of approaches to incorporating finite momentum resolution effects into theory have been adopted [72]. These have ranged from ignoring finite res olution effects altogether (i.e. no folding of the theory) to using the uniformly-weighted planar grid (UW-PG) method [41,66,74,75] with an assumed uniform angular resolu tion function over the estimated instrumental acceptance angles 2 #o i A0 at the nominal instrumental settings Carlo techniques, it was shown [72] that at lower values of 14.4 eV), Xe 5p (binding energy 12.6 eV), H20 lbi (binding energy 12.2 eV), and H2S 2bi (binding energy 10.5 eV). These examples cover a factor of two in binding energy and exhibit a considerable variation in the position of the maximum and the half-width of their respective momentum distributions. The folded calculations are compared with the respective experimentally measured intensities [28,33,73] as a function of pnom- The reasons for the choice of pnom rather than pavg at the present juncture are discussed below, but again it should be stressed that the overall conclusions of the present work and also those of reference [72] regarding incorporation of momentum resolution effects, for comparison of theory and experiment, are independent of the particular choice of momentum scale. 4.2 Conceptual Framework The main objective of the present study is to assess the effectiveness of the recently pro posed momentum-averaged Gaussian-weighted resolution folding method [72] in compar ison with the earlier used uniformly-weighted planar-grid [41,66,74,75] and Ap (Gaus sian) [76,77] resolution folding methods by systematic application to a wide range of examples for which high quality calculations and experimental data are available. This section will provide the conceptual framework for such a comparison. In particular, the experimental resolution function will be defined in a way which facilitates comparison of the three resolution folding methods. The various interrelated quantities involved in the comparison of EMS experiments and theory (see section 4.1 above and section 4.2 following) are illustrated in fig. 4.1. The goal of any resolution folding procedure is to incorporate the effects of finite experimental resolution into the theoretical prediction of the experimental results. As such, if the theory is correct the output of any resolution folding procedure should be Chapter 4. Modeling Finite Resolution Effects 46 Different Spaces n EMS Experiment and Theory Argon 3p 15.8eV RESOLUTION 0 FOLDING Momentum (au) s(?o)= (c) x l/i(p)R(p;^o)dp- c Resolution folded •a N n I i I r- 0 10 20 30 E i o COMPARER Nominal Momentum (au) Nominal Momentum (au) Figure 4.1: Schematic summarizing terminology and showing the relationship between theoretical and experimental quantities in EMS. See text, section 4.2, for details. Chapter 4. Modeling Finite Resolution Effects 47 as close as possible to the experimentally measured quantity. Since the experimental quantity of primary interest in the present paper, namely the EMS signal (fig. 4.1b), is measured in "angle-space" as a function of a nominal machine angle Hx,y,z,...;xo,yo,ZQ..\X ,y ,Z . . .', XQ, y0, ZQ . . .) (4.1J will denote the conditional probability density of detecting x = x', y = y', z = z'... when the spectrometer is set to x — x'0, y = y'0, z = z'Q... . Since this notation is cumbersome, it will be abbreviated, when no confusion is expected to arise, to R(x,y,z...;x0,yo,z0...) (4.2) meaning the conditional probability density of detecting x,y,z,... given xo,yo,zQ, Note that with this convention R becomes a different function for each new set of variables. This conditional probability density is in fact a probability density and has units of x~1y~1z~l... because / R(x,y,z...;x0,y0,z0.. .)dx dy dz... = 1. (4.3) Conditional probabilities can be used to provide a concise mathematical description of experimental resolution effects. The (perfect resolution) theory of the EMS reaction is discussed in detail in chapter 2. Briefly, for appropriately chosen electron scattering kinematics, the perfect resolution Chapter 4. Modeling Finite Resolution Effects 48 EMS cross section is demonstrated to be proportional to H(p), the square of the spher ically averaged momentum distribution of a generalized (or ion-neutral) overlap orbital (see fig. 4.1a) n(p) = J\(p¥N~v I *w)|2 da? (4.4) under kinematic conditions for which the half-off-shell Mott scattering cross section is effectively constant. Here \P^ and $(JV_1) are the electronic wavefunctions for the N- electron target species and (N-l)-electron product ion respectively and p represents a plane wave spin orbital with momentum p which is equal to — q. If correlation and relaxation effects are small enough, then the target Hartree-Fock approximation gives a good description of the generalized overlap orbital, and in this case, H(p) oc J \l>(p)\2 2 2 p = y/[2pi COS0X - p0] + [2p2 sin62 sin ($/2)] (4.6) where 2 2 P = f( (#i = e2 = e = 45°). The function f((/>,e) plays a central role in the calculations presented in this paper. In an actual experiment, several physical effects [66,72], including a finite collision volume and finite acceptance angles, dictate that a range of momenta, rather than a Chapter 4. Modeling Finite Resolution EiFects 49 single momentum p, will be sampled when the spectrometer is set to the nominal angles JW = /(^oA) (4.8) Consequently the predicted EMS signal is the sum over the ideal signal for each p times the conditional probability R(p; E R & d 4 9 S( ' H(p)R(p; (some of the results in section 4.4.2 are graphed in terms of It should be emphasized that the use of pnom is (like ^o) a convenient simple assign ment of the momentum scale based on the nominal instrumental settings (o,6o). In reality, as discussed above, there is an (asymmetric) spread of momenta sampled at each ( f°° Pavg = / P R(P14>o) dp, (4.11) Jo is more representative of the distribution of momentum sampled at each experimen tal angle Pnom-space and not in paVg-space. Thus far, very little has been said about the explicit form of the resolution func tion R(p;<^o). This section is concluded by briefly reviewing three previously proposed EMS resolution folding methods which will be compared within the framework of the as sumptions listed above. They differ primarily in their ansatz for the resolution function R(p; A 2 E P = FEIe-<*b>-H The parameter a was determined from a best (but not perfect) fit of essentially Hartree- Fock limit (or CI) theory to experiment for the 3p orbital of argon [72,77]. (Note that essentially Hartree-Fock limit theory and CI level theory give effectively the same result for the momentum distribution of Ar 3p [53,79].) Chapter 4. Modeling Finite Resolution Effects 52 Both the uniformly-weighted planar-grid [41,66,74,75] and momentum-averaged Gauss ian-weighted [72] methods are more realistic in that they account for resolution loss di rectly in (cf>, 0)-space, which corresponds to the actual experimental situation. The meth ods are similar except for the shapes of the resolution functions (see also section 4.4.3). Both assume s( R(P; <^o, 0o) = / % - f( uw pG VVf pG except £ - (0;0o) = 1/(2A0) for \0-0o\ < A0 and R - ( Gy/ PG nate transformation to pavg-space. The angle resolution functions, R ~ (0] 0o) and .RGW-PG^. <^O), are given by (normalized) Gaussians with standard deviations of 0 = 1.0° and (as discussed above) each would have a different pavg-scale. Hence the GW-PG method Chapter 4. Modeling Finite Resolution Effects 53 will be used in lieu of the full original MAGW procedure when comparing the various resolution folding methods. It should be noted that the GW-PG formulation contains all the angular-resolution folding details of the MAGW method and is used here with the same acceptance angles (A 4.3 Computational Details It is important to ensure that any artifacts due to numerical errors are sufficiently small so that they may be neglected when comparing TMPs calculated using different resolution folding methods with experiment. In the course of performing this study, two different numerical methods have been investigated (Monte Carlo and "analytic") for calculating TMPs, and it has been found that each has its own special merits. The Monte Carlo method was used by Bawagan and Brion [72]. The principal nu merical errors in the Monte Carlo method result from statistical errors caused by the use of random numbers. Bawagan and Brion used a large set of random numbers in or der to keep these errors small. However, their histograms [72] of the resolution function R(p] In addition, the smooth graphs of the resolution function R(p; 4.3.1 Monte Carlo Method As recently emphasized by Titus [80], the Monte Carlo procedure is particularly well suited for a conceptually straightforward propagation of uncertainty. His ideas apply equally well to the present problem, except that the "uncertainty" to be propagated is a resolution function instead of an experimental "error". A streamlined Monte Carlo version of resolution folding is presented here because it has been found to be adequate for most purposes and simple in execution. However, the analytic method (see below) is superior for the calculation of the resolution function R(p; Select binding energy T for use a GaussiaI n ran calculate averagI e inten dom number generator sity s((f>0) to generate #,• T I calculate p, using equa ( continue V tion 4.6 f stop J T calculate II(p,) (equa tion 4.4) 4 continue J © Figure 4.2: Flowchart of the Monte Carlo method. See text, section 4.3.1, for details. Chapter 4. Modeling Finite Resolution Effects 56 Note that it is relatively simple to use a wide variety of response functions by simple modifications of the program. In this way, all different shapes and types of resolution function may be studied. The Monte Carlo method has been applied as a crosscheck on the implemention of the analytic method and is found to compare quite favorably with the more accurate analytic method. 4.3.2 Analytic Method It is well-known that analytical or traditional approaches are often more accurate than Monte Carlo methods for calculating many quantities, and this also applies to the calcu lation of EMS resolution functions. In particular, the Monte Carlo integration described above has an error which scales as n-1/2 with the number n of sets of random (^, 0)- points (ref. [82] pg. 480) while Simpson's rule integration (used in the analytic method described below) involves an error which scales as n~4 with the number of grid points n (ref. [82] pg. 480). Consequently, it is possible to carry out essentially "numerical error free" resolution folding using analytical or traditional methods. An additional benefit is that smooth, essentially numerical error-free resolution functions R(p\ (e.g. pavg, or most probable momentum) can be calculated as needed (see below). The "analytic" procedure consists of the direct integration of eq. 4.14. This requires a change of coordinates followed by an integration by quadrature. More specifically, eq. 4.7 is solved for 2 xpj J$min p\sin 0sing(p,0) Chapter 4. Modeling Finite Resolution Effects 57 Nom Avg Nom Avg p = 2° = 0° Nom Avg 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 Nom Avg Nom Avg 0 5 = 10° 1.0 0.6 75" 1.8 2.2 2.6 3.0 Momentum (au) Figure 4.3: Momentum resolution functions for EQ = 1215.8 eV and (i.e. the maximum of R(p; 4.4 Results The GW-PG, UW-PG, and Ap (Gaussian) methods are compared in this section. Ac cording to the fundamental resolution folding equation 4.10, the most direct comparison of theory and experiment is in ^o-space. However, for the reasons discussed above, it is usually desirable to consider the calculations and experimental results on a momentum like scale, which usually to this point has been assigned using the nominal momentum (eq. 4.8). Since the effects of several different types of resolution function are to be contrasted on the same graph, the nominal momentum will also be used to establish the momentum scale in the present work because pnom (unlike pavg) is independent of the resolution functions being compared. The main objective is first to find which of Chapter 4. Modeling Finite Resolution Effects 59 the previously proposed resolution functions brings theory into closest agreement with experiment, and in particular to investigate whether the good agreement found for H20 lbi and Ar 3p using the MAGW method [72] extends to a wider range of atomic and molecular orbitals. For reasons stated above, this will be carried out with the GW-PG part of the MAGW formulation. An explicit comparison of the GW-PG and MAGW methods is then made in order to clarify the significance of any differences between the two methods. And, finally, a brief investigation is made of the use of non-Gaussian angular resolution functions in planar grid-type methods. 4.4.1 Evaluation of the Ap, UW-PG, and GW-PG Models The highest occupied orbitals of the noble gases Ne, Ar, Kr, and Xe, as well as those of the hydrides H2O and H2S, were chosen for study because of their range of orbital ener gies and shapes as well as the high quality of available experimental data and theoretical calculations. The experimental data for the noble gases are taken from ref. [73]. The calculated (CI) distribution for neon is from ref. [53], and those for argon, krypton, and xenon (near Hartree-Fock limit SCF) are from [83]. It should be noted that CI and near Hartree-Fock limit SCF calculations for Ar 3p give essentially identical results [53,79]. The experimental data and CI calculations for H20 (109-G(CI) and 140-G(CI) calcula tions) and H2S (122-G(CI) calculation) are taken from refs. [28] and [33] respectively. Although all the distributions are p-type4, they differ significantly in the sharpness and position of their maxima and are a more demanding test of the resolution functions employed than would be the comparatively featureless s-type MDs. The GW-PG, UW-PG, and Ap resolution folding methods for the above range of atoms and molecules are compared in fig. 4.4. All functions are dimensioned as in ref. 4A p-type (unfolded) momentum distribution is one with zero intensity at zero momentum; s-type momentum distributions have their maximum intensity at zero momentum. Chapter 4. Modeling Finite Resolution Effects 60 (d) Xe 5p 12.6 eV a « 7 TO Perfect Resolution n « ZTZ UW-PG U V GW-PG I Experiment M v i- (b) Ar 3p H20 1b, ? 15.8 eV 12.2 eV 1 f \ f V c) 1 2 3 (c) (f) Kr 4p H2S 2b, 14.4 eV 10.5 eV J n Y \ 7 JL * # ^**«*i -4-'» '*», ». 1 3 0 1 Nominal Momentum (au) Figure 4.4: Comparison of experimental data and GW-PG resolution-folded near SCF limit or CI theory (see text, section 4.4.1, for details), (a) Ne 2p (CI), (b) Ar 3p (SCF). (c) Kr 4p (SCF). (d) Xe 5p (SCF). (e) H20 lbx (109/140-G(CI)). (f) H2S 2bx (122-G(CI)). Chapter 4. Modeling Finite Resolution Effects 61 [72]: i.e. A0 = 1.0° and A 4.4.2 Comparison of the GW-PG and MAGW Methods The GW-PG resolution folding method has some features common to both the MAGW method [72] and the UW-PG method [41,66,74,75]. For example, the GW-PG and MAGW methods differ only in their choice of x-axis (momentum) variable for use in comparing folded theory and experiment (section 4.2). The MAGW method, like the GW-PG method, calculates a folded theoretical intensity distribution as a function of (f>0 (i.e. s(^o), see eq. 4.10). Bawagan and Brion [72] chose to compare folded theory and experiment as a function of pavg, the average momentum (eq. 4.11), since they considered this to be a more realistic choice for a single value representative of the location of the experimentally sampled spread of momentum than the nominal momentum (eq. 4.8). However, in the present GW-PG method, a pnom scale has been selected for the momen tum scale to permit comparison of different folding methods and resolution functions on a common graph. As emphasized in section 4.2 and demonstrated for example in the case of Ar 3p (fig. 4.5 a-c) and H20 (fig. 4.5 d-f), these different scale choices (i.e. <^0, Pnom, or paVg) merely result in different horizontal deformations of the graph comparing theory and experiment while leaving the vertical dimensions (i.e. intensity at a given Ar 3p H20 1b, 15.8 eV 12.2 eV GW-PG GW-PG (mar SCF limit) (109/140-G(O)) I Experiment J KG 10 20 , 30 40 10 20 30 40 a>a (degrees) fa (degrees) 1- (b) « Ar 3p H20 1b, ?• 15.8 eV 12.2 eV s / \ it 1 >*• / \ k I %* o-/ V 1 2 0 1 2 . . 3 , Nominal Momentum (au) Nominal Momentum (au) (c) (0 Ar 3p H20 1b, 15.8 eV 12.2 eV / \ / \ \ i %< i i " 1 1 2 , 2 , v Average Momentum (au) Average Momentum (au) Figure 4.5: Comparison of GW-PG resolution-folded calculations (near Hartree-Fock limit SCF for Ar 3p and 109/140-G(CI) for H20 lbx) using three different independent variables (i.e. 4.4.3 Non-Gaussian Angular Resolution Functions The GW-PG (and also MAGW [72]) and the UW-PG methods used in the present work differ only in the shapes of the the GW-PG method. It is noteworthy that all three types of resolution function yield essentially identical values of the average momentum at any given value of 4.5 Conclusions The present work has demonstrated the importance of adequate consideration of the instrumental angular acceptance angles and response functions in the instrumental reso lution function when folding theoretical calculations of electron momentum distributions for comparison with EMS experiments. Such folding procedures are essential for all spectrometers if correct evaluation and design of atomic and molecular wavefunctions generated using high levels of theoretical quantum chemistry is to be achieved. It has also been clearly shown that commonly employed folding procedures employing constant width Gaussian functions directly in momentum space will lead to incorrect results, and thus possibly incorrect conclusions, particularly at lower momenta where the transfor mation between angular and momentum space is significantly non-linear. The GW-PG method and associated computer code used in the present work are readily applicable to all other EMS spectrometers. All that is needed is to input the appropriate polar and azimuthal acceptance angles, the electron energies and the response function. The present work suggests that the standard deviation of the response functions Chapter 4. Modeling Finite Resolution Effects 66 should be given by the acceptance angles. The analysis of a wide range of previous results for atoms and molecules using the procedures developed in this work has shown the close correspondence between EMS measurements and GW-PG folded high level quantum mechanical calculations. These findings are consistent with the Plane Wave Impulse Approximation model being a highly satisfactory interpretive reaction model for general studies using EMS. In the case of the HOMO orbitals of the highly polar second row hydrides such as the lbi orbital of H2O, the existing outstanding discrepancies between the highest levels of theory and exper iment [28] have been essentially eliminated at current levels of measurement precision when the GW-PG folding method is employed. For all other orbitals of the other atomic and molecular targets reanalyzed to date, the already quite good level of agreement be tween theory and experiment is even further improved when the identical GW-PG folding formulation is used. These results strongly suggest that the GW-PG angular resolution folding procedures provide an effective basis for the reliable comparison of experiment and theory. Application of the present procedures should permit the evaluation and de sign of new and highly accurate molecular wavefunctions for use in theoretical quantum chemistry and computer aided chemical applications. Chapter 5 Evaluation of Existing EMS Models: A Study of Acetylene 5.1 Introduction Acetylene (ethyne, C2H2) has long been regarded as an interesting benchmark molecule both because of its electronic structure and because it is the simplest hydrocarbon in the alkyne homologous series and thus one of the smallest and simplest triply bonded molecules. It is also of interest because of its unusual chemical reactivity, for example in the formation of metal acetylides. In addition polymers, such as polyacetylenes, are of growing importance due to their relatively high electrical conductivity. A detailed knowl edge of the electronic structure of C2H2, and in particular the momentum distributions of the valence electrons is of importance in understanding such chemical and physical phenomena. This prototype triply bonded molecule is also of considerable fundamental interest from the standpoint of theoretical quantum chemistry. In particular it is impor tant to obtain sufficiently accurate molecular wavefunctions suitable for calculations in all regions of phase space including the lower momentum (i.e. larger distance as weighted by the Fourier transform) regions which can play an important role in directing chemical reactivity. In these respects it is useful to obtain as detailed a knowledge as possible of the electron density distributions of the valence electrons in both position and momentum space. The ionization potentials and binding energy spectrum of C2H2 have been well char acterized by photoelectron spectroscopy at UV and X-ray energies using Hel [84], Hell 67 Chapter 5. Evaluation of Existing EMS Models: A Study of Acetylene 68 [85,86], Al Ka [87,88], and synchotron radiation [89-91]. In addition to the four verti cal IPs [84-86] due to the main 1TTU (11.4 eV), 3 (l7Tu) and C Is core orbitals, while the question of the valence shell satellite spectrum was not addressed. Calculations and data for the two orbitals were reported in the form of scattering cross-sections in polar coordinates rather than as momentum distributions. The various limitations and uncertainties in the earlier EMS studies [60,96-99] and the varied conclusions from the calculations [88,89,93-95] discussed above provided the orig inal motivations for the present work. A much improved experimental EMS study of the valence orbitals of C2H2 has also been reported by Weigold et al. [92] as the present study was approaching completion. While the experimental findings are generally in good agreement with the present data, only a single SCF calculation of the momentum distribution using a basis set of modest quality is shown by Weigold et al. [92]. In contrast, a wide range of quantum chemical momentum distribution calculations, em ploying much better basis sets within the Roothaan SCF, second order Green function, and CI methods, are included in the present study. Furthermore, a detailed experimental study and analysis of the inner valence region of the binding energy spectrum has been undertaken in the present work. 5.2 Experimental Method A detailed description of the symmetric non-coplanar EMS spectrometer has been given in chapter 3, and the theory necessary for interpretation of the calculations presented in this chapter has been discussed in chapters 2 and 4. The gas sample of C2H2 was obtained from a commercial cylinder (Matheson) and was used without further purification. No impurities were evident in the binding energy spectrum. -»«-*». Chapter 5. Evaluation of Existing EMS Models: A Study of Acetylene 71 5.3 Computational Details 5.3.1 Preliminary Details In order to compare the theoretical calculations with the experimental momentum profile, the full overlap expression (equation 2.7) or the THFA expression (equation 2.14) is folded with the experimental angular resolution functions according to the Gaussian Weighted- Planar Grid method and the resulting spherically averaged cross sections placed on the nominal momentum scale to give the Theoretical Momentum Profile, discussed in chapter 4 and reference [100]. The theoretical momentum profile has been referred to in earlier EMS publications as either the Overlap Distribution if correlation and relaxation are included (see equation 2.7) or the Momentum distribution if the THFA (see equation 2.14) has been applied. It is important to note that the quantity referred to above as the TMP is both spherically averaged and resolution folded. As such it is suitable for direct comparison with the experimentally determined average cross-section for randomly oriented molecules (i.e. the XMP). Quantum chemical calculations were carried out in Vancouver using GAUSSIAN76 [101] and a second-order Green function program developed by Casida, and in Blooming- ton by Davidson and Maxwell using the MELD [102] suite of programs. All calculations used the experimental D^ geometry [103]: R(CC) = 1.203 A, R(CH) = 1.061 A. All results reported in this work were calculated using basis sets of cartesian Gaussian- type and these basis sets have been given short names to facilitate discussion. However, the long names2 of the basis sets are given in Table 5.1. The distinction between the different types of basis sets used in quantum chemical calculations have been reviewed 2 The long notation {nt8,npp,ndd,njf/m,sympp) -*• [n't8,npp,n'dd,n'ff/m't8,m'pp] means n, prim itive Gaussian-type s functions on carbon contracted to n's Gaussian-type s functions, np primitive Gaussian-type p functions on carbon contracted to n'p Gaussian-type p functions, ..., m, primitive Gaussian-type s functions on hydrogen contracted to m', Gaussian-type s functions on hydrogen, Chapter 5. Evaluation of Existing EMS Models: A Study of Acetylene 72 Table 5.1: Total energies of C2H2 calculated using several basis sets and methods. The basis sets are more fully described in the text (section 5.3). Basis Set Total Energy Short Name Long Name Size (Hartree) Ref. STO-3G (6s3p/3s) -• [2slp/ls] 12 -75.8529 [104] DZ (10s5p/4s) -* [4s2p/2s] 24 -76.7992 [105] 34CGTO (Ils6p/Ss) -• [Ss3p/3s] 34 -76.8322 [106] 60CGTO (19s7pld/6slp) -» [7s4pld/3slp] 60 -76.8476 [107] 186CGTO (23sl2p3dlf/10s3p2d) -• [I4sl0p3dlf/10s3p2d] 186 -76.8555 [49] and this work Hartree-Fock Limit -76.860 [108] GF2 (60-CGTO) (19s7pld/6slp) -+ [7s4pld/3slp] 60 Not Applicable 186-G(CI) (23sl2p3dlf/10s3p2d) -* [14sl0p3dlf/10s3p2d] 186 -77.169443 [49] and this work 3The estimated non-vibrating, non-relativistic total energy is -77.3338 Hartree. in ref. [101]. The basis sets used in this work are briefly described below. STO-3G A minimal or single £ basis set [104], DZ A [4s2p/ls] basis set of effectively double ( quality due to Snyder and Basch [105]. 34CGTO A split-valence basis set of roughly triple ( quality for the valence orbitals due to Dunning [106]. 60CGTO A better than triple £ basis set with 3 p-type polarization functions on the hydrogens and 5 d-type polarization functions on the carbons. The original basis set was taken from the work of Salez and Veillard [107]. Contraction 10 from table 3 of that paper was used for the carbon s functions, contraction 8 from table 8 for the carbon p functions, and contraction 5 from table 9 for the hydrogen s functions. Note that the carbon s function contraction was applied to both the Is and 2s carbon functions in ref. [107] (i.e. a general instead of the usual segmented contraction was used.) The polarization exponents were a For hydrogen, a (10s3p2d) basis set was contracted to [6s3pld]. This same basis set was used previously in calculations on CH4 [49]. The Roothaan-Hartree-Fock self-consistent-field (RHF SCF) energies for neutral acety lene calculated using these different basis sets have been collected in table 5.1. The 186CGTO SCF energy (-76.8555 Hartree) compares very well with the near SCF limit energy of -76.8496 Hartree reported previously [108] and lies very close to the estimated Hartree-Fock limit of-76.860 Hartree [HO]4. 5.3.2 Post-Target Hartree-Fock Calculations and the Effects of Correlation and Relaxation Although the THFA is adequate for many purposes, it neglects correlation and relax ation effects present in the GOA. Recent studies such as the work of Bawagan et al. for H20 [28] show that correlation and relaxation must sometimes be included to obtain good agreement between experimental and theoretical momentum profiles. There are essentially two ways to do this. One way is to carry out two configuration interaction calculations (one on the ion and one on the parent) and then calculate the generalized overlap directly from the eq. 2.7. The other way is to carry out a single Green function calculation. Although these two methods can be shown to be equivalent under certain conditions [54], these conditions are almost never met in practice. This is particularly true in the present work where the CI calculations are of a very high quality while the Green function calculations were only carried out at the simplest (GF2) level. Neverthe less, it was of interest to see to what extent the simpler, less elaborate, Green function calculations would be able to produce results similar to the more elaborate CI calculations in the case of C2H2. The CI calculations reported here were carried out in Bloomington by Davidson and 41 Hartree = 1 a.u. of energy = me4/h? Chapter 5. Evaluation of Existing EMS Models: A Study of Acetylene 74 Maxwell using the MELD suite of programs [102]. For each basis set, a preliminary SCF calculation was carried out on the parent neutral acetylene. The results of this calcula tion were then used to construct K orbitals [111] for the parent from which singles and doubles CI configurations were constructed for both the parent and ion CI wavefunctions. First a frozen-core single reference singles and doubles configuration interaction (CI(SD)) calculation was performed on each state. All configurations from this calculation with coefficients larger than a cut-off ranging from 0.22 to 0.0325, depending upon the state, were kept as references in a second frozen-core multi-reference singles and doubles con figuration interaction calculation (MR-SDCI) on the parent. A larger cut-off was needed for ionic states with holes nearer to the core to keep the calculation manageable. The 2<7j hole required a 2-root calculation (to extract both the "main peak" and the "first satellite" in the binding energy spectrum), and in this case all configurations from the second root with coefficients greater than 0.045 were included along with all those from the first root with a cut-off of 0.0325. Second-order Rayleigh-Schrodinger perturbation theory was used to preselect the im portant configurations for the above calculations. Epstein-Nesbet partitioning was used. That is, the zero-order Hamiltonian was denned by diagonalizing the electronic Hamil- tonian over the reference space. This was followed by a second-order RSPT calculation of the correlation energy for all configurations outside the reference space. Only configu rations having a RSPT contribution to the energy exceeding 10-7 Hartree were kept for the subsequent variational calculation. The MR-SDCI energy for neutral acetylene was determined to be -77.16944 Hartree (accounting for 66% of the estimated total correlation energy or 85% of the estimated valence-valence correlation energy - see table 5.1). Vertical ionization potentials were then found by calculating energy differences between the parent and ion states. The resultant VIPs are collected in table 5.2. Chapter 5. Evaluation of Existing EMS Models: A Study of Acetylene 75 Table 5.2: Vertical ionization potentials for the principal ionic states of CaHj. Spectro scopic factors (S$j)) are given in parenthesis for the post-Target Hartree-Fock calculations (i.e. GF2 and MRSD-CI) and for experiment. Vertical Ionization Potentials (eV) 1 1 1 Method (ITT,,)- (3a,)- (2<7«)- (2CT,)"1 SCF/60CGTO 11.18 18.59 20.92 28.08 SCF/186CGTO 11.2 18.6 20.9 28.0 GF2/60CGTO 11.13(0.906) 16.34(0.861) 18.55(0.825) 23.67(0.718) 0.5 GF2/60CGTO5 11.16 17.46 19.73 25.87 MRSD-CI/186CGTO 11.3(0.817) 17.2(0.817) 19.2(0.740) 24.6(0.602) EMS, this work 11.406 (~1) 16.77 (~1) 18.97 (~0.83) 23.5 (0.5) 5Since the GF2 approximation has been found to reproduce trends in correlation and relaxation effects for similar molecules when uniform quality basis sets are employed, improved agreement with experiment can be obtained by including an empirical scaling factor (here 0.5) based upon studies of related molecules (ref. [54] and unpublished results). 6 The energy scale was calibrated using the PES value [84-86] for ionization from the l7ru orbital. 7 It should be noted that the XMPs (figures 5.3 and 5.4) measured for the 3(7g and 2 The second-order Green function calculations reported here were performed by Casida with a second-order Green function program developed at UBC. These calculations are much simpler than the CI calculations, but are expected to give results which are quali tatively similar to the CI results if the same basis set is used for both calculations [54,63]. In particular, GF2 calculations can be shown to be equivalent to low order wavefunction RSPT calculations [63]. However, unlike in wavefunction calculations, there is an explicit link in Green function calculations between VIPs and the corresponding generalized over lap orbitals. The ionization potentials (I) and the generalized overlaps (ip), are discussed in chapters 2 and 6. That is, they satisfy equation 2.12. In the present study, the self- energy is small and is treated as a perturbation. The ionization potential is given by the diagonal elements of the self-energy matrix in the basis of parent canonical Hartree-Fock Chapter 5. Evaluation of Existing EMS Models: A Study of Acetylene 76 orbitals -/ = e, + S,„(-/) (5.1) while the generalized overlap requires the off-diagonal elements, *, = *•+ iffc5*^. (5-2) * ei ~ e* The norm of the generalized overlap (i.e. the spectroscopic factor S = (V'l^')) is obtained from The self-energy describes all the many-body effects on ionization not present in the original Hartree-Fock theory. These effects are too complicated to treat exactly and approximations are required. The GF2 approximation used here is simply the lowest order, simplest approach. As discussed in ref. [55,56], the GF2 approximation may be roughly thought of as describing an electron moving in the dynamically polarized field of the other electrons. Since any systematic errors in the diagonal matrix elements of the self-energy are expected to be reflected in similar errors in the off-diagonal elements of the self-energy matrix, correlation and relaxation effects in ionization potentials and their corresponding generalized overlaps are clearly linked. This offers an important cross-check on the quality of the GF2 approximation. 5.4 Results and Discussion In the independent particle picture the ground state of C2H2 has the electron configura tion 2 2 2 2 4 (1*,)\1*1 (2 Binding energy spectra and momentum distributions will be discussed with respect to this configuration. The ionization potentials of the valence electrons are summarized in table 5.2. 5.4.1 Binding Energy Spectra Figures 5.1 (a) and (b) show the long range binding energy spectra (7-55 eV) of acetylene obtained at relative azimuthal angles of 20 30 40 50 60 Binding Energy (eV) Figure 5.1: The long range binding energy spectrum of acetylene at (a) 5.4.2 Experimental and Theoretical Momentum Profiles for the Valence Or- bitals of C2H2 The measured experimental momentum profiles are compared in figures 5.2 - 5.5 with theoretical momentum profiles calculated with the wide range of basis sets and methods presented in section 5.3. All TMPs have been folded with the experimental angular (or momentum) resolution using the GW-PG method discussed in chapter 4 and reference [100]. The XMP measurements corresponding to the 1TTU and 2 XMP for the 1TTU orbital was normalized to the (best fitting) resolution folded 186-G(CI) calculation by least squares fitting below 1.5 a.u.. This single normalization of experiment and (186-G(CI)) theory for the l7ru orbital then permits all four orbital measurements and all corresponding resolution folded calculations (table 5.1) to be quantitatively compared on a common intensity scale. Also shown on figures 5.2 - 5.5 are the respective momentum and position density maps for an oriented C2H2 molecule, calculated using the Salez and Veillard 60-CGTO basis and the Roothaan Hartree-Fock method. The side panels of the density maps show the density profiles corresponding to the respective dashed lines on the maps. Chapter 5. Evaluation of Existing EMS Models: A Study of Acetylene 80 In comparing the measured and calculated momentum distributions the following gen eral observations can be made. First, the STO-3G minimum basis set gives a calculated distribution which is in rather poor agreement with experiment for the 1TTU orbital (figure 5.2). In contrast the other SCF calculations using the Snyder and Basch [105], Dunning 34-CGTO [106], Salez and Veillard 60-CGTO [107], and 186-CGTO basis sets all give a quite reasonable and fairly similar description for the shapes in the case of each XMP (see figures 5.2 - 5.5). This is in marked contrast to the situation for the outer valence orbitals of the row 2 hydrides such as NH3 [42], H20 [28], and HF [50,71], which exhibit considerable changes in the TMPs as the basis set is improved. However the situation for C2H2 is quite similar to that for the non-polar saturated hydride CH4 [49], where SCF calculations at the double zeta level and above give very good descriptions of the XMPs. It should also be noted that both the intensities and shapes of the liru (fig. 5.2) and 3(7g (fig. 5.3) orbitals are accounted for in a very satisfactory manner, particularly by the better basis sets. Therefore in these cases the pole strength is essentially unity and thus the independent particle description is evidently quite adequate for these momentum dis tributions. It is clear, however, that appreciable experimental intensity is missing from the XMPs of the 2cru and in particular the 2crg orbitals. Evidently the pole strengths are less than unity for the main 2 The individual momentum distributions and the "missing intensity" in the 2 Consider first the l7ru (HOMO) orbital of acetylene (figure 5.2). It is clear that all calculations except the STO-3G reproduce the p-type shape of this orbital quite well, although the simple Snyder and Basch [105] and also the Dunning 34-CGTO [106] Chapter 5. Evaluation of Existing EMS Models: A Study of Acetylene 81 C2H2 1 7TU 4c 11.4 eV 3 5c 5c 186-G(CI) 5 186-GTO 4c 60-GTO (GF2) 4 60-GTO 3 34-GTO 2 Snyder and Basch r 1 ST0-3G Area of BES peak I Experiment 0 J 2 Nominal Momentum (a.u.) MOMENTUM DENSITY POSITION DENSITY CgHg JL 1TT„ JM^ 1 . .I 1 i.i.i • 11 -^r -so s • -•-SO 3 • (0.11.) (0.0.) Figure 5.2: Experimental and theoretical momentum profiles for the lirn (HOMO) orbital of acetylene, measured at a binding energy of 11.4 eV. The spherically-averaged calcu lations have been resolution-folded using the GW-PG procedure described in chapter 4 and reference [100]. A common intensity scale for all measurements and calculations in figures 5.2 - 5.5 has been established from the peak areas in the binding energy spectrum as described in section 7 of the text. The square on the diagram is derived from the binding energy spectrum at The measured and calculated momentum distributions together with the density maps for the strongly bonding 3ag orbital are shown in figure 5.3. The dominant transverse components in the momentum density map reflect the strongly a bonding character of this orbital clearly seen in the (more familiar) charge density maps. The measured XMP indicates that the 3 4c 3 4 30 - \ 5c C2H2 3(7g 16.2 eV 2 01 1 - l\ ft 5c 186-G(CI) a5 20 5 1B6-GT0 4c 60-GTO (GF2) 4 60-GTO 3 34-GTO 2 Snyder and Basch 1 STO-3G J2 10 \ i Area of BES peak Experiment • ^y I 0 0 \ 3 Nominal MomentuTm (a.u.~ •) MOMENTUM DENSITY POSITION DENSITY C^Hg A 3fff -3- -6 w-3 0 3 (o.u.) Figure 5.3: Experimental and theoretical momentum profiles for the 3(Tg orbital of acety lene, measured at an energy of 16.2 eV. A common intensity scale for all measurements and calculations in figures 5.2 - 5.5 has been established from the peak areas in the bind ing energy spectrum as described in section 5.4.2 of the text. The square on the diagram is derived from the binding energy spectrum at 20- c 5c 186-G(CI) 5 186-GT0 4c 60-GTO (GF2) 4 60-GTO 10- 3 34-GTO 2 Snyder and Basch 1 STO-3G o 1 Area of BES peak 5- en I Experiment 0 *^r- 0 T I Nominal Momentum (a.u.) MOMENTUM DENSITY POSITION DENSITY CgHg Cgrig 2cr„ 2au : A »" n r: -3 0 -S9O 3 C (a.u.) Figure 5.4: Experimental and theoretical momentum profiles for the 2 60 50 >0- 186-GC 186-GT0' c 60-GT0 (GF2) 60-GT0 34-GTO c30 Snyder and Basch ST0-3G •-P20 I 23.5eV peak only JO I (21 to 55eV) - 0.21(2 0 ii ix xi 0 Nominal Momentum (a.u.) MOMENTUM DENS ITY POSITION DENSITY CgHg A 2 i 3 0 > -3 ;-fH|>1 - 1 <=r> -3036 (a.u.) r igure 5.5^ Experimental and theoretical momentum profiles for the 2 1 associated with 2c-" ionization since all the intensity for the largely s-type 3 5.1) and represent the estimated intensities for the 2 First, the 2 (^o = 0.5° and 2crg peak. It can be seen from figure 5.5 that the experimental intensity multiplied by 2.05 (the solid triangles) is generally quite well reproduced in shape and magnitude by Chapter 5. Evaluation of Existing EMS Models: A Study of Acetylene 88 all calculations. The fact that the 186-CGTO and the 186-G(CI) calculations are slightly higher than experiment may indicate that a small portion of the overall 2c"1 intensity is located at even higher binding energies above the limit of the present experimental data (see figure 5.1). The above analysis clearly indicates that the small balance of the 1 2c" intensity, over and above the main 2cru peak at 18.9 eV is located together with 1 the 2(7~ satellite intensity to higher energy above the main 2ag peak at 23.5 eV. In the following section angular distributions are studied in the satellite spectrum to investigate the energy localization of the 2er~1 satellite intensity. 5.4.3 Detailed Studies of the Binding Energy Spectra and Momentum Dis tributions in the Inner Valence Region The inner valence region of the binding energy spectrum of acetylene and the origin of the satellite structure have been the focus of numerous earlier experimental and theoretical studies. A variety of earlier PES [89,91] and XPS [87,88,113,114] experiments have revealed partially resolved satellite structure at binding energies in the 26 to 31 eV region. The low-resolution EMS study of C2H2 reported by Dixon et al. [60] had also shown a considerable concentration of highly unresolved satellite intensity above 25 eV and out to 47 eV. In the most recent and highest resolution XPS study, Svensson, Zdansky, et al. [87] reported five satellite peaks at 26.6, 28.0, 29.9, 31.2, and 33.4 eV in the inner-valence region. The positions of these structures have been used as an aid in the interpretation and deconvolution of the angular resolved EMS inner valence binding energy spectra obtained at lower energy resolution in the present work and which are discussed below. As mentioned in the introduction (section 5.1), several widely different theoretical interpretations of the origins of the satellite structures in the inner valence region of the binding energy spectrum of acetylene have been published [88,89,93-95]. Bradshaw et al. [89] used the 2-particle hole Tamm-Dancoff approximation (2ph-TDA) Chapter 5. Evaluation of Existing EMS Models: A Study of Acetylene 89 and a [9s 5p + 4s/4s 2p + 2s] basis set in Green function calculations which predicted four satellites at 26.24, 27.21, 28.85, and 29.45 eV of eru, crs, crg, and crg symmetries respectively. Muller et al. [88], using complete active space multiconfiguration SCF (CASSCF) calculations with a [9s 5p / 5s] basis set contracted to [4s 2p / 2s], have predicted two satellites in this region, at 26.6 and 27.81 eV, both of crg-type symmetry. Chong [94] has used the HAM/3 semiempirical method to predict four satellites at 26.91, 28.53, 30.96, and 33.01 eV, of (Tg, Binding Energy (eV) Figure 5.6: Binding energy spectra in the inner valence region (25 - 34 eV) of acetylene, (a) High resolution XPS spectrum as reported by Svensson, Zdansky, tt al. [87], (b) EMS spectra summed over all angles, (c) - (i) EMS spectra at The 2cru intensity may be spread out over many small poles in the binding energy spec trum. However, significant p-type contributions in localized energy regions can only be investigated by a more detailed analysis of many-angle EMS binding energy spectra such as those shown in figure 5.6. It should be noted that essentially all the 3crg and a major portion (83%) of the 2au ionization strength has already been accounted for. Therefore since the 23.5 eV peak is due to a portion (~50%) of the 2 5.6(a)) has been fitted with Gaussian peaks of 2.1, 2.5, and 2.7 eV fwhm respectively at each of these three energies. The individual spectra (figure 5.6(b) - (h)) were then fitted with the same template of peak widths and energies, with the heights being determined by the fitting program. From figure 5.6 it is clear that the three peaks fitted to the EMS angular resolved spectra correspond respectively to peaks 1, 2, and (3+4) of the high resolution XPS spectrum [87]. The extra unfitted intensity in the EMS spectra above ~32 eV clearly corresponds to peak 5 at 33.4 eV in the high resolution XPS spectrum [87]. The binding energy spectra shown in figure 5.6 have been further analyzed in two slightly different but complementary ways in order to try to ascertain whether any sig nificant p-type (i.e. 2 31.Oil eV, and 33.Oil eV are shown as a function of Veillard 60-CGTO SCF calculations of the cross-section plotted on the 2<7g orbital, separately height normalized to the data in each case. These results clearly indicate that the dominant portion of the satellite intensity in the 25 - 34 eV region is due to the 2a-"1 process. The 2a~x satellite intensity is therefore presumably spread out in many small poles somewhere over the 25 - 55 eV satellite spectrum. These findings are entirely consistent with the conclusions resulting from the consideration of the measured and calculated momentum profiles. Similar general conclusions have also been reported in the very recent EMS study of C2H2 by Weigold et al. [92]. The more precise location of the relatively small "missing" portion of the 2a"1 intensity may possibly be revealed Chapter 5. Evaluation of Existing EMS Models: A Study of Acetylene 93 Inner Valence Angular Distributions 31.0 eV Peak (IH+IV) cp0 (degrees) Figure 5.7: Angular distributions in the inner valence region (25 - 34 eV) of acetylene, (a - d) Areas of sections and (e - g) areas of fitted Gaussians as a function of X tribution from 2crg ionization in the satellite region above 25 eV with ~17% of the 2 5.5 Conclusions The present work has shown that initial (ground) state electron correlation effects are not a determining factor in the valence orbital momentum distributions of the hydrocarbon C2H2. Good SCF wavefunctions of DZ or better quality give an adequate description and should be suitable for describing phenomena involving the low momentum (outer spatial) regions of the electron distribution. This finding is similar to that observed for the saturated hydrocarbon CH4 and the atom Ne, but is in marked contrast to the situation for the HOMOs of the highly polar second row hydrides NH3, H20, and HF, where electron correlation effects must be included in order to reproduce the measured momentum distributions. These differences may be associated with the fact that (i) the electronegativities of N, O, and F are high compared to those of C and H (ii) C2H2, CH4, and Ne are non-polar whereas NH3, H2O, and HF are highly polar molecules (iii) All valence orbitals in C2H2, and CH4 have bonding (or anti-bonding) character, and are somewhat delocalized over the entire molecular framework (see figures 5.2 - 5.5), whereas the HOMOs of NH3, H20, and HF are essentially non-bonding molecular orbitals localized on the hetero atom (i.e. lone pairs in the Valence Bond description). _1 Final state correlation effects are found to play a significant role in the (2cru) and -1 (2crg) inner valence ionization processes. These effects are manifested in the missing intensity (i.e. S& < 1) in the inner valence momentum distributions and also in the extensive satellite structure present in the inner valence binding energy spectrum. The j) -1 x si values of 0.83 and 0.50 indicated by the intensities of the main (2 -1 C2H2 ionization are dominantly associated with (2crg) processes. Chapter 6 Density Functional Theory At this point it is necessary to introduce some of the concepts of density functional theory, as these are required for use in subsequent chapters. This chapter will introduce the DFT formalism and present its computational implementation and the assumptions involved. Only the closed-shell formalisms will be discussed, as all molecules discussed in this thesis are closed-shell. Modern density functional theory has been shown by Casida [118] to provide, in the limit of an exact exchange and correlation potential, the variationally best local approximation to the non-local self energy of Green Function theory (see also chapter 7). Computationally, however, its implementation is very similar to that of conventional Roothaan-Hartree-Fock [16], so DFT will be presented from that perspective in this chapter. The relationship of DFT to Green Function theory will be discussed in chapter 7 as it pertains to relating the Kohn-Sham orbital to the EMS cross-section. 6.1 Hartree-Fock Theory This section will present the necessary background for an understanding of Hartree- Fock theory (and, subsequently, density functional theory). To better understand the approximations used in conventional DFT, the more familiar Hartree-Fock total energy and Hamiltonian will be presented first. The Hartree-Fock total energy may be written 96 Chapter 6. Density Functional Theory 97 in the following form in the spatial molecular orbital representation: 2H 2J K 6 1 EHF = (*HF I H | 9HF) = E i + E ( V ~ V)' C - ) i occ. i,j occ Hi, Jij, and Kij are the kinetic energy and all potential energies, the Coulomb integral, and the electron exchange integral, respectively. They have the following forms: Hi = J V* (r)[-|v2 + v(r))1>i(r)dr (6.2) Jij = j J HriWi{rx)j-rj{r2)^{r2)dndr2 (6.3) Kij = J j rMWji^j-^Wjir^dridn (6.4) In the absence of any external field, the v(r) term in equation 6.2 is simply the electron- nuclear attraction (sometimes referred to as the external potential in DFT). It has the form »(r) = -ET' ^ o where "a" are the nuclei in the molecule. A quick inspection of equations 6.3 and 6.4 shows that, for any pair of electrons in a single orbital ^i(r), Ka = «7,-,-, which is the reason that the sum in equation 6.1 may include the i—j terms. The exchange operator is central to the difference between DFT and Hartree-Fock and will appear again later. If equation 6.1 is minimized with respect to the occupied orbitals (subject to the con straint that fij>i(r)il)j(r)dr = 6{j), the Hartree-Fock differential equations are obtained: Hi(r) = E fe- (6-6) j occ Here, p = -±V* + v + g (6.7) and the Coulomb and exchange operator g is given by g = 2j - k (6.8) Chapter 6. Density Functional Theory 98 A A The operators j and k have the explicit forms J(n)/(rtf - E / tffoWMr-ZCri)** (6-9) and *(ri)/(n) = E f+iMfM—Hri)dr2. (6.10) If the orbitals (and hence the Fock operator as well) in equation 6.6 are subjected to a unitary transformation (the matrix e is hermitian, and so permits this), then a set of orbitals may be chosen which diagonalizes the matrix e. Multiplication of the transformed form of equation 6.6 by ip*(r) and integration gives "orbital energies" as the diagonal elements of the (Hermitian) matrix e. 6.1.1 Computational Implementation Because the Roothaan-Hartree-Fock method employs linear combinations of atomic or bitals, it is implemented as a matrix eigenvalue problem. Specifically, a set of coefficients C is desired which diagonalizes the Fock matrix, F. That is, a matrix of atomic orbital coefficients is desired which satisfies FC = CE. (6.11) The basis functions used in computational problems are usually not orthogonal, however, and so the overlap matrix, S, is also introduced: FC = SCE. (6.12) As the Fock operator is constructed from the coefficients upon which it depends, equations 6.11 and 6.12 (the equation solved depends on the form chosen for the atomic orbitals) must be solved by iterative methods until self-consistency (within some tolerance) is achieved. Chapter 6. Density Functional Theory 99 6.2 Density Functional Theory The Hartree-Fock formalism has two very well-known shortcomings: It neglects cor relation beyond that predicted by mean-field theory, and the exchange and Coulomb potentials are difficult to calculate, as they (formally) increase the complexity of the calculation as N4, where N is the number of basis functions in the calculation. Density functional theory addresses both of these limitations by adopting a different form for the exchange potential, and by including correlation in the DFT analogue to the Fock operator. The first calculation resembling those of modern DFT was carried out by Slater in 1951 [22], who sought at the time to simplify the calculation of the exchange operator in equation 6.10 above by simply substituting the expression for the exchange potential of a free electron gas (due to Dirac [23]) in its place. It was not until the work of Kohn and Sham in 1966 [25] that Slater's approach was linked to density functional theory as proposed by Hohenberg and Kohn in 1964 [24]. Density functional theory as proposed by Hohenberg and Kohn focused not on the wave- function, but on the electron density, and methods for computational implementation were not readily apparent if only the electron density were to be used. Kohn and Sham proposed and formally justified the use of one-electron orbitals in the construction of the electron density. In conjunction with this, they employed a different partitioning of the total energy, expressing it (in a similar manner to Hohenberg and Kohn) as a functional of the electron density: E[p) = Ta[p) + J[p) + Exc[p) + J p(r)v(x)d(v). (6.13) Equation 6.13 is exact. In it, Ts[p] is the (approximate) Sham kinetic energy (of the same form as the Hartree-Fock kinetic energy), J[p] is the classical repulsion energy between different charge densities (of the same form as equation 6.3, and //a(r)u(r)c?(r) is the Chapter 6. Density Functional Theory 100 energy from any external potentials present, usually just the electron-nuclear attraction energy (as in equation 6.2). The principal difference between the DFT energy and the Hartree-Fock energy comes in the term Exc[p], which represents the exchange and cor relation energy, as well as the difference between the Sham kinetic energy and the true kinetic energy of the system. Indeed, if the expression for the exchange energy given in equation 6.4 above were used for an approximate Exc in equation 6.13 above, equation 6.13 would simply be an alternative expression for the Hartree-Fock energy. Minimiza tion of equation 6.13 subject to the constraint that the density must integrate to the number of electrons in the system under study gives the Kohn-Sham equations: htdi = E fe- (6-14) J occ Equation 6.14 bears a striking resemblance to equation 6.6, and indeed many of the constituent components of the Kohn-Sham "Fock operator" are of exactly the same form: 2 FKs=-^ + vefJ. (6.15) Here, ue// is the external potential v(r), the Coulomb repulsion operator, and the ex change and correlation potential. The external potential and Coulomb repulsion operator have exactly the same form as in the Fock operator defined in equation 6.7. The exchange and correlation potential is new and will be discussed further below. Just as for equation 6.6, equation 6.14 may be subjected to a unitary transformation which diagonalizes the matrix e, thus yielding orbital energies as its diagonal elements: (hH[p] + Ml; P]) ti*(l) = ef VP(1) • (6.16) The equation has been re-expressed in terms of the electron density for later purposes. Chapter 6. Density Functional Theory 101 6.3 Computational Implementation Density functional theory is, as was suggested in section 6.2, implemented in a manner similar to Hartree-Fock. The equation to be solved is of exactly the same form as equation 6.12 for the Hartree-Fock case, only the "Fock matrix'' to be diagonalized differs from that of Hartree-Fock theory by the form of the exchange and correlation potential. Before the Kohn-Sham equations may be solved, however, two key additional approx imations must be made in the calculation of the "Fock matrix". Specifically, a functional form for the exchange and correlation potential must be chosen and, as a consequence of the functional form chosen (see the next section), either a numerical grid or an auxiliary basis set must be chosen to aid in the evaluation of the chosen form for the exchange and correlation potential. 6.3.1 Approximations to the Exchange-Correlation Potential There exists no known closed form for the exchange and correlation potential. As a result, approximations must be made and a functional form for the potential chosen before any calculations may be performed. These approximations fall into two general classes: local and non-local density approximations. Local density approximations (LDAs) are those which use a single point in space to calculate the exchange and correlation potential, while non-local density approximations use a region of space and not just a single point in the calculation of the potential. This section will discuss some common approximations used, with an emphasis placed on the local density approximation, as this is the most common functional type for which results are reported in the DFT literature. Extensions to the local density approximation will be discussed in a separate section. The results presented in this thesis have been calculated using three approximate functionals, the first two of which are local in nature: Xa [25], VWN [119], and a combination of a non-local exchange Chapter 6. Density Functional Theory 102 potential by Becke [120] with a (non-local) correlation potential by Perdew [121]. Each of these will be discussed below. 6.3.2 The Xa Potential The Xa potential is the closed form for the exchange potential (or energy, depending on the value of a used) of a homogeneous electron gas. It was derived by Dirac in 1930 [23] as an extension to Thomas-Fermi theory [19], and first used by Slater in his approximate Hartree-Fock calculations in 1951 [22]. It has the form ^(r) = -|a{f/>(r)}1/3. (6.17) Slater used a value of 1 for a, to match the exchange-correlation potential for the homo geneous electron gas. Kohn and Sham [25] used 2/3, to match the exchange-correlation energy. Others have allowed a to vary so that the total energy of the density functional theory matched that of a Hartree-Fock calculation for the same system. The Xa potential is a local potential, as the exchange energy at a given point in space depends only on the electron density at that point. The Xa potential as proposed above is very simple to implement and not compu tationally intensive, and so was very commonly used in the early DFT literature. It has disadvantages, however, and these will be discussed briefly here. First, it neglects correlation beyond the mean-field theorem. As a physical model, therefore, this makes it at best no better a choice for an exchange potential than the Hartree-Fock exchange operator. Second, the exchange potential in the Xa approximation falls off too rapidly at large interelectron distances. This causes an electron in the large r region of space to feel the full (non-self-interaction-corrected) potential of the Coulomb repulsion term. This has the effect that the charge distribution becomes more diffuse than it would be with the proper self-interaction-corrected potential. This is discussed further in chapter Chapter 6. Density Functional Theory 103 7. Third, the Xa potential tends to overbind atoms in molecules. Fourth, and finally, the X„ potential is not physically very realistic. The potential energy it produces varies more smoothly than in real atomic and molecular systems. As a consequence, not all properties derived from the density are properly predicted (see chapter 9). 6.3.3 Local Potentials Including Correlation Local functionals including correlation have been proposed by several different authors. The local functional of primary interest in this thesis is due to Vosko, Wilk, and Nusair [119], who simply used the X« potential and added a functional form for the correlation potential which was derived from the work of Ceperly and Alder [122]. Ceperly and Alder performed quantum Monte Carlo calculations on homogeneous electron gases at varying gas densities to obtain the correlation energy for a homogeneous electron gas, from which Vosko et al. derived their correlation functional by performing a Pade (least squares) fit. The resulting VWN functional has the advantage that it now includes correlation, and (perhaps in large measure due to this) yields reliable one-electron properties in the larger r region of space (see chapter 9). However, the VWN exchange and correlation potential still tends to overbind molecules, and still falls off too rapidly at large r, indicating that it is still not self-interaction corrected. 6.3.4 Non-local Exchange and Correlation Potentials The problem of incorrect asymptotic exchange and correlation potential behaviour has been addressed by many workers. Most proposed solutions involve the use of so-called gradient corrections to the potential, meaning that derivatives of the charge density are added to the potential expression of interest in a manner analogous to a Taylor series expansion. When gradients of the density are added in this manner, the resulting potential is said to be nonlocal, as the derivative terms have the effect that the calculated Chapter 6. Density Functional Theory 104 potential reflects not just the charge density at one point in space but in a region about that point. An example of this is the nonlocal Xa/g (exchange only) potential due to Herman and co-workers [123,124], which has the form 1/3 6 18 »..,« = [-§« + WG(P)] {^W} . ( - ) where G(p) is a function involving gradients of the electron density. The parameter fiwa s determined by least squares fit to the Hartree-Fock exchange energy [125]. The nonlocal approach of primary interest to this thesis, however, is that taken by Becke [120]. Becke added gradient corrections and performed a Pade (least squares) fit to determine gradient coefficients, and ensured that the exchange energy density, defined by Ex = Jex(p(r))p(r)dr (6.19) (ex (p(r)) is the exchange energy density), showed the correct behaviour for large r. The same cannot be said with certainty for the exchange potential which arises from this method (as the functional derivative of the exchange energy), however. The correlation potential (treated independently of the exchange potential) of primary interest to this thesis was proposed by Perdew [121], from wave vector arguments outside the scope of this thesis. Just as for Becke's exchange potential, Perdew's correlation potential required coefficients for the gradient expansion which were derived using least squares fits. It is interesting to note that in both the local (VWN) and non-local (Becke and Perdew) cases, the correlation potential was treated as an independent quantity from the exchange potential, so that the resulting exchange and correlation potential was simply the sum of the two potentials: vxc = vx + vc. (6.20) The problem of electron self-interaction has been addressed on its own by, for example, Perdew and Zunger [126], and by workers addressing the long-range behaviour of the Chapter 6. Density Functional Theory 105 exchange potential. As no explicitly self-interaction-corrected functionals were used in the work presented in this thesis, they will not be discussed further. 6.3.5 Evaluation of the Exchange and Correlation Potential Any functional form of those discussed above gives rise to integrals involving fractional powers of the electron density. This in turn gives rise, given that deMon (the density functional theory program used to obtain the results presented in this thesis) uses orbital basis sets (see chapter 2), to terms involving cube roots of sums of Gaussians. Integrals over such terms are impossible analytically, and so a numerical approximation must be made to facilitate their evaluation. One of the most common methods employed, and the method used by deMon, is to evaluate the exchange and correlation potential at a series of points about the molecule called a grid. Once this is done, a secondary basis set, called an auxiliary basis set, is fitted to the calculated potential at these points via least squares. The auxiliary basis sets used in deMon both consist of Gaussian primitives, and are collectively described by the notation (j,k;m,n), where j and k are the number of s-type and sets of s-,p-,d-type GTOs for the charge density fit, and m and n represent sizes of similar types of bases for the exchange-correlation fit. Each set of s-,p-,d-type GTOs has a common exponent for all £ values. Fitting the potential in this manner reconverts the exchange and correlation potential to a simple sum over Gaussians, and so provides for simpler integral evaluation. The Coulomb potential is computationally demanding as well, so the charge density is also fitted with a (different) set of auxiliary functions to expedite integral evaluation. The fit quality and precision of the calculated total energy therefore depend both upon the quality of the grid and on the choice of auxiliary basis sets. Generally, larger auxiliary basis sets will provide more accurate results, as will finer grids, though special attention must be paid to the former as inappropriately chosen auxiliary functions will hinder Chapter 6. Density Functional Theory 106 convergence. Chapter 7 EMS from DFT: The Target Kohn-Sham Approximation 7.1 Introduction Highly accurate Dyson orbitals can be obtained from configuration interaction calcula tions, and the MDs of these Dyson orbitals are in quantitative agreement with experi ment. However, this level of accuracy is computationally demanding, nominally scaling as 0(N5) or worse, where N is the size of the basis set. This limits the practical utility of CI for routine EMS calculations to very small molecules. One way around this problem is to use Hartree-Fock to approximate Dyson orbitals, through the "target Hartree-Fock approximation". The quality of the resulting MDs is, of course, lower than that obtained from CI, but the accuracy is usually quite adequate for the needs of EMS. Since HF scales nominally as 0(N4), this allows EMS calculations for larger molecules than is possible with CI but, especially in view of the need for extended basis sets for calculating MDs, this is still too restrictive to treat many problems of interest to electron momentum spec- troscopists. Thus a computationally less demanding method is needed in order to realize the advantages of EMS for larger molecules. In this chapter, the extension of DFT to the calculation of approximate EMS cross-sections is described. Density functional theory would seem to be a natural choice, since Kohn-Sham DFT calculations which use an auxiliary basis of M oc TV functions scale nominally as 0(iV3), and the quality of the results for a wide variety of properties is generally comparable to or better than Hartree-Fock [127-141]. However, using KS DFT to calculate MDs for 107 Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 108 EMS is not simply a matter of calculating yet another property from DFT, because it involves using KS DFT to approximate Dyson orbitals. In their formulation of DFT [25], Kohn and Sham introduced orbitals purely as a convenient mathematical construct for simplifying calculations rather than as physically meaningful quantities. The observation that Kohn-Sham orbital energies provide rather poor estimates of ionization potentials and bandgaps only seemed to emphasize the fictitious nature of these orbitals. However, there is a formally different approach to DFT in which the true Kohn-Sham orbitals (i.e. those that would correspond to the exact exchange-correlation potential) arise as approximations to Dyson orbitals [118,142-144]. Unfortunately, this formal connection says nothing about the quality of the approximation, which must therefore be assessed computationally, as must the effect of the further approximation involved in using Kohn- Sham orbitals obtained from approximate functionals. The formal sense in which the KS orbitals approximate Dyson orbitals (or equivalently Hartree-Fock orbitals, in the exchange-only case), as well as the available numerical evidence, is reviewed in the next section. The primary goal of this chapter is to investigate DFT as a potential computational tool for use in EMS. Yet in assessing the quality of MDs obtained from KS orbitals by comparison with those from high quality CI calculations and with experiment, the present work also makes a small contribution to the larger question of the quality of the approximation of Dyson orbitals by KS orbitals. 7.2 Theoretical Background The formal justification for using Kohn-Sham orbitals as approximate Dyson orbitals for use in the theory of electron momentum spectroscopy is presented in this section. First, however, the relevant properties of Dyson orbitals in the context of the quasiparticle Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 109 equation will be reviewed. The Kohn-Sham density-functional formalism is then summa rized, and the section ends with a discussion of the sense in which Kohn-Sham orbitals approximate Dyson orbitals. 7.2.1 Dyson's equation and target approximations The direct method for obtaining Dyson orbitals and ionization potentials by solving Dyson's quasiparticle equation [58] is reviewed here, as is the important notion of a target approximation. A wavefunction-based explanation of Dyson's quasiparticle equation may be found in ref. [54]. Dyson's quasiparticle equation, (hH[p] + £xc("ij) simultaneously describes vertical ionization of the molecule M, M -+ M+ + e- 7 (7.2) •r *f -1) and electron attachment, M + e~ —-• M~ (7.3) *f> #("+!) ' Here, = -Iv> -1- ii(r \4- f P&) r-r2 is the usual Hartree hamiltonian expressed in terms of the density, p, and v is just the nuclear attraction potential (plus the external field, if any). The term Sxc(u>) is the exchange-correlation (xc) self-energy operator which includes many-body effects such as correlation and relaxation [63] and dynamic polarization [55,56] as well as a correction for the self-interaction error in the Coulomb part of the Hartree hamiltonian. Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 110 Dyson's equation is a generalized eigenvalue problem whose solutions fall into two classes. If the /th ionization potential and electron affinity are denoted by 2/ and Ai, respectively, the "ionization solutions" satisfy u>/ = -J/, (7.5) 1 N Ml) = VNjJ..j¥?- >(2,3r--,N)*i \h2,-.-,N)d2--.dN, (7.6) while the "electron affinity solutions" satisfy wj = -Ai, (7.7) V>/(l) = V/JVTT//---/^2,3,.--,JV^^ (7.8) There are an infinite number of Dyson orbitals, ^>j, and their norms, i.e. the spectroscopic factors Si, may be calculated from the energy derivative of the xc self-energy, 1 Si = {il>i\i>i) = ftMft,("j)hhr~ (7.9) A Dyson's equation must be solved self-consistently because the xc self-energy, Sxc(w/), is a function of the orbital energy, w/, and the Hartree hamiltonian, hii[p], depends upon the orbitals through ionization p(i)= E HMi)l2. (7.io) The HF equation is a limiting case of Dyson's equation where the self-energy is ap proximated by its exchange-only part, HF F (2)X(2) SX X(1) = - ^E °> (1) / ^ d2, (7.11) defined here by its action on an arbitrary function x Since this self-energy is independent of w, the Dyson orbitals are normalized to unity (Eq. (7.9)) in the HF approximation. It Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 111 then follows (Eq. (7.10)) that the integral of the density, p, gives the number of ionization solutions. Hence, since the density integrates to the number of electrons, there must be exactly one ionization solution (i.e. exactly one occupied orbital) for each electron. The approximation of the ionization potentials by the negative of the Hartree-Fock orbital energies gives the Koopmans' ionization potential, F W,«ef . (7.12) The frozen orbital approximation consists of taking the Dyson orbitals to be HF orbitals, 0,-««&P\ (7-13) Note that, at this level of approximation, the index /, which labels the many-electron state of the daughter ion formed can be replaced by the label i, of the orbital out of which ionization occurs. The FOA is qualitatively incorrect. Since it gives only N Dyson orbitals, all normalized to unity, it cannot account for the existence of "satellites", nor for the intensities, in the binding energy spectrum (see chapter 2). While the HF approximation has some value in describing outer valence ionization, more elaborate and accurate self-energy approximations [116,145-149] are needed to describe the complex many-body phenomena seen in inner valence ionization spectra [28,33,42,47-53]. The corrections to the HF independent-particle picture of ionization which arise from the use of better self-energy approximations may be analyzed using simple Rayleigh-Schrodinger perturbation theory [63,150]. Any independent particle Schrodinger equation, hfa = ti where, u> = A-&j/. (7.16) Note that w = vxc if the zero-order hamiltonian is the Kohn-Sham hamiltonian, and that w = S^F if the zero-order hamiltonian is the Hartree-Fock hamiltonian and differences between the true and Hartree-Fock densities are ignored. Once again, the index J labels a many-electron ion state while the index i labels an orbital. The nonlinear nature of Dyson's equation means that many Is can correspond to the same i. For each zero-order solution, {e,-, <£,}, the first-order equation, 4° = « + ( can be solved for a set of ujj. This corresponds to the experimentally observed frac tionation of principal ionization transitions into many "satellite" processes. Each value of uy can then be used to generate a first-order solution for the corresponding Dyson orbital and spectroscopic factor, & m * + gV*l^>-»•*>, (7.18) s? = [I - wtu»P)i*>r\ (7-i9) where #0 = 4<>/y/sP (7.20) is a renormalized Dyson orbital. Since the derivative (0i|££e(w/ )I0«) < 0, Eq. (7.19) indicates that the spectroscopic factors are less than one, though spectroscopic factors near unity are typical for the outer valence region where the self-energy is normally a relatively slowly varying function of energy. Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 113 Most of these features are preserved in the simpler "target approximation" commonly used to analyze EMS experiments. This simply assumes that the first-order terms in Eq. (7.18) can be neglected, with the result that the renormalized Dyson orbital, $}', is approximately equal to 0j«VSF*. (7.21) Since the HF approximation is the zero-order approximation generally used for this pur pose, this is usually called the "target Hartree-Fock approximation" [29]. An important consquence of the target approximation follows from the fact that the occupation num ber, n[x], of any orbital, x, can be evaluated as an expectation value of the one-electron reduced density matrix, 7, n[X] = JJ X*(l)7(l, I')X(I') dUV , (7.22) using the expression for the density matrix in terms of Dyson orbitals, ionization 7(1,1')= E iMWi(l'). (7-23) j Combining the target approximation (Eq. (7.21)) with these expressions gives ionization »[*]- E 5}°, (7-24) which is close to unity in most cases, ionization E fif an. (7.25) This allows the experimental determination of spectroscopic factors in the following fash ion: First a zero-order orbital, <^,-, is assigned to each observed binding energy by com parison of the shape of the XMP at that binding energy with that from the theoretically calculated MD. The spectroscopic factors, Sj , can then be assigned from the relative Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 114 intensities of the XMPs associated with the same zero-order orbital, using the "normal ization" relation (7.25). Although Eq. (7.21) is reminiscent of the FOA, the target approximation is in fact qualitatively closer to the first order solution of Dyson's equation. In the FOA, each occupied orbital, fa, corresponds to a single observable ionization transition with a spec troscopic factor of unity and an ionization potential (—£»)• In contrast, the target ap proximation allows the ionization process to be decomposed into several transitions, u>j , given by Eq. (7.17), each with spectroscopic factors less than unity (Eq. (7.19)), con sistent with the picture arising from the perturbation analysis of Dyson's equation and with the experimentally observed facts. 7.2.2 Kohn-Sham density-functional theory Kohn-Sham density-functional theory is a computationally less demanding alternative to conventional ab initio electronic structure methods. The KS formalism is briefly summarized here with emphasis on similarities and differences between the KS equation and Dyson's equation. More complete reviews of DFT can be found in refs. [158-160]. Basing their method on an existence proof due to Hohenberg and Kohn [24], Kohn and Sham [25] proposed determining the total energy, E, and density, p, of a system of N interacting electrons in an external potential, v, by minimizing a functional which, in its spin-density variant, is written 2 S P E[P) = -i |>F|V |# > + / v(rt)p(l) dl + l/J -^- dU2 + EM, (7.26) where the KS orbitals, /»(!) = EltfW- (7-27) «=1 Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 115 Note that p(l) depends on both the space and spin coordinates of electron 1, hence Exc[p] = Exc[p\ />*] is a spin-density functional. The four terms in Eq. (7.26) represent respectively, (i) the kinetic energy of a fictitious system of noninteracting electrons with orbitals quasiparticle equation. However, while the KS equation is the orbital equation for a fic titious system of noninteracting particles moving in the (mathematically) local potential (w + / />(2)/r12 d2 + uxc), Dyson's equation refers to quasiparticles moving in a potential with an orbital energy-dependent mathematically nonlocal contribution, £xc(k>j). Conse quently, there are only N orthonormal KS orbitals, while the Dyson orbitals are neither orthogonal, nor normal, nor finite in number. Nevertheless, the two equations share a few properties in common which are worth pointing out. Both have orbital solutions whose densities sum to the exact total density (Eqs. (7.10) and (7.27).) This alone indicates that the KS and Dyson orbitals can differ by no more than a phase factor in regions of space dominated by only a single orbital from each set. In particular, this must be true for the highest occupied molecular orbital in the large r limit, since all other orbitals must die off more quickly [69,179], |^H0M0(1)|2 - U (?-28) A corollary [179] which follows from examination of the asymptotic behavior of the charge density is that both the KS and Dyson HOMO energies must equal the negative of the first ionization potential provided the orbital energy zero is chosen so that the xc potential vanishes asymptotically. The equality of the KS and Dyson HOMO energies is an example of a property which holds for exact DFT but which is generally not found in practical DFT calculations with the approximate functionals presently available. 7.2.3 Approximate Dyson orbitals from DFT The computational advantages of DFT make it an attractive possibility for the calculation of MDs for use with EMS. However, the question of principle regarding the use of KS orbitals to approximate Dyson orbitals should be addressed first. This is properly done in the framework of exact DFT. Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 117 It is widely appreciated that, in the KS approach to DFT, the orbitals of a fictitious noninteracting system were introduced purely as a formal device to facilitate the treat ment of the kinetic energy functional. Thus, Kohn and Sham did not consider these orbitals to be physically meaningful. This view seemed to be corroborated by the fact that the eigenvalues of the KS equation were found not to be good approximations for the ionization potentials and electron affinities, a problem which is most pronounced when approximate functionals are used but which persists in exact KS DFT [164,175,180]. However, there is another approach to DFT in which the KS equation is derived as an approximation to Dyson's quasiparticle equation. This is an outgrowth of Slater's initial concept [22] of vx as a (mathematically) local approximation to the (mathematically) nonlocal exchange operator in Hartree-Fock. From this point of view, there is no reason to think that the KS orbitals should be devoid of physical significance. In this approach, one starts with Dyson's equation and finds the mathematically local (in both space and time) potential which best approximates Sxc(u>), in a well-defined variational sense. The resulting potential is known as the optimized effective potential. In their seminal paper, Sharp and Horton [142] and subsequently Talman and Shadwick [143] gave an expression for this OEP in the exchange-only case. Casida [118] has recently generalized this approach to treat the correlated case as well. In both cases, the resulting OEP can 1 be identified with the exact KS vxc (or vx) derived by Sham and Schluter [118,144]. Thus, the KS equation is the variationally best mathematically local approximation to Dyson's quasiparticle equation, so in this sense it is natural to consider KS orbitals and orbital energies as approximate Dyson orbitals and orbital energies. Note that, unlike the total energy and charge density, the individual Dyson orbitals and orbital energies 1This identification holds within the linear response approximation to the Sham-Schliiter equation (see ref. [118].) This approximation is routinely used in calculations of the exact vxc (or vx), and its acceptance is implicit in the acceptance of OEP calculations as giving the exact KS x(c) potential [143,164,165,167-178,180-183] Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 118 are not among the quantities which are obtained exactly in "exact" KS DFT; it is the true KS orbitals and orbital energies which here arise as approximations to the Dyson orbitals and orbital energies. The question then becomes whether this approximation is good enough to be useful. The shortcomings of the KS orbital energies as approximate Dyson orbital energies are well known. Indeed, as Perdew and Levy [184] have pointed out, this is to be expected on the basis of derivative discontinuities in the xc energy functional. However, this difficulty with the orbital energies does not imply that the KS orbitals are necessarily poor approximations to the Dyson orbitals. While it may seem counter-intuitive that the KS orbitals could be good approximations to the Dyson orbitals when this is not true for the corresponding orbital energies, the qualitative description of the mathematical localization process given in appendix C suggests that, even if the Dyson orbitals were directly proportional to KS orbitals, the corresponding eigenvalues should differ in order to produce an orbital-independent mathematically local potential vxc. Unfortunately, these formal considerations do not give any a priori statement as to the quality of the approximation of Dyson orbitals by KS orbitals, nor any limiting case in which it becomes exact. In order to obtain an idea of the quality of this approximation, the available numerical data from exact DFT calculations is examined here. First the exchange-only case. In this case, one measure of the quality of the OEP (or exact KS) orbitals as approximate HF orbitals is how well they minimize the HF energy expression. That is, the difference between the HF energy expression evaluated with OEP orbitals and the true HF energy provides a measure of the severity of the (mathematically) local approximation. Explicit calculations on atoms show that this energy difference ranges from about 50 ppm for the lightest atoms to about 5 ppm for atoms with atomic numbers near 50 [170,175]. This and other calculated properties [167,168] such as impulse Comp- ton profiles, atomic form factors, and dipole polarizabilities, as well as orbital properties Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 119 [170] such as (r), (r2), (1/r), and (1/r3) indicate that the exchange-only Kohn-Sham or bitals are remarkably close to HF orbitals. A similar conclusion was reached by Zhao and Parr [185,186] in their direct comparison of exact (exchange-only) atomic KS orbitals with the corresponding HF orbitals. Although much less information is available in the exchange- correlation case, some information is available from Green function calculations on solids [151,152,154], though this is not from exact DFT. These begin with KS DFT calculations using the local density approximation for the xc energy as the zero-order description, and then follow up by solving Dyson's quasiparticle equation using the GW (i.e. Green function, G, times screened interaction, W) approximation [187] for the xc self-energy. Overlaps of 99.9% between the KS and renormalized Dyson orbitals are found in these calculations. Thus the numerical evidence is that KS orbitals can serve as good approximations for renormalized Dyson orbitals. Now let us clarify the precise sense in which KS orbitals are intended to be approx imate Dyson orbitals. Just as in HF, there are only N KS orbitals, as opposed to an infinite number of Dyson orbitals, and the considerations in Subsection 7.2.1 on the frozen orbital versus target approximations apply equally to DFT and HF. Thus what is being proposed is a target Kohn-Sham approximation (TKSA) in which each ionization Dyson orbital is taken to be proportional to an occupied (canonical) Kohn-Sham orbital (Eq. (7.21)). The TKSA is expected to break down when the perturbation (SXC(U>J) — vxc) becomes too large. However, just as with the target Hartree-Fock approximation, this breakdown is most likely to occur for energies, wj, where the self-energy is varying rapidly, in which case the spectroscopic factor will be small (Eq. (7.19)) and the transition will be difficult to observe experimentally. As was remarked in Chapter 2, Dyson orbitals may be calculated either directly, as solutions of the quasiparticle equation, or indirectly via the generalized overlap for mula (2.7). Since both approaches yield the same result in the limit of an exact treatment Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 120 of many-body effects, either may be used as a starting point for approximations to the Dyson orbital. The former (quasiparticle equation) approach has been chosen for use, since it lends itself more readily to approximation via DFT. The latter (generalized over lap) approach would have required introducing an JV-particle wavefunction in the DFT treatment, as well as doing DFT calculations on excited states of the daughter ion, with all its attendant formal and practical difficulties. Or, in order to avoid the problems of excited states, one would need some sort of Koopmans-like theorem for DFT, relating the generalized overlap to the parent KS orbitals, but no such theorem has been proven to date. In contrast, the approach taken in this paper is based entirely on the quasi particle equation. The self-energy is approximated through a mathematical localization (OEP) procedure which yields the KS equation. Thus the (canonical) KS orbitals are just the solutions of this approximate quasiparticle equation, and, in this sense, approximate Dyson orbitals. This point of view has the advantage that the aforementioned problems with the generalized overlap approach are not encountered. Although, in the case of an exact self-energy, the generalized overlap yields the same Dyson orbital as is obtained by solving the quasiparticle equation, this need not always be the case for an approximate self-energy. The generalized overlap and the solution of the quasiparticle equation are then equally justified, but possibly different, approxima tions to the true Dyson orbital. Nevertheless, it is interesting, for purposes of clarifying the physical picture implied by a given self-energy approximation, to try to understand the relationship between the generalized overlaps and the solutions of the quasiparticle equation, within this self-energy approximation. This relationship is not yet clear in the case of DFT. In addition to the TKSA whose formal justification has just been discussed, two ad hoc DFT approximations of Dyson orbitals are also considered in this paper, in order to illuminate different points, and each of which might naively appear to offer some Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 121 advantages. The first ad hoc approach is based on analogy with Hartree-Fock. In the Hartree- Fock approximation, Koopmans' theorem provides the relation between the solutions of the quasiparticle equation and the generalized overlaps. Since Hartree-Fock consists of approximating the exchange-correlation self-energy, Exc, by the HF exchange operator, Sx, the (canonical) HF orbitals, being solutions of this approximate quasiparticle equa tion, approximate Dyson orbitals. Koopmans' original formulation [188] of the theorem that bears his name concerns a special case in which the generalized overlap is also equal to the canonical HF orbital — namely when the problem is restricted to the space of the occupied molecular orbitals of the parent species. Then the orbital describing the hole in the daughter (N — l)-electron HF wavefunction is identical to a canonical HF orbital of the TV-electron parent. Although no variant of Koopmans' theorem has been proven for Kohn-Sham density functional theory, it is interesting to ask what happens if the Kohn- Sham problem of the daughter is solved in the restricted space of the parent occupied KS orbitals. The orbital describing the resulting hole is termed the "Kohn-Sham Koopmans' hole" (KSKH) orbital. Since these KSKH orbitals are related to the canonical KS orbitals by a unitary transformation, they are still KS orbitals, though not necessarily canonical ones. Symmetry arguments alone suffice to show that for any occupied orbital whose symmetry representation is unique among the occupied orbitals (termed "lone symmetry states" in ref. [63]), this KSKH orbital is identical to the canonical KS orbital. The question is how similar these orbitals are in the absence of such symmetry constraints, and whether this hole (i.e. the KSKH orbital) provides any better description of the Dyson orbital than does the canonical KS orbital. This KS Koopmans' hole approach still involves the formal difficulty of calculating excited states in DFT (see e.g. ref. [158] pg. 204 and ref. [159] pg. 32). Nevertheless, the analogy with HF is interesting. The second ad hoc approach returns to the quasiparticle equation (rather than the Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 122 generalized overlap) point of view, and is aimed at improving the KS eigenvalues as approximations to the eigenvalues of Dyson's equation. The large r behavior of orbitals plays an important role in their MDs, and this behavior is intimately related to the eigenvalue in Dyson's equation. In the atomic case, the large r limit for this orbital is given by [69,179] V>/(r This relationship also holds between the large r behavior of the KS orbitals and their eigenvalues. While this is no problem for the HOMO where the Dyson equation eigenval ues and the exact KS eigenvalues are both equal to minus the first ionization potential, the eigenvalues and hence the asymptotic behavior of the other Dyson and KS orbitals are expected to differ. Note also that even the HOMO KS eigenvalue is rarely a good approx imation to minus the first ionization potential in practical calculations using approximate functionals. The Slater-Janak transition state method [26,189], which consists of solving the KS equations with half an electron removed from the iih orbital, is a well known way to modify the KS equation so that the ith eigenvalue provides a better approximation to the ith. principal ionization potential. (See e.g. ref. [190] for an assessment of the tran sition state method for calculating ionization potentials.) If "better" eigenvalues, in the sense of giving better ionization potentials, imply better orbital behavior in the asymp totic region, then using this "transition orbital" (i.e. the half-occupied orbital resulting from a transition state calculation) might be a useful device for calculating MDs for use with EMS. This will be referred to this as the "transition orbital method" (TOM). The discussion in this section has been in terms of exact DFT. However, since it is of primary interest to examine the question of the practical utility of the approximations discussed for calculating MDs for EMS, the computational investigation of the approx imations presented here necessarily uses approximate functionals. It should be kept in Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 123 mind that this introduces a further approximation beyond that involved in approximating Dyson orbitals by KS orbitals. 7.3 Computational details All density-functional calculations reported in this paper were carried out with the pro gram deMon [191-193] and will be compared with the highly accurate multireference singles and doubles configuration interaction calculations of Davidson and coworkers [28,42,49,50,52,53,57], who used the program MELD [102]. In order to avoid compli cating the comparison between KS, CI, and HF MDs with differences arising from the geometries, the present work uses the geometries of Davidson et al. The present calculations use the (4,4;4,4) (heavy atoms) and (3,1;3,1) (hydrogen) auxiliary basis sets (see chapter 6), which are the best presently available in the deMon library file. The grid used was extrafine, random, and the self-consistent field convergence criterion was five successive energy differences of less than 10-8 a.u. The ion calculations done for the KSKH and TOM orbitals were started from the parent density, and were converged to five successive energy differences of less than 10-6 a.u. The polarizabilities reported in this work were calculated by finite difference, with a field strength of ±0.0005 a.u. using the polarizability option in deMon. The MDs (spherically averaged orbital momentum distributions) were calculated from the orbitals using the in-house HEMS (H-compiler-optimized Electron Momentum Spectroscopy) program of reference [77]. Several exchange and correlation functionals were employed in this study; these will be distinguished using the abbreviations LDAx, LDAxc, B88x, and B88x+P86c. The "x" refers to the exchange functional while V refers to the correlation functional, so that the LDAx and B88x functionals are exchange-only approximations. The LDA neglects any dependence on the gradients of the density, and is therefore Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 124 most justified for slowly varying densities, though its range of applicability is actually much wider than this would suggest. The next higher level of approximation consists of adding gradient correction terms to the LDA exchange-correlation energy density. One of the most important shortcomings of the LDA is that it tends to overbind molecules [194], and many gradient-corrected functionals were designed with this in mind. In principle, gradient corrections should also improve the results for other properties besides binding energies, though in practice this varies depending on the functional and the property in question. Since the large r behavior of orbitals is very important for the calculation of properties such as polarizabilities, and MDs for EMS, it is worth noting that the LDAx and LDAxc potentials and energy densities violate the known asymptotic behavior of the exact functionals. The 1988 gradient-corrected exchange functional of Becke [120] is an improvement in this regard, being specifically designed so that the exchange energy density, ex, has the correct asymptotic behavior, although the same cannot be said for the resulting B88x potential. However, the asymptotic behavior of the B88x potential is an improvement on that of the LDAx potential. The gradient corrections used in the present work are the B88x functional, for exchange, and the 1986 correlation functional of Perdew [121] (P86c). The P86c functional is based upon a comparison of wave-vector analyses for the correlation energy density of molecules and the homogeneous electron gas [121]. The orbital basis sets used in this work are described below. All consist of contractions of Gaussian-type orbitals, and use sets of six cartesian d-functions. STO-3G. This minimal basis set, due to Pople [104], is of single zeta quality, and consists of a single contraction of 3 Gaussian primitives for each atomic orbital. STO-3G#. The # symbol designates the addition of a single set of diffuse p- functions, to the oxygen basis set in the H2O calculation. The exponent used is that given by Casida and Chong [69] in their study of the relation between MDs and large r Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 125 behavior. NM. This basis set, designed by Neuman and Moscowitz [195], is of roughly triple zeta plus polarization quality on oxygen and double zeta plus polarization quality on hydrogen. It consists of a (10s6p2d) —• [5s3pld] contraction on oxygen and a (4slp) —• [2slp] contraction on hydrogen. TZP. This consists of the (7111/411/1*) (i.e. [4s3pld]) (oxygen) and (41/1*) (i.e. [2slp]) (hydrogen) library basis sets of deMon. Here, TZP stands for "triple zeta plus polarization", although it would be better described as "valence triple zeta plus polar ization" on oxygen and "double zeta plus polarization" on hydrogen. ANO—. The atomic natural orbital basis sets of reference [196] were truncated to d-functions on the heavy atoms for use in the present study, since deMon is not yet able to use higher angular momentum atomic orbitals. To maintain balance in the basis sets, the hydrogen set was truncated to p-functions. The resulting basis sets consist of 5 s-contractions, 4 p-contractions, and 3 d-contractions on the heavy atoms, and 4 s- and 3 p-contractions on hydrogen. NHF—. The near Hartree-Fock quality 109-GTO basis set of reference [28], for water, was truncated to d-functions on oxygen and p-functions on hydrogen. Sets of six cartesian d-functions were used in the present study, instead of the sets of five d-functions used in ref. [28]. The resulting basis set consisted of 92 contractions of GTOs. TZP+, ANO+, NHF+ . The "+" indicates augmentation of the substrate (TZP, ANO—, or NHF—) basis with the field-induced polarization (FIP) functions of Zeiss et al. [197] as described in ref. [139] (s and d on the heavy atoms and p on hydrogen). However, due to linear-dependencies encountered in the case of ANO+, only the d and not the s FIP for the heavy atoms was added in forming the ANO+ and NHF+ basis sets. The ANO+ basis sets were used for most of the calculations reported in this work. Table 7.1 shows the basis set dependence of the total energies, dipole moments, and Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 126 polarizabilities obtained for H20, using deMon with the LDAxc functional. The total energy decreases as the basis set size increases and appears to be converging to about -75.913 hartrees. This is well above the best experimental estimate of the total energy, namely -76.440 hartrees. Although the KS equations are derived by a variational mini mization of an energy functional and finite basis calculations of the total energy should converge from above to the KS value for the functional used, the use of approximate functionals in practical DFT programs means that, unlike the case in HF and CI cal culations, the total energy may fall either above or below the true nonrelativistic value. Hence comparisons of DFT total energies with total energies from other methods cannot be used as an indication of basis set convergence. However, comparison of DFT total energies calculated using different basis sets but the same functional is useful for this purpose. With the exception of the out of plane component of the polarizability tensor, the dipole moment and polarizability components seem to be converging to values in reasonably good agreement with experiment. Since the dipole moment and polarizabil ity can be defined in terms of derivatives of the total energy with respect to an applied electric field, the fact that the LDAxc results for these properties are better than for the total energy is consistent with the well-known fact that density functional calculations often give much better relative than absolute energies. Figure 7.1 shows the basis set convergence of the TKSA MD for the lbi orbital of water. The CI MD of Davidson et al, [28] is included only as a reference for the reader's convenience. As seen from Fig. 7.1, the convergence properties of the MDs generally reflect those observed in Table 7.1 for the energy, dipole moment, and components of the polarizability tensor, except that the addition of FIP functions to the basis is more efficacious for the dipole moment and polarizability than for MDs, which is not surprising since these functions were designed for electrical response properties. Nevertheless, since a good description of the large r region is important for MDs, the FIPs, being diffuse Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 127 Table 7.1: Basis set dependence of the total energy, dipole moment, and principle com ponents of the dipole polarizability of H20. The molecule is oriented in the (a;,z)-plane with its dipole aligned along the 2-axis. The basis sets are listed in order of decreasing total energy. The basis sets used in the deMon calculations are described in the text; those used in the MELD and H0ND08 calculations are described in the appropriate references. The abbreviation "NA" in the table stands for "not available." Polarizability ( au) 1 Method/Basis set Size Dipole Moment (au) Otxx <*»» Otzz Tot. Energy (au) deMon calculations2 LDAxc/STO-3G 7 0.6807 4.778 0.040 2.163 -74.7331 LDAxc/STO-3G# 10 0.9533 4.644 0.182 3.047 -74.8820 LDAxc/NM 36 0.8104 7.989 4.634 6.498 -75.8868 LDAxc/TZP 29 0.8532 7.895 4.057 6.653 -75.8996 LDAxc/TZP+ 42 0.7455 10.290 9.942 10.114 -75.9031 LDAxc/ANO- 61 0.7489 9.327 7.068 8.703 -75.9113 LDAxc/ANO+ 73 0.7334 10.430 9.890 10.234 -75.9127 LDAxc/NHF- 92 0.7597 9.508 9.533 9.032 -75.9129 LDAxc/NHF+ 104 0.7366 10.434 10.574 10.391 -75.9132 HOND08 calculations15 HF/[10s7p4d/7s4p] 93 0.780 9.16 7.91 8.46 -76.0654 MELD calculations1 HF/99CGTO 99 0.7893 NA NA NA -76.0669 HF/109CGTO 109 0.7891 NA NA NA -76.0671 HF/140CGTO 140 0.7794 NA NA NA -76.0673 CI/99CGTO 99 0.7439 NA NA NA -76.3736 CI/109CGTO 109 0.7455 NA NA NA -76.3761 CI/140CGTO 140 0.7356 NA NA NA -76.3963 Experimental values 0.727s 10.31s 9.556 9.91s -76.43967 1 Number of contractions of Gaussian-type orbitals. 2 Present work. The total energy is the "analytic energy" calculated using both charge density and exchange-correlation fitting functions. 3 Taken from ref. [139]. 4 All MELD calculations are taken from ref. [57] with the exception of the 109CGTO calculation which is from ref. [28]. 5 Ref. [198]. 6 Ref. [199]. 7 From ref. [57]. Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 128 0.08 Momentum (au) Figure 7.1: Effect of basis set on the TKSA MD for the lbi orbital of H20, using the LDAxc functional and the basis sets described in the text. The CI/109CGTO MD of reference [28] (dashed curve) is shown for comparison purposes. All MDs have been calculated from orbitals which are normalized to unity. Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 129 functions, do have a significant effect on the MD when added to the TZP and ANO— basis sets (to give TZP+ and ANO+). However this effect of the diffuse FIPs on the MD is almost imperceptible for the larger NHF— basis set. Although no attempt was made to go beyond the NHF+ to try to fully saturate the basis, the MDs appear to be reasonably well converged. Judging from the convergence of the total energy, dipole moment, polarizabilities, and MDs, the ANO+ basis seems to be of good quality, while being only moderately large, and offering a uniform level of description for the different molecules studied. This level of convergence should be quite adequate for the purpose of the present study, which is to investigate the viability of using KS orbitals to approximate Dyson orbitals for calculating MDs. The ANO+ basis set will therefore be used throughout the rest of this study. It is worth noting that the ST0-3G# basis gives remarkably good MDs for such a small basis set. This is typical for this basis. This is because the ST0-3G# basis set was created from the overly small ST0-3G basis set by adding diffuse functions specifically designed to increase the ability of this basis set to describe the large r behavior, which is of primary importance for calculating MDs [69]. Since the resultant basis set is still quite small, it provides an efficient approximate method for the calculation of MDs for large molecules. On the other hand, Table 7.1 shows that the same cannot be said for the dipole moment, polarizability, and total energy, for which a good description in the small r as well as the large r, region is needed. One further step is necessary in order to compare theoretical MDs with experiment, namely the finite resolution of the experimental apparatus must be taken into account. To accomplish this, the procedure described in chapter 4 and ref. [100] is used to resolution- fold MDs in order to compare with XMPs. In addition, since absolute-intensity XMPs are not yet obtainable from EMS experiments, the XMPs for different orbitals in the Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 130 same molecule are normalized only relative to one another, but not on an absolute scale. In order to compare experiment and theory, the arbitrary normalization constant is fixed by scaling the data such that the height of the XMP agrees with the resolution-folded CI MD for one orbital in each molecule. It is now well established [30] that the CI MDs of Davidson et al. show excellent agreement with experiment when resolution folded using the GW-PG procedure discussed in chapter 4 and references [72,100]. This is illustrated in Figure 7.2 for the lbi orbital of H20. Thus these high quality CI MDs can be used as a reference against which to judge the MDs obtained from DFT approximations to Dyson orbitals. 7.4 Results and discussion This section presents the results of the assessment of the target Kohn-Sham approxima tion for the calculation of MDs, within the LDA, for 18 orbitals in 6 small molecules and atoms where high quality HF and CI calculations as well as experimental data are avail able for comparison. The experimental results used here are those of the University of British Columbia EMS group, while the CI MDs are those of Davidson et al. at the Uni versity of Indiana. Two interesting ad hoc approximations to the Dyson orbital are also considered, in subsection 7.4.2. Finally, the TKSA is returned to in the final subsection in order to make a preliminary investigation of the effect of approximate functionals. The calculation of spectroscopic factors requires a perturbative treatment going be yond the basic KS or HF calculation (see Eq. (7.19)) and is thus computationally more involved. For purposes of using MDs to assign peaks in EMS spectra, theoretically cal culated spectroscopic factors are not necessary. Thus calculate spectroscopic factors are not calculated, but the comparisons are made in terms of the renormalized Dyson or bitals ^y' which have the spectroscopic factors divided out (Eq. (7.20)). In the THFA or Chapter 7. EMS from DFT: The Target Kohn-Shaxn Approximation 131 8 H20 1 bn 6-i 12.2 eV C CD ^4H CD > D 1532 0 o 3 Nominal Momentum (au) Figure 7.2: Comparison of the experimental XMP and resolution-folded CI/109CGTO MD for the lbx orbital of H2O. Both the experimental data and CI MD have been taken from ref. [28], but the resolution folding of the CI MD has been done according to the prescription given in chapter 4 of the present work and ref. [100] for a more appropriate comparison between theory and experiment. Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 132 TKSA, ffi is just the (canonical) HF or KS orbital fc (Eq. (7.21)), while the experimen tally determined (via Eq. (7.25)) and CI spectroscopic factors are divided out of their respective XMPs or MDs. In view of the resolution-folding and height scaling involved in comparing theory with experiment, and the size of the error bars on the experimental data, the cleanest test of the DFT orbital MDs for these small molecules is in comparison with the CI MDs. Comparison with the experimental results is, of course, also important, since the adequacy of the DFT results for use in EMS peak assignments is of primary concern. 7.4.1 Target Kohn-Sham Approximation in the LDA A practical investigation of the TKSA necessarily involves the use of an approximate exchange-correlation functional. The present study focuses primarily on the TKSA in the local density approximation, though the effect of the functional is investigated for H20 in Subsection 7.4.3. The present subsection deals with the results using the LDAxc functional. In order to test the quality of the KS orbital MDs obtained from the TKSA (within the LDA), five small molecules, HF, H2O, NH3, CH4, C2H2, and one atom, Ne, were chosen, as they have been well studied both experimentally and theoretically by EMS [28,42,49,50,52,53,57,100]. DFT MDs were calculated for all their valence orbitals. H2O is a particularly interesting test molecule because it is more difficult to obtain a quantitative description of the MD for the HOMO, and to a lesser extent the 3ai orbital, than for the outer valence orbitals of many other molecules. Figures 7.3 and 7.4 show the calculated momentum distributions for all the valence orbitals of water and acetylene respectively. In addition to the KS orbital MDs, these figures contain the results from the KSKH and TOM orbital approximations which will be Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 133 0.08 .•£•0.06 m c o 0.04 0.02 - 0.00 Momentum (au) 1 • -J Momentum (au) 0.4 (d) a H20 2a, .£0.3 . v/ -KS.KSKH in c 0.2 D z 0.0 .2 3 Momentum (au) 1 ' Momentu1 ' m '(au ) Figure 7.3: Comparison of several DFT approximations for MDs with target Hartree-Fock and CI MDs, for the valence orbitals of H2O. The DFT calculations used the LDAxc functional and the ANO+ basis set. The CI calculations are the 109CGTO basis set calculations of ref. [28], while the HF calculations are those of ref. [57], using the 99CGTO basis set, except for the 2ai orbital, where only the 84CGTO calculation of ref. [200] was available (at the HF level the 84CGTO, 99CGTO, and 109CGTO MDs are indistinguishable from one another). The curve labels are: target Hartree-Fock approx imation (HF), configuration interaction (CI), target Kohn-Sham approximation (KS), Kohn-Sham Koopmans' hole (KSKH), and transition orbital method (TOM). Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 134 0.18 u.o - (o) (b) hKSKH C2H2 17TU .0.4- C2H2 3(Tg .-(0* 1°-12 c •2 0.3- 'A £ •p 0) TOMVV N r* .§0.2 • "5 0.06 • 0 § °0.1 • 0.00 0 V1 .2 . ' 3 0.0 ^ 1 .2. 3 Momentum (au) Momentum (au) V.jLO • (c) a ^0.20 • C2H2 2au ) •gO.15 • TL\ / K •v \, J0.10 - TA |0.05 - / u.uu -s V1 • '.2 . 3 1 ' 2" Momentum (au) Momentum (au) Figure 7.4: Comparison of several DFT approximations for MDs with target Hartree-Fock and CI MDs for the valence orbitals of C2H2. The DFT calculations used the LDAxc functional and the ANO+ basis set. The Hartree-Fock and CI calculations are the 186CGTO calculations from [52]. The curve labels are: target Hartree-Fock approximation (HF), configuration interaction (CI), target Kohn-Sham approximation (KS), Kohn-Sham Koopmans' hole (KSKH), and transition orbital method (TOM). Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 135 discussed in the next subsection. Looking at the KS results, it is evident that the positions and shapes of the KS, HF, and CI MDs are all similar, though there are significant differences between the shape and height of each calculation. In particular, it appears that the KS MDs match the CI MDs very closely for shape, but not height. In contrast, the HF MDs do not match the CI MDs as closely for shape as do the KS MDs. This is also the case for the other molecules studied, so for the remaining molecules, only the peak heights and positions are reported here. The results are shown in Fig. 7.5 and Table 7.2. It is evident from Figs. 7.3-7.5 and Table 7.2 that the overall quality of the TKSA in the LDA is quite similar to that of the THFA for the orbital MDs of these molecules if both overall MD shape and height are considered. However, in all cases shown in table 7.2 it is clear that the CI peak position (but not height) is reproduced more accurately by the KS MDs than by the HF MDs. Overall (considering both MD height and shape), for some orbitals the KS MDs are a little better than the HF, while for others the HF is a little better, and for many orbitals the magnitude of the error in the KS peak height parallels that for HF, being larger when correlation is more important. For both the KS and HF, the largest error in the peak height occurs for the 2aa orbital of acetylene. The errors in the heights of the KS MDs, compared to CI, range from 1% to 6%, with the exception of the 2o~g orbital of acetylene with an error of 17% and the 2a\ orbital of CH4 with an error of only 0.4%. In comparison, the peak height errors in the HF MDs are in the range 2% to 11%, except for four orbitals (CH4 lt2, NH3 2a\ and le, and H20 162) having errors of 0.3% or less. The average error of 4.6% in the peak heights for all the orbitals is the same for the KS and HF. The errors, again relative to the CI, in the positions of the peaks, i.e. in the value of the momentum for which the MD has its maximum, are often less than the estimated uncertainty in the determination of this value and the average error (excluding those orbitals for which symmetry constrains the Chapter 7. EMS from DFT: The Target Kobn-Sbam Approximation 136 0.8 MD Peak Hei«ght s A 0.6 - o IS) o0.4 - X A 0.2 - oHF AKS 0.0 0.0 0.2 0.4 0.6 0.8 CI Figure 7.5: Correlation plot of the MD peak heights obtained in the target Kohn-Sham approximation (KS) and in the target Hartree-Fock approximation (HF) with configu ration interaction (CI). Values are from Table 7.2 Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 137 Table 7.2: Absolute peak heights and positions of the maxima of MDs for the valence orbitals of CH4, NH3, H20, HF, Ne, and C2H2. Positions should be considered accurate to only ±0.025 a.u. and heights to within 1 % . KS/LDAxc: Target Kohn-Sham ap proximation using the LDAxc functional and ANO-f- basis set. HF: Target Hartree-Fock approximation. CI: Multireference singles and doubles configuration interaction. Posi tions and heights are given in atomic units. KS/LDAxc HF1 CI1 Orbital Position Height Position Height Position Height CH4 2ai 0.0 0.7100 0.0 0.6923 0.0 0.7074 1*3 0.567 0.1288 0.600 0.1228 0.567 0.1231 NH3 2ai 0.0 0.4451 0.0 0.4381 0.0 0.4370 le 0.633 0.09037 0.633 0.08704 0.633 0.08709 3ai 0.533 0.1185 0.533 0.1049 0.533 0.1121 H20 2aj 0.0 0.2947 0.0 0.2851 0.0 0.3137 lbs 0.667 0.06473 0.700 0.06288 0.700 0.06307 3ai 0.633 0.07277 0.650 0.06438 0.633 0.06877 lb! 0.600 0.07061 0.650 0.06113 0.600 0.06643 HF la 0.0 0.1976 0.0 0.1872 0.0 0.2077 3 ^he HF and CI calculations are taken from the following references: CH4 [49], NH3 [42], HF [50], Ne [53], C2H2 [52]. For H20, the CI calculations are the 109CGTO basis set calculations of ref. [28], while the HF calculations are those of ref. [57], using the 99CGTO basis set, except for the 2ai orbital, where only the 84CGTO calculation of ref. [200] was available (at the HF level the 84CGTO, 99CGTO, and 109CGTO MDs are indistinguishable from one another). Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 138 maximum to occur at zero momentum) is 1.4% and 1.9% for KS and HF respectively. It is interesting to note that, while the magnitude of the errors in the KS and HF MDs are quite similar, they usually differ from the corresponding CI MDs in opposite directions. The maximum of the MD is shifted, if at all, to higher momentum in the HF, and to lower momentum in the KS, MDs. Out of the 18 orbitals treated here, the height of the MD from KS is greater than that from CI for 14 orbitals, and the HF MD height is less than the CI MD height for 15 orbitals. (In only one case is the height from KS less than that from HF.) While both KS and HF orbitals approximate Dyson orbitals, and apparently about equally well for the MDs, they do so in different ways. Of course the quantitative aspects of the errors in the TKSA vs. THFA would be expected to depend on the basis sets used in the respective calculations. For example, the peak height errors might be somewhat larger in the "LDA limit" than those reported here. However, in view of the level of convergence of the MDs shown in Fig. 7.1, the basic conclusion that the TKSA (in the LDA) and the THFA are of roughly comparable quality for MDs when considering peak height and MD shape, seems unlikely to be altered by further improvements in the basis set. Figure 7.6 shows the resolution-folded KS, HF, and CI MDs in comparison with the experimental data for H2O. The arbitrary normalization of the experimental XMPs has been fixed by scaling the data such that the height of the peak matches the CI MD for the lbi orbital. Since practical applications of the DFT MDs where no CI calculation is available would require a similar scaling, except that the data would be normalized to a KS orbital, a fourth curve has been included in which the KS MDs have been scaled (by a factor of 0.94) such that the lbi orbital peak height matches that of the CI (and thus the experiment). Figures 7.7, 7.8, 7.9, and 7.10 show the same type of figure for CH4, NH3, HF, and Ne, with the important difference that in these latter figures the normalized KS MD has been left off the figure so that a comparison of the KS, HF, and Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 139 o- 3 (b) Vs 1 J H20 1b, H20 3di 12.2 eV 6- 15.0 eV infr • i a c i a 2 HF 2 HF 3 KS c4- 3 KS 4 KS>0.94 4 KStO.94 { M> 2 j J ftO $ ««° " i j W ^4i i* •• Nit; 0 2 . ' 3 1 i 3 Nomina1 l Momentum (au) °o Nominal Momentum (au) o- t-u- (c) (d) ul 1.4 H20 1b2 H20 2a, 6- 3v Ij 18.6 eV 30- 'ITT 32.2 eV sfil cVi • 'in 2J&J r 1 a 4 1 a V , 2 HF c 2 HF V 3 KS \S 3 KS 14- \ 4 KS-0.84 c20- 4 KS.0.84 0 L% | Dfi t > • .1 { HiO 12- •*> IK 0 «10- tK il 0 0 () 2 . 3 () \ ' M 2 * . 3 Nomina1 l Momentum (au) Nominal Momentum (au) Figure 7.6: Comparison of target Kohn-Sham approximation (KS), target Hartree-Fock approximation (HF), and CI TMPs with XMPs for the valence orbitals of H20. The theoretical MDs are those of Fig. 7.3, but here they have been resolution-folded according to the prescription given in ref. [100] in order to compare with experiment. Two KS TMPs are shown for each orbital. The first retains the absolute intensity from the original calculation, while the second is multiplied by 0.94 (for all orbitals) so that the heights of the CI and KS MDs for the lbi orbital match (see text). Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 140 40- CH4 1t2 3X1 / i ij^ 14.13 eV Z1'2 1 146-G(CI) r 2 146-CTO 20- 3 DFT [543f;430f] > I Area of BES peak £ Experiment cm®1 0 - -C M 0 ~r 3 Nominal Momentum (au) 80 - T CH4 2at 1 60-- 23.05 eV w c - J Area of BES peak D i Experiment ©20H- a: \B - \i ^^£ii l'»t •• 0 -C M -) \ 3 c Nominal Momentum (au) Figure 7.7: Comparison of target Kohn-Sham approximation (KS), target Hartree-Fock approximation (HF), and CI TMPs with XMPs for the valence orbitals of CH4. The theoretical MDs have been resolution-folded according to the prescription given in chapter 4 and reference [100] in order to compare with experiment. The KS TMPs shown for each orbital have been scaled (by the same factor for each orbital) to match the height of the CI TMPs. The factor chosen was derived from the lt2 calculation (see text). The experimental data as well as the CI and HF calculations are taken from reference [49]. Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 141 15 NH3 3a, 10.9 eV 1 12«-G(CI) 2 126-GTO r jf 3 OFT [S43ft430f] 2/ X Exporimnt 2 i» r 1 or V ^***—~J_ 1 0 \ 2 3 Nominal Momentum (au) au- ' 20- NH3 1e T NH3 2a, 16.6 eV 40- 27.8 eV k,, -£> £> 'Il"ftP "15- 1 10B-Cfcl) 230- 12»-G(a) 2 106-OTO 2 12«-5tO c 3 OFT [543f;«30f] c DFT [5*3f;430f] ©10- \ I Bcparimant §20- Exparimant ,> I D o \ or 5- V 3 Qi10- JS V. Vj •C M U« , °( 3 °() ' \ 2 . ' 3 I ' Nomina\ l Momentum (au) Nominal Momentum (au) Figure 7.8: Comparison of target Kohn-Sham approximation (KS), target Hartree-Fock approximation (HF), and CI TMPs with XMPs for the valence orbitals of NH3. The theoretical MDs have been resolution-folded according to the prescription given in chapter 4 and reference [100] in order to compare with experiment. The KS TMPs shown for each orbital have been scaled (by the same factor for each orbital) to match the height of the CI TMPs. The factor chosen was derived from the 3ai calculation (see text). The experimental data as well as the CI and HF calculations are taken from reference [42]. Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 142 0.05 HF 17T 0.04 16.1 eV \7/~ c0.03 1 148-0(CI) s 2 1W-PI0 2 3 DFT [S4Jft«oa f/2 Vv" 10.02 3 \V W I Exp«rim«nt 0.06 0.25 HF 3er HF 2CT 19.9 eV 39.7 eV 1 #1 1 ne-cfta) i i«-G(a) r z 14«-d5TO 2 i*e-.oto 2 # 3 DFT [543f;430t] 3 DFT [543fi430f] 10.02 \. T I Experiment CD / a: i 0.00 •C M 0 1 3 Nominal Momentum (au) T " ¥ Nominal Momentum (au) Figure 7.9: Comparison of target Kohn-Sham approximation (KS), target Hartree-Fock approximation (HF), and CI TMPs with experimental XMPs for the valence orbitals of HF. The theoretical MDs have been resolution-folded according to the prescription given in chapter 4 and reference [100] in order to compare with experiment. The KS TMPs shown for each orbital have been scaled (by the same factor for each orbital) to match the height of the CI TMPs. The factor chosen was derived from the \ir calculation (see text). The experimental data as well as the CI and HF calculations are taken from reference [50]. Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 143 Ne 2p 21.5 eV 1 106-G(CI) 2 106-CTO 3 DFT [543f;430f] Experiment 7. "1—' I 7~~5" Nominal Momentum (au) Ne 2s 48.5 eV 1 106-G(CI) 2 106-CTO 3 DFT [543f:+30f] J)9 X Experiment Kl 0 \ 5 ^3^ Nominal Momentum (au) Figure 7.10: Comparison of target Kohn-Sham approximation (KS), target Hartree-Fock approximation (HF), and CI TMPs with XMPs for the valence orbitals of Ne. The theoretical MDs have been resolution-folded according to the prescription given in chapter 4 and reference [100] in order to compare with experiment. The KS TMPs shown for each orbital have been scaled (by the same factor for each orbital) to match the height of the CI TMPs. The factor chosen was derived from the 2p calculation (see text). The experimental data as well as the CI and HF calculations are taken from reference [53]. Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 144 CI MDs may be made for shape only. While such rescaled MDs are appropriate from the point of view of practical applications, they do not give an accurate indication of the relative quality of the different theoretical models since the most pronounced difference between the theoretical MDs is in the peak heights. The TKSA MDs in Figures 7.6-7.10 are certainly of good quality from the point of view of applications to EMS, as for all orbitals shown it would appear that the KS MD reproduces the shape of the CI MD quite well. While both the shape and the height of the orbital MD are sensitive to the large r behavior of the orbital, this relationship is stronger for the shapes than for the heights [69]. Thus a tendency to give good MD shapes may be an indication of the quality of the large r behavior of the KS orbitals. This would be consistent with the previously observed [139] good quality of DFT dipole moments, polarizabilities, and hyperpolarizabilities, since these properties also depend upon a good description in the large r region (see also chapter 9). Although it is well known that the presently available functionals do not give the correct asymptotic behavior of the exchange-correlation potential, Umrigar and Gonze's calculations of the exact exchange-correlation potential [201,202] for He and Ne suggest that, for physically relevant large distances, the approximate potentials are off by an almost constant shift, whereas the behavior close to the nucleus is more problematic. 7.4.2 Ad hoc DFT approximations for Dyson orbitals Two interesting but ad hoc approximations for Dyson orbitals were presented at the end of Subsection 7.2.3. The TOM approximation consists of using the transition orbital obtained in Slater's transition state method, while the KSKH approximation involves car rying out a KS DFT calculation on the daughter ion in the restricted space of the parent occupied molecular orbitals to obtain the orbital for the hole and using this "KSKH" or bital. Just as for the canonical KS orbitals, both the KSKH and TOM orbitals are used Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 145 to approximate Dyson orbitals via the target approximation. The results for H2O and acetylene, using the LDAxc functional, are shown in Figures 7.3 and 7.4, respectively. Consider first the KSKH approximation. For H2O, the l&i and 162 orbitals are lone symmetry states, so these KSKH orbitals should be identical to the canonical KS orbitals. This was indeed found to be the case in the calculations, which constitutes a useful check. Mixing of the three parent a\ states is allowed by symmetry, so the KSKH orbitals can differ from the canonical KS orbitals for these states. Nevertheless, Fig. 7.3 shows that the 3ai and 2ai KSKH orbital MDs are quite similar to the canonical KS orbital MDs. For acetylene (Fig. 7.4), the liru orbital is a lone symmetry state, and the KSKH and canonical KS orbitals were again found to be identical. But for the other states the KSKH and canonical KS orbital MDs look quite different. However, except for the case of ionization from the HOMO, these KSKH calculations involve excited states of the ion. Unfortunately, significant symmetry breaking, i.e. obtaining orbitals which do not belong to the irreducible representations of the symmetry group for the molecule, sometimes occurs in KS DFT calculations for excited state configurations, and the KSKH calculations presented here are no exception. This is readily apparent in the KSKH orbital MD for the 2cru orbital of acetylene which should vanish at zero momentum, by symmetry, but which does not. Symmetry breaking is apparent for the 3o-g, as well as the 2au, KSKH orbital, from Table 7.3, which shows the dipole moment, in centre of mass coordinates, of the daughter ion. No symmetry breaking is evident for the 2ag orbital, though this KSKH orbital is viewed with caution. The KSKH orbital MD for this orbital differs markedly from the canonical KS orbital MD. Considering all the orbitals of both molecules, the question remains open as to how much of the observed difference between the KSKH and canonical KS orbitals is attributable to problems with the excited state DFT calculations and how much is a real effect. It is clear, however, that, due to the vagaries of excited state DFT calculations, the KSKH approach is not reliable enough Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 146 Table 7.3: Symmetry breaking in DFT calculations on excited states of C2H2 as indicated by the nonzero dipole moment. The dipole moment is that of the daughter ion formed by removing an electron (half an electron in the case of the transition orbital method) from the orbital indicated. All calculations were performed with the origin at the centre of inversion symmetry for the molecule. Dipole Moment (a.u.) Orbital Transition Orbital Method Kohn-Sham Koopmans' Hole Orbitals lir^ (LOOOO OlJOOO 3(7g 0.0021 4.2457 2au 0.6884 3.2158 2 to be useful as an ad hoc DFT approximation to Dyson orbitals, and the results don't appear to be particularly better than those for the canonical KS orbitals anyway. The TOM also involves DFT calculations using excited state configurations, and some symmetry breaking in the TOM calculations on acetylene is apparent from the dipole moments (Table 7.3), though to a much lesser extent than in the KSKH calculations. Nevertheless, the transition state method often provides excellent estimates of ionization potentials [133,190,203]. Recall that the motivation for considering the TOM here is that the MD is sensitive to the asymptotic behavior of the orbital, which is in turn related to its eigenvalue. Thus one might think to improve the MDs by improving the orbital energies. For all the valence orbitals of H20 and C2H2 (Figs. 7.3 and 7.4) the height of the TOM orbital MD is less than the height of the corresponding KS orbital MD. A rough qualitative explanation of why this is so can be given in terms of the ionization potentials, since this is the primary difference between the KS and TOM approxima tions. As illustrated in Table 7.4, while the transition state method eigenvalue is a good approximation to the negative of the orbital ionization potential, the negative of the KS orbital energy gives an ionization potential which is substantially too low. A higher ion- Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 147 Table 7.4: Comparison of the negative of Kohn-Sham orbital energies at full (KS orbital energy) and half (transition state method) occupancy with experimental ionization po tentials, for C2H2 and H2O, using the LDAxc functional and the ANO+ basis set. The experimental values are taken from ref. [52], for C2H2, and from ref. [204]. for H20. The ionization potential "of the 2ai orbital" of H2O is not well defined, due to the severe breakdown of the one-electron picture in this region of the binding energy spectrum. Ionization Potentials (eV) Orbital KS orbital energy Transition state method Experiment C2H2 l7Tu 7.31 11.70 11.40 3 ization potential would generally be expected to correspond to a more contracted orbital in position space (as is consistent with Eq. (7.29), for neutrals), which leads to a loss of MD amplitude at low momentum, by the well known Fourier correspondence between position and momentum space distributions [69]. Although the KS orbital MDs do tend to be too high, this "correction" is too extreme. In all but one of the 8 orbitals in Figs. 7.3 and 7.4, the TOM MD is further from the CI MD than is the KS MD, and there is a marked disagreement between the TOM and CI MDs for the liru orbital of acetylene and all the valence orbitals of water. These results for the TOM MDs serve to illustrate that, even for the purpose of calculating MDs, a property which is particularly sensitive to the large r behavior of the orbital and hence to its eigenvalue, approximating the Dyson orbital is not synonymous with approximating its eigenvalue. Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 148 7.4.3 Effect of the Functional The results in the previous two subsections, on the TKSA and on the ad hoc KSKH and TOM approximations, are all at the level of the local density approximation, LDAxc (LDA including both exchange and correlation). The present subsection returns to the TKSA and gives a preliminary investigation of the effect of the functional, for H2O. The effect of gradient corrections, as well as the separate contributions of the exchange and correlation functionals, are considered. Among the well-known problems with the local density approximation are self-inter action errors, overbinding of molecules, and an exchange-correlation (xc) potential which falls off too rapidly, asymptotically. One might expect that this last point would be important for MDs. The physical effect of an xc potential which falls off too rapidly is that an electron in the large r region feels the full non-self-interaction corrected Hartree potential for N electrons instead of the correct potential for (N — 1) electrons. This makes the electron less bound than it should be, its position space orbital too diffuse, and the corresponding MD have too large an amplitude at small p. This is consistent with the TKSA results reported in Subsection 7.4.1. Given the sensitivity of MDs to the large r region, the 1988 exchange-correction of Becke (B88x) [120] was chosen as an alternate exchange functional for use in this study. This was combined with the 1986 correlation correction of Perdew (P86c) [121]. The Becke functional gives the correct asymptotic behavior of the exchange-energy density, ex(r), which is expected to improve the asymptotic behavior of the exchange-potential, vx(r). Nevertheless, it should be noted that the B88x potential still does not have the correct asymptotic form. Figure 7.11 shows the TKSA MDs for the valence orbitals of water as well as the corresponding THFA and CI MDs. The KS orbitals have been calculated using the LDAx and B88x exchange-only functionals and the LDAxc and Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 149 u.ua - (o) BM« BBOx+PBec LDta> H 0 1b, ^ 2 cO.06 • <3 a HF \ N |0.03 - t o 2 •C M ) 1 3 1 Momentum (au) ' Ts Momentum (au) 0.09 0.40- (d) H20 2a, UM» UMXC.B88K 'B88x+P8Sc HF 0.00 1 1 ^/^ ' T, Momentum (au) Momentum (au) Figure 7.11: Effect of the exchange-correlation functional on the target Kohn-Sham ap proximation MDs for the valence orbitals of water. The curve labels are: exchange-only local density approximation (LDAx); exchange-correlation local density approximation (LDAxc); exchange-only functional using Becke's 1988 gradient correction (B88x); ex change-correlation functional using Becke's 1988 gradient correction for exchange and Perdew's 1986 gradient correction for correlation (B88x+P86c); target Hartree-Fock ap proximation (HF); configuration interaction (CI). Dashed curves are used for MDs cal culated with gradient-corrected functionals. The density functional calculations use the AN0+ basis set. The HF and CI calculations, taken from refs. [28,57,200], are those shown in Fig. 7.3. Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 150 B88x+P86c exchange-correlation functionals. The effect of the B88x and P86c gradient corrections on these MDs is to lower the peak heights slightly. This constitutes an improvement for the three outer-valence orbitals, but not for the inner-valence orbital where the LDA (x and xc) MDs were already below the CI. As mentioned in subsection 7.2.3, the exact exchange-only KS orbitals have been calculated for atoms and have been found to be remarkably similar to HF orbitals. This would lead one to expect that the exchange-only TKSA should give very similar MDs to the THFA MDs, if the functional were exact. However the difference between the LDAx TKSA MDs and the THFA MDs shown in Fig. 7.11 is dramatic. This difference is probably primarily due to deficiencies in the LDAx functional. Nevertheless, in the absence of MDs calculated from the exact KS orbitals, the possibility cannot be ruled out that the difference might be real and not an artifact of the functional used. The B88x functional is expected to improve the description of the asymptotic behavior of the KS exchange-potential. However this gradient correction moves the MDs for the exchange-only KS orbitals only a little closer to the THFA MDs. An interesting point to note about the curves in Fig. 7.11 is the effect of including correlation in the functional. This acts to reduce the height of the TKSA MDs, whereas the CI MDs for these orbitals have a larger amplitude than do the corresponding THFA MDs. Since a more diffuse (contracted) orbital generally corresponds to a higher (lower) amplitude MD, the correlation functionals used here appear to result in a contraction of the outer orbitals and hence of the overall charge density, while the addition of electron correlation to Hartree-Fock acts to make the valence Dyson orbitals, and hence the overall charge density, more diffuse. This is an indication that electron correlation is being treated differently in DFT than in conventional ab initio electronic structure methods. This same effect has been previously observed [205] for dipole polarizabilities, another property which is sensitive to the large r behavior of the charge density. Table 7.5 shows Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 151 Table 7.5: Effect of exchange-correlation functional on the dipole moments and polariz- abilities of H20, using the ANO+ basis set. The molecule is oriented in the (a;,z)-plane with the dipole along the .z-axis. The notations "x" and "c" refer respectively to the in clusion of exchange and correlation in the functional. HOND08 near Hartree-Fock-limit results have been included for comparison purposes. Polarizability (a.u.) Method Dipole Moment (a.u.) Local density approximationoCxx s yy * LDAx1 0.7204 10.985 10.673 10.906 LDAxc2 0.7334 10.430 9.890 10.234 Gradient-corrected functionals B88x3 0.6997 10.855 10.369 10.661 B88x3+P86c4 0.7135 10.364 9.779 10.110 Experimental values5 0.727 10.31 9.55 9.91 HOND08 calculations6 HF 0.7802 9.16 7.91 8.46 1 Original Xa = 2/3 functional of Kohn and Sham [25]. 2 As parameterized by Vosko, Wilk, and Nusair [119]. 3 Becke's 1988 gradient correction for exchange [120]. 4 Perdew's 1986 gradient correction for correlation [121]. 5 See the footnotes for Table 7.1. 6 Taken from Ref. [139]. the experimental and Hartree-Fock dipole polarizabilities of water as well as polarizabil- ities calculated using the same basis sets and density functionals used elsewhere in the present study. It is evident that the exchange-only calculations overestimate the dif- fuseness and hence the polarizabilities while the exchange-correlation density functionals give much more reasonable results. An explanation was proposed in ref. [205] where it was noted that correlation man ifests itself through two competing effects. On the one hand, an admixture of excited state configurations in the many-electron wavefunction tends to make the electron density more diffuse hence increasing the polarizability. On the other hand, electron correlation enhances the ability of the electrons to avoid each other hence minimizing the effects Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 152 of electron repulsion and leading to a more contracted charge density. Present density functional appear to emphasize the latter effect for MDs and polarizabilities, although the former effect should apparently be the dominant one for these properties. This picture seems physically reasonable, providing it is refined by recognizing that which one of these two effects of correlation dominates may vary from one region of space to another. This is easiest to see for atoms. Since electron correlation lowers the total electronic energy, by the virial theorem, it must also increase the kinetic energy. One way for this to happen is for the electrons to move closer to the nucleus in the energetically important core region. At the same time, this contraction of the core increases the screening of the nuclear charge, thereby allowing the outer, valence parts of the charge density to expand. This general picture is consistent with findings on the exact correlation potential for two electron atoms [202,206]. Umrigar and Gonze [202] have compared this exact correlation potential with correlation potentials calculated from the LDA and various state-of-the-art functionals and find that the correlation potentials from the approximate functionals have on average the wrong sign in comparison with the exact result. However, this tends to counteract errors in the exchange potential, so the net effect is a cancellation of errors in the exchange and correlation functionals. Thus, in spite of the shortcomings of the correlation functionals, the xc functionals do generally improve upon the exchange-only calculation. 7.5 Conclusions This chapter introduces the target Kohn-Sham approximation for use in the analysis of electron momentum spectroscopy experimental momentum profiles. Instead of approx imating Dyson orbitals as being proportional to canonical Hartree-Fock orbitals as is done in the well known target Hartree-Fock approximation, the TKSA approximates Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 153 Dyson orbitals as being proportional to Kohn-Sham density-functional orbitals. Since density-functional calculations are computationally less demanding than Hartree-Fock calculations, the TKSA provides a more efficient way to calculate Dyson orbitals for use in conjunction with EMS. The theoretical foundation for this approximation has been discussed, and an assessment of the quality of the spherically averaged orbital momentum distributions thus obtained. Density functional theory has developed from two complementary points of view. In Kohn and Sham's formulation of DFT [25], the true total energy is obtained via an (exact) functional of the charge density, and a set of orbitals whose charge densities sum to the true total charge density was introduced as a physically fictitious mathematical device to facilitate representation of the kinetic energy. Sham and Kohn [207] noted the differences between the KS and Dyson equations early on in the history of modern DFT. While their discussion focused on the eigenvalues of the equation rather than the orbitals, it did serve to emphasize that solutions of Dyson's equation cannot be calculated exactly within the framework of Kohn-Sham DFT even if the exact exchange-correlation functional were known. The other point of view, dating back to Slater [22], and rigorously formulated by Sharp and Horton [142], Talman and Shadwick [143], and Casida [118], consists of finding the variationally best mathematically local approximation to Dyson's quasiparticle equation (or equivalently to the HF equation, in the exchange-only case). Here the KS orbitals and orbital energies are naturally seen as approximations to the Dyson (or HF) orbitals and orbital energies. The work of Sham and Schluter [144] on the exact KS exchange-correlation potential shows that (within the linear response approximation to the Sham-Schluter equation) these two, formally different approaches both lead to the same Kohn-Sham equation. The present work is based on the second point of view. Although the idea of KS Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 154 orbitals and orbital energies as approximate Dyson orbitals and orbital energies is of ten viewed with skepticism due to the poor quality of this approximation for the orbital energies, this by no means implies a corresponding problem with the orbitals. Indeed, ex amination of the nature of the mathematical localization process, in appendix C, clarifies why the KS eigenvalues should be shifted even if the orbitals are quite good approxi mations to the Dyson orbitals. The various theoretical considerations involved in the approximation of Dyson orbitals by Kohn-Sham orbitals, including the introduction of the target Kohn-Sham approximation, were presented in Section 7.2. The rest of this discussion is devoted to investigating the practical utility of the TKSA for the calculation of MDs for use in EMS. This necessarily involves a second approx imation, namely the use of approximate functionals. Results using the local density approximation for the Kohn-Sham exchange-correlation functional have been presented for 18 orbitals in 6 small molecules and atoms where high quality CI and HF calculations as well as experimental data are available for comparison. The quality of these results for the TKSA is comparable to that of the THFA. It is interesting to note that while the magnitude of the error in the TKSA peak heights and positions is generally similar to that in the THFA, the direction of the error in the TKSA is often opposite (for height) to that in the THFA. Both KS and HF orbitals approximate Dyson orbitals, but they do so in different ways. The effect of using different approximations for the exchange-correlation functional was considered for H2O. From this it appears that part of the observed difference between the TKSA and the CI MDs may be due to the approximate functionals used rather than to the underlying approximation of Dyson orbitals by KS orbitals via the TKSA. Just how much of the error is due to which of these two approximations is an interesting question whose answer will require the calculation of MDs from exact KS orbitals. The success of the target Kohn-Sham approximation introduced here leads to much Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 155 interesting work which remains to be done. Certainly, the calculation of exact exchange- correlation potentials, for molecules as well as for atoms, which would allow assessment of the quality of the TKSA itself, separate from the question of the approximate func tional, would be of fundamental interest. Since EMS probes primarily the large r region, calculation of other orbital-dependent (as opposed to total) properties, as well as direct comparison of the Dyson and KS orbitals in the TKSA, would be very useful in evalu ating the overall quality of this approximation. This is particularly important since, as has been pointed out above, while the formal considerations [118] make it clear that the KS orbitals are approximations to Dyson orbitals, there is no a priori statement as to how good an approximation this is. For the same reason, the extension of this study to a larger number of molecules, and especially to molecules whose MDs have more compli cated shapes, is essential to obtaining an accurate picture both of the overall quality of the TKSA and of its utility for EMS. Work in this direction has already been undertaken by the University of British Columbia EMS group, and interesting results for acetone [208] as well as the amines, ethylene, and glycine have been obtained [209]. In all cases quite good agreement (for shape) has been observed between the experimental and theoretical momentum distributions. From a practical point of view, a more extensive investigation of the relative merits of various approximate functional for calculating MDs would also be useful for EMS. Such efforts are presently underway [209], As the energy resolution and signal to noise ratio of EMS experiments improves, EMS is able to handle larger molecules with increasing accuracy. However, the analysis of these experiments places more severe demands upon the accuracy of the theoretical models. One of the primary advantages of EMS over photoelectron spectroscopy is the ability to assign binding energy spectra on the basis of comparisons between experimental momentum profiles and calculated spherically averaged orbital momentum distributions. Chapter 7. EMS from DFT: The Target Kohn-Sham Approximation 156 Of course this advantage can only be realized for molecules where the theoretical cal culations can be done. The quality of the MDs obtained in the target Hartree-Fock ppro ximation has proven to be generally adequate for this purpose. Unfortunately how ever, the HF calculations have become the limiting factor in determining the size of the molecules whose MDs are feasible to calculate routinely for use with EMS. Since DFT is computationally less demanding than HF, the present finding that the shape of the MDs in the target Kohn-Sham approximation is comparable to that of the THFA or CI MDs is of considerable practical utility for EMS. Chapter 8 Calculation of Ionization Potentials from Density Functional Theory 8.1 Introduction Vertical ionization potentials of small molecules can be accurately computed by the Green function method [210], Rayleigh-Schrodinger perturbation theory [211], configuration interaction [212], and other techniques [213]. For larger molecules, one is forced to sacrifice accuracy by using smaller basis sets. Typical basis sets are double-zeta Slater- type orbitals (STOs) or contracted Gaussian-type orbitals (CGTOs). In contrast, density functional methods use a single configuration. Consequently, one can employ better basis sets and treat larger molecules. Twenty years ago, Slater introduced the transition-state (TS) method [26] Consider an electronic process such as ionization or excitation at fixed molecular geometry. Let us first define k 2 3 4 s E(X) = £ X Ek = E0 + XEX + X E2 + X E3 + X E4 + X Es + • • • (8.1) k where E(0) and E(l) correspond to the initial and final states respectively, and A is assumed to be a continuous variable. In other words, the endothermicity sought is given by AE = E(l) - E(0) = Ex + E2 + E3 + E4 + EB + • • • (8.2) Now, if the derivative 2 3 4 F(X) = dE/dX = Ei + 2XE2 + 3X E3 + 4A E4 + 5X ES + •-• (8.3) 157 Chapter 8. Calculation of Ionization Potentials from Density Functional Theory 158 is defined, then, in Slater's original TS concept, AE is approximated by F(h = E! + E2 + 3£3/4 + £4/2 + 5E6/16 + • • • (8.4) with an error of STS = F(h -AE = -E3/4 - E4/2 - 11£5/16 + • • • (8.5) For ionization of an electron from molecular orbital <}>k, for example, F(|) is equal to — e/t when 0.5 electron has been removed from fa. In the HAM/3 semi-empirical MO method, Asbrink et dl. [214] proposed the so-called diffuse ionization (DI) model, which is essentially an average of Slater's TS method [26], In the DI model, half of an electron is removed evenly from the valence MOs and the negative of the resulting orbital energies correspond to calculated VIPs. This allows all the valence VIPs to be obtained in a single calculation. The objective of this work was to test the accuracy of such a simple procedure using density functional theory. In the generalized transition-state (GTS) method, Williams et al. [215] proposed the use of F(l) = Ei+ 4£2/3 + 4£3/3 + 32£4/27 + 80£5/81 + • • • (8.6) Therefore, AE can be approximated by F(0) + 3F(|)] /4 = Ei + E2 + E3 + 8E4/9 + 20£5/27 + • • • (8.7) with an error of only SEGTS = -E4/9 - 7£5/27 + • • • (8.8) which is expected to be smaller than STSE. Consequently, it is also desired to test the accuracy of the DI version of such a simple extension of Slater's TS concept. Chapter 8. Calculation of Ionization Potentials from Density Functional Theory 159 8.2 Density Functional Computations All calculations were performed at the experimental equilibrium geometry [103,216]. A basis set of excellent quality was used, consisting of [543f;430f] atomic natural orbitals [196], [543f] for first-row atoms and [430f] for hydrogen. The numerical fitting grid chosen is extra-fine and non-random and the auxiliary functions are those recommended [217], namely (4,4;4,4) for first-row atoms and (3,1;3,1) for hydrogen. Although there is an option to use the non-local exchange-correlation potential vxc, the local density approximation vxc of Vosko et al. [119] was employed exclusively in this study. The approach outlined in the introduction can be applied in two ways. In what is labelled "restricted" in this work, the MOs for a are the same as those for ft electrons. The C2H2 molecule, with five valence MOs and ten valence electrons can be used as an example to illustrate the various approaches which were studied in this work. In the restricted transition state (rTS) method, 0.25 a electrons and 0.25 ft electrons are removed from the MO k of interest, and the VIP is approximated by the negative of the orbital energy of fa. In the restricted diffuse ionization (rDI) model (which is the model used in the HAM/3 semi-empirical method [214]), the electron removal is spread out over all the valence MOs. In other words, each valence MO of C2H2 has 0.95 a and 0.95 /? electrons. The restricted generalized diffuse ionization (rGDI) model is just the extension of the rDI model using Eqn. 8.7. On the other hand, in the 'unrestricted' calculations, the MOs for a electrons are different from those for /? electrons. In the unrestricted transition-state (uTS) method, 0.5 electron is removed from the spin-orbital of interest and the negative of that orbital energy is regarded as the approximate VIP. For the unrestricted diffuse ionization model (uDI), each valence MO for the a orbitals of C2H2 has 0.9 electrons. Finally, in the unrestricted generalized diffuse ionization model (uGDI), | a electron is ionized evenly Chapter 8. Calculation of Ionization Potentials from Density Functional Theory 160 over the valence MOs for the a electrons. In a preliminary study [128], the method labelled rDI here was tested on the VIPs of OF2, HNF2, and CH2F2, with encouraging results. This chapter presents a more detailed study of some of the other approaches and includes more molecules in the test sample. 8.3 Results and Discussion In table 8.1, detailed results are presented for the VIPs of the C2H2 molecule. For C2H2, the VIPs can be predicted particularly well by removing 0.5 electrons from the Z(Tg MO and looking at the -e*.. Moreover, the entries in table 8.1, lend support to the DI model of Asbrink et al. [214], The overall trend confirms the earlier finding of Baerends and Ros [218] and of Sambe and Felton [219,220] that, for the same charge, ground and excited states yield very similar eigenvalues. It can also be seen that the extension by Eqn. 8.7 has very little effect on the results. However, it would not be appropriate to draw definite conclusions from a comparison of only four VIPs. In any case, the DI model is worth further study because it can save much CPU time by giving all VIPs in a single calculation. In table 8.2, the various density functional approximations are compared to VIPs mentioned above with the observed values [85,86,221-229] and with those calculated by the HAM/3 semi-empirical MO method [94,230,231] and by Rayleigh-Schrodinger perturbation theory [232-236]. There are now 38 VIPs for nine small molecules and the comparison is more meaningful. The average absolute deviations of various calculated values from experiment confirm the tentative conclusions from Table 8.1. Moreover, the average absolute deviation is adversely affected by the VIPs above 20 eV. If those were excluded from the analysis, then the average absolute deviation would decrease by as much as 0.1 eV as indicated. Chapter 8. Calculation of Ionization Potentials from Density Functional Theory 161 Table 8.1: Ionization potentials of C2H2 as calculated using different methods in density functional theory. See the text for a detailed description of the methods used. r/uDI: re stricted/unrestricted diffuse ionization; r/uTS: restricted/unrestricted transition-state; r/uGDI: restricted/unrestricted generalized diffuse ionization. Molecular orbital of interest Method l7Tu 3 j = 3cTg 11.66 17.10 18.75 23.16 0.24 j = 2 Approximate uTS method with 0.5 electrons removed from spatial MO j, where j = l7Tu 11.70 16.51 18.12 23.18 0.33 j = 3 1 Bieri and Asbrink [86]. Chapter 8. Calculation of Ionization Potentials from Density Functional Theory 162 Table 8.2: Ionization potentials of several molecules as calculated by various methods within density functional theory. See text for details. rTS: restricted transition state; rDI: restricted diffuse ionization; rGDI: restricted generalized diffuse ionization; uTS: unrestricted transition state; uDI: unrestricted diffuse ionization; uGDI: unrestricted generalized diffuse ionization. deMon density functional method3 Molecule Orbital Obs. HAM/31 RSPT2 rTS rDI rGDI uTS uDI uGDI s 8 H20 lbi 12.62* 12.93 12.06 13.80 13.47 13.57 13.13 13.14 13.23 3ai 14.74 15.39 13.93 15.90 15.46 15.56 15.23 15.15 15.23 lb2 18.51 18.28 18.72 19.82 19.51 19.59 19.27 19.21 19.28 2a, 32.27 32.88 32.02 32.13 31.66 31.75 31.65 31.31 31.39 2b 10.98 10.739 11.156 11.29 11.14 11.18 10.89 10.94 10.99 H2CO 2 lbi 14.5 14.83 14.73 15.62 15.26 15.30 15.17 15.07 15.11 5ai 16.1 16.47 16.19 16.58 16.07 16.12 16.11 15.87 15.91 lb2 17.0 17.35 17.59 17.25 17.01 17.05 16.92 16.85 16.89 4ai 21.4 21.24 21.87 20.93 20.32 20.36 20.51 20.17 20.21 2bi 13.2610 13.459 13.29* 13.30 12.97 13.00 12.97 12.83 12.85 OF2 6ai 16.17 16.02 16.33 16.48 16.24 16.28 16.36 16.10 16.13 4b2 16.32 15.87 16.54 16.10 15.89 15.92 15.89 15.74 15.76 la2 16.47 16.29 16.69 16.23 16.12 16.15 16.09 15.97 16.00 lbj 18.68 18.83 18.55 18.40 18.32 18.35 18.18 18.19 18.22 5aj 19.50 19.50 18.90 19.81 19.54 19.57 19.47 19.42 19.44 3b2 20.90 20.94 19.99 21.43 21.19 21.22 21.12 21.05 21.08 11 s 12 N2 3 13 F2 llTg 15.83 16.38* 15.66" 15.88 15.84 15.89 15.61 15.62 15.66 liru 18.80 19.39 18.69 19.09 19.14 19.19 18.86 18.93 18.98 Z In Chong's earlier study [231], the VIPs of sixteen 22-electron molecules were calcu lated with the HAM/3 semi-empirical MO method and compared with experiment. In this work, the same sixteen molecules were selected and the rDI and uDI models applied to them. The results are summarized in Table 8.3. In total, then, 106 VIPs for 25 closed-shell molecules have been computed using the deMon density functional program [191-193] and compared the results with experiment and the results from the HAM/3 semi-empirical MO method. The accuracy is very similar, with an average absolute deviation of about 0.4 eV, slightly better when the Chapter 8. Calculation of Ionization Potentials from Density Functional Theory 163 deMon density functional method Molecule Orbital Obs. HAM/3 RSPT rTS rDI rGDI uTS uDI uGDI HF Iff 16.19" 16.829 15.5318 17.17 17.07 17.19 16.68 16.67 16.78 3 HCN l?r 13.8018 14.029 13.48" 14.34 14.26 14.31 14.10 14.07 14.12 5(7 14.15 14.68 13.49 14.23 14.13 14.18 14.01 13.91 13.95 4a 19.68 20.58 20.33 21.36 19.96 20.00 20.56 19.80 19.83 s 12 C2H2 liru 11.49" 11.59 11.05 11.91 11.84 11.88 11.70 11.68 11.72 Zffg 16.7 17.18 17.00 17.10 16.83 16.87 16.74 16.65 16.68 2 20 21 22 CH3CO 2bi 9.8 9.70 9.14 10.30 10.14 10.17 9.99 10.03 10.06 2b2 14.2 14.38 14.23 14.50 14.39 14.40 14.23 14.25 14.27 lbi 15.0 15.28 15.56 16.04 15.50 15.53 15.63 15.37 15.40 lbs 16.3 16.45 16.61 16.38 16.27 16.29 16.18 16.14 16.16 7ai 16.8 16.75 16.70 16.52 16.45 16.48 16.32 16.34 16.37 6ai 18.2 18.26 18.44 18.38 17.45 17.48 17.71 17.31 17.34 Average absolute 0. 0.33 0.40 0.57 0.45 0.47 0.39 0.43 0.43 deviation for all 38 VIPs Average ab 0. 0.34 0.36 0.52 0.37 0.40 0.31 0.32 0.33 solute deviation for 32 VIPs excluding those above 20 eV 1 Semi-empirical HAM/3 molecular orbital method. 2 Perturbation corrections to Kootmans' theorem using double-zeta Slater-type orbital or contracted Gaussian-type orbital basis set. 3 With local Vse and large [543fj430f] basis set. See text. *Potts and Price [221] 5Chong [94] "Turner et al. [225] "Bieri et al. [85] 8Chong et al. [232] 12Chong and Takahata [233] "Pridh and Asbrink [228] rBrundle et al. [222] "Cornford et al. [226] "Bieri and Asbrink [86] *von Niessen et al. [223] "Chong et al. [234] 20Hall et al. [229] 9Chong [230] "Shaw and Thomas [227] 21 Chong [231] 10Cornford et al. [224] "Mukherjee and Chong [235] 22Chong [236] inner valence VIPs are excluded. Because of the large sample size this value of 0.4 eV is not expected to change even if other experimental photoelectron spectra and/or slightly different assignments for some of the spectra were used. While the HAM/3 method is far less demanding computationally, it suffers from two drawbacks: it is a heavily parameterized method, and has not be extended beyond the five elements H, C, N, 0, and F. On the other hand, density functional methods, including the parameterization in the local vxc of Vosko et al. [119] are generally more acceptable Chapter 8. Calculation of Ionization Potentials from Density Functional Theory 164 Table 8.3: Comparison of vertical ionization potentials (VIPs) for sixteen 22-electron molecules. See the text for a detailed description of the methods used. r/uDI: re stricted/unrestricted diffuse ionization; DFT* Molecule Orbital Obs. HAM/32 rDI uDI FCN 2ir 13.653 14.07 13.62 13.49 7 FCCH 2TT 11.26* 11.70 11.43 11.32 lff 17.8 17.57 17.26 17.15 7 NNO 2* 12.89* 12.48 13.43 13.29 7 HCNO 2ir 10.836 10.99 11.30 11.16 lir 15.92 17.14 16.93 16.80 7a 17.48 18.84 17.72 17.56 6 HNCO 2a" 11.627 11.84 12.13 12.00 9a' 12.30 12.61 12.61 12.47 8a' 15.8 16.17 16.17 16.04 la" 15.8 16.19 16.24 16.11 7a' 17.50 17.59 17.15 16.99 6a' 19.24 20.37 19.95 19.83 HN3 2a" 10.70* 10.48 11.40 11.28 9a' 12.2 11.95 12.67 12.55 8a' 15.47 15.48 15.39 15.26 7a' 16.7 17.29 17.43 17.31 la" 17.4 17.41 17.53 17.40 6a' 20.1 20.94 20.22 20.10 9 H2NCN 9a' 10.65 10.76 10.89 10.77 2a" 12.50 12.37 12.30 12.19 8a' 12.98 13.40 12.63 12.50 7a' 14.23 14.13 14.32 14.21 la" 18.8 18.36 17.62 17.51 6a' 19.6 19.28 20.67 20.54 10 CH3CN 2e 12.18 12.16 12.30 12.18 7ai 13.11 13.10 12.54 12.41 le 15.5 15.29 15.44 15.34 6ai 17.4 16.87 16.79 16.68 11 12 CH3NC 7a, 11.24 ' 11.80 10.80 10.70 2e 12.46 12.24 12.55 12.43 le 16.14 15.79 15.88 15.78 Chapter 8. Calculation of Ionization Potentials from Density Functional Theory 165 DFT1 Molecule Orbital Obs. HAM/32 rDI uDI CH3CCH 2e 10.3710 10.27 10.35 10.25 le 14.4 14.58 14.27 14.17 7ai 15.5 15.01 15.01 14.90 6ai 17.2 17.64 16.60 16.49 6 CH2N2 2bj 9.00 8.97 9.65 9.55 2b2 14.13 13.88 14.14 14.03 7ai 15.13 15.05 14.84 14.71 lbj 16.93 15.94 16.29 16.16 lb2 16.93 17.37 17.37 17.26 6ai 18.5 18.70 17.81 17.70 Diazirine 3b! 10.75" 10.75 11.16 11.05 2b2 13.25 13.05 13.14 13.03 6ax 14.15 14.30 14.07 13.94 5ai 16.5 16.36 16.67 16.55 lb2 17.5 17.01 17.32 17.21 2bi 20.3 19.19 19.39 19.27 4ai 21.5 22.37 21.26 21.14 13 Cyclopropene 2b2 9.86 10.01 9.99 9.90 3bj 10.89 11.08 11.13 11.02 6a! 12.7 12.82 12.45 12.34 lb2 15.09 15.28 14.56 14.46 Sai 16.68 16.65 16.25 16.14 2bi 18.3 19.32 17.64 17.54 4ai 19.6 20.30 18.49 18.38 Allene 2e 10.02" 10.06 10.35 10.26 le 14.75 15.11 14.60 14.50 4b2 14.75 15.20 14.90 14.79 3ai 17.3 17.57 16.48 16.37 Avg. Abs. Dev. 0. 0.42 0.39 0.41 ^hong [231] 3 With local VIC and large [543f;430f] basis set. See text. 3Bieri [237] 7Cradock et al. [238] "Lake and Thompson [239] 4Haink et al. [240] 8Lee et al. [241] 12 van Piggelen and Worrell et al. [242] sBrundle and Turner [243] 9Stafast and Bock [244] "Robin et al. [245] 6astide and Maier [246] 10 Frost et al. [247] "Thomas and Thompson [248] to chemists and physicists, and can be applied to any molecule for which good basis sets are available. The deMon program used in this work is one of several density functional methods being developed currently. It has the advantage that the calculations can even be performed on a microcomputer [127]. Chapter 9 Testing the Accuracy of the DFT Electron Density 9.1 Introduction Because DFT is a relatively new topic it has, in its present form, not undergone any extensive evaluation of the nature and accuracy of the electron density which it pre dicts. This is starting to change, however, as recent work concerned with the dipole moments [127,134,249-251], dipole polarizabilities and hyperpolarizabliities [128,136, 137,139,252-254] spin densities [255,256], and infrared intensities [129-131,134,257- 261] (related to normal coordinate derivatives of the dipole moment) has been carried out, with encouraging results. It is not enough, however, to know that DFT predicts accurate dipole moments as well as electric field and nuclear coordinate derivatives of the dipole moment. If DFT is to be extended to other applications (for instance to electron momentum spectroscopy [262]), an accurate assessment must be made of other regions of the charge density as well. Electron momentum spectroscopy [29] will provide an accurate assessment of the longer-range characteristics of the charge density. However, what is really required is a full assessment of the charge density over many regions of space, from very close in to an atomic centre to larger distances away from the molecule. It is the purpose of this chapter to perform a preliminary survey of this kind. That is, one-electron properties will be calculated which weight different regions of the electron density, from very close to or at the nucleus to farther out spatially. Eight small molecules 166 Chapter 9. Testing the Accuracy of the DFT Electron Density 167 were chosen for this study: NH3, PH3, H20, H2S, HF, HC1, N2, and CO. These same molecules were also chosen for a benchmark study of one-electron properties calculated by multireference singles and doubles configuration interaction by Feller et al. in 1987 [57]. Because the same set of molecules is used as was presented in that study (and the same geometries), the properties reported in this work should, if calculated with sufficient accuracy, match the very accurate calculations of Feller et al. [57]. A subset of the properties reported in Feller et al. is reported in this work. Specifically, the dipole moment, quadrupole moment, and diamagnetic susceptibility will be used to assess the quality of the intermediate and outer spatial part of the charge density. To investigate the accuracy of the electron density in regions closer to the nucleus, the electron density at the nucleus, electric field gradient, and electric field (with the corresponding Hellmann- Feynman forces) will be compared to experimental values (as available) as well as to the configuration interaction calculations of Feller et al. The rest of this chapter is organized as follows. Section 9.2 will discuss the choice of basis set used for the work reported in this study, and describe some of the considera tions necessary when importing deMon output into other quantum chemistry programs (HONDO calculations were carried out by Dupuis at Cornell). The region of the charge density which each property emphasizes will also be discussed. Finally, section 9.3 de scribes the results of the computations carried out, and section 9.4 describes further work that would follow naturally from this study. 9.2 Computational Details All density-functional calculations reported in the present work were carried out with the program deMon [191-193]. As the results of these calculations were to be compared with the benchmark calculations of Feller et al. [57], all calculations carried out in the present Chapter 9. Testing the Accuracy of the DFT Electron Density 168 work employed the geometries reported in Feller et al. [57]. The basis sets employed in that work were not available, however, and so the best available alternatives were used. These are discussed below. In order to maintain a high level of accuracy and to enforce the high symmetry in the small molecules studied here, an extrafine nonrandom grid was used in all calculations. To further ensure precision in the calculated properties, all calculations carried out using deMon were converged to five successive energy differences of less than 10-8 a.u.. The coefficients (which had to be rearranged to match the order required for HONDO), basis sets, and molecular geometries from these calculations were output in a form ready for input into HONDO 8.4 for calculation of the one-electron properties [263]. It was necessary that HONDO be modified for this study. Normally, when a basis set and MO coefficients are read in by HONDO, the resulting MOs are re-orthonormalized before being run through the properties calculations. For this study, however, it was necessary that the orthonormalization step in HONDO be deactivated because of the different treatment of near-linear basis set dependencies between HONDO and deMon. When the orthonormalization step was skipped, the calculated dipole moments from deMon and HONDO agreed exactly. Special attention was paid to the auxiliary functions used in this study. Because properties which depended on the charge density both near to and far from the nucleus were to be evaluated, and because basis sets with large exponents (spatially "tight") were to be used, it was important to ensure that the auxiliary functions used fit the charge density generated as closely as possible. To accomplish this, an additional large-exponent auxiliary function was added to both the charge density and exchange-correlation sets. The resulting auxiliary basis functions were thus labelled (5,4;5,4) and (6,4;6,4) for the first and second period elements, and (4,1;4,1) for hydrogen. Table 9.1 lists the auxiliary basis functions. Chapter 9. Testing the Accuracy of the DFT Electron Density 169 Table 9.1: Auxiliary basis functions for used in conjunction with the ANO and Dunning basis sets for one-electron property calculations. For carbon to fluorine, for example, these sets were derived from the (4,4;4,4) sets from deMon by deleting the highest-exponent primitive in the original sets and generating two new higher-exponent primitives based on the observed geometric progression within the set. The basis function groups are given in the (top-down) order j, k, m, and n, where these are as defined in chapter 6. Hydrogen Carbon Nitrogen Oxygen Fluorine Phosphorus Sulphur Chlorine 225.0 5197.5 7229.25 8977.5 10867.5 52420.0 65540.0 81920.0 37.5 1155.0 1606.5 1995.0 2415.0 10484.0 13108.0 16384.0 7.5 330.0 459.0 570.0 690.0 2621.0 3277.0 4096.0 1.5 94.3 131.0 163.0 197.0 655.0 819.0 1024.0 27.0 37.5 46.5 56.3 164.0 205.0 256.0 41.0 51.0 64.0 0.3 9.92 13.8 17.0 21.0 13.0 16.0 20.0 2.2 3.06 3.8 4.6 2.6 3.2 4.0 0.63 0.88 1.08 1.33 0.64 0.8 1.0 0.18 0.25 0.31 0.38 0.16 0.2 0.25 75.0 1732.5 2409.75 2992.5 3622.5 17473.0 21847.0 27307.0 12.5 3S5.0 535.5 665.0 805.0 3495.0 4369.0 5461.0 2.5 110.0 153.0 190.0 230.0 874.0 1092.0 1365.0 0.5 31.4 43.7 54.0 65.7 218.0 273.0 341.0 9.0 12.5 15.5 18.8 55.0 68.3 85.0 13.6 17.1 21.0 0.1 3.3 4.6 5.66 7.0 4.3 5.3 6.7 0.73 1.02 1.27 1.53 0.85 1.07 1.3 0.21 0.29 0.36 0.44 0.21 0.27 0.33 0.06 0.083 0.1 0.13 0.053 0.067 0.083 deMon, like all density functional programs, requires that a choice for the functional form for the evaluation of exchange-correlation potential and energy densities be made. The local density functional approximation is the simplest and most widely used approx imation in the density functional literature, and was the one chosen for the majority of the results presented in this study for that reason. The parameterization used for the LDA functional is given by Vosko, Wilk, and Nusair [119], as an exact form for it is unknown. The properties reported in this work are the dipole moment, quadrupole moment, diamagnetic susceptibility, electron density at the nucleus, electric field and Hellmann- Feynman forces at the nucleus, and the electric field gradients at the nucleus. These properties emphasize different regions of the electron density, and so can be ordered Chapter 9. Testing the Accuracy of the DFT Electron Density 170 according to the particular region of electron density which contributes most to their values. All reported properties include nuclear contributions with the exception of the diamagnetic susceptibility and the electron density at the nucleus. The electron density at the nucleus, 8, is perhaps the property which depends the most on an accurate representation of the electron density near the nucleus, since it is a direct measure of the density right at the nucleus (but does not include any contributions from the nuclei themselves). Because the electron density has units of electrons per unit volume, it can be said, loosely speaking, that S weights the electron density according to 1/r3. The components of the electric field gradient, q, also weight very heavily the region very close to the nucleus, as they are calculated as the (traceless) expectation values of, for example, |(3z2 — r2)/rs (for the zz component). The directional nature of these components provide extra information too, indicating the accuracy of the charge density in different directions out from the nucleus. Dimensionally, the components of q weight the electron density as 1/r3. The components of the electric field at the nucleus and the corresponding Hellmann- Feynman forces also provide an indication of the accuracy of the electron density close to the nucleus, though these properties do not weight the density near to the nucleus as much as the electric field gradients. This is because electric field components are given by the expectation value of the operator (for example) zjr3 (for the z component), and so weight the electron density by a factor of 1/r2. The Hellmann-Feynman forces are obtained from the electric field components by multiplying the charge on the nucleus by the electric field at that nucleus, and summing each component over the nuclei. The components of the dipole moment, £» <7« ' rt> are a good measure of the overall accuracy of the electron density, as they place equal weight on all its regions. In terms of moments, the dipole moment is a test of the accuracy of the first moment of the electron Chapter 9. Testing the Accuracy of the DFT Electron Density 171 density, as it measures the distance between the centres of electronic and (constant for fixed geometry) nuclear charge in a molecule. The coordinate systems for the molecules studied in this work have been chosen to lie along the major axis of symmetry for each of the molecules, and so for the polar molecules there is only one non-zero component. The diamagnetic susceptibility (the expectation value of r2) and individual quadru- pole moment components are the most spatially diffuse properties reported in this work, weighting the electron density according to r2. The quadrupole moments are reported in 2 their traceless form, so that 0XX, for example, is really not just the expectation value of x but the expectation value of |(3a;2 — r2). This is important and will be discussed further later. The quadrupole moments include nuclear contributions, though the diamagnetic susceptibilities do not. 9.2.1 Preliminary Considerations The primary purpose of this study is to assess the reliability of the charge density calcu lated by DFT within the LDA by calculating molecular properties which weight different regions of the charge density. In order to accomplish this, it was necessary to eliminate, as far as possible, any inaccuracies resulting from the use of inadequate basis sets, as the basis sets used in ref. [57] were neither all readily available nor suited for use in deMon. It was necessary, therefore, to verify that the basis set space was adequately covered, and that results were well converged. To do this, the one-electron properties of water were calculated using six different basis sets (of varying size and quality), all within the LDA. The results are presented in table 9.2, and the basis sets used to obtain them are described below. The augmented correlation-consistent polarized valence double, triple, quadruple, and quintuple zeta (aug-cc-pV[D,T,Q,5]Z) basis sets due to Dunning and co-workers [264,265] were chosen to investigate the convergence of the calculated properties with systematic Chapter 9. Testing the Accuracy of the DFT Electron Density 172 Table 9.2: One-electron properties of water as calculated using several different orbital basis sets. In the table, dz, tz, qz, and 5z refer to the aug-cc-pvnz basis sets of references [264] and [265], modified (as described in the text) for use in this work. The ANO basis sets are also described in the text. The extended auxiliary basis sets (described in the text) were used to calculate all the orbital basis set results presented here. All calculations were performed within the LDA approximation. The Hartree-Fock and MR-SDCI results (shown for comparison) are taken from reference [57]. Experimental numbers (where available) are taken from the references indicated. dz tz qz 5z ANO Hartree-Fock MR-SDCI Obs. 0.7348 0.7372 0.7356 0.7360 0.7334 0.7794 0.7356 0.727±0.002[198] Oyy 1.9097 1.9478 1.9565 1.9571 1.9126 1.8986 1.9000 1.96 ±0.01 [266] (r3) 20.2274 20.1696 20.1513 20.1464 20.0717 19.3958 19.6648 18. ± 2.[267] (So) 294.56 294.35 301.12 303.45 302.01 309.09 309.11 0.3694 0.4081 0.4269 0.4396 0.4345 0.4549 0.4579 -0.0810 -0.0452 -0.0215 -0.0162 -0.0175 -0.0028 0.0000 0.0 0.0158 0.0111 0.0072 0.0064 0.0055 0.0122 -0.0005 0.0 (ivlri) 0.0255 0.0186 0.0125 0.0114 0.0101 0.0125 -0.0009 0.0 Electric field gradients for Oxygen IBB -0.2011 -0.1817 -0.1892 -0.1884 -0.1881 -0.24 ± 0.02[268] ICC -1.3577 -1.4340 -1.4936 -1.4932 -1.5029 -1.8341 -1.7165 -1.67 ±0.01 [268] Electric field gradients for Hydrogen IBB -0.0342 -0.0379 -0.0371 -0.0373 -0.0376 9CC -0.2374 -0.2547 -0.2471 -0.2455 -0.2457 improvements to the basis sets. Because deMon cannot use f or higher angular momentum basis functions, it was necessary for these basis sets to be modified for use in this study, The basis sets were therefore all truncated at the d functions on the heavy atoms, and at the p functions for hydrogen (to maintain balance in the basis sets). The atomic natural orbital basis sets of reference [196] were also chosen for property calculation in this study. They were altered in two ways for use here. First, since deMon is not yet able to use f or higher angular momentum atomic orbitals, all ANO basis sets were truncated at the d functions on the heavy atoms. To maintain balance in the basis sets, the hydrogen set was truncated at the p functions. The resulting sets were then augmented with field induced polarization functions (d only on the heavy atoms and p only on hydrogen [197]) to create the final functions. The basis sets thus constructed consist of 6 s contractions, 5 p contractions, and 4 d contractions on phosphorus through chlorine, 5 s contractions, 4 p contractions, and 4 d contractions on carbon through Chapter 9. Testing the Accuracy of the DFT Electron Density 173 fluorine, and 4 s and 4 p contractions on hydrogen. These basis sets were used for most of the results reported in this work. Table 9.2 shows the one-electron properties obtained for H2O with deMon and the LDA functional as a function of the basis set used. The dipole moment appears to be relatively well converged to approximately 0.734 a.u., in good agreement with both the MR-SDCI value and experiment. This result is consistent with earlier studies of the dipole moment and polarizabilities [127,128,134,136,137,139,249-254], which also demonstrated very good agreement between calculated and observed dipole moments. The values for (r2) show convergence properties similar to the calculated dipole mo ments. The value appears to be reasonably well converged for all basis sets, and approx imately 2% higher than the MR-SDCI value. Agreement with experiment is less easy to gauge because of the very approximate nature of the available experimental value. The quadrupole moments are less well converged, with the aug-cc-pvnz sets appearing to converge to an answer (1.96) in excellent agreement with the available experimental value, while the ANO basis set gives an answer approximately 2% too low, though still in better agreement with experiment than the MR-SDCI value. All components of the electric field gradients appear to be reasonably well converged as well, for any of the basis sets larger than the Dunning triple zeta. The electron density at the oxygen nucleus is not nearly as well converged as the other properties, however, and in general is not as well-predicted by DFT as the other properties, particularly for the smaller basis sets. Even taking an average value for the larger bases, there still appears to be approximately a 2% deviation from the MR-SDCI value, a rather greater disagreement than the dipole moments. The electron density at hydrogen follows similar trends. Probably one of the more revealing properties in evaluation of basis set performance is the electric field at the nuclei. This is the electrostatic electric field felt by each nucleus Chapter 9. Testing the Accuracy of the DFT Electron Density 174 due to the other nuclei and the electron density, and should be zero when the Hellmann- Feynman Theorem is obeyed (as it is for exact Kohn-Sham density functional theory [139]). The closer the method and basis set come to being "correct", the closer the answer should come to zero, given the right geometry. It is evident from examination of the electric field calculations in table 9.2 that an increase in basis set size results in a very definite decrease in the electric field, with the DFT calculations reaching a minimum for the ANO basis set (for hydrogen) and the aug-cc-pv5z set (for oxygen). Both of these results are well above the calculated MR-SDCI values, however. Given the results above, the choice of basis set is rather easily narrowed to two: Dunning's aug-cc-pv5z, and the ANO basis set. Both sets appear to predict properties with equal reliability, and both predict relatively small electric fields. Studies of molecules other than H2O also demonstrate that the two basis sets yield equivalent properties, so only one set was chosen (the ANO set) for the results in the remainder of this study. 9.3 Results and Discussion All Hartree-Fock and MR-SDCI results quoted in this work are taken from reference [57]. Given the accuracy of the quoted MR-SDCI results in comparison with the available experimental numbers, it is felt that the MR-SDCI calculations should provide a good measure of the accuracy of the DFT results where no experiment is available. The properties will be presented in an order based on the degree to which they weight the density at or near the nucleus, with the properties which place the highest weight on the density near the nucleus being presented first. Consider the electron densities at the nucleus (denoted Sx, where "X" is the name of the nucleus for which the density is being evaluated) shown in table 9.3. In all cases shown (except for the chlorine atom in hydrogen chloride and the sulphur atom in H2S), Chapter 9. Testing the Accuracy of the DFT Electron Density 175 Table 9.3: Electron densities at the nuclei as calculated by DFT and compared to Hartree-Fock and MR-SDCI. The ANO orbital basis sets and extended auxiliary ba sis sets (described in the text) were used to generate all the DFT results presented here. The Hartree-Fock and MR-SDCI calculations are taken from reference [57]. This work Hartree-Fock MR-SDCI NH3 N 198.50 202.04 201.80 H 0.4446 0.4614 0.4624 PH3 P 2161.05 2162.45 2162.69 H 0.4016 0.4180 0.4160 H20 0 302.01 309.09 309.11 H 0.4345 0.4549 0.4579 H2S S 2641.44 2639.86 2639.46 H 0.4006 0.4121 0.4160 HF F 435.286 445.120 451.112 H 0.4072 0.4127 0.4209 HC1 CI 3206.41 3186.17 3186.20 H 0.3799 0.3866 0.3932 CO 0 302.840 309.198 308.935 C 123.306 123.690 123.428 N2 N 199.7920 203.3783 203.0902 Chapter 9. Testing the Accuracy of the DFT Electron Density 176 the densities are predicted lower than either the Hartree-Fock or MR-SDCI values. Some times the differences are reasonably minor, such as for the carbon in carbon monoxide, which compares quite well with the Hartree-Fock and MR-SDCI values. This, however, is probably coincidence given the rather large deviations from MR-SDCI values for the fluorine atom in hydrogen fluoride. In general, though, two trends become clear on further examination of the calculated densities. The first is that DFT provides (within the LDA) unreliable predictions of the electron densities at hydrogen in any of the hydrides presented here. Even in the best case (for the hydrogen in hydrogen chloride) the calculated density is still not within one percent of either the Hartree-Fock or MR-SDCI values. In addition, in all cases shown the Hartree-Fock calculation always predicts the electron density at the hydrogen more reliably than DFT under the LDA. This same trend holds for the second and third period atoms as well, where Hartree-Fock is still closer to the MR-SDCI value than are the LDA values, although here at least the electron density at the sulphur nucleus in hydrogen sulphide as predicted by LDA agrees with the Hartree-Fock and MR-SDCI values to three significant figures (this, however, is probably just coincidence). Given these results then, it is readily evident that the LDA should not be used when quantitative predictions of electron densities at nuclei are desired. Table 9.4 shows the electric field gradients calculated using DFT within the LDA, compared to available experiment, Hartree-Fock, and MR-SDCI calculations. Because most field gradients are usually obtained in the inertial frame of the molecule, the results given are presented in this fashion as well. In addition, because the charge gradients must sum to zero, only qsB and qcc are pesented. However, reference [57] presents their results in centre of mass coordinates for the the heavy atoms of the molecules studied, and as principal values for hydrogen, so comparison with the hydrogen results as presented in this work is not straightforward. However, because each of the molecules presented here Chapter 9. Testing the Accuracy of the DFT Electron Density 177 Table 9.4: Field gradients of several small molecules as calculated by DFT within the LDA. The ANO orbital basis sets and extended auxiliary basis sets (described in the text) were used for all DFT calculations. The Hartree-Fock and MR-SDCI calculations are taken from reference [57], and the observed (experimental) numbers are taken (where available) from the references indicated. All values are presented in the inertial frame of the molecule, so comparison of some numbers with those in reference [57] is not possible. In these cases the DFT results are presented simply for reference. In addition, the sign of the Hartree-Fock and MR-SDCI field gradients given in reference [57] for HF, HC1, and CO have all been changed to reflect the geometry given in that work (which was used for the calculations presented here); presumably an uncorrected typographical error in the original work led to the geometries being inconsistent with the signs of the calculated field gradients. This Work Hartree-Fock MR-SDCI Obs. NH3 Electric field gradients for N IBB -0.4294 qcc -0.4294 -0.9561 -0.9106 Electric field gradients for in-plane H This Work Hartree-Fock MR-SDCI Obs. Electric field gradients for H qBB -0.0376 qcc -0.2457 H2S Electric field gradients for S qBB -0.4147 -0.6[269] qcc -1.8693 -2.6350 -2.5821 -3.1 [269] Electric field gradients for H qBB -0.0626 qcc -0.0894 HF qcc,F -2.6881 -2.8484 -2.7373 qcc,H -0.5756 -0.5158 -0.5342 -0.527[270] HC1 qcc,ci -3.2127 -3.5999 -3.5283 -3.64[271] qcc,H -0.3120 -0.2801 -0.2829 CO qcc,c 0.9685 1.1710 1.0145 qcc,o 0.7130 0.7280 0.7229 0.794[57] N2 qcc -1.1387 -1.3644 -1.2120 has its principal axes' origin coincident with the centre of mass, it is possible to translate the values for the heavy atoms presented in ref. [57] (usually qyy) to correspond with the principal axis values which are obtained from deMon. The best which may be said about the results presented in table 9.4 is that they are qualitative only. The best agreement of the density functional theory results with either experiment, Hartree-Fock, or MR-SDCI is to within approximately 10%, and usually it is much poorer than this. Clearly DFT within the LDA is inadequate for prediction of field gradients. Likely causes for the poor agreement observed will be discussed below, but further studies employing different functional and, perhaps, calculation methods, are clearly necessary. Chapter 9. Testing the Accuracy of the DFT Electron Density 179 The electric fields and total Hellmann-Feynman forces (all should be zero) presented in table 9.5 are not predicted much more accurately than are the other "tight" properties. In particular, the total Hellmann-Feynman forces are much larger than their Hartree-Fock or MR-SDCI counterparts. However, it seems that here the electric fields (and hence the Hellmann-Feynman forces) for hydrogen are predicted much closer to zero than for the heavier atoms in any given molecule. This is in contrast to the Hartree-Fock and MR- SDCI results presented, for which the heavy atom contributions to the electric field are generally smaller than those for the hydrogen. Because of this, the overall Hellmann- Feynman forces as predicted by DFT using the LDA are much larger (usually an order of magnitude or more) than either Hartree-Fock or MR-SDCI. The poor predictive power of DFT for the "tighter" properties most likely has several causes, each of which bears further investigation but is beyond the scope of this study. The most obvious of these is the very approximate nature of the functional itself. Clearly the LDA is not "right", and possibly the use of a non-local functional in its place may improve the quality of the DFT predictions. Second, as was outlined in section 9.2, deMon employs auxiliary basis functions to fit the charge density and the exchange-correlation potential, in order to facilitate the evaluation of necessary integrals. Clearly this fitting of what is an approximate density already (due to the nature of the functional involved) will not improve the quality of any of the calculated properties in regions where they are already very sensitive to the quality of the orbital basis functions and the functional used. The dipole moments as calculated using the DFT within the LDA are shown in table 9.6. All the calculated LDA dipole moments demonstrate very good agreement with the experimental and MR-SDCI values. However, the Hartree-Fock calculations (which in all cases employed the same basis set as was used in the MR-SDCI calculation) do not. The most notable case is the well-known discrepancy between the Hartree-Fock and Chapter 9. Testing the Accuracy of the DFT Electron Density 180 Table 9.5: Electric field and total Hellmann-Feynman force at the nucleus calculated by DFT within the LDA and compared to Hartree-Fock and MR-SDCI. The ANO orbital basis sets and extended auxiliary basis sets (described in the text) were used for all the DFT results presented here. The Hartree-Fock and MR-SDCI results are taken from ref erence [57]. The Hellmann-Feynman forces were calculated by multiplying the individual field components on each nucleus by the charge of the nucleus and then summing the components for each nucleus. Calculated values This work Hartree-Fock MR-SDCI NH3 0.0133 0.0002 0.0012 0.0032 0.0088 0.0023 -0.0001 -0.0047 -0.0009 Total Hellmann-Feynman forces: z 0.0931 -0.0139 0.0057 PH3 -0.0273 0.0044 0.0040 <«*/4> -0.0008 0.0007 -0.0029 -0.0002 0.0038 -0.0010 Total Hellmann-Feynman forces: z -0.4104 0.0774 0.0570 H20 -0.0175 -0.0028 0.0000 0.0055 0.0122 -0.0005 ±0.0101 0.0125 -0.0009 Total Hellmann-Feynman forces: z -0.1290 0.0020 -0.0010 HaS 0.0290 -0.0044 -0.0035 -0.0064 -0.0016 0.0017 Table 9.6: Dipole moments of several small molecules as calculated by DFT within the LDA. The ANO orbital basis sets and extended auxiliary basis sets (described in the text) were used for all DFT calculations. The Hartree-Fock and MR-SDCI results are taken from reference [57], and the observed (experimental) results are taken from the references indicated. The sign of the Hartree-Fock and MR-SDCI dipole moments given in reference [57] for HF, HC1, and CO have all been changed to reflect the geometry given in that work (which was used for the calculations reported here); presumably an uncorrected typographical error in the original work led to the geometries being inconsistent with the calculated dipole moments. Calculated values This work Hartree-Fock MR-SDCI Obs. NH3 -0.6065 -0.6359 -0.6130 -0.5789[272] PH3 0.2719 0.2638 0.2353 0.227±0.005[273] H20 0.7334 0.7794 0.7356 0.727±0.002[198] H2S -0.4118 -0.4297 -0.4081 -0.401 [274] HF 0.7050 0.7566 0.7192 0.7068[270, 275] HC1 0.4503 0.4767 0.4570 0.441 ± 0.02[276] CO 0.0927 -0.1045 0.0400 0.0481 [277] MR-SDCI values for the dipole moment of CO. It is worth noting that, like MR-SDCI, density functional theory obtains the correct sign for the dipole moment, whereas the Hartree-Fock calculation does not. (This, however, is not indicative of a poor description of CO by Hartree-Fock, but rather is more likely due to the small dipole moment of CO [57].) In addition, with the exception of PH3, in all cases shown the dipole moment as calculated by density functional theory is both closer to experiment and to the MR-SDCI result than the Hartree-Fock calculation from which the MR-SDCI result was obtained. The results presented above are not surprising, however; several previous studies [127,128,134,136,137,139,249-254] have all also indicated the uniform high quality of the dipole moments predicted by density functional theory, even within the local density approximation as was employed in this study. Clearly then the LDA is providing an over all accurate value for the first moment of the charge distribution. It is also worth noting Chapter 9. Testing the Accuracy of the DFT Electron Density 182 Table 9.7: (r2) for several small molecules as calculated by DFT within the LDA and compared to Hartree-Fock, MR-SDCI, and experiment. The ANO orbital basis sets and extended auxiliary basis sets (described in the text) were used for all DFT calculations. The Hartree-Fock and MR-SDCI calculations are taken from reference [57], and the experimental results (where available) are taken from the references indicated. Note also that Reference [57] does not give the correct values for the MR-SDCI or Hartree-Fock values for PH3; the correct values were obtained through private communication with E.R. Davidson. Calculated values This work Hartree-Fock MR-SDCI Obs. NH3 27.1209 26.5927 26.7063 25.5 ±1 [278] PH3 55.4047 55.6863 55.2424 H20 20.0717 19.3968 19.6489 18. ± 2[267] H2S 43.6698 43.7875 43.5463 / HF 14.3439 13.7235 13.9177 HC1 34.3279 34.1387 34.0287 CO 40.2257 39.8900 39.7910 N2 39.2865 39.1452 38.9108 38.09 ± 0.03[279] that studies of the nuclear position [129-131,134,257-261] and electric field derivatives [128,136,137,139,252-254] of the dipole moment have demonstrated similar success. In slight contrast to the overall excellence of the predicted dipole moments in table 9.6, the diamagnetic susceptibilities (expectation values for ( r2 )) shown in table 9.7 are not as well predicted by DFT as they are by Hartree-Fock when compared to the MR-SDCI calculations and available experimental values. Specifically, it appears that for the second period molecules (ammonia, water, hydrogen fluoride, carbon monoxide, and nitrogen), all the values of ( r2 ) are overestimated (when compared to the corresponding MR- SDCI value) by approximately five percent. This is consistent with the results of chapter 7, and reference [262], where, for all second-period hydrides which had non-bonding orbitals (ammonia, water, hydrogen fluoride) that the heights of the outer valence Kohn- Sham momentum distributions were always predicted greater (also by approximately five Chapter 9. Testing the Accuracy of the DFT Electron Density 183 Table 9.8: Quadrupole moments of several small molecules calculated by DFT within the LDA and compared to Hartree-Fock, MR-SDCI, and experiment. The ANO basis sets and extended auxiliary basis sets (described in the text) were used for all DFT results presented here. The Hartree-Fock and MR-SDCI calculations are taken from reference [57], and the experimental values (where available) are taken from the references indicated. 9ZZ is shown in all cases except H20, for which 9yy is given. Calculated values This work Hartree-Fock MR-SDCI Obs. NH3 -2.2010 -2.1592 -2.1828 -2.42 ± 0.04[272] PH3 -1.6179 -1.7634 -1.6761 H20 1.9126 1.8986 1.9000 1.96±0.01[266] H2S -2.8627 -2.9013 -2.5821 HF 1.6756 1.7338 1.6972 1.75±0.02[271] HC1 2.8225 2.8081 2.7339 2.78 ± 0.09[271] CO -1.4705 -1.5355 -1.5219 N2 -1.1289 -0.9285 -1.0905 -1.09 ±0.02(279] percent) than their Hartree-Fock and MR-SDCI analogues in momentum space. In contrast to this, however, the values of ( r2 ) for the third period hydrides (PH3 and H2S) presented in this work are very well-predicted by the LDA functional employed in the present study, with the values falling nearly exactly halfway between the Hartree- Fock and MR-SDCI values in either case. This is not surprising given the results of the EMS studies of these two molecules [33,48] in which it was found that the calculated momentum distributions were relatively unaffected by the inclusion of correlation in the calculation. It would seem therefore that when the accuracy of calculated properties is affected significantly by the level of correlation incorporated into the calculation, the LDA exchange-correlation approximation of Vosko, Wilk, and Nusair used in this study [119] cannot be used if very accurate predictions of results are desired. The quadrupole moments (given in traceless form - that is, 0XX + 9yy + 9ZZ = 0) shown in table 9.8 present nearly the opposite case to the values for the diamagnetic Chapter 9. Testing the Accuracy of the DFT Electron Density 184 susceptibilities. In other words, in general it would seem that the DFT results for the second-period hydrides are in similar or better agreement with experiment than the MR- SDCI or corresponding Hartree-Fock calculations. One example of this is the water molecule, where 6ZZ as calculated by DFT is actually closer (by a small margin) to the experimental value than is the MR-SDCI value. The only possible exception to this trend is carbon monoxide, whose MR-SDCI and Hartree-Fock values are very similar and approximately four percent higher than the DFT value, although the experimental value is unavailable for comparison. At first it might seem incongruous that the values of ( r2 ) predicted for these molecules are all larger than expected but that the individual quadrupole moment components themselves are not. (Of course the quadrupole moments presented here are not compo nents of ( r2 ) since the quadrupole moments include the sum over the nuclei but ( r2 ) does not.) Recall that the quadrupole moments are being presented in their traceless forms, which requires that all the quadrupole moments are calculated using expressions of the 2 2 form (the xx-component is shown) 0XX = | (3s — r ), so that 9XX + 6yy + 6ZZ = 0. Given the general high quality of the values thus calculated, what is likely occurring is that the xx, yy, and zz components by themselves are predicted too large, and the overestimation of the value for ( r2 ) compensates for this. These results are not inconsistent with the results for the third-period hydrides. Table 9.8 shows the DFT values for PH3 and H2S closer to the MR-SDCI values than the corresponding Hartree-Fock values, while the DFT value for HC1 is predicted somewhat higher than the Hartree-Fock value, but in about as good agreement with experiment as the MR-SDCI value (as was the value for ( r2 )). Clearly for the molecules presented here accurate estimates of the (traceless) quadrupole moments are possible with DFT when using the LDA. Chapter 9. Testing the Accuracy of the DFT Electron Density 185 Table 9.9: One-electron properties of water as calculated using different model exchange and correlation functionals. For the DFT calculations, the ANO orbital basis set and (5,4;5,4)/(4,l;4,l) auxiliary basis sets (see text for details) were used. In the table, "LDA" refers to calculations carried out using the local density approximation as parameterized by Vosko, Wilk, and Nussair [119], while "Non-local" refers to calculations carried out using the 1988 gradient-corrected exchange potential of Becke [120] and the 1986 gra dient-corrected correlation potential of Perdew [121]. The Hartree-Fock and MR-SDCI results are taken from reference [57]. Experimental numbers (where available) are taken from the references indicated. LDA non-local Hartree-Fock CI Obs. {**) 0.7334 0.7134 0.7794 0.7356 0.727±0.002[19J 9yv 1.9126 1.8718 1.8986 1.9000 1.96 ±0.01 [266] (r2> 20.0717 19.9408 19.3958 19.6648 18. ± 2.[267] (So) 302.01 305.66 309.09 309.11 fa) 0.4345 0.4547 0.4549 0.4579 <«*/r&> -0.0175 -0.0323 -0.0028 0.0000 0.0 ± 0.001 <«*/«*> 0.0055 0.0051 0.0122 -0.0005 0.0 ± 0.001 3 (w/r H) 0.0101 0.0103 0.0125 -0.0009 0.0 ± 0.001 Electric field gradients for Oxygen qBB -0.1881 -0.2066 -0.24 ± 0.02[268] gcc -1.5029 -1.5651 -1.8341 -1.7165 -1.67±0.01[268] Electric field gradients for Hydrogen qBB -0.0376 -0.0374 qcc -0.2457 -0.2448 9.3.1 Non-local Functionals As a preliminary investigation of the effects of a non-local functional on the calculated properties, a calculation was run for the water molecule using the ANO basis set and the gradient-corrected exchange and correlation potentials of Becke [120] and Perdew [121] respectively. The results are shown in table 9.9, along with the LDA, Hartree-Fock, MR-SDCI, and experimental results. As can be seen from the table, the spatially less diffuse properties, though still not in very good agreement with experiment, are each corrected in the proper direction. In Chapter 9. Testing the Accuracy of the DFT Electron Density 186 particular, the electron density at the oxygen nucleus is brought significantly closer to agreement with the Hartree-Fock and MR-SDCI values, and the density at the hydrogen nucleus is brought into near perfect agreement with the Hartree-Fock value. Whether or not the agreement for hydrogen is coincidence can only be determined through studies of other molecules, however. The electric field gradients are improved as well, with the value for oxygen coming into closer agreement with experiment than before. With no numbers with which to compare them, no conclusive statement can be made about the quality of the hydrogen electric field gradients, however. The electric fields are the only property which does not appear to benefit from the switch to the non-local functional. The electric fields at hydrogen appear to be largely unaffected, and those for oxygen are made significantly worse. The properties which weight the outer spatial regions of the electron density are generally not affected much by the use of the gradient-corrected functionals. The dipole moment is decreased, but still remains in very good agreement with experiment, although now not on par with the MR-SDCI results. The diamagnetic susceptibility is lowered somewhat, consistent with the lowering of the momentum distribution seen when this combination of functionals is used to calculate the approximate EMS cross-section [262]. Finally, the quadrupole moment component presented in the table seems largely unaf fected by the change to a non-local functional. 9.4 Conclusions This work has reported the results for a number of properties for a number of small molecules. It has shown that reasonable confidence may be placed in the values calculated for the spatially more diffuse properties, such as the dipole and quadrupole moments, and diamagnetic susceptibilities reported here. It has also shown, however, that less spatially Chapter 9. Testing the Accuracy of the DFT Electron Density 187 diffuse properties, such as the electron density at the nucleus, may not be calculated reliably at all using the LDA. There are a number of possible explanations for the poor results obtained for the "tight" properties. One is self-interaction. It is well known that the approximate ex change potential used in local functionals does not fully cancel the self-interaction intro duced by the coulomb repulsion operator. This will cause the electron cloud to become more spatially diffuse, as the electrons are pushed farther apart than they would normally be. This is consistent with results obtained for calculated momentum distributions [262], which are observed to be too high at low momentum. One other possible cause for the poor "tight" properties is the lack of gradient correc tions in the LDA. At a nuclear centre, the density is high and hence the gradients of the density must also be high. To accurately calculate the energy due to these high gradients at the nuclear centres, gradient corrections are needed. Adding one of the many possible choices for a gradient-corrected energy functional yields results which seem to be trend ing in the right direction, both for the "tight" properties as well as for the calculated momentum profiles [262]. It would be useful, therefore, to monitor the electron densities at nuclear centres when different gradient corrections to the exchange and correlation potential are tested, since once the closest match to the best available value for that nuclear centre has been found, the other properties should also improve. One last possible source for error may simply be that the LDA functional used was not obtained using sufficient high electron density points in the original fit. In their work, Vosko et al. [119] used eight points to fit the functional form, and there is a possibility that there were not enough points to adequately represent the correlation energy at higher electron densities. A new functional generated from a greater number of points, and a greater number of points in the high density region, would clarify this. Chapter 10 Conclusions This thesis has proposed two new models for use in electron momentum spectroscopy: a new model for experimental angular resolution and a new method for the calculation of the experimental cross section, the target Kohn-Sham approximation. Both models have subsequently been extensively tested both here, and in other work [30,51,71,72,208] and have been found to perform well under a variety of situations. Existing theoretical methods for calculating approximate cross-sections have also been tested, and these also seemed to perform well for the system (acetylene) studied. This does not mean, however, that there is no room for improvement in these models; certainly the resolution function is not completely representative of the spectrometers being used. The Mott scattering cross-section might be factored in, as might more realistic modeling of the electron optics in the spectrometer, to facilitate shapes for the resolution functions which reflect the actual range of momenta seen by the spectrometer in an experiment. For now, though, the methodology prescribed (the use of Gaussians in 9 and (f>) seems to perform well enough that these modifications very likely would not make a large difference to the calculated cross-section if they were incorporated. Existing theoretical methods for calculation of the cross-section also have their limita tions. Hartree-Fock, though providing accurate cross-sections for acetylene and methane, fails to do so for water, ammonia, or hydrogen fluoride [28,42,71]. It has been found that, for these molecules, an accurate treatment of correlation in CI calculations using highly saturated (extended) basis sets is required to provide good agreement of theory 188 Chapter 10. Conclusions 189 with experiment. However, these methods for calculation of the overlap (Green functions and configu ration interaction calculations) are not without their limitations. Any post Hartree-Fock calculation will have the limitation that, although providing more accurate answers to electronic structure problems, they are generally computationally very intensive. The least so, Green function calculations, scale as JV5 (where N is the number of basis func tions used in the calculation), and other methods are more intensive than this. Therefore, until computers increase greatly in their capabilities (a factor of 100,000 in computing power will only allow a system 10 times larger than those presently solvable to be treated) or a breakthrough in mathematical or computational techniques is made, a new, less com putationally intensive method for electronic structure calculations is needed. Such a method, Kohn-Sham density functional theory, has been proposed for use in EMS in this thesis. Testing revealed that the best method for obtaining EMS cross- sections from DFT was to use the Kohn-Sham orbitals themselves. Provided that the exchange and correlation functional used in the calculation is reasonably accurate, the orbitals obtained from the DFT calculations should be reasonable approximations to their mathematically nonlocal counterparts from Green function and CI methods. In order to test the approximate functionals used more extensively, molecular prop erties weighting various parts of the electron density have been calculated and compared with configuration interaction results and results from experiment (where available). It was found that the local density functional available in deMon was adequate for the cal culation of long-range properties, but unreliable for the calculation of properties which depended on accurate densities near the nucleus. This could be due to errors in the functional, or to the local density approximation in general. Further testing is required to determine which of these might be the cause. List of Abbreviations 2ph-TDA 2 particle-hole Tamm-Dancoff Approximation ADC Analogue-to-Digital Converter ADC(n) Adiabatic Diagrammatic Construction to order n ANO Atomic Natural Orbital BE Binding Energy BES Binding Energy Spectrum CASSCF Complete Active Space multiconfiguration Self-Consistent Field CFD Constant Fraction Discriminator CGTO Contracted Gaussian-Type Orbital CI Configuration Interaction CI(SD) Singles and Doubles Configuration Interaction DFT Density Functional Theory, usually Kohn-Sham Density Functional Theory DI Diffuse Ionization rDI spin-restricted Diffuse Ionization uDI spin-unrestricted Diffuse Ionization DWIA Distorted Wave Impulse Approximation DZ Double Zeta EMS Electron Momentum Spectroscopy FIP Field-Induced Polarization FOA Frozen Orbital Approximation FWHM Full Width at Half Maximum 190 List of Abbreviations GDI Generalized Diffuse Ionization rGDI spin-restricted Generalized Diffuse Ionization uGDI spin-unrestricted Generalized Diffuse Ionization GF Green Function GF2 Second order Green Function GOA Generalized Overlap Approximation GTO Gaussian-Type Orbital GTS Generalized Transition State rGTS spin-restricted Generalized Transition State uGTS spin-unrestricted Generalized Transition State GW-PG Gaussian Weighted Planar Grid HAM/3 Hydrogen Atoms in Molecules, version 3 HOMO Highest Occupied Molecular Orbital IP Ionization Potential KS Kohn-Sham KSKH Kohn-Sham Koopmans' Hole LDA Local Density Approximation MAGW Momentum Averaged Gaussian Weighted MCSCF MultiConfiguration Self-Consistent Field MD Momentum Distribution, excluding resolution folding. 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Appendix A Quadrature for the Analytic Method The "analytic" method involves one integral which must be done numerically, namely i ; *te M = ~?r L n2 ,,„„ „•„ IT; ^ #> (A.i) The quadrature is complicated by the fact that the integrand diverges at the boundries (Omin and 0max) which are defined by the condition g(p, 0) = 0. Nevertheless Simpson's rule integration can be used after first parameterizing 6 with a new coordinate z - e-z + e+z \ • > and then transforming the integral A.l to an integral over z. One check on the accuracy of the resolution function R(p; 1= 1°° R(p; 207 Appendix B Contributions to the Width of the Momentum Resolution Function The contributions of the different experimental variables to the standard deviation ( o*(/(*i. x2,..., xn)) * IX|£) V(/(«i)) (B.l) «=i OXi where The sizes of specific contributions at EQ = 1215.8 eV are tabulated in table B.l. The standard deviation a(p) of R(p\ Hence it is reasonable to neglect finitep0,px,p2 resolution and to fix 0i = 02 in the GW-PG and UW-PG models. For comparison purposes, the standard deviation of the GW-PG resolution function R(p; 208 Appendix B. Contributions to the Width of the Momentum Resolution Function 209 Table B.l: Contributions of the various dependent variables to the standard deviation of the momentum p used in the resolution function R(p; fa) for 0X = 02 = 45°,<£0 = 8.0°, Ei = E2 = 600eV, and E0 = 1215.8e^. The standard devia tions in the momenta po, p\, and p2 are derived from the estimated standard deviations of the corresponding energies EQ (a = 0.3397 eV, i.e. 0.80 eV fwhm), Ei (a = 0.4247 eV, i.e. 1.0 eV fwhm), and E2 { variable name, x standard deviation, cr(x) term, {df/dxfa\x) 0.7° 3.2469 x 10-3 a.u. Si (eq. 4.6) 1.0° 2.3271 x lO"4 a.u. -4 02 (eq. 4.6) 1.0° 1.2960 x 10 a.u. 0 (eq. 4.7) 1.0° 7.0965 x lO-4 a.u. Po 1.3208 x 10"3 a.u. 1.5110 x 10"8 a.u. 3 -8 Pi 2.350 x 10- a.u. 9.5668 x 10 a.u. 3 -8 P2 2.350 x 10" a.u. 5.3279 x 10 a.u. p (eq. 4.6) 0.0601 a.u. 3.6094 x 10-3 a.u. p (eq. 4.7) 0.0629 a.u. 3.9567 x 10"3 a.u. identical results. Also shown are the estimates of o o -40 X Calculations of the Width of the GW-PG h30 i Resolution Function h20^ V.;».,.,-'-'.»*-"-t" j*" *y~r^-^ —- CD • 100,000 point Monte Carlo -10 E . Analytic X . d\ — 92 estimate . 0,5* 62 estimate o CL QL 0 < 10 20 30 cpo (degrees) Figure B.l: The standard deviation of the momentum resolution function as a function of 0 for 0X = 02 = 45°, Ei = E2 = 600 eV, and E0 = 1215.8 eV. See Appendix B and table B.l for further details. Appendix C Orbital Energy Shifts The purpose of this appendix is to discuss in more detail why the orbital energies from the Kohn-Sham equation and ionization potentials from Dyson's equation should differ even if the Kohn-Sham orbitals are good approximations to renormalized Dyson orbitals. To this end, let us start by assuming that the target Kohn-Sham approximation (analogous to the target Hartree-Fock approximation), Eq. (7.21), holds. That is, Mi) * JsPUi). (C.i) where fc is a Kohn-Sham orbital. In this approximation, Dyson's equation (7.1) becomes 0 [hB + fUw} )] Ml) « 4'V.(1) • (C2) Subtracting this from the KS equation (6.16) gives a closed expression for the xc potential, •^>"^SP + <«-«P>. (C-3) in the target KS approximation. The first term on the RHS is analogous to Slater's orbital-dependent localization procedure [22] for the Hartree-Fock exchange operator and determines vxc up to an orbital-dependent additive constant. (That it is only an additive constant which is undetermined is evident from the fact that the second term on the RHS is a (possibly orbital-dependent) constant, while the LHS is orbital independent.) Hence the difference between the KS and Dyson orbital energies provides the orbital-dependent constant shift which is needed to yield an orbital-independent vxc. This orbital-dependent 211 Appendix C. Orbital Energy Shifts 212 shift can also be regarded as a change of energy zero needed to make the RHS of Eq. (C.3) go to zero asymptotically. Since the KS and Dyson HOMO orbital energies are equal when the energy zero is chosen such that uxc goes to zero asymptotically, the differences between the KS and Dyson orbital energies must shift the energy zeros of the other orbital-dependent localizations of the self-energy so that all the resultant potentials go to zero asymptotically. Of course this argument is strictly correct only if the target Kohn-Sham approxi mation holds exactly. Since it is approximate, the expression (C.3) for vxc should be expected to contain some residual orbital-dependence. This can be removed by an or bital averaging procedure, as was done by Slater [22] in the exchange-only case. Left multiplying Eq. (C.3) by Sf'V«(l)P and summing over all orbitals i and final states J, using the relations (7.25) and (7.27), gives Ml) ^ + jjg , (C4) upon division by p(l). It is interesting to note that in the quasiparticle approximation (as defined in ref. [118]), where S} = 1 and there is only one state / corresponding to each orbital i, this becomes Uxc(1) pJT) + pJT) ' (c-5) which is the same equation that was derived in ref. [118] in a different way, starting from the OEP equation for the exact vxc and making the quasiparticle approximation and an average orbital energy approximation, but without invoking the target Kohn-Sham approximation. The first term on the RHS of Eq. (C.5) is the xc version of Slater's original (orbital-independent) local exchange potential [22] while the second term includes an orbital energy correction. In the exchange-only case, Eqs. (C.4) and (C.5) are identical, and reduce to the excellent approximation of Krieger, Li, and Iafrate [172,175].