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Calculation of electron binding energies

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Calculation of electron binding energies
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Kurtz, Henry Allan, 1950-
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N electrons ( jstor )
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Chemistry thesis Ph. D
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Thesis--University of Florida.
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Bibliography: leaves 92-94.
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Typescript.
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Vita.
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by Henry Allan Kurtz.

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CALCULATION OF ELECTRON BINDING ENERGIES


By

HENRY ALLAN KURTZ












A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY






UNIVERSITY OF FLORIDA


1977
















ACKNOWLEDGEMENTS


The support and encouragment of my chairman, Prof. Yngve Ohrn, is gratefully acknowledged. During my stay at the Quantum Theory Project, he was of unfaltering assistance.

I would also like to thank all memebers of the Quantum Theory

Project for providing a very stimulating atmosphere in which to work. Many valuable discussions were held with the faculty, post-docs and, in particular, other students.

I would also like to express my great appreciation for the constant support of my wife Bette Ackerman.




















ACKNOWLED LIST OF T LIST OF F ABSTRACT. INTRODUCT CHAPTER 1

1.1 1.2 1.3 1.4 1.5
1.6 CHAPER 2

2.1 2.2
2.3 CHAPTER 3


3.1 3.2
3.3 3.4
3.5 3.6 3.7


TABLE OF CONTENTS



GEMENTS ...... ....................

ABLES ...... .....................

IGURES ...... ....................



ION ....... ........... ........

IONIZATION ENERGIES ................

Discussion of Problem ...............
Relaxation Effects .... ..............
Correlation Effects ..................
Approximation of the Self-energy ......... Application to the Water Molecule ........ Application to the Neon Atom ...........

POSITIVE ELECTRON AFFINITIES ...........

Discussion of Problem ...............
Analysis of Effects ..................
Application to LiH .... ..............

NEGATIVE ELECTRON AFFINITIES ..........

Resonances ...... .................
Methods for Calculating Cross Sections . . . Choice of Potential .................
Relation to Other Methods - Helium Test Case Beryllium ...... ..................
Magnesium ...... ..................
Evaluation of Resonance Parameters .......


Page

* .ii

.iv



� vi



.4

.4
�6 .13 � .17t
* .20 .25

* .35

.35 .36 .39

.44

.44 .45
* .49 .51 .55
* .68


APPENDIX A THIRD ORDER RSPT ON THE ASCF IONIZATION ENERGY . . APPENDIX B THIRD ORDER RSPT ON THE TOM IONIZATION ENERGY . . . APPENDIX C EXPANSION OF ASCF ENERGY IN TERMS OF TOM QUANTITIES BIBLIOGRAPHY ......... ..........................

BIOGRAPHICAL SKETCH ........ ......................

iii















LIST OF TABLES


Table Page 1 Comparison of Third Order Difference Terms for Water (eV) 14 2 14 CGTO Basis Used for Water 22 3 Water Basis I Results (eV) 23 4 Water Basis II Results (eV) 24 5 Ionizations in 30-40 eV range 30 6 Comparison of Results (eV) 31 7 Neon STO Basis 33 8 Neon Results (eV) 34 9 LiH 19 STO Basis 40 10 LiH 13 CGTO Basis 41 11 LiH Electron Affinity Results (eV) 42 12 Helium Basis Sets 54 13 HelO Results 56 14 He20 Results 57 15 Be Basis Functions 62 16 Be Static-exchange Results 64 17 Be Static-exchange plus Polarization Results 65 18 Magnesium Basis 69 19 Mg Static-exchange Results 70 20 Mg Static-exchange plus Polarization Results 71 21 Resonance Parameters 80


















LIST OF FIGURES


Figure Page 1 Plot of Relaxation and Correlation Corrections for Water 8 2 Plot of Relaxation and Correlation Corrections for Furan 10 3 Plot of -I(ASCF) + Zji(E) Versus Energy for Water using Basis I 27 4 Plot of -I(ASCF) + E~i(E) Versus Energy for Water using Basis II 29 5 Plot of Phase Shift Versus Energy for Helium 59 6 Comparison of Helium Phase Shifts with Other Calculations 61 7 Plot of Cross Section Versus Energy for Be 67 8 Plot of Cross Section Versus Energy for Mg 73 9 Plot of Phase Shift Versus Energy for Be 76 10 Plot of Phase Shift Versus Energy for Mg 78
















Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


CALCULATION OF ELECTRON BINDING ENERGIES

By

Henry Allan Kurtz

December 1977

Chairman: N. Y. Ohrn
Major Department: Chemistry

Contributions from relaxation and correlation effects are

examined for ionization energies and electron affinities. A comparison is made of the ASCF and TOM methods showing that they are not the same through third order in Rayliegh-Schrddinger perturbation theory as had previously been thought. A direct method of obtaining values for ionization energies and electron affinities is proposed which includes the complete relaxation correction and an approximate correlation correction. This method is used to evaluate ionization energies for water and neon and to evaluate the electron affinity of lithium hydride. The results compare very favorably with both the experimental and other theoretical calculations.

Negative electron affinities are also studied. These quantities are given by electron scattering resonance energies. A method is derived to compute Harris phase shifts from which the resonance parameters can be obtained. The potentials used are static-exchange

vi






vii


potentials with and without the inclusion of an approximate polarization potential. Polarization effects are shown to be very important in applications to some representative Group IIA atoms.






Chairman
















INTRODUCTION


The strength with which an electron is bound in a system,

referred to as the binding energy, is a quantity which has recieved a great deal of interest in recent years, both from a theoretical and experimental viewpoint. Knowledge of such a quantity can be very useful in the study of chemical analysis and bonding. This fact has been demonstrated e.g. by K. Siegbahn in the development of ESCA (Electron Spectroscopy for Chemical Analysis). The study of electron binding energies can be broken down into the study of two main processes, electron detachment and electron attachment. The quantities measured in these processes are called the ionization energies and electron affinities respectively.

The ionization energy is the amount of energy required to

completely remove an electron from a system. This can be obtained theoretically from the total energy differences between the N electron (usually neutral) initial state and the appropriate N-I electron state. For atomic systems, this definition is unambiguous, but for molecular systems the ionization energies will depend on the internuclear arrangements. When such a condition exists, there are two main types of ionization energies of interest exper2mentally and theoretically. These are known as the adiabatic and vertical ionization energies. The adiabatic ionization energy is given by the energy difference between states with each in its own equilibrium










geometry and the vertical ionization energy is obtained for a fixed nuclear arrangement, usually the equilibrium geometry of the initial N electron state. In Chapter I of this work, several methods of use today for calculating vertical ionization energies are explored. An attempt is made to analyze the various improvements and effects presently considered and to propose an alternative method which provides an accurate and efficient procedure by which ionization energies can be calculated. Applications of this method are given for both atomic and molecular systems.

The concept of electron affinity, or electron attachment, is very closely related to that of the ionization energy, or electron detachment, in that each can be thought of in terms of total energy differences. Electron affinities are usually defined as the total energy difference between the N electron parent system and the N+1 electron final system. This sign convention for electron affinities is such that if a positive result is obtained then the N+1 electron system is of lower energy and, hence, stable with respect to the initial system. As was the case for molecular ionization energies, there is the same distinction between adiabatic and vertical processes when referring to molecular electron affinities, with the vertical process still usually obtained at the geometry of the N electron parent. It is of interest to note that for atomic systems, the electron affinity of a given N electron state must be the same as an ionization energy of the resulting N+1 electron state. Chapter 2 is devoted to the analysis of methods used for obtaining positive electron affinities. It is shown that the same method that was developed for ionization










energies is also applicable to electron affinities and an example is given of such a calculation.

As of yet the possibility that the total energy of the N+1

electron final system will lie above that of the N electron parent, thus yielding a negative electron affinity, has not been mentioned. This possibility usually lies well outside the range of methods discussed so far and into the category of scattering processes. For most cases, the idea of a negative electron affinity is meaningless because the N+1 electron state is not stable and can, therefore, have any energy desired, depending only on the energy of the free electron. In some instances though, the concept of a negative electron affinity can have a meaning. Such a case is when, due to the shape of the potential which the "free" electron feels, the N+1 electron state has a non-negligible lifetime. The temporary negative ion state formed in this manner is knows as a resonance and can be related to a specific energy value. The energy at which a resonance of this type occurs can then be used as the definition of the negative electron affinity. In Chapter 3 this type of binding is treated. A short review of the theory of electron-atom scattering resonances is given and a simple procedure is developed for the calculation of electron-atom scattering cross sections, from which such resonances can be obtained. This method is analyzed in terms of existing related methods and then applied to some representative Group IIA atoms to determine values of their negative electron affinities.
















CHAPTER 1
IONIZATION ENERGIES

1.1 Discussion of Problem

As stated previously, the ionization energy of a system is the energy required to completely remove an electron. This can be given theoretically as the difference between the N electron and the N-1 electron system total energies. Experimentally, ionization energies can be determined via photoelectron spectroscopy. In such an experiment the system is bombarded by photons of a known energy, or equivalently frequency, and the kinetic energy distribution of the ejected electrons is measured. The electron binding energy I (ionization energy) can then be computed by Einstein's relation I = hmo - E e

where w is the incident photon frequency and Ee is the ejected electron's kinetic energy. Both adiabatic and vertical energies can be detected by this technique. The adiabatic energy is measured from the band head in a spectrum and the vertical is measured relative to the band maximum. Several good reviews of photoelectron spectroscopy have been given by Siegbahn et al. (1967 and 1969), Turner et al. (1970), Eland (1974), and Price (1974).

One of the earliest and still most widely used methods for obtaining theoretical vertical ionization energies is known as Koopmans' theorem (Koopmans, 1933). The "Koopmans' theorem" states that the ionization energy is given by the negative of the Hartree-










Fock orbital energy, -6. It can be derived by first assuming a Hartree-Fock description of the N electron system and then allowing the same orbitals to approximate the N-I electron state Hartree-Fock solutions. Discussions of the errors involved in the "frozen orbital" method have been given by Mulliken (1949) and Lbwdin (1955a).

One seemingly obvious way of improving an ionization energy

calculation is by improving the methods used to calculate total energies. Approaches of this type are called indirect methods since the quantity of interest (the ionization energy) is obtained from the results of two separate total energy calculations. Ab initio methods commonly used to calculate total energies for indirect methods include the Hartree-Fock method, Configuration Interaction, MC-SCF, or a variety of perturbation theory methods. All of these have the drawback that they require a high degree of accuaracy since, in the last step, two very large numbers must be subtracted to obtain what is usually a quite small number. There also can be other even more serious problems with some indirect methods in ensuring that each state is calculated within the same level of approximation. This problem is addressed in greater detail in a later section.

A method capable of calculating the ionization energy directly via a single calculation would circumvent most of the problems mentioned for indirect calculations. Such methods are called direct methods. The two most common direct methods are the transition orbital method (Goscinski et al., 1973) and the electron propagator, or onebody Green's function (Linderberg and Ohrn, 1973). In following sections, the relationships between these methods and the more common indirect methods are discussed.










1.2 Relaxation Effects

In the Koopmans' ionization approximation, the orbitals obtained for the N electron state are assumed to remain unchanged by the ionization process. Clearly the potential felt by the remaining electrons is now different and the orbitals should be allowed to change (via reorganization or relaxation). One method of obtaining these reorganization effects is to perform an independent Hartree-Fock calculation for the N-I electron state. The procedure by which the ionization energy is obtained from separate Hartree-Fock calculations is known as ASCF (Bagus, 1965). This technique can be used to define what is called the relaxation correction to Koopmans' theorem as the difference between the ASCF result and the Koopmans' result.

It should be mentioned at this point that the Hartree-Fock

calculations referred to in this work are unrestricted Hartree-Fock calculations. Other types of Hartree-Fock methods, such as Roothaan's (Roothaan, 1960) or a projected Hartree-Fock (L~wdin, 1955b), may give different results and other interpretations from those discussed here.

Further non-relativistic corrections to the Koopmans' results not included in the relaxation correction are grouped together and called correlation corrections. These corrections can then be defined as the difference between the non-relativistic experimental result and the ASCF results. Figures 1 and 2, for water and furan respectively, show the relative importance of the relaxation and correlation corrections over an energy range from core ionization to outer valence ionization energies. It is evident from these figures that for core ionizations, the relaxation corrections can be quite large. In fact, they account for almost the entire difference between























Figure 1 Plot of relaxation and correlation corrections
for water at each ionization energy. The corrections are shown near the respective
energies with the correlation on the left and
the relaxation on the right.



















(AO) S5JOu3


0"0 oo 0"0 O'






0'






















C








I .. ..






I U1




















Plot of relaxation and correlation corrections for furan at each ionization energy. The corrections are shown near the respective energies with the correlation on the left and the relaxation on the right. The values were obtained from Hehenberger (1977).


Figure 2









2 0.0
2o o -..


0.0r 1'
I j





5.0 25.
En


.0
ergy (eV)


290.0


540.0










the experimental and Koopmans' ionization energies and, therefore, the ASCF ionization energy will be very accurate. Although small the correlation contributions may be also important for the core in the calculation of chemical shifts. In the intermediate and valence regions, the correlation effects can be as important as the relaxation effects and possibly more important. Therefore, in these regions the ASCF ionization energy may not be very good. It is very interesting to note that for the systems examined here, the relaxation and correlation corrections for the outer valence level have almost equal magnitude but are opposite in sign. In cases such as this the original Koopmans' energy will be a better approximation than ASCF to the ionization energy.

To arrive at an explicit expression for the relaxation contribution, Rayleigh-Schridinger perturbation theory can be used. The details of this procedure and its resulting expressions are given in Appendix A.

So far, only the indirect ASCF method has been discussed for calculation of ionization energies with the inclusion of relaxation effects. There is also a direct method designed to include the major relaxation contributions known as the transition operator method (TOM) (Goscinski et al., 1973). The ionization energy is given by this method as the eigenvalue, cx, of the "transition operator"
FT = h + E +
X a

To obtain an expression for the TOM correction to Koopmans' theorem which can be compared with the ASCF expression, Rayleigh-Schrddinger perturbation theory can again be used. Goscinski et al. (1975) pointed out that when perturbation theory is used to expand the transition










operator eigenvalue using the N electron ground state Hartree-Fock solutions as a reference, then the expressions obtained will differ from the ASCF expression in third order. But if the N and N-I electron state total energy expressions are expanded with the TOM solutions as a reference, the ASCF expression will be equal to the TOM value through third order.

To demonstrate the techniques used and to check the results of Goscinski et al. (1975), both of these procedures were carried out. The first method, expanding the TOM eigenvalue, is demonstrated in Appendix B. By comparing the results obtained with the ASCF expression of Appendix A, the difference through third order is ASCF - 4 =-
x a (-)(l,k~a (F1-a)(k ca'

F ~
a
l*a (clca)'
This expression supports the statement by Goscinski that this procedure gives a non-zero difference in the third order expressions. The next procedure, where the total energies are expanded in TOM quantities and then subtracted, is demonstrated in Appendix C. The results found there can be stated as


(ASCF)T _ CT




a ( _ )2
lia a










where the tildes refer to TOM quantities. This result is clearly in contradiction with the result of Goscinski, who states that such a difference should be zero. It should be noted that if all the TOM orbitals and energies in the above expression are expanded in terms of the N electron state Hartree-Fock orbitals and energies, and then all terms of order higher than third are discarded, we arrive at the first expression obtained above.

Even though this work shows that the ASCF and TOM ionization energies are not identical, they have been found in the past to lie very close numerically. This should mean that the difference terms found here are fairly small. To demonstrate that this is true, in Table 1 the third order difference terms expressed in the Hartree-Fock orbitals are compared with the TOM and ASCF values obtained for the water molecule. For water, the difference terms are clearly small and will not cause a significant effect, with the possible exception of the la, state.



1.3 Correlation Effects

Both of the previous methods discussed, TOM and ASCF, have

included what was called relaxation effects but neither method was designed to take into account the remaining effects, called correlation effects. Correlation effects derive their name from the fact that in the Hartree-Fock approximation an electron sees the average, not the instantaneous, potential due to the other elect-ons. This means that there is no correlation between the motions of the electrons. As was shown in Figures 1 and 2, these neglected effects in theoretically determined ionization energies can be substantial, especially for the valence region.

























Table 1

Comparison of Third Order Difference
for Water (eV)


Ionization

la, 2a1 lb2 3a1 lbI


ASCFa

540.8 34.6 17.8 13.0 11.0


aASCF values from Goscinski et al.


TOM 3rd order diff.


540.64 34.51 17.78 12.86 10.88


0.186 0.008 0.013 0.020 0.020


(1975)


Terms










There are several methods designed to include correlation

effects in the calculation of total energies, thereby leading to indirect methods of calculating ionization energies. Perhaps the most widely used of those is the method of configuration interaction

(C). A very good review has recently been given on this subject by Shavitt (1977).

Calculating energy differences between states, particularly between states with different numbers of electrons, can be a very difficult matter using CI. In such cases, not only does one have to worry about choosing an orbital basis but a choice must also be made of many-electron functions (determinants or projected determinants) to be used. If possible, it would be preferrable to use all the many-electron functions that could be generated from a chosen orbital basis. This would then guarantee that all possible correlation (obtainable with the orbital basis) had been included. Unfortunately, such Cl's would be so large that they are impossible to perform in almost all cases of interest. Therefore, one must make selections of the many-electron basis resulting in approximations to the correlation energy. If extreme care is not taken in selecting the functions then the correlation effects may not be included for each state in a balanced manner.

In recent years, a direct method for calculation of ionization

energies has emerged in quantum chemistry. This technique is known as the electron propagator (Linderberg and Ohrn, 1973). Reviews of this subject have been given by Hedin and Lundqvist (1969) in solid-state physics, Csanak et al. (1971) in atomic physics, Ohrn (1976) and Cederbaum and Domcke (1977) in molecular theory. To illustrate how










ionization energies are obtained, a short survey of some of the more important formulas and aspects of electron propagator theory will be given.

The electron propagator in spectral representation is given as


G (E) = lim F( fs(p) f*(q) + gs(p) gs(q)
p- s" E + EN IVI .
p+ E + in E+ + -s - in


where gs and f are the overlap amplitudes defined by
S


gs(x) = g (i)'u(x) with g (i) = and fs(x) = fs(i)'u(x) with f (i) =
15

These amplitudes can also be calculated via CI and could possibly give a mode of comparison between the two methods which is more sensitive than an energy criteria (Kurtz et al., 1976).

From the above expression for G(E), one can see immediately that the ionization energies are obtained as the poles of G(E) or the zeros of G-1(E). One way of obtaining G is by solving the Dyson equation


G = Go + G�EG


which can be rewritten


G-1 = (GO)-I


In these equations Go is the propagator obtained for the unperturbed reference state. Usually this is the Hartree-Fock state and










(G-')ij = (E - �i)6 ij

which leads to

(G-l)i = (E - ci)6ij - Z .(E)

The terms Eij(E) are matrix elements of what is known as the sel f-energy.

If we assume we have an expression for Zij (E), then the

problem is to solve for the zeros of the above energy dependent equation. One method of solving the equation is by a direct poleresidue search as described by Purvis and Ohrn (1974).

If only the diagonal element of G-1 is considered, it is evident that for E to be a solution it must obey the following relation:

E = ci + ii(E)

Solutions of this equation will also lead to improved values of ionization energies and this method has become known as the "quasiparticle" method.



1.4 Approximation of the self-energy

As was shown in the last section, to obtain improved ionization energies with respect to the Koopmans' value, an expression for Eij(E) is needed. A general expression for the self-energy has been given by Purvis and Ohrn (1974) in terms of a superoperator formalism (Goscinski and Lukman, 1970) as


E(E) = (aif)(fI(Ei - )L�-1(fIRn )


The superoperators I and H are defined for any operator X by IX = X and HX = [X,H]_ (where H is the second quantized Hamiltonian)









and the binary product is defined as


(XiXj) = I 1 +

where IN> is a suitable reference state.

If the manifold f is chosen to be the set of fermion-like

operators {aata a }, where the field operators are associated
P b% a bap q
with the Hartree-Fock spin orbitals, this leads to

( = E) (M-1a) Ilip

apq a' apq,a'p'q'
+ 2 Z (M-l b a bp
abp p' abp, a'b' >
a'b'


where M = (fl(Ei - H)lf). In the expression for M, the Fock superoperator, F, can be substituted for the Hamiltonian superoperator, H. This is known as the Born collision approximation and causes M to be diagonal. The resulting self-energy is

E= +
a E + a - E - C P E + c - a - b
pq P q ab p a


Setting E = �i in the expression for E ii (E) gives E =i + E ii(ei)


which is just the Rayleigh-Schr~dinger perturbation theory expression for the correction to Koopmans' theorem. The alternative way of improving the Koopmans' value using the expression for ii(E) was called the "quasi-particle" approximation and has been applied using










the second order expression given above (Hohlneicher et al., 1972). Both of these methods are well-known and give reasonable values for valence electrons.

Unfortunately, neither of the two previous methods are known to give accurate values for ionization energies of core electrons. The main reason for this is that the relaxation contributions are very large for core electrons and can not be adequately described by second, or even third, order perturbation theory of this type.

It was noted in second order (Pickup and Goscinski, 1973),

and then extended to third order (Born et al., 1978), that Zii( i) can be conveniently separated into the Rayleigh-Schrbdinger expressions for the relaxation and correlation corrections to Koopmans' theorem. This can be represented as

R C C
E = i + Zii.(.i ) + ii(i)


The first two terms in the above equation can be identified as the Rayleigh-Schrbdinger approximation to the ASCF ionization energy given in Appendix A. If instead of the approximate value, the exact value is used, the following equation is obtained
C
E = -I(ASCF) + Eii( i)


This expression gives the correlation correction to the fully relaxed ionization energy and is the same in second order as the one given by Purvis and Ohrn (1976). This equation should now overcome the problem with the simple Rayleigh-Schrbdinger expression and should provide excellent core ionizations. Since -I(ASCF) is just a numerical value, it is possible to approximate it with the TOM










value. Even though the TOM ionization energy can be obtained from a single calculation, there is not any direct computational advantage in using it since a ground state calculation must always be done in order to obtain reference orbtials and energies.

An alternative approach to the one above is arrived at by

making the relaxation-correlation separation in the original energy dependent expression for E,


E .(E) = ER (E) + EC (E)
13 ij ij

If it is assumed that 1 + IR(E) can still be approximated by the ASCF ionization energy, the following equation for E is obtained


E = -I(ASCF) + E~i(E)


This approximation will be referred to as the "fully-relaxed quasi-particle" approximation by analogy to the previously mentioned "quasi-particle" approximation. The expression for the second order correlation correction is


( .Il + II2
ai E + ca - Cp - Cq ati E + cp - Ea - Eb
p,q bti
p




1.5 Application to the Water Molecule

In order to make a comparison of the approximations to the ionization energy which have been discussed, they are applied to the water molecule. For this comparison two different basis sets were used to describe the molecule. The first, called basis I, is










a 14 CGTO combination of Dunning's oxygen basis (Dunning, 1970) and Huzinaga's hydrogen basis (Huzinaga, 1965). This basis is shown in Table 2. The other basis, basis II, is comprised of basis I plus a set of d-type GTO's on the oxygen with unit exponents and a set of p-type GTO's on each hydrogen, also with unit exponents. This augmented basis is used to give some check on the basis set effects of the approximations and to include functions which span all four irreducible symmetry representations of the C2v molecule. The results obtained with each approximation are shown in Tables 3 and

4 for basis I and II respectively.

In the quasi-particle expression for the ionization energy,


E = ci + Eii(E) ,


if the "exact" self-energy expression is used, information about all ionizations can be obtained, not just the principle value related to the ith Koopmans' energy. Even in the relaxed quasi-particle method with a second order approximate ZC, other ionization energies can be found with varying degrees of accuracy. These are approximate values for the "shake-up" processes which correspond to simultaneous ionization and excitation.

The method used to solve the relaxed quasi-particle equation for all the states listed in Tables 3 and 4, except the 2aI states, is simple iteration with the Koopmans' value as a starting point. Convergence to 10-10 Hartrees was always very quick. For the 2aI states, though, the iterative method converged to a value different than the one listed or it diverged. To explain this apparent anomaly it is helpful to look a a plot of E versus -I(ASCF) + Z .(E) given II




















Table 2

14 CGTO Basis Used for Water



orbital exponent coefficient

sH 17.370 0.032828
2.6273 0.23121 0.58994 0.81724 s'H 0.16029 1.0 sO 7816.5 0.002031 1175.8 0.015436 273.19 0.073771 81.17 0.2476 27.184 0.61183 3.4136 0.2412 s'O 9.5322 1.0 s-0 0.9398 1.0 s 0 0.2846 1.0 pO 35.183 0.01958 7.904 0.12419 2.3051 0.39473
0.7171 0.6273 pO 0.2137 1.0















Table 3

Water Basis I Results (eV)


ASCFa

540.8 34.6 17.8 13.0 11.0


TOM

540.64 34.51 17.78 12.86 10.88


2nd order
RSPT

529.77 33.12 18.01
12.64 10.52


aGoscinski et al. (1975) bAlml6f (1972)


state

la1 2a, lb2 3a1 lb1


KT

559.44 37.04 19.52 15.43 13.78


2nd corr
ASCF

541.0 34.4 18.7 13.8 11.8


2nd corr
TOM

540.81 34.26 18.66 13.61 11.65


ASCF g-part.

541.0 32.8 18.7 13.8 11.8


TOM g-part.

540.85 32.73 18.67 13.63 11.68


Expb

540.2 32.2 18.6 14.7 12.6















Table 4

Water Basis II Results (eV)


2nd order
RSPT

529.72 31.81 17.98 13.28 10.87


2nd corr
ASCF

540.69 32.78 19.14 14.30 12.06


2nd corr
TOM

540.55 32.89 18.63 14.18 11.92


state

la1 2a1 lb2 3a1 lb2


KT

559.39 36.62 19.34 15.66 13.67


ASCF

540.49 33.86 17.93 13.15 10.89


TOM

540.35 33.97
17.42 13.03
10.75


aAlml6f (1972)


ASCF q-part.

540.72 32.31 18.72 14.33 12.09


TOM q-part.

540.58 32.36 18.64 14.21 11.95


Expa

540.2 32.2 18.6 14.7 12.6










if Figure 3 for basis I and Figure 4 for basis II. It can be seen
C
that because of the poles of Eii(E) which exist in the region shown, there are actually three solutions. Two of the solutions are quite close (one of which is the primary solution) and this is the source of the problems in the iterative procedure. Table 5 shows the values obtained for the principle ionization and two shake-ups in the region.

Table 6 shows a comparison of the results of this work with those of Cederbaum et al. (1973) and Hubac and Urban (1977). The results of Cederbaum were obtained by solving the Dyson equation with second and third order self-energies. The basis used was the same as basis II except the hydrogen p-like GTO's had exponents of 0.75. The results of Hubac and Urban are Rayleigh-Schr6dinger results for both second and third order using a basis similar to basis I. The results of the ASCF-relaxed quasi-particle method in second order are clearly better than the previous second order results and compare favorably with the third order results. It is worth noting that neither Cederbaum nor Hubac and Urban have reported any results for the la1 ionization where relaxation effects dominate.



1.6 Application to the Neon Atom

As a further test of the methods discussed for the calculation of ionization energies, the neon atom is chosen. Since it is an atom, it has the advantage when comparing with experiment of not having vibrational fine structure. There has also been a great deal of related work done on neon (Purvis and Ohrn, 1976) and, therefore, it provides a good check. The basis set used for the
























Figure 3 Plot of -I(ASCF) + Zi(E) versus energy for
water obtained with basis I. A second order
self-energy was used and the ASCF-relaxed
quasi-particle solutions are the intersections
with the 450 line.














/
/
I,




















/
/1


N


7
/


/


/



/



'I


N


N


LO 1


LIV)I


'I


"N


U)1


























S.
'0 2:I


/
/
























Plot of -I(LSCF) + Eqi(E) versus energy for water obtained with basis II. A second order self-energy was used and the ASCF-relaxed quasi-particle solutions are the intersections with the 450 line.


Figure 4






































(O d~


I
I


L()





























Table 5

Ionizations in 30-40 eV range



designation basis I basis II

principle 2a, 32.76 32.31
shake-up 34.62 33.66 shake-up 36.39 36.96






























Table 6

Comparison of Results (eV)


State Ced2a


2aI lb2
3a1 lbI


32.93 17.70 13.18 10.92


Ced3b

35.10 19.22 15.18 13.03


HU2C

33.38 17.93 12.62 10.48


HU3d

35.22
19.42 14.74 12.75


BIe

32.8 18.7 13.8 11.8


aSecond order results of Cederbaumn (1973). bThird order results of Cederbaum (1973). CSecond order results of Hubac and Urban (1977). dThird order results of Hubac and Urban (1977). eASCF-relaxed quasi-particle results with basis I. fASCF-relaxed quasi-particle results with basis II.


BIIf

32.31 18.72 14.33 12.09










neon calculations is shown in Table 7 and the results obtained are shown in Table 8.

As can be seen, excellent results were obtained for both the second order corrected TOM and the TOM-relaxed quasi-particle approximations. These results are also compared to the second order corrected TOM approximation of Purvis and Ohrn (1976) and they compare very well. The discrepancy in this work for the 2s level can be explained as a basis set effect. This discrepancy is removed in the work of Purvis and Ohrn (1976) where they used the same basis except that a 2s orbital replaced the is (8.91407) orbital.

























Table 7

Neon STO Basis



orbital exponent

Is 8.91407
Is 9.5735 ls 15.4496 2s 7.7131 2s 4.7746 2s 2.8462 2s 1.0000 2p 4.4545 2p 2.3717 2p 1.470 2p 0.750 3d 2.80















Table 8

Neon Results (eV)


TOM

868.82 49.17 20.53


2nd order corr. TOM

869.07 47.64
21.58


TOM-rel ax.
g-part.


aKoopmans' Theorem bpurvis and hrn (1976)


State

ls
2s 2p


KTa

892.18 52.64 23.11


869.07 47.76 21.56


p& b


869.12 48.21 21.56


869.28
48.19 21.02
















CHAPTER 2
POSITIVE ELECTRON AFFINITIES


2.1 Discussion of Problem

A positive electron affinity represents the decrease in total

energy a system attains with the addition of an electron. Analogous to ionization energies, there are two different types of electron affinities, vertical and adiabatic. The adiabatic electron affinities are also known as the thermodynamic electron affinities. Methods of obtaining electron affinities can be broken down into three classes: experimental, theoretical, and empirical fitting. There are numerous methods which fall into the classes of experimental and empirical fitting and a discussion of these is outside the scope of this work. A very good discussion of many of these methods is given by Steiner (1972).

One problem common to almost all methods, theoretical and nontheoretical, is the very small size of electron affinities. The largest known electron affinity is 3.82 eV for CN (Berkowitz et al., 1969). This presents special problems for the indirect theoretical methods where the N and N+1 electron total energies are used.

In the next section, an analysis of the contributions to the electron affinity is made in a similar manner to that used for ionization energies. An attempt is made to obtain explicit expressions for these contributions as before.










2.2 Analysis of Effects

The simplest theoretical method for calculating electron

affinities is Koopmans' theorem. The theorem is the same as it was for ionization energies, except now the negative of the first unoccupied orbital energy is used. Relaxation and correlation corrections can again be defined in terms of the ASCF electron affinity. The perturbation expressions derived in Chapter 1 for ionization energies can be applied to electron affinities if the N+1 electron state is used as the reference state. Clearly this is not what is desired. It would be most useful if expressions in terms of the N electron reference state were obtained so that the corrections to the Koopmans' electron affinity could be calculated with the same quantities as are used to calculate corrections to the Koopmans' ionization energy.

To find an expression for the relaxation correction to Koopmans' electron affinity, Rayleigh-Schr6dinger perturbation theory is again used. The following treatment will parallel closely what is done in Appendix A.

The expression for the ASCF electron affinity is


-EA(ASCF) = EN+I - EN
HF,x HF

x + Z a - Cal - Z - {<5bjlb> - }.
a a a,b

In the above expression and the following derivation, the "tilded" quantities will refer to the N+1 electron state. When these N+1 electron state quantities are expanded in terms of the N electron state quantities and only terms of third order or less are kept, the









following expression is obtained

-EA(ASCF) = + E(I) + 6(2) + C(3)
X X X X

+ { C(1) + C(2) + C(3)} a a a
{ + +


+ + + + + + + + }


a E { + + + + }

- , { +
a,b

+ + } In order to simplifiy this expression, the form of the perturbation must be known. This can be found by considering the relation between the Fock operators

F N+ = FN + < JJR> + Z{< ll > - }
X a

By using this perturbation to find expressions for a and c(, it can be shown that they will cancel many of the integral terms as they did in the ionization expression. It can also be easily shown that E(2) and 6(3) vanish through third order. This leaves
x x









S(2) (3) - Z { -E(SF {La2 + LE } b Z lab>
F a ax a a,b

+ + + 1


By inserting into this expression the RSPT equations for the orbital and orbital energy corrections and simplifing, the following result is obtained

-EA(ASCF) = L - I I 12 x a (Ek - La) k~a
{ + ' :
a,b (Ek - La)(El - Lb) lb
+ (Lk - 6a)(cl - Lb)

( k - a)(El - Lb)

(k - C) - Lb)

+ E a (El - La)(k - La) k,l~a a
a (cI - La)2 1*a
If this expression is now compared to the one obtained for -I(ASCF) in Appendix A, it is seen to be identical except for the signs on some of the terms. It is interesting to note that the number of times the perturbation appears in a term can be correlated with










the sign difference. If appears an even number of times, the signs are different, and if it appears an odd number of times, the signs are the same.

It is fortunate that this sign change occurs in the ASCF

expression, at least in second order. By subtracting the second order relaxation expression from the full second order self-energy obtained previously, the correlation correction to the relaxed electron affinity is obtained. It is given by

l + Z ,l2
x a C + C - C - C ab E + C - C - C
p,qtx x a p q p x x p a b

It should be noted that this is the same correlation expression that was used to obtain ionization energies. The only difference is that now the virtual orbtial sums are restricted instead of the occupied sums. This means that the program written to evaluate the RSPT correlation correction and the relaxed quasi-particle correction for ionization energies can also be used to obtain the similar quantities for electron affinities.



2.3 Application to LiH

The molecule LiH was chosen as a test case mainly because it is well characterized by previous work (Jordan et al., 1976). Two differenct basis sets were used to perform the calculations at an equilibrium distance of 3.015 a.u. One basis is a 19 STO basis, shown in Table 9, and the other is a 13 CGTO basis, shown in Table 10. The results of the calculations are shown in Table 11.


















Table 9

LiH 19 STO Basis



orbital exponent

IsLi 4.6990 IsLi 2.5212 2sLi 1.2000 2sLi 0.7972 2sLi 0.6000 2sLi 0.3000 2poLi 2.7500 2poLi 1.2000 2poLi 0.7369 2poLi 0.6000 2poLi 0.3000 2pfLi 0.7369 2plTLi 0.3500 IsH 1.5657 IsH 0.8877 2sH 1.3765 2sH 0.4000







41







Table 10

LiH 13 CGTO Basis



orbital exponent coefficient

sLi 642.41895 0.00214
96.79849 0.01621 22.09109 0.07732 6.20107 0.24579 1.93512 0.47019 0.63674 0.34547
sLi 2.19146 0.03509 0.59613 0.19123 sLi 0.07455 1.00000 sLi 0.02079 0.39951 0.00676 0.70012 sLi 0.08948 1.00000 pLi 2.19146 0.00894 0.59613 0.14101 0.07455 0.94535 pLi 0.08948 0.15559 0.02079 0.60768 0.00676 0. 39196 sH 18.73110 0.03349 2.82539 0.23473 0.64012 0.81376 sH 0.16128 1.00000

















Table 11

LiH Electron Affinity Results (eV)


TOM

0.236


2nd corr.
ASCF

0.256 0.287


ASCF-relax.
g-part.

0.256 0.287


aKoopmans' theorem value.


basis

13CGTO 19STO


KTa

0.204 0.198


ASCF

0.237 0.238










These results agree very well with the result of 0.2986 eV

of Jordan et al. (1976) using Simon's EOM method (Simons and Smith, 1973). The fact that the STO result is better than the GTO result is evidence that the STO basis is a much more complete basis in the region of the maximum of the charge distribution of LiH-. As another check on the quality of the basis sets, the dipole moment was calculated from the Hartree-Fock orbtials. A value of 6.11 D was obtained for the GTO basis and 6.02 D for the STO basis, compared to an experimental value of 5.88 D.
















CHAPTER 3
NEGATIVE ELECTRON AFFINITY



3.1 Resonances
As was mentioned previously, the study of negative electron affinities is embodied in the study of electron scattering. In recent years, considerable amounts of work have been done on the calculation of resonances in electron scattering with both atomic and molecular targets and several excellent reviews have been given by Burke (1968), Taylor (1970), Schulz (1973), and Nesbet (1975). Following the definitions of Taylor (1970), resonances (also called "temporary negative ions" or "compound states") can be separated into three main classifications; core excited types I and II and single particle resonances.

Core excited type I (C.E. I) resonances are also called Feshbach or hole-particle resonances. Such resonances are formed when the incoming electron excites a target electron into an excited orbital, lessening the screening of the nuclear field. The electron then gets trapped in this potential and the resonance acts as a bound state relative to the excitation threshold. These resonances can be viewed as being caused by the excited state having a positive electron affinity. C.E. I resonances are usually the narrowest, thereby longest lived, of the three types.

Core excited type II (C.E.II) resonances are very similar to C.E. I resonances. The difference is that they lie above the

44










excitation threshold. The resonance is formed by trapping the incoming electron inside a centrifugal barrier set up by the combination of its angular momentum component and the potential well induced by the electron when it excites and polarizes the target. Since such a resonance is caused by the shape of the potential, it is given the classification of a shape resonance, and because it depends on the 1(1+1)/(2r2) term in the potential, we expect to see resonances only for p and higher partial waves.

The final type of resonances are called single particle resonances. They, like C.E. II resonances, are shape resonances. The difference from C.E. II resonances lies in that there is no target excitation. Energetically, single particle resonances lie just above the ground state and well below excitation thresholds. It is these single particle resonances that can be identified as defining the negative electron affinity of a system. Until recently, this type of shape resonance has been very difficult to detect, both experimentally and theoretically. In the last year or so, Burrow et al. (1976) have performed low energy electron transmission experiments in which they have detected such resonances in the Group IIA elements Mg, Zn, Cd, and Hg.



3.2 Methods for Calculating Cross Sections

In order to detect broad single particle shape resonances

theoretically, the electron scattering cross section must be calculated. Most methods of use in scattering calculations today can be divided into two main types, numerical integration methods and algebraic methods. The numerical integration methods are exemplified by the










close-coupling method (Burke, 1968). The subject of algebraic methods has been extensively reviewed by Truhlar et al. (1974) and Harris and Michels (1971). For bound state calculations an analogous classification of methods can be made. In this case, the numerical Hartree-Fock method (Froese, 1963) is the analogue of the close-coupling method and the matrix Hartree-Fock methods are the analogues of the algebraic methods. Experience from bound state problems has shown that as the complexity of the problem is increased in going from atomic to molecular systems, the algebraic methods are much more useful than the numerical integration methods. It is anticipated that a similar trend will exist in scattering problems as the complexity increases, for example by having many coupled equations to solve or by using non-spherical potentials.

The early algebraic methods were given by Hulth~n (1944), Kohn (1948), and again Hulth~n (1948) (also given by Rubinow (1955)). All three of these methods had the problem of giving spurious resonances, called pseudoresonances. In 1967, Harris caused renewed interest with the proposal of a method for single channel scattering (Harris, 1967). Nesbet (1968) was the first to analyze the source of the spurious results in the early methods and proposed the anomaly-free method for single channel scattering. The anomaly-free method was later extended to multichannel scattering (Nesbet, 1969) and also Harris and Michels generalized the Harris method and extended it to multichannel problems in the minimum-norm method (Harris and Michels, 1969). The method chosen in this work for the calculation of shape resonances is the simple Harris method. The reasons for this choise are brought out in the following brief derivation of the method.










As has been demonstrated (Hedin and Lundqvist, 1969) the

wavefunction for the scattered electron is a solution of the formally exact equation


h(x)'(x) + f(x,x';E)T(x')dx' = ET(x)


where E is a complex nonlocal potential known as an "optical potential" and h is a single particle operator composed of the kinetic energy and nuclear attraction operators. Since only electron-atom scattering is to be considered, the central potential allows a separation of variables in spherical polar coordinates to be made. By also making a partial wave expansion of T (Roman, 1965), the above equation reduces to the following radial equation

{dr2+ 12r+ + V(r)}l(r) = E4l(r)
dr2 2

where E = 12k2 with k being the momentum of the scattered electron. For conveniance, the notation

d2 1(1+11
L = d2 + 2r2 + V(r)


is used. It should be kept in mind that L, through V(r), is in general nonlocal, energy dependent, and complex.

The function fI is assumed to be of the form sI + tcI + Zbiui, where sl and cI asymptotically are solutions to the radial equation with V = 0, namely the regular and irregular spherical Bessel functions Jl(kr) and nl(kr). The explicit forms of s, and c, are discussed later. The set of functions {ui} is any suitable L2 set of basis functions which go to zero both asymptotically and at the orgin. By using










this form for I, t can be identified as tanSl, where 61 is the phase shift of the Ith partial wave.

In order to solve for 61, the condition that the projection of (L - E) 1 on the set of L2 functions vanish,


(uiIL - EJp1) = 0 for all ui


is required to be satisfied. Upon substitution for I' this leads to


(uiIL - EfsI) + t(uiIL - EIcl) + Ebj(uiIL - Eiuj) = 0


A vast simplification can be obtained by choosing the L2 functions to be eigenfunctions of the operator L. This assumption then gives


(uilL - ElsI) + t(uilL - EICl) + bi(Ei - E) = 0 .


By fixing the value of E to be equal to the eigenvalue corresponding to ui and rearranging, the sought after expression for 61 is obtained as


tan61 = _ (uiIL - EilsI)
(uiIL - Eilcl)

This is the expression given for the Harris phase shift (Harris, 1967).

One major advantage in calculating Harris phase shifts, over other algebraic methods, is that only integrals containing one continuum function need be calculated. A very obvious disadvantage to this method is that phase shifts can be determined only at energies corresponding to the positive eigenvalues of L. If the system being considered is small enough so that a large basis set can be used, the restriction to eigenvalues of L creates no problem.










The partial cross section can be calculated from the phase shift via the relation
47T
= i-sin261 (21 + 1)


A total elastic cross section can then be obtained by summing the partial cross sections

a = aI.


Therefore, all that is now needed to calculate elastic cross sections is a form for the potential V(r) in L.



3.3 Choice of Potential

One simple choice for a potential is the Hartree-Fock (or

static-exchange) potential. This corresponds to letting the operator L be the Fock operator,


L = h + Z<.i>.
1 1

With this potential, the expansion functions ui and scattering energies Ei are the virtual Hartree-Fock orbitals pi and orbital energies ei. Finding the phase shifts at the positive orbital energies now corresponds to the evaluation of the following integrals


( ijTl - EilS) - (fllIs) + 14jILqiS) where S is either sl or c,.

The functions s1 and cI, as mentioned earlier, should go

asymptotically as j and nI respectively, but they should also go to zero at the origin. Therefore, the actual forms used for these










functions are


Sl = Jl(kr)

Cl = jl+l(kr) + (]{I)jI+2(kr)


This particular choice of cI is due to Armstead (1968) and it has the advantage that all the integrals needed will only involve regular spherical Bessel functions. A description of how to evaluate the necessary integrals is given by Harris and Michels (1971).

In the static-exchange method, the target potential is only

weakly affected by the incoming electron through the exchange terms. In addition to exchange effects, polarization effects have been shown to be very important in electron scattering (Truhlar et al., 1971). For elastic scattering on atoms an important polarization effect is the charge induced-dipole polarization, which for large r behaves as


V (r)'tpo1 2r'

Such a form for the polarization potential can not be used because of the unphysical singularity at the origin and several alternate forms have been suggested. The one chosen for this work is

Vpol(r) -O - r
V~0 r4 r>ro

-cmr r 2r'


where a is the static polarizability and ro is a cutoff parameter. The role of ro is to cut off the attractive polarization potential before it penetrates the charge distribution of the atom. More is said later about how to pick the parameter ro.










Using the above potential, the operator L is now L = F + Vpo1


With this choice of L, we can no longer use the Hartree-Fock orbitals and orbital energies as the expansion functions and scattering energies. To obtain new functions and energies, the eigenvalue problem LC = EC must be solved. Using the Hartree-Fock orbtials as a basis for solving this problem leads to the simplification of L such that L = e + V , where c is a diagonal matrix of Hartree-Fock orbital energies and ( = (4iIVpolIbj). Once the solutions have been found in terms of the Hartree-Fock orbitals, they can be back transformed to the primitive basis giving the desired orbitals and energies to proceed in finding the phase shifts.



3.4 Relation to Other Methods - Helium Test Case

In order to evaluate the results of the method developed, the relationships between it and two other methods, namely those of Harris and Michels (1971) and Yarlagadda et al. (1973), are explored. In the method of Harris and Michels, the target is described by the N electron determinantal (CI) solutions of the full N electron Hamiltonian,


=Z a IDN>
p lip It

The scattering solution is then given by solutions to an N+1 electron Hamiltonian, HN+1, written as N+1 1 1
HN+1 - E = (HN - Ey) + (T 1-k2) + X )
[(H 1 j=2 rj r,










The solution is assumed to be of the form (s + t c + Ebni i, where il
the functions s' c and qi are N+1 electron determinantal functions given by


s = E a14ISDN> ,

11 P
c = E ,,FDN


and nri = E biIDN+I>, Ill P

where ICDN> and ISDN> are N+1 electron determinants. The equation for the Harris phase shift is then


- tan6 =

target is represented by a single determinant of occupied Hartree-Fock orbitals, Xa, represented by

N
IDHF> ,


and the scattering solution is formed by adding sI, cI, and Xp to make the N+1 electron functions



@= ICl DnF> = c]DN > and p IXp DHF> where Xp is a virtual Hartree-Fock orbital. Under these conditions the integral








2NISN
{ - +NZl}/(N+1),
p1 p r, I +X=xx 1xs1}/I +

This can be immediately recognized as our previous expression.

Yarlagadda et al. (1973) used the same one electron non-local potential equation used here as a starting point but included a correlated self-energy operator. In their treatment Z is given by


E(x,x';E) = EHF(x,x') + Z<(x,x';E) + Z>(x,x';E) ,


where E< and Z> are generalized polarization potentials given in terms of the generalized response function in the random-phase approximation (RPA). By dropping all terms in E except ZHF' the present static-exchange method is obtained. The polarization potential used in this work can be viewed as a local, energy independent approximation to the potential of Yarlagadda et al.

To test the programs that were written for the calculation of Harris phase shifts, a series of tests were performed on the helium atom. This system was chosen because it is a very simple system to obtain target wavefunctions for and because of the wealth of prior work, both theoretical and experimental. In the calculations performed two different basis sets were used for the Hartree-Fock description of the helium atom. The first, called HelO, is a ten STO basis comprised of the five optimized STO's of Clementi (1965) plus five more diffuse STO's of higher principal quantum number ana the other, called He20, is a twenty STO extension of basis He1O. Both of these basis sets are shown in Table 12.

There are several reasons for using extended basis sets. One main reason is that they give more virtual orbitals and energies at
















Table 12

Helium Basis Sets


Basis HlelO exponent

1.4300 2.4415 4.0996 6.4843 0.7978




1.000
0.500 0.250



1.000 0.500


Basis He20 exponent

1.4300
2.4415 4.0996 6.4843 0.7978
0. 100 0.050 0.075 0.010 1.000
0.500 0.250 0.100
0.075 0.025 1.000 0.500 0.250 0.100
0.050


orbital










which phase shifts and cross sections can be calculated. The nature of the basis sets used is such that many of the virtual eigenvalues are clustered at low energies, thereby giving a better description of the shape of the cross section curve in this region. The use of extended basis sets, like those used here, may also improve the calculations by giving a more accurate description of the long range behavior of the static-exchange potential.

Tables 13 and 14 show the static-exchange results obtained with the basis sets mentioned. These results are also shown graphically in Figure 5 and can be seen to agree very well. The discrepancy at low energy can possibly be explained by the better long range description of the helium is orbital provided by basis He20.

Figure 6 shows a comparison of the results obtained in this work with some previous calculations.



3.5 Beryllium
Since the ultimate goal of this work is the calculation of

shape resonances in the group IIA elements, a likely place to begin is beryllium. From a theoretical standpoint, this is a very good starting point because of its small size. Unfortunately, it is difficult to do the necessary experiments on beryllium because of health hazards.

In the calculations to be described here, Clementi's (1965) extended basis was used to describe the target Is aid 2s orbitals. The virtual orbitals and energies were obtained from three different Hartree-Fock calculations each using a different set of 2po functions. These basis sets are shown in Table 15. Only po functions were




























energy (eV)


0.1537 0.7728 2.3729 6.1827 15.3040 38.7226 106.7075 350.8599 1896.1131


Table 13 HelO Results


phase shift (rad)


3.0136 3.8155 2.5609 2.2320 1.8277 1.3881
0.9928 0.6607 0.3152


cross section
(au2)


18.1326 22.7077 21.6837
17.2258 10.4505
4.2695 1.1240 0.1835 0. 0037















Table 14 He20 Results


energy (eV)

0.0004 0.0023 0.0071 0.0177 0.0384 0.0776 0.1515 0.2923 0.5639 1.0970 2.1696 4.3910 9.1601 19.9452 46.5337 120.9710 381.3872 2008.1614


phase shift
(rad)


3.1350 3.1243 3.1097 3.0905 3.0655 3.0329 2.9895 2.9304 2.8489 2.7355 2.5771 2.3578 2.0652 1.7040 1.3127 0.9571 0.6457 0.3088


cross section
(au2)


17.4187 22.4827 24.3011 25.2611 25.7255 25.9196 25.9036 25.7048 25.2345 24.3211 22.5510 19.4053 14.4625 8.4209 3.4348 0.9446 0.1623 0.0079

































Figure 5 Plot of phase shift versus energy for helium. The
A curve was obtained using the HelO basis and the *
curve was obtained using the He20 basis.











3.02




















1.5- ~~ 0.41.-3 .
Enrg-(V

























Plot of phase shift versus energy. The curves were obtained from: Harris and Michels (1971) ( + ); Yarlagadda et al. (1973) (< ); Present work with HelO basis ( A ); Present work with He20 basis ( *


Figure 6












Phase Shift (rad.)
'-a U'
0 0







/
4'
7/.'


~

'A'
I,
44

'/ ~" 'I ~
1. IA'
/1/!
m ff4
I
CD !? '/ a,'

41/
CD
I-.
01
o
/ I.
'1'

/

/

I

/ 'n~ 'I

* /











0 ffi. - -0

19


- ~~1~~ -


- ~ I














Table 15

Be Basis Functions Target orbitals


exponent


3.4703 6.3631 0.7516
0.9084 1.4236 2.7616


ls coef.

0.91792 0.08742 0.00147
-0.00267 0.00222 0.00597


Virtual orbitals


basis I


1.900 1.400 0.900 0.700 0.450 0.350 0.220
0.170 0.110 0.085
0.055 0.025


basis 2

2.000 1.500 1.000 0.750 0.500 0.375 0.250 0.180 0.120 0.090 0.060 0.030


2s coef.

0.17065 0.01469
-0.11551
-0.67835
-0.30265 0.09232


basis 3

2.100 1.600 1.100 0.800 0.550 0.400 0.280 0.190 0.130 0.095 0.065 0.035










used because the shape resonance should occur as a 2 state and, therefore, will show up only in the p-wave cross section. Three different basis sets were used to provide more virtual orbitals in a useful energy range without doing a very large calculation. One advantage in doing these calculations on beryllium is there are no po orbitals in the occupied Hartree-Fock set of orbitals. Therefore, Po functions can be added to the basis or their exponents can be varied and there will be no changes in the static-exchange potential except through the exchange terms which are very small. The results of the calculations are given in Table 16.

Calculations were also performed using the polarization

potential mentioned previously. Since this potential has an arbitary parameter ro, some method of arriving at its value is needed. In the work discussed here r0 is picked as the value at which the potential (1+1)1/2r2 + VpoI is zero. The expression for ro is then = c 12
ro { -+)


For beryllium, the static polarizability used is 47 a.u. and ro is chosen as 5 Bohrs. The results obtained with the polarization potential included are shown in Table 17. A plot of the low energy portion of the cross section curve is shown in Figure 7 for both the static-exchange and static-exchange + polarization calcuiations. As can be seen, the addition of the polarization potential causes a dramatic change in the cross section. The reliability of this change is discussed further during the evaluation of the resonance parameters.














Table 16

Be Static-exchange Results


E (eV)


0.004202
0.005451 0.006798
0.018394 0.022614 0.027191 0.054174 0.064984 0.076655
0.133786 0.158757 0.185447 0. 300029 0.352412 0. 406992 0.608721 0.701153 0.795346
1.143305 1.316582 1.505334 2.252472
2.658461 3.105742 5.017304 5.965638 6.997279 11.693033 13.847953
16.178037' 30.325205 35.674818 41.487912 99.405096 117.676441 137.705365


phase shifts
(rad.)


0. 000139 0. 000207 0. 000290 0. 001577 0. 002151 0. 002840 0. 008019 0. 010721 0. 013949 0. 035220 0.046136 0.059121 0.128540 0.169153 0.216708 0.440872
0.554051 0.670897 1.029797
1.157449 1.257652 1.442995
1.469213 1.477906 1.422467
1.401906 1.380897
1.315351 1.294340
1.275813 1.150602 1.131722 1.112447 0.890287
0.853701 0.819420


cross sections
(au2)

0.002375 0.004039 0.006353 0.069319 0.104979 0.152096 0.608875 0.907111 1.301892 4.753734 6.872079 9.655860
28.090688 41.247963 58.263549 153.435384 202.499887 249.247327 329.638437 326.720581 308.394359 224.010669 190.950717 163.727877 99.995392 83.548150
70. 689548 41.064003 34.279256 29.024436 14.098987 11.779206 9.942484 3.117141 2.476104 1.988966














Table 17


Be Static-exchanqe plus Polarization Results


E(eV)


0.004192 0.005435 0.006773 0.018233 0.022364
0.026821 0.052723 0.062787
0.073415 0.121498 0.140120 0.158628 0.231334 0.261811 0.294031 0.431053 0.503026
0.582814 0.921154 1.098676 1.292951 2.050532
2.454522 2.895756 4.731597 5.639251 6. 625188
11.208313 13.366430 15.710701 29.944744 35.301277
41.124872 99.174761 117.456041 137.493739


phase shifts
(rad.)


0.003042 0.004018 0.005135
0.019675 0.025801 0.032850 0. 083696
0.108474 0.138261 0.335596 0. 435851 0.549157 1. 049741 1. 244082
1.415030 1.818372 1.897914 1.945246 1.964526 1.951096 1.930785
1.860148 1.821383 1. 784643 1. 666117 1. 629738 1.600228 1.580295 1.520797 1.450245 1.034357 1.125939 1.270586 1.119829 1.085710 0.868290


cross sections
(au2)


1.132535 1.523699 1.996677
10.888717 15.263814 20.628901 67.988909 95.746635 132.703602 457.866409 652.437972 880.948064 1667.766610 1757.304698 1702.425504 1118.444717 914.376183 762.322092 474.862546 402.517857 347.471062 229.770366 196.115766 169.147309 107.418995 90.637887 77.351077 45.757406 38.277120 32.174991 12.654530 11.838870 11.381315 4.189378 3.417380 2.173016




























Figure 7 Plot of cross section versus energy for beryllium.
The A curve was calculated using the static-exchange
potential and the * curve was calculated using the
static-exchange plus polarization potential.

























































2.5


Energy (eV)


1800.


1000.01


0.0


5.0










3.6 Magnesium

The next group IIA element is magnesium and, unlike beryllium, experimental work has been done on it. Unfortunately from a theoretical standpoint, this system is much more difficult to calculate because of its larger number of electrons.

Again, as was done for beryllium, Clementi's (1965) extended

basis sets were used to describe the target Is, 2s, 2p, and 3s orbitals. In order to obtain virtual orbitals and energies, this basis was augmented by three different sets of p0 orbitals. The basis is shown in Table 18. Unlike the beryllium case, where the additional PO functions have no effect on the target potential, in magnesium they do. The additional p0 functions will mix with the occupied 2po function and cause a slight breaking of the degeneracy of the 2p functions. Thus, as the additional po functions are varied the occupied 2po function changes and, therefore, the potential changes. While in general, this would seem to create problems, in practice the changes are so slight that the potential can be regarded as constant provided no major changes are made in the additional p0 functions. The results of these static-exchange calculations are shown in Table 19.

Calculations were also done using the polarization potential

with a = 81. and ro = 6.3. These results are given in Table 20 and a comparison plot of the cross section with and without the polarization potential is given in Figure 8. Again, notice the dramatic effect of the polarization potential.















Tabl e 18

Magnesium Basis


Target orbitals:


exponent

12.0 13.5552 9.2489 6.5517 4.2008 2.4702 1.4331 0.8783

exponent


ls

0.96539 0.03767 0.01583
-0.00212 0.00104
-0.00014 0.00005 0.00001


6.0
7. 9884 5.3197 3.7168 2.5354


Virtual orbitals:


exponent


1.00 0.30
0.325
0.35


0.51090 0.08276 0.32223 0.21178 0.03591



npc, (n=3,4,5,6,7)


all calc. basis set basis set basis set


-0.24382
-0.00504 0.10176 0.39908 0.55719 0.04715
-0.00783 0.00197


0.04695 0.00166
-0.02391
-0.07764
-0.13548
-0.00710 0.50699 0.57016

















Table 19

Mg Static-exchange Results


phase shifts
(rad.)


0.021150 0.025704 0.030670 0.153096 0.184554 0.217919 0.515179 0.592669 0.666490 0.994143 1.057389 1.106756 1.213852 1.220699 1.220718 1.142453 1.122871 1.102427
0.931352 0.908184 0.885543
0.651735 0.631349
0.611290 0.357590 0.339243 0.320871 2.956181 2.935589 2.914877


cross sections
(au')

2.803479 3.647776 4.631591
47.377459 61.136029 76.568734 237.727264 276.075872 308.626046 378.713888 369.561864 354.638997 258.454327 234.703816 213.788917 128.551696 115.479886 104.257131 51.565066 46.410493 41.901138 14.864938 13.340873 11.976657 2.353288 2.037852 1.754699 0.247567 0.293486 0.341272


F (eV)


0.081827 0.092882 0.104139 0.251770 0.282524 0.313107 0.523734 0.579667 0.635199 0.951745 1.053091 1.156575 1.742237 1.928260 2.116931 3.301539 3.608436 3.917077 6.404669 6.869169 7.338163 12.695070 13.393855 14.106339 26.702131 27.871870 29.076516 70.411131 73.122476 75.936550

















Table 20

Mg Static-exchange plus Polarization Results


E (eV)


0.071849 0.079643 0.087114 0.181427 0.198396 0.215599 0.387777 0.436002 0.485986 0.808969 0.912347 1.018105 1.612432 1.801153
1.991531 3.154180 3.458330 3.763056
6.144810 6.600492 7.061153 12.352179 13.054273 13.769579 26.396296 27.567770 28.774576 70.227328 72.940730 75.756894


phase shifts
(rad.)


0.187897
0.218614 0.249523 1.041606 1.157026 1.259715 1.745007 1.776453 1.794933 1.788866 1.770552 1.749677 1.618644
1.584771 1.552758
1.381273 1. 351105 1. 322886
1. 133358 1. 107281 1. 081434 0. 782344 0. 756306 0.730963 0. 413249 0.393949 0.373839 2.932335 2.901953
2.889071


cross sections
(au2)

249.080902 302.912990 359.038448
2106.574817 2167.357707 2156.112829 1282.953281 1127.335632 1003.262067 604.353123 540.051458 487.839174 317.369196 284.711727 257.461588 156.841341 141.267270 128.094564 68.491260 62.174431 56.588127 20.635094 18.502856 16.601043 3.133697 2.741167 2.377253 0.315174 0.396148 0.422635
































Figure 8 Plot of cross section versus energy for magnesium.
The A curve was calculated using the static-exchange
potential and the * curve was calculated using the
static-exchange plus polarization potential.
























































5.0


Energy (eV)


2000.0


1000.0


0.0


0.0










3.7 Evaluation of Resonance Parameters

The shape of the cross sections for Be and Mg given in Figures

7 and 8 shows evidence for the existence of a resonance but from them it is not obvious how to obtain the resonance parameters. These parameters are the resonance energy Er and the width F. To evaluate Er and r it is easier to consider the shape of the phase shift curves. Figure 9 shows a plot of 6 versus E for Be and Figure 10 shows a similar plot for Mg.

One way of obtaining resonance information via phase shifts is through a study of time delay (Bohm, 1951; Wig ner, 1955). The time delay of an outgoing particle due to the presence of a potential is given by

At = 2h d6
dE

The energy of maximum time delay is identified as the resonance energy and the resonance width is then given as

2 d
= dtE E=Er



To actually solve for the values of Er and F from the phase shift curves, the values of the first and second derivative curves are needed. The second derivative is used because it is simpler to find the zero of it than the maximum of the first derivative. In order to evaluate the derivative curves some form must be chosen for the phase shift curve. In this work a cubic spline was used to fit the phase shift values. First and second derivatives are then evaluated at the known energies and these quantities are also fit via cubic splines to provide continuous curves.
































Figure 9 Plot of phase shift versus energy for beryllium. The
A curve is from calculations using the static-exchange
potential and the * curve is from calculations using
the static-exchange plus polarization potential.










3.0 1.5


0.0


Energy (eV)


7.5


5.0
































Figure 10 Plot of phase shift versus energy for magnesium. The
A curve is from calculations using the static-exchange
potential and the * curve is from calculations using
the static-exchange plus polarization potential.










V
3.0




















-~
.t-J
L F

H


C') K
K

H
I'

II
I / 4
2
U
H /
I




0.0
0.0 2.5 5.0 7.5


Energy4) (RV)










The resonance parameters obtained from the Be and Mg calculations are shown in Table 21. Also given in this table are the experimental results of Burrow et al. (1976) and the model potential results of Hunt and Moiseiwitsch (1970). The first thing that should be noticed is the large change in the parameters when the polarization potential is added. Because of this, it is obvious that for the systems studied here polarization effects are extremely important. These effects can not be ignored if accurate values are sought.

With the polarization potential used in this work, it is not possible to evaluate the quality of the effects included. This is because the potential depends on the parameter ro and no completely justifiable method for obtaining its value is given. Further studies should be made to explore how the effects change as ro changes and also to explore other types of polarization potentials.





























Table 21

Resonance Parameters


staticexchange

0.769 1.611

0.460 1.374


s-e pl us Vpe!

0.195 0.283

0.161
0.238


aHunt and Moiseiwitsch (1970) bBurrow et al. (1976)


Be Er
F

Mg Er
Fr


H&Ma

0.60 0.22 0.37 0.10


Exp. b




0.15 0.14
















APPENDIX A
THIRD ORDER RSPT ON THE ASCF IONIZATION ENERGY

As was previously stated, the ASCF ionization energy is the

difference between two Hartree-Fock energies. If we use the notation that j and c. ("untilded" quantities) are the Hartree-Fock orbtials and orbital energies of the N electron state and j and i. ("tilded" quantities) are the same for the N-I electron state, we obtain the following expression for the ionization energy


-I(ASCF) = EN - EN-1
HF HF,x

= { a - U } - { a - 2E }
a a,b a:x a,box
= Lx +a(a - a) - ( - s { - ).
axa a a*x a,b~x

In the preceding equation, as throughout the paper, the notation used is: a,b,c,d = occupied spin orbitals; p,q,r,s = unoccupied spin orbitals; i,j,k,l = unspecified spin orbitals.

By assuming the N electron Hartree-Fock ground state to be the

reference state for a perturbation expansion of the N-I electron state, we arrive at the following expression for the ionization energy through third order


-I(ASCF) = [x (1) C (2) C (3) X aox a a a anx

+ E { + +
a,b*x

+ + } 81










+ b{ +
a,bsx

+ + }


Now in order to evaluate the corrections to the orbtials and orbital energies, we must consider the relation between the N electron state Fock operator and the N-I electron state Fock operator FN = ho + E
0 a

and FN-I = ho + E< > x a~x

where x is the label of the removed spin orbtial and where stands for the coulomb minus exchange operator. This gives the relation


FN-1 = FN - + E { - }
x atx

The perturbation is given by V1 + V2, where VI = - and V2 =a { - }. The perturbation V2 is somewhat unusual because it is defined in terms of both unperturbed and perturbed orbitals. Therefore, to obtain expression up to a given order in electron interaction we must apply our perturbation in an iterative fashion.

Before we apply this result to the expression for the ionization energy, we can simplify the expression by noting that

a 1)M = E = -E + Z { - } atx atx atx a,box = x + E { +
ab a,bAx
+ + + }










This cancels some of our previous terms and leaves


-I(ASCF) = x {(2) + 6(3)} + 2 Z {
-ayx a a a,btx

+ + + }


To illustrate how to deal with the perturbation, we shall work through an example in detail. Let us consider the case of the second order energy correction


S(2= ,
a*x a aix (ck - C ) k~a

Substituting the perturbation we get


z I12 (V :VI term)
kaa Ck - Ca


+ , { - } (V1:V2 term)
kpa ck - Ea


+ . { - l (V term)
a:$x (2:VItem
kTa Ek - Ea

Using the expansion for 5 and keeping terms through third order we now obtain

- I12 + . { + }
a~x �k _ a,b~x Ck _ Ca
ka -a k~a


+ . { + }
a,bx Ek _ ca
k ia









- _y ll2 a*x Ek - Ea k:a


+ Z { +
a,b:x (Lk - La)(Ll - cb)
k~a lfb

+ +
(ck - La)(El - cb)

Similarly, we obtain from the other energy expression the following terms



(Ek - La)(Ll - Lb)


(Ek - ca)(6l - 6b)

parts of the ionization


(3) ]+ XLax> > ), = I> -_Ld .. ... L 0. . + �a~x a a~x (61 - La)(ck - ca) ax (LI - Ea)2
1,k:a Ia

and


2E { +
a ,btx
a fx {

a,btx (C - C )(CL - Cb)
k~a k a b
Itb

+
(6k - La)(-1 - cb)

Now by combining the terms we ha expression for the ionization energy a third order in electron interaction.


+ + }

+
(Ek - La)(Ll - Lb)



(Ck - La)(cl - Lb)

,e obtained, we arrive at the it the ASCF level through


-I(ASCF) : Lx + . jl1
XaVx
k*a k - Ea
s {
a,bx (Lk - a)(Ll - b) (Lk - a)(Ll - Lb)
k*a lb













a~x 1, ksa


(ck - ca)(cl - Eb)


asx
l~a


(cI - ca)2
















APPENDIX B
THIRD ORDER RSPT ON THE TOM IONIZATION ENERGY

The TOM ionization energy is given by the eigenvalue cx of the transition operator defined by


FT = ho + E + !
x amx

In this section we will obtain an expression for EX by using RSPT with the Hartree-Fock N electron ground state as our reference. In order to define the perturbation, we must consider the relation between the Fock operators


Fx F= - + ! + E { - } aix

Keeping all terms through third order, the perturbation V can be expressed as


V = - + + Z { - } a;x

= - + E { + +
a~x

+
+ } Now we can proceed to evaluate c = E + C(1) + C(2) + C(3) x x x x x
First let us consider E(I)
x

=

= - + E { +


+ + + }









Using the expressions for a(I) and a(2)

a() 1<11V a> a*a cI - ca


a(2) = Z 1<1 V k>
l,k~a(Ll -a)(Lk-La)


- E 1<11V a>
IPa (cl1-a)2


- 21 a
14a (cl-ca)


we obtain


- -z { a~ x 1la



(lI - La)


+
(61 - ca)


+ . a~x (LI - La)(Lk - La)
1 ,kfa


+
(L1 - La)(k - La)


(cI - Ca)2


- +
(c 1 - c a) 2


<1IVlk> }
(L1 - La)(Lk - La)


(c1 - La)2



(cI - f-a)2


If at this point we note that c(2) and 43) are zero through third order, we see that by adding c to L(1) and substituting for V, we arrive at the expression for L .


x x + E
ax 1lPa


lI2
l - La


-I2z { a,bx (c1 - Ea)(Lk - Eb) 19,a
kr b
+
(6l1 - a)(k - cb)


+
(cI - �a)(Lk - Lb)

+ }
(cl - a)(k - b)


+aFx{
a~x 1, kta


a; x

- 2-z {
ax 14a







88

+ 3/4 E
atx (cl - ca)(Ek - ca)
l,kra

- 3/4 E
ax (CI - ca)2
1lia
















APPENDIX C
EXPANSION OF ASCF ENERGY IN TERMS OF TOM QUANTITIES

In order to expand the ASCF ionization energy is terms of the TOM orbitals and orbital energies, we shall first expand EN and EN-i in terms of the TOM quantities and then subtract the expressions
x
obtained.

Expansion of EN

In this section quantities with a tilde will represent the N electron state Hartree-Fock solutions. The expression for the total energy is


EN = a-
a a,b

and the perturbation to be used is defined by
FN = FT + ' + E {
- }
x atx

Therefore, the expression for EN through third order is
E N = F[ +C()+6(2) + (3)

a a a a a

- E { + +
a,b

+ + + + + +


+ + +










+ + + }


Using the expression for c (1) to simplify, we arrive at

EN = Z {C + C(2) + (3)} + '2 - Z
a a a a a,b

- Fb{ + +

+ }

Expansion of ENi

The expression for EN-1 is given by
X

N-i
E = E Ea - E ,
atx a,bFx

where the tilded quantities now refer to the N-I electron state Hartree-Fock solutions. The perturbation is now defined by
N-I
Fx =1 F- =T xl Ix> + E [ - }
x x atx

N
Following the procedure used for E , we obtain


EN- = {6a + (2)+ E3)} - - E
arx a,box

2 { + +
a,bix

+ }


Difference EN N-1


In considering the difference between the equations for EN and
EN-1
x, we see the leading terms give and all first order terms
x x
cancel. If we also note that the leading term in the perturbation, VI, is ! for the N electron energy and -! for the N-I electron energy, we immediately see that the four intergral terms of









each expression cancel. The expression for c(2) in terms of V1 is
a
- 12 + { +
la;a ca a - E:a) (ck - cb) (ck - Eb) k; b
From this we can see that V, will appear an even number of times and,
(2)
therefore, the terms in ca cancel.

This leaves only terms in c(3) to be considered. The expression for e(3) is

a ( C
l,k a C- (-1 - Fa)2 Since there is an odd number of VI s in a the terms in the N and N-I electron energy expressions will not cancel as previous terms had but will add together.

The result for the differences is then


-I(ASCF) = EN - EN-1
x

SCx + E < lxlIkx> a (cI - ca)(Ek - ca) l,kra

- 1
a
10a (C1 - ca)'















BIBLIOGRAPHY
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Full Text

PAGE 1

CALCULATION OF ELECTRON BINDING ENERGIES By HENRY ALLAN KURTZ A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OE FLORIDA 1977

PAGE 2

ACKNOWLEDGEMENTS The support and encouragment of my chairman, Prof. Yngve Ohrn, is gratefully acknowledged. During my stay at the Quantum Theory Project, he was of unfaltering assistance. I would also like to thank all memebers of the Quantum Theory Project for providing a very stimulating atmosphere in which to work. Many valuable discussions were held with the faculty, post-docs and, in particular, other students. I would also like to express my great appreciation for the constant support of my wife Bette Ackerman.

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TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii LIST OF TABLES iv LIST OF FIGURES v ABSTRACT vi INTRODUCTION 1 CHAPTER 1 IONIZATION ENERGIES 4 1.1 Discussion of Problem 4 1.2 Relaxation Effects 6 1.3 Correlation Effects 13 1.4 Approximation of the Self-energy 17 1.5 Application to the Water Molecule 20 1.6 Application to the Neon Atom 25 CHAPER 2 POSITIVE ELECTRON AFFINITIES 35 2.1 Discussion of Problem 35 2.2 Analysis of Effects 36 2.3 Application to LiH 39 CHAPTER 3 NEGATIVE ELECTRON AFFINITIES 44 3.1 Resonances 44 3.2 Methods for Calculating Cross Sections 45 3.3 Choice of Potential 49 3.4 Relation to Other Methods Helium Test Case 51 3.5 Beryllium 55 3.6 Magnesium 68 3.7 Evaluation of Resonance Parameters 74 APPENDIX A THIRD ORDER RSPT ON THE ASCF IONIZATION ENERGY 81 APPENDIX B THIRD ORDER RSPT ON THE TOM IONIZATION ENERGY 86 APPENDIX C EXPANSION OF ASCF ENERGY IN TERMS OF TOM QUANTITIES . . .89 BIBLIOGRAPHY 92 BIOGRAPHICAL SKETCH 95

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LIST OF TABLES Table Page 1 Comparison of Third Order Difference Terms for Water (eV) 14 2 14 C6T0 Basis Used for Water 22 3 Water Basis I Results (eV) 23 4 Water Basis II Results (eV) 24 5 Ionizations in 30-40 eV range 30 6 Comparison of Results (eV) 31 7 Neon STO Basis 33 8 Neon Results (eV) 34 9 Li H 19 STO Basis 40 10 LiH 13 CGTO Basis 41 11 LiH Electron Affinity Results (eV) 42 12 Helium Basis Sets 54 13 HelO Results 56 14 He20 Results 57 15 Be Basis Functions 62 16 Be Static-exchange Results 64 17 Be Static-exchange plus Polarization Results 65 18 Magnesium Basis 69 19 Mg Static-exchange Results 70 20 Mg Static-exchange plus Polarization Results 71 21 Resonance Parameters 80 i v

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LIST OF FIGURES Figure P a 9e 1 Plot of Relaxation and Correlation Corrections for Water 8 2 Plot of Relaxation and Correlation Corrections for Furan 10 3 Plot of -I(ASCF) + £jj(E) Versus Energy for Water using Basis I 27 4 Plot of -I(ASCF) + E^.(E) Versus Energy for Water using Basis II 29 5 Plot of Phase Shift Versus Energy for Helium 59 6 Comparison of Helium Phase Shifts with Other Calculations 61 7 Plot of Cross Section Versus Energy for Be 67 8 Plot of Cross Section Versus Energy for Mg 73 9 Plot of Phase Shift Versus Energy for Be 76 10 Plot of Phase Shift Versus Energy for Mg 78 v

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Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALCULATION OF ELECTRON BINDING ENERGIES By Henry Allan Kurtz December 1977 Chairman: N. Y. Ohrn Major Department: Chemistry Contributions from relaxation and correlation effects are examined for ionization energies and electron affinities. A comparison is made of the ASCF and TOM methods showing that they are not the same through third order in Rayl iegh-Schrodinger perturbation theory as had previously been thought. A direct method of obtaining values for ionization energies and electron affinities is proposed which includes the complete relaxation correction and an approximate correlation correction. This method is used to evaluate ionization energies for water and neon and to evaluate the electron affinity of lithium hydride. The results compare very favorably with both the experimental and other theoretical calculations. Negative electron affinities are also studied. These quantities are given by electron scattering resonance energies. A method is derived to compute Harris phase shifts from which the resonance parameters can be obtained. The potentials used are static-exchange vi

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potentials with and without the inclusion of an approximate polarization potential. Polarization effects are shown to be very important in applications to some representative Group 1 1 A atoms. u A. UA J Chairman

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INTRODUCTION The strength with which an electron is bound in a system, referred to as the binding energy, is a quantity which has recieved a great deal of interest in recent years, both from a theoretical and experimental viewpoint. Knowledge of such a quantity can be very useful in the study of chemical analysis and bonding. This fact has been demonstrated e.g. by K. Siegbahn in the development of ESCA (Electron Spectroscopy for Chemical Analysis). The study of electron binding energies can be broken down into the study of two main processes, electron detachment and electron attachment. The quantities measured in these processes are called the ionization energies and electron affinities respectively. The ionization energy is the amount of energy required to completely remove an electron from a system. This can be obtained theoretically from the total energy differences between the N electron (usually neutral) initial state and the appropriate N-l electron state. For atomic systems, this definition is unambiguous, but for molecular systems the ionization energies will depend on the internuclear arrangements. When such a condition exists, there are two main types of ionization energies of interest experimental ly and theoretically. These are known as the adiabatic and vertical ionization energies. The adiabatic ionization energy is given by the energy difference between states with each in its own equilibrium 1

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geometry and the vertical ionization energy is obtained for a fixed nuclear arrangement, usually the equilibrium geometry of the initial N electron state. In Chapter 1 of this work, several methods of use today for calculating vertical ionization energies are explored. An attempt is made to analyze the various improvements and effects presently considered and to propose an alternative method which provides an accurate and efficient procedure by which ionization energies can be calculated. Applications of this method are given for both atomic and molecular systems. The concept of electron affinity, or electron attachment, is very closely related to that of the ionization energy, or electron detachment, in that each can be thought of in terms of total energy differences. Electron affinities are usually defined as the total energy difference between the N electron parent system and the N+l electron final system. This sign convention for electron affinities is such that if a positive result is obtained then the N+l electron system is of lower energy and, hence, stable with respect to the initi system. As was the case for molecular ionization energies, there is the same distinction between adiabatic and vertical processes when referring to molecular electron affinities, with the vertical process still usually obtained at the geometry of the N electron parent. It is of interest to note that for atomic systems, the electron affinity of a given N electron state must be the same as an ionization energy of the resulting N+l electron state. Chapter 2 is devoted to the analysis of methods used for obtaining positive electron affinities. It is shown that the same method that was developed for ionization

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3 energies is also applicable to electron affinities and an example is given of such a calculation. As of yet the possibility that the total energy of the N+l electron final system will lie above that of the N electron parent, thus yielding a negative electron affinity, has not been mentioned. This possibility usually lies well outside the range of methods discussed so far and into the category of scattering processes. For most cases, the idea of a negative electron affinity is meaningless because the N+l electron state is not stable and can, therefore, have any energy desired, depending only on the energy of the free electron. In some instances though, the concept of a negative electron affinity can have a meaning. Such a case is when, due to the shape of the potential which the "free" electron feels, the N+l electron state has a non-negl igible lifetime. The temporary negative ion state formed in this manner is knows as a resonance and can be related to a specific energy value. The energy at which a resonance of this type occurs can then be used as the definition of the negative electron affinity. In Chapter 3 this type of binding is treated. A short review of the theory of electron-atom scattering resonances is given and a simple procedure is developed for the calculation of electron-atom scattering cross sections, from which such resonances can be obtained. This method is analyzed in terms of existing related methods and then applied to some representative Group 1 1 A atoms to determine values of their negative electron affinities.

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CHAPTER 1 IONIZATION ENERGIES 1.1 Discussion of Problem As stated previously, the ionization energy of a system is the energy required to completely remove an electron. This can be given theoretically as the difference between the N electron and the N-l electron system total energies. Experimentally, ionization energies can be determined via photoelectron spectroscopy. In such an experiment the system is bombarded by photons of a known energy, or equivalently frequency, and the kinetic energy distribution of the ejected electrons is measured. The electron binding energy I (ionization energy) can then be computed by Einstein's relation I = hio E , e where co is the incident photon frequency and E g is the ejected electron's kinetic energy. Both adiabatic and vertical energies can be detected by this technique. The adiabatic energy is measured from the band head in a spectrum and the vertical is measured relative to the band maximum. Several good reviews of photoelectron spectroscopy have been given by Siegbahn et al . (1967 and 1969), Turner et al . (1970), Eland (1974), and Price (1974). One of the earliest and still most widely used methods for obtaining theoretical vertical ionization energies is known as Koopmans' theorem (Koopmans, 1933). The "Koopmans 1 theorem" states that the ionization energy is given by the negative of the Hartree4

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5 Fock orbital energy, -e. It can be derived by first assuming a Hartree-Fock description of the N electron system and then allowing the same orbitals to approximate the N-l electron state Hartree-Fock solutions. Discussions of the errors involved in the "frozen orbital" method have been given by Mul liken (1949) and Lowdin (1955a). One seemingly obvious way of improving an ionization energy calculation is by improving the methods used to calculate total energies. Approaches of this type are called indirect methods since the quantity of interest (the ionization energy) is obtained from the results of two separate total energy calculations. Ab initio methods commonly used to calculate total energies for indirect methods include the Hartree-Fock method, Configuration Interaction, MC-SCF, or a variety of perturbation theory methods. All of these have the drawback that they require a high degree of accuaracy since, in the last step, two very large numbers must be subtracted to obtain what is usually a quite small number. There also can be other even more serious problems with some indirect methods in ensuring that each state is calculated within the same level of approximation. This problem is addressed in greater detail in a later section. A method capable of calculating the ionization energy directly via a single calculation would circumvent most of the problems mentioned for indirect calculations. Such methods are called direct methods. The two most common direct methods are the transition orbital method (Goscinski et al., 1973) and the electron propagator, or onebody Green's function (Linderberg and Ohrn, 1973). In following sections, the relationships between these methods and the more common indirect methods are discussed.

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6 1.2 Relaxation Effects In the Koopmans' ionization approximation, the orbitals obtained for the N electron state are assumed to remain unchanged by the ionization process. Clearly the potential felt by the remaining electrons is now different and the orbitals should be allowed to change (via reorganization or relaxation). One method of obtaining these reorganization effects is to perform an independent Hartree-Fock calculation for the N-l electron state. The procedure by which the ionization energy is obtained from separate Hartree-Fock calculations is known as ASCF (Bagus, 1965). This technique can be used to define what is called the relaxation correction to Koopmans' theorem as the difference between the ASCF result and the Koopmans' result. It should be mentioned at this point that the Hartree-Fock calculations referred to in this work are unrestricted Hartree-Fock calculations. Other types of Hartree-Fock methods, such as Roothaan's (Roothaan, 1960) or a projected Hartree-Fock (Lowdin, 1955b), may give different results and other interpretations from those discussed here. Further non-relativistic corrections to the Koopmans' results not included in the relaxation correction are grouped together and called correlation corrections. These corrections can then be defined as the difference between the non-relativistic experimental result and the ASCF results. Figures 1 and 2, for water and furan respectively, show the relative importance of the relaxation and correlation corrections over an energy range from core ionization to outer valence ionization energies. It is evident from these figures that for core ionizations, the relaxation corrections can be quite large. In fact, they account for almost the entire difference between

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in c -o o c : •Ira +-> u CD CD 4-> CD JZ > 4s1— •rCD S+-> i — o L) u • CD CD >> CLJZ sz CD tn 4-> o CD •r— CD Sd 4-> C O 03 CD CD i — _£Z c CD C 4-> o • So •r+-> s•r— S4-> JZ o +-> 03 03 CD o 03 CD i — •i— N £Z CD s~ "O •r— Sc c C sCD 03 o £ o Jd •r— O u +-> c _c o _c in CD £Z • 1 — o JZ O +-> 03 CD 4-> 03 CD c X 03 _cz o 03 4-> 4-> •1 — r — 03 in •i — 4-> CD cz > 03 S~ So X cu *1 — in 03 4— 4-> 1 > CD i — O 03 o •i — CD 3: CD cn d -P s~ d O ssC D CD i — o o sz JZ Q_ 4o CD -P CD d zs CD

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8 (/\3) SU014D3*U03 Energy (eV)

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to c T3 CD o c S•I— 03 (D 4-> o CD CD 4-3 CD SZ > 4to •i — i — Z3 o CJ i — cj • CD CD 03 >> Q-JC > c cn CO 4) o CD CD • •r— CD Sc JZ . — >. -M c o 1 — r^ 03 CD > — S•t— S4-> JZ o 4-> 03 03 CD SCJ 03 a) i — •i — CD N c CD SCD T3 •r— C C c S +-> •( — o i — 03 to •r4—3 SCD c > 03 4S~ C o X 03 •r— CO 03 "O 4CD i — CD O 13 CJ •i — CD C 4CD CD S•r— 4-> SS03 O S-. SCD CD 4-> i — o o C JZ _Q O4u CD 4-> O OJ CD S=3 CD •i— Li_

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10 o IT) (A3) * suotq.D0jj.o3 25.0 290.0 540. Energy (eV)

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11 the experimental and Koopmans' ionization energies and, therefore, the ASCF ionization energy will be very accurate. Although small the correlation contributions may be also important for the core in the calculation of chemical shifts. In the intermediate and valence regions, the correlation effects can be as important as the relaxation effects and possibly more important. Therefore, in these regions the ASCF ionization energy may not be very good. It is very interesting to note that for the systems examined here, the relaxation and correlation corrections for the outer valence level have almost equal magnitude but are opposite in sign. In cases such as this the original Koopmans' energy will be a better approximation than ASCF to the ionization energy. To arrive at an explicit expression for the relaxation contribution, Rayleigh-Schrodinger perturbation theory can be used. The details of this procedure and its resulting expressions are given in Appendix A. So far, only the indirect ASCF method has been discussed for calculation of ionization energies with the inclusion of relaxation effects. There is also a direct method designed to include the major relaxation contributions known as the transition operator method (TOM) (Goscinski et al . , 1973). The ionization energy is given by this method as the eigenvalue, e , of the "transition operator" A f" 1 " = h + Z + % . x a To obtain an expression for the TOM correction to Koopmans' theorem which can be compared with the ASCF expression, Rayleigh-Schrodinger perturbation theory can again be used. Goscinski et al . (1975) pointed out that when perturbation theory is used to expand the transition

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12 operator eigenvalue using the N electron ground state Hartree-Fock solutions as a reference, then the expressions obtained will differ from the ASCF expression in third order. But if the N and N-l electron state total energy expressions are expanded with the TOM solutions as a reference, the ASCF expression will be equal to the TOM value through third order. To demonstrate the techniques used and to check the results of Goscinski et al . (1975), both of these procedures were carried out. The first method, expanding the TOM eigenvalue, is demonstrated in Appendix B. By comparing the results obtained with the ASCF expression of Appendix A, the difference through third order is ASCF -T i = kZ <1 x | I kx> a 1 , k*a (E r E a )( V £ a> _ j, £ l 1 ,fc*a (^l“^a)(^k”^a) j, ^ a l*a 2

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where the tildes refer to TOM quantities. This result is clearly in contradiction with the result of Goscinski, who states that such a difference should be zero. It should be noted that if all the TOM orbitals and energies in the above expression are expanded in terms of the N electron state Hartree-Fock orbitals and energies, and then all terms of order higher than third are discarded, we arrive at the first expression obtained above. Even though this work shows that the ASCF and TOM ionization energies are not identical, they have been found in the past to lie very close numerically. This should mean that the difference terms found here are fairly small. To demonstrate that this is true, in Table 1 the third order difference terms expressed in the Hartree-Fock orbitals are compared with the TOM and ASCF values obtained for the water molecule. For water, the difference terms are clearly small and will not cause a significant effect, with the possible exception of the la^ state. 1.3 Correlation Effects Both of the previous methods discussed, TOM and ASCF, have included what was called relaxation effects but neither method was designed to take into account the remaining effects, called correlation effects. Correlation effects derive their name from the fact that in the Hartree-Fock approximation an electron sees the average, not the instantaneous, potential due to the other elections. This means that there is no correlation between the motions of the electrons. As was shown in Figures 1 and 2, these neglected effects in theoretically determined ionization energies can be substantial, especially for the valence region.

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14 Table 1 Comparison of Third Order Difference Terms for Water (eV) Ionization ASCF a TOM 3 r< ^ order la i 2a i 540.8 540.64 0.186 34.6 34.51 0.008 lb 2 17.8 17.78 0.013 3a i Ibi 13.0 12.86 0.020 11.0 10.88 0.020 a ASCF values from Goscinski et al . (1975)

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15 There are several methods designed to include correlation effects in the calculation of total energies, thereby leading to indirect methods of calculating ionization energies. Perhaps the most widely used of those is the method of configuration interaction (Cl). A very good review has recently been given on this subject by Shavitt (1977). Calculating energy differences between states, particularly between states with different numbers of electrons, can be a very difficult matter using Cl. In such cases, not only does one have to worry about choosing an orbital basis but a choice must also be made of many-electron functions (determinants or projected determinants) to be used. If possible, it would be preferrable to use all the many-electron functions that could be generated from a chosen orbital basis. This would then guarantee that all possible correlation (obtainable with the orbital basis) had been included. Unfortunately, such Cl's would be so large that they are impossible to perform in almost all cases of interest. Therefore, one must make selections of the many-electron basis resulting in approximations to the correlation energy. If extreme care is not taken in selecting the functions then the correlation effects may not be included for each state in a balanced manner. In recent years, a direct method for calculation of ionization energies has emerged in quantum chemistry. This technique is known as the electron propagator (Linderberg and Ohrn, 1973). Reviews of this subject have been given by Hedin and Lundqvist (1969) in solid-state physics, Csanak et al. (1971) in atomic physics, Ohrn (1976) and Cederbaum and Domcke (1977) in molecular theory. To illustrate how

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ionization energies are obtained, a short survey of some of the more important formulas and aspects of electron propagator theory will be given. The electron propagator in spectral representation is given as G (E) pq 1 im E( n+0 s Mp) ff(q) E + TJ+T + in 9s(P) 9s(q) E + EN-1 E N in where and f s are the overlap amplitudes defined by 9c(x) = S g.(i)-u(x) with g (i) = ° 1 j b I and f s (x) = ? f s (i)*u(x) with f g (i) = . These amplitudes can also be calculated via Cl and could possibly give a mode of comparison between the two methods which is more sensitive than an energy criteria (Kurtz et al . , 1976). From the above expression for G(E), one can see immediately that the ionization energies are obtained as the poles of G(E) or the zeros of G~^(E). One way of obtaining G is by solving the Dyson equation G = G 0 + G°ZG , which can be rewritten G -1 = (G 0 )" 1 Z . In these equations G° is the propagator obtained for the unperturbed reference state. Usually this is the Hartree-Fock state and

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17 (G°" 1 ) . 1J (E £ • )6 . . 1 1J which leads to ij ' (E E i )a i: z ij (E) ' The terms £• -(E) are matrix elements of what is known as the sel f-energy. If we assume we have an expression for I^(E), then the problem is to solve for the zeros of the above energy dependent equation. One method of solving the equation is by a direct poleresidue search as described by Purvis and Ohrn (1974). If only the diagonal element of G -*is considered, it is evident that for E to be a solution it must obey the following relation: E e. j + ^ -j i ( E ) • Solutions of this equation will also lead to improved values of ionization energies and this method has become known as the "quasiparticle" method. 1.4 Approximation of the self-energy As was shown in the last section, to obtain improved ionization energies with respect to the Koopmans 1 value, an expression for I . • ( E ) is needed. A general expression for the self-energy has been ' *J given by Purvis and Ohrn (1974) in terms of a superoperator formalism (Goscinski and Lukman, 1970) as I( E) = (a|Hf)(f|(EI H)|f) _1 (f|Ha) The superoperators I and H are defined for any operator X by IX = X and HX = [X,H]_ (where H is the second quantized Hamiltonian)

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18 and the binary product is defined as (X n |Xj) = , where |N> is a suitable reference state. If the manifold f is chosen to be the set of fermion-like ‘f* + operators {a^a^, a b a p a q}> where the field operators are associated with the Hartree-Fock spin orbitals, this leads to E. .(E) = ig Z E (M _1 'J apq a p'q* apq.a'p'q h e e abp p' a 1 (M -1 ) abp ,a ' b 1 p ' : j P ' II a ' b 1 > . A /\ where M = (f | (El H)|f). In the expression for M, the Fock superA operator, F, can be substituted for the Hamiltonian superoperator, A H. This is known as the Born collision approximation and causes M to be diagonal. The resulting self-energy is E. .(E) = % E ij a pq !+ p p r a b p b q + J 2 E p ab E + e p ' £ a " £ b Setting E = in the expression for E^ . (E) gives E = £i + E ii (e i ) , which is just the Rayleigh-Schrodinger perturbation theory expression for the correction to Koopmans' theorem. The alternative way of improving the Koopmans' value using the expression for E^. (E) was called the "quasi-particle" approximation and has been applied using

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the second order expression given above (Hohl neicher et al . , 1972). Both of these methods are well-known and give reasonable values for valence electrons. Unfortunately, neither of the two previous methods are known to give accurate values for ionization energies of core electrons. The main reason for this is that the relaxation contributions are very large for core electrons and can not be adequately described by second, or even third, order perturbation theory of this type. It was noted in second order (Pickup and Goscinski, 1973), and then extended to third order (Born et al . , 1978), that E^( ) can be conveniently separated into the Rayleigh-Schrodinger expressions for the relaxation and correlation corrections to Koopmans 1 theorem. This can be represented as e. + l I?.(e.) + Z L .(e.) n v i ' n v i The first two terms in the above equation can be identified as the Rayleigh-Schrodinger approximation to the ASCF ionization energy given in Appendix A. If instead of the approximate value, the exact value is used, the following equation is obtained E = -I (ASCF) + ^.(e.) . This expression gives the correlation correction to the fully relaxed ionization energy and is the same in second order as the one given by Purvis and Ohrn (1976). This equation should now overcome the problem with the simple Rayleigh-Schrodinger expression and should provide excellent core ionizations. Since -I(ASCF) is just a numerical value, it is possible to approximate it with the TOM

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20 value. Even though the TOM ionization energy can be obtained from a single calculation, there is not any direct computational advantage in using it since a ground state calculation must always be done in order to obtain reference orbtials and energies. An alternative approach to the one above is arrived at by making the relaxation-correlation separation in the original energy dependent expression for £, Z..(E) = E R .(E) + Z C .(E) . iJ U If it is assumed that e.. + E R .j(E) can still be approximated by the ASCF ionization energy, the following equation for E is obtained E = -I (ASCF) + Z?.(E) . This approximation will be referred to as the "fully-relaxed quasi-particle" approximation by analogy to the previously mentioned "quasi-particle" approximation. The expression for the second order correlation correction is z9 . (E) n ' ' a?*i ^ p»q + e= £ n e. a/i b*i P 1 | E + Cn " £ Ct 1.5 Application to the Mater Molecule In order to make a comparison of the approximations to the ionization energy which have been discussed, they are applied to the water molecule. For this comparison two different basis sets were used to describe the molecule. The first, called basis I, is

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21 a 14 CGTO combination of Dunning's oxygen basis (Dunning, 1970) and Huzinaga's hydrogen basis (Huzinaga, 1965). This basis is shown in Table 2. The other basis, basis II, is comprised of basis I plus a set of d-type GTO's on the oxygen with unit exponents and a set of p-type GTO's on each hydrogen, also with unit exponents. This augmented basis is used to give some check on the basis set effects of the approximations and to include functions which span all four irreducible symmetry representations of the C 2v molecule. The results obtained with each approximation are shown in Tables 3 and 4 for basis I and II respectively. In the quasi-particle expression for the ionization energy, E = ci + E ii (E) , if the "exact" self-energy expression is used, information about all ionizations can be obtained, not just the principle value related to the i Koopmans 1 energy. Even in the relaxed quasi-particle method r with a second order approximate E , other ionization energies can be found with varying degrees of accuracy. These are approximate values for the "shake-up" processes which correspond to simultaneous ionization and excitation. The method used to solve the relaxed quasi-particle equation for all the states listed in Tables 3 and 4, except the 2a^ states, is simple iteration with the Koopmans' value as a starting point. Convergence to 10 ^ Hartrees was always very quick. For the 2a^ states, though, the iterative method converged to a value different than the one listed or it diverged. To explain this apparent anomaly r it is helpful to look a a plot of E versus -I(ASCF) + EV. (E) given

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22 Table 2 14 CGTO Basis Used for Water orbital exponent coefficient sH 17.370 0.032828 2.6273 0.23121 0.58994 0.81724 s'H 0.16029 1.0 sO 7816.5 0.002031 1175.8 0.015436 273.19 0.073771 81.17 0.2476 27.184 0.61183 3.4136 0.2412 s'O 9.5322 1.0 s~0 0.9398 1.0 CO O 0.2846 1.0 pO 35.183 0.01958 7.904 0.12419 2.3051 0.39473 0.7171 0.6273 P'0 0.2137 1.0

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23 JZi CM CM LO LO CL X O CM CO •^t* CM LU *3CO 1 1 i H i — l LO 4-> LO CO CO 00 2C co LO LO LO O rd 1— CL O CM CO CO 1 — 1 1 «dco i — 1 1 — 1 1 — t oi LO 4-> Lu O CO r-M co oo O rd OD CL H CM CO CO t — 1 ho co co 1 — 1 CD *o fO H H r—H ** — c LO CM m i — s=3 SCO L0 o Lu o r — co co CD o O CD CC oo H ^ co CO » — ( r— TD <1 t — I JO » — i C LO rd CM t— L0 •r — 00 Srd CD CQ ~o NCM CM 1— «-h o lo LO So Q. CD 00 CT) CO CO CM o 4-> TD CL CM CO r-H r— 1 r-H rd C LO CM 1 — 1 co LO co LO LO co oo o 1— o r-CM o OO T 1 r-H 1 — 1 LO rd Lu O 00 <1 CD +-> ro 4-> LO co LO 00 O O o CO r—H CO r-H r-H r-H -N LO LO cr> r — 1 > — xCM OO co o LO • • • • • • 1 cr> CT> LO co rd LO co i — 1 r-H r-H ^ -* LO -4-> CM CD CT, • 1 — r-H * 1 — r-H CM r-H r-H (/) rd rd JD rd JD £= 4— r—H CM » — I CO r—H •r:o o (/) O r— CD C
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24 rc CL X LxJ C\J C\J UD fN UD O (\J CO C\J *vt" OO r—4 rH \ — I LO 4-> CO CD i— H LO ? LO OO LO CXI cr> o 03 1— CL o CXI CO T— H 1 OO 1 — 1 r-H r-H C r LO 4-> (XI r— H C\J OO CTi Ll. C. OO OO O c_> 03 • • • • • CO CL o CXI co vf (XJ > u o i — =3 cLO c. cn co * 3 o LO CD o Ll. lo t-H oo O a: CJ CJ CD LO O CXI CXI i — h -4 T D ex on so Cl. CD OO CX> r — 4 oo o 4 -> TO cc CXI OO t-H i — 1 T 1 03 c LO 3 CXI LO CXI oo LO OO CTi o o i— o OO oo o OO T 1 « t r-H LO cn lo OO LO CO Li_ co cr> i — < co CJ OO c OO oo o cx ^ lo CO CO CO CD 14D ^ I o'* lo cr» lo co LO OO t-H »H t — I LO C\J r^ CO CD 4-> 03 4 J LO 1 — < T— 4 C\J ' — < C\J 03 03 -Q 03 -Q HCCIHOOH :o 1 < fl 3

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25 if Figure 3 for basis I and Figure 4 for basis II. It can be seen C that because of the poles of E-j-j(E) which exist in the region shown, there are actually three solutions. Two of the solutions are quite close (one of which is the primary solution) and this is the source of the problems in the iterative procedure. Table 5 shows the values obtained for the principle ionization and two shake-ups in the region. Table 6 shows a comparison of the results of this work with those of Cederbaum et al. (1973) and Hubac and Urban (1977). The results of Cederbaum were obtained by solving the Dyson equation with second and third order self-energies. The basis used was the same as basis II except the hydrogen p-like GTO's had exponents of 0.75. The results of Hubac and Urban are Rayleigh-Schrodinger results for both second and third order using a basis similar to basis I. The results of the ASCF-relaxed quasi-particle method in second order are clearly better than the previous second order results and compare favorably with the third order results. It is worth noting that neither Cederbaum nor Hubac and Urban have reported any results for the la^ ionization where relaxation effects dominate. 1.6 Application to the Neon Atom As a further test of the methods discussed for the calculation of ionization energies, the neon atom is chosen. Since it is an atom, it has the advantage when comparing with experiment of not having vibrational fine structure. There has also been a great deal of related work done on neon (Purvis and Ohrn, 1976) and, therefore, it provides a good check. The basis set used for the

PAGE 33

CO c d o d CD •r— O “O 4-> 4ST3 O O CD (D >> X CO CD "O 03 d Sd i — 0J CD O CD 4-> d O d d CD CD 1 •1 — to U_ CO o CD =3 s~ CD • CD CD > — < _c d +-> 03 CO UJ *r"O CO c d *rn3 03 o Or-JD •1 — W "O 4-> JZ CD 3 + 4-> to i — • •r— 3 O CD ^ ' > to d Ll_ to •r— CJ T3 03 CD i — UO CD 3 i — < d > "r~ Ln i — i ro CD 4-> 1 4-> d d _Q CD 03 CD 4O d CL _d O CD 1 -M S1 •i — +-> CD 4to _£Z O 4-> i — 03 4~> • — 03 cl' 3 • i — Q3 to cr 3 CO CD s3 CD Ll_

PAGE 34

-1.15 27 LO I — I I (3)^2 + (J3SV) IEnergy (Hart.

PAGE 35

to sc; CD o S"O •r— o -4-> 4— O •o CJ CD CD >5-0 X to CD C= 03 sSo ( — CD CD U CD 4-3 C CD c CD tO 1 •1 — Ll_ to c O CD Z3 U1 sz CO <\ 4-> S~ • CD » — » CD CD > • .c S4-> 03 " — • to UJ •«TD to ^ to C c •1— 03 03 o Ot-JD •rW -o 4-> JZ CD Z3 4-*-> to i — • •rZ3 O CD CO C L u to •i — C__> “O 03 CD i — t/3 CD 5 i — <1 cz oo ** — " •( — >> •I — LO 1 — i fO cn+J 1 4-> Ss~ -Q CD 03 CD 4O £Z CL JZ O CD 1 4-> S~ 1 •i — 4-> CD 4CO JZ O 4-> r— 03 M i — 03 CD 3 •i — Q^ c /) cr 2 ^ 3 * CD %Z3 cn u_

PAGE 36

29 ( 3 ) ( 30GV ) I Energy (Har

PAGE 37

30 Table 5 Ionizations in 30-40 eV range designati on basis I basis II principle 2a ^ shake-up shake-up 32.76 34.62 36.39 32.31 33.66 36.96

PAGE 38

31 Table 6 Comparison of Results (eV) State Ced2 a Ced3 b HU2 C HU3 d BI e BI I f 2a l lbo 32.93 35.10 33.38 35.22 32.8 32.31 17.70 19.22 17.93 19.42 18.7 18.72 3a-. 13.18 15.18 12.62 14.74 13.8 14.33 1 b l 10.92 13.03 10.48 12.75 11.8 12.09 a Second order results of Cederbauin (1973). b Third order results of Cederbaum (1973). ^Second order results of Hubac and Urban (1977). d Third order results of Hubac and Urban (1977). •“ASCF-relaxed quasi-particle results with basis I. ' ASCF-rel axed quasi-particle results with basis II.

PAGE 39

32 neon calculations is shown in Table 7 and the results obtained are shown in Table 8. As can be seen, excellent results were obtained for both the second order corrected TOM and the TOM-relaxed quasi-particle approximations. These results are also compared to the second order corrected TOM approximation of Purvis and Ohrn (1976) and they compare very well. The discrepancy in this work for the 2s level can be explained as a basis set effect. This discrepancy is removed in the work of Purvis and Ohrn (1976) where they used the same basis except that a 2s orbital replaced the Is (8.91407) orbital.

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33 Table 7 Neon STO Basis orbital exponent Is Is Is 2s 2s 2s 2s 2p 2p 2p 2p 3d 8.91407 9.5735 15.4496 7.7131 4.7746 2.8462 1.0000 4.4545 2.3717 1.470 0.750 2.80

PAGE 41

34 -Q : o Q_ CO 03 CXJ C\J i — I O 03 CO * — * CO C\J CO CXJ « — i CO • r-H CV1 LO CL • • X CD CO i— H LU CO ^0 CXJ CO , X 03 • N CO CO r— -P ONLO CD S• • • 03 03 r-H 1 CL CO CsJ !>* | oo — o cr > hCD v — CO CO -P CD i — i — ZJ CD O N «sf cc _Q CO a hO CO LO 03 CD S• * . 1— cc o • 03 t — i SCO *vf CXJ c -O L CO o sz o CD CXJ o iZ cxi oo CO H LO o CO 03 o f— co cxj CO CO » — 1 o sO Z5 ^ CL ro -Q

PAGE 42

CHAPTER 2 POSITIVE ELECTRON AFFINITIES 2.1 Discussion of Problem A positive electron affinity represents the decrease in total energy a system attains with the addition of an electron. Analogous to ionization energies, there are two different types of electron affinities, vertical and adiabatic. The adiabatic electron affinities are also known as the thermodynamic electron affinities. Methods of obtaining electron affinities can be broken down into three classes: experimental, theoretical, and empirical fitting. There are numerous methods which fall into the classes of experimental and empirical fitting and a discussion of these is outside the scope of this work. A very good discussion of many of these methods is given by Steiner (1972). One problem common to almost all methods, theoretical and nontheoretical, is the very small size of electron affinities. The largest known electron affinity is 3.82 eV for CN (Berkowitz et al . , 1969). This presents special problems for the indirect theoretical methods where the N and N+l electron total energies are used. In the next section, an analysis of the contributions to the electron affinity is made in a similar manner to that used for ionization energies. An attempt is made to obtain explicit expressions for these contributions as before. 35

PAGE 43

36 2.2 An al ysis of Effects The simplest theoretical method for calculating electron affinities is Koopmans' theorem. The theorem is the same as it was for ionization energies, except now the negative of the first unoccupied orbital energy is used. Relaxation and correlation corrections can again be defined in terms of the ASCF electron affinity. The perturbation expressions derived in Chapter 1 for ionization energies can be applied to electron affinities if the N+l electron state is used as the reference state. Clearly this is not what is desired. It would be most useful if expressions in terms of the N electron reference state were obtained so that the corrections to the Koopmans 1 electron affinity could be calculated with the same quantities as are used to calculate corrections to the Koopmans' ionization energy. To find an expression for the relaxation correction to Koopmans' electron affinity, Rayl eigh-Schrodinger perturbation theory is again used. The following treatment will parallel closely what is done in Appendix A. The expression for the ASCF electron affinity is -EA(ASCF) = E N+1 E N HF,x HF = e + E{e e } E JsE{ }. x a a a a a ,b In the above expression and the following derivation, the "tilded" quantities will refer to the N+l electron state. When these N+l electron state quantities are expanded in terms of the N electron state quantities and only terms of third order or less are kept, the

PAGE 44

37 following expression is obtained -EA(ASCF) = e + + e ^ xx x x + E {e^ + e^ 2 ) + e^ 3 h a a a a g ( + + + + + + + + + + } 1 { + + d , D + + } hi { + a ,b +
+ + E{ } x a By using this perturbation to find expressions for and cl X it can be shown that they will cancel many of the integral terms as they did in the ionization expression. It can also be easily shown that and e^ 3 ) vanish through third order, x x This leaves

PAGE 45

38 -EA(ASCF) = e + £{e^ + e^} %£ {
X a a a a ,b + + + ob^ 1 ^ | |a^b>} By inserting into this expression the RSPT equations for the orbital and orbital energy corrections and simplifing, the following result is obtained -EA(ASCF) = e x E 1 | 2 a (c k e a ) k*a + ?i£ a,b k*a l*b { <1 x | |bx> (c k ' e aHei “ e b ) < kx | |ax> + (c k e a )(e-| e b ) <1 x | | bx>< kx | |ax> (e k £ a ) + E • a k , 1 *a < 1 x | | kxxkx | |ax> < E 1 ' E a )(e k E a> <1 x | | ax> a (e-j e a ) 2 l*a If this expression is now compared to the one obtained for -I(ASCF) in Appendix A, it is seen to be identical except for the signs on some of the terms. It is interesting to note that the number of times the perturbation appears in a term can be correlated with

PAGE 46

39 the sign difference. If appears an even number of times, the signs are different, and if it appears an odd number of times, the signs are the same. It is fortunate that this sign change occurs in the ASCF expression, at least in second order. By subtracting the second order relaxation expression from the full second order self-energy obtained previously, the correlation correction to the relaxed electron affinity is obtained. It is given by r C _ ^ y I r . .. ^ | | 2 L xx 2 ^ ah d £+£-£-£ a,u £ + £ £ £, p,q*x x a p q p* x x p a b It should be noted that this is the same correlation expression that was used to obtain ionization energies. The only difference is that now the virtual orbtial sums are restricted instead of the occupied sums. This means that the program written to evaluate the RSPT correlation correction and the relaxed quasi-particle correction for ionization energies can also be used to obtain the similar quantities for electron affinities. 2.3 Application to LiH The molecule LiH was chosen as a test case mainly because it is well characterized by previous work (Jordan et al . , 1976). Two differenct basis sets were used to perform the calculations at an equilibrium distance of 3.015 a.u. One basis is a 19 STO basis, shown in Table 9, and the other is a 13 CGTO basis, shown in Table 10. The results of the calculations are shown in Table 11.

PAGE 47

40 Table 9 Li H 19 STO Basis orbital exponent Is L i IsLi 2s L i 2sLi 2sLi 2sLi 2poLi 2poLi 2paLi 2poLi 2poLi 2pTrLi 2piTLi IsH IsH 2sH 2sH 4.6990 2.5212 1.2000 0.7972 0.6000 0.3000 2.7500 1.2000 0.7369 0.6000 0.3000 0.7369 0.3500 1.5657 0.8877 1.3765 0.4000

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41 Table 10 Li H 13 CGTO Basis orbital exponent coefficient sLi 642.41895 0.00214 96.79849 0.01621 22.09109 0.07732 6.20107 0.24579 1.93512 0.47019 0.63674 0.34547 sLi 2.19146 0.03509 0.59613 0.19123 sLi 0.07455 1.00000 s Li 0.02079 0.39951 0.00676 0.70012 sLi 0.08948 1.00000 pLi 2.19146 0.00894 0.59613 0.14101 0.07455 0.94535 pLi 0.08948 0.15559 0.02079 0.60768 0.00676 0.39196 sH 18.73110 0.03349 2.82539 0.23473 0.64012 0.81376 sH 0.16128 1.00000

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42 Table 11 LiH Electron Affinity Results (eV) basis KT a ASCF TOM 2 nd corr. ASCF ASCF-rel ax q-part. 13CGT0 0.204 0.237 0.236 0.256 0.256 19ST0 0.198 0.238 0.287 0.287 a Koopmans' theorem value.

PAGE 50

43 These results agree very well with the result of 0.2936 eV of Jordan et al . (1976) using Simon's EOM method (Simons and Smith, 1973). The fact that the STO result is better than the GTO result is evidence that the STO basis is a much more complete basis in the region of the maximum of the charge distribution of LiH". As another check on the quality of the basis sets, the dipole moment was calculated from the Hartree-Fock orbtials. A value of 6.11 D was obtained for the GTO basis and 6.02 D for the STO basis, compared to an experimental value of 5.88 D.

PAGE 51

CHAPTER 3 NEGATIVE ELECTRON AFFINITY 3.1 Resonances As was mentioned previously, the study of negative electron affinities is embodied in the study of electron scattering. In recent years, considerable amounts of work have been done on the calculation of resonances in electron scattering with both atomic and molecular targets and several excellent reviews have been given by Burke (1968), Taylor (1970), Schulz (1973), and Nesbet (1975). Following the definitions of Taylor (1970), resonances (also called "temporary negative ions" or "compound states") can be separated into three main classifications; core excited types I and II and single particle resonances. Core excited type I (C.E. I) resonances are also called Feshbach or hole-particle resonances. Such resonances are formed when the incoming electron excites a target electron into an excited orbital, lessening the screening of the nuclear field. The electron then gets trapped in this potential and the resonance acts as a bound state relative to the excitation threshold. These resonances can be viewed as being caused by the excited state having a positive electron affinity. C.E. I resonances are usually the narrowest, thereby longest lived, of the three types. Core excited type II (C.E. II) resonances are very similar to C.E. I resonances. The difference is that they lie above the 44

PAGE 52

45 excitation threshold. The resonance is formed by trapping the incoming electron inside a centrifugal barrier set up by the combination of its angular momentum component and the potential well induced by the electron when it excites and polarizes the target. Since such a resonance is caused by the shape of the potential, it is given the classification of a shape resonance , and because it depends on the l(l+l)/(2r 2 ) term in the potential, we expect to see resonances only for p and higher partial waves. The final type of resonances are called single particle resonances. They, like C.E. II resonances, are shape resonances . The difference from C.E. II resonances lies in that there is no target excitation. Energetically, single particle resonances lie just above the ground state and well below excitation thresholds. It is these single particle resonances that can be identified as defining the negative electron affinity of a system. Until recently, this type of shape resonance has been very difficult to detect, both experimentally and theoretically. In the last year or so, Burrow et al . (1976) have performed low energy electron transmission experiments in which they have detected such resonances in the Group 1 1 A elements Mg, Zn, Cd, and Hg. 3.2 Methods for Calculating Cross Sections In order to detect broad single particle shape resonances theoretically, the electron scattering cross section must be calculated. Most methods of use in scattering calculations today can be divided into two main types, numerical integration methods and algebraic methods. The numerical integration methods are exemplified by the

PAGE 53

46 close-coupling method (Burke, 1968). The subject of algebraic methods has been extensively reviewed by Truhlar et al . (1974) and Harris and Michels (1971). For bound state calculations an analogous classification of methods can be made. In this case, the numerical Hartree-Fock method (Froese, 1963) is the analogue of the close-coupling method and the matrix Hartree-Fock methods are the analogues of the algebraic methods. Experience from bound state problems has shown that as the complexity of the problem is increased in going from atomic to molecular systems, the algebraic methods are much more useful than the numerical integration methods. It is anticipated that a similar trend will exist in scattering problems as the complexity increases, for example by having many coupled equations to solve or by using non-spherical potentials. The early algebraic methods were given by Hulth£n (1944), Kohn (1948), and again Hulthen (1948) (also given by Rubinow (1955)). All three of these methods had the problem of giving spurious resonances, called pseudoresonances. In 1967, Harris caused renewed interest with the proposal of a method for single channel scattering (Harris, 1967). Nesbet (1968) was the first to analyze the source of the spurious results in the early methods and proposed the anomaly-free method for single channel scattering. The anomaly-free method was later extended to multichannel scattering (Nesbet, 1969) and also Harris and Michels generalized the Harris method and extended it to multichannel problems in the minimum-norm method (Harris and Michels, 1969). The method chosen in this work for the calculation of shape resonances is the simple Harris method. The reasons for this choise are brought out in the following brief derivation of the method.

PAGE 54

47 As has been demonstrated (Hedin and Lundqvist, 1969) the wavefunction for the scattered electron is a solution of the formally exact equation h(x)t(x) + /£(x,x' ;E)V(x' )dx 1 = Ef(x) , where T, is a complex nonlocal potential known as an "optical potential" and h is a single particle operator composed of the kinetic energy and nuclear attraction operators. Since only electron-atom scattering is to be considered, the central potential allows a separation of variables in spherical polar coordinates to be made. By also making a partial wave expansion of T (Roman, 1965), the above equation reduces to the following radial equation { dr 2 + + v ( r ) }< M r ) = E( h( r ) ’ where E = hk 2 with k being the momentum of the scattered electron. For conveniance, the notation di 10 + 11 L . 2 + d r^ 2r “ + V(r) is used. It should be kept in mind that L, through V(r), is in general nonlocal, energy dependent, and complex. The function is assumed to be of the form s-j + tc-j + Sb-jU^ , where s-j and c-| asymptotically are solutions to the radial equation with V = 0, namely the regular and irregular spherical Bessel functions j -] ( kr ) and n-|(kr). The explicit forms of s-j and c-| are discussed later. 2 The set of functions { u ^ } is any suitable L set of basis functions which go to zero both asymptotically and at the orgin. By using

PAGE 55

48 this form for , t can be identified as tan^ , where 5-j is the phase shift of the 1^ partial wave. In order to solve for 6-| , the condition that the projection of (L E)i on the set of functions vanish, ( u n 1 L E 1 4>-j ) = 0 for all , is required to be satisfied. Upon substitution for , this leads to ( Ui |L E | s-, ) + t( Ui |L E|c-|) + Zb J -(u i |L E|uj) = 0 . A vast simplification can be obtained by choosing the functions to be eigenfunctions of the operator L. This assumption then gives ( u i | L E | S] ) + t ( u -j | L E | c -, ) + b i (E i E) = 0 . By fixing the value of E to be equal to the eigenvalue corresponding to u^ and rearranging, the sought after expression for 6-j is obtained as ( u i | L E -j | c i ) This is the expression given for the Harris phase shift (Harris, 1967). One major advantage in calculating Harris phase shifts, over other algebraic methods, is that only integrals containing one continuum function need be calculated. A very obvious disadvantage to this method is that phase shifts can be determined only at energies corresponding to the positive eigenvalues of L. If the system being considered is small enough so that a large basis set can be used, the restriction to eigenvalues of L creates no problem.

PAGE 56

49 The partial cross section can be calculated from the phase shift via the relation 4'jy o-| = psin 2 6 ] (21 + 1) . A total elastic cross section can then be obtained by summing the partial cross sections o = E oi. 1 1 Therefore, all that is now needed to calculate elastic cross sections is a form for the potential V(r) in L. 3.3 Choice of Potential One simple choice for a potential is the Hartree-Fock (or static-exchange) potential. This corresponds to letting the operator L be the Fock operator, L = h + £<(f>.j | | With this potential, the expansion functions uand scattering energies are the virtual Hartree-Fock orbitals <|>. and orbital energies e^. Finding the phase shifts at the positive orbital energies now corresponds to the evaluation of the following integrals ( i | T i E ^ | S ) (^||-|S) + I k-jS) , where S is either s-| or c-| . The functions Sj and c-| , as mentioned earlier, should go asymptotically as j ^ and n^ respectively, but they should also go to zero at the origin. Therefore, the actual forms used for these

PAGE 57

50 functions are S] = j] (kr) c 1 = J'i+i(kr) + (^l)j 1+2 (kr) . This particular choice of c-| is due to Armstead (1968) and it has the advantage that all the integrals needed will only involve regular spherical Bessel functions. A description of how to evaluate the necessary integrals is given by Harris and Michels (1971). In the static-exchange method, the target potential is only weakly affected by the incoming electron through the exchange terms. In addition to exchange effects, polarization effects have been shown to be very important in electron scattering (Truhlar et al . , 1971). For elastic scattering on atoms an important polarization effect is the charge induced-dipole polarization, which for large r behaves as Such a form for the polarization potential can not be used because of the unphysical singularity at the origin and several alternate forms have been suggested. The one chosen for this work is -a V pol( r ) 2 r 4 r> r where a is the static polarizability and r Q is a cutoff parameter. The role of r Q is to cut off the attractive polarization potential before it penetrates the charge distribution of the atom. More is said later about how to pick the parameter r Q .

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51 Using the above potential, the operator L is now L = F + V pol With this choice of L, we can no longer use the Hartree-Fock orbitals and orbital energies as the expansion functions and scattering energies. To obtain new functions and energies, the eigenvalue problem LC = EC must be solved. Using the Hartree-Fock orbtials as a basis for solving this problem leads to the simplification of L such that L = e + V , where e is a diagonal matrix of Hartree-Fock _ _ _p orbital energies and (V ) — = I V p 0 i | . ) • Once the solutions have been found in terms of the Hartree-Fock orbitals, they can be back transformed to the primitive basis giving the desired orbitals and energies to proceed in finding the phase shifts. 3.4 Relation to Other Methods Helium Test Case In order to evaluate the results of the method developed, the relationships between it and two other methods, namely those of Harris and Michels (1971) and Yarlagadda et al . (1973), are explored. In the method of Harris and Michels, the target is described by the N electron determinantal (Cl) solutions of the full N electron Hamiltonian, , N T= Z a | D > . y pf p The scattering solution is then given by solutions to an N+l electron Hamiltonian, H^ + j, written as Vi E E r> + < T i *’> + ' ~>

PAGE 59

52 The solution is assumed to be of the form

+ Eb-n. , where S C i I 1 the functions S , , and n-j are N+l electron determinantal functions given by and E a J |SD N > , y yr 1 y X a „ 1 CDfj> , y yr i y 5 X b . |d n+1 > , y yi 1 y where |CD^> and |SD^> are N+l electron determinants, y 1 y for the Harris phase shift is then The equation s > tan<5 = — — i I Hn+ 1 " E i 1 4 , c > This method reduces to the one used here provided that the target is represented by a single determinant of occupied Hartree-Fock orbitals, Xa> represented by r = |d hf > , and the scattering solution is formed by adding s-j , c-| , and Xp to make the N+l electron functions and ^s l s l D HF > 5 C hV ’ Ix d Dhf> ’ where Xp is a virtual Hartree-Fock orbital. Under these conditions the integral reduces to

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53 This can be immediately recognized as our previous expression. Yarlagadda et al . (1973) used the same one electron non-local potential equation used here as a starting point but included a correlated self-energy operator. In their treatment E is given by Z(x,x';E) = E hf (x,x') + E < (x,x';E) + E > (x,x';E) , where E < and ^ are generalized polarization potentials given in terms of the generalized response function in the random-phase approximation (RPA). By dropping all terms in E except E , the Hr present static-exchange method is obtained. The polarization potential used in this work can be viewed as a local, energy independent approximation to the potential of Yarlagadda et al . To test the programs that were written for the calculation of Harris phase shifts, a series of tests were performed on the helium atom. This system was chosen because it is a very simple system to obtain target wavefunctions for and because of the wealth of prior work, both theoretical and experimental. In the calculations performed two different basis sets were used for the Hartree-Fock description of the helium atom. The first, called HelO, is a ten STO basis comprised of the five optimized STO's of Clementi (1965) plus five more diffuse STO's of higher principal quantum number ana the other, called He20, is a twenty STO extension of basis HelO. Both of these basis sets are shown in Table 12. There are several reasons for using extended basis sets. One main reason is that they give more virtual orbitals and energies at

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54 Table 12 Hel ium Basis Sets Basis HelO orbi ta 1 exponent Basis He20 exponent Is 1.4300 1.4300 Is 2.4415 2.4415 Is 4.0996 4.0996 Is 6.4843 6.4843 Is 0.7978 0.7978 Is 0.100 Is 0.050 Is 0.075 Is 0.010 2s 1.000 1.000 2s 0.500 0.500 2s 0.250 0.250 2s 0.100 2s 0.075 2s 0.025 3s 1.000 1.000 3s 0.500 0.500 3s 0.250 3s 0.100 3s 0.050

PAGE 62

55 which phase shifts and cross sections can be calculated. The nature of the basis sets used is such that many of the virtual eigenvalues are clustered at low energies, thereby giving a better description of the shape of the cross section curve in this region. The use of extended basis sets, like those used here, may also improve the calculations by giving a more accurate description of the long range behavior of the static-exchange potential. Tables 13 and 14 show the static-exchange results obtained with the basis sets mentioned. These results are also shown graphically in Figure 5 and can be seen to agree very well. The discrepancy at low energy can possibly be explained by the better long range description of the helium Is orbital provided by basis He20. Figure 6 shows a comparison of the results obtained in this work with some previous calculations. 3.5 Beryllium Since the ultimate goal of this work is the calculation of shape resonances in the group 1 1 A elements, a likely place to begin is beryllium. From a theoretical standpoint, this is a very good starting point because of its small size. Unfortunately, it is difficult to do the necessary experiments on beryllium because of health hazards. In the calculations to be described here, dementi's (1965) extended basis was used to describe the target Is and 2s orbitals. The virtual orbitals and energies were obtained from three different Hartree-Fock calculations each using a different set of 2p Q functions. These basis sets are shown in Table 15. Only p Q functions were

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56 Table 13 HelO Results energy (eV) phase shift (rad) cross section (au 2 ) 0.1537 3.0136 18.1326 0.7728 3.8155 22.7077 2.3729 2.5609 21.6837 6.1827 2.2320 17.2258 15.3040 1.8277 10.4505 38.7226 1.3881 4.2695 106.7075 0.9928 1.1240 350.8599 0.6607 0. 1835 1896.1131 0.3152 0.0087

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57 e nergy (eV ) 0.0004 0.0023 0.0071 0.0177 0.0384 0.0776 0.1515 0.2923 0.5639 1.0970 2.1696 4.3910 9.1601 19.9452 46.5337 120.9710 381.3872 2008.1614 Table 14 He20 Results phase shift cross section (rad) (au 2 ) 3.1350 17.4187 3.1243 22.4827 3.1097 24.3011 3.0905 25.2611 3.0655 25.7255 3.0329 25.9196 2.9895 25.9036 2.9304 25.7048 2.8489 25.2345 2.7355 24.3211 2.5771 22.5510 2.3578 19.4053 2.0652 14.4625 1.7040 8.4209 1.3127 3.4348 0.9571 0.9446 0.6457 0.1623 0.3088 0.0079

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Figure 5 Plot of phase shift versus energy for helium. The A curve was obtained using the HelO basis and the * curve was obtained using the He20 basis.

PAGE 66

59 o oo (pej) ^.liis eseqd Energy (eV)

PAGE 67

CD r— -C “O 03 I— c O 03 4-5 » • O • CD CD CXJ >> OO zn CD cn •i— 03 : c S^ST3 -C CD S~ X) +-> JZ C 03 03 *r4-5 CD =C cn 5 • 1— 03 $ cn « • i — 13 E S00 O 03 O u s>“ 5 O CD 4£ > X3 • * 4-5 ^ C 4-5 -»-> CD CD C 4— C + in CD •r— •i — CD cn SZ ra — LCD cn 4-> Q_ s_Q * — >. Q_ CD o i — I • 00 r*** • c\ • 03 CD cn ^ s / — . sz %*“ • 't ( Cl CD V\ <1 * 4cn v — " O cn i — CD CD CO l n cn 4-> > JZ •i — •r— O Scj cn 00 cn i — =3 •i — rH 03 03 Q_ u E s — JD JD co CD S13 cn •r— Li_

PAGE 68

61 o CO f-H (•pea) WLi|S Energy (eV)

PAGE 69

62 Table 15 Be Basis Functions Target orbitals STO exponent Is coef. 2s coef. Is 3.4703 0.91792 0.17065 Is 6.3631 0.08742 0.01469 2s 0.7516 0.00147 -0.11551 2s 0.9084 -0.00267 -0.67835 2s 1.4236 0.00222 -0.30265 2s 2.7616 0.00597 0.09232 Virtual orbital s STO basis 1 basis 2 basis 3 2p0 1.900 2.000 2.100 2p0 1.400 1.500 1.600 2p0 0.900 1.000 1.100 2p0 0.700 0.750 0.800 2p0 0.450 0.500 0.550 2p0 0.350 0.375 0.400 2p0 0.220 0.250 0.280 2p0 0.170 0.180 0.190 2p0 0.110 0.120 0.130 2p0 0.085 0.090 0.095 2p0 0.055 0.060 0.065 2p0 0.025 0.030 0.035

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63 2 used because the shape resonance should occur as a P state and, therefore, will show up only in the p-wave cross section. Three different basis sets were used to provide more virtual orbitals in a useful energy range without doing a very large calculation. One advantage in doing these calculations on beryllium is there are no p 0 orbitals in the occupied Hartree-Fock set of orbitals. Therefore, p 0 functions can be added to the basis or their exponents can be varied and there will be no changes in the static-exchange potential except through the exchange terms which are very small. The results of the calculations are given in Table 16. Calculations were also performed using the polarization potential mentioned previously. Since this potential has an arbitary parameter r , some method of arriving at its value is needed. In the work discussed here r is picked as the value at which the o potential (l + l)l/2r 2 + V .j is zero. The expression for r Q is then For beryllium, the static polarizability used is 47 a.u. and r Q is chosen as 5 Bohrs. The results obtained with the polarization potential included are shown in Table 17. A plot of the low energy portion of the cross section curve is shown in Figure 7 for both the static-exchange and static-exchange + polarization calculations. As can be seen, the addition of the polarization potential causes a dramatic change in the cross section. The reliability of this change is discussed further during the evaluation of the resonance parameters.

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64 Table 16 Be Static-exchange Results phase shifts cross E(eV) (rad. ) ( 0.004202 0.000139 0 . 0.005451 0.000207 0 . 0.006798 0.000290 0 . 0.018394 0.001577 0 . 0.022614 0.002151 0 . 0.027191 0.002840 0 . 0.054174 0.008019 0 . 0.064984 0.010721 0 . 0.076655 0.013949 1. 0.133786 0.035220 4. 0.158757 0.046136 6. 0.185447 0.059121 9. 0.300029 0.128540 28. 0.352412 0.169153 41. 0.406992 0.216708 58. 0.608721 0.440872 153. 0.701153 0.554051 202. 0.795346 0.670897 249. 1.143305 1.029797 329. 1.316582 1.157449 326. 1.505334 1.257652 308. 2.252472 1.442995 224. 2.658461 1.469213 190. 3.105742 1.477906 163. 5.017304 1.422467 99, 5.965638 1.401906 83, 6.997279 1.380897 70, 11.693033 1.315351 41, 13.847953 1.294340 34, 16.178037' 1.275813 29 30.325205 1.150602 14 35.674818 1.131722 11, 41.487912 1.112447 9, 99.405096 0.890287 3 117.676441 0.853701 2 137.705365 0.819420 1 sections au 2 ) 002375 004039 006353 069319 104979 152096 608875 907111 301892 753734 872079 655860 090688 247963 263549 435384 499887 247327 638437 720581 394359 010669 950717 727877 995392 548150 689548 064003 279256 024436 098987 779206 942484 117141 476104 988966

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65 Be Static E(eV) 0.004192 0.005435 0.006773 0.018233 0.022364 0.026821 0.052723 0.062787 0.073415 0.121498 0.140120 0.158628 0.231334 0.261811 0.294031 0.431053 0.503026 0.582814 0.921154 1.098676 1.292951 2.050532 2.454522 2.895756 4.731597 5.639251 6.625188 11.208313 13.366430 15.710701 29.944744 35.301277 41.124872 99.174761 117.456041 137.493739 Table 17 exchange plus Polarization Results phase shifts cross sections (rad. ) (au 2 ) 0.003042 1.132535 0.004018 1.523699 0.005135 1.996677 0.019675 10.888717 0.025801 15.263814 0.032850 20.628901 0.083696 67.988909 0.108474 95.746635 0.138261 132.703602 0.335596 457.866409 0.435851 652.437972 0.549157 880.948064 1.049741 1667.766610 1.244082 1757.304698 1.415030 1702.425504 1.818372 1118.444717 1.897914 914.376183 1.945246 762.322092 1.964526 474.862546 1.951096 402.517857 1.930785 347.471062 1.860148 229.770366 1.821383 196.115766 1.784643 169.147309 1.666117 107.418995 1.629738 90.637887 1.600228 77.351077 1.580295 45.757406 1.520797 38.277120 1.450245 32.174991 1.034357 12.654530 1.125939 11.838870 1.270586 11.381315 1.119829 4.189378 1.085710 3.417380 0.868290 2.173016

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Figure 7 Plot of cross section versus energy for beryllium. The A curve was calculated using the static-exchange potential and the * curve was calculated using the static-exchange plus polarization potential.

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Cross Section (au 2 ) 67 Energy (eV)

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68 3.6 Magnesium The next group 1 1 A element is magnesium and, unlike beryllium, experimental work has been done on it. Unfortunately from a theoretical standpoint, this system is much more difficult to calculate because of its larger number of electrons. Again, as was done for beryllium, dementi's (1965) extended basis sets were used to describe the target Is, 2s, 2p, and 3s orbitals. In order to obtain virtual orbitals and energies, this basis was augmented by three different sets of p Q orbitals. The basis is shown in Table 18. Unlike the beryllium case, where the additional p Q functions have no effect on the target potential , in magnesium they do. The additional p Q functions will mix with the occupied 2p 0 function and cause a slight breaking of the degeneracy of the 2p functions. Thus, as the additional p 0 functions are varied the occupied 2p 0 function changes and, therefore, the potential changes. While in general, this would seem to create problems, in practice the changes are so slight that the potential can be regarded as constant provided no major changes are made in the additional p Q functions. The results of these static-exchange calculations are shown in Table 19. Calculations were also done using the polarization potential with a = 81. and r 0 = 6.3. These results are given in Table 20 and a comparison plot of the cross section with and without the polarization potential is given in Figure 8. Again, notice the dramatic effect of the polarization potential.

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69 Table 18 Magnesium Basis Target orbitals : exponent Is 12.0 0.96539 13.5552 0.03767 9.2489 0.01583 6.5517 -0.00212 4.2008 0.00104 2.4702 -0.00014 1.4331 0.00005 0.8783 0.00001 exponent 6.0 0.51090 7.9884 0.08276 5.3197 0.32223 3.7168 0.21178 2.5354 0.03591 2s 3s -0.24382 0.04695 -0.00504 0.00166 0.10176 -0.02391 0.39908 -0.07764 0.55719 -0.13548 0.04715 -0.00710 -0.00783 0.50699 0.00197 0.57016 Virtual orbitals : exponent np 0 (n-3,4,5,6,7) 1.00 0.30 0.325 0.35 all calc, basis set 1 basis set 2 basis set 3

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70 Table 19 Mg Static-exchange Results E(eV) phase shifts (rad. ) cross 0.081827 0.021150 2 0.092882 0.025704 3 0.104139 0.030670 4 0.251770 0.153096 47 0.282524 0.184554 61 0.313107 0.217919 76 0.523734 0.515179 237 0.579667 0.592669 276 0.635199 0.666490 308 0.951745 0.994143 378 1.053091 1.057389 369, 1.156575 1.106756 354, 1.742237 1.213852 258, 1.928260 1.220699 234, 2.116931 1.220718 213, 3.301539 1.142453 128, 3.608436 1.122871 115, 3.917077 1.102427 104, 6.404669 0.931352 51. 6.869169 0.908184 46. 7.338163 0.885543 41. 12.695070 0.651735 14. 13.393855 0.631349 13. 14.106339 0.611290 11. 26.702131 0.357590 2. 27.871870 0.339243 2. 29.076516 0.320871 1 . 70.411131 2.956181 0 . 73.122476 2.935589 0 . 75.936550 2.914877 0 . sections (au 2 ) .803479 .647776 631591 377459 136029 568734 727264 075872 626046 713888 561864 638997 454327 703816 788917 551696 479886 257131 565066 410493 901138 864938 340873 976657 353288 037852 754699 247567 293486 341272

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71 Table 20 Mg Static-exchange plus Polarization Results phase shifts cross section; E(eV) (rad. ) (au 2 ) 0.071849 0.187897 249.080902 0.079643 0.218614 302.912990 0.087114 0.249523 359.038448 0.181427 1.041606 2106.574817 0.198396 1.157026 2167.357707 0.215599 1.259715 2156.112829 0.387777 1.745007 1282.953281 0.436002 1.776453 1127.335632 0.485986 1.794933 1003.262067 0.808969 1.788866 604.353123 0.912347 1.770552 540.051458 1.018105 1.749677 487.839174 1.612432 1.618644 317.369196 1.801153 1.584771 284.711727 1.991531 1.552758 257.461588 3.154180 1.381273 156.841341 3.458330 1.351105 141.267270 3.763056 1.322886 128.094564 6.144810 1.133358 68.491260 6.600492 1.107281 62.174431 7.061153 1.081434 56.588127 12.352179 0.782344 20.635094 13.054273 0.756306 18.502856 13.769579 0.730963 16.601043 26.396296 0.413249 3.133697 27.567770 0.393949 2.741167 28.774576 0.373839 2.377253 70.227328 2.932335 0.315174 72.940730 2.901953 0.396148 75.756894 2.889071 0.422635

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Figure 8 Plot of cross section versus energy for magnesium. The A curve was calculated using the static-exchange potential and the * curve was calculated using the static-exchange plus polarization potential.

PAGE 80

Cross Section (au 2 ) 73 Energy (eV)

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74 3.7 Evaluation of Resonance Parameters The shape of the cross sections for Be and Mg given in Figures 7 and 8 shows evidence for the existence of a resonance but from them it is not obvious how to obtain the resonance parameters. These parameters are the resonance energy E r and the width F. To evaluate E r and T it is easier to consider the shape of the phase shift curves. Figure 9 shows a plot of 6 versus E for Be and Figure 10 shows a similar plot for Mg. One way of obtaining resonance information via phase shifts is through a study of time delay (Bohm, 1951; Wigner, 1955). The time delay of an outgoing particle due to the presence of a potential is given by At 9h d6 2 h xThe energy of maximum time delay is identified as the resonance energy and the resonance width is then given as 2 r d6 dE E=Ev To actually solve for the values of E r and T from the phase shift curves, the values of the first and second derivative curves are needed. The second derivative is used because it is simpler to find the zero of it than the maximum of the first derivative. In order to evaluate the derivative curves some form must be chosen for the phase shift curve. In this work a cubic spline was used to fit the phase shift values. First and second derivatives are then evaluated at the known energies and these quantities are also fit via cubic splines to provide continuous curves.

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Figure 9 Plot of phase shift versus energy for beryllium. The A curve is from calculations using the static-exchange potential and the * curve is from calculations using the static-exchange plus polarization potential.

PAGE 83

0 5 oi o. 76 r 2.5 Energy (eV) 5.0 7.5

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Figure 10 Plot of phase shift versus energy for magnesium. The A curve is from calculations using the static-exchange potential and the * curve is from calculations using the static-exchange plus polarization potential.

PAGE 85

Phase Shift (rad. 78 0.0 2.5 5.0 7.5 Energy (eV)

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79 The resonance parameters obtained from the Be and Mg calculations are shown in Table 21. Also given in this table are the experimental results of Burrow et al . (1976) and the model potential results of Hunt and Moiseiwitsch (1970). The first thing that should be noticed is the large change in the parameters when the polarization potential is added. Because of this, it is obvious that for the systems studied here polarization effects are extremely important. These effects can not be ignored if accurate values are sought. With the polarization potential used in this work, it is not possible to evaluate the quality of the effects included. This is because the potential depends on the parameter r Q and no completely justifiable method for obtaining its value is given. Further studies should be made to explore how the effects change as r Q changes and also to explore other types of polarization potentials.

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Table 21 Resonance Parameters staticexchange s-e plus Vl H&M a Exp. b Be E r 0.769 0.195 0.60 r r 1.611 0.283 0.22 — Mg E r 0.460 0.161 0.37 0.15 r r 1.374 0.238 0.10 0.14 a Hunt and Moiseiwitsch (1970) ^Burrow et al . (1976)

PAGE 88

APPENDIX A THIRD ORDER RSPT ON THE ASCF IONIZATION ENERGY As was previously stated, the ASCF ionization energy is the difference between two Hartree-Fock energies. If we use the notation that j and e. ("untilded" quantities) are the Hartree-Fock orbtials and orbital energies of the N electron state and j and c. ("tilded" J quantities) are the same for the N-l electron state, we obtain the following expression for the ionization energy -I(ASCF) = E^ E^"l 1 1 HF HF,x = { Ee, %E
} { Ee } a a,b a*x a,b*x = e Y + E (e„ e ) Z %E ( ). x a*x a a a * x a,b*x In the preceding equation, as throughout the paper, the notation used is: a,b,c,d = occupied spin orbitals; p,q,r,s = unoccupied spin orbitals; i , j , k , 1 = unspecified spin orbitals. By assuming the N electron Hartree-Fock ground state to be the reference state for a perturbation expansion of the N-l electron state, we arrive at the following expression for the ionization energy through third order -I (ASCF) = e E {e^ e^ e^} E x a*x a a a a tx + E ( + + a ,b*x + + } 81

PAGE 89

82 + £ { + + ob^ 1 ^ I a ^ 1 ^ b>} Now in order to evaluate the corrections to the orbtials and orbital energies, we must consider the relation between the N electron state Fock operator and the N-l electron state Fock operator F^ = h_ + £
o a and F^ * = h n + £ , x 0 a*x " where x is the label of the removed spin orbtial and where stands for the coulomb minus exchange operator. This gives the relation pN-1 _ pN _ + j { } . x a*x The perturbation is given by Vj + V^, where = - and V 2 %^x { }. The perturbation is somewhat unusual because it is defined in terms of both unperturbed and perturbed orbitals. Therefore, to obtain expression up to a given order in electron interaction we must apply our perturbation in an iterative fashion. Before we apply this result to the expression for the ionization energy, we can simplify the expression by noting that £ = £ = -£ + £ { } a*x a a*x a#x a,b*x = -£ + £ { + a^x a,b*x + + + } .

PAGE 90

83 This cancels some of our previous terms and leaves -I(ASCF) = e Y £ {e^ + e^} + hi {o ^ 1 V 1 ) | |ab> x a?x a a a ,b?x + + + ) . To illustrate how to deal with the perturbation, we shall work through an example in detail. Let us consider the case of the second order energy correction (2) = _ y
< k 1 V | a> a *x a a*x (c. _ e \ k*a k aj Substituting the perturbation we get £ I [ 2 a?x k*a e. ( V i : term) + E a*x k^a { ) e k " e a (VpV 2 term) + £ { ] < kx | |ax> a?x p _ p k?a e k a (V 2 '-^i term) Using the expansion for 6 and keeping terms through third order we now obtain -£ 1 [ + E ( + < kb [ |ab^)>} a*x p. _ p a ,b*x k*a E ° k / a + Y. a ,b*x k?a ( + } Ci, e =

PAGE 91

84 -I a*x k?a | : e k " e a + ^ ^ + < kb | |a1> a k^a X (ek ' £ a )(e l " £b) (£|< " £a)(Cl " £b) lfb < kx | [ax> + <1 x | [ bxxkx [ 1 ax> ( e k ' e a )(E l " G b) : a )( c i “ Similarly, we obtain from the other parts of the ionization energy expression the following terms c (3) _ y . x M kxxkx 1 1 axx a x 1 1 1 x> + <1 x | [ ax>< ax | | a xxax 1 |lx> I _ a*x a a*x _ £ ) a*x _ e ) 2 l,k« l l a " k a ' 1M l ' and { + + + } a jb^x ^ r + ft k* l*b a,b*x (e. e )(e-i e.) k*a k a < E k ’ E a ) + < kb | | al >< 1 x | [ bx> < kx 1 [ax> (e k e a )(e-, “ %) ( £ k " £ a )(e l " e b) Now by combining the terms we have obtained, we arrive at the expression for the ionization energy at the ASCF level through third order in electron interaction. -I (ASCF) = e + £ x a*x | 1 hi ( k*a ' b a < kl 1 | ab> <1 x | [ bx> a,b^x (e k e )(e 1 e b ) k*a l*b < E k ' E a ,

PAGE 92

85 + E a*x 1 ,k*a <1 x | | kxxkx | | axxax | 1 1 x> (ci EgXE^ ' E b> z a*x 1 ?a (e 1 E a> :

PAGE 93

APPENDIX B THIRD ORDER RSPT ON THE TOM IONIZATION ENERGY The TOM ionization energy is given by the eigenvalue e v of X the transition operator defined by F T = h n + Z + Js . x 0 a*x In this section we will obtain an expression for by using RSPT with the Hartree-Fock N electron ground state as our reference. In order to define the perturbation, we must consider the relation between the Fock operators x> + ' 2 + Z { ) . a*x Keeping all terms through third order, the perturbation V can be expressed as V = + h + Z { } arx -h + Z ( + + a*x + + } . Now we can proceed to evaluate e = e + . X ^ X X X First let us consider X e* = = -% + Z ( + a?x + + + } r> r

PAGE 94

87 Using the expressions for and a^ ,(D = 1<1 1 V I a> Ua e a ( (2) _ ^ 1<1 | V | k>< k | V | a> ^ 1<1 j V | a>
1 ,k*a(£-|-e )(e k -e ) l?a (s^-eg) 2 , v a<1 | V | a> ta (e r e a ) 2 ’ we obtain ( 1 ) £ ^ + <1 ] V [ a> a?x 1 ^a (e l ' e a> (E 1 E a> + a•x (e-, e )(e. e ) 1 , k^a I a k a + y. { + <1 1 V [ k> ^ afx (e n ej(e u ej 1 , k*a (cn ej(e. e ) <1 [ V j a> a*x {e, e ) 2 (e-, e ) 2 1 *a 1 a 1 a , ^ r <1 | V|a> '2 a*x l?a a> (e, e ) 2 ' 1 a' (e l r 'a )2 }. If at this point we note that and ep) are zero through third order, we see that by adding e v to ep) and substituting for V, we arrive at the expression for e . 5 = £ + £ a*x e, l*a 1 a j ^ + + <1 k | | ab> + <1 b | | a k>< kx | [bx> } (e l £ a )(£ k " e b> (e-, ej(e u e K )

PAGE 95

88 + 3/4 E a*x 1 ,k?a xl | | xa>< Mil X> (c, C a )(e k C a ) 3/4 l a?x (e l*a xa><1 x | | a x>

PAGE 96

APPENDIX C EXPANSION OF ASCF ENERGY IN TERMS OF TOM QUANTITIES In order to expand the ASCF ionization energy is terms of the N TOM orbitals and orbital energies, we shall first expand E and N-l E in terms of the TOM quantities and then subtract the expressions x obtained. Expansion of E N In this section quantities with a tilde will represent the N electron state Hartree-Fock solutions. The expression for the total energy is E N = le, hZ a a,b and the perturbation to be used is defined by F^ = F T + % + Z {
} . x a^x Therefore, the expression for E^ through third order is f N v , . (1) + (2) ^ (3), E = Z { e , + e + e + e' } a a a a a ,(D HZ { + + a,b ,(D> + «h(D + + + + + + + ob^ 1 ^ I I ab ^ 1 ^ > + 89

PAGE 97

90 + + | |ab> + %E
a a a a a ,b hZ, { + + a , b + } . Expansion of Ep* X The expression for Ep* is given by X E N_1 = E e, HZ , x a*x d a,b*x where the til ded quantities now refer to the N-l electron state Hartree-Fock solutions. The perturbation is now defined by 1 = F* '2 + E { ) . x x m a *x N Following the procedure used for E , we obtain E^”* = E (e a + ep) + eph ^ HZ a?x a,b*x hZ { + a,b?x + } . n . rr r N _N -1 Difference E E„ X N In considering the difference between the equations for E and N-l E , we see the leading terms give e and all first order terms x x cancel. If we also note that the leading term in the perturbation, V^, is '2 for the N electron energy and J 2 for the N-l electron energy, we immediately see that the four intergral terms of

PAGE 98

91 each expression cancel. The expression for e(2) in terms of Vi is 6 _£ |
1 2 + £ ^<1 k 1 labxbjV^ k> + <1 b | |ak> ^ l«a E 1 ‘ c a ^a (q " E a) < E k ' c b> < E k ' E b> k*b From this we can see that will appear an even number of times and, ( 2 ) therefore, the terms in 1 cancel. This leaves only terms in ep) to be considered. The expression for ep) is a ^ <1 lV 1 |k> ^ <1 | V 1 1 a> l,kM (e l £ a )(e k • E a> ,2a < E 1 ' E a> 2 Since there is an odd number of Vi's in eO) the terms in the N i a and N-l electron energy expressions will not cancel as previous terms had but will add together. The result for the differences is then -I(ASCF) = E N E N_1 x = e x + hi a 1 ,k?a <1x1 [kxxkxj 1 ax> < E 1 E a ) < E 1 E a )2

PAGE 99

BIBLIOGRAPHY Almlof, J., Univ. of Stockholm Inst, of Physics Report 72-09 (1972). Armstead, R. L., Phys. Rev. 171 , 91 (1968). Bagus, P. S., Phys. Rev. 139, 619 (1965). Berkowitz, J., Chupka, W. A., and Walter, T. A., J. Chem. Phys. 50, 1497 (1969). Bohm, D., Quantum Theory , New York: Prentice-Hall, 1951. Born, G., Kurtz, H. A., and Ohrn, Y., J. Chem. Phys. 00, 0000 (1977). Burke, P. G., Adv. in Atomic Mol. Phys. 4_, 173 (1968). Burrow, P. D. , Michejda, J. A., and Comer, J., J. Phys. B 9, 3225 (1976). Cederbaum, L. S., and Domcke, W. , Adv. Chem. Phys. 36, 205 (1977). Cederbaum, L. S., Hohlneicher, G., and von Niessen, W., Chem. Phys. Lett. 18, 503 (1973). Clementi, E., Tables of Atomic Functions , San Jose: IBM, 1965. Csanak, Gy., Taylor, H. S., and Yaris, R. , Adv. Atomic Mol. Phys. 7, 278 (1971). Dunning, T. H., J. Chem. Phys. !T3, 2823 (1970). Eland, J. H. D., Photoelectron Spectroscopy , London: Butterworth, 1974. Froese, C., Can. J. Phys. 41, 1895 (1963). Goscinski, 0., Hehenberger, M. , Roos, B., and Siegbahn, P., Chem. Phys. Lett 33, 427 (1975). Goscinski, 0. and Lukman, B., Chem. Phys. Lett. 7, 573 (1970). Goscinski, 0., Pickup, B. T. , and Purvis, G. D., Chem. Phys. Lett 22, 167 (1973). ~ Harris, F. E., Phys. Rev. Lett. 19, 173 (1967). 92

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BIOGRAPHICAL SKETCH Henry Allan Kurtz was born July 3, 1950, in Fort Myers, Florida. He attended public schools in Fort Myers until graduating from Fort Myers High School in 1968. From 1968 until 1972 he attended Georgia Institute of Technology were he received his B.S. degree. From September 1972 until present he has been a graduate student in the Quantum Theory Project and the Department of Chemistry at the University of Florida. 95

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. N. Yngve dht^n, Chairman Professor* of Chemistry and Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. David A. Micha Professor of Chemistry and Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. r A Xu y 'Ll Charles E. Reid Associate Professor of Chemistry I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. John R. Sabin Professor of Physics and Chemistry I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. C ka lie ^ L\ Lite. L ^ Charles P. Luehr Associate Professor of Mathematics

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This dissertation was submitted to the Graduate Faculty of the Department of Chemistry in the College of Arts & Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1977 Dean, Graduate School