Exchange-Only Density Funct for Semi-Infinit e Jellium · 2004-11-29 · Exchange-Only Density Funct ional Theory for a Semi-Infinit e Jellium Surface by Fred Nastos A thesis submitted

Exchange-Only Density Funct ional Theory for a Semi-Infinit e Jellium Surface

by

Fred Nastos

A thesis submitted to the Department of Physics

in conformity with the requirements for

the degree of Master of Science

Queen's University

Kingston, Ontario: Canada

October 2000

@ Red Nastos 2000

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Abstract

The optimized effective potential (OEP) provides the best local approximation to the ex-

change potential within density functional theory. We have investigated the OEP for the

semi-infinite jellium mode1 of a metal surface and have obtained the asymptotic form of

the potential far into the vacuum. The development follows the approach of Krieger e t

al. (Phys. Rev. A 45, 101 (1992)) for the asymptotic £rom of the OEP for a finite system

by d e a h g directly with the integral equation for the OEP but with the different geome-

try and with the compfication of a continuum of states. We show that the OEP has an

image-like asymptotic form, like the classical image potential of - 1/42 but with a coefficient

which depends on the jeilium density. In addition, our asymptotic form contains the entire

asymptotic form of the Slater potential for the jellium surface, showing that the result of

Solomatin and Sahni (Phys. Lett. A 212, 263 (1996)) obtained fkom an approximation to

the OEP is incorrect.

We dso investigate the exchange hole, for the jellium surface, and analytically show

that as an electron moves farther away kom the surface the exchange hole becomes more

delocalized. Our analysis allows us to determine some exact properties, and prove that a

significant portion of the exchange hole is between the surface and a distance into the metal

equal to the classical image charge location.

Finally, we also present a procedure for efficiently calculating the surface exchange en-

ergy for the semi-infinite jellium surface. Our method applies to arbitrary local effective

potentials, and thus is more general than the published methods used for mode1 effective

potentials. This will allow the calculation of the exact OEP and the correspondhg surface

energy for the jeilium surface.

Acknowledgement s

1 would like to first acknowledge my supervisor, Dr. Malcolm J. Stott? who provided much

insight over the course of this thesis. I would like to thank him for many discussions that

went beyond the direct scope of my work, and for sharing his enthusiasm for physics in gen-

eral. 1 would Like to thank him for arranging a visit to Spain, while he was on leave at the

University of Valladolid. 1 would also Like to thank him for introducing me to reference [56].

1 a m sure it will provide many enjoyable nights of reading in the future.

I would also like to thank Dr. Ulf von Barth for collaboration on the material in Chapter

4, and for much insight during his stay at Queen's University, Kingston.

Many thanks are due to Prof. Eugene Zaremba for interesting discussions, and for his help

in obtaining the jellium surface Kohn-Sham LDA code of Prof. Ansgar Liebsch, to whom 1

am also indebted and express my gratitude.

1 woiild like to thank my office-mate Lkaro Calderin and his wife Maribel Baguer: who

allowed me to move into their spare room for the final month of this work, and 1 would also

like to thank L5zaro for many conversations about our respective work.

1 wodd like to salute the other member of the electronic structure group, Bing Wang. 1

appreciate the many interesting conversations about his work and other issues.

The remainder of the 511 group should also be acknowledged, as well as ot her graduate

students in the department. Thank you for maay exchanges regarding everything kom

tricky textbook probIems: to computing issues, to issues reIated to research. Thanks to

Katsumoto Ikeda for providing much entertainment and maay Iaughs while we shared an

apartment for the final year.

1 would like to thank the department staff, including Margaret Morris for timely reminders

regarding many administrative tasks, and S teve Gillen for attending to many network and

printer woes.

1 am also grateful for hancial support fkom Queen's University and the Department of

P hysics ,

Thanks go to my family, in particular to my parents for support and understanding.

Finally, 1 want to thank my wonderful partner Erin Montgomery, for many telephone calls,

weekend visits, and countless emails. Being apart for the past year has not been the easiest,

but we made it!

Contents

- - Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Contents.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

1 Introduction 1

1.1 The uniform background model: jellium . . . . . . . . . . . . . . . . . . . . 2

1.2 Exchange: the Hartree-Fock approximation . . . . . . . . . . . , . . . . . . 4

1.3 Density functional theory - . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 The optimized effective potential method . . . . . . . . . . . . . . . . . . . 12

1.5 The jellium surface model . . . , . . . . . . . . . . . . . . . . . . . . . . . . 15

2 The Asymptotic Form of the OEP 22

2.1 Comment on previous studies . . . . . . . . . . . . . . , . . . . . . . . . . . 22

2.2 Integral equation for the OEP in the jellium surface model . . . . . . . - . . 27

2.3 Asymptotic expansions of the orbitals: part 1 . . . . . . . . - . . . + - . . . 34

2.4 The asymptotic expansion of the OEP . . . . . . . . . , . . . . . . . . . . . 34

2.5 Asymptotic expansions of the orbitals: part 2 . . . . . . . , . . . . . . . . . 43

CONTENTS vii

3 The Exchange Hole for Asymptotic EIectron Positions 45

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.1 The asymptotic density and density matrix . . . . . . . . . . . . . . 47

3.2.2 The exchange fiole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.3 Density profile of the exchange hole . . . . . . . . . . . . . . . . . . 51

3.2.4 PIanar integrated exchange hole . . . . . . . . . . . . . . . . . . . . 54

3.2.5 Asymptotic form of the inverse radius of the exchange hole . . . . . 59

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Closing 59

4 The surface energy and the OEP 61

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction 61

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Surface energy 62

4.3 The surface exchange energy a, . . . . . . . . . . . . . . . . . . . . . . . . . 65

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Future Work 70

5 Conclusions 71

A Results used in Chapter 2 75

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. l Derivation of Eq . (2.21) 75

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Derivation of Eq . (2.23) 76

. . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Derivation of Eqs . (2.30, 2.31) 76

. . . . . . . . . . . . . . . . . A.4 Expansion of orbitals in the asymptotic limit 78

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Derivation of Eq . (2.34) 81

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Evaluation of RkF ( z , z') 82

. . . . . . . . . . . . . . . . . . . A.7 Are there more leading terms in the OEP? 83

... CONTENTS vin

33 Derivations for Chapter 4 85

B. l Derivation of Eqs . (4.7). (4.8) and (4.9) . . . . . . . . . . . . . . . . . . . . 85

B.2 Limiting forms of I(k, k', Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

B.3 Evaluation of IB1I2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

B.4 Evaluation of B2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

B.5 Evaluation of IB3I2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

B.6 DeaLing with the poles in BI x [B2 + B3]* . . . . . . . . . . . . . . . . . . . 92

Bibliography 95

Vita 100

List of Figures

1.1 Schematic of the pseudopotential in simple metals . . . . . . . . . . . . . . 4

1.2 Typical density profile for a jellium slab . . . . . . . . . . . . . . . . . . . . 18

1.3 Various effective potentiak for the jellium surface . . . . . . . . . . . . . . . 20

. . . 2.1 Cornparison of limits giving rise to finite and extended jeUium systems 24

2.2 A contour plot of the Kohn-Mattson function F(b. b') . . . . . . . . . . . . . 33

3.1 Contour plot of the exchange hole for an electron at z = 8 a . u . . . . . . . . 49

3.2 Contour plot of the exchange hole for an electron at z = 20 a . u . . . . . . . 50

3.3 Normalized profile of the exchange hole density . . . . . . . . . . . . . . . . 52

3.4 Planar integrated exchange hole density . . . . . . . . . . . . . . . . . . . . 56

3.5 Totalexchangeholechargelocatedbetweenthesurfacemd-z . . . . . . . 58

4.1 Contour plot of i(k. k'. 0.1) for k~ = 0.48 a . u . . . . . . . . . . . . . . . . . . 67

B.l Mode1 potentials represented by Eq . (B.l). . . . . . . . . . . . . . . . . . . . 90

B.2 Phases corresponding to some mode1 potentials represented by Eq . (B.1) . . 91

B.3 Variation B3 as a hnction of Q for different k and k' values . . . . . . . . . 93

List of Tables

1.1 Cornparison of HF and OEP total ground state energies for atoms . . . . . 14

Chapter 1

Introduction

At the heart of modern electronic structure theory is the W c u l t quantum -wany-body

problem. One fruitful approach to thîs problem for large systems is density functional

theory (DFT) [l] which formally replaces the wavefunction as the basic variable by the more

physical electron density. A critical test for al1 many-body theories are surfaces, as the loss

of translat ional invariance renders the electron gas highly nonuniform. Consequently, many

methods suitable to the b u k region begin to fail. A subset of DFT that is gaining more

popularity in the treatment of inhomogeneous systems is the optimized effective potential

(OEP) method. This thesis focuses on the application of the OEP to a simple model of a

surface, and this chapter lays the theoretical foundation, and describes the model.

1.1 The uniform background model: jellium 2

1.1 The uniform background model: jellium

In the time-independent case, the Harniltonian for N electrons in coordinate representation

is given by l

V& (ri) is a spin independent extemal potential experienced by an electron at ri usua.lly

due to the nuclei of the system, and v(ri, r j ) is the Coulomb electron-electron interaction

given by v ( r i , T ~ ) = Iri - ~ ~ 1 - l . For anything but the simplest cases involving just a few

particles, solving the time-independent Schrodïnger equation,

for the rnany-body wa~efunct ion~~ @(xi, xz, . . . xN), and energy, E, directly is an impos-

sible task. In fact for more than a few dozen or so particles it is a safe prediction that we

will never be able to solve this equation, for the simple reason that the storage required

to represent the wavefunction grows as sN where s is the storage per space dimension [2].

This fact though has not impeded the development of various ingenious approximations for

solving Eq. (1.2) - One approximation begins by separating the electrons into two classes: the inert core

electrons which are bound to the nuclei, and the chemicaily active valence electrons on

which we will focus our attention. In the uniform background model of a solid, also known

as the jellium model, one then replaces the assumed rigid ion cores of the underlying crystal

lattice by a srneated out uniform distribution of positive charge. The valence electrons are

spread throughout the jellium and interact with the positive background via the Coulomb

'We use atomic units throughout, so that R = me = le1 = 1. The unit of the energy is the Hartree = 27.21 eV, and of distance the Bohr radius, ao = 0.529 A.

2 ~ e have compressed the real and spin space coordinates into one so that xi (ri; s i )

1.1 The uniforrn background model: jellium 3

interaction. In an infinite crystal the translational symmetry of this Iattice representation

results in a uniform external potential, that is Vea ( r ) is a constant, and hence, plane-wave

states for the electrons in a single-particle scheme. Consequently the electron density, n(r),

is uniform and can be related to the Fermi wave vector, kF. by n(r) = n = ,4$/(37r2).

Alternatively, the density can also be specified by the r, parameter, defined as the radius of

a sphere which on average contains a single electron, through the relation kF = (97r/4) /r,.

Typical met& have an associated r, in the range of 2 < r, < 6 with sodium corresponding

to r, = 3.93.

HistoricaUy, the jellium model has been important in understanding the properties of

simple rnetals, which comprise elements in groups 1 - IV of the periodic table. Because the

electron states in the jellium mode1 have a plane wave nature: the reliability of the model

for this class of materials hinges on the occurrence of simi1a.r behaviour for the conduction

electrons in the solid.

To understand how the jeilium model can properly describe these metals, one must

turn to pseudopotential theory [3]. By separating out the plane wave components hom the

valence electron wavebctions it can be s h o m that an effective Schrodinger eqiiation for

the valence levels arises whose associated potential, the pseudopote~tial~ is the attractive

periodic potential of the lattice of ions plus a repulsive term. By construction, the energy

eigenvalues are res tricted to correspond exactly to the valence eiectron energies. Unlike the

transit ion met als, whose propert ies are s trongly influenced by the d-orbit al electrons, the

atoms of simple metals have s and p valence electrons surroundhg a closed-shell noble gas

configuration ion. The resulting periodic pseudopotential for the simple metals is relatively

flat, varying slowly over the unit cell, except for a core sized region centered on the ions. A

typical one dimensional representation of the pseudopotential for a simple metal is shown

in Fig. (1.1). Even though the uniform background model is unredistic for anything other

t han the simple metals, and has unphysical characteristics such as being uiechanically unsta-

1.2 Exchange: the Hartree-Fock approximation 4

Figure 1.1: Schematic of the pseudopotential and delocalized electron clouds in one dimension for simple met als

ble [4], it is still popular because it d o w s one to study the quantum many-body effects for

an inhomogeneous system closely and precisely, providing a benchmark for approximations

that hope to be implemented in various electronic structure schemes.

1.2 Exchange: the Hartree-Fo ck approximation

The Hartree-Fock (HF) approximation arnounts to an independent particle scheme wit h the

required Fermion symmetry imposed on the wavefunction [5]. Denoting the single particle

orbitals by $ i ( ~ j ) , it is easy to see that by virtue of the definition of the determinant (det)

the wavefunction constructed korn the ansatz

satisfies the required symmetry. Namely that Pijq = -@, where Pij is the transposition

operator whose effect is to E p the coordinate labels i and j of the wavefunction 9. The

expectation value of the Hamiltonian with respect to the state given by Eq. (1.3) for the


unpolarized case3 is

where n(r ) = 2 x:cc $; ( T ) $ ~ ( Y ) is the density, and n(r; r') = 2 C,SCC % ( r ) q (r') the single-

particle density matrix. The factor of two here accounts for the spin, and the 'occ' denotes

that the sum is over the doubly-occupied Ievels. On examination of the above equation we

see that the expectation value of the electron-electron interaction term of the Hamiltonian

has resulted in two terms. One is the classical electrostatic energy of the average electron

density distribution, often calied the direct term. The second term is an interaction energy

between electrons of the same spin, known as the exchange energy, Er

An immediate avenue to optimize the choice of orbitals is through the Raleigh-Ritz

variational principle, wtiich states that the minimization of the expectation value of the

Hamiltonian, with respect to the parameters in the wavefunction gives the optimal values

for those parameters. Minimizing Eq. (1.4) with respect to the orbitals while maintaining

normalizat ion implies solving the variational equation,

where ei are the Lagrange undetermined mutlipliers. The solution of Eq. ( 1 . 5 ) gives rise to

3 ~ h i s case of an even number of electrons with N/2 electrons spin-up and N / 2 spin-domm is cailed the restricted Hartree-Fodc method [6].


the single-particle equations for the so called Hartree-Fock orbitals and eigenvalues,

(-:oz + VeYt ( r ) + d 3 n f v ( r T ) J

In addition to the usual independent-particle terms: the left hand side of Eq. (1.6) has an

additional non-local term. In a self-consistent calcdation thïs term must be evaluated for

each orbital at every step of the calcdation, hence requiring more effort than for an inde-

pendent particle calcdation. For the homogeneous electron gas, where the single particle

states are plane waves the total energy c m be determined exactly within the Hartree-Fock

approximation. In this case the energy per particle is E, = -0.458/rs [7].

We can rewrite the exchange energy in such a way as to give it a more physical inter-

pretation- First we define the exchange hole

n,(r;rf) =

density as

which is the depletion of charge at T' given an electron at r . This quantity has the feature

of satisfying the perfect screening sum rule

and so supports a quasi-particle picture where there is a depletion of charge around each

electron amounting to one unit of charge which screens the interactions between particles.

A physical quantity of interest is the inverse radius of the exchange hole given by


Using this definition we see that the exchange energy can be written in the form

Written like this, the exchange energy is clearly negative. For historical reasons GrL ( r ) i s

also c d e d the Slater potential, since Slater realized that &-L(r) provided a local potential

that could be used as an approximation for the non-local term in the HF' equations [8j.

In a more complete solution of the many-body Schrodinger equation the charge cIoud

surrounding an electron contains correlations due to the electron-electron repulsions as well

as exchange effects, whereas in the HF approximation there is spatial correlation between

parallel spin electrons only. To take this correlation into account one approach is to represent

the wavefunction by a linear combination of Slater determinants, since the set of aU Slater

determinants form a complete set. Other approaches to correlation go under the name of

diagrammatic techniques or the method of Green's functions [9]. The difTerence between the

HF energy and the true total ground state energy is deiîned as the correlation energy, E,,

and is always negative since the variational nature of the Hartree-Fock method guarantees

t hat it overestimates the true energy.

One would like to be able to calcuiate the exchange energy rather precisely because i t

is a large part of the total energy for many systems. Unfortunately, the issues raised above

related to the nonlocal potential make this a difficult task. As we are about to see in the

next few sections, Kohn-Sham density functional theory dong ci-ith the optimized effective

potential method provides the means to accurately calculate the exchange energy through

the introduction of a local potential.

1.3 Density functional theory 8

1.3 Density functional theory

The later part of the 20th century has seen Kohn-Sham density functional theory become the

authoritative choice for electronic structure calculations. Its foundation is the Hohenberg-

Kohn (HK) theorem [IO], which we now introduce here in the static, non-degenerate case.

For more general formulations the reader is referred to the literature [l]. The &st statement

of the HK theorem is t hat the externd potential of a system V,, (r ) is, apart fiom an additive

constant, a unique functional of the electron density n ( r ) . The proof dernonstrates t hat an

inconsistency arises if one assumes that h o dXerent external potentials can result in the

same density distribution. The impLications of this are profound: since the external potential

f i e s the Hamiltonian H , the ground state ly is also a functional of the density. Moreover,

since 9 determines the properties of the system, the ground state expectation value of any

observable Ô is therefore also a unique functional of the electron density. In other words

there exists a functional O[n(r)] for any operator O such that O[n(r)] = (P[n]1Ô1@[n]).

The focus in the HK theorem is on the ground state energy functional

where a rninimization principle and the notion of a universal functional for the many-

body problem are introduced. From the Rayleigh-Ritz variational principle and the first

part of the HK theorem, the ground state energy of a system may be determined via the

minimization of the functional E[n] over the class of densities that corne fiom local external

potentials. Since the wavefunction @(r) c m be determined from the density n( r ) , then

so can the interaction and kinetic energies. Therefore, we can separate out the external

potential energy as follows


where FHK[n] is a universai functional of the electron density alone, and the second term

accounts for the interaction with the external field. The hinctional FR&] contains the

kinetic, and electron-electron interactions, thus it has inherited al1 the complexity of the

many-body problem.

Unfortunately, HK is an existence theorem, and does not prescribe how one may deter-

mine the functionals- What can be done though, is to take one step back from the goal

of a density based scheme and separate out contributions to FHK[n] that can be evaluated

accurately fkom an independent particle picture. This is the idea behind the method of

Kohn and Sham.

Kohn-Sham DFT

Kohn and Sham showed that as a consequence of the HK theorem oile can map the problem

of interacting electrons to an auxiliary system of non-interacting electrons whose density is

the same as in the true interacting problem [Il]. This will ailow the determination of the

groundstate energy through Eq. (1.12), with the errors depending only on approximations

in FHK[n]. To see how this cornes about, h s t note that the classical Coulomb energy of

the electrons, and the khetic energy corresponding to a system of non-interacting particles

with the same electron density can be extracted from the universal functional FHrc[n]: so

that the total energy can be written as

where

is the electrostatic potential of the system, and T,[n] is the kinetic energy density functional

for a system of noninteracting electrons with density n ( r ) . The final term in Eq. (1.13) is


known as the exchange and correlation energy. It shouid be noted that that since we have

used the kinetic energy corresponding to a non-interact ing wavefunction ins tead of the

true wavefunction, there will be a kinetic contribution to Ex,[n] and the potential energy

accounting for this ciifference. Of course, all concerns about the many-body effects have

moved to Ex, [n] .

From the HK theorem the stationary property of the energy functional gives

where p. is the chernical potential determined by enforcing the partide number N to be

h e d : 1 d3r n(r ) = N. Now: if we were to apply the variational principle to a system of

non-interacting electrons in an external potential, Veff ( r ) , that gave the same density, n ( r ) ,

as the interacting system then the result, analogous to Eq. ( L E ) , would be equivalent if

This implies that associated with each interacting system, there is an effective potential

that can be utilized in the auxiliary non-interacting system by solving the associated Euler-

Lagrange equat ions

for the Kohn-Sham orbitals, +Fs(r), with the density,

identical to the interacting density. The ground state energy can be determined by using


Eq. (1-13), with

i= 1 J

The set of equations contained in Eqs. (l.l6), (l.l7), and (1.18) are known as the Kohn-

S ham equations. Their self-consistent solution toget her with various approximations for

E,,[n], provides the most cornmon approach now taken in electronic structure calculations.

Local density approximation

The functional E,,[n] contains all the many-body effects and is unknown. One approxima-

tion, valid for slowly varying densities, is to assume that everywhere the density locally may

be regarded as uniform. This amounts to making the local density approximation (LDA),

where & , , ( n ( r ) ) is the energy per particle for a homogeneous electron gas of density n.

The most popular choice for E,&) is the parameterized version presented by Perdew and

Zunger [12] of the results from the Monte-Carlo calculations of Ceperley and Alder [13].

B y introducing an exchange-correlat ion hole: n,, (T ; Y'), somewhat analogous t O the

average exchange hole in Eq. (l.ï), the exchange-correlation energy can be expressed as

and so supplies an interpretation that Ex, is the interaction between an electron and the

charge in its exchange-correlation hole [14]. The success of the LDA can be traced back

to two important points concerning the behaviour of the exchange-correlat ion hole. Firs t ,

from the coupling constant integration technique as applied to DFT it is seen that E,,[n]

depends o d y on the spherical average of the exchange-correlation hole as opposed to the

1.4 The optimized effective potential method 12

detailed structure of the hole [15]. By considering the pair correlation function for the

homogeneous electron gas, ghom(r; ri): defined as the probability of h d i n g a particle at r'

given that there is a particle at r , Gunnarson et al- have constructed a representation of

the exchange-correlation hole within the LDA, nkpA (r; r i ) , and verzed that its spherical

average reproduces the spherical average of the exact hole in hydrogen like atoms, and in

the exchange-only limit for heavier atoms [16].

Secondly, withui the LDA the exchange-correlation hole correctly satisfies the sum rule

J d3ri nkpA ( r ; ri) = - 1, analogous to the exchange hole sum r d e in Eq. (1.10). This implies

that even though the LDA does not provide an accurate representation of the exchange-

correlation hole there is a systematic cancellation of error [15].

An improvement to the LDA is the generalized gradient approximation where one re-

places the energy density for a uniform gas, &,,(n(r)), with an energy density depending

on the electron density, and its gradients (n(r) , V ( r ) , . . .) [17] .

Even though the LDA and GGA are popular methods for electronic structure calcu-

rations they do have some serious shortcornings- For example it is known that the LDA

overes t imates the binding energies of molecules and solids, and underes t irnates semiconduc-

tor band gaps [18]. The GGA overestimates lattice constants, and also gives incorrect band

gaps. In addition, both the LDA and GGA do not sat ise some formal requirements such

as giving the correct asymptotic form for the Kohn-Sham potential V,,(r) for a localized

system, or the discontinuities in the functional derivat ive wit h respect to particle number

of the true E,,[n].

1.4 The optimized effective potential method

Since the exchange energy term, Eq. (1.10), c m be calculated exactly fiom single particle

orbitals as was the independent particle kinetic energy, Ts[n], and is usually a more sig-

nificant contribution to the total energy than the correlation effects, one might consider


expressing Vx (r ) exactly for inclusion into the Kohn-S ham scheme, and treat ing correlat ion

as a correction to be applied subsequently. To take this approach we fkst need to tailor

the definition of exchange to a context relevant to DFT, From this point we define the

exact exchange energy functional to be the exchange term of Eq. (1.8) evaluated with the

Kohn-Sham orbitals that give rise to the density n(r) via Eq- (1.18). In other words:

OCC OCC , + I ~ ~ (r ) +;KS (rf)+Fs (T') @FS (T) Ir - r l [

7 (1.22)

where n ( r ) is given by Eq. (1.18).

To determine the Kohn-Sham exchange potential, V,KS ( T ) , requires the evaluation of

the funct ional derivative

However, we only know how to express the exchange energy in terms of the orbitals, and

since their functional derivatives with respect to density are unknown, it is not possible to

evaluate t his potential directly fiom its definition. An alternate approach, first suggested

before the developrnent of DFT by Sharp and Horton [19] in a study of the Slater exchange

potential, is to use orbitals derived fiom a local potential which minimize the HF expectation

value. They found that the optimized effective potential (OEP), V&(r), can be obtained as

the solution of an integral equation which we shall derive later. Although self-consistently

solving single particle Schrodinger equations with a local effective potential is more straight-

forward than a HF calculation, a disadvantage is that since the OEP is obtained Tom the

functionai derivative with respect to a local potential it is a more constrained variation, and

hence the OEP total energies must always lie above the lW energies.

An explicit calculation using the OEP method had to wait until the work of Talman

and Shadwick [ml, who applied this idea to the case of spherical atorns by solving the

integral equation presented by Sharp and Horton. The results for atoms are remarkably


Table 1.1: Cornparison of HF and OEP total ground state energies, in a. u, for various atoms. Values taken from [21].

Atom HF OEP

nate, giving total energies to within parts per hundred thousand of the HF .ml .ues. The

reason for this accuracy is not known. Since we do know that the Hartree-Fock equations

come 6-om a minimization over the choice of single-particle orbitak, this means rthat to Erst

order nny difference of &$J fiom the exact orbitals wiil lead to an error in the expectation

value of the Hamiltonian of O(b~+5~). This alone though does not seem to explain the

remarkable success of the OEP. To give more credit to the OEP method, iït was Iater

realized that the approach in fact represents an exact exchange-only Kohn-Sham DFT

method within the linear response approximation [22]. More recently, the O E F has been

applied to semiconductors [23, 241, polymers [25] and srnall molecu1es [26].

An interesting feature is that for atoms, and h i t e systems in general, the O E P gives the

correct asymptotic form for the potential experienced by an electron far from the system.

F'rom classical arguments we know that when a charge is removed fiom a systenr a distance

r away, the resulting potential due to remaining charges asymptotically faus off as - I / r .

We should expect that for a finite system, since the average exchange hole is CO-ed by the

boundaries, t hat the exchange potential should also behave asymp tot ically like - 1 /r . For

finite systems the OEP reproduces this result [SOI, and consequently shows that correlation

does not change the long range behaviour. This can be understood physically since the

averaged exchange hole satisfies the correct sum rule, and is localized on the system.

1.5 The jelliurn surface mode1 15

For extended systems, such as a metal surface, the situation has not been so clear-

cut. Classical arguments give that VeR(z) fAr fkom a metal surface to be -1/4z where z is

the distance fkom the surface [27]. The same result is expected to come fiom a quantum

mechanical analysis as shown by Bardeen [28], and from DFT as shown by Almbladh and

von Barth [29]. Since the exchange-correlation hole in a metal is no longer c o f i e d , a

long-lasting problem has been to determine at what level of approximations a quantum

mechanical analysis would give the correct asymptotic form. To address this issue, in

Chapter 2 we present an andyticd derivation of the asymptotic form for the metal surface?

directly based on the OEP method.

In the early days of DFT it was assumed that for a metal surface exchange effects played

a lesser role than correlation in the asymptotic form of the potentiai. Sham stated that

a rnany-body analysis resulted in a -1/z2 dependence of the potential, and this was later

supported by the calculations of Eguiluz et al. [30] performed on finite slabs of jellium. More

recently, Sahni haç studied the asymptotic form of the Slater potential, as an approximation

to the exchange potential, for the metal surface. He has s h o w that the Slater potential

has an image like potential, but instead of 1/4, the coefficient depends on the Fermi energy

and work function of the metal. In Chapter 2 we shall taise a closer look at the exchange

potential, and obtain the asymptotic form of the OEP for the jellium model of a metâl

surface, but first we wiU discuss the details of this model.

1.5 The jellium surface model

Introduction of a surface into the jellium model, Sec- (1.1) , can be achieved by considering a

haif-space filled with positively charged jellium in contact with the vacuum. The background

1.5 The jeilium surface mode1 16 -

density can be written as

5 for r < 0: (1 .24)

O f o r r > O .

The abrupt jelliurn edge at r = O defhes the surface of the mode1 rnetd. Far into the metal

the electron density, n(z), canceIs the background so we have that hm,,-, n ( z ) = n, In

the vacuum region a potential barrier is established and n(z) decays exponentially.

To define the electron states we place the jellium in a large box of cross-sectional area

A. At the end of the box, deep in the bulis, where a = -L we place an infinite barrier, and

at the perimeter of the cross-sectional area we use periodic boundaty conditions. The edge

of the positive background deep in the bu&, cannot be at -L since we must allow room for

the electrons to decay out past the surface. By requiring charge neutrality,

one can show that the shift due to the infinite barrier is JIB = 3 5 r / ( 8 h ~ ) , [31, 321. The

volume of the jellium slab, R, is A(L - SIB). We consider semi-infinite jelliurn, and thus

take the L, A + ca limit, so that the set of discrete states becomes continuous. In this limit

the surface at z = -L becomes incidental, and we make reference to it only when necessary-

We deal with the OEP method throughout this thesis, and so the electrons are considered

as independent particles, interacting wit h some effective potential Veff (2). The single particle

orbitals, .Stk (r), separate into

where p is a radiai vector parallel to the surface, and k is the state label. Abusing the

notation, we call the component of k which is perpendicdar to the surface k, and the

vector component pardel to the surface k,, , i e. k = (k: k,,) , and 1 k [ = (k2 + ki) ' j 2 . If we

1.5 The jellium surface mode1 17

set KR(-m) = O the eigenvalues, ek, for the state lCk are

The Fermi energy, E ~ , defined as the energy of the highest occupied state, is just +kg.

The electron density, n ( z ) , can be calculated using Eq. (1.18) and in this geornetry it

simplifies to

A feature of the electron density due to the surface are Friedel oscillations. These long

range oscillations arise because an electron deep in the bulk experiences a uniform potential

VeK(-cm) and so Ilk(z) has the form,

( z ) - s i n - for r « -1, (1.29)

where the phase shifts, y ( k ) , are due to the surface at z = 0, and are uniquely determined

by the conditions that y(0) = O and enforcing continuity. Using Eq. (1.29) the electron

density far into the metal is

A figure representative of a typical prome is given in Fig. (1-2).

This jellium surface model has played an important role in the development of our

understanding the nature of a metal surface, and has added to o u understanding of in-

homogeneous electronic systems. In addition, it Lias become a testing ground for various

methods of including many-body effects into electronic structure caiculations. One of the

eariiest studies of the jellium surface model was by F'renkel [34], in the context of the

Thomas-Fermi approximation [35]. The motivation for his study was to qualitat ively ex-


- -

- -

- -

- 1 I

I

-L -20 O Position from jeilium surface (a. u.)

Figure 1.2: A typical density profile for the jellium slab corresponding to r, = 4. The surface of interest is at z = O. At z = -L we have positioned an infinite barrier. The dotted line represents the positive background. For the solution at the edge deep in the b u k , we have used the exact density for the infinite barrier model, for the surface region the density cornes fiom a self-consistent LDA calculation [33].

1.5 The jellium surface model 19

plain the peculiar behaviour observed by Sommerfeld [36] that the work function of the

metals increased directly with increasing Fermi energies. Of the ma.ny inadequacies of the

Thomas-Fermi approximation a serious one is the lack of Fkiedel oscillations due to the

surface.

Subsequent developments allowed more accurate methods to be applied for the jellium

surface model. Once the exchange potential was better understood [8], Bardeen reported

an independent particle calculation in which the Coulomb interaction was evaluated self

consistently, but the exchange energy was determined from a fked effective exchange po-

tential 1371. To this date a full self-consistent Hartree-Fock calculation for a semi-infinite

or even finite jellium model has yet to be performed- After the work of Bardeen there

was little theoretical progress in surface studies until the appearance of DFT, and perhaps

appreciable computing power! In a series of seminal papers Lang and Kahn performed the

k t fully self-consistent calculations of a metal surface within the LDA [38: 39, 401. Their

calculations gave surface energies, work functions, and dipole barriers that compared well

with experimental results, in addition to realistic potentials and densities. This LDA ap-

proach to the jellium surface model is still used today to study, among other things, topics

such as harmonic generation [41] and quantum-size effects [42] at metal surfaces.

As we saw above: exchange effects can be calculated exactly within a KS DFT kame-

work. One approach, adopted by early researchers because of it s relative comput ational

ease was to choose an analytically solvable model effective potential. That is, represent the

exact one-dimensional effective potential by some model potential. The only models that

have been used to this date are the infinite barrier [43], finite-step barrier [44]: and Iinear

ramp barrier [45] models cf. Fig. (1.3).

We present in the second part of the thesis a method to niunerically calculate trie

surface energy for a semi-infinite jellium slab of any model effective potential, including

ones more physically realistic than those aforementioned. The pursuit of an optimized

1.5 The jellium surface model 20

Figure 1.3: Typical effective potential profiles that have been employed in the literature to model the jellium surface metal. The top three profiles are the idnite-barrier, finite- step, and linear-ramp model potentials respectively, and the lowest figure is a self consistent Kohn-S ham LDA potential [33]. The hatched region represents the positive background, whose surface is a t z = 0.


effective potential is still ongoing. Only one solution of the integrai equation for jellium

slabs exists in the Literature and this is for finite jellium slabs [46]. A finite slab calculation

is somewhat lacking because of the close proximity of the two surfaces to each other, and

ideaily, one would like to be able to i den te the contributions to surface properties fiom a

single edge. One attempt at an OEP calcuiation for semi-infinite jellium exists [47], as does

an attempt at a HF calculation [48], similar in spirit to the calculation of Bardeen. Both

these calculations assume a iinear ramp model, and treat the ramp dope as a variational

parameter. A more complete minimization should deal with more parameters so that the

effective potential, and wavefunctions can show full structure.

Chapter 2

The Asymptotic Form of the OEP

In this chapter we obtain the asymptotic form of the OEP far into the vacuum. Our

discussion begins by commenting on a previous study and we show fiom a simple argument

t hat the published analysis is highly questionable. We t hen derive the integral equation for

the OEP of the jellium surface, and show how an expansion about the Fermi level, of the

orbitals far into the vacuum can be used to obtain the asymptotic form of the OEP. We first

consider orbitals derived from an asymptotically flat potential, and discover that they give

rise to an image-like potential dserent fiom the previously reported result. We conclude

by demonstrating that our asymptotic form is the correct self-consistent asymptotic form

of the OEP.

2.1 Comment on previous studies

Talman and Shadwick [2O], and later, Krieger et al- [49] obtained the optimized effective

potential (OEP) for finite systems. They discovered that asyrnptotically the OEP gives the

correct potential for atoms, namely that V,O(r) - - l / r . Their derivation hinged on the

fact that for a finite system, the electron states form a discrete set. In the KS scheme,

electrons in an atom decay exponentially int O the vacuum, wit h the energet icaUy highest

2.1 Comment on previous studies 23

occupied orbital decaying the slowest- In the asymptotic region only this orbital contributes

significantly to the density, which simpMes the analysis somewhat. The metal surface is

an extended system, and the same ideas hinging on discrete sets of states no longer apply.

We should first clearly distinguish the a e r e n c e between b i t e and extended systems.

For the moment consider a simple finite system, the neutral jellium sphere of radius R,

with an electron a distance z radially away from the surface- Of course, this means that the

electron is a distance r + R away fiom the centre of the system. If we take the a >> R limit

then we have the same class of system Talman and Shadwick considered: a h i t e system

with only one electron far into the vacuum. To mimic the surface of an extended metd

within the jellium sphere model, one can take the R + CU and z + ca limits, since this

takes the sphere's volume to infini@, but keeping S T / ~ T << z « R, see Fig. (2.1). In this

Limiting case the curvature of the sphere goes to zero, and we reproduce a pIanar surface.

Because the buk of the metal is extended, we no longer have a discrete set of states.

Lnstead, there is a continuum of states, and for any r far into the vacuum there is a group

of occupied states near the Fermi level that must be considered. This is the idea we take

in the jellium slab approach: far away kom the surface we consider a continuous set of

orbitals. This allows for an expansion of the orbitals about the Fermi level, which as we

will see, dows the asymptotic form of the optimized effective potential to be determined.

In principle, it may be possible to approach the sirface case through the jellium sphere

analogy made above by considering spherical jellium and taking the appropriate limits,

but the ordering of the levels and the degeneracies are complicated making the approach

unattractive.

The asymptotic form of the exchange potential for a jellium surface has also been studied

by Solomatin and Sahni [50, 511. By considering the orbitals for the finite-linear ramp model,


Figure 2.1: Cornparison of the two lirnits giving rise to finite or extended systems. a) The finite jellium sphere in the top figure gives rise to a discrete set of occupied orbitals at the position r. If z is large enough there wili o d y be one occupied orbital. b) If we consider arbitrarily large jellium spheres, as in the bottom diagram, but we keep r < R then to the electron a t z the sphere begins to look like a surface. Even though we can still take the large z limit, since z is always significantly smaUer than R we are not in the regime where only one orbital exists a t z, but instead there is a continuum.


they first determined the asymptotic form of the Slater potential'. Their result is,

~ + 2 a ! I n c i ! 4-y~) = - for z » 1. m ( l + CG) '

They then argued that their result was independent of the mode1 used for the surface,

and claïmed that Eq. (2.1) is the correct self-consistent Slater potential. By considering

results in [49], and [52], they then extended their arguments to claim that the exact density

functional exchange potential, Vy(z) , and the optimized effective potential, V$ (2) , both have

the same asymptotic form of +&-'(-'(r), for the jellium surface. These arguments are not

the most convincing, considering that the results from [49] pertain to discrete systems and

that the other result was used within [52] to End an asymptotic form proportional to - l /z2

for the j ellium surface. In a later report [51] Solomatin and Sahni supported their result

by applying the definition of the exchange potential, Vx ( r ) = dEx/6n(r), to the exchange

energy, Ex, expressed in terms of the Slater potential as in Eq. (1.10),

Immediately taking the functional derivative gives

Solomatin and Sahni then suggested that the second term above was negligible compared

to the h s t - We believe that this clairn is wrong, since it has been demonstrated that

the leading contribution to the OEP is the entire Slater potential [19, 49, 531, not half of

it. Addïtionally, it has been noted [54] that a more careful evaluation of the functional

derivative taken above demonstrates that the second term in Eq. (2.2), contains another

'We use the term Slater potentid, instead of inverse radius of exchange hole Eq. (1.9), to follow the authors of [SOI.


term of (r) . The reason for this is in the non-locality of the exchange potential. To

demonstrate this for the jeliium surface, we can mite the total exchange energy as

where we have defined

and 2 is the unit vector along the axis perpendicular to the surface. To obtain the asymptotic

potential, variations of Ex are taken with respect to 6n(z), for z large. If we introduce a

constant, c, such that O < c < z, and use the symmetry in h(z1, zrf) we can rewrite the

integrals of Eq. (2.3) as

1 = 00

E, = 5 1, dn' n(zf ) LL dz" n(zff) h(zl , z") + Lm di r n (d ) /__ di" n(zM) h(zl, z")

(2-5) - Lm dzr n(d) lm dz" n(zf')h(zl, zrl).

Since we are interested in the potential far from the surface, the variations in the density

are taken out in the vacuum. The Erst term contains only integrals over the buk where the

density will not vary, and so it will not contribute to the asymptotic form of the potential.

The third term is the interaction energy between the electrons out in the vacuum, and is

an exponentially srnall contribution, and so we neglect it. The second term represents the

interaction of the electron in the vacuum, with the bulk metal. Functional differentiation

gives,

00 6 00

V~ (z) = lm dzt n(zl) lx (z, zl) + dzl n (zt 1 - (2-6)

Substituting back in for h(z, 2') shows that the first term is the Slater potential at z. This

2.2 Integral equation for the OEP in the jellium surface mode1 27

demonstrates that, as for finite systems, the full Slater potential, and not one half of it,

gives a leading contribution to the long range exchange potential. Unlike finite systems

though, the Slater potential is not the entire asymptotic form. The proper way to approach

exchange only DFT, as emphasized in [21], is through the OEP method. In what follows

we examine the optimized effective potential for the jelliurn surface, and veri@ that in the

asymptotic region the entire Slater potential is present, as well as demonstrate the existence

of other image-like terms.

2.2 Integral equation for the OEP in the jellium surface

rnodel

Following the procedure Talman and Shadwick used for the spherically syrnmetric case of

atoms, we derive the integral equation for the OEP particular to the jellium surface model,

Although we give a fairly detailed outline below, some resdts are confined to Appendix A,

and we make reference to them as necessary. Defining the exact exchange energy functional

a s the Hartree-Fock energy using Kohn-Sham orbitais, as specified in the introduction, the

exact exchange potential within DFT is given by

We do not know how to perform this derivaiive, since Ex is d e b e d in terms of orbitals, and

there is no method yet to obtain the functional derivative of those with respect to density.

Alternatively, we find the optimized effective potential, which is the local potential,

Vs(r), that when used in a single-particle Schrodinger scheme yields eigenfunctions whose

Slater determinant minimizes the expect at ion value, E, of the Hamiltonian.

Using the completeness of the eigenfunctions we can express the functional derivative

2.2 Integral equation for the OEP in the jeilium surface mode1 28

of the total energy with respect to variations in the effective potential as

where the sum is over all k. The above expansion, as Talmann and Shadwick pointed

out, can be simplified using the Euler-Lagrange equation and perturbation theory. When

GE/JVen(z) = O is solved, the minimum is achieved, and we Say that the effective potentia1

and its wavefunctions are optimized. To keep them distinct fkom KS or HF quantities we

denote them with a superscript o.

As mentioned, the above orbitals satisfjr a Schrodinger-iike single-particle equation,

and have the same form as Eq. (l.26),

If we consider the potential, Veff(z), to be perturbed about the optimized potential, so

that Veff(z) = Qff(z) + 6V,R(r), then f7om perturbation theory we immediately have that

to f is t order the wavefiinction is aItered £rom its optimized value, $k(r), by

where,

and the prime on the summation denotes that the sum skips k' = k. Upon functional

2.2 Integral equation for the OEP in the jeilium surface mode1 29

derivation, Eq, (2.9) gives

which has the effect of projecting states into a subspace orthogonal to $f ( r ) . Note that the

left hand side of Eq. (2.11) appears in the integrand of Eq- (2.7). By considering the total

energy, we can h d an expression for the other factor in the integrand. When the effective

potential is optimked, the total energy of the electrons is given by

where the h a 1 term is the HF term

OCC OCC

E,HF[{$Z}J = - C C JJd3rd3rr ar (TI$:? 17' (rr)l/>i - rfl (r1ME1 (4 (2.13) k k'

and

is the density evaluated with the optimized orbitals. Taking the functional derivative of the

energy expression: Eq. (2 .12) : and using Eq. (2.8) we get

where we have used the definition


Now we define the local exchange potential, V,O ( r ' ) : to be the clifference between the effective

potential, and the combined external and Hartree potentials,

This definition lets us mi te Eq. (2-15) as

1 SE - 2 Wi(r l ) = [eh - V: (2) + ( r ' ) ] $r ( r ' ) .

It should be noted that Eq. (2.18) is valid only for occupied k states, and that vf ( r ' ) is the

usual non-local HF exchange potential. Now we use Eq. (2-11) and Eq. (2.18) in Eq. (2.7),

dong with the identity

t O arrive a t

/ d3r' [V: (2') - ug (T')]G* ( r ' , r ) $ r ( r ' ) + C ( r ) = O. (2.20)

This expression has dso been derived by others [19: 20, 461, and represents the general form

of the integral equation for V,O(z). We now tailor this to the jelliurn surface model.

In Sec. (A.l) we show that in the large A limit we can rework Eq. (2.16) to be

so that, as we expect from symmetry arguments, this HF-like term, u:(r1), is really only a

function of z'. Next, we use the jellium surface orbitals of Eq. (1.26), to rewrite the integral

equation, Eq. (2.20),


Employing in Eq- (2.22) the identity,

derived in Sec. (A.2), where

immediat ely gives

The k sum over occupied states consists of a sum over k l l states and a sum over the k

states. The range of summation is restricted so that ki + k2 5 kg, and we wiil always

perform the k l l sums first. We represent the set of all occupied k l l states by a disk of radius

(k; - k2)1/2 in the k l l plane, which we denote by the symbol D. In the large A M t the k l l

states become more dense, and the sum can be replaced by an integral,

Similady, in the large L Iimit, the f i sum can be replaced by a one dimensional integral,

Performing the integral over the k l l states in Eq.(2.25) we arrive at the integral equation


for the semi-infinite jellium surface model:

where we have defined

Furthermore, we show in Sec. (A.3) that the kll sum in Eq. (2.29) can be performed, so that

u: (2) satisfies

where " JI (bx) Jl (b'x)

F (b, 6') = bb' /i dz z,/-

is a function &st introduced by Kohn and Mattson [55] with

bf = (g - kf2) 12" - zf 1 , and Ji (x) is the first order Bessel function. Later, we will require

the asymptotic form of F(b, b') and so to better visualize the function we provide a contour

plot of F(b, b') in Fig. (2.2).

The derivation of Eq. (2-28) could also have proceeded by considering the energy as

a function of just the orbitals perpendicular to the surface, ?,bk(z)? instead of the entire

orbitals. In that case Eq. (2.7) would have been different, involving just a sum over the

k states. This alternate derivation is more cumbersome, but leads to the same integral

equat ion.


Figure 2.2: A contour plot of the Kohn-Mattson function F(b, b')

2.3 Asymptotic expansions of the orbitals: part 1 34

2.3 Asymptotic expansions of the orbitals: part 1

Our approach in determining the asymptotic form of the effective potential kom Eq. (2.28)

requires the form of the orbitals $~k(z) far into the vacuum. We do not know the exact form

of the potential in the vacuum, since this is what are determining, so we first consider a

potential that is asymptotically flat, that is V&(z > 2) = constant for some large value

2. Some mode1 potentials that fall under this category wodd be the finite-step potential

and the finite-linear-ramp potential. We use the fact that the Fermi-levei orbital decays

exponentidy less fast than any other occupied orbital, so as shown in Sec. (A.4) the orbitals

for these potentials can be expanded about the Fermi level as follows

where a' = eF/W, W is the work function, and Ak = l+ - k.

Eq. (2.32) demonstrates that for any asymptotic position z, there is a band of orbitals

concentrated about the Fenni level which contribute to n(z ) . The thickness of this band

is châracterized by the decay length, Ak, = l/az, and so depends on the exact position z.

As z grows, the band slowly becomes thinner, since Ak, depends inversely on r. Later we

will find it important to consider the expansion associated with other classes of potentials,

ancl the resulting asymptotic form for the orbitals.

2.4 The asymptotic expansion of the OEP

To extract the asymptotic form of the OEP for the jellium surface we consider Eq. (2.28)

in the large z limit. Whenever an asymptotic orbital, at z appears, we use the orbital

expansion of Eq. (2.32) directly in the integral equation, and divide out the Fermi level

orbital. We drop the XAk term in the orbital expansion, since we are interested only in the

2.4 The asyrnptotic expansion of the OEP 35

leading term of the asymptotic expansion. Applying this once gives

Even though the k s u . should be performed only for a range of values just below kF we

have left the k sum over ail occupied states since the exponential damping ensures that the

leading contribution will corne kom the terrns within a band of thickness Ak, below the

Fermi level. The exponentials also guide us in our solution, acting as a reservoir of factors

of Ako; every order of Ak occurring in the integrand will pull a factor of Ak, from the

exponential.

To isolate the effective potential we operate with [ek, - h(z)] . This will have the effect

of extracting V,O (z) from inside the integral. In Sec. (A.5): we show that

Using this in Eq. (2.33), and integrating over the delta function we find that in the z > 1

limit ,

AL 00 I = e--/ dzr [ p ( l i ~ - k 2 ) ~ ~ ( z f ) -U~?(Z~)] [$J;( r 2 0

-00

11 +&)- (2-35) k < k ~

The above tri& has introduced another orbital at z so we again use the expansion of

2.4 The asymptotic expansion of the OEP 36

Eq. (2.32) for this orbital, and cancel the resulting Fermi level orbital to leave

At this point we should comment that if V:(z) is a solution to this equation, then trying

V',(z) + C where C is an arbitrary constant, as a solution in Eq. (2.36) immediately gives

The orthonormality of the orbitals then gives us that V,O(z) + C is also a solution to

Eq. (2.36), and we have the option of choosing C as we see fit,

We proceed to evaluate the terms in Eq- (2.36). Replacing the k sum by an integral,

the first term on the left h m d side of Eq. (2-36) becomes

which is easy to evaluate. Using Ak, = l / ( a z ) , the leading order term in z? of (2.38) is

The f i s t term on the right hand side of Eq. (2.36),

is also easily dealt with. In the bulk region, the potential V,O (z) is very flat, so we chose a

position in the metal: z = -a, where a > O, where the wavefunctions have their asymptotic


form. In the vacuum region, the potential slowly approaches a ~ o n s t â n t ~ ~ but the orbital5

decay exponential away from the surface, since they are in a classically forbidden region.

Splitting up the 2 integral at z' = -a this term is less than

where V,O is determhed up to a constant. Since 1@;(zf) l 2 has a 2/L normalkation factor,

the fbst term in the square brackets is of the order V,O(-L), and the second term is of the

order a/L . The remaining sum over k states leads to precisely the same sum as in the term

we previously considered, Eq. (2.38) - Thus the effect of Eq- (2.40) is a contribution to the

height of the barrier relative to the bulk value. Therefore it plays no role in the asymptotic

form, and so we ignore it3. Our integral equation, Eq. (2.36), in the r + oo E t has been

reduced to

The two terms on the right hand side are more difficult to work with. We first consider the

term

which, using Eq. (2.30) : can be expressed as

-2 Oo $$ (z")9kt (z)$Ji, (2") F (b , 6' ) - e / _ _ c ~ k<kF kl<kF $Ji (4 I Z - z"I3 '

'We have not forgotten about the assumption that the potentiai is flat in the lacuum! This is just a more general consideration.

3The next highest order contribution from this term can readily be seen to be of order l/z2. Since the h o k sums are identical, the relative expansion to higher orders of Ako on each side is the same, and the first order terms cancel.


Inserting the asymptotic forms for the orbitals4 in z and converting the k sums into integrals

gives,

The variables z and 2'' are coupled in the integrand by F(b, b')/lr - FI3, so we look for an

asymptotic expansion of F (b, b') . We

b =

know that

and that k and k' are restricted to be very near kF by virtue of the exponential factors

appearing in Eq. (2.45), so that b - J-12 - z"l: and b' - d-1~ - r" 1 - Since

Ali - l /az , we see that b - JG, and similarly for bf. Therefore b and 6' are both large

in F (b, 6'): " JI (bx) JI (b'z)

F(b, b') = bb' dr x , / W -

We know that Ji (bx) decays like cos(bz + 7r/4)/& for large bx. As b becomes larger. the

contribution to the integral cornes fiom smaller values of x so we keep the upper bmit a t oo

and take the small x expansion of the denominator in the integrand of Eq. (2.47) to obtain

" JI (bz ) JI (b'x) F(b, b') - bb' 1 dx

x

This integral can be found in Watson [56] to be5

b: F(b, b') - - , b< = rnin(b, bf). 2

(2 -49)

4 ~ h e orbital in the denominator of Eq. (2.44) should be of no concern, since we are in the large r region where there are no nodes present in the orbitals.

5?Tote that (2.49) can be verified by inspection of Fig. (2.2).


Returning to Eq. (2.45), substituting for b and b', and reordering the integrals we have

where

and Ak< is the lesser of dk and Ak'. RkF (2; z') plays another role later in this report, when

we discuss the exchange hole density. To evaluate RkF (z, 2') we note that the surface region

is small compared to the actual thickness of the crystal, and the orbitals exponentially decay

outside the metal, so the important contributions fkom the integral over z' in Eq. (2.50)

will corne fiom the bulk region where the orbitals have the form dF sin(kz - ~ ( k ) ) . The

result obtaïned in Sec (A.6), is

1 ~ ' (2" - 3a2z2) sin (2kFZf - 2y(kF)) RkF (17 z') = 4az(a2z2 + ta) +

4(a2z2 + zQ ) (2.52)

az(3zR - a2z2) COS (2kFz' - 2y(kF)) + 4(a2z2 + z")~

For h e d z the first tenn rnonotonicaLly decays with increasing -2, and the second and

third terms oscillate about zero. When used in Eq. (2-50), only the first term survives in

the large z limit, and we find

It is worth noting that if we equate the right hand side of Eq. (2.53) to (2.39), and neglect

the other terms in the integral equation, V,O(z) becomes the asymptotic form of the Slater

potential, as derived in [50] for the finite linear ramp model. As part of the next chapter

we will show how to derive the asymptotic form of the Slater potential, or inverse exchange


hole radius, by using our expansion of the orbitals about the Fermi level.

The h d term of Eq. (2.42),

is dl that remains to be considered- Treating it in a similar way as the previous term? we

&st use Eq. (2.30) to rewrite the term as

We see that only the k sum, and not the k' sum, is confined to just below the Fermi level.

Since the integrand of the z' and z" integrals are darnped by the lz' - zl'l factor: we are

only interested in the region around z' z z". The function F(b, b') has an expansion here

also. For values of k near kF we find that

since 4 k - Ak, oc: l/z. It will become apparent shortly that we must keep 6 exactly. We

may use the small argument expansion of J I ,

.fi (bx) N bx/2,

with Eq. (2.56), to obtain

The integral can be evaluated [57] so that the final term of our integral equation, Eq. (2.42),


becomes

We stiU need to isolate the k dependence contained in the orbitals, since we wish to perform

the leftmost sum. To accomplish this, we can expand the orbitals about the Fermi level:

The validity of this expansion for ail z wiU not be proved, but we note that in the asymptotic

regions in the vacuum, and bulk, the orbitds have known analytic forms which allow for this

expansion. Using Eq. (MO), and performing the k sum we find that the final term, (2.54),

is asymptoticdy equal to

where,

The first term in Eq. (2.61) adds a constant to V,O(z), but the second term makes another

contribution to the asymptotic form.

The realization of this constant term, and the constant contained in (2.40) leads us to

wonder if there are yet more contributions of order l/z to the OEP when we include the

higher order terms in Eq. (2.34). Considering the next term in Eq. (2.34) introduces another


expression to be considered,

Llk 00 d Ake-.*. di ' (k~Akv:(r') - $(il)) $J; (2') bG* (z', i) - a-G'E (zt3 z)) .

k < k ~ dz

We cannot rigorously evaluate this, because in Gg(z', z) the k' sum extends over a l l possible

k' including the unoccupied states- However, it is possible to see that this may contribute a

further asymptotic term by assumiog that since the denominator of Gg(zr, r) is ek, - eki the

major contribution comes fiom near the singularity, k' h: k. We relegate further discussion

on this to Sec. (-4.7) since the argument is sketchy.

The final result for the asymptotic expansion of the OEP for the jellium surface is

plus possible additional terms. This shows that for orbitals which have an expansion as

in Eq. (2.32) the resulting effective potential is image like, but does not have the correct

coefficient 1/4. Instead, the coefficient depends on the details of the barrier, and hence the

orbitals, and of course the mean jellium density. Additionally, we have seen that consistent

with our initial argument, the entire Slater potential contributes to the Ieading tenn, not

half of it.

In the final term of the integral equation the leading order z dependence appears because

of considerations in higher orders of Ak. This implies that we must investigate the details

of the asymptotic potential, to see if the changes induced by a more accurate potential will

affect our result for the asymptotic form.


2.5 Asymptotic expansions of the orbitals: part 2

The result of the previous section indicates that the asymptotic expansion for the potential

in the vacuum is image iike, with a coefficient that depends on the details of the potential.

For the derivation though we employed an expansion of the orbitals for a potential that

was quickly approaching a constant in the vacuum region. If instead, we use an image-like

potential of the form V(z) = V(l - /3/z), where V = eF + W : and ,û > O, as we show in

Sec. (A.4) the orbitals resulting fiom this are

where d = 2(V - E ~ ) , B = 2VP, and D (k) is a normalkation constant. The expansion

about the Ferrni-level is aow

where q = Da(1 + a2) /2 . The important thing to notice is that the correction to the

Fermi level orbital now contains a l n ( z m ) term, which is of b e d value throughout the

derivation. The term, (1 - q Ak ln(z@)) should thus be treated as a correction Linear in

dk, but we have already considered these, and we have shown that they amount to a l/z2

contribution. Therefore, the correction to the OEP, when going kom the Bat potentials, to

the image type is of order (ln(r) concluding that the self-consistent form of the OEP

is image like, and contains the entire asymptotic form of the Slater potential as opposed to

half of it as previously reported.

In the introduction we mentioned that Eguiluz et al. [30] numerically determined that the

exchange potential from a finite jellium slab decays faster than - l/z. Sahni has argued that

using his analysis, a finite slab would indeed give rise to a -1/z2 dependence for the Slater


potential, but we have not verihed that for the OEP. The classicai image potential for a h i t e

jellium slab remains -1/4z, so the conclusion of Sahni leads to some interesting properties

for the exchange-correlation hole in finite slabs, since the part of it due to correlations

must now compensate for the disappearance of the image contribution from the exchange

hole. We try to gain some physical inçight into this resdt in the next chapter where we

employ the same kind of orbital expansion, about the Fermi level, to study the exchange

hole. Additionally, our numerical met hod to calculate the exchange energy and OEP for the

semi-infinite jeiiïum surface, which we introduce in Chapter 4 may provide some numerical

clues to solve this connundrum.

Chapter 3

The Exchange Hole for Asymptotic

Electron Positions

We try to gain physical insight into the resuits of the previous chapter by studying the

exchange hole due to an electron asymptotically fas from the surface. We derive some prop-

erties of the exchange hole analytically that have not been tractable to previous numerical

st udies.

3.1 Introduction

The exchangecorrelation hole, n,,(r; r'), is the depletion of charge around an electron at

r due to Fermi-Dirac statistics and Coulomb repuhion. As an electron is removed from

a metal surface, it is believed that its exchange-correlation hole should be left behind at

the surface and result in the classical image charge experienced by the electron in the

asymptotic Lirnit [58]. The part of the exchange-correlation hole due to the Fermi-Dirac

3.1 Introduction 46

stat istics, n, (r; r') , is c d e d the exchange hole or Fermi hole given by,

and only affects electrons of parallel spin. It also satisfies the perfect screening sum rule, and

hence integrates to unity. For electrons deep in the bulk region of the jellium the exchange

hole has the uniform gas form

where ji (z) is the &st-order spherical Bessel function given by ji (x) = (sin x - x cos x) /x2.

For Slater determinant wavefunctions, the physical sigrScance of the exchange hole density

is revealed by its relation to the pair function, I'(r;rf), given by [59]

The earliest study we were able to h d on the exchange hole of an asymptotic electron

in the jellium surface mode1 was an analytical approach within the infinite and finite barrier

modeIs of the surface, reported by Juretschke [60]. He noticed that as the electron position

along an axis perpendicular to the sutface became larger, the exchange hole took on a

Iaminar structure like successive disks iined up along the axis, but he concluded that the

exchange hole remained within the surface region as expected for the exchange-correlation

hole. More recently, studies have been performed by Sahni and Bohnen [61], and Harbola

and Sahni [62], in which they implemented the finitestep and linear ramp models for the

jeIlium surface. Their numerical studies for various electron positions both in the bulk and

vacuum regions convincingly demonstrated t hat as the electron position moved from wit hin

the metal to the vacuum region the exchange hole is left behind in the metal, and as the

3.2 Results and discussion 47

electron is taken further out, the exchange hole becomes delocalized throughout the bulk

of the crystal taking on the laminar structure noticed by Juretschke. The physical reason

behind the delocalization is that when the electron is far out from the metal, only the most

energetic bulk electrons can extend out far enough to interact with this electron. In tbis

chapter we ver@ the numerical results of [62] with mode1 independent results based on

the orbital expansion employed in the previous chapter, and we investigate some analytic

details of the exchange hole for an asymptotic electron inaccessible to numerical studies.

3.2 Results and discussion

3.2.1 The asymptotic density and density matrix

As shown in Sec. (A.4), the one-dimensional orbital riinning perpendicular to the surface,

?,bk(z), for large z, can be expanded about the Fermi level orbital ?,bkF(z) as follows,

Using this expansion, the asymptotic

Eq. (1.28) to be

n(z) =

We see £rom Eq. (3.4) that unlike the

form of the density, n ( x ) , is easily calculated using

atomic case, where the ground state density decays

like e - 2 ~ r [l], where M is a positive constant, the metal surface density decays faster. Our

result is diffèrent from the result: z-' e x p ( - l m z ) , for solid surfaces given by Almbladh

and von Barth [29], but is in agreement with the earlier result of Zaremba and Kohn [63].

We believe the difYerence is because the authors of [29] did not properly account for the

degeneracy due to momentum states parallel to the surface.

To arrive at the exchange hole we also require the single-part ide density matrix, n(r; r') ,


in addition to the density. Performing the same types of integrals as were needed in the

previous chapter, the single-particle density matriu for large z becomes

3.2.2 The exchange hole

Using Eq. (3.4) and Eq. (3.5) we can determine the exchange hole. Without loss of generalik

we set p = O, so that r = ZZ and consider the exchange hole density:

From Eq. (3.6) we can immediately gain some insight into the behaviour of the exchange

hole for electron positions far into the vacuum. Since Ak is of order l/z, the argument of the

Bessel function, (2ksdk)tpr, is also small unless p' is of order zl/*. For small arguments,

J I (x) x / 2 , and so it is clear that a t each position, z', the exchange hole is laterally spread

out over a radius of about z1I2. Of course, the details of the nx will depend on the rest of

the expression in Eq. (3.6). Using some typical values for simple metals, and numerically

evaluating Eq. (3.6)) we display a portion of the exchange hole due to an electron far into

the vacuum in Figures (3.1) and (3.2), which clearly show the laminar structure of the

exchange hole within the bulk. We can also infer £rom the figures that the hole is extended

into the solid, and dimishes rapidly into the bulk region. The curved, bean-like structure is

consistent with the findings of Kohn and Mattsson [55] for the exchange hole of an electron

near the surface, evaluated for the linear ramp model, We can now investigate how the

exchange hole varies with z.


-20 - 10 O 10 20

Radial position p'

Figure 3.1: Contour plot of the negative exchange hole for an electron a distance z = 8 a.u. f?om the surface. We have chosen the parameters a = fi, kF = 0.489 a-u., and y ( b ) = 0.955-


O

Radial position p'

Figure 3.2: Contour plot of the negative exchange hole for an electron a distance z = 20 a-u- fIom the surface. We have chosen the parameters a = fi, kF = 0.489 a-u., and y ( k ) = 0.955-


3.2.3 Density profile of the exchange fiole

For p not too large, the argument of Ji, in Eq. (3.6) is small, and the limiting form of the

Bessel function reduces the exchange hole to

n,(z; zr) can be interpreted as the profile of the exchange hole density along a Line containing

the asymptotic electron and running perpendicular to the surface. For z' in the bulk region,

the orbitals are roughly f l f k ( z ) , where fk(z) = sin(kzt - y ( k ) ) , and so we can evaluate

the k integral for this region to obtain,

where = -y(&) Y c = kpz; x = k z r , and E is the bulk densityl. This damped siniisoidal

behaviour with respect LO z' of the exchange hole profile has been obtained nurnericaily

within the finite ramp, and linear ramp models [61, 621. From Eq. (3.8) we clearly sec that

deep in the bulk the exchange hole profile decays like .- (1 -cos(2x - 2 7 - ~ ) ) / x ~ once -x » <. We provide sample illustrations of this profile in Fig. (3 .3) . Our plot is representative of the

figures in [62], where the authors also noted the strong z dependence2 by presenting their

calculations over a range of three z values, and that the minima of the exchange hole become

evenly spaced with a separation of T/?Q for -x >> C,. Because of their numerical approach

they were not able to see the rapid decay experienced by the exchange hole once -x >> C, - - -

' ~ h i s specific choice of dirnensionless units is only to make a connection with the figures of [62]. W e also inform the reader that the sign converition for our work is difFerent than that of [62] so that our exchange hole is the negative of t h e h

'1n [61] and [62] the authors refer to the density profile as the exhange hole cross-section.

3.2 Resuits and discussion 52

EIectron at <=8

1 Electron at <=ZO

Position dong crystal a i s , z (dimensionless units)

Figure 3 -3: Normalized profile of the negative exchange hoIe density, -n, ( r , 2') / (n/2) as a function of z' for three different z values. We have chosen the norrnalization and units for cornparison with the calculations of [62]: a2 = 2, C = b z , x = bz', and p = 0.955.


being Zimited by their numerics to x > -20. On cornparison with the corresponding figures

in [62], we see that there is poor agreement for C = 8, the values dinering by about a factor

of 2- This is probably due to = 8 not being an accurate representation of an asymptotic

position which we have assumed. In the Iower two graphs, where = 20 and C = 50, we have

a much better fit with the results from [62]- The zeroes occur at the same positions, and the

densities match closely3. This implies that the asymptotic region is reached around C = 20.

We can now try to comect this with the numerical resdts of Eguiluz et al., [30]. Ji the

dimensionless unit s used above the jellium slab t hickness studied in [30] was approximately

L = 25, and the asymptotic positions were taken to about C = 8. By comparing our exact

results for the asymptotica,lly distant electron to the numerical ones of [62], we see that

this position is not in the asymptotic regime, and so the results of [30] for the exchange

potential should not immediately be expected to apply in the asymptotic limit.

We are now in a position to examine in more detail the form of the exchange hole for an

electron far into the vacuum. One way to describe the behaviour of a distribution, such as

a charge density, is through its moments. We wiU first concern outselves with the average

position along the z' axis of the exchange hole profile,

The denorninator of Eq. (3.9) is just the integrated vaiue of n,(z, 2') and in the large z

31n their study, Harbola and Sahni investigated both the step and 1inea.r ramp models of a surface. We attach more significance to their step model calculations rather than their linex ramp mode1 ones, since the linear ramp model potentiai is unbounded in the t + oo Iimit, not fiat like the step potential or the exact potential. This results in an incorrect asymptotic behaviour of the orbitah, and consequently incorrect behaviour for the density. The exchange hole depends crucially on the density for inhomogeneous syçtems, since it appears in the denominator of Eq- (3.6). This anomdous vacuum behaviour of the orbitals was noticed in [62] but they still attached physical significance to their results.


J dz' nx(z, z') - -lq7/27raz-

This just shows that as we consider electrons further and further away from the surface, the

magnitude of the exchange hole on the axis p' = O diminishes slowly. This is clearly seen in

Fig. (3.3). The numerator of Eq. (3.9) turns out to be be a constant, kF/n2, in the large z

limit, so that 7 = -2azlrr. Since the exchange hole in the large r limit factors into 2 and

p' coordinates, this supports our contention that the radial extent h a . a z ' / ~ dependence.

We can also look at how the exchange hole prome spreads along this axis as < is increased

by b d i n g the mean square deviation, aif2, given by

After some algebra, we h d that = ( z ~ r ) ~ (1 -4/7r2), which demonstrates that the profXe

spreads dong the zr axis proportional to the distance into the vacuum of the electron. From

the above we have seen that the exchange hole charge near the axis defined by the electron in

the vacuum moves further into the bulk as the electron becomes asymptotic. To incorporate

the charge off the axis we c m examine the planar integrated exchange hole.

3.2.4 Planar integrated exchange hole

An argueably more usefd quantity than the profile of the exchange hole, is the planar

integrated exchange hole, (n,(z; z ' ) ) , given by

(n,(z; t')) = / d2p' n, ( z ; r i ) . J

4 ~ h i s cornes just from the leading te- alone. The trigonometric t e m of the exchange hole profile introduce Si and Ci functions in t. It so happens, that when these t e m s are considered in the as-ymptotic - -

Iimit the contributions kom them decay exponentially with large z.


We can infer that (n,(z; z' ) )dz gives the total charge contained in a plane perpendicular

to the zf axis of thickness d d . The integral of (nx(z ;z ' ) ) over aU 2 of course gives unity

since this integral is the perfect screening sum rule discussed in section Sec. (1.2). Using

Eq. (3.6), and integrating over p' we find that

This integral is the function RkF (z , z') that we encountered in our study of the asymptotic

form (see Eq. (2.51)). Taking the result £iom Sec. (A.6) we find that the planar averaged

exchange hole density within the bulk region for large z is

a ~ s (z2 - 3a2C2) sin (22 - 237) + a2c2 (3x2 - ct2c2) COS (22 - 2 7 ~ )

(a2C2 + x2)2

where the variables have the same definitions as before. Integrating this function over z'

for large z gives unity and a small contribution coming fi-om the trigonometric terms that

decay with z, which provides a check on this result. In [62] the behaviour of the exchange

hole deep in the bulk was determined to behave as l/x2. This is evident hom Eq. (3.14),

which shows more precisely that this behaviour occurs only when -x >> <. In Fig. (3.4) we

display the behaviour of the planar integrated exchange hole with respect to z'. Again, our

analytical results compare well with the numerical ones of Harbola and Sahnis for large <. It is interesting that we have a much better agreement for the = 8 plot then we did for

the profile of the exchange hole. We do not understand why this is the case.

Since the planar integrated exchange hole behaves like l / z R , for very large -z l , relevant

moments such as the center of mass of the exchange hole are inaccessible since the required

'.Again, we only compare to the finite-step model calulations and not the linear ramp model ones in [62], since these later calulations display an incorrect l / z f behaviour deep in the bulk, as can be verified from inspection of the figures in [62].


Electron at <=8

Position dong crystal ax is , x (dimensionless units)

Figure 3.4: Planar integrated exchange hole density, - (n,(z, z r ) ) / ( 3 b /7ï) as a function of z' for three different a values. We have chosen the normalization m d units for cornparison with the calculations of [62]: cr2 = 2; C = ] î F i , x = k F i , and = 0.955.


integrals are singular. Harbola and Sahni interpreted this to mean that there was always

charge a t the far surface, a t z' = -W. Because the moments are inaccessible, Harbola and

Sahni instead decided to numerically study the quantity

which is a nice idea, since it gives the total amount of exchange hole charge within a distance

into the bulk equal to the position of the electron. They argued that in the large z limit

t his quantity would go like 1 - 1/z7 and that formally - lim,,, 5-' dz' (n, (z; 2')) = 1. We

will shortly explain why this is incorrect.

Using the same physical parameters as in [62] we calculate the q~~ant i ty , (3.15), for our

analytical representation of the planar integrated exchange hole6, and display the result in

Fig. (3.5). Our figure is similar to the same quantity in [62], showing a large variation near

the origin, and tapering off to a constant for large r. The fact that this tends to a constant

not equal to one is odd, since it contradicts the above claim of Harbola and Sahni. This

confusion has its source in the long range form of the planar integrated exchange hole deep in

the bulk, whose leading order term, can be found from Eq. (3.14) to be -2kFa</n(a2 c2+x2).

Although, the function behaves like -1/z2 for large -x the properties are critically different

near the lower Lmit of the integral, where -x ==: <. Calculating the quantity in (3.15) for

large z, the trigonometric terms become negligble, and only the leading term contibutes.

The result is that 00

dz' (n,(z; 2')) = 2 a r c t an ( l / a ) /~ . (3.16)

For a = a, which we used above for the figures, the limit is 0.3918. This compares well

with the limiting form of the plot in Fig. (3.5), and with the figure presented by Harbola

and Sahni for the same parameters. This implies that there is always a significant portion,

%ve terminate the integral at O instead of infkity. This is allowable because the contribution from z' > O is exponentially small


10 20 30 40 50

Electron position (dimensionless units)

Figure 3.5: Total exchange hole charge located between the surface and -z into the bulk. We plot the negative to compare with the figure in [62].

3.3 Closing 59

about 40%, of the exchange hole within the region between the position of the electron in

the vacuum and the position of its classical image charge in the bulk.

3.2.5 Asyrnptotic form of the inverse radius of the exchange hole

Finally, we note that our method allows an altemate derivation to that in [50] of the inverse

radius of the exchange hole for asymptotic electron positions. Using the large z form of the

average exchange hole given by Eq. (3.6), and the dehit ion of xL(r), Eq. (1.9) gives

Performing the p' integration and treating the Kohn-Mattson function, Eq. (2.31), as we

did in Sec. (2.4) gives

Keeping only the leading term in z gives the desired result,

which is identical to the result of Solomatin and Sahni [50].

3.3 Closing

A possible use of our exchange hole results could be in a self-consistent calculation of the

Slater potential. As mentioned in [64]? where the Slater potential was studied within the

finite-step model, a self consistent calculation of the Slater potential requires the evaluation

of the exchange hole deep in the bulk at every iteration. This is a time consuming process,

3.3 Closing 60

and knowing the exact behaviour of the exchange hole deep in the bulk may assist in the

computations.

Above, we have shown how the behaviour of the exchange hole is coupled to the electron

position when it is asymptotically far from a metal surface, and we have determined some

exact properties of the exchange hole. This though does not seem to shed any additional

light on the origin of the asymptotic potential derived in Chapter 2, than what has been

suggested before by Solomatin and Sahni [50]. Their suggestion was thzt since the exchange-

correlation hole is localized to the surface, in the bulk region the correlation hole is positive

and cancels the exchange hole. However, the correlation hole must integrate to zero so

there must be positive charge localized at the surface resdting in a surface charge region

independent of the physicd parameters of the metal.

Chapter 4

The surface energy and the OEP

4.1 Introduction

A recent paper by Kohn and Mattson [55], in which they used the Airy gas model to study

the edge electron gas hm renewed interest in the jellium surface model as a testing ground

for DFT. For example, Vitos et al. have used the Airy gels model for the jellium surface to

generate approximate local density functionals for bot h the single particle kinet ic energy [65]

and exchange energy [66]. Uniike the kinetic energy functional which can be checked against

an abundance of exact results, the exchange energy has a more select group to be tested

against. To test their exchange functional, Vitos et al. used some simple atoms, and the

OEP resdts of Mohammed and Sahni (473, who minimized the energy with respect to one

parameter in the linear ramp model.

There have been some inconsistent results for the surface energy of jellium. Li et al.,

in their Monte-Ca10 study [67], provided a cornparison of the literature values for the

surface energy of jellium and a wide range of exchange o d y results for the surface energies

is displayedl. Although, some of the discrepancies can be accounted for, a rigorous test for

'For the density corresponding to r, = 2 Sahni and Ma found an upper value to the exchange-only syface encrgy of -0.091 ev/A2 [48], Li et al. determined -0.141 ev/A2 [67], and the LDh d u e is -0.061 e ~ / . 4 - [38].

4.2 Surface energy 62

DFT approximations demands accurate self-consistent exact-exchange only calculations.

In this chapter we report our work towards establishg the semi-infinite jelliurn surface

mode1 as a benchmark for the exchange energy of DFT calculations. Our goal is to obtain

the OEP and surface energy for the jellium surface. One way to obtain the OEP is to solve

the integral equation, Eq. (2.28). This has oniy been accomplished for finite jellium slabs,

and it is unclear how to solve it for the semi-infinite slab. Moreover, even performing this

feat will not give the surface energy. Our proposal is to return to the definition of the OEP

and establish a procedure to directly rninimize the surface energy with respect to the local

potential. This will give the surface energy and OEP simultaneously, as well as allow us

to investigate the shape of the minimum. We will outline our approach here, reserving the

details for Appendix B, and explain how our scheme c m be implernented.

4.2 Surface energy

Consider a fmite slab of jellium with a large length and cross-sectional area A. The surface

energy, a, is the energy per unit area, required to split the slab along a plane, Say down

the centre, and to t hen move the slabs infinitely far apart. We focus on just one half of the

slab .md disregard the other which is at infiniS.. DFT states that we can express the total

energy of the system, E, as the sum

where the three terms in order of appearance represent the single-particle kinetic energy,

the electrostat ic energy, and the exchange-correlat ion energy. We are interested in exchange

only DFT, so we neglect correlations for now2. Each component of the energy has two con-

tributions, a bulk term that scales with the volume of the sample, Q, and a surface term that

*One may inchde a correlation energy densi@ functional easily since we are in pursuit of a potential to be used in a single particle scheme, and the density is easily calculated.


scales with the surface area, A. We can thus write o = os + os + fa,, where the terms repre-

sent the surface analogues of those in Eq. (4.1). Since we are interested in non-interacting

particles in an effective

surface kinet ic energy3

external potential, we c m use the result of Huntington [68], for the

o s 3

where we have fixed Veff[n; -a>] = O. Over the range of metal densities os is negative due

to the spreading of the electron density into the vacuum when the surface is formed. The

electrostatic contribution to the surface energy is given by

Extracting the surface contribution to the exchange energy is a more arduous task. In

many jellium mode1 applications the exchange energy is evaluated within the local density

approximation,

where ~ , ( n ( z ) ) is the exchange energy per particle of a uniforrn gas of density n(z ) . To

irnplement the OEP method we need to determine the surface contribution of the exact

exchange energy calculated using the orbitals ob tained from an arbitrary local effective

potential. Once this can be calculated efficiently, minimizing the surface energy over al1

possible effective potentials gives the optimized effective potential.

In preparation for this minimization we have adapted a KS-LDA program for the serni-

infinite jellium surface mode1 to solve the LDA problem by minimizing the energy with

respect to the effective potential. Since variations in the potential do not necessarily con-

3 ~ h i s form of the surface kinetic energq- assumes that the jellium surface is at z = O- TVe refer the reader to [44] and [69] for the situtaion where the positive background is zlIowed to vary in order to neutralize the system


serve the total charge, the code was modified to shift the effective potential, amd electron

density dong the z axis in order to compensate for any excess or depletion of charge. Our

method is then equivalent to minimizing the quantity

where 6742) = n ( z ) - n+ ( z ) , and eF is the lagrange multiplier for this problem. The second

term in (4.5) accounts for the penalty in the energy incurred by removing or adding particles

£rom the bulk region in order to change the number of particles at the surface. Mary

powerfuI minimization techniques require knowing the gradients of the energy with respect

to the potential, but since there seems to be no easy way to calculate these for our problem

we have resorted to the simplex method [TOI which unfortunately scales terribly with the

number of input parameters. Ideaily: we would like to place the effective potential on a grid,

and treat the values of the potential on the grid as the variational parameters. Alternatively,

if this is asking too much of present day computing power, we could parameterize the

potential, by some combination of functions4, and determine a good fit for the OEP.

We have found that by considering localized variations about the self-consistent potential

at various locations in the bu& region we can reproduce the LDA potential to approximately

2% depending on the exact position. The corresponding surface energies M e r £rom the self-

consistent ones by about 1 part per million. In the vacuum region, our procedure is less

precise, since the ground-state energy is very insensitive to variations of the potential in

this region. For example, we have found that a 10% perturbation of the barrier height

o d y alters the energy in the 9th digit. We have also performed a rninirnization using a

parameterized function. With three variational parameters we can obtain surface energies

that are within 0.5% of the self-consistent energy, but barrier heights that are significantly

4Although we have not expiicitly decided on such a pararneterization, the results Erom Chapter 2 indicate that the choice shouId maintain a VV + r~ - /3/r dependence in the vacuum.

4.3 The surface exchange energy O, 65

lower than the published self-consistent KS-LDA ones. We now turn to our method of

extracting the surface contribution of the exchange energy.

4.3 The surface exchange energy oX

The spin-compensated eschange energy of a system is given by

OCC OCC ,. ,.

Kohn and Mattson showed how, in principle, one can evaluate this quantity in real space for

the jellium surface, but their method involves a four fold integral and the generation of their

universal function, Eq. (2.31). Similady, Ma and Sahni [45] have approached this quantity

directly in real space, but their analysis was geared to the linear ramp model fkom the start

and so in addition to dealing with five-fold integrals it is not clear fkom their analysis how to

extract the surface contribution for an arbitraq effective potential which is our goal. In the

calculation reported by Mahan [44] the surface energy was calculated using orbitals fiom a

finite barrier, and the bulk and surface terms were separated just as Harris and Jones [43]

did previously for the infinite barrier model, so again, the separation is dependent on the

analytical forms of the orbitals. We need to devise a method to separate the contributions

for numerical orbitals kom an arbitrary effective potential. We take an alternate route hem

the previous studies, first Fourier transforrning the Coulomb potential to decouple the r

and r' coordinates. After doing this: the result fkom Sec. (B.1), for Eq. (4.6) is

where

4.3 The surface exchange energy 0, 66

and

with T = k: - k2, = kg - k@ and F~ = (T + T' + Q2)2 - 4TT'. We see that this

has reduced the original expression to the evaluation of B(k, fi', Q), and then the 3-fold

integration. We present a plot of I(k, ktz Q) for Q = 0.1 in Fig. (4.1). Since the Q integral

must be evaluated over the entire domain, in Sec. (B -2) we give the asymptotic expansions

of I(k, k', Q) for large Q, which will d o w us to treat the integral with respect to Q more

precisely for large Q values. In Sec. (B.2) we also see that for small Q, I(k, k', Q) has a

logarithmic singdarity, so we will need to treat the functions analytically around there.

To facilitate the separation of the exchange energy into a b u k term and a surface term

for arbitrary effective potentials we introduce two positive quantities, a and b. These serve

to partition the r-axis into three regions: (i) the deep b d k region, z E [-L, -a]: where the

orbitals are described by their asymptotic buik form and (ii) the surface region, z E [-a: b]:

around the jelIium edge where the orbitals go fiom an oscillatory form to an exponential

decay. At the endpoint of this region, where z = b the orbitais are negligib1e5. The boundazy

point between the two regions, z = -a, must be chosen far enough into the bulk to obtain

the desired accuracy. This is achieved when the results are independent on the choice of

a. This partition allows the separation of energy terms we are after, by providing a frame

on which to pIace our orbitals and subtract off the corresponding bulk forms. We now

5A third region, r > b, we have omitted on purpose. If we wish, we can use our results from the previouç chapter to calculate the eschange energy contribution corresponding to this region since we know both the inverse radius of the =change hole, and the density for large z. As we have seen many times already the eschange energy can be written a s the integral over the product of these t-ivo quantities. This shows that the contribution from large z is just ~ F ( K + 2a In a)/(47r4cr2(1 + a')) [T dz 1 f k F ( ~ ) 1 2 / ~ 3 . Clearly this integrai is Mnishingly s m d since fk, - eq( -dm) , so we wïli neglect it. I t may be possible to include higher order terms to obtain a result that may be appreciable, and actually assist in the calculation, but we have not investigated thk yet.

4.3 The surface exchange energy O, 67

Figure 4.1: Contour plot of I(k, k', 0.1) for h~ = 0.48. For difFerent Q values i(k, k', Q ) show the same features: quarter circle isolines in the neighbourhood of the origin, and straight isolines near the Fermi level.

4.3 The surface exchange energy a, 68

introduce the quantity @ ( k , t 'a) defined by

where the quantities y(k) are just the asymptotic phases of the numericdy determined

orbitals, and

for z < O

( 1 + <z)e-cZ for z > 0.

Defining drp(k, k', r) by

separates IB (lc, kt, Q) 1 into an analytic contribution which will contain the bu& energy term

and a numerical contribution. The quantity IB(k, k t , Q)I2 can then be written as

IB(k, k', Q ) ] ~ = ( ~ 1 1 ~ + 1 ~ 2 1 ~ + 1~31' + 292 {B2 x B3*} + 2% { B l x [B2 + B3J') , (4.13)

where

B k L &ip(k, k', z )e iQz , 8 2 = l D O dz @(k, t', z)eiQzl

and

This separates Eq. (4.7) into a sum of five terms, al1 of which need to be treated slightly

differently. Our particular choice for g ( z ) when z > O keeps @(k1 k t , z ) and its first derivative

continuous at z = O. We originally considered g ( z ) = 0 , for r > 0 , but discovered that the

resulting discontinuity in 69(k, k' , z ) , after perforrning the z integral, introduced oscillatory

terms in B3 that decayed too slowly with respect to Q for efficient numerical integration-

4.3 The surface exchange energy a, 69

We note that if we were to consider the infinite barrier mode1 with the barrier positioned

at z = 0, using the same approach, the terms B2, B3, and B4 wouid all be zero, leaving

only B1 to be evaluated.

The first and last terms on the right hand side of Eq. (4.13) both involve BI: so in

Sec. (B.3) we analyze BI more carefulIy, showing that 1 B1I2 contains the bulk contribution

to the energy, and that it can be simply extracted. Evaluating B3 is a little more involved

since it requires the numericdy determined orbitals. We are using the Numerov-Milnes

method to calculate the orbitds, which is accurate to order h4, but more accurate methods

that are stiil fast codd be employed [70]. In Sec. (B.5) we demonstrate the quanti@

1 ~ 3 1 ~ using an effective potential more realistic than either the £bite-baxrier or Linear-ramp

models and a typicd parameter for the simple rnetals. We also show the subsequent steps

in calculating the contribution to Ex Tom B3, discussîng further deta,&. The term B2

can be determined analytically, and the result is presented Sec. (B.4). The fourth term,

2X{B2 x B3*), is well behaved and can be added to the numericd term, 1B3I2 + II32l2,

to be multiplied by I(k, k', Q) and then integrated over k, k' and then Q, but we have not

implemented this yet.

Finally, the last term in Eq. (4.13) is, possibly, the most difEcult to deal with. First,

we note that B1 has terms containing the factor ( - I ) ~ + * ~ ' Q ~ / ( Q + kp), where k, is some

combination of *k + kt. However, there should be no L dependence in the surface terms.

In Sec. (B.6) we show that each term does indeed give rise to an L dependent surface

contribution, but when considered together the contributions nicely cancel, allowing us to

discard these terms. The rest of the term still needs to be computed, which requires careful

handling of dl of the singularities which appear.

4.4 Future Work 70

4.4 Future Work

Above, we have outlined our proposal for calculating the exchange contribution to the

surface energy. We have provided an algorithm which appears to be more eEcient than

previously reported ones. In addition, our routine is applicable to any effective potential

for the semi-fite jellium model of a surface. Once the program is complete, and has

been tested against previous model effective potentiak the routine for the surface exchange

energy can be introduced into the Kohn-S ham algorithm, replacing the LDA evaluation

for the exchange. Before performing a full minimixation, we would h s t investigate the

dependence of the surface energy and the separate terms on some general features of the

effective potentiai. For instance, we would examine the dependence of the energy terms on

the work function or the width of the potential as it goes hem bulk to vacuum form. These

preliminaq calculations will allow us to identify the parts of the potential on which to focus

in the full minimization. If our procedure turns out to be efiicient and accurate, we may

even be able to perform a fidl Hartree-Fock calculation by concentrating on the orbitals

and considering those as the variational parameters. Still though: work needs to be done

to obtain the exchange energy. Some terms of IB(k, k',Q)I2, still need to be computed and

integrated, but we believe that we have presented the important d e t d s in the procedure.

Chapter 5

Conclusions

An important system in the study of electronic structure is the jellium surface, for in ad-

dition to providing a mode1 for studying the simple metals, it provides a system on which

one can closely investigate many-body effects. Kohn-Sham densi@ functional theory (DFT)

provides a rigorous fkamework on which to base these cdculations, but relies on approxima-

tions for incorporating many-body effects [Il]. An interest ing met hod to introduce exchange

effects accurately into the KS scheme is the optimized effective potential (OEP) method,

where the exchange energy is calculated exactly using orbitals fiom a local potential that

minimizes the total energy [201. In this thesis we have examined the jellium surface in the

exchange-only approximation provided by the OEP method. We have answered the long

standing question of the asymptotic form of the OEP out in the vacuum, derived some exact

results for the exchange hole of an electron far into the vacuum, and devised a scheme to

calculate the exact OEP for semi-infinite jellium.

Our first result deals with the asymptotic form of the OEP in the vacuum region of a

semi-infinite jellium surface. Tt is well known that in all cases, the KS effective potential,

VeE(r), should have as its limiting form, fax away from the system, the classical image re-

sult [29]. For f i t e systems, it was discovered long ago that the OEP reproduces the correct

CONCLUSIONS 72

image form of -l/r, thus demonstrating that the long range form of the KS potential is

purely an exchange effect [20]. For a long time it was believed that, for metal surfaces: the

KS exchange potential should decay like - l / r2 for t h e metal surface system, leaving corre-

Iations to âccount for the entire classical image result of - 1 / k for the jellium surface [52].

More recently, Solomatin and S a h i have conjectured that, for the jellium surface, the

asymptotic form of the OEP is in fact image-like and 5s exactly half the asymptotic form of

the Slater potential, whïch they have derived exactly within the simple linear ramp mode1

of the surface potential [50]. To settle this confusion we have taken a carefuI and direct

look at the OEP starting fkom its definition- We have derived the integral equation that

governs the OEP and have used an expansion for the occupied orbitals around the Fermi-

level to extract the asymptotic form of the self-consistent, OEP. Our result shows that the

previous studies are in error. In fact, the OEP for t he jellium surface in the asymptotic

region contains the entire Slater potential, not half of it, as well as other image-Like terms.

The exact value of the coefficient for this Mage-like behaviour depends on the d e t d s of the

potential, and so remains unknown. We have ako prcrvided an argument directly fkom the

definition of the KS exchange potential, which clearly demonstrates the presence of the en-

tire Slater potential. Our results contradict reported murnerical work on finite jellium slabs,

and we suspect that a thorough analytical analysis orn finite slabs will reveal the source of

the discrepancy.

The exchange-correlation hole provides a physical quantity of interest, since it is directly

related to the charge distribution around an electrorn. We have investigated the part of

the exchange-correlation hole due to the Fermi-Dirac statistics, the exchangehole, for an

electron asymptotically far away f o m the surface. The structure of the exchange hoIe can

be determined analytically in this region, and we have formally shown that as an electron

moves farther away &om the surface into the vacuum, tlhe exchange hole becomes delocalized

throughout the crystal. Our results are comparable t o the numerical study of Sahni and

CONCLUSIONS 73

Bohnen [61], and HarboIa and Sahni [62], where they attempted to investigate the same

asymptotic limit using numerical techniques for model effective potentials. In fact, o u

results can be considered as an exact, model independent proof of their findings. We have

also been able to determine some quantities that evade a numerical analysis. For example,

in addition to deriving some moments related to the exchange hole, we have derived an

expression that gives the total amount of exchange hole charge from the asymp totic electron

position to an equal distance into the metal. We find this quantity to be a sigrtifkant portion

of the total exchange hole, approximately 40%.

The final part of the thesis concerns itself with the surface energy of the jellium surface.

An accurate exchange-only cdculation for the jellium surface has not been reported in the

literature, to our knowledge- One possible method: solving the integral equation for the

OEP, has only been performed for m t e slabs [46], and it is not clear how to extend the

solution to the serai-infinite case. We propose to solve the OEP via its variational principle,

replacing the LDA exchange approximation with the exact exchange expression, and then

minimixing the total quantity over all possible effective potentials. To prepare for this

task we have investigated the solution of the KS-LDA problem via minimization of the

effective potential, and have concluded that the effective potential within the buLk region

can be obtained easily, but in the surface region a more refined calculation is required. For

an exact exchange calculation, the LDA exchange energy must be repIaced by the exact

exchange expression. Since all reported exact exchange-only calculations for the semi-

infinite system use a model potential that permits the separation through their analytical

orbitals we have developed a method to calculate the exchange energy for an arbitrary local

effective potential. We believe our met hod is more efficient t han previously implemented

methods, but this efficiency comes at a cost of requiring a more intricate procedure- Initial

steps towards realizing the calculation have been taken, and we believe that we will attain

our goal of providing an accurate OEP and surface energy.

CONCLUSIONS 74

For future work, the numericd value of the coefficient for the asymptotic expansion of

the OEP should be determined. Since the coefficient depends on the details of the potential,

the exact value shodd be accessible once the OEP is obtained. It will be interesting to see

how close the asymptotic form of the OEP is to the exact asymptotic form of - 1/&. S i d a r

investigations should be performed for the finite jellium slab case, to determine whether the

asymptotic form of the OEP is image-Like as in the serni-idhite case, and to investigate the

behaviour of the exchange hole.

Appendix A

Results used in Chapter 2

A.1 Derivation of Eq. (2.21)

Substituting in the form of the surface mode1 orbitals, Eq. (1.24), allows the evaluation of

the radial integral:

A.2 Derivation of Eq. (2.23) 76

A.2 Derivation ofEq. (2.23)

~e wish to evaluate 5 d2p' Gk (r" r )e -ikll -pf . Using Eq. (2.10) and the separation of the

pIanar orbitals, Eq. (1.26): gives

Which gives the required result on defining Gk (z', z) as in the text.

A.3 Derivation of Eqs. (2.30, 2.31)

Together, Eqs. (2.21), (2.29), and (2.26) give

$*(z'b*(z') = -' +& (2') d r df ( ï ) $ g f (ï) / d2kll ( 2 ~ ) ~ kf<kF

lCll

Focusing on the rightmost two integrals we defbe

A.3 Derivation of Eqs. (2.30, 2-31) 77

Making the substitution q = Ir - zllkll and q' = Ir - z'lki, into the integrand, Eq. (A.1)

becomes

where b = 11 - $1 d k $ - k2 and similady for 6'. Now, using the Fourier transforrn pair

we can write

and using the relation

gives

We h d f (x) by evaluating Eq. (A.2)

- - 1 x2 with E = -

1 + E C O S ~ 2 3-22

so that " JI (bz) JI (b'z)

~ ( b , b' ) = bb' dz z J ï T - 2 '

A.4 Expansion of orbitak in the asymptotic fimit 78

A.4 Expansion of orbitals in the asymptotic limit

Flat potential

Here we show that in the vacuum region far from the surface the orbitds can be expanded

about the Fermi level. For a potential that is flat in the large z lirnit the Schrodinger

equation we need to solve is

where the primes denote derivatives with respect to z. In the large z limit we are only

interested in an interval of k-states just below the Fermi level. Rom - 2ek = 2b nk -

Ak2, where Ak = kF - k7 we trivially have

where we have used a2 = W. htroducing the normalkat ion constant D (k) , and neglect-

ing terms of higher order t han Ak in the exponential, the solution, $ J ~ ( z ) , can be expanded

about the Fermi level Lke so,

A k -- = i>kF (z)e Ako (1 - XAk + 0(dk2) ) ,

A.4 E ~ a n s i o n of orbitals in the asvm~totic limit 79

where X = ~(k~)-'(aD(k)/akl~=~~)- For the &te barrier mode1 this expansion becomes

Image potential

We dso use an expansion of the wavefunction when the effective potential has an image

form. Introducing the parameter V - EF + W and a positive image constant 6, the potential

far fiom the surface can be described by V(z ) = V(l - P / z ) - The corresponding Schrodinger

equation far into the vacuum is

We c m make the following transformations

to obtain

which we identifS. as the Whittaker equation [71]. Its solution which converges at infinity

we denote by W ( K , x) and has the asymptotic form

Introducing the normalkation constant D ( k ) , the orbitals far in the vacuum are

$k(z ) = ~ ( k ) e - & ( 2 & ) 5 [1+ 0 ( 1 / z ) ] ,

A.4 Expansion of orbitals in the asymptotic Iimit 80

or, in terms of the original variables

The Fermi-level orbital is

Now we proceed to expand this about the Fermi-Ievel. Using Eq. (A.3),

Expanding t his,

Defining qAk = Da(1 + a 2 ) d k / 2 = V P C / ~ , and dropping the where = &-orm- O(l/z) term we can further simplify this

In terms of the Fermi-level orbital

A.5 Derivation of Eq. (2.34) 81

We see that for = O this gives back the expansion for the case of a flat potential, as we

should expect-

A.5 Derivation of Eq. (2.34)

Now we can s i m p w [€kF - h ( z ) 1 Gg (2, z ) ,

A.6 Evaluation of R, (2, 2') 82

FinaUy, we have

Keeping only the leading term on the right reduces this to Eq. (2.34)

A.6 Evaluation of RkF (2,~')

In t h s section we provide an evaluation of

where d k < is the lesser of Ak and Akf . Transfonning the integration variable gives

k~ A k -- d k' -- R . ~ ( Z , Z ~ ~ = 1 dAke ""O q5EF-Ak (2) lkF ddik' e yl'&-4k. (z') dk<.

We take the upper limit of the integrals, b, to oo, because when 2r is asymptotic the

exponential factors keep only the contributions coming from Ak < Ak,. Expanding the

integrand gives

The two terms on the right hand side can be shown to be equivalent, so we only con-

sider twice the first one. The form of the orbitals deep in the bulk is J2/Lsin(hz' -

~ ( b ) - Ak[Zf + ~'(b)]) . We drop y' (kF ) since it is srnall compared to zr, and changing the

A.7 Are there more leading terms in the OEP? 83

integration variable to x = A k / A k o gives

where k z f = kFzr - 'Y(&). After performùig these integrals one finds

Akz Z ' A ~ ~ ( ( Z ' A ~ , ) ~ - 3 ) sin ( 2 z ' k ~ - 2y(h) ) RkF k 2' 1 = 4 ( 1 + ( ~ ' d k ~ ) ~ ) + 4(1 + ( z ' ~ l k ~ ) ~ ) ~

(2.52) z ' ~ k ~ ( 3 ( d d k ~ ) * - 1) COS ( 2 1 b - 2y(k~)) + 4 ( 1 + ( z ' A ~ , ) ~ ) ~

As a verifkation, we have pefiormed the integrals, with y'(,+) included for the finite barrier

mode1 orbitals, and have arrived at the same conclusion.

A.7 Are there more leading terms in the OEP?

Here we show that term (2 .63) ,

b k d di' ( A ( z ) - v ( z ) ) ( z ) [ G (2 , ) - a - ( ) ,

k < k ~ d r 1

(2.63)

may, under various assumptions, contribute a l / z type term to the OEP. The factor in

square brackets becomes

The wave-vectors, k , are restricted to just around the Fermi wave-vectors. If we assurne

that the major contributions in the sum corne fiom k' h: k, that is, when the denominator

is small, the orbitals labelled by k' will still have an energy below the barrier, and so will

decay. This allows us to expand them about kF, and take the derivative with respect to z

A-7 Are there more leading terms in the OEP? 84

so that

We know the form of the orbitals at the Fermi level, and so (2.63) becornes,

If k = kF in the denominator of the k' sum, the entire term may very well contribute to

the l/z dependence since the k sum is restricted to near &, and since we have established

that vk(zf)~k(z') contributes to the integrand an order of Ak. This is o d y a rough sketch

though, and requires a more careful andysis.

Appendix B

Derivat ions for Chapter 4

B.1 Derivation of Eqs. (4.7), (4.8) and (4.9)

To calculate the exchange contribution to the surface energy we will first use the 3-dimensional

Fourier representation of the Coulomb potential in cylindrical coordinates given by

where the integral is over all Q-space. Q has been decomposed int O a coordinat e perpen-

dicular to the surface, Q, and a radial coordinate parallel to the surface QI,. Substituthg

this representation into the exchange energy expressiont Eq. (4.6), gives

B.l Derivation of Eqs. (4.7), (4.8) and (4.9) 86

Using the form of the jelIium surface orbitah, Eq- (1.26), in this expression we have

and integrating over the pardel real space components, using

gives

Converting the k sums into integrals using Eq. (2.26) and Eq. (2 .27) , and rearranging

slight ly, gives

with

The two kll integrals may be performed to reduce this expression to a closed form. Com-

pleting the angular integrals h s t and using the transformations ki = x7 kt = y gives

I(k, k', Q) = lT dz lT' dg 1

J(x i- y + Q 2 ) 2 - 4 x y 7 (W

where T = k; - k2 and T' = k$ - ka. These double integral can be found in [57], or [72]

I 2

rnay be used, to give Eq. (4.9). It is clear that dz e ' ~ ' ( Z ) $ J ~ ( Z ) 1 is an even function

13.2 Lirniting forrns of I(k, k', Q) 87

of Q, as is I(k, k', Q), so Eq- (B.6) reduces to Eq. (4.7)

B -2 Limiting forms of I(k, k', Q )

It can be shown that in the s m d Q limit Eq. (4.9) has the series expansion

when k = k'. For k' > k the expansion is

(B. IO)

Exchanging T and T' in this series gives the result for k > kt- I t is now easy to see that

the Q integral needs special case near Q = O. We may d s o need the asymptotic form of

I(k: k', Q) for large Q which is given by

Ln order to successfidly evaluate the k and kt integrals in Eq- (4.7) we must i d e n t e and

deal with any awkward regions. When k = kF7 I(k, k', Q) = ( l n ( ~ * +Tt) - ~ Q ~ ) T + 0(T2), so there is nothing to concern us in this region.

B.3 Evaluation of IB1I2

The first part of IB(k, kt, Q) I 2 that we consider is 1 ~ 1 1 ~ . The factor B1 is given by

Perfonning the integral gives

where k1 = k + k f , k2 = k - kf t y* = y ( k ) + y ( k f ) and 7 2 = y(k) - ? ( k f ) . Since the

wavefunctions go to zero at z = -L7 -ive have that kL + y (k ) = rn r , and k f L + ~ ( k ' ) = n.ir

where m and n are integers, giving

After performing some algebra; for JB 112 we find

where C.C. denotes the complex conjugate of the term immediately before it. It has been

shown that the first two terrns, containing only the second-order poles, give a contribution

to the total energy of -M3/EF;n/47ï [73]. Since the jellium slab is slightly shorter than the

length L, this gives the bulk energy, plus a trivial contribution to the surface energy. We

can now set aside those h o terms in our analysis. When 7 1 = y 2 = 0: Eq. (B.15) reduces

to the infinite barrier result, that has been thoroughly studied [43], and should provide a

B.4 Evaluation of B2 89

check that tbis term is being dedt with accurately-

B.4 Evaluation of B2

Explicitly, B2 is given by

(B. 16)

Performing the integral gives

where P = iQ - 6. We omit presenting IB2I2 for obvious reasons.

B.5 Evaluation of IB3I2

Here we demonstrate some details about

potential we choose a Woods-Saxon form

described by

the calculation of IB3I2. As a model effective

modified by two small Gaussian perturbations

(B. 18)

where w describes the width of the surface region in which the potential varies. The second

and third terms describe smali perturbations in the potential that are typical of self consis-

tent jeilium calculations. Using some typicd parameters, we display this model potential

in Fig. (B.1). We choose /& = 0.48 (a-u.) so that the work function is comparable to

the exchange only results of Mahan [44], but we note that the LDA gives a work function

that varies appreciably with r, [38]. For our model potential we display the asymptotic

phase in Fig. (B.2) and compare it to the trivial analytic result fkom the finite step. We

B.5 Evaluation of IB3I2 90

-20 -10 O 10

Position, z, a.u.

Figure B.1: Mode1 potentials represented by Eq. (B.1). The thick dashed Iine corresponds to a = b = O and is the typical Woods-Saxon type potential. For the thin soiid line we have set a = 0.2, and b = 0.01, and the potential is more representative of the exact potential exhibiting wiggles in the bulk region. The solid horizontal line is set at the Fermi energy.

B.5 Evaluation of [ ~ 3 1 91

- Modified Woods-Saxon banierp=0.2 bS.0 1 1 4

-

-

-

-

-

-

-

-

-

-

- \ 0 -

- -. 0 \

- -.

- - . - + O

-0.4 t 1 1 1 I 1 I 1

O O. 1 0.2 0.3 0.4 O -5

Wavevector, k, (a.u.)

-. - 0 Woods-Saxon type barrier 0

- - Finite-step barrier

0 0

0 0

0 0

/ 0

0 0

0 0

0 0

/

/ /

1- 1

/ \ / - \ /

\ / \. - \ /

\ 0

Figure B.2: Phases corresponding to some mode1 potentials represented by Eq. (B.1). The top phase corresponds to the finite barrier result, with the same work function as . The bottom phase is a ty-pical Woods-Saxon type potential, arrived at by taking a = b = O in Eq. (8.1). The middle phase, is for a = 0.2 and b = 0.01. It is interesting to note the large variations in the phase for different potentials.

B.6 Dealing with the poles in B1 x [B2 + 833' 92

see that there is substantial deviation f?om the finite-step mode1 result. m e n a = b = O

the phase shows a significant curvature when compared to the finitestep phase, due to the

h i t e spread of the barrier. As we consider more structure in the potential, increasing a, the

phase exhibits higher order k dependence. Using the phases we can perform the algorithm

presented in the text, and in Fig. (8.3) we present the real and imaginary components of

B 3 as well as the quantity 1 ~ 3 1 ~ for combinations of two k values as a function Q. Since

the function I(k, kt, Q) behaves as .- 1/Q2 for large Q , the function I ~ 3 1 ~ 1 ( k , k', Q) can be

numerically integrated dong Q by brute force.

B.6 Dealing with the poles in B1 x [B2 + B3]*

In the text we saw that B ( k , k', Q ) separated into the sum of five contributions that could

be treated independently, Eq. (4.13). We examine the term given by 2 8 { B l x [B2 + B3]*},

focussing on the p z t s containing L, the length of the system. Since we have separated out

the b d k contributions to the total energy, we expect our results to be independent of L,

but before we c m discard terms with the factor ( - l ) m f n e i Q L /(Q + k,) we need to look

carefidy at possible contributions fkom the poles. To do this we go back to the expression,

Eq. (B.5), for the exchange energy before the 5, s u s were performed, but only include this

one term of IB(k, k', Q)I2. We consider the Q integral fhst so the contribution becomes

where f ( Q ) is (B2 + B ~ ) / ( Q ~ + lkll - k[112). For each Q integral on the right hand side.

we note that for fixed L the integrand remains finite as the denominator tends to zero, so

we can separate the integrand and take the principal values (Y). The dQ integral then

B.6 Dealing with the poles in BI x [82 + 2331' 93

1 1 0.6 - I -

- - h-=O.176,kW.176 - k=û. 176, ke=O.352 -

- k0.352, k -S .352 0.4 -

- -

C I - 0.2 - -

- -

0-

-0.2 I I 1 I 1 l I

- - - - - -

I f I I -

O 1 Q 3 4

Figure B.3: Variation B3 as a hinction of Q for different h and k' values, corresponding to the potential given by Eq. (B.18) with a = 0.2 and b = 0.01. The top and middle figures display the real and imaginary parts of B3 respectively, whereas the bottom figure displays 1 ~ 3 1 ~ .

B.6 Dealinn with the poles in B1 x iB2 + B31* 94

becomes

and evaluating the second term on the right hand side we obtain in the large L Iimit

Where we have expanded f (Q) about kp in powers of 1/L. We can insert these results back

into Eq. (B.19), to arrive a t

2 OCC OCC 2 ~ ~ 1 ~2 + ~ 3 1 * i = -lx{C(-l)p+l

Q 2 + l k i l - k i 1 2 A p=l k kt

We see from the definitions of B2 and B3 that f (kp) = f * (-kJ, and so the term in the sum

due to the poles is purely imaginary. Since we require only the real part we can continue

the analysis taking just the principal value for the dQ integral in Eq. (B.19) and neglecting

the tenns containing L.

Bibliography

[l] R. M. Dreizler and E. K. U. Gross, Densitg Funetional the on^: An Approach to the

Quantum Mang-Body Pro blem (S pringer-Verlag, New-York, 1990).

[2] W. Kohn, Rev. Mod. Phys. 71, 1253 (1999).

[3] W. E- Pickett, Cornputer Physics Reports 9? 115 (1989).

[4] N- Wiser and M. Cohen, J. Phys. C 2, 193 (1968).

[5] V. Fock, Zeits. f. Physik 75, 622 (1932).

[6] E. K. U. Gross, E. Runge, and 0. Heinonen, Many-Particle Theory (IOP Publishing

Ltd., New York, 1991).

[7] N. W. Ashcroft and N. D. Mermin, Solid State Physics (Harcourt Brace College Pub-

lishers, Toronto, 1976).

181 J. Slater, Rev. Mod. Phys. 6, 209 (1934).

[9] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-

Hill, Inc., Toronto, 1971).

[IO] P. Hohenberg and W. Kohn, Phys. Rev. 136B, 864 (1964).

[Il] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).

BIBLIOGRAPHY 96

[12] J. P. Perdew and ,4. Zunger, Phys. Rev. B 23, 5048 (1981).

[13] D. Ceperley and B. Alder, Phys. Rev. Lett 45, 566 (1980).

[14] D. C. Langreth and J- P. Perdew, Solid State Commun. 17: 1425 (1975).

[15] 0. Gunnarson and B. 1. Lundqvist, Phys. Rev. B 13, 4274 (1976).

[16] 0. Gunnarson, M. Jonson, and B. 1. Lundqvist, Phys. Rev. B 20, 3136 (1979).

[17] J. P. Perdew and Y. Wasg, Phys. Rev. B 33, 8800 (1986).

[18] J. P. Perdew and M. Levy, Phys. Rev. Lett. 51, 1884 (1983).

[19] R. Sharp and G. K. Horton, Phys. Rev. 90, 317 (1953).

[30] J. Talman and W. Shadwick, Phys. Rev. A 14, 36 (1976).

[21] Y. Wang e t al., Phys. Rev. A 41: 78 (1990).

[22] V. Sahni, J. Gruenebaum, and J. P. Perdew, Phys. Rev. B 26, 4371 (1982).

[23] M. Stadele e t al., Phys. Rev. Lett. 79, 2089 (1997).

[24] M. Stadele e t al., Phys. Rev. B 59, 10031 (1999).

[25] P. Süle, S. Kurth, and V. E. Van Doren, J. Chern. Phys. 112, 7355 (2000).

[26] A. Gorling, Phys. Rev. Lett. 83, 5459 (1999).

[27] J. D. Jackson, Classical Electrodyzanaics, 3rd ed. (John-Wiley, New-York, 1999).

[28] J. Bardeen, Phys. Rev. 58, 727 (1940).

[29] C.-O. Almbladh and U. von Barth, Phys. Rev. B 31, 3231 (1985).

[30] A. G. Eguiluz, M. Heinrichsmeier, A. Fleszar, and W. Hanke, Phys. Rev. Lett. 68,

1359 (1992).

[31] A. Sugiyama, J- Phys. Soc. Japan 15, 965 (1960).

[32] D. C. Langreth, Phys. Rev. 5, 2842 (1972).

[33] We would like to thank A. Liebsch who generously made available his cornputer code

for the KS LDA jeIlium surface. (1999).

[34] J. E'renkel, Zeits. f. Physik 51: 232 (1928).

[35] L. Thomas, Proc. Cambridge Phil. Soc. 23, 542 (1927).

[36] N. Sommerfeld, Naturwiss. 15: 825 (1927).

[37] J. Bardeen, Phys. Rev. 49, 653 (1936).

[38] N. D. Lang and W. Kohn, Phys. Rev. B 1 , 4555 (1970).

[39] N. D. Lang and W. Kohn, Phys. Rev. B 3, 1215 (1971).

[40] N. D. Lang and W. Kohn, Phys. Rev. B 7, 3541 (1973).

[41] W. L. Schaich, Phys. Rev. B 61, 10478 (2000).

[42] 1. Sarria, C. Henriques, C. Fiolhais, and J. Pitarke, Phys. Rev. B 62. 1699 (2000).

[43] J. Harris and R. O. Jones, J. Phys. F 4, 1170 (1974).

[44] G. D. Mahan, Phys. Rev. B 12, 5885 (1975).

[45] C. Q. M a and S. V., Phys. Rev. B. 20: 2291 (1979).

[46] E. Krotscheck, W. Kohn, and G.-X. Qian, Phys. Rev. B 32, 5693 (1985).

[47] A.-R. E. Mohammed and V. Sahni, Phys. Rev. B 31, 4879 (1985).

[48] V. Sahni and C . Q. Ma, Phys. Rev. B 22, 5987 (1980).

[49] J. B. Krieger, Y. Li: and G . J. Iafrate: Phys- Rev. A 45, 101 (1992).

[50] A. Solomatin and V. Sahni, Phys. Lett. A 212, 263 (1996).

[51] A. Solomatin and V. Sahni, Phys. Rev. B 56, 3655 (1997).

[52] L. Sham, Phys. Rev. B 32, 3876 (1985).

[53] L. KIeinman, Phys. Rev. B 49, 14197 (1994).

[54] O. Gritsenko, R. van Leeuwen, E. van Lenthe, and E. J. Baerends, Phys. Rev. A 51:

1944 (1995).

[55] W. Kohn and A. E. Mattsson, Phys- Rev. Lett. 81, 3487 (1998).

[56] G. N. Watson, A Treatise on the Theorr~ of Bessel functions, 2nd ed. (Cambridge

University Press, Cambridge, England, 1958).

[57] 1. S. Gradshteyn and 1- M. Ryzhik, in Table of Integrals, Series, and Products, fifth

ed., edited by A. JefFrey (Academic Press, Toronto, 1994).

[58] 1. D. Moore and N. H. March, A m . Phys. (N.Y.) 97, 136 (1976).

[59] N- H. March, W. H. Young, and S. Sampanthar, The &fang-Body Problem in Quantum

Mechanics (Cambridge University Press, London, 1967).

[60] H. J. Juretschke, Phys. Rev. 92, 1140 (1953).

[Cil] V. S M and K.-P. Bohnen, Phys. Rev. B 31, 7651 (1985).

[62] M. K. Harbola and V. Sxhni, Phys- Rev. B 37, 745 (1988).

[63] E. Zaremba and W. Kohn, Phys. Rev. B 15, 1769 (1977).

[64] M. K. Harbola and V. S;thni, Phys. Rev. B 36, 5024 (1987).

BIBLIOGRAPHY 99

[65] L. Vitos, B. Johansson, J- Kollk, and H. L. Skriver, Phys. Rev. A 61: 52511 (2000).

[66] L. Vitos, B. Johansson, J. KOU&, and H. L. Skriver, LANL preprint,

http://xyz.lanl.gov/abs/cond-mat/0005292-

[67] X--P. Li, R. J. Needs, R. M. Martin, and D. M. Ceperly, Phys. Rev. B 45, 6124 (1992).

[68] H. B. Huntington, Phyç. Rev- 81: 1035 (1951).

[69] V. Sahni, C. Q. Ma, and J. S. Flamholz, Phys. Rev. B 18, 3931 (1978).

[70] W. Press, S. Teukolsky, W. Vetterhg, and B. P. Flanne% Numerical Recipes (Cam-

bridge University Press, London, 1992).

[71] D . Zwillinger, Handbook of Diffenntial Equations, 3rd ed. (Academic Press, Toronto,

1998) -

[72] Maple6, Waterloo Maple.

[73] p. c- Ulf von Barth.

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