Subsystem Functionals in Density Functional Theory

Subsystem Functionals in Density Functional Theory

Thesis for the degree of Teknologie licentiat

Subsystem Functionals in

Density Functional Theory

Towards a New Class of Exchange-Correlation Functionals

Rickard Armiento

Theory of Materials, Department of Physics

Royal Institute of Technology, SE-106 91 Stockholm, Sweden

Stockholm, Sweden 2002

Stockholm framl¨agges till offentlig granskning f¨or avl¨aggande av teknisk licencihus 11, Albanova universitetscentrum, Stockholms centrum f¨or fysik, astronomi och bioteknik, Roslagstullsbacken 11, Stockholm.

Cover illustration : An expression describing how the Mathieu gas density deviates from a uniform electron density for the z-point where the density gradient is zero, as a function of two parameters, ¯ p ¯p

2

λ.

ISBN 91-7283-416-1

TRITA-FYS-2002:48

ISSN 0280-316X

ISRN KTH/FYS/--02:48--SE

° Rickard Armiento, November 2002

Printed by Universitetsservice US AB, Stockholm 2002

Abstract

This thesis presents results relevant to the development of improved methods for theoretical calculations of properties of materials, molecules and atoms. The work is performed within the theoretical framework of the density functional theory, and is aimed towards new approximations of the exchange-correlation energy per particle, i.e., new ‘functionals’. Two main contributions are given in this area: a) results and requirements for using approximations based on different starting points in different parts of a system, i.e., subsystem functionals, b) properties important for a starting point of a subsystem functional aimed at interior-like regions with slowly varying electron densities. A main result of the investigation is that the exchange energy per particle cannot be modeled with an ordinary (analytical) power expansion.

Key words: density functional theory, exchange energy, exchange-correlation functional, subsystem functional, Mathieu gas model.

Sammanfattning

Detta arbete presenterar resultat inom utvecklandet av nya metoder f¨or teoretiska ber¨akningar av egenskaper hos material, molekyler och atomer. Arbetet imationer av utbytes-korrelationsenergi per partikel, d.v.s. nya “funktionaler”.

ka grunder i olika delar av ett och samma system, s. k. subsystemfunktionaler, b) resultat viktiga f¨or grunden till en subsystemfunktional f¨or inner-regioner med bytesenergin per elektron inte kan beskrivas av en vanlig (analytisk) serieutveckling.

Nyckelord: t¨athetsfunktionalsteori, utbytesenergi, utbytes-korrelationsfunktional, subsystemfunktional, Mathieugasmodell.

iii

iv

Preface

This thesis describes scientific work for the degree of Teknologie licentiat carried out in the group of Theory of Materials at the Department of Physics, Royal

Institute of Technology. The work is a part of a very large ongoing effort: the development of improved methods for electron density functionals to be used in density functional theory (DFT) calculations. The main part of the time spent on this project has been used to develop a trustworthy and sufficiently optimized computer program for calculations on a quantum mechanical model system, which has been used to investigate properties of DFT quantities. Apart from the results obtained by using this program, efforts have also been made to carefully document the main construction of the program (this documentation is placed in the appendices of the paper in part II of the thesis). In addition, since the main results are argued from unanticipated numerical data, a clear description of the data treatment and a careful error analysis have been necessary to make the results convincing.

The thesis begins with a brief and general introduction to density functional theory in chapter 2 and 3. Chapter 4 presents the ideas of the subsystem functional approach towards improved DFT functionals. This is followed, in chapter 5, by an introductory treatment of the Mathieu gas model system, which is used as a model system for obtaining the data results of the thesis, modeling the behavior of DFT quantities. Chapter 6 investigates the power expansion of the kinetic energy density by using numerical data from the Mathieu gas model system. Finally, chapter 7 briefly discusses the purpose and contents of the paper comprising part II of the thesis.

The paper in part II has the title “Subsystem functionals in density functional theory: Investigating the exchange energy per particle”. All data presented there were calculated by me, using a program written by me (although Ann Mattsson had performed some preliminary computations and her results provided a starting point). The data analysis and the writing of the main part of the paper were done jointly with Ann Mattsson, but the appendices were written by me.

v

vi

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii

Sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii

Preface

Contents v vii

1 Introduction

1.1

Some Comments on Units . . . . . . . . . . . . . . . . . . . . . .

2 Density Functional Theory

2.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

The Thomas–Fermi Model . . . . . . . . . . . . . . . . . . . . . .

9

2.3

The Hohenberg–Kohn Theorems . . . . . . . . . . . . . . . . . .

10

7

7

3

4

3 DFT Calculations 13

3.1

The Kohn–Sham Scheme . . . . . . . . . . . . . . . . . . . . . . .

13

3.2

The Kohn–Sham Orbitals . . . . . . . . . . . . . . . . . . . . . .

14

3.3

The Exchange Correlation Energy . . . . . . . . . . . . . . . . .

15

3.4

Local Density Approximation, LDA . . . . . . . . . . . . . . . .

16

3.5

Gradient Expansion Approximation, GEA . . . . . . . . . . . . .

16

3.6

Generalized-Gradient Approximations, GGAs . . . . . . . . . . .

18

3.7

The Chemical Potential . . . . . . . . . . . . . . . . . . . . . . .

19

4 Subsystem Functionals 21

4.1

General Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

4.2

Current Status of the Subsystem Functional Approach . . . . . .

22

4.3

Related Efforts . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

5 The Mathieu Gas as a DFT Model System 25

5.1

Definition of the MG . . . . . . . . . . . . . . . . . . . . . . . . .

25

5.2

Electron Densities of the MG Model . . . . . . . . . . . . . . . .

25

5.3

Exploring the Parameter Space of the MG . . . . . . . . . . . . .

26

5.3.1

The Limit of Slowly Varying Densities . . . . . . . . . . .

27

5.3.2

The Free Electron Gas Limit . . . . . . . . . . . . . . . .

27 vii

viii

Contents

5.3.3

The Harmonic Oscillator Limit . . . . . . . . . . . . . . .

28

6 Investigation of the Kinetic Energy Density 29

6.1

Definition of the Kinetic Energy Density . . . . . . . . . . . . . .

29

6.2

The KE Density in the MG . . . . . . . . . . . . . . . . . . . . .

30

6.3

Analyzing the KE Density . . . . . . . . . . . . . . . . . . . . . .

30

7 Concluding Remarks and Comments on the Paper

Acknowledgments

Bibliography

33

35

37

II Paper 39

Thread carefully, here there be monsters.

A traditional warning on old maps, designating unexplored territory.

Chapter 1

Introduction

Current theories in physics, most notably quantum mechanics, provide a mathematical description of the world on a very small scale. They describe matter as composed by particles and their interactions. So far, measurements on the physical world seem to agree very well with quantum mechanics and it is believed that enough of the physical world has been captured in the theory to make it possible to describe physical properties also on larger scales, by using descriptions involving more and more particles. However, when trying to construct and study such a larger scale quantum mechanical description, the mathematics becomes too complicated to solve exactly; investigations must be done by approximative methods.

The study of quantum mechanics through approximations has proven very successful and has turned into a means of deriving unknown properties of compounds and materials. As a result, a new branch of physics has developed which can be described as using computers to perform ‘experiments’ and ‘measurements’ upon the mathematical framework of quantum mechanics.

Density Functional Theory (DFT) is a formulation of quantum mechanics that utilizes a view of the particles known as electrons as a compound sea, focusing on the density of electrons as a fundamental property. This is opposed to so called wave-functional approaches that keep track of all individual electrons. The gain of DFT over the wave-functional approaches is that the density of electrons is a three-dimensional property, whereas the wave-function used to keep track of the electrons is of a dimensionality proportional to the number of electrons in the system (which usually is very large). The price paid for the comparably simpler DFT description is that it gives up the ability to describe certain properties of a system (most of which are related to the individual movement of electrons). However, for other properties the DFT formulation has been shown to be as fundamental

1 as a wave-functional based approach.

When the DFT formulation is transformed into a form

2 suitable for computer calculations, the most difficult quantum mechanical behavior of the sea of electrons is put into the exchange-correlation energy functional, E xc

. This is

3

4

Original Kohn and

Mattsson approach

Edge

Interior

R

1

R

2

Chapter 1. Introduction

Generalized idea

R

3

R

5

R

4

Figure 1.1.

The generalized idea of dividing a system into subsystems, applying different functionals to the different parts. The left figure refers to the approach presented by Kohn and Mattsson in Ref. 4.

usually a part of minor size, but it is the only part that has to be approximated relative to an explicit wave-functional solution. Several approximations of E xc suitable for computer calculations have been developed, but there is a demand for even more accurate ones. This thesis presents work towards such improved approximations.

The main underlying idea of the work presented here is that a system can be split into several regions, and in each region a different way of approximating

E xc can be used. Each such approximation could then be specifically designed for the part of the system it is applied to. This is not possible if one and the same approximation is used throughout the whole system. The approach described here is based on the locality, or ‘near-sightedness’, of a system of electrons, which is a property that previously has been discussed in other works.

3, 4

Kohn and Mattsson

4 have also presented efforts towards splitting a system into specific interior and edge parts. Hence, the present approach is based on a generalization of their ideas, as illustrated in Fig. 1.1.

The main scientific contributions of this thesis can be summarized as:

1. Establishment of fundamental results and requirements for using different approximations of E xc in different parts of a system.

2. Investigation of properties of an approximation of E xc varying electron densities.

useful for slowly

1.1

Some Comments on Units

The form of certain equations depends on the choice of units in which they are expressed. This work uses SI units, but to avoid confusion because of the differing practices of units common in this field of physics, a brief summary of some unit systems follows.

1.1. Some Comments on Units

5

Hartree atomic units

The bohr unit is introduced as a length based on quantities common for calculations on atomic scales. The hartree is then defined as the Coulomb repulsion between two electrons separated by one bohr.

a

0

=

4π²

0

¯h

2 m e e

2 c

= 1 bohr, e

4π²

2 c

0 a

0

= 1 hartree.

(1.1)

But when speaking of hartree atomic units, one usually takes the hartree unit to be dimensionless (i.e., 1 hartree = 1) and additionally sets

¯h

2

2m e

=

1

2

.

(1.2)

Some presentations stop here, because this is enough to get rid of the most common prefactors of quantum mechanical equations, simplifying them significantly.

However, when numerical values are cited, it is not unusual also to make some other quantities dimensionless and equal to 1. Most commonly one lets e c

= 1, which equivalently makes 1/(4π²

0

) = 1. It is also common to additionally set

¯h = 1, which equivalently makes m e

= 1. After this last step, numerical values of quantities of other dimensions than energy and length are usually marked as given in “a.u.”, designating atomic units, but giving the values with no unit at all is also formally correct.

Rydberg atomic units

The rydberg atomic units are based on the same ideas as the hartree atomic units but define the rydberg as the electron energy of the hydrogen atom

1

2 m e e

4 c

(4π²

0

) 2 ¯h

2

= 1 rydberg.

(1.3)

It is found that 1 hartree = 2 rydberg. Within the rydberg atomic units one takes the rydberg to be dimensionless (i.e., 1 rydberg = 1) and also sets

¯h

2

2m e

= 1.

(1.4)

As for hartree atomic units, some presentations stop here, but again it is common when citing values to also let other quantities be dimensionless and equal to one

(usually the same quantities as for hartree atomic units).

SI and cgs units

The cgs and SI system of units are based on similar ideas within dimensions of mass, time and length but differ significantly in the area of electromagnetism. It exists at least two different conventions for the cgs system in this area, cgs-emu

6

Chapter 1. Introduction and cgs-esu. In cgs-esu the charge unit has been chosen to simplify equations involving interactions between static electric charges by fixing the constant in

Coulomb’s law to one, giving ²

0

= 1/4π. In cgs-emu the conventions are chosen to simplify equations involving moving charges by fixing the permeability of vacuum

µ

0

= 1 thus giving ²

0

= 1/c

2

. In general, when converting equations from the cgs systems to the SI system, one needs to insert carefully chosen constant factors as combinations of 4π, ²

0 and c (or µ

0

= 1/²

0 c

2

) for all electromagnetic variables present. The result of this conversion is, of course, dependent on the kind of cgs system one is working with, which, if unknown, usually must be determined by observing the appearance of a Coulomb factor in an equation where it is known that such a factor should appear.

Chapter 2

Density Functional Theory

Density functional theory (DFT) has already been briefly introduced in chapter 1. The main idea is to regard the total particle density as the primary quantity from which properties of the system can be obtained. This chapter begins with some fundamental concepts and then introduces the theoretical framework.

For a more extensive treatment of density functional theory the reader is referred to text books on the subject.

5, 6

2.1

Background

The starting point is the time independent Schr¨odinger equation, which is the equation implicitly defines all possible states, Ψ, of a system, and their related energies, E:

(2.1)

For electrons moving around positive-charged nuclei in the kind of systems we intend to treat here, it is common to assume that this equation can be separated into independent electronic and nuclei parts (this is the Born-Oppenheimer approximation,

7 valid when electrons can be said to enter equilibrium on a time scale sufficiently shorter than the time scale on which the nuclei move). This separates our states into independent states for nuclei, φ, and electrons ψ with respectively energies E n and E e

. The Hamiltonian is split into corresponding

H n

H e

. We choose to put the interaction energy between nuclei and electrons into the electronic part. This gives

Ψ = φψ, H = ˆ n

H e

, (2.2)

H n

φ = E n

φ, H e

ψ = E e

ψ.

(2.3)

Using this separation it is comparably easy to handle the nuclei part. Our efforts are thus directed towards the electronic part, which here can be seen as a system

7

8

Chapter 2. Density Functional Theory of multiple interacting electrons moving in a static external potential created by the electric field of nuclei.

H e

, is conventionally split into a sum of three contributions: the kinetic energy of the

H e

(2.4)

Let the location of electron i be denoted r i

, the total number of electrons in the system N , and the potential energy field originating from the nuclei, in which the electrons are moving, v(r). In a wave-function based description of the system electronic states, ψ = ψ(r

Hamiltonian become:

1

, r

2

, ..., r

N

), the contributions to the

µ

¯h

2

2m e

N

X

2 i

, i=1

µ e

2 c

4π²

0

N

X i<j

1

|r i

− r j

|

,

N

X v(r i

).

i=1

(2.5)

(2.6)

(2.7)

Taking the expectation value for the energy gives an explicit expression

E e

Z Z

...

Z

µ ¯h

2

2m

N

X

ψ

2 i

ψ + i=1

µ e

2 c

4π²

0

N

X i<j

|ψ|

2

|r i

− r j

|

+

+

N

X

|ψ|

2 v(r i

)

!

dr

1 dr

2

...dr

N

.

i=1

(2.8)

Here T , U and V are introduced as the individual, scalar, expectation values of the respective operators.

The expression for E e can be simplified by introducing the electron density, n(r). The relation between n(r) and the wave function is n(r

1

) = N

Z Z

...

Z

|ψ(r

1

, r

2

, ..., r

N

)|

2 dr

2 dr

3

...dr

N

.

(2.9)

The expression on the right hand side is similar to a wave function normalization expression, but with one of the integrals left out, i.e., leaving one index free. It

2.2. The Thomas–Fermi Model

9 is not important which of the indices of the wave function is left out, as the electrons are indistinguishable, and a proper wave function must reflect this.

When looking at the three terms in the expression for the expectation value of the energy, Eq. (2.8), we see that only one term is easily rewritten in terms of the density, the term for the external potential, V :

V =

Z Z

...

Z

N

X v(r i

) |ψ|

2 dr

1 dr

2

...dr

N i=1

=

=

1

N

N

X

Z i=1 n(r i

)v(r i

)dr i

=

Z n(r)v(r)dr.

(2.10)

The other two terms in Eq. (2.8) cannot be rewritten in this simple way. In the kinetic energy term, T , the derivative operator between the wave functions makes it non-straightforward to put it into a ‘density’-form with the two wave functions next to each other, and in the term of the internal potential energy,

U , the two different particle locations present within the integrations prevent a simple term by term integration.

The dependencies found above can be stated in an explicit way using functionals:

E e

= T [ψ] + U [ψ] + V [v, n] = F [ψ] + V [v, n].

(2.11)

This is just a simplified notation for the energy of a general electron wave function, i.e., Eq. (2.8). As indicated by the designation F [ψ], that functional is independent of the external potential v.

A possible way of solving an electron energy problem is to search for the ψ that minimize E e

. This approach leads to wave-function based methods, i.e., methods that are based on finding a minimizing wave-function in a space of functions of a dimensionality proportional to the number of electrons in the system. In contrast, the DFT approach is to rewrite F [ψ] so as to depend only on the electron density n(r). Below it is shown how a minimization of the threedimensional n(r) constitutes an alternative way of solving the electron energy problem. The first thing to be considered is the existence of a F [n] functional.

2.2

The Thomas–Fermi Model

Early attempts to approximate F [ψ] in functionals of only the density were performed by Thomas and Fermi.

8, 9

Quantum mechanical calculations using

A functional is an object that acts on a function to produce a scalar. It is written as an ordinary function except that the function it acts on is placed within bracket parentheses, [ ].

For a good review on functionals, see Ref. 5

10

Chapter 2. Density Functional Theory assumptions about the distribution and the interaction between electrons, give the following approximation for the kinetic energy:

T ≈ T

TF

[n] =

3

10

(3π

2

)

2/3

µ e

2 c

4π²

0

¶ Z n

5/3

(r)dr.

(2.12)

A simplistic approximation of the internal potential energy, U , is given by the expression for a classically repulsive gas (i.e., completely ignoring the quantum interactions between electrons described by the Pauli principle):

U ≈ J[n] =

1 µ

2 e

2 c

4π²

0

¶ Z Z n(r

1

)n(r

2

) dr

1 dr

2

.

|r

1

− r

2

|

This gives the Thomas–Fermi model:

E e

≈ T

TF

[n] + J[n] +

Z n(r)v(r)dr.

(2.13)

(2.14)

The result is a first approximation of a F [n] functional, i.e., F [ψ] ≈ T

TF

[n]+J[n].

2.3

The Hohenberg–Kohn Theorems

The early efforts to find energy density functionals (the Thomas–Fermi model and extensions along the same idea) were all based on ‘sensible’ approximations which gave decent results when applied to certain real problems. There is a great conceptual difference between these heuristic ‘guesses’ and the elegant theoretical framework following the work of Hohenberg and Kohn

1

(HK).

The first of the two HK theorems tells us that the ground state electron density n(r) determines the potential of a system, v(r), within an additive constant

(which only sets the absolute energy scale). The proof is outlined in the following. Assume two different system potentials, v a

(r) and v b

(r). They must, if they differ by more than an additive constant, give rise to two different ground states in the Schr¨odinger equation, ψ a and ψ b

. Let us assume that the states are non-degenerate and that they give the same electronic density, n(r). The

H a as the Hamiltonian for the system with potential v a

(r))

(2.15) E a

= hψ a

| ˆ a

|ψ a i < hψ b

H a

|ψ b i = F [ψ b

] + V [v a

, n], and consequently

E b

< F [ψ a

] + V [v b

, n].

Adding the two equations, and recollecting the terms, gives

(2.16)

E a

+ E b

< E b

+ E a

.

(2.17)

This last relation is clearly a contradiction. The conclusion is that for systems without degenerate ground states, two different potentials cannot give the same

2.3. The Hohenberg–Kohn Theorems

11 ground state electron density. This means that it is theoretically possible to obtain all ground state properties of a system from the ground state electron density, n(r). Based on this, the ground state wave function can be considered as a functional of the ground state density ψ[n], which proves the existence of an unapproximated energy functional, E e

[n]:

E e

[n] = h ψ[n] | ˆ e

| ψ[n] i.

(2.18)

Thus the electronic energy functional, F [n], must also exist as a functional of the ground state electron density.

It was later shown

10–13 that it is possible to construct a (somewhat artificial) explicit definition of F [n]:

F [n] = min

ψ→n

(2.19) where the minimum is taken over all ψ that give an electron density of n. This is commonly referred to as the constrained search formulation. In this form the

DFT formalism does not require a non-degenerate ground state.

The second of the two Hohenberg–Kohn theorems provides a variational principle for the total energy combination

F [n] + V [v, n]. If we denote a trial density

H e e

| ψ[n] i = F [n] + V [v, n], (2.20) where the minimum is known to occur only for:

ψ[˜ ⇒ n = n.

(2.21)

The existence of this variational principle is the reason to why it is so useful to have an approximation of F [n]. Given a good enough approximation, the electronic energy can be obtained by what seems to be a rather uncomplicated minimization of a three dimensional function n(r). Hence, we have found a method of handling the difficult part of Eq. (2.3).

In practice, however, complications do enter here, since the formal constrained search definition of F [n], Eq. (2.19), is very complicated and actually returns the scheme to a minimization with respect to wave-functions of high

Note that the combination used here, F [n] + V [v, n], is not equivalent to the E e

[n] shown to exist in Eq. (2.18). The difference is that the latter uses the external potential that corresponds to the inserted trial density, whereas the former uses an explicit dependence on the real external potential of the actual system. It is possible to see the difference in formal terms by defining

E e

[n] as F [n] + V [v(r, [n]), n]. A clarification to why this E e

[n] cannot be used in a variational

E[n], as all trial densities should give higher energies than the true density. But, in a different this problem, because the explicit dependence on the real external potential v(r) differentiates between the two test cases.

12

Chapter 2. Density Functional Theory order. It is not feasible to perform a constrained search in practice, because then nothing would be gained over traditional wave-function based methods. Instead, the above findings are just a formal establishment of the existence and possible use of a general F [n], giving little or no hint on how to approximate it.

Consequently, approximations of F [n] must be constructed using more elaborate methods than simply ‘just approximating the constrained search’.

Chapter 3

DFT Calculations

Having established the theoretical foundation of DFT, it is time to consider its practical use.

3.1

The Kohn–Sham Scheme

In 1965 it became clear how to turn the theoretical framework of density functional theory into a practical computational tool as Kohn and Sham

2 proposed a method for computing the most important part of the kinetic energy functional to good accuracy. The idea was to rewrite the problem of many interacting electrons to make it possible to use an analogue with a system of many noninteracting electrons.

The first step is to restructure the components of the functional F [n] slightly, introducing T s as the kinetic energy of a system of non-interacting electrons and

J[n] as it was defined in the section about Thomas–Fermi theory, Eq. (2.13):

F [n] = T s

[n] + J[n] + E xc

[n], (3.1) where the exchange correlation energy, E xc

[n] consequently is defined to make this relation exact, i.e.,

E xc

[n] = F [n] − T s

[n] − J[n].

(3.2)

This component of F [n] does, in some sense, take care of the non-classical corrections to the energy, and is the only part for which approximations are needed.

Using the above expression for the energy, it is possible to recast the system of interacting electrons into a system of non-interacting electrons. The basic idea is to use the non-interacting kinetic energy as the kinetic energy term for the system, and thus to hide all other energy contributions in the potential term. To do this correctly it is necessary to examine the underlying variational problem, and

13

14

Chapter 3. DFT Calculations carefully designing a new potential function, the KS-effective potential v eff

(r).

This potential function is found to be (where δ denotes a functional derivative): v eff

(r) = v(r) +

µ e

2 c

4π²

0

¶ Z v xc

(r) =

δE xc

[n]

,

δn(r) n(r

0

)

|r − r 0 | dr

0

+ v xc

(r), (3.3)

(3.4) where v xc is the exchange correlation potential.

The effective potential function is designed with the primary purpose of giving the electron density of the real system from orbitals found by solving a corresponding non-interacting one particle orbital Schr¨odinger equation, the KSorbital Schr¨ :

µ

¯h

2

2m

2

+ v eff

ψ i

= ² i

ψ i

.

(3.5)

The orbitals obtained, ψ i

, are the KS-orbitals. Their corresponding energies,

² i are the KS-orbital energies. The density of the real system is then, by construction of v eff

, obtained as the density of this non-interacting orbital system

(accounting with a factor of two for a double degeneracy in each orbital because of the electron spin): n(r) = 2

N

X

|ψ i

(r)|

2

.

(3.6) i

The total energy of the real system is obtained by inserting this density into the total energy functional, F [n] + V [v, n]. The presented set of equations,

Eqs. (3.3)–(3.6), are usually referred to as the KS equations.

However, the definition of v eff includes the electron density, which is the quantity we are solving for. Thus, for a normal quantum chemistry problem where electrons are moving in a known external potential generated by the nuclei, one has to solve the KS-equations in iterative steps. Classically this is done by cyclically going through the equations, each time using the last obtained density to construct a new v eff

, until self consistency is attained. There also exist more refined approaches that converge faster towards the right density (and arguably more reliably).

3.2

The Kohn–Sham Orbitals

It is important to note that the KS-orbitals, ψ i in Eq. (3.5) is not the actual electronic orbitals in the interacting multi electron system. The KS-orbital

Schr¨odinger equation is obtained from a mathematical trick and is designed to give the correct density in Eq. (3.6), not to give the correct orbital description of the system. In the same way there is no reason to expect any simple interpretation of the KS-orbital energies, ² i

, in Eq. (3.5). It is, despite this, quite common

3.3. The Exchange Correlation Energy

15 to take the KS-orbitals and energies as approximations for the real orbitals and energies and the results are usually surprisingly good.

3.3

The Exchange Correlation Energy

To be able to use the Kohn–Sham scheme above one needs an explicit expression for the exchange correlation energy, E xc

. It is clear that E xc cannot, in any sense, be a local quantity, as it is independent of space coordinates and is equally affected by all changes throughout the system. For further analysis, and to develop useful approximations, E xc is split into a sum of contributions that, arguably, will be local. This gives the exchange correlation energy density, ² xc defined in every point of space, ² xc

([n]; r), and is assumed to show some kind of

‘locality’ in the sense of being mostly dependent on the part of the density which is close to r. It is defined implicitly as giving the exchange correlation energy by integration:

Z

E xc

[n] = n(r)² xc

([n]; r)dr.

(3.7)

The implicit definition of ² xc

([n]; r) makes several choices possible, since the only requirement is that it should integrate to the total exchange correlation energy. This freedom of choice is important for the subsystem functional approach and will be discussed more in chapter 4. Following the formalism of Kohn and

Mattsson,

4 the most common choice will be described using the inverse radius of the exchange correlation hole , R

− 1 xc

, to which ² xc

([n]; r) is simply related as:

² irxh xc

([n]; r) = −

µ e

2 c

4π²

0

¶ 1

2

R

1 xc

([n]; r),

R

− 1 xc

= −

Z n xc

(r; r

0

) dr

0

,

|r − r 0 |

(3.8)

(3.9) where n xc

(r; r

0

) is the conventional exchange correlation hole density.

In this work the primary focus is on the exchange part of ² xc

([n]; r). The quantity n xc can be split into the exchange hole density, n x

, and the correlation hole density

, n c

, as follows: n xc

= n x

+ n c

,

² irxh x

² irxh xc

= ² irxh x

([n]; r) =

µ e

2 c

+ ² irxh c

,

4π²

0

¶ Z n x

(r; r

0

) dr

0

,

|r − r 0 |

² irxh c

([n]; r) =

µ e

2 c

4π²

0

¶ Z n c

(r; r

0

|r − r

0

|

) dr

0

.

(3.10)

(3.11)

(3.12)

(3.13)

It is a matter of conventions if one regards n xc

(r; r

0

) as a well defined quantity, or if one considers Eqs. (3.8) and (3.9) as definitions and regards n xc

(r; r

0

) as the source of nonuniqueness. Both viewpoints have been present in recent works of different authors. We will only use n xc

(r; r

0

) to mean the conventional, unique, definition.

16

Chapter 3. DFT Calculations

Here ² x is the exchange energy per particle and ² c the correlation energy per particle

. The exchange hole can be expressed unapproximated in the KS orbitals: n x

(r; r

0

) = −

1

2

1

(r; r

0 n(r)

)|

2

,

ρ

1

(r; r

0

) = 2

X

ψ

ν

(r)ψ

ν

(r

0

),

ν

(3.14)

(3.15) where n x

(r; r

0

) is the exchange hole density and ρ

1

(r; r

0

) the first-order spinless density matrix .

There are two important relations involving n x and n c

, valid for all r:

Z n x

(r; r

0

)dr

0

= −1,

Z n c

(r; r

0

)dr

0

= 0.

(3.16)

3.4

Local Density Approximation, LDA

The first and most simple approximation of the exchange correlation energy is the local density approximation

(LDA), proposed by Kohn and Sham

2 in their first work on DFT. LDA sets ² xc

E xc in every space point r, with density n(r), equal to per electron of a system with a constant v eff

(a uniform electron gas) chosen to make the density of the uniform system equal to n(r). It is straightforward to derive the exchange part of LDA; the KS orbitals for a constant effective potential (plane waves) are inserted into Eq. (3.8), which gives a constant ² irxh x

, that can be parameterized in the uniform electron density to give:

²

LDA x

(n(r)) = −e

2

3

¡3π

2 n(r)

¢

1/3

.

(3.17)

The correlation part of LDA was first estimated by Wigner

14 cently numerically computed by Monte Carlo calculations

15 and more re-

(there are several different parameterization of the Monte Carlo data, for example the one by

Perdew and Zunger

16

).

LDA was constructed as a suitable approximation for systems with very slowly varying electron densities, but it was found remarkably successful for wider use. It is still often used for electronic structure calculations of solid materials, although this is outside its formal region of validity. The accepted explanation to the success of LDA is that, apart from just being a ‘sane’ approximation of the real exchange-correlation energy, it also fulfills several constraints which it is known that the real exchange-correlation energy functional must fulfill.

3.5

Gradient Expansion Approximation, GEA

Even in the earliest works of DFT there was some work done towards improving the LDA by extending it through a gradient expansion approximation (GEA).

3.5. Gradient Expansion Approximation, GEA

17

In every space point LDA just uses the local value of density. The basic idea behind the GEA is that this can be seen as using the first term in a power expansion in the local variation of the density. Hence the idea is to construct an approximation of ² xc as a power expansion in the density and its derivatives. For a proper power expansion to second order, taking several known symmetries into account, terms proportional to two derivatives are needed. These derivatives are usually put into dimensionless quantities. The dimensionless gradient s =

|∇n(r)|

2(3π

2

)

1/3 n

4/3

(r)

.

(3.18) and the dimensionless Laplacian q =

2 n(r)

4(3π

2

)

2/3 n

5/3

(r)

.

(3.19)

However, based on the freedom of choice of the definition Eq. (3.7) it is possible to use integration by parts to eliminate the q-term. The resulting expression for the GEA is

²

GEA x

(n(r), |∇n(r)|) = ²

LDA x

(n(r))

µ

1 +

10

81 s

2

, (3.20)

In a truly slowly varying system, the GEA should perform well, but outside of its area of formal validity the GEA is found to be unsatisfactory when applied in computations. The fact that it often is less accurate than the LDA is somewhat disappointing. However, GEA has successfully been used in the derivation of modern nonempirical functionals as the limit of low-density variation, this approach has given very useful functionals (eg. certain GGAs,

17, 18 see Sec. 3.6).

There was much ado before the correct gradient coefficient, 10/81, was established. First Sham performed

19 a calculation based on the correlation methods of Ma and Brueckner

Gross and Dreizler

21

20 and obtained a value of 7/81. Another calculation of also gave the same result; but empirical results

22 indicated that the value was too low. Antoniewicz and Kleinman obtained 10/81, and after some suggestion of Perdew and Wang,

23

Kleinman and Lee

24 numerically demonstrated that the cause of the confusion was an order of limits problem between the Yukawa screening factor ¯

Y and the wave vector of the density variation K. The problem is nicely exemplified by a (here slightly modified) proposed guess of a possible explicit form by Perdew and Wang:

² x k

Y

) =

7

81

+

3

81

K

2

K

2

2

Y

.

(3.21)

It is found that lim k

Y

0 lim

K→0

² x k

Y

) = 7/81 (Sham result), lim

K→0

(3.22) lim

Y

→ 0

² x k

Y

) = 10/81 (Antoniewicz–Kleinman result).

(3.23)

18

Chapter 3. DFT Calculations

The plots of Kleinman and Lee

24 indicate that the qualitative behavior of this toy model is not far from the truth. Given this, it is evident that the ‘right’ answer is the Antoniewicz–Kleinman result, because in a true Coulomb system the Yukawa screening factor is identically zero, and hence must always be smaller than the wave vector of the density variation, that only tends to zero as we approach a slowly varying density.

However, in the successive papers of Antoniewicz, Kleinman and Lee there appear some comments regarding whether the right gradient coefficient may be dependent on how the correlation energy term is obtained, i.e., that perhaps the errors in the Sham exchange are cancelled by errors in the Ma–Brueckner correlation (since they are obtained in similar ways this is not too unlikely).

Whether this issue was finally resolved or not is unknown to the author.

3.6

Generalized-Gradient Approximations,

GGAs

An abstract definition of a Generalized-Gradient Approximation (GGA) is simply a quantity that given the local value of the density and its gradient, s

2

, produces an approximation of the exchange-correlation energy per particle, i.e.:

E

GGA xc

=

Z

²

GGA xc

(n(r), s

2

)dr.

(3.24)

Roughly the idea is that a GGA should be constructed to produce good results for a wide range of density variations; perhaps using the GEA as a constraint for slowly varying densities, and some other known property for more rapidly varying densities. Further constraints are commonly taken from the exact conditions that the real exchange-correlation energy functional is known to fulfill (previously mentioned when explaining the success of LDA). The aim is a simple expression on closed form, which can be used for real calculations.

One usually distinguishes between empirical and non-empirical GGAs. Empirical GGAs use experimental or otherwise known results to adjust the parameters in the expression, whereas non-empirical GGAs should only use parameters derived from basic physical (quantum mechanical) principles. The common sense is that it is possible to obtain very accurate results using an empirical GGA adjusted for some precise class of systems; but that same GGA may perform poorly if applied to something that is slightly outside this class. In contrast, a good non-empirical GGA can be applied to a wide variety of problems. Because of this, to use an empirical GGA when there is no other (eg. experimental) result to compare with seems (at least to a physicist) somewhat dangerous.

Currently a large number of good GGAs are in use within quantum chemistry, materials physics and other areas. But expressions have also been developed with more complicated input parameters, such as the Laplacian q and kinetic energy densities.

3.7. The Chemical Potential

19

3.7

The Chemical Potential

We will be referring frequently to the chemical potential, µ, in the following chapters. The chemical potential is a term from statistical mechanics, interpreted as the energy needed to add a particle to a multi particle system. In density functional theory it enters with the requirement that the density should integrate to the total number of particles. In an electron system at zero temperature the chemical potential is half way between the energy of the highest occupied orbital and the lowest unoccupied. It can be seen that this also holds for the KS-orbital energies, and thus the KS-orbital energies fulfill:

² i

≤ µ ∀ occupied i.

(3.25)

20

Chapter 4

Subsystem Functionals

The subsystem functional approach was mentioned in the introduction, but will now be discussed in more detail. Note however, that the paper in part II of this thesis also presents the subsystem functional approach in some detail.

4.1

General Idea

The subsystem functional approach is mostly based on the idea of locality (nearsightedness) of the electron behavior, which has previously been discussed in other works.

3, 4

The general idea is that in most electron systems, the main effect on a given electron is caused by the closest of the other electrons, and thus its behavior should be governed by local or semi-local properties of the electron cloud. There are, however, some well known situations where this statement is known not to be true, eg., superconducting systems.

The starting point is the implicit definition of the exchange correlation energy per particle:

Z

E xc

= n(r)² xc

(r)dr.

(4.1)

The integration, to be taken over the whole system, may be decomposed into integrations over several separate regions:

E xc

=

Z

R

1 n(r)² xc

(r)dr +

Z

R

2 n(r)² xc

(r)dr + ... +

Z

R

N n(r)² xc

(r)dr.

(4.2)

This general idea was illustrated in chapter 1 in Fig. 1.1.

Approximations of ² xc which can be applied in this way, i.e., using different approximations in each individual region, are referred to as subsystem functionals . Obviously, a subsystem functional may not be derived based on the assumption that individual integrations of Eq. (4.2) are taken over the whole system.

Rather, when integrated over only a part of a system, a subsystem functional must give a valid approximation of the integrated value of some exact ² xc in that

21

22

Chapter 4. Subsystem Functionals part. Another view of this is that, given the freedom of choice of ² xc in Eq. (3.7) it is necessary for all subsystem functionals applied to the same system to approximate integrated values of precisely the same choice of ² xc

. This requirement is needed to ensure that the correct total exchange correlation energy results from the sum of contributions from the different parts. It is demonstrated in the paper in part II of this thesis how this requirement is enforced if all subsystem functionals applied to a system are taken to approximate one common, explicitly defined, ² xc

, such as, for example, the irxh-based definition of Eq. (3.8).

All the ideas presented here have been expressed in the coordinate space, but it is also possible to consider a partition in k-space (i.e., by performing a Fourier-transform of Eq. (4.1) before partitioning the integral). This is just briefly mentioned here, and has not been formally investigated. Such an investigation must probably begin with a careful examination of what concept of locality should be employed, giving an idea of the kind of systems where this approach can be successfully used.

4.2

Current Status of the Subsystem

Functional Approach

Modern functionals in use today (eg. GGAs) have usually been derived utilizing integration by parts and other methods that assume that they are to be used throughout the whole system, and thus cannot be used in only a partial region. Because of this, for the subsystem functional scheme to be successful, it is necessary to create a GGA-like functional conforming to the subsystem functional requirements. Much of the paper in part II of the thesis is devoted to this approach; i.e., trying to unravel necessary information to create such a functional.

We have currently not discussed the non-trivial problem of performing the partitioning of a system. If a wide range of subsystem functionals was available, they could either be carefully combined by computational scientists targeting some specific system, or be composed into more general functionals applicable to a general set of problems, such as systems with electronic edges (which was the aim of the work of Kohn and Mattsson

4 referred to in the introduction in chapter 1). However, it seems fruitless to speculate in how to construct useful constraints on the partitioning before characteristics of specific subsystem functionals are known.

4.3

Related Efforts

Here the subsystem functional approach has been presented on an abstract level, but several efforts that can be seen as related have been used in actual DFT calculations. For example, Vitos et al. developed the Local Airy Gas (LAG)

25 functional. Roughly, it corresponds to using the Airy Gas exchange energy per

4.3. Related Efforts

23 particle and the LDA correlation energy per particle in the edge region, while using LDA exchange and correlation energies per particle in the interior region.

This functional is briefly discussed in the paper in part II of this thesis. Another example is so called hybrid functionals. They have been successfully employed in calculations, which are combining very different functionals within one system.

However, usually hybrid functionals do not partition the system into regions, but rather combine a number of functionals to be used simultaneously throughout the whole system.

24

Chapter 5

The Mathieu Gas as a DFT

Model System

Much of the scientific work in this thesis is based on the numerical study of a simple model system, the Mathieu Gas (MG). This chapter introduces the MG and discusses its usefulness as a DFT model system.

5.1

Definition of the MG

The MG can be viewed as a family of electron densities parameterized by two from a noninteracting system of electrons moving in an external potential v eff F,u z)).

(5.1)

Here µ is the chemical potential and k

F,u is the Fermi wave vector of a uniform electron gas with µ as its chemical potential. By solving the corresponding nondensity, are obtained.

5.2

Electron Densities of the MG Model

Solving the MG effective potential system for the KS orbitals is significantly simpler than solving a general non-interacting electron system, since the system is essentially one-dimensional. The one-dimensional nature of the system makes the KS-orbital Schr¨odinger equation separable, which reduce the Fermi edge to just a point (indicating where the one-dimensional integration should stop). In a real, non-separable three-dimensional system, the Fermi edge may describe a complicated curve, and may not be as easily treated.

25

26

Chapter 5. The Mathieu Gas as a DFT Model System

The two constant x and y dimensions are separately handled, giving a KS orbital of the form:

ψ

ν

(x, y, z) = √

L

1

1

L

2 e i(k

1 x+k

2 y) ϕ

η

(z), (5.2) where ν specifies k

1

, k

2 and η; L

1

L

2 is the x, y area of the system and will approach infinity; and k i

L i

= 2πm i

(i = 1, 2, m i integer), and finally ϕ

η

(z) is the one-dimensional z-direction KS orbital. The resulting one dimensional eigenvalue equation now becomes:

µ ¯h

2

2m d

2 dz

2

+ v eff

(z) − ²

η

¶ ϕ

η

(z) = 0, (5.3)

Inserting the MG v eff gives a Mathieu differential equation with known solutions

(see Ref. 26 for definitions of the Mathieu function symbols, se

η ϕ

η

(z) =

½ (1/

(1/

L) ce

L) se

η

η

(pk

F,u

(pk

F,u p

2 p

2

) if η > 0

) if η < 0

, ce

η

, a

η and b

η

)

(5.4)

²

η

= p

2 a

η p

2 b

η

(5.5) n(r) = k

3

F,u

2

Z

˜

− ˜

L|ϕ

η

(z)|

2

µ

1 −

²

η

µ

¶ dη, (5.6) where ˜

η

≤ µ.

However, we are still left with the issue of finding suitable numerical routines for calculating the values of the Mathieu functions ce and se. In fact, the method of obtaining explicit numerical values for the Mathieu functions is to go back to the differential equation, Eq. (5.3), and solve it by standard matrix methods; very much like a generalized DFT-program would. This method is presented in some detail in the paper in part II of the thesis. Furthermore, when the KS orbitals are known they can also be inserted into the equations for the exchange part of the exchange-correlation energy per particle, Eq. (3.12), (3.14) and (3.15).

This is essentially how the data was obtained for the plots in Figs. 7–12 of the paper in part II.

From a careful analysis of the energy expression, Eq. (5.5), it is seen that the MG shows a rudimentary energy-band structure. The index η acts as a dimensionless scaled wave-vector, that indexes this band structure, very much like the wave vector in an extended Brillouin zone-scheme.

5.3

Exploring the Parameter Space of the MG

interest in the pursuit of a functional working for such densities, we have also found it very useful to investigate some specific limits in the MG as they are model systems in their own right.

5.3. Exploring the Parameter Space of the MG

27

5.3.1

The Limit of Slowly Varying Densities

From the construction of the MG family of densities it follows that the limit of of this limit makes it challenging to analyze the evaluated numerical data in a consistent way. This is one of the larger issues with the data analysis of DFT quantities in the MG gas. In the present work this was handled by plotting calculated data versus a combination of the two parameters, that essentially equalized similar behavior of the curves. The creation of such a parameter is based on the observation that the shape of the curves of most DFT quantities are closely related to the energy structure of the non-interacting electron system used to construct the MG family of densities. A suitable choice of such an energy related parameter was constructed as:

α =

²

µ − ²

η

1

η

2

− ²

η

1

+ |η

1

|, (5.7) where, if µ is inside a z-dimension energy band, ² band. If µ is not inside an energy band, ²

η

1

η

1 is the lowest energy in this is the lowest energy in the band which contains the z-dimension energy state with highest energy ≤ µ. Furthermore,

²

η

2 is the lowest possible energy of all z-dimension energy states within bands that only contain energies > µ. By construction η

1 and η

2 are integer.

The parameter α describes the position of the chemical potential relative to the lower band edges, that is, the lowest energies of the energy bands in the z dimensional energy band structure. The parameter α differs from η in that it indexes values of the chemical potential both within and between the energy bands in the z dimension, making it useful throughout the parameter space of the MG.

5.3.2

The Free Electron Gas Limit

potential. This makes the solutions to the MG differential equation approach plane wave solutions to a free electron (FE) gas: ϕ

η

(z) =

²

η

1

L

3 exp(iη ¯

= µη

2

¯

2

.

z), (5.8)

(5.9)

Hence, in this limit the MG densities show similarities to a system with an infinite, weakly perturbed, system. Disregarding the one-dimensionality of the

MG effective potential, by moving away from the FE limit in the right direction, the MG should, in a very abstract sense, appear much like an infinite, slowly varying, crystal.

28

Chapter 5. The Mathieu Gas as a DFT Model System

We also note that specifically in the FE limit, the α parameter can be simplified to

α

F E

= p

2

+ N (N + 1)

2N + 1

, N =

¹ 1 º

.

(5.10)

5.3.3

The Harmonic Oscillator Limit

p

2

→ ∞. The MG effective potential approach an harmonic oscillator (HO), but this might not be as obvious as was the case for the FE limit. The situation might be more clear through an analysis of the energy structure, which in this limit becomes:

² n

= µ q

2¯ p

2

(2n + 1) .

(5.11)

This clearly describes equally spaced energy levels, with a distance of µ p2¯λ¯p

2

, which is similar to a text book HO system.

The corresponding KS orbitals are: ϕ n

(z) =

à k

F,u

( p2¯λ¯p

2

)

1/2

π2 n n!

!

1/2

H n

(( q

2¯ p

2

)

1/2

× exp (−[( q

2¯ p

2

)

1/2 z]

2

/2), (5.12) where H n

(x) are Hermite polynomials

26 and n = 0, 1, 2, . . . .

Finally, in the HO limit the α parameter can be simplified to:

α

HO

=

1

2 p2¯λ¯p

2

1

2

.

(5.13)

The HO limit has turned out to be very useful in the work presented in this thesis, since it is possible to do computations for this system, i.e., as a model system in its own right instead of employing a limiting procedure in the

MG. Comparing MG data in the HO limit and data for the pure HO gives an extra check on the numerics. Also, in a very abstract sense, the HO limit can be said to mimic an atomic-like system, as it effectively is of finite size in the z dimension. Hence, this limit and the ’crystal like’ FE limit previously mentioned are fundamentally different.

Chapter 6

Investigation of the Kinetic

Energy Density

This chapter investigates the power expansion of the kinetic energy density.

Although this power expansion is already well known, the procedure provides an independent evaluation of our numerical methods. The study also serves as a simplified example of the methods of the investigation of the exchange energy per particle presented in the paper in part II of the thesis.

6.1

Definition of the Kinetic Energy Density

The kinetic energy density τ (r) is a localized version of the total kinetic energy

(KE) of the non-interacting KS electron system. It is designed with the same approach that led to the exchange-correlation energy per particle, Eq. (3.7). An implicit definition, much in the same style as Eq. (3.7), is (T s was introduced in

Eq. (3.1))

T s

[n] =

Z

τ (r)dr.

(6.1)

The conventional definition of τ (r) for a spin unpolarized system is

τ (r) =

X

|∇ψ

ν

(r)|

2

,

ν

(6.2) where the sum is taken over all occupied orbitals. It is known

27–29 second order gradient expansion of this quantity is that the

τ

τ exp

LDA

(r) = τ

(r) =

LDA

µ

1 +

5

27 s

2

+

3

10

(3π

2

)

2/3 n(r)

5/3

.

20

9 q

, (6.3)

(6.4)

29

30

Chapter 6. Investigation of the Kinetic Energy Density

6.2

The KE Density in the MG

The KS orbitals corresponding to the MG family of densities can be inserted into Eq. (6.2) to compute numerical values of the kinetic energy density. It is expected that the computed values in the limit of slowly varying densities should behave accordingly to Eq. (6.3). This is investigated here by evaluating curves p

2

= 0.8 and plot them versus 1/α in the limit 1/α → 0; i.e., the limit of slowly varying densities.

Given Eq. (6.3), the following limits are expected for the MG:

For s

2

= 0, 1/α → 0 :

For q = 0, 1/α → 0 :

µ τ (r)

τ

LDA

µ τ (r)

τ

LDA

− 1

/q →

− 1

/s

2

20

9

,

5

27

.

(6.5)

(6.6)

It is simple to find a path where s

2

= 0 as this is always fulfilled at the minimum point of the effective potential, i.e., z = 0, because of symmetries. To find a path where q = 0, a search is implemented in the program that finds a z where this is fulfilled for every α-point to be evaluated.

6.3

Analyzing the KE Density

Plotting evaluated data for the two limits discussed above gives Figs. 6.1 and

6.2. We first note that the limiting behaviors predicted in Eqs. (6.5) and (6.6) are correctly obtained in the 1/α → 0 limits. Apart from this expected result, it is interesting to comment on the behavior at higher 1/α. In the paper in part II of the thesis a careful study of the behavior of the electron density, the density

Laplacian and the exchange energy per particle is performed. These quantities were observed to be strongly influenced by the energy structure of the MG, most clearly seen at values of α where the chemical potential is placed between energy bands

. We observe that the KE density also is affected by the energy structure, as the same values of α describes locations of rapid change (1/α = 1/2 and

1/α = 1/3 in Fig. 6.1; the curve in Fig. 6.2 also changes rapidly at the first point, but it has only a small ’bump’ at the second point). However, note that the variation of the KE density is on a very small scale. Plotting for yet higher values of 1/α should give a more relevant picture of the behavior of the KE density outside the domain of slowly varying densities.

Refer to the paper for more details on the MG energy structure.

6.3. Analyzing the KE Density

2.23

2.22

2.21

2.2

0

2.27

2.26

2.25

2.24

MG−tau, s

2

=0

KE expansion coeff: 20/9

0.1

0.2

0.3

1/

α

0.4

0.5

0.6

0.7

Figure 6.1.

The quantity (τ (r)/τ

LDA

λ/¯

2

= 0.8 and s

2

= 0.

In the limit of slowly varying densities, 1/α → 0, this quantity approaches the

Laplacian coefficient in the kinetic energy density power expansion, Eq. (6.3), as is expected.

31

0.4

0.35

0.3

0.25

0.2

0.15

0

0.6

0.55

0.5

0.45

MG−tau, q=0

KE expansion coeff: 5/27

0.1

0.2

0.3

1/

α

0.4

0.5

0.6

0.7

Figure 6.2.

The quantity (τ (r)/τ

LDA − 1)/s

2

λ/¯

2

= 0.8 and q = 0.

In the limit of slowly varying densities, 1/α → 0, this quantity approaches the gradient coefficient in the kinetic energy density power expansion, Eq. (6.3), as is expected.

32

Chapter 7

Concluding Remarks and

Comments on the Paper

The paper in part II of this thesis both presents the subsystem functional approach and examines properties of a suggested form of a subsystem functional for subsystems with slowly varying electron densities. The paper goes into details of the Mathieu Gas to make a careful data analysis possible. The exchange energy per particle is calculated and presented, giving a consistent picture of the behavior of the DFT quantities of the MG system. However, when investigating the exchange energy per particle in this system it is shown that it cannot be modeled by an ordinary power expansion. This introduces an obstacle for the subsystem functional approach, since it prevents a simple starting point for building further GGA-like functionals.

As mentioned in the conclusions of the paper, several options for future directions are possible. From work done after the paper was published, it seems that the most fruitful of these options are the parameterization of the non-analytical behavior and the approach that splits the exchange–correlation energy in some alternative way to what is traditional. We are currently working in these directions.

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Acknowledgments

There are a lot of people, without whom, this thesis would not be a reality.

The supervisor of my research projects, Ann Mattsson; your almost constant aid and supervision has truly made me feel that we are working together, towards a common goal. All the scientific and not-so-scientific discussions with you, and your husband Thomas Mattsson, ranging between all things from Earth to Sky, have probably affected the way I look at the world. Thanks to you both for receiving me in Albuquerque; my stays there have been most pleasant.

Professor G¨oran Grimvall, who has provided me with complete assistance in all these matters that must be handled during the time as a graduate student; planning trips and funds, creating time schedules, handling formal matters, and some occasional discussions about my work; this has altogether given me a very supportive environment in which I do my work. And, also many thanks for his excellent efforts to proof-read my manuscripts (although, he has surely not made me eliminate all the occurrences of strangely located addendums within parenthesis, the sudden interposed sentences located here and there, numerous splling errurs missed punctuation, and other inaccuracies...)

All the friendly people at the department with whom I have had the delight to discuss a lot of interesting subjects, often over lunch; thank you for all these nice occasions. Thanks also to Tore Ersmark for suggesting a sushi lunch once in a while as a nice alternative to the more ordinary lunch food.

Thanks to professor John Perdew and the members of his group for receiving me at Tulane University in New Orleans. My visit there was both interesting and fruitful.

Thanks also to Askell Kjerulf and the people at KTH-lan (most notably Anders Hillbo) for being sane points of aid in an otherwise totally chaotic computer environment in which I have been appointed the questionably honorable title of

‘System administrator’. While I am at it; thanks to the open source community in general and Linus Torvalds in particular; your efforts have made my work that much simpler, now referring both to system administration and research work.

Thanks to my family for being there for me, and, when I have been most engulfed in my work, having checked up on me to verify that I was okay.

Last, but certainly not least, I would like to send my warmest heartily thanks to my love Maria Tengner; thank you for your constant support in adversity and success alike. I love you.

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Part II

Paper

Phys. Rev. B 66 165117 (2002)

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