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Master’s Thesis
Density Functional Theory for
Systems with Electronic Edges
Rickard Armiento
Theory of Materials, Department of Physics
Royal Institute of Technology, SE-100 44 Stockholm, Sweden
Stockholm, March 2000
Typeset in LATEX
Examensarbete inom ämnet fysik för avläggande av civilingenjörsexamen inom
utbildningsprogrammet Teknisk Fysik (F).
ISBN 91-7170-538-4
TRITA-FYS-8042
ISSN 0280-316X
ISRN KTH/FYS/TEO/R--00/4--SE
c Rickard Armiento, March 2000
Printed in Sweden by KTH Högskoletryckeriet, Stockholm 2000
Abstract
This thesis presents a number of results for basic quantum mechanical models
intended to be used in the development of density functional theory for systems
with edges. Following previous work, the edges are handled by dividing the
system into an edge region and an interior region, treating them differently. After
a brief introduction to density functional theory, some known results for the edge
region are presented. The formalism used for the edge region is then extended to
handle a simple model for the interior region based on the harmonic oscillator.
The purpose of the investigation of this model system is to find an expansion
for the density dependence of the exchange energy density, which is a central
task in density functional theory. To find this expansion, a number of quantities
are computed in the limit of low curvatures (the density, the Laplacian of the
density, the exchange hole and the inverse radius of the exchange hole). The
main result is that the exchange energy density seems to behave unanalytically
in the studied limit, making the previously proposed expansions invalid. The
suggested solution is to introduce a logarithmic term in the expansion.
Key words: edge system, density functional theory, exchange energy, harmonic
oscillator
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Preface
This thesis is a part of my diploma work at Theory of Materials, Theoretical Physics, Department of Physics at the Royal Institute of Technology. The
diploma work consisted of three parts. The first part was to understand the
structure of density functional theory. This was done by reading a large amount
of material on the subject. The second part was to gain knowledge about how
this could be applied to systems with electronic edges. This was primary done by
a very careful study of a paper by Walter Kohn and my supervisor Ann Mattsson
[1]. The third part was to do a number of calculations on a model system to
examine further possibilities for this work. I have tried to reflect all three parts
in this thesis.
The thesis begins with a brief introduction to density functional theory, which
is covered in chapter 2. After this comes a short discussion of the fundamental
idea of dividing a system into interior and edge regions which is followed by a
number of important results for edge systems, this is found in chapter 3. Then,
in chapter 4, the focus changes to interior regions, which is the primary area for
my own work. Here, the main subject is the study of a simple model system
for the interior region. Some surprising results for the low curvature limit of
the model system are found and, to some degree, discussed and analyzed. The
chapter ends with a conclusion in the form of a suggestion of a possible new form
for the expansion of the exchange energy density as a functional of the density.
My diploma work is a small part of a very large project; the construction of
an improved density functional for systems with edge regions. The purpose of
the diploma work has been to provide valuable information for the construction
of this functional and my hopes are that further research will be able to make
use of the information found during my work.
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Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contents
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1 Introduction
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2 Density Functional Theory
2.1 Background . . . . . . . . . . . . . . . . . . . . . . .
2.2 The Thomas–Fermi Model . . . . . . . . . . . . . . .
2.3 The Hohenberg–Kohn Theorems . . . . . . . . . . .
2.3.1 Degenerate Ground States . . . . . . . . . . .
2.3.2 The v-Representability Problem . . . . . . .
2.4 The Kohn–Sham Method . . . . . . . . . . . . . . .
2.4.1 The Kohn–Sham Orbitals . . . . . . . . . . .
2.4.2 The Exchange Correlation Energy . . . . . .
2.4.3 Local Density Approximation, LDA . . . . .
2.4.4 Generalized-Gradient Approximations, GGAs
2.4.5 Local Gradient Expansions . . . . . . . . . .
2.5 Some Comments About the Chemical Potential . . .
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3 Edge Systems
3.1 Background : Dividing a System . . . . . . . . . . . . . . . . . .
3.2 The General Edge Gas . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Density of the General Edge Gas . . . . . . . . . . . . . .
3.2.2 Inverse Radius of the Exchange Hole for the General Edge
Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The Airy Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Density of the Airy Gas . . . . . . . . . . . . . . . . . . .
3.3.2 Exchange Hole and Inverse Radius of the Exchange Hole
for the Airy Gas . . . . . . . . . . . . . . . . . . . . . . .
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viii
Contents
4 Interior Systems
4.1 The Uniform Electron Gas . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Density of the Uniform Gas . . . . . . . . . . . . . . . . .
4.1.2 Inverse Radius of the Exchange Hole for the Uniform Gas
4.2 The Hermite Electron Gas . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Background : Why we Examine a Harmonic Potential . .
4.2.2 Density of the Hermite Gas . . . . . . . . . . . . . . . . .
4.2.3 Discussion of Problems with Finite Systems . . . . . . . .
4.2.4 Inverse Radius of the Exchange Hole for the Hermite Gas
4.2.5 The Laplacian of the Hermite Gas . . . . . . . . . . . . .
4.2.6 The Exchange Hole Density of the Hermite Gas . . . . . .
4.3 Summarizing and Analyzing the Results for the Hermite Gas . .
4.3.1 A Study of One Dimensional Systems . . . . . . . . . . .
4.3.2 Suggesting a Form for a Local Gradient Expansion . . . .
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Acknowledgments
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References
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Chapter 1
Introduction
Our modern picture of what a material is, on the smallest scale, has its foundation
in the early 20th century with the discovery of the atom and the development of
quantum mechanics. This picture describes a material as consisting of electrons
and atomic nuclei behaving according to the probabilistic predictions of quantum
theory. This description gave rise to two competing approaches on how to find
the energy of a many electron system. The first approach was to solve the
quantum mechanical Schrödinger equation for the movement of the individual
electrons and to derive the energies from this procedure, using a many particle
wave function of a dimensionality proportional to the number of electrons in the
system (which usually is a very large number). In contrast, the other approach
focused on the total electron density as a fundamental variable and tried to find
a direct relation between this density and the energy of the system. Initially,
starting with the work of Thomas and Fermi in 1927 [2], this was seen as a way
of obtaining crude approximations to the Schrödinger equation method. This
status was greatly improved, however, by Hohenberg and Kohn [3] who in 1964
showed how this approach was theoretically capable of finding as exact energies
as the ones obtained from the Scrödinger equation. The resulting theoretical
framework was called density functional theory and can be regarded as a method
equivalent to solving the Scrödinger equation for the energy but avoids many of
the difficulties associated with many electron wave functions.
The theorems by Hohenberg and Kohn show that it should be possible to
obtain the energy of a many electron system from the electron density alone, but
the exact relationship remains unknown. In the attempts to obtain an expression
for this relation one usually has to introduce approximations into the otherwise
unapproximated density functional theory. The Thomas–Fermi theory can be
taken as a first approximation, and several good approximations have followed
since then, but they all have shortcomings and no one is refinable to arbitrary
precision.
1
2
Chapter 1. Introduction
This thesis is in the line of improved approximations for the electron energy
in density functional theory. The underlying idea is, following Kohn and Mattsson [1], that a system may be split into edge regions and interior regions, where
different approximations are applied in the different regions. My work in this extensive project has been to examine details of fundamental quantum mechanical
systems, where the primary model system has been based on a simple harmonic
potential. The purpose has been to provide valuable properties for such systems
for use in the construction of a general energy density functional for the interior
region.
Chapter 2
Density Functional Theory
The density functional theory has already been presented in a historical way
in the introduction. The idea is, as mentioned there, to regard the total particle density as the primary quantity from which properties of the system can
be calculated. This chapter begins with some fundamental concepts and then
introduces the theoretical framework. For more extensive treatments of density
functional theory the reader is recommended the books [4] and [5].
2.1
Background
Our primary interest in the study of quantum mechanical systems will be the
energy of multi-electron systems in different external potentials, for example, the
potential of an atomic nucleus. The starting point will be the time independent
Schrödinger equation, which is the eigenvalue equation for the energy operator
(the Hamiltonian, Ĥ). This equation defines all possible states of the system, Ψ,
and their related energies, E:
ĤΨ = EΨ .
(2.1)
The energy operator, Ĥ, for a multi electron gas is often split into three parts:
the kinetic energy of the electrons, T̂, the internal potential energy (the repulsion
between the different electrons), Û, and finally the external potential energy (the
attraction between the electrons and atomic nucleus), V̂:
Ĥ = T̂ + Û + V̂ .
(2.2)
The kinetic energy of the electrons, T̂ and the internal potential energy, Û constitutes the (total) internal electronic energy, F̂:
F̂ = T̂ + Û
⇒
Ĥ = F̂ + V̂ .
(2.3)
Following usual notation, we denote the location of electron i as ri , the
total number of electrons in the system as N , and the potential energy field in
3
4
Chapter 2. Density Functional Theory
which the electrons are moving as v(r). The different energy quantities in the
Hamiltonian can now be expressed explicitly:
T̂
=
Û
=
−
N
X
~2 2
∇
2m i
i=1
N
X
i<j
V̂
=
N
X
(2.4)
e2
|ri − rj |
(2.5)
v(ri ) .
(2.6)
i=1
Taking the expectation value gives for the energy:
= hΨ|Ĥ|Ψi = hΨ|T̂ + Û + V̂|Ψi =

ZZ
Z
N
N
X
~2 2 X
e2
∇i +
+
=
... Ψ∗ −
2m
|ri − rj |
i=1
i<j
!
N
X
v(ri ) Ψ dr1 dr2 ...drN .
+
E
(2.7)
i=1
We would like to express this energy, E, in terms of the electron density, n(r).
The electron density is related to the wave function as:
ZZ
Z
2
... |Ψ(r1 , r2 , ..., rN )| dr2 dr3 ...drN .
(2.8)
n(r1 ) = N
The right hand expression is quite similar to a wave function normalization
expression. The difference is that one of the integrals is left out, leaving one
index free. It is not important which one of the indices of the wave function we
leave as the electrons are indistinguishable, and the wave function must reflect
this.
If we now look at the three terms in the expression for the expectation value
of the energy, equation (2.7), we see that only one term is easily rewritten in
terms of the density, the term for the external potential, V . This is done as
follows:
ZZ
V
=
...
Z X
N
v(ri ) Ψ∗ Ψ dr1 dr2 ...drN =
i=1
=
Z
N Z
1 X
v(ri )n(ri )dri = v(r)n(r)dr .
N i=1
(2.9)
The other two terms in equation (2.7) cannot be rewritten in this simple
way. In the term for the kinetic energy, T , the nabla operator between the wave
2.3. The Hohenberg–Kohn Theorems
5
functions makes it impossible to put it into the “density”-form, with the two
wave functions next to each other. In the term for the internal potential, U ,
there are two different particle locations involved in the integration, preventing
us from integrating out the density. If we denote functional dependence with
bracket parentheses, [ ] we can express this as:
E
= T [Ψ] + U [Ψ] + V [n] = F [Ψ] + V [n] .
(2.10)
We note that the F -functional is independent of the external potential and thus
can be applied to all energy problems once it has been found. In the following
sections we will try to rewrite F [Ψ] to only be dependent on the electron density,
n(r). By using the obtained functional, F [n], in the energy expression we then
get an energy functional, E[n].
2.2
The Thomas–Fermi Model
There were some early attempts to approximate the kinetic and internal potential
energy as functionals of only the density. The most well known approach is
probably the one by Thomas and Fermi [2]. Quantum mechanical calculations
using assumptions about the distribution and the interaction between electrons,
give this approximation for the kinetic energy (where CF is a constant determined
by the calculation):
Z
(2.11)
T [Ψ] ≈ TT F [n] = CF n5/3 (r)dr .
We can also make a simple approximation of the internal potential energy, U
by using the expression for a classically repulsive gas (i.e. completely ignoring
electron correlations):
U [Ψ] ≈ J[n] =
e2
2
ZZ
n(r1 )n(r2 )
dr1 dr2 .
|r1 − r2 |
(2.12)
This gives the Thomas–Fermi model:
ZZ
Z
Z
n(r1 )n(r2 )
e2
dr1 dr2 .(2.13)
E ≈ ET F [n] = CF n5/3 (r)dr − n(r)v(r)dr +
2
|r1 − r2 |
2.3
The Hohenberg–Kohn Theorems
The early efforts to find energy 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 analytical
framework following the work of Hohenberg and Kohn [3].
6
Chapter 2. Density Functional Theory
Hohenberg and Kohn proved two well known theorems, known as the HKtheorems. The first 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 and thus is uninteresting).
If we have two different system potentials, v1 (r) and v2 (r) they must, if they
differ more than an additive constant, give rise to two different ground states
in the Schrödinger equation, Ψ1 and Ψ2 . We assume that the states are nondegenerate and that they give the same electronic density, n(r). If we use the
quantum principle of variation we obtain:
Z
(2.14)
E1 < hΨ2 |H1 |Ψ2 i = hΨ2 |H2 |Ψ2 i + (v1 (r) − v2 (r)) n(r)dr
Z
E2 < hΨ1 |H2 |Ψ1 i = hΨ1 |H1 |Ψ1 i + (v2 (r) − v1 (r)) n(r)dr . (2.15)
Addition gives:
E1 + E2 < E2 + E1 .
(2.16)
The last relation is clearly a contradiction, and thus it can be concluded that
for systems without degenerate ground states, two different potentials cannot
give the same ground state electron density. This means that if we know the
ground state electron density, n(r) we could also obtain all other properties of
the system. The first useful conclusion is that the ground state wave function can
be considered as a functional of the ground state density: Ψ[n]. It is also clear
that this shows the existence of an unapproximated energy functional, E[n]:
E[n]
=
F [n] + V [n] = h Ψ[n] |Ĥ| Ψ[n] i .
(2.17)
The second of the two Hohenberg–Kohn theorems provide a variational principle for the energy functional, using the density as the variational quantity
instead of the wave function. However, a subtle problem enters. Assume the
density functions n(r) and ñ(r) are different. If we let n be the exact density and use ñ as a trial density we would like the variational principle to say
E[ñ] > E[n], telling us that all trial densities give higher energies than the true
density. But if ñ is the exact density and we use n as a trial density we would
instead expect E[ñ] < E[n].
The problem is that when inserting a trial density, ñ into the energy functional, E[ñ], the potential term V [ñ] will use the potential function connected
to the trial density, v[ñ], and not the potential of the real system. This was no
problem as long as we only considered inserting the correct density, but now it
seems like a good idea to simply redefine the potential term to always use the
real system potential:
E[v0 , n]
V [v0 , n]
= F [n] + V [v0 , n]
Z
=
v0 (r)n(r)dr .
(2.18)
(2.19)
Hohenberg and Kohn proved that this energy functional is minimized by the
true ground state density if we only consider densities which are normalized to
2.3. The Hohenberg–Kohn Theorems
7
the correct number of electrons and for which there exist a potential energy
function giving the density. If we note the trial density by ñ(r), the quantum
mechanical principle of variation directly gives (remember that we now use the
correct potential, v0 , in the Hamiltonian, Ĥ, on both sides):
E[v0 , ñ] = h Ψ[ñ] |Ĥ| Ψ[ñ] i ≥ h Ψ[n] |Ĥ| Ψ[n] i = E[v0 , n] .
(2.20)
Where the minimum is known to occur for and only for:
Ψ[ñ] = Ψ[n]
⇒
ñ = n .
(2.21)
In the following we will revert to the previous notation, using E[n] = E[v0 , n],
V [n] = V [v0 , n], making the dependence on the potential implicit. The important
thing is that F [n] remains independent of v0 and still is a general functional.
2.3.1
Degenerate Ground States
It has been shown that the density functional formalism can be extended to
handle degenerate ground states [6]. In this case there are several ground state
wave functions (this is the definition of degeneracy) and one has to redefine the
map Ψ[n] to give any of the possible wave functions generating the density n(r).
Using this it is also possible to restate the variational principle, with the only
difference that the minimum occurs for (and only for) any one of the ground
state densities associated with the potential of the system.
2.3.2
The v-Representability Problem
One of the constraints on the density functional variational principle was that
only densities which actually correspond to some potential function are used as
trial functions. We call the densities which have this property for v-representable
densities. When trying to implement a density minimization, this requirement
provides a problem; there is no simple constraint to put on the trial densities to
make sure we only use v-representable densities in our search for the minimum.
A way of handling the v-representability problem is to extend the functionals
to be defined for non v-representable densities in such a way that the minimization principle still holds. This has been done in several different ways, but all of
them tend to make the formal definition of F [n] very complicated.
The problem of v-representability is often ignored in practical calculations,
and this has not yet provided any problems. The reason could be that, although
non v-representable densities exist [7], they might be “rare” enough to make
it unlikely that the implemented search algorithm accidently tries one which
disturbs the search.
8
2.4
Chapter 2. Density Functional Theory
The Kohn–Sham Method
In 1965 it became clear how to turn the theoretical framework of density functional theory into a practical computational tool as Kohn and Sham [8] 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
non-interacting electrons.
First we rewrite the energy functional, equation (2.18) slightly, introducing
Ts 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, equation (2.12):
E[n]
= F [n] + V [n] =
= Ts [n] + J[n] + V [n] + Exc [n]
= F [n] − Ts [n] − J[n] .
Exc [n]
(2.22)
(2.23)
The exchange correlation energy, Exc is defined by the above expressions. It
does, in some sense, take care of the non classical corrections to the energy.
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 a new kinetic energy term
and to “stuff away” all other energy parts into the potential term. To do this it
is necessary to examine the underlying variational problem, carefully designing
a new potential term, veff , to take care of the terms in question. In this way it
is found that the potential should be defined as (where the δ denote functional
derivatives, see [4] for an excellent review on functionals):
Z
veff (r)
= v(r) + e
vxc (r)
=
2
n(r0 )
dr0 + vxc (r)
|r − r0 |
δExc [n]
.
δn(r)
(2.24)
(2.25)
We will call veff the KS-effective potential and vxc the exchange correlation potential.
The effective potential function was designed with the purpose of providing
the correct density from orbitals found by solving its corresponding non interacting one particle orbital Schrödinger equation, which we will call the KS-orbital
Schrödinger equation:
~2 2
∇ + veff
−
2m
ψi = i ψi .
(2.26)
The orbitals obtained, ψi , will be called the KS-orbitals, their corresponding
energies, i will be called KS-orbital energies. The density for the real system is
2.4. The Kohn–Sham Method
9
obtained as the density of this non interacting orbital system (accounting with a
factor two for a double degeneracy in each orbital because of the electron spin):
n(r)
=
2
N
X
2
|ψi (r)| .
(2.27)
i
The total energy for the real system is obtained by inserting this density into
equation (2.22) above.
For a normal quantum chemistry problem, where electrons are moving in
some kind of known potential, one has to solve the KS-equations iteratively for
self-consistence. The reason is that the effective potential, veff is dependent on
the density, which we are solving for.
2.4.1
The Kohn–Sham Orbitals
It is important to note that the KS-orbitals, ψi in equation (2.26) have very
little to do with the actual electronic orbitals in the interacting multi electron
system. The KS-orbital Schrödinger equation is obtained from a mathematical
trick and is designed to give the correct density in equation (2.27), 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, i , in equation
(2.26). It is, despite this, quite common to take the KS-orbitals and energies
as approximations for the real orbitals and energies and the results are usually
surprisingly good.
2.4.2
The Exchange Correlation Energy
To be able to use the Kohn–Sham scheme above one needs an explicit expression
for the exchange correlation energy, Exc . We know that Exc cannot be a local
quantity; it is equally affected by all changes throughout the system. Because of
this we introduce an exchange correlation energy density, xc in every point of
space, xc ([n]; r), which we assume should show some kind of “locality” in the
sense of being mostly dependent on the part of the density which is close to r.
We define it to give the exchange correlation energy by integration:
Z
n(r)xc ([n]; r)dr .
(2.28)
Exc [n] =
We note that the definition of the exchange correlation energy density makes
it possible to choose xc [n] in several ways, since the only requirement is that it
should integrate to the total exchange correlation energy. We will, following the
formalism of Kohn and Mattsson [1], use the choice based on the inverse radius
−1
, to which our xc [n] is simply related as:
of the exchange correlation hole, Rxc
xc ([n]; r)
=
−
e2 −1
R ([n]; r)
2 xc
(2.29)
10
Chapter 2. Density Functional Theory
Z
−1
Rxc
= −
nxc (r; r0 ) 0
dr .
|r − r0 |
(2.30)
We call nxc (r; r0 ) the exchange correlation hole density.
−1
by simply splitting
In this work we will focus on the exchange part of Rxc
nxc into the exchange hole density, nx , and the correlation hole density, nc , as
follows:
nxc
−1
Rxc
=
=
Rx−1
=
Rc−1
=
nx + nc
Rx−1 + Rc−1
Z
nx (r; r0 ) 0
dr
−
|r − r0 |
Z
nc (r; r0 ) 0
dr .
−
|r − r0 |
(2.31)
(2.32)
(2.33)
(2.34)
Here Rx−1 is the inverse radius of the exchange hole and Rc−1 is the inverse radius
of the correlation hole. For the exchange part there are explicit expressions:
1 |ρ1 (r; r0 )|2
2
n(r)
X
∗
2
ψj,kx ,ky (r)ψj,k
(r0 ) .
x ,ky
nx (r; r0 )
= −
(2.35)
ρ1 (r; r0 )
=
(2.36)
j,kx ,ky
We call nx (r; r0 ) the exchange hole density and ρ1 (r; r0 ) the first-order spinless
density matrix.
We also note the important relations, valid for all r:
Z
Z
nc (r; r0 )dr0 = 0 .
(2.37)
nx (r; r0 )dr0 = −1,
2.4.3
Local Density Approximation, LDA
The simplest approximation of the exchange correlation energy is the local density
approximation (LDA) proposed by Kohn and Sham [8]. LDA uses the inverse
−1
:
radius of the exchange hole of the uniform electron gas, Rxc,LDA
LDA
[n]
Exc
=
−
e2
2
Z
−1
(n(r))dr .
n(r)Rxc,LDA
(2.38)
In this expression only the exchange part is known analytically. The correlation part was first estimated by Wigner [9] and more recently computed to good
accuracy by Monte Carlo calculations [10] (there are also several different parameterization of the Monte Carlo data, for example the one by Perdew and
Zunger [11]).
2.4. The Kohn–Sham Method
2.4.4
11
Generalized-Gradient Approximations, GGAs
One can try to improve the LDA expression for the exchange correlation energy,
equation (2.38), by introducing a dependence on the gradient of the electron
density. This would formally look like:
Z
GGA
Exc
=
GGA
(n(r), |∇n(r)|) .
xc
(2.39)
have been proposed and are used
A large number of good expressions for GGA
xc
within quantum chemistry, materials physics and other areas today.
2.4.5
Local Gradient Expansions
As was pointed out above, the definition of the exchange correlation energy
density, equation (2.28) makes several choices for xc [n] possible. This means that
for different GGAs which are not derived from the same analytical expressions
) there is no way of
(or uses transformations based on the integration of GGA
xc
knowing what choice of xc they represent. This freedom makes it only possible
GGA
integrated over the whole system and
to compare different end results of Exc
.
not to compare individual values of different GGA
xc
The general idea in this work will be to divide a system into an edge region
and an interior region and to provide different exchange correlation energy densities, xc [n] for the integration over the different regions. This makes it impossible
to use traditional GGAs for the interior region as the connection between the
different regions clearly is based on integrating the different exchange correlation
densities over only parts of the system.
To be able to handle the exchange energy of the interior region we have to
create a gradient expansion known to be based upon the inverse exchange hole
radius, Rx−1 and attempt to identify the coefficients, Cn . The gradient series is
obtained by assuming locality in the density dependence of Rx−1 and also that it is
possible to approximate the local behavior of the density with a Taylor expansion
with only a few terms. In this way, and using some symmetry arguments, the
following expansion can be justified:
Rx−1 (n(r), |∇n(r)|, ∇2 n(r), ...) =
|∇n(r)|2
∇2 n(r)
−1
(n(r)) 1 + C1 8/3
+ C2 5/3
+ ... .(2.40)
= Rx,LDA
n (r)
n (r)
As this is an expansion for the inverse radius of the exchange hole it will, in
contrast to ordinary GGAs, be locally comparable to the local density approximation. We call a gradient expansion with this property for a local gradient
expansion.
12
2.5
Chapter 2. Density Functional Theory
Some Comments About 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:
i ≤ µ ∀ occupied i .
(2.41)
Chapter 3
Edge Systems
3.1
Background : Dividing a System
In a multi electron system there is a great difference between the interior region,
where the potential energy is smaller than all the electron energies and the
exterior region where the opposite relationship holds. A clear difference is seen,
for example, in the wave function; in the interior region the wave functions are
oscillatory and in the exterior region they decay exponentially. We will further
develop this idea by defining the edge surface as all r for which (where veff (r)
and µ are defined in chapter 2):
veff (r)
= µ.
(3.1)
The main idea is to develop different exchange energy densities in different regions. We will begin with the edge system, modeling the region close to the
edge surface. We will closely follow Kohn and Mattsson [1] to obtain several
interesting results for this system.
3.2
The General Edge Gas
We define an edge system as a system where the KS-effective potential varies
slowly along two dimensions (which we will call the x and y directions) and
more rapidly along a third dimension (the z direction). It is important to note
that the potential in question is the KS-effective potential and not the actual
system potential.
If we take a limit of slow variations for the x and y directions we reach a
constant effective potential. We take Lx and Ly to be the length of the system
in these dimensions, but they will eventually approach infinity.
The three dimensional KS-orbital Schrödinger equation (2.26) for the general edge system can be separated in the usual way to give the wave functions
13
14
Chapter 3. Edge Systems
and energies (denoting the solutions for the z-part as ϕj and enumerating the
solutions in the x and y directions by positive and negative integers, mx , my ):
~2 2
∇ + veff (z) ψi (x, y, z) = i ψi (x, y, z) ⇒
(3.2)
−
2m
ψj,kx ,ky
=
x
=
kx
=
1
ϕj (z) p
ei(kx x+ky y)
Lx Ly
~2 2
k
2m x
2πmx
,
Lx
(3.3)
(3.4)
and similar for ky .
(3.5)
The different solutions for the z-part of the Schrödinger equation, ϕj fulfill:
~2 d2
+
v
(z)
ϕj = j ϕj .
(3.6)
−
eff
2m dz 2
We know from equation (2.41) that the total energy of all occupied orbitals
must be less than the chemical potential µ:
x + y + j ≤ µ ∀ occupied x, y, j .
(3.7)
We can use this to determine the “maximum wave vector” for the plane waves
in x- and y-directions, which we denote kj . It is clear why this varies with j; the
higher the energy of the z-part of the wave-function, the lower is the maximum
possible energy for the x and y-parts, as the sum of the three energies must be
lower than the chemical potential. Thus kj is given by:
~2 2
(k + ky2 ) + j ≤ µ
2m x
3.2.1
⇒
kj2 =
2m(µ − j )
.
~2
(3.8)
Density of the General Edge Gas
Using the expression for the density for a KS-orbital system, equation (2.27), it
is possible to get a more explicit expression for the density of the edge gas:
X
X
m
|ψj,kx ,ky (r)|2 = 2
|ϕj (z)|2
(µ − j ) .
(3.9)
n(r) = 2
2
2π~
j
Occupied
orbitals
It is also possible to modify this expression to include a temperature correction in the KS-orbital system by including a Fermi–Dirac distribution, f () =
1/(1 + e(−µ)/kB T ), in the expression for the density:
X
|ψj,kx ,ky (r)|2 f (x + y + j ) =
n(r, T ) = 2
All
orbitals
3.3. The Airy Gas
15
= kB T
m X
2
(µ−j )/kB T
|ϕ
(z)|
ln
e
+
1
.
j
π~2
(3.10)
All j
The expression for the density at zero temperature, equation (2.27), is obtained
from this expression in the appropriate limit.
3.2.2
Inverse Radius of the Exchange Hole for the
General Edge Gas
We start from the exchange hole density, nx (r, r0 ), equation (2.35) and insert
the wave function found above for the general
p edge gas, equation (3.3). Using
the notation x̃ = x − x0 , ỹ = y − y 0 and ρ̃ = x̃2 + ỹ 2 we obtain:
nx (r, r0 ) =
2
X
1 1
∗ 0 kj J1 (kj ) −
ϕj (z)ϕj (z )
.
2n(z) π j
ρ̃
(3.11)
This expression can be inserted into the definition of Rx−1 , equation (2.33), to give
an expression for the inverse radius of the exchange hole for a general z-potential
(using ∆z = |z − z 0 |):
Rx−1 (r)
=
1
πn(z)
Z
dz 0
XX
j
ϕj (z)ϕ∗j (z 0 )ϕ∗j 0 (z)ϕj 0 (z 0 ) ×
j0
1
× 3 g(kj ∆z, kj 0 ∆z)
∆z
Z ∞
J1 (st)J1 (s0 t)
0
0
dt √
.
g(s, s ) = ss
t 1 + t2
0
3.3
(3.12)
(3.13)
The Airy Gas
The simplest possible edge system is a system where the effective potential in
the z-direction is linear. This is also the limiting case of all edge systems where
the effective potential in the z-direction is sufficiently slowly varying. Thus this
is in a way a first order approximation we would like to use for the edge region
of a divided system. The potential is (taking L to be very large):
veff (z) =
−F z, z < L
∞,
z≥L.
(3.14)
To simplify the expressions we will use an energy scale such that the chemical
potential, µ, fulfills:
µ = 0.
(3.15)
16
Chapter 3. Edge Systems
Inserting the potential function in the z-direction part of the KS-orbital
Schrödinger equation (3.6), gives:
~2 d2
−
F
z
−
lim ψn (z) = 0, ψn (L) = 0 .
(3.16)
−
n ψn = 0,
z→ −∞
2m dz 2
The normalized solutions to this equation are (denoting by Ai the Airy function
satisfying limz→∞ Ai(z) = 0, its derivate as Ai0 and its zeros as an ):
ϕn (z) =
n
=
z
n 1
√ 0
Ai − −
l
lAi (an )
− an − F L,
= F l,
(3.17)
l=
~2
2mF
1/3
.
(3.18)
The index n denotes the different energy levels, starting from the lowest. The
important orbitals for the edge region are those which have energies close to the
chemical potential, µ. Because of this we will reorder the orbitals. We introduce
the label j, where the orbital energy decreases with j and the orbital with j = 0
is the highest occupied orbital. The reordered energies are denoted j and their
respective orbitals ϕj . This reordering and approximations valid for orbitals
with energies sufficiently close to the chemical potential, µ, gives:
ϕj (z)
j
z
j π 1/2
−
Ai
−
l
(Ll)1/4
1/2
l
≈ −j
π .
L
≈
(3.19)
(3.20)
These orbitals and energies defines the Airy gas.
3.3.1
Density of the Airy Gas
To compute the density of the Airy gas we insert the obtained expressions for
orbitals and energies, equations (3.19) and (3.20), into the expression derived for
the density of the edge system, equation (3.9) (the index a is for Airy):
= l−3 n0,a (z/l)
(3.21)
Z ∞
1
Ai2 (ζ 0 − ζ)ζ 0 dζ 0 =
n0,a (ζ) =
2π 0
1
= 13π ζ 2 Ai2 (−ζ) + ζAi02 (−ζ) − Ai(−ζ)Ai0 (−ζ) . (3.22)
2
na (z)
In figure (3.1) there is a graph of this expression, and in figure (3.2) this graph
is compared to some systems with finite z-orbital occupation numbers which
have been calculated directly from the exact solutions for the linear potential,
equations (3.17) and (3.18).
3.3. The Airy Gas
17
0.035
0.03
0.025
n0,a(ζ)
0.02
0.015
0.01
0.005
0
−1
−0.8
−0.6
−0.4
−0.2
0
ζ
0.2
0.4
0.6
0.8
1
Figure 3.1. The density of the Airy gas, n0,a (ζ)
0.035
Airy
50
20
10
5
0.03
0.025
0.02
n
0,a
(ζ)
1
0.015
0.01
0.005
0
−1
−0.8
−0.6
−0.4
−0.2
0
ζ
0.2
0.4
0.6
0.8
1
Figure 3.2. The density for different linear potential systems with finite number
of occupied orbitals as functions of the dimensionless distance in the z-direction.
From below the graphs are for 1, 5, 10, 20 and 50 occupied z-orbitals. The
uppermost graph is the density of the Airy gas and we see how the linear potential
systems approach this with increasing number of occupied orbitals.
18
3.3.2
Chapter 3. Edge Systems
Exchange Hole and Inverse Radius of the Exchange
Hole for the Airy Gas
Kohn and Mattsson have also continued the study of the Airy gas by numerical
computations of the exchange hole, nx (r, r0 ) from equation (3.11) and the inverse
radius of the exchange hole, Rx−1 (r) from equation (3.12). The obtained results
make it possible to create an estimate for an exchange energy density. The
results are found in their paper [1].
Chapter 4
Interior Systems
In this chapter we move the focus from the edge region, studied in the previous
chapter, to the interior region of a system. We will use a model for the interior region quite similar to the one used for the edge, with a constant effective
potential in two dimensions and a more general in the third. Because of this
it is possible to reuse a number of equations for the general edge system from
the previous chapter. However, to study the interior properties we will choose
an energy scale such that the effective potential minimum has zero energy. This
makes the chemical potential a positive value, in contrast to the treatment for
the Airy gas in the previous chapter where we decided to set it equal to zero.
4.1
The Uniform Electron Gas
The simplest example of an interior KS-orbital system is the uniform electron
gas with the potential V (r) = constant < µ. Similar to how we handled the x
and y dimensions in the edge system we take Lx , Ly and Lz to be the length of
the system in the different directions, but they will eventually approach infinity.
The wave functions and energies of this system are known to be (enumerating
the solutions in the x, y and z directions by positive and negative integers, mx ,
my and mz ):
ψkx ,ky ,kz
=
x
=
kx
=
1
p
ei(kx x+ky y+kz z)
Lx Ly Lz
~2 2
k
2m x
2πmx
,
Lx
(4.1)
(4.2)
and similar for ky and kz .
(4.3)
It is possible to insert a constant z-dependence into the equations found
in the previous chapter for a general z-potential, but as we now have a more
symmetric problem, we get more explicit expressions if we start over from the
19
20
Chapter 4. Interior Systems
definitions. First we need a new definition of the “maximum k-vector” for this
more symmetric system. We previously called it kj , but now, as there will be no
kind of “j-dependence”, we denote it kF :
x + y + j ≤ µ
4.1.1
~2 2
(k + ky2 + kz2 ) ≤ µ
2m x
⇒
⇒
kF2 =
2mµ
.
~2
(4.4)
Density of the Uniform Gas
It is possible to calculate the density of the uniform electron gas directly from
the definition (The index u is for uniform):
nu (r)
=
X
2
Occupied
kx , ky , kz
3
ψkx ,ky ,kz (r)2 = kF .
3π 2
(4.5)
If we introduce a Fermi–Dirac distribution into the definition of the density
of the KS-orbital system, similar to what we did in equation (3.10), it is also
possible to obtain a temperature corrected expression:
nu (r, T )
4.1.2
=
3
nu (r)
2
kB T
µ
3/2 Z
√
∞
0
1+
x
ex−µ/kB T
dx .
(4.6)
Inverse Radius of the Exchange Hole for the
Uniform Gas
We proceed by calculating the inverse radius of the exchange hole for the uniform
electron gas. To make use of the symmetry we, rather than using the expression
for the general z-potential inverse radius of the exchange hole, equation
(3.12),
p
start from the density matrix, ρ1 , equation (2.36). Using ∆r = x̃2 + ỹ 2 + z̃ 2
we obtain:
ρ1,u (r; r0 ) =
=
2
Lx Ly Lz
1
√
2
X
ei(kx x̃+ky ỹ+kz z̃) =
(4.7)
kx ,ky ,kz
kF
π∆r
3/2
J3/2 (kF ∆r) .
(4.8)
When r0 → r this expression approaches, as expected, the correct density, equation (4.5).
Inserting the expression for ρ1 into the definition of the exchange hole density,
equation (2.35), gives directly:
nx,u (r; r0 )
= −
kF3
J2 (kF ∆r) .
4π 3 ∆r3 n(r) 3/2
(4.9)
4.2. The Hermite Electron Gas
21
Using this in the definition for the inverse radius of the exchange hole, equation
(2.33) we get:
−1
(r)
Rx,u
=
kF4
3kF
.
=
n(r)2π 3
2π
(4.10)
In section 2.4.3 we stated that the local density approximation uses the inverse radius of the exchange correlation hole of the uniform gas. To find the
−1
(n(r)) we must recast the result in
expression for the exchange part, Rx,LDA
equation (4.10) to be a function of density. Using the relation between the density and kF found in equation (4.5) we find:
−1
(n(r))
Rx,LDA
4.2
4.2.1
=
1/3
3
3π 2 n(r)
.
2π
(4.11)
The Hermite Electron Gas
Background : Why we Examine a Harmonic
Potential
We now introduce a simple model for the interior region of a system, possible
to use in connection to the Airy gas for the edge region treated in the previous
chapter. The model is one of the simplest possible with a curvature, consisting
of a harmonic oscillator potential in one direction and a constant potential in the
other two. We need a potential with a curvature because our primary interest
is how this curvature affects the inverse radius of the exchange hole. In this
way we would like to find the coefficients in a local gradient expansion of Rx−1
(this was mentioned in section 2.4.5). This analysis might also clarify some
unexpected results obtained for a system with a cosinusoidal potential studied
by my supervisor, Ann Mattsson (these results are not yet published).
The treatment presented here will be limited to the center point of the harmonic oscillator model system. As the symmetry forces the gradient of the
density to be zero in this point it should be possible to obtain the coefficient for
the term containing the Laplacian of the density in the local gradient expansion,
equation (2.40). Additionally, the symmetry also makes this point the simplest
to compute. It should be a natural extension of this work to repeat the analysis
for a point where the Laplacian is zero and in this way try to find the coefficient
of the term containing the gradient of the density.
4.2.2
Density of the Hermite Gas
We will, as was said above, use a constant potential in the x and y directions
and a harmonic potential in the z direction:
veff (z) =
mω 2 2
z .
2
(4.12)
22
Chapter 4. Interior Systems
The z-part of the Schrödinger equation, equation (3.6), can along with the proper
boundary conditions be solved for the normalized wave functions and energies
(where Hn are the Hermite polynomials):
r
1/2
r
mω 2
mω 1
mω
z
e− ~ z /2
H
n
~π 2n n!
~
1
~ω n +
.
2
ψn (z) =
=
n
(4.13)
(4.14)
It is possible to insert the wave functions for the z-direction into the expression derived for the density of the general z-potential system, equation (3.9) (the
index h is for Hermite and bxc is the floor-function, the largest integer ≤ x):
X
|ψkx ,ky ,n (r)|2 ⇒
nh (z) = 2
kx ,ky ,n
nh (σ) =
kF3
2π
r
N
w X 1
2
H
2π n=0 2n n! n
w=
~ω
,
µ
N=
r
wσ 2
w
1
σ e− 2
(4.15)
1−w n+
2
2
1
1
−
,
w 2
σ = kF z .
(4.16)
In the above we use w as the curvature parameter, which is proportional to the
square root of the curvature of the potential and N is the number of occupied
z-orbitals. We turn this expression into a dimensionless form to make it easier
to apply numerical techniques by dividing with the density for free electrons,
equation (4.5). A plot of this expression, nh (σ)/nu , is found in figure (4.1).
We will only study the point σ = z = 0, as was said in the beginning of this
chapter, giving:
√ 3/2
π w
(6/w − 3 − 4Q)(2Q + 1)!
nh (0)
=
(4.17)
nu
2
2
4Q (Q!)2
1
1
N
−
=
.
(4.18)
Q =
2
2w 4
Now we wish to expand this in a series of the curvature. It is important to
remember that the curvature is proportional to the curvature parameter squared,
w2 . We will, despite this, do the expansion in w, as we will find some terms of odd
order in w (what this means for the expansion will be analyzed later). Because
of this we would like to try the following:
nh (0)
nu
=
C0 + C1 w + C2 w2 + ... .
(4.19)
We expect to find C0 = 1 as the Hermite gas should behave as the uniform
gas to the zero:th order. By calculating the expression nh (0)/nu for smaller and
4.2. The Hermite Electron Gas
23
smaller values of w we clearly see that this is the case, as shown in figure (4.2).
As this coefficient now is known we can make a graph of (nh (0)/nu − C0 )/w
to obtain C1 , which is found to be zero. The plot for C2 is found in figure
(4.3), where a problem becomes evident; the C2 -coefficient does not converge to
a single value. The origin of this problem and some solutions will be covered in
the next section.
4.2.3
Discussion of Problems with Finite Systems
We recall that the energy levels are continuous in the two plane wave dimensions
but discrete for the harmonic oscillator. This means that it is possible for the
chemical potential to be anywhere between two energy levels in the harmonic
oscillator, in contrast with the case for the one dimensional harmonic oscillator.
Some investigations tell us that the problems with the C2 coefficient seem to be
related to a different behavior of the system depending on how close the chemical
potential is to the next energy level of the harmonic oscillator. To examine this
we introduce a parameter α such that:
Q =
N
2
1
1
−
=
2w 4
⇒
1
Q≤
2
w=0.66, N=1
⇒
1.2
nh(σ)/nu
1
0.8
nh(σ)/nu
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
−1
−0.5
0
0.5
σ/(2/w)
1
0
−1.5
1.5
−1
w=0.02, N=50
−0.5
0
0.5
σ/(2/w)
1
1.5
1
1.5
All values of w, N
1.2
1.2
nh(σ)/nu
1
0.8
nh(σ)/nu
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
−1.5
1
1
−
w 2
w=0.22, N=4
1.2
0
−1.5
−1
−0.5
0
0.5
σ/(2/w)
1
1.5
0
−1.5
−1
−0.5
0
0.5
σ/(2/w)
Figure 4.1. The density of the Hermite gas, nh (σ)/nu for different values of w.
The σ scale has been divided by the distance to the classical turning points, 2/w,
making the classical turning points ±1 for all the plots.
24
Chapter 4. Interior Systems
1.03
1.02
1.01
1
nh(0)/nu
0.99
0.98
0.97
0.96
0.95
0.94
0.93
0
0.1
0.2
0.3
0.4
0.5
w
0.6
0.7
0.8
0.9
1
Figure 4.2. A plot of nh (0)/nu as a function of w = ~ω/µ. When ω → 0 this
approach the first coefficient, C0 , in the expansion of the density, equation (4.19).
We clearly see that C0 = 1.
0.1
0.05
0
(nh(0)/nu−1)/w2
−0.05
−0.1
−0.15
−0.2
−0.25
−0.3
−0.35
0
0.1
0.2
0.3
0.4
0.5
w
0.6
0.7
0.8
0.9
1
Figure 4.3. A plot of (nh (0)/nu − 1)/w2 as a function of w. When w → 0
this should approach the third coefficient, C2 , in the expansion of the density,
equation (4.19). The value, however, does not seem to converge. The dotted line
is the mean value per peak, inserted here to be compared with figure (4.8).
4.2. The Hermite Electron Gas
⇒
1
2
Q =
25
1
1
− −α ,
w 2
0≤α<2.
(4.20)
The parameter α, according to the above definition, describes to what degree the
chemical potential lies between the levels of the harmonic oscillator. When α = 0
the chemical potential is exactly on an even level. When α = 1 the chemical
potential lies exactly on an odd level. If we keep α constant while taking the
limit w → 0 we consider series of systems with the chemical potential (for every
system in the series) equally positioned between two energy levels. The reason
we would like to look at this limit is that it might converge, even though the
ordinary limit does not.
Inserting the above expression for Q(α, w) into the expression for the density,
equation (4.17), gives:
3
3 2
3 2 1 3
5
1
nh (0)
2
= 1 + − + α − α w + − α + α − α w3 +
nu
16 4
8
4
8
8
3
33 2
9
9 4
29
− α−
α + α3 −
α w4 + ... .
(4.21)
+
512 64
128
32
128
To see how the limit varies with α we present a plot of the value of the second
coefficient, C2 (α) as a function of α in figure (4.4). We might be worried about
the existence of a term proportional to w3 in the expansion above, since w is
proportional to the square root of the curvature. The existence of such a term
0.1
0.05
0
−0.05
C2(α)
−0.1
−0.15
−0.2
−0.25
−0.3
−0.35
0
0.2
0.4
0.6
0.8
1
α
1.2
1.4
1.6
1.8
2
Figure 4.4. A plot of the third term, C2 (α), in the Taylor expansion of the
density as a function of the parameter α as defined in equation (4.20). This is,
in some sense, all the “possible” limits of C1 when ω → 0.
26
Chapter 4. Interior Systems
should thus give an unanalytical behavior of the density in the limit of small
curvatures.
Averaging
Our aim is to find how the density behaves for low curvatures. We believe
that the origin of the problematic behavior in the low curvature limit is the
strong correlation between the wave functions of different energy levels in the
harmonic system. This behavior should probably disappear for systems not at
zero temperature (we will see what a correction for this does soon) or for systems
with a “real” potential showing a more fractal structure and thus a much less
degree of symmetry. This means that the “wiggliness” in the density graph,
figure (4.3), caused by these effects is not present for a more real system and
must be removed in some way if we are to get a useful result.
One way of removing the unwanted behavior is to simply take an average
over the coefficient C2 (α) in the interval 0 − 2:
C2,mean
=
1
2
Z
0
2
C2 (α)dα = −
1
.
16
(4.22)
In addition, when taking the average of the parameter in C3 (α) (the term
proportional to w3 ) which worried us above, we get C3,mean = 0.
Finite Temperatures
We will now try another method to get rid of the density “wiggliness”; we extend
the KS-orbital system with a temperature correction. The idea is that much of
the problem with making C2 converge seems to be related to the heavy oscillations induced by the truncations made above. The truncations are there because
the Fermi–Dirac distribution takes the form of a step function when making calculations at zero temperature. Thus, introducing a finite temperature might
“pick out” the average behavior we are interested in.
We add temperature corrections to our KS-orbital system by directly applying the expressions found above for densities with temperature, equations (3.10)
and (4.6) and inserting the wave functions for the harmonic potential. If we introduce the dimensionless temperature, τ = kB T /µ it is possible to numerically
compute nh (0, τ )/nu (0, τ ) for smaller and smaller values of w. From this treatment it is possible to conclude that the temperature corrected third coefficient,
C2,temp , seems to converge to the same value as found with the averaging method
above, C2,mean , as can be seen in figure (4.5). Additionally, if we numerically
compute the expression for C3,temp we see that this value also seems to converge
to the same value as obtained by the averaging method, C3,mean = 0, as can be
seen in figure (4.6).
4.2. The Hermite Electron Gas
27
0.05
−0.05
(nh(0,τ)/nu−1)/w
2
0
−0.1
−0.15
−0.2
0
0.1
0.2
0.3
0.4
0.5
w
0.6
0.7
0.8
0.9
1
Figure 4.5. A plot of (nh (0, τ )/nu (0, τ ) − 1)/w2 as a function of w for τ = 0.05.
When ω → 0 this approaches the third coefficient, C2 ≈ −0.06. Note that it is
first below w . τ , i.e. ~ω . kB T , the “wiggliness” seems to disappear.
0.4
0.3
0.1
2
((nh(0,τ)/nu−1)/w + 1/16)/w
0.2
0
−0.1
−0.2
−0.3
−0.4
0
0.1
0.2
0.3
0.4
0.5
w
0.6
0.7
0.8
0.9
1
Figure 4.6. A plot of ((nh (0, τ )/nu (0, τ ) − 1)/w2 − 1/16)/w as a function of w
for τ = 0.05. When ω → 0 this approaches the coefficient C3 ≈ 0.
28
Chapter 4. Interior Systems
4.2.4
Inverse Radius of the Exchange Hole for the
Hermite Gas
It is possible to derive an expression for the inverse radius for the exchange
hole for the Hermite electron gas using the expression for Rx−1 for the general
z-potential, equation (3.12):
−1
(0)
Rx,h
nu
= 3kF w
nh (0)
×
Z
∞ bN/2c
X
dσ
0
e 2n
H
n,n0 =0
r
r
wσ 2
w
w
e
0
σ H2n
σ e− 2 ×
2
2
1
g(k̃2n σ, k̃2n0 σ)
σ3
(−1)n
H2n (z),
for integer n
n!22n
s
1
1−w n+
k̃n =
2
Z ∞
J1 (st)J1 (s0 t)
dt √
.
g(s, s0 ) = ss0
t 1 + t2
0
e 2n (z)
H
=
(4.23)
(4.24)
(4.25)
(4.26)
This must be computed numerically, in contrast to the analytical calculations
of the density, and the accuracy of the numerical methods becomes very critical
and requires some careful considerations.
Some Notes on Numerical Methods
−1
(0) can be done in any order. Both
The two integrals in the expression for Rx,h
ways have been tried, but the universality of the g2 function suggests doing this
first, because successful optimizations can later be applied to other z-potential
systems. The g2 integral is heavily oscillating, and to be able to compute this
integral at an acceptable speed, the following was used:
Z ∞
Z ∞
J1 (s1 t)J1 (s2 t)
J1 (s1 t)J1 (s2 t)
√
dt =
dt +
2
t2
t t +1
0
0
Z ∞
1
1
−
J1 (s1 t)J1 (s2 t) √
dt . (4.27)
+
t t2 + 1 t2
0
Where (with K and E as the complete elliptic integrals of the first and second
kinds respectively and s1 and s2 is chosen such that s1 > s2 ):
Z ∞
s2
s2
J1 (s1 t)J1 (s2 t)
2
2
2
2
2
dt
=
+
s
)E
−
s
)K
(s
−
(s
. (4.28)
1
2
1
2
t2
3s2 π
s1
s1
0
If s1 = s2 we replace this by:
Z
0
∞
(J1 (st))2
4s
.
dt =
t2
3π
(4.29)
4.2. The Hermite Electron Gas
29
The integrand in the second integral on the right hand side in equation (4.27)
decrease much faster than the original integrand and is thus possible to integrate
much faster.
The outermost integral in the expression for Rx−1 , equation (4.23), was converged to a relative accuracy of 10−7 to cope with the cancellation induced by
removing the zeroth term (the leading 1) in equation (4.30) below. The innermost integral was converged to a relative accuracy of 10−10 to cope with
summation errors due to the many terms in the double summation in equation
(4.23).
A number of different integration methods were examined, and a recursive
ten point Gauss–Legendre integration [12] was found to be the fastest. The
Bessel functions were computed by a rational approximation for low values and
a cosinus-sinus expansion for high values [12], [13]. The elliptic integrals were
computed by the algorithms provided in [12].
Analyzing the Results
We would like to expand the local exchange energy in the curvature. However,
as we already know the density expansion, we can instead study the expansion
of:
−1
=
R̃x,h
nh (0) −1
Rx,h = C0 + C1 w + C2 w2 + ... .
nu
(4.30)
The reason to why we would like to study this expression rather than the ex−1
is that its appearance is more similar to the density, and this
pansion of Rx,h
will make the results much more clear. Once we find a decent expansion it is
possible to divide it with the expansion of the density to obtain the expansion
for the real local exchange energy.
The limit for the first coefficient, C0 is shown in figure (4.7) and it clearly
converges to one as expected, which means that to first order the behavior is
similar to the uniform electron gas. We first assume that C1 is zero, as was the
case for the density, because the curvature is proportional to w2 and we expect
to find an analytical expansion in the curvature. Thus we plot the limit for the
third term, C2 . The curve is found in figure (4.8) and it looks quite similar to
the density plot, figure (4.3). However, this time the values also seem to drift
away towards infinity. This is a more serious problem which, as we will see, is
not possible to handle by taking mean values.
We begin the analysis by taking mean values of each peak similar to the way
we did for the density (note that w is defined in the same way as for the density,
so each point of a peak still corresponds to a parameter α, ranging from 0 to
2). This gives the dashed line in figures (4.8) and (4.9), and as expected from
the behavior of the unaveraged curve, the mean value curve seems to diverge for
low values of w. This result indicates a lower term in the expansion. The first
natural conclusion would be that the coefficient C1 is nonzero, and to examine
this possibility we plot the average values divided with w (and not with w2 as
30
Chapter 4. Interior Systems
1.04
1.03
1.02
1.01
−1
(Rh (0)/Ru )
1
−1
0.99
0.98
0.97
0.96
0.95
0.94
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
w
−1
−1
Figure 4.7. A plot of R̃x,h
(0)/Rx,u
as a function of w. When ω → 0 this
approaches the first coefficient, C0 = 1, as expected.
in the graph for the coefficient C2 ), this plot is found in figure (4.10). At first
it looks like the value converges to a value just above zero. However, when we
examine the part of the graph for low w more closely (in the rightmost plot in
the same figure), we find that the curve seems to be turning, as if the value had
not yet converged.
Attempts to fit polynomials to the obtained curves give poor results where
the coefficients have a tendency to become quite unbalanced, to make high order terms affect the curve at very low values and in this way give the bending
behavior. A standard log-log plot of the data, figure (4.11), also indicates a
behavior far from the behavior of wλ with λ an arbitrary constant (which becomes a straight line in a log-log plot). We realize that the shape more resembles
a “logarithmic behavior” and try to fit curves with logarithmic terms. In this
way we obtain very good fit with only two fitted parameters using the expression
D1 w2 ln(w2 ) + D2 w2 . This expression is shown in figures (4.12) and (4.13) which
also contain a polynomic curve to compare with. The good results suggest that
our interpretation of the behavior is correct.
As mentioned in the analysis above, the best fit found suggests the expansion:
−1
(0) =
Rx,h
D1 w2 ln(w2 ) + D2 w2 + ... .
(4.31)
Where the coefficients were found to be D1 ≈ −0.014 and D2 ≈ −0.068. The
results indicate that the local exchange energy as a function of curvature seems
to be unanalytic in the limit for zero curvature and, to be more specific, it seems
like the studied derivative diverges in a logarithmic fashion.
4.2. The Hermite Electron Gas
31
0.3
0.2
0
u
(R−1(0)/R−1−1)/w2
0.1
h
−0.1
−0.2
−0.3
−0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
w
−1
−1
Figure 4.8. A plot of ((R̃x,h
(0) − 1)/Rx,u
)/w2 as a function of w. The dotted
line is the mean value per peak. A closer look at the mean value per peak is
found in figure (4.9).
0.05
0.04
0.03
Average/w2
0.02
0.01
0
−0.01
−0.02
−0.03
−0.04
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
w
−1
−1
Figure 4.9. A plot of the mean values per peak of ((R̃x,h
(0) − 1)/Rx,u
)/w2 as
a function of w. When ω → 0 we expect this to approach a value to use for,
C2,mean . However, the value seems to diverge towards infinity.
32
Chapter 4. Interior Systems
−3
−4
x 10
2
x 10
9
0
8
Average/w
Average/w
−2
−4
7
6
−6
5
−8
−10
0
0.1
0.2
w
0.3
4
0.4
0
0.02
0.04
0.06
w
−1
−1
Figure 4.10. A plot of the mean values per peak of (R̃x,h
(0) − 1)/Rx,u
divided
by w, as a function of w. On a large scale the curve seems to converge just above
zero (left graph). A closer look reveals a sharp bend in the curve (right graph).
−3
Average/w
10
−4
10
−2
−1
10
10
w
−1
−1
Figure 4.11. A log-log plot of the mean value per peak of (R̃x,h
(0) − 1)/Rx,u
divided with w, as a function of w. The purpose of this plot is to reveal possible
wλ , λ arbitrary, behavior. However, the slope varies heavily with w.
4.2. The Hermite Electron Gas
33
−3
5
x 10
0
Average/w
−5
−10
−15
−20
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
w
−1
−1
Figure 4.12. A plot of the mean value per peak of (R̃x,h
(0) − 1)/Rx,u
divided
by w as a function of w (solid line) and two fitted curves. The best of the fitted
curves is the suggested two-parameter logarithmic fit (dashed). The other curve
(dash-dotted) is a fitted polynomial of second degree (three parameters). Both
fits were done using a least square algorithm on all points below 0.1 .
−4
9.2
x 10
9
8.8
8.6
Average/w
8.4
8.2
8
7.8
7.6
7.4
7.2
0.015
0.02
0.025
0.03
0.035
w
0.04
0.045
0.05
0.055
Figure 4.13. A closer look on the graph in figure (4.12); a plot of the mean value
−1
−1
(0) − 1)/Rx,u
divided with w as a function of w (solid line) and
per peak of (R̃x,h
two fitted curves. The best of the fitted curves is the suggested two-parameter
logarithmic fit (dashed). The other curve (dash-dotted) is a fitted polynomial of
second degree (three parameters).
34
Chapter 4. Interior Systems
4.2.5
The Laplacian of the Hermite Gas
The dimensionless Laplacian, encountered in the local gradient expansion in
section 2.4.5, is defined as:
L
=
∇2 n(r)
.
n5/3 (r)
(4.32)
For the Hermite gas we wish to expand this as:
∇2 nh (r) = C0 + C1 w + C2 w2 + ... .
Lh =
5/3
nh (r) r=0
(4.33)
A graph of Lh is found in figure (4.14). An analytical expansion gives C0 = 0
and C1 = 0. The C2 -coefficient, however, does not converge to a single value,
similar to the density and the local exchange energy. It is possible to do an
analytical α-expansion similar to the one we did for the density, and the result
is:
C2
= −
35/3 π 4/3
(1 − |α − 1|) .
2
(4.34)
This expression for C2 is shown in figure (4.15). The mean value of C2 =
−35/3 π 4/3 /4 ≈ −7.18.
0
−0.2
−0.4
h
L (0)
−0.6
−0.8
−1
−1.2
−1.4
0
0.05
0.1
0.15
0.2
w
0.25
0.3
0.35
0.4
Figure 4.14. A plot of the Laplacian of the Hermite gas, Lh (0), as a function
of w = ~ω/µ.
4.3. Summarizing and Analyzing the Results for the Hermite Gas
35
0
2
C (α)
−5
−10
−15
0
0.2
0.4
0.6
0.8
1
α
1.2
1.4
1.6
1.8
2
Figure 4.15. A plot of the third term, C2 in the expansion of the Laplacian of
the density of the Hermite gas, equation (4.33), as a function the parameter α.
4.2.6
The Exchange Hole Density of the Hermite Gas
The exchange hole density, nx (r, r0 ), equation (2.35) can be interesting to study
to understand the behavior of a system as it describes, in a way, the “spreading”
of an electron from the space point r to the point r0 .
To study the exchange hole density we plot nx,h (0, r0 ) as a two dimensional
surface, where the arguments are the dimensionless distance
in the z-direction,
p
σ = kF z and in the plane wave dimensions, ρ = kF x2 + y 2 . We plot this
surface for different values on the parameter w in figure (4.16).
In the surface plots it is possible to see how the exchange hole density of the
harmonic oscillator system shows a more and more “circle”-like behavior when
the oscillator becomes wider and wider. This behavior is associated with the
exchange hole density of a uniform gas.
4.3
Summarizing and Analyzing the Results for
the Hermite Gas
We have seen how the local exchange energy seems to show unanalytic behavior
in the limit of low curvatures. We tried to cover for this unanalytic behavior by
introducing a logarithmic term in the Taylor expansion. The logarithmic term
we inserted was of the form w2 ln(w2 ). The characteristic of this expression is
36
Chapter 4. Interior Systems
w = 0.05
w = 0.08
40
0.07
30
30
0.004
σ
40
20
20
10
10
0.0001
1e−06
10
20
30
ρ
40
50
10
20
30
ρ
40
50
10
20
30
40
50
10
20
30
40
50
1e−08
1e−10
40
40
30
30
σ
1e−13
1e−18
20
20
10
10
1e−25
w = 0.15
w = 0.30
Figure 4.16. The exchange hole density, −nx,h (0, r0 )/nu as a function of the
(dimensionless) distance in the z-dimension, σ = kF z, and in the plane wave
dimensions, ρ = kF x2 + y 2 . The white lines in the plots mark the classical
turning points for the harmonic oscillator. The color scale is very non-linear.
p
that the second derivative diverges. To better understand this behavior we will
take a brief look at one dimensional systems.
4.3.1
A Study of One Dimensional Systems
The first, and most obvious, system to study is the one dimensional quantum
harmonic oscillator. The potential is v(z) = −mω 2 z 2 /2, where z is the only
available dimension. We obtain for the inverse radius of the exchange hole:
bN/2c
Rx−1 (0)
4 mω X (−1)n+n
×
=
n(0) ~π
4n+n0 n!n0 !
n,n0
r
r
Z ∞
2
1
mω
mω
H2n
z H2n0
z e−mωz /~ dz . (4.35)
×
z
~
~
0
0
We first consider a system where N = 0. In this case the expression becomes:
Z
4 mω ∞ 1 −mωz2 /~
e
dz .
(4.36)
Rx−1 (0) =
n(0) ~π 0 z
The obtained integral is known to diverge. (This can be seen by changing the low
integration limit to a, performing the integration analytically, and then examine
the limit a → 0.) This divergence remains for N > 0, which can be realized from
4.3. Summarizing and Analyzing the Results for the Hermite Gas
37
the fact that the term of lowest order in the even Hermite polynomials is always
a constant.
The situation is similar for the one dimensional infinite quantum well. The
potential is:
0, −L ≤ z ≤ L
v(z) =
.
(4.37)
∞, |z| > L
We obtain for the inverse radius of the exchange hole:
Z L
1 2 πz 2 π(1 + bN/2c)z
1
csc
sin
dz .
Rx−1 =
L2 n(0) 0 z
2L
L
(4.38)
This expression can also be seen to diverge for finite L by studying the behavior
of the integrand in the limit of low z.
The divergences for the inverse radius of the exchange hole for one dimensional systems do not mean we should expect it to diverge for three dimensional
systems as well. Indeed, the divergences in the analysis of the three dimensional harmonic system were in the second derivative, and apart from this, Rx−1
was well behaved. The introduction of more dimensions is known to give more
integrations, resulting in a smoother behavior.
The divergences found for one dimensional systems were both of the kind:
Z b
Z b
1
1
dz = lim
dz = lim (ln(b) − ln(a)) .
(4.39)
a→0
a→0
z
z
0
a
It is possible to do some “schematical calculations” for the behavior of this
divergence when integrated another two times (for two extra dimensions):
ZZ
3
1
(4.40)
ln(z)dz = z 2 ln(z 2 ) − z 2 .
4
4
The expression found here closely resembles the expansion we found for the
inverse radius of the exchange hole for the harmonic oscillator system, equation
(4.31).
This discussion does not, of course, constitute any kind of “proof” for the
behavior of Rx−1 for three dimensional systems. Rather, it is an attempt to give a
motivation to why it is possible to find unanalyticity in our calculations. Judging
from this study, the occurrence of a divergence in the second derivative of the
inverse radius of the exchange hole should perhaps not be very surprising.
4.3.2
Suggesting a Form for a Local Gradient Expansion
We recall the result from section 4.2.5 for the Laplacian in the limit of low
curvatures:
∇2 nh (r) 1
≈ βw2 ⇒ w2 ∼ Lh (0) .
(4.41)
Lh (0) = 5/3
β
n (r) h
r=0
In this expansion we found β ≈ −7.18.
38
Chapter 4. Interior Systems
We also found an expansion for the local exchange energy in section 4.2.4:
−1
−1
(0) ≈ Rx,u
1 + C1 w2 ln(w2 ) + C2 w2 .
(4.42)
Rx,h
Where we found C1 ≈ −0.014 and C2 ≈ −0.068 .
We will now do a heuristic attempt to parameterize this expansion by identifying the occurring w2 as coming from the Laplacian. This gives (we introduce
an absolute value inside the logarithm to make our expansion defined also for
positive Laplacians):
C1
C1
−1
−1
Lh (0) ln |Lh (0)| + C2 −
ln |β| Lh (0) . (4.43)
(0) ≈ Rx,u
1+
Rx,h
β
β
We would like to recast this expression in the form of equation (2.40), with
−1
−1
−1
outside the parentheses. The difference between Rx,u
and Rx,LDA
is
Rx,LDA
−1
−1
that Rx,LDA uses the “real” density, whereas Rx,u uses the uniform density. If we
correct for this using equation (4.11) we obtain (where we introduced γ = −1/16
as the coefficient in the expansion of the harmonic density, equation (4.22)):
C1
−1
−1
Lh (0) ln |Lh (0)| +
(4.44)
Rx,h (0) ≈ Rx,LDA (nh (0)) 1 +
β
γ
C1
ln |β| −
Lh (0) . (4.45)
+ C2 −
β
3
It is important to note, as was said in the beginning of this chapter, that this
work only studies the Laplacian part of the local gradient expansion. To be able
to generalize our results to be possible to use for other systems one needs the
coefficient for the dimensionless gradient, as it occurred our original suggestion
for a local gradient expansion, equation (2.40):
Gh (r) =
|∇n(r)|2
.
n8/3 (r)
(4.46)
The coefficient for this gradient term could be obtained from a point where the
Laplacian is zero (and the gradient is not), repeating much of the analysis in
this work. It is also possible to argue, based on the result found here, that it
is not at all clear that the expansion should be an analytical function of the
gradient of the density. To cover for this possibility we write our suggestion for
a general local gradient expansion on the following form (with A and B as scalar
coefficients and Z as an unknown function):
−1
(n(r)) 1 + Z (G(r)) + A L(r) ln |L(r)| + B L(r) .
(4.47)
Rx−1 ≈ Rx,LDA
The result of this work is thus, as summarized here, a suggestion for how it
might be possible to handle the Laplacian in a local gradient expansion. However,
there is much work needed to examine the validity of this suggestion. The local
gradient expansions suggested here should be regarded mostly as a hint, telling
us that the quite frequent assumptions of analytical behavior within density
functional theory clearly need some consideration.
Acknowledgments
First and foremost I would like to express my deepest gratitude to my supervisor
Ann Mattsson for giving me the opportunity to do my diploma work in a very
exiting field of science. Our many sessions and meetings have not only provided
me with a tremendous amount of aid in my work on this project, but they have
also enlightened me within a large number of other, related and unrelated, fields.
My feeling is also that our many discussions, especially the ones on the Zurich
trams, concerning subjects ranging from Swedish politics to fractal compression
algorithms, have brought us very close. Without your aid, Ann, this diploma
work wouldn’t even be a fraction of what it now is.
I would also like to thank professor Göran Grimvall for becoming my formal
supervisor, for recommending me Ann Mattsson’s project as a suitable area for
a diploma work, for arranging financial means making it possible for me to join
Ann on the trip to Zurich to meet Kohn, and for helping me with a number of
other formal matters (as how to correctly compose a travel economy calculation).
I would like to thank professor Walter Kohn for welcoming me to join Ann in
her trip to Zurich to visit him and to discuss our work with him. The discussions
with Kohn were very enlightening and have given a number of important physical
insights incorporated in this thesis.
I would like to thank all the friendly people at Theoretical Physics with whom
I have had the delight to eat lunch with, and (in the same time) discussed a lot
of interesting subjects.
Among the previous crowd I would like to give special thanks to:
• Anders Vestergren, for providing me with his master’s thesis which I have
made extensive use of as a template for what a master’s thesis should look
like and also for providing a large number of funny jokes.
• Tommy Ohlsson, for creating the original LATEX-template upon which I
have based the layout of this thesis.
• Nils Sandberg, for letting me use his computer for extensive numerical
work (I am not sure he really knows to what extent I have been using this
hospitality ...).
39
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Acknowledgments
My last thank within the department of Theoretical Physics is to professor
Håkan Snellman for being an excellent mentor for me in the mentor project
which I entered during my first year of studies at Engineering Physics at the
Royal Institute of Technology. We have had a lot of inspiring lunch sessions
since then, which very well could be the reason I ended up doing this diploma
work within Theoretical Physics.
I would also like to thank all the other friends I have made here at the Royal
Institute of Technology during my studies. I feel an urge to mention Marios
Nikolaou who has been a very close friend during basically all my time here, and
who probably inspired me to study just a little harder for every new exam (as
his results always, a bit annoyingly, seemed to end up just a little better than
mine). I would also like to mention Tore Ersmark for dragging me away from the
computer to eat dinner at least some evenings when I have been working late.
I would like to thank my dear but perhaps a bit aged friend Harry (A Dell
Optiplex GXMT 5166 computer) on which I have been working since the Ultra
Sparc I begun my work on was reclaimed by its rightful owner due to a Y2K
problem with the computer he was using. Oh–I almost forgot–I would also like
to thank the open source community in general and Linus Torvalds in particular,
for providing me with excellent free software to run on Harry.
Last, but certainly not least, I would like to send my warmest heartly thanks
to my love Maria Tengner; you have been very understanding for my late nights
at Theoretical Physics, and you have always been there for me.
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