Download First Principles Investigation into the Atom in Jellium Model System

First Principles Investigation into the
Atom in Jellium Model System
Andrew Ian Duff
H. H. Wills Physics Laboratory
University of Bristol
A thesis submitted to the University of Bristol in
accordance with the requirements of the degree of
Ph.D. in the Faculty of Science
Department of Physics
March 2007
Word Count: 34, 000
Abstract
The system of an atom immersed in jellium is solved using density functional theory
(DFT), in both the local density (LDA) and self-interaction correction (SIC) approximations, Hartree-Fock (HF) and variational quantum Monte Carlo (VQMC). The main aim
of the thesis is to establish the quality of the LDA, SIC and HF approximations by comparing the results obtained using these methods with the VQMC results, which we regard
as a benchmark. The second aim of the thesis is to establish the suitability of an atom in
jellium as a building block for constructing a theory of the full periodic solid.
A hydrogen atom immersed in a finite jellium sphere is solved using the methods
listed above. The immersion energy is plotted against the positive background density of
the jellium, and from this curve we see that DFT over-binds the electrons as compared
to VQMC. This is consistent with the general over-binding one tends to see in DFT
calculations. Also, for low values of the positive background density, the SIC immersion
energy gets closer to the VQMC immersion energy than does the LDA immersion energy.
This is consistent with the fact that the electrons to which the SIC is applied are becoming
more localised at these low background densities and therefore the SIC theory is expected
to out-perform the LDA here.
DFT is used within the framework of the effective medium theory (EMT) to calculate
Wigner-Seitz radii for solids made up of atoms up to and including the 4d transition
metals. The EMT uses, as a building block, calculations of the constituent atom of the
solid immersed in infinite jellium. The calculated Wigner-Seitz radii are found to reproduce
the same trends observed in the experimental Wigner-Seitz radii as a function of atomic
number.
To my Family
Acknowledgments
Thanks to my supervisor James Annett and also to Balazs Györffy.
Authors Declaration
I declare that the work in this thesis was carried out in accordance with the regulations of
the University of Bristol. The work is original except where indicated by special reference
in the text and no part of the thesis has been submitted for any other degree. Any
views expressed in the thesis are those of the author and in no way represent those of
the University of Bristol. The thesis has not been presented to any other University for
examination either in the United Kingdom or overseas.
SIGNED: .........................................
DATE: ..........................
Contents
1 Introduction
1
2 Solving the Many-Electron Schrödinger equation
2.1
17
The Many-Electron Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1
Single-Electron Theories . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.2
The Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2
Hartree Fock Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3
Density Functional Theory
2.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.1
Minimising the Energy Functional . . . . . . . . . . . . . . . . . . . 21
2.3.2
The Kohn-Sham Equations . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.3
Self-Consistent Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.4
The Exchange-Correlation Energy and Potential . . . . . . . . . . . 29
2.3.5
Self-Interaction Correction . . . . . . . . . . . . . . . . . . . . . . . . 32
Variational Quantum Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.1
A Variational Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.2
The Monte Carlo Technique . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.3
The Variational Quantum Monte Carlo Method . . . . . . . . . . . . 36
2.4.4
Metropolis Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.5
Equilibration and Serial Correlation . . . . . . . . . . . . . . . . . . 39
2.4.6
The Choice of the Trial Wavefunction . . . . . . . . . . . . . . . . . 40
2.4.7
Updating the Slater Determinants . . . . . . . . . . . . . . . . . . . 41
2.4.8
Calculating the Local Energy . . . . . . . . . . . . . . . . . . . . . . 42
2.4.9
Cusp Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.10 Correlated Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.11 Blocking Analysis to Calculate Error on Mean . . . . . . . . . . . . 48
xi
xii
CONTENTS
2.4.12 Calculating the Probability Density . . . . . . . . . . . . . . . . . . 50
2.4.13 HF Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 An Atom in Infinite Jellium Solved using DFT
3.1
3.2
3.3
3.4
3.5
3.6
53
Solving the Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.1
The Radial Schrödinger Equation . . . . . . . . . . . . . . . . . . . . 53
3.1.2
The Electron Density . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.3
Potential Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1.4
Criterion for Convergence . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.5
Simplifying the Coulomb Potential for the Case of Spherical Symmetry 58
Scattering States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.2
Boundary Conditions on Scattering States . . . . . . . . . . . . . . . 61
3.2.3
Matching to the Boundary Condition
3.2.4
Normalisation of Scattering States . . . . . . . . . . . . . . . . . . . 64
3.2.5
Calculating the Scattering State Density . . . . . . . . . . . . . . . . 65
3.2.6
Friedel Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.7
Friedel Sum Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2.8
Properties of the Phase-Shift . . . . . . . . . . . . . . . . . . . . . . 70
. . . . . . . . . . . . . . . . . 62
Numerical Algorithm for Solving the Radial Schrödinger Equation . . . . . 73
3.3.1
Radial Schrödinger Equation Solutions in the Limits r → 0 and r → ∞ 73
3.3.2
The Runge-Kutta Algorithm . . . . . . . . . . . . . . . . . . . . . . 74
3.3.3
Bound State Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3.4
Scattered State Calculation . . . . . . . . . . . . . . . . . . . . . . . 78
The Immersion Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.4.1
Derivation of Immersion Energy . . . . . . . . . . . . . . . . . . . . 78
3.4.2
Finite Radius Corrections . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4.3
Numerical Parameters and Error Analysis . . . . . . . . . . . . . . . 83
3.4.4
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
The Effective Medium Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5.1
Background Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Cerium Solved using the LDA and SIC . . . . . . . . . . . . . . . . . . . . . 97
CONTENTS
xiii
3.6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.6.2
Cerium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.6.3
Spin-polarised LDA for Cerium . . . . . . . . . . . . . . . . . . . . . 100
3.6.4
Imposing Orthogonality when Applying SIC . . . . . . . . . . . . . . 102
3.6.5
SIC-LDA for Cerium . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.6.6
Magnetic Solution of Cerium . . . . . . . . . . . . . . . . . . . . . . 108
4 Hydrogen Immersed in a Finite Jellium Sphere
4.1
4.2
111
Hydrogen in Finite Jellium Spheres using the LDA . . . . . . . . . . . . . . 111
4.1.1
Energy of An Atom in a Finite Jellium Sphere . . . . . . . . . . . . 112
4.1.2
Filling of Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.1.3
Applying SIC to a Hydrogen Atom in a Finite Jellium Sphere . . . . 115
4.1.4
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Hydrogen in Finite Jellium Spheres using VQMC . . . . . . . . . . . . . . . 126
4.2.1
The Choice of the Trial Wavefunction . . . . . . . . . . . . . . . . . 126
4.2.2
Calculating the Local Energy . . . . . . . . . . . . . . . . . . . . . . 129
4.2.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5 Conclusions
145
A Local Kinetic Energy Calculation for Atom in Jellium
149
xiv
CONTENTS
List of Tables
4.1
Total energies of hydrogen in 10-electron jellium spheres . . . . . . . . . . . 132
4.2
Total energies of 10-electron jellium spheres . . . . . . . . . . . . . . . . . . 135
4.3
Immersion energies of hydrogen in 10-electron jellium spheres . . . . . . . . 136
xv
xvi
LIST OF TABLES
List of Figures
1.1
The probability of finding two electrons a separation |r| apart from one
another for parallel and anti-parallel spins. The system is an electron gas
solved using Hartree-Fock theory, and shows how the correlation between
electrons due to exchange is captured by the theory (see the reduced probability of two same spin electrons being close to one another) but the correlation due to the Coulomb interaction is not (no reduced probability in
the different spin case) [1].
1.2
. . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Wigner-Seitz radii for transition metal elements as calculated by Moruzzi
et al [2] using LDA for a full periodic solid (circles) and the experimental
values (crosses).
1.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Bulk moduli for transition metal elements as calculated by Moruzzi et al [2]
using the LDA for a full periodic solid (circles) and the experimental values
(crosses). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
7
Band-gaps predicted by the LDA (triangles) are too small by up to 3eV
compared to the experimental values (diamonds and circles) [3]. Squares
are the GW approximation.
1.5
. . . . . . . . . . . . . . . . . . . . . . . . . .
8
The model used. The full crystal is approximated as a positive ion of charge
Z surrounded by the smeared out effective charge of all the surrounding ions.
The assumption of spherical symmetry is made in the final step, which is
consistent with our omission of details regarding the shape of the unit cell.
Ω is the atomic volume, rW S is the Wigner-Seitz radius, nbs is the number
of bound states per atom and nval is the number of valence electrons per
atom. Charges are in units of the electron charge, e. . . . . . . . . . . . . . 11
xvii
xviii
1.6
LIST OF FIGURES
The background density, n̄i , in a given cell i is made up of the sum of the
density tails of all the other atoms, averaged over cell i. This picture applies
to the EAM and the EMT. Figure taken from a paper by Yxklinten et al [4]. 14
2.1
Re-blocking analysis for hydrogen immersed in a 10-electron jellium sphere
of density 0.03a−3
B . The error on the mean levels off at just under 0.0031eV
and therefore this is the error we quote on the total energy. . . . . . . . . . 49
3.1
Phase-shifts (top panel) and the corresponding density of states (lower
panel) for a cerium atom embedded in jellium of rs = 5.3 . . . . . . . . . . 71
3.2
The l = 0 phase-shifts for a hydrogen atom immersed in infinite jellium of
−3
background densities 0.01a−3
B (top panel) and 0.05aB (bottom panel).
3.3
. . 72
The quantity dUoutwards (r = rmatch )/dr − dUinwards (r = rmatch )/dr (as
described in the main text) for l = 0 is plotted as a function of energy. The
system is a Technetium atom immersed in jellium of background density
−3
, and is non-magnetic, so the curve applies for both spin-up and spin0.03aB
down electrons The l = 0 bound state energies are at the points where the
curve crosses the x-axis, I.e. at: −744.939a.u., −103.763a.u., −17.363a.u.
and −1.845a.u..
3.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Determining the parameter ∆V req , for a SI-corrected cerium atom in infinite jellium of density n0 = 0.01a−3
B . Immersion energy is plotted against
log(∆V req ), and error bars (in green) are placed at different values of the
convergence.
3.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Density plots for hydrogen in infinite jellium of density 0.005a−3
B . Values for
rmax equal to 24.676aB , 25.326aB and 25.963aB are shown. Only the second
choice of rmax gives the correct form for the density oscillation (the peak
of the last oscillation is at the same height as the penultimate oscillation).
The values of the Friedel sum for these choices are 0.98, 1.00 and 1.02
respectively, showing that selecting rmax to get the correct density profile
is equivalent to selecting rmax to satisfy the Friedel sum.
. . . . . . . . . . 86
LIST OF FIGURES
3.6
xix
Determining the parameter lnum . This value has to be large enough so that,
for a given rmax , the density is correctly realised at all radii. The above
are results for a cerium atom immersed in jellium of density 0.01aB , with
rmax ≈ 20aB . The red curve corresponds to the actual calculated density,
the green curve to the theoretical density (Eq. (3.2.40) ). We see that only
the final choice of lnum (= 20) gives the correct density profile, and therefore
this is the value that we use. . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.7
Immersion energy versus background density curves for atoms with atomic
numbers 1 to 18 as obtained by our calculations. Elements P, S and O
are excluded because of difficulty in obtaining converged solutions for these
elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.8
Immersion energy versus background density curves for atoms with atomic
numbers 1 to 18 as calculated by Puska et al [5]. Elements P and S were
excluded because of unsatisfactory convergence of solutions.
3.9
. . . . . . . . 92
Squares are experimental Wigner-Seitz radii, blue diamonds are our neutral
sphere radii, crosses are neutral sphere radii as calculated by Yxklinten et
al [4].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.10 Cohesive energy versus neutral sphere radii for 4d transition metals. . . . . 98
3.11 Experimental phase diagram of cerium [6] . . . . . . . . . . . . . . . . . . . 100
3.12 Experimental results showing the molar volume of cerium against the pressure applied to the sample [7] . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.13 Phase-shifts for (non-magnetic) ground-state solutions of a cerium atom embedded in jellium of different densities. From top to bottom, rs = 1.81aB ,
rs = 3.24aB , rs = 5.30aB . The red, green, blue and magenta curves correspond respectively to l = 0, l = 1, l = 2 and l = 3 (and are also labelled in
the bottom panel).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.14 Angular momentum resolved density of states for the ground-state solution
of a cerium atom embedded in jellium of rs = 5.3 . . . . . . . . . . . . . . . 104
3.15 Energy of the 4f bound state for a SI-corrected cerium atom immersed in
jellium, as a function of the background jellium density. The points are
calculated energies, and the line is extrapolated to zero energy. . . . . . . . 108
xx
LIST OF FIGURES
3.16 Phase-shifts for the LDA solution of a cerium atom immersed in infinite
jellium for a variety of background densities. From top to bottom, n0 =
−3
−3
−3
0.04a−3
B , n0 = 0.03aB , n0 = 0.02aB and n0 = 0.01aB . The separation
of the spin-up and spin-down phase-shifts as the background density is
increased corresponds to the formation of a magnetic moment on the cerium
atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.1
Plots of (spin-up) bound state energies for a 338-electron jellium sphere and
a hydrogen atom in a 338-electron jellium sphere, along with the (spin-up)
potentials for these systems. The background densities of the jellium are
−3
0.03 a−3
B (upper panel) and 0.008 aB (lower panel). The bound states are
shown as lines, with the lengths of these lines corresponding to the angular
momentum (l=0 is the shortest and l=9 is the longest). . . . . . . . . . . . 119
4.2
Plots of immersion energy versus number of electrons for jellium spheres.
−3
−3
Densities of 0.001a−3
B , 0.007aB and 0.03aB are considered. The lines are
the values of the immersion energy for the infinite jellium system. The
largest size of jellium sphere used in these plots is a 138-electron jellium
sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.3
Plots of immersion energy versus background density for a hydrogen atom
immersed in jellium spheres of size 10 and 50 electrons. Also plotted is the
immersion energy curve for a hydrogen atom in infinite jellium. . . . . . . . 122
4.4
Plots of immersion energy versus background density for a hydrogen atom
immersed in jellium using different exchange-correlation functionals. The
top panel is for a 10 electron jellium sphere and bottom panel is for a 50
electron jellium sphere.
4.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Plots of atom induced densities for hydrogen in finite jellium spheres of
background density 0.01a−3
B , with 10, 50 and 338 electrons (top, middle
and bottom panel respectively). Also plotted is the atom induced density
for a hydrogen atom in infinite jellium at the same background density. . . 126
4.6
The total density for a hydrogen atom in a 106-electron jellium sphere and
for a hydrogen atom in infinite jellium for a background density ≈ 0.004a−3
B . 127
LIST OF FIGURES
4.7
xxi
Plots of spin-up potentials (upper panel) for a 338-electron jellium sphere
for different background densities. The energy of the 1s bound state is also
included for each potential, and is plotted as a straight-line on the left of
the graph. The lower panel shows the expectation value of the radius of the
(spin-up) 1s electron for the different background densities. See main text
for discussion.
4.8
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Total energy of a hydrogen atom immersed in a 10-electron jellium sphere
for different background densities
4.9
. . . . . . . . . . . . . . . . . . . . . . . 133
Total energy of a 10-electron jellium sphere for different background densities
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.10 Immersion energies for a hydrogen atom immersed in a 10-electron jellium
sphere for different background densities
. . . . . . . . . . . . . . . . . . . 134
4.11 Local energy distribution for a VQMC calculation of a hydrogen atom in a
10-electron jellium sphere of density 0.03a−3
B . . . . . . . . . . . . . . . . . . 137
4.12 Re-blocking analysis for hydrogen immersed in a 10-electron jellium sphere
of density 0.03a−3
B . The error on the mean levels off at just under 0.0031eV
and therefore this is the error we quote on the total energy. . . . . . . . . . 138
4.13 Electron density of a hydrogen atom in a 10-electron jellium sphere of background density 0.03a−3
B using HF, LDA, SIC and VQMC
. . . . . . . . . . 139
4.14 Electron density of a hydrogen atom in a 10-electron jellium sphere of back−3
using HF, LDA, SIC and VQMC. Note in the top
ground density 0.03aB
graph curves for HF, LDA and SIC coincide. . . . . . . . . . . . . . . . . . 140
4.15 Electron density of a hydrogen atom in a 10-electron jellium sphere of background density 0.002a−3
B using HF, LDA, SIC and VQMC
. . . . . . . . . 141
4.16 Electron density of a hydrogen atom in a 10-Electron jellium sphere of
background density 0.002a−3
B using HF, LDA, SIC and VQMC. Note in the
top graph curves for HF, LDA and SIC coincide. . . . . . . . . . . . . . . . 142
4.17 The electron density across a slab of jellium as calculated by Li and Needs
et al [8]. The origin is at the centre of the slab.
. . . . . . . . . . . . . . . 143
xxii
LIST OF FIGURES
Chapter 1
Introduction
In this thesis, we will solve the system of an atom immersed in jellium. That is, we
will calculate the ground-state energy and density of a system of N electrons sitting in
an external potential set up by an ion of charge, Z, and a uniform positively charged
background of density, n0 . Our motivation for studying this model system is that it can
be used as a building block from which to construct a theory of a full periodic solid. In
this introduction we will expand upon this motivation and we will detail the techniques
which we use to solve the model system.
Solving the atom-in-jellium system formally requires us to solve the time-independent
Schrödinger equation (in the Born-Oppenheimer approximation [9], so that the nucleus is
held fixed)

N
X

i=1
−
~2
2me
∇2i +
N
1X
2
i6=j

e2
4π0 |ri − rj |
+ vext (r1 , r2 , ..., rN ) Ψ(r1 , r2 , ..., rN )
= EΨ(r1 , r2 , ..., rN )
(1.0.1)
where the external potential, vext (r), is given by
vext (r) = −
Z
N
e2 X Z
e2
n0
−
dr0
4π0
ri 4π0
|r − r0 |
(1.0.2)
i=1
Solving Eq. (1.0.1) is made difficult by the second term, which describes the Coulomb
repulsion between electrons. In practice, approximate methods of solving the equation
are often employed. In this thesis, we will use three approximate methods to solve the
equation.
1
2
Introduction
The first of our methods, and one which was historically the first serious attempt at
solving systems of interacting electrons is the method of Hartree-Fock (HF) [10, 11, 12, 13].
Our second method is the widely used density functional theory (DFT) [14, 15, 16]. The
third method is a stochastic method known as variational quantum Monte Carlo (VQMC)
[17, 18].
The three methods form a hierarchy of increasing accuracy, with HF then DFT and
then VQMC in ascending order of accuracy. It is the treatment of the correlation between
electrons, i.e. their interaction with one another as a function of electron separation, that
determines the accuracy of the methods.
Electron correlation has two physical origins. Firstly, electrons of the same spin will
be forced to stay apart from one another due to the Pauli exclusion principle (PEP) [19].
This is called exchange correlation. Secondly, the Coulomb repulsion between electrons
will also encourage separation. This is Coulombic correlation.
HF only includes exchange correlation, and so is the least accurate of the methods
considered here. DFT improves on HF by including Coulombic correlation, however, the
correlation is still only included in an approximate manner. VQMC is the most accurate
method, treating correlation to a high degree of accuracy, provided one is able to make a
good enough physically-informed guess at the form of the wavefunction.
There now follow brief introductions to these methods. These introductions are greatly
expanded upon in subsequent sections and are only intended as brief overviews.
Hartree-Fock
HF is a variational theory in which one attempts to express the true ground-state
wavefunction as a determinant of single-electron functions (I.e. functions of the position
and spin coordinates of only one electron). The determinant changes sign on interchange
of particle coordinates, thereby satisfying the PEP.
The method is actually based on a prior variational theory known as Hartree theory
[10, 11, 12] in which the wavefunction is written as a single product of these single-electron
functions. This wavefunction however did not satisfy the PEP and so V. Fock [13] wrote a
new wavefunction which changed sign upon particle interchange. This new wavefunction
was later identified as a Slater determinant [20, 21].
Using HF one minimises hΨ|Ĥ|Ψi/hΨ|Ψi (where the Hamiltonian, Ĥ, is the operator
on the left-hand side of Eq. (1.0.1)), with respect to the single-electron functions, or
Introduction
3
Figure 1.1: The probability of finding two electrons a separation |r| apart from one another
for parallel and anti-parallel spins. The system is an electron gas solved using HartreeFock theory, and shows how the correlation between electrons due to exchange is captured
by the theory (see the reduced probability of two same spin electrons being close to one
another) but the correlation due to the Coulomb interaction is not (no reduced probability
in the different spin case) [1].
’orbitals’, as they are known. This results in N Schrödinger-like eigenvalue equations of
these orbitals, which have to be solved in a self-consistent manner to obtain the HF energy
and wavefunction. In accordance with the variational principle (see Section 2.1.2) , the HF
energy is then an upper bound to the true ground-state energy and the HF wavefunction
is an approximation to the true ground-state wavefunction.
The weakness of the method is in the writing of the wavefunction as a determinant of
single-electron functions. In practice this is a poor approximation to the true wavefunction,
and results in solutions which do not include Coulombic correlation (see Fig. 1.1).
An example of the failure of HF to properly include correlations can be found in the
dissociation of a hydrogen molecule. The HF wavefunction for a hydrogen molecule gives
a finite probability of finding both electrons on the same atom. This is even the case in
the limit of dissociation, i.e. when the two atoms are pulled infinitely far apart from one
another. This is clearly unphysical, as in this limit, the solution should just be that of two
4
Introduction
hydrogen atoms. We find the HF energy in this limit to be a substantial overestimate of
the exact energy, which should equal twice the energy of the isolated hydrogen atom. HF
gets it wrong because it fails to include Coulombic correlations, which would prohibit the
occupation of the same atom by both electrons in the limit of dissociation.
Density Functional Theory
The second method we will use to solve the Schrödinger equation is DFT [14, 16]. In
DFT, one proves that the ground-state energy, E0 , of a system of N interacting electrons in
an external potential, vext (r), is a unique functional of the ground-state density, n0 (r). To
this end a functional, Ev [n(r)] is constructed which has the property that it is minimised
by n0 (r) and that the ground-state energy equals the value of the functional at this point,
i.e. E0 ≡ Ev [n0 ].
Writing n(r) =
PN
2
i=1 |φi (r)| ,
and functionally differentiating Ev with respect to φ∗i (r),
one obtains N single-particle Schrödinger-like equations known as Kohn-Sham equations
[15]. The equations must be solved self-consistently and the self-consistent n(r) is then (in
principle) the true ground-state density, n0 (r), and the ground-state energy is obtained
by inserting this into Ev [n(r)].
Ev [n(r)] contains a universal part, i.e. one which just depends on n(r) and not vext (r).
This universal part is split up into a ’single-particle’ kinetic energy term (see more later),
a classical Coulomb energy term and the exchange-correlation energy, Exc [n(r)]. DFT is
not an exact theory, because although it can be proved that there is an exact functional
for Exc [n(r)], in reality it is not known what it is. It is for this reason that the electron
correlation discussed earlier is only treated approximately within DFT. This approximate
treatment of the correlation is still a big improvement over the exchange-only correlation
included in HF theory however.
Reducing the 3N -dimensional many-particle equation into N 3-dimensional equations
is a dramatic simplification, and makes solving a system within DFT computationally very
efficient. Furthermore, a particularly simple approximation to the exchange-correlation
functional, known as the local density approximation (LDA), produces results in surprisingly good agreement with experiment. In this approximation, one assumes that the
exchange-correlation energy can be written in the form
Introduction
5
Z
Exc [n(r)] =
n(r)xc (n(r))dr
(1.0.3)
where xc (n(r)) is the exchange-correlation energy per electron of a homogeneous electron
gas of uniform density n(r).
The LDA has been used with great success in calculating structural properties of solids,
such as Wigner-Seitz radii and bulk moduli. Fig. 1.2 and Fig. 1.3 show LDA calculations
for Wigner-Seitz radii and bulk moduli for transition metal elements (using a full periodic
solid in the calculation) by Moruzzi et al [2]. Experimental Wigner-Seitz radii are included
alongside these calculated results, and there is good agreement between the two. In fact, in
more recent calculations using the LDA, Wigner-Seitz radii and bulk moduli are predicted
to lie within 1% and 10% respectively, of their experimental values [22]. Despite this good
agreement, the LDA is known to systematically overbind materials, predicting cohesive
energies and bulk moduli that are too large and lattice constants that are too small [23].
Another well established deficiency of the LDA is that it predicts band-gaps that are too
small. Fig. 1.4 illustrates this, with band-gaps that are too small by up to 3eV.
An unphysical element of the LDA is that a given electron interacts with itself via the
Coulomb interaction. A scheme known as the self-interaction correction (SIC) [24] corrects
for this. Using SIC, an immediate improvement over the LDA is that for the system of an
atom, the potential appearing in the Kohn-Sham equations now tends to −1/r as r → ∞,
instead of exponentially decaying as is the case for the LDA solution. This is consistent
with the fact that far away from the ion, a given electron sees an ion which is screened by
all but one of the electrons in the atom.
As an example of an application of SIC, Lüders et al solved the cerium metal using a
KKR method with the LDA and SIC [25, 26]. Experimentally, the cerium metal shows
a phase transition as a function of pressure, from a gamma phase at low pressure to an
alpha phase at higher pressure. This gamma-alpha phase transition is accompanied by
a 15% volume collapse. One theory which attempts to explain this phase transition is
known as the Mott transition model [27]. In this theory, there is a localised f -electron in
the gamma phase, which becomes itinerant in character in the alpha phase.
In their SIC solution, Lüders et al find that the f -electron of cerium is bound, whereas
in their LDA solution the electron is a valence electron. Within the Mott transition model
of the gamma-alpha transition, these two solutions correspond to gamma and alpha phases
respectively. Lüders et al plotted total energy curves for these two solutions as a function
6
Introduction
Figure 1.2: Wigner-Seitz radii for transition metal elements as calculated by Moruzzi et
al [2] using LDA for a full periodic solid (circles) and the experimental values (crosses).
Introduction
7
Figure 1.3: Bulk moduli for transition metal elements as calculated by Moruzzi et al [2]
using the LDA for a full periodic solid (circles) and the experimental values (crosses).
8
Introduction
Figure 1.4: Band-gaps predicted by the LDA (triangles) are too small by up to 3eV
compared to the experimental values (diamonds and circles) [3]. Squares are the GW
approximation.
Introduction
9
3
of volume. The minimum of the energy curve for the LDA was at 23.4Å whereas the
3
minimum for the SIC curve was at 29.9Å . Experimentally, the alpha phase has a volume
3
3
28.2Å and the SIC has a volume 34.7Å . So Lüders et al slightly underestimate both
volumes, but successfully provide evidence supporting the Mott-transition model of the
gamma-alpha transition.
This is just one example of an application of SIC. Other important examples include
calculations of valencies and lattice constants as a function of atomic number across the
rare-earth elements [28]
Quantum Monte Carlo
Quantum Monte Carlo (QMC) methods use random numbers to solve the Schrödinger
equation. We will use a method known as VQMC [17, 18]. In this method, one makes a
guess at a trial wavefunction, using the physics of the situation to aid them with the guess.
The expectation value of the Hamiltonian for this trial wavefunction is then evaluated
using a random-walk technique known as the Metropolis algorithm [29]. One then varies
the trial wavefunction (or more specifically, parameters within the wavefunction) in order
to minimise this expectation value. The minimised expectation value is then an upper
bound on the exact ground-state energy of the system. With a good trial wavefunction,
this upper bound can be made very close to the exact ground-state energy.
One application of QMC has been in the calculation of cohesive energies. The cohesive
energy of a solid is the energy required, at zero temperature, to separate all of the constituent atoms of the solid infinitely far apart. Within VQMC, this calculation requires
trial wavefunctions sufficiently accurate to describe both the solid, and the constituent
atom. Because these two systems are very different, the calculation constitutes a challenging test of the theory. Accordingly it was seen as an impressive success for the theory
when Fahy et al [30, 31] calculated cohesive energies for tetrahedrally bonded Carbon and
Silicon. VQMC was used to obtain cohesive energies of 7.27eV/atom and 4.82eV/atom
respectively, which are in good accord with the experimental values of 7.37eV/atom and
4.62eV/atom. For comparison, the LDA overestimates these energies, giving cohesive
energies of 8.61eV/atom and 5.28eV/atom respectively.
The effectiveness of VQMC relies entirely upon the quality of the trial wavefunction.
Normally one would use this method as a driver for another QMC method known as
diffusion QMC (DQMC) [17, 32]. The results from this method are in principle exact,
10
Introduction
barring an issue known as the sign problem [17]. We will limit ourselves to VQMC however,
partly because published results on jellium spheres [33] (not too dissimilar to the hydrogen
in a jellium sphere system which we will be studying - see later) show that VQMC and
DQMC results are quite similar.
The Atom-in-Jellium Model
Our motivation for considering the system of an atom immersed in jellium is that
it can be used to model a condensed matter system. This idea comes in the first place
from just considering pure jellium (i.e. a uniform electron gas with a charge neutralising
background) as a model for a solid. In this case we would regard the positive background
as the smeared out effective charge of the ions in the solid, and the negative charge as the
conduction electron density.
If each atom occupies a volume, 43 πa3 , where a is the Wigner-Seitz radius, and if each
of these atoms contributes Nv valence electrons to the solid, then with an electron density
n = 1/ 34 π(rs aB )3 , we have the relation
4
3
3 πa
3
4
3 π(rs aB )
∴
= Nv
1
a = Nv 3 rs aB
(1.0.4)
where aB is the Bohr radius. It turns out that the energy of an electron gas as a function
of rs minimises at 4.8, showing that even for such a simple system, the lattice parameter
predicted is of the correct order of magnitude - namely, of the order of the Bohr radius.
Sodium for example, which is a typical alkali metal, has rs = 3.96.
As an intermediate step between pure jellium and a full periodic solid, we model
the system of a single atom immersed in jellium. The positive background density then
represents the smeared out effective charges of all of the other ions of the solid, as shown
in Fig. 1.5. In this model of the solid, the Schrödinger equation has two classes of solution.
There are a discrete set of bound states, and a continuum of positive energy ’scattering
state’ solutions. The bound states and the atom-induced scattering states (the change in
the number of scattering states upon adding the atom to the jellium) of this system are
interpreted as the bound and valence electrons per atom of the full periodic solid.
In order to use this model of an atom in jellium to make predictions about real solids,
we need an expression for the energy of the solid. Clearly, the total energy of this model
Introduction
11
Figure 1.5: The model used. The full crystal is approximated as a positive ion of charge Z
surrounded by the smeared out effective charge of all the surrounding ions. The assumption
of spherical symmetry is made in the final step, which is consistent with our omission of
details regarding the shape of the unit cell. Ω is the atomic volume, rW S is the WignerSeitz radius, nbs is the number of bound states per atom and nval is the number of valence
electrons per atom. Charges are in units of the electron charge, e.
12
Introduction
is infinite and so we need some kind of energy per atom. This is where this simple
view of the solid encounters difficulties, as it turns out that there is no straightforward
way of constructing an energy per atom. For example, one could try to calculate an
energy per atom by calculating the energy in the region immediately surrounding the
central atom (where there is no positive background charge). However, the energy is
a non-local quantity which has, for example, contributions due to electrons within this
region interacting with electrons outside the region. One would have to arbitrarily cut off
these non-local contributions in a somewhat physically unsatisfactory manner. In fact, an
attempt has been made [34] to construct an energy for the solid using such an approach.
The work had some success, for example reproducing the trends in the Wigner-Seitz radius
across row six of the periodic table. However, we will not use this model, and will instead
find a more physically sensible method of constructing a total energy of the solid from the
system of an atom in jellium.
The approach we will follow is based on the effective medium (EM) approach [35, 36]
or the equivalent quasiatom [37] theory. The EM approach is often referred to in the
literature as the effective medium theory (EMT), however, another theory which we shall
introduce shortly also goes by the same name. Therefore, in order to avoid confusion we
will reserve the EMT abbreviation for the latter, and simply refer to the former as the
EM.
The EM and quasiatom theories are concerned with adding an impurity atom into
an inhomogeneous electron gas (referred to as the host system). Stott and Zaremba [37]
proved that the energy of the impurity in this host is a functional of the density of the
host before the impurity has been added, I.e.: E ≡ FZ,R [nhost ], where Z is the charge of
the impurity, R is its position and nhost (r) is the unperturbed charge density of the host.
One makes the assumption that only the unperturbed host density immediately surrounding the impurity is important in the calculation of the energy of the impurity. Applying this assumption in the simplest possible way results in one replacing the energy of
the impurity in the inhomogeneous host with the energy of the impurity in a homogeneous
electron gas of background density nhost (R).
Using the EM approach, Nørskov [36] has been successful in calculating heats of solution for light interstitial impurities such as hydrogen and helium. Following the above
approach, one replaces the immersion energy of the impurity in the inhomogeneous host,
with the immersion energy of the impurity in jellium of background density nhost (R) (the
Introduction
13
immersion energy is the total energy of the impurity in the host minus the energies of the
separate impurity and host). The heat of solution, which is the change in energy when
one mole of hydrogen gas is absorbed by the solid, is then obtained by subtracting the
binding energy of the hydrogen molecule per atom from the immersion energy. Note that
in these calculations, the background density of the jellium was not in fact just taken to
be the density of the host at the point R. Instead it was chosen as some average of the
host electron density over the volume to be occupied by the impurity.
The embedded atom method (EAM) [38] is based on the EM approach. In this method,
we are interested in calculating the cohesive energy of a solid. Each atom in the solid is
viewed as being embedded in the electron gas set up by the remaining atoms of the solid.
For an atom i, we denote this as n̄i (r). We assume that this electron density is a linear
superposition of the electron densities from each of the atoms at sites Rj (where j 6= i),
which we label ∆nj (|r−Rj |). In addition, we assume that ∆ni is just equal to the electron
density of the atom at lattice site i in free space. Furthermore, we spherically symmetrise
these atomic densities. Therefore we have
n̄i (r) =
X
∆nj (|r − Rj |)
(1.0.5)
j6=i
Following the EM approach, we then replace n̄i (r) in the atomic cell i with its value
at Ri . Therefore each atom sits in a homogeneous electron gas set up from the sum of the
density tails from all other atoms. This is illustrated in Fig. 1.6. The cohesive energy is
then written as
Ec =
X
i
∆E(n̄i (Ri )) +
1X
Uij (Rij )
2
(1.0.6)
i6=j
where ∆E(n0 ) is the immersion energy of an atom immersed in jellium of background
density n0 . The second term describes the electrostatic interactions between the atoms.
This term is not known exactly, and is in practice determined from experimental data,
making the method semi-empirical overall. This cohesive energy must be minimised as a
function of the atomic positions. Notice that because we have fixed the density around
each atom to equal the density of the constituent atom in a vacuum, the theory does not
allow the electron densities at each site to alter in order to lower the cohesive energy.
EAM has enjoyed success in many bulk and surface problems. Problems such as
phonons [39], thermodynamic functions and melting points [40, 41] and surface ordered
14
Introduction
Figure 1.6: The background density, n̄i , in a given cell i is made up of the sum of the
density tails of all the other atoms, averaged over cell i. This picture applies to the EAM
and the EMT. Figure taken from a paper by Yxklinten et al [4].
alloys [42, 43], to name but a few, have been treated using the method. For a full discussion
of the applications, see the review of EAM by Daw [44].
A theory due to Jacobsen et al, referred to as the effective medium theory (EMT)
[45, 46], also proceeds along a similar line of thought to the EAM. Again, each atom is
viewed as being embedded in an electron gas set up by the electron densities from all other
atoms. This theory however, is derived fully from first principles within the framework
of the LDA, and unlike the EAM doesn’t require experimental parameters to specify the
theory. This theory is described in some detail later in this thesis, but for now we just
quote the main results.
The cohesive energy for EMT is similar to Eq. (1.0.6), except that the second term
is replaced with a Coulomb interaction term which describes the attraction between the
Hartree potential of a given atom with the sum of the electron densities from all other
atoms impinging on the atomic cell in question. We find a cohesive energy per atom of
Ec (n̄)
= ∆E(n̄) + n̄
N
Z
r=s Z ∞
r=0
r0 =0
∆n(r0 ) 0 Z
dr −
|r − r0 |
r
dr
(1.0.7)
where ∆n(r) is the atom-induced density for an atom with atomic number Z immersed
in jellium of density n̄. The quantity s is referred to as the neutral sphere radius and is
Introduction
15
defined as
Z
r=s
n(r)dr = Z
(1.0.8)
r=0
The theory requires only the atomic number, Z, of the constituent atom of the solid
as the input parameter. We minimise Ec (n̄)/N with respect to n̄ and the corresponding
s is then the Wigner-Seitz radius of the solid as predicted by the theory. Calculations of
these Wigner-Seitz radii, as well as other cohesive properties of solids, such as the bulk
moduli and cohesive energies are in good agreement with experimental results [45, 47, 4].
Other applications of the theory include calculations of the phonon dispersion relations
and surface properties [45].
Calculations
We will perform calculations on the atom-in-jellium system for a variety of atoms and
across a range of positive background densities. In the first three applications we will use
the LDA and the SIC, and for the fourth application we will solve using the LDA, SIC,
HF and VQMC. These calculations will be performed using DFT and VQMC computer
programs written in Fortran by the author of this thesis.
Our first application will be to use the LDA to calculate the immersion energy as a
function of background density for elements from the first three rows of the periodic table.
Comparing these results to existing calculations in the literature will allow us to check
that the DFT computer program works correctly.
We will then proceed to use the EMT to verify previously reported calculations of
Wigner-Seitz radii for solids as a function of the atomic number of the constituent atom
of the solid for the 2p, 3p and 3d series of elements. In addition, new results will be
obtained in the form of Wigner-Seitz radii for solids made up of atoms from the 4d series
of elements.
In another application we will use the atom-in-jellium model, solved within DFT, to
model the alpha and gamma phases of bulk cerium. As we have discussed, Lüders et al
calculated LDA and SIC solutions for cerium for which the f -electron was delocalised in
the former and localised in the later. Within the Mott-transition model of cerium, these
correspond to the alpha and gamma phases of cerium respectively. We too will use LDA
and SIC solutions to model the alpha and gamma phases of cerium, but this time within
16
Introduction
our atom-in-jellium model.
The central result of the thesis will be to solve the system of a hydrogen atom immersed
in jellium within the theoretical frameworks of HF, LDA, SIC and VQMC. The latter will
be used as a benchmark against which the accuracy of the preceding methods will be
tested. The aim is to calculate the electron density, the total energy and the immersion
energy as functions of the positive background density. Positive background densities in
−3
the range 0.001a−3
B to 0.03aB will be considered.
We will have to consider an atom in a finite jellium sphere instead of our model system
of an atom in infinite jellium. This is because in order to solve the problem with QMC
the system must be of finite size (or must be periodic). Ideally, we would like our jellium
spheres to be as large as possible, so we can use our results to make inferences about the
atom in infinite jellium system. In practice, we will be limited to sizes for which the QMC
calculation time is not prohibitively long.
In fact, VQMC calculations have already been attempted on the system of a hydrogen
atom in jellium [48]. These calculations however resulted in immersion energies which
differed significantly from the LDA results. The reason cited by the authors for this
mismatch was that the trial wavefunction was not of optimal form.
In our calculations, we will fix the number of electrons in the finite jellium sphere.
Therefore the radius of the sphere will vary as we vary the background density. In order
to decide on the number of electrons in the jellium sphere, we carry out a study, within the
LDA, of the dependence of the immersion energy (for a particular background density)
on the number of electrons in the jellium sphere. We carry out this study for a range
of background densities, allowing us to select a value for the number of electrons which
yields an immersion energy versus background density curve which best approximates the
same curve for the hydrogen atom in infinite jellium. We then solve this system of a
hydrogen atom in a jellium sphere using HF, LDA, SIC and VQMC, and compare the
results obtained using these different methods.
Chapter 2
Solving the Many-Electron
Schrödinger equation
2.1
The Many-Electron Problem
The time-independent Schrödinger equation for an N-electron system in an external potential, vext (r1 , r2 , ..., rN ), within the Born-Oppenheimer approximation [9] (all nuclei held
fixed), is

N
X

i=1

N
~2 2 1 X
e2
−
∇ +
+ vext (r1 , r2 , ..., rN ) Ψ(r1 , r2 , ..., rN )
2me i
2
4π0 |ri − rj |
i6=j
= EΨ(r1 , r2 , ..., rN )
(2.1.1)
Note that for the rest of this thesis atomic units are used (~2 /m=e2 /4π0 =1). The
external potential, vext , could be for example the Coulomb attraction between the electrons
and an ion of charge Z:
vext = −
i=N
X
i=1
Z
ri
(2.1.2)
Eq. (2.1.1) is very difficult to solve analytically due to the second term, which describes
the Coulomb repulsion between the electrons. In this thesis we will solve the equation using
three approximate methods which are Hartree Fock (HF) theory, density functional theory
(DFT) and variational quantum Monte Carlo (VQMC).
17
18
Solving the Many-Electron Schrödinger equation
In this chapter we will discuss these methods in detail. First however, we introduce
the concept of the single-electron theory in Section 2.1.1 and then the variational principle
in Section 2.1.2.
2.1.1
Single-Electron Theories
HF and DFT are single-electron theories. In a single-electron theory each electron has its
own Schrödinger-like equation
1 2
− ∇i + V (r) φi (r) = i φi (r)
2
(2.1.3)
and orbital φi (r). The total electron density of the system is written as a sum over the
single-electron densities, ni (r)
n(r) =
X
ni (r) =
X
i
|φi (r)|2
(2.1.4)
i
The potential, V (r), contains a term in which electron i interacts via the Coulomb
interaction with the charge density of all of the other electrons. For example, in the
Hartree approximation (which pre-dates HF theory) we have
V (r) = vext (r) +
XZ
dr0
j6=i
nj (r0 )
|r − r0 |
(2.1.5)
The single-electron equations are solved self-consistently to obtain a set of φi (r). These
orbitals are then plugged into Eq. (2.1.4) to calculate the ground-state electron density
as predicted by the theory. The ground-state energy is obtained by inserting the orbitals
into some energy functional defined within the theory.
So, the physical picture of a single-electron theory is one in which each electron occupies
its own orbital and interacts with the other electrons only through a mean-field generated
by these electrons.
2.1.2
The Variational Principle
The variational principle [49] states that the expectation value of the Hamiltonian for any
given state will always be greater than or equal to the expectation value of the Hamiltonian
for the ground-state (i.e., the ground-state energy). I.e.:
2.2 Hartree Fock Theory
19
hΨ|Ĥ|Ψi ≥ hΨ0 |Ĥ|Ψ0 i = E0
(2.1.6)
where |Ψi is an arbitrary state ket and |Ψ0 i is the ground-state state ket. These are
normalised as hΨ|Ψi = 1. The equality holds when |Ψi = |Ψ0 i.
To prove the result, let us write |Ψi in terms of eigen-kets of the time-independent
Schrödinger equation:
|Ψi =
X
ci |Φi i
(2.1.7)
i
substituting into hΨ|Ĥ|Ψi we get:
hΨ|Ĥ|Ψi =
X
c∗i cj hΦi |Ĥ|Φj i =
ij
X
c∗i cj hΦi |Φj iEj =
ij
X
|ci |2 Ei
(2.1.8)
i
where Ei are the energies of the time-independent Schrödinger equation: Ĥ0 |Φi i = Ei |Φi i
(E0 is the ground state and E1 , E2 , etc are excited states of increasing energy). The fact
P
that the |Φi > are orthonormal means that to have < Ψ|Ψ >= 1 we need i |ci |2 = 1.
This normalisation condition and the above equation tell us that:
hΨ|Ĥ|Ψi ≥ E0
(2.1.9)
proving the variational principle.
2.2
Hartree Fock Theory
In HF theory [10, 11, 12, 13], one expresses the wavefunction as a determinant of singleparticle orbitals
φ1 (x1 ) φ1 (x2 ) ... φ1 (xN )
φ2 (x1 ) φ2 (x2 ) ... φ2 (xN )
Ψ(x1 , x2 , ..., xN ) = ..
..
.
.
φN (x1 ) φN (x2 ) ... φN (xN )
(2.2.1)
here, x includes the position and the spin. This is the simplest way of including a signchange in the wavefunction when the positions and spins of any two particles are exchanged, as is required in order to satisfy the Pauli exclusion principle:
20
Solving the Many-Electron Schrödinger equation
Ψ(x1 , ..., xi , ..., xj , ..., xN ) = −Ψ(x1 , ..., xj , ..., xi , ..., xN )
(2.2.2)
The expectation value of the Hamiltonian, hΨ|Ĥ|Ψi, where
Ĥ =
N
X
i=1
N
X
1
1
+ vext (r1 , r2 , ..., rN )
− ∇2i +
2
|ri − rj |
(2.2.3)
i6=j
is then minimised with respect to each orbital. Lagrange multipliers, Ei , are introduced
in order to ensure the orbitals are normalised.
∂
∂φ∗i
hΨ|Ĥ|Ψi −
X
!
Z
|φi |2
Ei
=0
(2.2.4)
i
where
hΨ|Ĥ|Ψi =
XZ
i
1X
δσi ,σj
2
i,j
φ∗i (r)
Z
1 2
1 X φ∗i (r)φi (r)φ∗j (r0 )φj (r0 )
− ∇ φi (r)dr +
drdr0 −
2
2
|r − r0 |
i,j
Z
φ∗i (r)φi (r0 )φ∗j (r0 )φj (r)
drdr0 +
|r − r0 |
Z X
|φi (r)|2 vext (r)dr
(2.2.5)
i
Here, σi is the spin associated with orbital i. The third term is referred to as the
exchange energy. Performing the minimisation, N so-called HF equations are obtained
[50].
1
− ∇2 +
2
Z
XZ
φ∗ (r0 )φi (r0 )φj (r)
n(r0 )
0
0 j
dr
+
v
(r)
φ
(r)
−
dr
δσi ,σj = Ei φi (r)
ext
i
|r − r0 |
|r − r0 |
j
(2.2.6)
where the electron density, n(r), is given by
n(r) = hΨ(r1 , r2 , ..., rN )|
X
δ(r − ri )|Ψ(r1 , r2 , ..., rN )i =
i
X
|φi (r)|2
(2.2.7)
i
These equations are solved to obtain the self-consistent set of φi (r). These orbitals
can then be plugged into hHiΨ in order to obtain the HF energy, which on account of the
variational principle is an upper-bound on the exact ground-state energy. Also, the HF
wavefunction, which is obtained by putting the self-consistent φi (r) into Eq. (2.2.1) is an
approximation to the true ground-state wavefunction.
2.3 Density Functional Theory
21
The physical picture of HF theory is the single-electron picture as described in Section
2.1. The various terms in Eq. (2.2.6) describe how an electron interacts with the other
electrons and the external potential. In particular, the second term is the Hartree term,
and describes the repulsion between electron i and the charge density of all of the other
electrons. Notice that the charge density of electron i is also included in this sum, but
is correctly cancelled off by the ith term of the sum in the fourth term of this equation.
This cancellation is necessary as we don’t want electron i to interact with itself. We will
see later that this correct cancellation of the electron ’self-interaction’ is not present in
the standard formulation of the local density approximation (LDA).
The third term in Eq. (2.2.6) describes the electron’s interaction with the external
potential. The fourth term is the exchange term, which has the effect of pushing samespin electrons apart from one another.
The HF energy, although qualitatively correct for many systems, is not sufficiently
accurate to make quantitative predictions. The shortcoming of the theory is in the ansatz
for the wavefunction, which does not adequately describe Coulombic correlations. In fact,
as we shall see later with VQMC, the ansatz should include factors which contain the
electron-electron separation, rij , in order to adequately describe Coulombic correlations.
In this thesis, our HF calculations will not use the self-consistency procedure outlined
above. In fact, the calculations will not strictly be HF calculations, but we will show that
they are approximately so.
The method will involve evaluating the expectation value of the Hamiltonian for a
Slater determinant of single-particle orbitals. However, these orbitals are not the selfconsistent HF orbitals described above. Instead they will be taken from LDA calculations.
These orbitals are close to the HF orbitals however, and because errors in the orbitals only
appear as the squares of these errors in the energy calculation, this small difference will not
markedly affect the energy. This procedure, which has been used before in the literature
[33], is described further in Section 2.4.13.
2.3
2.3.1
Density Functional Theory
Minimising the Energy Functional
In DFT [14], the density replaces the wavefunction as the basic variable for solving the
Schrödinger equation. Using DFT, we can calculate the ground-state density and energy
22
Solving the Many-Electron Schrödinger equation
by minimising a functional of the density. When this functional is a minimum, the value of
the functional equals the ground-state energy, and the density is the ground-state density.
The theory can also be used to calculate excited states, although we will not do so in this
thesis.
The derivation of the functional which we minimise to yield the ground-state solution
centres on two parts. First we have to prove that a given ground-state density, n0 (r), can
only be generated by a single form of the external potential vext (r) (plus some arbitrary
additive constant). This is the first Hohenberg-Kohn theorem. We then use this functional
dependence of the external potential on the ground-state density to construct an energy
functional of the density. We then prove that the functional has the properties described
above. This is the second Hohenberg-Kohn theorem.
We will now prove these two theorems. First let us write out the Hamiltonian in second
quantised form
Ĥ = T̂ + V̂ + Û
(2.3.1)
where T̂ ,V̂ and Û are the kinetic, external potential and electron-electron repulsion terms,
which are given by
Z
T̂ =
1
ψ̂ † (r) − ∇2 ψ̂(r)dr
2
Z
V̂ =
Û =
1
2
Z
ψ̂ † (r)vext (r)ψ̂(r)dr
1
ψ̂ † (r)ψ̂ † (r0 )ψ̂(r0 )ψ̂(r)drdr0
|r − r0 |
(2.3.2)
(2.3.3)
(2.3.4)
where ψ̂ † (r) and ψ̂(r) are electron creation and annihilation operators respectively. We
will now prove that the ground-state density
n0 (r) = hΨ0 |ψ̂ † (r)ψ̂(r)|Ψ0 i
(2.3.5)
is a unique functional of vext (r), i.e., n0 (r) ≡ n0 [vext (r)](r). We start with a potential,
vext (r), which has a ground-state solution, Ψ0 , and a ground-state density, n0 (r). Let us
0 (r), which has a ground-state solution, Ψ0 , gives
assume that a different potential, vext
0
rise to the same ground-state density, n0 (r). Now, Ψ00 6= Ψ0 , since they are ground-state
2.3 Density Functional Theory
23
solutions to different Schrödinger equations. Denoting the Hamiltonians as Ĥ and Ĥ 0 ,
0 (r) cases
and the ground-state energies as E0 and E00 (corresponding to the vext (r) and vext
respectively), and using the variational principle
E00 = hΨ00 |Ĥ 0 |Ψ00 i < hΨ0 |Ĥ 0 |Ψ0 i = hΨ0 |(Ĥ + Vˆ0 − V̂ )|Ψ0 i
(2.3.6)
Therefore
E00 < E0 +
Z
0
(vext
(r) − vext (r))n0 (r)dr
(2.3.7)
If we interchange the primed and un-primed quantities in Eq. (2.3.6) (and remember
that hΨ0 |ψ̂ † (r)ψ̂(r)|Ψ0 i = hΨ00 |ψ̂ † (r)ψ̂(r)|Ψ00 i = n0 (r)), we instead obtain
E0 < E00 +
Z
0
(vext (r) − vext
(r))n0 (r)dr
(2.3.8)
Adding together Eq. (2.3.7) and Eq. (2.3.8) gives
E0 + E00 < E0 + E00
(2.3.9)
0 (r) give
which shows that the initial assumption that the two potentials, vext (r) and vext
rise to the same ground-state density, n0 (r), was not correct. Hence we have shown that
the external potential (to within a constant) is a unique functional of the ground-state
density. Thus we have proved the first Hohenberg-Kohn theorem.
Furthermore, since the ground-state density is also trivially a unique functional of the
external potential, then we have established that there is a one-to-one mapping between
ground-state density and external potential:
n0 (r) vext (r) ± const
(2.3.10)
We now proceed to prove the second Hohenberg-Kohn theorem. We observe that if the
external potential is known, then this completely specifies the Hamiltonian. Therefore the
ground-state wavefunction, Ψ0 is also a functional of n0 (r). I.e. Ψ0 ≡ Ψ0 [n0 ]. Consider
the following expectation value of the Hamiltonian (where Ĥ is give by Eq. (2.3.1) for
some vext (r))
Ev [n(r)] = hΨ[n(r)]|Ĥ|Ψ[n(r)]i
(2.3.11)
24
Solving the Many-Electron Schrödinger equation
We can regard this functional as taking a density, n(r), determining the external
0 (r) for which this density is the ground-state density (which will not in
potential vext
general equal vext (r)), putting this into the Schrödinger equation in order to calculate
the ground-state wavefunction for this potential, Ψ(r), and then using this to evaluate
hΨ|Ĥ|Ψi. We can write this symbolically as
0
vext
(r)
| {z }
n(r) →
Ψ(r)
| {z }
The external potential for which
The wavefunction obtained by
n(r) is the ground-state density
0
inserting vext
(r) into
for a system of N-interacting
Eq. (2.3.3) and solving for
electrons.
the ground-state of the
→ hΨ|Ĥ|Ψi
Hamiltonian in Eq. (2.3.1).
(2.3.12)
From now on, Ψ[n(r)] can simply be read as, ’the ground-state wavefunction of a
system of N -interacting electrons, for which the ground-state density is n(r)’. However,
we must bear in mind that in order to make this mapping, there must exist an external
0 (r) for which n(r) is the ground-state density for a system of N-interacting
potential vext
electrons. We say that the n(r) must be V-representable.
Now, if we put n0 (r) into Ev , I.e. the ground-state density of a system of N -interacting
electrons in an external potential vext (r), then we obtain
Ev [n0 (r)] = hΨ0 |Ĥ|Ψ0 i = E0
(2.3.13)
where Ψ0 (r) and E0 are the ground-state wavefunction and energy of a system of N interacting electrons in an external potential vext (r).
Furthermore, if n(r) 6= n0 (r), then the wavefunction returned by Ψ[n(r)] will not equal
Ψ0 . Therefore by the variational principle we will have
Ev [n(r)] > E0
(2.3.14)
Hence we have constructed a functional, Ev [n(r)], which is minimised by the groundstate density, n0 (r), of a system of N -interacting electrons in an external potential vext (r),
and which equals the ground-state energy of this system at that point: Ev [n0 ] = E0 .
Written out fully, our functional is
2.3 Density Functional Theory
25
Ev [n(r)] = hΨ[n(r)]|T̂ |Ψ[n(r)]i + hΨ[n(r)]|Û |Ψ[n(r)]i
Z
+ vext (r)hΨ[n(r)]|ψ̂ † (r)ψ̂(r)|Ψ[n(r)]idr
Z
= T [n(r)] + U [n(r)] + vext (r)n(r)dr
(2.3.15)
In this derivation of the second Hohenberg-Kohn theorem, the density has to be Vrepresentable, otherwise the minimal property of the functional cannot be guaranteed. An
alternative derivation has the requirement that the density only has to be N-representable
[51], which places a less stringent constraint on the form that the density can take. Also,
the above derivation assumes a non-degeneracy of the ground-state solution. The proof
can easily be generalised so that this assumption need not be made.
2.3.2
The Kohn-Sham Equations
We now show how the minimisation of the functional in Section 2.3.1 can be transformed
into a problem involving N -separate equations, which must be solved in a self-consistent
manner to yield the ground-state energy and density. This is the Kohn-Sham formulation
of DFT [15].
The energy functional of Section 2.3.1 is reproduced here
1
hΨ[n(r)]|ψ̂ † (r) − ∇2 ψ̂(r)|Ψ[n(r)]idr+
2
Z
Z
1
ψ̂ † (r)ψ̂ † (r0 )ψ̂(r0 )ψ̂(r)
0
hΨ[n(r)]|
|Ψ[n(r)]idrdr + vext (r)n(r)dr
2
|r − r0 |
Z
Ev [n(r)] =
(2.3.16)
where the functional dependence of the Ψ on n(r) is described in Section 2.3.1, Eq. (2.3.12).
The first step is to re-write the functional as
Ev [n(r)] = Ts [n(r)] +
1
2
Z
n(r)n(r0 )
drdr0 +
|r − r0 |
Z
vext (r)n(r)dr + Exc [n(r)]
(2.3.17)
Here we have split off from the kinetic energy term, the ’single-particle’ kinetic energy,
Ts , which is the kinetic energy of a system of N non-interacting electrons with a groundstate density n(r). The remaining part of the kinetic energy goes into the new Exc term,
which is called the exchange-correlation energy. We have also pulled out the classical
Coulomb term from the Coulomb energy, and put the remainder of the Coulomb energy
into the Exc term.
26
Solving the Many-Electron Schrödinger equation
The single-particle kinetic energy, Ts , can be written as
1
Ts [n(r)] =
2
Z
1 2
hΨ [n(r)]| − ∇ |Ψni [n(r)]idr
2
ni
(2.3.18)
where Ψni is the ground-state wavefunction of a system of N non-interacting electrons with
a ground-state density n(r). Notice that we are still entitled to write the wavefunction
as a functional of the density, because the first Hohenberg-Kohn theorem holds for both
the interacting and the non-interacting electron case. This is because the only term we
change in the Hamiltonian to go from interacting to non-interacting is:
1
2
Z
1
ψ̂ † (r)ψ̂ † (r0 )ψ̂(r0 )ψ̂(r)drdr0 → 0
|r − r0 |
(2.3.19)
which doesn’t affect the derivation of the first Hohenberg-Kohn theorem.
P
Let us make the mathematical transformation, n(r) = i |φi (r)|2 . We will refer to
these φi (r) as ’single-particle orbitals’, for reasons which will become apparent. We now
perform a functional differentiation of Ev [n(r)] with respect to φ∗i (r), with the constraint
that the single-particle orbitals be orthonormal. In fact we will only impose the constraint that the orbitals be normalised to one, since as we will see later, the orbitals will
automatically be orthogonal to one another.
δ
∗
δφi (r)
Z
n(r)n(r0 )
0
drdr + vext (r)n(r)dr
|r − r0 |
!
X Z
+Exc [n(r)] −
Ei |φi (r)|2 dr = 0
1
Ts [n(r)] +
2
Z
(2.3.20)
i
where the Ei are lagrange multipliers arising from the normalisation constraint. Performing the differentiation:
δTs [n] 1
+
δφ∗i (r) φi (r)
Z
n(r0 )
δExc [n]
0
dr +
+ vext (r) φi (r) = Ei φi (r)
|r − r0 |
δn
(2.3.21)
where we have used the chain-rule
δ
δφ∗i (r)
=
δn(r) δ
δ
= φi (r)
∗
δφi (r) δn(r)
δn(r)
(2.3.22)
and the relations
δ
δf (r)
Z
f (r0 )g(r0 )dr0 = g(r)
(2.3.23)
2.3 Density Functional Theory
27
and
δ
δf (r)
Z
g(r0 )dr0 = 0
g(r) 6= g[f (r)](r)
where
(2.3.24)
Let us simplify the first term of Eq. (2.3.21). We know that the ground-state solution
to the non-interacting Schrödinger equation is of the form
φ1 (r1 ) φ1 (r2 )
φ2 (r1 ) φ2 (r2 )
1
ni
Ψ (r1 , r2 , ..., rN ) = √ ..
..
N
.
.
φN (r1 ) φN (r2 )
φ1 (rN ) · · · φ2 (rN ) ..
..
.
.
· · · φN (rN ) ···
(2.3.25)
and that this gives rise to the density
Z
n(r) =
dr1 dr2 ...drN
N
X
δ(r − ri )|Ψ(r1 , r2 , ..., rN )|2 =
i=1
X
|φi (r)|2
(2.3.26)
i
where we have used the fact that the single-particle orbitals are normalised to one.
Inserting this form for the wavefunction into Eq. (2.3.18) allows us to evaluate Ts
1 X 2 ni
Ts [n] = hΨ [n]|−
∇i |Ψ [n]i =
2
ni
i
Z
dr
X
i
φ∗i (r)
1 2
− ∇ φi (r)
2
(2.3.27)
Performing a functional differentiation on this term and inserting it into Eq. (2.3.21)
gives us our final result
1
− ∇2 +
2
Z
n(r0 )
0
dr + Vxc [n] + vext (r) φi (r) = Ei φi (r)
|r − r0 |
(2.3.28)
where Vxc [n] = δExc [n]/δn is called the exchange-correlation potential. These are the
Kohn-Sham equations, and the φi (r) are known as Kohn-Sham orbitals. The result has
been derived for a non spin-polarised system, but a more general derivation gives us
1
− ∇2 +
2
Z
n(r0 )
0
σ ↑ ↓
dr
+
V
[n
,
n
]
+
v
(r)
φσi (r) = Eiσ φσi (r)
ext
xc
|r − r0 |
(2.3.29)
σ [n↑ , n↓ ] = δE [n↑ , n↓ ]/δnσ .
where Vxc
xc
So we have transformed our system of N -interacting electrons into the single-electron
picture (see Section 2.1). In this picture, each electron interacts with a mean field set up
28
Solving the Many-Electron Schrödinger equation
by all of the other electrons and with the external potential, as described by the above
non-interacting Schrödinger-like equations. The total electron density is then equal to the
sum of all of the individual electron densities.
The transformation is exact, and given the correct analytic form for the exchangecorrelation energy, Exc [nσ ], we can calculate the exact ground-state density and energy.
The catch is that we don’t know the correct analytic form for the exchange-correlation
energy and so in practice have to make a guess at it (see Section 2.3.4).
As we discussed earlier, the minimisation of Ev [n(r)] was under the twin conditions
that the orbitals are normalised to one and that they are orthogonal to one another. We
imposed the former from the outset, but we did not impose the second constraint, saying
at the time that it would be automatically satisfied. We see now that this is the case,
because all orbitals are derived from the same eigenvalue equation (which has a Hermitian
operator on the left-hand side) and therefore must be mutually orthogonal.
2.3.3
Self-Consistent Solutions
In Section 2.3.2 we derived the form of the Kohn-Sham equation which we need to solve
when working in the Kohn-Sham formulation of DFT. This Schrödinger-like equation is
written
1 2
σ
− ∇ + V (r) φσi (r) = Eiσ φσi (r)
2
(2.3.30)
and the potential is given by
V σ (r) =
Z
n(r0 )
σ
dr0 + vext (r) + Vxc
(n↑ (r), n↓ (r))
|r0 − r|
(2.3.31)
where the electron density, n(r), is given by
n(r) =
N
X
|φi (r)|2
(2.3.32)
i=1
σ (n↑ (r), n↓ (r)) is an approximation to the exact exchange-correlation potential (see
Vxc
section 2.3.4).
In order to solve the Kohn-Sham equation, and therefore obtain the orbitals, we need
to know the potential, V σ (r). However this potential is a functional of the density and
therefore of the orbitals. The problem must therefore be treated self- consistently.
2.3 Density Functional Theory
29
First we guess a V ↑ (r) and V ↓ (r), and solve for the Kohn-Sham orbitals. From these
orbitals we construct new densities
nσ (r) =
X
|φσi (r)|2
(2.3.33)
i
New V σ (r) are calculated using these densities and the procedure is repeated until
convergence is achieved.
If the system is magnetic, V ↑ (r) and V ↓ (r) will converge to different values, while for
non-magnetic systems V ↑ (r) = V ↓ (r).
2.3.4
The Exchange-Correlation Energy and Potential
The Exchange-Correlation Energy
The most widely used approximation to the exchange-correlation energy functional is the
LDA, which was proposed in the original DFT paper by Hohenberg and Kohn [14].
One assumes that the exchange correlation energy density is a local quantity, and
that this energy density at a particular point is equal to the exchange-correlation energy
density of a homogeneous electron gas of the density at that point. Therefore the exchangecorrelation energy is written as
LDA
Exc
[n(r)]
Z
=
drxc (n(r))n(r)
(2.3.34)
where xc (n) is the exchange-correlation energy per electron for a homogeneous electron
gas of density n.
The local spin density approximation (LSDA) [52, 53] is a straightforward generalisation of this approximation to include spin. In this approximation we have
LSDA σ
Exc
[n (r)] =
Z
drxc (nσ (r))n(r)
(2.3.35)
where xc (nσ ) is the exchange-correlation energy per electron for a homogeneous electron
gas with electron density n↑ for spin-up electrons and n↓ for spin-down electrons. We will
use the LSDA throughout this thesis, but henceforth will simply refer to it as the LDA.
We know that the exchange energy in HF gives a good account of exchange correlations.
We therefore include this energy explicitly as part of the exchange-correlation functional.
The exchange energy is
30
Solving the Many-Electron Schrödinger equation
Ex [nσ (r)] = −
1X
2
Z
i,j,σ
φσi (r)∗ φσi (r0 )φσj (r0 )∗ φσj (r)
drdr0
|r − r0 |
(2.3.36)
and in the LSDA we find that the exchange energy per electron (writing xc = x + c ) is
"
3
x (rs , χ) = −
4π
9π
4
1
3
3
+ (21/3 − 1)
4π
9π
4
1
#
3
f (χ) /rs
(2.3.37)
where rs is defined by 43 πrs (r)3 n(r) = 1 and where χ is the spin polarisation
χ(r) =
n↑ (r) − n↓ (r)
n↑ (r) + n↓ (r)
(2.3.38)
and f (χ) is defined by
4
f (χ) =
4
(1 + χ(r)) 3 + (1 − χ(r)) 3 − 2
(2.3.39)
1
2(2 3 − 1)
The remainder of the exchange-correlation energy is referred to as the correlation energy. To calculate the correlation energy one can use exact analytic results at rs → 0
and rs → ∞ and construct an interpolation formula to connect between the two. The
Gunnarsson-Lundqvist exchange-correlation functional [54] follows this approach. Alternatively, the correlation part can be calculated by using an interpolation formula to connect
QMC results which have been calculated for 2 < rs < 100. The Perdew-Wang [55] and
Perdew-Zunger [56] functionals both follow this approach. In this thesis all three of these
exchange-correlation functionals are used.
The Perdew-Zunger functional [56] uses calculations of the correlation energy per electron for a homogeneous electron gas as calculated by Ceperley and Alder [57]. The correlation energy per electron was calculated using the diffusion QMC technique for a finite
volume system with periodic boundary conditions imposed. Calculations were performed
for various volumes, and the final correlation energy per electron was obtained by extrapolation to infinite volume.
The Exchange-Correlation Potential
The quantity which appears in the Schrödinger equation, the exchange-correlation potential, is
↑/↓
Vxc
(r)
δExc [n↑ , n↓ ] =
δn↑/↓ (r) n↓/↑
(2.3.40)
2.3 Density Functional Theory
31
Notice that when differentiating with respect to the spin-up density, the spin-down
density is held constant, and vice-versa. From here on we will drop the explicit reference
to this, but it should be remembered when reading the derivation. Eq. (2.3.35) into
Eq. (2.3.40) for spin-up electrons gives
Exc [n↑ (r0 ) + ηδ(r0 − r), n↓ (r0 )] − Exc [n↑ (r0 ), n↓ (r0 )]
η
Writing f (n↑ (r), n↓ (r)) = xc n↑ (r), n↓ (r) n(r) we get
↑
Vxc
(r)
↑
Vxc
(r)
δExc [n]
=
= lim
η→0
δn↑ (r)
1
= lim
η→0 η
Z
(2.3.41)
Z
0
↑ 0
↓ 0
dr f n (r ) + ηδ(r − r), n (r ) − dr f n (r ), n (r )
(2.3.42)
0
↑
0
0
↓
0
We now Taylor expand the function in the first integral about n↑ (r0 )
f n↑ (r0 ) + ηδ(r0 − r), n↓ (r0 ) = f n↑ (r0 ), n↓ (r0 ) +
df n↑ (r0 ), n↓ (r0 )
0
ηδ(r − r)
+ O(η 2 ) + · · ·
dn↑ (r0 )
(2.3.43)
Dropping terms containing η to powers greater than one, this gives
↑
Vxc
(r)
1
= lim
η→0 η
(Z
)
df n↑ (r0 ), n↓ (r0 )
dr ηδ(r − r)
dn↑ (r0 )
0
0
df n↑ (r), n↓ (r)
d dxc
↑
↓
=
(n
(r),
n
(r))n(r)
= xc + n(r) ↑
=
xc
↑
↑
dn (r)
dn (r)
dn (r)
(2.3.44)
An analogous expression is obtained for spin down, giving the general formula
σ
Vxc
(r) = xc + n(r)
dxc
dnσ (r)
(2.3.45)
The xc in Eq. (2.3.45) consists of an exchange and correlation part
xc n↑ (r), n↓ (r) = x n↑ (r), n↓ (r) + c n↑ (r), n↓ (r)
(2.3.46)
Hence we can write
dc
dx
σ
+ c + n(r) σ
Vxc
(r) = x + n(r) σ
dn (r)
dn (r)
{z
}
|
{z
} |
Vcσ (r)
Vxσ (r)
(2.3.47)
We will consider the exchange contribution to this potential in the next section.
32
Solving the Many-Electron Schrödinger equation
The Exchange Contribution to the Exchange-Correlation Potential
From Eq. (2.3.37) we have
#
" 1
1
dx (r)
3 9π 3
3 9π 3
1/3
f (χ)
=−
+ (2 − 1)
4π 4
dnσ (r)
4π 4
1
3
4
↑
↓
π(n + n )
−
3
1
1 1
3 9π 3 df (χ) dχ
(2 3 − 1)
rs
4π 4
dχ dnσ (r)
d
× σ
dn (r)
(2.3.48)
where
dχ d
dχ
≡
↓=
↑
↑
dn (r)
dn (r) n
dn↑
dχ
dχ d
≡
↑=
↓
↓
dn (r)
dn (r) n
dn↓
n↑ − n↓
n↑ + n↓
n↑ − n↓
n↑ + n↓
=
1
n↑ − n↓
1
−
= − (χ − 1) (2.3.49)
↑
↓
↑
↓
2
n
n +n
(n + n )
=−
1
n↑ − n↓
1
−
= − (χ + 1) (2.3.50)
↑
↓
↑
↓
2
n
n +n
(n + n )
hence
(χ ∓ 1)
dχ
=−
n
dn↑/↓
(2.3.51)
Therefore we have
1
1
dx (r)
1
1
3 9π 3 df (χ) 1
= x (r)
+ (χ ∓ 1)(2 3 − 1)
dnσ (r)
3
n(r)
4π 4
dχ rs n(r)
(2.3.52)
Therefore the exchange part of the exchange-correlation potential is
Vxσ (r)
2.3.5
1
1
3 9π 3 df (χ) 1
4
= x (r) + (χ ∓ 1)(2 3 − 1)
dχ rs
4π 4
3
(2.3.53)
Self-Interaction Correction
The energy functional which is minimised in DFT (Section 2.3.2, Eq. (2.3.17) ) contains
a repulsive Coulomb term
1
U [n] =
2
Z
Z
dr
dr0
n(r)n(r0 )
|r0 − r|
Splitting the density into orbital spin densities using
(2.3.54)
2.3 Density Functional Theory
33
n(r) =
X
nσα (r) =
α,σ
X
|φσα (r)|2
(2.3.55)
α,σ
(where α label the orbitals), gives
1
U [n] =
2
X Z
α,σ,α0 ,σ 0
Z
dr
σ
σ0 0
0 nα (r)nα0 (r )
dr
|r0 − r|
(2.3.56)
Notice that this term contains the interaction of a given orbital spin charge density
with itself. This self-interaction is physically spurious, and should not be present. In fact,
if the exact exchange-correlation energy were known, then these self-interaction terms
would cancel exactly with terms in the exchange-correlation energy
1
2
|
Z
Z
dr
nσ (r)nσ (r0 )
exact σ
dr0 α 0 α
+Exc
[nα , 0] = 0
|r − r|
{z
}
(2.3.57)
=U [nσ
α]
The exchange-correlation energy is approximated however, and so this cancellation is
not exact. In the self-interaction correction (SIC) scheme [24], we add extra terms to the
energy functional in order to make the cancellation exact. The energy functional in this
scheme is
Z
Ev [n] = Ts [n] +
vext (r)n(r)dr + U [n] + Exc [n↑ , n↓ ] −
X
(U [nσα ] + Exc [nσα , 0]) (2.3.58)
α,σ
Minimisation of the energy functional yields a Schrödinger equation with an orbital
dependent potential
1 2
α,σ
− ∇ + VSIC (r) φσα (r) = Eασ φσα (r)
2
α,σ
VSIC
(r)
σ
Z
= V (r) −
nσα (r) 0
σ
dr − Vxc
(nσα (r), 0)
|r0 − r|
(2.3.59)
(2.3.60)
The fact that the potential is now orbital dependent means that, unlike in the LDA,
the orbitals are no longer orthogonal to one another. We need the orbitals to be orthogonal
to one another, as this is specified in the Kohn-Sham theory. I.e. we require
Z
φσα ∗ (r)φσα0 (r)dr = 0
(2.3.61)
34
Solving the Many-Electron Schrödinger equation
for all α and α0 .
One approach would be to re-derive the Kohn-Sham equations using additional Lagrange multipliers which force orbitals to be orthogonal to one another [58]. Alternatively
we can make the orbitals orthogonal to one another by hand after each iteration in the
self-consistency cycle. We take the latter, simpler approach, and we use the Gram-Schmidt
orthogonalisation [59] method for this.
The method of Gram-Schmidt orthogonalisation is most naturally applied to systems
consisting of a discrete set of states. The first step in the procedure is to orthogonalise
the state with the second lowest energy against the state with the lowest energy:
φσ,orth
(r)
E2
=
1
N (1)σ
φσE2 (r)
Z
−
0 σ
0
0 σ
φσ∗
E1 (r )φE2 (r )dr φE1 (r)
(2.3.62)
where the energy eigenvalues, Ei , are used as labels for the orbitals, and where E1 < E2 .
R
The normalisation factor N (1)σ ensures that |φσ,orth
(r)|2 dr = 1. The two states are
E2
now orthogonal to one another, as can be seen by pre-multiplying with φσ∗
E1 (r), and then
integrating over r to give zero. The next step is to orthogonalise the state with the third
lowest energy against the first two states:
φσ,orth
(r)
E3
=
1
N (2)σ
Z
φσE3 (r)
Z
−
0 σ
0
0 σ
φσ∗
E1 (r )φE3 (r )dr φE1 (r)−
φσ,orth∗
(r0 )φσE3 (r0 )dr0 φσ,orth
(r)
E2
E2
where E1 < E2 < E3 and the normalisation factor N (2)σ ensures that
(2.3.63)
R
|φσ,orth
(r)|2 dr = 1.
E3
In this way, we now have a set of three orbitals which are all orthogonal to each other.
This procedure is repeated until all orbitals are orthogonal to one another.
2.4
2.4.1
Variational Quantum Monte Carlo
A Variational Theory
The VQMC method [17, 18] is as follows. Given an N particle system, we choose some trial
wavefunction ΨT (R) (where R contains the set of vectors {r1 , r2 , ..., rN }) and calculate
hΨT |Ĥ|ΨT i
(2.4.1)
2.4 Variational Quantum Monte Carlo
35
A Monte Carlo method is then used to calculate this integral, i.e. one for which random
numbers form an intrinsic part of the algorithm. The method is described in Section 2.4.2
By the variational principle we have
hΨT |Ĥ|ΨT i ≥ E0
(2.4.2)
So, just as with HF theory, one could vary ΨT (R) in order to minimise the expectation
value hΨT |Ĥ|ΨT i and quote this as an upper bound to the exact ground-state energy. In
VQMC one actually uses a slightly different procedure, as we will discuss later in Section
2.4.3.
2.4.2
The Monte Carlo Technique
Given an integral
Z
b
f (x)dx
(2.4.3)
a
a non-analytic method of calculating the integral is to use the Monte Carlo approach. In
this approach we have
Z
n
b
f (x)dx ≈
a
(b − a) X
f (xi )
n
(2.4.4)
i=1
where n is large and where the xi are taken from a uniform probability distribution in the
range a to b.
Importance Sampling
If we were to use the above method directly, then for many integrals the number of terms
in the sum required to calculate the integral to a given accuracy would be too large to
make this a useful method. This would in particular be the case for multi-dimensional
integrals.
To improve the efficiency of the method, one has to use importance sampling. This is
where we sample xi (or xi in more than one dimension) not from a uniform probability
distribution but from a distribution that is weighted preferentially in regions where f (x)
is large. To see how this works we write
36
Solving the Many-Electron Schrödinger equation
Z
b
Z
f (x)dx =
a
a
b
n
f (x)
1 X f (xi )
dx =
|n→∞
g(x)
g(x)
n
g(xi )
(2.4.5)
i
In this case the xi in the sum are sampled from the probability distribution, g(x) and
R
g(x) is normalised as, g(x)dx = 1. g(x) must be positive everywhere in order for it to
be used as a probability distribution. The best choice of g(x) (in terms of reducing the
number of terms we need in the summation) is |f (x)|.
2.4.3
The Variational Quantum Monte Carlo Method
Returning to VQMC, we use the Monte Carlo technique described above to calculate
hΨT |Ĥ|ΨT i. Written out fully, we want to calculate:
1
hΨT |Ĥ|ΨT i =
N
Z
Ψ∗T (R)ĤΨT (R)dR
(2.4.6)
R
where N = |ΨT (R)|2 dR. Notice that we have re-written the trial wavefunction: ΨT (R) →
√
ΨT (R)/ N so that it is automatically normalised to one. In this definition the ΨT itself
need not be normalised. This 3N -dimensional integral is calculated using the Monte Carlo
technique, but first has to be re-written in such a way so that we can introduce importance
sampling
1
N
Z
Ψ∗T (R)ĤΨT (R)dR =
Z
|ΨT (R)|2 ĤΨT (R)
dR
N
ΨT (R)
(2.4.7)
we calculate this integral by turning it into a sum:
Z
n
1 X ĤΨT (R)
|ΨT (R)|2 ĤΨT (R)
dR ≈
= Ē
N
ΨT (R)
n
ΨT (R)
(2.4.8)
i=1
where n is large. The values of R in the summation are taken from the probability
distribution |ΨT (R)|2 /N . We will refer to these Rs as configurations. The quantity being
summed over, (ĤΨT )/ΨT , is known as the local energy.
The method then is to generate a set of configurations according to the probability distribution |ΨT (R)|2 /N and then calculate the local energy for each of these configurations.
The mean average of the local energy is then quoted as an upper bound to the ground√
state energy. The error on this upper bound is approximately σl.e. / n − 1, where σl.e. is
the standard deviation of the local energy (see Section 2.4.11 for a discussion of why the
error is not exactly equal to this quantity). One can therefore calculate this upper bound
2.4 Variational Quantum Monte Carlo
37
to as high a level of accuracy as is required by increasing the number of configurations.
To obtain an approximation to the true ground-state wavefunction one could then vary
ΨT (R) in order to minimise this upper bound on the ground-state energy.
In practice, we do not minimise this upper bound but instead minimise σl.e. [60, 61],
which is given by the equation
n
2
σl.e.
1X
=
n
i=1
ĤΨT (Ri )
− Ē
ΨT (Ri )
!2
(2.4.9)
To see how this works, consider the Schrödinger equation for the ground-state wavefunction
ĤΨ0 (R) = E0 Ψ0 (R)
(2.4.10)
We see that the local energy for the exact ground-state wavefunction is, E0 , i.e. a
constant. The standard deviation of the local energy is therefore zero for the exact groundstate wavefunction. The standard deviation of the local energy will not be zero for some
arbitrary trial wavefunction and so by varying the wavefunction in order to minimise the
standard deviation one therefore has a procedure for getting closer to the exact groundstate wavefunction.
One reason why this method is preferable to minimising the upper bound on the
ground-state energy is that there is a known lower bound to this standard deviation, i.e.
zero, which gives one a better gauge as to how close one is to the ground-state solution.
We will discuss a second reason in Section 2.4.10.
2.4.4
Metropolis Algorithm
We now need a method which generates configurations distributed according to the probability distribution |ΨT (R)|2 /N (where R is the set of electron position vectors, {r1 , · · · , rN }).
The Metropolis algorithm [29] is used for this purpose.
In the Metropolis algorithm, we start with a given particle configuration R. We
term this set of 3N coordinates a ’walker’. In the algorithm’s simplest form, a random
move is then proposed from a probability distribution, Ptrial (R → R0 ), taking the walker
from configuration R to configuration R0 . This R0 can be any other configuration in the
3N -dimensional space. This move is then either accepted or rejected according to some
acceptance probability Pacceptance (R → R0 ). Another move is then made.
38
Solving the Many-Electron Schrödinger equation
In order that the configurations generated this way correctly sample the probability
distribution, P (R) (= |ΨT (R)|2 /N in our case), we need the following relationship to be
satisfied:
P (R0 )
P (R → R0 )
=
P (R0 → R)
P (R)
(2.4.11)
where P (R → R0 ) is the total probability of a move taking place from R to R0 , i.e.:
P (R → R0 ) = Ptrial (R → R0 )Pacceptance (R → R0 )
(2.4.12)
P (R0 → R) = Ptrial (R0 → R)Pacceptance (R0 → R)
(2.4.13)
We also have
If we choose our Ptrial such that Ptrial (R → R0 ) = Ptrial (R0 → R), then dividing
Eq. (2.4.12) by Eq. (2.4.13) and using Eq. (2.4.11), we find
Pacceptance (R → R0 )
P (R0 )
=
P (R)
Pacceptance (R0 → R)
(2.4.14)
A form for the acceptance probability which satisfies this equation, and the one that
we use is
P (R0 )
Pacceptance (R → R ) = min 1,
P (R)
0
(2.4.15)
At first, the configurations generated using this method will not reflect the probability
distribution that we’re trying to sample. However after a large number of moves have been
made, the Metropolis algorithm will begin correctly sampling the probability distribution.
At this point we say the random walk has reached equilibrium.
In the algorithm as described above, a walker can move from a given configuration R
to any other configuration R0 in a single move. It can be proven straightforwardly that
the Metropolis algorithm is still valid if this is not the case, provided that it is possible in
a finite number of moves for any configuration to be reached from any other configuration.
If this is the case, we say the random walk is ergodic.
In our implementation, we do not move all of the electrons in the configuration at once,
but instead use an electron-by-electron approach whereby each electron is moved one at a
time. One electron in the configuration (electron i) is chosen and displaced according to
2.4 Variational Quantum Monte Carlo
39
a probability distribution, Ptrial (r → r0 ), centred on that particle (the form for which is
discussed in the next section) . The move is then accepted or rejected according to
P (r0 )
Pacceptance (r → r ) = min 1,
P (r)
0
(2.4.16)
Each electron in the configuration is then given the opportunity to move in this manner,
until all electrons have been cycled through. At this point a ’configuration move’ has been
completed. The process is then repeated starting again with the first electron.
In this implementation, it is not possible for all configurations to be reached in a single
move. However, after a finite (albeit very large) number of moves, any configuration can
be reached, and so the Metropolis Algorithm is still valid.
2.4.5
Equilibration and Serial Correlation
Once the configurations have been generated by the Metropolis algorithm, we are ready
to evaluate the local energy for these configurations. However we must be careful on two
fronts. Firstly as mentioned earlier, we must allow the Metropolis algorithm to equilibrate
before we start using configurations for local energy calculation. In practice, and depending
on the number of electrons in the system, this requires us to throw away the first 1000 or
so configurations.
The second consideration is that once equilibration is complete, we must be sure to use
configurations that are sufficiently far apart from one another that they are statistically
independent. That is, we need to avoid serial correlation in the local energy, and also in
other measured quantities such as the electron density (see Section 2.4.12). This means
calculating the measured quantities only for every ncorr th configuration, and throwing the
other configurations away.
One should choose Ptrial (r → r0 ) in order to minimise this ncorr , and thereby improve
the efficiency of the algorithm. For our purposes, we will not worry too much about using
the most efficient form for Ptrial (ri , r) (see [62] for a more efficient form). In fact we will
use a very simple probability distribution, namely a box surrounding the electron:
Ptrial (r → r0 ) =













x − L/2 < x0 < x + L/2
1
L3
if y − L/2 < y 0 < y + L/2
z − L/2 < z 0 < z + L/2
0
otherwise
(2.4.17)
40
Solving the Many-Electron Schrödinger equation
We vary L in order to minimise ncorr . This is achieved by optimising the average
probability of the acceptance of a move. If L is too small, then this average acceptance
probability is very high, but because the moves are small, there is strong serial correlation.
Conversely, if L is too large then too few moves are accepted, and again we have strong
serial correlation. It turns out that the optimum choice for L corresponds to an acceptance
probability ≈ 0.4.
With this choice, we find that for a 10 electron system, a choice of ncorr = 50 (5
complete configuration moves) is usually enough to remove serial correlations.
Note that we chose the electron-by-electron algorithm described in the previous section
in favour of the full configuration move algorithm because this algorithm allows a smaller
choice for ncorr .
2.4.6
The Choice of the Trial Wavefunction
The effectiveness of the VQMC method depends entirely on the quality of the trial wavefunction. The wavefunction must have the correct symmetry under particle exchange and
must contain as much of the physics of the system as possible.
Since we are concerned with electrons, the trial wavefunction must be anti-symmetric
under exchange of any two electrons. The simplest wavefunction which has this feature is
the Slater determinant:
φ1 (r1 ) φ1 (r2 ) ... φ1 (rN )
φ2 (r1 ) φ2 (r2 ) ... φ2 (rN )
ΨT (r1 , · · · , rN ) = ..
..
..
..
.
.
.
.
φN (r1 ) φN (r2 ) ... φN (rN )
(2.4.18)
where φi (rj ) are single-particle orbitals, and N is the total number of electrons. Provided
we are not calculating the expectation values of spin-dependent operators, then we can
separate this determinant into a spin-up and spin-down part:
ΨT (r1 , · · · , rN ) = D↑ (r1 , · · · , rN/2 )D↓ (rN/2+1 , · · · , rN )
where
(2.4.19)
2.4 Variational Quantum Monte Carlo
41
φ1 (r1 )
φ1 (r2 )
φ2 (r2 )
φ2 (r1 )
D↑ (r1 , · · · , rN/2 ) = ..
..
.
.
φN/2 (r1 ) φN/2 (r2 )
φ1 (rN/2 ) · · · φ2 (rN/2 ) ..
..
.
.
· · · φN/2 (rN/2 ) ···
(2.4.20)
and
φN/2+1 (rN/2+1 ) φN/2+1 (rN/2+2 )
φN/2+2 (rN/2+1 ) φN/2+2 (rN/2+2 )
↓
D (rN/2+1 , · · · , rN ) = ..
..
.
.
φN (rN/2+1 )
φN (rN/2+2 )
···
···
..
.
···
φN/2+1 (rN ) φN/2+2 (rN ) (2.4.21)
..
.
φN (rN ) where we have defined orbitals 1 to N/2 to be spin-up and orbitals N/2 + 1 to N to
be spin-down. Making this separation into spin-up and spin-down parts speeds up the
calculation of the trial wavefunction within the code.
To improve the trial wavefunction, we should try to include effects due to the Coulombic correlation between electrons. This can be achieved by making the wavefunction small
whenever any two electrons get close to one another. The following wavefunction incorporates this:

ΨT (r1 , · · · , rN ) = exp −

X
1≤i<j≤N
arij  ↑
D (r1 , · · · , rN/2 )D↓ (rN/2+1 , · · · , rN )
1 + brij
(2.4.22)
The new term is called the Jastrow factor [63]. The parameter, a, is fixed by the
electron-electron cusp condition (see later) and b is a parameter which can be varied in
order to minimise the standard deviation of the local energy.
This form for the Jastrow factor is suitable for calculations of atoms [60], provided the
orbitals in the Slater determinant satisfy the nuclear cusp condition (see later). We use
a more elaborate form of trial wavefunction when we discuss our calculations of atoms in
jellium. This is described in detail in Section 4.2.1.
2.4.7
Updating the Slater Determinants
For each move, we have to calculate the acceptance probability of an electron i moving
from a position ri to a position r0i (Eq. (2.4.15) ). This means calculating the quantity
42
Solving the Many-Electron Schrödinger equation
σ |2
|ΨT (r1 , r2 , ..., r0i , ..., rN )|2
|Dnew
≈
σ |2
|ΨT (r1 , r2 , ..., ri , ..., rN )|2
|Dold
(2.4.23)
The Slater determinant Dσ (where σ is the spin of electron i) therefore needs to be
calculated for the trial configuration in which electron i has been moved to the position
r0i . Dσ will already have been calculated for the case where the electron is at ri from
the previous move. Because only one electron has moved, the determinant will only
change by the elements of one column. Therefore it would be wasteful to re-calculate the
whole determinant. In fact there is an algorithm which allows the efficient updating of a
determinant for the case where only one column has been changed [31].
In fact, the method updates the inverse of the transpose of the determinant according
to
σ T −1
(Dnew
)jk
=

σ T )−1 /q σ


(Dold

jk

if k = i

hP
i


N
σ T )−1 φ (r0 ) /q σ if k 6= i
 (Dσ T )−1 − (Dσ T )−1
(D
l=1
old jk
old ji
old lk l i
N
N
j=1
j=1
X
X
Dσ
σ T −1
σ
σ T −1
q = new
=
(D
)
(D
)
=
(Dold
)ji φj (r0i )
ji
new
old ji
σ
Dold
σ
(2.4.24)
where the old and new labels on Dσ denote whether the determinant has been calculated
with electron i in position r or r0 . For the first move, (DσT )−1 must be calculated explicitly.
From then on however it is updated using the above equations. Crucially, the quantity
σ |2 /|D σ |2 , in order to evaluate the acceptance probability, is
we need to calculate, |Dnew
old
automatically generated by this updating algorithm, and is q σ2 .
2.4.8
Calculating the Local Energy
The local energy (ĤΨT )/ΨT , is to be calculated for configurations R which sample the
probability distribution |ΨT (R)|2 /N . The Hamiltonian has the form:
X 1 Ĥ =
− ∇2i + V (r1 , r2 , · · · , rN )
2
(2.4.25)
i
The potential energy part of the local energy is just V (r1 , r2 , · · · , rN ). The kinetic
energy part, − 21 ∇2i ΨT /ΨT , however, requires a little thought. The Jastrow factors in ΨT
2.4 Variational Quantum Monte Carlo
43
means that our calculation of the local kinetic energy will be more accurate if we first take
the logarithm of ΨT . Calculating grad and grad squared of ln ΨT gives us
∇i ΨT
ΨT
∇i ΨT 2
∇2 ΨT
−
∇2i ln ΨT = i
ΨT
ΨT
∇i ln ΨT =
∴−
1 ∇2i ΨT
1
1
= − ∇2i ln ΨT − (∇i ln ΨT )2
2 ΨT
2
2
Introducing the quantities Ti = − 14 ∇2i ln ΨT and Fi =
√1 ∇i ln ΨT ,
2
(2.4.26)
the local kinetic
energy can be written
X 1 ∇2 ΨT X
i
−
=
2Ti − Fi2
2 ΨT
i
(2.4.27)
i
As an aside, the quantities Ti and Fi are useful numerically. If we average these
quantities over a large number of configurations, then we can write
Z
1X 2
1
Ti = −
∇i ln ΨT ≡ −
drΨ2T ∇2i ln ΨT
4
4
i
i
Z
X
1X
1
2
2
Fi =
(∇i ln ΨT ) ≡
drΨ2T (∇i ln ΨT )2
2
2
X
i
(2.4.28)
(2.4.29)
i
Then using ∇i ln ΨT = ∇i ΨT /ΨT and integration by parts:
−
1
4
Z
drΨ2T ∇2i ln ΨT = −
1
4
Z
drΨT ∇2i ΨT +
1
4
Z
dr(∇i ΨT )2
Z
Z
Z
1
1
1
1
2
2
= [− ΨT ∇i ΨT ] − −
dr(∇i ΨT ) + dr (∇i ΨT ) =
dr(∇i ΨT )2
4
4
4
2
(2.4.30)
and
1
2
Z
drΨ2T (∇i ln ΨT )2
1
=
2
Z
dr(∇i ΨT )2
(2.4.31)
Hence when summed over a large number of configurations, n, we have:
X
i
Or dividing through by n:
Ti =
X
i
Fi2
(2.4.32)
44
Solving the Many-Electron Schrödinger equation
< Ti >=< Fi2 >
(2.4.33)
Making sure these quantities are equivalent to one another within statistical error, is
a useful test when de-bugging code.
2.4.9
Cusp Conditions
Let us consider the local energy for a system of electrons in the presence of an ion of
charge Z. This local energy will contain the following terms
1
1
1
Z
Z
− ∇2ri − ∇2rj +
− −
2
2
rij
ri rj
ΨT (r1 , ..., ri , ..., rj , ..., rN )/ΨT (r1 , ..., ri , ..., rj , ..., rN )
(2.4.34)
where particles i and j are arbitrarily chosen particles.
In order to optimise our trial wavefunction (and thereby bring it closer to the true
ground-state wavefunction) we need to minimise the standard deviation of this quantity.
We therefore wish to avoid any divergences in this quantity. As we have written it, one
such divergence occurs as ri or rj approach zero. We can prevent this divergence by
designing the trial wavefunction so that the kinetic energy term diverges in the opposite
sense to the divergent Coulombic term whenever ri or rj approaches zero. If the trial
wavefunction is designed in this way then it is said to satisfy the nuclear cusp condition
[64, 65].
Similarly, when rij → 0 the repulsive Coulomb energy diverges. Also, if the electrons i
and j have the same spin, then as rij → 0, the determinental part of the wavefunction on
the denominator tends to zero, providing another origin for divergence. To prevent this,
the trial wavefunction is constructed so that again, the kinetic energy term diverges in the
opposite sense to the Coulomb term. If the wavefunction is designed in such a way, then
it is said to satisfy the electron-electron cusp condition [64, 65].
Nuclear Cusp Condition
To prevent the local energy blowing up as ri → 0, we need
1
−
2
2 ∂
ri ∂ri
Z
−
ri
ΨT (r1 , ..., ri , ..., rN )|ri →0 = 0
(2.4.35)
2.4 Variational Quantum Monte Carlo
45
We have omitted the angular parts of the Laplacian, and also the second order differential operator of the radial part of the Laplacian, as these terms do not diverge as
ri → 0. The differential operator will act on the orbitals in the determinental part of the
wavefunction, namely φ1 (ri ), ..., φN (ri ), and so we require that
−
1
2
2 ∂
ri ∂ri
Z
ri
−
φ(ri )|ri →0 = 0
(2.4.36)
The nuclear cusp condition is therefore
∂φj (ri )
|ri →0 = −Z
∂ri
(2.4.37)
In our calculations, the orbitals we will be using will be the LDA orbitals calculated
within the atom in jellium model. These orbitals automatically satisfy Eq.( 2.4.37).
Electron-Electron Cusp Condition
The equation which must be satisfied in order to stop the local energy diverging as rij → 0
is
1
1
1
− ∇2ri − ∇2rj +
2
2
rij
ΨT (r1 , ..., ri , ..., rN )|rij →0 = 0
(2.4.38)
Our first step in deriving the electron-electron cusp condition is to make the following
change of variables
r = ri − rj
R=
ri + rj
2
(2.4.39)
It is easy to show (use Cartesian coordinates) that
1
1
1
− ∇2ri − ∇2rj = −∇2r − ∇2R
2
2
4
(2.4.40)
Now, with a wavefunction of the form
ΨT = exp(−u(r))D↑ D↓
(2.4.41)
where
u(r) =
ar
1 + br
(2.4.42)
46
Solving the Many-Electron Schrödinger equation
we must satisfy
−∇2r [exp(−u(r))D↑ D↓ ] + 1r
|r→0 → 0
exp(−u(r))D↑ D↓
(2.4.43)
Therefore
(
∂ 2 u(r)
+
−
∂r2
∂u
∂r
2
2 ∂u
∂u ∇D↑ D↓ ∇2 D↑ D↓ 1
−
− 2 r̂. ↑ ↓ +
+
r ∂r
∂r D D
r
D↑ D↓
)
|r→0 → 0
(2.4.44)
where we have dropped terms that are not divergent. For anti-spin electrons, D↑ D↓ will
not in general be zero, and so the only terms which blow up in the above expression are
the third and sixth. Therefore, using that
∂u
∂r |r→0
→ a , we need a = − 12 .
For same-spin electrons, we use that fact that, due to the Pauli principle, Dσ (r) =
ar + O(r3 ) (where σ is the spin of the two electrons). Therefore, in addition to the third
term, the fourth term also diverges. We see that in this case we need a = − 41 .
2.4.10
Correlated Sampling
When minimising the standard deviation of the local energy with respect to the Jastrow
parameters, one encounters the problem that the standard deviation does not approach a
minimum in a smooth manner. There is a large statistical noise associated with the fact
that for each different set of Jastrow parameters, we are using a new set of configurations
from which to calculate the local energies. We can get around this problem by using
the same set of configurations to calculate the local energies for all choices of Jastrow
parameters, a technique known as correlated sampling [17, 66, 67]. Essentially this allows
the calculation of the difference in the standard deviation between two or more sets of
Jastrow parameters to a greater accuracy than the calculation of the standard deviation
itself.
Recall from Section 2.4.3 that the expectation value of the Hamiltonian is calculated
as
Z
< ΨT |Ĥ|ΨT >=
n
|ΨT (R)|2 ĤΨT (R)
1 X ĤΨT (R)
dR ≈
N
ΨT (R)
n
ΨT (R)
(2.4.45)
i=1
where N =
R
|ΨT (R)|2 dR. Here, one calculates the local energy for a set of configurations,
Ri , which sample the probability distribution |ΨT (R)|2 /N . One then calculates the mean
average of these local energies.
2.4 Variational Quantum Monte Carlo
47
Let us introduce a set of Jastrow parameters, {α0 }, from which we generate configu(0)
rations, Ri . Then for a ΨT which has a different set of Jastrow parameters, {α}, we can
write
R
< ΨT |Ĥ|ΨT >=
T (R)
|ΨT (R)|2 ĤΨ
ΨT (R) dR
R
=
|ΨT (R)|2 dR
Pn ≈
|ΨT (R)|2
N (0)
J(R)
J (0) (R)
(0)
R
|ΨT (R)|2
N (0)
2
ĤΨT (R)
ΨT (R) dR
2
J(R)
J (0) (R)
dR
2
ĤΨT (Ri )
ΨT (Ri )
Pn J(Ri ) 2
i=1 J (0) (Ri )
i=1
J(Ri )
J (0) (Ri )
(0)
R
= Ē
(2.4.46)
where J is the Jastrow factor, the (0) superscript denotes a quantity calculated using the
Jastrow parameters, {α0 }, and where the configurations in the final step are generated
from a wavefunction with the Jastrow parameters, {α0 }.
Similarly the variance of the local energy can be written
Pn 2
σl.e.
=
i=1
J(Ri )
J (0) (Ri )
2 h
ĤΨT (Ri )
ΨT (Ri )
Pn J(Ri ) 2
i=1 J (0) (Ri )
− Ē
i2
(2.4.47)
The procedure for minimising σl.e. using correlated sampling is as follows. We guess a
set of Jastrow parameters, and then generate a set of configurations using the Metropolis
algorithm. These parameters are then the {α0 } parameters in the above description of
correlated sampling. A new set of parameters, {α}, are then chosen and σl.e. is calculated
using Eq. (2.4.47). An unconstrained minimisation algorithm (I.e. one which does not
require derivatives of the quantity being minimised) is used to vary {α} in order to minimise σl.e. . The algorithm we use is the E04CCF subroutine from the NAG library [68].
Once an optimal set of {α} have been obtained using this algorithm, we generate new
configurations using these optimal Jastrow parameters and re-calculate σl.e. (for which
we can use Eq. (2.4.9), since there is no correlated sampling for this calculation), which
(1)
is recorded as σl.e. . The optimisation procedure is then repeated using these optimised
Jastrow parameters as the new {α0 } for the correlated sampling.
The optimisation procedure is repeated three or four times. The optimisation run
(i)
which produced the lowest value of σl.e. corresponds to the solution which is the closest to
the true ground-state solution. The energy and the electron density from this optimisation
run are then quoted as the VQMC results for the system under consideration.
48
Solving the Many-Electron Schrödinger equation
It is important to use a large enough number of configurations (n in the above equa(i)
tions) when performing the correlated sampling. This way, the total energy and σl.e.
obtained in different optimisation runs will be similar to one another. If too few configurations are used, then there can be a large difference in these quantities over different
optimisation runs.
(i)
One of the benefits of minimising σl.e. as opposed to the total energy is that fewer
(i)
configurations are needed in order to get steady values of σl.e. and the total energy over
adjacent optimisation runs.
2.4.11
Blocking Analysis to Calculate Error on Mean
Recall that once we have optimised our trial wavefunction, we quote the mean of the local
energies as our VQMC energy. The error of this energy is therefore the error on the mean
of the local energies. If the local energies were serially un-correlated then this error would
be
σ
√ l.e.
n−1
(2.4.48)
where σl.e. is the standard deviation of the local energies and n is the number of local
energies over which the average is taken. However, even though we have worked hard to
remove serial correlation (see Section 2.4.5), some serial correlation will still remain.
In order to remove the effect of the serial correlation from the error on the mean, we
use the ’blocking method’ [69]. In this method, if we have n values, xi , to average over
(local energies in our case), then we perform a series of transformations on these values
according to:
(j)
xi
1 (j−1)
(j−1)
= (x2i−1 + x2i )
2
(2.4.49)
(j)
(0)
where j is the transformation number, i runs from 1 to n/2j (= ntot ) and xi
this way the number of values of
(j)
xi
= xi . In
halves after each transformation. Notice that the
(j)
mean, x̄, of the xi remains the same after each transformation. If σl.e. is the standard
(j)
deviation of the xi
is
(j)
at transformation number j, then the error on the mean of these xi
PSfrag replacements
2.4 Variational Quantum Monte Carlo
49
Error on Mean of the Local Energy
0.0032
0.00315
0.0031
0.00305
0.003
0.00295
0.0029
2
0
4
6
8
Transformation Number
12
10
Figure 2.1: Re-blocking analysis for hydrogen immersed in a 10-electron jellium sphere of
density 0.03a−3
B . The error on the mean levels off at just under 0.0031eV and therefore
this is the error we quote on the total energy.
(j)
σmean
=q
(j)
σl.e.
(j)
ntot − 1


=
1
2
(j)
ntot
1 X
1
(j)
(j)
(ntot − 1) ntot
(j) 2
(xi

− x̄2 )
(2.4.50)
i=1
There is an error attached to this error on the mean which is approximated as
2
(j)
ntot − 1
!1
4
(j)
σmean
(2.4.51)
In order to use this blocking technique to calculate the true error on the mean of the
data points xi , we plot the error on the mean as a function of transformation number. We
find that the error on the mean increases to begin with, and then reaches a plateau. It is
the value of the error on the mean at the plateau that one reads off as the true error on
the mean. A typical example of this plot (taken from our own results reported in Chapter
4) is shown in Fig. 2.1.
50
Solving the Many-Electron Schrödinger equation
2.4.12
Calculating the Probability Density
We wish to calculate the probability density, n(r), for finding an electron at a position r:
Z
n(r) =
dr1 dr2 ...drN
N
X
δ(r − ri )
i=1
|ΨT (r1 , r2 , ..., rN )|2
N
(2.4.52)
In fact, because the electrons sample the probability distribution |ΨT (r1 , ..., rN )|2 /N
via the Metropolis algorithm, the VQMC method has a built-in method for doing this. One
partitions space into small boxes of volume Vi , each centred on an ri , and then counts the
number of electrons, Ni , that enter each of these boxes over the course of the simulation.
The (discretised) probability density at a point r is then
Ni /Vi
n(r) = ni = P
j Nj
(2.4.53)
where the position vector r lies within the box centred at ri , and where ni is normalised
P
so that i ni Vi = 1.
Later on when we apply VQMC to systems with atoms and jellium spheres we will
be interested in calculating the radial probability density. In this case the ’boxes’ are
spherical shells, with box i lying in the region between radii ri−1 and ri , with r0 = 0.
Therefore the above equation becomes
n(r) = ni =
4
3
3 π(ri
Ni
P
− ri−1 3 ) j Nj
(2.4.54)
where Ni is the number of electrons that have entered shell i over the course of the
simulation.
It is important that these shells are small enough that the density is accurately calculated. However, they must not be so small that only a few electrons enter them over the
simulation run. If this were the case then the statistical noise in the probability density
would be too large.
2.4.13
HF Calculations
In Section 2.2 we discussed how our HF calculations will not mirror the exact self-consistent
procedure outlined in that section. Instead we said that we would evaluate the expectation
value of the Hamiltonian using the wavefunction of a Slater determinant of LDA orbitals.
We see now that this can be achieved within the framework of VQMC. One simply uses
2.4 Variational Quantum Monte Carlo
51
the same procedure as for VQMC, except that the wavefunction has no Jastrow part, and
there is no minimisation of σl.e. . In this way one obtains an expectation value of the energy
which is approximately equal to the HF energy.
52
Solving the Many-Electron Schrödinger equation
Chapter 3
An Atom in Infinite Jellium
Solved using DFT
In this chapter, the system of an atom in infinite jellium is solved within the Kohn-Sham
formulation of density functional theory (DFT), for which the local density (LDA) and
self-interaction correction (SIC) approximations are used.
Sections 3.1, 3.2, 3.3 and 3.4 develop the DFT for the purposes of solving the system
of an atom in infinite jellium. This theoretical background is also relevant to the DFT
results presented in the next chapter (which are for an atom in a finite jellium sphere).
In Section 3.4, results are presented for immersion energies across the first three rows of
the periodic table. The effective medium theory (EMT) is derived in Section 3.5, and our
calculations of the Wigner-Seitz radii using this theory for solids up to the 4d transition
metals are presented. Finally in Section 3.6 the SIC is applied to the system of a cerium
atom in jellium, and an attempt is made to use this solution to model the alpha and
gamma phases of bulk cerium.
3.1
3.1.1
Solving the Schrödinger Equation
The Radial Schrödinger Equation
We reproduce the Kohn-Sham equations (Eq. (2.3.29) )
1 2
σ
− ∇ + V (r) φσi (r) = Eiσ φσi (r)
2
53
(3.1.1)
54
An Atom in Infinite Jellium Solved using DFT
where σ is the spin ↑ or ↓. In this thesis, we consider the system of an atom in infinite
jellium and the system of an atom in a finite jellium sphere. For the atom in infinite
jellium, with an atom of charge Z and jellium with positive background density, n0 , the
potential, V σ (r), is
Z
σ
V (r) =
n(r0 ) − n0 0 Z
n0 n0
σ
dr − + Vxc
(n↑ (r), n↓ (r)) − Vxc ( , )
|r0 − r|
r
2 2
(3.1.2)
where integrals in r are over all space unless the limits are stated explicitly, r = |r|
σ (n↑ (r), n↓ (r)) is the exchange-correlation potential in the LDA. In particular we
and Vxc
use forms due to Perdew and Zunger [56], Perdew and Wang [55] and Gunnarsson and
Lundqvist [54].
For the case of an atom immersed in a finite jellium sphere
σ
Z
V (r) =
n(r0 )
dr0 −
|r0 − r|
Z
r 0 =Rjell
r0 =0
n0
Z
σ
dr0 − + Vxc
(n↑ (r), n↓ (r))
|r0 − r|
r
(3.1.3)
3 n = N , where N is the number
where Rjell is the radius of the jellium sphere and 43 πRjell
0
of electrons in the jellium sphere. There will be N + Z solutions to the Kohn-Sham
equation for the case of the finite jellium sphere, providing overall charge neutrality.
Notice the arbitrary additive constant to the potential for the atom in infinite jellium,
σ (n /2, n /2). This term guarantees that V σ (r) → 0 as |r| → ∞. This will be
namely −Vxc
0
0
a desirable property when we come to calculate the positive energy scattering states later
on.
The electron density will be spherically symmetric for all systems considered. This
will either be imposed as an approximation or will be exact, on account of complete filling
of atomic orbitals. The fact that the electron density is spherically symmetric means that
the potential will also be spherically symmetric. For a spherically symmetric potential, we
σ
can make a separation of variables in the wavefunction: φσi (r) = Ylm (θ, φ)RE
(r), and
i lm
obtain the radial Schrödinger equation
1 ∂2
2 ∂
l(l + 1)
σ
σ
σ
−
+
+
+ V (r) REl
(r) = EREl
(r)
2 ∂r2 r ∂r
2r2
(3.1.4)
where l labels the eigenstates of L2
L2 Ylm (θ, φ) = l(l + 1)Ylm (θ, φ)
(3.1.5)
3.1 Solving the Schrödinger Equation
55
and where m takes the values −l, −l + 1, · · · , l. Because m does not appear in the radial
σ (r) will have a degeneracy of 2l + 1. Defining
Schrödinger equation, a given solution REl
σ (r) = rRσ (r) we can see that:
UEl
El
d2
2 d
+
2
dr
r dr
σ (r)
UEl
d
=
r
dr
σ
σ 0
−UEl
UEl
+
r2
r
2
+
r
σ
σ 0
−UEl
UEl
+
r2
r
σ 0
σ
σ 0
σ 00
σ
σ 0
σ 00
UEl
UEl
UEl
UEl
UEl
UEl
UEl
−
+
−
−
2
+
2
=
r3
r2
r
r2
r3
r2
r
Putting this into the Schrödinger equation gives
2
σ (r)
1 1 d2 UEl
l(l + 1) σ
σ
σ
+
REl (r) + V σ (r)REl
(r) = EREl
(r)
2
2 r dr
2r2
Multiplying by r gives
−
=
(3.1.6)
(3.1.7)
σ (r)
1 d2 UEl
l(l + 1) σ
σ
σ
+
UEl (r) + V σ (r)UEl
(r) = EUEl
(r)
(3.1.8)
2 dr2
2r2
This is the form for the radial Schrödinger equation that we will use most commonly
−
throughout this thesis.
For the atom immersed in infinite jellium, the equation above has two classes of solution. There are solutions which have negative energy eigenstates and solutions which have
positive energy eigenstates. The former decay exponentially with distance from the origin
and are therefore described as bound-states. These states form a discrete set. The latter
class of solution form a continuous set in the energy eigenvalue, and for reasons which will
become apparent, are referred to as scattering-states. These states have an infinite radial
extent.
For the atom in a finite jellium sphere, all of the solutions to the radial Schrödinger
equation are bound states and form a discrete set of solutions.
3.1.2
The Electron Density
For a system with a discrete set of bound states, such as an atom or an atom in a finite
jellium sphere, the electron density is simply
σ
n (r) =
bound
X
2
|Ylm (θ, φ)|
n,l,m
σ 2 (r)
Un,l
r2
(3.1.9)
The electron density for an atom in infinite jellium however has an additional term
due to a contribution from the scattering states
56
An Atom in Infinite Jellium Solved using DFT
σ
n (r) =
bound
X
2
|Ylm (θ, φ)|
n,l,m
σ 2 (r)
Un,l
r2
1
+ 2
2π
Z
kF
0
X
(2l + 1)k 2
l
Ulσ 2 (k, r)
dk
r2
(3.1.10)
1
where, kF , is the Fermi wave-number and is given by kF = (3π 2 n0 ) 3 . The form for the
scattering state contribution to this density is derived in section 3.2.5.
Provided all atomic sub-shells are either completely occupied or empty, we can use
Pm=l
2l+1
2
m=−l |Ylm (θ, φ)| = 4π to re-write the first term as
bound
σ 2 (r)
Un,l
1 X
(2l + 1)
4π
r2
(3.1.11)
n,l
For the case where sub-shells are only partially occupied, we impose spherical symmetry
by making the approximation:
X
|Ylm (θ, φ)|2 =
m
mnum
4π
(3.1.12)
where mnum are the number of m values occupied in the sub-shell. For example, for a
carbon atom in free space (therefore no scattering states) we would have
n↑ (r) =
i
1 h ↑2
↑ 2
↑ 2
U
(r)
+
U
(r)
+
2U
(r)
2s
2p
4πr2 1s
(3.1.13)
i
1 h ↓2
↓ 2
U
(r)
+
U
(r)
2s
4πr2 1s
(3.1.14)
n↓ (r) =
3.1.3
Potential Mixing
σ(i)
For a given iteration, we take the spin-up and spin-down potentials, Vin (r), (i is the
iteration number) solve the radial Schrödinger equation, Eq. (3.1.8), and calculate the
spin-densities, nσ (r). From these we generate new spin-up and spin-down potentials,
σ(i)
Vout (r), using either Eq. (3.1.2) or Eq. (3.1.3) depending on whether our system is an
atom in infinite jellium or an atom in a finite jellium sphere. We can then use these
σ(i+1)
potentials for the next iteration, i.e. Vin
σ(i)
(r) = Vout (r), and repeat the whole process
again. The process is repeated until the potentials no longer change from one iteration
to the next (I.e. they have ’converged’), at which point we have achieved self-consistency
and have solved the radial Schrödinger equation.
3.1 Solving the Schrödinger Equation
σ(i+1)
In practice, using Vin
57
σ(i)
(r) = Vout (r), results in the charge density moving from
low to high radius over subsequent iterations (’charge sloshing’), which results in poor
convergence. To avoid this, one scheme for generating new potentials is
σ(i+1)
Vin
σ(i)
σ(i)
(r) = αVout (r) + (1 − α)Vin (r)
(3.1.15)
where α is a mixing fraction, and satisfies 0 < α < 1. This method, known as linear
mixing, leads to improved convergence. However in many cases, α, has to be made very
small (∼ 10−3 or smaller) in order to avoid charge-sloshing. A far better method of mixing
the potentials is the Broyden method [70, 71]. In this method, potentials from previous
iterations are used to generate the new potential
σ(i+1)
Vin
σ(i)
σ(i)
σ(i−1)
(r) = f [Vout , Vin , Vin
, ...]
(3.1.16)
The method is described no further in this thesis, but is described in detail by Johnson
[70]. In the DFT computer program written for this thesis, the Broyden mixing algorithm
is incorporated by using an existing subroutine originally written by D. D. Johnson.
As an alternative to mixing potentials, one could instead mix densities. The Broyden
method would generate new spin densities for the next iteration using the output spin
densities from the current iteration and the input spin densities from the current iteration
and previous iterations
σ(i+1)
nin
σ(i+1)
The potential Vin
σ(i)
σ(i)
σ(i−1)
(r) = f [nout , nin , nin
, ...]
(3.1.17)
σ(i+1)
(r) can then be obtained by substituting nin
(r) into Eq. (3.1.2)
or Eq. (3.1.3). Depending on the circumstances, mixing densities rather than potentials
might be more efficient. In this thesis we only use potential mixing.
3.1.4
Criterion for Convergence
One criterion for the convergence of the potentials (and therefore self-consistency) is that
the quantities
∆V
σ
=
Z σ(i)
σ(i)
Vout (r) − Vin (r) r2 dr
(3.1.18)
fall below a specified level, which we take to be ∼ 10−10 in this thesis (see Section 3.4.3
for a discussion of how we chose this value). The exact form of the integrand has a certain
58
An Atom in Infinite Jellium Solved using DFT
arbitrariness and we include the r2 factor to try and reflect the increasing number of
electrons in shells at larger radii.
3.1.5
Simplifying the Coulomb Potential for the Case of Spherical Symmetry
The Hartree term in the Schrödinger equation can be simplified by using the fact that the
density is spherically symmetric.
Z
n(r0 )
1
dr0 =
0
|r − r|
r
Z
r
0
02
0
Z
n(r )4πr dr +
0
∞
n(r0 )4πr0 dr0
(3.1.19)
r
To prove the relation we use Gauss’s theorem (in S.I. units):
I
1
E·dS =
ε0
Z
dτ ρ(r)
(3.1.20)
where E is the electric field vector and ρ(r) is the charge density (ρ(r) = en(r)). The
integral on the left is a surface integral, and dS is a vector which at a given position points
away from the surface of integration and has a magnitude equal to the infinitesimal area
of the surface at that position. The integration on the right is taken over the volume
enclosed by the surface.
We now apply Gauss’s Theorem to the case of a spherically symmetric charge distribution. We ask the question, what is the electric field at a radius R due to this charge
distribution. We choose the surface of integration to be the surface of a sphere of radius
R centred on r = 0. The left side of Gauss’s Theorem, Eq. (3.1.20), gives us
I
I
E·dS =
Er dSr = Er (R)4πR2
(3.1.21)
And the right side gives
1
ε0
Z
1
dτ ρ(r) =
ε0
R
Z
ρ(r)4πr2 dr
(3.1.22)
0
Therefore equating the two sides and using ρ(r) = en(r)
e
Er (R) =
4πε0 R2
Or in atomic units
Z
0
R
n(r)4πr2 dr
(3.1.23)
3.1 Solving the Schrödinger Equation
59
Z
1
eR2
Er (R) =
R
n(r)4πr2 dr
(3.1.24)
0
The Coulomb potential at R due to this charge distribution is
1
V (R) =
eR
Z
0
R
1
n(r)4πr dr −
e
2
R
Z
n(r)4πrdr + const
(3.1.25)
0
as is proved below
∂V (R)
=
∂R
Z R
Z
∂
1
1 R
−
n(r)4πr2 dr −
n(r)4πrdr + const =
∂R eR 0
e 0
Z R
Z R
Z R
1
1 ∂
1 ∂
2
2
n(r)4πr dr −
n(r)4πr dr +
n(r)4πrdr =
eR2 0
eR ∂R
e ∂R
0
0
Z R
Z R
1
1
1
1
2
2
n(r)4πr
dr
−
n(R)4πR
+
n(R)4πR
=
n(r)4πr2 dr
(3.1.26)
eR2 0
eR
e
eR2 0
Er (R) = −
This gives us the correct radial component of the electric field. Setting R = 0 in the
expression for V (R) causes the first two terms on the right side to equal zero (if we assume
that n(r → 0) → const) which tells us that the constant is V (0). In S.I. units V (0) is
Z
∞
V (0) =
∞
Z
edn(r)
4πε0 r
dV (r) =
0
0
(3.1.27)
Using dn(r) = n(r)4πr2 dr and switching to atomic units gives
V (0) =
1
e
Z
∞
n(r)4πrdr
(3.1.28)
0
If we now insert this into our equation for V (R), and use the fact that the Coulomb
energy for an electron is eV (R), we obtain
1
eV (R) =
R
Z
R
Z
2
∞
n(r)4πrdr
n(r)4πr dr +
0
(3.1.29)
R
as required.
The Coulomb Potential of the Positive-Background of the Jellium
The Coulomb potential for the attraction to the positive background density of the
jellium, for the infinite jellium case is
Z
n0
1
dr0 =
0
|r − r|
r
Z
r
02
0
Z
n0 4πr dr +
0
r
∞
n0 4πr0 dr0
(3.1.30)
60
An Atom in Infinite Jellium Solved using DFT
The proof for this is identical to the proof given above.
In our code, we only integrate the radial Schrödinger equation outwards to some finite
(but large) rmax . Correspondingly, n(r) is only calculated out to this radius. For the
atom in infinite jellium, beyond this radius we assume that the electron density is equal
to n0 . Therefore when Eq. (3.1.29) and Eq. (3.1.30) are substituted into Eq. (3.1.2), the
integrals in the second term of these equations between the limits rmax and ∞ will cancel.
Therefore the infinities on the integrals in Eq. (3.1.29) and Eq. (3.1.30) should be replaced
with rmax . Eq. (3.1.30) becomes
Z
1
r
r
n0 4πr02 dr0 +
0
Z
rmax
r
4
n0 4πr0 dr0 = πr2 n0 + 2π(rmax 2 − r2 )n0
3
(3.1.31)
In the case of the finite jellium sphere, where the jellium only extends out to a radius
Rjell , we have
Z
r 0 =Rjell
r 0 =0


n0
dr0 =
0

|r − r|
4
2
3 πr n0
+ 2π(Rjell 2 − r2 )n0 r ≤ Rjell
3
1 4
n0 r > Rjell
r 3 πRjell
(3.1.32)
The proof for the expression for r ≤ Rjell is the same as that given above. The
expression for r > Rjell is proved firstly by showing that the corresponding field Er
satisfies Gauss’s Theorem, and secondly by showing that the potential as defined by the
above equations is continuous. The latter is clearly true, leaving us to prove the former.
Using Er (r) = −∂V (r)/∂r, and remembering that the quantity above equals eV (r) we
obtain
1
Er (r) = −
e
−1
r2
4
πRjell 3 n0
3
(3.1.33)
In S.I. units this reads
Er (r) =
e 4
πRjell 3 n0
4πε0 r2 3
(3.1.34)
Therefore
1 4
πRjell 3 n0 e
ε0 3
I
Z
1
→ E·dS =
dτ ρ(r)
ε0
Er (r)4πr2 =
(3.1.35)
3.2 Scattering States
61
where the surface of integration is a sphere of radius r centred on r = 0 and the charge
density is a sphere of positive charge of radius Rjell and density n0 . Hence we have shown
that the field obeys Gauss’s theorem, and so have proved Eq. (3.1.32).
3.2
Scattering States
3.2.1
Introduction
We will now discuss how to solve the radial Schrödinger equation to obtain the scattering
state solutions for the case of an atom immersed in infinite jellium. In Section 3.2.2 we
derive the boundary conditions on these solutions for large r. Then in Section 3.2.3 we
discuss how to use these boundary condition to solve the radial Schrödinger equation.
We then discuss the normalisation of these scattering states in Section 3.2.4. The
scattering state contribution to the electron density is calculated in Section 3.2.5, and
its asymptotic behaviour at large r is calculated in Section 3.2.6. Finally, in the closing
sections of our discussion of scattering states, we will discuss a few of the mathematical properties of the phase-shift, which is a quantity which will emerge over subsequent
sections.
3.2.2
Boundary Conditions on Scattering States
We want to find the boundary condition on R(r) as r → ∞. In this limit, remembering
that we chose V (r → ∞) = 0 in Section 3.1.1, the radial Schrödinger equation, Eq. (3.1.4),
becomes
∂2
2 ∂
+
2
∂r
r ∂r
l(l + 1)
2
σ
−
+ k REl
(r) = 0
r2
(3.2.1)
where we have defined, k 2 = 2E. If we then make the change of variables ρ = kr, then we
have
∂2
2 ∂
+
2
∂ρ
ρ ∂ρ
l(l + 1)
+ 1−
ρ2
Rlσ (ρ) = 0
(3.2.2)
σ (r) = Rσ (ρ) because the operator on the left side of the equation
We have re-written Rkl
l
σ (r) must
now only depends on ρ (and not on k and r separately), and so the solution Rkl
also only depend on ρ. This equation is called the spherical Bessel differential equation,
and has the solutions:
62
An Atom in Infinite Jellium Solved using DFT
l
jl (ρ) = (−ρ)
nl (ρ) = −(−ρ)l
1 ∂
ρ ∂ρ
l
1 ∂
ρ ∂ρ
sin ρ
ρ
l
cos ρ
ρ
(3.2.3)
(3.2.4)
which are called spherical Bessel functions and spherical Neumann functions respectively
[72, 73].
Hence the solutions to Eq. (3.2.1) take the form
Rk,l (r) = Bl (k)jl (kr) + Cl (k)nl (kr)
(3.2.5)
where we have suppressed the σ and will continue to do so over the next few sections.
Therefore scattering state solutions to the radial Schrödinger equation have the following
asymptotic form
Rk,l (r → ∞) → Bl (k)jl (kr) + Cl (k)nl (kr)
3.2.3
(3.2.6)
Matching to the Boundary Condition
When tasked with writing a computer program to solve the radial Schrödinger equation,
the boundary condition in Eq. (3.2.6) becomes
Rk,l (rmax ) = Bl (k)jl (krmax ) + Cl (k)nl (krmax )
(3.2.7)
where rmax is some suitably chosen large value of r. As we shall discuss in more depth
later (Section 3.3.2), our procedure for calculating the scattering state solutions will involve starting at some small radius rmin and then propagating the Uk,l (r) = rRk,l (r)
outwards to the radius rmax by numerically integrating the radial Schrödinger equation.
The propagated solution is then matched onto the boundary condition of Eq. (3.2.7).
Rk,l (rmax ) = Bl (k) [jl (krmax ) − tan δl (k)nl (krmax )]
(3.2.8)
0
Rk,l
(rmax ) = Bl (k) kjl0 (krmax ) − k tan δl (k)n0l (krmax )
(3.2.9)
where the primes denote differentiation with respect to kr. Notice that we have defined
the quantity δl (k), known as the phase shift, by
3.2 Scattering States
63
tan δl (k) = −
Cl (k)
Bl (k)
(3.2.10)
0 , we proceed by first matching the logarithmic
Instead of matching Rk,l and then Rk,l
derivative of Rk,l
0 (r)
Rk,l
∂
ln Rk,l (r) =
∂(kr)
Rk,l (r)
(3.2.11)
and then matching Rk,l itself. Matching the logarithmic derivatives (dividing Eq. (3.2.8)
by Eq. (3.2.9) ) gives
Rk,l (rmax )
jl (krmax ) − tan δl (k)nl (krmax )
=
0
Rk,l (rmax )
k[jl0 (krmax ) − tan δl (k)n0l (krmax )]
(3.2.12)
Re-arranging gives
R
tan δl (k) =
(r
)
(r
)
max
jl (krmax ) − k Rk,l
j 0 (krmax )
0 (r
max ) l
k,l
R
max
nl (krmax ) − k Rk,l
n0 (krmax )
0 (r
max ) l
(3.2.13)
k,l
0 (r
0
We now need to replace the Rk,l (rmax ) and Rk,l
max ) with Uk,l (rmax ) and Uk,l (rmax ),
since these are the quantities that we propagate outwards using the radial Schrödinger
equation.
0
0
Uk,l (r) = rRk,l (r) → Uk,l
(r) = Rk,l (r) + rRk,l
(r)
0 (r)
Uk,l
0
1 Rk,l (r)
→
= +
Uk,l (r)
r Rk,l (r)
→
Rk,l (r)
0 (r) =
Rk,l
1
0 (r)
Uk,l
Uk,l (r)
(3.2.14)
−
1
r
Putting this into Eq.( 3.2.13) gives
tan δl (k) =
kjl0 (krmax ) −
U0
kn0l (krmax ) −
U0
k,l (rmax )
Uk,l (rmax )
k,l (rmax )
Uk,l (rmax )
−
−
1
jl (krmax )
nl (krmax )
rmax
1
rmax
(3.2.15)
This equation allows us to calculate the phase shift, δl (k), given the propagated values
0 (r
Uk,l
max )
and Uk,l (rmax ). The value of tan δl (k) ranges from −∞ to +∞, and hence for
a given l this equation can be solved for tan δl (k) for any k. This means that there are a
continuous set of scattering state solutions in k.
64
An Atom in Infinite Jellium Solved using DFT
Having calculated tan δl (k), we calculate the phase shifts using
δl (k) = tan−1 (tan δl (k)) + nπ,
nZ
(3.2.16)
To make these δl (k) uniquely defined we demand that δl (0) = 0 and insist that δl (k)
is continuous in k.
With the logarithmic derivatives matched, we must now match the propagated Rk,l (r)
to its asymptotic form at rmax
Rk,l (rmax ) = Bl (k)[jl (krmax ) − tan δl (k)nl (krmax )]
(3.2.17)
We set the normalisation for Rk,l (r) by choosing
Rk,l (rmax ) =
Bl (k)
[cos δl (k)jl (krmax ) − sin δl (k)nl (krmax )]
cos δl (k)
| {z }
taken to be = 1
(3.2.18)
The question of the scattering state normalisation is discussed further in the next
section.
3.2.4
Normalisation of Scattering States
In section 3.2.3 we set the normalisation of the scattering states by demanding that
Rk,l (rmax ) = cos δl (k)jl (krmax ) − sin δl (k)nl (krmax )
(3.2.19)
The spherical Bessel and Neumann functions have the asymptotic property
sin(x − lπ/2)
x
cos(x − lπ/2)
nl (x → ∞) → −
x
jl (x → ∞) →
(3.2.20)
(3.2.21)
Therefore our boundary condition is
Rl (krmax ) =
sin(krmax + δl (k) − lπ/2)
krmax
(3.2.22)
This choice of normalisation was arbitrary, and in practice we could have chosen other
normalisations. The only point at which the normalisation will matter will be when we
calculate the scattering state density.
3.2 Scattering States
65
To calculate the scattering state density (see Section 3.2.5), we will impose hard-wall
boundary conditions on the scattering states at some large radius rhw . This will result in
a discrete set of scattering states each of which will be normalised to one over the range
r = 0 to r = rhw . We can then calculate the scattering state density using the formula
P
nscatt (r) = i |φi (r)|2 . In the final step we will then let rhw → ∞ in order to obtain the
scattering state density for our infinite jellium system. In this section we will derive the
prefactor to Eq. (3.2.22) which is required for such a normalisation.
A scattering state normalised to 1 in the range r = 0 to r = rhw satisfies
r=rhw
Z
N
Rl 2 (k, r)r2 dr =
r=0
Z
r=rasym
N
2
r=rhw
Z
2
sin(kr + δl (k) − lπ/2)2
dr = 1
k2
Rl (k, r)r dr + N
r=0
r=rasym
(3.2.23)
The rasym here is chosen to be the radius after which R(k, r) takes on its asymptotic
form. The sin term in the above equation integrates to (rhw − rasym )/2. As rhw → ∞, the
second term on the right-hand side dominates the first and we get the result
N =
2k 2
rhw
(3.2.24)
This means that the correct normalised form for the scattering states, for large r, when
hard-wall boundary conditions are imposed at rhw , is
r
Rk,l (r) =
3.2.5
2
rhw
k (cos δl (k)jl (kr) − sin δl (k)nl (kr))
(3.2.25)
Calculating the Scattering State Density
We introduced the charge density for scattering state electrons in section 3.1.1, Eq. (3.1.10)
as
nscatt (r) =
1
2π 2
Z
kF
0
X
(2l + 1)k 2
σ,l
Ulσ 2 (k, r)
dk
r2
(3.2.26)
Here we motivate its form. The starting point is the equation
nscatt (r) =
X
|φσl,m (k, r)|2
k,l,m,σ
We take it that the Rlσ (k, r) are independent of m. Hence
(3.2.27)
66
An Atom in Infinite Jellium Solved using DFT
X
|φσl,m (k, r)|2 =
m
2l + 1 σ 2
Rl (k, r)
4π
(3.2.28)
Now we impose hard-wall boundary conditions on R(k, r) at some large radius rhw .
This was discussed in section 3.2.4, where the correct normalised form for the scattering
state solutions to the radial Schrödinger equation were found to be
σ
Rk,l
(r)
r
2
k (cos δlσ (k)jl (kr) − sin δlσ (k)nl (kr))
(3.2.29)
rhw
where r is large but r < rhw . With the hard-wall boundary conditions the k values have
→
to satisfy
2 sin(krhw + δlσ (k) − lπ/2)
=0
(3.2.30)
rhw
rhw
= lπ/2 − δlσ (k) + nπ, where n is an integer. In the limit that rhw → ∞ the sum
Rlσ (krhw )
I.e, krhw
r
=
over k in Eq. (3.2.27) will become an integral
X
Z
→
0
k
kF
dk
∆k
(3.2.31)
where ∆k is the spacing between k points and is given by ∆k = π/rhw . If we separate out
the normalisation factor N from R(k, r) (and from now on regard the normalisation of
R(k, r) as being that Rlσ (kr → ∞) → sin(k + δlσ (k) − lπ/2)/k) then we obtain the correct
expression
n
scatt
(r) =
X (2l + 1) Z
σ,l
4π
0
kF
1
dk 2k 2 σ 2
Rl (k, r) = 2
π/rhw rhw
2π
Z
0
kF
X
σ,l
(2l + 1)k 2
Ulσ 2 (k, r)
dk
r2
(3.2.32)
The Fermi wavenumber, kF , must be chosen so that the scattering state density is
equal to the background density, n0 , at large radius. At large radius the system is just
the free electron gas. We therefore use the standard result for the Fermi wavenumber for
1
a free electron gas, which is kF = (3π 2 n0 ) 3 for an electron gas of uniform density n0 .
3.2.6
Friedel Oscillations
The charge density, n(r), tends to a constant, n0 , as r → ∞. However, n(r) oscillates
sinusoidally about n0 as a function of r for large r. These oscillations are known as Friedel
oscillations and are derived in this section.
3.2 Scattering States
67
The charge density due to the scattering states is
n
scatt
Z
1
(r) = 2
2π
kF
0
X
(2l + 1)k 2 Rlσ 2 (k, r)dk
(3.2.33)
σ,l
where Rlσ are normalised according to
Rlσ (k, r → ∞) →
sin(kr + δlσ (k) − lπ/2)
kr
(3.2.34)
We want to evaluate ∆nscatt (r) = nscatt (r) − n0 , where n0 can be written as
1
n0 = 2
2π
Z
kF
0
X
(2l + 1)k 2 jl 2 (kr)dk
(3.2.35)
σ,l
Hence as r → ∞
∆n
scatt
1
(r) → 2
2π
kF
Z
0
X
σ,l
sin(kr + δlσ (k) − lπ/2)2 sin(kr − lπ/2)2
(2l + 1)
−
dk
r2
r2
(3.2.36)
Using cos(2x) = 1 − 2 sin2 (x) gives
∆nscatt (r) →
1
2π 2
Z
0
kF
X
(2l + 1)
σ,l
1
[cos(2kr − lπ) − cos(2kr + 2δlσ (k) − lπ)] dk
2r2
(3.2.37)
Performing the integration over k, and using kr δlσ (k)
in order to ignore the k-
dependence of δlσ (k) in this integration, one obtains
∆n
scatt
1 X
1 sin(2kF r − lπ) sin(2kF r + 2δlσ (k) − lπ)
(r) → 2
(2l + 1) 2
−
(3.2.38)
2π
2r
2r
2r
σ,l
Re-writing the sin terms respectively as sin(2kF r−lπ+δlσ (k)−δlσ (k)) and sin(2kF r−lπ+
δlσ (k)+δlσ (k)) and using sin(x+y) = sin(x) cos(y)+cos(x) sin(y) with x as 2kF r−lπ+δlσ (k)
and y as −δlσ (k) or δlσ (k) gives
∆nscatt (r) →
1
1 X
(2l + 1) 3 [sin(2kF r − lπ + δlσ (kF )) cos(−δlσ (kF ))+
2
2π
4r
σ,l
cos(2kF r − lπ + δlσ (kF )) sin(−δlσ (kF )) − sin(2kF r − lπ + δlσ (kF )) cos(δlσ (kF ))
68
An Atom in Infinite Jellium Solved using DFT
− cos(2kF r − lπ + δlσ (kF )) sin(δlσ (kF ))]
(3.2.39)
giving
∆nscatt (r) → −
1 1 X
(2l + 1)(−1)l cos(2kF r + δlσ (kF ) sin(δlσ (kF ))
4π 2 r3
(3.2.40)
σ,l
Therefore for large r, the charge density oscillation has a period π/kF .
3.2.7
Friedel Sum Rule
The Friedel sum rule [74] states that if we add an impurity atom to jellium, then the
electron charge density will increase around the impurity in order to completely screen
the positive charge of the impurity. In other words, the change in the number of scattering
states plus the number of bound states formed due to the impurity equals the charge of
the impurity, i.e:
∆Nscatt + Zb = Z
(3.2.41)
where Z is the atomic number of the impurity atom and Zb is the number of the bound
P
states. We will prove that ∆Nscatt = π1 σ,l (2l + 1)δlσ (kF ), and so:
1X
(2l + 1)δlσ (kF ) = Z − Zb
π
(3.2.42)
σ,l
The equation applies when we are considering an atom of charge Z immersed in infinite
jellium. From now on, whenever we use the term Friedel sum rule, we will be referring to
this equation. This is an important equation, which is checked numerically as the program
runs.
To prove the relation: ∆Nscatt =
1
π
P
σ,l (2l
+ 1)δlσ (kF ), consider the expression for a
σ (r) for large r
scattering state Rk,l
σ
Rk,l
(r → ∞) →
1
lπ
sin(kr + δlσ (k) − )
kr
2
(3.2.43)
Applying hard-wall boundary conditions at some large radius, rhw , gives
krhw + δlσ (k) −
lπ
= nπ
2
In the case of a system with zero potential everywhere (pure jellium) we have
(3.2.44)
3.2 Scattering States
69
σ
Rk,l
(r) = cos δlσ (k)jl (kr) + sin δlσ (k)nl (kr)
(3.2.45)
for all r. But nl (kr)|r→0 → ∞ and therefore we must have δlσ (k) = 0, in which case we
have
krhw −
lπ
= nπ
2
(3.2.46)
Re-arranging Eq. (3.2.44) and Eq. (3.2.46) in terms of n, we see that for the lth partial
wave the number of scattering states in the range [0, k] is
Nlσ ≡ n =
1
lπ
(krhw + δlσ (k) − )
π
2
for V (r) 6= 0
(3.2.47)
Nlσ ≡ n =
1
lπ
(krhw − )
π
2
for V (r) = 0
(3.2.48)
The number of states in the range [k, k + dk] is therefore
dNlσ
dn
rhw
1 dδlσ
dk ≡
dk = (
+
)dk
dk
dk
π
π dk
dNlσ
dn
rhw
dk ≡
dk =
dk
dk
dk
π
for V (r) 6= 0
for V (r) = 0
(3.2.49)
(3.2.50)
Therefore the change in the number of scattering states for the lth partial wave in the
range [k, k + dk] upon adding the atom to an infinite jellium system is
d∆Nlσ
1 dδlσ
dk =
dk
dk
π dk
(3.2.51)
Therefore the total change in the number of scattering states is
XXZ
σ
l,m
Z
d∆Nlσ
dδ σ (k)
1 X kF
dk
dk =
dk(2l + 1) l
=
dk
π
dk
0
σ,l
1X
(2l + 1)(δlσ (kF ) − δlσ (0))
π
σ,l
which, if we take δlσ (0) = 0 gives the correct result.
(3.2.52)
70
An Atom in Infinite Jellium Solved using DFT
3.2.8
Properties of the Phase-Shift
Eq. (3.2.51) tells us that the atom-induced density of states is given by
d∆Nlσ
(2l + 1) dδlσ
=
dk
π
dk
(3.2.53)
A typical plot of the phase-shifts and the corresponding density of states for a cerium
atom (solved using the LDA) embedded in jellium of rs = 5.3 is given in Fig. 3.1. The
abrupt jump in the l = 3 phase-shift by π corresponds to a very narrow, almost bound
state-like density of states containing 2l + 1 electrons. We refer to this as a resonance.
The fact that most of this resonance lies above the Fermi-level means that only a few of
the electrons in this resonance are included in the calculation.
Levinson’s Theorem
Levinson’s Theorem [75] states that
nb = (2l + 1)δlσ (0)
(3.2.54)
Here, nb is the number of bound states of angular-momentum l and spin σ, and the
phase-shift is fixed by δlσ (kF ) = 0.
We give an example of Levinson’s Theorem from our own calculations in Fig. 3.2. The
figure shows the l = 0 phase-shift for a hydrogen atom immersed in jellium of background
−3
densities 0.01a−3
B and 0.05aB . The system is non-magnetic and so the phase-shift is the
same for both spins.
For the 0.01a−3
B background density the 1s bound state is occupied. In accordance
with Levinson’s Theorem the phase-shift equals π at zero energy (when we define the
phase-shift to equal zero at infinite energy). For the background density of 0.05a−3
B , the
effective attractive power of the ion has been reduced, and these states are no longer
bound. Correspondingly the phase-shift is zero at zero energy and a resonance has formed
in the continuum of scattering states.
3.2 Scattering States
71
50
dN l ()
−1
(eV
)
dNscatt/dE/ev-1
d
40
Fermi level
30
l=3
20
l=2
10
0
0.355 0.36 0.365 0.37 0.375 0.38 0.385 0.39 0.395 0.4
sqrt(2*epsilon)/a.u.
√
2 (a.u)
Figure 3.1: Phase-shifts (top panel) and the corresponding density of states (lower panel)
for a cerium atom embedded in jellium of rs = 5.3
72
An Atom in Infinite Jellium Solved using DFT
4
Fermi level
3.5
2
2.5
3
1.5
1
0.5
0
4
1.2 1.4 1.6 1.8
2
1
0.2 0.4 0.6 0.8
0
Fermi level
3.5
2
2.5
3
1.5
1
0.5
0
1.5
2
2.5
3
3.5
1
0.5
0
Figure 3.2: The l = 0 phase-shifts for a hydrogen atom immersed in infinite jellium of
−3
background densities 0.01a−3
B (top panel) and 0.05aB (bottom panel).
3.3 Numerical Algorithm for Solving the Radial Schrödinger Equation
3.3
73
Numerical Algorithm for Solving the Radial Schrödinger
Equation
3.3.1
Radial Schrödinger Equation Solutions in the Limits r → 0 and
r→∞
Ignoring for a moment the self-consistency problem posed by the radial Schrödinger equation, let us ask how we can, for a given V σ (r), solve the radial Schrödinger equation to
σ (r). As we shall see, this requires us to know the analytic
obtain the radial solutions, UEl
σ (r) and their derivatives dU σ (r)/dr in the limits r → 0 and r → ∞.
values of UEl
El
For the case of an atom in jellium we have
σ (r)
1 d2 UEl
l(l + 1) σ
−
+
UEl (r) +
2 dr2
2r2
Z
rmax
+
r
Z r
1
(n(r0 ) − n+ (r0 ))4πr02 dr0
r 0
Z
σ
σ
σ
(n(r ) − n (r ))4πr dr − + Vxc (r) UEl
(r) = EUEl
(r)
r
0
+
0
0
0
(3.3.1)
σ (r)/r
We look first at the limit r → 0. The two divergent terms in this case are −ZUEl
σ (r)/2r 2 . This means that in the limit r → 0 we have
and l(l + 1)UEl
1
2
σ (r)
d2 UEl
dr2
=







σ (r) if l = 0
− Zr UEl
(3.3.2)
l(l+1) σ
UEl (r)
2r 2
if l > 0
σ (r) = re−Zr as is shown below
In the first case the solution is UEl
1 d2 (re−Zr )
1 d −Zr
1
1
1
=
(e
− Zre−Zr ) = − Ze−Zr − Ze−Zr + Z 2 re−Zr
2
2
dr
2 dr
2
2
2
(3.3.3)
σ (r)/r as required. The solution for
As r → 0 this result tends to −Ze−Zr = −ZUEl
σ (r) = r l+1 which is proved below
l > 0 is UEl
1 d2 (rl+1 )
1
l(l + 1) σ
= l(l + 1)rl−1 =
UEl (r)
2 dr2
2
2r2
as required. We will worry about the normalisation of these solutions later.
In summary
(3.3.4)
74
An Atom in Infinite Jellium Solved using DFT
σ
UEl
(r → 0) →

−Zr if l = 0


 re



(3.3.5)
rl+1 if l > 0
Let us now look at the r → ∞ limit. In this limit V (r) → 0 and l(l + 1)/r2 → 0.
Therefore
−
σ (r) = e−
The solution is UEl
−
2
1d
2
√ √
2 −Er
e−
r
2
dr
σ
1 d2 UEl
σ
= EUEl
(r)
2 dr2
√ √
2 −Er /r
1 d
=−
2 dr
−
(3.3.6)
as is shown below
e−
√ √
2 −Er
r2
!
√ √
2 −E −√2√−Er
−
e
r
(3.3.7)
Evaluating this expression and ignoring terms except for the 1/r term gives
1
=
2
√ √
√ √
2 −E √ √
E √√
σ
(− 2)( −E)e− 2 −Er = e− 2 −Er = EUEl
(r)
r
r
(3.3.8)
as required.
3.3.2
The Runge-Kutta Algorithm
In the computer program, the 4th order Runge-Kutta algorithm [76] is used to solve the
radial Schrödinger equation. In order to apply this algorithm, the radial Schrödinger
equation is split into two equations
−
1 dAσEl (r) l(l + 1) σ
σ
σ
+
UEl (r) + V σ (r)UEl
(r) = EUEl
(r)
2 dr
2r2
σ (r)
dUEl
= AσEl (r)
dr
(3.3.9)
(3.3.10)
σ (r) and Aσ (r) at some r, the Runge-Kutta algorithm
Given starting values of UEl
El
σ (r) and Aσ (r) outwards or inwards to some other value of r.
propagates UEl
El
σ (r) and Aσ (r) at some small value of r, r
Consider the case where we start with UEl
min
El
(typically of the order 10−4 ). We would then like to propagate this solution outwards, i.e.,
σ (r) and Aσ (r) for larger values of r. In the previous section we found
determine UEl
El
3.3 Numerical Algorithm for Solving the Radial Schrödinger Equation
σ
UEl
(r → 0) →

−Zr if l = 0


 re



75
(3.3.11)
rl+1 if l 6= 0
Using these limits we can write
σ
UEl
(rmin )
=

−Zrmin


 rmin e



AσEl (rmin )
=
if l = 0
(3.3.12)
rmin l+1 if l 6= 0

−Zrmin − Zr
−Zrmin

min e

 e



if l = 0
(3.3.13)
(l + 1)rmin l if l 6= 0
σ (r ) and
The Runge-Kutta algorithm uses Eq. (3.3.9) and Eq. (3.3.10) to calculate UEl
2
AσEl (r2 ), where r2 = rmin + δr and δr is some small increment. This procedure is repeated
σ (r) and Aσ (r) outwards to some specified value of r.
until we have propagated UEl
El
σ (r) and Aσ (r) inwards, and will do so from r = r
We can also propagate UEl
max . In
El
σ
σ (r
this case initial starting values UEl
max ) and AEl (rmax ) are required.
Using outwards or inwards propagation, or a combination of the two, the quantities
σ (r) and Aσ (r) can be calculated over a set of r values. We choose these r points
UEl
El
to follow a logarithmic scale for small r and then a linear scale for larger r. We choose
a logarithmic scale for small r because the potential is varying rapidly here and so we
have to take smaller steps in order to reduce the error associated with the Runge-Kutta
algorithm. In our code, the r points are given by
rmin , rmin
rint +
rint
rmin
1
n0 −1
, rmin
rint
rmin
2
n0 −1
, · · · , rmin
rint
rmin
n0 −2
n0 −1
, rint ,
2
n1 − 1
1
(rmax − rint ), rint + (rmax − rint ), · · · , rint +
(rmax − rint ), rmax
n1
n1
n1
where rint is the value of r after which the r-points become linear, n0 is the number of
logarithmic points and n1 is the number of linear points.
3.3.3
Bound State Calculation
We would like to calculate bound states using the Runge-Kutta method described above.
σ (r)
Since we are dealing with bound states the E are negative. First we propagate UEl
76
An Atom in Infinite Jellium Solved using DFT
outwards from rmin to some rmatch (usually taken to be ≈ 1aB ) using the method described
σ (r) inwards from r
above. Then we propagate UEl
max to rmatch The inwards solution is
then re-scaled so that the inwards and outwards solutions taken together are continuous.
The energy E is then varied until the gradients of the two solutions match at rmatch . The
σ (r) combine
resulting E is then a bound state energy, and the outwards and inwards UEl
to give the full bound state solution.
σ (r
σ
The outward integration uses the starting values UEl
min ) and AEl (rmin ) given in the
previous section. For the inward integration, we have already found that
σ
UEl
(r
→ ∞) =
e−
√ √
2 −Er
r
(3.3.14)
σ (r
σ
The starting values of UEl
max ) and AEl (rmax ) are therefore
σ
UEl
(rmax ) =
AσEl (rmax )
e−
√ √
2 −Ermax
rmax
√ √
√ √
√ √
2 −Ee− 2 −Ermax
e 2 −Ermax
=−
−
rmax
rmax 2
(3.3.15)
(3.3.16)
For the first iteration in the self-consistency cycle, we start at some minimum value of
E and then scan upwards until we reach E = 0. We do this initially for the case where
l = 0. For each value of E we re-scale the inwards solution so that the two solutions are
continuous at rmatch . As we scan across E, we keep track of the quantity dUoutwards (r =
rmatch )/dr − dUinwards (r = rmatch )/dr. A typical plot of this quantity versus energy will
look like that shown in Fig. 3.3.
The E values where dUoutwards (rmatch )/dr − dUinwards (rmatch )/dr = 0 are bound state
σ (r)/r are the bound state solutions. When E crosses
energies, and the corresponding UEl
these points, the bisector method is implemented in order to home in on the exact energy.
Note that as the energy range is scanned, the change of sign of dUoutwards (rmatch )/dr −
dUinwards (rmatch )/dr is looked for in order to locate the bound states. However, the
asymptotes also cause the sign of this quantity to change, and so the code has to be smart
enough to skip past these.
For l = 0, the first bound state we reach will be the 1s bound state, followed by the
2s bound state, etc. After we have scanned upwards and have reached zero energy, we
move onto l = 1 and repeat the process. The process is repeated until all l values for
which there are bound state solutions have received a bound state scan. In addition, if
the solution is magnetic, then the whole process must be repeated for both spins.
3.3 Numerical Algorithm for Solving the Radial Schrödinger Equation
77
2
1
0
2
-1
1
0
-2
-1
-2
-3
-3
-4
-4
-5
-1000
-5
-20
-900
-800
-700
-15
-10
-600
-500
-5
-400
0
-300
-200
-100
0
Energy / a.u.
Figure 3.3: The quantity dUoutwards (r = rmatch )/dr − dUinwards (r = rmatch )/dr (as described in the main text) for l = 0 is plotted as a function of energy. The system is a Technetium atom immersed in jellium of background density 0.03a−3
B , and is non-magnetic, so
the curve applies for both spin-up and spin-down electrons The l = 0 bound state energies
are at the points where the curve crosses the x-axis, I.e. at: −744.939a.u., −103.763a.u.,
−17.363a.u. and −1.845a.u..
78
An Atom in Infinite Jellium Solved using DFT
If linear mixing is being used to generate the new potentials, then we can exploit
the fact that the bound state energies will not move a great deal in-between successive
iterations. Instead of scanning from some minimum value up to zero, we instead start the
bound state search for each bound state at the energy of the same bound state from the
previous iteration.
If we are using Broyden mixing, then in general the potential changes significantly in
between successive iterations. The bound state energies of the previous iteration cannot
now be assumed to provide reliable starting points for the new bound state search. When
using Broyden mixing then, a full energy sweep is required for each iteration.
Notice that instead of propagating the solution outwards and also inwards, we could
have just propagated the solutions outwards from rmin to rmax . In this case, bound states
could be found by varying the energy in order to satisfy the boundary condition given
by Eq. (3.3.15). This method would be less accurate than the method described above
however. This is because the error which accumulates in the Runge-Kutta algorithm will
be larger if we require a propagation of the solution over a larger distance, as would be
the case if we just considered outwards propagation.
3.3.4
Scattered State Calculation
As discussed in Section 3.2.3, the procedure for calculating the scattering states is to start
at some small radius rmin , and use the radial Schrödinger equation to propagate the U (r)
outwards to rmax . The fourth order Runge-Kutta algorithm is used to achieve this. The
logarithmic derivative of the R(r) for the propagated solution is then matched to the
asymptotic form for R(r) at rmax . This specifies the phase-shift, δl (k). The propagated
R(r) is then matched to the asymptotic R(r) at rmax , setting the normalisation of the
scattering state.
3.4
3.4.1
The Immersion Energy
Derivation of Immersion Energy
The form of the energy functional which must be minimised to yield the ground-state
solution (Eq. (2.3.17), with Ts given by Eq. (2.3.27) ) is as follows
3.4 The Immersion Energy
79
E[n↑ , n↓ ] =
XZ
1
φσi (r)∗ (− ∇2 )φσi (r) dr+
2
{z
}
i,σ
|
1
2
|
Z
n(r)n(r0 )
|r − r0 |
{z
=Ecoulomb
=Ekinetic
Z
drdr0 + drvext (r)n(r)dr +Exc [n↑ , n↓ ]
{z
}
} |
(3.4.1)
=Eexternal
where vext (r) is the external potential due to the positive background of the jellium and
the atom.
In this section we derive the immersion energy, which is the total energy of the combined atom-in-jellium system minus the energies of the jellium and the atom as if they
existed separately from one another. I.e.:
Eimm = Ecombined − Ejellium −Eatom
|
{z
}
(3.4.2)
=∆E
Let us separate ∆E (as defined above) into kinetic, Coulomb and exchange-correlation
components
∆E = ∆T + ∆C + ∆Exc
(3.4.3)
where we have combined the Ecoulomb and Eexternal terms defined earlier into the one C
term. For the kinetic term, let us consider the bound states and scattering states separately
and write
b.s
s.s
b.s
s.s
∆T = Tcombined
+ Tcombined
− Tjellium
− Tjellium
(3.4.4)
b.s
Tcombined
can be calculated from the Kohn-Sham equations
b.s
Tcombined
=
X
Eiσ
XZ
−
nσb.s (r)V σ (r)dr
(3.4.5)
σ
i,σ
where Eiσ are the bound state energies of the combined system and nσb.s (r) and V σ (r) are
the bound state densities and potentials for the combined system. There are no bound
b.s
states for pure jellium and so Tjellium
= 0.
s.s
s.s
and Tjellium
The Kohn-Sham equations can also be used to find expressions for Tcombined
T
s.s
=
XZ
σ
k=kF
k=0
Z
σ
∗
φ (k, r)
1 2
dN σ
− ∇ φσ (k, r)dr
dk
2
dk
(3.4.6)
80
An Atom in Infinite Jellium Solved using DFT
where
dN σ
dk dk
is the number of spin σ scattering states in the range k, k + dk. The
Kohn-Sham equations for scattering states
k2
1 2
σ
− ∇ + V (r) φσ (k, r) = φσ (k, r)
2
2
(3.4.7)
are rearranged in terms of ∇2 and inserted into the equation for T s.s to give
T s.s =
XZ
σ
k=kF
k=0
X
k 2 dN σ (k)
dk −
2
dk
σ
Z
V σ (r)
Z
k=kF
k=0
|
|φσ (k, r)|2
{z
=nσ
s.s (r)
dN σ (k)
dk dr
dk
}
(3.4.8)
Therefore
s.s
Tcombined
−
s.s
Tjellium
=
XZ
σ
k=kF
k=0
X
k 2 d∆N σ (k)
dk −
2
dk
σ
Z
nσs.s (r)V σ (r)dr
(3.4.9)
where ∆N σ (k) = N σ (k)combined − N σ (k)jellium .
Using the equation
σ
d∆N σ (k)
1 X dδl,m (k)
dk =
dk
dk
π
dk
(3.4.10)
l,m
allows us to write the first term as
XZ
σ
Writing k =
√
k=kF
k=0
Z
σ
k 2 d∆N σ (k)
1 X k=kF k 2 dδl,m (k)
dk =
dk
2
dk
π
2
dk
k=0
(3.4.11)
σ,l,m
2E and proceeding using integration by parts
Z
Z
σ
1 X E=EF dδl,m (E)
1 X
1 X E=EF σ
σ
=
E
dE =
EF δl,m (EF ) −
δl,m (E)dE
π
dE
π
π
E=0
E=0
σ,l,m
σ,l,m
σ,l,m
(3.4.12)
Putting all the contributions to ∆T together gives
∆T =
X
i,σ
Eiσ
−
XZ
nσ (r)V σ (r)dr+
σ
Z
1 X E=EF σ
1 X
σ
EF δl,m
(EF ) −
δl,m (E)dE
π
π
E=0
σ,l,m
σ,l,m
(3.4.13)
3.4 The Immersion Energy
81
where we have used nσ (r) = nσb.s (r) + nσs.s (r).
We look next at the Coulomb energy.
∆C = Ccombined − Cjellium
(3.4.14)
Cjellium is straightforward. For infinite jellium, the negative charge density at all points
equals the positive background charge: n(r) = n0
Therefore for pure jellium
Cjellium
1
=
2
Z
n(r)n(r0 )
drdr0 −
|r − r0 |
Z
n(r)n0
1
drdr0 +
0
|r − r |
2
Z
n20
drdr0 = 0
|r − r0 |
(3.4.15)
Notice the inclusion again of the background self-repulsion energy (the third term)
which makes Cjellium = 0. We will include this term in both Cjellium and Ccombined
however, resulting in its cancellation, and so including it was not strictly necessary.
The Coulomb term for the combined system is
Ccombined =
(n(r) − n0 ) (n(r0 ) − n0 )
drdr0
|r − r0 |
Z
Z
− dr (n(r) − n0 )
r
1
2
Z
(3.4.16)
Again, the background self-repulsion energy is included and in addition so is the energy
due to the ion-background charge repulsion. We therefore have
Z Z
(n(r0 ) − n0 ) 0 Z
1
∆C =
dr −
(n(r) − n0 ) dr
2
|r − r0 |
r
(3.4.17)
The final term to consider is the exchange-correlation term
combined
jellium
∆Exc = Exc
− Exc
(3.4.18)
For the pure jellium, we have Exc (n0 /2, n0 /2) and for the combined system we have
Exc n↑ (r), n↓ (r) . Therefore in total we have
n0 n0
∆Exc = Exc n↑ (r), n↓ (r) − Exc ( , )
2 2
Putting all these results together
Eimm =
X
i,σ
Eiσ −
XZ
σ
nσ (r)V σ (r)dr+
(3.4.19)
82
An Atom in Infinite Jellium Solved using DFT
Z
σ
1 X σ σ
1 X E=EF σ
EF δl,m (EF ) −
δl,m (E)dE+
π
π
E=0
σ,l,m
σ,l,m
Z Z
1
(n(r) − n0 ) 0 Z
dr −
(n(r) − n0 ) dr+
2
|r − r0 |
r
n0 n0
Exc n↑ (r), n↓ (r) − Exc ( , ) − Eatom
2 2
3.4.2
(3.4.20)
Finite Radius Corrections
The integrals over r in the above expression for the immersion energy read
−
XZ
σ
Z
r=∞
1
2
r=0
Z
Z
r0 =∞
r 0 =0
r=∞ r=∞
nσ (r)V σ (r)4πr2 dr+
r=0
(n(r) − n0 )
Z
2
4πr0 dr0 −
|r − r0 |
r
!
n(r)εxc (n↑ (r), n↓ (r)) − n0 εxc (
r=0
(n(r) − n0 ) 4πr2 dr+
n0 n0 , ) 4πr2 dr
2 2
(3.4.21)
In the code, we replace the upper limits of the integrations with r = rmax . We assume
that the charge density is equal to n0 and that the potential is zero for r ≥ rmax . The
assumption that the charge density is constant after rmax neglects Friedel oscillations
however. The true density outside rmax oscillates about n0 as a function of r with a
period π/kF . The potential V σ (r) also contains this Friedel oscillation. All of the terms
in the above expression are bilinear in the Friedel oscillation, except for the last. A first
order correction to the immersion energy, as calculated in the code, to correct for our
approximations of n(r) = n0 and V σ (r) = 0 beyond rmax is therefore
(correction)
∆Exc
Z
r=∞
=
n(r)εxc (n↑ (r), n↓ (r)) − n0 εxc (
r=rmax
n0 n0 , ) 4πr2 dr
2 2
(3.4.22)
The exchange-correlation energy density is first expanded about the jellium density
εxc (n↑ (r), n↓ (r)) = εxc (
n0 dεxc (n↑ (r), n20 ) n0 n0
, ) + (n↑ (r) − )
↑
2 2
2
dn↑
n (r) =
(n↓ (r) −
which using
n0 dεxc ( n20 , n↓ (r)) )
↓
2
dn↓
n (r) =
n0
2
n0
2
+
(3.4.23)
3.4 The Immersion Energy
83
dεxc (n↑ (r), n20 ) ↑
dn↑
n (r) =
n0
2
=
dεxc ( n20 , n↓ (r)) ↓
dn↓
n (r) =
(3.4.24)
n0
2
gives us
εxc (n↑ (r), n↓ (r)) = εxc (
dεxc (n↑ (r), n20 ) n0 n0
, ) + (n(r) − n0 )
↑
2 2
dn↑
n (r) =
n0
2
(3.4.25)
Inserting Eq. (3.4.25) into Eq. (3.4.22) and rearranging gives
Z
r=∞
n0 n0
, )4πr2 dr+
2 2
r=rmax
Z r=∞
dεxc (n↑ (r), n20 ) n(r)(n(r) − n0 )
4πr2 dr
↑
dn↑
n (r) = n20
r=rmax
(correction)
∆Exc
(n(r) − n0 )εxc (
=
(3.4.26)
The second term is re-written using n(r) = n0 + δn(r)
Z
r=∞
2
Z
r=∞
(n0 + δn(r))δn(r)4πr2 dr =
n(r)(n(r) − n0 )4πr dr =
r=rmax
r=rmax
Z
r=∞
n0
(n(r) − n0 )4πr2 dr + O(δn(r)2 )
(3.4.27)
r=rmax
Ignoring second order terms in δn(r) and combining the two terms
(correction)
∆Exc
=
dεxc (n↑ (r), n20 ) n0 n0
εxc ( , ) + n0
↑
2 2
dn↑
n (r) =
!Z
n0
2
r=∞
(n(r) − n0 )4πr2 dr
r=rmax
(3.4.28)
Finally, using
R r=∞
r=0
n(r)4πr2 dr = Z +
(correction)
∆Exc
R r=∞
r=0
n0 4πr2 dr, we obtain
dεxc (n↑ (r), n20 ) n0 n0
= εxc ( , ) + n0
↑
2 2
dn↑
n (r) =
Z r=rmax
2
Z−
(n(r) − n0 )4πr dr
!
n0
2
×
(3.4.29)
r=0
This correction to the immersion energy is included in our calculations.
3.4.3
Numerical Parameters and Error Analysis
In the next section we will discuss our calculations of immersion energies for various atomin-jellium systems. Before that, we will briefly cover some of the numerics involved in these
calculations.
84
An Atom in Infinite Jellium Solved using DFT
A number of numerical parameters need to be set when performing these calculations.
For an atom in infinite jellium, these include lnum , the maximum angular momentum value
used in calculating the scattering state solutions. There are also numerical parameters to
do with the radial mesh, which include the minimum radius, rmin , the maximum radius,
rmax , the radius at which the r-mesh changes from logarithmic to linear, rint , and the
(log)
(lin)
number of points used in the logarithmic and linear parts of the mesh, rnum and rnum
respectively. We also have to set numerical parameters relating to the k-mesh, including
the minimum k value, kmin , and the number of points used for each angular momentum
(l=0)
(l=1)
(l=l
)
value, knum , knum , ..., knumnum . Another numerical parameter is ∆V req , which is the
value below which the quantity defined in Eq. (3.1.18) must pass in order that we can
claim to have achieved a self-consistent solution. For the case of an atom in a finite
(l)
jellium sphere, we do not have the lnum and knum parameters.
We will describe how these numerical parameters are set by using the example of a
SI-corrected cerium atom in infinite jellium of density n0 = 0.01a−3
B . First we consider
∆V req , which we choose so that the error due to lack of complete convergence is sufficiently
small. Fig. 3.4 shows that if we set ∆V req = 10−10 then we obtain an error of one fifth of
a meV, which is suitable for our purposes.
We set rmax so that approximately two of the Friedel oscillations described by Eq. (3.2.40)
are present in a plot of (n(r) − n0 )r3 against r (Fig. 3.6 shows examples of such plots).
For the system under consideration, this occurs with rmax ∼ 20aB .
In the case of a non-SI corrected system, the precise value of rmax has to be chosen
so that the Friedel sum described by Eq. (3.2.42) is satisfied. In practice we find that as
we vary rmax , the left-hand side of the equation oscillates sinusoidally about the correct
value as given by the right-hand side of the equation. This oscillation occurs with the
wavelength of the Friedel oscillation. In practice then, having chosen an rmax which yields
approximately two Friedel oscillations, we use a bisector algorithm in order to home in on
a choice of rmax which satisfies Eq. (3.2.44) to some specified accuracy. This accuracy is
chosen to be ±10−5 , which corresponds to an error-bar in the immersion energy of ∼ 10−5 .
In fact, it turns out that this fine-tuning of rmax in order to satisfy the Friedel sum
is equivalent to choosing an rmax so that (n(r) − n0 )r3 correctly reproduces the Friedel
oscillations at r = rmax . Fig. 3.5 illustrates this, by showing the density profiles for
a number of choices of rmax , only one of which coincides with the correct theoretical
prediction.
3.4 The Immersion Energy
85
32.6118
32.6116
Immersion Energy/eV
32.6114
32.6112
32.6110
32.6108
32.6106
32.6104
32.6102
32.6100
-11.5
-11
-10.5
-10
-9.5
-9
log(convsu)
-8.5
-8
-7.5
-7
Figure 3.4: Determining the parameter ∆V req , for a SI-corrected cerium atom in infinite
req ), and
jellium of density n0 = 0.01a−3
B . Immersion energy is plotted against log(∆V
error bars (in green) are placed at different values of the convergence.
86
An Atom in Infinite Jellium Solved using DFT
placements
0.1
rmax =24.676aB
rmax =25.326aB
rmax =25.963aB
0.08
(n(r) − n0 )r3
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
0
5
10
15
20
25
r / aB
Figure 3.5: Density plots for hydrogen in infinite jellium of density 0.005a−3
B . Values for
rmax equal to 24.676aB , 25.326aB and 25.963aB are shown. Only the second choice of
rmax gives the correct form for the density oscillation (the peak of the last oscillation is at
the same height as the penultimate oscillation). The values of the Friedel sum for these
choices are 0.98, 1.00 and 1.02 respectively, showing that selecting rmax to get the correct
density profile is equivalent to selecting rmax to satisfy the Friedel sum.
3.4 The Immersion Energy
87
For a SI-corrected system, Eq. (3.2.42) does not hold. This is because as we shall see
later in Section 3.6.4, when we apply SIC to a bound state, the scattering states of the
same l, m and σ are no longer orthogonal to the bound state (as is required in the KohnSham formulation of DFT). We must therefore orthogonalise the scattering states against
the SI-corrected bound state. This leads to a change in the atom-induced scattering state
density and therefore, the relation
d∆Nlσ
dk dk
=
σ
1 dδl
π dk dk,
which was used in the proof of the
Friedel sum, no longer holds.
Therefore when using SIC, instead of choosing rmax to satisfy the Friedel sum, we
choose rmax instead so that (n(r) − n0 )r3 correctly reproduces the Friedel oscillations at
r = rmax . This method of choosing rmax is less accurate than the method of satisfying the
Friedel sum. Ideally we would like to construct an equation similar to Eq. (3.2.42) but for
SI-corrected systems, which would then enable us to follow the same procedure as for the
non-SIC case.
Having chosen rmax , the parameter lnum is chosen so that the density is calculated
correctly for all values of r. We check that this is the case by comparing the quantity
(n(r) − n0 )r3 to the theoretical Friedel oscillations (Eq. (3.2.40) ). The two curves should
roughly coincide at large radius. If however, the density drops off beyond some radius,
this tells us that we should increase lnum . Once the density is correctly calculated at all
radii, further increases in lnum will yield no further improvement in accuracy. In the case
of the cerium atom in jellium, we find that for an rmax value of 20aB we need lnum = 20.
Fig. 3.6 shows how we determined this value.
In order to obtain an immersion energy with an error-bar less than 1meV , we choose
(log)
(lin)
rint , rnum and rnum to be 1aB , 800 and 200 respectively. It turns out that these parameters
(lin)
work well with all systems considered in this thesis (although the particular choice of rnum
(l)
depends on rmax ). We choose knum = 240 for all values of l except for l = 3, which contains
a narrow resonance. Here we put in an extra 100 or so points around the resonance to
(l)
enable a more accurate calculation. A choice of knum = 240 is good for all systems
considered in this thesis, although extra k values are required for cases where there is a
resonance which is particularly narrow (up to 1000 extra points in some cases).
3.4.4
Results
Fig. 3.7 shows our immersion energy versus background density curves for atoms with
atomic numbers 1 to 18 immersed in jellium. We check our results against those by Puska
88
An Atom in Infinite Jellium Solved using DFT
lnum=10
2
(n(r)-n0)r3
1.5
1
0.5
0
-0.5
0
5
10
rmax/aB
15
20
15
20
lnum=13
2
(n(r)-n0)r3
1.5
1
0.5
0
-0.5
0
5
10
rmax/aB
3.4 The Immersion Energy
89
lnum=15
2
(n(r)-n0)r3
1.5
1
0.5
0
-0.5
0
5
10
rmax/aB
15
20
15
20
lnum=20
2
(n(r)-n0)r3
1.5
1
0.5
0
-0.5
0
5
10
rmax/aB
Figure 3.6: Determining the parameter lnum . This value has to be large enough so that,
for a given rmax , the density is correctly realised at all radii. The above are results for
a cerium atom immersed in jellium of density 0.01aB , with rmax ≈ 20aB . The red curve
corresponds to the actual calculated density, the green curve to the theoretical density
(Eq. (3.2.40) ). We see that only the final choice of lnum (= 20) gives the correct density
profile, and therefore this is the value that we use.
90
An Atom in Infinite Jellium Solved using DFT
et al [5], which are reproduced here in Fig. 3.8. For the sake of the comparison we use the
same exchange-correlation functional as used by Puska et al, namely that by Gunnarsson
and Lundqvist [54]. Our curves are in good agreement with those by Puska et al.
Our calculations show two types of immersion energy versus background density curve.
The first type of curve rises almost linearly and has no minimum. The second type of curve
has a negative minimum and then proceeds to rise almost linearly.
Inert atoms exhibit the first type of curve. For these atoms, the positive immersion
energies correspond to the repulsive interaction of these atoms with any type of electronic
environment. The second type of curve occurs for atoms which form negative ions when
added to jellium in the limit that the background density of the jellium tends to zero.
For example, hydrogen binds two electrons as the background density tends to zero (and
indeed across the entire range of background densities considered on our graphs) and
therefore forms a negative ion in this limit. The immersion energy curve for hydrogen
is thus of the second type. Helium however, has an immersion energy curve of the first
type. This is because Helium also binds two electrons, however in this case this results in
a neutral atom.
For both types of curve, the immersion energy increases in an approximately linear
manner for large enough background densities. This is because the extra states introduced
by higher jellium densities have to be made orthogonal to one another. Therefore electrons
are pushed further from the ion and the reduction in energy associated with proximity to
the ion is lost for these electrons.
3.5
3.5.1
The Effective Medium Theory
Background Theory
The EMT [45] is a theory which uses the atom in jellium model as a building block from
which to construct a full theory of a condensed matter system. The only input parameter
in the theory is the atomic number, Z. The theory has been successfully put to use in the
calculation of cohesive properties of solids, amongst other uses. In particular, the theory
reproduces trends in the experimental lattice constants, bulk moduli and cohesive energies
across the periodic table [45, 47, 4].
Working within the framework of the LDA (in its non spin-dependent form), we start
by writing down the potential, V (r), appearing in the Kohn-Sham equations for a system
3.5 The Effective Medium Theory
14
10
8
6
4
2
-2
0.01
0.02
n0 / aB-3
4
2
0.03
0
60
F
Ne
Na
Mg
20
15
10
5
0
0.01
0.02
n0 / aB-3
0.03
0.01
0.02
n0 / aB-3
0.03
Al
Si
Cl
Ar
50
Immersion Energy / eV
Immersion Energy / eV
25
6
-2
0
30
8
0
0
35
B
C
N
10
Immersion Energy / eV
Immersion Energy / eV
12
H
He
Li
Be
12
91
40
30
20
10
0
-5
-10
-10
0
0.01
0.02
n0 / aB-3
0.03
0
Figure 3.7: Immersion energy versus background density curves for atoms with atomic
numbers 1 to 18 as obtained by our calculations. Elements P, S and O are excluded
because of difficulty in obtaining converged solutions for these elements.
92
An Atom in Infinite Jellium Solved using DFT
Figure 3.8: Immersion energy versus background density curves for atoms with atomic
numbers 1 to 18 as calculated by Puska et al [5]. Elements P and S were excluded because
of unsatisfactory convergence of solutions.
3.5 The Effective Medium Theory
93
of ions of charge Z at lattice positions, Ri
Z
V (r) =
n(r0 ) −
0
i Zδ(r
0
|r − r |
P
− Ri )
dr0 + Vxc (n(r))
(3.5.1)
We next propose an ansatz for the ground-state density of this system of ions, which
is a sum of overlapping densities each centred on a site Ri
n(r) =
X
∆ni (r)
(3.5.2)
i
The specific form of ∆ni will be dealt with later on. Inserting this form into the above
equation yields
V (r) =
X Z ∆ni (r0 ) − Zδ(r0 − Ri )
dr0 + Vxc (n(r))
|r − r0 |
i
Z
=
∆ni
(r0 )
− Zδ(r0 − Ri ) 0 X
dr +
∆Φj (r) + Vxc (n(r))
|r − r0 |
(3.5.3)
j6=i
where i labels the Wigner-Seitz (WS) cell corresponding to site Ri (which we denote with
ai ) in which r lies, and where we have introduced the quantity, ∆Φi (r)
Z
∆Φi (r) =
∆ni (r0 ) − Zδ(r0 − Ri ) 0
dr
|r − r0 |
(3.5.4)
Next, we introduce the quantity, ∆Vi (r)
Z
∆Vi (r) =
∆ni (r) − Zδ(r0 − Ri ) 0
dr + Vxc (n̄i + ∆ni ) − Vxc (n̄i )
|r − r0 |
(3.5.5)
Let us choose ∆ni to be the atom-induced density of an atom immersed in a homogeneous electron gas of background density n̄i , with the atom centred on Ri . With this
choice, the above quantity is just the Kohn-Sham potential for the same system. Furthermore, ∆Φi (r), is the Hartree potential for this system.
Re-writing Eq. (3.5.3) in terms of ∆Vi (r) gives
V (r) = ∆Vi (r) − Vxc (n̄i + ∆ni ) + Vxc (n̄i ) +
X
∆Φj (r) + Vxc (n(r))
(3.5.6)
j6=i
Now, let us set n̄i to be the sum of the density tails from all other cells, spatially
averaged over cell ai . I.e.:
94
An Atom in Infinite Jellium Solved using DFT
X
n̄i = h
∆nj (r)iai
(3.5.7)
j6=i
Next we make the approximation that the sum over the density tails does not vary
much over cell ai , I.e.
n̄i ≈
X
∆nj (r)
(3.5.8)
j6=i
in this case terms 2 and 5 of Eq. (3.5.6) cancel. Furthermore, if we assume that n̄i does not
vary much from cell to cell, then the third term is just a constant and can be neglected.
Next, just as for the density tails, we assume that the sum of the tails of the Hartree
potentials from all other cells does not vary much over cell ai . Therefore the average of
this quantity over cell ai is
Φ̄i ≈
X
∆Φ(r)
(3.5.9)
j6=i
Again, we assume that this quantity does not vary much from cell to cell. Therefore
term 4 in Eq. (3.5.6) is also dropped as a constant. We therefore have
V (r) = ∆Vi (r) in cell ai
(3.5.10)
So in a given cell, ai , the potential is fixed to that of an atom immersed in a homogeneous electron gas of background density n̄i .
Let us now construct the LDA energy functional for the periodic array of ions. With
P
vext (r) = − i Z/|r − Ri | this is
1
E[n, v] = T [n, v] +
2
Z
X
n(r)n(r0 )
0
drdr
−
|r − r0 |
Z
Zn(r)
dr + Exc [n]
|r − Ri |
i
(3.5.11)
inserting Eq. (3.5.2) gives
1X
E[n, v] = T [n, v] +
2
ij
Z
X
∆ni (r)∆nj (r0 )
0
drdr
−
|r − r0 |
ij
Z
Z∆nj (r)
dr + Exc [n] (3.5.12)
|r − Ri |
The quantity we are interested in is the cohesive energy of the solid
3.5 The Effective Medium Theory
95
∆E[n, v] = E[n, v] −
X
E atom
(3.5.13)
i
where E atom is the energy of the constituent atom of the solid in a vacuum. We refer the
reader to the original paper [45] for the rest of the derivation, and only present an outline
of the remaining steps here.
The above binding energy is re-written in terms of the immersion energy of an atom
in an electron gas, Eimm (n̄i ), and a number of additional terms. Many of these terms can
be neglected when we make the so-called atomic sphere approximation (ASA). The ASA
involves approximating the WS cells as so-called atomic spheres. These atomic spheres
have the same volume and charge as the WS cell. They are thus charge neutral:
Z
r=s
n(r)dr = Z
(3.5.14)
r=0
where s is known as the neutral sphere radius, and is interpreted as the WS radius when
using the ASA. The final ∆E[n, v] is
∆E[n, v] =
X
Ec (n̄i ) + ∆E1−el
(3.5.15)
i
where
Z
r=s Z
Ec (n̄) = Eimm (n̄) + n̄
r=0
∆n(r0 ) 0 Z
dr −
|r − r0 |
r
dr
(3.5.16)
which is the cohesive energy per atom. Here ∆n(r) is the atom-induced density centred
on the origin. The second term is attractive and has the effect of lowering Ec (n̄). It can
be viewed as the attraction of the sum of the density tails from all other cells (n̄) with the
Hartree potential from cell ai .
The ∆E1−el term is the sum of the change of the one-electron energy eigenvalues when
we go from the homogeneous electron gas to the real host. This change occurs because
of covalent bonding, hybridisation and effects due to wavefunction orthogonalisation. A
number of ways have been proposed to include this term [77, 78, 79, 80, 81], however in
our calculations we will neglect this term. Our reason for neglecting the term is that the
focus of our results is on reproducing the minima seen in the experimental WS radius as
a function of atomic number. As we shall see, the first two terms in Eq. (3.5.15) will be
sufficient for this purpose.
96
An Atom in Infinite Jellium Solved using DFT
5
Wigner Seitz Radius / aB
4.5
4
3.5
3
2.5
2
1.5
1
0
5
10
15
20
25
30
Atomic Number
35
40
45
50
Figure 3.9: Squares are experimental Wigner-Seitz radii, blue diamonds are our neutral
sphere radii, crosses are neutral sphere radii as calculated by Yxklinten et al [4].
The procedure in EMT is to solve for some arbitrarily chosen n̄, solve the self-consistent
problem of an atom in a homogeneous gas in order to obtain ∆n(r) and then evaluate
Ec (n̄). One then varies n̄ in order to minimise Ec (n̄). This minimum value of Ec (n̄) is
then the cohesive energy per atom of the solid, and the corresponding value of s is the
WS radius as predicted by the theory.
3.5.2
Results
The EMT described in Section 3.5.1 is used to calculate WS radii for solids with constituent
elements up to the 3d transition metals. These calculations have already been performed
by Yxklinten et al [4]. We compare these published results with our own calculations.
In addition we present new results for the 4d transition metal elements which have not
previously been calculated. We present our WS radii as well as those of Yxklinten et al
and also the experimental WS radii in Fig. 3.9.
3.6 Cerium Solved using the LDA and SIC
97
We see that our results for elements up to Zinc are in good agreement with those of
Yxklinten et al. Furthermore, these results correctly reproduce the same minima seen in
the experimental WS radii as we fill a particular atomic sub-shell (1p, 2p and 3d).
In addition, our new results for the 4d transition metals also show the same minimum
as is shown by the experimental WS radii. This enhances our confidence in the suitability
of the EMT for describing properties of condensed matter systems.
We have not calculated the WS radii for all elements up to the 4d transition metals
due to difficulty in obtaining convergence in some cases. This difficult was due to the
scattering state resonance being on the verge of crossing over to become a bound state.
This occurred for Z = 47 and Z = 48, for example, where the d-resonance becomes very
low in energy (due to the increasing nuclear attraction) and consequently takes on a very
abrupt step-like form in the phase-shift. One has to increase the number of energy values
for which the scattering states are calculated in order to correctly model this step-like
function, which slows down the code substantially. However, in principle and given more
time, there is no reason why the WS radii cannot also be calculated for these trickier
elements. At present however it is sufficient that we have demonstrated that the minima
in the WS radii are correctly reproduced by the theory.
Fig. 3.10 shows the cohesive energy versus neutral sphere radius curves which are used
to obtain the EMT predictions for the WS radii. As we discussed in Section 3.5, the
neutral sphere radius at the minimum of this curve for a particular value of Z is the WS
radius, as predicted by the theory, for a solid made up of this constituent atom. This
minimum can be seen to move left as we increase Z from 39 until we get to Z = 44, at
which point the minimum starts to move to the right.
3.6
3.6.1
Cerium Solved using the LDA and SIC
Introduction
In this section we solve for the system of a cerium atom embedded in infinite jellium using
the LDA and SIC. As we have discovered, using a theory such as the EMT enables the
solution to this system to be used as a building block out of which the solution to the full
periodic solid can be constructed (albeit in an approximate manner).
From this viewpoint, the bound states of the atom-in-jellium system (which are well
localised) can be interpreted as the bound states localised around each atom in the full
98
An Atom in Infinite Jellium Solved using DFT
6
Yttrium
Zirconium
Niobium
Molybdenum
Technetium
Ruthenium
Rhodium
4
Cohesive Energy / eV
2
0
-2
-4
-6
-8
-10
-12
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
Neutral Sphere Radius / aB
3.8
4
Figure 3.10: Cohesive energy versus neutral sphere radii for 4d transition metals.
4.2
3.6 Cerium Solved using the LDA and SIC
99
crystal. Similarly, the atom-induced scattering states (which are more delocalised than the
bound-states) can be taken to be the valence electrons per atom in the full periodic solid.
We will use this interpretation of the bound and scattering states of the atom-in-jellium
system in order to use this system to model the alpha and gamma phases of cerium.
This is similar to work done by Lüders et al [25]. They used full periodic LDA and
SIC solutions to model these phases of cerium, and in fact were able to construct energy
curves and calculate equilibrium atomic volumes for these phases (as discussed in Chapter
1). In our calculations, we have not yet constructed energy curves or calculated atomic
volumes. In order to achieve this we could use a theory such as the EMT.
3.6.2
Cerium
Cerium is interesting because it is the first element in the periodic table to contain an
f -electron. It also exhibits a pressure-induced isostructural (fcc to fcc) phase transition
(see Fig. 3.11). There is a large 15% volume collapse [26] under this phase transition.
The high volume phase is the gamma phase and the low volume phase is the alpha phase.
There are local magnetic moments on the f -electron sites in the gamma phase but not in
the alpha phase. Experimental data (see Fig. 3.12) shows how the molar volume varies
with pressure.
In the Mott transition model of this phase transition [27], the 4f electrons are localised in the gamma phase and itinerant in character in the alpha phase. The bonding
contribution due to the extra itinerant electrons in the alpha phase then accounts for the
volume collapse. The disappearance of the local magnetic moment on the f -electron sites
is also straight-forwardly explained by the delocalisation of the f -electron. In this model
of the phase transition, the gamma phase has a valence of three, whereas the alpha phase
corresponds to an intermediate valence between three and four.
In the work by Lüders et al [25], LDA and SIC solutions for the full periodic solid were
used to model the alpha and gamma phases of cerium respectively. We also follow this
approach, and in Sections 3.6.3 and 3.6.5 solve our atom-in-jellium model using the LDA
and SIC.
100
An Atom in Infinite Jellium Solved using DFT
Figure 3.11: Experimental phase diagram of cerium [6]
3.6.3
Spin-polarised LDA for Cerium
In free space a cerium atom consists of the bound states : [Xe] 4f 1 , 5d1 , 6s2 . Some of these
bound states will become scattering states when we embed the atom in jellium, reflecting
the weaker hold the cerium atom has on the electrons. We see this in our calculations of
a cerium atom in jellium solved using the spin-polarised LDA. For example, when solving
for a cerium atom embedded in jellium with rs = 1.8 we find that the 4f , 5d and 6s bound
states enter the continuum and become scattering states. This corresponds to a valency
of four in the real solid, with no 4f bound state electron localised to each atom. Therefore
this solution corresponds approximately to the alpha phase of cerium (the valence for the
alpha phase is in between three and four however, and so this correspondence is not exact).
As we lower the jellium density (increase rs ) we expect some of the atom-induced
scattering states to fall down in energy and reappear again as bound states. This lowering
of the background density is analogous to decreasing the external pressure. Therefore one
might expect that the f -electron would form a bound state first, in line with the Mott
transition model. However this is not the case, and it is the 6s bound state which is the
first to form as we lower the background density.
3.6 Cerium Solved using the LDA and SIC
101
Figure 3.12: Experimental results showing the molar volume of cerium against the pressure
applied to the sample [7]
102
An Atom in Infinite Jellium Solved using DFT
The reason for this is because in the LDA, the effective potential for a given spin (the
total potential, V σ (r) plus the centrifugal term, l(l + 1)/2r2 ) is the same for all values of
m. Therefore if there is a bound state solution to the l = 3 radial Schrödinger equation for
a given spin, σ, then that bound state exists for all seven values of the magnetic quantum
number, m. Therefore all seven electrons must occupy the bound state. However because
the Coulomb repulsion energy associated with a seven electron bound state is prohibitively
large, such a state would never form. An s bound state however, holds just one electron
and therefore there is no such barrier to the formation of such a bound state, as indeed
happens, as we lower the background density.
Figure 3.13 shows plots of phase-shifts for cerium in jellium at different rs . According
to Levinson’s theorem, the formation of an s bound state would manifest itself in a jump
in the l = 0 phase-shift by π at zero energy (if we fix the value of the phase-shift to be
zero at infinite energy). As a precursor to this, we see a hump that develops in the l = 0
phase-shift as rs is increased. When this hump reaches zero energy, the phase-shift will
jump by π.
Figure 3.14 shows the scattering density of states for a cerium atom in jellium of
rs = 5.3aB . The f -resonance is extremely narrow. The peak corresponding to the l=2
scattering state is markedly less so.
In Section 3.6.5 we will solve the system using SIC. First however, we need some extra
background theory.
3.6.4
Imposing Orthogonality when Applying SIC
In section 2.3.5 we discussed how the Kohn-Sham orbitals calculated from the SI-corrected
Schrödinger equation are no longer automatically orthogonal to one another. Eq. (2.3.61)
Z
σ
φσ∗
α (r)φα0 (r)dr = 0
(3.6.1)
now has to be imposed by Gram-Schmidt orthogonalisation.
Note that orbitals with different l, m or σ values will automatically be orthogonal to
one another on account of the angular or spin part of the wavefunction.
Z
0
σ
φσ∗
l,m (r)φl0 ,m0 (r)dr =
Z
Z
σ∗
σ0
2
∗
Rl,m (r)Rl0 ,m0 (r)r dr × Yl,m
(θ, φ)Yl0 ,m0 (θ, φ) sin θdθdφ × δσ,σ0
3.6 Cerium Solved using the LDA and SIC
103
4
δl ()
deltal(epsilon)/a.u.
3
2
1
0
-1
-2
-3
0
0.5
1
1.5
2
2.5
3
√sqrt(epsilon)/a.u.
2 (a.u)
4
3
δl (²)
2
1
0
-1
PSfrag replacements
-2
-3
0
0.2
0.4
0.6
√ 0.8 1 1.2
2² (a.u)
1.4
1.6
Figure 3.13: Phase-shifts for (non-magnetic) ground-state solutions of a cerium atom
embedded in jellium of different densities. From top to bottom, rs = 1.81aB , rs = 3.24aB ,
rs = 5.30aB . The red, green, blue and magenta curves correspond respectively to l = 0,
l = 1, l = 2 and l = 3 (and are also labelled in the bottom panel).
104
An Atom in Infinite Jellium Solved using DFT
50
dN l ()
−1
(eV
)
dNscatt/dE/ev-1
d
40
Fermi level
30
l=3
20
l=2
10
0
0.355 0.36 0.365 0.37 0.375 0.38 0.385 0.39 0.395 0.4
sqrt(2*epsilon)/a.u.
√
2 (a.u)
Figure 3.14: Angular momentum resolved density of states for the ground-state solution
of a cerium atom embedded in jellium of rs = 5.3
3.6 Cerium Solved using the LDA and SIC
Z
=
0
σ∗
Rl,m
(r)Rlσ0 ,m0 (r)r2 dr × δl,l0 δm,m0 δσ,σ0
105
(3.6.2)
With this in mind, we only need to impose orthogonality between states of the same
l, m or σ values. Since our wavefunctions are spherically symmetric, this means a set of
equations which look like
UEorth
(r) = UE3 (r) −
3
Z
Z
UE1 (r0 )UE3 (r0 )dr0 UE1 (r)−
UE2 (r0 )UE3 (r0 )dr0 UE2 (r)
(3.6.3)
where orbitals are labelled with their energy eigenvalues, and where E1 < E2 < E3 , etc.
In the case of cerium, we will apply the SIC to one electron in the 4f bound state
(with magnetic quantum number m0 and spin σ 0 ). Therefore this bound state will no
longer be orthogonal to the f -scattering states with m = m0 and spin σ 0 . We therefore
have to impose orthogonality between the 4f bound state and these scattering states.
Note that the magnetic quantum number m0 is arbitrary because of our approximation
that the density is spherically symmetric.
We use Gram-Schmidt orthogonalisation to orthogonalise the bound state and the
scattering states with respect to one another
orth
Uscatt
(k, r)
Z
= Uscatt (k, r) −
Uscatt (k, r0 )Ubound (r0 )dr0 Ubound (r)
(3.6.4)
This orthogonalisation must be performed for all k values between 0 and kF . Notice
that the normalisation factor for the scattering states cancels out. Using scattering state
solutions normalised according to Uscatt (k, r → ∞) → sin(kr + δlσ (k) − lπ/2)/k therefore
orth (k, r) with the same normalisation. These new scattering states can then
generates Uscatt
be inserted straightforwardly into Eq. (3.1.10) in order to calculate the new density.
Notice that we have orthogonalised the scattering states against the bound state and
not the other way around. This is because making the bound state orthogonal to the
scattering states would result in a bound state with scattering state character, and the
bound state would no longer satisfy the appropriate boundary condition at large r.
Now that we have orthogonalised the l = 3, m = m0 , spin σ 0 scattering states against
the SI-corrected bound state, we have to convince ourselves that the l = 3, m = m0 , spin
σ 0 scattering states are still orthogonal to one another. Here follows a proof that this is
the case.
106
An Atom in Infinite Jellium Solved using DFT
As before, let us impose hard-wall boundary conditions at rhw , and normalise each of
the scattering states to one in the range r = 0 to r = rhw . From Section 3.2.4 this means
p
orth (k, r). We have
putting the factor 2/rhw in front of the Uscatt
Z r
r
2
rhw
orth
kUscatt
(k, r)
2
2
rhw
orth 0
k 0 Uscatt
(k , r)dr =
Z
Uscatt (k, r)Uscatt (k 0 , r)dr+
rhw
Z
Z
Z
Ubound (r)Ubound (r)dr − 2
Uscatt (k, r)Ubound (r)dr Uscatt (k 0 , r0 )Ubound (r0 )dr0
kk
0
(3.6.5)
We are interested in taking the limit rhw → ∞ since this corresponds to our infinite
jellium system. All integrals containing Ubound (r) will be finite, and so terms containing
such integrals will not survive when we take this limit. Therefore we have
Z r
2
rhw
r
orth
kUscatt
(k, r)
2
rhw
k
0
orth 0
Uscatt
(k , r)dr
= kk
0
2
Z
rhw
Uscatt (k, r)Uscatt (k 0 , r)dr (3.6.6)
This is just the overlap integral for scattering states before the SIC has been imposed,
which we know to be zero. Hence we have demonstrated that the l = 3, m = m0 , σ = σ 0
scattering states are still orthogonal to one another.
3.6.5
SIC-LDA for Cerium
We apply SIC to one electron in the 4f bound state. The procedure is to achieve selfconsistency with respect to three potentials: V σ (r) and V SIC (r). The latter is used when
calculating the lowest lying l = 3 bound state, I.e. the 4f bound state. To begin with, we
make a guess for this potential by choosing a potential which yields an f bound state. As
discussed in Section 3.1.2, we approximate this bound state as being spherically symmetric
by calculating the density contribution from the state as
nSIC (r) =
2 (r)
USIC
4πr2
(3.6.7)
The new SIC potential is then calculated using Eq. (2.3.60). The potentials are then
made self-consistent.
A point should be raised here regarding the f -electrons which have not been SIcorrected. From Fig. 3.14 we see that these electrons (which number more than the single
3.6 Cerium Solved using the LDA and SIC
107
electron which we have SI-corrected) feature in a very pronounced resonance. These
electrons are therefore very well localised in space, and therefore feature a strong selfinteraction. They should therefore be treated within the SIC theory. However, in the
implementation of SIC used in this thesis (which is the original formulation put forward
by Perdew and Zunger [24]) only bound-states can be SI-corrected, and therefore the electrons in the f resonance cannot be SI-corrected. Alternative formulations of SIC, such
as the local-SIC formulation by Lüders et al [25] do include SI-correction of continuum
states. Therefore, a future improvement to these calculations would be to SI-correct these
resonant electrons within a SIC formulation such as this.
The ground-state solution for SIC, for a certain range of background densities, is found
to contain an f bound state. This solution contains three atom-induced scattering state
electrons and a localised f electron and therefore corresponds to the gamma phase of
cerium. As the background density is increased, this bound state enters the continuum of
scattering states. This is shown in Fig. 3.15. Notice that the highest background density
for which we have placed a calculated bound state energy is 0.014a−3
B . This is because
we encountered convergence problems above this density. However, the extrapolated line
clearly shows a cross-over at approximately 0.0155a−3
B .
One could try to interpret the cross-over point at which the 4f bound state becomes
a scattering state as the cross-over point between the alpha and gamma phases of cerium.
However, this would be erroneous. To see why, simply consider the change in volume
across this transition using the simple model discussed in Chapter 1, Fig. 1.5. In this
model, the background density equals the effective charge of an ion smeared out over the
atomic unit cell, I.e. n0 = Nv /Ω, where Nv is the number of valence electrons per atom
and Ω is the atomic unit cell volume. The change in volume as we go from gamma to
alpha phases using this model would be
∆Ω =
4
3
−
0.0155 0.0155
in units of a−3
B . This is positive, when it should be negative!
In order to properly calculate the volume change across the phase-transition, one needs
to use a theory such as the EMT to construct curves of energy against atomic volume for
the two phases. The minima of the total energy curves will then give the equilibrium atomic
volume of the two phases, and the volume change per atom is then just the difference
between these equilibrium atomic volumes. This procedure was followed by Lüders et al
108
An Atom in Infinite Jellium Solved using DFT
Energy of 4f bound-state / eV
0
rag replacements
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
0.01
0.011
0.012
0.013
Density /
0.014
0.015
0.016
a−3
B
Figure 3.15: Energy of the 4f bound state for a SI-corrected cerium atom immersed in
jellium, as a function of the background jellium density. The points are calculated energies,
and the line is extrapolated to zero energy.
[25], as discussed in Chapter 1, and their results were in good agreement with experiment.
3.6.6
Magnetic Solution of Cerium
The SIC solution is magnetic on account of the spin-up 4f bound state. In addition, the
number of atom-induced spin-up scattering state electrons will be different to the number
of atom-induced spin-down scattering state electrons, providing a further contribution to
the magnetism.
In the LDA solution, the bound states are the same for both spin-up and spin-down
electrons. However, the number of atom-induced spin-up and spin-down scattering state
electrons are different, meaning that we still see the formation of a magnetic moment.
This can be seen in Fig. 3.16. Here the spin-up and spin-down phase-shifts start to
move apart as the background density is decreased. The f resonance increases in energy
for the minority spin and decreases for the majority spin. For the minority spin, this
causes more electrons in this resonance to move past the Fermi energy and therefore be
excluded from the calculation. For the majority spin, the opposite occurs, with more
3.6 Cerium Solved using the LDA and SIC
electrons being brought below the Fermi-level and into the calculation.
109
110
An Atom in Infinite Jellium Solved using DFT
Figure 3.16: Phase-shifts for the LDA solution of a cerium atom immersed in infinite
jellium for a variety of background densities. From top to bottom, n0 = 0.04a−3
B , n0 =
−3
−3
0.03a−3
B , n0 = 0.02aB and n0 = 0.01aB . The separation of the spin-up and spin-down
phase-shifts as the background density is increased corresponds to the formation of a
magnetic moment on the cerium atom.
Chapter 4
Hydrogen Immersed in a Finite
Jellium Sphere
In this chapter, a comparison is made of the ground-state solutions of a hydrogen atom immersed in jellium as calculated by the local density (LDA) and self-interaction correction
(SIC) approximations of density functional theory (DFT), Hartree-Fock (HF) and variational quantum Monte Carlo (VQMC). In order to perform the calculation with VQMC,
it is necessary to replace the infinite jellium background with one which is finite in size.
Jellium spheres are chosen, which we centre on the atom. Ideally, the jellium sphere must
be large enough so that it reasonably well approximates infinite jellium. In this way our
results can be used to try to say something about the infinite jellium case. To this end,
in Section 4.1 a study is made using the LDA of the dependence of the immersion energy
and the atom-induced density on the size of the jellium sphere.
Having chosen a size for the jellium sphere, the system is solved using VQMC in Section
4.2. A comparison is made of the curves of the immersion energy and total energy versus
background density as calculated using HF, LDA, SIC and VQMC. The electron density
curves as calculated using the different methods are also compared.
4.1
Hydrogen in Finite Jellium Spheres using the LDA
In this section we solve the hydrogen atom in a finite jellium sphere using the LDA. We
fix the positive background density of the jellium and gradually increase the number of
electrons in the jellium sphere. As we do so, we calculate quantities such as the immersion
111
112
Hydrogen Immersed in a Finite Jellium Sphere
energy and the atom-induced density. We investigate the dependency of these quantities
on the size of the jellium sphere, and in particular seek to establish how large our jellium
sphere needs to be in order that our system of an atom in a jellium sphere approximates
the system of an atom immersed in infinite jellium.
Similar calculations have been performed by previous authors [82, 83]. In particular,
the atom-induced density and immersion energy have been plotted as a function of the
radius of the jellium sphere for a single choice of background density. We improve on these
results by, in the case of the immersion energy curves, plotting the immersion energy as
a function of jellium sphere radius for a variety of background densities. We also include
more points in these plots, enabling us to correctly identify the Friedel oscillations which
occur in these plots - an important feature which was overlooked by the previous authors.
4.1.1
Energy of An Atom in a Finite Jellium Sphere
In order to calculate the immersion energy for an atom in finite jellium we need to subtract
the total energy of an N -electron jellium sphere and the energy of a hydrogen atom from
the total energy of a hydrogen atom in an N electron jellium sphere. To do this, we
need to derive the energy of an atom immersed in a finite jellium sphere in the LDA. The
starting point is the DFT energy functional (Eq. (2.3.17), with Ts given by Eq. (2.3.27) ).
↑
↓
E[n , n ] =
XZ
i,σ
|
1
2
|
Z Z
n(r)n(r0 )
|r − r0 |
{z
=ECoulomb
1 2 σ
φσ∗
i (r)(− ∇ )φi (r)dr +
2
{z
}
=Ekinetic
Z
drdr0 + vext (r)n(r)dr +Exc [n↑ , n↓ ]
{z
}
} |
(4.1.1)
=Eexternal
where vext (r) is the external potential due to the positive background of the jellium and
the atom.
We can use the Kohn-Sham equations
1 2
σ
− ∇ + V (r) φσi (r) = Eiσ φσi (r)
2
to re-write the kinetic energy term (just as in Section 3.4.1)
Ekinetic =
XZ
i,σ
φσ∗
i (r)
Z
X
1 2
σ
σ
σ
2 σ
− ∇ φi (r)dr =
Ei − |φi (r)| V (r)dr
2
i,σ
(4.1.2)
(4.1.3)
4.1 Hydrogen in Finite Jellium Spheres using the LDA
113
where in the first term on the right-hand side we have used the normalisation of the single
P
particle orbitals. Using nσ (r) = i |φσi (r)|2 we have
Ekinetic =
X
Eiσ
XZ
−
nσ (r)V σ (r)dr
(4.1.4)
σ
i,σ
The external potential for the system is
Z
vext (r) = − −
r
Z
r 0 =Rjell
r0 =0
n0
dr0
|r − r0 |
(4.1.5)
therefore
Z
r=∞ Z r0 =Rjell
Eexternal = −
r 0 =0
r=0
n(r)n0
drdr0 − Z
|r − r0 |
Z
r=∞
r=0
n(r)
dr
r
(4.1.6)
We also add on the self-repulsion of the positive background charge and also the
repulsion between the ion and the positive background:
1
2
Z
r=Rjell
Z
r 0 =Rjell
n20
drdr0 + Z
|r − r0 |
r0 =0
r=0
Z
r=Rjell
r=0
n0
dr
r
(4.1.7)
We will calculate the first of these terms explicitly. In doing so we will use the result
derived in Section 3.1.5:
Z
r0 =Rjell
r 0 =0
|r0
n0
4
dr0 = πr2 n0 + 2π(Rjell 2 − r2 )n0
− r|
3
for r ≤ Rjell
(4.1.8)
Using this result
1
2
1
2
Z
Z
r=Rjell
2
4πr n0
r=0
Z
r=Rjell
r=0
r0 =Rjell
r 0 =0
r=0
r=Rjell
2πn20
Z
n20
drdr0 =
|r − r0 |
4 2
2
2
πr n0 + 2π(Rjell − r )n0 dr =
3
2
16
2
5
(− πr4 + 2πRjell
r2 )dr = π 2 n20 Rjell
3
15
(4.1.9)
3 n /3 and substituting in for R
Using N = 4πRjell
0
jell in the above expression
=
16 2
π
15
1
4π 5/3
3
1
5
n0 3 N 3
Then using 43 πrs 3 n0 = 1 to eliminate n0 gives the final result
(4.1.10)
114
Hydrogen Immersed in a Finite Jellium Sphere
1
2
Z
r=Rjell
Z
r0 =Rjell
r 0 =0
r=0
n20
3 N 5/3
0
drdr
=
|r − r0 |
5 rs
(4.1.11)
Putting all these terms together, the DFT energy functional we use for an atom in
finite jellium is
↑
↓
E[n , n ] =
X
Eiσ
−
i,σ
Z
r=∞ Z r0 =Rjell
r=0
r 0 =0
XZ
σ
1
drn (r)V (r) +
2
n(r)n0
drdr0 − Z
|r − r0 |
σ
Z
r=∞
r=0
σ
Z Z
n(r)
dr + Z
r
Z
Exc [n↑ , n↓ ]
n(r)n(r0 )
drdr0 −
|r − r0 |
r=Rjell
r=0
n0
3 N 5/3
dr +
+
r
5 rs
(4.1.12)
Note that the energy functional for a jellium sphere with no atom is obtained by setting
Z = 0 in the above expression. Similarly, the energy functional for an atom with no jellium
is obtained by setting n0 = N = 0.
4.1.2
Filling of Orbitals
Electrons in the ground-state of a jellium sphere occupy those angular momentum orbitals
which result in the lowest total energy, and which also comply with the Pauli exclusion
principle (PEP). This is in accordance with the fact that in DFT the energy functional
must be minimised in order to obtain the ground-state. Following this prescription, as we
put electrons into the jellium sphere the orbitals fill up as 1s2 , 2p6 , 3d10 , 2s2 , 4f 14 , 3p6 ,
5g 18 , 4d10 , 6h22 , 3s2 , 5f 14 . Electrons fill up a given orbital so as to maximise the total
spin of that orbital, in accordance with Hund’s second rule [84]. This means that for a
given orbital, the spin-up electrons will be filled first, followed by the spin-down electrons
(or vice-versa).
When we are considering an atom in a jellium sphere, we follow the same procedure,
putting electrons into orbitals to minimise the energy and adhering to the PEP. If however
we know the orbitals of the constituent atom and jellium, we may follow a short-cut which
is illustrated in the following example.
In work by Kurkina et al [85], an Iron atom is added to a 10-electron jellium sphere.
Separately, the orbital configurations are
Fe : 1s2 2s2 2p6 3s2 3p6 4s2 3d6
4.1 Hydrogen in Finite Jellium Spheres using the LDA
115
Jellium : 1s2 2p6 3d2
From these we take the fully filled orbitals from both separate systems, I.e.: 1s2 , 2s2 ,
2p6 , 3s2 , 3p6 and 4s2 . These orbitals appear in the combined system. However notice that
1s2 and 2p6 appear in both the Iron atom and the jellium, but that they can only appear
once in the combined system. Therefore, eight electrons are set to one side, along with
all the remaining electrons for the atom and the jellium which did not sit in full orbitals.
These electrons, which are the eight electrons from the aforementioned 1s2 and 2p6 , the
3d6 from the Iron atom and the 3d2 from the jellium sphere, are then placed in orbitals
so as to minimise the energy whilst complying with the PEP.
For the hydrogen atom in a jellium sphere, we employ this ’shortcut’, and occupy all
orbitals which are fully occupied in the jellium sphere. The remaining electrons from the
jellium sphere, and the electron from the hydrogen are then placed in orbitals which give
the lowest total energy.
4.1.3
Applying SIC to a Hydrogen Atom in a Finite Jellium Sphere
For a hydrogen atom in a finite jellium sphere, we SI-correct the lowest energy spin-up
and spin-down s states. The idea is that the two lowest energy states will be the most
localised, and so the most in need of the SIC. Also, as we increase the size of the jellium
sphere, and so approach the infinite jellium limit, it will be these two states which will
become the analogues of the two bound states in our system of a hydrogen atom in infinite
jellium. And we know that for a hydrogen atom in infinite jellium, only the bound states
need SI-correcting, as the scattering states do not show any pronounced resonances.
We need to orthogonalise all of the higher lying s states of a given spin against the 1s
bound-state of that spin being SI-corrected. For this purpose, we use the Gram-Schmidt
orthogonalisation as described in section 2.3.5.
For a 10-electron jellium sphere, for example, we have 1s and 2s electrons. Therefore
the 2s orbital for a given spin has to be orthogonalised against the 1s orbital of the same
spin
φσ,orth
(r) =
2s
where I σ =
1.
R
1
(φσ (r) − I σ φσ1s (r))
N σ 2s
φσ2s (r)φσ1s (r)dr and N σ is a normalisation factor to ensure
(4.1.13)
R
|φσ,orth
(r)|2 dr =
2s
116
Hydrogen Immersed in a Finite Jellium Sphere
The SIC energy functional of Eq. (2.3.58) is used for our SIC calculations. When
calculating the kinetic energy one must take care over the l = 0 contribution for spin-up
and spin-down electrons. For a 10-electron jellium sphere the contribution due to the 2s
electron of spin σ is given by
Z
1
(r)dr
φσ,orth
(r)(− ∇2 )φσ,orth
2s
2s
2
If we insert Eq. (4.1.13) and use the Kohn-Sham equations
1
σ
σ σ
(− ∇2 + VSIC
(r))φσ1s (r) = E1s
φ1s (r)
2
1
σ σ
(− ∇2 + V σ (r))φσ2s (r) = E2s
φ2s (r)
2
then we obtain
Z
1
φσ,orth
(r)(− ∇2 )φσ,orth
(r)dr =
2s
2s
2
Z
Z
1
σ2
σ
σ
2 σ
σ
σ
(1 − I )E2s − |φ2s (r)| V (r)dr + I
φσ2s (r)φσ1s (r)(VSIC
(r) + V σ (r))dr−
N σ2
I
σ2
Z
σ
|φσ1s (r)|2 VSIC
(r)dr
which can be calculated straightforwardly in the code. Alternatively one may explicitly
differentiate φσ,orth
(r) by using
2s
1 U (r)00
l(l + 1)
1
− ∇2 R(r)Ylm (θ, φ) = −
Ylm (θ, φ) +
R(r)Ylm (θ, φ)
2
2 r
2r2
(where U (r) = rR(r)) which gives
Z
1
φσ,orth
(r)(− ∇2 )φσ,orth
(r)dr
2s
2s
2
1
=
2
Z
1
=−
2
Z
σ,orth
σ,orth
U2s
(r)U2s
(r)00 dr
σ,orth
σ,orth
U2s
(r)0 U2s
(r)0 dr
where we have performed integration by parts in the final step. In the code, both approaches can be applied and are found to give the same results.
4.1 Hydrogen in Finite Jellium Spheres using the LDA
4.1.4
117
Results
Energy Levels
For a hydrogen atom in a 338-electron jellium sphere of background density 0.008a−3
B , we
find that the orbitals are from lowest to highest energy (in the notation 1s↑ ≡ (1, 0)↑ , etc)
(1, 0)↑ (1, 0)↓ (2, 0)↑ (2, 0)↓ (2, 1)↑ (2, 1)↓ (3, 2)↑ (3, 2)↓ (4, 3)↑ (4, 3)↓ (3, 1)↑ (3, 0)↑ (3, 1)↓ (3, 0)↓
(5, 4)↑ (5, 4)↓ (4, 2)↑ (4, 2)↓ (6, 5)↑ (6, 5)↓ (5, 3)↑ (5, 3)↓ (7, 6)↑ (7, 6)↓ (4, 1)↑ (4, 1)↓ (4, 0)↑ (4, 0)↓
(8, 7)↑ (8, 7)↓ (6, 4)↑ (6, 4)↓ (5, 2)↑ (5, 2)↓ (9, 8)↑ (9, 8)↓ (7, 5)↑ (7, 5)↓ (6, 3)↑ (6, 3)↓ (5, 1)↑ (5, 1)↓
(10, 9)↑ (5, 0)↑ (8, 6)↑ (8, 6)↓
where all orbitals are completely filled. For comparison, the orbitals for the same (nonmagnetic) jellium sphere with no atom, which we find to be in agreement with existing
results [86], are
(1, 0)(2, 1)(3, 2)(2, 0)(4, 3)(3, 1)(5, 4)(4, 2)(6, 5)(3, 0)(5, 3)(7, 6)(4, 1)(8, 7)(6, 4)(5, 2)
(4, 0)(9, 8)(7, 5)(6, 3)(5, 1)(10, 9)(8, 6)
The orbitals are the same except for an extra 5s orbital in the case of the atom in the
jellium sphere. Notice that for the atom in jellium, there is an extra spin-up and spin-down
electron at the low energy end of the bound states. These s electrons are closely localised
to the hydrogen atom, and are the analogues of the bound state s electrons of the atom
in infinite jellium.
Fig. 4.1 shows our calculations of the structure of the energy levels of the atom in
jellium and also of the jellium sphere. Comparing the energy levels of the atom in a
jellium sphere with those of the jellium sphere by itself, we see that only the l = 0 bound
states are significantly different between the two. This makes sense, as the addition of the
atom will only change the density in the region immediately surrounding the atom, and
the l = 0 states are the main contributor to the density in this region.
For the background density 0.008a−3
B , we see that the 2s bound state is lower than
the 2p bound state for the atom in jellium. In fact, if the background density is lowered
further, then the energy of the 2s bound state will eventually equal that of the 1s bound
118
Hydrogen Immersed in a Finite Jellium Sphere
state in the pure jellium sphere. At this point, the structure of the energy levels for the
atom in jellium will be approximately the same as that of the jellium sphere, except for
two additional 1s electrons lying below the jellium energy levels. This is consistent with
the fact that as the background density tends to zero, the atom in jellium solution will
tend towards a straightforward superposition of the solutions of the constituent atom and
the jellium.
Conversely, if the background density is increased, then the 1s states contributed by
the hydrogen atom will begin to become more jellium-like in character. This can be seen
in the plot for the background density 0.03a−3
B , which shows the s levels increasing in
energy relative to the jellium sphere s levels as compared to the 0.008a−3
B plot. If we were
to increase the background density further (and also increase the number of electrons in
the jellium sphere in order to stop the jellium sphere shrinking) then eventually the energy
levels for the atom in jellium and for the jellium would begin to tend towards one another,
with the exception of an additional high lying s electron in the atom in jellium case.
Immersion Energy Curves
Fig. 4.2 illustrates the dependence of the immersion energy on the size of the jellium
−3
sphere for three background densities. The densities considered are 0.001a−3
B , 0.007aB
−3
. The most obvious feature of these plots is the small scale bunching of immerand 0.03aB
sion energies, with the immersion energy increasing or decreasing slightly as a particular
angular momentum shell is filled. A larger scale feature is that the immersion energy
oscillates around the value for the infinite jellium. These are the Friedel oscillations described in Section 3.2.6, and are the same wavelength in each of the plots. The wavelength
predicted by the theory is
∆R
1 π
1
π
π
=
=
1 =
1 =
2
3π
rs
rs kF
rs (3π 2 n0 ) 3
( 4/3π ) 3
4π 2
9
13
= 1.637
(4.1.14)
which is in good agreement with the wavelength as read off from the graphs in Fig. 4.2.
The amplitude of these Friedel oscillations becomes smaller as the size of the jellium sphere
is increased. This tells us that we can make the immersion energy arbitrarily close to the
infinite jellium immersion energy by making the jellium sphere very large.
We want the atom in finite jellium to approximate the atom in infinite jellium. To this
end, the oscillation of the energy about the infinite jellium value is undesirable. Even for
PSfrag replacements
4.1 Hydrogen in Finite Jellium Spheres using the LDA
0
119
Jellium Potential
Atom in Jellium Potential
Potential, Energy / a.u.
-0.1
-0.2
-0.3
-0.4
-0.6
40
30
20
10
0
r / aB
0
10
20
30
40
30
40
Jellium Potential
Atom in Jellium Potential
-0.05
Potential, Energy / a.u.
PSfrag replacements
-0.5
-0.1
-0.15
-0.2
-0.25
-0.3
-0.35
40
30
20
10
0
10
20
r / aB
Figure 4.1: Plots of (spin-up) bound state energies for a 338-electron jellium sphere and
a hydrogen atom in a 338-electron jellium sphere, along with the (spin-up) potentials for
these systems. The background densities of the jellium are 0.03 a−3
B (upper panel) and
0.008 a−3
B (lower panel). The bound states are shown as lines, with the lengths of these
lines corresponding to the angular momentum (l=0 is the shortest and l=9 is the longest).
120
Hydrogen Immersed in a Finite Jellium Sphere
n0 = 0.001a−3
B
PSfrag replacements
Immersion Energy / eV
0
-0.5
-1
-1.5
-2
-2.5
0
1
2
1
2
3
4
5
6
3
4
5
6
4
5
6
-0.5
-1
-1.5
-2
PSfrag replacements
PSfrag replacements
Immersion Energy / eV
0
Rjell /rs
n0 = 0.007a−3
B
-2.5
-3
0
Rjell /rs
n0 = 0.03a−3
B
3
Immersion Energy / eV
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
0
1
2
3
Rjell /rs
Figure 4.2: Plots of immersion energy versus number of electrons for jellium spheres.
−3
−3
Densities of 0.001a−3
B , 0.007aB and 0.03aB are considered. The lines are the values of
the immersion energy for the infinite jellium system. The largest size of jellium sphere
used in these plots is a 138-electron jellium sphere
4.1 Hydrogen in Finite Jellium Spheres using the LDA
121
an atom in a jellium sphere of 138 electrons, the size of the Friedel oscillation is of the
order of 1eV.
Our approach will therefore be to select a jellium sphere size with immersion energies
for the atom in jellium which are approximately equal to those of the infinite jellium
across the range of background densities considered. I.e. where the oscillation crosses
the asymptotic value. To this end, and for the purposes of the comparison of HF, DFT
and VQMC methods, we choose a 10-electron jellium sphere. We will also consider a
50-electron jellium sphere, although this will just be for consideration within the LDA.
The 10 and 50 electron jellium spheres corresponds to values of Rjell /rs of 2.15 and 3.68
respectively. From Fig. 4.2 then, we see that the corresponding energies lie as close to the
asymptotic values as we can get them.
Plots of the immersion energy versus background density for these sizes of jellium
sphere, along with a plot of the infinite jellium immersion energy, are shown in Fig. 4.3.
We see that both the 10 and 50-electron curves feature a minimum, but that the 50-electron
curve matches the form of the infinite jellium curve more closely.
Fig. 4.4 shows the dependence of the immersion energy curve for a 10 and 50-electron
jellium sphere on the choice of the exchange-correlation potential. Note that in Section 4.2,
where we compare the DFT results to the VQMC results, we will use the Perdew-Zunger
functional.
We will refrain from presenting the SIC results until Section 4.2.
Atom induced Density Profiles
As we increase the number of electrons in the jellium sphere, we expect the atom induced
density to approach that of the hydrogen atom in infinite jellium. This is indeed the
−3
case, and we have verified this for background densities in the range 0.001a−3
B to 0.03aB .
Fig. 4.5 shows our calculations of the atom-induced densities for a hydrogen atom in
jellium spheres with a background density 0.01a−3
B . We look at jellium spheres with 10, 50
and 338 electrons. We also plot the atom-induced density for a hydrogen atom in infinite
jellium. We see clearly that as we increase the number of electrons in the jellium sphere,
the atom-induced density approaches the limiting atom-induced density of a hydrogen
atom in infinite jellium.
The total density requires a far greater size of jellium sphere before it begins to approach the total density of the atom in infinite jellium solution. This is shown in Fig. 4.6,
122
Hydrogen Immersed in a Finite Jellium Sphere
10-Electron jellium sphere
50-Electron jellium sphere
Infinite jellium
1.5
Immersion Energy / eV
PSfrag replacements
0
2
1
0.5
0
-0.5
-1
-1.5
-2
0
0.005
0.01
0.015
n0 /a−3
B
0.02
0.025
0.03
Figure 4.3: Plots of immersion energy versus background density for a hydrogen atom
immersed in jellium spheres of size 10 and 50 electrons. Also plotted is the immersion
energy curve for a hydrogen atom in infinite jellium.
0
-0.5
-1
-2
0
0.005
0.01
0.02
0.015
n0 /a−3
B
0.02
0
0.015
n0 /a−3
B
2
PSfrag replacements
-1.5
0.025
0.03
0.025
0.03
Gunnarsson-Lundqvist
Perdew-Zunger
Perdew-Wang
1.5
Immersion Energy / eV
123
Gunnarsson-Lundqvist
Perdew-Zunger
Perdew-Wang
0.5
Immersion Energy / eV
0
1.5
2
1
PSfrag replacements
4.1 Hydrogen in Finite Jellium Spheres using the LDA
1
0.5
0
-0.5
-1
-1.5
-2
0
0.005
0.01
Figure 4.4: Plots of immersion energy versus background density for a hydrogen atom
immersed in jellium using different exchange-correlation functionals. The top panel is for
a 10 electron jellium sphere and bottom panel is for a 50 electron jellium sphere.
124
Hydrogen Immersed in a Finite Jellium Sphere
where we have plotted the density for a hydrogen atom in a 106-electron jellium sphere for
background density ≈ 0.004a−3
B , alongside the corresponding plot for the infinite jellium.
Here we see that the density of the atom in finite jellium is quite different from the density
of the atom in infinite jellium (also plotted). We find that this is also the case for much
larger jellium spheres. The reason why the atom-induced density more readily approaches
that of the infinite jellium as we increase the size of the jellium sphere is because the effect
of the edge of the jellium sphere on the solution is cancelled out as we subtract the jellium
solution from the atom in jellium solution.
Potential for Finite Spheres
In Fig. 4.7 we plot the spin-up potential for a hydrogen atom in a 338-electron jellium
sphere for a range of background densities. The potential is the strongly attractive potential of the ion close to the origin. It then flattens out into an approximate plateau at
larger radii. The plateau corresponds to a region where the ion is almost fully screened,
and the potential is approximately that of bulk jellium. As we approach the radius of the
jellium sphere, the potential increases to zero.
The larger the radius of the jellium sphere, the closer the potential should be to the
potential of a hydrogen atom in infinite jellium. As we increase this jellium sphere radius
(which corresponds to decreasing the background density of the jellium), the plateau
becomes larger and closer to zero potential. This is consistent with the fact that the
potential for a hydrogen atom in infinite jellium consists of an ionic part, which then
decays to zero.
The potential plot also includes the energy of the 1s spin-up bound state for each
background density. For high background densities, we see that this bound state lies above
the plateau of the potential. This bound state is analogous to a scattering state resonance
in the case of the hydrogen in infinite jellium. Then as we decrease the background
density, and the state is drawn closer to the ion, the energy of the bound state falls below
the plateau of the potential. This occurs at n0 = 0.015a−3
B . At this point, the bound state
is analogous to the same bound state appearing in the infinite jellium calculations.
Fig. 4.7 also includes a plot of the expectation value of the radius of the 1s spin-up
electron as a function of the background density of the jellium. We see that as we lower
the density from 0.03a−3
B the expectation value increases. This is just because we are
increasing the size of the jellium sphere, and the electron is spreading out to fill this
4.1 Hydrogen in Finite Jellium Spheres using the LDA
125
PSfrag replacements
Atom Induced Density × r 2 / a−1
B
0.07
Infinite jellium
Finite jellium
Jellium sphere radius
0.06
0.05
0.04
0.03
0.02
0.01
0
-0.01
-0.02
0
5
10
15
20
25
30
Radius / aB
PSfrag replacements
Atom Induced Density × r 2 / a−1
B
0.07
Infinite jellium
Finite jellium
Jellium sphere radius
0.06
0.05
0.04
0.03
0.02
0.01
0
-0.01
-0.02
0
5
10
15
Radius / aB
20
25
30
126
Hydrogen Immersed in a Finite Jellium Sphere
lacements
Atom Induced Density × r 2 / a−1
B
0.07
Infinite jellium
Finite jellium
Jellium sphere radius
0.06
0.05
0.04
0.03
0.02
0.01
0
-0.01
-0.02
0
5
10
15
20
25
30
Radius / aB
Figure 4.5: Plots of atom induced densities for hydrogen in finite jellium spheres of background density 0.01a−3
B , with 10, 50 and 338 electrons (top, middle and bottom panel
respectively). Also plotted is the atom induced density for a hydrogen atom in infinite
jellium at the same background density.
increase in volume. However, when we get to n0 = 0.015a−3
B , the expectation value starts
to decrease. This is because the electron is now the analogue of the bound state in the
atom in infinite jellium system and is therefore more strongly localised to the ion.
4.2
4.2.1
Hydrogen in Finite Jellium Spheres using VQMC
The Choice of the Trial Wavefunction
The form of the trial wavefunction used in our atom in jellium calculations is the same
as that used by Sottile et al [33] in their calculations of jellium spheres (they did not
include an embedded atom), except for a ’multipolar’ term which we do not include in our
calculations. The trial wavefunction consists of a single-body contribution and a two-body
PSfrag replacements
4.2 Hydrogen in Finite Jellium Spheres using VQMC
0.006
127
Finite jellium
Infinite jellium
Density / a−3
B
0.005
0.004
0.003
0.002
0.001
0
0
5
10
15
20
25
30
r / aB
Figure 4.6: The total density for a hydrogen atom in a 106-electron jellium sphere and for
a hydrogen atom in infinite jellium for a background density ≈ 0.004a−3
B .
contribution (the first and the second exponential respectively):
"
ΨT (r1 , · · · , rN ) = exp
N
6
X
X
i=1

× exp 
X
1≤i<j≤N
1
2
n=1
αn(i) j0
nπri
Rjell
!#

aij rij + bij rij 2
aij rij + bij rij 2

+
1 + cij (ri )rij + dij rij 2 1 + cij (rj )rij + dij rij 2
×D↑ (r1 , · · · , rN/2 )D↓ (rN/2+1 , · · · , rN )
(4.2.1)
where Rjell is the radius of the jellium sphere, and cij is given by
2
cij (ri ) = c0ij + c1ij arctan[(ri2 − Rjell
)/2 4 Rjell ]
(4.2.2)
The ij subscripts on the parameters denote a dependence on whether the spins of
(i)
electrons i and j are parallel or anti-parallel, and the i superscript on αn denotes a
dependence on the spin of electron i. The parameters aij are again determined by the
electron-electron cusp condition (see Section 2.4.9). The other 15 parameters are variational parameters that are varied in order to minimise the standard deviation of the local
energy.
PSfrag replacements
128
Hydrogen Immersed in a Finite Jellium Sphere
0
Background density / a−3
B
0.008
0.01
0.015
0.02
0.025
0.03
0.035
-0.2
V (r)
-0.4
-0.6
-1
0
5
10
15
20
PSfrag replacements
-0.8
25
30
35
40
r/aB
4.4
4.35
< r1s > /aB
4.3
4.25
4.2
4.15
4.1
4.05
0.005
0.01
0.015
0.02
Background density /
0.025
0.03
a−3
B
Figure 4.7: Plots of spin-up potentials (upper panel) for a 338-electron jellium sphere for
different background densities. The energy of the 1s bound state is also included for each
potential, and is plotted as a straight-line on the left of the graph. The lower panel shows
the expectation value of the radius of the (spin-up) 1s electron for the different background
densities. See main text for discussion.
4.2 Hydrogen in Finite Jellium Spheres using VQMC
129
We expect the above Jastrow factor to work well for our system of an atom embedded
in a jellium sphere, even though it was constructed in the first place for calculations of
jellium spheres without an embedded atom. We think this because firstly, away from the
atom, our atom in jellium solution will tend towards that of the pure jellium. Therefore,
for electrons in this region, the above form for the Jastrow factor is clearly appropriate.
In addition, the Jastrow factor is also suitable for calculations of an atom. We see this by
(i)
setting c1ij = αn = 0 and obtaining the following form for the Jastrow factor

exp 
X
1≤i<j≤N

aij rij + bij rij 2

1 + c0ij (ri )rij + dij rij 2
(4.2.3)
which has been used previously for atomic calculations [60].
So the capability of the Jastrow factor to be able to describe the atom and the jellium
separately gives one hope that it will also give a good description of a combined system
of the two.
The orbitals in the Slater determinant are taken to be the Kohn-Sham orbitals from
LDA calculations of the same atom in finite jellium system. As we have discussed, in
the LDA calculations we impose spherical symmetry by insisting that the electron density
contribution of an open shell is spherically symmetric (the closed shells are automatically
spherically symmetric). In the VQMC calculations, spherical symmetry is not imposed
and the angular part of orbitals in an open-shell (Yl,m (θ, φ)) can make the electron density
non-spherically symmetric overall.
In fact we find that the lowest energy VQMC solution is obtained by choosing values of
m for the open-shell electrons such that the wavefunction is real. For example, hydrogen in
a 10-electron jellium sphere contains the following full shells (in the notation 1s↑ ≡ (1, 0)↑ ,
etc)
(1, 0)↑ (1, 0)↓ (2, 0)↑ (2, 0)↓ (2, 1)↑ (2, 1)↓
in addition to an extra electron in the (3, 2)↑ orbital. Here we find that choosing m = 0
for this electron results in the lowest VQMC energy.
4.2.2
Calculating the Local Energy
Recall from section 2.4.8 that the local energy is of the form
130
Hydrogen Immersed in a Finite Jellium Sphere
X 1 ∇2 ΨT ĤΨT
i
=
−
+ V (r1 , r2 , · · · , rN )
ΨT
2 ΨT
(4.2.4)
i
In this section we calculate this quantity for our system of an atom in a finite jellium
sphere.
Local Potential Energy
The potential energy part of the local energy is
V (r1 , r2 , · · · , rN ) =
X Z Z n+ (r) X
1
− −
dr +
ri
|r − ri |
|ri − rj |
(4.2.5)
i<j
i
where
n+ (r0 )
Z
|r0 − r|
0
dr =



3
2 − r2 )
(Rjell
2rs3
3
Rjell 3 1
rs
r
r < Rjell
(4.2.6)
r ≥ Rjell
Local Kinetic Energy
The kinetic energy part of the local energy is
X
X 1 ∇2 ΨT X 1
1
i
−
=
− ∇2i ln ΨT −
(∇i ln ΨT )2
2 ΨT
2
2
i
i
In order to calculate
∇2i ln ΨT
(4.2.7)
i
and ∇i ln ΨT we need to substitute in our form for ΨT .
We have
"
ΨT (r1 , · · · , rN ) = exp
N
6
X
X
n=1
i=1

exp −
X
1≤i<j≤N
1
2
αn(i) j0
nπri
Rjell
!#
2
2

aij rij + bij rij
aij rij + bij rij

+
2
1 + cij (ri )rij + dij rij
1 + cij (rj )rij + dij rij 2
D↑ (r1 , · · · , rN/2 )D↓ (rN/2+1 , · · · , rN )
(4.2.8)
where Rjell is the radius of the jellium sphere, and cij is given by
2
cij (ri ) = c0ij + c1ij arctan[(ri2 − Rjell
)/2 4 Rjell ]
(4.2.9)
Therefore
6
X
1
1
∇i ln ΨT = ↑ ∇i D↑ + ↓ ∇i D↓ + ∇i
αn(i) j0
D
D
n=1
nπri
Rjell
−
4.2 Hydrogen in Finite Jellium Spheres using VQMC
1X
∇i
2
j6=i
aij rij + bij rij 2
aij rij + bij rij 2
+
1 + cij (ri )rij + dij rij 2 1 + cij (rj )rij + dij rij 2
131
1
1
1
1
∇i D↑ )2 + ↑ ∇2i D↑ − ( ↓ ∇i D↓ )2 + ↓ ∇2i D↓ +
↑
D
D
D
D
6
X
nπri
−
∇2i
αn(i) j0
Rjell
n=1
X
aij rij + bij rij 2
aij rij + bij rij 2
1
2
∇i
+
2
1 + cij (ri )rij + dij rij 2 1 + cij (rj )rij + dij rij 2
(4.2.10)
∇2i ln ΨT = −(
(4.2.11)
j6=i
The calculation of these derivatives are discussed in more detail in Appendix A.
4.2.3
Results
In our calculations, we found that the Jastrow parameters c1ij and 4 were both prone to
become very large over the course of the optimisation. From Eq. (4.2.2) we see that a large
value for 4 makes the argument of the arctan become very small. Therefore the arctan
itself approaches the value of its argument. Hence the two separate Jastrow parameters,
c1ij and 4 combine to a single effective Jastrow parameter, c1ij /4.
This is born out in the optimisation procedure, where we see the ratio c1ij /4 tend
towards a constant. The optimisation procedure then, seems to be rejecting the extra
variational freedom afforded by the arctan. In order to let the Jastrow parameters tend
towards constant values (and stop c1ij and 4 shooting off to infinity) we simply set 4
equal to a very high value (say 100,000) and then let c1ij vary freely along with all the
other Jastrow parameters.
Total Energies and Immersion Energies
We report VQMC solutions for a hydrogen atom immersed in 10-electron jellium spheres
−3
of background densities 0.001a−3
B through to 0.03aB . Total energies were calculated and,
along with HF, LDA and SIC results, are presented in Table 4.1. Energies for the same
jellium spheres but without the hydrogen atom are reported in Table 4.2. Finally, the
immersion energies are reported in Table 4.3. The data from these tables are reproduced
in graphical form in Fig. 4.8, Fig. 4.9 and Fig. 4.10.
In calculating the immersion energy for HF, VQMC and SIC the exact value of the
hydrogen atom energy is used, namely −13.606eV . This is appropriate as calculations
using HF and SIC both give this value and in the case of VQMC, the calculation is correct
132
Hydrogen Immersed in a Finite Jellium Sphere
to a very high level of precision. For the LDA calculation, the atom energy is found to be
−13.030eV.
The distribution of local energies for the VQMC calculations are found to be Gaussians
centred approximately on the total energy of the system. This is shown in Fig. 4.11 for
a hydrogen atom in jellium of density 0.03a−3
B , and is an indication that the statistics of
the VQMC simulation are good.
The re-blocking analysis described in Section 2.4.11 is applied in Fig. 4.12 to the
solution of a hydrogen atom immersed in a 10-electron jellium sphere of density 0.03a−3
B .
We find an error in the total energy of 0.0031eV. This error is sufficiently close to the
error obtained without the re-blocking analysis however, that in practice, we just quote
the un-reblocked error as the error to the total energy.
Table 4.1: Total energies of hydrogen in 10-electron jellium spheres
HF
Density/a−3
B
Energy/eV
σl.e.
0.002
-26.41773 ± 0.00711
12.216
0.003
-26.30472 ± 0.00771
13.336
0.004
-25.82384 ± 0.00842
14.557
0.005
-25.20595 ± 0.00868
15.001
0.01
-21.19465 ± 0.01039
17.972
0.02
-12.55116 ± 0.01237
21.384
0.03
-4.26587 ± 0.01368
23.663
VQMC
LDA
SIC
σl.e.
Energy/eV
Energy/eV
-31.61214 ± 0.00188
3.249
-32.384
-32.658
0.003
-31.85682 ± 0.00196
3.385
-32.55
-32.808
0.004
-31.68001 ± 0.00202
3.499
-32.317
-32.582
0.005
-31.26107 ± 0.00208
3.602
-31.870
-32.155
0.01
-27.93851 ± 0.00232
4.019
-28.417
-28.825
0.02
-19.88415 ± 0.00266
4.595
-20.249
-20.838
0.03
-11.87088 ± 0.00291
5.032
-12.187
-12.906
Density/a−3
B
Energy/eV
0.002
The total energy curves calculated using the different methods are all of the same shape
and feature minima at roughly the same locations. The LDA, SIC and VQMC curves lie
4.2 Hydrogen in Finite Jellium Spheres using VQMC
0
Hartree-Fock
VQMC
LSDA
SIC
-5
Energy (eV)
133
-10
-15
-20
-25
-30
-35
0
0.005
0.01
0.015
0.02
0.025
0.03
n0 (a−3
B )
Figure 4.8: Total energy of a hydrogen atom immersed in a 10-electron jellium sphere for
different background densities
10
Hartree-Fock
VQMC
LSDA
Energy / eV
5
0
-5
-10
-15
-20
0
0.005
0.01
0.015
0.02
0.025
0.03
n0 (a−3
B )
Figure 4.9: Total energy of a 10-electron jellium sphere for different background densities
134
Hydrogen Immersed in a Finite Jellium Sphere
Immersion energy (eV)
1.5
Hartree-Fock
VQMC
LSDA
SIC
1
0.5
0
-0.5
-1
-1.5
-2
0
0.005
0.01
0.015
0.02
0.025
0.03
n0 (a−3
B )
Figure 4.10: Immersion energies for a hydrogen atom immersed in a 10-electron jellium
sphere for different background densities
4.2 Hydrogen in Finite Jellium Spheres using VQMC
135
Table 4.2: Total energies of 10-electron jellium spheres
HF
Density/a−3
B
Energy/eV
σl.e.
0.001
-11.19489 ± 0.00429
7.415
0.002
-11.65007 ± 0.00528
9.127
0.003
-11.43242 ± 0.00631
10.905
0.004
-10.94642 ± 0.00662
11.219
0.005
-10.36378 ± 0.00692
11.962
0.01
-6.75033 ± 0.00842
14.564
0.02
0.81546 ± 0.01032
17.840
0.03
7.95062 ± 0.01155
19.973
VQMC
LDA
Density/a−3
B
Energy/eV
σl.e.
Energy/eV
0.001
-16.06998 ± 0.00055
0.947
-16.543
0.002
-16.99168 ± 0.00072
1.247
-17.519
0.003
-17.07582 ± 0.00083
1.437
-17.609
0.004
-16.80527 ± 0.00093
1.600
-17.353
0.005
-16.40632 ± 0.00098
1.695
-16.920
0.01
-13.21880 ± 0.00130
2.256
-13.786
0.02
-6.07881 ± 0.00165
2.854
-6.645
0.03
0.84458 ± 0.00190
3.285
0.284
within 1eV of one another in the atom in jellium total energy graph and similarly the
LDA and VQMC curves are within 1eV of one another in the jellium sphere total energy
graph. HF gives total energy curves for the atom in jellium and for the jellium sphere
which are rigidly shifted by about 5eV above the VQMC curves.
We note that our VQMC total energy curve for the jellium sphere is slightly different
to the type obtained by Sottile et al [33]. They observed VQMC results which were higher
than the LDA results for low densities. Then above background densities of about 0.01a−3
B
the VQMC results were found to be slightly lower than the LDA results. This is probably
because we neglected to include the ’multipolar’ term in the trial wavefunction which
Sottile et al used in their calculations. In their work, this term was found to improve the
VQMC solution for the larger background densities.
136
Hydrogen Immersed in a Finite Jellium Sphere
Table 4.3: Immersion energies of hydrogen in 10-electron jellium spheres
HF
VQMC
LDA
SIC
Density/a−3
B
Energy/eV
Energy/eV
Energy/eV
Energy/eV
0.002
-1.16196 ± 0.01239
-1.01477 ± 0.00260
-1.835
-1.534
0.003
-1.26661 ± 0.01402
-1.17531 ± 0.00279
-1.913
-1.593
0.004
-1.27172 ± 0.01504
-1.26905 ± 0.00295
-1.934
-1.623
0.005
-1.23648 ± 0.01559
-1.24905 ± 0.00306
-1.920
-1.630
0.01
-0.83864 ± 0.01882
-1.11402 ± 0.00362
-1.601
-1.433
0.02
0.23908 ± 0.02268
-0.19965 ± 0.00431
-0.574
-0.588
0.03
1.38920 ± 0.02524
0.89023 ± 0.00481
0.559
0.416
When the total energy curves are subtracted from one another to give the immersion
energy curves, the large error in the HF calculations cancel out. We find all four curves lie
within 1eV of one another and all curves feature a minimum at approximately 0.004a−3
B .
Notice that the LDA results for the immersion energy curve are slightly lower than the
VQMC results, which means that the LDA is slightly overbinding the atom to the jellium
relative to VQMC.
We see that the VQMC immersion energy curve below 0.005a−3
B is more closely followed
by the SIC curve than by the LDA curve. This is consistent with the fact that the 1s
bound states are becoming more localised at these low background densities. Therefore
the self-interaction of the bound states is increasing, and so one would expect the SIC to
give better results than the LDA for these background densities.
Notice that the LDA and SIC solutions for the total energy and the immersion energy do not coincide at large background densities. For an atom in infinite jellium, we
would expect the curves to coincide because the 1s electrons would become increasingly
delocalised at larger background densities and so their self-interaction would tend to zero.
However for a finite jellium sphere with a fixed number of electrons, at large background
densities the jellium sphere becomes smaller, and so the 1s electrons (along with all of the
other electrons) actually become more localised. This was illustrated earlier in Fig. 4.7
for a 338-electron jellium sphere.
This increasing localisation causes the self-interaction of these electrons to increase.
The SIC energies therefore increasingly diverge from the LDA energies at large background
densities. This is not a physical effect to do with immersing an atom in infinite jellium, but
PSfrag replacements
4.2 Hydrogen in Finite Jellium Spheres using VQMC
137
10000
No. Configurations
8000
6000
4000
2000
0
-30
-25
-20
-15
-10
Local Energy / eV
-5
0
5
Figure 4.11: Local energy distribution for a VQMC calculation of a hydrogen atom in a
10-electron jellium sphere of density 0.03a−3
B
is instead a side-effect of our approximating the infinite jellium as a finite jellium sphere.
Densities
The electron density was also calculated for the hydrogen atom in the 10-electron jellium
−3
sphere for background densities of 0.002a−3
B and 0.03aB . These are shown in Fig. 4.13,
Fig. 4.14, Fig. 4.15 and Fig. 4.16.
Notice that the ’HF’ curves in the density plots are not actually HF solutions. They
correspond to VQMC calculations using only the Slater determinant part of the trial
wavefunction with no Jastrow factor. Unlike the energy, the density curves obtained
in this way cannot be said to be approximately equal to those obtained using HF. The
purpose of including the ’HF’ density curves is just as a numerical check. With no Jastrow
factor, and with a Slater determinant containing LDA orbitals, the density obtained using
VQMC should be the same as that obtained using the LDA. From our calculations we see
that this is the case.
The SIC densities are almost identical to those of the LDA, albeit slightly more de-
138
Hydrogen Immersed in a Finite Jellium Sphere
PSfrag replacements
Error on Mean of the Local Energy
0.0032
0.00315
0.0031
0.00305
0.003
0.00295
0.0029
0
2
4
6
8
Transformation Number
10
12
Figure 4.12: Re-blocking analysis for hydrogen immersed in a 10-electron jellium sphere
of density 0.03a−3
B . The error on the mean levels off at just under 0.0031eV and therefore
this is the error we quote on the total energy.
PSfrag replacements
4.2 Hydrogen in Finite Jellium Spheres using VQMC
0.6
139
LDA
SIC
Variational QMC
HF
0.5
−3
Density / aB
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
Radius / aB
Figure 4.13: Electron density of a hydrogen atom in a 10-electron jellium sphere of background density 0.03a−3
B using HF, LDA, SIC and VQMC
localised. The VQMC densities are significantly more delocalised compared to the LDA
and SIC densities. For the background density 0.03a−3
B , the VQMC density at the atom
is approximately 10% lower than the LDA density, whereas for the background density
0.002a−3
B this difference increases to approximately 15%.
Diffusion Quantum Monte Carlo for Exact Results
So far we have framed our results with the view that the VQMC results are the most
accurate of the methods used. If this were so, then our conclusions would be that the
VQMC results have corrected for the over-binding present in the LDA and that the SIC
results for the immersion energy are closer than the LDA results to the exact results for
low background densities of the jellium.
However, as we have discussed, the VQMC total energy is only an upper bound to
the exact ground-state energy. How close this upper bound is to the true ground-state
energy depends on the quality of the trial wavefunction. Application of the in-principle
exact diffusion quantum Monte Carlo (DQMC) method would result in a lower energy
PSfrag replacements
140
Hydrogen Immersed in a Finite Jellium Sphere
0.005
LDA
SIC
Variational QMC
HF
Density / a−3
B
0.004
0.003
0.002
0
5
7
6
PSfrag replacements
0.001
8
Radius / aB
0.8
9
10
LDA
SIC
Variational QMC
HF
0.7
−3
Density / aB
0.6
0.5
0.4
0.3
0.2
0
0.05
0.1
0.15
0.2
0.25
0.3
Radius / aB
0.35
0.4
0.45
0.5
Figure 4.14: Electron density of a hydrogen atom in a 10-electron jellium sphere of background density 0.03a−3
B using HF, LDA, SIC and VQMC. Note in the top graph curves
for HF, LDA and SIC coincide.
PSfrag replacements
4.2 Hydrogen in Finite Jellium Spheres using VQMC
0.4
141
LDA
SIC
Variational QMC
HF
0.35
−3
Density / aB
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
Radius / aB
Figure 4.15: Electron density of a hydrogen atom in a 10-electron jellium sphere of background density 0.002a−3
B using HF, LDA, SIC and VQMC
than that obtained by VQMC.
In fact, work by Sottile et al [33] has shown DQMC results for an 8-electron jellium
sphere (with no atom) which were, for all background densities, rigidly shifted downwards
by around 0.2eV relative to the VQMC results. This suggests that if we were to perform
DQMC calculations, then the DQMC immersion energy curve would be different to that
of the VQMC.
However, the immersion energy curve is a difference between the total energy of the
atom in jellium and the jellium sphere. Therefore some of the change in the energy in
going from VQMC to DQMC would cancel out in the immersion energy, just as it did for
the HF results. This holds out hope that the DQMC immersion energy curve might also
lie above the LDA curve, and therefore illustrate the over-binding of the LDA. Obviously
further calculations using DQMC would be required to test this hypothesis.
Our conclusion that the SIC immersion energy is closer than that of the LDA to the
exact immersion energy for low background densities also hinges on the DQMC immersion
energy curve being similar to that of the VQMC. Again, without further calculations we
can’t be sure of this. However the similarity in the upturn of the VQMC and SIC curves for
PSfrag replacements
142
Hydrogen Immersed in a Finite Jellium Sphere
0.005
LDA
SIC
Variational QMC
HF
Density / a−3
B
0.004
0.003
0.002
0
0
5
PSfrag replacements
0.001
10
Radius / aB
0.4
15
20
LDA
SIC
Variational QMC
HF
0.35
−3
Density / aB
0.3
0.25
0.2
0.15
0.1
0.05
0
0.2
0.4
0.6
Radius / aB
0.8
1
Figure 4.16: Electron density of a hydrogen atom in a 10-Electron jellium sphere of background density 0.002a−3
B using HF, LDA, SIC and VQMC. Note in the top graph curves
for HF, LDA and SIC coincide.
4.2 Hydrogen in Finite Jellium Spheres using VQMC
143
Figure 4.17: The electron density across a slab of jellium as calculated by Li and Needs
et al [8]. The origin is at the centre of the slab.
low background densities and the inability of the LDA to describe this upturn is certainly
compelling.
The density calculated using DQMC will also be different to that of the VQMC. As
an example, Li and Needs et al [8] have calculated the electron density in a slab of jellium
using the LDA, VQMC and DQMC. They found the DQMC and LDA curves to be in
good accord with one another, but the VQMC curve to be significantly different in form.
The fact that the VQMC and DQMC results were substantially different in this work
might indicate a relative lack of accuracy in their trial wavefunction as compared to the
more recent trial wavefunction used in this thesis. However, at the very least, their results
indicate that application of DQMC to our system will result in an electron density different
to the VQMC electron density and probably closer to the LDA electron density.
144
Hydrogen Immersed in a Finite Jellium Sphere
Larger Jellium Spheres
We discussed earlier how we would like to choose a jellium sphere size so that the solution of
the atom immersed in a jellium sphere approximates that of the atom in infinite jellium.
We specifically chose a 10-electron jellium sphere because this was the smallest size of
jellium sphere which produced an immersion energy curve approximately equal to that of
the infinite jellium. A 50-electron jellium sphere was also considered purely within the
LDA, and was shown to give an even better approximation to the immersion energy curve
of the atom in infinite jellium.
Given more time, we would like to perform calculations for a 50-electron jellium sphere
using HF, VQMC and SIC. If our results for this size of jellium sphere were similar to
those reported here for the 10-electron jellium sphere, then we could try and extrapolate
our findings to the atom in infinite jellium case.
We have not had time to perform the VQMC calculations on a 50-electron jellium
sphere. However, to demonstrate that the closeness of the DFT and VQMC results for
our 10-electron sphere was not simply due to a fortuitous selection of jellium sphere size,
we have performed a VQMC calculation for a hydrogen atom in a 20-electron jellium
sphere of background density 0.03a−3
B . We find that the total energy is
Energy/eV
σl.e.
LDA
-15.482
HF
1.592 ± 0.236
33.322
VQMC
-14.799 ± 0.047
6.684
Again we see that the LDA and VQMC results lie within 1eV of each other. The
work by Sottile et al [33] indicates that the same calculation for a jellium sphere without
the embedded hydrogen atom would also give LDA and VQMC results within 1eV of
one another. Therefore, the agreement between the LDA and VQMC total energies and
immersion energy for this jellium sphere are just as good as for the 10-electron jellium
sphere.
Chapter 5
Conclusions
Hydrogen in a Finite Jellium Sphere
For a hydrogen atom immersed in a finite jellium sphere using the local density approximation (LDA) of density functional theory (DFT), it has been demonstrated that
as the radius of the sphere is increased, the immersion energy oscillates around the immersion energy of a hydrogen atom in infinite jellium. The period of this oscillation was
found to be that of the Friedel oscillation of the system. It has also been shown that
the atom-induced density of the atom in finite jellium tends towards that of the atom in
infinite jellium as the size of the jellium sphere is increased.
The system of a hydrogen atom in a 10-electron jellium sphere was chosen for calculations of the LDA and the self-interaction correction (SIC) approximations of DFT,
Hartree-Fock (HF) and variational quantum Monte Carlo (VQMC). This size of jellium
sphere was chosen because it was found to be the smallest sphere for which the immersion
energy versus background density curve reasonably well approximated that of the atom in
infinite jellium.
The immersion energy versus background density curves for the LDA and SIC show an
over-binding of the atom to the jellium relative to the VQMC result. Viewing the VQMC
as a benchmark, this is consistent with the general overbinding seen in DFT. In addition,
for low background densities, the SIC immersion energy curve more closely matches that
of the VQMC than does the LDA immersion energy curve. Again, viewing VQMC as a
benchmark, this is consistent with the fact that the SIC is expected to be more accurate
than the LDA for systems with more strongly localised electrons (as is the case for the
145
146
Conclusions
low background densities).
However, the VQMC results are not exact. In order to claim an overbinding of the
DFT results relative to the exact result, and similarly to establish that the SIC is more
exact than the LDA for low background densities, one would have to replace the VQMC
results with diffusion quantum Monte Carlo (DQMC) results.
The DQMC calculations would lower both the total energy of the atom in jellium and
the total energy of the jellium relative to the VQMC results. Because the immersion energy
is calculated as the difference between these energies, part of the change in going from
VQMC to DQMC will cancel out when we calculate the immersion energy. Furthermore,
Sottile et al found a rigid shift of only −0.2eV in the DQMC energies of an 8-electron
jellium sphere relative to the VQMC energies. This holds out the possibility that the
DQMC immersion energy will also lie above the LDA immersion energy, and also that the
SIC immersion energy will still give a better account than the LDA immersion energy of
the DQMC immersion energy for low background densities.
The system of a hydrogen atom in a finite jellium sphere was intended as an approximation to a hydrogen atom in infinite jellium. If further calculations were performed on
a 50-electron jellium sphere, or on even larger jellium spheres, then we could extrapolate
our conclusions to the system of a hydrogen atom in infinite jellium. We propose this
extension as a future work.
In the LDA and SIC, the fact that one has to approximate the exchange-correlation
functional means that the theory is no longer variational. It is not possible then to claim
that the approximation (LDA or SIC) which yields the lowest total energy is the more
exact solution. However, we propose that one can test which of the two approximations is
the more exact by performing a VQMC calculation using a trial wavefunction consisting
of a Jastrow factor and a Slater determinant, which would contain orbitals taken either
from LDA or SIC. Then, because VQMC is a variational theory, whichever of the LDA or
the SIC orbitals yields the lowest energy will then tell us whether the LDA or the SIC is
the more exact theory.
Effective Medium Theory Calculations
Our new results show that the experimental minimum in the Wigner-Seitz radius
across the 4d transition metals is correctly reproduced by the effective medium theory
(EMT). Previously reported results for the Wigner-Seitz radius for elements below the 4d
Conclusions
147
transition metals were also re-calculated in this thesis. Taken together with results for
other cohesive properties such as the bulk moduli and cohesive energies, reported elsewhere
in the literature [45, 47, 4], this supports the idea that the atom-in-jellium model is useful
as a model for the full condensed matter system.
Cerium Calculations
The system of a cerium atom immersed in jellium has been solved using the LDA
and SIC. The bound state and atom-induced scattering state electrons of this system
are interpreted as the bound and valence electrons per atom of bulk cerium. With this
interpretation, it has been shown that for certain ranges of the background density of the
jellium, the LDA and SIC solutions can be used to model the alpha and gamma phases of
cerium respectively.
In order to use these solutions to yield quantitative predictions, one would have to
construct total energy curves for the solid as a function of atomic volume for both phases.
A theory such as the EMT could be used for this purpose. The minima of these curves
would then yield the equilibrium atomic volumes of the two phases, which could then be
compared with the experimental values.
We would like to improve our implementation of SIC so that it treats electrons in
the f -resonance as well as the bound state f electron. We would also like to remove the
approximation that the SI-corrected f bound state is spherically symmetric, as the angular
dependence of this single localised f electron is very large, and could markedly affect the
results.
148
Conclusions
Appendix A
Local Kinetic Energy Calculation
for Atom in Jellium
In this Appendix we discuss how the local kinetic energy is calculated within the code
for the case of an atom in a finite jellium sphere. From Section 4.2.2, this requires us to
calculate the quantities
∇i D
σ
, ∇2i Dσ , ∇i
6
X
αn(i) j0
n=1
1X
∇i
2
j6=i
1X 2
∇i
2
j6=i
nπri
Rjell
, ∇2i
6
X
αn(i) j0
n=1
nπri
Rjell
aij rij + bij rij
aij rij + bij rij 2
+
1 + cij (ri )rij + dij rij 2 1 + cij (rj )rij + dij rij 2
2
,
aij rij + bij rij 2
aij rij + bij rij 2
+
1 + cij (ri )rij + dij rij 2 1 + cij (rj )rij + dij rij 2
,
(A.0.1)
In the computer program, the derivatives of the Slater determinants are calculated
numerically using either Cartesian or spherical polar coordinates. In the former we have
∇i Dσ (ri ) =
Dσ (xi , yi + δ, zi ) − Dσ (xi , yi , zi )
Dσ (xi + δ, yi , zi ) − Dσ (xi , yi , zi )
î +
ĵ+
δ
δ
Dσ (xi , yi , zi + δ) − Dσ (xi , yi , zi )
k̂
δ
and
∇2i Dσ (ri ) =
Dσ (xi + δ, yi , zi ) − 2Dσ (xi , yi , zi ) + Dσ (xi − δ, yi , zi )
+
δ2
149
(A.0.2)
150
Appendix A. Local Kinetic Energy Calculation for Atom in Jellium
Dσ (xi , yi + δ, zi ) − 2Dσ (xi , yi , zi ) + Dσ (xi , yi − δ, zi )
+
δ2
Dσ (xi , yi , zi + δ) − 2Dσ (xi , yi , zi ) + Dσ (xi , yi , zi − δ)
(A.0.3)
δ2
where δ is a small displacement (∼ 10−4 ) and î, ĵ and k̂ are unit vectors in the x, y and
z directions respectively. If we instead use spherical polar coordinates we have
∇i Dσ (ri ) =
1 ∂Dσ ˆ
1
∂Dσ
∂Dσ
rˆi +
θi +
φ̂i
∂ri
ri ∂θi
ri sin θi ∂φi
(A.0.4)
and
∇2i Dσ (ri ) =
∂ 2 Dσ
2 ∂Dσ
cos θi ∂Dσ
1 ∂ 2 Dσ
1
∂ 2 Dσ
+
+
+
+
ri ∂ri
∂ri2
ri2 ∂θi2
ri2 sin θi ∂θi
ri2 sin2 θi ∂φ2i
(A.0.5)
where again, rˆi , θˆi and φ̂i are unit vectors. In these expressions, the ∂D/∂ri , etc, are
calculated by constructing the determinant
R1 (r1 )Yl1 m1 (θ1 , φ1 ) · · ·
R2 (r1 )Yl2 m2 (θ1 , φ1 ) · · ·
∂D/∂ri = ..
.
RN (r1 )YlN mN (θ1 , φ1 ) · · ·
R1 (rN )Yl1 m1 (θN , φN ) · · · R2 (rN )Yl2 m2 (θN , φN ) ..
..
.
.
∂RN (ri )
YlN mN (θi , φi ) · · · RN (rN )YlN mN (θN , φN ) ri
(A.0.6)
∂R1 (ri )
Yl1 m1 (θi , φi )
ri
∂R2 (ri )
Yl2 m2 (θi , φi )
ri
···
where li and mi denote the l and m values of orbital i. In fact, this determinant is
calculated using the method described in Section 2.4.7, which allows a quicker calculation
for the case where only one column of the determinant has changed.
The ∂Ri (r)/∂r and ∂ 2 Ri (r)/∂r2 can be calculated directly by fitting a spline to the
Ri (r) and reading off the first and second order derivatives at the required radius. The
∂Ylm /∂θ, ∂ 2 Ylm /∂θ2 and ∂Ylm /∂φ are calculated analytically. Calculating the derivatives
of Dσ using Cartesian and spherical polar coordinates both give the same results in the
code.
We now move onto the remaining quantities of Eq. (A.0.1). These are calculated either
numerically or analytically in the code. The third and fourth terms can be calculated
analytically be using the fact that j0 (x) = sin(x)/x
∇i
6
X
n=1
αn(i) j0
nπri
Rjell
=
6
X
n=1
αn(i)
nπri xi Rjell
nπri xi
1
cos(
) −
sin(
)
x̂i + . . .
Rjell ri2 nπ
Rjell
ri 3
Appendix A. Local Kinetic Energy Calculation for Atom in Jellium
∇2i
6
X
n=1
αn(i) j0
nπri
Rjell
=
6
X
nπri
αn(i) sin(
)
Rjell
n=1
−nπ
ri Rjell
151
(A.0.7)
The fifth term in Eq. (A.0.1) can be written
1X
∇i
2
j
1X
2
j
(
aij rij + bij rij 2
aij rij + bij rij 2
+
1 + cij (ri )rij + dij rij 2 1 + cij (rj )rij + dij rij 2
=
2 dc(ri )
(arij −1 + 2b − adrij + bc(ri )rij )(xi − xj ) − (a + brij )rij
dxi
2 )2
(1 + c(ri )rij + drij
)
x̂+
1 X (arij −1 + 2b − adrij + bc(rj )rij )(xi − xj )
x̂ + . . .
2 )2
2
(1
+
c(r
)r
+
dr
j
ij
ij
j
(A.0.8)
4c1 4 Rjell xi
dc(ri )
=
2 + (r 2 − R2 )4
dxi
4 42 Rjell
i
jell
(A.0.9)
where
The remaining term in Eq. (A.0.1) can be written
aij rij + bij rij 2
aij rij + bij rij 2
1X 2
∇i
+
=
2
1 + cij (ri )rij + dij rij 2 1 + cij (rj )rij + dij rij 2
j
( −1
−2
X arij
(−arij
− ad + bc(ri ))(xi − xj )2
+ 2b − adrij + bc(ri )rij
+
−
2 )2
2 )2 r
2(1
+
c(r
)r
+
dr
2(1
+
c(r
)r
+
dr
i
ij
i
ij
ij
ij
ij
j
2
(a + brij )rij
(a + brij )(xi − xj ) dc(ri )
d2 c(ri )
−
−
2
2
(1 + c(ri )rij + drij )2 dxi
2(1 + c(ri )rij + drij )2 dx2i
dc(ri )
−1
2
(arij + 2b − adrij + bc(ri )rij )(xi − xj ) − (a + brij )rij dxi
2 )3
(1 + c(ri )rij + drij
(xi − xj ) dc(ri )
× (c(ri ) + 2drij )
+
rij +
rij
dxi
−1
arij
+ 2b − adrij + bc(rj )rij
2 )2
2(1 + c(rj )rij + drij
+
−2
(−arij
− ad + bc(rj ))(xi − xj )2
2 )2 r
2(1 + c(rj )rij + drij
ij
)
−1
+ 2b − adrij + bc(rj )rij )(c(rj ) + 2drij )(xi − xj )2
(arij
+ ...
2 )3
rij (1 + c(rj )rij + drij
−
(A.0.10)
where
2 )3
32c1 4 Rjell x2i (ri2 − Rjell
4c1 4 Rjell
d2 c(ri )
=
−
2 + (r 2 − R2 )4
2 + (r 2 − R2 )4 )2
dx2i
4 42 Rjell
(4 42 Rjell
i
i
jell
jell
(A.0.11)
152
Appendix A. Local Kinetic Energy Calculation for Atom in Jellium
Both analytic and numerical differentiation for these quantities give the same results
in the code.
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