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 Gy¨orﬀy.

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 Schr¨odinger equation 17 2.1 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 ...... 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 2.4 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 53 3.1 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 3.2 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 ...... 62 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 3.3 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 3.4 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 3.5 The Effective Medium Theory ...... 90 3.5.1 Background Theory ...... 90 3.5.2 Results ...... 96 3.6 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 111 4.1 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 4.2 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]...... 3

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)...... 6

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)...... 7

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...... 8

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, rWS 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 LIST OF FIGURES

1.6 The background density,n ¯i, 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 −3 of density 0.03aB . 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 −3 background densities 0.01aB (top panel) and 0.05aB (bottom panel). . . 72

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 −3 0.03aB , 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...... 77

3.4 Determining the parameter ∆V req, for a SI-corrected cerium atom in inﬁ- −3 nite jellium of density n0 = 0.01aB . Immersion energy is plotted against log(∆V req), and error bars (in green) are placed at diﬀerent values of the convergence...... 85

−3 3.5 Density plots for hydrogen in inﬁnite jellium of density 0.005aB . 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 xix

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...... 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...... 92

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]...... 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 em-

bedded 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 −3 0.04aB , 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 −3 0.03 aB (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 −3 Densities of 0.001aB , 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...... 123

4.5 Plots of atom induced densities for hydrogen in ﬁnite jellium spheres of −3 background density 0.01aB , 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 −3 for a hydrogen atom in inﬁnite jellium for a background density ≈ 0.004aB . 127 LIST OF FIGURES xxi

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...... 128 4.8 Total energy of a hydrogen atom immersed in a 10-electron jellium sphere for different background densities ...... 133 4.9 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 −3 10-electron jellium sphere of density 0.03aB ...... 137 4.12 Re-blocking analysis for hydrogen immersed in a 10-electron jellium sphere −3 of density 0.03aB . 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 back- −3 ground density 0.03aB using HF, LDA, SIC and VQMC ...... 139 4.14 Electron density of a hydrogen atom in a 10-electron jellium sphere of back- −3 ground density 0.03aB using HF, LDA, SIC and VQMC. Note in the top graph curves for HF, LDA and SIC coincide...... 140 4.15 Electron density of a hydrogen atom in a 10-electron jellium sphere of back- −3 ground density 0.002aB using HF, LDA, SIC and VQMC ...... 141 4.16 Electron density of a hydrogen atom in a 10-Electron jellium sphere of −3 background density 0.002aB 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 Schr¨odinger equation (in the Born-Oppenheimer approximation [9], so that the nucleus is held ﬁxed)

  N 2 N 2 X ~ 2 1 X e  − ∇i + + vext(r1, r2, ..., rN ) Ψ(r1, r2, ..., rN ) 2me 2 4π0|ri − rj| i=1 i6=j

= EΨ(r1, r2, ..., rN ) (1.0.1) where the external potential, vext(r), is given by

2 N 2 Z e X Z e n0 v (r) = − − dr0 (1.0.2) ext 4π r 4π |r − r0| 0 i=1 i 0 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|Hˆ |Ψi/hΨ|Ψi (where the Hamiltonian, Hˆ , 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 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].

’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 Schr¨odinger 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].

PN 2 ∗ Writing n(r) = i=1 |φi(r)| , and functionally diﬀerentiating Ev with respect to φi (r), one obtains N single-particle Schr¨odinger-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 sur- prisingly 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 of volume. The minimum of the energy curve for the LDA was at 23.4A˚3 whereas the minimum for the SIC curve was at 29.9A˚3. Experimentally, the alpha phase has a volume 28.2A˚3 and the SIC has a volume 34.7A˚3. So L¨uderset 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 chal- lenging 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. 4 3 If each atom occupies a volume, 3 πa , 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 4 3 n = 1/ 3 π(rsaB) , we have the relation

4 πa3 3 = N 4 3 v 3 π(rsaB) 1 ∴ a = Nv 3 rsaB (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 Schr¨odinger 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, rWS 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. 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. Ap- plying 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 asn ¯i(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

X n¯i(r) = ∆nj(|r − Rj|) (1.0.5) j6=i

Following the EM approach, we then replacen ¯i(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

X 1 X E = ∆E(¯n (R )) + U (R ) (1.0.6) c i i 2 ij ij i 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,n ¯i, 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

Z r=s Z ∞ 0 Ec(¯n) ∆n(r ) 0 Z = ∆E(¯n) +n ¯ 0 dr − dr (1.0.7) N r=0 r0=0 |r − r | r where ∆n(r) is the atom-induced density for an atom with atomic number Z immersed in jellium of densityn ¯. 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(¯n)/N with respect ton ¯ 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 −3 the range 0.001aB 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 Schr¨odinger equation

2.1 The Many-Electron Problem

The time-independent Schr¨odinger 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 2 N 2 X ~ 2 1 X e  − ∇i + + vext(r1, r2, ..., rN ) Ψ(r1, r2, ..., rN ) 2me 2 4π0|ri − rj| i=1 i6=j

= EΨ(r1, r2, ..., rN ) (2.1.1)

2 2 Note that for the rest of this thesis atomic units are used (~ /m=e /4π0=1). The external potential, vext, could be for example the Coulomb attraction between the electrons and an ion of charge Z:

i=N X Z v = − (2.1.2) ext r i=1 i 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 Schr¨odingerequation

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 Schr¨odinger-like equation

1 − ∇2 + V (r) φ (r) = φ (r) (2.1.3) 2 i i i i and orbital φi(r). The total electron density of the system is written as a sum over the single-electron densities, ni(r)

X X 2 n(r) = ni(r) = |φi(r)| (2.1.4) i 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

Z 0 X nj(r ) V (r) = v (r) + dr0 (2.1.5) ext |r − r0| j6=i

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

X |Ψi = ci|Φii (2.1.7) i substituting into hΨ|Hˆ |Ψi we get:

(E0 is the ground state and E1, E2, etc are excited states of increasing energy). The fact P 2 that the |Φi > are orthonormal means that to have < Ψ|Ψ >= 1 we need i |ci| = 1. This normalisation condition and the above equation tell us that:

hΨ|Hˆ |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 single- particle orbitals

φ1(x1) φ1(x2) ... φ1(xN )

φ2(x1) φ2(x2) ... φ2(xN ) Ψ(x1, x2, ..., xN ) = (2.2.1) . . . .

φN (x1) φN (x2) ... φN (xN ) here, x includes the position and the spin. This is the simplest way of including a sign- change in the wavefunction when the positions and spins of any two particles are ex- changed, as is required in order to satisfy the Pauli exclusion principle: 20 Solving the Many-Electron Schr¨odingerequation

Ψ(x1, ..., xi, ..., xj, ..., xN ) = −Ψ(x1, ..., xj, ..., xi, ..., xN ) (2.2.2)

The expectation value of the Hamiltonian, hΨ|Hˆ |Ψi, where

N N ˆ X 1 2 X 1 H = − ∇i + + vext(r1, r2, ..., rN ) (2.2.3) 2 |ri − rj| i=1 i6=j is then minimised with respect to each orbital. Lagrange multipliers, Ei, are introduced in order to ensure the orbitals are normalised.

! ∂ X Z hΨ|Hˆ |Ψi − E |φ |2 = 0 (2.2.4) ∂φ∗ i i i i where

Z Z ∗ ∗ 0 0 X 1 1 X φi (r)φi(r)φj (r )φj(r ) hΨ|Hˆ |Ψi = φ∗(r) − ∇2 φ (r)dr + drdr0− i 2 i 2 |r − r0| i i,j Z ∗ 0 ∗ 0 Z 1 X φi (r)φi(r )φj (r )φj(r) X δ drdr0 + |φ (r)|2v (r)dr (2.2.5) 2 σi,σj |r − r0| i ext i,j 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].

Z 0 Z ∗ 0 0 1 n(r ) X φj (r )φi(r )φj(r) − ∇2 + dr0 + v (r) φ (r) − dr0 δ = E φ (r) 2 |r − r0| ext i |r − r0| σi,σj i i j (2.2.6) where the electron density, n(r), is given by

X X 2 n(r) = hΨ(r1, r2, ..., rN )| δ(r − ri)|Ψ(r1, r2, ..., rN )i = |φi(r)| (2.2.7) i 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 same- spin 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 self- consistent 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 Density Functional Theory

2.3.1 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 Schrödingerequation 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

Hˆ = Tˆ + Vˆ + Uˆ (2.3.1) where Tˆ,Vˆ and Uˆ are the kinetic, external potential and electron-electron repulsion terms, which are given by

Z 1 Tˆ = ψˆ†(r) − ∇2 ψˆ(r)dr (2.3.2) 2

Z † Vˆ = ψˆ (r)vext(r)ψˆ(r)dr (2.3.3)

1 Z 1 Uˆ = ψˆ†(r)ψˆ†(r0)ψˆ(r0)ψˆ(r)drdr0 (2.3.4) 2 |r − r0| where ψˆ†(r) and ψˆ(r) are electron creation and annihilation operators respectively. We will now prove that the ground-state density

† n0(r) = hΨ0|ψˆ (r)ψˆ(r)|Ψ0i (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 0 assume that a different potential, vext(r), which has a ground-state solution, Ψ0, gives 0 rise to the same ground-state density, n0(r). Now, Ψ0 6= Ψ0, since they are ground-state 2.3 Density Functional Theory 23 solutions to different Schrödinger equations. Denoting the Hamiltonians as Hˆ and Hˆ 0, 0 0 and the ground-state energies as E0 and E0 (corresponding to the vext(r) and vext(r) cases respectively), and using the variational principle

0 0 ˆ 0 0 ˆ 0 ˆ ˆ0 ˆ E0 = hΨ0|H |Ψ0i < hΨ0|H |Ψ0i = hΨ0|(H + V − V )|Ψ0i (2.3.6)

Therefore

Z 0 0 E0 < E0 + (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 ˆ† ˆ 0 ˆ† ˆ 0 that hΨ0|ψ (r)ψ(r)|Ψ0i = hΨ0|ψ (r)ψ(r)|Ψ0i = n0(r)), we instead obtain

Z 0 0 E0 < E0 + (vext(r) − vext(r))n0(r)dr (2.3.8)

Adding together Eq. (2.3.7) and Eq. (2.3.8) gives

0 0 E0 + E0 < E0 + E0 (2.3.9)

0 which shows that the initial assumption that the two potentials, vext(r) and vext(r) give 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 Hˆ is give by Eq. (2.3.1) for some vext(r))

Ev[n(r)] = hΨ[n(r)]|Hˆ |Ψ[n(r)]i (2.3.11) 24 Solving the Many-Electron Schr¨odingerequation

We can regard this functional as taking a density, n(r), determining the external 0 potential vext(r) for which this density is the ground-state density (which will not in general equal vext(r)), putting this into the Schr¨odinger equation in order to calculate the ground-state wavefunction for this potential, Ψ(r), and then using this to evaluate hΨ|Hˆ |Ψi. We can write this symbolically as

0 ˆ n(r) vext(r) → Ψ(r) → hΨ|H|Ψi | {z } |{z} The external potential for which The wavefunction obtained by

0 n(r) is the ground-state density inserting vext(r) into for a system of N-interacting Eq. (2.3.3) and solving for electrons. the ground-state of the 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 potential vext(r) for which n(r) is the ground-state density for a system of N-interacting 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|Hˆ |Ψ0i = 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 ground- state 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

In this derivation of the second Hohenberg-Kohn theorem, the density has to be V- representable, 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

Z 1 E [n(r)] = hΨ[n(r)]|ψˆ†(r) − ∇2 ψˆ(r)|Ψ[n(r)]idr+ v 2 1 Z ψˆ†(r)ψˆ†(r0)ψˆ(r0)ψˆ(r) Z hΨ[n(r)]| |Ψ[n(r)]idrdr0 + v (r)n(r)dr (2.3.16) 2 |r − r0| ext 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

1 Z n(r)n(r0) Z E [n(r)] = T [n(r)] + drdr0 + v (r)n(r)dr + E [n(r)] (2.3.17) v s 2 |r − r0| ext xc 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 ground- state 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 Schr¨odingerequation

The single-particle kinetic energy, Ts, can be written as

1 Z 1 T [n(r)] = hΨni[n(r)]| − ∇2 |Ψni[n(r)]idr (2.3.18) s 2 2 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 Z 1 ψˆ†(r)ψˆ†(r0)ψˆ(r0)ψˆ(r)drdr0 → 0 (2.3.19) 2 |r − r0| which doesn’t affect the derivation of the first Hohenberg-Kohn theorem. P 2 Let us make the mathematical transformation, n(r) = i |φi(r)| . 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 φi (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.

Z 0 Z δ 1 n(r)n(r ) 0 ∗ Ts[n(r)] + 0 drdr + vext(r)n(r)dr δφi (r) 2 |r − r | Z ! X 2 +Exc[n(r)] − Ei |φi(r)| dr = 0 (2.3.20) i where the Ei are lagrange multipliers arising from the normalisation constraint. Perform- ing the diﬀerentiation:

Z 0 δTs[n] 1 n(r ) 0 δExc[n] ∗ + 0 dr + + vext(r) φi(r) = Eiφi(r) (2.3.21) δφi (r) φi(r) |r − r | δn where we have used the chain-rule

δ δn(r) δ δ ∗ = ∗ = φi(r) (2.3.22) δφi (r) δφi (r) δn(r) δn(r) and the relations

δ Z f(r0)g(r0)dr0 = g(r) (2.3.23) δf(r) 2.3 Density Functional Theory 27 and

δ Z g(r0)dr0 = 0 where g(r) 6= g[f(r)](r) (2.3.24) δf(r) Let us simplify the ﬁrst term of Eq. (2.3.21). We know that the ground-state solution to the non-interacting Schr¨odinger equation is of the form

φ1(r1) φ1(r2) ··· φ1(rN )

φ (r ) φ (r ) ··· φ (r ) ni 1 2 1 2 2 2 N Ψ (r1, r2, ..., rN ) = √ (2.3.25) N ......

φN (r1) φN (r2) ··· φN (rN ) and that this gives rise to the density

Z N X 2 X 2 n(r) = dr1dr2...drN δ(r − ri)|Ψ(r1, r2, ..., rN )| = |φi(r)| (2.3.26) i=1 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 Z X 1 T [n] = hΨni[n]|− ∇2|Ψni[n]i = dr φ∗(r) − ∇2 φ (r) (2.3.27) s 2 i i 2 i i i

Performing a functional diﬀerentiation on this term and inserting it into Eq. (2.3.21) gives us our ﬁnal result

1 Z n(r0) − ∇2 + dr0 + V [n] + v (r) φ (r) = E φ (r) (2.3.28) 2 |r − r0| xc ext i i i 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 Z n(r0) − ∇2 + dr0 + V σ [n↑, n↓] + v (r) φσ(r) = Eσφσ(r) (2.3.29) 2 |r − r0| xc ext i i i σ ↑ ↓ ↑ ↓ σ where Vxc[n , n ] = δExc[n , n ]/δn . 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 Schrödingerequation 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 exchange- σ correlation energy, Exc[n ], 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 Schr¨odinger-like equation is written

1 − ∇2 + V σ(r) φσ(r) = Eσφσ(r) (2.3.30) 2 i i i and the potential is given by

Z n(r0) V σ(r) = dr0 + v (r) + V σ (n↑(r), n↓(r)) (2.3.31) |r0 − r| ext xc where the electron density, n(r), is given by

N X 2 n(r) = |φi(r)| (2.3.32) i=1 σ ↑ ↓ Vxc(n (r), n (r)) is an approximation to the exact exchange-correlation potential (see 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

σ X σ 2 n (r) = |φi (r)| (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 exchange- correlation energy is written as

Z LDA Exc [n(r)] = 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 generalisa- tion of this approximation to include spin. In this approximation we have

Z LSDA σ σ Exc [n (r)] = drxc(n (r))n(r) (2.3.35)

σ where xc(n ) 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 Schr¨odingerequation

Z σ ∗ σ 0 σ 0 ∗ σ 1 X φi (r) φi (r )φj (r ) φj (r) E [nσ(r)] = − drdr0 (2.3.36) x 2 |r − r0| i,j,σ and in the LSDA we ﬁnd that the exchange energy per electron (writing xc = x + c) is

" 1 1 # 3 9π 3 3 9π 3 (r , χ) = − + (21/3 − 1) f(χ) /r (2.3.37) x s 4π 4 4π 4 s

4 3 where rs is deﬁned by 3 πrs(r) n(r) = 1 and where χ is the spin polarisation

n↑(r) − n↓(r) χ(r) = (2.3.38) n↑(r) + n↓(r) and f(χ) is deﬁned by

4 4 (1 + χ(r)) 3 + (1 − χ(r)) 3 − 2 f(χ) = 1 (2.3.39) 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. Alter- natively, 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 extrap- olation to infinite volume.

The Exchange-Correlation Potential

The quantity which appears in the Schr¨odinger equation, the exchange-correlation potential, is

↑ ↓ ↑/↓ δExc[n , n ] Vxc (r) = (2.3.40) δn↑/↓(r) n↓/↑ 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

↑ 0 0 ↓ 0 ↑ 0 ↓ 0 ↑ δExc[n] Exc[n (r ) + ηδ(r − r), n (r )] − Exc[n (r ), n (r )] Vxc(r) = = lim (2.3.41) δn↑(r) η→0 η

↑ ↓ ↑ ↓ Writing f(n (r), n (r)) = xc n (r), n (r) n(r) we get

Z Z ↑ 1 0 ↑ 0 0 ↓ 0 0 ↑ 0 ↓ 0 Vxc(r) = lim dr f n (r ) + ηδ(r − r), n (r ) − dr f n (r ), n (r ) (2.3.42) η→0 η

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) ηδ(r0 − r) + O(η2) + ··· (2.3.43) dn↑(r0) Dropping terms containing η to powers greater than one, this gives

(Z ↑ 0 ↓ 0 ) ↑ 1 0 0 df n (r ), n (r ) Vxc(r) = lim dr ηδ(r − r) η→0 η dn↑(r0)

df n↑(r), n↓(r) d d = = (n↑(r), n↓(r))n(r) = + n(r) xc (2.3.44) dn↑(r) dn↑(r) xc xc dn↑(r) An analogous expression is obtained for spin down, giving the general formula

d V σ (r) = + n(r) xc (2.3.45) xc xc dnσ(r)

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

d d V σ (r) = + n(r) x + + n(r) c (2.3.47) xc x dnσ(r) c dnσ(r) | {z } | {z } σ σ Vx (r) Vc (r) We will consider the exchange contribution to this potential in the next section. 32 Solving the Many-Electron Schr¨odingerequation

The Exchange Contribution to the Exchange-Correlation Potential

From Eq. (2.3.37) we have

" 1 1 # d (r) 3 9π 3 3 9π 3 x = − + (21/3 − 1) f(χ) dnσ(r) 4π 4 4π 4

1 d 4 3 × π(n↑ + n↓) − dnσ(r) 3

1 1 1 3 9π 3 df(χ) dχ 3 (2 − 1) σ (2.3.48) rs 4π 4 dχ dn (r) where

dχ dχ d n↑ − n↓ 1 n↑ − n↓ 1 ≡ = = − = − (χ − 1) (2.3.49) dn↑(r) dn↑(r) n↓ dn↑ n↑ + n↓ n↑ + n↓ (n↑ + n↓)2 n

dχ dχ d n↑ − n↓ 1 n↑ − n↓ 1 ≡ = = − − = − (χ + 1) (2.3.50) dn↓(r) dn↓(r) n↑ dn↓ n↑ + n↓ n↑ + n↓ (n↑ + n↓)2 n hence

dχ (χ ∓ 1) = − (2.3.51) dn↑/↓ n Therefore we have

1 dx(r) 1 1 1 3 9π 3 df(χ) 1 3 σ = x(r) + (χ ∓ 1)(2 − 1) (2.3.52) dn (r) 3 n(r) 4π 4 dχ rsn(r) Therefore the exchange part of the exchange-correlation potential is

1 3 σ 4 1 3 9π df(χ) 1 Vx (r) = x(r) + (χ ∓ 1)(2 3 − 1) (2.3.53) 3 4π 4 dχ rs

2.3.5 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 Z Z n(r)n(r0) U[n] = dr dr0 (2.3.54) 2 |r0 − r| Splitting the density into orbital spin densities using 2.3 Density Functional Theory 33

X σ X σ 2 n(r) = nα(r) = |φα(r)| (2.3.55) α,σ α,σ (where α label the orbitals), gives

Z Z σ σ0 0 1 X n (r)n 0 (r ) U[n] = dr dr0 α α (2.3.56) 2 |r0 − r| α,σ,α0,σ0 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 Z Z nσ (r)nσ (r0) dr dr0 α α +Eexact[nσ , 0] = 0 (2.3.57) 2 |r0 − r| xc α | {z } σ =U[nα] 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 ↑ ↓ X σ σ Ev[n] = Ts[n] + vext(r)n(r)dr + U[n] + Exc[n , n ] − (U[nα] + Exc[nα, 0]) (2.3.58) α,σ

Minimisation of the energy functional yields a Schr¨odinger equation with an orbital dependent potential

1 − ∇2 + V α,σ (r) φσ (r) = Eσφσ (r) (2.3.59) 2 SIC α α α

Z nσ (r) V α,σ (r) = V σ(r) − α dr0 − V σ (nσ (r), 0) (2.3.60) SIC |r0 − r| xc α 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 σ ∗ σ φα (r)φα0 (r)dr = 0 (2.3.61) 34 Solving the Many-Electron Schr¨odingerequation for all α and α0. One approach would be to re-derive the Kohn-Sham equations using additional La- grange 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:

1 Z φσ,orth(r) = φσ (r) − φσ∗ (r0)φσ (r0)dr0φσ (r) (2.3.62) E2 N (1)σ E2 E1 E2 E1 where the energy eigenvalues, Ei, are used as labels for the orbitals, and where E1 < E2. The normalisation factor N (1)σ ensures that R |φσ,orth(r)|2dr = 1. The two states are E2 now orthogonal to one another, as can be seen by pre-multiplying with φσ∗ (r), and then E1 integrating over r to give zero. The next step is to orthogonalise the state with the third lowest energy against the ﬁrst two states:

1 Z φσ,orth(r) = φσ (r) − φσ∗ (r0)φσ (r0)dr0φσ (r)− E3 N (2)σ E3 E1 E3 E1 Z φσ,orth∗(r0)φσ (r0)dr0φσ,orth(r) (2.3.63) E2 E3 E2 where E < E < E and the normalisation factor N (2)σ ensures that R |φσ,orth(r)|2dr = 1. 1 2 3 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 Variational Quantum Monte Carlo

2.4.1 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 |Hˆ |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 |Hˆ |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 |Hˆ |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

n Z b (b − a) X f(x)dx ≈ f(x ) (2.4.4) n i a 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 Schr¨odingerequation

Z b Z b n f(x) 1 X f(xi) f(x)dx = g(x) dx = | (2.4.5) g(x) n g(x ) n→∞ a a i i

In this case the xi in the sum are sampled from the probability distribution, g(x) and g(x) is normalised as, R 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 |Hˆ |ΨT i. Written out fully, we want to calculate:

1 Z hΨ |Hˆ |Ψ i = Ψ∗ (R)Hˆ Ψ (R)dR (2.4.6) T T N T T R 2 where N = |ΨT (R)| 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

Z Z 2 ˆ 1 ∗ ˆ |ΨT (R)| HΨT (R) ΨT (R)HΨT (R)dR = dR (2.4.7) N N ΨT (R) we calculate this integral by turning it into a sum:

Z 2 ˆ n ˆ |ΨT (R)| HΨT (R) 1 X HΨT (R) dR ≈ = E¯ (2.4.8) N Ψ (R) n Ψ (R) T i=1 T where n is large. The values of R in the summation are taken from the probability 2 distribution |ΨT (R)| /N . We will refer to these Rs as configurations. The quantity being summed over, (Hˆ ΨT )/ΨT , is known as the local energy. The method then is to generate a set of configurations according to the probability dis- 2 tribution |ΨT (R)| /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 ˆ !2 1 X HΨT (Ri) σ2 = − E¯ (2.4.9) l.e. n Ψ (R ) i=1 T i To see how this works, consider the Schr¨odinger equation for the ground-state wavefunction

Hˆ Ψ0(R) = E0Ψ0(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 ground- state 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 ground- state 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 proba- 2 bility distribution |ΨT (R)| /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 0 move is then proposed from a probability distribution, Ptrial(R → R ), 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 0 acceptance probability Pacceptance(R → R ). Another move is then made. 38 Solving the Many-Electron Schrödingerequation

In order that the conﬁgurations generated this way correctly sample the probability 2 distribution, P (R) (= |ΨT (R)| /N in our case), we need the following relationship to be satisﬁed:

P (R → R0) P (R0) = (2.4.11) P (R0 → R) P (R) where P (R → R0) is the total probability of a move taking place from R to R0, i.e.:

0 0 0 P (R → R ) = Ptrial(R → R )Pacceptance(R → R ) (2.4.12)

We also have

0 0 0 P (R → R) = Ptrial(R → R)Pacceptance(R → R) (2.4.13)

0 0 If we choose our Ptrial such that Ptrial(R → R ) = Ptrial(R → R), then dividing Eq. (2.4.12) by Eq. (2.4.13) and using Eq. (2.4.11), we ﬁnd

0 0 P (R ) Pacceptance(R → R ) = 0 (2.4.14) P (R) Pacceptance(R → R) A form for the acceptance probability which satisﬁes this equation, and the one that we use is

P (R0) P (R → R0) = min 1, (2.4.15) acceptance P (R) 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

0 a probability distribution, Ptrial(r → r ), 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) P (r → r0) = min 1, (2.4.16) acceptance P (r) 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 ncorrth configuration, and throwing the other configurations away. 0 One should choose Ptrial(r → r ) 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:

 0  x − L/2 < x < x + L/2   1 0 0  L3 if y − L/2 < y < y + L/2 Ptrial(r → r ) = (2.4.17) 0  z − L/2 < z < z + L/2   0 otherwise 40 Solving the Many-Electron Schr¨odingerequation

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 ) = (2.4.18) ......

φN (r1) φN (r2) ... φN (rN ) 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 ) (2.4.19) where 2.4 Variational Quantum Monte Carlo 41

φ1(r1) φ1(r2) ··· φ1(rN/2)

φ (r ) φ (r ) ··· φ (r ) ↑ 2 1 2 2 2 N/2 D (r1, ··· , rN/2) = (2.4.20) ......

φN/2(r1) φN/2(r2) ··· φN/2(rN/2) and

φN/2+1(rN/2+1) φN/2+1(rN/2+2) ··· φN/2+1(rN )

φ (r ) φ (r ) ··· φ (r ) ↓ N/2+2 N/2+1 N/2+2 N/2+2 N/2+2 N D (rN/2+1, ··· , rN ) = (2.4.21) ......

φN (rN/2+1) φN (rN/2+2) ··· 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 Coulom- bic correlation between electrons. This can be achieved by making the wavefunction small whenever any two electrons get close to one another. The following wavefunction incor- porates this:

  X arij ↑ ↓ ΨT (r1, ··· , rN ) = exp −  D (r1, ··· , rN/2)D (rN/2+1, ··· , rN ) 1 + brij 1≤i

2.4.7 Updating the Slater Determinants

For each move, we have to calculate the acceptance probability of an electron i moving 0 from a position ri to a position ri (Eq. (2.4.15) ). This means calculating the quantity 42 Solving the Many-Electron Schr¨odingerequation

0 2 σ 2 |ΨT (r1, r2, ..., ri, ..., rN )| |Dnew| 2 ≈ σ 2 (2.4.23) |ΨT (r1, r2, ..., ri, ..., rN )| |Dold| 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 0 σ ri. D 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

  (Dσ T )−1/qσ if k = i  old jk σ T −1  (Dnew )jk =  h i  σ T −1 σ T −1 PN σ T −1 0 σ  (Dold )jk − (Dold )ji l=1(Dold )lk φl(ri) /q if k 6= i

N N Dσ X X qσ = new = (Dσ T )−1(Dσ ) = (Dσ T )−1φ (r0 ) (2.4.24) Dσ old ji new ji old ji j i old j=1 j=1 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 σ 2 we need to calculate, |Dnew| /|Dold| , in order to evaluate the acceptance probability, is automatically generated by this updating algorithm, and is qσ2.

2.4.8 Calculating the Local Energy

The local energy (Hˆ ΨT )/ΨT , is to be calculated for conﬁgurations R which sample the 2 probability distribution |ΨT (R)| /N . The Hamiltonian has the form:

X 1 Hˆ = − ∇2 + V (r , r , ··· , r ) (2.4.25) 2 i 1 2 N i

The potential energy part of the local energy is just V (r1, r2, ··· , rN ). The kinetic 1 2 energy part, − 2 ∇i 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

∇iΨT ∇i ln ΨT = ΨT 2 2 2 ∇i ΨT ∇iΨT ∇i ln ΨT = − ΨT ΨT

2 1 ∇i ΨT 1 2 1 2 ∴ − = − ∇i ln ΨT − (∇i ln ΨT ) (2.4.26) 2 ΨT 2 2 Introducing the quantities T = − 1 ∇2 ln Ψ and F = √1 ∇ ln Ψ , the local kinetic i 4 i T i 2 i T energy can be written

2 X 1 ∇ ΨT X − i = 2T − F 2 (2.4.27) 2 Ψ i i i T 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

X 1 X 1 Z T = − ∇2 ln Ψ ≡ − drΨ2 ∇2 ln Ψ (2.4.28) i 4 i T 4 T i T i i X 1 X 1 Z F 2 = (∇ ln Ψ )2 ≡ drΨ2 (∇ ln Ψ )2 (2.4.29) i 2 i T 2 T i T i i

Then using ∇i ln ΨT = ∇iΨT /ΨT and integration by parts:

1 Z 1 Z 1 Z − drΨ2 ∇2 ln Ψ = − drΨ ∇2Ψ + dr(∇ Ψ )2 4 T i T 4 T i T 4 i T 1 1 Z Z 1 1 Z = [− Ψ ∇ Ψ ] − − dr(∇ Ψ )2 + dr (∇ Ψ )2 = dr(∇ Ψ )2 (2.4.30) 4 T i T 4 i T 4 i T 2 i T and

1 Z 1 Z drΨ2 (∇ ln Ψ )2 = dr(∇ Ψ )2 (2.4.31) 2 T i T 2 i T Hence when summed over a large number of conﬁgurations, n, we have:

X X 2 Ti = Fi (2.4.32) i i Or dividing through by n: 44 Solving the Many-Electron Schr¨odingerequation

2 < Ti >=< Fi > (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 2 1 2 1 Z Z − ∇ri − ∇rj + − − ΨT (r1, ..., ri, ..., rj, ..., rN )/ΨT (r1, ..., ri, ..., rj, ..., rN ) 2 2 rij ri rj (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 ∂ Z − − ΨT (r1, ..., ri, ..., rN )|ri→0 = 0 (2.4.35) 2 ri ∂ri ri 2.4 Variational Quantum Monte Carlo 45

We have omitted the angular parts of the Laplacian, and also the second order dif- ferential 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 ∂ Z − − φ(ri)|ri→0 = 0 (2.4.36) 2 ri ∂ri ri The nuclear cusp condition is therefore

∂φj(ri) |ri→0 = −Z (2.4.37) ∂ri 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 2 1 2 1 − ∇ri − ∇rj + ΨT (r1, ..., ri, ..., rN )|rij →0 = 0 (2.4.38) 2 2 rij Our ﬁrst step in deriving the electron-electron cusp condition is to make the following change of variables

r = ri − rj r + r R = i j (2.4.39) 2 It is easy to show (use Cartesian coordinates) that

1 1 1 − ∇2 − ∇2 = −∇2 − ∇2 (2.4.40) 2 ri 2 rj r 4 R Now, with a wavefunction of the form

↑ ↓ ΨT = exp(−u(r))D D (2.4.41) where

ar u(r) = (2.4.42) 1 + br 46 Solving the Many-Electron Schr¨odingerequation we must satisfy

−∇2[exp(−u(r))D↑D↓] + 1 r r | → 0 (2.4.43) exp(−u(r))D↑D↓ r→0 Therefore

( ) ∂2u(r) ∂u2 2 ∂u ∂u ∇D↑D↓ ∇2D↑D↓ 1 − + − − 2 ˆr. + + | → 0 (2.4.44) ∂r2 ∂r r ∂r ∂r D↑D↓ D↑D↓ r r→0 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 ∂u 1 the third and sixth. Therefore, using that ∂r |r→0 → a , we need a = − 2 . 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 1 term, the fourth term also diverges. We see that in this case we need a = − 4 .

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 2 ˆ n ˆ |ΨT (R)| HΨT (R) 1 X HΨT (R) < Ψ |Hˆ |Ψ >= dR ≈ (2.4.45) T T N Ψ (R) n Ψ (R) T i=1 T R 2 where N = |ΨT (R)| dR. Here, one calculates the local energy for a set of conﬁgurations, 2 Ri, which sample the probability distribution |ΨT (R)| /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

ˆ |Ψ(0)(R)|2 2 ˆ R 2 HΨT (R) R T J(R) HΨT (R) |ΨT (R)| dR (0) (0) Ψ (R) dR ˆ ΨT (R) N J (R) T < ΨT |H|ΨT >= R = (0) |Ψ (R)|2dR |Ψ (R)|2 2 T R T J(R) dR N (0) J(0)(R)

2 ˆ Pn J(Ri) HΨT (Ri) i=1 J(0)(R ) Ψ (Ri) ≈ i T = E¯ (2.4.46) 2 Pn J(Ri) (0) i=1 J (Ri) 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

2 h ˆ i2 Pn J(Ri) HΨT (Ri) − E¯ i=1 J(0)(R ) Ψ (Ri) σ2 = i T (2.4.47) l.e. 2 Pn J(Ri) (0) i=1 J (Ri)

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 Schr¨odingerequation

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. (2.4.48) n − 1 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:

1 x(j) = (x(j−1) + x(j−1)) (2.4.49) i 2 2i−1 2i

j (j) (0) where j is the transformation number, i runs from 1 to n/2 (= ntot) and xi = xi. In (j) this way the number of values of xi halves after each transformation. Notice that the (j) mean,x ¯, of the xi remains the same after each transformation. If σl.e. is the standard (j) (j) deviation of the xi at transformation number j, then the error on the mean of these xi is PSfrag replacemen

2.4 Variational Quantum Monte Carlo 49 ts

0.0032

0.00315 Energy 0.0031 cal Lo

the 0.00305 of

Mean 0.003 on

Error 0.00295

0.0029 0 2 4 6 8 10 12 Transformation Number Figure 2.1: Re-blocking analysis for hydrogen immersed in a 10-electron jellium sphere of −3 density 0.03aB . The error on the mean levels oﬀ at just under 0.0031eV and therefore this is the error we quote on the total energy.

1  (j)  2 (j) n σ 1 1 tot 2 (j) l.e.  X (j) 2  σmean = q = (xi − x¯ ) (2.4.50) (j) (n(j) − 1) n(j)  ntot − 1 tot tot i=1

There is an error attached to this error on the mean which is approximated as

! 1 2 4 σ(j) (2.4.51) (j) mean ntot − 1

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 Schrödingerequation

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 2 X |ΨT (r1, r2, ..., rN )| n(r) = dr dr ...dr δ(r − r ) (2.4.52) 1 2 N i N i=1 2 In fact, because the electrons sample the probability distribution |ΨT (r1, ..., rN )| /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 (2.4.53) j Nj where the position vector r lies within the box centred at ri, and where ni is normalised P so that i niVi = 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 n(r) = n = i (2.4.54) i 4 3 3 P 3 π(ri − ri−1 ) j Nj 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 Schr¨odingerequation 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 Solving the Schr¨odinger Equation

3.1.1 The Radial Schr¨odinger Equation

We reproduce the Kohn-Sham equations (Eq. (2.3.29) )

1 − ∇2 + V σ(r) φσ(r) = Eσφσ(r) (3.1.1) 2 i i i

53 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 n(r0) − n Z n n V σ(r) = 0 dr0 − + V σ (n↑(r), n↓(r)) − V ( 0 , 0 ) (3.1.2) |r0 − r| r xc xc 2 2 where integrals in r are over all space unless the limits are stated explicitly, r = |r| σ ↑ ↓ and Vxc(n (r), n (r)) is the exchange-correlation potential in the LDA. In particular we 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

0 Z 0 Z r =Rjell σ n(r ) 0 n0 0 Z σ ↑ ↓ V (r) = 0 dr − 0 dr − + Vxc(n (r), n (r)) (3.1.3) |r − r| r0=0 |r − r| r

4 3 where Rjell is the radius of the jellium sphere and 3 πRjelln0 = N, where N is the number 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 σ namely −Vxc(n0/2, n0/2). This term guarantees that V (r) → 0 as |r| → ∞. This will be 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 can make a separation of variables in the wavefunction: φσ(r) = Y (θ, φ)Rσ (r), and i lm Eilm obtain the radial Schrödinger equation

1 ∂2 2 ∂ l(l + 1) − + + + V σ(r) Rσ (r) = ERσ (r) (3.1.4) 2 ∂r2 r ∂r 2r2 El El where l labels the eigenstates of L2

2 L Ylm(θ, φ) = l(l + 1)Ylm(θ, φ) (3.1.5) 3.1 Solving the SchrödingerEquation 55 and where m takes the values −l, −l + 1, ··· , l. Because m does not appear in the radial σ Schrödinger equation, a given solution REl(r) will have a degeneracy of 2l + 1. Defining σ σ UEl(r) = rREl(r) we can see that:

d2 2 d U σ (r) d −U σ U σ 0 2 −U σ U σ 0 + El = El + El + El + El = dr2 r dr r dr r2 r r r2 r U σ U σ 0 U σ 00 U σ 0 U σ U σ 0 U σ 00 2 El − El + El − El − 2 El + 2 El = El (3.1.6) r3 r2 r r2 r3 r2 r Putting this into the Schr¨odinger equation gives

1 1 d2U σ (r) l(l + 1) − El + Rσ (r) + V σ(r)Rσ (r) = ERσ (r) (3.1.7) 2 r dr2 2r2 El El El Multiplying by r gives

1 d2U σ (r) l(l + 1) − El + U σ (r) + V σ(r)U σ (r) = EU σ (r) (3.1.8) 2 dr2 2r2 El El El 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

bound σ 2 X Un,l (r) nσ(r) = |Y (θ, φ)|2 (3.1.9) lm r2 n,l,m 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

bound 2 σ Z kF σ2 X Un,l (r) 1 X U (k, r) nσ(r) = |Y (θ, φ)|2 + (2l + 1)k2 l dk (3.1.10) lm r2 2π2 r2 n,l,m 0 l

2 1 where, kF , is the Fermi wave-number and is given by kF = (3π 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 2 2l+1 m=−l |Ylm(θ, φ)| = 4π to re-write the ﬁrst term as

bound σ 2 1 X Un,l (r) (2l + 1) (3.1.11) 4π r2 n,l For the case where sub-shells are only partially occupied, we impose spherical symmetry by making the approximation:

X mnum |Y (θ, φ)|2 = (3.1.12) lm 4π m 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

1 h 2 2 2 i n↑(r) = U ↑ (r) + U ↑ (r) + 2U ↑ (r) (3.1.13) 4πr2 1s 2s 2p

1 h 2 2 i n↓(r) = U ↓ (r) + U ↓ (r) (3.1.14) 4πr2 1s 2s

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) σ(i) potentials for the next iteration, i.e. Vin (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 SchrödingerEquation 57

σ(i+1) σ(i) In practice, using Vin (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) σ(i) σ(i) Vin (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) σ(i) σ(i) σ(i−1) Vin (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) σ(i) σ(i) σ(i−1) nin (r) = f[nout , nin , nin , ...] (3.1.17) σ(i+1) σ(i+1) The potential Vin (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

Z σ σ(i) σ(i) 2 ∆V = Vout (r) − Vin (r) r 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 Sym- metry

The Hartree term in the Schr¨odinger equation can be simpliﬁed by using the fact that the density is spherically symmetric.

Z 0 Z r Z ∞ n(r ) 0 1 0 02 0 0 0 0 0 dr = n(r )4πr dr + n(r )4πr dr (3.1.19) |r − r| r 0 r To prove the relation we use Gauss’s theorem (in S.I. units):

I 1 Z E·dS = dτρ(r) (3.1.20) ε0 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 2 E·dS = ErdSr = Er(R)4πR (3.1.21)

And the right side gives

1 Z 1 Z R dτρ(r) = ρ(r)4πr2dr (3.1.22) ε0 ε0 0 Therefore equating the two sides and using ρ(r) = en(r)

Z R e 2 Er(R) = 2 n(r)4πr dr (3.1.23) 4πε0R 0 Or in atomic units 3.1 Solving the Schr¨odingerEquation 59

Z R 1 2 Er(R) = 2 n(r)4πr dr (3.1.24) eR 0 The Coulomb potential at R due to this charge distribution is

1 Z R 1 Z R V (R) = n(r)4πr2dr − n(r)4πrdr + const (3.1.25) eR 0 e 0 as is proved below

∂V (R) E (R) = − = r ∂R ∂ 1 Z R 1 Z R − n(r)4πr2dr − n(r)4πrdr + const = ∂R eR 0 e 0 Z R Z R Z R 1 2 1 ∂ 2 1 ∂ 2 n(r)4πr dr − n(r)4πr dr + n(r)4πrdr = eR 0 eR ∂R 0 e ∂R 0 Z R Z R 1 2 1 2 1 1 2 2 n(r)4πr dr − n(R)4πR + n(R)4πR = 2 n(r)4πr dr (3.1.26) eR 0 eR e eR 0 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 ∞ Z ∞ edn(r) V (0) = dV (r) = (3.1.27) 0 0 4πε0r Using dn(r) = n(r)4πr2dr and switching to atomic units gives

1 Z ∞ V (0) = n(r)4πrdr (3.1.28) e 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 Z R Z ∞ eV (R) = n(r)4πr2dr + n(r)4πrdr (3.1.29) R 0 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 Z r Z ∞ n0 0 1 02 0 0 0 0 dr = n04πr dr + n04πr dr (3.1.30) |r − r| r 0 r 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 Schr¨odinger 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 r Z rmax 1 02 0 0 0 4 2 2 2 n04πr dr + n04πr dr = πr n0 + 2π(rmax − r )n0 (3.1.31) r 0 r 3 In the case of the ﬁnite jellium sphere, where the jellium only extends out to a radius

Rjell, we have

0  Z r =Rjell 4 2 2 2 n0 0  3 πr n0 + 2π(Rjell − r )n0 r ≤ Rjell 0 dr = (3.1.32) r0=0 |r − r| 1 4 3  r 3 πRjell n0 r > Rjell

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 −1 4 E (r) = − πR 3n (3.1.33) r e r2 3 jell 0 In S.I. units this reads

e 4 3 Er(r) = 2 πRjell n0 (3.1.34) 4πε0r 3 Therefore

2 1 4 3 Er(r)4πr = πRjell n0e ε0 3 I 1 Z → E·dS = dτρ(r) (3.1.35) ε0 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 Schr¨odinger equation, Eq. (3.1.4), becomes

∂2 2 ∂ l(l + 1) + − + k2 Rσ (r) = 0 (3.2.1) ∂r2 r ∂r r2 El where we have deﬁned, k2 = 2E. If we then make the change of variables ρ = kr, then we have

∂2 2 ∂ l(l + 1) + + 1 − Rσ(ρ) = 0 (3.2.2) ∂ρ2 ρ ∂ρ ρ2 l σ σ We have re-written Rkl(r) = Rl (ρ) because the operator on the left side of the equation σ now only depends on ρ (and not on k and r separately), and so the solution Rkl(r) must 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

1 ∂ l sin ρ j (ρ) = (−ρ)l (3.2.3) l ρ ∂ρ ρ

1 ∂ l cos ρ n (ρ) = −(−ρ)l (3.2.4) l ρ ∂ρ ρ 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 Schr¨odinger equation have the following asymptotic form

Rk,l(r → ∞) → Bl(k)jl(kr) + Cl(k)nl(kr) (3.2.6)

3.2.3 Matching to the Boundary Condition

When tasked with writing a computer program to solve the radial Schr¨odinger 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 Schr¨odinger 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 0 0 Rk,l(rmax) = Bl(k) kjl(krmax) − k tan δl(k)nl(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

Cl(k) tan δl(k) = − (3.2.10) Bl(k) 0 Instead of matching Rk,l and then Rk,l, we proceed by ﬁrst matching the logarithmic derivative of Rk,l

0 ∂ Rk,l(r) ln Rk,l(r) = (3.2.11) ∂(kr) Rk,l(r) 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 = 0 0 (3.2.12) Rk,l(rmax) k[jl(krmax) − tan δl(k)nl(krmax)] Re-arranging gives

Rk,l(rmax) 0 jl(krmax) − k 0 jl(krmax) Rk,l(rmax) tan δl(k) = (3.2.13) Rk,l(rmax) 0 nl(krmax) − k 0 n (krmax) Rk,l(rmax) l 0 0 We now need to replace the Rk,l(rmax) and Rk,l(rmax) with Uk,l(rmax) and Uk,l(rmax), since these are the quantities that we propagate outwards using the radial Schr¨odinger equation.

0 0 Uk,l(r) = rRk,l(r) → Uk,l(r) = Rk,l(r) + rRk,l(r) U 0 (r) 1 R0 (r) → k,l = + k,l Uk,l(r) r Rk,l(r) R (r) 1 → k,l = (3.2.14) 0 U 0 (r) Rk,l(r) k,l − 1 Uk,l(r) r Putting this into Eq.( 3.2.13) gives

0 0 Uk,l(rmax) 1 kjl(krmax) − U (r ) − r jl(krmax) tan δ (k) = k,l max max (3.2.15) l U 0 (r ) 0 k,l max 1 kn (krmax) − − nl(krmax) l Uk,l(rmax) rmax

This equation allows us to calculate the phase shift, δl(k), given the propagated values 0 Uk,l(rmax) 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

−1 δl(k) = tan (tan δl(k)) + nπ, n Z (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

Bl(k) Rk,l(rmax) = [cos δl(k)jl(krmax) − sin δl(k)nl(krmax)] (3.2.18) cos δl(k) | {z } taken to be = 1 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) j (x → ∞) → (3.2.20) l x cos(x − lπ/2) n (x → ∞) → − (3.2.21) l x Therefore our boundary condition is

sin(krmax + δl(k) − lπ/2) Rl(krmax) = (3.2.22) krmax 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 2 nscatt(r) = i |φi(r)| . 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

Z r=rhw 2 2 N Rl (k, r)r dr = r=0

Z r=rasym Z r=rhw 2 2 2 sin(kr + δl(k) − lπ/2) N Rl (k, r)r dr + N 2 dr = 1 (3.2.23) r=0 r=rasym k

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

2k2 N = (3.2.24) rhw 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 2 Rk,l(r) = k (cos δl(k)jl(kr) − sin δl(k)nl(kr)) (3.2.25) rhw

3.2.5 Calculating the Scattering State Density

We introduced the charge density for scattering state electrons in section 3.1.1, Eq. (3.1.10) as

1 Z kF X U σ2(k, r) nscatt(r) = (2l + 1)k2 l dk (3.2.26) 2π2 r2 0 σ,l Here we motivate its form. The starting point is the equation

scatt X σ 2 n (r) = |φl,m(k, r)| (3.2.27) k,l,m,σ σ We take it that the Rl (k, r) are independent of m. Hence 66 An Atom in Inﬁnite Jellium Solved using DFT

X 2l + 1 |φσ (k, r)|2 = Rσ2(k, r) (3.2.28) l,m 4π l m

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 Schr¨odinger equation were found to be

r σ 2 σ σ Rk,l(r) → k (cos δl (k)jl(kr) − sin δl (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

r σ σ 2 sin(krhw + δl (k) − lπ/2) Rl (krhw) = = 0 (3.2.30) rhw rhw σ I.e, krhw = lπ/2 − δl (k) + nπ, where n is an integer. In the limit that rhw → ∞ the sum over k in Eq. (3.2.27) will become an integral

X Z kF dk → (3.2.31) ∆k k 0 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 σ σ R(k, r) as being that Rl (kr → ∞) → sin(k + δl (k) − lπ/2)/k) then we obtain the correct expression

Z kF 2 Z kF σ2 scatt X (2l + 1) dk 2k σ2 1 X 2 Ul (k, r) n (r) = Rl (k, r) = 2 (2l + 1)k 2 dk 4π π/rhw rhw 2π r σ,l 0 0 σ,l (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 2 1 a free electron gas, which is kF = (3π 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

1 Z kF X nscatt(r) = (2l + 1)k2Rσ2(k, r)dk (3.2.33) 2π2 l 0 σ,l σ where Rl are normalised according to

sin(kr + δσ(k) − lπ/2) Rσ(k, r → ∞) → l (3.2.34) l kr scatt scatt We want to evaluate ∆n (r) = n (r) − n0, where n0 can be written as

1 Z kF X n = (2l + 1)k2j 2(kr)dk (3.2.35) 0 2π2 l 0 σ,l Hence as r → ∞

1 Z kF X sin(kr + δσ(k) − lπ/2)2 sin(kr − lπ/2)2 ∆nscatt(r) → (2l + 1) l − dk 2π2 r2 r2 0 σ,l (3.2.36) Using cos(2x) = 1 − 2 sin2(x) gives

1 Z kF X 1 ∆nscatt(r) → (2l + 1) [cos(2kr − lπ) − cos(2kr + 2δσ(k) − lπ)] dk 2π2 2r2 l 0 σ,l (3.2.37) σ Performing the integration over k, and using kr δl (k) in order to ignore the k- σ dependence of δl (k) in this integration, one obtains

σ 1 X 1 sin(2kF r − lπ) sin(2kF r + 2δ (k) − lπ) ∆nscatt(r) → (2l + 1) − l (3.2.38) 2π2 2r2 2r 2r σ,l

σ σ Re-writing the sin terms respectively as sin(2kF r−lπ+δl (k)−δl (k)) and sin(2kF r−lπ+ σ σ σ δl (k)+δl (k)) and using sin(x+y) = sin(x) cos(y)+cos(x) sin(y) with x as 2kF r−lπ+δl (k) σ σ and y as −δl (k) or δl (k) gives

1 X 1 ∆nscatt(r) → (2l + 1) [sin(2k r − lπ + δσ(k )) cos(−δσ(k ))+ 2π2 4r3 F l F l F σ,l

σ σ σ σ cos(2kF r − lπ + δl (kF )) sin(−δl (kF )) − sin(2kF r − lπ + δl (kF )) cos(δl (kF )) 68 An Atom in Inﬁnite Jellium Solved using DFT

σ σ − cos(2kF r − lπ + δl (kF )) sin(δl (kF ))] (3.2.39) giving

1 1 X ∆nscatt(r) → − (2l + 1)(−1)l cos(2k r + δσ(k ) sin(δσ(k )) (3.2.40) 4π2 r3 F l F l F 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 1 P σ states. We will prove that ∆Nscatt = π σ,l(2l + 1)δl (kF ), and so:

1 X (2l + 1)δσ(k ) = Z − Z (3.2.42) π l F b 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. 1 P σ To prove the relation: ∆Nscatt = π σ,l(2l + 1)δl (kF ), consider the expression for a σ scattering state Rk,l(r) for large r

1 lπ Rσ (r → ∞) → sin(kr + δσ(k) − ) (3.2.43) k,l kr l 2

Applying hard-wall boundary conditions at some large radius, rhw, gives

lπ kr + δσ(k) − = nπ (3.2.44) hw l 2 In the case of a system with zero potential everywhere (pure jellium) we have 3.2 Scattering States 69

σ σ σ Rk,l(r) = cos δl (k)jl(kr) + sin δl (k)nl(kr) (3.2.45)

σ for all r. But nl(kr)|r→0 → ∞ and therefore we must have δl (k) = 0, in which case we have

lπ kr − = nπ (3.2.46) hw 2

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

1 lπ N σ ≡ n = (kr + δσ(k) − ) for V (r) 6= 0 (3.2.47) l π hw l 2

1 lπ N σ ≡ n = (kr − ) for V (r) = 0 (3.2.48) l π hw 2

The number of states in the range [k, k + dk] is therefore

dN σ dn r 1 dδσ l dk ≡ dk = ( hw + l )dk for V (r) 6= 0 (3.2.49) dk dk π π dk

dN σ dn r l dk ≡ dk = hw dk for V (r) = 0 (3.2.50) dk dk π

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∆N σ 1 dδσ l dk = l dk (3.2.51) dk π dk

Therefore the total change in the number of scattering states is

X X Z d∆N σ 1 X Z kF dδσ(k) dk l dk = dk(2l + 1) l = dk π dk σ l,m σ,l 0

1 X (2l + 1)(δσ(k ) − δσ(0)) (3.2.52) π l F l σ,l

σ which, if we take δl (0) = 0 gives the correct result. 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∆N σ (2l + 1) dδσ l = l (3.2.53) dk π dk

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

σ nb = (2l + 1)δl (0) (3.2.54)

Here, nb is the number of bound states of angular-momentum l and spin σ, and the σ phase-shift is ﬁxed by δl (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 −3 densities 0.01aB and 0.05aB . The system is non-magnetic and so the phase-shift is the same for both spins.

−3 For the 0.01aB 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 −3 phase-shift to equal zero at infinite energy). For the background density of 0.05aB , 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

40 Fermi level ) 1

− 30 eV ( l=3 )

20 ( l d dNscatt/dE/ev-1 l=2 dN 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 3

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Figure 3.2: The l = 0 phase-shifts for a hydrogen atom immersed in inﬁnite jellium of −3 −3 background densities 0.01aB (top panel) and 0.05aB (bottom panel). 3.3 Numerical Algorithm for Solving the Radial Schr¨odingerEquation 73

3.3 Numerical Algorithm for Solving the Radial Schr¨odinger Equation

3.3.1 Radial Schr¨odinger Equation Solutions in the Limits r → 0 and r → ∞

Ignoring for a moment the self-consistency problem posed by the radial Schr¨odinger equation, let us ask how we can, for a given V σ(r), solve the radial Schr¨odinger equation to σ obtain the radial solutions, UEl(r). As we shall see, this requires us to know the analytic σ σ values of UEl(r) and their derivatives dUEl(r)/dr in the limits r → 0 and r → ∞. For the case of an atom in jellium we have

2 σ Z r 1 d UEl(r) l(l + 1) σ 1 0 + 0 02 0 − 2 + 2 UEl(r) + (n(r ) − n (r ))4πr dr 2 dr 2r r 0

Z rmax 0 + 0 0 0 Z σ σ σ + (n(r ) − n (r ))4πr dr − + Vxc(r) UEl(r) = EUEl(r) (3.3.1) r r

σ We look ﬁrst at the limit r → 0. The two divergent terms in this case are −ZUEl(r)/r σ 2 and l(l + 1)UEl(r)/2r . This means that in the limit r → 0 we have

 − Z U σ (r) if l = 0 2 σ  r El 1 d UEl(r)  2 = (3.3.2) 2 dr   l(l+1) σ 2r2 UEl(r) if l > 0

σ −Zr In the ﬁrst case the solution is UEl(r) = re as is shown below

1 d2(re−Zr) 1 d 1 1 1 = (e−Zr − Zre−Zr) = − Ze−Zr − Ze−Zr + Z2re−Zr (3.3.3) 2 dr2 2 dr 2 2 2

−Zr σ As r → 0 this result tends to −Ze = −ZUEl(r)/r as required. The solution for σ l+1 l > 0 is UEl(r) = r which is proved below

1 d2(rl+1) 1 l(l + 1) = l(l + 1)rl−1 = U σ (r) (3.3.4) 2 dr2 2 2r2 El as required. We will worry about the normalisation of these solutions later. In summary 74 An Atom in Inﬁnite Jellium Solved using DFT

 re−Zr if l = 0  σ  UEl(r → 0) → (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

1 d2U σ − El = EU σ (r) (3.3.6) 2 dr2 El √ √ σ − 2 −Er The solution is UEl(r) = e /r as is shown below

√ √ 2 e− 2 −Er √ √ √ √ ! 1 d r 1 d e− 2 −Er 2 −E √ √ − = − − − e− 2 −Er (3.3.7) 2 dr2 2 dr r2 r

Evaluating this expression and ignoring terms except for the 1/r term gives

√ √ 1 2 −E √ √ √ √ E √ √ = (− 2)( −E)e− 2 −Er = e− 2 −Er = EU σ (r) (3.3.8) 2 r r El 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 Schr¨odinger equation. In order to apply this algorithm, the radial Schr¨odinger equation is split into two equations

1 dAσ (r) l(l + 1) − El + U σ (r) + V σ(r)U σ (r) = EU σ (r) (3.3.9) 2 dr 2r2 El El El

dU σ (r) El = Aσ (r) (3.3.10) dr El

σ σ Given starting values of UEl(r) and AEl(r) at some r, the Runge-Kutta algorithm σ σ propagates UEl(r) and AEl(r) outwards or inwards to some other value of r. σ σ Consider the case where we start with UEl(r) and AEl(r) at some small value of r, rmin (typically of the order 10−4). We would then like to propagate this solution outwards, i.e., σ σ determine UEl(r) and AEl(r) for larger values of r. In the previous section we found 3.3 Numerical Algorithm for Solving the Radial Schr¨odingerEquation 75

 re−Zr if l = 0  σ  UEl(r → 0) → (3.3.11)   rl+1 if l 6= 0 Using these limits we can write

 r e−Zrmin if l = 0  min σ  UEl(rmin) = (3.3.12)   l+1  rmin if l 6= 0  e−Zrmin − Zr e−Zrmin if l = 0  min σ  AEl(rmin) = (3.3.13)   l  (l + 1)rmin if l 6= 0 σ The Runge-Kutta algorithm uses Eq. (3.3.9) and Eq. (3.3.10) to calculate UEl(r2) and σ AEl(r2), where r2 = rmin + δr and δr is some small increment. This procedure is repeated σ σ until we have propagated UEl(r) and AEl(r) outwards to some speciﬁed value of r. σ σ We can also propagate UEl(r) and AEl(r) inwards, and will do so from r = rmax. In σ σ this case initial starting values UEl(rmax) and AEl(rmax) are required. Using outwards or inwards propagation, or a combination of the two, the quantities σ σ UEl(r) and AEl(r) can be calculated over a set of r values. We choose these r points 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

1 2 n0−2 rint n0−1 rint n0−1 rint n0−1 rmin, rmin , rmin , ··· , rmin , rint, rmin rmin rmin

1 2 n1 − 1 rint + (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 Since we are dealing with bound states the E are negative. First we propagate UEl(r) 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 above. Then we propagate UEl(r) inwards from rmax 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 resulting E is then a bound state energy, and the outwards and inwards UEl(r) combine to give the full bound state solution. σ σ The outward integration uses the starting values UEl(rmin) and AEl(rmin) given in the previous section. For the inward integration, we have already found that

√ √ e− 2 −Er U σ (r → ∞) = (3.3.14) El r σ σ The starting values of UEl(rmax) and AEl(rmax) are therefore

√ √ − 2 −Ermax σ e UEl(rmax) = (3.3.15) rmax √ √ √ √ √ √ − 2 −Ermax 2 −Ermax σ 2 −Ee e AEl(rmax) = − − 2 (3.3.16) rmax rmax 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 energies, and the corresponding UEl(r)/r are the bound state solutions. When E crosses 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 Schr¨odingerEquation 77

§ 2

¦ -1 ¤ ¥ ¡ 1 ¤ £ ¢ 0 ¡ -2 -1 -2 -3 -3 -4 -4 -5 -20 -15 -10 -5 0

-5 -1000 -900 -800 -700 -600 -500 -400 -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 Tech- −3 netium atom immersed in jellium of background density 0.03aB , 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 Schr¨odinger 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 The Immersion Energy

3.4.1 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

X Z 1 E[n↑, n↓] = φσ(r)∗(− ∇2)φσ(r) dr+ i 2 i i,σ | {z } =Ekinetic 1 Z n(r)n(r0) Z drdr0 + drv (r)n(r)dr +E [n↑, n↓] (3.4.1) 2 |r − r0| ext xc | {z } | {z } =Ecoulomb =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 (3.4.2) | {z } =∆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

Z b.s X σ X σ σ Tcombined = Ei − nb.s(r)V (r)dr (3.4.5) i,σ σ σ σ σ where Ei are the bound state energies of the combined system and nb.s(r) and V (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 The Kohn-Sham equations can also be used to ﬁnd expressions for Tcombined and Tjellium

X Z k=kF Z 1 dN σ T s.s = φσ(k, r)∗ − ∇2 φσ(k, r)dr dk (3.4.6) 2 dk σ k=0 80 An Atom in Inﬁnite Jellium Solved using DFT

dN σ where dk dk is the number of spin σ scattering states in the range k, k + dk. The Kohn-Sham equations for scattering states

1 k2 − ∇2 + V σ(r) φσ(k, r) = φσ(k, r) (3.4.7) 2 2 are rearranged in terms of ∇2 and inserted into the equation for T s.s to give

X Z k=kF k2 dN σ(k) X Z Z k=kF dN σ(k) T s.s = dk − V σ(r) |φσ(k, r)|2 dk dr (3.4.8) k=0 2 dk k=0 dk σ σ | {z } σ =ns.s(r)

Therefore

X Z k=kF k2 d∆N σ(k) X Z T s.s − T s.s = dk − nσ (r)V σ(r)dr (3.4.9) combined jellium 2 dk s.s σ k=0 σ

σ σ σ where ∆N (k) = N (k)combined − N (k)jellium. Using the equation

σ σ d∆N (k) 1 X dδl,m(k) dk = dk (3.4.10) dk π dk l,m allows us to write the ﬁrst term as

Z k=kF 2 σ Z k=kF 2 σ X k d∆N (k) 1 X k dδl,m(k) dk = dk (3.4.11) 2 dk π 2 dk σ k=0 σ,l,m k=0 √ Writing k = 2E and proceeding using integration by parts

Z E=EF σ Z E=EF 1 X dδl,m(E) 1 X 1 X = E dE = E δσ (E ) − δσ (E)dE π dE π F l,m F π l,m σ,l,m E=0 σ,l,m σ,l,m E=0 (3.4.12) Putting all the contributions to ∆T together gives

Z X σ X σ σ ∆T = Ei − n (r)V (r)dr+ i,σ σ

1 X 1 X Z E=EF E δσ (E ) − δσ (E)dE (3.4.13) π F l,m F π l,m σ,l,m σ,l,m E=0 3.4 The Immersion Energy 81

σ σ σ where we have used n (r) = nb.s(r) + ns.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

1 Z n(r)n(r0) Z n(r)n 1 Z n2 C = drdr0 − 0 drdr0 + 0 drdr0 = 0 (3.4.15) jellium 2 |r − r0| |r − r0| 2 |r − r0| 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

1 Z (n(r) − n )(n(r0) − n ) C = 0 0 drdr0 combined 2 |r − r0| Z Z − dr (n(r) − n ) (3.4.16) r 0 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 1 Z (n(r0) − n ) Z ∆C = 0 dr0 − (n(r) − n ) dr (3.4.17) 2 |r − r0| r 0 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

n n ∆E = E n↑(r), n↓(r) − E ( 0 , 0 ) (3.4.19) xc xc xc 2 2 Putting all these results together

Z X σ X σ σ Eimm = Ei − n (r)V (r)dr+ i,σ σ 82 An Atom in Inﬁnite Jellium Solved using DFT

E=Eσ 1 X 1 X Z F Eσ δσ (E ) − δσ (E)dE+ π F l,m F π l,m σ,l,m σ,l,m E=0 Z 1 Z (n(r) − n ) Z 0 dr0 − (n(r) − n ) dr+ 2 |r − r0| r 0 n n E n↑(r), n↓(r) − E ( 0 , 0 ) − E (3.4.20) xc xc 2 2 atom

3.4.2 Finite Radius Corrections

The integrals over r in the above expression for the immersion energy read

X Z r=∞ − nσ(r)V σ(r)4πr2dr+ σ r=0

Z r=∞ Z r0=∞ ! 1 (n(r) − n0) 02 0 Z 2 0 4πr dr − (n(r) − n0) 4πr dr+ r=0 2 r0=0 |r − r | r Z r=∞ ↑ ↓ n0 n0 2 n(r)εxc(n (r), n (r)) − n0εxc( , ) 4πr dr (3.4.21) r=0 2 2

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 σ period π/kF . The potential V (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 σ approximations of n(r) = n0 and V (r) = 0 beyond rmax is therefore

Z r=∞ (correction) ↑ ↓ n0 n0 2 ∆Exc = n(r)εxc(n (r), n (r)) − n0εxc( , ) 4πr dr (3.4.22) r=rmax 2 2 The exchange-correlation energy density is ﬁrst expanded about the jellium density

↑ n0 ↑ ↓ n0 n0 ↑ n0 dεxc(n (r), 2 ) εxc(n (r), n (r)) = εxc( , ) + (n (r) − ) + ↑ ↑ n0 2 2 2 dn n (r) = 2

n0 ↓ n dεxc( , n (r)) (n↓(r) − 0 ) 2 (3.4.23) ↓ ↓ n0 2 dn n (r) = 2 which using 3.4 The Immersion Energy 83

↑ n0 n0 ↓ dεxc(n (r), ) dεxc( , n (r)) 2 = 2 (3.4.24) ↑ ↑ n0 ↓ ↓ n0 dn n (r) = 2 dn n (r) = 2 gives us

↑ n0 ↑ ↓ n0 n0 dεxc(n (r), 2 ) εxc(n (r), n (r)) = εxc( , ) + (n(r) − n0) (3.4.25) ↑ ↑ n0 2 2 dn n (r) = 2 Inserting Eq. (3.4.25) into Eq. (3.4.22) and rearranging gives

Z r=∞ (correction) n0 n0 2 ∆Exc = (n(r) − n0)εxc( , )4πr dr+ r=rmax 2 2 Z r=∞ ↑ n0 dεxc(n (r), 2 ) 2 n(r)(n(r) − n0) 4πr dr (3.4.26) ↑ ↑ n0 r=rmax dn n (r) = 2

The second term is re-written using n(r) = n0 + δn(r)

Z r=∞ Z r=∞ 2 2 n(r)(n(r) − n0)4πr dr = (n0 + δn(r))δn(r)4πr dr = r=rmax r=rmax Z r=∞ 2 2 n0 (n(r) − n0)4πr dr + O(δn(r) ) (3.4.27) r=rmax Ignoring second order terms in δn(r) and combining the two terms

↑ n0 ! Z r=∞ (correction) n0 n0 dεxc(n (r), 2 ) 2 ∆E = εxc( , ) + n0 (n(r) − n0)4πr dr xc ↑ ↑ n0 2 2 dn n (r) = 2 r=rmax (3.4.28) R r=∞ 2 R r=∞ 2 Finally, using r=0 n(r)4πr dr = Z + r=0 n04πr dr, we obtain

↑ n0 ! (correction) n0 n0 dεxc(n (r), 2 ) ∆E = εxc( , ) + n0 × xc ↑ ↑ n0 2 2 dn n (r) = 2

Z r=rmax 2 Z − (n(r) − n0)4πr dr (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 atom- in-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=lnum) req value, knum , knum , ..., knum . Another numerical parameter is ∆V , 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 −3 SI-corrected cerium atom in infinite jellium of density n0 = 0.01aB . 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) 3 are present in a plot of (n(r) − n0)r 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 3 is equivalent to choosing an rmax so that (n(r) − n0)r 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

32.6114

32.6112

32.6110

32.6108

32.6106 Immersion Energy/eV

32.6104

32.6102

32.6100 -11.5 -11 -10.5 -10 -9.5 -9 -8.5 -8 -7.5 -7

log(convsu)

Figure 3.4: Determining the parameter ∆V req, for a SI-corrected cerium atom in infinite −3 req jellium of density n0 = 0.01aB . Immersion energy is plotted against log(∆V ), and error bars (in green) are placed at different values of the convergence. 86 An Atom in Infinite Jellium Solved using DFT

PSfrag replacements 0.1 rmax=24.676aB 0.08 rmax=25.326aB rmax=25.963aB 0.06

0.04 3 r ) 0

n 0.02 − )

r 0 ( n ( -0.02

-0.04

-0.06

-0.08 0 5 10 15 20 25

r / aB

−3 Figure 3.5: Density plots for hydrogen in inﬁnite jellium of density 0.005aB . 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 Kohn- Sham 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 σ σ d∆Nl 1 dδl density and therefore, the relation dk dk = 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 3 choose rmax instead so that (n(r) − n0)r 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 3 (n(r) − n0)r 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

1.5

3 1 )r 0

(n(r)-n 0.5

-0.5 0 5 10 15 20

rmax/aB

lnum=13 2

1.5

3 1 )r 0

(n(r)-n 0.5

-0.5 0 5 10 15 20

rmax/aB 3.4 The Immersion Energy 89

lnum=15 2

1.5

3 1 )r 0

(n(r)-n 0.5

-0.5 0 5 10 15 20

rmax/aB

lnum=20 2

1.5

3 1 )r 0

(n(r)-n 0.5

-0.5 0 5 10 15 20

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 The Eﬀective Medium Theory

3.5.1 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 91

14 12 H B 12 He C Li 10 N 10 Be 8 8 6 6 4 4 2 2 Immersion Energy / eV Immersion Energy / eV 0 0

-2 -2 0 0.01 0.02 0.03 0 0.01 0.02 0.03 -3 -3 n0 / aB n0 / aB

35 60 F Al 30 Ne Si Na 50 Cl 25 Mg Ar 40 20 15 30

10 20 5 10 0 Immersion Energy / eV Immersion Energy / eV 0 -5 -10 -10 0 0.01 0.02 0.03 0 0.01 0.02 0.03 -3 -3 n0 / aB n0 / aB

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 n(r0) − P Zδ(r0 − R ) V (r) = i i dr0 + V (n(r)) (3.5.1) |r − r0| xc 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

X n(r) = ∆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

Z 0 0 X ∆ni(r ) − Zδ(r − Ri) V (r) = dr0 + V (n(r)) |r − r0| xc i Z 0 0 ∆ni(r ) − Zδ(r − Ri) X = dr0 + ∆Φ (r) + V (n(r)) (3.5.3) |r − r0| j xc 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 ∆n (r0) − Zδ(r0 − R ) ∆Φ (r) = i i dr0 (3.5.4) i |r − r0|

Next, we introduce the quantity, ∆Vi(r)

Z ∆n (r) − Zδ(r0 − R ) ∆V (r) = i i dr0 + V (n ¯ + ∆n ) − V (n ¯ ) (3.5.5) i |r − r0| xc i i xc i

Let us choose ∆ni to be the atom-induced density of an atom immersed in a homogeneous electron gas of background densityn ¯i, with the atom centred on Ri. With this choice, the above quantity is just the Kohn-Sham potential for the same system. Further- more, ∆Φi(r), is the Hartree potential for this system.

Re-writing Eq. (3.5.3) in terms of ∆Vi(r) gives

X V (r) = ∆Vi(r) − Vxc(n ¯i + ∆ni) + Vxc(n ¯i) + ∆Φj(r) + Vxc(n(r)) (3.5.6) j6=i

Now, let us setn ¯i 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 n¯i = 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.

X n¯i ≈ ∆nj(r) (3.5.8) j6=i in this case terms 2 and 5 of Eq. (3.5.6) cancel. Furthermore, if we assume thatn ¯i 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

X Φ¯ i ≈ ∆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 densityn ¯i. 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 Z n(r)n(r0) X Z Zn(r) E[n, v] = T [n, v] + drdr0 − dr + E [n] (3.5.11) 2 |r − r0| |r − R | xc i i inserting Eq. (3.5.2) gives

Z 0 Z 1 X ∆ni(r)∆nj(r ) X Z∆nj(r) E[n, v] = T [n, v] + drdr0 − dr + E [n] (3.5.12) 2 |r − r0| |r − R | xc ij ij i

The quantity we are interested in is the cohesive energy of the solid 3.5 The Eﬀective Medium Theory 95

X ∆E[n, v] = E[n, v] − Eatom (3.5.13) i where Eatom 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(¯ni), 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

X ∆E[n, v] = Ec(¯ni) + ∆E1−el (3.5.15) i where

Z r=s Z 0 ∆n(r ) 0 Z Ec(¯n) = Eimm(¯n) +n ¯ 0 dr − dr (3.5.16) r=0 |r − r | r 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(¯n). It can be viewed as the attraction of the sum of the density tails from all other cells (¯n) 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

4.5

4 B

3.5

2.5

Wigner Seitz Radius / a 2

1.5

1 0 5 10 15 20 25 30 35 40 45 50 Atomic Number 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 chosenn ¯, solve the self-consistent problem of an atom in a homogeneous gas in order to obtain ∆n(r) and then evaluate

Ec(¯n). One then variesn ¯ in order to minimise Ec(¯n). This minimum value of Ec(¯n) 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 Cerium Solved using the LDA and SIC

3.6.1 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 4 Niobium Molybdenum 2 Technetium Ruthenium Rhodium 0

-2

-4

-6 Cohesive Energy / eV

-8

-10

-12 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2

Neutral Sphere Radius / aB Figure 3.10: Cohesive energy versus neutral sphere radii for 4d transition metals. 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 L¨uders 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 L¨uders 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 Schr¨odinger 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 Schr¨odinger equation are no longer automatically orthogonal to one another. Eq. (2.3.61)

Z σ∗ σ φα (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 σ∗ σ0 φl,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

1 )

( 0 l δ

deltal(epsilon)/a.u. -1

-2

-3 0 0.5 1 1.5 2 2.5 3 √sqrt(epsilon)/a.u.2 (a.u)

) 1 ² ( l

δ 0

-1

PSfrag replacements -2

-3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 √2² (a.u)

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

40 Fermi level ) 1

− 30 eV ( l=3 )

20 ( l d dNscatt/dE/ev-1 l=2 dN 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 105

Z σ∗ σ0 2 = Rl,m(r)Rl0,m0 (r)r dr × δl,l0 δm,m0 δσ,σ0 (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

Z orth 0 0 0 UE3 (r) = UE3 (r) − UE1 (r )UE3 (r )dr UE1 (r)− Z 0 0 0 UE2 (r )UE3 (r )dr 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 σ0). Therefore this bound state will no longer be orthogonal to the f-scattering states with m = m0 and spin σ0. 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

Z orth 0 0 0 Uscatt(k, r) = Uscatt(k, r) − Uscatt(k, r )Ubound(r )dr 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 σ solutions normalised according to Uscatt(k, r → ∞) → sin(kr + δl (k) − lπ/2)/k therefore orth generates Uscatt(k, r) with the same normalisation. These new scattering states can then 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 σ0 scattering states against the SI-corrected bound state, we have to convince ourselves that the l = 3, m = m0, spin σ0 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 putting the factor 2/rhw in front of the Uscatt(k, r). We have

Z r r 2 orth 2 0 orth 0 kUscatt(k, r) k Uscatt(k , r)dr = rhw rhw Z 0 2 0 kk Uscatt(k, r)Uscatt(k , r)dr+ rhw Z Z Z 0 0 0 0 Ubound(r)Ubound(r)dr − 2 Uscatt(k, r)Ubound(r)dr Uscatt(k , r )Ubound(r )dr (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 r Z 2 orth 2 0 orth 0 0 2 0 kUscatt(k, r) k Uscatt(k , r)dr = kk Uscatt(k, r)Uscatt(k , r)dr (3.6.6) rhw rhw rhw 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 = σ0 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 self- consistency 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

U 2 (r) nSIC (r) = SIC (3.6.7) 4πr2 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 SI- corrected. 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 self- interaction. 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 L¨uders 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 −3 for which we have placed a calculated bound state energy is 0.014aB . This is because we encountered convergence problems above this density. However, the extrapolated line −3 clearly shows a cross-over at approximately 0.0155aB . 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

4 3 ∆Ω = − 0.0155 0.0155 −3 in units of aB . 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

-0.02

-0.04

-0.06

-0.08

-0.1 Energy of 4f bound-state / eV

-0.12 PSfrag replacements 0.01 0.011 0.012 0.013 0.014 0.015 0.016 a 3 Density / 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 109 electrons being brought below the Fermi-level and into the calculation. 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 −3 jellium for a variety of background densities. From top to bottom, n0 = 0.04aB , n0 = −3 −3 −3 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. 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) ).

X Z 1 E[n↑, n↓] = φσ∗(r)(− ∇2)φσ(r)dr + i 2 i i,σ | {z } =Ekinetic 1 ZZ n(r)n(r0) Z drdr0 + v (r)n(r)dr +E [n↑, n↓] (4.1.1) 2 |r − r0| ext xc | {z } | {z } =ECoulomb =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 + V σ(r) φσ(r) = Eσφσ(r) (4.1.2) 2 i i i to re-write the kinetic energy term (just as in Section 3.4.1)

X Z 1 X Z E = φσ∗(r) − ∇2 φσ(r)dr = Eσ − |φσ(r)|2V σ(r)dr (4.1.3) kinetic i 2 i i i i,σ i,σ 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 σ P σ 2 particle orbitals. Using n (r) = i |φi (r)| we have

Z X σ X σ σ Ekinetic = Ei − n (r)V (r)dr (4.1.4) i,σ σ The external potential for the system is

0 Z r =Rjell Z n0 0 vext(r) = − − 0 dr (4.1.5) r r0=0 |r − r | therefore

0 Z r=∞ Z r =Rjell Z r=∞ n(r)n0 0 n(r) Eexternal = − 0 drdr − Z dr (4.1.6) r=0 r0=0 |r − r | r=0 r We also add on the self-repulsion of the positive background charge and also the repulsion between the ion and the positive background:

0 Z r=Rjell Z r =Rjell 2 Z r=Rjell 1 n0 0 n0 0 drdr + Z dr (4.1.7) 2 r=0 r0=0 |r − r | r=0 r We will calculate the ﬁrst of these terms explicitly. In doing so we will use the result derived in Section 3.1.5:

0 Z r =Rjell n0 0 4 2 2 2 0 dr = πr n0 + 2π(Rjell − r )n0 for r ≤ Rjell (4.1.8) r0=0 |r − r| 3 Using this result

0 Z r=Rjell Z r =Rjell 2 1 n0 0 0 drdr = 2 r=0 r0=0 |r − r |

Z r=Rjell 1 2 4 2 2 2 4πr n0 πr n0 + 2π(Rjell − r )n0 dr = 2 r=0 3

Z r=Rjell 2 2 4 2 2 16 2 2 5 2πn0 (− πr + 2πRjellr )dr = π n0Rjell (4.1.9) r=0 3 15 3 Using N = 4πRjelln0/3 and substituting in for Rjell in the above expression

16 2 1 1 5 = π n0 3 N 3 (4.1.10) 15 4π 5/3 3 4 3 Then using 3 πrs n0 = 1 to eliminate n0 gives the ﬁnal result 114 Hydrogen Immersed in a Finite Jellium Sphere

0 Z r=Rjell Z r =Rjell 2 5/3 1 n0 0 3 N 0 drdr = (4.1.11) 2 r=0 r0=0 |r − r | 5 rs Putting all these terms together, the DFT energy functional we use for an atom in ﬁnite jellium is

X X Z 1 ZZ n(r)n(r0) E[n↑, n↓] = Eσ − drnσ(r)V σ(r) + drdr0− i 2 |r − r0| i,σ σ

0 Z r=∞ Z r =Rjell Z r=∞ Z r=Rjell 5/3 n(r)n0 0 n(r) n0 3 N 0 drdr − Z dr + Z dr + + r=0 r0=0 |r − r | r=0 r r=0 r 5 rs ↑ ↓ Exc[n , n ] (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, 5g18, 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 : 1s22s22p63s23p64s23d6 4.1 Hydrogen in Finite Jellium Spheres using the LDA 115

Jellium : 1s22p63d2

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

1 φσ,orth(r) = (φσ (r) − Iσφσ (r)) (4.1.13) 2s N σ 2s 1s σ R σ σ σ R σ,orth 2 where I = φ2s(r)φ1s(r)dr and N is a normalisation factor to ensure |φ2s (r)| dr = 1. 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 φσ,orth(r)(− ∇2)φσ,orth(r)dr 2s 2 2s

If we insert Eq. (4.1.13) and use the Kohn-Sham equations

1 (− ∇2 + V σ (r))φσ (r) = Eσ φσ (r) 2 SIC 1s 1s 1s 1 (− ∇2 + V σ(r))φσ (r) = Eσ φσ (r) 2 2s 2s 2s then we obtain

Z 1 φσ,orth(r)(− ∇2)φσ,orth(r)dr = 2s 2 2s

1 Z Z (1 − Iσ2)Eσ − |φσ (r)|2V σ(r)dr + Iσ φσ (r)φσ (r)(V σ (r) + V σ(r))dr− N σ2 2s 2s 2s 1s SIC

Z σ2 σ 2 σ I |φ1s(r)| VSIC (r)dr which can be calculated straightforwardly in the code. Alternatively one may explicitly σ,orth diﬀerentiate φ2s (r) by using

1 1 U(r)00 l(l + 1) − ∇2R(r)Y (θ, φ) = − Y (θ, φ) + R(r)Y (θ, φ) 2 lm 2 r lm 2r2 lm

(where U(r) = rR(r)) which gives

Z 1 1 Z φσ,orth(r)(− ∇2)φσ,orth(r)dr = − U σ,orth(r)U σ,orth(r)00dr 2s 2 2s 2 2s 2s

1 Z = U σ,orth(r)0U σ,orth(r)0dr 2 2s 2s 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 117

4.1.4 Results

Energy Levels

−3 For a hydrogen atom in a 338-electron jellium sphere of background density 0.008aB , 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 (non- magnetic) 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. −3 For the background density 0.008aB , 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 −3 in the plot for the background density 0.03aB , which shows the s levels increasing in −3 energy relative to the jellium sphere s levels as compared to the 0.008aB 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 −3 sphere for three background densities. The densities considered are 0.001aB , 0.007aB −3 and 0.03aB . The most obvious feature of these plots is the small scale bunching of immersion 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

1 ∆R 1 π 1 π π 4π2 3 = = = = = 1.637 (4.1.14) 1 3π2 1 rs rs kF rs 2 3 3 9 (3π n0) ( 4/3π ) 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 replacemen

4.1 Hydrogen in Finite Jellium Spheres using the LDA 119 ts

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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 −3 these systems. The background densities of the jellium are 0.03 aB (upper panel) and −3 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). 120 Hydrogen Immersed in a Finite Jellium Sphere

3 n0 = 0.001aB− 0

-0.5 eV /

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-1.5 0 1 2 3 4 5 6 Rjell/rs Figure 4.2: Plots of immersion energy versus number of electrons for jellium spheres. −3 −3 −3 Densities of 0.001aB , 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 −3 case, and we have verified this for background densities in the range 0.001aB to 0.03aB . Fig. 4.5 shows our calculations of the atom-induced densities for a hydrogen atom in −3 jellium spheres with a background density 0.01aB . 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 PSfrag replacemen ts 0

2 10-Electron jellium sphere 50-Electron jellium sphere 1.5 Inﬁnite jellium

1 eV / 0.5

Energy 0

-0.5

Immersion -1

-1.5

-2 0 0.005 0.01 0.015 0.02 0.025 0.03 3 n0/aB−

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. PSfrag replacemen

4.1 Hydrogen in Finite Jellium Spheres using the LDA 123 1.5 ts 2 0

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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 −3 background density ≈ 0.004aB , 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 −3 the plateau of the potential. This occurs at n0 = 0.015aB . 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 −3 the density from 0.03aB 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

0.07 Inﬁnite jellium 0.06 Finite jellium 1

− B Jellium sphere radius a

/ 0.05 2 r

× 0.04 y 0.03

Densit 0.02

0.01 Induced 0 tom A -0.01

-0.02 0 5 10 15 20 25 30

Radius / aB PSfrag replacements

0.07 Inﬁnite jellium 0.06 Finite jellium 1

− B Jellium sphere radius a

/ 0.05 2 r

× 0.04 y 0.03

Densit 0.02

0.01 Induced 0 tom A -0.01

-0.02 0 5 10 15 20 25 30

Radius / aB 126 Hydrogen Immersed in a Finite Jellium Sphere PSfrag replacements

0.07 Inﬁnite jellium 0.06 Finite jellium 1

− B Jellium sphere radius a

/ 0.05 2 r

× 0.04 y 0.03

Densit 0.02

0.01 Induced 0 tom A -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 back- −3 ground density 0.01aB , 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.

−3 increase in volume. However, when we get to n0 = 0.015aB , 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 Hydrogen in Finite Jellium Spheres using VQMC

4.2.1 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 replacemen

4.2 Hydrogen in Finite Jellium Spheres using VQMC 127 ts

0.006 Finite jellium Inﬁnite jellium 0.005

3 0.004 − B a /

y 0.003

Densit 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 −3 a hydrogen atom in inﬁnite jellium for a background density ≈ 0.004aB . contribution (the ﬁrst and the second exponential respectively):

" N 6 !# X X nπri Ψ (r , ··· , r ) = exp α(i)j T 1 N n 0 R i=1 n=1 jell   2 2 X 1 aijrij + bijrij aijrij + bijrij × exp  2 + 2  2 1 + cij(ri)rij + dijrij 1 + cij(rj)rij + dijrij 1≤i

0 1 2 2 cij(ri) = cij + cij arctan[(ri − 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 replacemen

128 Hydrogen Immersed in a Finite Jellium Sphere ts

0 3 Background density / aB− 0.008 0.01 -0.2 0.015 0.02 0.025 0.03 -0.4 0.035 ) r ( V -0.6 PSfrag

-0.8 replacemen

-1 0 5 10 15 20 25 30 35 40 ts r/aB 4.4

4.35

4.3 B

/a 4.25 > s 1

r 4.2 < 4.15

4.1

4.05 0.005 0.01 0.015 0.02 0.025 0.03 3 Background density / aB−

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 1 (i) setting cij = αn = 0 and obtaining the following form for the Jastrow factor

  2 X aijrij + bijrij exp   (4.2.3) 1 + c0 (r )r + d r 2 1≤i

(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

ˆ 2 HΨT X 1 ∇ ΨT = − i + V (r , r , ··· , r ) (4.2.4) Ψ 2 Ψ 1 2 N T i T 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

X Z Z n+(r) X 1 V (r , r , ··· , r ) = − − dr + (4.2.5) 1 2 N r |r − r | |r − r | i i i i

 2 + 0 3 2 r Z  3 (R − ) r < Rjell n (r ) 0 2rs jell 3 dr = 3 (4.2.6) |r0 − r| Rjell 1 r ≥ R  rs r jell Local Kinetic Energy The kinetic energy part of the local energy is

2 X 1 ∇ ΨT X 1 X 1 − i = − ∇2 ln Ψ − (∇ ln Ψ )2 (4.2.7) 2 Ψ 2 i T 2 i T i T i i 2 In order to calculate ∇i ln ΨT and ∇i ln ΨT we need to substitute in our form for ΨT . We have

" N 6 !# X X nπri Ψ (r , ··· , r ) = exp α(i)j T 1 N n 0 R i=1 n=1 jell   2 2 X 1 aijrij + bijrij aijrij + bijrij exp − 2 + 2  2 1 + cij(ri)rij + dijrij 1 + cij(rj)rij + dijrij 1≤i

↑ ↓ 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

0 1 2 2 cij(ri) = cij + cij arctan[(ri − Rjell)/2 4 Rjell] (4.2.9)

Therefore

6 1 1 X nπri ∇ ln Ψ = ∇ D↑ + ∇ D↓ + ∇ α(i)j − i T D↑ i D↓ i i n 0 R n=1 jell 4.2 Hydrogen in Finite Jellium Spheres using VQMC 131

2 2 1 X aijrij + bijrij aijrij + bijrij ∇i 2 + 2 (4.2.10) 2 1 + cij(ri)rij + dijrij 1 + cij(rj)rij + dijrij j6=i 1 1 1 1 ∇2 ln Ψ = −( ∇ D↑)2 + ∇2D↑ − ( ∇ D↓)2 + ∇2D↓+ i T D↑ i D↑ i D↓ i D↓ i 6 X nπri ∇2 α(i)j − i n 0 R n=1 jell 2 2 1 X 2 aijrij + bijrij aijrij + bijrij ∇i 2 + 2 (4.2.11) 2 1 + cij(ri)rij + dijrij 1 + cij(rj)rij + dijrij j6=i The calculation of these derivatives are discussed in more detail in Appendix A.

4.2.3 Results

1 In our calculations, we found that the Jastrow parameters cij 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, 1 1 cij and 4 combine to a single effective Jastrow parameter, cij/4. 1 This is born out in the optimisation procedure, where we see the ratio cij/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 1 towards constant values (and stop cij and 4 shooting off to infinity) we simply set 4 1 equal to a very high value (say 100,000) and then let cij 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 −3 of background densities 0.001aB 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 −3 a hydrogen atom in jellium of density 0.03aB , 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 −3 solution of a hydrogen atom immersed in a 10-electron jellium sphere of density 0.03aB . 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 −3 Density/aB 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 −3 Density/aB Energy/eV σl.e. Energy/eV Energy/eV 0.002 -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

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 133

0 Hartree-Fock VQMC -5 LSDA SIC -10

-15

-20 Energy (eV) -25

-30

-35 0 0.005 0.01 0.015 0.02 0.025 0.03 n a 3 0 ( 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 5 LSDA

-5

Energy / eV -10

-15

-20 0 0.005 0.01 0.015 0.02 0.025 0.03 n a 3 0 ( B− )

Figure 4.9: Total energy of a 10-electron jellium sphere for diﬀerent background densities 134 Hydrogen Immersed in a Finite Jellium Sphere

1.5 Hartree-Fock VQMC 1 LSDA SIC 0.5

-0.5

-1

Immersion energy (eV) -1.5

-2 0 0.005 0.01 0.015 0.02 0.025 0.03 n a 3 0 ( 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 −3 Density/aB 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 −3 Density/aB 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 −3 than the LDA results for low densities. Then above background densities of about 0.01aB 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 −3 Density/aB 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 −3 within 1eV of one another and all curves feature a minimum at approximately 0.004aB . 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. −3 We see that the VQMC immersion energy curve below 0.005aB 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 replacemen

4.2 Hydrogen in Finite Jellium Spheres using VQMC 137 ts

10000

8000

6000

Conﬁgurations 4000 No.

2000

0 -30 -25 -20 -15 -10 -5 0 5 Local Energy / eV Figure 4.11: Local energy distribution for a VQMC calculation of a hydrogen atom in a −3 10-electron jellium sphere of density 0.03aB 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 −3 sphere for background densities of 0.002aB 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 replacemen ts

0.0032

0.00315 Energy 0.0031 cal Lo

the 0.00305 of

Mean 0.003 on

Error 0.00295

0.0029 0 2 4 6 8 10 12 Transformation Number Figure 4.12: Re-blocking analysis for hydrogen immersed in a 10-electron jellium sphere −3 of density 0.03aB . 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 replacemen

4.2 Hydrogen in Finite Jellium Spheres using VQMC 139 ts

0.6 LDA SIC Variational QMC 0.5 HF

0.4 3 − B a /

y 0.3 Densit 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 back- −3 ground density 0.03aB using HF, LDA, SIC and VQMC localised. The VQMC densities are signiﬁcantly more delocalised compared to the LDA −3 and SIC densities. For the background density 0.03aB , the VQMC density at the atom is approximately 10% lower than the LDA density, whereas for the background density −3 0.002aB 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 replacemen 140 Hydrogen Immersed in a Finite Jellium Sphere ts

0.005 LDA SIC Variational QMC HF 0.004 3

− B 0.003 a / y

0.002 Densit PSfrag 0.001 replacemen

0 5 6 7 8 9 10

Radius / aB ts 0.8 LDA SIC Variational QMC 0.7 HF

0.6 3 − B a /

y 0.5 Densit 0.4

0.3

0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Radius / aB Figure 4.14: Electron density of a hydrogen atom in a 10-electron jellium sphere of back- −3 ground density 0.03aB using HF, LDA, SIC and VQMC. Note in the top graph curves for HF, LDA and SIC coincide. PSfrag replacemen

4.2 Hydrogen in Finite Jellium Spheres using VQMC 141 ts

0.4 LDA SIC 0.35 Variational QMC HF 0.3

3 0.25 − B a /

y 0.2

Densit 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 back- −3 ground density 0.002aB 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 replacemen 142 Hydrogen Immersed in a Finite Jellium Sphere ts

0.005 LDA SIC Variational QMC HF 0.004 3

− B 0.003 a / y

0.002 Densit PSfrag 0.001 replacemen

0 0 5 10 15 20

Radius ts / aB 0.4 LDA SIC Variational QMC 0.35 HF

0.3 3 − B

a 0.25 / y 0.2 Densit

0.15

0.1

0.05 0 0.2 0.4 0.6 0.8 1

Radius / aB Figure 4.16: Electron density of a hydrogen atom in a 10-Electron jellium sphere of back- −3 ground density 0.002aB 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 −3 sphere of background density 0.03aB . 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

6 6 X nπri X nπri ∇ Dσ, ∇2Dσ, ∇ α(i)j , ∇2 α(i)j , i i i n 0 R i n 0 R n=1 jell n=1 jell 2 2 1 X aijrij + bijrij aijrij + bijrij ∇i 2 + 2 , 2 1 + cij(ri)rij + dijrij 1 + cij(rj)rij + dijrij j6=i 2 2 1 X 2 aijrij + bijrij aijrij + bijrij ∇i 2 + 2 (A.0.1) 2 1 + cij(ri)rij + dijrij 1 + cij(rj)rij + dijrij j6=i 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

Dσ(x + δ, y , z ) − Dσ(x , y , z ) Dσ(x , y + δ, z ) − Dσ(x , y , z ) ∇ Dσ(r ) = i i i i i i ˆi + i i i i i i ˆj+ i i δ δ Dσ(x , y , z + δ) − Dσ(x , y , z ) i i i i i i kˆ (A.0.2) δ and

Dσ(x + δ, y , z ) − 2Dσ(x , y , z ) + Dσ(x − δ, y , z ) ∇2Dσ(r ) = i i i i i i i i i + i i δ2

149 150 Appendix A. Local Kinetic Energy Calculation for Atom in Jellium

Dσ(x , y + δ, z ) − 2Dσ(x , y , z ) + Dσ(x , y − δ, z ) i i i i i i i i i + δ2 Dσ(x , y , z + δ) − 2Dσ(x , y , z ) + Dσ(x , y , z − δ) i i i i i i i i i (A.0.3) δ2 where δ is a small displacement (∼ 10−4) and ˆi, ˆj and kˆ are unit vectors in the x, y and z directions respectively. If we instead use spherical polar coordinates we have

σ σ σ σ ∂D 1 ∂D 1 ∂D ∇iD (ri) = rî + θî + φî (A.0.4) ∂ri ri ∂θi ri sin θi ∂φi and

2 σ σ 2 σ σ 2 σ 2 σ ∂ D 2 ∂D 1 ∂ D cos θi ∂D 1 ∂ D ∇i D (ri) = 2 + + 2 2 + 2 + 2 2 2 (A.0.5) ∂ri ri ∂ri ri ∂θi ri sin θi ∂θi ri sin θi ∂φi where again,r î, θî and φî are unit vectors. In these expressions, the ∂D/∂ri, etc, are calculated by constructing the determinant

∂R1(ri) R1(r1)Yl1m1 (θ1, φ1) ··· r Yl1m1 (θi, φi) ··· R1(rN )Yl1m1 (θN , φN ) i ∂R2(ri) R2(r1)Yl m (θ1, φ1) ··· Yl m (θi, φi) ··· R2(rN )Yl m (θN , φN ) 2 2 ri 2 2 2 2 ∂D/∂ri = ......

R (r )Y (θ , φ ) ··· ∂RN (ri) Y (θ , φ ) ··· R (r )Y (θ , φ ) N 1 lN mN 1 1 ri lN mN i i N N lN mN N N (A.0.6) 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. 2 2 The ∂Ri(r)/∂r and ∂ Ri(r)/∂r 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 2 2 ∂Ylm/∂θ, ∂ Ylm/∂θ 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

6 6 X nπri X nπri xi 1 nπri xiRjell ∇ α(i)j = α(i) cos( ) − sin( ) xˆ + ... i n 0 R n R r2 nπ R r 3 i n=1 jell n=1 jell i jell i Appendix A. Local Kinetic Energy Calculation for Atom in Jellium 151

6 6 X nπri X nπri −nπ ∇2 α(i)j = α(i) sin( ) (A.0.7) i n 0 R n R r R n=1 jell n=1 jell i jell The ﬁfth term in Eq. (A.0.1) can be written

2 2 1 X aijrij + bijrij aijrij + bijrij ∇ + = 2 i 1 + c (r )r + d r 2 1 + c (r )r + d r 2 j ij i ij ij ij ij j ij ij ij

( 2 dc(ri) ) 1 X (arij−1 + 2b − adrij + bc(ri)rij)(xi − xj) − (a + brij)rij dx i xˆ+ 2 (1 + c(r )r + dr2 )2 j i ij ij

1 X (arij−1 + 2b − adrij + bc(rj)rij)(xi − xj) xˆ + ... (A.0.8) 2 (1 + c(r )r + dr2 )2 j j ij ij where

1 dc(ri) 4c 4 Rjellxi = 2 2 2 2 4 (A.0.9) dxi 4 4 Rjell + (ri − Rjell) The remaining term in Eq. (A.0.1) can be written

2 2 1 X aijrij + bijrij aijrij + bijrij ∇2 + = 2 i 1 + c (r )r + d r 2 1 + c (r )r + d r 2 j ij i ij ij ij ij j ij ij ij ( −1 −2 2 X arij + 2b − adrij + bc(ri)rij (−arij − ad + bc(ri))(xi − xj) + − 2(1 + c(r )r + dr2 )2 2(1 + c(r )r + dr2 )2r j i ij ij i ij ij ij

2 2 (a + brij)(xi − xj) dc(ri) (a + brij)rij d c(ri) 2 2 − 2 2 2 − (1 + c(ri)rij + drij) dxi 2(1 + c(ri)rij + drij) dxi (ar−1 + 2b − adr + bc(r )r )(x − x ) − (a + br )r2 dc(ri) ij ij i ij i j ij ij dxi 2 3 (1 + c(ri)rij + drij) (xi − xj) dc(ri) × (c(ri) + 2drij) + rij + rij dxi −1 −2 2 arij + 2b − adrij + bc(rj)rij (−arij − ad + bc(rj))(xi − xj) 2 2 + 2 2 − 2(1 + c(rj)rij + drij) 2(1 + c(rj)rij + drij) rij −1 2 ) (arij + 2b − adrij + bc(rj)rij)(c(rj) + 2drij)(xi − xj) 2 3 + ... (A.0.10) rij(1 + c(rj)rij + drij) where

2 1 1 2 2 2 3 d c(ri) 4c 4 Rjell 32c 4 Rjellxi (ri − Rjell) 2 = 2 2 2 2 4 − 2 2 2 2 4 2 (A.0.11) dxi 4 4 Rjell + (ri − Rjell) (4 4 Rjell + (ri − Rjell) ) 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. Bibliography

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