First Principles Investigation into the Atom in Jellium Model System. Andrew Ian Duff

Transcription

1 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

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3 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.

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5 To my Family

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7 Acknowledgments Thanks to my supervisor James Annett and also to Balazs Györffy.

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9 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:...

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11 Contents 1 Introduction 1 2 Solving the Many-Electron Schrödinger equation The Many-Electron Problem Single-Electron Theories The Variational Principle Hartree Fock Theory Density Functional Theory Minimising the Energy Functional The Kohn-Sham Equations Self-Consistent Solutions The Exchange-Correlation Energy and Potential Self-Interaction Correction Variational Quantum Monte Carlo A Variational Theory The Monte Carlo Technique The Variational Quantum Monte Carlo Method Metropolis Algorithm Equilibration and Serial Correlation The Choice of the Trial Wavefunction Updating the Slater Determinants Calculating the Local Energy Cusp Conditions Correlated Sampling Blocking Analysis to Calculate Error on Mean xi

12 xii CONTENTS Calculating the Probability Density HF Calculations An Atom in Infinite Jellium Solved using DFT Solving the Schrödinger Equation The Radial Schrödinger Equation The Electron Density Potential Mixing Criterion for Convergence Simplifying the Coulomb Potential for the Case of Spherical Symmetry Scattering States Introduction Boundary Conditions on Scattering States Matching to the Boundary Condition Normalisation of Scattering States Calculating the Scattering State Density Friedel Oscillations Friedel Sum Rule Properties of the Phase-Shift Numerical Algorithm for Solving the Radial Schrödinger Equation Radial Schrödinger Equation Solutions in the Limits r 0 and r The Runge-Kutta Algorithm Bound State Calculation Scattered State Calculation The Immersion Energy Derivation of Immersion Energy Finite Radius Corrections Numerical Parameters and Error Analysis Results The Effective Medium Theory Background Theory Results Cerium Solved using the LDA and SIC

13 CONTENTS xiii Introduction Cerium Spin-polarised LDA for Cerium Imposing Orthogonality when Applying SIC SIC-LDA for Cerium Magnetic Solution of Cerium Hydrogen Immersed in a Finite Jellium Sphere Hydrogen in Finite Jellium Spheres using the LDA Energy of An Atom in a Finite Jellium Sphere Filling of Orbitals Applying SIC to a Hydrogen Atom in a Finite Jellium Sphere Results Hydrogen in Finite Jellium Spheres using VQMC The Choice of the Trial Wavefunction Calculating the Local Energy Results Conclusions 145 A Local Kinetic Energy Calculation for Atom in Jellium 149

14 xiv CONTENTS

15 List of Tables 4.1 Total energies of hydrogen in 10-electron jellium spheres Total energies of 10-electron jellium spheres Immersion energies of hydrogen in 10-electron jellium spheres xv

16 xvi LIST OF TABLES

17 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] 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) 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) 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 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, r W S is the Wigner-Seitz radius, n bs is the number of bound states per atom and n val is the number of valence electrons per atom. Charges are in units of the electron charge, e xvii

18 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] Re-blocking analysis for hydrogen immersed in a 10-electron jellium sphere of density 0.03a 3 B. The error on the mean levels off at just under eV and therefore this is the error we quote on the total energy Phase-shifts (top panel) and the corresponding density of states (lower panel) for a cerium atom embedded in jellium of r s = The l = 0 phase-shifts for a hydrogen atom immersed in infinite jellium of background densities 0.01a 3 B (top panel) and 0.05a 3 B (bottom panel) The quantity du outwards (r = r match )/dr du inwards (r = r match )/dr (as described in the main text) for l = 0 is plotted as a function of energy. The system is a Technetium atom immersed in jellium of background density 0.03a 3 B, and is non-magnetic, so the curve applies for both spin-up and spindown electrons The l = 0 bound state energies are at the points where the curve crosses the x-axis, I.e. at: a.u., a.u., a.u. and 1.845a.u Determining the parameter V req, for a SI-corrected cerium atom in infinite jellium of density n 0 = 0.01a 3 B. Immersion energy is plotted against log( V req ), and error bars (in green) are placed at different values of the convergence Density plots for hydrogen in infinite jellium of density 0.005a 3 B. Values for r max equal to a B, a B and a B are shown. Only the second choice of r max 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 r max to get the correct density profile is equivalent to selecting r max to satisfy the Friedel sum

19 LIST OF FIGURES xix 3.6 Determining the parameter l num. This value has to be large enough so that, for a given r max, the density is correctly realised at all radii. The above are results for a cerium atom immersed in jellium of density 0.01a B, with r max 20a B. 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 l num (= 20) gives the correct density profile, and therefore this is the value that we use 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 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 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] Cohesive energy versus neutral sphere radii for 4d transition metals Experimental phase diagram of cerium [6] Experimental results showing the molar volume of cerium against the pressure applied to the sample [7] Phase-shifts for (non-magnetic) ground-state solutions of a cerium atom embedded in jellium of different densities. From top to bottom, r s = 1.81a B, r s = 3.24a B, r s = 5.30a B. 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) Angular momentum resolved density of states for the ground-state solution of a cerium atom embedded in jellium of r s = 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

20 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, n 0 = 0.04a 3 B, n 0 = 0.03a 3 B, n 0 = 0.02a 3 B and n 0 = 0.01a 3 B. 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 Plots of (spin-up) bound state energies for a 338-electron jellium sphere and a hydrogen atom in a 338-electron jellium sphere, along with the (spin-up) potentials for these systems. The background densities of the jellium are 0.03 a 3 B (upper panel) and a 3 B (lower panel). The bound states are shown as lines, with the lengths of these lines corresponding to the angular momentum (l=0 is the shortest and l=9 is the longest) Plots of immersion energy versus number of electrons for jellium spheres. Densities of 0.001a 3 B, 0.007a 3 B and 0.03a 3 B 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 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 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 Plots of atom induced densities for hydrogen in finite jellium spheres of background density 0.01a 3 B, with 10, 50 and 338 electrons (top, middle and bottom panel respectively). Also plotted is the atom induced density for a hydrogen atom in infinite jellium at the same background density The total density for a hydrogen atom in a 106-electron jellium sphere and for a hydrogen atom in infinite jellium for a background density 0.004a 3 B. 127

21 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 Total energy of a hydrogen atom immersed in a 10-electron jellium sphere for different background densities Total energy of a 10-electron jellium sphere for different background densities Immersion energies for a hydrogen atom immersed in a 10-electron jellium sphere for different background densities Local energy distribution for a VQMC calculation of a hydrogen atom in a 10-electron jellium sphere of density 0.03a 3 B Re-blocking analysis for hydrogen immersed in a 10-electron jellium sphere of density 0.03a 3 B. The error on the mean levels off at just under eV and therefore this is the error we quote on the total energy Electron density of a hydrogen atom in a 10-electron jellium sphere of background density 0.03a 3 B using HF, LDA, SIC and VQMC Electron density of a hydrogen atom in a 10-electron jellium sphere of background density 0.03a 3 B using HF, LDA, SIC and VQMC. Note in the top graph curves for HF, LDA and SIC coincide Electron density of a hydrogen atom in a 10-electron jellium sphere of background density 0.002a 3 B using HF, LDA, SIC and VQMC Electron density of a hydrogen atom in a 10-Electron jellium sphere of background density 0.002a 3 B using HF, LDA, SIC and VQMC. Note in the top graph curves for HF, LDA and SIC coincide 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

22 xxii LIST OF FIGURES

23 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, n 0. Our motivation for studying this model system is that it can be used as a building block from which to construct a theory of a full periodic solid. In this introduction we will expand upon this motivation and we will detail the techniques which we use to solve the model system. Solving the atom-in-jellium system formally requires us to solve the time-independent Schrödinger equation (in the Born-Oppenheimer approximation [9], so that the nucleus is held fixed) N 2 2 i + 1 2m e 2 i=1 N e 2 4πɛ 0 r i r j + v ext(r 1, r 2,..., r N ) Ψ(r 1, r 2,..., r N ) i j where the external potential, v ext (r), is given by v ext (r) = e2 4πɛ 0 = EΨ(r 1, r 2,..., r N ) (1.0.1) N i=1 Z r i e2 4πɛ 0 n 0 r r dr (1.0.2) 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

24 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 Ψ Ĥ Ψ / Ψ Ψ (where the Hamiltonian, Ĥ, is the operator on the left-hand side of Eq. (1.0.1)), with respect to the single-electron functions, or

25 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

26 4 Introduction hydrogen atoms. We find the HF energy in this limit to be a substantial overestimate of the exact energy, which should equal twice the energy of the isolated hydrogen atom. HF gets it wrong because it fails to include Coulombic correlations, which would prohibit the occupation of the same atom by both electrons in the limit of dissociation. Density Functional Theory The second method we will use to solve the Schrödinger equation is DFT [14, 16]. In DFT, one proves that the ground-state energy, E 0, of a system of N interacting electrons in an external potential, v ext (r), is a unique functional of the ground-state density, n 0 (r). To this end a functional, E v [n(r)] is constructed which has the property that it is minimised by n 0 (r) and that the ground-state energy equals the value of the functional at this point, i.e. E 0 E v [n 0 ]. Writing n(r) = N i=1 φ i(r) 2, and functionally differentiating E v with respect to φ i (r), one obtains N single-particle Schrödinger-like equations known as Kohn-Sham equations [15]. The equations must be solved self-consistently and the self-consistent n(r) is then (in principle) the true ground-state density, n 0 (r), and the ground-state energy is obtained by inserting this into E v [n(r)]. E v [n(r)] contains a universal part, i.e. one which just depends on n(r) and not v ext (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, E xc [n(r)]. DFT is not an exact theory, because although it can be proved that there is an exact functional for E xc [n(r)], in reality it is not known what it is. It is for this reason that the electron correlation discussed earlier is only treated approximately within DFT. This approximate treatment of the correlation is still a big improvement over the exchange-only correlation included in HF theory however. Reducing the 3N-dimensional many-particle equation into N 3-dimensional equations is a dramatic simplification, and makes solving a system within DFT computationally very efficient. Furthermore, a particularly simple approximation to the exchange-correlation functional, known as the local density approximation (LDA), produces results in surprisingly good agreement with experiment. In this approximation, one assumes that the exchange-correlation energy can be written in the form

27 Introduction 5 E xc [n(r)] = n(r)ɛ xc (n(r))dr (1.0.3) where ɛ 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

28 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).

29 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).

30 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.

31 Introduction 9 of volume. The minimum of the energy curve for the LDA was at 23.4Å 3 whereas the minimum for the SIC curve was at 29.9Å 3. Experimentally, the alpha phase has a volume 28.2Å 3 and the SIC has a volume 34.7Å 3. So Lüders et al slightly underestimate both volumes, but successfully provide evidence supporting the Mott-transition model of the gamma-alpha transition. This is just one example of an application of SIC. Other important examples include calculations of valencies and lattice constants as a function of atomic number across the rare-earth elements [28] Quantum Monte Carlo Quantum Monte Carlo (QMC) methods use random numbers to solve the Schrödinger equation. We will use a method known as VQMC [17, 18]. In this method, one makes a guess at a trial wavefunction, using the physics of the situation to aid them with the guess. The expectation value of the Hamiltonian for this trial wavefunction is then evaluated using a random-walk technique known as the Metropolis algorithm [29]. One then varies the trial wavefunction (or more specifically, parameters within the wavefunction) in order to minimise this expectation value. The minimised expectation value is then an upper bound on the exact ground-state energy of the system. With a good trial wavefunction, this upper bound can be made very close to the exact ground-state energy. One application of QMC has been in the calculation of cohesive energies. The cohesive energy of a solid is the energy required, at zero temperature, to separate all of the constituent atoms of the solid infinitely far apart. Within VQMC, this calculation requires trial wavefunctions sufficiently accurate to describe both the solid, and the constituent atom. Because these two systems are very different, the calculation constitutes a challenging test of the theory. Accordingly it was seen as an impressive success for the theory when Fahy et al [30, 31] calculated cohesive energies for tetrahedrally bonded Carbon and Silicon. VQMC was used to obtain cohesive energies of 7.27eV/atom and 4.82eV/atom respectively, which are in good accord with the experimental values of 7.37eV/atom and 4.62eV/atom. For comparison, the LDA overestimates these energies, giving cohesive energies of 8.61eV/atom and 5.28eV/atom respectively. The effectiveness of VQMC relies entirely upon the quality of the trial wavefunction. Normally one would use this method as a driver for another QMC method known as diffusion QMC (DQMC) [17, 32]. The results from this method are in principle exact,

32 10 Introduction barring an issue known as the sign problem [17]. We will limit ourselves to VQMC however, partly because published results on jellium spheres [33] (not too dissimilar to the hydrogen in a jellium sphere system which we will be studying - see later) show that VQMC and DQMC results are quite similar. The Atom-in-Jellium Model Our motivation for considering the system of an atom immersed in jellium is that it can be used to model a condensed matter system. This idea comes in the first place from just considering pure jellium (i.e. a uniform electron gas with a charge neutralising background) as a model for a solid. In this case we would regard the positive background as the smeared out effective charge of the ions in the solid, and the negative charge as the conduction electron density. If each atom occupies a volume, 4 3 πa3, where a is the Wigner-Seitz radius, and if each of these atoms contributes N v valence electrons to the solid, then with an electron density n = 1/ 4 3 π(r sa B ) 3, we have the relation 4 3 πa3 4 3 π(r sa B ) 3 = N v a = N v 1 3 rs a B (1.0.4) where a B is the Bohr radius. It turns out that the energy of an electron gas as a function of r s 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 r s = 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 In this model of the solid, the Schrödinger equation has two classes of solution. There are a discrete set of bound states, and a continuum of positive energy scattering state solutions. The bound states and the atom-induced scattering states (the change in the number of scattering states upon adding the atom to the jellium) of this system are interpreted as the bound and valence electrons per atom of the full periodic solid. In order to use this model of an atom in jellium to make predictions about real solids, we need an expression for the energy of the solid. Clearly, the total energy of this model

33 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, r W S is the Wigner- Seitz radius, n bs is the number of bound states per atom and n val is the number of valence electrons per atom. Charges are in units of the electron charge, e.

34 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 F Z,R [n host ], where Z is the charge of the impurity, R is its position and n host (r) is the unperturbed charge density of the host. One makes the assumption that only the unperturbed host density immediately surrounding the impurity is important in the calculation of the energy of the impurity. Applying this assumption in the simplest possible way results in one replacing the energy of the impurity in the inhomogeneous host with the energy of the impurity in a homogeneous electron gas of background density n host (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 n host (R) (the

35 Introduction 13 immersion energy is the total energy of the impurity in the host minus the energies of the separate impurity and host). The heat of solution, which is the change in energy when one mole of hydrogen gas is absorbed by the solid, is then obtained by subtracting the binding energy of the hydrogen molecule per atom from the immersion energy. Note that in these calculations, the background density of the jellium was not in fact just taken to be the density of the host at the point R. Instead it was chosen as some average of the host electron density over the volume to be occupied by the impurity. The embedded atom method (EAM) [38] is based on the EM approach. In this method, we are interested in calculating the cohesive energy of a solid. Each atom in the solid is viewed as being embedded in the electron gas set up by the remaining atoms of the solid. For an atom i, we denote this as n i (r). We assume that this electron density is a linear superposition of the electron densities from each of the atoms at sites R j (where j i), which we label n j ( r R j ). In addition, we assume that n i 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 n i (r) = j i n j ( r R j ) (1.0.5) Following the EM approach, we then replace n i (r) in the atomic cell i with its value at R i. 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 The cohesive energy is then written as E c = E( n i (R i )) i U ij (R ij ) (1.0.6) where E(n 0 ) is the immersion energy of an atom immersed in jellium of background density n 0. 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 i j

36 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 E c ( n) N = E( n) + n r=s r=0 ( r =0 n(r ) r r dr Z ) dr (1.0.7) r where n(r) is the atom-induced density for an atom with atomic number Z immersed in jellium of density n. The quantity s is referred to as the neutral sphere radius and is

37 Introduction 15 defined as r=s r=0 n(r)dr = Z (1.0.8) The theory requires only the atomic number, Z, of the constituent atom of the solid as the input parameter. We minimise E c ( n)/n with respect to n 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

38 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 the range 0.001a 3 B to 0.03a 3 B 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]. differed significantly from the LDA results. These calculations however resulted in immersion energies which mismatch was that the trial wavefunction was not of optimal form. The reason cited by the authors for this 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.

39 Chapter 2 Solving the Many-Electron Schrödinger equation 2.1 The Many-Electron Problem The time-independent Schrödinger equation for an N-electron system in an external potential, v ext (r 1, r 2,..., r N ), within the Born-Oppenheimer approximation [9] (all nuclei held fixed), is N 2 2 i + 1 2m e 2 i=1 N e 2 4πɛ 0 r i r j + v ext(r 1, r 2,..., r N ) Ψ(r 1, r 2,..., r N ) i j = EΨ(r 1, r 2,..., r N ) (2.1.1) Note that for the rest of this thesis atomic units are used ( 2 /m=e 2 /4πɛ 0 =1). The external potential, v ext, could be for example the Coulomb attraction between the electrons and an ion of charge Z: i=n Z v ext = (2.1.2) r i i=1 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

40 18 Solving the Many-Electron Schrödinger equation In this chapter we will discuss these methods in detail. First however, we introduce the concept of the single-electron theory in Section and then the variational principle in Section Single-Electron Theories HF and DFT are single-electron theories. In a single-electron theory each electron has its own Schrödinger-like equation ( 1 ) 2 2 i + V (r) φ i (r) = ɛ i φ i (r) (2.1.3) and orbital φ i (r). The total electron density of the system is written as a sum over the single-electron densities, n i (r) n(r) = i n i (r) = i φ i (r) 2 (2.1.4) The potential, V (r), contains a term in which electron i interacts via the Coulomb interaction with the charge density of all of the other electrons. For example, in the Hartree approximation (which pre-dates HF theory) we have V (r) = v ext (r) + j i dr n j(r ) r r (2.1.5) The single-electron equations are solved self-consistently to obtain a set of φ i (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 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.:

41 2.2 Hartree Fock Theory 19 Ψ Ĥ Ψ Ψ0 Ĥ Ψ 0 = E 0 (2.1.6) where Ψ is an arbitrary state ket and Ψ 0 is the ground-state state ket. These are normalised as Ψ Ψ = 1. The equality holds when Ψ = Ψ 0. To prove the result, let us write Ψ in terms of eigen-kets of the time-independent Schrödinger equation: Ψ = i c i Φ i (2.1.7) substituting into Ψ Ĥ Ψ we get: Ψ Ĥ Ψ = ij c i c j Φ i Ĥ Φ j = ij c i c j Φ i Φ j E j = i c i 2 E i (2.1.8) where E i are the energies of the time-independent Schrödinger equation: Ĥ 0 Φ i = E i Φ i (E 0 is the ground state and E 1, E 2, etc are excited states of increasing energy). The fact that the Φ i > are orthonormal means that to have < Ψ Ψ >= 1 we need i c i 2 = 1. This normalisation condition and the above equation tell us that: proving the variational principle. Ψ Ĥ Ψ E 0 (2.1.9) 2.2 Hartree Fock Theory In HF theory [10, 11, 12, 13], one expresses the wavefunction as a determinant of singleparticle orbitals Ψ(x 1, x 2,..., x N ) = φ 1 (x 1 ) φ 1 (x 2 )... φ 1 (x N ) φ 2 (x 1 ) φ 2 (x 2 )... φ 2 (x N ).. φ N (x 1 ) φ N (x 2 )... φ N (x N ) (2.2.1) here, x includes the position and the spin. This is the simplest way of including a signchange in the wavefunction when the positions and spins of any two particles are exchanged, as is required in order to satisfy the Pauli exclusion principle:

42 20 Solving the Many-Electron Schrödinger equation Ψ(x 1,..., x i,..., x j,..., x N ) = Ψ(x 1,..., x j,..., x i,..., x N ) (2.2.2) The expectation value of the Hamiltonian, Ψ Ĥ Ψ, where Ĥ = N i + i=1 N i j 1 r i r j + v ext(r 1, r 2,..., r N ) (2.2.3) is then minimised with respect to each orbital. Lagrange multipliers, E i, are introduced in order to ensure the orbitals are normalised. where φ i ( Ψ Ĥ Ψ i E i φ i 2 ) = 0 (2.2.4) Ψ Ĥ Ψ = i φ i (r) ( 12 ) 2 φ i (r)dr φ i (r)φ i (r)φ j (r )φ j (r ) r r drdr i,j 1 2 i,j φ i (r)φ i (r )φ j δ (r )φ j (r) σi,σ j r r drdr + Here, σ i is the spin associated with orbital i. φ i (r) 2 v ext (r)dr (2.2.5) i The third term is referred to as the exchange energy. Performing the minimisation, N so-called HF equations are obtained [50]. ( n(r ) ) r r dr + v ext (r) φ i (r) j where the electron density, n(r), is given by dr φ j (r )φ i (r )φ j (r) r r δ σi,σ j = E i φ i (r) (2.2.6) n(r) = Ψ(r 1, r 2,..., r N ) i δ(r r i ) Ψ(r 1, r 2,..., r N ) = i φ i (r) 2 (2.2.7) These equations are solved to obtain the self-consistent set of φ i (r). These orbitals can then be plugged into H Ψ 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 φ i (r) into Eq. (2.2.1) is an approximation to the true ground-state wavefunction.