My PhD thesis - Condensed Matter Theory - Imperial College London

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Chapter 9 A new calculation of the jellium surface energy The stated aim of this <strong>thesis</strong> is to investigate methods of improving the accuracy of surface calculations, with particular reference to quantum Monte Carlo simulations. The standard test system for surface calculations is the unbounded jellium slab, described in chapter 5 and referred to throughout this work; the disagreement between existing QMC results for this system and those obtained using other methods was discussed in chapter 5. The techniques investigated here should make it possible to carry out a more accurate calculation of the jellium surface energy. The MPC interaction, while not reducing finite-size errors, makes simulations much more efficient; the plasmon normal mode theory, which produced a Jastrow factor more appropriate to the bounded jellium slab, nevertheless also led to the discovery of the efficient shortranged two-body term with the accompanying analytic one-body term, helping to overcome the problems of trial wave function optimisation. However, there are other avenues to explore in the attempt to achieve a more accurate simulation. Two of these will be described in this chapter; finally, a new calculation of the surface energy for one particular density will be presented. 166
CHAPTER 9. ENERGY A NEW CALCULATION OF THE JELLIUM SURFACE 9.1 Improved orbitals The single-electron orbitals which make up the Slater determinants in the QMC trial wave function must be obtained from some prior calculation. Typically, this is an LDA calculation; the earlier QMC simulations of the jellium slab relied on LDA wave functions. As described in chapter 2, a key component of any density-functional theory calculation is the exchange-correlation potential V XC . It was shown by Lang and Kohn [48] that outside a metal surface, the correct asymptotic form of the potential is image-like: 1 V XC (z) = − 4(z − z 0 ) . (9.1) Here z is the coordinate normal to the surface and z 0 is the position of the image plane. However, the LDA gives a potential which decays exponentially. Recently, Eguiluz and coworkers investigated the form of V XC at a metal surface from first principles [19, 20]; their potential reproduced the asymptotic form given in equation (9.1), and matched the conventional LDA value inside the metal. Having the correct image tail in the potential is important for studying several processes relevant to experiment: Eguiluz cites binding energies and lifetimes of image-potential-bound surface states, tunnelling currents in the scanning-tunnelling microscope, and resonant-tunnelling rates for ion-surface collisions as examples. It is not clear whether it will prove equally important when calculating the ground-state energy in QMC. To investigate this, density-functional theory calculations were carried out using a version of V XC containing the image potential, with two different positions 1 for the image plane. The resulting wave functions were tested in VMC simulations, and compared with the traditional LDA wave functions. Figure 9.1 shows the original and modified forms of the potential. In the vacuum 1 The two image-plane positions (relative to the slab edge) were z 0 = 0.72 and z 0 = 1.49; these values were obtained by Eguiluz, the first by fitting to the image tail, the second from the linear response. 167
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Improving the accuracy of surface s
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Acknowledgements I am greatly indeb
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CONTENTS 3.3.4 Importance-sampled d
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CONTENTS 10 Conclusions 181 A The q
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LIST OF FIGURES 5.3 The LDA energy
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LIST OF FIGURES 8.4 The x- and z-co
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LIST OF FIGURES 8.20 Electron densi
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List of Tables 8.1 The function F k
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Chapter 1 Introduction In recent de
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CHAPTER 1. INTRODUCTION A successfu
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CHAPTER 2. MECHANICS THE SIMPLIFICA
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CHAPTER 3. QUANTUM MONTE CARLO METH
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Chapter 4 Errors in QMC simulations
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CHAPTER 4. ERRORS IN QMC SIMULATION
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CHAPTER 5. THE JELLIUM SLAB The jel
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CHAPTER 5. THE JELLIUM SLAB 0.08 0.
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CHAPTER 5. THE JELLIUM SLAB The DMC
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CHAPTER 5. THE JELLIUM SLAB -9.125
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CHAPTER 5. THE JELLIUM SLAB -0.3 Su
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CHAPTER 5. THE JELLIUM SLAB 0.14 be
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CHAPTER 5. THE JELLIUM SLAB At the
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CHAPTER 7. THE ELECTRONIC GROUND-ST
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My PhD thesis - Condensed Matter Theory - Imperial College London

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