My PhD thesis - Condensed Matter Theory - Imperial College London

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CHAPTER 9. ENERGY A NEW CALCULATION OF THE JELLIUM SURFACE 0.2 0 0 Potential -0.2 -0.4 V KS (no image tail) V KS (z 0 =0.72) V KS (z 0 =1.49) V XC (no image tail) V XC (z 0 =0.72) V XC (z 0 =1.49) electron density -0.05 n(z) s/2 s z Figure 9.1: The effect of including the image tail in V XC . The simple LDA form (shown for comparison) is matched to the image potential outside the slab. The test system has density parameter r s = 2.07 and slab width s = 17.64248; the simulation cell contains 600 electrons. In addition to the exchange-correlation potential V XC , the Kohn-Sham potential V KS = V Ha +V B +V XC is also plotted. The positions of the image planes are indicated by ticks on the z-axis. region, the LDA potential decays exponentially, while the modified potentials have the form described in equation (9.1). The electron density is plotted as a reference; on this scale, the difference between the three density curves corresponding to the different exchange-correlation potentials is not observable. The effect on the single-electron orbitals is small; the greatest difference is observed in the higher sub-bands, because these extend further into the vacuum region. The highest occupied orbital for the current test system (the sixth sub-band) is plotted in figure 9.2. Note that there is a very small correction to the nodes. The overall correction to the wave function is small, and this is reflected in the VMC energies listed in table 9.1. The wave functions calculated with the image plane at z 0 = 1.49 give a slightly higher energy, and will not be used in the QMC 168
CHAPTER 9. ENERGY A NEW CALCULATION OF THE JELLIUM SURFACE 0.4 0.004 0.2 0.002 ϕ LDA ∆ϕ 0 0 -0.2 ϕ 0.72 - ϕ LDA ϕ 1.49 - ϕ LDA ϕ LDA -0.002 s/2 s z Figure 9.2: The effect of the image tail on the highest occupied single-electron sub-band. The original orbital is shown, along with the difference between this and the modified orbitals (magnified one hundred times - note the two scales). simulations to follow. Those obtained with the image plane at z = 0.72 give the same VMC energy as the conventional LDA wave functions, although the variance appears to be slightly lower (less moves are required for a given accuracy); these wave functions will be used from now on. 9.2 Alternative k-point sampling In all the simulations carried out so far, strictly periodic boundary conditions have been applied to the wave function in the xy-plane. The system is homogeneous in the xy-direction, which means that the DFT orbitals must have the following form: φ nk‖ (r) = u n (z)e ik ‖·r ‖ . (9.2) 169
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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 6. THE MODIFIED PERIODIC CO
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CHAPTER 7. THE ELECTRONIC GROUND-ST
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Page 207 and 208: Bibliography [1] P. H. Acioli and D
Page 209 and 210: BIBLIOGRAPHY [19] A. G. Eguiluz and
Page 211 and 212: BIBLIOGRAPHY [39] P. R. C. Kent, R.
Page 213 and 214: BIBLIOGRAPHY [59] H. J. Monkhorst a
Page 215: BIBLIOGRAPHY [80] C. J. Umrigar, K.
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My PhD thesis - Condensed Matter Theory - Imperial College London

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