My PhD thesis - Condensed Matter Theory - Imperial College London

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CHAPTER 9. ENERGY A NEW CALCULATION OF THE JELLIUM SURFACE -0.5 -1 -1.5 ε slab (mHa) -2 -2.5 -3 -3.5 MPC Q2D Ewald 3D Ewald -4 0.005 0.01 0.015 0.02 0.025 0.03 0.035 s/N Figure 9.7: Comparing the MPC and Ewald interactions when s = 18.4851. ± 0.042 respectively. From equation (9.6), these points should lie on a straight line when plotted against 1/s; figure 9.8 shows that this is in fact the case. The gradient of the line passing through these points is -28.5 ± 2.3, which gives the surface energy of jellium as -0.384 ± 0.031 mHa bohr −2 or -600 ± 50 erg cm −2 . This figure brings the QMC result into line with the other calculations referred to in chapter 5. In contrast, the values of σ quoted in table 9.5 were calculated using the method of Li et al. [53]: no Coulomb finite-size correction was applied, and the slab energy was simply combined with Ceperley and Alder’s fixed-node value 3 for ɛ bulk in equation (9.6). This demonstrates that, although the Coulomb finite-size error acts to decrease the energy (see the three calculations with s = 11.7783 in table 9.5), the greatest error is positive and comes from comparing the results of bulk and slab simulations. The quality of the nodal surface of the trial wave function is better in the homogeneous electron gas than in the jellium slab. 3 The fixed-node bulk calculation of Ceperley and Alder referred to by Li is unpublished; the quoted energy per electron is ɛ bulk = −0.2017 eV per electron. 178
CHAPTER 9. ENERGY A NEW CALCULATION OF THE JELLIUM SURFACE Slab width Number of electrons ɛ slab (mHa) σ (erg cm −2 ) 11.7783 332 -9.18 ± 0.13 -440 ± 30 11.7783 466 -8.901 ± 0.097 -370 ± 20 11.7783 588 -8.818 ± 0.088 -350 ± 20 15.1317 454 -8.486 ± 0.094 -340 ± 30 18.4851 572 -8.15 ± 0.15 -290 ± 60 Table 9.5: Slab energies per electron calculated in DMC, for various slab widths and system sizes. The surface energy estimates listed in the final column are included to demonstrate the errors introduced by combining the results of bulk and slab calculations without due care. To carry out a more accurate DMC calculation of the surface energy using the fitting method applied above to the VMC results would require significant computational resources. The largest systems studied using VMC and used in the final surface energy calculations contained around 3000 electrons: in smaller systems, there remains the difficult problem of correcting the Coulomb finite-size error. 179
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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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CHAPTER 8. APPLYING THE PLASMON NOR
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Page 185 and 186: APPENDIX A. THE QUASI-2D EWALD SUM
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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
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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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