My PhD thesis - Condensed Matter Theory - Imperial College London

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Chapter 5 The jellium slab Historically, the electron gas (jellium) has been the proving-ground for electronic structure methods [12, 27]; 1 without the additional complication of real ions, it provides the first basic test. The jellium slab, or quasi-2D electron gas, is the simplest surface system available, and is used as the subject of or test system for most of the work contained in this <strong>thesis</strong>. 5.1 Defining the system A homogeneous electron gas is specified by the single parameter r s . This is the radius of a sphere of size equal to the average volume of space per electron, which implies that the electron density is given by the relation n = 3 . (5.1) 4πrs 3 The electron gas may be used as a crude model for a metal. Typical metallic densities correspond to the range 1 < r s < 4; in this work, the density parameter appropriate to aluminium (r s = 2.07) has been used. 1 Accurate studies of the electron gas are important in their own right, because (among other reasons) they provide the information on which exchange-correlation functionals (and hence densityfunctional theory calculations) are based. 70
CHAPTER 5. THE JELLIUM SLAB The jellium slab is an electron gas with finite extent in one of the three spatial dimensions. There are different ways to constrain the electrons to a slab: two will be described here. 5.1.1 Constraining the electrons The first and more usual way to define the slab is to set up the following background charge density: ⎧ ⎪⎨ 3/(4πrs) 3 0 < z < s ρ b (z) = (5.2) ⎪⎩ 0 otherwise. This fixes the slab width s. In a QMC simulation, a finite number of electrons must be used. The number of electrons then determines the size of the simulation cell in the xy-direction; the cell extends from −∞ to ∞ in the z-direction. The integral of the electron density over z is equal to that of the positive background density; this is another way of saying that the system is charge-neutral. The electrons are constrained by the attractive potential of the positive background, and pushed apart by their own mutual repulsion and kinetic energy. A typical electron density profile for this form of the jellium slab is shown in figure 5.1. Some electrons spill out of the slab into the vacuum region; standing wave oscillations caused by reflection of electron waves from the confining potential are evident, decaying from the edge of the slab towards the centre [47]. The oscillations are more pronounced when a small number of electrons is used. It is also possible to constrain the electrons further, by imposing infinite barriers at z = 0 and at z = s. Electrons are no longer allowed to spill out into the vacuum region; the resultant density is shown in figure 5.2. This is the infinite barrier model; the term jellium slab is usually reserved for the unbounded system, and this convention will be applied here. 71
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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 8. APPLYING THE PLASMON NOR
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CHAPTER 9. ENERGY A NEW CALCULATION
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Chapter 10 Conclusions The original
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CHAPTER 10. CONCLUSIONS pling and o
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APPENDIX A. THE QUASI-2D EWALD SUM
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APPENDIX B. THE CUSP CONDITIONS may
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APPENDIX B. THE CUSP CONDITIONS wit
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APPENDIX C. INTEGRATING THE CUSP FU
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APPENDIX C. INTEGRATING THE CUSP FU
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APPENDIX D. RECONSTRUCTING A PROBAB
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Bibliography [1] P. H. Acioli and D
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BIBLIOGRAPHY [19] A. G. Eguiluz and
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BIBLIOGRAPHY [39] P. R. C. Kent, R.
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BIBLIOGRAPHY [59] H. J. Monkhorst a
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BIBLIOGRAPHY [80] C. J. Umrigar, K.
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My PhD thesis - Condensed Matter Theory - Imperial College London

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