Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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Let us estimate how large is this large number using Fourier analysis. In order to describe features which vary on a length scale δ, one needs Fourier components up to q max ∼ 2π/δ. In a crystal, our wave vectors q = k + G have discrete values. There will be a number N P W of plane waves approximately equal to the volume of the sphere of radius q max , divided by the volume Ω BZ of the BZ: N P W ≃ 4πq3 max , Ω BZ = 8π3 3Ω BZ Ω . (10.8) The second equality follows from the definition of the reciprocal lattice. A simple estimate for diamond is instructive. The 1s orbital of the carbon atom has its maximum around 0.3 a.u., so δ ≃ 0.1 a.u. is a reasonable value. Diamond has a ”face-centered-cubic” lattice with lattice parameter a 0 = 6.74 a.u. and primitive vectors: ( 1 a 1 = a 0 2 , 1 ) ( 1 2 , 0 , a 2 = a 0 2 , 0, 1 ) ( , a 3 = a 0 0, 1 2 2 , 1 ) . (10.9) 2 The unit cell has a volume Ω = a 3 0 /4, the BZ s a volume Ω BZ = (2π) 3 /(a 3 0 /4). Inserting the data, one finds N P W ∼ 250, 000 plane wave, clearly too much for practical use. It is however possible to use a plane wave basis set in conjunction with pseudopotentials: an effective potential that “mimics” the effects of the nucleus and the core electrons on valence electrons. The true electronic valence orbitals are replaced by “pseudo-orbitals” that do not have the orthogonality wiggles typical of true orbitals. As a consequence, they are well described by a much smaller number of plane waves. Pseudopotentials have a long history, going back to the 30’s. Born as a rough and approximate way to get decent band structures, they have evolved into a sophisticated and exceedingly useful tool for accurate and predictive calculations in condensed-matter physics. 10.3 Code: cohenbergstresser Code cohenbergstresser.f90 1 (or cohenbergstresser.c 2 ) implements the calculation of the band structure in Si using the pseudopotentials published by M. L. Cohen and T. K. Bergstresser, Phys. Rev. 141, 789 (1966). These are “empirical” pseudopotentials, i.e. devised to reproduce available experimental data, and not derived from first principles. Si has the same crystal structure as Diamond: a face-centered cubic lattice with two atoms in the unit cell. In the figure, the black and red dots identify the two sublattices. The side of the cube is the lattice parameter a 0 . In the Diamond structure, the two sublattices have the same composition; in the zincblende structure (e.g. GaAs), they have different composition. 1 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/cohenbergstresser.f90 2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/cohenbergstresser.c 76
The origin of the coordinate system is arbitrary; typical choices are one of the atoms, or the middle point between two neighboring atoms. We use the latter choice because it yields inversion symmetry. The two atoms in the unit cell are thus at positions d 1 = −d, d 2 = +d, where ( 1 d = a 0 8 , 1 8 , 1 ) . (10.10) 8 The lattice parameter for Si is a 0 = 10.26 a.u.. 3 Let us re-examine the matrix elements between plane waves of a potential V , given by a sum of spherical symmetric potentials V µ centered around atomic positions: V (r) = ∑ ∑ V µ (|r − d µ − R n |) (10.13) n µ With some algebra, one finds: 〈b i,k |V |b j,k 〉 = 1 ∫ V (r)e −iG·r dr = V Si (G) cos(G · d), (10.14) Ω Ω where G = G i − G j . The cosine term is a special case of a geometrical factor known as structure factor, while V Si (G) is known as the atomic form factor: V Si (G) = 1 ∫ V Si (r)e −iG·r dr. (10.15) Ω Ω Cohen-Bergstresser pseudopotentials are given as atomic form factors for a few values of |G|, corresponding to the smallest allowed modules: G 2 = 0, 3, 4, 8, 11, ..., in units of (2π/a 0 ) 2 . The code requires on input the cutoff (in Ry) for the kinetic energy of plane waves, and a list of vectors k in the Brillouin Zone. Traditionally these points are chosen along high-symmetry lines, joining high-symmetry points shown in figure and listed below: Γ = (0, 0, 0), X = 2π a 0 (1, 0, 0), W = 2π a 0 (1, 1 2 , 0), K = 2π a 0 ( 3 4 , 3 4 , 0), L = 2π a 0 ( 1 2 , 1 2 , 1 2 ), 3 We remark that the face-centered cubic lattice can also be described as a simple-cubic lattice: a 1 = a 0 (1, 0, 0) , a 2 = a 0 (0, 1, 0) , a 3 = a 0 (0, 0, 1) (10.11) with four atoms in the unit cell, at positions: ( d 1 = a 0 0, 1 2 , 1 ) ( 1 , d 2 = a 0 2 2 , 0, 1 ) ( 1 , d 3 = a 0 2 2 , 1 ) 2 , 0 , d 4 = (0, 0, 0) . (10.12) 77
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Lecture notes Numerical Methods in
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3.3 Code: crossection . . . . . . .
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Introduction The aim of these lectu
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is an important and well-known libr
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Chapter 1 One-dimensional Schrödin
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Under the assumption of Eq.(1.8), E
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point its value is still ∼ 60% of
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an equation involving finite differ
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eigenvalue) to force the code to pe
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1.3.3 Laboratory Here are a few hin
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The Hamiltonian can thus be written
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In the following we will use q 2 e
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The dominating term close to the nu
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whose goal is to give an estimate,
Page 29 and 30: Chapter 3 Scattering from a potenti
Page 31 and 32: 3.2 Scattering of H atoms from rare
Page 33 and 34: 1. Solution of the radial Schrödin
Page 35 and 36: • In the limit of a weak potentia
Page 37 and 38: By modifying ψ as ψ + δψ, the e
Page 39 and 40: This demonstrates Eq.(4.19), since
Page 41 and 42: that the same equations, for a fini
Page 43 and 44: eported here are immediately genera
Page 45 and 46: BLAS 5 (collected here: dgemm.f 6 )
Page 47 and 48: known as generalized eigenvalue pro
Page 49 and 50: The Hamiltonian operator for this p
Page 51 and 52: Chapter 6 Self-consistent Field A w
Page 53 and 54: (the variations of all other normal
Page 55 and 56: nucleus. Even in open-shell atoms,
Page 57 and 58: Chapter 7 The Hartree-Fock approxim
Page 59 and 60: Now that we have the expectation va
Page 61 and 62: number of possible Slater determina
Page 63 and 64: Let us now introduce a further vari
Page 65 and 66: The index R is a reminder that both
Page 67 and 68: Molecular orbitals theory can expla
Page 69 and 70: Verify how sensitive the result is
Page 71 and 72: potential, we can hope to obtain a
Page 73 and 74: with wave vector k will be purely k
Page 75 and 76: Fast Fourier Transform In the simpl
Page 77 and 78: truncate it, it is convenient to co
Page 79: A reciprocal lattice of vectors G m
Page 83 and 84: Chapter 11 Exact diagonalization of
Page 85 and 86: The only nonzero matrix elements fo
Page 87 and 88: The code requires the number N of s
Page 89 and 90: Appendix A From two-body to one-bod
Page 91 and 92: Appendix B Accidental degeneracy an
Page 93 and 94: and [J 2 1 , J z] = [J 2 2 , J z] =
Page 95 and 96: part and symmetric coordinate part)
Page 97 and 98: and the corresponding energy is E =
Page 99 and 100: D.3.1 Laboratory • observe the ef
Page 101: • Relative error: |a − b| < ɛa
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Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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