Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

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Notice that the FT is now a periodic function in the variable G, with period G n = 2πn/a! This shouldn’t come as a surprise though: the FT of a periodic function is a discrete function, the FT of a discrete function is periodic. It is easy to verify that the potential in real space an be obtained back from its FT as follows: V (x) = n−1 ∑ j=0 yielding the inverse FT in discretized form: f j e iG jx , (9.20) V j = n−1 ∑ m=0 f m exp[2π jm n i]. (9.21) The two operations of Eq.(9.19) and (9.19) are called Discrete Fourier Transform, or DFT (not to be confused with Density-Functional Theory!). One may wonder where have all the G vectors with negative values gone: after all, we would like to calculate f j for all j such that |G j | 2 < E c (for some suitably chosen value if E c ), not for G j with j ranging from 0 to n − 1. The periodicity of DFT in both real and reciprocal space however allows to refold the G j on the “right-hand side of the box”, so to speak, to negative G j values, by making a translation of 2πn/a. 9.2 Code: periodicwell Let us now move to the practical solution of a “true”, even if model, potential: the periodic potential well, known in solid-state physics since the thirties under the name of Kronig-Penney model: V (x) = ∑ n v(x − na), v(x) = −V 0 |x| ≤ b 2 , v(x) = 0 |x| > b 2 (9.22) and of course a ≥ b. Such model is exactly soluble in the limit b → 0, V 0 → ∞, V 0 b →constant. The needed ingredients for the solution in a plane-wave basis set are almost all already found in Sec.(4.3) and (4.4), where we have shown the numerical solution on a plane-wave basis set of the problem of a single potential well. Code periodicwell.f90 1 (or periodicwell.c 2 ) is in fact a trivial extension of code pwell. Such code in fact uses a plane-wave basis set like the one in Eq.(9.10), which means that it actually solves the periodic Kronig-Penney model for k = 0. If we increase the size of the cell until this becomes large with respect to the dimension of the single well, then we solve the case of the isolate potential well. The generalization to the periodic model only requires the introduction of the Bloch vector k. Our base is given by Eq.(9.10). In order to choose when to 1 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/periodicwell.f90 2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/periodicwell.c 72
truncate it, it is convenient to consider plane waves up to a maximum (cutoff) kinetic energy: ¯h 2 (k + G n ) 2 ≤ E cut . (9.23) 2m Bloch wave functions are expanded into plane waves: ψ k (x) = ∑ n c n b n,k (x) (9.24) and are automatically normalized if ∑ n |c n| 2 = 1. The matrix elements of the Hamiltonian are very simple: H ij = 〈b i,k |H|b j,k 〉 = δ ij ¯h 2 (k + G i ) 2 2m + 1 √ aṼ (G i − G j ), (9.25) where Ṽ (G) is the Fourier transform of the crystal potential. Code pwell may be entirely recycled and generalized to the solution for Bloch vector k. It is convenient to introduce a cutoff parameter E cut for the truncation of the basis set. This is preferable to setting a maximum number of plane waves, because the convergence depends only upon the modulus of k +G. The number of plane waves, instead, also depends upon the dimension a of the unit cell. Code periodicwell requires in input the well depth, V 0 , the well width, b, the unit cell length, a. Internally, a loop over k points covers the entire BZ (that is, the interval [−π/a, π/a] in this specific case), calculates E(k), writes the lowest E(k) values to files bands.out in an easily plottable format. 9.2.1 Laboratory • Plot E(k), that goes under the name of band structure, or also dispersion. Note that if the potential is weak (the so-called quasi-free electrons case), its main effect is to induce the appearance of intervals of forbidden energy (i.e.: of energy values to which no state corresponds) at the boundaries of the BZ. In the jargon of solid-state physics, the potential opens a gap. This effect can be predicted on the basis of perturbation theory. • Observe how E(k) varies as a function of the periodicity and of the well depth and width. As a rule, a band becomes wider (more dispersed, in the jargon of solid-state physics) for increasing superposition of the atomic states. • Plot for a few low-lying bands the Bloch states in real space (borrow and adapt the code from pwell). Remember that Bloch states are complex for a general value of k. Look in particular at the behavior of states for k = 0 and k = ±π/a (the “zone boundary”). Can you understand their form? 73
Page 1 and 2:
Lecture notes Numerical Methods in
Page 3 and 4:
3.3 Code: crossection . . . . . . .
Page 5 and 6:
Introduction The aim of these lectu
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is an important and well-known libr
Page 9 and 10:
Chapter 1 One-dimensional Schrödin
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Under the assumption of Eq.(1.8), E
Page 13 and 14:
point its value is still ∼ 60% of
Page 15 and 16:
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
Page 23 and 24:
In the following we will use q 2 e
Page 25 and 26: The dominating term close to the nu
Page 27 and 28: 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: Fast Fourier Transform In the simpl
Page 79 and 80: A reciprocal lattice of vectors G m
Page 81 and 82: The origin of the coordinate system
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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