Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

More documents

Recommendations

Info

Chapter 9 Electrons in a periodic potential The computation of electronic states in a solid is a nontrivial problem. A great simplification can be achieved if the solid is a crystal, i.e. if it can be described by a regular, periodic, infinite arrangement of atoms: a crystal lattice. In this case, under suitable assumptions, it is possible to re-conduct the solution of the many-electron problem (really many: O(10 23 )!) to the much simpler problem of an electron under a periodic potential. Periodicity can be mathematically formalized in a simple and general way in any number of dimensions, but in the following we will consider a one-dimensional model case, that still contains the main effects of periodicity on electrons. 9.1 Crystalline solids Let us consider an infinite periodic system of “atoms”, that will be represented by a potential, centered on the atomic position. This potential will in some way – why and how being explained in solid-state physics books – the effective potential (or crystal potential) felt by an electron in the crystal. We will consider only valence electrons, i.e. those coming from outer atomic shells. Core electrons, i.e. those coming from inner atomic shells, are tightly bound to the atomic nucleus: their state is basically atomic-like and only marginally influenced by the presence of other atoms. We assume that the effects of core electrons can be safely included into the crystal potential. The pseudopotential approach formalizes the neglect of core electrons. The assumption that core electrons do not significantly contribute to the chemical binding and that their state does not significantly change with respect to the case of isolated atoms is known as frozen-core approximation. This is widely used for calculations in molecules as well and usually very well verified in practice. We also consider independent electrons, assuming implicitly that the crystal potential takes into account the Coulomb repulsion between electrons. The aim of such simplification is to obtain an easily solvable problem that still captures the essence of the physical problem. With a judicious choice of the crystal 66
potential, we can hope to obtain a set of electronic levels that can describe the main features of the crystal. A rigorous basis for such approach can be provided by Hartree-Fock or Density-Functional theory. In the end, the basic step is to solve to the problem of calculating the energy levels in a periodic potential. We haven’t yet said anything about the composition and the periodicity of our system. Let us simplify further the problem and assume a one-dimensional array of atoms of the same kind, regularly spaced by a distance a. The atomic position of atom n will thus be given as a n = na, with n running on all integer numbers, positive and negative. In the jargon of solid-state physics, a is the lattice parameter, while the a n are the vectors of the crystal lattice. The system has a discrete translational invariance, that is: it is equal to itself if translated by a or multiples of a. Called V (x) the crystal potential, formed by the superposition of atomic-like potentials: V (x) = ∑ n V n(x − a n ), the following symmetry holds: V (x + a) = V (x). Such symmetry plays a very important role in determining the properties of crystalline solids. Our one-dimensional space (the infinite line) can be decomposed into finite regions (segments) of space, of length a, periodically repeated. A region having such property is called unit cell, and the smallest possible unit cell is called primitive cell. Its definition contains some degree of arbitrarity: for instance, both intervals [0, a[ and ] − a/2, +a/2] define a valid primitive cell in our case. 9.1.1 Periodic Boundary Conditions Before starting to look for a solution, we must ask ourselves how sensible it is to apply such idealized modelling to a real crystal. The latter is formed by a macroscopically large (in the order of the Avogadro number or fractions of it) but finite number of atoms. We might consider instead a finite system containing N atoms with N → ∞, but this is not a convenient way: translational symmetry is lost, due to the presence of surfaces (in our specific 1D case, the two ends). A much more convenient and formally correct approach is to introduce periodic boundary conditions (PBC). Let us consider the system in a box with dimensions L = Na and let us consider solutions obeying to the condition ψ(x) = ψ(x + L), i.e. periodic solutions with period L >> a. We can imagine our wave function that arrives at one end “re-enters” from the other side. In the one-dimensional case there is a simple representation of the system: our atoms are distributed not on a straight line but on a ring, with atom N between atom N − 1 and atom 1. The advantage of PBC is that we can treat the system as finite (a segment of length L in the one-dimensional case) but macroscopically large (having N atoms, with N macroscopically large if a is a typical interatomic distance and L the typical size of a piece of crystal), still retaining the discrete translational invariance. Case N → ∞ describes the so-called thermodynamical limit. It is to be noticed that a crystal with PBC has no surface. As a consequence there is no “inside” and “outside” the crystal: the latter is not contemplated. This is the price to pay for the big advantage of being able to use translational symmetry. In spite of PBC and of translational symmetry, the solution of the Schrödinger equation for a periodic potential does not yet look like a simple problem. 67
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
Page 7 and 8:
is an important and well-known libr
Page 9 and 10:
Chapter 1 One-dimensional Schrödin
Page 11 and 12:
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
Page 17 and 18:
eigenvalue) to force the code to pe
Page 19 and 20: 1.3.3 Laboratory Here are a few hin
Page 21 and 22: 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: Verify how sensitive the result is
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 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
show all

Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?