Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

Fast Fourier Transform
In the simple case that will be presented, the matrix elements of the Hamiltonian,
Eq.(9.13), can be analytically computed by straight integration. Another
case in which an analytic solution is known is a crystal potential written as a
sum of Gaussian functions:
This yields
V (x) =
N−1 ∑
p=0
〈b i,k |V |b j,k 〉 = 1 a
v(x − pa), v(x) = Ae −αx2 . (9.14)
∫ L/2
−L/2
Ae −αx2 e −iGx dx (9.15)
The integral is known (it can be calculated using the tricks and formulae given
in previous sections, extended to complex plane):
∫ ∞
√ π
e −αx2 e −iGx /4α
dx =
α e−G2 (9.16)
−∞
(remember that in the thermodynamical limit, L → ∞).
For a generic potential, one has to resort to numerical methods to calculate
the integral. One advantage of the plane-wave basis set is the possibility to
exploit the properties of Fourier Transforms (FT), in particular the Fast Fourier
Transform (FFT) algorithm.
Let us discretize our problem in real space, by introducing a grid of n points
x i = ia/n, i = 0, n − 1 in the unit cell. Note that due to periodicity, grid points
with index i ≥ n or i < 0 are “refolded” into grid points in the unit cell (that
is, V (x i+n ) = V (x i ), and in particular, x n is equivalent to x 0 . Let us introduce
the function f j defined as follows:
f j = 1 ∫
V (x)e −iGjx dx, G j = j 2π , j = 0, n − 1. (9.17)
L
a
Apart from factors, this is the usual definition of FT (and is nothing but matrix
elements). We can exploit periodicity to show that
f j = 1 a
∫ a
0
V (x)e −iG jx dx. (9.18)
Note that the integration limits can be translated at will, again due to periodicity.
Let us write now such integrals as a finite sum over grid points (with
∆x = a/n as finite integration step):
f j = 1 a
= 1 n
= 1 n
n−1 ∑
m=0
n−1 ∑
m=0
n−1 ∑
m=0
V (x m )e −iG jx m
∆x
V (x m )e −iG jx m
V m exp[−2π jm n i], V i ≡ V (x i). (9.19)
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