CASINO manual - Theory of Condensed Matter

and the Dirac delta function is
δ(r − r ′ ) = 1 ∑
e −iG·(r−r′) . (284)
Ω
The 3N-dimensional Fourier transform of the wave function is
∫
1
˜Ψ(k 1 , . . . , k N ) =
Ω N Ψ(r 1 , . . . , r N ) e ik1·r1 . . . e ik N ·r N
dr 1 . . . dr N (285)
∑
Ψ(r 1 , . . . , r N ) = ˜Ψ(k1 , . . . , k N ) e −ik1·r1 . . . e −ik N ·r N
. (286)
k 1,...,k N
G
33.1.2 Translational averaging
Consider a function f(r, r ′ ) whose Fourier transform is
˜f(k, k ′ ) = 1 Ω 2 ∫
f(r, r ′ ) e ik·r e ik′·r ′ dr dr ′ . (287)
The translational average of f(r, r ′ ) is
f T (r) = 1 Ω
∫
f(r ′′ , r ′′′ ) δ(r ′′ − r ′′′ − r) dr ′′ dr ′′′ . (288)
The Fourier transform of the translational average is
˜f T (k) = 1 ∫
f T (r) e ik·r dr (289)
Ω
= 1 Ω 2 ˜f(k, −k). (290)
Therefore the Fourier transform of f(r, r ′ ) and the Fourier transform of its translational average are
related by
˜f(k, −k) = Ω 2 ˜f T (k). (291)
The operations of translational averaging and Fourier transforming commute.
The function f(r, r ′ ) has the periodicity of the simulation cell,
f(r, r ′ + R) = f(r, r ′ ), (292)
where R is a translation vector of the simulation cell lattice. Equation (292) implies that f T (r) is
also periodic,
f T (r + R) = f T (r). (293)
The Fourier transforms of f T (r) and f(r, r ′ ) are therefore nonzero only on the reciprocal lattice vectors
of the simulation cell lattice.
33.1.3 Rotational averaging
The rotational average of a function is
f(r) =
∫
1
4πr 2
f(r ′ ) δ(|r ′ | − r) dr ′ . (294)
The rotationally averaged function f(r) is no longer periodic. The Fourier transform of the rotational
average is
˜f(k) = 1 ∫
f(r) e ik·r dr (295)
Ω
= 1 Ω
∫ ∞
0
f(r) 4πr 2 sin(kr)
kr
dr. (296)
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