CASINO manual - Theory of Condensed Matter

The operations of rotational averaging and Fourier transforming commute because
∫
∫
1
4πk 2 f(r) e ik′·r δ(|k ′ | − k) dk ′ 1
dr =
Ω
4πr 2 Ω f(r′ ) δ(|r ′ | − r) e ik·r dr ′ dr. (297)
In practise, two-point correlation functions such as the pair correlation function and density matrix
are calculated by summing over the contributions from pairs of particles whose separation is defined
by the minimum image convention. We accumulate contributions from all pairs of particles whose
separation is less than or equal to the radius L WS of the sphere inscribed within the WS cell of the
simulation cell. We then accumulate
˜f WS (k) = 1 ∫
f(r) Θ(L WS − r) e ik·r dr (298)
Ω
= 1 Ω
∫ ∞
0
f(r) Θ(L WS − r) 4πr 2 sin(kr)
kr
dr, (299)
which is the convolution of ˜f(k) with the Fourier transform of the Heaviside function. Rather than
perform the deconvolution it is probably better just to calculate the translational average in reciprocal
space on the reciprocal lattice vectors of the simulation cell, averaging over the length of the vectors
as the calculation proceeds.
A spherical average can be performed to obtain
˜f T (G) = 1 ∑N G
N G
where N G is the number of reciprocal lattice vectors of length G.
accumulated as the calculation proceeds.
i=1
˜f T (G i ), (300)
The spherical average can be
33.2 Density and spin density
K eywords: density, spin density
33.2.1 Periodic systems
The charge density operator for spin α is
ρ α (r) = ∑ i,α
δ(r − r i,α ). (301)
The Fourier transform of the charge density operator is
˜ρ α (k) = 1 ∫ ∑
e ik·r δ(r − r i,α ) dr (302)
Ω
i,α
= 1 ∑
e ik·ri,α . (303)
Ω
The charge density for spin α is
n α (r) =
i,α
∫
|Ψ| 2 ρ α (r) dR
∫
|Ψ|2 dR
(304)
=
∫
|Ψ|
2 ∑ i,α δ(r − r i,α) dR
∫ . (305)
|Ψ|2 dR
The Fourier transform of the charge density is
ñ α (k) = 1 ∫
Ω
n α (r) e ik·r dr. (306)
So the charge density may be accumulated by summing exp(iG · r) for each primitive-cell G vector
after each single electron move from r ′ −→ r. At the end of the simulation, we then divide by the
total weight (e.g., the number of accumulation steps in VMC) to get the average of each n(G). The
Fourier coefficients are normalized such that n(G = 0) is the number of electrons in the primitive cell
(which is obtained from the above average n(G) through division by the number of primitive cells in
the simulation cell).
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