CASINO manual - Theory of Condensed Matter

we see that the error due to this approximation is second order in (ρ − ρ A ), and in addition the
operator (v E − f) becomes very small as the size of the simulation cell goes to infinity. The error term
is therefore usually small and is neglected although it could be calculated after the simulation. We
use the MPC expressions of Eqs. (129) and (130) in both VMC and DMC calculations.
The first term of the Hamiltonian of Eq. (129) is evaluated in real space and the second term in
Fourier space. The third term is a constant which is evaluated in reciprocal space at the start of the
calculation. Introducing the Fourier transformed quantities,
f G = 1 ∫
f(r)e iG·r dr , (132)
Ω
ρ G = 1 Ω
∫
WS
WS
ρ(r)e iG·r dr , (133)
where Ω is the volume of the cell, and noting that the Fourier transform of the Ewald interaction is
4π/(ΩG 2 ), we have
Ĥ e−e = ∑ f(r i − r j ) + Ω ∑ ∑
[ ] 4π
ΩG 2 − f G ρ A,G e −iG·ri − Ω ∑ f G=0 ρ A,G=0 − C , (134)
i>j
i
i
G≠0
where
C = Ω2
2
∑
G≠0
[ ] 4π
ΩG 2 − f G ρ ∗ A,Gρ A,G − Ω2
2 f G=0ρ ∗ A,G=0ρ A,G=0 (135)
The calculation of f G is achieved using the following scheme developed by Randy Hood. The integrand
in Eq. (132) diverges at the origin and we separate out the divergent behaviour by writing
f(r) = g(r) + h(r) (136)
where
g(r) =
h(r) =
{ y(r) r < L
1/r r > L ,
{
1/r − y(r) r < L
0 r > L ,
(137)
(138)
(139)
and L is the radius of the largest sphere which is contained within the WS cell and y(r) is chosen to
be
y(r) = − r2
2L 3 + 3
2L , (140)
so that both g and h have continuous first derivatives at r = L. The Fourier transform of h(r) is
calculated analytically as
h G = 1 Ω
∫ L ∫ +1
0
= 4π
ΩG 2 +
from which the G = 0 value can be extracted as
−1
( 1
r + r2
2L 3 − 3 )
2πr 2 e iGrcosθ dcosθ dr
2L
]
12π [
ΩL 2 G 4 cosGL − sinGL
GL
, (141)
h G=0 = 2πL2
5Ω , (142)
The Fourier transform of g(r) must be evaluated numerically. The gradient of g(r) is discontinuous
at the boundary of the WS cell. The errors in the Fourier components obtained using the fast Fourier
transform (FFT) method therefore fall off slowly, going approximately as N −2
grid , where N grid is the
number of FFT grid points along each lattice vector. It is assumed that the FFT Fourier coefficients
of g satisfy
g N grid
G
= a G N −4
grid + b GN −2
grid + c G, (143)
where the {a G }, {b G } and {c G } are constants. Note that c G = g ∞ G , the value of the Fourier coefficient
in the limit of an infinite grid. FFTs are carried out at three different grid sizes in order to determine
the {g ∞ G }. 144