CASINO manual - Theory of Condensed Matter

In the infinite system limit, the sum over G should be replaced by an integral. The leading order
finite-size error is due to the omission of the G = 0 contribution in the third term in Eq. (246) [17].
Assuming that ũ αα (k) has the same form at different system sizes, the missing contribution, which is
our finite-size correction, is given by
∑N s
∫
N Q
α
∆T = −
4(2π) 3 4πk 2 × k 2 ū αα (k) dk (247)
m α
α=1
in three dimensions, where ū(k) is the spherical average of ũ(k) and Q is the radius of the sphere in
reciprocal space that has the same volume as that associated with each G vector, (2π) 2 /Ω. Hence
Q = (6π 2 /Ω) 1/3 . Note that we have also reintroduced the particles masses in Eq. (247).
The corresponding expression in two dimensions is
0
∆T = − ∑ α
∫
N Q
α
4(2π) 2 m α
0
2πk × k 2 ũ αα (k) dk, (248)
where Q = 2 √ π/A is the radius of the circle in reciprocal space that has the same area as that
associated with each G vector, (2π) 2 /A, where A is the area of the simulation cell.
In one dimension, the correction to the kinetic energy is
∆T = − ∑ α
∫
N Q
α
4(2π)m α
0
2k 2 ũ αα (k) dk, (249)
where Q = π/L is half the length associated with each G vector, 2π/L, where L is the length of the
simulation cell.
29.2 Fourier transformation of CASINO’s two-body Jastrow factor
29.2.1 The two-body Jastrow factor
casino’s two-body Jastrow factor consists of two terms: u and p [45]. For any given spin-type, u is
of the form
∑N u
u(r) = (r − L u ) C Θ(L u − r) α l r l , (250)
where L u , C and {α} are parameters, Θ is the Heaviside function, and r is the magnitude of electron–
electron distance evaluated within the minimum-image convention. (Throughout this section we omit
the spin indices α and β.)
The Fourier transformation of the two-body Jastrow factor is just the sum of the Fourier transforms
of u and p. Note that the Fourier transform of u is spherically symmetric, whereas spherical averaging
may have to be performed for the Fourier transform of p.
l=0
29.2.2 Fourier transformation of p
The p term in the Jastrow factor is
p(r) = ∑ ∑
a A cos(G A · r) = ∑
A
A
G + A
∑
G A
1
2 a A exp(iG A · r), (251)
where A denotes a set of symmetry-equivalent simulation-cell G vectors and ‘+’ means that, if G is
included in the sum, −G is excluded [45]. So the Fourier coefficient for G A is Ωa A /2 in 3D, Aa A /2
in 2D and La A /2 in 1D.
29.2.3 Fourier transformation of u in 3D
Suppose k ≠ 0. Then
ũ(k) =
∫
Ω
u(r) exp(−ik · r) dr
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