CASINO manual - Theory of Condensed Matter

= 4π k
= 4π k
∫ Lu
0
∑N u
∑
C
α l
l=0 m=0
ru(r) sin(kr) dr
( ∫ C
Lu
u )
m)(−L C−m r m+l+1 sin(kr) dr. (252)
Let I n (k) = ∫ L u
0
r n sin(kr) dr. Then I 0 = [1 − cos(kL u )]/k and I 1 = −L u cos(kL u )/k + sin(kL u )/k 2 ,
and, for n ≥ 2,
I n (k) = 1 k
[ n
k
0
(
L
n−1
u sin(kL u ) − (n − 1)I n−2 (k) ) ]
− L n u cos(kL u ) . (253)
Hence we can rapidly evaluate I n (k) for all n required (that is, up to n = N u + C + 1).
For k = 0 we have
∑N u
ũ(0) = 4π
∑
C
α l
l=0 m=0
( C C−m Ll+m+3 u
u )
m)(−L
l + m + 3 . (254)
29.2.4 Fourier transformation of u in 2D
An analytic expression for ũ(k) is not available in two dimensions. We therefore evaluate ũ(k) numerically
using a fast Fourier transform.
29.2.5 Fourier transformation of u in 1D
ũ(k) =
∫ L/2
−L/2
u(|x|) exp(−ikx) dx
∑N u ∫ Lu
= 2 α l (x − L u ) C x l cos(kx) dx
l=0
∑N u
= 2
0
∑ C
α l
l=0 n=0
( C
n)
(−L u ) C−n J n+l (k), (255)
where L is the length of the simulation cell and J n = ∫ L u
x n cos(kx) dx. Suppose k ≠ 0. Then
0
J 0 (k) = sin(kL u )/k, J 1 (k) = L u sin(kL u )/k + [cos(kL u ) − 1]/k 2 and, for n ≥ 2
J n (k) = 1 k
[ n
k
(
L
n−1
u cos(kL u ) − (n − 1)J n−2 (k) ) ]
+ L n u sin(kL u ) . (256)
Hence we can rapidly evaluate J n (k) for all n required (from n = 0 to n = N u + C). If k = 0 then
J n = L n+1
u /(n + 1).
29.3 Fitting form for the long-ranged two-body Jastrow factor (3D)
29.3.1 The ‘RPA-Kato’ Jastrow factor
Consider the infinite-system ‘RPA-Kato’ two-body Jastrow factor [11] for pairs of particles of type α,
which satisfies the Kato cusp condition and has the long-ranged 1/r decay predicted by the random
phase approximation (RPA) [79],
u αα (r) = − A α
r [1 − exp(−r/F α)] , (257)
where A α is a free parameter and Fα 2 = c α A α is determined by the cusp conditions, where c α =
2/(qαm 2 α ). The Fourier transformation of this two-body Jastrow factor is
( )
1
ũ αα (k) = −4πA α
k 2 − 1
k 2 . (258)
+ 1/c α A α )
176