CASINO manual - Theory of Condensed Matter

Note that this has the k −2 divergence predicted by the RPA [79]. If the value of ũ αα (k) is known at
a single point k then the parameter A α may be evaluated as
A α =
−k 2 ũ αα (k)
4π + c α k 4 ũ αα (k) . (259)
If one inserts this form for ũ αα into Eq. (247), noting that it is already spherically symmetric, then
one obtains
(
∑N s
N α Q
(√ ∆T =
− tan−1 c α A α Q ) )
√ . (260)
2πm α c α c α cα A α
α=1
Making use of the Taylor expansion of tan −1 , it is found that the leading-order correction to the
kinetic energy is
∆T =
∑N s
α=1
πN α A α
Ωm α
+ O(N −2/3 ), (261)
where the first term is independent of N, so the error in the kinetic energy per particle falls off as
O(N −1 ).
29.3.2 Application to the HEG
The Jastrow factor of Eq. (257) for a HEG of density parameter r s has A α = 1/ω p where ω p = √ 3/r 3 s is
the plasma frequency [11]. Note that Ω = 4πr 3 sN/3 = 4πN/ω 2 p, and m α = 1. Hence the leading-order
correction to the kinetic energy is ∆T = ω p /4, as found by Chiesa et al. [17].
29.3.3 A simpler fitting form for the long-ranged two-body Jastrow factor
Suppose
(
Aα
ũ αα (k) = −4π
k 2 + B )
α
, (262)
k
for small k, which has the divergent terms predicted by the RPA. Then Eq. (247) gives
[
∑N s
πN α
∆T =
A α + 3B ( )
α 6π
2 1/3
]
. (263)
m α Ω 4 Ω
α=1
29.4 Fitting form for the long-ranged two-body Jastrow factor (2D)
Suppose
ũ αα (k) = − a α
k 3/2 − b α
k , (264)
for small k, which has the k −3/2 divergence predicted by the RPA in 2D (see e.g. Ref. [80]). Then
∆T =
∑N s
α=1
(√
N α 2π 1/4 √ )
a α πbα
+
m α 5A 5/4 3A 3/2
(265)
where the leading term is O(N −1/4 ), so that the error in the kinetic energy per electron falls off as
O(N −5/4 ).
29.5 Applying the correction scheme in practice
The steps carried out by casino in order to calculate the kinetic-energy correction for a 3D system
are as follows:
1. Calculate the Fourier transformation of u αα (r) + p αα (r) in the Jastrow factor for each spin α
and perform spherical averaging over G vectors of equal length.
2. For each α, determine the parameter A αα by inserting the value of ũ αα + ˜p αα at the smallest
nonzero star of G vectors into Eq. (259).
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