CASINO manual - Theory of Condensed Matter

Suppose the ground-state trial wave function Ψ can be written as the product of a part involving
long-ranged two-body correlations u αβ and a part consisting of everything else, Ψ s [17, 14]:
⎛
∑N s
Ψ(R) = Ψ s (R) exp ⎝
∑N s
N β
α=1 β=α+1 i=1 j=1
∑N α
∑
∑N s
u αβ (r iα − r jβ ) +
α=1
N∑
α−1
i=1
∑N α
j=i+1
⎞
u αα (r iα − r jα ) ⎠ , (238)
where u αβ (r) has the periodicity of the simulation cell and u αβ (r) = u αβ (−r). Note that, throughout
this section, we use u to denote the two-body Jastrow factor; in fact this is the sum of the u and p
terms in casino’s Jastrow factor.
Let the Fourier transform of u αβ be
∫
ũ αβ (G) =
where the {G} are the simulation-cell reciprocal lattice vectors. Let
Ω
u αβ (r) exp(−iG · r) dr, (239)
∑N α
˜ρ α (G; R) = exp(−iG · r iα ) (240)
i=1
be the Fourier transform of the density operator ρ α (r; R) = ∑ N α
i=1 δ(r − r iα). Then
⎛
Ψ(R) = Ψ s (R) exp ⎝ 1 N∑
s−1
∑N s
∑
ũ αβ (G)˜ρ ∗
Ω
α(G; R)˜ρ β (G; R)
α=1 β=α+1 G
)
+ 1 ∑N s
∑N α
∑
ũ αα ˜ρ ∗
2Ω
α(G; R)˜ρ β (G; R) − 1 ∑N s
∑
N α ũ αα (G)
2Ω
α=1 i=1 G
α=1 G
⎛
⎞
= Ψ s (R) exp ⎝ 1 ∑N s
∑N s
∑
ũ αβ (G)˜ρ ∗
2Ω
α(G; R)˜ρ β (G; R) + K⎠ , (241)
where K is independent of R.
α=1 β=1 G≠0
If we assume that only electrons are present, the ‘TI’ kinetic-energy estimator [11] may be written as
T (R) = −1
4 ∇2 log(Ψ) = T s (R) − 1
8Ω
where T s = −∇ 2 log(Ψ s (R))/4 [14].
It can easily be shown that
∑N s N s
∑ ∑
ũ αβ (G)∇ 2 [˜ρ ∗ α(G; R)˜ρ β (G; R)] , (242)
α=1 β=1 G≠0
∇ 2 [˜ρ ∗ α(G; R)˜ρ β (G; R)] = −2|G| 2 [˜ρ ∗ α(G; R)˜ρ β (G; R) − N α δ αβ ] . (243)
Hence the kinetic energy is
⎛
⎞
〈T (R)〉 = 〈T s 〉 + 1 ∑ ∑N s
∑N s
∑N s
|G| 2 ⎝ ũ αβ (G) 〈˜ρ ∗
4Ω
α(G; R)˜ρ β (G; R)〉 − N α ũ αα (G) ⎠ . (244)
G≠0
α=1 β=1
The Fourier transform of the translationally averaged structure factor is
˜S αβ (k) = 1 N
α=1
(〈˜ρα (k; R)˜ρ ∗ β(k; R) 〉 − 〈˜ρ α (k; R)〉 〈˜ρ ∗ β(k; R) 〉) . (245)
The charge density has the periodicity of the primitive lattice and therefore 〈˜ρ α (k; R)〉 is only nonzero
for G vectors of the primitive lattice. In particular, the second term in the Fourier-transformed
structure factor is zero for small k, which is the regime of relevance here. Hence
〈T 〉 = 〈T s 〉 + N 4Ω
∑ ∑N s
∑N s
|G| 2
G≠0
α=1 β=1
ũ αβ (G) ˜S ∗ αβ(G) − 1
4Ω
∑ ∑N s
|G| 2 N α ũ αα (G). (246)
G≠0
α=1
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