CASINO manual - Theory of Condensed Matter

33.4 Structure factor and spherically averaged structure factor
K eywords: structure factor, struc factor sph
One may define the following correlation function,
S ′ αβ(r, t; r ′ , t ′ ) = 〈Ψ|ρ α (r, t) H ρ β (r ′ , t ′ ) H |Ψ〉, (363)
where ρ is the density operator, and the subscript H denotes that we are using the Heisenberg picture.
In fact one normally considers a modified correlation function
S αβ (r, t; r ′ , t ′ ) = 〈Ψ|ρ α (r, t) H ρ β (r ′ , t ′ ) H |Ψ〉 − n α (r)n β (r ′ ), (364)
because S tends to zero as |t − t ′ | → ∞, so that the Fourier transform of S exists in the time domain.
S is known as the structure factor. We are mostly interested in the static structure factor, which is
Eq. (364) evaluated at equal times,
S αβ (r, r ′ ) = 〈Ψ|ρ α (r) H ρ β (r ′ ) H |Ψ〉 − n α (r)n β (r ′ ), (365)
which has the form of a covariance. The covariance might show a lower variance than the individual
terms in Eq. (365), so it might be better to evaluate them together. Note that as |r − r ′ | → ∞ we
expect the electrons to be uncorrelated, so that in this limit S αβ (r, r ′ ) → 0. The static structure
factor can be written as
∫
|Ψ|
2 ∑
S αβ (r, r ′ i,α
) =
δ(r i,α − r) ∑ j,β δ(r j,β − r ′ ) dR
∫ − n α (r)n β (r ′ ). (366)
|Ψ|2 dR
33.4.1 Relationship between S αβ and g αβ
The static structure factor is related to the pair correlation function by
S αβ (r, r ′ ) = n α (r)n β (r ′ ) [g αβ (r, r ′ ) − 1] + n α (r)δ αβ δ(r − r ′ ), (367)
see Dreizler and Gross, page 276. Using Eq. (339) we find that S αβ (r, r ′ ) satisfies
∫
S αβ (r, r ′ ) dr dr ′ = 0. (368)
The Fourier transform of S αβ is
˜S αβ (k, k ′ ) = 1 Ω 2 ∫
S αβ (r, r ′ ) e ik·r e ik′·r ′ dr dr ′ (369)
= 〈˜ρ α (k)˜ρ β (k ′ )〉 − ñ α (k)ñ β (k ′ ). (370)
The translational average of S αβ (r, r ′ ) is
Sαβ(r) T = 1 ∫
S αβ (r ′′ , r ′′′ ) δ(r ′′ − r ′′′ − r) dr ′′ dr ′′′ (371)
Ω
= 1 ∫
|Ψ|
2 ∑ ∑
i,α j,β δ(r i,α − r j,β − r) dR
∫ − 1 ∫
n α (r ′ )n β (r ′ − r) dr ′ . (372)
Ω
|Ψ|2 dR
Ω
The first term in Eq. (372) is the translational average of n α (r)n β (r ′ )g αβ (r, r ′ ) + n α (r)δ αβ δ(r − r ′ ),
while the second is the translational average of −n α (r)n β (r ′ ). Sαβ T (r) satisfies
∫
Sαβ(r) T dr = 0. (373)
The Fourier transform of Sαβ T (r) is
˜S T αβ(k) = 1 Ω
∫
S T αβ(r) e ik·r dr (374)
= 1 Ω 2 〈˜ρ α(k)˜ρ β (−k)〉 − 1 Ω 2 ñα(k)ñ β (−k) (375)
= 1 Ω 2 ˜S αβ (k, −k), (376)
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