CASINO manual - Theory of Condensed Matter

33.3.3 Homogeneous and isotropic systems
Suppose we specialize to a homogeneous and isotropic system where g αβ (r, r ′ ) = g αβ (|r−r ′ |) = g αβ (r),
and n α (r) = N α /Ω, where Ω is the volume of the simulation cell. Performing a translational average
of g αβ we obtain
g αβ (r) = 1 ∫
g αβ (r ′′ , r ′′′ ) δ(r ′′ − r ′′′ − r) dr ′′ dr ′′′ (340)
Ω
=
Ω
N α N β
∫
|Ψ|
2 ∑ i,α
∑
j,β≠i,α δ(r i,α − r j,β − r) dR
∫ . (341)
|Ψ|2 dR
Performing a rotational average we obtain
∫
1
g αβ (r) =
4πr 2 g αβ (r ′ ) δ(|r ′ | − r) dr ′ (342)
∫
Ω |Ψ|
2
∑ ∑
i,α j,β≠i,α
=
δ(|r i,α − r j,β | − r) dR
∫
4πr 2 . (343)
N α N β |Ψ|2 dR
The rotationally and translationally averaged pair correlation functions could be evaluated by collecting
in bins, see Sec. 33.3.5.
The sum rule of Eq. (339) gives
N β
Ω
33.3.4 Translational and rotational averaging of g αβ (r, r ′ )
∫
[g αβ (r) − 1] 4πr 2 dr = −δ αβ . (344)
The translational average of g αβ (r, r ′ ) is
gαβ(r) T = 1 ∫
g αβ (r ′′ , r ′′′ ) δ(r ′′ − r ′′′ − r) dr ′′ dr ′′′ (345)
Ω
= 1 ∫ |Ψ|
2 ∑ i,α δ(r i,α − r ′′ ) ∑ j,β≠i,α δ(r j,β − r ′′′ ) δ(r ′′ − r ′′′ − r) dr ′′ dr ′′′
∫ dR (346)
Ω |Ψ|2 dR n α (r ′′ ) n β (r ′′′ )
= 1 ∫ |Ψ|
2 ∑ ∑
i,α j,β≠i,α δ(r i,α − r j,β − r)
∫ dR. (347)
Ω |Ψ|2 dR n α (r i,α ) n β (r j,β )
The rotational average of gαβ T (r) is
g T R
αβ (r) =
=
1
4πr 2 Ω
1
4πr 2 Ω
∫
gαβ(r T ′ ) δ(|r ′ | − r) dr ′ (348)
∫ |Ψ|
2 ∑ ∑
i,α j,β≠i,α δ(|r i,α − r j,β | − r)
∫ dR. |Ψ|2 dR n α (r i,α ) n β (r j,β )
(349)
As well as the above averages one can calculate the pair correlation g αβ (r, r ′ ) where an electron of
spin α is fixed at a particular position r. Suppose we write g αβ (r ′ , r ′ ) as
∫
|Ψ|
2 ∑
g αβ (r, r ′ i,α
) =
δ(r i,α − r) ∑ j,β≠i,α δ(r j,β − r ′ ) dR
∫ ∑ |Ψ|
2
i,α δ(r (350)
i,α − r) dR n β (r ′ )
= N ∫
α(N β − δ αβ ) |Ψ| 2 δ(r 1,α − r)δ(r 2,β − r ′ ) dR
∫ . (351)
N α |Ψ|2 δ(r 1,α − r) dR n β (r ′ )
We now define the probability distribution |Ψ F | 2 as
∫
|Ψ F | 2 = |Ψ| 2 δ(r 1,α − r) dr 1,α . (352)
Writing Eq. (350) in terms of |Ψ F | 2 we obtain
∫
|Ψ
g αβ (r, r ′ F | 2 δ(r 2,β − r ′ ) dr 2 . . . dr N
) = (N β − δ αβ ) ∫
|ΨF | 2 dr 2 . . . dr N n β (r ′ )
(353)
= n αβ(r, r ′ )
n β (r ′ , (354)
)
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