CASINO manual - Theory of Condensed Matter

where G is a reciprocal lattice vector of the simulation cell.
After each configuration move, we can evaluate a contribution to two out of the four components of
˜ρ sdm for each electron. Given that the ith electron has spin value s i , we can record ˜ρ sdm (G; s, s i ) for
both values of s by evaluating the ratio of the wave functions for those two values.
Casino evaluates the Fourier components of the spin-density matrix if the spin density keyword is
set to T in a noncollinear-spin calculation. The Fourier components are stored in the expval.data
file. The plot expval utility reads the Fourier components in the expval.data file and enables the
user to plot the components of the spin-density matrix along lines in real space. It also asks the user
if he or she wishes to plot the components of the magnetization density along lines in real space.
33.3 Reciprocal-space and spherical real-space pair-correlation functions
K eywords: pair corr, pair corr sph
The two-particle density matrix is defined as
∫
γ (2)
Ψ
αβ (r, r′ ; r ′′ , r ′′′ ∗ (r, r ′ , r 3 , . . . , r N )Ψ(r ′′ , r ′′′ , r 3 , . . . , r N ) dr 3 . . . dr N
) = N α (N β − δ αβ ) ∫ , (328)
Ψ∗ (r 1 , r 2 , . . . , r N )Ψ(r 1 , r 2 , . . . , r N ) dR
where r and r ′′ are α-spin electron coordinates and r ′ and r ′′′ are β-spin electron coordinates. The
diagonal elements of the two-particle density matrix,
∫
γ (2)
Ψ
αβ (r, r′ ; r, r ′ ∗ (r, r ′ , r 3 , . . . , r N )Ψ(r, r ′ , r 3 , . . . , r N ) dr 3 . . . dr N
) = N α (N β − δ αβ ) ∫ , (329)
Ψ∗ (r 1 , r 2 , . . . , r N )Ψ(r 1 , r 2 , . . . , r N ) dR
are of special interest. The normalization has been chosen so that
∫
γ (2)
αβ (r, r′ ; r, r ′ ) dr dr ′ = N α (N β − δ αβ ). (330)
The sum over spin indices gives
where N = N α + N β .
∑
N α (N β − δ αβ ) = N(N − 1), (331)
αβ
The pair correlation functions, g αβ (r, r ′ ), are related to the diagonal elements of the two-particle
density matrix by
γ (2)
αβ (r, r′ ; r, r ′ ) = n α (r) n β (r ′ ) g αβ (r, r ′ ). (332)
The pair correlation functions are given by
g αβ (r, r ′ ) =
1
n α (r)n β (r ′ )
= N α(N β − δ αβ )
n α (r)n β (r ′ )
∫
|Ψ|
2 ∑ i,α δ(r i,α − r) ∑ j,β≠i,α δ(r j,β − r ′ ) dR
∫ (333)
|Ψ|2 dR
∫
|Ψ| 2 δ(r i,α − r) δ(r j,β − r ′ ) dR
∫ . (334)
|Ψ|2 dR
33.3.1 The total or spin-averaged pair correlation function
g(r, r ′ ) = ∑ α,β
n α (r)n β (r ′ )
n(r)n(r ′ )
g αβ (r, r ′ ). (335)
33.3.2 Properties of the pair correlation functions
The pair correlation functions satisfy the following properties:
∫
g αβ (r, r ′ ) ≥ 0 (336)
g αβ (r, r ′ ) = g βα (r ′ , r) (337)
g αβ (r, r ′ ) = 0 for α = β, r = r ′ (338)
n β (r ′ ) [g αβ (r, r ′ ) − 1] dr ′ = −δ αβ . (339)
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