CASINO manual - Theory of Condensed Matter

The spin-density matrix is a two-by-two matrix which is the generalization of the spin density in a
noncollinear-spin system (see Sec. 36). It is diagonal for the case of collinear spins, with the two
diagonal elements being equal to the spin densities.
Let x i ≡ {r i , s i }, where r i and s i are the spatial and spin coordinates of electron i.
notation ∫ f(x) dx ≡ ∑ ∫
s f({r, s}) dr.
The spin-density matrix is defined as
We use the
ρ sdm (r; s, s ′ ) = N ∫ Ψ ∗ ({r, s ′ }, x 2 , . . . , x N )Ψ({r, s}, x 2 , . . . , x N ) dx 2 . . . dx
∫ N
. (320)
|Ψ|2 dx 1 . . . dx N
Note that ρ sdm (r; s, s ′ ) = ρ ∗ sdm (r; s′ , s).
The diagonal elements of the spin-density matrix, which are real, give the spin density:
The magnetization density operator is defined as
ρ s (r) = ρ sdm (r; s, s). (321)
ˆM(r) = 2 ∑ i
δ(r − r i )ŝ i , (322)
where ŝ i is the spin operator for electron i. Let M(r) = 〈 ˆM(r)〉.
So
M z (r) = 2 ∑ i
∫
Ψ ∗ δ(r − r i )ŝ iz Ψ dx 1 . . . dx N
∫
|Ψ|2 dx 1 . . . dx N
= 2N ∑ ∫
s 1 Ψ ∗ ({r, s 1 }, x 2 , . . . , x N )ŝ 1z Ψ({r, s 1 }, x 2 , . . . , x N ) dx 2 . . . dx N
∫
|Ψ|2 dx 1 . . . dx N
∫
|Ψ({r, ↑}, x2 , . . . , x N )| 2 dx 2 . . . dx N − ∫ |Ψ({r, ↓}, x 2 , . . . , x N )| 2 dx 2 . . . dx N
= N
∫
|Ψ|2 dx 1 . . . dx N
= ρ sdm (r; ↑, ↑) − ρ sdm (r; ↓, ↓) = ρ ↑ (r) − ρ ↓ (r). (323)
We also have
M x (r) = 2 ∑ ∫
i Ψ ∗ δ(r − r i )ŝ ix Ψ dx 1 . . . dx
∫ N
|Ψ|2 dx 1 . . . dx N
∑ ∫
s
= N 1 Ψ ∗ ({r, s 1 }, x 2 , . . . , x N )[ŝ 1+ + ŝ 1− ]Ψ({r, s 1 }, x 2 , . . . , x N ) dx 2 . . . dx N
∫
|Ψ|2 dx 1 . . . dx N
[∫
Ψ ∗ ({r, ↑}, x 2 , . . . , x N )Ψ({r, ↓}, x 2 , . . . , x N ) dx 2 . . . dx N
= N
∫
|Ψ|2 dx 1 . . . dx N
+ ∫ Ψ ∗ ]
({r, ↓}, x 2 , . . . , x N )Ψ({r, ↑}, x 2 , . . . , x N ) dx 2 . . . dx
∫ N
|Ψ|2 dx 1 . . . dx N
= ρ sdm (r; ↓, ↑) + ρ sdm (r; ↑, ↓) = 2Re [ρ sdm (r; ↓, ↑)] , (324)
where we have used the raising and lowering properties of the ladder operators ŝ + = ŝ x + iŝ y and
ŝ − = ŝ x − iŝ y . In a similar fashion,
To evaluate the spin-density matrix in VMC, we rewrite it as
∫
|Ψ| 2 δ(r − r i ) δ si,s ′
ρ sdm (r; s, s ′ ) = ∑ i
〈 ∑
=
i
M y (r) = 2Im [ρ sdm (r; ↓, ↑)] . (325)
Ψ(...,{r i,s},...)
Ψ(...,{r i,s ′ },...) dx 1 . . . dx N
∫
|Ψ|2 dx 1 . . . dx N
〉
Ψ(. . . , {r i , s}, . . .)
δ(r − r i ) δ si,s ′ Ψ(. . . , {r i , s ′ . (326)
}, . . .)
In practice we gather the Fourier components of the spin-density matrix, which are
〈 〉
˜ρ sdm (G; s, s ′ ) = 1 ∑ Ψ(. . . , {r i , s}, . . .)
exp(−iG · r i ) δ si,s
Ω
′ Ψ(. . . , {r
i
i , s ′ , (327)
}, . . .)
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