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190 APPENDIX A. DETERMINANTS AND PFAFFIANS IN VMC
Moreover, we can further order the a i,j terms with respect to the index i :
|φ〉 = ∑ ( )
∑
a(R 1 ,P(j 1 ))...a(R N ,P(j N ′ ))(−1)sign(P) (−1) sign(P′ )
J I
c † R 1 ↑ ...c† R N ↑ c† R ′ 1 ↓.....c† R ′ N ↓ |0〉
(A.6)
Using that j k = P ′ (R k ′ ), we get:
|φ〉 = ∑ J
( ∑
I
a(R 1 ,P(P ′ (R ′ 1)))...a(R N ,P(P ′ (R ′ N)))(−1) sign(P) (−1) sign(P′ )
)
c † R 1 ↑ ...c† R N ↑ c† R ′ 1 ↓.....c† R ′ N ↓ |0〉
(A.7)
This leads to the final result that the projection of the state |φ〉 on the configuration
〈α| is a determinant:
〈α|φ〉 =
∑ P
a(R 1 , P(R ′ 1 ))...a(R N, P(R ′ N ))(−1)sign(P)
= det ( {a(R i ,R ′ j )})
This calculation can be extended to the case of a polarized fermionic configuration:
|α〉 = c † R 1 σ R1
...c † R 2N σ R2N
|0〉
(A.8)
And the terms in the wave-function |φ〉 that are not killed by the projection are
given by:
=
|φ〉 =
∑
{i 1 ...i 2N }
∑
{{i 1