CASINO manual - Theory of Condensed Matter

18 Wave-function updating
Consider the Slater wave function
Ψ S (R) = D ↑ (r 1 , . . . , r N↑ )D ↓ (r N↑ +1, . . . , r N ) , (83)
where D ↑ and D ↓ are Slater determinants for the spin-up and spin-down electrons respectively. We
will need to calculate the ratio of the new wave function to the old when, for example, the ith spin-up
electron is moved from r old
i to r new
i . The wave-function ratio can be written as
q ↑ = D↑ (r 1 , r 2 , . . . , r new
i , . . . , r N )
D ↑ (r 1 , r 2 , . . . , r old
i , . . . , r N ) . (84)
A direct calculation of the determinants in q ↑ at every move by LU decomposition is time-consuming,
and instead we use an updating method. We define the Slater matrix D ↑ via
D ↑ jk = ψ j(r k ) , (85)
where ψ j is the jth one-electron orbital of the spin-up Slater determinant and r k is the position of
the kth spin-up electron. The transpose of the inverse of D ↑ , which we call D ↑ , may be expressed in
terms of the cofactors and determinant of D ↑ ,
D ↑ jk = cof(D↑ jk )
det(D ↑ ) . (86)
The move of electron i changes only the ith column of D ↑ and so does not affect any of the cofactors
associated with this column. The new Slater determinant may be expanded in terms of these cofactors
and the result divided by the old Slater determinant to obtain
q ↑ = det(D↑,new )
det(D ↑,old ) = ∑ j
ψ j (r new
i )D ↑,old
ji . (87)
If the D ↑ matrix is known, one can compute q ↑ in a time proportional to N.
Evaluating D ↑ using LU decomposition takes of order N 3 operations; but once the initial D ↑ matrix
has been calculated it can be updated at a cost proportional to N 2 using the formulae
in the case where k = i, and
when k ≠ i.
D ↑,new
jk
= D ↑,old
jk
D ↑,new
ji
= 1 q ↑ D↑,old ji , (88)
− 1 q ↑ D↑,old ji
∑
m
ψ m (r new
i )D ↑,old
mk , (89)
19 Evaluating the local energy
The local energy is given by
E L (R) =
N∑
− 1 N∑
N∑
2 Ψ−1 (R)∇ 2 i Ψ(R)+ V (R)+ Ψ −1 ps
(R) ˆV
nl,i Ψ(R)+V CPP(R)+
i=1
i=1
i=1
N∑
v e−e (R) , (90)
where the terms are the kinetic energy, the local part of the external potential energy, the nonlocal part
of the potential energy, the core polarization potential energy (if present) and the electron–electron
interaction energy. The evaluation of the kinetic energy is discussed in Sec. 19.1. The evaluation of
the nonlocal energy is discussed in Sec. 19.2, while the core-polarization potential energy is discussed
in Sec. 19.3. The local part of the external potential energy is divided into a short range part around
each ion, which is evaluated directly, and a long range Coulomb part which is evaluated using the
Ewald potential in periodic systems (see Sec. 19.4) or simply as a sum of 1/r potentials in finite
systems. The electron–electron interaction energy is evaluated either using the Ewald interaction or
the MPC interaction (see Sec. 19.4.4) in periodic systems, or simply as a sum of 1/r potentials in
finite systems.
i>j
137