CASINO manual - Theory of Condensed Matter

19.1 Evaluating the kinetic energy
The kinetic part of the local energy, K, can be expressed as a sum of contributions from each electron,
N∑ N∑
K = K i = − 1 2 Ψ(R)−1 ∇ 2 i Ψ(R) . (91)
i=1
i=1
Because of the exponential form of the Jastrow factor, it is convenient to re-express K i in terms of
the logarithm of Ψ. We define
T i = − 1 4 ∇2 i (ln |Ψ|) = − 1 ∇ 2 i Ψ
4 Ψ + 1 4
( ) 2 ∇i Ψ
, (92)
Ψ
and the drift vector F i ,
Therefore
F i = 1 √
2
∇ i (ln |Ψ|) = 1 √
2
∇ i Ψ
Ψ . (93)
K i = 2T i − |F i | 2 . (94)
In VMC an integration by parts shows that
〈K〉 = 〈|F| 2 〉 = 〈T 〉 , (95)
where the angle brackets denote averages over the variational distribution, |Ψ(R)| 2 . Equation (95)
provides a useful consistency check for VMC calculations but note that it does not hold exactly
within DMC, except in the limit of perfect importance sampling. In VMC the kinetic energy may
be evaluated using any of the three estimators in Eq. (95). casino automatically uses 〈K〉 for the
evaluation of the total energy, because this normally leads to the lowest variance. However, the lowest
variance of the kinetic energy itself is often obtained from 〈T 〉. In DMC the three estimators are not
exactly equivalent and 〈K〉 should always be used as the kinetic-energy estimate.
For the Slater-Jastrow wave function of Eq. (83) we have
∇ i (ln |Ψ|) = ∇ iD σi
+ ∇ i J , (96)
D σi ( ) 2
∇ 2 i (ln |Ψ|) = ∇2 i Dσi ∇i D σi
−
+ ∇ 2
D σi D σi i J . (97)
The terms involving Slater determinants may be evaluated by expanding D σi in terms of the cofactors
of the ith column of the Slater matrix D σi . If electron i has spin up, for example, the required
expansion is
D ↑ = det (D ↑ ) = ∑ ψ j (r i ) cof(D ↑ ji ) . (98)
j
Since all the cofactors appearing in this equation are independent of r i , we obtain
∇ i D ↑
D ↑ = ∑ j
∇ 2 i D↑
D ↑ = ∑ j
(∇ i ψ j (r i )) D ↑ ji , (99)
(
∇
2
i ψ j (r i ) ) D ↑ ji . (100)
When moving electron i from r old
i
to r new
i , it is useful to be able to evaluate the kinetic energy at the
new position before updating the D matrix. Since the cofactors in Eq. (98) are independent of r i ,
Eqs. (99) and (100) become
where q ↑ = D ↑,new /D ↑,old .
∇ i D ↑,new
D ↑,new = 1 ∑
q ↑ (∇ i ψ j (r new
i )) D ↑,old
ji , (101)
j
∇ 2 i D↑,new
D ↑,new = 1 ∑ (
∇
2
q ↑ i ψ j (r new
i ) ) D ↑,old
ji , (102)
j
138