16.11 Efficient Optimization of the Trial Wave Function 529
16.11Efficient Optimization of the Trial Wave Function
Energy minimization requires the evaluation of the derivative of the trial wave function with
respect to the variational parameters. The computational cost of this operation depends, of
course, on the algorithm selected. In practice, evaluatingthe derivatives of the trial wave
function with respect to the variational parameters analitically is possible only for small sys-
tems (two to four electrons). On the other hand, the numerical solution needs the repetead
evaluation of the trial wave function (the product of a Slater determinant by a Jastrow func-
tion) with respect to each variational parameter. As an example, consider using a central
difference scheme to evaluate the derivative of the Slater determinant part with respect to a
parameterα,
dΨSD
dα
=
ΨSD(α+∆ α)−ΨSD(α−∆ α)
2 ∆ α
+O(∆ α^2 ).The reader should note that for the Slater determinant part we need to compute the expres-
sion above two times per Monte Carlo cycle per variational parameter. Computing a deter-
minant is a highly costly operation. Moreover, the numerical accuracy in the solution will
depend on the choice of the step size∆ α.
In the following we suggest a method to efficiently compute the derivative of the energy
with respect to the variational parameters. It derives fromthe fact that the energy derivative
is equivalent to
∂E
∂cm
= 2
[〈
EL
∂lnΨTcm
∂cm〉
−E
〈
∂lnΨTcm
∂cm〉]
,
or more precisicely,
∂E
∂cm= 2
{
1
N
N
∑
i= 1[
(EL[cm])i(
∂lnΨTc
∂cm)
i]
−
1
N^2
N
∑
i= 1(EL[cm])iN
∑
j= 1(
∂lnΨTc
∂cm)
j}
, (16.40)
and becauseΨTcm=ΨSDcmΨJcm, we get that
lnΨTcm=ln(ΨSDcmΨJcm) =ln(ΨSDcm)+ln(ΨJcm)
=ln(ΨSDcm↑ΨSDcm↓)+ln(ΨJcm)
=ln(ΨSDcm↑)+ln(ΨSDcm↓)+ln(ΨJcm).Then,
∂lnΨTcm
∂cm
=
∂ln(ΨSDcm↑)
∂cm+
∂ln(ΨSDcm↓)
∂cm+
∂ln(ΨJcm)
∂cm, (16.41)
which is a convenient expression in terms of implementationin an object oriented fashion
because we can compute the contribution to the expression above in two separated classes
independently, namely the Slater determinant and Jastrow classes.
Note also that for each of the derivatives of concerning the determinants above we have,
in general, that
∂ln(ΨSDcm)
∂cm=
∂ΨSDcm
∂cm
ΨSDcm↑
For the derivative of the Slater determinant yields that ifAis an invertible matrix which
depends on a real parametert, and ifddAt exists, then