Computational Physics - Department of Physics

16.9 Evaluating the∇ΨC/ΨCRatio 525

r≡











0 r 1 , 2 r 1 , 3 ··· r 1 ,N

. 0 r 2 , 3 ··· r 2 ,N
. 0
rN− 1 ,N
0 0 0 ··· 0











. (16.19)

This applies tog=g(ri j)as well.

16.8 Computing the Correlation-to-correlation Ratio

For the case where all particles are moved simlutaneously, all thegi jhave to be reevaluated.
The number of operations for gettingRCscales asO(N^2 ). When moving only one particle
at a time, say thekth, onlyN− 1 of the distancesri jhavingkas one of their indices are
changed. It means that the rest of the factors in the numerator of the Jastrow ratio has a
similar counterpart in the denominator and cancel each other. Therefore, onlyN− 1 factors
ofΨCnewandΨCcuravoid cancellation and

RC=

ΨCnew

ΨCcur=

k− 1

∏

i= 1

gnewik

gcurik

N

∏

i=k+ 1

gnewki

gcurki. (16.20)

For the Padé-Jastrow form

RC=

ΨCnew

ΨCcur

=e

Unew

eUcur

=e∆U, (16.21)

where

∆U=

k− 1

∑

i= 1

(

fiknew−fikcur

)

+

N

∑

i=k+ 1

(

fkinew−fkicur

)

(16.22)

One needs to develop a special algorithm that iterates only through the elements of the
upper triangular matrixgthat havekas an index.

16.9 Evaluating the∇ΨC/ΨCRatio

The expression to be derived in the following is of interest when computing the quantum
force and the kinetic energy. It has the form

∇iΨC

ΨC

=^1

ΨC

∂ΨC

∂xi

,

for all dimensions and withirunning over all particles. From the discussion in section 16.8,
for the first derivative onlyN− 1 terms survive the ratio because theg-terms that are not
differentiated cancel with their corresponding ones in thedenominator. Then,

1

ΨC

∂ΨC

∂xk

=

k− 1

∑

i= 1

1

gik

∂gik

∂xk

+

N

∑

i=k+ 1

1

gki

∂gki

∂xk

. (16.23)