528 16 Improved Monte Carlo Approaches to Systems of Fermions
The substitution of the last result in Eq. (16.36) gives∇k·(
1
gik
∇kgik)
=−
1
g^2 ik(
∂gik
∂rik) 2
+
1
gik[(
d− 1
rik)
∂gik
∂rik+
∂^2 gik
∂r^2 ik]
.
Inserting the last expression in Eq. (16.35) and after division byΨCwe get,∇^2 kΨC
ΨC=
(
∇kΨC
ΨC) 2
+
k− 1
∑
i= 1−
1
g^2 ik(
∂gik
∂rik) 2
+
1
gik[(
d− 1
rik)
∂gik
∂rik+∂^2 gik
∂r^2 ik]
+
N
∑
i=k+ 1−^1
g^2 ki(
∂gki
∂rki) 2
+^1
gki[(
d− 1
rki)
∂gki
∂rki+∂
(^2) gki
∂rki^2
]
. (16.37)
For the exponential case we have
∇^2 kΨPJ
ΨPJ=
(
∇kΨPJ
ΨPJ) 2
+
k− 1
∑
i= 1−
1
g^2 ik(
gik
∂fik
∂rik) 2
+
1
gik[(
d− 1
rik)
gik
∂fik
∂rik+∂
∂rik(
gik
∂fik
∂rik)]
+
N
∑
i=k+ 1−^1
g^2 ki(
gik∂fki
∂rki) 2
+^1
gki[(
d− 1
rki)
gki∂fki
∂rki+ ∂
∂rki(
gki∂fki
∂rki)]
.
Using
∂
∂rik(
gik∂fik
∂rik)
=∂gik
∂rik∂fik
∂rik
+gik∂(^2) fik
∂r^2 ik
=gik
∂fik
∂rik
∂fik
∂rik
+gik
∂^2 fik
∂rik^2
=gik
(
∂fik
∂rik) 2
+gik∂(^2) fik
∂rik^2
and substituting this result into the equation above gives rise to the final expression,
∇^2 kΨPJ
ΨPJ
=
(
∇kΨPJ
ΨPJ) 2
+
k− 1
∑
i= 1[(
d− 1
rik)
∂fik
∂rik+
∂^2 fik
∂r^2 ik]
+
N
∑
i=k+ 1[(
d− 1
rki)
∂fki
∂rki+
∂^2 fki
∂rki^2]
. (16.38)
Again, for thelinear Padé-Jastrow, we get in this case the closed-form result∂^2 fi j
∂r^2 i j=−
2 ai jβi j
( 1 +βi jri j)^3