526 16 Improved Monte Carlo Approaches to Systems of Fermions
An equivalent equation is obtained for the exponential formafter replacinggi jbyexp(gi j),
yielding:
1
ΨC
∂ΨC
∂xk=
k− 1
∑
i= 1∂gik
∂xk+
N
∑
i=k+ 1∂gki
∂xk, (16.24)
with both expressions scaling asO(N).
Later, using the identity
∂
∂xi
gi j=−
∂
∂xj
gi j (16.25)on the right hand side terms of Eq. (16.23) and Eq. (16.24), weget expressions where all the
derivatives act on the particle are represented by thesecondindex ofg:
1
ΨC∂ΨC
∂xk=
k− 1
∑
i= 11
gik∂gik
∂xk−
N
∑
i=k+ 11
gki∂gki
∂xi, (16.26)
and for the exponential case:
1
ΨC∂ΨC
∂xk=
k− 1
∑
i= 1∂gik
∂xk−
N
∑
i=k+ 1∂gki
∂xi. (16.27)
16.9.1Special Case: Correlation Functions Depending on the Relative Distance
For correlation forms depending only on the scalar distancesri j, we note that
∂gi j
∂xj=
∂gi j
∂ri j∂ri j
∂xj=
xj−xi
ri j∂gi j
∂ri j, (16.28)
after substitution in Eq. (16.26) and Eq. (16.27) we arrive at
1
ΨC∂ΨC
∂xk=
k− 1
∑
i= 11
gikrik
rik∂gik
∂rik−
N
∑
i=k+ 11
gkirki
rki∂gki
∂rki. (16.29)
Note that for the Padé-Jastrow form we can setgi j≡g(ri j) =ef(ri j)=efi jand
∂gi j
∂ri j
=gi j
∂fi j
∂ri j. (16.30)
Therefore,
1
ΨPJ
∂ΨPJ
∂xk=
k− 1
∑
i= 1rik
rik∂fik
∂rik−
N
∑
i=k+ 1rki
rki∂fki
∂rki, (16.31)
where
ri j=|rj−ri|= (xj−xi)eˆ 1 + (yj−yi)eˆ 2 + (zj−zi)eˆ 3 (16.32)
is the vectorial distance. When the correlation function isthelinear Padé-Jastrow, we set
fi j=
ai jri j
( 1 +βi jri j)