16.10 Computing the∇^2 ΨC/ΨCRatio 527
which yields the closed-form expression
∂fi j
∂ri j=ai j
( 1 +βi jri j)^2. (16.34)16.10Computing the∇^2 ΨC/ΨCRatio
For deriving this expression we note first that Eq. (16.29) can be written as
∇kΨC=k− 1
∑
i= 11
gik
∇kgik+N
∑
i=k+ 11
gki
∇kgki.After multiplying byΨCand taking the gradient on both sides we get,
∇^2 kΨC=∇kΨC·(k− 1
∑
i= 11
gik
∇kgik+N
∑
i=k+ 11
gki
∇kgki)
+ΨC∇k·( N
∑
i=k+ 11
gki
∇kgki+N
∑
i=k+ 11
gki
∇kgki)
=ΨC
(
∇kΨC
ΨC) 2
+ΨC∇k·( N
∑
i=k+ 11
gki
∇kgki+N
∑
i=k+ 11
gki
∇kgki)
. (16.35)
Now,
∇k·(
1
gik∇kgik)
=∇k(
1
gik)
·∇kgik+1
gik∇k·∇kgik=−1
g^2 ik
∇kgik·∇kgik+1
gik∇k·(
rik
rik∂gik
∂rik)
=−^1
g^2 ik(∇kgik)^2+
1
gik[
∇k(
1
rik∂gik
∂rik)
·rik+(
1
rik∂gik
∂rik)
∇k·rik]
=−
1
g^2 ik(
rik
rik∂gik
∂rik) 2
+^1
gik[
∇k(
1
rik∂gik
∂rik)
·rik+(
1
rik∂gik
∂rik)
d]
=−^1
g^2 ik(
∂gik
∂rik) 2
+^1
gik[
∇k(
1
rik∂gik
∂rik)
·rik+(
1
rik∂gik
∂rik)
d]
, (16.36)
withdbeing the number of spatial dimensions.
Moreover,
∇k(
1
rik∂gik
∂rik)
=
rik
rik∂
∂rik(
1
rik∂gik
∂rik)
=
rik
rik(
−
1
r^2 ik∂gik
∂rik+1
rik∂^2 gik
∂r^2 ik