Chemistry Reference
In-Depth Information
δ
E
[ ( )]
( )
ρ
r
XC
v
[ ( )]
ρ
r
≡
.
(9.24)
XC
δρ
r
.
.
Due. to. the. variational. nature. of. the. approximated. density,. the. partial. derivatives.
with.respect.to.the.density.matrix.elements.can.be.found.as
ρ
∂
∂
( )
r
∂
∂
x
P
∑
=
k
k
( ).
r
(9.25)
P
μν
μν
.
.
k
With.the.deinition.of.
x
k
‾
.from.the.variational.itting,
∑
∑
−
1
x
=
G
l
μν
P
μν
,
(9.26)
k
k l
.
μ ν
,
l
.
it.follows.for.the.derivative.of.the.itting.coeficients
∂
∂
x
P
∑
k
−
1
=
G
l
μν
.
(9.27)
k l
μν
.
.
l
Inserting.this.expression.into.the.above.derivative.of.the.approximated.density.yields
∂
∂
ρ
( )
r
∑
−
1
=
μν
l G k
l k
( ).
r
(9.28)
P
μν
.
k l
,
.
Thus,. we. ind. for. the. partial. derivative. of. the. (local). exchange-correlation. energy.
functional.with.respect.to.the.density.matrix.elements
∂
E
[ ( )]
ρ
r
∑
∫
XC
=
μν
l G
−
1
k
( )
r
v
[ ( )]
ρ
r
d
r
XC
l k
∂
P
μν
k l
,
∑
−
1
ρ
=
μν
l G
k v
[ ] .
(9.29)
XC
l k
.
k
,
l
.
For.convenience.in.the.notation,.we.now.introduce.a.new.set.of.itting.coeficients,.
which.we.name.exchange-correlation.itting.coeficients,.as
∑
1
−
ρ
z
≡
G
l v
[ ] .
.
(9.30)
XC
K
k l
.
l
The.ADFT.Kohn-Sham.matrix.elements.are.then.given.by
∂
∂
E
P
∑
(
)
(9.31)
K
=
=
H
+
μν
k
x
+
z
.
.
μν
μν
k
k
μν
.
k
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