Chemistry Reference
In-Depth Information
1
2
n r n r
r
( ) (
ʹ
)
∑
∫
∫
∫
E
[ ]
n
=
2
ε
−
drdr
ʹ
+
E n
[ ]
−
μ
( ) ( )
n n r dr E
+
.
. (8.12)
KS
i
xc
xc
io
n ion
−
|
−
r
ʹ
|
.
i
For. the. moment. we. avoid. the. integration. over.
k
. for. periodic. systems. of. Equation.
8.9..In.the.work.of.Foulkes.and.Haydock.[5],.the.self-consistent.density.is.written.as
.
n r
( )
=
n r
( )
+
δ
n r
( ),
(8.13)
.
0
where
n
0
.is.a.reference.density
δ
n
.is.the.difference.between.the.reference.density.and.the.self-consistent.density,.
where.this.last.density.is.assumed.to.be.small.in.some.sense
For.simplicity,.
n
0
.is.taken.to.be.a.sum.of.atomic.densities.(or.some.other.more.reason-
able.ansatz),.
n r
=
∑
..In.the.following.we.expand.the.energy.of.Equation.
8.12.at.a.reference.density.
n
0
.up.to.second.order.in.δ
n
,
atoms
0
( )
n r
i
( )
i
1
2
n r n r
r
( )
(
ʹ
)
∑
∫
∫
∫
0
0
0
E
( )
n
=
ε
−
drdr
ʹ
+
E n
[
]
−
μ
[
n n dr
]
+
E
KS
i
xc
0
xc
0
0
ion ion
−
|
−
r
ʹ
|
i
⎛
⎜
2
⎞
⎟
1
2
1
E
n r n r
δ
∫
∫
xc
+
+
δ
n r n r dr
( )
δ
(
ʹ
)
dr
ʹ
.
(8.14)
|
r
−
r
ʹ
|
δ
( )
δ
(
ʹ
)
.
In. this. equation,. the. energies.
ε
i
0
. are. the. eigenvalues. of. the. non-self-consistent.
Kohn-Sham.equation.(Equation.8.11).when.
n
(
r
).=.
n
0
(
r
)..The.non-self-consistent.TB.
approximation.may.be.recovered.from.Equation.8.14.for.
E
KS
.by.neglecting.the.last.
term.involving.δ
n
.and.deining.the.repulsive.TB.energy.of.Equation.8.9.as
1
2
n r n r
r
( )
(
ʹ
)
∫
∫
∫
0
0
−
drdr
ʹ
+
E n
[
]
−
μ
[
n n dr E
]
+
.
.
(8.15)
xc
0
xc
0
0
ion ion
−
|
−
r
ʹ
|
.
If.the.reference.charge.density.is.a.superposition.of.atomic-like.neutral.charge.den-
sities,. this. last. term. can. be. approximated. by. a. sum. of. two. center. contributions. as.
long.as.the.exchange.correlation.part.is.expanded.in.a.cluster.series..The.neglect.of.
three-center.contributions.can.be.justiied.by.screening.arguments..Since.the.atomic.
charge. density. corresponds. to. a. neutral. atom,. the. three-center. electron-electron.
interaction.mostly.is.canceled.by.the.ion-ion.repulsion..Due.to.screening,.these.two.
centers.can.be.assumed.short-range.as.in.classical.TB.
Clearly,.as.long.as.δn.is.negligible,.all.energy.contributions.depend.only.on.
n
0
;.
thus.this.scheme.is.equivalent.to.the.non-self-consistent.Harris.functional.[6],.pro-
viding.a.conceptual.framework.for.the.energy.functional.of.Chadi,.Equation.8.9,.and.
a.recipe.for.the.evaluation.of.TB.parameters.from.irst.principles.calculations.
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