Chemistry Reference
In-Depth Information
of. the. density. functional. framework,. ultimately. with. good. approximations. for. the.
exchange-correlation.terms.
The.irst.of.these.got.rid.of.the.mufin-tin.potential..In.a.key.paper,.Dunlap.et.al..
[11].worked.out.how.to.bring.Gaussian.functions.into.the.formalism..With.the.usual.
Linear.Combinational.Atomic.Orbit.(LCAO).approximations,.the.Xα.equations.are.
cast.into.matrix.form.as
∑
μν
1
2
∑
∑
E
=
P H
+
P P
μν στ
+
E
X
[ ( )].
ρ
r
(9.12)
μν
μν στ
α
μ ν
,
μ ν
,
σ τ
,
.
.
Here
P
μ
v
.and.
P
στ
.denote.elements.of.the.density.matrix
H
μ
v
.denotes.an.element.of.the.core.Hamiltonian.matrix
The.atomic.orbitals.used.to.expand.the.molecular.orbitals.(MOs).are.indicated.by.
the.Greek.letters.μ,.
v
,.σ,.and.τ..The.last.term.represents.the.Xα.approximation.to.the.
exchange.energy.given.by
1 3
/
9
8
⎛
⎜
3
11
⎞
⎟
∫
4 3
/
E
X
α
ρ
[ ( )]
r
= −
α
ρ
( )
r
d
r
.
(9.13)
.
.
For. the. (contracted). four-center. electron. repulsion. integral. (ERI),. we. have. intro-
duced.the.following.short-hand.notation:
μ
(
r
ʹ
) (
ν
r
ʹ
)
∫∫
(9.14)
μ
ν σ ≡
τ
σ
( ) ( )
r
τ
r
d
r
ʹ
d
r
.
r
− ʹ
r
.
.
The. calculation. of. these. integrals. represents. the. computational. bottleneck. in. the.
above.energy.expression..To.maintain.eficiency,.Dunlap.et.al..incorporated.the.vari-
ational.itting.of.the.Coulomb.potential.with.auxiliary.functions.[11-13]..It.is.based.
on.the.minimization.of.the.following.error:
[
]
−
[
]
ρ
( )
r
−
ρ
( )
r
ρ
(
r
ʹ −
)
ρ
(
r
ʹ
)
1
2
∫∫
ε
2
=
d
r
ʹ
d
r
.
(9.15)
r
− ʹ
r
.
.
In.the.LCGTO.approximation,.the.orbital.density.ρ(
r
).is.expanded.in.terms.of.atomic.
orbitals
∑
P
ρ
( )
r
=
μ
( ) ( ).
r
ν
r
(9.16)
μν
μ ν
,
.
.
For. the. expansion. of. the. approximated. density.
~
(
r
),. auxiliary. functions. are. intro-
duced.. With. these. functions,. the. following. linear. expansion. of. the. approximated.
density.is.obtained:
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