Density Functional Calculations of Metal Clusters: Structure, Dynamics, and Reactivity - Metallic Systems: A Quantum Chemist's Perspective - page 281

with

∑ 1

−

ρ

F = G m f | [ ] | l . (9.40)

XC

k l k m

.

m

.

With.the.above.expression.for.the.perturbed.Kohn-Sham.matrix,.we.ind.for.the.SCP.
density.matrix

occ uno

H

( λ )

∑ ∑

( )

( λ ) ia

P = 2 c c + c c

μν μ ν i a μ a ν i

ε − ε

i a

i a

occ uno ( λ )

μν
ε

k x

∑ ∑ ∑

( )

k

+ 2 c c + c c

μ ν i a μ a ν i

− ε

i a

i a k

occ uno ( λ )

μν k F x

∑ ∑ ∑

( )

k l

l

(9.41)

+ 2 c c + c c ν .

μ ν i a μ a i

ε − ε

i a

.

i a k l

On.the.other.hand,.the.derivative.of.the.itting.equation.system.with.respect.to.the.
perturbation. parameter. λ,. assuming. perturbation-independent. basis. and. auxiliary.
function.sets,.yields

.

∑ ∑

( λ ) ( λ )

G x = P μν || m .

(9.42)

μν

mk

k

k μ ν ,

. .

Inserting.the.above.expression.for. P μ ( ) .into.the.perturbed.itting.equation.system,.we.
obtain.a.system.of.equations.for.the.perturbed.Coulomb.itting.coeficients.only.[37]

occ uno

( λ )

〈 m ia H || 〉

∑ ∑ ∑

( λ ) ia

G x = 4

mk

k

ε − ε

i a

k i a

occ uno

∑ ∑ ∑

〈 m ia ia k || 〉〈 || 〉

( λ )

+ 4 x

ε − ε

k

i a

i a k

occ
∑

uno

〈 m ia ia k || 〉〈 || 〉

∑ ∑

(9.43)

( λ )

+ 4 F x .

k l

l

ε − ε

i a

.

i a k

In. order. to. simplify. the. further. notation,. we. now. introduce. the. Coulomb. and.
exchange-correlation.coupling.matrices. A .and. B ,.as.well.as.the.perturbation.vector.
b (λ) ..Their.elements.are.deined.as

.

occ uno

∑ ∑

〈 k || ia ia l 〉〈 || 〉

A ≡ , (9.44)

k l

ε − ε

i a

.

i a

.