Density Functional Calculations of Metal Clusters: Structure, Dynamics, and Reactivity - Metallic Systems: A Quantum Chemist's Perspective - page 280

∑ μ ν

( λ ) ( λ ) .

K = c c K (9.33)

ia i a μν

μ ν ,

. .

Based. on. the. above. deined. ADFT. Kohn-Sham. matrix,. we. ind. for. the. perturbed.
Kohn-Sham.matrix

≡ ∂
∂

K

∑

( )

( λ ) μν ( λ ) ( λ ) ( λ )

(9.34)

K = H + μν k x + z .

μν μν

k k

λ

. .

k

The.perturbed.exchange-correlation.itting.coeficients.are.given.by

∑ 1

( λ ) − ( λ )

ρ

z = G l | v [ ] . (9.35)

XC

kl

k

. .

l

Since.υ XC [ ~ ].is.a.(local).functional.of.the.(approximated).density,.it.follows

ʹ

δ v [ ( )]

(

ρ r ∂ ρ ( r )

∫∫

( λ ) XC

ʹ

l | v [ ] ρ = l ( ) r d d r r

XC

ʹ

δρ r ) ∂ λ

∑

(9.36)

( λ )

= l f [ ρ] m x m .

XC

.

m

.

Compared.to.the.standard.LCGTO.kernel.integral.〈μν| f XC [ρ]|στ〉,.the.scaling.of.the.
ADPT. kernel. integral. 〈 l ‾ | f XC [ ~ ]| ‾ 〉. is. reduced. by. almost. two. orders. of. magnitude..
The. newly. appearing. exchange-correlation. kernel. f XC [ ~ ]. is. deined. as. the. second.
functional.derivative.of.the.exchange-correlation.energy

2

δ E [ ] ρ

ʹ ≡

XC

f [ ( ), ( ρ r ρ r )]

) .

(9.37) XC

ʹ

δρ ( ) r δρ ( r

. .

For.pure.density.functionals,.the.arguments.of.the.approximated.densities.are.col-
lapsed.and.thus.we.ind

2 2

δ E [ ] ρ δ E [ ] ρ δ v XC [ ]

( )

ρ

ʹ ≡

XC

− ʹ =

XC

f [ ( ), ( ρ r ρ r )] δ ( r r ) = .

(9.38)

XC

ʹ

2

δρ ( ) r δρ ( r ) δ ρ ( ) r δρ r

. .

With.the.explicit.form.for.the.perturbed.exchange-correlation.itting.coeficients,.we.
can.rewrite.the.perturbed.Kohn-Sham.matrix.in.terms.of.perturbed.Coulomb.itting.
coeficients.only

⎛ ⎞

∑ ∑

( λ ) ( λ ) ( λ ) ( λ )

K = H + μν k x + μν k F x ,

⎜ ⎟ (9.39) μν μν

k l

k l

⎝ ⎠

k k l ,

. .