Density Functional Calculations of Metal Clusters: Structure, Dynamics, and Reactivity - Metallic Systems: A Quantum Chemist's Perspective - page 270

Thus,.there.is.enough.information.in.ρ( r ).to.ix.the.total.energy.through.the.universal.
functional. F [ρ( r )],

+ ∫

E [ ( )] ρ r = F [ ( )] ρ r ρ ( ) ( ) r v r d r . (9.4)

. .

Here,. v ( r ). denotes. the. external. potential. of. the. system,. which,. in. the. absence. of.
external.ields,.is.given.by.the.nuclear.coniguration..The.unknown.universal.func-
tional. F [ρ( r )].can.further.be.divided.into.the.kinetic.energy. T [ρ( r )].and.the.electron-
electron. interaction. energy. V ee [ρ( r )].. The. Hohenberg-Kohn. formalism. is. exact. and.
includes.all.the.dificulties.and.subtleties.of.the.many-body.problem..The.challenge.
is.to.formulate.useful.approximations.
The. trail. to. applications. was. blazed. the. following. year. by. Kohn. and. Sham. [9].
who.proposed.a,.still.exact,.reformulation.of.DFT.that.introduced.the.genius.idea.of.
a.model.reference.system.in.which.the.electrons.behave.independently.of.each.other..
The.effective.one-electron.Hamiltonian.of.such.a.noninteracting.system.is.given.by

1
2

ˆ

2

(9.5)

h = − ∇ + v ( ). r

s s

. .

The.introduced.effective.Kohn-Sham.potential.is.chosen.such.that.the.density.of.the.
noninteracting. reference. system. just. matches. the. ground-state. density. of. the. real.
system..Because.of.the.noninteracting.nature.of.the.reference.system,.it.possesses.a.
one-determinantal.ground-state.wave.function

1

Ψ s =

! det

ϕ ϕ , , … , ϕ . (9.6)

1 2 N

N

. .

With.this.wave.function,.the.kinetic.energy.in.the.reference.system.can.be.calculated.
as.the.following.expectation.value

1
2

∑

2

(9.7)

T s [ ( )] ρ r = − ϕ | ∇ ϕ .

i i

. .

i

Of. course,. T s [ρ( r )]. is. not. the. exact. kinetic. energy. of. the. real. system.. However,. it.
accounts.for.the.major.part..To.proceed.further,.Kohn.and.Sham.deined.a.universal.
functional. F s [ρ( r )].for.the.noninteracting.system.very.much.in.the.same.spirit.as.in.
the.Hohenberg-Kohn.theorem.as

F [ ( )] ρ r = T [ ( )] ρ r + J [ ( )] ρ r + XC E [ ( )]. ρ r (9.8)

.

s s s

.

Here,. J [ρ( r )].denotes.the.classical.Coulomb.repulsion.between.the.electrons,.and.the.
new.deined.exchange-correlation.functional. E XC [ρ( r )].collects.all.nonclassical.inter-
actions.between.the.electrons,.i.e.,.exchange.and.correlation,.as.well.as.the.(small).
difference.between.the.exact.and.the.Kohn-Sham.kinetic.energy..The.corresponding.
Kohn-Sham.potential.is.then.given.by