Some Practical Considerations for Density Functional Theory Studies of Chemistry at Metal Surfaces - Metallic Systems: A Quantum Chemist's Perspective - page 167

Properties of the system

G(Ψ)

F (ρ)

Ψ(ρ)

ρ(Ψ)

Will always be
approximate, even with
the “divine” functional.

ρ

Ψ SE

Ψ KS Ψ MF

Schrödinger view DFT view Mean field view

E XC

Electron

Kohn-Sham particle
(non-interacting) Interaction

Effective potential

External potential

ρ ( r' ) δ E XC [ρ ( r )] AM05, LDA, GGA,

Meta-GGA, hybrids

v eff ( r ) = v ( r ) + dr' +

r - r' δρ ( r )

FIGURE 6.2 The.exact.functional.would.give.the.exact.density,.ρ,.and.thus.the.exact.prop-
erties. via. density. functionals,. F [ρ].. The. unknown. wave. function,. density. functional,. ψ[ρ],.
would.in.this.case.give.all.properties.that.can.be.calculated.from.the.Schrödinger.equation.
exactly..However,.the.mean.ield.view.of.using.the.KS.orbitals.as.approximate.electron.wave.
functions.would.be.approximate..Only.the.density.and.properties.calculated.via.density.func-
tionals.are.guaranteed.to.be.exact..The.quality.of.a.functional.can.thus.not.be.judged.by.how.
well.it.reproduces.wave.function.derived.properties.except.where.properties.of.the.exact.KS.
wave.functions.are.known.

nuclear.positions.are.independent.of.the.electrons..Even.this.simpliied.problem.is.
exponentially. dificult. mainly. due. to. the. dificulty. of. accounting. for. many. mutu-
ally.interacting.electrons.explicitly.with.a.wave.function..DFT.is.a.tool.to.calculate.
the.many-electron.density.that.results.from.a.full.solution.of.the.quantum.problem..
In. its. most. common. form,. DFT. relies. on. two. ideas.. First,. the. Hohenberg-Kohn.
(HK).theorem.provides.a.one-to-one.correspondence.between.potentials,.ν( r ),.and.
electronic.densities,. n ( r ),.implying.that.the.density.alone.contains.the.same.infor-
mation. as. the. potential. [5].. Since. the. potential. gives. a. formal. route. to. the. exact.
wave. function. through. the. Schrödinger. equation,. so. must. the. density.. Second,.
the. Kohn-Sham. (KS). scheme. provides. a. route. to. density. by. using. an. auxiliary.
Schrödinger-like.system.[6].to.represent.the.kinetic.energy..A.small.but.important.
contribution.to.the.total.energy,.the.exchange-correlation.energy. E XC [ n ],.is.all.that.
remains.to.be.approximated..Over.the.years,.increasing.sophistication.and.experi-
ence. have. been. built. into. ever. more. complicated. functionals. to. approximate. this.