Some Practical Considerations for Density Functional Theory Studies of Chemistry at Metal Surfaces - Metallic Systems: A Quantum Chemist's Perspective - page 171

functional.. Formally. in. DFT. for. a. inite. system,. inding. excited. states. amounts. to.
identifying.the.resonances.of.the.physical.density-density.response.function.in.the.
time-dependent.DFT.[11]..This.quantity.can.be.related.to.integrals.over.pairs.of.KS.
excitations. including. a. many-body. exchange-correlation. kernel. and. noninteracting.
auxiliary.KS.response.functions..Within.the.MFT.picture,.the.KS.orbital.eigenval-
ues.are.equated.to.actual.excited.states.of.the.system.and.KS.noninteracting.response.
functions. are. interpreted. as. physical. response. functions.. In. certain. contexts,. the.
mean-ield.and.density.functional.representations.may.not.differ.much..The.advan-
tage.of.the.former.is.that.given.a.highly.accurate.kernel.and.KS.orbitals,.the.excited.
states.would.tend.to.the.fully.interacting.exact.solutions..The.advantage.of.the.latter.
is.that.excitations.can.be.understood.in.terms.of.familiar.orbital.objects..This.orbital.
picture.allows.plotting.of.electrons.and.holes.as.well.as.useful.descriptions.in.terms.
of.single.and.double.excitations..These.distinctions.have.ambiguous.meaning.in.the.
exact.KS.picture.
The.exact.KS.functional.as.a.pure.functional.of.the.density.may.never.be.known..
Since. knowledge. of. the. exact. functional. implies. knowledge. of. the. exact. solution.
of.the.many-particle.interacting.Schrödinger.equation,.the.exact.functional.would.
likely. be. intractable.. The. power. and. beauty. in. DFT. is. that. tractable. approximate.
functionals.for.the.energy.do.exist.and.can.provide.accuracy.comparable.to.much.
more. expensive. solutions. of. the. Schrödinger. equation. for. many. systems.. Future.
improvements. in. exchange-correlation. energy. functionals. will. likely. extend.
this. trend.. However,. reliable. functionals. for. quantities. such. as. band. gap. are. still.
unknown..It.may.be.speculated.that.a.band.gap.functional.would.post-process.the.
exact.KS.band.gap,.providing.the.proper.band.gap..In.cases.where.the.density.func-
tional.for.a.physical.property.is.unknown,.it.makes.practical.sense.to.interpret.the.
DFT.results.as.MFT.
In.what.follows,.we.restrict.ourselves.to.the.most.important.approximate.ingredi-
ent. in. practical. KS. calculations,. the. exchange-correlation. functionals.. This. is. the.
critical.ingredient.in.inding.the.KS.system..We.will.give.the.reader.a.notion.of.the.
origins.of.several.classes.of.functionals.
In.general,.DFT.functionals.need.to.incorporate.information.obtained.from. out-
side . the. density. functional. framework,. for. example,. by. analysis. of. the. Scrödinger.
equation.[12].
6.3.1 l ocal d enSity a PProximation
The.LDA.was.one.of.the.earliest.functionals.introduced.[6]..It.is.constructed.by.refer-
ence.to.a.solvable.reference.system.through.solution.of.the.many-body.Schrödinger.
equation,.the.uniform.many-electron.system..The.idea.is.that.the.energy.density.of.
an. arbitrary. electron. density. distribution. should. be. the. same. locally. as. a. uniform.
reference. system. with. the. same. density.. Despite. being. completely. local,. the. LDA.
succeeds.for.many.applications.and.especially.in.solid-state.applications..It.is.often.
used.as.a.benchmark.functional.expected.to.be.highly.predictive.and.reliable.if.not.
always.accurate.
Improving. upon. the. LDA,. in. fact,. has. proven. dificult.. The. built-in. compat-
ibility. between. exchange. and. correlation. has. advantages.. When. the. exchange. and.