Some Practical Considerations for Density Functional Theory Studies of Chemistry at Metal Surfaces - Metallic Systems: A Quantum Chemist's Perspective - page 168

exchange-.correlation.contribution.to.the.energy,. E XC ..Our.agenda.here.is.to.give.the.
reader.a.taste.of.the.rationale.behind.various.classes.of.functionals.for. E XC .and.when.
to.trust.or.distrust.them.
Before. discussing. the. practical. approximations. to. exchange-correlation,. it. is.
instructive. to. consider. what. an. exact. KS. solution. using. the. exact. E XC . would. pro-
vide.. In. practice,. the. KS. equations. are. solved. instead. of. the. full. many-electron.
Schrödinger.equation..The.KS.equations.have.the.familiar.form.of.the.independent-
electron.Schrödinger.equation.with.an.effective.potential.and.this.has.led.to.a.great.
deal.of.misappropriate.interpretation.of.the.auxiliary.quantities.of.the.theory..On.one.
hand,.an.exact.solution.of.the.KS.equations.using.the.exact. E XC .could.provide.the.
same.accurate.prediction.of.observables.that.a.solution.of.the.electron.Schrödinger.
equation. could. provide,. albeit. not. always. in. a. direct. fashion.. Thus,. DFT. has. the.
extraordinary. promise.to.replace.a.much.more.cumbersome. problem.with.a.much.
simpler.one..On.the.other.hand,.a.great.deal.of.work.has.established.understanding.
regarding.the.solution.of.the.noninteracting.Schrödinger.equations..Even.prior.to.the.
formal.introduction.of.DFT,.researchers.were.developing.effective.potential.theories.
to. provide. ever. more. accurate. treatments. of. many-electron. systems,. for. example,.
X-α.theory.[7]..However,.the.similarities.to.the.noninteracting.Schrödinger.problem.
cause.confusion..Given.the.exact.functional,.DFT.is.not.a.mean-ield.theory.(MFT).
of.electrons.but.rather.an.exact.theory.of.the.electronic.density..There.are.two.fun-
damental.ideas.underlying.DFT.in.practice..First.is.that.the.proof.of.a.theory.can.be.
constructed.to.reproduce.the.same.density.of.electrons.as.quantum.mechanics.can..
The.second.is.that,.given.this.density,.useful.functionals.can.be.created.to.compute.
observables.in.the.same.way.the.expectation.values.of.the.many-body.wave.function.
can.be.used.to.extract.observables..In.practical.KS.calculations,.these.two.ideas.are.
merged.through.the.postulated.energy.functional.used.to.ind.the.density..However,.
in. principle,. the. two. ideas. are. discrete.. Keeping. this. principle. in. mind. will. allow.
users.to.better.determine,.in.particular.applications,.which.properties.are.meaning-
ful.and.why.certain.functionals.appear.to.fail.or.succeed.
In. an. effective. potential. theory. or. MFT,. the. derivation. of. the. theory. is. physi-
cally.motivated.but.approximate..Take.for.example.the.HF.theory.in.which.the.elec-
trons.interact.with.each.other.via.a.Coulomb.interaction..The.antisymmetry.of.the.
many-body.wave.function.with.respect.to.particle.interchanges.has.energetic.conse-
quences..The.resulting.wave.function.solution.is.accurate.when.one.Slater.determi-
nant.dominates.and.Hartree.and.exchange.account.for.the.majority.of.the.relevant.
physics.beyond.the.external.potential..Since.HF.is.based.on.a.physically.motivated.
model. of. the. structure. of. the. many-electron. wave. function,. it. is. perfectly. sensible.
to. interpret. the. results. by. analyzing. properties. of. the. HF. wave. function. solution..
For. example,. ionization. potentials. can. formally. be. related. to. HF. eigenvalues. via.
Koopmans'.theorem..Additionally,.it.would.make.sense.to.apply.perturbation.theory.
to.the.resulting.HF.wave.function..The.point.is.that.in.the.MFT,.the.resulting.wave.
function.has.formal.signiicance.
The. crucial. difference. between. an. effective. potential. theory,. like. HF. theory,.
and.exact.KS.DFT.is.that.using.the.exact.KS.potential.would.give.the.exact.elec-
tronic.density.rather.than.the.exact.electronic.wave.function..In.the.KS.paradigm,.
the.whole.machinery.of.a.Schrödinger-like.equation.is.auxiliary..The.KS.orbitals.in.