Some Practical Considerations for Density Functional Theory Studies of Chemistry at Metal Surfaces - Metallic Systems: A Quantum Chemist's Perspective - page 172

correlation. are. independently. constructed,. as. is. often. done. in. practice,. and. then.
assembled.into.a.functional,.the.result.will.often.lack.this.compatibility..The.LDA's.
exactness.when.dealing.with.at.least.one.physical,.albeit.ideal,.system.ensures.that.
its.predictions.are.typically.at.least.reasonable.
A.natural.extension.to.the.LDA.is.to.include.gradients.of.density,. n ( r ),.into.the.
functional.form,.resulting.in.a.gradient.expansion.approximation.(GEA).form,

∫

GEA

E [ ] n = n ( ) r ε ( ( ) n r ,| ∇ n ( ) ) r | d r . (6.1)

XC XC

.

Lamentably,.the.GEA.does.not.provide.systematically.better.results.than.the.LDA,.
and. a. more. sophisticated. generalized. gradient. approximation. (GGA. [14]). needs. to.
be.employed..This.is.because.density.gradients.in.real.solids.and.molecules.can.vary.
strongly.and.the.expansion.must.be.controlled..Several.approaches.were.developed.
to.overcome.these.challenges.and.an.assortment.of.these.will.be.discussed.shortly..
GGAs. allowed. suficient. accuracy. for. DFT. to. be. applied. to. molecules. as. well. as.
solids.

6.3.2 f unctionalS from m odel S yStemS , the am05 f unctional
Functional.approximations.based.on.the.method.of.reference.systems.builds.upon.
the. strengths. of. the. LDA.. Namely,. the. exchange. and. correlation. energy. expres-
sions.come.from.a.solvable. model system ,.the.uniform.electron.gas..The.resulting.
approximation. is. therefore. internally. consistent. in. its. modeling. of. exchange. and.
correlation,.making.the.combined.quantity.more.widely.transferrable.and.accurate.
than. the. individual. components.. We. refer. to. this. feature. as. compatible . exchange.
and.correlation..Compatibility.is.sometimes.dismissed.as.a.strong.cancellation.of.
errors.between.the.exchange.and.correlation.parts.of.contributions,.but.this.fails.to.
recognize. the. important. physics. behind. treating. the. exchange-correlation. energy.
consistently.
Kohn.and.Mattsson.discussed.the.creation.of.an.XC.functional.from.a.surface-
oriented. model. system. and. its. possible. combination. with. other. treatments. [15,16]..
The.approach.was.formalized.and.generalized.in.the.subsystem.functional.scheme.
by.Armiento.and.Mattsson.[17]..The.idea.is.to.create.separate.functionals.suitable.for.
different.model.systems.and.to.merge.them.using.a.density.functional.index.[18].that.
locally.determines.the.nature.of.the.system..These.ideas.culminated.in.the.AM05.
functional. [14,19].. The.functional. invokes.two.model. systems:. for.regions.that.are.
locally.bulk-like,.the.uniform.electron.gas;.and.for.regions.that.are.locally.surface-
like,.the.Airy.gas.[15].and.jellium.surfaces.[20].
For.bulk-like.electron.regions,.LDA.with.local.density.correlation.according.to.
the. Perdew-Wang. parameterization. [21]. based. on. Ceperly-Alder. data. [22]. is. used..
Thus,.the.exchange-correlation.energy.per.particle.in.the.interior.is

interior LDA LDA

ε [ ] n = ε ( ) n + ε ( ) n , . (6.2)

.

XC X C