Chemistry Reference
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6.3.5  h yBrid  f unctionalS
A.hybrid.functional.combines.local.or.semi-local.density-based.exchange.with.some.
HF. exchange.. This. approach. is. pragmatically. motivated. by. the. beneits. of. adding.
DFT.correlation.to.HF.and.nonlocal.exchange.in.DFT..A.more.formal.motivation.
has.also.been.given.by.Becke.[35].and.further.discussed.by.Perdew.et.al..[36]..Since.
many.quantum.chemistry.codes.had.been.designed.to.do.both.HF.and.DFT.calcula-
tions,.combining.the.two.into.a.hybrid.form.was.readily.implemented..The.reader.is.
cautioned.that.hybrid.functionals.are.often.coded.as.a.mixture.of.DFT.and.HF.rather.
than.a.proper.formulation.involving.a.local.potential..It.is.often.argued.that.the.ener-
gies.differ.by.a.small.amount.in.the.two.approaches;.however,.the.unoccupied.KS.
spectrum.may.differ.substantially.
A.comprehensive.analysis.of.the.performance.of.PBE0.[36,37].and.HSE06.[38].
for.solids.was.recently.given.by.Paier.et.al..[38]..A.major.limitation.of.hybrid.schemes.
is.a.steep.increase.in.computational.cost,.especially.if.the.KS.local.potential.is.found..
At.present,.their.applicability.to.large.systems.and/or.long.molecular.dynamics.sim-
ulations.is.unfortunately.limited.by.computational.resources.
B3LYPs.poor.performance.on.solids.is.discussed.in.Ref..[39]..Since.any.amount.
of.Fock-like.exchange.will.be.unscreened,.many.of.the.beneits.of.using.DFT.on.met-
als.are.lost.when.using.hybrid.functionals.
6.3.6  f unctional  B aSed on the  a diaBatic  c onnection  r elationShiP
We.have.mentioned.that.the.exact.KS.exchange-correlation.functional.is.unknown,.
but. this. is. not. entirely. true.. The. exchange-correlation. functional. is. not. explicitly.
known.as.a.functional.of.the.density..However,.the.exact.functional.is.equivalent.to.
the.exact.many-body.solution.of.the.problem.and.this.is.known.to.the.extent.that.the.
exact.solution.can.be.formally.written.as.a.perturbative.expansion..There.are.subtle-
ties.about.the.convergence.of.a.truncated.version.of.the.full.many-body.expansion,.
but.formally,.the.interacting.ground-state.energy.can.be.found.through.ininite.order.
perturbation.theory.in.the.electron.interaction..A.similar.approach.can.be.applied.in.
DFT.[40]..However,.the.density.is.ixed.to.be.the.density.of.the.interacting.system.
and.the.resulting.curve.is.called.the.adiabatic.connection.relationship..In.this.case,.
the.energies.and.orbitals.used.in.the.expansion.are.not.the.HF.ones.but.rather.the.
KS. ones.. These. are. obtained. as. the. orbitals. from. a. single. local. potential.. It. turns.
out.that.the.local.potential.requirement.constrains.the.expansion.so.that.terms.that.
depend.on.the.local.potential.contribute.in.second.and.higher.order..The.resulting.
series.is.called.the.Goerling-Levy.expansion.and.provides.a.useful.representation.of.
the.exact.exchange-correlation.functional..The.exact.exchange-correlation.energy.
is.obtained.by.an.integration.of.this.expansion.along.the.adiabatic.connection.curve..
To.irst.order,.the.functional.is.the.exact.exchange.functional.or.a.Fock-like.exchange.
term.using.KS.orbitals..This.explains.why.approximate.functionals.including.some.
fraction.of.exact.exchange.tend.to.be.more.accurate..The.second-order.term.looks.
much.like.a.second-order.Moller-Plesset.(MP2).or.an.RPA.term.with.a.correction.
involving.the.irst-order.local.exchange.potential.
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