Some Practical Considerations for Density Functional Theory Studies of Chemistry at Metal Surfaces - Metallic Systems: A Quantum Chemist's Perspective - page 173

where
n .is.the.electronic.density
ε C .is.the.correlation.contribution
ε X .is.the.exchange.contribution

At.edges.or.regions.of.space.where.the.density.abruptly.changes,.an.exchange.energy.
per.particle.derived.from.the.Airy.gas.system.[14,15].(the.local.Airy.approximation).
is.used.together.with.a.scaled.PW.LDA.correlation:

edge LAA LDA

ε [ ] n = ε ( n s , + ) γ ε ( ) n γ = . 0 8098 . , (6.3)

.

XC X C

,| ∇ | = |∇ |

n

where. s ( ) ( 2 .is.a.dimensionless.scaled.gradient.also.used.in.many.

other. GGA-type. functionals. with. k F ( n ).=.(3π 2 . n ) 1/3 ,. the. local. Thomas. Fermi. wave.
number.
The.interpolation.index,. X s

= s n n

k n n

F

1 α .with.α.=.2.804.depends.only.on. s .and.deter-
mines. the. ratio. of. edge. versus. interior. functional. to. use. at. each. point.. Thus,. the.
AM05.local.energy.per.particle.is

( ) =

1

+ s

AM05 interior edge

ε ( n s , ) = ε ( ) n X s ( ) + ε ( n s , )( 1 − X s ( )) . . (6.4)

.

XC XC XC

The. AM05. functional. can. be. seen. as. a. consistent. improvement. over. LDA. in. the.
sense. that. it. reproduces. the. exact. XC. energy. for. two . types. of. model. systems,. the.
uniform. electron. gas. and. the. jellium. surfaces;. two. situations. with. fundamentally.
different. physics.. The. subsystem. functional. approach. suggests. that. a. sequence. of.
improvements. are. possible. by. maintaining. proper. descriptions. of. these. exact. XC.
model. systems,. and. incorporating. additional. model. systems. into. a. systematically.
improved.functional..An.additional.functional.based.on.an.exactly.solvable.model.
system.should.be.introduced.to.treat.qualitatively.unique.aspects.of.the.density,.just.
as.AM05.introduces.gradients.paired.with.exact.results.for.edges.

6.3.3 c onStraint -B aSed c onStruction , PBe and tPSS f unctionalS
When.LDA.was.analyzed.to.understand.its.success.for.nonuniform.electron.densi-
ties,. it. was. observed. that. the. LDA. fulills. a. number. of. exact. constraints. that. the.
exact.XC.functional.must.satisfy..Several.works,.most.prominently.of.Perdew.and.
coworkers,. focus. on. this. observation. and. argue. that. improved. functionals. can. be.
constructed.by.retaining.the.constraints.LDA.fulills.and.imposing.additional.physi-
cally.motivated.constraints..A.framework.for.the.development.of.functions.has.been.
given.by.Perdew.et.al..[13].and.termed.Jacob's.Ladder.[13].
One. of. the. most. prominent. examples. of. a. GGA. XC. functional. is. the. popular.
and.successful.PBE.[23]..For.atoms,.molecules,.and.transition.metals,.PBE.offers.a.
signiicant.improvement.over.LDA..The.construction.explicitly.enforces.seven.con-
straints..Some.of.these.are.argued.to.be.energetically.signiicant.and.satisied.by.the.
LDA..However,.despite.PBE.being.constructed.to.satisfy.the.important.constraints.
of.LDA,.it.does.not.always.improve.the.results.with.respect.to.LDA.for.solids..For.