Chemistry Reference
In-Depth Information
LCAO. codes. have. a. different. shortcoming. in. the. basis. set. in. that. they. do. not.
always.describe.very.well.the.slow.fall-off.in.the.density.at.a.metal.surface..To.repro-
duce.the.slow.attenuation.of.the.density.at.the.surface,.it.is.particularly.important.to.
get.an.accurate.calculation.of.the.work.function,.which.might.require.very.diffuse.
orbitals.on.the.metal.atoms..Very.diffuse.orbitals.overlap.with.other.basis.functions.
in.bulk.metal.and.could.trigger.linear.dependence.in.periodic.calculations..To.get.
good.work.functions,.loating.orbitals.(FOs),.basis.functions.disconnected.from.any.
nuclear. center,. can. be. used. effectively. [60].. The. speciication. of. FOs. for. a. given.
system.is.somewhat.arbitrary..We.have.found.that.it.is.suficient.to.use.a.set.of.FO.
with.just.the.outermost.gaussian.of.each.angular.momentum.in.the.metal.atom.basis,.
at.the.bulk.position.of.the.irst.missing.layer..For.the.Er.(0001).surface,.we.add.one.
extra.layer.of.ghost.atoms.on.top.of.the.surface.with.FO.consisting.of.single-gaussian.
s-orbitals,.p-orbitals,.and.d-orbitals,.a.total.of.nine.basis.functions.per.ghost.atom..
A.ghost.atom.is.a.set.of.basis.functions,.known.as.loating.basis.functions,.at.a.posi-
tion.where.there.is.no.atom;.there.is.no.atomic.potential.associated.with.the.basis.
function.centers.
The.slab.formation.energy.is.simply.the.difference.of.the.total.energy.of.the.slab.
model.and.the.energy.of.the.same.number.of.atoms.in.the.perfect.bulk,.in.the.limit.of.
in.initely.thick.slabs..Assuming.the.two.surfaces.are.equivalent,.the.surface.energy.
E
surface
.for.a.given.slab.with.
n
slab
.atoms.is.half.the.slab.formation.energy:
=
1
E
2
(
E
−
n E
)
.
.
(6.5)
surface
slab
slab
bulk
.
This.formula.requires.a.reference.energy.
E
bulk
.of.the.metal.atom.in.the.bulk..This.
value.cannot.be.simply.chosen.as.the.bulk.energy.computed.in.the.preliminary.bulk.
calculation,.as.the.slab.calculation.of.a.bulk.atom.in.the.center.of.the.slab.and.the.
bulk.calculation.are.not.compatible..An.incompatibility.multiplied.by.the.number.of.
atoms.in.the.slab.
n
slab
.will.not.lead.to.convergence.of.the.surface.formation.energy.as.
the.thickness.of.the.slab.is.increased.[61].
Fiorentini. and. Methfessel. [62]. speciied. a. bulk. energy. from. a. it. to. the. incre-
mental.change.in.energy.as.the.slab.thickness.is.increased..With.this.deinition,.the.
surface.energy.converges.by.construction..Da.Silva.et.al..[63].showed.that.if.inde-
pendent. bulk. and. slab. calculations. were. both. performed. with. very. high. accuracy,.
converged. energy. cutoffs,. and.
k
-sampling,. one. could. obtain. a. convergent. surface.
energy..Both.of.these.involve.rather.expensive.calculations,.as.one.needs.rather.thick.
slabs. or. highly. converged. total. energy. calculations. to. demonstrate. the. necessary.
numerical.convergence.
A.different.method.to.obtain.a.compatible.bulk.energy.for.a.convergent.surface.
energy.calculation.is.to.construct.a.bulk.calculation.with.a.unit.cell.compatible.with.
the. speciic. slab-surface. calculation,. using. a. bulk. reference. energy. derived. from.
a. bulk. unit. cell. that. is. a. structural. incremental. building. block. of. the. slab. super-
cell,. with. equivalent. numerics.. Construction. of. this. bulk. super-cell. is. illustrated.
in.Figure.6.5.(d)-(f).for.the.low.index.surfaces.of.the.fcc.crystal..The.central.cell.
highlighted.in.each.is.a.building.block.that.is.inserted.to.make.the.slab.thicker,.but.
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