Some Practical Considerations for Density Functional Theory Studies of Chemistry at Metal Surfaces - Metallic Systems: A Quantum Chemist's Perspective - page 193

is. n ot.a.variational.quantity.with.respect.to.the.basis.set..The.total.energy.can.vary.
minimally.with.changes.in.the.FOs,.but.the.work.function.will.vary.strongly.with.
changes.in.the.true.FOs..It.is.possible.to.converge.highly.accurate.work.functions.
with.a.LCAO.code,.even.with.complex.surface.chemistry.[60],.but.it.is.not.straight-
forward.. The. work. functions. quoted. in. Table. 6.3. have. not. been. speciically. veri-
ied.for.convergence.through.augmentation.with.FOs,.but.can.be.used.to.assess.the.
convergence.of.the.slab.model.with.thickness.and.BZ.sample..We.see.that.the.work.
function.converges.even.more.rapidly.than.the.energy.or.layer.relaxation,.both.with.
slab.thickness.and.with.surface.BZ.sampling.
This.exercise.has.identiied.the.minimal.characteristics.necessary.in.a.slab.model.
to.serve.as.an.adequate.substrate.for.a.surface.chemistry.simulation..In.the.case.of.
the.Er(0001).surface,.the.slab.thickness.should.be.at.least.seven.layers.and.it.may.
sufice.to.use.a.surface. k -sampling.as.small.as.8 2 .to.get.adequately.converged.results..
This.is.somewhat.less.than.might.be.deduced.from.examining.the.convergence.of.
the.bulk.properties.with. k -sampling..This.is.useful,.as.the.smaller.the.sampling,.the.
less.expensive.the.calculation.and.the.more.atoms.can.be.inserted.into.a.slab.model,.
either.in.a.thicker.slab.to.converge.the.surface.calculation.better,.or.in.making.the.
surface.super-cell.larger.and.more.realistic.in.the.sense.that.the.impurity.is.further.
from.its.copies.

6.4.7 S urface S uPer -c ellS and c hemiStry at S urfaceS
These. small. 1.×.1. slab. models. are. primitive. cells. for. constructing. larger. surface.
super-cells. with. which. to. examine. longer-ranged. surface. phenomena.. Examples.
include.such.phenomena.as.adsorption.of.less.than.one.monolayer,.investigation.of.
the. initial. steps. of. oxidation,. or. surface. reconstructions. that. entail. more. than. the.
simple.relaxation.described.above..The.most.famous.of.these.is.perhaps.the.recon-
struction. of. the. silicon. (111). surface,. which. takes. a. 7.×.7. reconstruction.. To. con-
struct.a.2.×.2.super-cell,.simply.double.the.lattice.vectors.in.the.surface.and.add.the.
atoms.enclosed.within.the.expanded.cell,.replicating.the.atoms.of.the.1.×.1.primitive.
cell.appropriately.offset.by.the.primitive.lattice.vectors..The.expanded.cell.requires.
a. reduced. BZ. sampling.. The. required. dimensions. of. the. k -grid. are. inversely. pro-
portional. to. the. real-space. dimensions. of. the. grid.. A. 2.×.2. super-cell. with. 4 2 . BZ.
sampling.is.formally.equivalent.to.the.1.×.1.primitive.cell.with.an.8 2 .sampling.and.
results. for. the. clean. surface. (energy,. relaxation,. and. work. function). should. be. the.
same..A.3.×.3.super-cell.with.a.4 2 .BZ.sampling.is.equivalent.to.a.12 2 .sampling.in.the.
1.×.1.primitive.cell.
With. the. deinition. of. the. periodic. slab. substrate. in. the. DFT. calculation,. the.
simulations.follow.the.same.path.as.for.typical.molecular.quantum.chemistry.simu-
lation.. In. order. to. compute. the. adsorption. of. a. chemical. species,. such. as. carbon.
monoxide.on.a.platinum.surface,.for.example,.one.would.add.a.CO.to.the.super-cell..
The.energy.of.adsorption.is.the.difference.between.the.total.energy.of.the.adsorbed.
system. and. the. sum. of. the. total. energies. of. the. isolated. molecule. and. the. clean.
substrate:

E = E − ( E + E ) . . (6.6)

.

ads CO/Pt ( 111 ) CO Pt ( 111 )