Some Practical Considerations for Density Functional Theory Studies of Chemistry at Metal Surfaces - Metallic Systems: A Quantum Chemist's Perspective - page 191

Er hcp(0001) slab formation energy (eV)

1.4900

1.4800

1.4700

1.4600

1.4500 k = 8 2

k = 10 2
k = 12 2
k = 14 2
k = 16 2

1.4400

1.4300

0 5 10 15 20 25 30

Slab layers

FIGURE 6.7 The. convergence. of. slab. formation. energy. as. a. function. of. slab. thickness,.
using. the. slab-compatible. bulk. reference. energy. appropriate. to. the. surface. k -grid,. which.
ranges. from. 8 2 . to. 16 2 .. The. surface. formation. energy. exhibits. greater. variation. with. layers.
than. the. bulk. reference. did. with. k -sample,. indicating. that. the. surfaces. are. still. interacting.
across.the.slab,.even.with.slabs.as.thick.as.27.layers..Starting.with.nine.layers.and.thicker,.the.
formation.energy.is.as.well.converged.as.it.will.be..The.10 2 .and.greater.surface. k -grids.are.
fully.converged,.while.the.8 2 . k -grid.yields.a.slab.formation.energy.only.0.01.eV.less.

through.16 2 .and.slightly.less,.0.732.eV,.with.the.8 2 . k -sample..Hence,.the.10 2 .sampling.
looks.well.converged.and.the.8 2 .is.nearly.converged.
The.virtue.of.using.the.approach.of.slab-compatible.bulk.reference.energies.is.the.
reduced. computational. expense.. One. does. not. need. highly. converged. calculations.
for.each.piece,.either.slabs.thick.enough.to.extract.a.reliable.bulk.limit.[62],.or.basis.
cutoffs. and. k -sampling. for. both. slab. and. bulk. calculations. suficient. to. absolutely.
converge.both.the.slab.and.bulk.independently.[63]..One.need.only.ensure.that.the.
slab.and.bulk.calculations.are.equivalently.converged..The.computed.slab-compati-
ble.Er.bulk.reference.energy.for.the.10 2 .×. N k , z .through.16 2 .×. N k , z .bulk.sampling.are.all.
within.1.meV.of.one.another.(although.at.27.layers,.this.1.meV.error.in.a.single.atom.
would.translate.into.a.total.27.meV.difference.in.the.corresponding.reference.energy),.
but.the.8 2 .×. N k , z .reference.energy.is.4-5.meV.less.than.any.of.these..Nonetheless,.the.
slab. formation. energy. for. the. 8 2 . BZ. sampling. is. almost. perfectly. lat. (converged).
out.to.at.least.27.layers,.verifying.that.the.slab-compatible.bulk.reference.procedure.
gives.a.valid.reference.energy.for.the.surface.energy.calculation..The.8 2 .×. N k , z .is.not.
a.highly.converged.calculation.of.the.total.energy.for.the.bulk,.nor.is.the.8 2 .sampling.
a.highly.converged.calculation.for.the.total.energy.for.the.slab,.but.in.tandem,.these.
give.highly.converged.calculations.of.slab.formation.because.their.convergence.with.
respect.to.the.computational.model.setup.is.compatible.
Other.properties.of.the.slab.are.also.well.converged.with.relatively.thin.slabs.and.
sparse.surface. k -grids..In.Table.6.3,.the.layer.relaxations.and.work.functions.for.the.