Tight Binding Methods for Metallic Systems - Metallic Systems: A Quantum Chemist's Perspective - page 242

In.Equation.8.35,.the.upper.index.0.denotes.the.reference.orbital.population.obtained.
from.the.parameterized.system,.and.the. C μ .are.the.TB.eigenvector.coeficients..The.
γ μ v . terms. in. Equation. 8.34. correspond. to. the. Coulomb-exchange-correlation. inte-
grals.over.atomic.orbitals,.which,.for.the.atomic.cases,.are.approximated.using.the.
Slater-Condon.parameters.of.the.neutral.atoms..All.terms.in.Equation.8.34.can.be.
estimated.in.a.non-self-consistent.fashion.making.use.of.the.density.matrix.obtained.
directly.from.the.reference.TB.Hamiltonian,.as.opposed.to.the.self-consistent.proce-
dure.used.in.standard.TB.
The.model.proposed.in.this.work.does.not.require.any.additional.parameters.as.
compared.to.conventional.TB.methods..Only.knowledge.of.the.reference.population.
P � 0 .is.required..This.reference.orbital.population.varies.with.the.chemical.environ-
ment,.and.is.usually.modeled.by.a.function.that.depends.on.the.Cartesian.coordinates.
of.the.system..Consequently,.the.reference. P � 0 .is.a.function.of.atomic.environment.
variables. given. by. the. geometric. coniguration. of. the. system.. For. an. atom. i ,. we.
deine. the. chemical. environment. function. in. terms. of. an. atomic. local. density,. as.
proposed.elsewhere.[10,11].
We. have. tested. our. TB. approximation. on. gold. nanoclusters. by. considering.
the.NRL-TB.method.as.the.reference.TB.Hamiltonian..In.this.method,.the.sec-
ond. term. of. Equation. 8.14. is. not. explicitly. considered. because. it. is. implicitly.
contained.in.the.irst.term.of.this.equation.when.the.eigenvalues.are.shifted.by.a.
.constant amount.to.it.the.total.DFT.energy.[31]..Furthermore,.the.parameteriza-
tion.of.this.TB.Hamiltonian.includes.a.wide.variety.of.crystal.structure.conigu-
rations.as.well.as.different.lattice.constants,.probing.a.large.number.of.different.
atomic.environments.during.the.TB.parameterization..This.leads.to.a.signiicant.
range.of.atomic.local.densities,.making.it.very.convenient.for.our.TB.implemen-
tation,. as. mentioned. previously.. In. the. case. of. gold,. the. NRL-TB. parameters.
were.obtained.from.a.itting.procedure.that.considered.local.density.approxima-
tion.(LDA).calculations.of.the.band.structure.of.different.crystal.conigurations.
(FCC, BCC,. and. SC). and. lattice. constants. varied. by. ±20%. of. the. equilibrium.
value.[24,52].
We.irst.tested.the.method.on.a.gold.dimer..Figure.8.1.depicts.the.corresponding.
binding. energy. per. atom. as. a. function. of. the. internuclear. distance. obtained. with.
the.NRL-TB.parameterization.mentioned.above,.with.(circles).and.without.(dashed.
curve).the.perturbative.correction.proposed.in.this.work..Results.obtained.from.DFT.
calculations.at.the.LDA.and.GGA.levels.combined.with.the.LANL2DZ.basis.set.are.
also.depicted.in.Figure.8.1.(solid.curve)..GGA.calculations.were.performed.using.the.
pure.gradient-corrected.exchange-correlation.functional.(PW91).
As.a.second.test,.we.assess.the.performance.of.the.method.on.the.prediction.of.
equilibrium. structures. of. small. gold. clusters.. The. structures. of. small. gold. cluster.
anions. and. cations. with. up. to. 13. atoms. have. been. inferred. through. a. comparison.
of.theoretical.and.experimental.collision.cross.sections.from.ion-mobility.measure-
ments.[53].or.electron.diffraction.data.[54]..A.remarkable.inding.of.this.study.was.
that. a. 2D-3D. structural. transition. for. Au n − . occurs. at. the. surprisingly. large. cluster.
sizes.of. n .=.11.and.12..A.similar.result.was.obtained.in.the.case.of.neutral.gold.clus-
ters.based.on.DFT.calculations.[55]..For.the.ionic.case,.the.result.was.later.conirmed.