Exploring the Borderland between Physics and Chemistry: Theoretical Methods in the Study of Atomic Clusters - Metallic Systems: A Quantum Chemist's Perspective - page 376

suffer.dynamic.rearrangements.leading.to.more.stable.structures..Thus,.an.important.
part.of.all.cluster.studies.is.to.ind.the.optimum.structures.in.the.coniguration.space.
spanned.by.the.nuclear.coordinates..In.the.next.section.we.discuss.some.of.the.most.
common.methods.used.to.optimize.the.geometry.of.clusters.
11.3 OPTIMIZATION METHODS APPLIED TO CLUSTER
STUDIES: FINDING THE MOST STABLE ISOMERS
In. general. terms,. the. problem. of. inding. stable. isomer. structures. is. equivalent. to.
the. problem. of. inding. minima. on. a. multidimensional. hypersurface.. Every. local.
minimum. represents. the. geometry. of. an. isomer. and. the. global. minimum. is. the.
most.stable.one.(i.e.,.the.geometry.with.the.lowest.total.energy)..The.total.energy.
is. a. function. of. the. nuclear. coordinates.. There. are. 3 N -6. variables. (3 N -5. in. the.
case.of.a.linear.molecule),.where. N .is.the.number.of.atoms..In.general,.geometry.
optimization.is.a.nontrivial.mathematical.problem.that.does.not.lend.itself.to.an.ana-
lytic.solution..Therefore,.it.is.necessary.to.use.numerical.techniques..There.are.two.
main.types.of.optimization.techniques,.those.that.follow.the.gradient.(of.the.total.
energy).to.reach.a.minimum.energy.and.stochastic.techniques..(Recently,.some.very.
interesting. techniques. based. on. quantum. molecular. dynamics. have. emerged,. but.
these.will.not.be.addressed.here.[21].).The.most.traditional.methods.used.in.quan-
tum. chemistry. are. gradient-following. techniques. and. are. implemented. in. almost.
all. computational. codes.. The. most. successful. gradient-following. techniques. are.
approximations.to.Newton's.method.and.are.termed.quasi-Newton.methods..These.
need.a.very.limited.number.of.energy.and.gradient.evaluations.compared.to.other.
approaches.and.are.therefore.very.eficient..However,.they.are.dependent.on.the.ini-
tial.geometry.and.locate.only.one.stationary.point.at.a.time..They.cannot.jump.from.
one. minimum. to. another.. Hence,. if. one. has. a. good. initial. guess. of. the. geometry.
of.the.most.stable.isomer,.a.quasi-Newton.optimization.is.likely.the.most.effective.
method..Nevertheless,.one.cannot.be.sure.to.have.reached.the.global.minimum..In.
the.case.of.optimization.of.“normal”.molecules,.where.the.chemical.bonding.rules.
reliably.predict.the.geometry.of.the.most.stable.isomer,.quasi-Newton.methods.are.
very.useful..However,.for.a.cluster.with.a.moderate.number.of.atoms,.for.instance.
Si 9 ,.the.chemical.rules.of.bonding.are.not.as.helpful.and.the.number.of.isomers.with.
energies.close.to.the.global.minimum.energy.is.high..For.Si 9 .there.are.at.least.14.
such.isomers.[22].
On.the.other.hand,.stochastic.methods.are.based.on.a.random.search.on.the.poten-
tial.energy.hypersurface.(PEH)..As.a.consequence,.they.are.able.to.jump.from.one.
minimum.to.another,.allowing.the.location.of.various.isomers.in.one.run..They.are.
also,.in.principle,.independent.of.the.initial.guess..However,.this.should.be.veriied.
by.performing.a.series.of.similar.optimizations.starting.at.different.locations.on.the.
PEH. of. the. system. under. study.. The. drawback. of. stochastic. methods. is. that. they.
generally. need. many. more. evaluations. of. the. energy. function. than. quasi-Newton.
methods,.which.makes.it.dificult.to.apply.them.at.a.high.level.of.electronic.struc-
ture. theory.. Usually,. one. starts. with. a. low. level,. in. many. cases. classical. molecu-
lar.dynamics,.and.then.proceeds.to.perform.high-level.calculations.of.the.minima.