Exploring the Borderland between Physics and Chemistry: Theoretical Methods in the Study of Atomic Clusters - Metallic Systems: A Quantum Chemist's Perspective - page 380

TABLE 11.2
Typical Simulated Annealing
Parameters
Parameter Li 5 Li 6 Li 7

Generated.structures 49 111 33

With.Δ E .<.0 14 10 8

With.Φ(Δ E ).<. P (Δ E ) 35 101 25

Finally.located.minima 2 3 2

tests.. If. the. energy. is. lowered,. Δ E .≤.0,. the. new. structure. is. accepted;. otherwise,.
when. Δ E .>.0,. the. structure. is. accepted. if. Φ(Δ E ).<. P (Δ E ). where. P (Δ E ).=.exp(−Δ E/
k B T ).is.the.Boltzmann.probability.distribution.function.and.Φ(Δ E ).=.|Δ E/E j |,.where.
E j .is.the.energy.of.the.structure.being.evaluated..(Most.simulated.annealing.algo-
rithms.are.based.on.the.Metropolis.sampling.method.that.simply.compares. P (Δ E ).
to.a.random.number.between.0.and.1.).If.neither.of.the.tests.is.satisied,.the.new.
structure.is.not.accepted,.the.parent.structure.is.subjected.to.another.random.modi-
ication,.and.the.acceptance.procedure.is.repeated..At.every.temperature.there.are.
a.maximum.number.of.generated.structures.that.satisfy.neither.of.the.two.accep-
tance.criteria..This.number.is.reduced.as.the.simulated.annealing.temperature.is.
decreased..The.method.for.reducing.the.temperature.is.called.the.quenching.sched-
ule;.the.simplest.method.is.one.that.decreases.the.temperature.by.a.constant.amount.
per.simulated.annealing.iteration..A.successful.run.generates.a.number.of.possible.
structures. of. different. energies.. Structures. with. energies. no. more. than. 0.02. har-
trees.above.the.lowest.energy.of.the.set.are.selected.for.a.inal.optimization..As.an.
example,. Table. 11.2. shows. typical. parameters. for. the. calculation. of. Li q . ( q .=.5-7).
clusters.
Notice. that. the. modiication. of. the. acceptance. criteria. in. principle. allows. for. a.
relatively.exhaustive.sampling.of.potential.energy.search.that.could.lead.to.the.gen-
eration.of.the.most.physically.relevant.structures..Figure.11.4.shows.the.results.of.a.
typical. simulated. annealing. optimization. of. the. Li 7 + . cluster.. Most. of. the. observed.
structures.are.sampled.early.on.in.the.simulated.annealing.procedure.as.a.result.of.
producing. Markov. chains. generated. at. high. temperatures,. which. allows. the. algo-
rithm.to.“jump”.over.potential.energy.barriers.with.relative.ease..As.the.annealing.
temperature. is. reduced,. the. number. of. accepted. structures. tends. toward. the. mini-
mum.energy.set.of.structures,.reducing.the.scope.of.the.sampling.
In.Ref..[33].the.procedure.described.above.was.applied.to.Li q −,.0,.+ .( q .=.5-7).clusters.
and.some.new.structures.were.reported..Most.of.the.isomers.agree.very.well.with.
previous. work. [34-36].. The. method. was.also. applied. to. the. more. complex. binary.
Li 5 Na.cluster.where.six.isomers.were.found.[33].

11.3.3 B ig B ang m ethod
The.“big.bang”.method.[37,38].is.the.simplest.of.the.three.methodologies.discussed.
in.this.chapter,.and.the.only.one.speciically.designed.for.geometry.optimization.of.