Density Functional Calculations of Metal Clusters: Structure, Dynamics, and Reactivity - Metallic Systems: A Quantum Chemist's Perspective - page 288

where. the. mapping. â n . is. calculated. after. the. random. rotations. are. applied. to. the.
coordinates.. In. this. way,. a. set. of. distances. are. obtained,. which. are. the. best. in.
the neighborhood.of.the.orientation.where.they.were.calculated..In.order.to.ensure.
the. reliability. of. this. approach,. enough. samples. of. the. rotation. space. are. needed..
For this.reason,.the.following.sample.algorithm.is.adapted..First,.we.make.a.sample.
P ˆ 1 ,.which.yields.the.mapping. â 1 .and.the.distance. d 1 ..This.is.initially.the.best.map-
ping. â * .with.its.associated.distance. d * ..In.a.next.step,.we.generate.a.new.(random).
sample. P ˆ 2 .and.analyze.it..Then.the.mapping.associated.with.the.smallest.distance.is.
preserved..The.process.is.successively.repeated.until.the.same.mapping.was.found.
50.times.in.a.row..This.has.worked.successfully.for.all.our.test.cases.
The.hierarchical.transition.state.inder.in.combination.with.the.automatic.align-
ment. procedure. has. proven. very. useful. for. the. study. of. metal. cluster. reactions..
Selected.examples.are.presented.in.Section.9.6.of.this.chapter.

9.2.5.4 Born-Oppenheimer Molecular Dynamics
Classical.molecular.dynamics.is.a.computer.simulation.technique.where.the.time.evo-
lution.of.a.set.of.interacting.atoms.is.followed.by.integrating.their.classical.equations.
of.motion..Given.a.set.of.initial.positions.and.velocities,.by.which.the.subsequent.time.
evolution.is.in.principle.completely.determined,.the.computer.calculates.a.trajectory.
in.a.6 N -dimensional.phase.space..In.this.way,.a.set.of.conigurations.according.to.an.
underlying.statistical.distribution.function.or.statistical.ensemble.is.obtained..Physical.
quantities.are.then.represented.by.averages.over.the.coniguration.set..Therefore,.the.
expectation. value. of. a. physical. quantity. can. be. obtained. by. the. arithmetic. average.
from.the.corresponding.instantaneous.values.of.this.quantity.along.the.simulated.tra-
jectory..In.the.limit.of.very.long.simulation.times,.i.e.,.many.trajectory.steps,.it.can.
be.expected.that.the.phase.space.is.fully.sampled..In.this.limit,.the.averaging.process.
yields. the. corresponding. thermodynamic. property.. In. practice,. the. simulations. are.
always.of.inite.length,.and.caution.must.be.taken.that.the.system.is.truly.equilibrated.
While. in. a. full. description. of. the. energetics. and. dynamics. of. a. system,. all.
constituents. (electronic. and. nuclear). should. be. treated. on. equal. footing. quantum.
mechanically;.physical.and.practical.considerations.motivate.calculations.within.the.
Born-Oppenheimer.approximation.[57]..This.approximation.introduces.a.separation.
between.the.time.scales.of.nuclear.and.electronic.motions..Moreover,.we.apply.the.
classical.equations.of.motion.for.the.propagation.of.the.nuclei..Therefore,.a.BOMD.
step.as.discussed.in.this.work.consists.of.solving.the.static.electronic.structure.prob-
lem,. i.e.,. solving. the. time-independent. Schrödinger. equation,. followed. by. propa-
gation. of. the. nuclei. via. classical. molecular. dynamics. [85].. The. resulting. BOMD.
method.is.deined.by

ˆ

H

e Ψ

= E Ψ , . (9.64)

.

ˆ

r ( )

m t = −∇ min Ψ H Ψ . . (9.65)

A A A e

.

Ψ

In.the.above-described.BOMD.step,.the.solution.of.the.electronic.Schrödinger.equa-
tion. (9.64). represents. the. computational. bottleneck.. The. computational. demand.