Structure of Liquid Metal Surfaces: A First Principles Perspective - Metallic Systems: A Quantum Chemist's Perspective - page 142

(coupled). subsystems:. the. systems. of. nuclei. and. electrons,. which. are. described. at.
different.levels.of.theory..The.system.of.nuclei.is.considered.as.a.set.of.classical.point.
particles.(described.by.appropriate.equations.of.motion),.with.a.component.of.force.
coming.from.the.system.of.electrons,.which.are.treated.quantum.mechanically..In.
this. formulation,. the. electrons. respond. instantaneously. (this. is. termed. “abiabatic-
ity”).to.any.changes.in.the.nuclear.positions..The.motions.of.the.coupled.electronic.
and.nuclear.systems.are.determined.numerically,.with.the.most.time-consuming.part.
of. the. calculation. being. computation. of. the. electronic. structure.. Once. calculated,.
the.electronic.structure.allows.determination.of.the.forces.on.the.nuclei.(using.the.
Hellman-Feynman.theorem.[51,52]).and.the.positions.of.the.nuclei.can.be.evolved.
using.the.pertinent.equations.of.motion..The.electronic.structure.is.recalculated.for.
this.new.set.of.nuclear.positions,.and.the.process.repeated..At.each.step.in.the.nuclear.
motion,.we.need.to.solve.the.problem.of.the.interacting.set.of.electrons.in.the.(ixed.
at.a.given.time.step).ield.of.the.nuclei..Though.the.Born-Oppenheimer.approxima-
tion.is.very.often.made.to.treat.the.nuclei.as.classical.particles,.effects.arising.from.
the.quantum.mechanics.of.the.nuclei.can.be.included,.in.particular.through.the.use.
of.path.integral.techniques.(see.Section.4.4.of.Ref..[50]).
The.molecular.dynamics.technique.provides.access.to.the.full.dynamical.behav-
ior.of.the.system,.an.advantage.over,.for.instance,.Monte.Carlo.techniques.that.can.
only.provide.time-averaged.information..This.comes.at.substantial.cost.in.compu-
tational. resources.. Ab. initio. MD. simulations. are. currently. limited. to. timescales.
from.tens.to.hundreds.of.picoseconds..Regardless.of.whether.the.computational.time.
required.to.use.molecular.dynamics.to.describe.the.nuclear.conigurations.is.more.
time-consuming. than. using. Monte. Carlo. sampling,. in. an. ab. initio. simulation,. the.
most.time-consuming.part.of.the.calculation.is.the.evaluation.of.the.forces.on.the.
atoms—this. is. because. the. electronic. structure. must. be. determined. at. each. step..
This.cost.can.be.mitigated.somewhat.by.using.itted.potentials.to.describe.the.inter-
actions.between.the.different.atoms.in.the.system.in.performing.empirical.molecular.
dynamics,. but. there. is. a. substantial. penalty. in. transferability. (the. prime. example.
being.that.changes.in.“bond”.types.and.arrangements.must.be.built.in.beforehand).
as.such.calculations.move.away.from.the.realm.of.irst.principles.
Many. methods. have. been. developed. to. perform. the. integration. of. the. nuclear.
equations.of.motion,.including.the.Gear.predictor-corrector.schemes.[53],.the.Verlet.
algorithms. [54]. (both. are. so-called. inite-difference. methods). and. the. split-step.
methods.[55]..The.widely.used.Verlet.algorithms.are.based.on.making.Taylor.series.
expansions.of.the.positions.and.velocities.of.the.atoms.at.time. t .+.δ t .in.terms.of.those.
at.time. t .and.the.forces.acting.on.the.nuclei,.with.δ t .the.time.interval.between.suc-
cessive.nuclear.conigurations.
5.3.1 d enSity f unctional t heory
An.immensely.popular.method.of.describing.a.system.of.interacting.electrons.that.
can. be. used. to. describe. the. valence. electrons. in. an. ab. initio. molecular. dynamics.
simulation. is. density. functional. theory. (DFT). [56,57].. For. reviews. focusing. on.
DFT,. the. reader. should. see,. for. instance,. Refs.. [58,59].. Here,. we. mention. only. in.
passing. that. metallic. systems. require. special. care,. since. there. is. discontinuity. in.