Structure of Liquid Metal Surfaces: A First Principles Perspective - Metallic Systems: A Quantum Chemist's Perspective - page 144

where. ρ. is. the. electronic. density,. the. {φ j }. are. a. set. of. Kohn-Sham. single-particle.
orbitals. with. associated. occupancies. { f j },. V ext ( r ). is. the. external. potential. acting. on.
the. electronic. system,. T e . is. the. kinetic. energy. of. the. non-interacting. Kohn-Sham.
orbitals,. E H .is.the.Hartree.energy,. E XC ,β is.the.exchange.correlation.energy.at.inite.
electronic.temperature. T elec =.1/β,.and. S .is.the.electronic.entropy..The.last.term.in.the.
second.part.of.Equation.5.3.gives.the.entropy.in.the.case.that.Fermi-Dirac.statistics.
are.used.to.describe.the.occupancies..In.this.approach,.the.occupancies.and.rotations.
within.the.subspace.of.occupied.orbitals.are.treated.separately.which,.gives.rise.to.
ill-conditioned,.nonlinear.degrees.of.freedom.in.the.minimization.problem.
Marzari. et. al.. [61,62]. subsequently. introduced. a. generalization. of. the. Mermin.
canonical. inite-temperature. DFT. known. as. ensemble. density. functional. theory.
(eDFT),.which.remedied.this.problem..In.this.approach,.the.free.energy.functional.
takes.the.form

∑

ˆ

F T [ ;{ },{ }] ϕ f = f ϕ T V + ϕ + E [ ] ρ + E [ ] ρ − T S [ { }] f ij . (5.4)

elec j ij ji i e ext j H XC , β elec

ij

.

= ∑

* ( )

ρ ϕ [{ },{ }]( ) f r f ϕ r ϕ ( ) r . (5.5)

j ij ji i j

.

ij

in.which.{ f ij }.is.the.set.of.“generalized”.occupancies..As.both.the.free.energy.and.
density. are. written. as. traces,. the. occupancies. and. rotations. within. the. occupied.
subspace.can.both.be.treated.consistently..This.allows.development.of.a.robust.and.
stable. algorithm. for. solving. the. minimization. problem,. the. full. details. of. which.
appear.in. Refs.. [61,62].. Closely.associated.with. the. ensemble. DFT.formulation. is.
the.“cold-smearing”.technique.[61,62]..In.this.scheme,.the.smearing.function.is.cho-
sen. so. that. the. systematic. errors. introduced. via. coupling. to. the. temperature. are.
minimized.
Application.of.the.ensemble.density.functional.theory.scheme.has.been.demon-
strated.in.simulations.of.various.metallic.systems.(including.surfaces):.studies.of.the.
microscopic. mechanisms. accompanying. the. sliding. of. grain. boundaries. in. alumi-
num.[63].and.the.properties.of.the.Al(110).surface.near.melting.[64].

5.3.2 o rBital -f ree m olecular d ynamicS
Recently,.a.number.of.liquid.metal.surfaces.have.been.studied.using.“orbital-free”.
DFT.methods.(described.in.detail.in.Ref..[65])..In.this.technique,.an.approximation.
for.the.kinetic.energy.operator.in.terms.of.the.density.is.introduced..Instead.of.cal-
culating.the.non-interacting.kinetic.energy.as.in.the.usual.orbital-based.Kohn-Sham.
formulation,.the.kinetic.energy.term.is.evaluated.as

T = T + β . T (5.6)

.

e W