Structure of Liquid Metal Surfaces: A First Principles Perspective - Metallic Systems: A Quantum Chemist's Perspective - page 145

where. T W .is.the.von.Weizsäcker.term:

1
8

∫

2

3 − 1

T

W [ ( )]

ρ r = d r ∇ ρ ( ) r ρ ( ) r . (5.7)

.

and

3
10

∫

3 5 3 2 / − β 2

T = d r ρ ( ) r k ( ) r . (5.8)

β

.

( )

∫

k

0 3 0

( ) r = ( 2 k ) d k w s ( ) s 2 k r − s . (5.9)

F β F

.

2 1 3 / β .

k ( ) r = 3 ( π ) ( ( )) ρ r (5.10)

.

where:
ρ e .=. N / V .( N .being.the.number.of.electrons.in.the.system.and. V .the.system.volume).
is.the.mean.electronic.density
k 0 .is.the.Fermi.wave.vector.for.the.mean.electronic.density.ρ e
w β . is. a. weight. function. chosen. so. that. both. the. linear. response. theory. and.
Thomas-Fermi.limits.are.recovered

As.the.relatively.time-consuming.evaluation.of.the.kinetic.energy.term.via.orbit-
als.in.a.full.formulation.of.DFT.is.avoided,.the.orbital-free.approach.can.be.applied.
to. systems. containing. greater. numbers. of. atoms,. as. well. as. making. it. possible. to.
collect. longer. sets. of. statistics. in. the. simulations.. Clearly,. questions. can. be. raised.
regarding.the.reliability.of.methods.in.which.the.kinetic.energy.in.DFT.is.approxi-
mated..This.is.expected.to.be.particularly.crucial.when.dealing.with.the.free.surface.
of.a.liquid.metal,.where.a.rapidly.varying.(i.e.,.decreasing.from.the.bulk.density.in.
the.liquid.to.close.to.zero.in.the.vacuum).is.an.important.characteristic.of.the.system.
A.number.of.simulations.have.applied.orbital-free.DFT.to.liquid.metal.surfaces,.
including.applications.to.Li,.Na,.K,.Rb,.Cs,.Mg,.Ba,.Al,.Tl,.and.Si.[65,66],.Ga.[67],.
as.well.as.the.alloys.Na:K.[68,69],.Li:Na.[68,69],.and.Na:Cs.[69]..In.these.simula-
tions,.slabs.containing.∼.2000.-.3000.atoms.were.considered..For.all.the.liquid.metal.
surfaces.simulated.using.the.orbital-free.MD.method,.oscillations.indicating.layer.
formation. were. observed. in. the. transverse. density. proiles.. With. increasing. tem-
perature,.both.the.range.and.amplitude.of.the.layering.oscillations.in.the.transverse.
density.proiles.across.the.surface.decreased.

5.3.3 e nSemBle dft-B aSed m olecular d ynamicS S imulationS
of l iquid m etal S urfaceS
5.3.3.1 Free Sodium Surfaces
Free. liquid. surfaces. of. sodium. have. been. modeled. with. molecular. dynamics.
with.the.forces.on.the.atoms.determined.using.ensemble.DFT.and.cold.smearing.
[70-74].. The. motivation. cited. for. undertaking. simulations. on. sodium. included.