Exploring the Borderland between Physics and Chemistry: Theoretical Methods in the Study of Atomic Clusters - Metallic Systems: A Quantum Chemist's Perspective - page 383

and.even.the.more.robust.natural.bond.order.method.can.fail.in.cases.of.extremely.
delocalized.bonds.that.are.common.in.atomic.clusters..One.of.the.most.useful.theo-
retical.tools.for.analyzing.bonding.in.atomic.clusters.is.the.ELF.
The.ELF.was.originally.proposed.by.Becke.and.Edgecombe.[42].and.is.deined.as

− 1

⎡

2

⎤

+ ⎛

T [ ( )]

[ ( )]

ρ
ρ

r ⎞

⎟

⎢
⎢

⎥
⎥

ELF( ) r = 1 . (11.2)

⎜ T r

0

⎣ ⎦

.

where. T [ρ( r ‾ )].=. T s [ρ( r ‾ )].−. T w [ρ( r ‾ )]isthe.kinetic.energy.density.difference.betweenthe.
kinetic.energy.density.of.the.noninteracting.system. T s [ρ( ˙ )].with.density.ρ( r ‾ ),.and the.
von.Weizsacker.kinetic.energy.density,. T w [ρ( r ‾ )]..In.addition,. T 0 [ρ( r ‾ )].indicates.the.
kinetic.energy.density.of.the.noninteracting.electron.gas..The.important.quantity.in.
Equation.11.2.is.the.term. T [ρ( r ‾ )].that.can.be.interpreted.[43].as.the.excess.kinetic.
energy.density.due.to.the.Pauli.exclusion.principle..A.simple.analysis.of.Equation.
11.2.indicates.that.the.function.ELF( r ‾ ).varies.between.0.and.1..The.interpretation.
of.the.function.is.that.the.region.of.the.space.where.the.ELF.has.a.value.close.to.1.
corresponds.to.the.regions.where.it.is.most.probable.to.ind.a.localized.electron.pair;.
regions.with.a.low.value.of.the.ELF.(≈0.5).correspond.to.regions.where.the.electrons.
are.delocalized..Hence,.the.interpretation.of.ELF.isosurfaces.allows.us.to.understand.
bonding.in.clusters..For.a.more.detailed.description.of.the.ELF,.see Ref..[42].
As.an.example.of.the.utility.of.the.ELF,.we.show.in.panel.(a).of.Figure.11.5.the.
position. of. the. atoms. of. one. of. the. isomers. of. the. cluster. of. Li 6 .. The. lines. in. this.

(a)

(b)

FIGURE 11.5 (a).Geometry.and.(b).ELF.isosurface.for.the.lithium.cluster.with.six.atoms.