Tight Binding Methods for Metallic Systems - Metallic Systems: A Quantum Chemist's Perspective - page 231

. 4.. Another.assumption.in.TB.theories.is.that.the.Hamiltonian.matrix..elements,.

H il , i ′ l ′ ,. only. depend. on. atomic. centers. in. which. orbitals. ( il ). and. ( i ′ l ′). are.
localized.. This. approximation. is. known. as. the. “two-center. approxima-
tion.”.We.note.that.the.Hamiltonian.also.contains.three.center.terms,.such.
as.the.Coulomb.terms,.which.are.neglected.in.TB.models..Slater.and.Koster.
showed.that.all.two-center.integrals.depend.only.on.the.angular.momentum.
of.the.orbitals.and.the.distance.between.the.atoms.on.which.these.orbitals.
are.located..For.an.atom.with.a.spd.orbital.basis.(e.g.,.a.metal),.10.indepen-
dent.parameters.are.required.for.the.computation.of.their.matrix.elements:.
ss σ,. sp σ,. pp σ, pp π,. sd σ,. pd σ,. pd π,. dd σ,. dd π,.and. dd δ..Using.these.param-
eters,.all.possible.integrals.within.a.spd.basis.can.be.computed.using.the.
Slater-Koster.scheme.

All. these. approximations. can. be. relaxed. and. improved. in. order. to. improve. the.
quality.of.the.TB.model.and,.hopefully,.the.accuracy.of.the.calculations..Although.
the.original.Slater-Koster.scheme.was.developed.for.the.calculation.of.the.band.
structure.of.periodic.solids,.it.was.not.obvious.how.to.generalize.the.problem.to.
the. calculation. of. the. total. energy. of. the. system.. In. 1979,. Froyen. and. Harrison.
proposed.an. r −2 .dependence.of.the.matrix.elements.of.the.Hamiltonian.with.vary-
ing. interatomic. distances. [21].. In. the. same. year,. Chadi. applied. this. method. to.
the.calculation.of.the.surface.energy.of.semiconductors.[22]..In.his.work,.Chadi.
proposed.writing.the.TB.total.energy.as.a.function.of.the.atomic.coordinates.in.
the.form.[22]

∑

dk

∑ >

∫

. (8.9)

E = E k ( ) + E ( , ) α β

TB j REP

3

( 2 π )

j α β

.

BZ

where.the.irst.term.on.the.right-hand.side.is.due.to.the.TB.band.energies.given.by.
Equation. 8.6,. the. summation. includes. all. occupied. electronic. eigenstates. and. the.
integration.is.over.the. k .points.in.the.irst.Brilloiun.zone.(BZ)..The.second.term.is.a.
contribution.from.all.atom.pairs.and.includes.a.short-range.repulsive.potential.that.
depends.on.atom.type..The.repulsive.interaction.is.a.parametric.functional.derived.
to.reproduce.the.cohesive.energies.and.elastic.constant.of.some.reference.crystalline.
materials.[23].
Total. energy. TB. calculations. depend. strongly. on. the. parameterization. scheme.
used. for. the. band. structure. and. the. repulsive. potential,. and. the. transferability. in.
many.cases.is.very.poor..Over.the.years,.non-orthogonal.TB.has.been.proposed.as.
one.important.means.to.achieving.more.transferable.parameterizations.[25,26].
In. order. to. reduce. the. effort. required. for. parameterization,. more. sophisticated.
and. eficient. TB. schemes. have. been. developed.. Of. particular. importance. are. the.
linear-mufin-tin-orbital.(LMTO).method.of.Andersen.and.Jepsen.[27].and.the.DFT.
parameterization.of.Cohen,.Mehl,.and.Papaconstantopoulos.[28].
In.earlier.work,.TB.schemes.were.used.as.an.interpolation.method.to.reproduce.
the.eigenvalue.spectrum.over.a.large.number.of. k .points..This.procedure.is.very.use-
ful.for.the.calculation.of.densities.of.states.and.Fermi.surfaces..The.determination.