Tight Binding Methods for Metallic Systems - Metallic Systems: A Quantum Chemist's Perspective - page 228

8.2 FOUNDATIONS OF THE TIGHT BINDING METHOD
In.the.classic.paper.by.Slater.and.Koster.[4],.they.proposed.a.simpliication.to.the.
linear.combination.of.atomic.orbital.(LCAO).method.with.the.objective.of.interpo-
lating.results.from.irst.principles.band.structure.calculations.of.periodic.systems..It.
is.important.to.understand.that.in.the.1950s,.it.was.impossible.to.perform. ab initio .
computations.of.even.the.simplest.solid..Along.with.the.dramatic.increase.in.CPU.
speed. and. memory. size. over. the. last. two. decades. and. the. emergence. of. eficient.
algorithms. for. electronic. structure. calculations. that. have. eliminated. many. of. the.
original. barriers. to. performing. ab initio . calculations,. the. Slater-Koster. method.
has.evolved.as.a.powerful.approach.to.treat.molecular.and.solid.systems,.especially.
for.metals.and.semiconductors,.permitting.studies.on.these.systems.that.would.be.
impossible.at.higher.levels.of.theory.
Although.the.original.Slater-Koster.scheme.was.proposed.for.the.calculation.of.
the.electronic.bands.of.periodic.solids,.over.time.these.ideas.were.generalized.to.the.
more.robust.TB.method.for.calculation.of.the.total.energy.and.other.electronic.prop-
erties..Among.the.important.improvements.to.the.Slater-Koster.scheme.of.particular.
historical. importance. are. the. universal. TB. method. of. Harrison. that. incorporates.
interatomic.distance.dependence.in.the.classical.matrix.elements.[21].and.the.inclu-
sion. of. a. short-range. repulsive. potential. to. the. classical. sum. over. states. by. Chadi.
[22]..These.additions.served.to.expand.the.range.of.applicability.and.conidence.of.
the.original.TB.methods.
During.the.1980s,.TB.models.were.widely.used.in.the.solid.state.community.with.
great.success..However,.the.relationship.of.these.TB.models.to.more.fundamental.
theories. has. never. been. clear. and. for. this. reason. these. models. are. used. as. itting.
and.interpolation.schemes.rather.than.as.quantitative.computational.tools..In.1989,.
Foulkes.and.Haydock.[5].generalized.the.ideas.of.Harris.[6],.clearly.demonstrating.
that.the.simple.TB.models.were.approximations.to.the.Kohn-Sham.version.of.DFT.
[23]..Following.the.insight.of.Foulkes.and.Haydock,.a.new.generation.of.DFT-based.
TB.methods.appeared.in.the.literature.
In. this. section. we. introduce. the. classical. TB. method. as. found. in. the. works. of.
Slater.and.Koster.[4].and.of.Chadi.[22]..Then.we.discuss.the.DFT.basis.of.the.TB.
method.. Knowledge. of. these. two. methods. is. essential. in. order. to. understand. the.
modern.realizations.of.the.TB.method.which.are.an.evolution.of.the.theory.given.in.
the.earlier.works.
8.2.1 c laSSical t ight B inding m ethod
In.this.section.we.briely.explain.the.classical.TB.method.for.a.crystalline.solid..
The. interested. reader. is. referred. to. the. topic. of. W.A.. Harrison. “Elementary.
Electronic.Structure”.for.a.more.comprehensive.introduction.to.the.fundamentals.
of.TB.theory.[3].
We. begin. by. describing. the. Slater-Koster. method,. which. is. derived. from. the.
original.LCAO.method.of.Bloch.for.electronic.bands.[24]..Consider.a.periodic.unit.
cell.containing. n .atoms,.where.the.lattice.vectors.are.denoted.by. R ,.and.the. i th.atom.
is.located.at.position. r i .in.the.unit.cell..Each.atom.has.a.set.of.atomic-like.orbitals.φ il ,.