Tight Binding Methods for Metallic Systems - Metallic Systems: A Quantum Chemist's Perspective - page 232

of.the.parameters.is.usually.done.by.a.nonlinear.least-squares.method.in.which.the.
target. function. to. be. minimized. is. the. difference. between. eigenvalues. from. irst.
principle.calculations.and.the.TB.model.
Reference. [29]. is. a. comprehensive. source. of. Slater-Koster. TB. parameters,.
including.cases.of.orthogonal.and.non-orthogonal.orbitals.and.two-.and.three-center.
approximations.to.the.Hamiltonian.elements.for.the.crystal.structure.of.53.elements..
This.topic.also.contains.technical.details.and.computer.programs,.as.well.as.a.short.
discussion.of.the.trends.in.band.structures.along.the.periodic.table.
The.strengths.of.classical.TB.provide.great.incentive.to.ind.ways.to.overcome.its.
shortcomings..Since.these.shortcomings.are.related.to.the.empirical.character.of.the.
TB.model.and.the.tedious.process.of.parameterization,.it.is.natural.to.seek.a.way.to.
overcome.these.dificulties.by.deriving.TB.from.a.irst.principles.theory.
Classical.TB.methods.are.computationally.eficient.for.several.reasons:.the.basis.
set.is.very.small.and.thus.the.dimension.of.the.eigenproblem.is.reduced.comparing.
with.most.irst.principles.methods.(especially.plane.wave.methods);.the.matrix.ele-
ments.are.rapidly.evaluated.compared.with.the.expense.of.evaluating.large.numbers.
of.gaussian.integrals.in. ab initio .or.DFT.methods;.in.general,.classical.TB.methods.
do.not.iterate.the.wavefunction.to.self-consistency.which.may.result.in.considerable.
savings;.the.range.of.interaction.is.always.short.and.thus.the.matrices.are.typically.
sparse.which.permits.use.of.linear.scaling.methodologies..Any.improvement.to.the.
TB.method.should.retain.these.properties.

8.2.2 d enSity f unctional B aSiS of t ight B inding m ethodS
We.now.take.a.brief.look.at.the.fundamental.theoretical.basis.underlying.TB.theories.
Within.the.pseudopotential.local.density.functional.approximation.(PP-LDA),.the.
Kohn-Sham.energy.of.the.system.with.density. n .is.written.as

1
2

n r n r
r

( ) ( ʹ )

∫ ∫ ∫

E [ ] n = T n [ ] + V ( ) ( ) r n r dr + drdr ʹ + E [ n ] + E ion ion , (8.10)

KS S PP ion − xc −

| − r ʹ |

.

where. this. expression. includes. the. kinetic. energy. of. the. ictitious. non-interacting.
system,. T S [ n ],. the. interaction. of. the. electron. with. the. pseudopotential. and. ions,.
the. Hartree. electron-electron. interaction,. the. local. exchange. correlation. energy,.
E n n r dr

∫ . and. the. ion-ion. repulsion,. respectively.. The. electron. density. is.
given.as.a.sum.over.the.occupied.orbitals, . n r

xc [ ] ( )

ʹ

= ∑

* ( )

( ) 2 φ r φ ( ). r

. These.orbitals.are .

i i

i

.eigenfunctions.of.the.1-electron.Kohn-Sham.equations,

⎡
⎢

1
2

n r
r

( ) ⎤

⎥

∫

2

. (8.11)

− ∇ + V ( ) r + dr ʹ μ + ( ) n φ ( ) r = ε φ ( ), r

PP ion − xc i i i

| − r ʹ |

.

( ) ( = . . The.total.energy.of.Equation.8.10.can.be.rewritten.using.the .
Kohn-Sham.orbital.energies.to.yield

δε n

xc

where . μ n

xc

δ n