Tight Binding Methods for Metallic Systems - Metallic Systems: A Quantum Chemist's Perspective - page 229

where.the.index. l .denotes.the.orbital.angular.quantum.number..Although,.in.general,.
atomic.orbitals.on.different.atoms.are.not.orthogonal,.classical.TB.theory.assumes.
that.they.are.orthogonal..Each.atomic-like.orbital.is.associated.with.a.Bloch.orbital.
Φ il ( k , r ),.which.is.a.linear.combination.of.the.φ il .in.all. N .cells.of.the.crystal.( N .∼.10 24 ).
weighted.by.a.phase.factor. e ik·R ,.where. k .is.the.Bloch.wave.vector,

N

1

∑

. (8.1)

ik R ⋅

Φ il ( , ) k r = ϕ ( r + − r R e ) .

il i

N

.

R

These.orbitals.satisfy.the.Bloch.theorem:.that.is,.if. T R .is.a.translational.operator.along.
the.lattice,.then. T R Φ il ( k , r ).=. e ik·R Φ il ( k , r )..The.crystal.orbitals.Ψ j ( k , r ).are.expressed.as.
a.linear.combination.of.Bloch.orbitals.as.follows:

= ∑

. (8.2)

Ψ ( , ) k r C k ( ) Φ ( , ), k r

j j il , il

.

il

where. C j , il ( k ). are. crystal. coeficients. to. be. determined.. Clearly,. Ψ j ( k , r ). must. also.
satisfy.the.Bloch.theorem..The.energy.of.the. j th.crystal.orbital,.as.a.function.of.the.
wave.vector,.is.given.by

∑
∑

* ( )

C k H ( ) k C

ʹ ʹ ( )

k

Ψ ( ) | k H | Ψ ( ) k

j il , il i l , ʹ ʹ j i l ,

j j

il i l , ʹ ʹ

E k ( ) = = , . (8.3)

j

* ( )

Ψ ( ) | k Ψ ( ) k

C k S ( ) k C ( ) k j j

j il , il i l , ʹ ʹ j i l , ʹ ʹ

.

il i l , ʹ ʹ

where
H .is.the.Hamiltonian.of.the.solid,.which.is.translationally.invariant
H il , i ′ l ′ ( k ). and. S il , i ′ l ′ ( k ). are. the. matrix. representation. of. the. Hamiltonian. in. the.
Bloch.orbital.basis.set.and.the.overlap.integral.matrix.between.Bloch.orbitals,.
respectively

These.are.deined.by

. (8.4)

H = Φ ( ) | k H | Φ ( ) , k

il i l , ʹ ʹ il i l ʹ ʹ

.

. (8.5)

S il i l = Φ ( ) | k Φ ( ) . k

, ʹ ʹ il i l ʹ ʹ

.

The.dimension.of.these.two.matrices.is.equal.to.the.number.of.orbitals.in.the.unit.
cell..In.the.case.of.an.orthogonal.basis.of.atomic.orbitals,.as.is.the.case.in.classical.
TB.theories,.the.overlap.matrix.is.the.identity.matrix..The.coeficients. C j , il .and.crys-
tal.energies.are.obtained.by.solving.the.matrix.eigenvalue.problem,

. (8.6)

H k C k ( ) ( ) = E k S k C k ( ) ( ) ( ).

.

Since.all.matrix.elements.depend.on.the.wave-vector. k ,.we.solve.this.equation.for.a.
number.of. k -points.in.the.irst.Brillouin.zone.(the.unit-cell.of.the.reciprocal.lattice).