Chemistry Reference
In-Depth Information
To.obtain.a.TB.expression.for.the.Hamiltonian.and.overlap.matrix,.we.write.the.
Bloch.orbitals.in.term.of.the.localized.atomic.orbitals,.obtaining
H
( )
k
=
ϕ
( ) |
r H
|
ϕ
( )
r
jl j l
,
ʹ ʹ
jl
j l
ʹ ʹ
∑
⎣
ik R
⋅
−
i
k R
⋅
⎦
+
e
ϕ
(
r R H
−
) |
|
ϕ
( )
r
+
e
ϕ
( ) |
r H
|
ϕ
(
r R
−
)
,
.
jl
j l
ʹ ʹ
jl
j l
ʹ ʹ
.
R
(8.7)
S
jl j l
( )
k
=
ϕ
( ) |
r
ϕ
( )
r
,
ʹ ʹ
jl
j l
ʹ ʹ
∑
⎣
ik R
⋅
− ⋅
ik R
⎦
+
e
ϕ
(
r R
−
) |
ϕ
( )
r
+
e
ϕ
jl
( ) |
r
ϕ
ʹ ʹ
−
(
r R
)
.
.
(8.8)
jl
j l
ʹ ʹ
j l
.
R
In.these.equations,.the.irst.term.on.the.right-hand.side.corresponds.to.the.Hamiltonian.
and.overlap.integrals.between.two.atomic.orbitals.that.are.located.in.the.same.unit.
cell..This.term.does.not.depend.on.the.wave.vector..The.second.term,.called.the.dis-
persion.term,.depends.on.the.Hamiltonian.and.overlap.integrals.between.two.atomic.
orbitals.located.in.two.unit.cells.separated.by.a.lattice.vector.
R
.
In. order. to. further. develop. TB. theory,. a. number. of. approximations,. some. of.
which.date.from.the.original.work.of.Slater.and.Koster.[1],.are.made:
.
1.. TB.methodologies.are.classiied.as.orthogonal.or.non-orthogonal.depend-
ing. on. the. orthogonality. of. the. atomic. orbitals. φ
jl
.. In. early. TB. methods,.
such. as. the. Slater-Koster. method. [4],. the. atomic. orbitals. are. taken. to. be.
orthogonal.. The. assumption. of. orthogonality. considerably. simpliies. the.
eigenvalue.problem.in.Equation.8.6,.because.the.overlap.matrix.is.the.iden-
tity.matrix..However,.it.has.become.clear.that.orthogonal.TB.theories.are.
less. transferable,. for. the. obvious. reason. that. the. atomic. orbitals. between.
two.atoms.are.not,.in.general,.orthogonal.
.
2.. It.is.assumed.in.the.TB.method.that.only.contributions.from.valence.orbital.
electrons.are.included.in.the.Hamiltonian.and.overlap.matrices..Thus.atoms.
containing.valence.d.orbitals,.as.is.the.case.for.metal.atoms,.have.a.valence.
basis.of.1s,.3p.(p
x
,.p
y
,.p
z
)-.and.5d.orbitals.(d
xy
,.d
yz
,.d
zx
,.d
x
2
-
y
2
,.d
3
z
2
-
r
2
).for.a.total.
of.nine.orbitals..Atoms.such.as.carbon.which.are.represented.by.a.sp.orbital.
basis.have.1s.and.3p.orbitals.for.a.total.of.four.orbitals..This.gives.rise.to.a.
signiicant.reduction.in.computational.expense.compared.to.other.schemes.
commonly.employed.in.irst.principles.calculations.
.
3.. Another. common. approximation. concerns. the. number. of. neighbor. cells.
included. in. the. summation. of. the. matrix. elements. of. Equations. 8.7. and.
8.8..Classical.TB.theories.are.dominated.by.the.so-called.“irst.neighbor”.
approximation,.in.which.the.summation.over.lattice.vector.is.extended.only.
over.adjacent.unit.cells..This.leads.to.a.signiicant.reduction.in.the.number.of.
non-zero.matrix.elements..This.property.can.be.exploited.to.study.larger.sys-
tems,.to.reduce.the.computer.time.required.to.compute.a.solution,.or.both.
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