Chemistry Reference
In-Depth Information
dk
dk
∑
(
∑
∫
∫
.
E
=
E k
( )
+
N V
=
⎣
E k
( )
+
V
⎦
.
(8.17)
TB
j
e
0
j
0
3
3
2
π
)
(
2
π
)
j
j
.
BZ
BZ
If. we. now. deine. shifted. eigenvalues. with. the. form.
E k
ʹ( ) ( = +
0
,. the. total.
energy. is. just. a. sum. over. the. shifted. occupied. eigenvalues.. Cohen,. Mehl,. and.
Papaconstantopoulos.[28].note.that.the.shifted.energies.are.in.some.sense.“univer-
sal”..That.is,.if.two.band.structure.calculations.are.suficiently.well.converged,.they.
will.have.the.same.total.energy.and.their.eigenvalues.will.differ.by.a.constant..Thus,.
in.the.NRL-TB.method,.the.total.energy.is.calculated.directly.from.the.shifted.eigen-
values.without.resorting.to.some.short-range.repulsive.potential.
In.the.NRL.methodology,.a.irst.principles.database.of.band.structures.and.total.
energies.is.constructed.for.several.crystal.structures.at.different.unit.cell.volumes..
Then,.a.nonlinear.parameterization.is.used.to.generate.non-orthogonal.two-center.
TB. Hamiltonian. and. overlap. matrix. elements. that. (approximately). reproduce. the.
energies.and.eigenvalues.of.the.database.
Of.particular.importance.to.the.success.of.the.NRL-TB.method.is.the.functional.
form.of.the.Hamiltonian.and.overlap.matrix.elements..It.is.assumed,.in.contrast.with.
other.DFT.based.TB.methods,.that.the.diagonal.terms.of.the.Hamiltonian.are.sensi-
tive.to.the.local.chemical.environment..For.single.element.systems,.each.atom.in.the.
crystal.is.associated.with.an.embedded.atom-like.density
E k
V
j
j
∑
exp(
.
(8.18)
2
ρ
=
−
λ
r F r
)
(
),
i
ij
ij
j
.
where.the.summation.extends.over.all.neighbors.of.atom.
i
.within.a.range.
r
cut
,.λ.is.a.
itting.parameter,.and.
F
(
r
).is.a.cut-off.function
θ
(
r
−
r
r
)
.
(8.19)
cut
F r
( )
=
⎡
⎢
−
r
⎤
⎥
⎛
⎜
⎞
⎟
cut
1
+
exp
.
and.θ(
z
).is.a.step.function..Typical.values.for.
r
cut
.lie.between.10.5
a
0
.and.16.5
a
0
.and.
ℓ
.
lies.between.0.25
a
0
.and.0.5
a
0
.
In.the.NRL-TB.method,.the.diagonal.on-site.terms.of.the.Hamiltonian.are.deined.
as.a.polynomial.expansion.in.the.embedded.density.by
2 3
/
4 3
/
2
.
(8.20)
H
=
a
+
b
ρ
+
c
ρ
+
d
ρ
.
.
il il
,
l
l
i
l
i
l
i
In. the. spirit. of. the. two-center. approximation,. the. non-diagonal. Hamiltonian. ele-
ments. are. expanded. in. an. exponential. polynomial. form. with. rotations. taken. into.
account.by.means.of.the.Slater-Koster.formalism.[4],
(
)
(
)
2
2
H
(
r
)
=
e
+
f
r
+
g
r
ex
p
−
h
r F r
(
).
.
(8.21)
il i l
,
ʹ ʹ
ii
ʹ
il i l
,
ʹ ʹ
il i l
,
ʹ ʹ
ii
ʹ
il i l
,
ʹ ʹ
il i l
,
ʹ ʹ
il i l
,
ʹ ʹ
ii
ʹ
ii
ʹ
.
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