Chemistry Reference
In-Depth Information
on. two. assumptions:. (i). using. an. ordinary,. ground. state. TB. calculation,. one. can.
obtain.information.similar.to.that.obtained.from.Kohn-Sham.DFT.for.the.difference.
between.the.occupied.and.virtual.orbitals,.and.(ii).the.TDDFT.Coulomb-exchange-
correlation. kernel. can. be. modeled. with. the. same. kind. of. approximation. as. in. the.
ground.state.TB..As.in.the.case.of.the.SCC-EHTB.model,.no.exhaustive.parameter-
ization.is.performed;.therefore,.we.expect.our.results.to.be.qualitatively.correct.and.
provide.a.guide.to.the.optical.properties.of.the.materials.studied.
In.our.work.we.use.the.method.proposed.by.Casidas.in.the.context.of.LR-TDDFT.
[67].. In. this. formalism,. the. excitation. energies,. ϖ
I
,. are. obtained. by. solving. the.
following.eigenvalue.problem.(for.closed.shells):
∑
⎣
⎦
2
I
2
I
.
.
(8.37)
ϖ δ δ δ
+
2
ϖ
K
ϖ
F
=
ϖ
F
ij
ik
jl
στ
jk
ij
σ τ
,
kl
kl
ij
α
I
kl
τ
ijkl
α
.
In.this.equation,.σ.and.τ.are.spin.indices,.
i
.and.
k
.denote.occupied.orbitals.(holes),.
j
.and.
l
.are.virtual.orbitals.(particles),.ϖ
ij
.=.ε
i
.−.ε
j
,.is.the.energy.difference.between.the.
one-particle.Kohn-Sham.orbitals,.and.
K
ij
σ,
kl
τ
.are.coupling-matrix.elements,.where
⎛
⎝
2
⎞
⎠
1
δ
δρδρ
E
*
( )
*
(
∫∫
xc
.
(8.38)
K
=
ψ
r
ψ
( )
r
+
ψ
r
ʹ
)
ψ
,
(
r drdr
ʹ
)
ʹ
.
⎜
⎟
ij
σ τ
,
kl
i
,
σ
j
,
σ
k
,
τ
l
τ
|
r
−
r
ʹ
ʹ
.
Using.the.same.approximation.employed.in.the.derivation.of.the.EHTB.formalism,.
and. deining. the. magnetization. as. the. difference. of. the. spin. densities.
m =
ρ
α
.−.ρ
β
,.
K
ij
σ,
kl
τ
.can.be.cast.in.the.form
∑
ij
kl
K
=
q q
⎣
γ
+
(
2
δ
−
1
)
m
⎦
ij
σ τ
,
kl
α β
αβ
στ
αβ
αβ
∑
∑
(
)
Γ ,
ij
kl
ij
kl
ij
kl
ij
kl
+
P P
+
P P
+
P P
+
P P
.
(8.39)
μ
v
μ
v
μ
v
v
μ
v
μ μ
v
v
μ
v
μ
μ
v
.
α
(
μ
> ∈
v
)
α
In.Equation.8.16.we.introduced.the.transition.density.matrix,
ij
,
.
(8.40)
P
=
C C
μ
v
μ
i
vj
.
and.the.Mulliken.transition.charges
1
2
∈
∑
(
)
ij
ij
ij
.
.
(8.41)
q
=
P S
+
P S
α
μ
n n n n
μ
μ
μ
μ α
.
n
Equation.8.16.also.contains.the.following.two.electrons.integrals,
⎛
⎜
⎞
⎟
1
δ
δρδρ
2
E
∫∫
F r
.
(8.42)
γ
α
xc
β
=
( )
− ʹ
+
F r drdr
(
ʹ
)
ʹ
,
αβ
r
r
ʹ
.
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