Tight Binding Methods for Metallic Systems - Metallic Systems: A Quantum Chemist's Perspective - page 247

2

δ
δ δ

E
m m F r drdr ∫∫

. (8.43)

α xc β

ʹ ʹ

m = F r ( ) ( ) ,

αβ

ʹ

.

⎛ ⎞

2

1 δ

δρδρʹ

E

∫∫

* ( ) * (

Γ μ

. (8.44)

xc

= φ r φ ( ) r φ r ʹ φ ) ( r drdr ʹ ) ʹ .

⎜ ⎟ μ

n n n

μ

r − r ʹ

⎝ ⎠

.

The.integral.in.Equation.8.43.is.equivalent.to.Equation.8.31.with.the.inal.density.ρ.
replacing.the.reference.density.ρ 0 ..If.the.charge.luctuation.is.suficiently.small.that.
it.can.be.neglected,.it.is.possible.to.use.the.values.of.the.ground.state.γ.integrals.as.
a. zero-order. approximation. in. Equation. 8.43.. Similarly,. the. integrals. in. Equation.
8.43.involve.atomic.quantities.that.can,.in.principle,.be.obtained.from.DFT.calcula-
tions.. However,. in. keeping. with. the. spirit. of. the. semiempirical. method,. we. com-
pute. these integrals. using. Slater-Condon. spectroscopic. parameters.. Furthermore,.
the. integral. in. Equation. 8.44. does. not. involve. a. long-range. Coulombic. term,. and.
therefore.is.approximated.by.an.on-site.parameter.obtained.from.atomic.DFT.cal-
culations..The.excitation.energies,. w I ,.are.obtained.from.the.eigenvalues.of.Equation.
8.37.. Despite. the. considerable. simpliication. introduced. in. the. evaluation. of. the.
TDDFT.response.kernel,.direct.diagonalization.for.systems.with.a.large.number.of.
particle.and.hole.states.remains.the.main.computational.bottleneck..In.our.program,.
we.make.use.of.the.DSYEVR.routine.implemented.in.LAPACK.(http://www.netlib.
org/lapack/).[86]..In.this.routine,.the.matrix.is.reduced.to.a.tridiagonal.form,.and.
the.eigenspectrum.is.computed.using.the.multiple.relatively.robust.representations.
method. where. Gram-Schmidt. orthogonalization. is. avoided. to. the. greatest. extent.
possible.
Once.the.excitation.energies.are.known,.the.corresponding.oscillator.strength.for.
the. I th .transition.(in.atomic.units).can.be.obtained.from.[67]

( )

2
3

* * *

I − 1 2 / I − 1 2 / I − 1 2 / I

. (8.45)

f = ϖ X S F + Y S F + Z S F .

I

.

where
( X , Y , Z ).represents.the.transition.dipole.vector
F I .are.the.eigenvectors.obtained.by.solving.Equation.8.37

In.Equation.8.45,. S .is.a.diagonal.matrix.deined.as

σ δ δ δ
ε

ik jl

. (8.46)

S ij = .

σ τ , kl

− ε

.

l k

This. formalism. yields. a. list. of. discrete. excitation. energies. with. their. associated.
dipole.transition.intensities..A.continuous.spectrum.can.be.drawn.by.convoluting.the.
discrete.spectrum.with.a.Gaussian.function.
Before.we.discuss.an.application.of.the.present.methodology,.it.is.useful.to.check.
the. validity. of. the. scheme.. We. used. our. method. to. reproduce. previous. results. of.