Tight Binding Methods for Metallic Systems - Metallic Systems: A Quantum Chemist's Perspective - page 239

⎧

⎪
⎪

α

F r R ( | − | ) if

if

μ = v , μ α ∈ ,

α

* ( )

φ r φ ( ) r = φ *( ) r φ ( ) r μ ≠ v , μ α ∈ , v ∈ α ,

⎨

μ v v

μ

⎪
⎪

1
2

⎣

α β

⎦

F r R ( | − | ) + F r R ( | − | ) S if μ ≠ v , μ α ∈ , v ∈ β ,

α β μ v

⎩

. (8.26)

where. F α (| r − R α |).is.a.spherical.radial.approximation.for.the.density.of.atom.α..Upon.
introduction.of.Equation.8.26,.the.second.term.of.Equation.8.14.becomes,

⎛ ⎞

1
2

∑ ∑ ∑

⎜
⎜

2

⎟
⎟

δ δ γ q q + δ P v Γ , .

α β αβ μ μ v

(8.27)

⎝ ⎠

αβ α ( μ > ∈ v ) α

.

where.we.deine.the.luctuation.in.the.Mulliken.charges.as

1
2 ∑ ∈ ∑

0

δ q = q − q = ( δ P S + δ P S ), .

(8.28)

α α α μ v μ v v μ v μ

.

μ α v

and

⎛
⎜

2

⎞
⎟

1 δ

δρ δρ

E

∫∫ F r

ʹ .

(8.29)

α xc β

γ = ( ) + F r drdr ( ʹ ) ,

αβ

| r − r ʹ ʹ

0 0

.

⎛
⎜

2

⎞
⎟

1 δ

δρ δρ

E

∫∫

* ( ) * (

. (8.30)

xc

Γ μ = φ r φ ( ) r + φ r ʹ ) φ ( r drdr ʹ ) ʹ.

v μ v μ v

| r − r ʹ ʹ

0 0

.

Although.the.integrals.in.Equations.8.29.and.8.30.could.be.computed. ab initio ,.in.the.
present.implementation.both.are.given.a.value.from.semi-empirical.approximations..
In.the.case.of.γ αβ .we.use.the.expression.[16],

1

. (8.31)

γ ( R ) = ,

αβ αβ

− 2

R

2

+ γ ( ) 0

αβ αβ

.

where. R αβ .is.the.distance.between.atomic.centers.α.and.β..The.value.of.this.inte-
gral.in.the.limit. R αβ .→.0.is.approximated.as.the.average.between.the.two.centers.
γ

1
2

+ ( ) ..The.γ αα (0).are.approximated.as.the.chemical.hardness.of.
the.neutral.atoms..For.the.evaluation.of.Γ μ v .integrals.we.use.the.spectroscopic.values.
of.the.Slater-Condon.parameters.as.in.the.ZINDO.semiempirical.method.[41,42].
The. last. term. of. Equation. 8.14. is. the. double. counting. term. and. the. ion-ion.
repulsion,. which. is. approximated. by. a. two. center. expansion. in. the. form.

( ) 0 γ ( ) 0 γ ( ) 0

αβ αα ββ

U αβ . .

∑

αβ