Tight Binding Methods for Metallic Systems - Metallic Systems: A Quantum Chemist's Perspective - page 251

8.5.3 m olecular c onductance
Computation.of.the.electrical.current.in.a.molecule.as.a.function.of.the.applied.volt-
age. is. of. fundamental. interest. in. studies. of. molecular. electronics.. The. most. rigor-
ous. approach. to. this. problem. involves. use. of. a. non-equilibrium. Green's. function.
[76].(NEGF).formalism.with.an. ab initio .electronic.structure.method.(such.as.DFT)..
Of.course,.a.TB.scheme.may.be.used.for.the.electronic.structure.part.of.the.NEGF.
method,.and.a.number.of.implementations.of.this.type.exist.[77]..The.reader.interested.
in.the.NEGF.method.and.it.application.is.encouraged.to.consult.the.literature.[76].
In.this.section.we.will.address.a.related,.but.considerably.simpler.problem..We.
will.compute.Green's.functions.at.zero.bias.voltage.for.benzene.dithiol.using.DFTB,.
EHTB,.and.DFT.at.the.B3LYP/cc-pvdz.level.of.theory..Since.accurate.computation.
of.the.Green's.function.is.essential.for.obtaining.reliable.results.from.NEGF.calcula-
tions,.study.of.this.quantity.will.yield.some.insight.into.the.appropriateness.of.using.
a.TB.method.with.the.NEGF.formalism.
The.Green's.function.of.interest.is.calculated.using.[78]

N N N

orb L R

c * S * c S

∑ ∑ ∑

i μ i μ j μ j μ

. (8.47)

G E ( ) = ,

( E − ε − is )

μ

.

μ = 1 i = 1 j = 1

where. G ( E ). is. the. Green's. function. at. energy. E ,. the. sums. are. over. the. number. of.
orbitals.( N orb ).and.the.atomic.centers.included.in.the.left-.and.right-hand.fragments.
( N L ,. N R ). of. the. molecule. (a. subset. of. orbitals. included. in. the. calculation),. c i μ . is. a.
molecular.orbital.coeficient,. S i μ .is.an.element.of.the.overlap.matrix,.ε μ .is.the.energy.
of.orbital.μ,.and. s .controls.the.magnitude.of.the.imaginary.term..Note.that.this.is.a.
modiication.of.the.original.method.presented.by.González.et.al..[78].as.we.sum.over.
all.orbitals.on.a.given.atomic.center.and.thus.include.the.overlap.in.the.computation..
Computation. of. the. Green's. function. requires. knowledge. of. the. molecular. orbital.
coeficients.and.the.overlap.matrix,.quantities.that.are.easily.extracted.from.many.
computational.codes.
The. Green's. function. was. calculated. in. the. manner. described. above. from. data.
extracted. from. the. DFTB. [79],. EHTB,. and. Gaussian. [80]. programs.. The. square.
modulus.of.the.Green's.function.versus.energy.is.presented.in.Figure.8.6..We.note.
that.the.TB.methods.show.reasonable.qualitative.agreement.for.the.peak.at.∼4.5.eV,.
but.miss.the.structure.above.10.eV..The.EHTB.results.show.the.largest.peak.closer.
to.4.5.eV.versus.the.DFTB.results,.but.also.show.more.structure.around.that.peak.
compared. to. the. B3LYP. results.. The. differences. between. the. TB. results. and. the.
DFT results.are.not.so.surprising.when.we.consider.that.the.TB.basis.set.is.much.
smaller.than.the.cc-pvdz.basis.set.used.for.the.B3LYP.calculation.and.thus.cannot.
show.the.same.level.of.detail..In.order.to.assess.the.effect.of.the.size.of.the.basis.
set,.we.performed.a.calculation.at.the.B3LYP/STO-3G.level.of.theory..The.Green's.
function.at.this.level.of.theory.differs.signiicantly.from.the.B3LYP/cc-pvdz.results.
and.from.the.TB.results.as.well..We.conclude.that.the.parameterization.of.the.TB.
methods. allows. them. to. reproduce. some. of. the. structure. of. the. Green's. function.
based.on.B3LYP/cc-pvdz.results,.and.that.the.spectra.show.less.detail..The.loss.of.