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occ
un
o
〈
k
||
ia ia m
G n f
〉〈
||
〉
∑
∑
∑
∑
−
1
B
≡
〈
|
[ ] |
ρ
l
〉 =
A F
,
(9.45)
mn
XC
k l
km
ml
ε
−
ε
i
a
.
i
a
m n
,
m
.
occ
uno
(
λ
)
〈
k
||
ia H
〉
∑
∑
(
λ
)
ia
b
≡
.
(9.46)
k
ε
−
ε
i
a
.
i
a
.
With.these.quantities,.Equation.9.43.can.be.recast.into
−
1
⎛
⎜
1
4
⎞
⎟
(
)
(
λ
)
(
λ
)
(
λ
)
(
λ
)
.
G
−
4
A
−
4
b x
=
4
b
⇔ =
x
G A B
−
−
b
(9.47)
.
.
Thus,.the.calculation.of.the.perturbed.itting.coeficients.boils.down.to.the.solution.
of.the.above.inhomogeneous.equation.system,.which.possesses.the.dimension.of.the.
auxiliary. functions.. A. direct. solution. is. feasible. even. for. large. systems.. With. the.
perturbed.itting.coeficients,.the.perturbed.Kohn-Sham.matrix.elements.can.be.cal-
culated.via.Equation.9.34..The.perturbed.density.matrix.elements.are.then.obtained.
by. the. corresponding. SCP. equation. (9.32).. Therefore,. the. inal. result. of. an. ADPT.
calculation.is.identical.to.the.result.of.a.coupled.perturbed.Kohn-Sham.calculation..
The.major.difference.is.the.improved.computational.performance.of.ADPT.calcula-
tions.due.to.the.use.of.the.perturbed.auxiliary.density.in.the.intermediate.steps.
9.2.5.3 Structure Optimization and Transition State Search
Central.to.the.theoretical.description.of.cluster.structures.is.the.Born-Oppenheimer.
approximation.[57],.which.enables.the.separation.of.electronic.and.nuclear.motions..
As.a.consequence,.a.potential.energy.can.be.assigned.to.each.cluster.structure..The.
resulting.potential.energy.surface.(PES).possesses.various.critical.points..Only.min-
ima.and.irst-order.saddle.points,.i.e.,.points.that.possess.one.negative.curvature.and.
all.other.curvatures.positive,. are.of. interest. to. us.. Because. the. PES.is.constructed.
from.solutions.of.the.electronic.Schrödinger.equation,.it.is.usually.not.available.in.
analytic.form..This.situation.contributes.considerably.to.the.challenge.of.the.theo-
retical.study.of.metal.clusters..Here,.theoreticians.face.the.formidable.task.of.locat-
ing.minima.on.the.PES.without.any.further.information.except.the.number.of.atoms.
in.the.metal.cluster..Thus,.it.does.not.come.as.a.surprise.that.(metal).cluster.science.
presents.a.strong.driving.force.for.the.development.of.local.and.global.optimization.
algorithms..Despite.considerable.effort,.global.minimization.algorithms.[58-61].for.
irst.principles.electronic.structure.methods.are.still.at.a.very.crude.developmental.
stage..Thus,.for.metal.clusters,.local.optimization.algorithms.based.on.quasi-Newton.
[62].restricted.step.methods.[63].are.most.often.employed..Here,.the.maximum.step.
length.is.restricted.to.a.trust.region.
h
..Within.the.quadratic.approximation,.the.mini-
mization.problem.then.reads
min (
q
p
)
subject to ʺ
|
p
|
h
.
(9.48)
p
.
.
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