Chemistry Reference
In-Depth Information
The.trust.region.is.a.hypersphere.deined.by.|
p
|.≤.
h
,.where.the.scalar.
h
.>.0.is.called.
the.trust.region.radius..The.model.function.
q
.is.deined.by.the.second-order.Taylor.
series.expansion.around.the.local.origin
+
1
2
T
T
q
(
p
)
=
E
+
g p
p Bp
,
(9.49)
.
.
where
E
.is.the.value.of.the.objective.function,.here.the.energy
g
.is.the.gradient.vector
B
.is.the.corresponding. Hessian. matrix.or.an. approximation. to.it,.each. of. them.
evaluated.at.the.expansion.point
If.the.solution.
p
.of.Equation.9.48.does.not.produce.a.suficient.decrease.in.the.energy,.
the.trust.region.is.too.large.and.needs.to.be.reduced..After.the.reduction.of.the.trust.
region. radius,. Equation. 9.48. is. solved. again.. This. procedure. is. repeated. until. an.
acceptable.decrease.in.the.energy.is.achieved.
In.order.to.solve.(9.48),.the.following.Lagrange.functional.is.introduced
1
2
1
2
(
)
T
T
T
2
L
(
p
, λ
)
=
E
+
g p
+
p Bp
+
λ
p p
−
h
.
(9.50)
.
.
Here,.λ.represents.the.undeined.Lagrange.multiplier..From.the.stationary.condition.
of.this.Lagrange.function,.we.ind.the.step.direction.as
−
1
p
=
−
(
B
+
λ
I
)
g
.
.
(9.51)
.
For.the.determination.of.the.Lagrange.multiplier.λ,.a.diagonal.representation.is.used.
[63-66]..In.our.experience,.the.restricted.step.algorithm.(RSA).is.very.well.suited.
for. the. local. optimization. of. metal. clusters.. Nevertheless,. it. must. be. kept. in. mind.
that.minimization.is.only.guaranteed.in.an.RSA.with.an.exact.Hessian.calculation..
However,.in.most.cases,.such.an.approach.is.not.computationally.eficient..For.this.
reason,.we.usually.apply.a.quasi-Newton.RSA.followed.by.a.frequency.analysis..In.
case. imaginary. frequencies. are. detected,. the. quasi-Newton. RSA. is. restarted. with.
the.initial.Hessian.from.the.frequency.analysis..Even.in.complicated.cases,.such.an.
approach.usually.converges.to.minima.structures.within.one.or.two.restart.runs.[67]..
For.metal.clusters,.it.is.often.advisable.to.use.a.level-shift.[68].during.the.structure.
optimization.. For. this. purpose,. a. dynamical. level-shift. procedure. is. implemented.
in. deMon2k.. It. reduces. the. shift. factor. according. to. the. SCF. convergence.. This.
approach. improves. considerably. the. SCF. convergence. behavior. of. metal. systems.
far.away.from.local.minima.without.jeopardizing.the.description.of.the.electronic.
structure.
By.and.large,.the.local.optimization.of.stable.metal.cluster.isomers.has.become.a.
routine.procedure.with.the.above-described.quasi-Newton.RSA..However,.the.cor-
responding.optimization.of.transition.states.still.involves.many.challenges..For.this.
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