Chemistry Reference
In-Depth Information
reason,.we.recently.developed.a.hierarchical.transition.state.inder.[38]..It.consists.
of.a.double-ended.saddle.interpolation.method.[69],.which.is.followed.by.an.uphill.
trust.region.optimization.[70,71]..To.enhance.the.stability.of.the.saddle.interpolation.
method,.we.included.the.distance.constraint.between.reactants.and.products.by.an.
undeined. Lagrange. multiplier. in. the. quadratic. model. function. rather. than. using.
modiied.gradients.as.suggested.elsewhere.in.the.literature.[72]..The.resulting.formu-
las.closely.resemble.their.trust.region.counterparts..Thus,.the.developed.algorithms.
for. the. solution. of. the. trust. region. subproblem. can. also. be. applied. to. the. double-
ended.saddle.interpolation.method..In.the.case.of.metal.clusters,.chemical.intuition.
most. often. fails. for. the. description. of. cluster. rearrangements.. For. this. reason,. the.
hierarchical. transition. state. inder. represents. signiicant. progress. because. it. needs.
only.reactant.and.product.structures..However,.these.structures.must.be.aligned.to.
maximum.coincidence.in.order.to.guarantee.the.success.of.the.hierarchical.transi-
tion.state.inder..For.this.reason,.an.automatic.alignment.procedure.for.reactant.and.
product.structures.[73].was.implemented.in.deMon2k.
The. basic. idea. of. this. automatic. alignment. is. the. selection. of. equivalent. atoms.
according.to.the.best.alignment.of.both,.reactant.and.product.molecules..In.the.lit-
erature,.several.methods.are.suggested.[74-78]..However,.to.the.best.of.our.knowl-
edge,.all.of.them.suffer.from.a.labeling.problem..In.other.words,.they.cannot.provide.
a. unique. measurement. of. the. distance. between. two. different. structures. with. arbi-
trary.atomic.labeling..For.this.reason,.we.developed.a.new.superimposing.algorithm,.
which.is.capable.of.performing.such.measurements.[73].
For.this.purpose,.we.represent.a.molecule.by.a.set.of.
N
.vectors.
r
A
.containing.the.
spatial.coordinates.of.
N
.nuclei..In.our.tests,.better.results.in.the.alignment.of.reac-
tants.and.products.were.achieved.using.mass-weighted.coordinates.instead.of.simple.
coordinates,.i.e.,.when.
r
A
.is.replaced.by.
m
A
.
r
A
..Deining.
R
A
.=.
m
A
.
r
A
,.the.two.sets.of.
vectors.that.represent.the.molecules.are
X
=
{
m m
r
,
r
,
m
r
,
…
,
m
r
} { ,
=
R R R
,
,
…
,
R
},
1 1
2 2
3 3
N N
1
2
3
N
.
(9.52)
ʹ
ʹ
ʹ
ʹ
X
ʹ =
{
m
r
ʹ
,
m
r
ʹ
,
m
r
ʹ
,
…
,
m
r
ʹ =
} { ,
R R R
ʹ
ʹ
,
ʹ
,
…
,
R
ʹ
}.
.
1 1
2
2
3 3
N N
1
2
3
N
The.deinition.of.the.metric.used.to.measure.the.separation.between.two.structures.
is.well.known.from.the.crystallographic.literature.[79]
N
∑
2
2
2
d
=
X
ʹ −
X
=
R
ʹ −
R
.
(9.53)
A
A
.
A
=
1
.
At.irst.glance,.it.seems.easy.to.superimpose.two.molecules.once.the.metric.deini-
tion.is.given..However,.this.is.not.the.case.because.the.order.of.
X
.and.
X
′.elements.
that.minimizes.the.distance.in.Equation.9.53.is.unknown..Fortunately,.the.task.of.
determining. the. optimum. assignment. between. the. elements. of. two. sets. is. a. well-
studied.problem.in.linear.optimization.theory.[80]..Kuhn.and.Munkres.[81,82].devel-
oped.a.specialized.algorithm,.related.to.the.simplex.method,.for.the.solution.of.the.
general. assignment. problem.. This. method. is. known. as. the. Hungarian. algorithm,.
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