Density Functional Calculations of Metal Clusters: Structure, Dynamics, and Reactivity - Metallic Systems: A Quantum Chemist's Perspective - page 285

which.is.more.eficient.than.the.simplex.method.and.can.be.used.without.modiica-
tion.in.our.problem..The.input.for.the.algorithm.is.the.so-called.cost.matrix. F .whose.
elements.are.given.in.our.case.by

2

F AB = R ʹ − R . .

(9.54)

B A

.

And.the.output.is.the.mapping.between.the.elements.of. X .and. X ′

a A ( ) = a , a ∈ { , 1 2 3 … , , , }, N a ≠ a if A ≠ B , (9.55)

.

A A A B

.

which.minimizes.the.total.cost

N N

∑ ∑ ʹ

2

2

F R − R = d .

(9.56) A a A , ( ) a A ( ) A

.

A = 1 A = 1

.

ʹ

Because.only.the.same.kind.of.atoms.can.be.matched,.the.restriction.

( ) .holds.
here..This.splits.the.set.of.coordinates.into.classes.with.the.same.mass..The.assign-
ment.algorithm.is.applied.separately.to.each.of.these.classes..To.simplify.the.nota-
tion,.the.operator. â .is.deined.by.its.ordering.action.on.the. X ′.elements.as.follows:

m = m

a i i

ˆ

a X � � � � ={ R , R , R , … , R � }. (9.57)

.

a ( ) 1 a ( ) 2 a ( ) 3 a N ( )

.

Now,.the.distance.associated.with.the.mapping. â .is

N

∑

2 2

2

ˆ

d = a X ʹ − X = R ʹ − R . (9.58)

a A ( ) A

.

A = 1

.

It. is. important. to. realize. that. the. mapping. â,. while. best. for. the. two. sets. of. coor-
dinates. X . and. X ′,. is. not. necessarily. the. overall. best. alignment. of. the. molecules,.
because.external.translations.and.rotations.can.be.applied.to.any.of.the.two.sets.of.
coordinates..Thus,.the.mapping.found.using.a.particular.orientation.is.the.best.only.
in.the.neighborhood.of.this.orientation..An.example.of.this.situation.is.illustrated.in.
Figure. 9.2,. where. two. two-dimensional. molecules. consisting. of. ive. atoms. in. dif-
ferent. orientations. are. shown.. The. irst. molecule. is. identiied. with. empty. circles.
whereas.the. second. molecule.is. identiied. with. solid. circles.. As. Figure. 9.2. shows,.
in.the.irst.orientation.(left.side.of.Figure.9.2),.the.optimum.mapping,.indicated.by.
connection.lines,.is.different.from.the.mapping.given.in.the.second.orientation.(right.
side.of.Figure.9.2),.where.the.second.molecule.is.rotated.by.120°.with.respect.to.the.
irst.molecule.that.is.maintained.ixed..By.rotating.the.second.molecule.by.90°.with.
respect.to.the.irst.one,.the.perfect.superposition.of.the.two.molecules.is.reached,.as.
can.be.anticipated.from.the.left.side.of.Figure.9.2.
With.this.discussion.in.mind,.we.have.implemented.the.following.alignment.algo-
rithm. in. deMon2k.. Because. the. translation. that. minimizes. the. distance. given. by.