Density Functional Calculations of Metal Clusters: Structure, Dynamics, and Reactivity - Metallic Systems: A Quantum Chemist's Perspective - page 286

120 ˚

5

1

5

5

2

4

4

4

1

1

3

4

2

3

1

3

2

3

2

5

FIGURE 9.2 Two-dimensional. example. of. mappings. with. different. orientations. of. the.
.molecule..(From.Vasquez-Perez,.J..et.al.,. J. Chem. Phys .,.131(12),.124126-1..With.permission.)

Equation.9.53.is.known.analytically.and.it.is.independent.of.the.mapping.used,.we.
apply.it.irst..Therefore,.the.two.sets.of.coordinates.are.transformed.initially.as

X = { , R R R , ,..., R } − R ,

1 2 3 N CM

. (9.59)

X ʹ = { 1 R R R ʹ , ʹ , ʹ ,..., R ʹ } − R ,

.

2 3 N CM

with

N N

1

∑ ∑

R = R and M = m .

CM A A

M

.

A = 1 A = 1

In.these.equations,. R CM .is.the.center.of.mass..On.the.other.hand,.the.optimum.rota-
tion. that. minimizes. Equation. 9.53. is. mapping. dependent,. but. can. easily.be. calcu-
lated.in.a.semi-analytical.fashion.using.the.quaternion.representation.of.rotations..In.
this.representation,.the.optimum.rotation.is.given.by.the.eigenvector. q .=.{ q 1 ,. q 2 ,. q 3 ,.
q 4 }.associated.with.the.lowest.eigenvalue.of.the.matrix.[80]

∑ ∑ ∑ ∑

⎛ ⎞

2 2 2

x + y + z y z − y z x z − x z x y − x y

m m m p m m p m p p m p m m p

⎜
⎜

⎟
⎟
⎟
⎟
⎟
⎟
⎟

∑ ∑ ∑ ∑

2 2 2

y z − y z y + z + x x y − x y x z − x z

⎜
⎜
⎜
⎜
⎜

p m m p p p m m m p p m m p p

∑ ∑ ∑ ∑

2 2 2

x z − x z x y − x y x + z + y y z − y z

m m p p

m p p m m m p p p p m

∑ ∑ ∑ ∑

2 2 2

x y − x y x z − x z y z − y z x + y + z

p m m p m m p p m m p p p p m

⎝ ⎠

. .

(9.60)

Here,.( x m ,. y m ,. z m ).=. R .−. R ′.and.( x p ,. y p ,. z p ).=. R .+. R ′..The.corresponding.rotation.matrix.
to. q .is.found.as.[63,79]