Chemistry Reference
In-Depth Information
⎛
2
2
2
2
⎞
q
+
q
+
q
+
q
2
(
q q
+
q q
)
2
(
q q
−
q q
)
1
2
3
4
2 3
1 4
2 4
1 3
⎜
⎜
⎜
⎟
⎟
⎟
ˆ
2
2
2
2
Q
=
2
(
q q
+
q q
)
q
+
q
+
q
+
q
2
(
q q
+
q q
)
.
(9.61)
2 3
1 4
1
3
2
4
3 4
1 2
2
2
2
2
2
(
q q
+
q q
)
2
(
q q
+
q q
)
q
+
q
+
q
+
q
⎝
⎠
2 4
1 3
3 4
1 2
1
4
2
3
.
.
The. orientation. associated. with. this. rotation. gives. the. least. distance. between. the.
molecules,.but.it.depends.on.the.mapping.used:
N
∑
ˆ
ˆ
2
ˆ
2
2
d
=
Qa
X
ʹ
−
X
=
Q
a A
R
ʹ
−
R
.
(9.62)
min
(
)
A
.
.
A
=
1
In.Figure.9.3,.the.distance.graphs,.according.to.Equation.9.53,.for.every.one.of.the.
120.permutations.of.the.example.showed.in.Figure.9.2.are.shown..The.distance,.as.
a.function.of.the.orientation.(in.degrees),.is.measured.in.arbitrary.length.units..The.
lower.contour.of.the.graphs.indicated.by.the.dashed.curve.in.Figure.9.3.is.the.distance.
given.by.Equation.9.58..As.Figure.9.3.shows,.the.best.orientation.in.this.case.is.found.
at. 90°. where. the. distance. between. the. two. molecules. is. zero.. Due. to. the. computa-
tional.demand.(factorial.scaling.with.the.number.of.atoms),.it.is.not.feasible.to.test.
all.mappings.in.order.to.ind.the.corresponding.minima.on.the.graphs..Therefore,.a.
probability-driven. approach. is. suggested.. In. this. approach,. one. of. the. molecules. is.
rotated.randomly,.next.the.optimal.mapping.is.computed.as.described,.and.inally.the.
molecules.are.aligned.with.the.rotation.matrices.
Q
ˆ
n
.according.to.Equation.9.62.
The. random. rotations. are. performed. by. random. uniformly. distributed. rotation.
matrices.
P
ˆ
n
.[83,84],.and,.therefore,.we.obtain
N
2
∑
2
ˆ
ˆ
ˆ
ˆ
ˆ
2
.
(9.63)
d
=
Q a P
X
ʹ
−
X
=
Q P
R
ʹ
−
R
.
n
n n n
n n
a A
(
)
A
.
A
=
1
70
60
50
40
30
20
10
45
90
135 180
Orientation (degrees)
225
270
315
360
FIGURE 9.3 (See color insert)
Distance.graphs.for.each.of.the.120.possible.mappings.in.
Figure.9.2.
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