1549380323-Statistical Mechanics Theory and Molecular Simulation

38 Classical mechanics

between the two hydrogens for a total of three holonomic constraint conditions. An am-
monia molecule (NH 3 ) could also be treated as a rigid pyramid by fixing the three NH
bond lengths and the three HH distances for a total of six holonomicconstraint con-
ditions. In a more complex molecule, such as the alanine dipeptide shown in Fig. 1.11,
specific groups can be treated as rigid. Groups of this type are shaded in Fig. 1.11.

H 3 C

C

O

N

H

C

H

H 3 C

O

C

H

N

H 3 C CH 3

C

O

N

H

H 3 CCC

CCH 3

H

N

O

C

C

H

Fig. 1.11 Rigid subgroups in a large molecule, the alanine dipeptide.

Of course, it is always possible to treat these constraint conditionsexplicitly using
the Lagrange multiplier formalism. However, since all the particles in arigid body
move as a whole, a simple and universal formalism can be used to treatall rigid
bodies that circumvents the need to impose explicitly the set of holonomic constraints
that keep the particles at fixed relative positions. Before discussing rigid body motion,
let us consider the problem of rotating a rigid body about an arbitrary axis in a fixed
frame. Since a rotation performed on a rigid body moves all of the atoms uniformly, it
is sufficient for us to consider how to rotate a single vectorrabout an arbitrary axis.
The problem is illustrated in Fig. 1.12. Letndesignate a unit vector along the axis of
rotation, and letr′be the result of rotatingrby an angleθclockwise about the axis.
In the notation of Fig. 1.12, straightforward geometry shows thatr′is the result of a
simple vector addition:

r′=

−→

OC+

−→

CS+

−→

SQ. (1.11.1)

Since the angle CSQ is a right angle, the three vectors in eqn. (1.11.1)can be expressed
in terms of the original vectorr, the angleθ, and the unit vectornaccording to

r′=n(n·r) + [r−n(n·r)] cosθ+ (r×n) sinθ, (1.11.2)

which can be rearranged to read

r′=rcosθ+n(n·r)(1−cosθ) + (r×n) sinθ. (1.11.3)

Eqn. (1.11.3) is known as therotation formula, which can be used straightforwardly
when an arbitrary rotation needs to be performed.
In order to illustrate the concept of rigid body motion, consider thesimple problem
of a rigid homonuclear diatomic molecule in two dimensions, in which each atom has