Physical Chemistry , 1st ed.

The benzene molecule has a diameterof a little over 3 Å across. The “electron

ring” is assumed to be slightly less than that, on average. This model predicts

a slightly larger diameter (2 radii, or4.7 Å in this case) than is the case.

However, given the approximations that were part of our assumptions in

applying this model to benzene, getting this close should be taken as a pos-

itive sign.

Two-dimensional rotational motion is the last system we consider where the

solution of the wavefunction is derived. Henceforth, major conclusions will be

presented instead of being derived explicitly. The systems considered to this

point have demonstrated sufficiently how the postulates of quantum mechan-

ics are applied to systems and how the results are obtained. After this, we will

concentrate more on the results and what they mean, rather than a step-by-

step derivation of the solutions. If you are interested in the mathematical de-

tails, consult a more advanced reference.

11.7 Three-Dimensional Rotations

It is a trivial step to expand rotation of a particle or rigid rotor to three di-

mensions. The radius of a particle from a center is still fixed, so three-dimen-

sional rotation describes motion on the surface of a sphere, as shown in Figure

11.11. However, in order to be able to describe the complete sphere, the coor-

dinate system is expanded to include a second angle . Together, the three co-

ordinates (r,,) define spherical polar coordinates.The definitions of these

coordinates are shown in Figure 11.12. In order to treat the subject at hand

more efficiently, several statements regarding spherical polar coordinates are

presented without proof (although they can be proven without much effort, if

desired).

There is a straightforward relationship between three-dimensional Cartesian

coordinates (x,y,z) and spherical polar coordinates (r,,). They are

xrsin cos 

yrsin sin  (11.40)

zrcos 

When one performs integrations in spherical polar coordinates, the form of

d and the limits of integration must be considered. The full form ofd for

integration over all three coordinates (which would be a triple integral, each

integral dealing independently with a single polar coordinate) is

d   r^2 sin dr dd (11.41)

In the case of 2-D integration over only and , the infinitesimal d is

d   sin dd (11.42)

Because two angles are defined, in order to integrate over all space just once,

one angle’s integration limits go from 0 to while the other angle’s integra-

tion limits range from 0 to 2(if both limits were 0 to 2, one would end up

covering all space twice). The accepted convention is that the integration lim-

its on are from 0 to 2, and the limits on go from 0 to . In cases where

integration in terms ofris considered, the integration limits are 0 to .

Finally, as in the case of the 2-D rotational motion, the form of the

Hamiltonian is different when using spherical polar coordinates. In the case

11.7 Three-Dimensional Rotations 341

m r



r

z

y

x

Figure 11.12 The definitions of spherical po-
lar coordinates r,, and . The coordinate ris the
distance between a point and the origin. The an-
gle is defined with respect to the projection of
the rvector in the xy plane, and is the angle that
this projection makes with the positive xaxis
(motion toward the positive yaxis being consid-
ered a positive angular value). The angle is the
angle between the rvector and the positive zaxis.

Figure 11.11 Three-dimensional rotations
can be defined as a mass moving along the sur-
face of a sphere having fixed radius r.