The Rotation of Symmetrical-top Molecules

The Rotation of Symmetrical-top Molecules.—A rigid body in which two of the three principal moments of inertia  

2 Every body has three axes the use of which permits the kinetic energy to be expressed in a particularly simple form. These are called the princijxd axes of inertia. The moment of inertia about a principal axis is defined by the expression fprHr, in which p is the density of matter in a given volume clement dr, r is the perpendicular distance of this element from the axis in question, and the integration is over the entire volume of the solid. For a discussion of this question see J. C. Slater and N. II. Frank, Introduction to Theoretical Physics, p. 94, McGraw-Hill Book Company, Inc., New York, 1933. 

In case that a molecule possesses an n-fold symmetry axis with n greater than 2 (such as ammonia, with a three-fold axis), then two principal moments of inertia about axes perpendicular to this symmetry axis are equal, and the 

Since we have considered only assemblages of point particles heretofore, we have not given the rules for setting up the wave equation for a rigid body. We shall not discuss these rules here1 but shall take the wave equation for the symmetrical top 

1 Since the dynamics of rigid bodies is based on the dynamics of particles, these rules must be related to the rules given in Chapter IV. For a discussion of a method of finding the wave equation for a system whose Hamiltonian is not expressed in Cartesian coordinates, see B. Podolsky, Phys. Rev. 32, 812 (1928), and for the specific application to the symmetrical top see the references below. 

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