Physical Chemistry , 1st ed.

Each integral can be solved separately. The first integral, over , is easily

shown to be



2 

 0

d 02  2  0  2 

The second integral, over , must be either integrated by parts or looked up

in an integrals table. This integral is included in Appendix 1.





 0

sin^3 d^13 cos (sin^2 2) 0 

 3 ^1 [(1)(0 2) (1)(0 2)] ^43 

If we combine all terms from the integral, we find

1,1

8

3



 2 

4

3

 1

confirming that the spherical harmonic wavefunction is indeed normalized.

Using the explicit forms of the spherical harmonics, one can use standard
trigonometric integrals to show that the wavefunctions are also orthogonal to
each other. That is,

*,m,md  0 unless  and mm (11.50)

The energy eigenvalues for 3-D rotational motion can be determined ana-
lytically by putting the spherical harmonics into the Schrödinger equation and
solving for the energy. It is a straightforward mathematical procedure (see ex-
ercise 11.33), but here we are more interested in the analytic expression for the
energy. It is

E E() 

(

2 I

1)^2

 (11.51)

The energy of 3-D rotational motion depends on the particle’s moment
of inertia, Planck’s constant, and . Since the total energy cannot have any
values other than these,the total energy is quantizedand depends on the
quantum number .It does notdepend on m. Therefore, each energy level
is (21)-fold degenerate.
The expression for the energy of a 3-D rotation is slightly different from the
energy levels of a 2-D rotation. Because of the 1 term in the numerator,
the energy of a 3-D rotation goes up slightly faster with the quantum number
than does the energy of a 2-D rotation versus m. This is illustrated by Figure
11.13, which diagrams the first eight energy levels of both the 2-D and 3-D
rotations.
A 3-D rigid rotoris a system of more than one particle having fixed rel-
ative positions (that is, a molecule) that is rotating in three-dimensional
space. The only change in any of the expressions derived above is the sub-
stitution of, the reduced mass, for m, the mass. (Be careful to not con-
fuse mass and rotational quantum number, both of which can be repre-
sented by m.) The wavefunctions, energy eigenvalues, and angular
momentum eigenvalues can be determined using the same expressions af-
ter substitution for .

11.7 Three-Dimensional Rotations 345