Page 288 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Problems 279

this non-rigid rotor. As the rotational energy increases, the internuclear
distance increases, resulting in an increased moment of inertia and conse-
quently a lower energy. Thus, this term is negative and increases as J increases.
The magnitude of the centrifugal distortion is in¯uenced by the value of the
force constant k as re¯ected by the factor ù ÿ2 in D. The last term contains both
quantum numbers n and J and represents a direct coupling between the
vibrational and rotational motions. This term contains two contributions: a
change in vibrational energy due to the centrifugal stretching of the molecule
and a change in rotational energy due to changes in the internuclear distance
from anharmonic vibrations. The constant term U(0) merely shifts the zero-
point energy of the nuclear energy levels and is usually omitted completely.
The molecular constants ù, B e , x e , D, and á e for any diatomic molecule may
be determined with great accuracy from an analysis of the molecule's vibra-
4
tional and rotational spectra. Thus, it is not necessary in practice to solve the
È
electronic Schrodinger equation (10.28b) to obtain the ground-state energy
å 0 (R).

Problems
10.1 Derive equation (10.47) as outlined in the text.
10.2 Derive equation (10.49) as outlined in the text.
10.3 Derive equation (10.50) as outlined in the text.
10.4 An approximation to the potential U(R) for a diatomic molecule is the Morse
potential
U(R) ˆÿD e (2e ÿa(RÿR e ) ÿ e ÿ2a(RÿR e ) ) ˆÿD e (2e ÿaq ÿ e ÿ2aq )
where a is a parameter characteristic of the molecule. The Morse potential has
the general form of Figure 10.2.
(a) Show that U(R e ) ˆÿD e , that U(1) ˆ 0, and that U(0) is very large.
(b) If the Morse potential is expanded according to equation (10.30), relate the
parameter a to ì, ù, and D e
(c) Relate the quantities x e , á e , and U (0) in equation (10.50) to ì, ù, and D e
for the Morse potential.
10.5 Another approximate potential U(R) for a diatomic molecule is the Rydberg
potential
U(R) ˆÿD e [1 ‡ b(R ÿ R e )]e ÿb(RÿR e ) ˆÿD e (1 ‡ bq)e ÿbq
where b is a parameter characteristic of the molecule.
(a) Show that U(R e ) ˆÿD e , that U(1) ˆ 0, and that U(0) is very large.

4 Comprehensive tables of molecular constants for diatomic molecules may be found in K. P. Huber and G.
Herzberg (1979) Molecular Spectra and Molecular Structure: IV. Constants of Diatomic Molecules (Van
Nostrand Reinhold, New York).