In chapter 1 we seek to calculate rotational energy level expressions for certain limiting situations known as Hund's coupling cases [1] (pp. 218-237), [2] (pp. 275-302) (footnote 1), and also for situations intermediate between the limiting coupling cases. Because any discussion of Hund's coupling cases involves a consideration of various angular momenta, it is convenient at this point to summarize the types of angular momenta considered and the notation to be used. This summary is presented in table 1. The first column of this table specifies the type of angular momentum. We consider here electronic orbital angular momentum, electronic spin angular momentum, and the angular momentum associated with the rotation of the nuclei of the diatomic molecule about the center of mass. We do not consider the nuclear spin angular momentum. Each angular momentum has associated with it a quantum number specifying its magnitude and a quantum number specifying its projection along the internuclear axis. For the purposes of this paper, the projection quantum numbers will be considered to be signed quantities.

Type of angular momentum | Operator | Quantum numbers | |
---|---|---|---|

Total | Projection | ||

Electronic orbital | L |
L |
Λ |

Electronic spin | S |
S |
Σ |

Rotational | R |
R |
. . . |

Total |
J = R + L + S |
J |
Ω = Λ + Σ |

Total-Spin | N = R + L |
N |
Λ |

It can be shown that the projection of the rotational angular momentum along
the internuclear axis is necessarily zero. Hence, we do not introduce a quantum
number for this projection. As a further consequence of this fact, the
projection of the angular momentum **N** along the internuclear axis is
equal to the projection of **L** along the axis, and the projection of
**J** is equal to the projection of **L + S**, as indicated
in the table.

In this monograph we shall define the various Hund's coupling cases in terms of two concepts: (i) the quantum number occurring in the expression for the rotational energy levels, and (ii) the good quantum numbers in the nonrotating-molecule problem. Table 2 describes Hund's coupling cases (a), (b), (c), and (d) in these terms.

Coupling case | Rotational energy level expression | Good quantum numbers in the nonrotating molecule (see text) |
Degeneracy in the nonrotating molecule |
---|---|---|---|

(a) | B J (J + 1) |
S, Σ
(&Omega = Λ + Σ) |
2 or 1 |

(b) | B N (N + 1) |
S, Σ
(Ω = Λ + Σ) |
S + 1) or
(2S + 1) |

(c) | B J (J + 1) |
Ω | 2 or 1 |

(d) | B R (R + 1) |
L, Λ, S, Σ
(Ω = Λ + Σ) |
L + 1) (2S +1) |

The second column of table 2 contains a rotational energy expression of
the form *BX* (*X* + 1), where *X* is one of the
quantum numbers from table 1. Rotational energies are
given by these simple expressions only for pure Hund's coupling cases.
Rotational energies for coupling cases intermediate between these pure coupling
cases are given by much more complicated expressions.

The third column of table 2 indicates which quantum numbers are good ones
in the nonrotating molecule for the various Hund's coupling cases. The
nonrotating-molecule quantum numbers shown in table 2 for Hund's cases
(a), (b), and (d) indicate those which *must* be good; other
nonrotating-molecule quantum numbers may, in fact, also be good ones. The
nonrotating-molecule quantum number shown in table 2 for Hund's
case (c) indicates the *only permissible* good quantum number.

The fourth column in table 2 indicates the degeneracy of the energy levels
in the nonrotating molecule. (This degeneracy need not be exact, but the nearly
degenerate states must be separated by energies small compared to
*B J*.) In Hund's cases (a) and (c) the only degeneracy in the
nonrotating molecule is that associated with the two values of Ω, i.e.,
± | Ω |. (States characterized by Ω = 0^{+} or
Ω = 0^{-} are, of course, nondegenerate.) In Hund's
case (b) the 2*S* + 1 values of Σ associated with a
given value of *S*, and the two values of Λ, i.e.,
± | Λ |, must give rise to a set of states, all of which
are degenerate in the nonrotating molecule. In Hund's case (d) the
2*L* + 1 values of Λ associated with a given value of
*L*, and the 2*S* + 1 values of Σ associated with a
given value of *S* must give rise to a set of states degenerate in the
nonrotating molecule.