Table of Contents of Rotational Energy Levels and Line Intensities in Diatomic Molecules

1. Calculation of Rotational Energy Levels

1.1. Hund's Coupling Cases (a), (b), (c), and (d)

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.

TABLE 1. The angular momenta involved in Hund's coupling cases

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.

TABLE 2. Brief description of four Hund's coupling cases

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) Λ, S, Σ (&Omega = Λ + Σ) 2 or 1
(b) B N (N + 1) Λ, S, Σ (Ω = Λ + Σ) 2(2S + 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) (2L + 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 2S + 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 2L + 1 values of Λ associated with a given value of L, and the 2S + 1 values of Σ associated with a given value of S must give rise to a set of states degenerate in the nonrotating molecule.

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