## 3. Calculation of Rotational Line Intensities

In this chapter we discuss the determination of spectral line intensities. We shall not be concerned, however, with the total intensity of an electronic transition, nor with the vibrational distribution of intensity in a given band system. Rather, we shall consider the rotational distribution of intensity within a given band [1] (pp. 204-211).

Optical transitions in diatomic molecules are said to be electric dipole allowed, magnetic dipole allowed, or electric quadrupole allowed if the transition moment matrix element is nonvanishing when the electric dipole operator, the magnetic dipole operator, or the electric quadrupole operator, respectively, is used [7] (pp. 79-111). (These classifications are not necessarily mutually exclusive.) The vast majority of observed transitions are electric dipole allowed, and we shall consider only that case in this monograph. However, the considerations below can be applied to magnetic dipole transitions after relatively minor changes: For example, all signs on the right-hand side of (eq 3.2) must be positive, and the six signs preceding the parentheses on the right-hand side of (eq 3.4) must be changed from +, +, -, -, -, - to -, -, +, +, -, -; these sign changes lead, of course, to some changes in the selection rules. A discussion of electric quadrupole transitions is quite different from the discussion for dipole transitions, since the quadrupole operator is a tensor of the second rank, rather than a vector.

As suggested above, the intensity of most optical transitions is governed by the value of matrix elements of the electric dipole moment operator [5] (pp. 272-282). We must therefore examine this operator in some detail. Classically, the dipole moment of a system of charges ei is a vector quantity given by an expression of the form Σi eiri, where ri is a vector from the point at which the dipole moment is being defined to the charge ei. Vector operators are sometimes a source of confusion in molecular spectroscopy, since they are often represented by their components in two different Cartesian axis systems, one fixed in the laboratory, the other fixed in the molecule. The dipole moment operator can also be resolved into components in either of these two axis systems. Matrix elements of the two sets of components have quite different interpretations.

We shall here follow the common, but not universal, practice of representing the laboratory-fixed components of the dipole moment operator by µX, µY, µZ, and the molecule-fixed components by µx, µy, µz. These two sets of components are related by an equation of the form

 (3.1)

where αRs, the direction cosine matrix [3] (pp. 10-11), [6] (pp. 285-6), [8], is found on the right-hand side of eqs (eq 2.3). (Note that the direction cosine matrix for linear molecules contains only two Eulerian angles.) The subscript R in (3.1) ranges over X, Y, Z; the subscript s over x, y, z. For most of the remainder of this chapter we shall discuss the determination and interpretation of matrix elements of the quantities occurring in eq (3.1).