## 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
*e*_{i} is a vector quantity given by an expression of the form
Σ_{i} *e*_{i}**r**_{i}, where
**r**_{i} is a vector from the point at which the dipole
moment is being defined to the charge *e*_{i}. 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
*µ*_{X}, *µ*_{Y},
*µ*_{Z},
and the molecule-fixed components by *µ*_{x},
*µ*_{y}, *µ*_{z}. These two sets of
components are related by an equation of the form

where α_{Rs}, 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).