The basic approach to calculations taken in this monograph derives from the fact that the Hamiltonian for a diatomic molecule [3] (pp. 3-34) can be written in the form

(1.1) |

where is
the vibrational-electronic part of the Hamiltonian, which does *not*
involve the rotational variables or the total angular momentum of the molecule,
and where
is the rotational part of the Hamiltonian, which does involve the rotational
variables and the total angular momentum, and which also involves many of the
coordinates and momenta occurring in . Since does not involve the rotational
variables or the total angular momentum of the molecule, it is convenient to
call the
Hamiltonian for the nonrotating molecule.

As a consequence of eq (1.1), the discussion to
follow will be divided into two parts, one dealing with the
*nonrotational* part of the problem, the other dealing with the
*rotational* part. In both parts we consider the concepts of approximate
Hamiltonian, limited set of basis functions, and matrix elements of the
Hamiltonian in the basis set. [As most spectroscopists know, energy levels are
commonly obtained either by evaluating perturbation-theory expressions
involving matrix elements of the Hamiltonian [4]
(pp. 151-179), or by solving the secular equation obtained from a
truncated Hamiltonian matrix [4] (pp. 191-198).]

Actually, there is some flexibility in the exact form of the Hamiltonian (1.1), depending on whether it is written in terms of laboratory-fixed or molecule-fixed spin functions. In this monograph we shall always use molecule-fixed spin functions [3] (pp. 12-14). Loosely speaking, this choice corresponds to using Hund's case (a) or case (c) basis set functions rather than Hund's case (b) or case (d) basis set functions.

The question of which basis set functions to use is of some importance, since
the same calculation in two different basis sets can appear quite different.
Some authors, for example, find it desirable to use Hund's case (b) basis
set functions when the rotational energy levels of the molecule under study are
very close to those of pure Hund's case (b). In my opinion, however, there
is less chance of confusion and error for the novice, if *all*
calculations are divided into a nonrotating part and a rotating part, so that
these parts can be dealt with separately. Such a division forces one to use a
basis set in which the quantum numbers of the nonrotating molecule are good,
and thus precludes the use of case (b) and case (d) basis sets.

In any case, the basis set functions used in this report can be written as simple products of the form

(1.2) |

where the functions are eigenfunctions of , i.e., wave functions for the nonrotating molecule problem, and where the functions are appropriate rotational wave functions for diatomic molecules [2 pp. 279-281]. (footnote 2)

Units can be somewhat confusing in spectroscopic calculations. The Hamiltonian
operator represents the energy of the molecule, and as such has dimensions
*ML*^{2}*T*^{ -2}*L*^{-1}, and, in particular, cm^{-1}. Conversion from
energy units to cm^{-1} is carried out by dividing energies by
*hc*, where *h* is Planck's constant (having dimensions
*ML*^{2}*T*^{ -1})*c* is
the velocity of light (having dimensions *LT*^{ -1}).*E* may represent *E*[*ML*^{2}*T*^{
-2}]*E*[cm^{-1}], the numerical values of the two
*E*'s being related by *E*[cm^{-1}] =*E*[*ML*^{2}*T*^{ -2}]
/ *hc*.*A***L · S** or *B***J**^{2}. These two
quantities occur in the Hamiltonian operator in such a way that they have the
dimensions of energy, i.e.,
*ML*^{2}*T*^{ -2}.*A* and *B*
have dimensions *M*^{ -1}*L*^{-2}.*A***L · S** and *B***J**^{2} are normally
represented by something like *A*ΛΣ or
*B J* (*J* + 1), where *A* and *B* have the
dimension cm^{-1}, and where Λ, Σ, and *J* are
dimensionless. The numerical values of the two *A*'s (or two *B*'s)
are related by *A*[cm^{-1}] =
*A*[*M*^{ -1} *L*^{-2}]
^{2} /
*hc*.*operator* may be
assumed to be in energy units, whereas *matrix elements* of the terms in
this operator may be assumed to be in cm^{-1}.