NBS Monograph 115 - 1.2 General approach to the calculations

1.2. General Approach to the Calculations

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

${\cal H} = {\cal H}_{\rm ev} + {\cal H}_{\rm r},$ (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. + is then called the Hamiltonian for the rotating 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 $|ev;r\rangle$ used in this report can be written as simple products of the form

$|{\rm ev}; {\rm r}\rangle = |{\rm ev}\rangle |{\rm r}\rangle,$ (1.2)

where the functions $|ev\rangle$ are eigenfunctions of $Hamiltonian_ev$ , i.e., wave functions for the nonrotating molecule problem, and where the functions $$|r\rangle$ 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 ML²T^-2 in terms of the fundamental quantities Mass, Length, and Time. Diatomic spectroscopists do not normally use energy units, however. Instead, they represent energy differences by their corresponding wave numbers, which have the dimension 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 ML²T^-1) and c is the velocity of light (having dimensions LT^-1). Confusion arises because the same symbol is normally used for the "same" quantity, regardless of the units in which it is expressed. For example, the symbol E may represent E[ML²T^-2] or E[cm^-1], the numerical values of the two E's being related by E[cm^-1] = E[ML²T^-2] / hc. A second source of confusion in units arises in discussing angular momentum couplings of the form AL · S or BJ². These two quantities occur in the Hamiltonian operator in such a way that they have the dimensions of energy, i.e., ML²T^-2. Since the matrix elements of angular momentum components are multiples of h bar . this requires that A and B have dimensions M^-1L^-2. In formulas used by diatomic spectroscopists, however, matrix elements of AL · S and BJ² are normally represented by something like AΛΣ 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] h bar ² / hc. In this monograph we shall follow common practice and represent the energy of the state, the spin-orbit coupling constant, the rotational constant, etc., by a single letter each, regardless of the units being used. As a general rule, terms of the Hamiltonian 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.