NBS Monograph 115 - 1.5. Nonrotating-Molecule Matrix Elements

1.5. Nonrotating-Molecule Matrix Elements

Matrix elements [5] (pp. 90-102 and 176-188) of the nonrotating-molecule Hamiltonian operator in the basis set |Ω⟩ have the following form

(1.6a)

(1.6b)

Since Ω is always a good quantum number in the nonrotating molecule, the exact eigenfunctions of the operator can be characterized by a value of Ω. It is convenient to choose the set of exact eigenfunctions of the nonrotating-molecule Hamiltonian to be the nonrotating-molecule basis set. Such exact eigenfunctions obey the relations given in (1.6). For the purposes of calculating rotational energy levels and rotational line intensities, the energy of each state |Ω⟩ is represented by an appropriate parameter, to be fit by comparison with experimental data.

Matrix elements of the nonrotating-molecule Hamiltonian operator in the basis set $|\Lambda S\Sigma\rangle$ can be taken to have the following form

(1.7a)

(1.7b)

(1.7c)

The two exact equalities in (1.7) are satisfied only if the functions involved in the matrix elements are exact eigenfunctions of . However, the basis set functions $|\Lambda S\Sigma\rangle$ defined above are not exact eigenfunctions of the nonrotating-molecule Hamiltonian, because spin-orbit coupling destroys the goodness of the three quantum numbers Λ, S, and Σ. In other words, because the spin-orbit interaction operator Σ_i ξ(r_i) l_i · s_i, does not commute with L_z, S², or S_z, this part of the Hamiltonian will mix together basis set functions characterized by different values of Λ, S, and Σ to form final nonrotating-molecule wave functions.

Nevertheless, when spin-orbit splittings between the various components of a given multiplet e.g., ²Π_1/2, ²Π_3/2) are small compared to separations between different multiplet groups (e.g., ²Σ, ⁴Σ, ²Π, ⁴Π, ²Δ, etc.), then the basis set functions $|\Lambda S\Sigma\rangle$ are very good approximations to the exact eigenfunctions of the nonrotating-molecule Hamiltonian. Under these circumstances it is convenient to shift our point of view a bit, and to consider the functions $|\Lambda S\Sigma\rangle$ to be these exact eigenfunctions. It is then necessary, however, to remember that the quantum numbers Λ, S, Σ are no longer perfectly good, i.e., the appropriate eigenvalue equations are only approximately satisfied. For example,

(1.8a)

(1.8b)

where |δ₁⟩ and |δ₂⟩ are small functions which vanish when the quantum numbers Λ and Σ are perfectly good. Because of the presence of small "left-over" functions like these |δ_i⟩, the diagonal matrix elements of the spin-orbit operator AL · S are only approximately equal to AΛΣ. Thus, precise energies of the multiplet components represented by the wave functions $|\Lambda S\Sigma\rangle$ deviate somewhat from the expression: constant +AΛΣ. However, the exact energies of the nonrotating- molecule problem can always be represented by a set of adjustable parameters in the rotational calculations.

A shift in point of view similar to that above is also desirable for the basis set functions $|\Lambda S\Sigma\rangle$ . If the functions $|\Lambda S\Sigma\rangle$ are taken to be the exact eigenfunctions of the nonrotating-molecule Hamiltonian, then we can write

(1.9a)

(1.9b)

where the quantum numbers L, Λ, S, Σ in these exact eigenfunctions of the nonrotating-molecule Hamiltonian are all slightly bad. The energies of these exact eigenfunctions can again be represented by appropriate adjustable parameters in the rotational calculations.