Table of Contents of Rotational Energy Levels and Line Intensities in Diatomic Molecules

1.4. Nonrotating-Molecule Basis Set

Like the Hamiltonian Hamiltonian ev the basis set functions $|ev\rangle$ for the nonrotating molecule are not considered in detail in calculations of rotational energy levels. Consequently, these basis set functions are often represented only formally, by symbols containing the quantum numbers used to describe the basis set, e.g., |Ω⟩, $|\Lambda S\Sigma\rangle$, or $|L\Lambda S\Sigma\rangle$ (see table 2).

At this point it is perhaps worthwhile to digress and consider the notion of good and bad quantum numbers. A basis set is said to be characterized by a good quantum number, if each function of the basis set is an eigenfunction of some particular operator, belonging to an eigenvalue which is some simple function of the good quantum number. For example, the quantum number Ω is said to be a good quantum number, if each function of the basis set satisfies an equation of the form

$(L_z+S_z) |\Omega\rangle = \Omega \hbar\, |\Omega\rangle.$ (1.3)

In this example, (Lz + Sz) is the operator, |Ω⟩ is the basis set function, Ω h bar is the eigenvalue, and Ω is the quantum number. A quantum number is not good (i.e., is bad) if equations of the form (1.3) are not satisfied by the functions of the basis set. For example, if

$L_z |\Omega\rangle \not= ({\rm constant}) \cdot |\Omega\rangle,$ (1.4)

then the quantum number associated with Lz, namely Λ, is not a good quantum number for the basis set.

We now consider the three types of nonrotating-molecule basis set functions suggested by column 3 of table 2 in more detail [1] (pp. 212-217). It turns out that the operator Lz + Sz commutes with Hamiltonian ev. Consequently, wave functions for the nonrotating molecule are always characterized by the good quantum number Ω, representing the projection of the sum of the electronic orbital angular momentum and the electronic spin angular momentum along the internuclear axis.

When spin-orbit interaction is very large, i.e., when the energy levels of the nonrotating molecule do not fall into recognizable multiplet groups, then the wave functions for the nonrotating molecule are characterized only by the good quantum number Ω. Under these circumstances it is convenient to use basis set functions which are also characterized by this single good quantum number. In eq (1.3), |Ω⟩ represents such a basis set function for the nonrotating molecule. The quantum number Ω can take on only integral values for molecules with an even number of electrons, and only half-integral values for molecules with an odd number of electrons. Both types of wave functions for the nonrotating molecule occur in degenerate pairs when Ω ≠ 0, the members of the pair being characterized by equal and opposite values of Ω. When Ω = 0, the nonrotating-molecule state is nondegenerate. Nonrotating-molecule basis set functions characterized only by Ω give rise to Hund's case (c) states in the rotating molecule.

When spin orbit interaction is not large, i.e., when the energy levels of the nonrotating molecule do fall into recognizable multiplet groups, then the wave functions for the nonrotating molecule are normally characterized by three almost good quantum numbers, namely: the value Λ of the projection of the total electronic orbital angular momentum along the internuclear axis (corresponding to the operator Lz), the value Σ of the projection of the total electronic spin angular momentum along the internuclear axis (corresponding to the operator Sz), and the value S of the total electronic spin angular momentum (corresponding to the operator S2). Under these circumstances it is convenient to use basis set functions $|\Lambda S\Sigma\rangle$ characterized by the three good quantum numbers Λ, S, and Σ. Since these basis set functions are eigenfunctions of Lz and Sz, belonging to the eigenvalues Λh bar and Σh bar, respectively, they are also eigenfunctions of the operator (Lz +  Sz), belonging to the eigenvalue Ωh bar, where

$\Omega = \Lambda + \Sigma ~ .  $ (1.5)

Because Ω is so simply related to Λ and Σ, it is often not explicitly indicated in the symbol for the basis set functions. (Note also, that in the true wave functions for the nonrotating molecule, the quantum number Ω is good, while Λ and Σ are only approximately good.) The recognizable multiplet groups of energy levels mentioned above consist of the 2(2S + 1) functions characterized by Λ = ± |Λ|, S = fixed value, and Σ = S, S - 1, S - 2, ..., -S (or consist of the (2S + 1) such functions when Λ = 0). As we shall see by example in sect. 1.9, nonrotating-molecule wave functions characterized by Λ, S, and Σ, give rise to Hund's case (a) rotational levels when the energy separations among the multiplet components of the nonrotating molecule are all large compared to B J, and give rise to Hund's case (b) rotational levels when these energy separations are all small compared to B J.

When spin-orbit interaction is not large, and when electrostatic interactions between the electrons and the axial field of the diatomic molecule [1] (pp. 323-330) are not large, e.g., when electrons are in Rydberg orbitals, then the wave functions of the nonrotating molecule are normally characterized by four almost good quantum numbers, namely: Λ, S, and Σ as above, and the value L of the total electronic orbital angular momentum (corresponding to the operator L2). In addition, nearly degenerate sets of such wave functions occur, consisting of the (2L + 1) (2S + 1) functions characterized by L = fixed value, S = fixed value, Λ = L, L - 1, ..., - L, and Σ = S, S - 1, ..., - S. Under these circumstances it is convenient to use basis set functions $|L\Lambda S\Sigma\rangle$ characterized by the four good quantum numbers L, Λ, S, and Σ. Nonrotating-molecule wave functions characterized by L, Λ, S, and Σ give rise to Hund's case (a) rotational levels when the energy separations in the nonrotating molecule among the various components of the 2S+1L complex are all large compared to B J. They give rise to Hund's case (b) rotational levels when the separations between nonrotating-molecule states of different |Λ| are all large compared to B J, while separations among the multiplet components within a given 2S+1Λ state are all small compared to B J. They give rise to Hund's case (d) rotational levels when the separations among the various components of the 2S+1L complex are all small compared to B J.

For any particular problem, it is necessary to choose which of the three basis sets for the nonrotating molecule will be used. From a calculational point of view, it is desirable to use the basis set with the most quantum numbers, since more quantum numbers mean that more matrix elements can be evaluated explicitly. However, if certain quantum numbers of the basis set are not good quantum numbers at all for the actual wave functions of the nonrotating molecule, then the advantage gained by being able to evaluate explicitly certain matrix elements involving these quantum numbers is offset by the fact that the values calculated for the matrix elements are incorrect. If the good quantum numbers of the basis set are almost good for the actual wave functions, then it is often convenient to allow for the slight errors made in the calculation of matrix elements by the introduction of a few small adjustable parameters. (See, for example, sect. 1.10, where B is replaced by B - ½ γ.) In the final analysis, basis set functions for the calculation of rotational energy levels are chosen by trial and error, to maximize agreement between the final calculations and the experimental observations.

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