NBS Monograph 115 - 1.6. Rotating-Molecule Hamiltonian

1.6. Rotating-Molecule Hamiltonian

The Hamiltonian operator corresponding to the rotational part of the problem in diatomic molecules [3] (pp. 6-16) must now be examined in some detail.

for any molecule can be written as a sum of three products, each product consisting of a rotational constant for the molecule and the square of one of the three components of the rotational angular momentum of the molecule [6] (pp. 273-284). Since there can be no rotational angular momentum of a diatomic molecule about its internuclear axis (the z axis), the third component of the rotational angular momentum is zero, and hence absent from

for diatomic molecules is thus written as

eq 1.10 (1.10)

where the rotational angular momentum is expressed in the second line of (1.10) as the total angular momentum (J) minus the electronic orbital and spin angular momenta (L and S). For the purposes of calculation, is often written in the form

(1.11)

where J_± = J_x ± iJ_y, L_± = L_x ± iL_y, and S_± = S_x ± iS_y.

The extremely thorough student will find that the total angular momentum operator J for linear molecules is somewhat peculiar, since its molecule-fixed components do not obey angular-momentum-type commutation relations. Furthermore, the operator for linear molecules does not have precisely the simple form indicated in (1.10). Nevertheless, it can be shown that correct results are obtained by ignoring the peculiarities associated with J in linear molecules and by treating J like the angular momentum operator defined for nonlinear molecules in [6].

The complete Hamiltonian describing the nonrotational and rotational parts of the problem is of course given by (1.1). It is the matrix of the Hamiltonian (1.1) which will ultimately be diagonalized to obtain molecular energy levels and molecular wave functions.

Loosely speaking, one can see in eq (1.10) the origin of the various entries in column 2 of table 2. Forgetting for a moment the absence of R_z in (1.10), and remembering that the eigenvalue associated with the sum of the squares of the components of an angular momentum operator has the form hbar ² X(X + 1), we note that: if the operators L and S in the rotational Hamiltonian (1.10) can both be ignored, then one might expect rotational energies to be given by B J(J + 1), since J is the quantum number associated with J²; if the operator L in (1.10) can be ignored, but the operator S cannot be, then one might expect rotational energies to be given by B N(N + 1), since N is the quantum number associated with (J - S)²; if neither L nor S in (1.10) can be ignored, then one might expect rotational energies to be given by B R(R + 1), since R is the quantum number associated with (J - L - S)².

Since the operators L and S in (1.10) affect the course of the rotational energy levels only through the four cross terms J_xL_x, J_yL_y, J_xS_x, J_yS_y, and since the selection rules for nonvanishing matrix elements of L_x, L_y, and S_x, S_y are ΔΛ and ΔΣ = ± 1, respectively, we see that the effects of L and/or S in (1.10) can be ignored when the separation between states of the nonrotating molecule satisfying the selection rules ΔΛ and/or ΔΣ = ± 1 is large compared to B J.