Pure precession approximation

This expression shows clearly that 1g depends on internuclear distance and that one-electron two-center integrals cannot be neglected in calculating matrix elements of 1q (Colbourn and Wayne, 1979). This situation is quite different from that for the off-diagonal spin-orbit parameters. The b parameters are given (Section 5.3.2) by  

It has been pointed out (Robbe and Schamps, 1976) that there is no general reason that the al+ and 1+ matrix elements should be simply proportional to each other. For al+, the contribution from the atom with the larger atomic spin-orbit parameter is dominant for 1+, the p orbitals of both atoms play equal roles. Table 5.11 compares some values for matrix elements of these two operators. For a+, the values of CO and SiO are similar, as are the values for CS and SiS. For b, the strong H-dependenceof this parameter prohibits any simple predictions. Even the sign changes when one passes from CS to CS+. 

The hypothesis of pure precession (Van Vleck, 1929) is often used in the estimation of A-doubling constants. These constants, in the o,p, q notation suggested by Mulliken and Christy (1931), are introduced into the effective Hamiltonian 

This discussion is intended to distinguish the levels of approximation often hidden behind the name pure precession. For clarity, only 2E 2II interactions will be discussed. The considerably more complicated 3E 3II case is treated by Brown and Merer (1979). 

and q constants may be defined in a form that is semiempirically evaluable by second-order perturbation theory (see the discussion of the Van Vleck transformation in Section 4.2). Using the notation suggested by Rostas, et al., (1974) and Merer, et al., (1975), 

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We can then make an even more drastic approximation and represent the molecular orbitals in these configurations by pure 2p atomic orbitals on the O atom. This approximation was called pure precession by Van Vleck [41] in this approximation the electrons in these outermost orbitals are in a spherically symmetric environment and they have a well defined value of the orbital angular momentum quantum number / (unity for a p orbital). In the pure precession approximation, we can derive very simple expressions for the g-factors [66], The values for OH predicted on the basis of this very simple model are given in table 9.4. The fact that they agree reasonably well with the experimental numbers suggests that the theoretical model is essentially correct. 

Once again using the pure precession approximation and the parameter values given above we obtain... 

Value obtained using the pure precession approximation. (Sections 3.5.4 and 5.5). Calculated value is —0.08 cm-1 (Lefebvre-Brion unpublished calculation). 

Finally, the pure precession approximation requires, in addition to the unique perturber, identical potential assumptions, that the interacting 2II and 2 states axe each well described by a single configuration, that these configurations are identical except for a single spin-orbital, and that this spin-orbital is a pure... 

For the OH radical, the values of p and q for the X2n (ground state can be attributed to a unique perturber interaction with the A2E+ (ow4) state. The pure precession approximation simply ignores the contribution of the atomic orbital to the per molecular orbital. For all hydrides, the oTsh orbital makes a negligible contribution to Hso and BL+ matrix elements. For OH, the H-atom contributions to a+ and b are 4 x 10 4% and 1%, respectively (Hinkley, et al., 1972). 

For valence states of nonhydride molecules, there is no reason to expect that the generalized pure precession approximation should be valid. In contrast, Rydberg orbitals, because of their large size and nonbonding, single-center, near-spherical, atomic-like character, are almost invariably well-described by the pure precession picture in terms of nl-complexes. 

Van Vleck introduced the pure precession approximation for diatomic molecules. The basic idea is that if the one-electron atomic orbital angular momentum is preserved in a diatomic molecule then the A-doubling and spin-rotation constants can be predicted, for example, in a 4pa(2l,+) and a 4p7r(2n) state. This pure precession picture can easily be generalized to... 

In writing this equation, we have made use of Van Vleck s pure precession hypothesis [12], in which the molecular orbital /.) is approximated by an atomic orbital with well-defined values for the quantum numbers n, l and /.. Such an orbital implies a spherically symmetric potential and its use is most appropriate when the electronic distribution is nearly spherical. Examples of this situation occur quite often in the description of Rydberg states. It is also appropriate for hydrides like OH where the molecule is essentially an oxygen atom with a small pimple, the hydrogen atom, on its side. Accepting the pure precession hypothesis allows the matrix elements of the orbital operators to be evaluated since... 

If the value of l is 1, o+ = 2l/2A. This is known as part of the pure precession hypothesis, which is approximately true only for hydrides, Rydberg states of an l > 0 complex, and certain highly ionic molecules (see Section 5.5). Although the pure precession hypothesis is frequently applied to homonuclear molecules, there is no justification for this. 

The idea of an effective Hamiltonian for diatomic molecules was first articulated by Tinkham and Strandberg (1955) and later developed by Miller (1969) and Brown, et al., (1979). The crucial idea is that a spectrum-fitting model (for example Eq. 18 of Brown, et al., 1979) be defined in terms of the minimum number of linearly independent fit parameters. These fit parameters have no physical significance. However, if they are defined in terms of sums of matrix elements of the exact Hamiltonian (see Tables I and II of Brown, et al., 1979) or sums of parameters appropriate to a special limiting case (such as the unique perturber approximation, see Table III of Brown, et al., 1979, or pure precession, Section 5.5), then physically significant parameters suitable for comparison with the results of ab initio calculations are usually derivable from fit parameters. 

The most interesting aspect of the rotational analyses of the A X, B-X and C X transitions are the values for the complete set of nine spin rotation parameters of the A, B, and C states (i aa, sbb, and f.cc for each state). If the orbitals containing the unpaired electron in these three states (Fig. 7) are approximate by a set of p orbitals (px, py, pz) then all nine spin-rotation constants can be easily estimated by pure precession relationships. 

For OH in its X H state, there are two A-doubling parameters, p and q. Using the explicit expressions given in equations (7.142) and (7.143), these parameters can be calculated very accurately using good quahty ab initio wave fimctions. A simple estimate can be made using the pure precession values for the relevant matrix elements which we have derived earlier. This approximation assumes that the A-doubling effects arise wholly from the perturbations with the A E+ state. For this approximation,... 

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