Intermediate coupling, deviation

The intermediate coupling, i.e., deviations from the Russell-Saunders approximation, was also treated by Bethe. We may here distinguish between systems with an even or an odd number of electrons. In spherical symmetry, J and 5 are integers, as is L, for an even number, while J and S are half integers 1/2, 3/2, 5/2, for odd electron numbers. The same intrinsic difference occurs in lower symmetries, e.g., for Oj the new quantum number r 7 has the same value for a given J as the function of L in Eq. (10). On the other hand, three values of viz., Fj e = 2), F e = 2), and Fg (e = 4), occur for octahedral systems with an odd number of electrons. Griffith proposes the names E, E, and U. ... 

Various materials, present by accident or design, can alter the course of reduction by arresting the reaction at an intermediate product or by causing the formation of coupled products (94,95). These deviations can range from only a small yield loss to the formation of a major product. The work of Kosak (56) on o-nitroanisole is instructive in this regard, where small amounts of... 

However, as was already emphasized for the non-relativistic transitions, all these sum rules are not rigorous due to interactions between the intermediate momenta and the presence of deviations from pure coupling schemes. 

For OH, experimental evidence indicates the intermediate case as illustrated in Figure 4, which shows the deviation of each rotational sublevel from its normalized Boltzmann value. Not surprisingly, the laser coupled state is overpopulated compared to other states. 

If the diabatic coupling matrix element, He, is -independent, this d/dR matrix element between two adiabatic states must have a Lorentzian H-depen-dence with a full width at half maximum (FWHM) of 46. Evidently, the adiabatic electronic matrix element We(R) is not - independent but is strongly peaked near Rc- Its maximum value occurs at R = Rc and is equal to 1/46 = a/4He. Thus, if the diabatic matrix element He is large, the maximum value of the electronic matrix element between adiabatic curves is small. This is the situation where it is convenient to work with deperturbed adiabatic curves. On the contrary, if He is small, it becomes more convenient to start from diabatic curves. Table 3.5 compares the values of diabatic and adiabatic parameters. The deviation from the relation, We(i )max x FWHM = 1, is due to a slight dependence of He on R and a nonlinear variation of the energy difference between diabatic potentials. When We(R) is a relatively broad curve without a prominent maximum, the adiabatic approach is more convenient. When We (R) is sharply peaked, the diabatic picture is preferable. The first two cases in Table 3.5 would be more convenient to treat from an adiabatic point of view. The description of the last two cases would be simplest in terms of diabatic curves. The third case is intermediate between the two extreme cases and will be examined later (see Table 3.6). 

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