Case of Intermediate Coupling Strength

Another example is in Li2 (Antonova, et al., 2000) where the vibrational, rotational, and isotope dependence of the linewidths is studied. However, the coupling term is taken as constant rather than a Lorentzian form, giving a not so good fit. 

The Golden Rule formula Eq. (7.5.16) for the FWHM and Eq. (7.5.9) for the level shift are expressed in terms of the unperturbed vibrational wavefunc-tions. For strong predissociations, this approximation becomes untenable. Exant methods exist that can determine both the linewidth and the level shift. One method consists of numerically solving the following coupled equations (Lefebvre-Brion and Colin, 1977 Child and Lefebvre, 1978)  

Even for this case of strong coupling, the adiabatic picture of two potential curves that avoid crossing is inappropriate. Child has introduced an intermediate coupling picture that takes advantage of both diabatic and adiabatic characteristics. The diabatic curve of the predissociated state is displayed (solid lines) in Fig. 7.36a The corresponding diabatic vibrational levels, Ed, are plotted versus J(J + 1) (solid lines) in Fig. 7.36b. 

The adiabatic picture provides an alternative point of view. In Fig. 7.36a the adiabatic curves are plotted as dashed lines. The upper adiabatic potential supports a set of adiabatic levels, Em. As the rotational constants for the levels of the adiabatic curve are smaller than those for the corresponding levels of the diabatic curve, the adiabatic levels (dashed lines on Fig. 7.36) will cross the diabatic levels. Note that there is no relationship between the numbering of vd and vad- 

By Child s semiclassical approach, it has been shown (Child, 1976) that the actual levels E (dotted-dashed lines in Fig. 7.36b) in the region of the curve crossing lie intermediate in energy between the two sets of approximate levels 

---

Much work has been done to predict the locations of the zeroes of the line width. Child (1976) has demonstrated that, in the case of intermediate coupling strength, a zero occurs in the linewidth whenever the energies of the diabatic and adiabatic levels coincide (and this conclusion is also valid in the case of weak coupling). However, until the work of Kim, et al., (1994) and recently of Cornett, et al., (1999), it seems that the connection between zeroes of predissociation linewidth and q reversal was not appreciated. 

In practice, one finds the case of intermediate coupling, and thus it is always necessary to describe a state by a superposition of several states, be it in the jj coupling scheme or the LS coupling scheme, which makes the opposite assumption as concerns the relative strength of Coulomb interaction and spin-orbit effects. Apart from the fact that a one-determinant approximation for a heavy system is usually more accurate in jj coupling, both coupling schemes are equivalent in a configuration interaction picture. 

This spectral density has a characteristic low-frequency behavior J((o) — rjo), where rj is the usual ohmic viscosity. The system-bath coupling strength can then be measured in terms of the dimensionless Kondo parameter K, and time scale of bath motions is described by a cutoff frequency (o. For many problems in low-temperature physics, this cutoff frequency is taken to be the largest frequency scale in the problem. In the case of electron transfer, the same spectral density with some intermediate value for is most appropriate for a realistic description of... 

---