Predissociation approximation methods

Previous fully quantum mechanical studies of predissociation phenomena in triatomic molecules do not, to our knowledge, use a Hamiltonian that has a non-zero total angular momentum. Tennyson et al[43, 44, 45, 46, 47, 48, 49, 50, 51] solve the same equations as we do but have not yet, to our knowledge, treated any predissociation problems. The adiabatic rotation approximation method of Carter and Bowman[52] plus a complex C2 modification have, on the other hand, been used to compute rovibrational energies and widths in the HCO[53, 54] and HOCl[55, 56, 57] molecules. This method is based upon the the Wilson and Howard[58], Darling and Dennison[59] and Watson[60] formalism. It is less transparent but the exact formalism in refs.[58, 59, 60] is equivalent to the one presented here and in ref [43]. While both we and Tennyson et al[43] include the exact Hamiltonian in our formalism the latter authors 152] use an approximate method which they have analysed and motivated. 

Many calculations of linewidths for the vibrational and rotational predissociation have been performed by various researchers in the last few years and both highly accurate and approximate methods have been used. In Section IV, we describe a complete calcdation we have carried out of the spectrum for rotational predissociation in Ne-HF[53], and also show results of... 

The two atomic molecule is discussed from the viewpoint of wave mechanics with the help of an approximation method. The nature of the eigenfunctions and the position of the terms, as well as the appearance of transition, are discussed. Some additional comments on the symmetry properties of the eigenfunctions, discussed in a previous paper, will be made in addition. PYom the investigation we first obtein an expression for the rotational quantum number dependency of the doublet splitting which derives from the possibility that the anguleu momentum of the electron with respect to the internuclear axis can be parallel or antiparallel. Secondly (we obtsdn] an interpretation for the deviations of single terms from the values computed from term formulae, indicated as perturbations as they have been empirically determined in many band spectra for certain combinations of electronic, vibrational and rotational quantum numbers. We finally obtain a description of the phenomenon of predissodationi discovered by Henri, namely an estimate of the lifetime of the predissociated molecule. As far as possible, the theory is compared to experiment and it is pointed out where an extension of the experimental material is desirable. 

From this starting point, the authors develop equations leading to the evaluation of the real symmetric K matrix to specify the scattering process on the repulsive surface and propose its determination by a variational method. Furthermore, they show explicitly the conditions under which their rigorous equations reduce to the half-collision approximation. A noteworthy result of their approach which results because of the exact treatment of interchannel coupling is that only on-the-energy-shell contributions appear in the partial linewidth. Half-collision partial linewidths are found not to be exact unless off-the-shell contributions are accidentally zero or (equivalently) unless the interchannel coupling is zero. The extension of the approach to indirect photodissociation has also been presented. The method has been applied to direct dissociation of HCN, DCN, and TCN and to predissociation of HCN and DCN (21b). 

Coulson derived a new method for computing P(t) that gives the Landau-Zener formula as a special case. He considered three cases (1) transition between two discrete states [approximate internal conversion when the molecule passes from one discrete (bound) state to another], (2) transition from a bound state to the continuum (predissociation), and (3) transitions between two states of the continuum (corresponding to scattering problems). Coulson omitted in his article the computational details for P(t) for case (1). For case (2) he gave... 

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) ... 

The combination of the Azimuthal method of Section He with close-coupling expansions in the vy=0, vy=l and v q=1 vibrations enables vibrational predissociation resonances to be predicted. Por the reasons given above these predicted resonances are quite broad for the Ne complex( -- 10" cm ) but are much narrower for Ar(- 10" cm" ). This prediction remains to be verified in experiments. Furthermore, methods similar to those described for Ne-HF, but with the inclusion of Ae sudden approximation, have been used very recently to predict the full infrared spectrum for Ne-C2H4[71]. 

---