Watson Hamiltonian equation

Contact Transformation for the Effective Hamiltonian.—The vibration-rotation hamiltonian of a polyatomic molecule, expressed in terms of normal co-ordinates, has been discussed in particular by Wilson, Decius, and Cross,24 and by Watson.27- 28 It is given by the following expression for a non-linearf polyatomic molecule, to be compared with equation (17) for a diatomic molecule ... 

Here, Aftot is the total mass of the nuclei. Note that when eliminating the motion of the center-of-mass we arbitrarily eliminated the first nuclear coordinate (we could eliminate any single coordinate y). Equation (1-13) can further be transformed to the body-fixed frame. When applying the so-called Eckart conditions35 one would get the standard Watson s Hamiltonian describing the nuclear motions in molecules36. 

We return now to the coupled-channels approach based on operator equations. The formalism is adequately covered in several books (see e.g. Goldberger and Watson, 1964 Newton, 1966 Levine, 1969) and we shall only present the main equations. Assume for simplicity that only two reaction channels a (for A + BC)and 6(AB + C) exist. The total Hamiltonian H may be split into two terms, a channel Hamiltonian Hc for free motion and a channel interaction Vc, with c a, b. If a is the initial free state and we want the scattering states in channel a, i.e. those in the absence of rearrangement, then the Lippmann-Schwinger equation gives... 

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. 

The many-body approach (impulsive models, Paddeev-Watson equations) represent a new direction in the development of chemical collision theory,in which the kinematic effects appear in addition to the molecular interactions. An excellent review of all these new methods is given by MICHA /89a/. In particular, the effective Hamiltonian methods have been considered by MICHA /89b/ with a detailed discussion of different approximations to optical potentials. 

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