A weak-mode representation of floppy molecules. II. Kinetic energy and total Hamiltonian matrix

Abstract

The total Hamiltonian matrix of a floppy molecule (i.e. in a bound state for which large-amplitude isomerization-like deformations are possible), is derived in an entirely analytical manner which rests on the basic idea that the weakest mode of internal deformation is separated from the other stronger modes. The latter are supposed to undergo throughout only small-amplitude vibrations with respect to equilibrium positions which vary adiabatically with the former. In using (i) for the potential energy function, a Taylor expansion in the strong-mode displacements around the weak-mode isomerization reaction pathway, and (ii) for the kinetic energy operator matrix representation, a basis that includes for the strong modes adiabatically displaced harmonic oscillator basis functions, the total Hamiltonian operator can be represented by a sparse matrix whose diagonal elements are very like analytical spectroscopic terms, and the off-diagonal elements are also analytical expressions in terms of the quantum numbers of the basis functions. This adiabatic representation, which is known to be efficient for numerical diagonalization, is also particularly interesting from the physical point of view insofar as it allows to bring very clearly to the fore the coupling scheme between the internal deformations. The J = 0 case (no angular momentum) is developed first. Then the case where J ≢ 0 is treated (also analytically), in using for the first time in dynamical calculations the principal-axis system as the body-fixed (BF) frame of reference. This last feature is important, since it allows to distinguish (at least formally, if not physically) the Coriolis contribution to the total energy from the rotational one.