Kinematics of the CS method for the treatment of molecular collisions

Abstract

This paper presents the kinematical part of a proposal for founding the CS method (for one definition, see V. Khare and D. J. Kouri [J. Chem. Phys. 72, 2017 (1980)]) for the quantum treatment of molecular collisions on a certain testable physical approximation scheme. That is, let the molecules be rigid diatoms A1A2 and B1B2, with internal angular momentum ja and jb, respectively; with relative position, momentum, and angular momentum r, p, and l=r×p; and with total angular momentum J=ja+jb+l. Then the motivating conjecture is that, in addition to commuting exactly with J, and as a result of dynamical properties not discussed here, the fully off-the-energy-shell T(E) operator approximately commutes with either r, r̂, or J ⋅ r̂=ω=‘‘the r-helicity.’’ The principal results obtained in the paper are these: First, the definitions of, and transformations between, certain complete sets of system basis states in which the r-helicity ω or the p-helicity λ=J ⋅ p̂ is diagonal are established by methods similar to those of Jacob and Wick [Ann. Phys. (N.Y.) 7, 404 (1959)]. Second, it is argued that in several papers in the literature of the CS method an explicitly or presumptively incorrect kinematical law was applied to derive the matrix elements of T(E) operators for atom–molecule collisions in a basis in which ω was said to be diagonal from the fully on-the-energy-shell matrix elements of a given T(E). It is clear from the contexts that the quantities tested for conservation could not have been the r-helicity in a quantum-mechanical sense. Thus, there is no foundation to the corresponding assertions in these papers that, even if the CS method works fairly well, r-helicity conservation is usually badly violated. Third, the Wigner–Mackey theory of induced representations of continuous groups and Schur’s lemma are applied to determine the limitations (analogous to the Wigner–Eckart theorem for single operators invariant under rotations) that commuting with J and with either of the sets of operators r, r̂, or ω, imposes on the matrix of a T(E) in a basis in which r is diagonal. Fourth, and finally, it is shown that the on-the-energy-shell matrix of a T(E) that commutes with J and r has the property that its matrix elements are zero unless the angular momentum transfer j′a+j′b−ja−jb (prime indicates post-, no prime indicates precollision) is perpendicular in a quantum sense to p′−p, a result that corresponds to one obtained for an analogous classical atom–molecule collision by V. Khare et al. [J. Chem. Phys. 74, 2275 (1981)].