International audienc

The system of two coupled Schrödinger differential equations (CE) arising in the close-coupling description of electron-atom scattering is considered. A “canonical functions” approach is used. This approach replaces the integration of CE for the eigenvector with unknown initial values by the integration of CE for “canonical” functions having well-defined initial values at an arbitrary origin r0(0 < r0 < ∞). The terms of the reactance matrix are then deduced from these canonical functions. The results are independent from the choice of r0. It is shown that many conventional difficulties in the numerical application are avoided, mainly that of the choice of the starting boundary variable rs∼0 which is automatically determined. The results of the present method are compared to those of other methods and show excellent agreement with previous accurate results

a b s t r a c tThe Canonical Function Method (CFM) is a powerful method that solves the radial Schrödinger equation for the eigenvalues directly, without having to evaluate the eigenfunctions. It is applied to various quantum mechanical problems in Atomic and Molecular physics in the presence of regular or singular potentials. It has also been developed to handle single and multiple channel scattering problems, where phaseshift is required for the evaluation of the scattering cross-section. Its controllable accuracy makes it a valuable tool for the evaluation of vibrational levels of cold molecules, a sensitive test of the Bohr correspondence principle and a powerful method for tackling local and non-local spin dependent problems.

The determination of the phase-shift 6,(E) (related to a central potential V ( r ) , a total energy E, and an angular momentum p) is considered. The "canonical functions" approach already used for the eigenvalue problem is adapted to that of 6. The conventional approach computes the radial wave functiony,(E; r) starting at r, -0 (with convenient initial values) and stepping on toward a large value r = R -m, where y , is matched to its asymptotic value y,,(R) -a sin(kRp 7r/2 + 6,) and 6 is deduced. The present approach starts at any "origin" ro, replaces the use of the wave functiony by that of the "canonical functions" a and p (well defined for given V , E, and p) and defines two functions q(r) and Q(r) in terms of a and p. When r 4 0, q(r) approaches a constant limit giving Q(ro), and thus the starting problem is avoided. Using this value Q(ro), the function Q(r) is generated for r > ro. The function Q(r) reaches a constant limit when r + m; this limit is precisely tan 8; thus, the "final" matching problem is avoided. The present method is applied to the Lennard-Jones potential function for low and high E and for low and high p . The comparison of the results of the present method with those of confirmed numerical methods show that the present method is competitive.