The Newton variational functional for the log-derivative matrix: Use of the reference energy Green’s function in an exchange problem

Abstract

The Newton Variational Principle for the log-derivative matrix (the Y-NVP) is studied in the context of a collinear exchange problem. In contrast to the integral equation methods that calculate the K or the T matrices directly, the matrix elements of the log-derivative Newton functional can be made independent of the scattering energy. This promises considerable savings in computational effort when state to state transition probabilities are calculated at several energies, since the matrix elements of the functional need be calculated only once. Green’s functions defined with respect to a reference energy, called the reference energy Green’s functions (or the REGFs), play a central role in the Y-NVP functional. The REGFs may be defined with or without reference to asymptotic channel energies. If channel dependent REGFs are used, the Y-NVP converges at the same rate as the GNVP for the K or T matrices, when the scattering energy is the same as the reference energy. On the other hand, channel independent REGFs permit even further reductions in computational effort. We use both types of REGFs in the present paper, and compare the rates of convergence. These comparisons show that the convergence rate of the method is not significantly altered by the type of REGF used. Further, we show that the Y-NVP is able to achieve rapid convergence of reactive transition probabilities over a large range of scattering energies, even when scattering resonances are present. An analysis of the computational effort required for each part of the calculation leads to the conclusion that a Y-NVP calculation using a channel independent REGF requires essentially only the same amount of computer time as a log-derivative Kohn (Y-KVP) calculation, while, presumably, offering faster convergence.