Group theory used to improve the efficiency of transfer-matrix computations

Transfer-matrix methodology is frequently used to deal with elastic scattering problems that require a solution of Schrödinger or homogeneous Maxwell equations in the continuous part of their spectra. As predicted by group theory, the basic states used for the expansion of the solutions can be separated into independent sets, thus enabling the scattering problem to be solved with a drastically improved efficiency. Depending on the peculiar symmetry in the problem, the basic states can present pairs of "conjugate sets," whose associated characters are complex conjugate of each other. When the potential energy takes strict real values, the transfer matrices corresponding to these conjugate sets have well-defined relationships that enable the transfer matrices of both conjugate sets to be computed from a single propagation step. This results in a further reduction of up to 50% of the total computation time. This paper presents the way group theory can be used systematically to improve the efficiency of transfer-matrix computations. In a first part, the basic states are separated into independent sets. Relationships between the transfer matrices corresponding to conjugate sets are then derived. The theory is finally illustrated by a simulation of electronic scattering by a C60 molecule in a projection configuration.