The Schwinger Multichannel (SMC) Method


The Schwinger Multichannel method is a modification of Julian Schwinger's original variational method for scattering problems. The modifications take into account the complexities that arise in scattering involving many-particle systems. In particular, the SMC method does not require continuum states of the target in order to form a proper representation of the Green's function. An important feature of the original Schwinger method that is preserved in the SMC method is that all required matrix elements are independent of the long range behavior of the trial functions.

The SMC method replaces the solution of Schrödinger's equation with the solution of a set of linear equations. Typically this set is modest in size (a few dozen to a few thousand equations), and its solution poses no difficulty. The computationally intensive steps are the construction of the coefficient and right-hand-side matrices that define the linear system. The elements of these matrices, which represent integrals involving many-particle states and operators, must be built up from elementary quantities describing the pairwise electron-electron and electron-nucleus interactions. Because both the number of such elementary quantities and the work needed to combine them into final matrix elements grow rapidly with the size of the molecule and the amount of detail included in the calculation, high-performance computing is essential to practical applications.

References

Julian Schwinger never bothered to publish his variational principle; the usual citation,

is merely the abstract of a conference talk. However, any good text on quantum scattering theory will explain it. The original references for the SMC method are