# 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,

- J. Schwinger, Phys. Rev.
**72**, 742 (1947)

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
- K. Takatsuka and V. McKoy, Phys. Rev. A
**24**, 2473 (1981)
- K. Takatsuka and V. McKoy, Phys. Rev. A
**30**, 1734 (1984)