The collision of an electron with a molecule A may be illustrated schematically as
where 
is the electron's initial kinetic energy and the momentum vector
points in its initial direction of travel; after the collision,
the electron travels along 
with kinetic energy 
. If 
differs from 
, the collision is said to be inelastic, and energy is
transferred to the target, leaving it in an excited state, denoted 
.
The quantity we seek is the probability of occurrence or cross section for
this process, as a function of the energies 
and 
and of the angle
between the directions 
and 
. (Since a gas is a very
large ensemble of randomly oriented molecules, orientational dependence of
these quantities for an asymmetric target A is averaged over in
calculations.)
The SMC procedure [Lima:90a], [Takatsuka:81a;84a], a multichannel extension of Schwinger's variational principle [Schwinger:47a], is a method for obtaining cross sections for low-energy electron-molecule collision processes, including elastic scattering and vibrational or electronic excitation. As such, it is capable of accurately treating effects arising from electron indistinguishability and from polarization of the target by the charge of the incident electron, both of which can be important at low collision velocities. Moreover, it is formulated to be applicable to and efficient for molecules of arbitrary geometry.
The scattering amplitude 
, a complex quantity whose
square modulus is proportional to the cross section, is approximated in the
SMC method as
where 
is an 
-electron interaction-free
wave function of the form
V is the interaction potential between the scattering electron and the
target, and the 
-electron functions 
are spin-adapted Slater
determinants which form a linear variational basis set for approximating the
exact scattering wave functions 
and
. The 
are elements of the
inverse of the matrix representation in the basis 
of the operator
Here P is the projector onto open (energetically accessible) electronic states,
is the 
-electron Green's function projected onto open
channels, and 
, where E is the total energy of the system and
H is the full Hamiltonian.
In our implementation, the 
-electron functions 
are
formed from antisymmetrized products of one-electron molecular
orbitals which are themselves combinations of
Cartesian Gaussian orbitals
commonly used in molecular electronic-structure studies. Expansion of
the trial scattering wave function in such a basis of exponentially
decaying functions is possible since the trial function of the SMC
method need not satisfy scattering boundary conditions asymptotically
[Lima:90a],
[Takatsuka:81a;84a]. All matrix elements needed in the evaluation of
can then be obtained analytically, except those
of 
. These terms are evaluated numerically via a
momentum-space quadrature procedure [Lima:90a],
[Takatsuka:81a,84a].
Once all matrix elements are calculated, the final step in the
calculation is solution of a system of linear equations to obtain the
scattering amplitude 
in the form given above.
The computationally intensive step in the above formulation is the evaluation of large numbers of so-called ``primitive'' two-electron integrals
for all unique combinations of Cartesian Gaussians 
,
, and 
, and for a wide range of 
in both
magnitude and direction. These integrals are evaluated analytically by
a set of subroutines comprising approximately two thousand lines of
FORTRAN. Typical calculations might require 
to 
calls
to this integral-evaluation suite, consuming roughly 80% of the total
computation time. Once calculated, the primitive integrals are
assembled in appropriate combinations to yield the matrix elements
appearing in the variational expression for 
.
The original CRAY code performs this procedure in two steps: first, a
repeated linear transformation to integrals involving molecular
orbitals, then a transformation from the molecular-orbital integrals to
physical matrix elements. The latter step is equivalent to an
extremely sparse linear transformation, whose coefficients are
determined in an elaborate subroutine with a complicated logical flow.