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.