Use of Approximate Functions in Evaluating the Born Matrix Element forH2+

Abstract

The Born matrix element for the $1s{\sigma }_g-2p{\sigma }_u$ transition in ${\mathrm{H}}_2^{+}$ at an internuclear separation $R$ of $2.0a_0$, is calculated using the linear-combination-of-atomic-orbitals (LCAO) and screened LCAO functions for ${\mathrm{H}}_2^{+}$. A comparison with the results found using the eigenfunctions and one other approximation is given. The $R$ dependence of the error for these approximations is also investigated. The behavior of the Born matrix element for large momentum transfer is shown to be accurately predicted by these approximate functions for intermediate and large $R$. The small-momentum-transfer behavior is less accurately predicted. The usefulness of these approximate functions for extremes in the momentum transfer is correlated with their local behavior in space. The calculation of the total cross section for scattering of an electron and of a hydrogen atom with these approximate functions is discussed and compared with the results for the eigenfunctions.