MOLECULAR PHOTOIONIZATION CROSS SECTIONS BY THE COMPLEX BASIS FUNCTION METHOD

Abstract

The photoionization cross section can be expressed as a particular matrix element of the resolvent of the molecular electronic Hamiltonian, \begin{equation} \sigma(w) = \frac{4\pi w}{c}\lim_{c\rightarrow 0} lm < \Psi_{0} |\mu (E_{0} + \omega - H + i \epsilon)^{-1} \mu|\Psi_{0}> \end{equation} where $\Psi_{0}$ is the ground state electronic wave function, $\mu$ is the dipole operator, $\omega$ is the frequency, c is the speed of light, and atomic units are employed. We have shown recently that by using a mixture of real and complex Gaussian basis functions to form a matrix representation of the Hamiltonian it is possible to use this expression to compute molecular photoionization cross sections from the results of a matrix $diagonalization.^{1}$ The working expression of this approach has the form \begin{equation} \lim (f,(E - H + i\epsilon)^{-1} f) = \sum^{N}{f = 1}\frac{(f, \Phi{i})(\Phi_{i}, f)}{E - E_{i}} \end{equation} where $f = \mu^\Psi{{D}}$, and the functions $\Phi{i}$, and associated eigenvalues, $E_{i}$, are from a finite-basis diagonalization of the Hamiltonian. This procedure is based on a variational principle for the matrix element from which the photoionization cross section is calculated, and can be applied in the presence of coupling between ionization channels. The interference between resonance features and the electron-ion scattering background is naturally incorporated by this method, and the method is easily implemented for polyatomic systems. Results are reported for K-shell and valence shell ionization of $N_{2}$ at the static-exchange level. Excellent agreement with most other calculations is obtained, especially those which explicitly compute the photoionization cross section from the electron-ion scattering wavefunction. The complex basis function technique is particularly successful in reproducing resonance features in these cross sections. A discussion of the variational behavior of the amplitude is given, and it is shown that the variational nature of these calculations provides a useful computational diagnostic.