Fitting continuum wavefunctions with complex Gaussians: Computation of ionization cross sections

Abstract

We implement a full nonlinear optimization method to fit continuum states with complex Gaussians. The application to a set of regular scattering Coulomb functions allows us to validate the numerical feasibility, to explore the range of convergence of the approach, and to demonstrate the relative superiority of complex over real Gaussian expansions. We then consider the photoionization of atomic hydrogen, and ionization by electron impact in the first Born approximation, for which the closed form cross sections serve as a solid benchmark. Using the proposed complex Gaussian representation of the continuum combined with a real Gaussian expansion for the initial bound state, all necessary matrix elements within a partial wave approach become analytical. The successful numerical comparison illustrates that the proposed all‐Gaussian approach works efficiently for ionization processes of one‐center targets.

A complex Gaussian approach to molecular photoionization

“…The resulting closed form expressions have been presented in ref. 26, and have been numerically validated with an rGTO representation of the exact states of the hydrogen atom 26 and of approximate states of small atoms. 54 We are currently working on the natural extension of the proposed complex Gaussian strategy to represent continuum states in two directions.…”

“…In this subsection, we give an overview of the cGTO representation method introduced in ref. 26, as a generalization of previous works on rGTOs. 20,21 We also provide a table of optimal exponents used later in Section 2.…”

“…The optimal exponents for l ¼ 1 have been already published in ref. 26 and were obtained with slightly different numerical parameters in the optimization code…”

“…24,25 In a recent work, we have developed an optimization method able to provide sets of complex Gaussians (i.e., GTOs with a complex exponent) able to reproduce accurately Coulomb waves, and whose scattering applications in the benchmark case of atomic hydrogen demonstrated their reliability. 26 The complex Gaussian-type orbitals (cGTOs) are defined as…”

“…33 In ref. 26, we have focused our efforts on the numerical method used to perform cGTOs optimization in order to improve their numerical efficiency as a continuum representation. cGTOs oscillate with a spatial frequency increasing with the radial coordinate r, a feature that is not physically satisfactory.…”