Optimization of complex slater‐type functions with analytic derivative methods for describing photoionization differential cross sections

Abstract

The complex basis function (CBF) method applied to various atomic and molecular photoionization problems can be interpreted as an L2 method to solve the driven-type (inhomogeneous) Schrödinger equation, whose driven term being dipole operator times the initial state wave function. However, efficient basis functions for representing the solution have not fully been studied. Moreover, the relation between their solution and that of the ordinary Schrödinger equation has been unclear. For these reasons, most previous applications have been limited to total cross sections. To examine the applicability of the CBF method to differential cross sections and asymmetry parameters, we show that the complex valued solution to the driven-type Schrödinger equation can be variationally obtained by optimizing the complex trial functions for the frequency dependent polarizability. In the test calculations made for the hydrogen photoionization problem with five or six complex Slater-type orbitals (cSTOs), their complex valued expansion coefficients and the orbital exponents have been optimized with the analytic derivative method. Both the real and imaginary parts of the solution have been obtained accurately in a wide region covering typical molecular regions. Their phase shifts and asymmetry parameters are successfully obtained by extrapolating the CBF solution from the inner matching region to the asymptotic region using WKB method. The distribution of the optimized orbital exponents in the complex plane is explained based on the close connection between the CBF method and the driven-type equation method. The obtained information is essential to constructing the appropriate basis sets in future molecular applications. © 2017 Wiley Periodicals, Inc.

Calculation of photoionization differential cross sections using complex Gauss‐type orbitals

“…As clarified in our previous study, [22] reflecting its mathematical structure, the solution to the driven equation shows two different behaviors originating from the system independent non-L 2 outgoing wave term and the system dependent L 2 real function term. The latter behavior arises from the inhomogeneous solution term; therefore, it is largely affected by the driven term lU i , whose contribution is represented by conventional real GTOs.…”

“…Here, we briefly discuss the relationship between and the Schrödinger equation. A more detailed discussion was given in our previous paper . Because is a real function, this inhomogeneous term vanishes if the imaginary part is taken for both sides of eq.…”

“…A more detailed discussion was given in FULL PAPER our previous paper. [22] Because lU i is a real function, this inhomogeneous term vanishes if the imaginary part is taken for both sides of eq. (15)…”

“…In fact, many years ago, McCurdy and Rescigno analyzed and developed a method to compute a continuum wave function using the CBF method; they successfully calculated the so‐called asymmetry parameter for the H atom photoionization problem, but with many basis functions . In a recent study, we optimized five or six cSTOs for the frequency dependent polarizability of H atom and found that the optimized cSTOs can accurately represent the solutions of the driven‐type equation in the molecular region . However, this optimization method does not work for cGTO basis functions.…”

“…[14] In a recent study, we optimized five or six cSTOs for the frequency dependent polarizability of H atom and found that the optimized cSTOs can accurately represent the solutions of the driventype equation in the molecular region. [22] However, this optimization method does not work for cGTO basis functions. In this study, we used an alternative method to represent the solutions of the driven-type equation in terms of several cGTOs by fitting the complex orbital exponents to the outgoing Coulomb functions.…”