Frank L. Yip

Hybrid orbital and numerical grid representation for electronic continuum processes: Double photoionization of atomic beryllium

Hybrid Orbital and Numerical Grid Representation for Electronic Continuum Processes: Double Photoionization of Atomic Beryllium F. L. Yip, 1 C. W. McCurdy, 1, 2 and T. N. Rescigno 1 Lawrence Berkeley National Laboratory, Chemical Sciences, and Ultrafast X-ray Science Laboratory, Berkeley, CA 94720 USA Departments of Applied Science and Chemistry, University of California, Davis, CA 95616 USA (Dated: April 20, 2010) A general approach for ab initio calculations of electronic continuum processes is described in which the many-electron wave function is expanded using a combination of orbitals at short range and the finite-element discrete variable representation(FEM-DVR) at larger distances. The orbital portion of the basis allows the efficient construction of many-electron configurations in which some of the electrons are bound, but because the orbitals are constructed from an underlying FEM- DVR grid, the calculation of two-electron integrals retains the efficiency of the primitive FEM-DVR approach. As an example, double photoionization of beryllium is treated in a calculation in which the 1s 2 core is frozen. This approach extends the use of exterior complex scaling (ECS) successfully applied to helium and H 2 to calculations with two active electrons on more complicated targets. Integrated, energy-differential and triply-differential cross sections are exhibited, and the results agree well with other theoretical investigations. I. INTRODUCTION Numerical grid methods combining the Finite-Element Discrete Variable Representation (FEM-DVR) approach with Exterior Complex Scaling (ECS) [1] have allowed effectively exact calculations of electron impact ioniza- tion [2] and double photoionization cross sections for two electron atomic [3] and molecular targets [4] over the last decade. The FEM-DVR approach has formed the basis of numerically accurate solutions of the time- dependent Schr¨ dinger equation for such systems under o the influence of ultrashort radiation pulses [5–7]. On the other hand the success of atomic and molecular electronic structure calculations on many-electron systems depends on expressing those wave functions in terms of configu- rations with orthogonal orbitals. The configuration in- teraction (CI) method and its variants are orbital based, and that is the standard approach and the natural lan- guage of electronic structure theory. Here we describe a method that has the advantages of both approaches. It exploits the finite element as- pect of the FEM-DVR to allow the construction of or- bitals from only the DVR polynomial basis in the first few finite elements, to describe electrons in bound or- bitals. The remainder of DVR grid representation is left untouched and describes the continuum portions of the wave function. We have previously demonstrated that a particular advantage of the DVR approach is that the two-electron integrals in the DVR basis are given by sim- ple formulas and are diagonal in the two pairs of indices that label them. That fact results in remarkable sim- plifications even when the DVR basis is transformed to atomic or molecular orbitals, as we will discuss below. As a demonstration of this numerical method we choose the relatively simple case of double photoioniza- tion (DPI) of the beryllium atom in which the orbital in the 1s 2 core remains frozen. Since the 1s 2 core electrons lie energetically far below the 2s valence shell and the mean value r of the 1s and 2s orbitals are very differ- ent [8], correlation effects between electrons occupying different shells are likely to be minimal. Thus, a rea- sonable approximation to double photoionization from beryllium allows for the valence electrons to be corre- lated separately from the closed-shell core electrons, and recent theoretical investigations on Be that have followed this approach to produce fully differential DPI cross sec- tions include convergent-close coupling (CCC) [9], time- dependent close coupling (TDCC) [10], and the hyper- spherical R-matrix method with semiclassical outgoing waves (HRM-SOW) [11]. Integral DPI cross sections have also been computed more recently using B-spline basis methods [12] as well as with the R-matrix-with- pseudostates (RMPS) method [13]. Each of these meth- ods has been modified to account for the presence of the core electrons in some way in order to apply these tech- niques to beryllium. Only limited experimental investi- gations of DPI from beryllium have been conducted [14– In this work, we calculate cross sections for DPI from beryllium at 20 eV above the double ionization thresh- old, and compare the resulting triply differential cross sections with those of other theoretical approaches using different core potential approximations and methods of solving DPI problem. In Section II we describe the hybrid orbital and FEM-DVR basis numerical method and the resulting implementation of the ECS approach to solving the double ionization problem. The method used previ- ously in ECS calculations on two-electron systems [2, 3] of extracting the amplitudes pertaining to double ionization from the scattering wave function must be modified in the presence of the field of the frozen core, and these details are also given in Section II. Section III presents the calcu- lated cross sections for DPI from beryllium, including the