Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

Abstract

Atomic structure of N-electron atoms is often determined by solving the Hartree-Fock equations, which are a set of integro-differential equations. The integral part of the Hartree-Fock equations treats electron exchange, but the Hartree-Fock equations are not often treated as an integro-differential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the Hartree-Fock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods). Because the Hartree-Fock equations are an iterative-refinement method, the inhomogeneous term relies on the previous guess of the wavefunction. In addition, there are numerical complications associated with solving inhomogeneous differential equations. This work uses matrix methods to solve the Hartree-Fock equations as an integro-differential equation. It is well known that a derivative operator can be expressed as a matrix made of finite-difference coefficients; energy eigenvalues and eigenvectors can be obtained by using linear-algebra packages. The integral (exchange) part of the Hartree-Fock equation can be approximated as a sum and written as a matrix. The Hartree-Fock equations can be solved as a matrix that is the sum of the differential and integral matrices. We compare calculations using this method against experiment and standard atomic structure calculations. This matrix method can also be used to solve for free-electron wavefunctions, thus improving how the atoms and free electrons interact. This technique is important for spectral line broadening in two ways: it improves the atomic structure calculations, and it improves the motion of the plasma electrons that collide with the atom.

The Fourth Workshop on Lineshape Code Comparison: Line Merging

“…The main difference from DWE is that the Griem, Blaha and Kepple [14] used in PPP is tailored for highly-charged Z and does not include exchange, while DWE does. As shown in [11,12], exchange makes a large difference for low Z-elements and this difference is exacerbated for high densities; in contrast plane or distorted wave calcuation differences are much smaller.…”

“…CONV has a number of options and relevant ones are discussed later on. Finally DWE [11,12] is a spectral Line broadening model that treats plasma electrons quantum mechanically using a second-order distorted wave (DW) treatment. Exchange between the plasma electrons and the bound electron(s) is included.…”

Current challenges in the physics of white dwarf stars

“…A series of improvements followed. The application of a different numerical method to solve the Hartree-Fock equations improved the accuracy of the wave functions of the bound and the perturber electrons used to calculate the transition probability [249]. The inclusion of correlations between the radiator and the plasma in the relaxation theory of electron broadening [250] resulted in a considerable improvement in the agreement with the line shift for n e = 2-10 × 10 16 cm −3 .…”

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

“…Most of these methods use LDA methods for exchange, but as demonstrated in the previous sections, these may not be the most accurate when exchange becomes important. This method of using an exact exchange treatment is now being implemented in the DWE line shape code [23,24] (which was used in the SLSP4 workshop), which improves upon the relaxation theory method of Junkel et al [25] to include exchange.…”