Preprint Article Version 1 This version is not peer-reviewed

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

Version 1 : Received: 7 March 2018 / Approved: 8 March 2018 / Online: 8 March 2018 (02:10:55 CET)

A peer-reviewed article of this Preprint also exists.

Gomez, T.; Nagayama, T.; Fontes, C.; Kilcrease, D.; Hansen, S.; Montgomery, M.; Winget, D. Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening. Atoms 2018, 6, 22. Gomez, T.; Nagayama, T.; Fontes, C.; Kilcrease, D.; Hansen, S.; Montgomery, M.; Winget, D. Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening. Atoms 2018, 6, 22.

Journal reference: Atoms 2018, 6, 22
DOI: 10.3390/atoms6020022

## Abstract

Atomic structure of N-electron atoms is often determined using the Hartree-Fock method, which is an integro-differential equation. The exchange term of the Hartree-Fock equations is usually treated as an inhomogeneous term of a differential equation, or with a local density approximation. This work uses matrix methods to solve for the Hartree-Fock equations, rather than the more commonly-used shooting method to integrate an inhomogeneous 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 computer linear-algebra packages. We extend the same technique to integro-differential equations, where a discretized integral can be written as a sum in matrix form. This method is compared against experiment and standard atomic structure calculations. We also can use this method for free-electron wavefunctions. This technique is important for spectral line broadening in two ways: improving the atomic structure calculations, and improving the motion of the plasma electrons that collide with the atom.

## Subject Areas

atomic structure; hartree fock; exchange; line broadening; scattering