Matrix Methods for Solving HartreeFock Equations in Atomic Structure Calculations and Line Broadening
Abstract
Atomic structure of Nelectron atoms is often determined by solving the HartreeFock equations, which are a set of integrodifferential equations. The integral part of the HartreeFock equations treats electron exchange, but the HartreeFock equations are not often treated as an integrodifferential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the HartreeFock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods). Because the HartreeFock equations are an iterativerefinement 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 HartreeFock equations as an integrodifferential equation. It is well known that a derivative operator can be expressed as a matrix made of finitedifference coefficients; energy eigenvalues and eigenvectors can be obtained by using linearalgebra packages. The integral (exchange) part of the HartreeFock equation can be approximated as a sum and written as a matrix. The HartreeFock 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 structuremore »
 Authors:

 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Univ. of Texas at Austin, Austin, TX (United States)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States); The Univ. of Texas at Austin, Austin, TX (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1457406
 Alternate Identifier(s):
 OSTI ID: 1483531; OSTI ID: 1511657
 Report Number(s):
 SAND20182057J; LAUR1823615
Journal ID: ISSN 22182004; ATOMC5; 660952; TRN: US1901372
 Grant/Contract Number:
 NA0003525; AC5206NA25396; NA0003843
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Atoms
 Additional Journal Information:
 Journal Volume: 6; Journal Issue: 2; Journal ID: ISSN 22182004
 Publisher:
 MDPI
 Country of Publication:
 United States
 Language:
 English
 Subject:
 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; atomic structure; hartree fock; exchange; line broadening; scattering; 79 ASTRONOMY AND ASTROPHYSICS; Hartree Fock, Exchange, Line Broadening, Scattering
Citation Formats
Gomez, Thomas, Nagayama, Taisuke, Fontes, Chris, Kilcrease, Dave, Hansen, Stephanie B., Montgomery, Mike, and Winget, Don. Matrix Methods for Solving HartreeFock Equations in Atomic Structure Calculations and Line Broadening. United States: N. p., 2018.
Web. doi:10.3390/atoms6020022.
Gomez, Thomas, Nagayama, Taisuke, Fontes, Chris, Kilcrease, Dave, Hansen, Stephanie B., Montgomery, Mike, & Winget, Don. Matrix Methods for Solving HartreeFock Equations in Atomic Structure Calculations and Line Broadening. United States. doi:10.3390/atoms6020022.
Gomez, Thomas, Nagayama, Taisuke, Fontes, Chris, Kilcrease, Dave, Hansen, Stephanie B., Montgomery, Mike, and Winget, Don. Mon .
"Matrix Methods for Solving HartreeFock Equations in Atomic Structure Calculations and Line Broadening". United States. doi:10.3390/atoms6020022. https://www.osti.gov/servlets/purl/1457406.
@article{osti_1457406,
title = {Matrix Methods for Solving HartreeFock Equations in Atomic Structure Calculations and Line Broadening},
author = {Gomez, Thomas and Nagayama, Taisuke and Fontes, Chris and Kilcrease, Dave and Hansen, Stephanie B. and Montgomery, Mike and Winget, Don},
abstractNote = {Atomic structure of Nelectron atoms is often determined by solving the HartreeFock equations, which are a set of integrodifferential equations. The integral part of the HartreeFock equations treats electron exchange, but the HartreeFock equations are not often treated as an integrodifferential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the HartreeFock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods). Because the HartreeFock equations are an iterativerefinement 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 HartreeFock equations as an integrodifferential equation. It is well known that a derivative operator can be expressed as a matrix made of finitedifference coefficients; energy eigenvalues and eigenvectors can be obtained by using linearalgebra packages. The integral (exchange) part of the HartreeFock equation can be approximated as a sum and written as a matrix. The HartreeFock 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 freeelectron wavefunctions, thus improving how the atoms and free electrons interact. Here, 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.},
doi = {10.3390/atoms6020022},
journal = {Atoms},
number = 2,
volume = 6,
place = {United States},
year = {2018},
month = {4}
}
Figures / Tables:
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