# 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 structuremore »

- Authors:

- Sandia National Lab. (SNL-NM), 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. (SNL-NM), Albuquerque, NM (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1457406

- Alternate Identifier(s):
- OSTI ID: 1483531

- Report Number(s):
- SAND-2018-2057J; LA-UR-18-23615

Journal ID: ISSN 2218-2004; ATOMC5; 660952

- Grant/Contract Number:
- NA0003525; AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Atoms

- Additional Journal Information:
- Journal Volume: 6; Journal Issue: 2; Journal ID: ISSN 2218-2004

- 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

### Citation Formats

```
Gomez, Thomas, Nagayama, Taisuke, Fontes, Chris, Kilcrease, Dave, Hansen, Stephanie B., Montgomery, Mike, and Winget, Don.
```*Matrix Methods for Solving Hartree-Fock 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 Hartree-Fock 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 Hartree-Fock 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 Hartree-Fock 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 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. 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 = {Mon Apr 23 00:00:00 EDT 2018},

month = {Mon Apr 23 00:00:00 EDT 2018}

}

Works referenced in this record:

##
Self-Consistent Equations Including Exchange and Correlation Effects

journal, November 1965

- Kohn, W.; Sham, L. J.
- Physical Review, Vol. 140, Issue 4A, p. A1133-A1138