Fisher Information-Based Optimization of Mapped Fourier Grid Methods
The mapped Fourier grid method (mapped-FGM) is a simple and efficient discrete variable representation (DVR) numerical technique for solving atomic radial Schrödinger differential equations. It is set up on equidistant grid points, and the mapping, a suitable coordinate transformation to the radial...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-10-01
|
| Series: | Atoms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2218-2004/12/10/50 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850205965779992576 |
|---|---|
| author | Sotiris Danakas Samuel Cohen |
| author_facet | Sotiris Danakas Samuel Cohen |
| author_sort | Sotiris Danakas |
| collection | DOAJ |
| description | The mapped Fourier grid method (mapped-FGM) is a simple and efficient discrete variable representation (DVR) numerical technique for solving atomic radial Schrödinger differential equations. It is set up on equidistant grid points, and the mapping, a suitable coordinate transformation to the radial variable, deals with the potential energy peculiarities that are incompatible with constant step grids. For a given constrained number of grid points, classical phase space and semiclassical arguments help in selecting the mapping function and the maximum radial extension, while the energy does not generally exhibit a variational extremization trend. In this work, optimal computational parameters and mapping quality are alternatively assessed using the extremization of (coordinate and momentum) Fisher information. A benchmark system (hydrogen atom) is employed, where energy eigenvalues and Fisher information are traced in a standard convergence procedure. High-precision energy eigenvalues exhibit a correlation with the extrema of Fisher information measures. Highly efficient mapping schemes (sometimes classically counterintuitive) also stand out with these measures. Same trends are evidenced in the solution of Dalgarno–Lewis equations, i.e., inhomogeneous counterparts of the radial Schrödinger equation occurring in perturbation theory. A detailed analysis of the results, implications on more complex single valence electron Hamiltonians, and future extensions are also included. |
| format | Article |
| id | doaj-art-958df274eddf4e33a2723b675f88a4ee |
| institution | OA Journals |
| issn | 2218-2004 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Atoms |
| spelling | doaj-art-958df274eddf4e33a2723b675f88a4ee2025-08-20T02:10:57ZengMDPI AGAtoms2218-20042024-10-0112105010.3390/atoms12100050Fisher Information-Based Optimization of Mapped Fourier Grid MethodsSotiris Danakas0Samuel Cohen1Atomic and Molecular Physics Laboratory, Physics Department, University of Ioannina, 45110 Ioannina, GreeceAtomic and Molecular Physics Laboratory, Physics Department, University of Ioannina, 45110 Ioannina, GreeceThe mapped Fourier grid method (mapped-FGM) is a simple and efficient discrete variable representation (DVR) numerical technique for solving atomic radial Schrödinger differential equations. It is set up on equidistant grid points, and the mapping, a suitable coordinate transformation to the radial variable, deals with the potential energy peculiarities that are incompatible with constant step grids. For a given constrained number of grid points, classical phase space and semiclassical arguments help in selecting the mapping function and the maximum radial extension, while the energy does not generally exhibit a variational extremization trend. In this work, optimal computational parameters and mapping quality are alternatively assessed using the extremization of (coordinate and momentum) Fisher information. A benchmark system (hydrogen atom) is employed, where energy eigenvalues and Fisher information are traced in a standard convergence procedure. High-precision energy eigenvalues exhibit a correlation with the extrema of Fisher information measures. Highly efficient mapping schemes (sometimes classically counterintuitive) also stand out with these measures. Same trends are evidenced in the solution of Dalgarno–Lewis equations, i.e., inhomogeneous counterparts of the radial Schrödinger equation occurring in perturbation theory. A detailed analysis of the results, implications on more complex single valence electron Hamiltonians, and future extensions are also included.https://www.mdpi.com/2218-2004/12/10/50DVRmapped Fourier grid methodSchrödinger equation solutionFisher information measureconvergence evaluation |
| spellingShingle | Sotiris Danakas Samuel Cohen Fisher Information-Based Optimization of Mapped Fourier Grid Methods Atoms DVR mapped Fourier grid method Schrödinger equation solution Fisher information measure convergence evaluation |
| title | Fisher Information-Based Optimization of Mapped Fourier Grid Methods |
| title_full | Fisher Information-Based Optimization of Mapped Fourier Grid Methods |
| title_fullStr | Fisher Information-Based Optimization of Mapped Fourier Grid Methods |
| title_full_unstemmed | Fisher Information-Based Optimization of Mapped Fourier Grid Methods |
| title_short | Fisher Information-Based Optimization of Mapped Fourier Grid Methods |
| title_sort | fisher information based optimization of mapped fourier grid methods |
| topic | DVR mapped Fourier grid method Schrödinger equation solution Fisher information measure convergence evaluation |
| url | https://www.mdpi.com/2218-2004/12/10/50 |
| work_keys_str_mv | AT sotirisdanakas fisherinformationbasedoptimizationofmappedfouriergridmethods AT samuelcohen fisherinformationbasedoptimizationofmappedfouriergridmethods |