Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T15:29:45.142Z Has data issue: false hasContentIssue false
  • English
  • Français

Efficient Algorithm for Many-Electron Angular Momentum and Spin Diagonalization on Atomic Subshells

Published online by Cambridge University Press:  15 January 2016

Christian B. Mendl
Christian B. Mendl*
Affiliation:
Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, 85748 Garching bei München, Germany
*
*Corresponding author. Email address:mendl@ma.tum.de (C. B. Mendl)
Get access

Abstract

We devise an efficient algorithm for the symbolic calculation of irreducible angular momentum and spin (LS) eigenspaces within the n-fold antisymmetrized tensor product ^n Vu, where n is the number of electrons and u = s,p,d,… denotes the atomic subshell. This is an essential step for dimension reduction in configuration-interaction (CI) methods applied to atomic many-electron quantum systems. The algorithm relies on the observation that each Lz eigenstate with maximal eigenvalue is also an L2 eigenstate (equivalently for Sz and S2), as well as the traversal of LS eigenstates using the lowering operators L_ and S_. Iterative application to the remaining states in ^nVu leads to an implicit simultaneous diagonalization. A detailed complexity analysis for fixed n and increasing subshell number u yields run time O(u3n–2). A symbolic computer algebra implementation is available online.

Keywords

Angular momentum and spin symmetryatomic many-electron quantum systemssymbolic computation
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 1 , January 2016 , pp. 192 - 204
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Froese Fischer, C., Brage, T., and Johnsson, P.. Computational atomic structure: An MCHF approach. CRC Press, 1997.Google Scholar
[2]Fraga, S., Karwowski, J., and Saxena, K. M. S.. Handbook of atomic data. Elsevier, 1976.Google Scholar
[3]Bauschlicher, C. W. and Taylor, P. R.. Benchmark full configuration-interaction calculations on H2O, F, and F¯. J. Chem. Phys., 85(5):2779–2783, 1986.Google Scholar
[4]Chaichian, M. and Hagedorn, R.. Symmetries in quantum mechanics: from angular momentum to supersymmetry. CRC Press, 1997.Google Scholar
[5]Friesecke, G. and Goddard, B. D.. Explicit large nuclear charge limit of electronic ground states for Li, Be, B, C, N, O, F, Ne and basic aspects of the periodic table. SIAM J. Math. Anal., 41(2):631–664, 2009.Google Scholar
[6]Friesecke, G. and Goddard, B. D.. Asymptotics-based CI models for atoms: properties, exact solution of a minimal model for Li to Ne, and application to atomic spectra. Multiscale Model. Simul., 7(4):1876–1897, 2009.Google Scholar
[7]Mendl, C. B. and Friesecke, G.. Efficient algorithm for asymptotics-based configuration-interaction methods and electronic structure of transition metal atoms. J. Chem. Phys., 133:184101, 2010.Google Scholar
[8]Mendl, C. B.. https://github.com/cmendl/irredLS, 2009–2014.Google Scholar
[9]Mendl, C. B.. The FermiFab toolbox for fermionic many-particle quantum systems. Comput. Phys. Commun., 182:1327–1337, 2011.Google Scholar
[10]Mendl, C. B.. FermiFab Matlab/Mathematica toolbox software, online at http://sourceforge.net/projects/fermifab, 2008–2014.Google Scholar
[11]Irwin, J. O.. On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson's type II. Biometrika, 19:225–239, 1927.Google Scholar
[12]Hall, P.. The distribution of means for samples of size N drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika, 19:240–245, 1927.Google Scholar