How to calculate atomic open-shell LS terms with tools designed to treat molecules

Dirk Andrae

Physical and Theoretical Chemistry, Institute of Chemistry and Biochemistry, Freie Universität Berlin, Germany

Initial version: 2014-Jan-27; last modification: 2017-Aug-28.

This web site provides a database with input files (and partly also output files) related to quantum chemical calculations in LS coupling (Russell-Saunders coupling [1]) for atoms or atomic ions with open-shell electron configurations. The LS terms of the most elementary cases encountered in the periodic table are exemplarily covered:

Input files are provided for use with the GAMESS electronic structure program [7]. After installation of this code on your computer you may feel free to download the input file for a case of interest from this website and perform the electronic structure calculation to get the corresponding output.

Important Notes:
(1) The main focus is on relevant data patterns (like degeneracies in orbital energies or LS term components), not so much on actually calculated numbers (like AO coefficients, values of orbital or total energies).
(2) The results in the outputs will show that perfect degeneracy, as expected from shell and/or LS term symmetry labels, can be achieved without use of the symmetry of the full rotation-reflection group. In fact, symmetry may be switched off completely (i. e. use of point group C1) without affecting the symmetry patterns expected to be seen among the data.
(3) In cases where more than one allowed LS term exists for certain combinations of L and S, the results obtained from the electronic structure calculation result from a full diagonalization of the Hamiltonian.
(4) The parity of LS terms is NOT indicated.

Cases of a single open-shell in LS coupling

The electron configuration s**1

A single LS term exists for this electron configuration.

The electron configurations p**n

The total sum of LS terms from all open-shell configurations p**n (0 < n < 6) is 11.

The electron configurations d**n

The total sum of LS terms from all open-shell configurations d**n (0 < n < 10) is 76.

The electron configurations f**n

The total sum of LS terms from all open-shell configurations f**n (0 < n < 14) is 647.


  1. E. U. Condon, G. H. Shortley, The Theory of Atomic Spectra, Cambridge University Press, Cambridge, 1935
  2. C. W. Nielson, G. F. Koster, Spectroscopic Coefficients for the p**n, d**n, and f**n Configurations, MIT Press, Cambridge, MA, 1963
  3. C. E. Moore, Atomic Energy Levels, Vol. 1 (H-V), Vol. 2 (Cr-Nb), Vol. 3 (Mo-La, Hf-Ac), U. S. Govt. Printing Office, Washington, 1971
  4. W. C. Martin, R. Zalubas, L. Hagan, Atomic Energy Levels - The Rare-Earth Elements, U. S. Govt. Printing Office, Washington, 1978
  5. A. Kramida, Yu. Ralchenko, J. Reader and NIST ASD Team, NIST Atomic Spectra Database (Version 5), National Institute of Standards and Technology, Gaithersburg, MD, 2013 [ available online at, accessed 2014-01-27 ]
  6. G. Racah, Phys. Rev. 62 (1942) 438-462
  7. (a) M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. Su, T. L. Windus, M. Dupuis, J. A. Montgomery, J. Comput. Chem. 14 (1993) 1347-1363; (b) M. S. Gordon, M. W. Schmidt, in: C. E. Dykstra, G. Frenking, K. S. Kim, G. E. Scuseria (editors), Theory and Applications of Computational Chemistry, Elsevier, Amsterdam, 2005, pp. 1167-1189; (c) General Atomic and Molecular Electronic Structure System (GAMESS,, 2017-Aug-28