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Mini-workshop on "Numerical challenges in relativistic quantum mechanics"
Organizers: Werner Kutzelnigg (Bochum), Eric Séré (Paris-Dauphine)


Pekka Pyykkö (Helsinki, Finland): "Aspects of Dirac-Coulomb systems"
Friday 19 April 2013, 09:30 - 10:15
Amphi Hermite


Department of Chemistry, University of Helsinki, P.O.B. 55, FIN–00014 Helsinki, Finland.
Tel: +358-9-191 50171. E-mail:

The Dirac-Coulomb problem was solved in 1928 by Darwin and by Gordon. The two-centre problems, such as H+ or the model system Th179+ can be solved numerically with a high precision [1]. For many-electron mean field models, such as Dirac-Fock (-Breit), the one-electron equations can be solved numerically by imposing the proper boundary conditions at R → 0 and R → ∞ [2,3]. A number of further physical contributions exist: 1) The finite nuclear size is for heavy atoms the largest one. It also allows one to continue the Periodic Table to about Z = 172. For a new shape of the PT, see [4]. 2) The Breit two-electron interaction and the quantum electrodynamical one-electron effects are of comparable magnitude [5]. The lowest-order QED terms are the vacuum polarisation and the self-energy ones. Attempts have been made to reproduce the latter by a local potential [6]. The various contributions can be compared against experiment on one- or few-electron
systems, such as Au78+ . For a recent review on molecular QED data, see [7]. An interesting deviation occurs between the proton radii, determined by electron scattering and by hyperfine structure in a muonic atom [8]. A part of the deviation could be understood by scaling the second-order hyperfine interaction [9] to the muonic case. A massive litterature of applications exists [10]. As an example, the available high-resolution microwave spectra on M-CN molecules (M=Cu-Au) provide an excellent testing ground for quantum chemistry. MP2 or CCSD(T) calculations with basis-sets up to cc-pVQZ, correlating all 19 VE of Au and including BSSE and SO corrections, are able to give M-C bond lengths to 0.6 pm, or better [11].

[1] O. Kullie, D. Kolb, J. Phys. B 36 (2003) 4361.
[2] J.-P. Desclaux, Comp. Phys. Comm. 9 (1975) 31.
[3] I. P. Grant, Comp. Phys. Comm. 11 (1976) 397.
[4] P. Pyykkö, Phys. Chem. Chem. Phys. 13 (2011) 161.
[5] P. Pyykkö, M. Tokman, L. Labzowsky, Phys. Rev. A 57 (1998) R689.
[6] P. Pyykkö, L.-B. Zhao, J. Phys. B 36 (2003) 1469.
[7] P. Pyykkö, Chem. Rev. 112 (2012) 371.
[8] A. Antognini et al., Science 339 (2013) 417.
[9] E. Latvamaa, L. Kurittu, P. Pyykkö, L. Tataru, J. Phys. B 6 (1973) 591.
[10] P. Pyykkö,; Relativistic Theory of Atoms and Molecules. III, Springer, Berlin (2000) (Lecture Notes in Chemistry 76, in all 10369 references).
[11] P. Zaleski-Ejgierd, M. Patzschke, P. Pyykkö, J. Chem. Phys. 128 (2008) 224303.