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


Trond Saue (Toulouse, France): "Exploring the limits of the no-pair approximation"
Friday 19 April 2013, 10:30 - 11:15
Amphi Hermite


Trond Saue
Laboratoire de Chimie et Physique Quantique (UMR 5626), CNRS/Université de Toulouse 3 (Paul Sabatier),
118 route de Narbonne, 31062 Toulouse, France
E-mail: This e-mail address is being protected from spambots. You need JavaScript enabled to view it. URL:

Exploring the limits of the no-pair approximation

“Sehr viel unglücklicher bin ich über die Frage nach der relativistischen Formulierung und über die Inkonsequenz der Dirac-Theorie... Also ich find’ die gengewärtige Lage ganz absurd und hab’ mich deshalb, quasi aus Verzweiflung, auf ein anderes gebiet, das der Ferromagnetismus begeben." W. Heisenberg (1928)

4-component relativistic quantum chemistry calculations based on the Dirac-Coulomb Hamiltonian, correlated or not, are today routinely carried out, yet this Hamiltonian has no bound solutions [1, 2]: Starting from the one-electron basis generated by a Hartree-Fock calculation, the reference Slater determinant “dissolves” into an in principle infinite number of degenerate determinants containing continuum solutions of both positive- and negative energy. Correlated calculations therefore invoke the no-pair approximations where negative-energy orbitals are excluded from the determinantal expansion. This correponds to the embedding of the Dirac-Coulomb Hamiltonian by projection operators onto the space of positive-energy orbitals


Although there has been attempts to go beyond the no-pair approximation [3, 4, 5], all such approaches imply that matter is not stable and has to be rejected on physical grounds. The best solution to the (projected) Dirac-Coulomb problem for a given basis therefore seems to be a full CI within the no-pair approximation. However, such a calculation employs projectors Λ+ defined with respect to the Hartree-Fock mean-field potential and are not optimized for the fully correlated potential. We will therefore argue that the optimal solution of the Dirac-Coulomb problem is an MCSCF calculation with a full CI-expansion within the no-pair approximation, but allowing rotations between positive- and negative-energy orbitals and thus full relaxation of projectors [6]. Benchmark calculation of the  helium isoelectronic series within this model will be presented and discussed.

[1] G. E. Brown and D. G. Ravenhall, Proc. Roy. Soc. London, 1951, A208, 552–559‘>.
[2] T. Saue, ChemPhysChem, 2011, 12, 3077.
[3] C. F. Bunge, R. Jauregui, and E. Ley-Koo, Int. J Quant. Chem., 1998, 70, 805.
[4] Y. Watanabe, H. Nakano, and H. Tatewaki, J. Chem. Phys., 2007, 126, 174105.
[5] Y. Watanabe, H. Nakano, and H. Tatewaki, J. Chem. Phys., 2010, 132, 124105.
[6] T. Saue and L. Visscher in Theoretical Chemistry and Physics of Heavy and Superheavy Elements, ed. S. Wilson and U. Kaldor; Kluwer, Dordrecht, 2003; p. 211.