
Miniworkshop on "Numerical challenges in relativistic quantum mechanics"
Organizers: Werner Kutzelnigg (Bochum), Eric Séré (ParisDauphine)
Trond Saue (Toulouse, France): "Exploring the limits of the nopair 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
Email:
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URL: http://dirac.upstlse.fr/saue
Exploring the limits of the nopair approximation
“Sehr viel unglücklicher bin ich über die Frage nach der relativistischen Formulierung und über die Inkonsequenz der DiracTheorie... 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)
4component relativistic quantum chemistry calculations based on the DiracCoulomb Hamiltonian, correlated or not, are today routinely carried out, yet this Hamiltonian has no bound solutions [1, 2]: Starting from the oneelectron basis generated by a HartreeFock 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 nopair approximations where negativeenergy orbitals are excluded from the determinantal expansion. This correponds to the embedding of the DiracCoulomb Hamiltonian by projection operators onto the space of positiveenergy orbitals
HDC→Λ+ HDC Λ+.
Although there has been attempts to go beyond the nopair 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) DiracCoulomb problem for a given basis therefore seems to be a full CI within the nopair approximation. However, such a calculation employs projectors Λ+ deﬁned with respect to the HartreeFock meanﬁeld potential and are not optimized for the fully correlated potential. We will therefore argue that the optimal solution of the DiracCoulomb problem is an MCSCF calculation with a full CIexpansion within the nopair approximation, but allowing rotations between positive and negativeenergy orbitals and thus full relaxation of projectors [6]. Benchmark calculation of the helium isoelectronic series within this model will be presented and discussed.
References
[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. LeyKoo, 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.