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


Stella Stopkowicz (Oslo, Norway): "Does Direct Perturbation Theory converge?"
Friday 19 April 2013, 15:00 - 15:45
Amphi Hermite


Stella Stopkowicz
Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry,
University of Oslo, P.O.Box 1033 Blindern, 0315 Oslo, Norway

Direct Perturbation Theory (DPT) [1] is an elegant approach to treat relativistic effects in a perturbative manner. The basic idea behind DPT is to transform the Dirac equation into a representation with a new metric and to employ an expansion in powers of c−2 with c as the speed of light. This expansion provides the correct non-relativistic limit and is in principle applicable in any order. Implementation of DPT within a quantum-chemical framework is facilitated if the theory is formulated in terms of energy derivatives, thereby making use of the close relationship between perturbation and derivative theory. Expressions suitable for an efficient implementation are obtained by exploiting the method of Lagrange multipliers [2]. In this manner, already available techniques for the analytic evaluation of energy derivatives can be used. In order to establish DPT as a standard tool, its convergence needs to be ensured. Kutzelnigg showed that DPT converges for the hydrogen-like ions up to a nuclear charge of Z = 137 [3]. Whether these findings can be generalized for many-electron atoms and molecules will be discussed for Hartree-Fock as well as for correlated methods.

[1] W. Kutzelnigg, in Relativistic Electronic Structure Theory. Part I. Fundamentals, (Elsevier, Amsterdam, 2002), p. 664, chapter 12
[2] S. Stopkowicz and J. Gauss, J. Chem. Phys. 134, 064114 (2011)
[3] W. Kutzelnigg, Z. Phys. D 11, 5-28 (1989)