Quasirelativistic theory I. Theory in terms of a quasi-relativistic operator

Abstract

In this first part of a general analysis of a quasi-relativistic theory, i.e. a relativistic theory for electrons only, the transformation of the Dirac operator to an operator with two-component spinor solutions is studied, while a forthcoming second part will be devoted to a transformation at matrix level.

We start with a simple derivation of the key relation between the upper (ϕ) and lower (χ) components of the Dirac bispinor (ψ) for both electrons and positrons. The three possible choices of a non-hermitian quasi-relativistic Hamiltonian L, a hermitian one with non-unit metric, and a hermitian one L ⁺ with unit metric are compared. The eigenfunctions of the first two are the upper components ϕ of ψ, while those of L ⁺ are the Foldy–Wouthuysen-type spinorsφ.

Some general properties of quasi-relativistic Hamiltonians and their eigenfunctions are discussed, especially the behaviour near the position of a nucleus. Exact solutions, and even variational ones are only obtained if the orbital basis describes the weak singularities at the positions of the nuclei correctly.

A new derivation of the quasi-relativistic version of direct perturbation theory (DPT) is given, followed by the theory of quasi-relativistic effective Hamiltonians, both non-hermitian and hermitian ones. The classical Foldy–Wouthuysen transformation is then presented as the singular limit of quasi-relativistic effective Hamiltonians with the model space extended to the entire space of positive-energy states. Finally, the problems that arise for a Douglas–Kroll transformation or the regular approximation at operator level are studied in detail. In part 2, it will be shown that everything becomes much simpler if one performs the transformation from relativistic to quasi-relativistic theory at matrix level.

¶This paper is dedicated to Andrzej Sadlej on the occasion of his 65th birthday.

Acknowledgements

The research reported here has been sponsored by Deutsche Forschungsgemeinschaft (DFG), Fonds der Chemie(VdCh), and the National Natural Science Foundation of China (Project Nos. 20573003 and 20333020), and MOST of China (Project No. 2006CB601103).

Notes

*Corresponding author. Email: [email protected]

¶This paper is dedicated to Andrzej Sadlej on the occasion of his 65th birthday.

Additional information

Notes on contributors

Wenjian Liu*

*Corresponding author. Email: [email protected]