Perturbation Theory Based on Quasi-Relativistic Hamiltonians

Perturbation Theory Based on Quasi-Relativistic Hamiltonians 

The Dirac equation with four spinor components demands large computational efforts to solve. Relativistic effects in electronic structure calculations are therefore usually considered by means of approximate one- or two-component equations. The approximate relativistic (also called quasi-relativistic) Hamiltonians consist of the nonrelativistic Hamiltonian augmented with additional 

One-component calculations or two-component calculations including also spin-orbit coupling effects provide a firm basis for the calculations of higher-order relativistic corrections by means of perturbation theory. Several quasi-relativistic approximations have been proposed. The most successful approaches are the Douglas-Kroll-Hess method (DKH) [1-7], the relativistic direct perturbation theory (DPT) [8-24], the zeroth-order regular approximation (ZORA) [25-48], and the normalized elimination of small components methods (NESC) [49-53]. Related quasi-relativistic schemes based on the elimination of the small components (RESC) and other similar nonsingular quasi-relativistic Hamiltonians have also been proposed [54-61]. 

In this Chapter, we will show how a whole family of one- and two-component quasi-relativistic Hamiltonians can conveniently be derived. The operator difference between the quasi-relativistic Hamiltonians and the Dirac equation can be explicitly identified and used in perturbation expansions. Expressions are derived for a direct perturbation theory scheme based on quasi-relativistic two-component Hamiltonians. The remaining difference between the variational energy obtained using quasi-relativistic Hamiltonians and the energy of the Dirac equation is estimated numerically by applying the direct perturbation theory ap- 

The transformation of the Dirac equation into two-component equations are also discussed in Chapter 11, whereas Chapter 12 deals with the relativistic direct perturbation theory. 

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General two-component methods have been discussed in various chapters of the first part of this book, for instance in chapter 11 on Two-Component Methods and the Generalised Douglas-Kroll Transformation by Wolf, Reiher and Hess [165], in chapter 12 by Kutzelnigg on Perturbation Theory of Relativistic Effects [166] and in chapter 13 by Sundholm on Perturbation Theory Based on Quasi-Relativistic Hamiltonians [167]. 

D. Sundholm, Perturbation theory based on quasi-relativistic Hamiltonians, in P. Schwerdtfeger (Ed.), Relativistic Electronic Structure Theory, Part 1, Fundamentals, Elsevier, Netherlands, 2002, pp. 763-798. 

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