Kramers-restricted configuration interaction

The alternative to the development of new algorithms to handle relativistic Hamiltonians is to search for a way to extend non-relativistic algorithms such that they can handle the additional couplings. Since most implementations are based on a restricted Hartree-Fock scheme the first step is to mimic the spin-restricted excitation operators used in the non-relativistic methods by Kramers restricted excitation operators. This can be done by employing the so-called X-operator formalism [37]. 

In the A -operator formalism the relations (35), (37) and (38) are built in to the excitation operators. A full definition is given by Jensen et al. [38] who introduce the pseudo quantum number Mk to play the role of the non-relativistic Ms quantum number. This number is defined as Mk = (M - M 2 with M and Mp the number of occupied unbarred and barred spinors, respectively. The analogue of the non-relativistic spin-restricted excitation operators are now those that preserve Mk, either by disallowing spin flips 

This form of the Hamiltonian shows explicitly the couplings between wave functions with different Mk values and makes possible to factorize occupation vectors in alpha- and beta-strings like done in non-relativistic Cl theory. The difference with non-relativistic theory is that calculations are not restricted to one value of Mk. Applied without further approximation the formalism gives therefore no dramatic reduction in operation count over the symmetry-adapted unrestricted scheme described in the previous section. An advantage of the formalism is, however, that it facilitates incorporation of the relativistic scheme in non-relativistic Cl or MCSCF implementations [35] and that the scheme gives a natural subdivision of the full Cl matrix. 

The major advantage of the scheme lies in the facile incorporation of approximate methods, though. If spin-orbit coupling effects are so small that they can be treated perturbatively, one uses the spinfree Hartree-Fock procedure to achieve the exact connection Ms = Mk. With this identification it is then e.g. possible to restrict the Cl space by a cut-off value of AMs or even to run the 

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The incorporation of electron correlation effects in a relativistic framework is considered. Three post Hartree-Fock methods are outlined after an introduction that defines the second quantized Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. Aspects that are considered are the approximations possible within the 4-component framework and the relation of these to other relativistic methods. The possibility of employing Kramers restricted algorithms in the Configuration Interaction and the Coupled Cluster methods are discussed to provide a link to non-relativistic methods and implementations thereof. It is shown how molecular symmetry can be used to make computations more efficient. 

T. Fleig, J. Olsen, C. M. Marian. The Generalized Active Space Concept for the Relativistic Treatment of Electron Correlation I. Kramers-restricted two-component configuration interaction. /. Chem. Phys., 114(11) (2001) 4775 790. 

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