Kramers-Restricted Algorithms

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. 

In the following we assume the use of an (effective) one particle Hamiltonian for which all eigenspinors come in degenerate pairs. This means that external magnetic fields are excluded and that a (Kramers-)restricted algorithm is used in the (D)HF step. 

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]. 

The Kramers-restricted form of the Hamiltonian that was used in Cl theory is not suitable for Coupled Cluster theory because it mixes excitation and deexcitation operators. One possibility is to define another set of excitation operators that keep the Kramers pairing and use these in the exponential parametrization of the wavefunction. This would automatically give Kramers-restricted CC equations upon rederivation of the energy and amplitude equations. A more pedestrian but simpler alternative is to start from the spin-orbital formulation and inspect the relations that follow from the Kramers relation of the two-electron integrals. This method does also readily give the relations between the Kramers symmetry-related amplitudes. We will briefly discuss the basic steps in this approach, a detailed description of a possible algorithm is given in reference [47],... 

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