Kramers-Restricted 2-Spinor Matrix Dirac-Hartree-Fock Equations

Kramers-Restricted 2-Spinor Matrix Dirac-Hartree-Fock Equations 

So far we have not taken time-reversal symmetry into account. From the preceding chapters, we expect that incorporating time-reversal symmetry in a Kramers-restricted Dirac-Hartree-Fock theory will result in a reduction of the work, and possibly also a reduction in the rank of the Fock matrix. The basis set we will use is a basis set of Kramers pairs. We develop the theory for a closed-shell reference, for which all Kramers pairs are doubly occupied.  

We partition both the molecular spinors and the basis set into Kramers pairs. This partitioning reduces to a partitioning of the large and small components into Kramers pairs, (rjrf, rlrf) for the molecular spinors and (x, xf) for the basis functions, where X is L or S. We cannot assume a priori that the molecular spinors for one of the Kramers partners can be expanded solely in the corresponding basis spinors, and we must therefore retain the full expansion until we have deduced the proper restrictions  

With the partitioning of the basis set, we can partition each block of each matrix— density, one-electron, and Fock—into four blocks. For the density matrix elements 

Pioneering work for closed-shell DHF theory was done by MalU and Oreg (1975) and by Hafner (1980). 

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Hence, the relativistic analog of the spin-restriction in nonrelativistic closed-shell Hartree-Fock theory is Kramers-restricted Dirac-Hartree-Fock theory. We should emphasize that our derivation of the Roothaan equation above is the pedestrian way chosen in order to produce this matrix-SCF equation step by step. The most sophisticated formulations are the Kramers-restricted quaternion Dirac-Hartree-Fock implementations [286,318,319]. A basis of Kramers pairs, i.e., one adapted to time-reversal s)mimetry, transforms into another basis under quatemionic unitary transformation [589]. This can be exploited not only for the optimization of Dirac-Hartree-Fock spinors, but also for MCSCF spinors. In a Kramers one-electron basis, an operator O invariant under time reversal possesses a specific block structure. 

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