Symmetry in the Kramers-Restricted Fock Matrix

The symmetry reductions in nonrelativistic methodology come from spin symmetry and from point-group symmetry. In relativistic methodology, time-reversal symmetry is the equivalent of spin symmetry, but it does not provide the same magnitude of reduction as does spin symmetry. This is due to the presence of spin-dependent terms in the Fock operator. Point-group symmetry is intimately connected with time-reversal symmetry in Kramers-restricted relativistic theory, as we saw in chapter 10. 

In the Dacre and Elder method, the reduction in the integral number comes from the relations between the symmetry-equivalent atoms. Consider two symmetry-equivalent atoms, A and B, each of which has two s /2 2-spinor basis functions, Xk and xx. which we label ka and kb, and A,a and Xb. The relations between the one-electron potential energy integrals (for example) that do not follow fi om Hermitian conjugation are 

Similar relations can be written for the odd-bar integrals. Relations like these can be used to reduce the number of unique integrals by a factor of 2 and the number of density matrix elements by a factor of 2. However, the odd-bar integrals are not necessarily zero because they are not integrals between symmetry functions, and all the terms in the expressions for the Fock matrix elements remain. The reduction in work comes from the fact that the sums are restricted to sums over symmetry-unique basis functions, and that only the symmetry-unique parts of the Fock matrix are constmcted. For matrix elements between functions on the symmetry axis, the reductions follow from the fact that they are symmetry functions, and hence an overall factor of 4 reduction in the work for the two-electron integrals is obtained. 

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