Expansion in Scalar Basis Sets

We have seen that the DHF equations are conveniently analyzed and manipulated in a 2-spinor form. The practical solution of these equations in a finite basis requires the calculation and handling of a large number of integrals. Some effort has been invested in calculating the integrals directly in a 2-spinor basis (Grant and Quiney 2002, Yanai et al. 2002). For nonrelativistic calculations, a substantial fraction of the developmental 

We start by considering the one-particle Dirac equation and proceed in a manner analogous to the free-particle solutions of chapter 7. The equation to solve is 

The kinetic energy matrices obey the relation TI = (ll y, so, despite appearances, the Dirac matrix in the scalar basis given above is Hermitian. 

In these equations, the ordering of the components is the conventional order— large components first, then small components, and within each component, a spin then spin. However, we could equally well have ordered by spin—a spin components first, and then fS spin components, and for each spin, large eomponent then small component. This ordering yields the matrix 

This form of the matrix equation displays the structure of the matrix representation of an operator that is symmetric under time reversal, given in (10.30) 

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