Practical Implications of Spin Separation

One important advantage of the modified Dirac equation is that, since the large component and the pseudo-large component have the same symmetry, we can use the same primitive basis set for both. However, if we want to use a contracted basis set, the contraction coefficients for these functions will differ. We will therefore distinguish the basis sets for the components when we expand them, which we do now  

We use the superscript P for the pseudo-large component, which represents both the initial letter, and also the fact that it is related to the small component through the momentum operator p. The one-electron modified Dirac equation in this basis set is then 

At the beginning of this chapter it was asserted that a considerable amount of work could be saved by performing a spin separation in the Dirac equation, using the spin-free part variationally, then treating the spin-dependent terms as a perturbation. We are now in a position to assess the reduction of work that would actually be obtained by performing a spin-free calculation. 

We also want to compare the amount of work in a spin-free modified Dirac calculation with that in a nonrelativistic calculation. One of the reasons given for the development of relativistic effective core potentials (RECPs) was the expense of four-component Dirac-Fock calculations. RECPs are often used in spin-free form, but the comparison is made with the spin-dependent unmodified Dirac approach. The comparison of the cost of a spin-free all-electron calculation with a nonrelativistic calculation will give a more realistic indication of the relative cost of the incorporation of spin-free relativistic effects. 

We will evaluate the numbers of integrals required for a calculation with the unmodified Dirac Hamiltonian, and compare them with the number of integrals required for a calculation with the spin-free modified Dirac Hamiltonian, and with the number required for a nonrelativistic calculation. The spin-free Hamiltonian is formed by summing all the spin-free terms defined above, but we will consider the Coulomb term and the Gaunt and Breit terms separately. For the purpose of this evaluation, we make the following assumptions and definitions  

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