In the preceding chapters, the theory for calculations based on the Dirac equation has been laid out in some detail. The discussion of the methods included a comparison with equivalent nonrelativistic methods, from which it is apparent that four-component calculations will be considerably more expensive than the corresponding nonrelativistic calculations—perhaps two orders of magnitude more expensive. For this reason, there have been many methods developed that make approximations to the Dirac equation, and it is to these that we turn in this part of the book. There are two elements of the Dirac equation that contribute to the large amount of work: the presence of the small component of the wave function and the spin dependence of the Hamiltonian. The small component is primarily responsible for the large number of two-electron integrals which, as will be seen later, contain all the lowest-order relativistic corrections to the electron–electron interaction. The spin dependence is incorporated through the kinetic energy operator and the correction to the electronic Coulomb interaction, and also through the coupling of the spin and orbital angular momenta in the atomic 2-spinors, which form a natural basis set for the solution of the Dirac equation. Spin separation has obvious advantages from a computational perspective. As we will show for several spin-free approaches below, a spin-free Hamiltonian is generally real, and therefore real spin–orbitals may be employed for the large and small components. The spin can then be factorized out and spin-restricted Hartree–Fock methods used to generate the one-electron functions. In the post-SCF stage, where the no-pair approximation is invoked, the transformation of the integrals from the atomic to the molecular basis produces a set of real molecular integrals that are indistinguishable from a set of nonrelativistic MO integrals, and therefore all the nonrelativistic correlation methods may be employed without modification to obtain relativistic spin-free correlated wave functions. In most cases, spin–free relativistic effects dominate the relativistic corrections to electronic structure. We will show later that in a perturbation expansion based on the nonrelativistic wave function, the spin-free effects for a closed-shell system enter in first order, whereas the spin-dependent effects make their first contribution in second order.
Evaluation of the one electron expectation values for different wave function of Be atom