Four Component Calculations

Although relativistic effects can be included by perturbative operators describing corrections to the non-relativistic wave function, this rapidly becomes cumbersome if higher order corrections are required, and it is then perhaps more satisfying to include relativistic effects by solving the Dirac equation directly. The simplest approximative wave function is a single determinant constructed from four-component one-electron functions, called spinors, having large and small components multiplied with the two spin functions. The spinors are the relativistic equivalents of the spin-orbitals in non-relativistic theory. With such a wave function, the relativistic equation corresponding to the Hartree-Fock equation is the Dirac-Fock equation, which in its time-independent form (setting p = p and m = 1 in eq. (8.8)) can be written as in

The requirement that the wave function should be stationary with respect to a variation in the orbitals, results in an equation that is formally the same as in non-relativistic theory, FC = SCe (eq. (3.51)). However, the presence of solutions for the positronic states means that the desired solution is no longer the global minimum (Figure 8.1), and care must be taken that the procedure does not lead to variational collapse. The choice of basis set is an essential component in preventing this. Since practical calculations necessarily use basis sets that are far from complete, the large and small component basis sets must be properly balanced. The large component corresponds to the normal non-relativistic wave function, and has similar basis set requirements. The small component basis set is chosen to obey the kinetic balance condition, which follows from

The use of kinetic balance ensures that the relativistic solution smoothly reduces to the non-relativistic wave function as c is increased. The presence of the momentum operator in eq. (8.39) means that the small component basis set must contain functions that are derivatives of the large component basis set, making the former roughly twice the size of the latter. This means that there are ~8 times as many large-small two-electron integrals and ~16 times as many small-small integrals, than there are large-large type integrals. A relativistic calculation thus requires roughly 25 times as many two-electron integrals compared with a non-relativistic calculation.

When the Dirac operator is invoked, the point charge model of the nucleus also becomes problematic. For a non-relativistic hydrogen atom, the orbitals have a cusp (discontinuous derivative) at the nucleus. However, the relativistic solutions have a singularity. A singularity is much harder to represent in an approximate treatment (such as an expansion in a Gaussian basis) than a cusp. Consequently, a (more realistic) finite-size nucleus is often used in relativistic methods. A finite nucleus model removes the singularity of the orbitals, which now assume a Gaussian type behaviour within the nucleus. Neither experiments nor theory, however, provide a good model for how the positive charge is distributed within the nucleus. The wave function and energy will of course depend on the exact form used for describing the nuclear charge distribution. A popular choice is either a uniformly charged sphere, where the radius is proportional to the nuclear mass to the 1/3 power, or a Gaussian charge distribution (which facilitates the calculation of the additional integrals) with the exponent depending on the nuclear mass. Note that this implies that the energy (and derived properties) depends on the specific isotope, not just the atomic charge, i.e. the results for say 37Cl will be (slightly) different from 35Cl. The difference between a finite and a point charge nuclear model is large in terms of total energy (~1au), however, the exact shape for the finite nucleus is not important. For valence properties, any "reasonable" model gives essentially the same results.

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