Dirac-Fock problem

It is of course possible to solve the Dirac—Fock problem with a linear combination of analytic orbitals. However, owing to the rapid variation of the orbitals near the nucleus it requires an awkwardly-large basis. If an analytic representation is convenient for a reaction calculation it may be obtained by a least-squares fit to a numerical orbital. 

Results of similar accuracy as relativistic TFDW are found with a simple procedure based on near-nuclear correction which leave space for further improvements. For the reasons mentioned at the end of previous section the direct way to improve the present approach seems to be the refinement of the near nuclear corrections, a problem that we have just tackled with success in the non-relativistic framework [31,32]. The aim was to describe the near-nuclear region accurately by means of using the quantum mechanical exact asymptotic expression up to of the different ns eigenstates of Schodinger equation with a fit of the semiclassical potential at short distancies to the exact asymptotic behaviour (with four terms) of the potential near the nucleus. The result is that the density below Tq becomes very close to Hartree-Fock values and the improvement of the energy values is large (as an example, the energy of Cs+ is improved from the Ashby-Holzman result of-189.5 keV up to -205.6, very close to the HF value of -204.6 keV). This result makes us expect that a similar procedure in the relativistic framework may provide results comparable to Dirac-Fock ones. 

There are many problems in e.g. catalysis in which relativity may play a deciding role in the chemical reactivity. These problems generally involve large organic molecules which cannot be handled within the Dirac Fock framework. It is therefore necessary to reduce the work by making additional approximations. Generally used approaches are based on the Pauli expansion or on the Douglas Kroll transformation [3]. 

Since the relativistic many-body Hamiltonian cannot be expressed in closed potential form, which means it is unbound, projection one- and two-electron operators are used to solve this problem [39], The operator projects onto the space spanned by the positive-energy spectrum of the Dirac-Fock-Coulomb (DFC) operator. In this form, the no-pair Hamiltonian [40] is restricted then to contributions from the positive-energy spectrum and puts Coulomb and Breit interactions on the same footing in the SCF calculations. 

The Hartree—Fock problem with the Dirac Hamiltonian (3.153) is called Dirac—Fock. The coordinate—spin representation of the orbital rj) is... 

Effective core potentials address the aforementioned problems that arise when using theoretical methods to study heavy-element systems. First, ECPs decrease the number of electrons involved in the calculation, reducing the computational effort, while also facilitating the use of larger basis sets for an improved description of the valence electrons. In addition, ECPs indirectly address electron correlation because ECPs may be used within DFT, or because fewer valence electrons may allow implementation of post-HF, electron correlation methods. Finally, ECPs account for relativistic effects by first replacing the electrons that are most affected by relativity, with ECPs derived from atomic calculations that explicitly include relativistic effects via Dirac-Fock calculations. Because ECPs incorporate relativistic effects, they may also be termed relativistic effective core potentials (RECPs). 

A different approach to the solution of the electron correlation problem comes from density functional theory (see Chapter 4). We hasten to add that in a certain approximation of relativistic density functional theory, which is also reviewed in this book, exchange and correlation functionals are taken to replace Dirac-Fock potentials in the SCF equations. Another approach, which we will not discuss here, is the direct perturbation method as developed by Rutkowski, Schwarz and Kutzelnigg (Kutzel-nigg 1989, 1990 Rutkowski 1986a,b,c Rutkowski and Schwarz 1990 Schwarz et al. 1991). 

Stassis and Deckman (1976b) have also given a completely relativistic formulation of the problem, and the reduced matrix elements for the (effective) multipole operators needed in this case have also been tabulated. In addition, of course, one needs to replace the radial wavefunction f(r) appearing in the expression for the by relativistic (Dirac-Fock) radial wave functions. 

---