Dirac-Fock approach

B. R Das, I. R Grant. Multiconfiguration Dirac-Fock approach to the fine-structure splitting in the 3s3p configuration of the magnesium sequence. /. Phys. B At. Mol. Phys., 19 (1986) L7-L11. 

For the calculations of relativistic density functions we used a multi-configuration Dirac-Fock approach (MCDF), which can be thought of as a relativistic version of the MCHF method. The MCDF approach implemented in the MDF/GME program [4, 27] calculates approximate solutions to the Dirac equation with the effective Dirac-Breit Hamiltonian [27]... 

There are two approaches to map crystal charge density from the measured structure factors by inverse Fourier transform or by the multipole method [32]. Direct Fourier transform of experimental structure factors was not useful due to the missing reflections in the collected data set, so a multipole refinement is a better approach to map charge density from the measured structure factors. In the multipole method, the crystal charge density is expanded as a sum of non-spherical pseudo-atomic densities. These consist of a spherical-atom (or ion) charge density obtained from multi-configuration Dirac-Fock (MCDF) calculations [33] with variable orbital occupation factors to allow for charge transfer, and a small non-spherical part in which local symmetry-adapted spherical harmonic functions were used. 

The relativistic coupled cluster method starts from the four-component solutions of the Drrac-Fock or Dirac-Fock-Breit equations, and correlates them by the coupled-cluster approach. The Fock-space coupled-cluster method yields atomic transition energies in good agreement (usually better than 0.1 eV) with known experimental values. This is demonstrated here by the electron affinities of group-13 atoms. Properties of superheavy atoms which are not known experimentally can be predicted. Here we show that the rare gas eka-radon (element 118) will have a positive electron affinity. One-, two-, and four-components methods are described and applied to several states of CdH and its ions. Methods for calculating properties other than energy are discussed, and the electric field gradients of Cl, Br, and I, required to extract nuclear quadrupoles from experimental data, are calculated. 

Table 2 Values of relativistic energies (E) and differences among relativistic and non-relativistic energies (AE) for neutral atoms in atomic units with the present approach using thefunctional given by Eq. (46) not including (1) or including (2) the term, compared to the results of Engel and Dreizler (ED) [23] using the relativistic Thomas-Fermi-Dirac- Weirsacker approach described in Section 2.6, and to Dirac-Fock values...

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]. 

One-center expansion was first applied to whole molecules by Desclaux Pyykko in relativistic and nonrelativistic Hartree-Fock calculations for the series CH4 to PbH4 [81] and then in the Dirac-Fock calculations of CuH, AgH and AuH [82] and other molecules [83]. A large bond length contraction due to the relativistic effects was estimated. However, the accuracy of such calculations is limited in practice because the orbitals of the hydrogen atom are reexpanded on a heavy nucleus in the entire coordinate space. It is notable that the RFCP and one-center expansion approaches were considered earlier as alternatives to each other [84, 85]. 

An even simpler approach to relativity, for heavy elements, is to use effective core potentials (ECPs) to represent the core electrons, taking the potentials from various compilations in the literature that explicitly include relativistic effects in the generation of the ECPs. References to such ECPs are given by Dyall et al. [103]. These relativistic ECPs (RECPs) allow the inclusion of some relativistic effects into a nonrelativistic calculation. Since ECPs will be treated in detail elsewhere, we will not pursue this approach further here. We may note, however, that recent comparisons with Dirac-Fock calculations suggest that the main weakness in the RECPs is not the treatment of relativity but the quality of the ECPs themselves [103]. Different RECPs gave spectroscopic constants with a noticeable scatter, compared to Dirac-Fock, but the relativistic corrections (difference between an RECP and the corresponding ECP value) were fairly consistent with one another. 

The so-called Hartree-Fock-Slater method is much more widely utilized, and is a hybrid of the Hartree and Thomas-Fermi-Dirac methods. In this method the direct part of the potential is calculated using the Hartree-Fock approach, whereas the exchange part is approximated by some statistical expression of the model of free electrons. The Slater potential is given by ... 

All-electron Dirac-Fock procedures are also capable of producing accurate molecular wave functions (76). However, it is not feasible at present to use this method as a standard tool for heavy-element systems because of the computational constraints. This approach can result in calculations of questionable value due to the use of inadequate basis sets and modest levels of Cl. Such calculations may not only be inaccurate, but, can be misleading if used for comparisons with REP calculations. Carefully selected benchmark calculations would be extremely useful for checking results of REP-based studies. 

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). 

Dreams is a program that has evolved as a Dirac-Fock code (Dyall 1994c Dyall et al. 1991a) and has been extended to the RMP2 approach for the estimation of correlation energies for closed and open-shell systems (Dyall 1994a). 

The programs described so far use basis-set expansions for the one-electron spinors. The fully numerical approach, which is still a challenging task for general molecules in nonrelativistic theory (Andrae 2001), has also been tested for Dirac-Fock calculations on diatomics (DtisterhOft etal. 1994,1998 Kullie etal. 1999 Sundholm 1987,1994 Sundholm et al. 1987 v. Kopylow and Kolb 1998 v. Kopylow et al. 1998 Yang et al. 1992). The finite-element method (FEM) was tested for Dirac-Fock and Kohn—Sham calculations by Kolb and co-workers in the 1990s. However, this approach has not yet been developed into a general method for systems with more than two atoms only test systems, namely few-electron linear molecules at some fixed intemuclear distance, have been studied with the FEM. Nonetheless, these numerical techniques are able to calculate the Dirac-Fock limit and thus yield reference data for comparisons with more approximate basis-set approaches. The limits of the numerical techniques are at hand ... 

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