Dirac—Fock calculations

The molecular orbitals in the nonrelativistic and one-component calculations and the large component in the Dirac-Fock functions were spanned in the Cd s Ap9d)l[9slp6d basis of [63] and the H (5s 2p)/[35 l/>] set [61]. Contraction coefficients were taken from corresponding atomic SCF calculations. The basis for the small components in the Dirac-Fock calculations is derived by the MOLFDIR program from the large-component basis. The basis set superposition error is corrected by the counterpoise method [64]. The Breit interaction was found to have a very small effect and is therefore not included in the results. 

Table 5 Comparison of equilibrium bondlengths for non relativistic ZORA and Dirac Fock calculations with the experimental value...

The inclusion of relativistic effects is essential in quantum chemical studies of molecules containing heavy elements. A full relativistic calculation, i.e. based upon Quantum Electro Dynamics, is only feasible for the smallest systems. In the SCF approximation it involves the solution of the Dirac Fock equation. Due to the four component complex wave functions and the large number of basis functions needed to describe the small component Dirac spinors, these computations are much more demanding than the corresponding non-relativistic ones. This limits Dirac Fock calculations, which can be performed using e.g. the MOLFDIR package [1], to small molecular systems, UFe being a typical example, see e.g. [2]. 

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]. 

The first two-step calculations of the P,T-odd spin-rotational Hamiltonian parameters were performed for the PbF radical about 20 years ago [26, 27], with a semiempirical accounting for the spin-orbit interaction. Before, only nonrelativistic SCF calculation of the TIF molecule using the relativistic scaling was carried out [86, 87] here the P,T-odd values were underestimated by almost a factor of three as compared to the later relativistic Dirac-Fock calculations. The latter were first performed only in 1997 by Laerdahl et al. [88] and by Parpia [89]. The next two-step calculation, for PbF and HgF molecules [90], was carried out with the spin-orbit RECP part taken into account using the method suggested in [91]. 

Later we performed correlation GRECP/NOCR calculations of the core properties in YbF [92], BaF [93], again in YbF [94] and in TIF [19]. In 1998, first all-electron Dirac-Fock calculations of the YbF molecule were also... 

We would like to emphasize here that the all-electron Dirac-Fock calculations on TIE and YbF are, in particular, important for checking the quality of the approximations made in the two-step method. The comparison of appropriate results, Dirac-Fock vs. RECP/SCF/NOCR, is, therefore, performed in papers [94, 19] and discussed in the present paper. 

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. 

Figure 21 Relativistic Dirac-Fock calculated data [29] of the valence ns radial electron probability densities Pns 2 for free Mg, Ca, Sr and Baas well as for the atoms encaged in C60, marked Mg, Ca, d>Sr and Ba, respectively. The domain of the C60 potential well is encompassed by the dashed vertical lines in this figure.

Table 2. Atomic ground-state configurations for the neutral elements 103 to 172 and 184 according to Mann (35) and Fricke and Waber 85, 60), using self-consistent Dirac—Fock calculations... 

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 numerical relativistic Dirac-Fock calculation for the uranium atom indicates that the spin-orbit averaged radii of the maximum radial density are 0.72 A, 0.86 A,... 

Table 4. Electron densities (in aa 3) for iron as derived from fully relativistic mixed configuration Dirac-Fock calculations. (Taken from Ref. 84)...

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 ... 

Dirac-Fock calculations were the standard four-component method for electronic structure calculations on molecules during the last decade. However, they are still very demanding or completely infeasible if applied to large unsymmetric molecules with several heavy atoms. In addition, taking properly care of electron correlation increases the computational effort tremendously. Future work will certainly continue the development of relativistic correlation methods, which will be far less expensive. 

A comparison of different methods was undertaken for the hydride of element 111 (Seth et al. 1996). The conclusion of this study was that Dirac-Fock calculations, all-electron DKH calculations and relativistic pseudopotential calculations give very similar results, showing that relativistic effects are also well described in the more approximate methods. A large relativistic bond length contraction of about 50 pm was found, which makes the bond length of (111)H even slightly shorter than that of AuH, which is 152.4 pm, with a relativistic effect of the order of 20 pm (see Kaldor and Hess 1994). 

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