Relativistic Hartree-Fock

DHF (Dirac -Hartree-Fock) relativistic ah initio method DHF (derivative Hartree-Fock) a means for calculating nonlinear optical properties... 

Fig. 18. Variation of the orbital energies of uranium, in atomic units, with charge, as determined by a Hartree-Fock relativistic calculation from Ref. [60]...

Hydride Hartree—Fock Relativistic Contribution Experimental R/... 

The analysis of atomic density functions can be furthered by comparing them in pairs. Specifically, the use of quantum similarity measures and indices as defined by Carbo [5] has shown that particular influences on the density functions can be estimated in this way. Here this feature is demonstrated by reviewing three case studies (1) the LS -term dependence of Hartree-Fock densities, (2) the comparison of atoms throughout the periodic table [6], and (3) the quantitative evaluation of the influence of relativistic effects, via a comparison of non-relativistic Hartree-Fock densities with Dirac-Hartree-Fock relativistic densities [7]. 

Ah initio calculation s can be performetl at th e Ilartree-Fock level of approximation, equivalent to a self-con sisten t-field (SCK) calculation. or at a post llartree-Fock level which includes the effects of correlation —defined to be everything that the Hartree-Fock level of appi oxiniation leaves out of a n on-relativistic solution to the Schrddinger ec nation (within the clamped-nuclei Born-Oppenh e-imer approximation ). 

There are also ways to perform relativistic calculations explicitly. Many of these methods are plagued by numerical inconsistencies, which make them applicable only to a select set of chemical systems. At the expense of time-consuming numerical integrations, it is possible to do four component calculations. These calculations take about 100 times as much CPU time as nonrelativistic Hartree-Fock calculations. Such calculations are fairly rare in the literature. 

There are several ways to include relativity in ah initio calculations more efficiently at the expense of a bit of accuracy. One popular technique is the Dirac-Hartree-Fock technique, which includes the one-electron relativistic terms. Another option is computing energy corrections to the nonrelativistic wave function without changing that wave function. 

In this review, we have mainly studied the correlation energy connected with the standard unrelativistic Hamiltonian (Eq. II.4). This Hamiltonian may, of course, be refined to include relativistic effects, nuclear motion, etc., which leads not only to improvements in the Hartree-Fock scheme, but also to new correlation effects. The relativistic correlation and the correlation connected with the nuclear motion are probably rather small but may one day become significant. 

The stability of gold(III) compared with silver(III) has been ascribed to relativistic effects causing destabilization of the 5d shell, where the electrons are less tightly held. Hartree-Fock calculations on AuX4 (X = F, Cl, Br) indicate that relativistic effects make a difference of 100-200 kJ mol-1 in favour of the stability of AuXJ (Table 4.12) [110]. 

Computed using Hartree-Fock methods and Including an approximate relativistic correction (24). 

As was the case with lanthanide crystal spectra (25), we found that a systematic analysis could be developed by examining differences, AP, between experimentally-established actinide parameter values and those computed using Hartree-Fock methods with the inclusion of relativistic corrections (24), as illustrated in Table IV for An3+. Crystal-field effects were approximated based on selected published results. By forming tabulations similar to Table IV for 2+, 4+, 5+ and 6+ spectra, to the extent that any experimental data were available to test the predictions, we found that the AP-values for Pu3+ provided a good starting point for approximating the structure of plutonium spectra in other valence states. However,... 

The Hartree-Fock iimit with a relativistic correction. 

If not otherwise stated the four-component Dirac method was used. The Hartree-Fock (HF) calculations are numerical and contain Breit and QED corrections (self-energy and vacuum polarization). For Au and Rg, the Fock-space coupled cluster (CC) results are taken from Kaldor and co-workers [4, 90], which contains the Breit term in the low-frequency limit. For Cu and Ag, Douglas-Kroll scalar relativistic CCSD(T) results are used from Sadlej and co-workers [6]. Experimental values are from Refs. [91, 92]. 

Figure4.7 Relativistic bond contractions A re for Au2 calculated in the years from 1989 to 2001 using different quantum chemical methods. Electron correlation effects Acte = te(corn) — /"e(HF) at the relativistic level are shown on the right hand side of each bar if available. From the left to the right in chronological order Hartree-Fock-Slater results from Ziegler et al. [147] AIMP coupled pair functional results from Stbmberg and Wahlgren [148] EC-ARPP results from Schwerdtfeger [5] EDA results from Haberlen and Rdsch [149] Dirac-Fock-Slater...

Hay, P. J., Martin, R. L., 1998, Theoretical Studies of the Structures and Vibrational Frequencies of Actinide Compounds Using Relativistic Effective Core Potentials With Hartree-Fock and Density Functional Methods ... 

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