Relativistic quantum-chemical calculations

The synthesis of the transactinides is noteworthy from a chemical and a nuclear viewpoint. From the chemical point of view, rutherfordium (Z = 104) is important as an example of the first transactinide element. From Figure 15.1, we would expect rutherfordium to behave as a Group 4 (IVB) element, such as hafnium or zirconium, but not like the heavy actinides. Its solution chemistry, as deduced from chromatography experiments, is different from that of the actinides and resembles that of zirconium and hafnium. More recently, detailed gas chromatography has shown important deviations from expected periodic table trends and relativistic quantum chemical calculations. 

In experiments with Os and Hs, performed under similar conditions, it was, however, found that Hs04 condenses at a higher temperature than 0s04 indicating that the Hs oxide is less volatile [10]. Because so few events were detected it is premature to say whether or not the experimental data are in disagreement with the predictions. It would be very interesting to discuss the influence of relativistic effects on the AHads of these molecules. This can be done after non-relativistic quantum-chemical calculations are performed. 

In the discussion of the rather unexpected chemical results [42], it was suggested that the chemical properties of the heaviest elements cannot reliably be predicted by simple extrapolations of trends within a group of elements , and that relativistic, quantum chemical calculations for compounds of Nb, Ta, Pa, and Db are needed to understand in detail the differences in the halide complexing of the group-5 elements . 

A remark should be made here with respect to the generation and adjustment of the widely used effective core potentials (ECP, or pseudopotentials) [85] in standard non-relativistic quantum chemical calculations for atoms and molecules. The ECP, which is an effective one-electron operator, allows one to avoid the explicit treatment of the atomic cores (valence-only calculations) and, more important in the present context, to include easily the major scalar relativistic effects in a formally non-relativistic approach. In general, the parameters entering the expression for the ECP are adjusted to data obtained from numerical atomic reference calculations. For heavy and superheavy elements, these reference calculations should be performed not with the PNC, but with a finite nucleus model instead [86]. The reader is referred to e.g. [87-89], where the two-parameter Fermi-type model was used in the adjustment of energy-conserving pseudopotentials. 

This was the first time that predictions of extraction behaviour of the heaviest elements based on relativistic quantum-chemical calculations were made, and also confirmed by specially designed experiments. Only by considering all possible equilibria in the aqueous phase including hydrolysis could this imexpected behaviour be predicted. Simple extrapolations of properties within the group would have shown the straightforward and, consequently, wrong trend. 

T. Yoshizawa, M. Hada. Relativistic quantum-chemical calculations of mag-netizabilities of noble gas atoms using the Douglas-Kroll-Hess method. Chem. Phys. Lett, 458 (2008) 223-226. 

Relativistic effects can be defined as the difference between results of relativistic and non-relativistic quantum chemical calculations, i.e., calculations with the speed of light at its correct value (c 137.036 a.u.) and the nonrelativistic limit (c oo), respectively. Since there are various choices of Hamiltonians, which may be applied in various quantum chemical approaches for the approximate solution of the Schrodinger equation, the magnitude of relativistic contributions somehow depends on the way they are evaluated [23]. 

AuH and Au2 serve as benchmark molecules to test the performance of various relativistic approximations. Figure 4.7 shows predictions for relativistic bond contractions of Au2 from various quantum chemical calculations over more than a decade. In the early years of relativistic quantum chemistry these predictions varied significantly (between 0.2 and 0.3 A), but as the methods and algorithms became more refined, and the computers more powerful, the relativistic bond contraction for Au2 converged and is now at 0.26 A. 

The growing importance of quantum-chemical calculations is dealt with in a short section, with emphasis on the consideration of relativistic effects, especially in systems containing mercury. These calculations aim at optimization of structures, determination of bond energies, simulation of spectra, and estimation of spectral parameters, independent of but complementary to experiments. 

There are three terms which appears in the first order relativistic expression the mass-velocity tehn, the Darwin term and the spin-orbit term[12]. Out of these terms the first two are comparatively easy to calculate, while the spin-orbit interaction term is more complicated. Fortunately, the spin-orbit interaction is often not too important for chemical properties, at least for the second row transition elements. It is therefore usual to neglect it in quantum chemical calculations. 

Several reviews have been written dealing with ab initio and semiempirical quantum chemical calculations including relativistic effects 1-3,17-20). 

First-principles quantum chemical calculations including relativistic effects have been carried out for dipole moments, polarizabilities, and first- and second-order hyperpolarizabilities for tellurophene. The estimated values were compared with the observed ones measured by the optical Kerr effects <2000SM185, 2003JMT207>. 

Relativistic effects may be also considered by other methods than pseudopotentials. It is possible to carry out relativistic all-electron quantum chemical calculations of molecules. This is achieved by various approximations to the Dirac equation, which is the relativistic analogue to the nonrelativistic Schrodinger equation. We do not want to discuss the mathematical details of this rather complicated topic, which is an area where much progress has been made in recent years and where the development of new methods is a field of active research. Interested readers may consult published reviews . A method which has gained some popularity in recent years is the so-called Zero-Order Regular Approximation (ZORA) which gives rather accurate results ". It is probably fair to say that... 

Quantum-chemical calculations of MH4 and MCI4 (M = C, Si, Ge, Sn, Pb) demonstrated that it is necessary to take into account the relativistic effects, which are proportional to the square of the atomic number of M and therefore essential when M = Ge, Sn, Pb . This was considered in calculations of the Me4- SnCl (n = 0-4) series . The mixing of orbitals of the unshared electron pairs of the chlorine atoms with the a(Sn—C) orbitals (a-n conjugation) intensifies with the rise of the number of methyl groups. On the contrary, increase in the number of chlorine atoms is accompanied by an increase in the population of the 5d-orbitals of the tin atom due to the d-n conjugation . The calculated HOMO energies and NMR chemical shifts of Me4- SnCl conform satisfactorily with experimental values. 

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