Relativistic bond lengths

The origin of relativistic bond-length contraction is still somewhat of an open question. First, the precise mechanism by which it occurs has not been determined unequivocally. In addition, the computed magnitudes often vary by as much a factor of two or more from calculation to calculation. 

To date, the best ab initio all-electron molecular calculations involving heavy elements are those of Lee and McLean, who published LCAO-MO SCF relativistic calculations on AgH and AuH (75). They reported relativistic bond-length contractions of 0.08 and 0.25 A, respectively, and increases in... 

It should be pointed out that Schwarz (20),using double perturbation theory,has demonstrated that it is possible to rationalize the relativistic bond length contraction in terms of the attractive Hellmann-Feynman force due to the relativistic change in electron density.In such an approach it would be necessary to analyze and get a physical picture of the relevant density changes... 

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). 

Relativistic bond length (A) and energy (eV) changes for selected lanthanide molecules... 

Interpretation 1 (traditional explanation) the change of the Hellmann-Feynman force dWdR) due to relativistic changes of the wavefunction dW/dr, i.e. orbital expansions and contraetions, eauses relativistic bond length changes ... 

In pseudopotential calculations one has a different nodal structure of the pseudovalence orbitals in comparison to the all-electron valence orbitals. (di WdR dr) and its expectation values are neither vanishing nor small. On the contrary, it is assumed and has numerically been shown for Au2 that the main contribution to the relativistic bond-length changes stem from relativistic corrections to the Hellmann-Feynman force resulting from the pseudopotential ... 

The results of HF- and DHF-OCE calculations for the tetrahedral molecules CeH4 and TI1H4 were compared by Pyykko and Desclaux (1978). For both molecules small relativistic bond-length expansions were found. By comparison to HfH4 and 104EH4 the values of the lanthanide and actinide contraction were established to be 0.19 A and 0.30 A, respectively (cf. also sect. 1.3). The lanthanide contraction was found to result for 86% from a nonrelativistic shell-structure effect and only for 14% from relativity. Results of similar calculations are available for YbHj (Pyykko 1979a). 

Relativistic and non-relativistic bond lengths of LaH, AcH, TmH, LuH and LrH. Reproduced with modification from Pyykkd (1979). 

IP. A. Christiansen, W. C. Ermler. Relativistic Bond Length and Atomic Orbital Contraction. Mol. Phys., 55 (1985) 1109-1111. 

Fig. 28 Nonrelativistic and relativistic bond lengths, (see also Table 13 for various values for RgH), dissociation energies, >e, and force constants, k, of the group-11 hydrides. Reprinted with permission from [194]. Copyright 1996 Elsevier...

PyykkO and Desclaux have calculated properties of several transition metal hydrides including (106)H6 by applying the Dirac-Fock one-center expansion method. The calculations show that relativistic effects become more important with increasing nuclear charge of the transition element. For example, the relativistic bond length contraction scales approximately as Z (Z is the nuclear charge). Schadel et al. carried out aqueous chemistry on element 106 (isotopes 265 and 266) which showed that the most stable oxidation state is -f6, and like Us homologs Mo and W, element 106 forms neutral... 

In many simple cases, in particular when the bonding is accomplished by the outer s and p electrons or suitable hybrids, we experience a relativistic bond length contraction. For the same column of the Periodic Table it is also proportional to Z (which is generally true also for other relativistic effects in the valence shell down a column of the Periodic Table), and can be over 10% for Au compounds. 

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