Relativistic Bond Length Contraction

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). 

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. 

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]. 

Pyykkb et al. [4] pointed out the Z -dependence of the relativistic hond-length contraction, but did not give its reason. Presumably, the Z -dependence of the bond length contraction can also he explained by the same reason. For homonuclear diatomic molecules containing heavy elements such as Ph, relativistic effects should be taken into account for both atoms when considering the atomic-numher dependence of the bond overlap population. 

Figure 5. Relativistic effects on bond lengths and binding energies of group 4 tctrahydrides XH. The bond length contraction (in A) and bond destabilization (in eV) were obtained as the difference between relativistic Dirac-Hartree-Fock calculations based on the Dirac-Coulomb-Gaunt Hamiltonian and corresponding nonrelativistic Hartree-Fock calculations [28,29].

Figure 6. Influence of relativistic corrections to the electron-electron interaction on the bond length contraction and bond destabilization of the group 4 tetrahydrides XH. The percentage of results obtained with the Dirac-Coulomb-Gaunt (DCG) Hamiltonian wrt. those obtained with the Dirac-Coulomb (DC) Hamiltonian has been derived from Dirac-Hartree-Fock calculations [28,29].

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). 

Table 46 shows the computed relativistic and non-relativistic bond lengths of AcH, TmH, LuH and LrH computed by Pyykko. Pyykko (1979) defined lanthanide and actinide contraction as... 

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

Snijders, J.G. and Pyykko, P. (1980) Is the relativistic contraction of bond lengths an orbital contraction effect Chemical Physics Letters, 75, 5-8. 

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