Bond length, relativistic effects

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

In this paper we present the first application of the ZORA (Zeroth Order Regular Approximation of the Dirac Fock equation) formalism in Ab Initio electronic structure calculations. The ZORA method, which has been tested previously in the context of Density Functional Theory, has been implemented in the GAMESS-UK package. As was shown earlier we can split off a scalar part from the two component ZORA Hamiltonian. In the present work only the one component part is considered. We introduce a separate internal basis to represent the extra matrix elements, needed for the ZORA corrections. This leads to different options for the computation of the Coulomb matrix in this internal basis. The performance of this Hamiltonian and the effect of the different Coulomb matrix alternatives is tested in calculations on the radon en xenon atoms and the AuH molecule. In the atomic cases we compare with numerical Dirac Fock and numerical ZORA methods and with non relativistic and full Dirac basis set calculations. It is shown that ZORA recovers the bulk of the relativistic effect and that ZORA and Dirac Fock perform equally well in medium size basis set calculations. For AuH we have calculated the equilibrium bond length with the non relativistic Hartree Fock and ZORA methods and compare with the Dirac Fock result and the experimental value. Again the ZORA and Dirac Fock errors are of the same order of magnitude. 

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]. 

Further aspects governing computational effort and accuracy are related to the explicit treatment of relativistic effects, which are pronounced for atoms with high atomic numbers, as their core electrons reach velocities that make the influence of special relativity significant. Contraction of bond lengths and a shifting of orbital energies are observed compared to the non-relativistic treatment. An efficient way to include a major fraction of these... 

In Table 5.3 we list MRCI+Q spectroscopic constants for N2 obtained with one of our largest basis sets. Of the remaining errors compared to experiment, we can estimate that about 0.001 A of the error in the bond length arises from relativistic effects [71]. These would also increase the frequency slightly. Explicit calculations... 

The accuracy of the various relativistic, non-relativistic, correlated and non-correlated methods in comparison with experimental results is shown in Table 1 for AuH, a sort of a test molecule (see also [63,64]). The data of Table 1 demonstrate the importance of relativistic and electron correlation effects. Thus, relativistic effects diminish the equilibrium bond length (Re) by 0.26 A (the HF-DF difference without correlation) or by 0.21 A (the HF+MP2 - DF+MP2 difference with correlation), and enlarge the binding energy (De) by 0.70 eV (the HF-DF difference without correlation) or by 2.21 eV (the HF+MP2 - DF+MP2 difference with correlation). Correlation diminishes Re on the DF level by 0.07 eV, but enhances De by 1.34 eV. Thus, even for AuH correlation amounts almost to 50% of the chemical bond strength. No additivity of correlation and relativistic effects was shown. 

Table 1. Accuracy of different molecular methods for AuH showing importance of relativistic and correlation effects. RL. is the equilibrium bond length and Pg is the binding energy...

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