Relativistic calculations of molecules relativity and bond lengths

In the introduction we remind the reader of the conceptual differences between the unboundedness of the Dirac hamiltonian, the Brown-Ravenhall and the basis set diseases, and of the methods to cure the latter. Then the theoretical interpretation of relativistic changes of bond lengths Δ_relR is worked out using only very elementary mathematics. The theoretical expression for Δ_relR can be split up into two contributions which are not individually invariant against unitary transformations.

The first contribution is given by the expectation value of the relativistic correction of the Hellmann-Feynman force, ⟨∂²V/∂R∂ ln c²⟩ It may be neglected in standard two- or four-component all-electron calculations. On the other hand it seems to be the dominant contribution in pseudopotential valence-only calculations, but has sometimes been overlooked in the literature.

The second contribution is the dominant one in all-electron calculations. It can be expressed in two alternative forms, which give rise to two complementary interpretations (IIa) and (IIb). They both differ from the one (I) appropriate for pseudopotential calculations. Whilst interpretation IIa is generally accepted in the literature, interpretation IIb although equivalent is often refused. The two interpretations differ in emphasizing the potential or kinetic energies, the bond or inner shell regions, the density changes due to relativity or due to overlapping, respectively.