Relativistic atomic orbital contractions and expansions: magnitudes and explanations

The magnitude of the relativistic contraction or expansion of atomic orbitals is usually obtained by a comparison of the expectation values of r in a Dirac-Fock calculation and in a Hartree-Fock calculation. As, however, the Dirac Hamiltonian is implicitly given in a different picture from the non-relativistic Schrodinger Hamiltonian, the operator r does not correspond to the same physical quantity in the two cases. A proper definition of relativistic AO contraction/expansion should use the same physical quantity in both the relativistic and non-relativistic cases; for instance experiments with photons measure matrix elements of Pcharger which is represented by the operator r in the Dirac picture and by U,,rU&, in the Schrodinger picture ( U,, is the Foldy-Wouthuysen transformation). Accordingly, the conventional values of the relativistic AO contraction consist of two contributions. One is due to the relativistic modification of the orbital; the other one is due to the different meanings of r in the Schrodinger and Dirac pictures. This latter difference turns out to be significant for Is AO, where it is 50%. The large relativistic contraction of valence s AO of heavy elements is investigated. Using perturbation theory or the resolution of the identity into projection operators, the orthogonality of the valence AO on the strongly contracted inner core orbitals is shown to have a slight valence-expanding effect, while mixing in of the higher continuum orbitals by the relativistic correction of the Hamiltonian is responsible for the overall contraction.