Relativistic theory of nuclear shielding in one-electron atoms 1. Theoretical foundations and first-order terms

The nuclear shielding in NMR spectroscopy is given in relativistic theory (Pyper, N. C., 1983, Chem. Phys. Lett., 96, 204) as the sum of the exact relativistic analogues of the non-relativistic diamagnetic and paramagnetic terms augmented by a further purely relativistic term. In this paper this theory, based on the Gordon decomposition of the Dirac current, is extended to encompass the point charge—point dipole model of the nucleus, the previous formulation (Pyper, 1983) being restricted to spatially extended descriptions of the nuclear charge and magnetization. For the point nucleus case, analytical expressions for both the purely relativistic shielding contribution and diamagnetic term are derived for all orbitals. The result for the diamagnetic shielding is derived by invoking the relativistic virial theorem, and thus depends only on the orbital energy and magnetic quantum number m. Therefore it shows, for given m, the same ‘accidental’ degeneracy between orbitals differing in the sign of the quantum number κ as the energy. The purely relativistic contribution to the shielding is shown to differ from the electronic contribution to the hyperfine energy by a factor of only m multiplied by fundamental constants. The present purely relativistic term is shown to equal the previous result (Pyper, 1983) even though appearing in a different form. For the interaction with a uniform magnetic field the relativistically exact Hamiltonian obtained from the Gordon decomposition is used with the virial theorem to derive a new expression for the g factor for all orbitals. The resulting simple expression depends only on the orbital energy in addition to the quantum numbers κ and m.