Generalization of Kato’s Cusp Conditions to the Relativistic Case

The behaviour of relativistic many-electron wave functions for arbitrary atoms or molecules at the Coulomb singularities of a point nucleus and of the electron interaction is derived. The wave functions have weak singularities of the type rv with v < 0. No singularity arises for extended nuclei. Solutions of the Klein-Gordon equation, the ‘square-root equation’ or of Sucher’s nopair equation with free-particle projectors’ have a different behaviour than those of the Dirac equation. While v is quadratic in Z/c for the Dirac equation, it is linear in Z/c for both the square-root operator and the free-particle no-pair-projected Dirac operator. For the latter two cases there is no smooth non-relativistic limit. For a Dirac-Coulomb Hamiltonian there is also a singularity of the type r _kl λ with λ < 0 at the point of coalescence of two electrons, while for the Dirac-Gaunt (or Dirac-Brown) operator one finds λ > 0 (in view of the higher singularity of the magnetic interaction one might have expected a more singular behaviour of the wave function in its presence), and for the Dirac Breit operator λ = 0. Like in the nuclear cusp, A is in these cases quadratic in 1/c, while it is linear in 1/c for the no-pair projected operators. In spite of the complexity of these results, they only affect an extremely small neighborhood of the point of coalescence of two electrons. A large region close to r₁₂ = 0 is governed by the behaviour in the non-relativistic limit, i.e. by Kato’s correlation cusp.