  It is shown that natural boundary conditions for non-relativistic wave
functions are of periodic or of homogeneous Robin type. Using asymptotic
central symmetry of Hamiltonian and theory of singular differential equations
the many-electron wave function is expanded in series both in the vicinity of
Coulomb singularities and at infinity. Hydrogenic angular dependence of three
leading terms of expansion about Coulomb singularities is found. Exact first-
and second-order cusp conditions are obtained demonstrating redundancy of
spherical average in Kato's cusp condition. Our first-order cusp condition
exhibits CP symmetry. Homogeneous Robin boundary conditions are obtained for
aperiodic many-electron systems from the expansions. Use of our explicit
boundary conditions improves both speed and accuracy of numerical calculations.
A confluent hypergeometric series defining arbitrarily high order cusp
conditions for the spherically averaged Hamiltonian is presented.
