Crater property in two-particle bound states: When and why

Crater has shown that, for two particles (with masses $m1$ and $m2)$ in a Coulombic bound state, the charge distribution is equal to the sum of the two charge distributions obtained by taking $m1→∞$ and $m2→∞,$ respectively, while keeping the same Coulombic potential. We provide a simple scaling criterion to determine whether an arbitrary Hamiltonian possesses this property. In particular, we show that, for a Coulombic system, fine structure corrections preserve this Crater property while two-particle relativistic corrections and/or hyperfine corrections may destroy it.