Asymptotic behavior of electron densities and computation of one-electron properties

The role of the asymptotic behavior of approximating sequences of electron densities ρn(r) in the calculation of one-electron properties is studied. Rigorous mathematical results in the frame of Hilbert spaces are used to prove the following facts: (i) Both the L2 convergence of wave functions ψn and the E convergence of the corresponding energies En guarantee the correctness of the limiting procedure limn→x ∫Ω s((overline)x(/overline)|ψn|2 d(overline)x(/overline) = ∫Ω s((overline)x(/overline))|ψ|2 d(overline)x(/overline) for the most frequently used operators s(x), Ω being any bounded region of the n-particle configuration space R3N; and (ii) the uniform boundedness of the sequence {ρn} together with both the L2 and E convergencies is sufficient to guarantee the correctness of the limiting procedure limn→x ∫∞0 s(r)ρnr2dr = ∫x0 s(r)ρr2 dr for most one-electron operators s(r) including the power moment operators rk which, for large k, are representative of the class of operators not relatively form-bounded by the Hamiltonian. The mathematical concept of uniform boundedness is used to give a characterization of the capability of {ρn} to reproduce the asymptotic behavior of the true electron density ρ and it is shown by means of numerical examples how a sequence {ρn} that does not reproduce the correct asymptotic behavior is not uniformly bounded and can give divergent expectation values of one-electron operators s(r) not relatively form-bounded by the Hamiltonian.