ASYMPTOTIC-BEHAVIOR OF ELECTRON-DENSITIES AND COMPUTATION OF ONE-ELECTRON PROPERTIES

The role of the asymptotic behavior of approximating sequences of elec tron densities rho(n)(r) in the calculation of one-electron properties is studied. Rigorous mathematical results in the frame of Hilbert spa ces are used to prove the following facts: (i) Both the L(2) convergen ce of wave functions psi(n) and the E convergence of the corresponding energies E, guarantee the correctness of the limiting procedure lim(n -->infinity)integral(Omega)s((x) over bar)\psi(n) \(2)d (x) over bar = integral(Omega)s((x) over bar)\psi \(2) d (x) over bar for the most frequently used operators s(x), Omega being any bounded region of the n-particle configuration space R(3N)and (ii) the uniform boundedness o f the sequence {p(n)} together with both the L(2) and E convergencies is sufficient to guarantee the correctness of the limiting procedure l im(n -->infinity)integral(0)(infinity) s(r)rho r(2)dr for most one-ele ctron operators s(r) including the power moment operators r(k) which, for large k, are representative of the class of operators not relative ly form-bounded by the Hamiltonian. The mathematical concept of unifor m boundedness is used to give a characterization of the capability of {rho(n)} to reproduce the asymptotic behavior of the hue electron dens ity rho and it is shown by means of numerical examples how a sequence {p(n)} that does not reproduce the correct asymptotic behavior is not uniformly bounded and can give divergent expectation values of one-ele ctron operators s(r) not relatively form-bounded by the Hamiltonian. ( C) 1996 John Wiley & Sons, Inc.