Bounds to average cosines ⟨cos θ12⟩ in the Hartree-Fock theory of atoms

We show that the expectation value (cos theta(12)) and its subshell-pair components <COS theta(12)>(nl,n'l') are bounded from below and above in the Hartree-Fock theory of atoms, where theta(12) is the interelectronic angle subtended by the position vectors r(1) and r(2) of a pair of electrons, and n and l are the principal and azimuthal quantum numbers. The lower bounds to (costheta(12))(nl,n'l') are -l/[2(2l - 1)(2l + 1)] for II' pairs with I = I'+ 1, while they are 0 for the other II' pairs, independently of n and n'. A weighted sum of these subshell-pair bounds gives a lower bound to <costheta(12)>. The upper bounds are 0 in all the cases. (C) 2004 Elsevier B.V. All rights reserved.