A general procedure for a nonarbitrary external localization of atomic and molecular orbitals given in the form of a general LCAO expansion is introduced by imposing an extremum principle on the sum of certain local orbital populations which are required to be localized in given regions of space, in particular around atoms or between pairs of atoms in the molecule. The localization, which is uniform insofaras these local electron populations are simultaneously maximized in all the available orbitals, yields well-defined inner-shell, lone-pair and bond orbitals. The orthogonal transformation which maximizes the localization function is obtained through an iterative sequence of 2 × 2 rotations between all N (N − 1)/2 possible pairs of molecular orbitals. Convergence was found to be excellent. The method turns out to be exceedingly simple, the coefficients in the LCAO expansion and the overlap integrals between the basic atomic orbitals only being required, and general enough to be valid for the localization of atomic orbitals as well of molecular orbitals, either for exact LCAO—SCF—MO's or for approximate LCAO—MO's constructed from nonorthogonal as well from orthogonal basic sets. The localized orbitals obtained in this paper starting from some unsymmetrically orthogonalized atomic orbitals and from the minimal-basis-set LCAO—SCF—MO wavefunctions given by Ransil for LiH, BH, NH, FH, LiF, BF, CO, Li2, Be2, N2, F2 prove to be very close to the energy localized orbitals recently obtained by Edminston and Ruedenberg by maximizing the sum of the orbital self-repulsion energies.

Two of the Hermitian eigenvalue equations resulting from the separation of the three-dimensional Schroedinger equation for H 2 + in spheroidals are solved perturbatively for the ground state by expanding the action in positive powers of the internuclear distance R near the united atom He + . The dispersion relations between the separation constants A and E e are seen to have rigorous analytic solutions, the third-order equation leading to an exact expansion for the inner determinantal equation up to R 10 . The explicit form for the expansion coefficients is determined up to n = 10, and is seen to contain up to the third power of (γ + ln 4R) logarithmic terms. Even if the general range of validity of the short-range R n -expansion is expected to be smaller than the corresponding long-range R −n -expansion, it is important to stress that such higher expansion coefficients are calculated exactly for the first time. These formulae give extremely accurate numerical results up to R ∼ = 0.3a 0 .

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A new translation method for Slater-type orbitals STOs is proposed involving exact translation of the regular solid harmonic part of the orbital followed by the series expansion of the residual spherical part in powers of the radial variable. The method is positively tested in the case of the overlap integral, showing good rate of convergence and great numerical stability under wide changes in the relevant molecular parameters.

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