Hybrid orbitals and the runge–lenz vector

Abstract

Matrix elements of the Runge–Lenz vector A are presented for those linear combinations of degenerate hydrogenic functions often referred to as hybrid orbitals. The uncertainties in the components of A for each type of wave-function are related to the distribution of classical Kepler orbits corresponding to each function. Matrix elements of A with respect to radially nodeless Slater functions are presented, as these functions are often used as a basis set in atomic and molecular calculations. The properties of A for a piecewise Coulombic central field are discussed in relation to the description of penetrating orbits in the old quantum theory. Simultaneous eigenfunctions of A and the Hamiltonian cannot be chosen for the piecewise Coulombic field because of a discontinuity in the radial derivative of the potential energy.