Hybrid orbitals and the runge–lenz vector

Abstract

AbstractsMatrix 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.On presente des elements de la matrice representant le vecteur de Runge-Lenz A dans une base consistant d'orbitales hybridtes de fonctions hydrogtnoydes. Les mesures d'indttermination des composantes de A pour chaque type de fonction d'onde sont assocites a la distribution des orbites de Kepler classiques correspondant a chaque fonction. On prtsente aussi des elements de la matrice de A par rapport B des fonctions de Slater sans naeuds. Les proprittts de A pour un champ Coulombien central par morceaux sont discuttes par rapport a la description des orbites phttrantes dans la vieille thtorie quantique. I1 n'est pas possible de choisir des fonctions propres simultantes de A et du Hamiltonien, a cause d'une discontinuitt dans la dtrivte radiale de I'knergie potentielle.

Quantum Alchemy: Transmutation of Atomic Orbitals

“…Now that we have obtained another set of operators that commute with the Hamiltonian, we can play the same games we did with angular momentum. The following results were derived by Lohr (7):…”