ANGULAR MOMENTUM QUANTUM NUMBERS IN SYMMETRIC $MOLECULES^{*}$

Abstract

“The generalized united-atom theory of molecules suggests that electronic states of symmetric molecules may be characterized by means of “atomic” quantum numbers. When the one-electron potential of a molecule is expanded in terms of coordinates of the center of symmetry, the angular terms may be considered to be perturbations on a basically spherical potential. For $H^{-}_{2}$ , the latter is a Coulomb potential with cut-off. Using Wannier’s $theory,^{1}$ the ground state of $H^{-}_{2}$ is shown to be 95% pure s in character. The quantum number l becomes increasingly pure with increase in energy. For cyclic polyenes, an analogous treatment in cylindrical coordinates indicates that the H\{u}Uckel ring quantum number is the magnetic quantum number $m_{i}$. The one-electron wave-functions for the low-lying levels have essentially the same ($p, $z) dependent part, and are distinguished by their $\phi$ dependence. Molecular symmetry splits the $m_{i}=\pm{N}$ levels for a cyclic polyene of 2N carbon atoms. The results are compared with LCAO and electron-gas theories. Additional perturbations due to substitution or cross-link formation are discussed in terms of the $\phi$ dependence.”““