Knowledge Bank

“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 $H2-$ , the latter is a Coulomb potential with cut-off. Using Wannier’s $theory,1$ the ground state of $H2-$ 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 $mi$. The one-electron wave-functions for the low-lying levels have essentially the same ($p, $z) dependent part, and are distinguished by their $\varphi$ dependence. Molecular symmetry splits the $mi=\pm N$ 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 $\varphi$ dependence.”““