Electronic Structures of Polyatomic Molecules and Valence. IV. Electronic States, Quantum Theory of the Double Bond

The possible types of electronic states of polyatomic molecules (assuming fixed nuclei and neglecting spin fine structure) are discussed and tabulated (Table I) with the help of simple group theory methods, applying results of Bethe and Wigner. A notation for electronic states ($\psi \text{'}\mathrm{s}$) and molecular orbitals ($\phi \text{'}\mathrm{s}$) for molecules having any type of symmetry to be found among the 32 crystal classes, is adopted; this is essentially the same as that used by Placzek for designating the vibrational states of molecules. It is shown how the possible $\psi \text{'}\mathrm{s}$ corresponding to any given electron configuration (set of $\phi \text{'}\mathrm{s}$) can be determined for any type of symmetry; for the more complicated cases, the results are tabulated (Table V). It is shown how all the selection rules for transitions between electronic states of molecules can be easily determined. Limitations resulting here from the application of the Franck-Condon principle are discussed. Extending work of Bethe, tables are given (Tables II-IV) showing how the various types of electronic states of atoms and of diatomic and polyatomic molecules ($S$, $P$, ${\Sigma }^{+}$, $\Delta A$, etc.) go over into various other types of states if the symmetry of the original system is decreased. Examples are given showing how electronic wave functions ($\psi \text{'}\mathrm{s}$) of molecules can be constructed which conform to the possible types (Table I) allowed by the symmetry of the nuclear skeleton, and which at the same time, with Slater's method, are antisymmetrical in the electrons (cf. section 2 and Eqs. (9-12)). It is shown that for molecules having all their electrons in closed shells or electron-pair bonds, zeroth approximation $\psi \text{'}\mathrm{s}$ which conform to the identical representation of the molecule's symmetry group (analogous to ${}_{}{}^1{}_{}{}^{}S$ of atoms and ${}_{}{}^1{}_{}{}^{}\Sigma _{}^{+}{}_{}{}^{}$ or ${{}_{}{}^1{}_{}{}^{}\Sigma _g^{}{}_{}{}^{}}^{+}$ of diatomic molecules) can be built up either by using electron-pair bonds or by using molecular orbitals. The approximate construction of molecular orbitals as linear combinations of atomic orbitals, in such a way that they conform to the types allowed by the symmetry of the molecule, is discussed and illustrated (cf. Eqs. (3, 8)). Several statements made in a previous paper (III) of this series, on the quantum theory of the double bond, are here justified by the methods mentioned above, thereby also providing examples of the application of the latter. Some additional details concerning the nature of the double bond are given. Finally, it is shown that the model of the double bond given in III should according to the theory be altered somewhat for the perp. form of the molecule, in a way which offers the possibility of improved agreement with experiment.