Stability Conditions for the Solutions of the Hartree–Fock Equations for Atomic and Molecular Systems. IV. A Study of Doublet Stability for Odd Linear Polyenic Radicals

Abstract

The doublet stability conditions for the restricted Hartree–Fock (HF) solutions of the simple open‐shell case (i.e., one electron in addition to the closed shell) are applied to the π‐electronic models of the odd linear polyenic radicals, using the Pariser–Parr–Pople‐type semiempirical Hamiltonian. It is found that symmetry‐adapted restricted HF solutions for linear polyenes with $\text{(4ν\hspace{0.17em}−\hspace{0.17em}1)}$ carbon atoms are always doublet stable while those with $\text{(4ν\hspace{0.17em}−\hspace{0.17em}1)}$ carbon atoms may be unstable for large enough coupling constants (i.e., the ratio of two‐electronic part of the Hamiltonian to the one‐electronic part). In fact the HF solutions for the latter systems are generally doublet unstable already for the currently used values of the semiempirical parameters or a very small lowering of the absolute value of the resonance integral $\beta$ (i.e., ∼0.1 − 0.3 eV) is sufficient to yield instability for any $\text{(4ν\hspace{0.17em}−\hspace{0.17em}1)}$ type polyene. The instability, in fact, occurs already for the allyl radical, which is studied in a considerable detail. The doublet stability of the solutions for the $\text{(4ν\hspace{0.17em}+\hspace{0.17em}1)}$ type polyenes is explained on the basis of symmetry properties of corresponding Kekulé structures. The symmetry‐nonadapted restricted HF solutions are calculated for illustration of their properties and possible multiplicities.