Theory of Isotropic Hyperfine Interactions in Pi-Electron Free Radicals. I. Basic Molecular Orbital Theory with Applications to Simple Hydrocarbon Systems

The general molecular orbital configuration interaction approach formulated by McConnell and others is used to express the hyperfine coupling matrix Q^A in terms of the properties of localized sigma bonds. [Q^A relates the coupling constant $a^A$ , A being ¹H or any first row element, to the pi‐electron spin‐density matrix ${\varrho }^{\pi }$ through the expression $a^A\mathrm{\hspace{0.17em}=}\mathrm{;trace}{\mathrm{(\varrho }}^{\pi }{\mathrm{Q}}^A)$ .] An important step is the inclusion of all overlap in the basis‐set inner‐ and valence‐shell atomic orbitals; failure to do this leads to artificial sensitivity of Q^A to sigma‐bonding details. An orthonormal set of localized sigma‐bond orbitals is generated from “equivalent” (two‐center) orbitals by a transformation involving various overlap matrices. An approximation to this transformation is used to examine the salient features of Q^A. The theory supports the first approximation for $a^A$ in terms of diagonal elements of Q^A and ${\varrho }^{\pi }{\mathrm{:\; a}}^A{{\mathrm{\approx Q}}_{\mathrm{AA}}^A}_{\mathrm{\rho AA}}^{\pi }{\mathrm{\hspace{0.17em}+\hspace{0.17em}\Sigma }}_X{Q_{\mathrm{XX}}^A}_{\mathrm{\rho XX}}^{\pi }$ , where ${}_{\mathrm{\rho AA}}^{\pi }$ and ${}_{\mathrm{\rho XX}}^{\pi }$ are the pi‐electron “spin densities” on atom A and its nearest neighbors. Contributions from inner‐shell orbitals and interactions between sigma bonds are very important. It is shown that $Q_{\mathrm{AA}}^A$ is particularly insensitive to environmental effects. The adjacent atom parameter $Q_{\mathrm{XX}}^A$ involves contributions from all substituents on A (if any) and on X. If A is a multivalent atom these contributions occur with both positive and negative signs so that the ratio $Q_{\mathrm{XX}}^A{\mathrm{\hspace{0.17em}/\hspace{0.17em}Q}}_{\mathrm{AA}}^A$ is quite small and can, in principle, be either positive or negative, contrary to previous predictions. Application of the theory to the CH₃ radical gives results which agree with experiment to well within the inherent uncertainties in the calculation. The CH₃ results are used to choose semiempirical “excitation energies” for a calculation of Q^C and Q^H in a hypothetical CH₂CH₂^±ion. For CH₂CH₂^± the large diagonal elements of Q̄^C and Q̄^H (which include a zero‐point “vibrational contribution”) are estimated as (in gauss) : ${\mathrm{Q̄}}_{\mathrm{CC}}^C\text{\hspace{0.17em}=\hspace{0.17em}+37.8\hspace{0.17em}±\hspace{0.17em}3.0}$ , ${\mathrm{Q̄}}_{\mathrm{C\prime C\prime }}^C\text{\hspace{0.17em}=\hspace{0.17em}−9.0\hspace{0.17em}±\hspace{0.17em}1.0}$ and ${\mathrm{Q̄}}_{\mathrm{CC}}^H\text{\hspace{0.17em}=\hspace{0.17em}−27.2\hspace{0.17em}±\hspace{0.17em}1.8}$ . The first and last of these can be compared with the values observed in CH₃, +38.3 and −23.0 G, respectively, showing that ${\mathrm{Q̄}}_{\mathrm{CC}}^C$ is quite insensitive to the substituents on C but that ${\mathrm{Q̄}}_{\mathrm{CC}}^H$ increases significantly with substitution on C. It is shown that these values of ${\mathrm{Q̄}}_{\mathrm{CC}}^C$ and ${\mathrm{Q̄}}_{\mathrm{C\prime C\prime }}^C$ give excellent agreement with experimental ¹³C data on selected aromatic radicals. The smaller elements of Q̄^C and Q̄^H are estimated (in gauss) : ${\mathrm{Q̄}}_{\mathrm{CC\prime }}^C{\text{\hspace{0.17em}=\hspace{0.17em}+1.8, Q̄}}_{\mathrm{CC\prime }}^H{\text{\hspace{0.17em}=\hspace{0.17em}−3.0, Q̄}}_{\mathrm{C\prime C\prime }}^H\text{\hspace{0.17em}=\hspace{0.17em}+1.8}$ . The estimate of the off‐diagonal element