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

Abstract

The general molecular orbital configuration interaction approach formulated by McConnell and others is used to express the hyperfine coupling matrix QA in terms of the properties of localized sigma bonds. [QA relates the coupling constant aA, A being 1H or any first row element, to the pi-electron spin-density matrix ϱπ through the expression aA =;trace (ϱπQA).] 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 QA 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 QA. The theory supports the first approximation for aA in terms of diagonal elements of QA and ϱπ: aA≈QAAAρAAπ + ΣXQXXAρXXπ, where ρAAπ and ρXXπ 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 QAAA is particularly insensitive to environmental effects. The adjacent atom parameter QXXA 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 QXXA / QAAA 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 QC and QH in a hypothetical CH2CH2±ion. For CH2CH2± the large diagonal elements of Q̄C and Q̄H (which include a zero-point “vibrational contribution”) are estimated as (in gauss) : Q̄CCC = +37.8 ± 3.0, Q̄C′C′C = −9.0 ± 1.0 and Q̄CCH = −27.2 ± 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 Q̄CCC is quite insensitive to the substituents on C but that Q̄CCH increases significantly with substitution on C. It is shown that these values of Q̄CCC and Q̄C′C′C give excellent agreement with experimental 13C data on selected aromatic radicals. The smaller elements of Q̄C and Q̄H are estimated (in gauss) : Q̄CC′C = +1.8, Q̄CC′H = −3.0, Q̄C′C′H = +1.8. The estimate of the off-diagonal element QCC′His in good agreement with empirical values which have been obtained for aromatic systems. Using CH3 as a model, the derivatives of QCCH and QCCC with respect to bond length and the effective nuclear charge of carbon are calculated and found to be an order of magnitude smaller than the results obtained by neglecting overlap.