Indirect proton hyperfine interactions in π-electron radicals are first discussed in terms of a hypothetical CH fragment which holds one unpaired π electron and two σ-CH bonding electrons. Molecular orbital theory and valence bond theory yield almost identical results for the unpaired electron density at the proton due to exchange coupling between the π electron and the σ electrons. The unrestricted Hartree-Fock approximation leads to qualitatively similar results. The unpaired electron spin density at the proton tends to be antiparallel to the average spin of the π electron, and this leads to a negative proton hyperfine coupling constant. The theory of indirect proton hyperfine interaction in the CH fragment is generalized to the case of polyatomic π-electron radical systems; e.g., large planar aromatic radicals. In making this generalization there is introduced an unpaired π-electron spin density operator, ρN, where N refers to carbon atom N. Expectation values of the spin density operator ρN are called ``spin densities,'' ρN, and can be positive or negative. In the simple one-electron molecular orbital approximation a π-electron radical always has a positive or zero spin density at carbon atom N, 0≤ρN≤1. In certain π-electron radical systems; e.g., odd-alternate hydrocarbon radicals, the spin densities at certain (unstarred) carbon atoms are negative when the effects of π—π configuration interaction are included in the π-electron wave function. The previously proposed linear relation between the hyperfine splitting due to proton N, aN, and the unpaired spin density on carbon atom N, ρN,aN=QρNis derived under very general conditions. Two basic approximations are necessary in the derivation of this linear relation. First, it is necessary that σ—π exchange interaction can be treated as a first-order perturbation in π-electron systems. Second, it is necessary that the energy of the triplet antibonding state of the C–H bond be much larger than the excitation energies of certain doublet and quartet states of the π electrons. This derivation of the above linear relation makes no restrictive assumptions regarding the degree of π—π or σ—σ configuration interaction. The validity of the above approximations is discussed and illustrated by highly simplified calculations of the proton hyperfine splittings in the allyl radical, assuming the π—π configuration interaction—and hence the negative spin density on the central carbon atom—to be small. Isotropic hyperfine interactions in molecules in liquid solution can also arise from spin-orbital interaction effects, and it is shown that these effects are negligible for proton hyperfine interactions in aromatic radicals.
Calculations of chemical shifts have been carried out using "locally dense" basis sets for the resonant atom of interest, and smaller, attenuated sets on other atoms in the molecule. For carbon, calculations involving a 6-311G(d) triply split valence set with polarization on the resonant atom and 3-21G atomic bases on other heavy atoms result in good agreement with experiment, and are virtually identical to those found employing the larger basis on all atoms. For species such as nitrogen, oxygen, and fluorine where standard balanced basis sets do not agree well with experiment, use of attenuated sets fail as well. The use of locally dense basis sets permits calculations previously impractical, and the successful application to carbon suggests that the chemical shift is most dependent on the local basis set, and less so on whether or not a balanced or unbalanced calculation is being carried out.
The locally dense basis set approach to the calculation of nuclear magnetic resonance shieldings is one in which a sufficiently large or dense set of basis functions is used for an atom or molecular fragment containing the resonant nucleus or nuclei of interest and fewer or attenuated sets of basis functions employed elsewhere. Provided the dense set is of sufficient size, this approach is capable of determining chemical shieldings nearly as well as a calculation with a balanced basis set of quality equal to the locally dense set, but with considerable savings of CPU time. Detailed comparisons are provided of locally dense and balanced calculations in the gauge including atomic orbital (GIAO) method for the individual principal values, the isotropic shieldings, and the tensor orientations for hydrogen, carbon, nitrogen, oxygen, fluorine, and phosphorus nuclei. It is seen that chemical functional groups can often define the appropriate molecular fragment to be taken locally dense. While the present test cases are for the most part small molecules, the value of the method is that it will allow calculations on systems that would otherwise presently be computationally expensive or inaccessible. 0 1993 by John Wiley & Sons, Inc.
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