ESR hyperfine coupling constants

We have previously defined the one-electron spin-density matrix in the context of standard HF methodology (Eq. (6.9)), which includes semiempirical methods and both the UHF and ROHF implementations of Hartree-Fock for open-shell systems. In addition, it is well defined at the MP2, CISD, and DFT levels of theory, which permits straightforward computation of h.f.s. values at many levels of theory. Note that if the one-electron density matrix is not readily calculable, the finite-field methodology outlined in the last section allows evaluation of the Fermi contact integral by an appropriate perturbation of the quantum mechanical Hamiltonian. 

For Eq. (9.35) to be useful the density matrix employed must be accurate. In particular, localization of excess spin must be well predicted. ROHF methods leave something to be desired in this regard. Since all doubly occupied orbitals at the ROHF level are spatially identical, they make no contribution to P only singly occupied orbitals contribute. As discussed in Section 6.3.3, this can lead to the incorrect prediction of a zero h.f.s. for all atoms in the nodal plane(s) of the singly occupied orbital(s), since their interaction with the unpaired spin(s) arises from spin polarization. In metal complexes as well, the importance of spin polarization compared to tire simple analysis of orbital amplitude for singly occupied molecular orbitals (SOMOs) has been emphasized (Braden and Tyler 1998). 

on the other hand, does optimize tire a and orbitals so tlrat they need not be spatially identical, and thus is able to account for both spin polarization and some small amount of configurational mixing. As a result, however, UHF wave functions are generally not eigenfunctions of the operator S-, but are contaminated by higher spin states. 

The challenge with unrestricted methods is tire simultaneous minimization of spin contamination and accurate prediction of spin polarization . The projected UHF (PUHF, see Appendix C) spin density matrix catr be employed in Eq. (9.36), usually with somewhat improved results. 

A complicating factor is tlrat each spitr density matrix element is multiplied by the corresponding basis function overlap at tire nuclear positions. The orbitals having maximal amplitude at the nuclear positions are tire core s orbitals, which are usually described with less flexibility than valence orbitals itr typical electronic structure calculations. Moreover, actual atomic s orbitals are characterized by a cusp at tire nucleus, a feature accurately modeled by STOs, but only approximated by the more commonly used GTOs. As a result, tlrere are basis sets in the literature tlrat systematically improve tire description of the core orbitals in order to improve prediction of h.f.s., e.g. IGLO-III (Eriksson et al. 1994) and EPR-III (Barone 1995). 

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Figure 3. INDO open shell hypersurface calculations for the hydrazine radical cation (23,24) contracted to 3 angular coordinates of freedom (cf. text) (A) INDO total energies vs. the B/w coordinate pair, and hypersurface maps for the dependence of the ESR hyperfine coupling constants, ajj (B and C) and ajj (D and E) on the dihedral angle w and the HNH bond angle a (B and D) or the out-of-plane bending angle B (C and E).

The ESR hyperfine coupling constants have been established experimentally (67MI20402) for the pyridinyl radical (134 R = H) and deuterated analogues, produced by y irradiation of a solid solution of pyridine in ethanol at 77 K, but the signs of the couplings are not known experimentally and are made solely on the basis of Huckel MO calculations. INDO MO calculations on this radical, together with the radical anions of quinoline, isoquinoline and acridine h ve also been carried out (740MR(6)5). 

Wood et al. (141) were able to estimate that the unpaired spin-density is mainly localized in the p-orbital of the nitrogen, while the nitrogen lone pair of electrons is largely in the 2s orbital. The ESR hyperfine coupling constants of some alkylamino and alkylimino radicals are given in Table 10. 

We have briefly considered some of the voluminous ESR studies of organic n-radicals and will now turn to the much fewer investigations of organic 6-radicals. In general the 6-radical has the unpaired electron mainly in an orbital with nonvanishing amplitude at the nucleus of the radical center. Thus ESR hyperfine coupling constants can be used to assign whether the radical is 6- or ir-type. 

The relationship between contact shift and esr hyperfine coupling constants found for the benzouranocene system does not apply straightforwardly to alkyluranocenes, perhaps because the alkyl groups... 

For some time to come, density functional methods will be the key to the study of NMR properties for transition metal compounds, as no other available quantum chemical method presently allows the necessary inclusion of electron correlation effects at manageable computational cost. Further progress in the development of exchange-correlation functionals should even widen the possible fields of application, in particular for the calculation of spin-spin coupling constants or of spin-orbit corrections to chemical shifts, which both involve Fermi-contact type contributions (the same outlook holds for the computation of ESR hyperfine coupling constants in transition metal compounds). 

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