ESR Hyperfine Calculations

The distribution of the unpaired electron(s) in a radical system strongly reflects the reactivity, and is furthermore a highly geometry-sensitive property. Thus, a theoretical approach that 

In this article, an overview will be presented of the applicability and limitations of such theoretical approaches, as well as a discussion of how it is possible to incorporate effects of the local surrounding, or the effects of molecular vibrations, in the radical hfcc. First, however, the mathematical formulae used for computing radical hyperfine parameters will be outlined. 

The first two terms describe the interactions between the electron and the field, and between the magnetic nuclei and the field, respectively, and are referred to as the electronic and nuclear Zeeman terms. The final term of equation (1) describes the magnetic hyperfine interaction between the magnetic moments of the electron and the nuclei. As seen from the equation this is a field-independent property, and will be the center of our attention in this present article. To derive the hyperfine components usually reported, the first step is to diagonalize the 3 x 3 hyperfine tensor T. This allows us to separate out the isotropic component. 

As this component depends on the value of the unpaired spin density evaluated at the position of the nucleus only, p -P(0), it is often referred to as a contact interaction, or Fermi contact term. This does not have a classical analog, but is a consequence of the quantum nature of the particles. The remainder of the hyperfine part of the spin Hamiltonian is a traceless tensor that describes the deviation from spherical 

The components of the total hyperfine tensor. A, observed in the ESR spectra, are obtained by adding Ajso to the Txx, Tyy, and components, respectively, of the dipolar tensor. For molecules with a well-defined axial symmetry it is common to report just one value of the dipolar hf tensor, Adip. This is related to the largest dipolar component (here assumed to be Tk), as Adip = 1 /2 It is also useful to compute the values perpendicular and parallel to a particular bond (here assumed to be along the z-axis). 

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Basis Sets Correlation Consistent Sets Circular Dichro-ism Electronic Complete Active Space Self-consistent Field (CASSCF) Second-order Perturbation Theory (CASPT2) Configuration Interaction Density Functional Applications Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Electronic Diabatic States Definition, Computation, and Applications ESR Hyperfine Calculations Magnetic Circular Dichroism of rt Systems Non-adiabatic Derivative Couplings Relativistic Theory and Applications Structure Determination by Computer-based Spectrum Interpretation Valence Bond Curve Crossing Models. 

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). 

Fig. 8.14 Calculation of the ESR hyperfine pattern (solid lines) of die CH CHg radical in solution.

The ESR spectra of the toluene extract of the coronary effluents collected during aerobic reperfusion are shown in Figs. 7a-e. The signal intensity increased during the early minutes of reperfusion and then declined. The ESR hyperfine parameters were calculated to be about on = 14.0 0.1G and a = 2.0 0.1 G. No ESR spectra were detected from the aqueous coronary effluents under otherwise identical conditions. Extraction with toluene was, therefore, absolutely crucial to detect an ESR signal from the PBN-adduct. At lower temperatures ( —100°C), the ESR signal intensity was increased with a slight loss in spectral resolution. Similar results have previously been reported [128]. 

As mentioned in Section V.B, it is usually assumed that Koa can be estimated from data representing the corresponding molecular carbonyl. The hyperfime fields and can be estimated theoretically from calculations for the molecule or experimentally from ESR hyperfine couplings in radicals that have been identified as CO" (an unpaired electron in the So) 141) or CO (an unpaired electron in the 2xr ) 142). From the calculated charge density at the nucleus for one direction of spin in the isolated CO molecule, is 398 kG at the carbon and 5.24 kG at the oxygen 143). The jr-hyperfine field is zero by virtue of symmetry on both sites. The experimental values for the 3C site are 7/bf, 363 kG and... 

The SAC-CI method is also useful for studying the ESR hyperfine splitting constants which characterize the electronic stmcture of radicals. It was shown that the STO-GTO expansion method is useful for calculating the HFSCs of radicals with the conventional GTO program the HFSCs are calculated with the STO basis set which satisfies the cusp condition at nuclei. 

The conventional, and very convenient, index to describe the random motion associated with thermal processes is the correlation time, r. This index measures the time scale over which noticeable motion occurs. In the limit of fast motion, i.e., short correlation times, such as occur in normal motionally averaged liquids, the well known theory of Bloembergen, Purcell and Pound (BPP) allows calculation of the correlation time when a minimum is observed in a plot of relaxation time (inverse) temperature. However, the motions relevant to the region of a glass-to-rubber transition are definitely not of the fast or motionally averaged variety, so that BPP-type theories are not applicable. Recently, Lee and Tang developed an analytical theory for the slow orientational dynamic behavior of anisotropic ESR hyperfine and fine-structure centers. The theory holds for slow correlation times and is therefore applicable to the onset of polymer chain motions. Lee s theory was generalized to enable calculation of slow motion orientational correlation times from resolved NMR quadrupole spectra, as reported by Lee and Shet and it has now been expressed in terms of resolved NMR chemical shift anisotropy. It is this latter formulation of Lee s theory that shall be used to analyze our experimental results in what follows. The results of the theory are summarized below for the case of axially symmetric chemical shift anisotropy. 

Alternatively, within the SOS-DFPT framework, Malkin et al. include the FC term by FPT (the SD term, which is neglected in this ansatz, has been found to be small " ). This approach has been motivated by experience with the calculation of ESR hyperfine couplings and with spin-spin coupling constants (cf. above). Based on the resulting spin-polarized Kohn-Sham orbitals, SOS-DFPT is used to account for the interplay between the external magnetic field and the spin-orbit operator. First applications include halomethanes, halomethyl cations, some other iodo compounds, and organomercury complexes. ... 

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). 

Figure 25 Schematic representation of the ESR lines calculated from simultaneous hyperfine interactions of a paramagnetic electron with C and Si nuclei. Terms in parentheses indicate the interaction types exhibiting each ESR fine. (From Ref. 124.)...

The ESR spectra of a large variety of sulfonyl radicals have been obtained photolytically in liquid phase over a wide range of temperature. Some selected data are summarized in Table 2. The magnitudes of hyperfine splittings and the observations of line broadening resulting from restricted rotation about the C—S bond have been used successfully in conjunction with INDO SCF MO calculations to elucidate both structure and conformational properties. Thus the spin distribution in these species is typical of (T-radicals with a pyramidal center at sulfur and in accord with the solid-state ESR data. 

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