Isotropic couplings

This coupling is referred to as the contact coupling since it appears when the unpaired electron has a finite probability to appear at the nucleus. This is expressed by the following equation, due to Fermi  

Here p(Xn) is the unpaired electron density at a nucleus located at X. The expression shows that the experimentally measurable hyperfine coupling constant a is directly proportional to the spin density p (Xn), related to the electronic structure of the radical. Some values for the free atoms are quoted in Table 1.1. 

Isotropic couplings occm for s-orbitals, while p-, d- and f-orbitals are characterised by a zero density at the nucleus (Fig. 1.9) and thus give no contribution to ao- The values of ao and Bo in the table are obtained when the unpaired electron is entirely localised to an s-orbital and a p-orbital, respectively. 

9 Angular dependence of (a) Is and (b) 2p orbitals. The probability to find the election within the contour is 0.9 

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Coupling between fluorine and a hydrogen, a carbon or another fluorine that may be separated by many bonds (four, five, six or more) can result from overlap of electronic orbitals occupied by lone pair electrons which are unshared and therefore not involved in normal covalent bonding. The term applied to this effect, through space is somewhat misleading, since all isotropic coupling must be transmitted in some way by electrons, either in bonds or in unshared pairs. 

Using different DFT functionals and basis sets (Focsan et al. 2008, Lawrence et al. 2008) it was confirmed that the isotropic ()-methyl proton hyperfine couplings do not exceed 9MHz for the carotenoid radical cation, Car-. DFT calculations of neutral carotenoid radicals, Car formed by proton loss (indicated by ) from the radical cation, predicted isotropic P-methyl proton couplings up to 16 MHz, a fact that explained the large isotropic couplings observed by ENDOR measurements for methyl protons in UV irradiated carotenoids supported on silica gel, Nafion films, silica-alumina matrices, or incorporated in molecular sieves (Piekara-Sady et al. 1991, 1995, Wu et al. 

The 59Co hyperfine matrix components must have identical signs in order that the average values match the observed isotropic couplings we assume the signs are negative since the isotropic couplings almost certainly arise from polarization of inner shell s orbitals (see below). 

Since 3dyz/4s admixture is symmetry-forbidden for these radicals, the Fermi contact contribution to the isotropic coupling, As, must be entirely from spin polarization,... 

From the isotropic coupling constant one may calculate C12, the fractional occupancy of the nitrogen s orbital, which is equal to AUo/Ao. The term Ao is the Fermi contact interaction for an unpaired electron in a pure nitrogen 2s orbital. For NO2 the value of ci2 = 56.5/550 = 0.103. The fraction of the unpaired electron associated with the N nucleus is then... 

The first such experiment (Fig. 21a), introduced independently by Gan [258] and Bodenhausen et al. [259], used 13C as the so-called spy nuclei in a 2D experiment similar to the HMQC scheme used in solution NMR [260], The transfer of magnetization occurred by a combination of weak ]JCN (through-bond) couplings and second-order quadrupolar-dipolar (through-space) cross-terms between 14N and 13C, which contain the isotropic and anisotropic terms l = 0, 2, and 4 and are referred to as residual dipolar splittings (RDS) [261-263], Under MAS, the 1 = 0 term results in isotropic coupling, which for NH3 groups is expected to be... 

The spin densities deduced from the formula aftft, = ( A1 + 2 A )/3 are presumed to be positive because of direct delocalization of the unpaired electron from the (a3da + b4s) hybrid orbital. It should be noted, however, that Aft1 and A , measured for rotating rings at T = 150 K, are not principal values of the proton hfs tensor, i.e. a , determined in this way does not exactly correspond to the isotropic coupling obtained by the common formula aftft, = (A" + A" + A )/3. 

Spin-Spin Coupling.—The relative roles of different coupling mechanisms in organophosphorus compounds have been studied theoretically.66 The spin-spin interaction constants for methylphosphine have been calculated by the MO-LCAO method.67 Unresolved isotropic coupling to boron caused broadening of the vinyl... 

One of the most important, pioneering works for calculating spin-spin coupling constants within the DFT framework was published by Malkin, Malkina et al.66 69 Their approach is frequently employed in the current literature to calculate coupling constants in medium sized molecules obtaining a reasonable accuracy.70 73 Malkin s method takes into account only three of the four terms that contribute to the isotropic couplings, namely the FC, the PSO and DSO contributions, while the SD contribution is neglected. 

Experimentally in an isotropic phase and without any partial orientation of the solute molecules, it is only possible to observe the average value or trace of the second rank J tensor. In the current literature this quantity is commonly referred to as the scalar coupling , which is a somewhat misleading expression. Obviously, it would be more correct to call it the isotropic coupling as can be seen in several papers however to follow the common usage in this work these expressions will be used indistinctly. 

Hz in complexes of [Hg Hg]2+ with crown ethers. In solid state minium, Pb304, the Pb4+-Pb2+ isotropic coupling, J(Pb, Pb), of 2.3 0.1 kHz is reported.457 The existence of this coupling is suggested to be a result of relativistically enhanced 6s- 6p promotion that seems to take place in heavy elements, which results in decreased hybridization efficiency and increased stability of lower oxidation states. It seems that such effects are to be expected for other relativistically bound molecular ions like [Hg2]2+. 

The authors also reconsidered radical 20, which, in the current work, was formed by x-irradiation and studied at 77 K. The experimentally determined hyperfine coupling tensors for the two methylene protons have the same principal values and agreed with the tensors found in the earlier work.16 These tensors are presumed to be the average of the two individual proton tensors. By assuming that C5 is sp2 hybridized the authors were able to calculate, from the observed average tensors, that Ami = 1.06 mT and Amai = 3.45 mT for the methylene protons. Using the individual tensors for the methylene protons and a 3.6 mT isotropic coupling for the C4(H) proton, a satisfactory simulation of the... 

The EPR spectra have always been interpreted2994 using an effective S = 2 spin Hamiltonian including the Zeeman term, /iBB g-S, and the hyperfine term, ICa-A-S, which describes the interaction of the unpaired electrons with the copper nucleus (7Cu = I). The spectra are very sensitive to the ratio between the isotropic coupling constant J and the local zero field splitting of nickel(II), Z)Ni.2982 In the limit J DNi it can easily be shown that the following relations hold ... 

Each of the resonances which are separated by the chemical shifts may be further split into several lines by spin-spin interactions. These can be described by an isotropic coupling Hamiltonian of the type... 

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