Electron nuclear distance

The physical interpretation of the anisotropic principal values is based on the classical magnetic dipole interaction between the electron and nuclear spin angular momenta, and depends on the electron-nuclear distance, rn. Assuming that both spins can be described as point dipoles, the interaction energy is given by Equation (8), where 6 is the angle between the external magnetic field and the direction of rn. 

In these equations S is the total electron spin of the paramagnetic ion, r is the electron-nuclear distance, co is the electron resonance frequency, and x i - and x 2 are the rate constants for the reorientation of the coupled magnetic moment vectors. They are related to other rate constants by the expressions ... 

Tie and Tae are the electron spin relaxation times and % is the rate constant for proton exchange. Thus, if this mechanism is dominant the observed linewidths, Aui[ = (71X2) ], must reflect the r dependence on the electron-nuclear distance. Hyperfine exchange relaxation, however, is given by the expressions ... 

It can be shown (Veseth, 1970) that all electron-nuclear distances, r ) can be referred to a common origin, and, neglecting only the contribution of spin-other-orbit interactions between unpaired electrons, the two-electron part of the spin-orbit Hamiltonian can be incorporated into the first one-electron part as a screening effect. The spin-orbit Hamiltonian of Eq. (3.4.2) can then be written as... 

This equation defines the spin-orbit coupling constant (r) as a function of the electron-nuclear distance, for which the approximate expression is... 

For a vanishing electron-nuclear distance we would expect an appearance of the Fermi contact term. However, due to the presence of k0, such a term disappears... 

Table 1 Computed energies of Li ground state (non-relativistic, Coulomb interaction only, infinite-mass nucleus) for various wavefunctions, in Hartree atomic units. This research is for a correlated exponential premultiplied by the electron-nuclear distance for the 2s electron, with the parameters given in Table 33...

The wavefunction described with the optimum a,y corresponds well with a somewhat perturbed lslj 2s configuration. Two of the electrons depend on the electron-nuclear distance with a values (screening parameters) that correspond to a partially screened interaction with the - -3-charged Li nucleus in a split-shell electron distribution. The third electron (that with pre-multiplying r) has an a value somewhat larger than for a hydrogenic 2s orbital, indicative of the fact that the inner-shell electrons do not completely shield the Li nucleus. The electron-electron a values all reflect the existence of electron-electron repulsion, with the effect most pronounced for the Is-ls interaction. All these observations are consistent with the notion that the exponentially correlated wavefunction gives an excellent zero-order description of the electronic structure of Li. 

The direct relation between the anisotropic part of the hyperfine coupling and the electron-nuclear distance is spoiled when there is spin density in orbitals other than 5 orbitals on this nucleus. Except for protons, alkali, and earth alkali ions, this is usually expected. In this situation, a quantum-chemical computation is mandatory before any attempt can be made to relate the hyperfine splitting to a distance. 

Purely anisotropic contributions + Ay + A = 0) to the hyperfine coupling result from spin density in p, d, or / orbitals on the nucleus and from the dipole-dipole interaction T between the electron and nuclear spin. If the electron spin is confined to a region that is much smaller than the electron-nuclear distance r n, both spins can be treated as point dipoles and the magnitude of T is proportional to In this case, T has axial symmetry and its principal values are given by T =Ty= T and = IT. Furthermore, if the spin density in p, d, and/orbitals on... 

The substantial increase in the donor strength is a reflection of the sharply decreased coulombic barrier (co) for electron detachment from an anion (co 0) compared with that for a neutral species (o) = e / re, where re represents the effective electron-nuclear distance). Anions therefore are more likely to participate in electron-transfer and EDA complex formation than the corresponding neutral (or cationic ) complexes. 

Often fliso is used to estimate the s-orbital spin population on the corresponding nucleus [9] (see, e.g., [10]). In Eq. (2) matrix T describes the anisotropic dipole-dipole coupling. The dominant contribution to dd, for nuclei other than protons, usually comes from the interaction of an electron spin in a p-, d-, or f-type orbital with the magnetic moment of the corresponding nucleus. By reference to suitable tables the dipole-dipole coupling can also be used to estimate the spin population in these orbitals [10]. For distances between the electron and nuclear spin greater than approximately 0.25 nm, the anisotropic part of the hyperfine interaction can be used to calculate the electron-nuclear distance and orientation with the electron-nuclear point-dipole formula ... 

Hyperfine Splitting. The isotropic value of the hfs depends on electron spin density at the nucleus in question. Since p orbitals have a node at the nucleus, it shows how much spin density resides in the atom s s orbitals. For the 2s orbital of carbon the proportionality constant is 1110 gauss/electron for the Is orbital of hydrogen, 507. ( ) The magnitude of the hfs amisotropy depends on how much the spin density in the neighborhood of the nucleus departs from spherical, and how much from axial symmetry. Since the through-space interaction is attenuated by the cube of the electron-nuclear distance, it samples primarily the immediate vicinity of the nucleus. For C-13 the anisotropic hfs commonly has an axis of symmetry and may be interpreted in terms of spin density in the 2p orbitals of the... 

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