Fermi-contact

The Fermi contact density is defined as the electron density at the nucleus of an atom. This is important due to its relationship to analysis methods dependent... 

Crystal can compute a number of properties, such as Mulliken population analysis, electron density, multipoles. X-ray structure factors, electrostatic potential, band structures, Fermi contact densities, hyperfine tensors, DOS, electron momentum distribution, and Compton profiles. 

Gaussian computes isotropic hyperfine coupling constants as part of the population analysis, given in the section labeled "Fermi contact analysis the values are in atomic-units. It is necessary to convert these values to other units in order to compare with experiment we will be converting from atomic units to MHz, using the following expressions ri6ltYg ... 

The spin Hamiltonian operates only on spin wavefunctions, and all details of the electronic wavefunction are absorbed into the coupling constant a. If we treat the Fermi contact term as a perturbation on the wavefunction theR use of standard perturbation theory gives a first-order energy... 

The only term surviving the Bom-Oppenheimer approximation is the direct spin-spin coupling, as all the others involve nuclear masses. Furthermore, there is no Fermi-contact term since nuclei cannot occupy the same position. Note that the direct spin-spin coupling is independent of the electronic wave function, it depends only on the molecular geometry. 

The and operators determine the isotropic and anisotropic parts of the hyperfine coupling constant (eq. (10.11)), respectively. The latter contribution averages out for rapidly tumbling molecules (solution or gas phase), and the (isotropic) hyperfine coupling constant is therefore determined by the Fermi-Contact contribution, i.e. the electron density at the nucleus. 

A completely different type of property is for example spin-spin coupling constants, which contain interactions of electronic and nuclear spins. One of the operators is a delta function (Fermi-Contact, eq. (10.78)), which measures the quality of the wave function at a single point, the nuclear position. Since Gaussian functions have an incorrect behaviour at the nucleus (zero derivative compared with the cusp displayed by an exponential function), this requires addition of a number of very tight functions (large exponents) in order to predict coupling constants accurately. ... 

The complete Hamiltonian of the molecular system can be wrihen as H +H or H =H +H for the commutator being linear, where is the Hamiltonian corresponding to the spin contribution(s) such as, Fermi contact term, dipolar term, spin-orbit coupling, etc. (5). As a result, H ° would correspond to the spin free part of the Hamiltonian, which is usually employed in the electron propagator implementation. Accordingly, the k -th pole associated with the complete Hamiltonian H is , so that El is the A -th pole of the electron propagator for the spin free Hamiltonian H . 

With these assignments at hand the analysis of the hyperfine shifts became possible. An Fe(III) in tetrahedral structures of iron-sulfur proteins has a high-spin electronic structure, with negligible magnetic anisotropy. The hyperfine shifts of the protons influenced by the Fe(III) are essentially Fermi contact in origin 21, 22). An Fe(II), on the other hand, has four unpaired electrons and there may be some magnetic anisotropy, giving rise to pseudo-contact shifts. In addition, there is a quintet state at a few hundred cm which may complicate the analysis of hyperfine shifts, but the main contribution to hyperfine shifts is still from the contact shifts 21, 22). 

The isotropic Fermi contact field B, which arises from a net spin-up or spin-down -electron density at the nucleus as a consequence of spin-polarization of -electrons by unpaired valence electrons [63] ... 

It is well-known that the hyperfine interaction for a given nucleus A consists of three contributions (a) the isotropic Fermi contact term, (b) the spin-dipolar interaction, and (c) the spin-orbit correction. One finds for the three parts of the magnetic hyperfine coupling (HFC), the following expressions [3, 9] ... 

The concept of spin-polarization has been found to be extremely useful for understanding the magnetic HFCs of organic radicals which are dominated by the Fermi contact contribution. The situation for transition metal complexes is rather different in several respects. The idea of spin-polarization is relatively simple and is best... 

In contrast to the EFG analyzed before, aU of these expectations from ligand field theory are largely confirmed by the DFT calculations. Despite the fact that the S = 2 state has a smaller prefactor for the isotropic Fermi contact term, the core polarization in the presence of four unpaired electrons is much larger and, consequently, the isotropic Fe-HFC is predicted to be roughly a factor of two larger in magnitude for as compared to 2g. Similarly, the dipolar MFCs are comparable for both spin states, which must be due to a considerable contribution from anisotropic covalency in the 5 = 2 species which partially compensates for the smaller prefactor. 

Evaluation of trends in /pp coupling constants in solid-state 31P NMR spectra of P-phospholyl-NHPs allowed one to establish an inverse relation between the magnitude ofM and P-P bond distances [45], The distance dependence of. /pp is in line with the dominance of the Fermi contact contribution, and is presumably also of importance for other diphosphine derivatives. At the same time, large deviations between lJvv in solid-state and solution spectra of individual compounds and a temperature dependence of lJ77 in solution were also detected (Fig. 1) both effects... 

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,... 

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