Fermi contact density

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 

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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. 

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. 

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] ... 

This is a measure of the unpaired electron spin density that is transferred from the paramagnet to the nucleus of the spin under observation. In the regime defined by eq 2, the NMR shift (d = (Acolcoo)) induced by the Fermi-contact interaction is directly proportional to... 

In these expressions, e and N refer to electron and nucleus, respectively, Lg is the orbital angular moment operator, rg is the distance between the electron and nnclens. In and Sg are the corresponding spins, and reN) is the Dirac delta fnnction (eqnal to 1 at rgN = 0 and 0 otherwise). The other constants are well known in NMR. It is worth mentioning that eqs. 3.8 and 3.9 show the interaction of nnclear spins with orbital and dipole electron moments. It is important that they not reqnire the presence of electron density directly on the nuclei, in contrast to Fermi contact interaction, where it is necessary. 

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. 

The predictions of the proton shifts is difficult the prediction is even more difficult in the case of heteroatoms. In the case of cr spin density like in a aliphatic amine (Table 2.2A), the 13C Fermi contact shift for the a-carbon of the aliphatic amine is upfield (A/h is negative) because of predominant spin polarization effects, whereas that for other carbon atoms is downfield, and rapidly attenuates with the number of bonds. A sizable downfield shift is experienced by the 14N nucleus when nitrogen is a donor atom. 

For 19F Fermi contact shift of fluorine bound to sp2 carbon atoms, spin density on the nucleus arises from spin polarization by as for analogous CC moieties, and from spin polarization by /Op, which occurs via direct delocalization through C—F ji bonding (Fig. 2.18). The hyperfine coupling is therefore... 

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