Electron-Nucleus Contact Term

We only briefly mention that a similar modification, i.e., a change from a Dirac delta distribution to an extended distribution, would be required for the spin-dependent electron-nucleus contact term, known as Fermi contact term, if the usual point-like nuclear magnetization distribution (the pointlike nuclear magnetic dipole approximation) is replaced by an extended nuclear magnetization distribution. 

In this last section we mention a few cases, where properties other than the energy of a system are considered, which are influenced in particular by the change from the point-like nucleus case (PNC) to the finite nucleus case (FNC) for the nuclear model. Firstly, we consider the electron-nuclear contact term (Darwin term), and turn then to higher quantum electrodynamic effects. In both cases the nuclear charge density distribution p r) is involved. The next item, parity non-conservation due to neutral weak interaction between electrons and nuclei, involves the nuclear proton and neutron density distributions, i.e., the particle density ditributions n r) and n (r). Finally, higher nuclear electric multipole moments, which involve the charge density distribution p r) again, are mentioned briefly. 

The electron-nucleus interaction, or Fermi contact term, arises from ... 

In Eq. (15), 8(rik) is the Dirac delta function which, when integrated with the wave function, gives the value of the wave function at rik = 0. The two terms in Eq. (15) are in reality two limiting forms of the same interaction. The first term is the ordinary dipole-dipole interaction for two dipoles that are not too close to each other. It is the proper form of M S1 to be applied to p, d, and / electrons which are not found near the nucleus. For s electrons, which have a finite probability of being at the nucleus, the first term is clearly inappropriate, since it gives zero contribution at large values of rik and does not hold for small values of rik. From Dirac s relativistic theory of the electron, it is found (4) that the second term in Eq. (15) is the correct form for Si when the electron is close to the nucleus. Thus the contribution toJT S] from s electrons is proportional to the wave function squared at the site of the nucleus and the second term in Eq. (15) is often called the contact term in the hyperfine interaction. 

This term is called the Fermi contact term. That part of the electron-nucleus magnetic-dipole interaction represented by (8.104) depends on the angular coordinates of the electron and is therefore anisotropic in contrast, the Fermi contact energy (8.108) is isotropic. The contact term plays an important role in the electron-coupled nuclear spin-spin interactions seen in the NMR spectra of liquids. 

The pseudocontact term Afipo,ar is due to the valence electrons of the shift reagent SR they cause an additional intramolecular field in the adduct S + SR, shielding or deshielding the nucleus i in the molecule S. The Fermi contact term Afon ac accounts for interaction between the nucleus i and the field of the unpaired electrons of the paramagnetic additive SR which may be delocalized within the adduct S + SR [103]. 

No explicit temperature dependence is included in the equations for R m and Rim, except for cases where Curie spin relaxation is the dominant term (Section 3.6). In the latter case, Curie paramagnetism has a T x dependence and therefore relaxation depends on T 2. The effect of temperature on linewidths determined by Curie relaxation is dramatic also because of the xr dependence on temperature, as shown in Eq. (3.8). All the correlation times modulating the electron-nucleus coupling, either contact or dipolar, are generally temperature dependent, although in different ways, and their variation will therefore be reflected in the values of Rim and Rim-... 

This hyperfine coupling is of two kinds An isotropic interaction arises from the possibility that the electronic wave-function, x , be non-zero at the nucleus, N. This is the Fermi contact term and the hyperfine coupling constant is given by ... 

Hi) In circular atoms, the Rydberg electron remains always very far from the nucleus. Hence, all the contact terms, which become significant corrections at the 10-AO level in the optical experiments and which depend upon the not-so-well known proton form factor, are in circular states completely negligible. Lamb-shift corrections are also very small for these states. From the point of view of Q. E. D. corrections, circular atoms are, by far, the best candidate for R metrology. 

In these expressions the index i runs over electrons and a runs over nuclei. The Fermi contact term describes the magnetic interaction between the electron spin and nuclear spin magnetic moments when there is electron spin density at the nucleus. This condition is imposed by the presence of the Dirac delta function S(rai) in the expression. The dipole-dipole coupling term describes the classical interaction between the magnetic dipole moments associated with the electron and nuclear spins. It depends on the relative orientations of the two moments described in equation (7.145) and falls off as the inverse cube of the separations of the two dipoles. The cartesian form of the dipole-dipole interaction to some extent masks the simplicity of this term. Using the results of spherical tensor algebra from the previous chapter, we can bring this into the open as... 

The expression for the anisotropic part of hyperfine coupling involves an integral over the spatial distribution of the unpaired electron, which is relatively easy to compute accurately even at a relatively low level of theory. The contact term, however, includes a delta-function that chips out the wave function amplitude at the nucleus point. The latter is quite difficult to compute both because standard Gaussian basis sets do not reproduce the wavefunction cusp at the nucleus point and because additional flexibility has to be introduced into the core part of the basis to account for the now essential core valence interaction. " ... 

For some other molecular properties the wave function must be accurate in the proximity of nuclei. Such a property is for example the hyperfine splitting caused by Fermi contact term which is proportional to the total spin density at the nucleus. Among the one-electron pro-... 

The occurrence of hyperfine coupling for a particular isotope indicates non-zero spin density at that nucleus in accordance with the Fermi contact term (which includes the electron wave function evaluated at the nucleus) [1]. The analysis in terms of spin density then requires at least a comparison with the (calculated [5]) isotropic hyperfine splitting constant ao, ideally, results from increasingly available open-shell quantum-chemical calculation procedures [64] are employed. 

Scalar couplings to metal nuclei are dominated by the Fermi contact term, and approximation on a similar level as for the chemical shifts leads to the expression in Eq. (5), with A being the mean triplet excitation energy, S(0) x the s-electron density at the nucleus X, and ttml the mutual polarizability of the orbital connecting the metal and the ligating atom L. ... 

A serious drawback to this model is its inability to account for observed Fermi contact interactions. The usual Slater determinant uses the same molecular orbital for spin up as for spin down and will therefore yield a Fermi contact contribution only if the orbital with the unpaired electron contains the s orbital of the atom concerned. Symmetry restrictions, however, prevent the presence of s orbitals in the molecular orbital for aromatic free radicals and many transition metal complexes nevertheless, the large isotropic terms observed in these systems require an extensive contribution from the Fermi contact term. This is explained by assuming the unpaired electrons polarize the inner-filled orbitals having s character to produce a small net unpairing of spin. Very small polarizations will produce large Fermi terms due to the large density of s orbitals in the vicinity of the nucleus. Theoretically, this problem is handled in... 

The theoretical contact term per electron y for some ions has been calculated (219, 220, 331). The theoretical contact term resulting from polarization of s electrons by all unpaired d electrons is Hg and equals 2S. The field felt by the electron due to this field at the nucleus, Also, and the field at the nucleus are related as follows... 

---