Fermi contact expression

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 ... 

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] ... 

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. 

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... 

It is instructive to consider in quantitative detail the analysis of a particular hy-perfine transition a simple example would seem to be the F = 2 <+ 1 transition in the level It = 1, N = 0, J = 1, which is observed at a frequency of 20.846 MHz for the v = 0 level. The expressions for the matrix elements of the magnetic and electric hy-perfine terms, (8.258), (8.259) and (8.270), show that for N =0 only the Fermi contact interaction is non-zero and the energies of the hyperfine levels are... 

Freund, Herbst, Mariella and Klemperer [112] expressed their magnetic hyperfine constants in the form originally given by Frosch and Foley [117]. As discussed elsewhere in this book, particularly in chapters 9, 10 and 11, we prefer to separate the different physical interactions, particularly the Fermi contact and dipolar interactions, in our effective Hamiltonian. This separation is usually made by other authors even when the effective Hamiltonian is expressed in terms of Frosch and Foley constants, because it is the natural route if the molecular physics of a problem is to be understood. Nevertheless since so many authors, particularly of the earlier papers, use the magnetic hyperfine theory presented by Frosch and Foley, we present in appendix 8.5 a detailed comparison of their effective Hamiltonian with that adopted in this book. The merit of the Frosch and Foley parameters is that they form the linear combination of parameters which is best determined (i.e. with least correlation) for a molecule which conforms to Hund s case (a) coupling. The values of the constants determined experimentally from the 7 LiO spectrum were therefore, in our notation (in MHz) ... 

The contact interaction is also referred to as the Fermi contact term. In a given atomic orbital basis -0, the isotropic hyperfine coupling constant (hfcc) for a particular nucleus N,, is given by the expression,... 

Observed linewidths of NMR signals in paramagnetic systems vary enormously and the conditions that govern the observed widths are considerably more complex than in diamagnetic systems. Swift (30) reviewed the problem some years ago. Relaxation times of spin-j nuclei are governed by dipolar and hyperfine exchange (Fermi contact) relaxation processes. The dipolar interaction is normally dominant except in some delocalized systems in which considerable unpaired spin density exists on nuclei far removed from the metal ions (e.g. Ti-radicals). Distinction between the two processes can be made by consideration of the different mathematical expressions involved. For dipolar relaxation when o)fx 1 (t = rate constant for rotation of the species containing the coupled pair and to, = nuclear resonance frequency) ... 

Such expressions are based on the dominance of the Fermi contact mechanism. (38) Negative values of /(C-C) are not possible on the basis of equation (7). 

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. ... 

Within a nonrelativistic calculation of the hyperfine fields in cubic solids, one gets only contributions from s electrons via the Fermi contact interaction. Accounting for the spin-orbit coupling, however, leads to contributions from non-s elections as well. On the basis of the results for the orbital magnetic moments we may expect that these are primarily due to the orbital hyperfine interaction. Nevertheless, there might be a contribution via the spin-dipolar interaction as well. A most detailed investigation of this issue is achieved by using the proper relativistic expressions for the Fermi-contact (F), spin-dipolar (dip) and orbital (oib) hyperfine interaction operators (Battocletti... 

Coupling constants are usually analysed at present in terms of a MO theory developed by Pople and Santry. They showed that coupling constants involving directly bonded atoms arise almost entirely from the Fermi contact interaction between nuclear moments and electron spins in 5 orbitals. Using the LCAO approximation and retaining only one-centre integrals, they derived the following expression for Ja-b—... 

Let us start with the field-free SO effects. Perturbation by SO coupling mixes some triplet character into the formally closed-shell ground-state wavefunction. Therefore, electronic spin has to be dealt with as a further degree of freedom. This leads to hyperfine interactions between electronic and nuclear spins, in a BP framework expressed as Fermi-contact (FC) and spin-dipolar (SD) terms (in other quasirelativistic frameworks, the hyperfine terms may be contained in a single operator, see e.g. [34,40,39]). Thus, in addition to the first-order and second-order ct at the nonrelativistic level (eqs. 5-7), third-order contributions to nuclear shielding (8) arise, that couple the one- and two-electron SO operators (9) and (10) to the FC and SD Hamiltonians (11) and (12), respectively. Throughout this article, we will follow the notation introduced in [58,61,62], where these spin-orbit shielding contributions were denoted... 

Further contributions beyond these, which were not previously considered in Ref. 20 and 21, expressions appearing here and also in Fukui et are the field-induced spin-orbit contributions (considered by Ref.22) and the term arising from the combination of mass-velocity, external field, Fermi contact operators (analysed by Ref. 9). In addition to these, there are the new terms derived in this work which have not appeared in the literature previously and remain to be treated quantitatively. 

There is also a delta function term in the expression for the magnetic field due to the nuclear spins. The cross terms between the delta function part and the dipolar part vanishes upon spherical averaging, and one is left with the delta function contribution to the nuclear spin-spin coupling constants. This term is called the Fermi contact contribution... 

The Fermi contact interaction is the major mechanism for J coupling. It is governed by the electron density at the nucleus (S,) and the nuclear gyromagnetic ratio according to the expression... 

The observed NMR shift, expressed as Am/Fermi contact hyperfine interaction in materials containing 3 /-transition metals is proportional to the unpaired electron spin density at the nucleus site. The magnitude of the interaction is directly proportional to the Fermi constant Ac and the time-averaged electron spin (5z) by... 

In interpreting the nmr shift, AB, obtained from the nmr spectra of paramagnetic molecules it is usual to consider two terms, the Fermi contact interaction where the contribution may be expressed(7) as... 

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