Operator Fermi contact

Many operators, like the Fermi-contact operator, changes both the spatial-and spin functions. 

Barone and coworkers250 also determined EPR hyperfine splittings nN of the radical 40 at the UMP2/DZ + P level of theory using the Fermi contact operator and a finite field method with an increment size of 0.001 a.u. Expectation values of aN, < aN >, at higher temperatures T were calculated by assuming a Boltzmann population of vibrational levels according to equation 23 ... 

We note from Figure 1 that scalar relativistic calculations (7) are entirely unable to reproduce the experimental trend for X = Br and I. Indeed, scalar relativistic and non-relativistic calculations gave almost identical results in this case (7). However, with the inclusion of spin-orbit/Fermi contact operators, the experimental trend is reproduced nicely (9). 

In the nonrelativistic limit, c —> oo, eq. (4.16c) yields the well-known Fermi-contact operator and the spin-dipolar interaction operator because... 

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. 

All these basis sets are essentially optimized for the calculation of electronic energies and are therefore able to represent the operators included in the field-free electronic Hamiltonian reasonably well. However, in the calculation of molecular electromagnetic properties it is necessary also to represent other operators such as the electric dipole operator, the electronic angular momentum operator, the Fermi-contact operator and more. Most of these basis sets are a priori not optimized for this and have to be extended. 

The Fermi contact operator acts both in the spin and in the spatial space of the electronic coordinates, often leading to a very different behavior of the response properties than for instance the operator. The Fermi contact operator represents the direct interaction... 

It is noted that the well-known nonrelativistic versions of the operators are obtained by formally letting /C 1 in equations (12.14b) to (12.14h). Afterwards, the derivatives of r /r] can be taken in equation (12.14f), which among other terms gives the Fermi contact operator in its usual form written in terms of 5 rN) distributions. In the ZORA form, the contact term is actually suppressed, which may be considered as a drastic form of picture change (PC) [21,23]. 

From the orbital density matrices considered in Section 2.7.1, we may calculate expectation values of singlet operators. For triplet operators such as the Fermi contact operator, a different set of density matrices is needed. Consider the evaluation of the expectation value for a one-electron triplet operator of the general form... 

One of the operators that describe the interaction between nuclear magnetic dipoles and the electrons is the Fermi contact operator... 

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 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. ... 

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. 

The singlet paramagnetic spin-orbital (PSO) operator (Equation (2.14)), the triplet Fermi contact (FC) operator... 

The first-order contribution of these hyperfine interactions to the effective electronic Hamiltonian involves the diagonal matrix elements of the individual operator terms over the electronic wave function, see equation (7.43). As before, we factorise out those terms which involve the electronic spin and spatial coordinates. For example, for the Fermi contact term we need to evaluate matrix elements of the type ... 

The inversion operation i which leads to the g/u classification of the electronic states is not a true symmetry operation because it does not commute with the Fermi contact hyperfine Hamiltonian. The operator i acts within the molecule-fixed axis system on electron orbital and vibrational coordinates only. It does not affect electron or nuclear spin coordinates and therefore cannot be used to classify the total wave function of the molecule. Since g and u are not exact labels, it was realised by Bunker and Moss [265] that electric dipole pure rotational transitions were possible in ll], the g/u symmetry breaking (and simultaneous ortho-para mixing) being relatively large for levels very close to the dissociation asymptote. The electric dipole transition moment for the 19,1 19,0 rotational transition in the ground electronic state was calculated... 

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