Operator spin-dipolar

There are two contributions - one from the Fermi contact and one from the spin-dipolar operator. The latter dipolar contribution is anisotropic and is averaged to... 

Hyperfine tensors are given in parts B and C of Table II. Although only the total hyperfine interaction is determined directly from the procedure outlined above, we have found it useful to decompose the total into parts in the following approximate fashion a Fermi term is defined as the contribution from -orbitals (which is equivalent to the usual Fermi operator as c -> < ) a spin-dipolar contribution is estimated as in non-relativistic theory from the computed expectation value of 3(S r)(I r)/r and the remainder is ascribed to the "spin-orbit" contribution, i.e. to that arising from unquenched orbital angular momentum. 

The dipolar spin-spin coupling operators are scalar operators of the form //1 . A 2 y.11. The tensorial structure of JCss becomes apparent if we write the Breit-Pauli spin-spin coupling operator as... 

We have seen that the dependence of Bn(R) on the vibrational coordinate causes a mixing of the vibrational level of interest with neighbouring levels. This mixing results in centrifugal distortion corrections to all the various parameters Xn(R) in the perturbation Hamiltonian 3C when combined in a cross term. The operator has the same form as in the original term, for example, (2/3) /6T 0(S, S) for the spin spin dipolar term, multiplied by N2. The coefficient which qualifies this term has the general form... 

Rumer basis, spin functions, 199 Spin-Dipolar (SD) operator, 251 method, 322 Wave function stability, 76... 

V operates solely on the inside of the parentheses. The nonrelativistic limit of this operator corresponds to the sum of the Fermi-contact (FC) and spin-dipolar (SD) terms of Ramsey s theory. Second-order differentiation of eq. (4.12) with respect to ptAj and psk gives... 

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

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... 

As mentioned above there are four main contributions to the nuclear spin-spin coupling constants the Fermi contact (FC), the paramagnetic spin-orbit (PSO), the spin-dipolar (SD) and the diamagnetic spin-orbit (DSO) contributions. The Fermi contact term is usually the most important of these and also the most sensitive to geometry changes [8]. The Fermi contact contribution arises from the interactions between the terms containing S(riM) and < (riN) in the operators Hon for nuclei N and M (see Eqn. (12)). 

The last term is the SD (spin-dipolar) contribution which arises from the interaction between the second term in the operators H011 for nuclei TV and M (see Eqn. (12)). In the case of the nuclear magnetic moments in the... 

The spin-spin dipolar interaction operator has the form expected for the interaction of two magnetic dipoles described by the operators (e / m)s and separated by r -. 

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... 

The situation for carbon-nitrogen couplings is complicated (636). Concerning one-bond carbon-nitrogen couplings it has been suggested that the Fermi contact term is dominant for all CN systems for which isosteric CC systems exist (156). In particular, in most CN systems with formal lone-pairs apart from the Fermi contact term the orbital and spin dipolar terms are operative and non-negligible (636,156). 

The Fermi contact, spin dipolar and their cross-term contain operators that include the electron spin operator S. Application of these operators on a singlet reference state 1 0° ) will give a linear combination of triplet states. The states thus... 

If one starts from a formally nonrelativistic Hamiltonian, third-order perturbation theory has to be used, as the spin-orbit operator has to be included in addition to the perturbations due to the nuclear magnetic moments and to the external magnetic field. As the spin-orbit operator permits spin polarization, a Fermi contact (FC) term and a spin-dipolar (SD) term also appear in the perturbed Hamiltonian and couple nuclear magnetic moment with electronic spin. 

Spin-adapted Configurations (SAC), 103 Spin-coupled Valence Bond (SCVB), 197 Spin-Dipolar (SD) operator, 251 Spin-orbit interaction, 209, 211 Spin-other-orbit interaction, 211 Spin-spin coupling, 251 Spin-spin interaction, 211 Spinorbital, 58 Spinors, 213... 

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