Finite nucleus quadrupole moment

Terms up to order 1/c are normally sufficient for explaining experimental data. There is one exception, however, namely the interaction of the nuclear quadrupole moment with the electric field gradient, which is of order 1/c. Although nuclei often are modelled as point charges in quantum chemistry, they do in fact have a finite size. The internal structure of the nucleus leads to a quadrupole moment for nuclei with spin larger than 1/2 (the dipole and octopole moments vanish by symmetry). As discussed in section 10.1.1, this leads to an interaction term which is the product of the quadrupole moment with the field gradient (F = VF) created by the electron distribution. 

One method of determining nuclear quadrupole moment Q is by measuring the quadrupole coupling constant, given by eqQ/h, where e is the charge of the electron and q the electric field gradient due to the electrons at the atomic nucleus. The extraction of Q depends on an accurately calculated q. As a test of our finite-field relativistic coupled cluster approach, preliminary results for Cl, Br, and I are presented. 

A number of calculations allow for internal consistency checks. For example, if either or B x.xx desired, one can use either total dipole moment or quadrupole moment expansions. These properties could not be calculated by uniform field computations alone. The finite-charge approach is easily carried out at all electronic structure levels (e.g., SCF, Cl, and MBPT) because it amounts to adding an extra nucleus, one with a negative or positive charge. It does not seem to be as widely employed as finite-field calculations. Examples include calculations on methane [77], Lij and ions [78], and HF [79]. 

A nuclear electric quadrupole moment arises from a non-spherical distribution of electrical charges in the nucleus. Because the energy of an electric quadrupole moment depends on its orientation with respect to an EFG, a randomly varying electric field gradient associated, for example, with molecular rotation having a finite amplitude of its spectral density at the Larmor speed can induce effective relaxation of the quad-rupolar nucleus. In liquids, it is possible to write down... 

The source of the electric field can be an externally applied field, or it can originate in the components of the nuclear potential that are not included in the internal component of the field (that is, the nuclear potential V). Such components arise from the nonspherical nature of the nucleus, the lowest-order term of which is the quadrupole moment. The implementation of a finite-nuclear model is quite straightforward we simply expand the nuclear charge distribution in a series ... 

Unlike the electric quadrupole moment, contributions to the magnetic interactions with the nucleus can come from (electron) spinors with j = 1/2, and thus the magnitude of the finite nucleus effect is likely to be larger than for the electric quadrupole moment. 

Atomic nuclei can be stretched like cigars (prolate shape) or compressed like discs (oblate shape). The deformation is described by the electric quadrii-pole moment Q (prolate Q > 0 oblate Q < 0). The principal interaction is, of course, the normal electrostatic (Coulomb) force on the charged nucleus monopole interaction). The differential interaction, which depends on the structure of the nucleus and on the valuation of the field across its finite extension, is of course very much smaller quadrupole interaction). It gives rise to an electric hyperfine structure. The energy contribution depends on the direction of the nuclear spin in relation to the electric field gradient. For the electric hyperfine interaction one obtains... 

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