Electric field gradient components

The distribution of bonding, or valence, electrons is the largest contributor to the electric field gradient. In a molecular frame of reference, we can define three orthogonal components of the electric field gradient, Vxx, Vyy, and Vzz, where Vxx + Vyy + Vzz = 0 and, by convention, VXX < V l < VZZ. From these electric field gradient components, we define two parameters ... 

The electric field gradient component produced along Oz by a pz electron is eqo and - eq0 along Ox and Oy. So, with the proposed orbitals and populations, the nitrogen atom quadrupole coupling constant in aC=N group is... 

The electric field gradient is again a tensor interaction that, in its principal axis system (PAS), is described by the tluee components F Kand V, where indicates that the axes are not necessarily coincident with the laboratory axes defined by the magnetic field. Although the tensor is completely defined by these components it is conventional to recast these into the electric field gradient eq = the largest component,... 

Here, I, I, and I are angular momentum operators, Q is the quadrupole moment of the nucleus, the z component, and r the asymmetry parameter of the electric field gradient (efg) tensor. We wish to construct the Hamiltonian for a nucleus if the efg jumps at random between HS and LS states. For this purpose, a random function of time / (f) is introduced which can assume only the two possible values +1. For convenience of presentation we assume equal... 

The Rh-Ir vector defines the z-axis, and the NN bridges define the x- and y-axes. The N and P atoms are solely electron donors, but the CO ligand involves a lot of re-back-bonding. Thus the quadrupole matrix, which has large components if there is an electric field gradient at the Ir nucleus, is rotated 45° compared with the g-matrix principal axes. 

The relativistic coupled cluster method starts from the four-component solutions of the Drrac-Fock or Dirac-Fock-Breit equations, and correlates them by the coupled-cluster approach. The Fock-space coupled-cluster method yields atomic transition energies in good agreement (usually better than 0.1 eV) with known experimental values. This is demonstrated here by the electron affinities of group-13 atoms. Properties of superheavy atoms which are not known experimentally can be predicted. Here we show that the rare gas eka-radon (element 118) will have a positive electron affinity. One-, two-, and four-components methods are described and applied to several states of CdH and its ions. Methods for calculating properties other than energy are discussed, and the electric field gradients of Cl, Br, and I, required to extract nuclear quadrupoles from experimental data, are calculated. 

The electric field gradient at an ion in an ionic compound is then given as the sum of contributions from individual components surrounding this ion (equation 20),... 

Including the spherical component, the electric field gradient can be interpreted as the second-moment tensor of the distribution p(r)/ r — r 5. 

Both the components of E and the elements of the electric field gradient as given by Eqs. (8.30) and (8.32) are with respect to the reciprocal-lattice coordinate system. A transformation is required if the values in the direct-space coordinate systems are needed. To obtain the elements of the traceless V tensor, the quantity — (47t/3)pe(r) = — (47r/3K) F(H) exp ( — 27tiHT) must be subtracted from each of the diagonal elements VEU. 

The operators for the potential, the electric field, and the electric field gradient have the same symmetry, respectively, as those for the atomic charge, the dipole moment, and the quadrupole moment discussed in chapter 7. In analogy with the moments, only the spherical components on the density give a central contribution to the electrostatic potential, while the dipolar components are the sole central contributors to the electric field, and only quadrupolar components contribute to the electric field gradient in its traceless definition. 

Due to the electric quadrupole interaction, the Mi = 1/2 and Mi = 3/2 components of the 7 = 3/2 state of 57Fe split up, giving rise to the quadrupole splitting. Derived from the interaction of the nuclear quadrupole moment with the electric field gradient at the iron nuclei, AEq provides information about the asymmetry of the electron density around the iron nucleus. The electric field gradient at the iron nucleus can be calculated to obtain AEq (97). Since both 6 and AEq are related to the electron density at the nucleus, basis sets with an enlarged flexibility at the core region... 

The three components of the electric field gradient tensor are related by Poisson s equation, as shown earlier. However, the electrons that have a finite probability density at the nucleus, the s and p1/2 electrons, have a spherically symmetric distribution around the nucleus and as such do not contribute to E2. Thus, in the computation of E2, the Un can be related by... 

Here 0dm is the angle between the director and the largest component, eq, of the electric field gradient tensor, and rj is the asymmetry parameter. The different coordinate systems used are shown in Figure 1. 

Classically, the electric field gradient at a nucleus is produced by the arrangement of charges (i.e., other nuclei and electrons) about that nucleus 66). If the nucleus is quad-rupolar, as in the case of 27A1, then the interaction of its nuclear quadrupole moment, eQ, with the largest component of the EFG tensor, V33, is defined as the quadrupolar coupling constant, CQ ... 

S nuclear quadrupole coupling constants have been determined from line width values in some 3- and 4-substituted sodium benzenesulphonates33 63 and in 2-substituted sodium ethanesulphonates.35 Reasonably, in sulphonates R — SO3, (i) t] is near zero due to the tetrahedral symmetry of the electronic distribution at the 33S nucleus, and (ii) qzz is the component of the electric field gradient along the C-S axis. In the benzenesulphonate anion, the correlation time has been obtained from 13C spin-lattice relaxation time and NOE measurements. In substituted benzenesulphonates, it has been obtained by the Debye-Stokes-Einstein relationship, corrected by an empirically determined microviscosity factor. In 2-substituted ethanesulphonates, the molecular correlation time of the sphere having a volume equal to the molecular volume has been considered. 

In the case of sulpholane,76 it has been found by ab initio calculations that the principal component of the electric field gradient at the 33S nucleus is perpendicular to the molecular plane containing the five ring moiety the intermediate component lies along the C2 axis. 

A membrane is a barrier that allows selective mass transport between two phases. It is selective since various components are able to pass through the membrane more efficiently than others. This makes membranes an appropriate means to separate a mixture of components. That is, a membrane is a permselective barrier between two phases that can be permeated owing to a driving force, such as pressure, concentration, or electric field gradient [18,19], The phases on either side of the membrane can be liquid or gaseous. 

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