Distribution electric field gradients

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. 

Figure 4.54 The effect of an electric field gradient (EFG) creating asymmetry in the electron distribution round a gold nucleus, leading to a quadrupole splitting in the Mossbauer spectrum. (Reproduced with permission from Gold Bull., 1982,15, 53, published by World Gold Council.)...

Fig. 4.8. (Below) A diagram of the bilipid layer membrane of a vesicle or a cell with (above) a typical lipid, phosphatidylcholine. Large molecules and ions cannot penetrate the membrane as illustrated by the ions surrounding and inside a cell, but the distribution is reversed in vesicles (see Chapter 7). The ions create chemical and electrical field gradients across the membrane.

The strength of the quadrupolar interaction is proportional to the quadrupole moment Q of a nucleus and the electric field gradient (EFG) [21-23]. The size of Q depends on the effective shape of the ellipsoid of nuclear charge distribution, and a non-zero value indicates that it is not spherically symmetric (Fig. 1). 

Fig. 1 (a) Schematic representation of the spherical and non-spherical charge distribution in a nucleus. The value of electric quadrupole moment Q for the quadrupolar nucleus depends on the isotope under consideration, (b) The quadrupolar interaction arises from the interaction of Q with surrounding electric field gradient (EFG)... 

The origin of the electric field gradient is twofold it is caused by asymmetrically distributed electrons in incompletely filled shells of the atom itself and by charges on neighboring ions. The distinction is not always clear, because the lattice symmetry determines the direction of the bonding orbitals in which the valence electrons reside. If the symmetry of the electrons is cubic, the electric field gradient vanishes. We look at two examples. 

In Formulas (19) and (20) the actual value of the quadrupole moment Q that occurs is not that of the bare nucleus, but the bare nucleus value multiplied by a parameter 1 — This is necessary because the electron distribution of the atom is distorted when the atom is in an electric field gradient, due to the interaction of the electrons with the field gradient. This distortion produces an additional gradient which is —7 times the... 

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

Peripheral contributions become important when short interatomic distances are involved, as, for example, for the EFG at nitrogen nuclei and especially at nuclei of hydrogen atoms. Since hydrogen has only one electron, the electric field gradient is mainly due to the density farther from the nucleus, and has therefore been described as less sensitive to the precise charge distribution (Tegenfeldt and Hermansson 1985). 

K. Schwarz, C. Ambrosch-Draxl and P. Blaha, Charge distribution and electric-field gradients in YBa2Cu307 j . Phys. Rev. B, 1990,42,2051-2061. 

Although 1 is one of the best investigated molecules, there is, apart from data concerning its electron density distribution, very little information available on its one-electron properties. In principle, accurate data could be obtained by correlation-corrected ab initio methods, but almost nothing has been done in this direction, which of course has to do with the fact that experimental data on one-electron properties of 1 are also rare, and therefore, it is difficult to assess the accuracy and usefulness of calculated one-electron properties such as higher multipole moments, electric field gradients, etc. 

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... 

The symmetry of the electron distribution about the nucleus, as reflected in the quadrupole splitting, can be divided into two parts (32, 53,54). First, the atomic electrons about the central nucleus may fill orbitals in such a manner that the resulting electron cloud produces an electric field gradient at the nucleus. Second, the electric charges external to the central atom from the neighboring atoms, ligands, or ions must also be considered in a calculation of the field gradient at the central atom nucleus. These two effects are expressed as ... 

Consider first the effect of the atomic electrons. A filled or half-filled electron shell has a spherically symmetric electron distribution, and as such gives rise to no electric field gradient (except through external deformation, i.e., Sternheimer antishielding). Thus, of all the atomic electrons, only the... 

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