The electric field gradient

The electric field gradient (EFG) is the tensor product of the gradient operator V = i d/dx + j d/dy + k 3/dz and the electric field vector E  

As in a Cartesian coordinate system the tensor product u v, of the vectors u and v, has the elements uxvp, the EFG tensor is a symmetric tensor with elements 

The EFG tensor elements can be obtained by differentiation of the operator in expression (8.5) for Ea to each of the three directions / . This procedure removes the spherical component, which does not affect the electric field, and yields the traceless result 

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

The definition of Eq. (8.8) and the result of Eq. (8.9) differ in that Eq. (8.8) does not correspond to a zero-trace tensor. The situation is analogous to the two definitions of the second moments, discussed in the preceding chapter. The trace of the tensor defined by Eq. (8.8) is given by 

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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,... 

Intramolecular quadrupolar 2 Reorientation of the electric field gradient principal axis Dominant for />1 (covalently bonded) [14]... 

Intermolecular quadrupolar 2 Fluctuation of the electric field gradient, moving multipoles Common for />1 In free Ions In solution [la... 

The PCM algorithm is as follows. First, the cavity siuface is determined from the van der Waals radii of the atoms. That fraction of each atom s van der Waals sphere which contributes to the cavity is then divided into a nmnber of small surface elements of calculable surface area. The simplest way to to this is to define a local polar coordinate frame at tlie centre of each atom s van der Waals sphere and to use fixed increments of AO and A(p to give rectangular surface elements (Figure 11.22). The surface can also be divided using tessellation methods [Paschual-Ahuir d al. 1987]. An initial value of the point charge for each surface element is then calculated from the electric field gradient due to the solute alone ... 

In Figure 6.4, the two electrodes are marked as cathode and anode, arising from the application of an external voltage between them. Before any discharge occurs, the electric-field gradient between the electrodes is uniform and is simply the applied voltage divided by the their separation distance, as shown in Figure 6.7. 

If the electrodes are moved closer together, the positive column begins to shorten as it moves through the Faraday dark space because the ions and electrons within it have a shorter distance through which to diffuse. Near the cathode, however, the electric-field gradient becomes steeper and electrons from the cathode are accelerated more quickly. Thus atom excitation through collision with electrons occurs nearer and nearer to the cathode, and the cathode glow moves down toward the electrode. 

The primary photochemical act, subsequent to near-uv light (wavelengths <400 nm) absorption by Ti02 particles, is generation of electron—hole pairs where the separation (eq. 3) into conduction band electrons (e g ) and valence band holes (/lyB ) faciUtated by the electric field gradient in the space charge region. Chemically, the hole associated with valence band levels is constrained at... 

Once an approximation to the wavefunction of a molecule has been found, it can be used to calculate the probable result of many physical measurements and hence to predict properties such as a molecular hexadecapole moment or the electric field gradient at a quadrupolar nucleus. For many workers in the field, this is the primary objective for performing quantum-mechanical calculations. But from... 

Just like the electric quadrupole moment, the electric field gradient matrix can be written in diagonal form for a suitable choice of coordinate axes. 

In a molecule, a given nucleus will generally experience an electric field gradient due to the surrounding electrons. The energy of interaction U between the nuclear quadrupole and the electric field gradient E is given by... 

It is usual to denote the electric field gradient at nuclear position N by q, which can be written as a 3 x 3 matrix... 

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. 

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 first derivative of the potential T at r = (0,0,0), taken as negative value, represents the ekctric field E, and the second derivative represents the electric field gradient tensor V at the nucleus,... 

When inserting into (4.5), the term ZeR will be multiplied with the elements of the electric field gradient tensor V. Fortunately, the procedure can be restricted to diagonal elements Vu, because V is symmetric and, consequently, a principal axes system exists in which the nondiagonal elements vanish, = 0. The diagonal elements can be determined by using Poisson s differential equation for the electronic potential at point r = 0 with charge density (0), AV = Anp, which yields... 

Spiering, H. The Electric Field Gradient and Quadmpole Interaction. In Long, G. (ed.) Mossbauer Spectroscopy Applied to Inorganic Chemistry, p. 79. Plenum, New York (1984)... 

Physical Interpretation of the Electric Field Gradient Tensor... 

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