Operator electric-field gradient

The inclusion of (nonrelativistic) property operators, in combination with relativistic approximation schemes, bears some complications known as the picture-change error (PCE) [67,190,191] as it completely neglects the unitary transformation of that property operator from the original Dirac to the Schrodinger picture. Such PCEs are especially large for properties where the inner (core) part of the valence orbital is probed, for example, nuclear electric field gradients (EEG), which are an important... 

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... 

There is a modest increase in the electrical conductance with an increase in the electric-field gradient, an effect that operates with both strong and weak electrolytes (the first Wien effect). More important in the present context is the marked increase in electrical conductance of weak electrolytes when a high-intensity electric field is applied (second Wien effect). The high field promotes an increase in the concentration of ion pairs and free ions in the equilibrium... 

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. 

In a rotating molecule containing one quadrupolar nucleus there is an interaction between the angular momentum J of the molecule and the nuclear spin momentum I. The operator of this interaction can be written as a scalar product of two irreducible tensor operators of second rank. The first tensor operator describes the nuclear quadrupole moment and the second describes the electrical field gradient at the position of the nucleus under investigation. 

In electro-osmosis (Fig. 5), when an externally applied electric field gradient operates across the wet clay, water is moved from the anode (the positive electrode) to the cathode (the negative electrode) that is, there is a movement of the liquid phase through the stationary solid phase (a clay, soil, capillary, or porous plug, etc.) in response to an applied electric field, as shown schematically in Fig. 6, taken from Probstein. ... 

In this Section, the derivation of useful expressions for the calculation of first-order properties at the quasi-relativistic level of theory will be outlined. The electric field gradient at the nucleus is chosen to represent first-order electrical properties. The relativistic corrections to the electric field gradients are large since the electric field gradient operator is proportional to r. The electric field gradient operator is thus mainly sampling the inner part of the electronic density distribution. 

In this case, the perturbation term is e.g. proportional to the tensor component of the electric field gradient operator (43) times the nuclear quadrupole moment or is actually multiplied by a perturbation-strength parameter, Q, which is linearly proportional to the nuclear quadrupole moment. 

By inserting the Hamilton operator (44) into the energy expression and differentiating it with respect to the perturbation-strength parameter Q, the expressions (46-48) for the calculation of the electric field gradient at the quasi-relativistic level of theory are obtained. 

In equation (54), the first term represents the nonrelativistic operator for the electric field gradient. The second and the third terms originate from the reorthonormalization of the large component and the last term appears due to the presence of the field gradient operator in the lower diagonal block of the original Dirac equation. 

In the previous Subsection, the derivation of expressions for the calculation of the electric field gradient at the quasi-relativistic level of theory has been outlined. Similar expressions must be used in order to obtain accurate values for other first-order electrical properties at quasi-relativistic level of theory. The expressions obtained in the present derivation show that at the quasi-relativistic level of theory, first-order properties must not be calculated as pure expectation values of the nonrelativistic property operator, but other operators also appear in the expressions. This is the so called picture-change effect previously discussed in several articles [71-76]. 

An overview of relativistic state-of-the-art calculations on electric field gradients (EFG) in atoms and molecules neccessary for the determination of nuclear quadrupole moments (NQM) is presented. Especially for heavy elements four-component calculations are the method of choice due to the strong weighting of the core region by the EFG operator and the concomitant importance of relativity. Accurate nuclear data are required for testing and verification of the various nuclear models in theoretical nuclear physics and this field represents an illustrative example of how electronic structure theory and theoretical physics can fruitfully interplay. Basic atomic and molecular experimental techniques for the determination of the magnetic and electric hyperfine constants A and B axe briefly discussed in order to provide the reader with some background information in this field. 

---