Field gradient operator

For MD and/or QM/MM geometry optimizations gradients of the energies are needed. They follow naturally from the energy expressions by replacing electrostatic potential and field operators by, respectively, the corresponding field and field gradient operators. 

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. 

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. 

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... 

For capillary zone electrophoresis (CZE) mass spectrometry coupling, another modification of an ESI interface has been developed. This interface uses a sheath flow of liquid to make the electrical contact at the CZE terminus, thus defining both the CZE and electrospray field gradients. This way, the composition of the electro sprayed liquid can be controlled independently of the CZE buffer, thereby providing operation with buffers that could not be used previously, e.g., aqueous and high ionic strength buffers. In addition, the interface operation becomes independent of the CZE flow rate. [62]... 

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 moments discussed in this chapter are sometimes referred to as the outer moments of the distribution, in contrast to the inner moments for which the powers of r in the operator 6 in Eq. (6.5) are negative. The electric field at the nucleus and the field gradient at the nucleus are examples of inner moments, which will be discussed in chapter 8. 

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. 

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