Effect of a uniform electric field

As a first illustration of the perturbation approach developed in Section 11.4, we consider the effect of a uniform electric field applied to a molecule in a non-degenerate ground state. The corresponding term in (11.3.10) will be H iec = Z,[- ( /)]. where (r,) is the electric potential at point r, due to the applied field. In terms of field components, which conveniently serve as perturbation parameters, we may write (cf. (11.4.6)) 

For a non-degnerate state the matrix Hew in (11.4.3) has only one element, and it follows immediately that 

The three terms in (11.5.1) give additive effects, and thus we find 

From (11.4.14), we obtain a second-order energy quadratic in the field components. Thus 

It is customary to write the quadratic energy term in the form 

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A new and efficient computational strategy has been presented, that simplifies the calculation of the vibrational frequencies of a molecular system adsorbed on moderate to large cluster models. This procedure is based on a certain hypothesis and assumption. Nevertheless, present results show that these do not affect the numerical accuracy of the calculated frequencies. An important consequence of this strategy is that largely simplifies the study of the effect of a uniform electric field on the frequencies of an adsorbed species. This is because it is not necessary to recalculate the normal coordinates at each value of the electric field. The method has been presented in connection to a cluster model representation of the surface, but it can be directly applied to periodical approaches without further modification. 

We shall now discuss the effect of a uniform electric field on the transition temperature between two phases at a given pressure. We consider a heterogeneous system of two phases and one component in the presence of the field. At equilibrium... 

It is instructive to compare these results with the behavior of the Is state. In the first place, the effect of a uniform electric field on the Is level is zero, to first order, because (ls i/ ls) vanishes for reasons of symmetry. Only when first-order corrections are made to the 1 s wavefunction are energy effects seen, and these occur in the second-order energy terms. Therefore, the Is state gives a second-order Stark effect, but no first-order effect. The Is state gives no first-order effect because the spherically symmetric zeroth-order wavefunction has no electric dipole to interact with the field. But the proper zeroth-order wave functions for some of the = 2 states, given by Eqs. (12-81) and (12-82), do provide electric dipoles in opposite directions that interact with the field to produce first-order energies of opposite signs. 

The above results evidently confirm the validity of a classical interpretation of the effect of a uniform electric field the polarization energy in (11.5.18) is actually associated with an induced electric moment, arising from a field-proportional distortion of the charge cloud, with components determined as in (11.5.21). 

From microwave spectra, in the presence of a uniform electric field applied to the gas (Stark effect), it is possible to obtain accurate measurements of dipole moments. 

In this paper we modify and extend this approach in several ways. In particular, we consider the magnetic fine structure effects in the presence of a uniform electric field F for ls2p Pj- excited states of helium. We introduce two separate differential polarizabihties to describe the quadratic part of the electric field splitting and three differential hyperpolarizabilities to describe the terms the order of in the fine-structure splitting of the atomic multiplet s2p Pj. We have developed a calculational approach that allows correct estimation of potential contributions due to continuum spectra to the dipole susceptibilities j3 and 7. In the next section we briefly outline our method. The details of the calculations of the angular and radial matrix elements have been described elsewhere [8,9] and are omitted here for brevity. Atomic units are used throughout. 

The electric polarization from internal continuum (EPIC) model has been developed to accurately predict the polarizability tensor of molecules ". The EPIC approach uses an intramolecular effective dielectric constant, together with associated atomic radii, to represent the detailed molecular polarizability. For a single atom of radius R in vacuum under the influence of a uniform electric field (E), the polarizability is given by the electric fleld prefactor of the induced dipole moment (Sin is the inner dielectric) ... 

The OMRC does not account for the orientation or structure of the probe species, nor does it account for the interconnectedness (i.e., the possibility of percolation in the gel whereby certain regions of the gel may be inaccessible to a particular species) of the matrix [361]. The OMRC also assumes a uniform electric field and does not consider the effects of the gel on the electric field. 

The linear velocity of the liquid developing under the effect of this force is zero directly at the solid surface, and increases to some maximum value v at the distance X = 8o from the surface. Solution regions farther out lack the excess charges that could come under the effect of the external electric field hence, there is no further increase in liquid velocity (Fig. 31.4). When the layer (Sg) is much thinner than the capillary radius, 5g r, the assumption can be made that the bulk of the solution moves with a uniform velocity v. 

The ab initio SCF cluster wavefunction has been used to investigate the bonding of CO and CN- on Cu,0 (5,4,1), (5 surface layer, 4 second layer and 1 bottom layer atoms), and to calculate their field dependent vibrational frequency shifts in fields up to 5.2 x 107 V/cm(46). A schematic view of the Cu10 (5,4,l)CO cluster is shown in Figure 8. In order to assess the significance of Lambert s proposal, that the linear Stark effect is the dominant factor in the field dependent frequency shift, the effect of the field was calculated by three methods. One is by a fully variational approach (i.e., the adsorbate is allowed to relax under the influence of the applied field) in which the Hamiltonian for the cluster in a uniform electric field, F, is given by... 

Among the recently published works, the one which showed that the cyclic structures of water clusters open up to form a linear structure above a certain threshold electric field value a was a systematic ab initio study on the effect of electric field on structure, energetics, and transition states of trimer, tetramer, and pentamer water clusters (both cyclic and acyclic) [36], Considering c/.v-butadiene as a model system, the strength and the direction of a static electric field has been used to examine the delocalization energy, the probabilities of some local electronic structures, the behavior of electron pairs, and the electronic fluctuations [37]. Another recent work performed by Rai et al. focused on the studies using the DFT and its time-dependent counterpart of effects of uniform static electric field on aromatic and aliphatic hydrocarbons [38],... 

A uniform electric field distribution across the sample is extremely important for achieving device quality materials. Unfortunately, real chromophore materials do not always behave as uniform insulator materials. We have already demonstrated that ionic impurities can dramatically reduce the effective electric field felt by chromophores. The presence of spatially and temporally varying nonuniform space charge distributions leads to nonuniform poling fields. The resulting nonuniform chromophore order can lead to light scattering. 

A dilute suspension of electrically and optically anisotropic colloidal particles becomes birefringent when subjected to an electric field. In random orientation the suspension is optically isotropic but when the grains align with a uniform electric field the suspension becomes anisotropic in particular the effective refractive index of the ordered suspension parallel to the field direction differs from its refractive index... 

The Stark effect on the magnetic fine structure occurs as a result of disturbance of atomic levels under the influence of the relativistic and correlation effects as well as the interaction with external electric field F. If the fields are weak enough the centre of multiplet is shifted and there occurs the splitting of sublevels of atomic multiplet n, L, J. The dipole moment induced in an atom by a uniform electric field F is for most purposes expressed as a linear function of F, but higher... 

Velkoff and Miller [336] investigated the effect of uniform and nonuniform electric fields on laminar film condensation of Freon-113 on a vertical plate. With screen grid electrodes providing a uniform electric field over the entire plate surface, a 150 percent increase in the heat transfer coefficient was obtained with a power expenditure of a fraction of one watt. Choi and Reynolds [337] and Choi [338] recently reported data for condensation of Freon-113 on the outside wall of an annulus in the presence of a radial electric field. With the maximum applied voltage of 30 kV, the average heat transfer coefficients for a 25.4-mm outside diameter by 12.7-mm inside diameter annulus were increased by 100 percent. 

Effective Conductivity of Dilute Mixtures. The simplest best-defined case is a cluster of spherical particles dispersed in a liquid and located in a uniform electrical field. If the particles have the same conductivity as the liquid, the potential around the particles will not be distorted, and the mixture conductivity is equal to that of the liquid. If the particles have a lower conductivity, the streamlines will diverge away from the particles, and the mixture conductivity will be lower than that of the liquid. If the particles have a higher conductivity, the streamlines will converge into the particle, and the mixture conductivity will be higher than that of the liquid. 

Electrophoresis, Fig. 1 Electrophoresis of a cation and an anion in a uniform electric field. The anode and cathode are assumed to extend infinitely in the vertical direction to avoid end effects. The positively charged cations move towards the negatively charged cathode, whereas the negatively charged anions move towards the positively charged anode... 

If molecules are introduced into a uniform electric field, two effects are observable which yield important information on the internal structure of the particles. 

In a uniform electric field, dipole-free molecules orient themselves with the axis of greatest polarizability aligned with the field (induction effect) ... 

Semi-dry transfer For the semi-dry transfer, the gel-membrane sandwich is placed between carbon plate electrodes. Semi-dry or horizontal blotting uses two plate electrodes (stainless steel or graphite/ carbon) for a uniform electrical field over a short distance, and sandwiches between these of up to six gel/membrane/filter paper assemblies, all well soaked in transfer buffer. The assembly is clamped or otherwise secured on its side and electrophoretic transfer effected in this position, using as transfer buffer only the liquid contained in the gel and filter papers or other pads in the assembly. 

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