Electric field perturbation

Other perturbations have been demonstrated. The pressure,, jump, similar to the T-jump in principle, is attractive for organic reactions where Joule heating may be impractical both because of the solvent being used and because concentrations might have to be measured by conductivity. Large (10 —10 kPa) pressures are needed to perturb equiUbrium constants. One approach involves pressurizing a Hquid solution until a membrane mptures and drops the pressure to ambient. Electric field perturbations affect some reactions and have also been used (2), but infrequentiy. 

The calculations of the and c constants lead to a system of linear equations similar to that of the SCF-CI method, but with three more lines and columns corresponding to the coupling of the polynomial function with the electric field perturbation. The methodology and computational details have already been discussed (1) we stress two points the role of the dipolar factor, the nature and the number of the exeited states to inelude in the summation. 

Electric discharge ozone generator, 77 798 Electric-field-induced second harmonic generation (EFISH), 20 515 Electric field intensity, exponents of dimensions, 8 585t Electric field perturbations, 74 616 Electric fields... 

The standard deviation of the Gaussian zones expresses the extent of dispersion and corresponds to the width of the peak at 0.607 of the maximum height [24,25]. The total system variance (ofot) is affected by several parameters that lead to dispersion (Eq. 17.22). According to Lauer and McManigill [26] these include injection variance (of), longitudinal (axial) diffusion variance (of), radial thermal (temperature gradient) variance (of,), electroosmotic flow variance (of,), electrical field perturbation (electrodispersion) variance (of) and wall-adsorption variance (of ). Several authors [9,24,27-30] have described and investigated these individual variances further and have even identified additional sources of variance, like detection variance (erf,), and others... 

Electrical field perturbation variance. It is well established in HPLC that when the injected sample solution has a different eluting strength than that of the mobile phase, peak deformation is bound to occur. A similar effect is observed in CE. Instead of eluting strengths, it concerns here differences in conductivity between the sample zone and the bulk electrolyte in the capillary [9,32], The conductivity (y, Ohm 1 m ) of a solution is given by the cumulative effect of the contributions of different ions ... 

Fig. 17.7. The effect of electric field perturbations, due to differences in conductivity between the sample- and the buffer electrolyte zone on the shape of the peaks, (a) conductivity distribution, (b) sample ions distribution, (c) electric field strength perturbations, and (d) effect on the peak shapes.

Any external electric field is minute in comparison with the internal field generated by the system of electrons and nuclei inside a molecule. The effect of the operator (8.4) is therefore always much smaller than the electronic energy of the molecule. In most cases, the effects of electric-field perturbations are also much smaller than the vibrational energy of the molecule. The interaction with an external DC field can thus be treated as a perturbation to the vibronic energy levels of molecules. 

A thorough discussion Is given of the field modulation technique, a new stationary relaxation method based on electric field perturbation of Ionic equilibria. Concomitantly the theory of electric field effect In Ionic systems Is reviewed especially stressing their Importance for conductance phenomena In low polar solutions. 

The applied electric field perturbs the orientational distribution function of the dipolar molecules. Dielectric relaxation due to classical molecular reorientational motions is a form of pure absorption spectroscopy whose frequency range of interest for materials, including polymers, is between 10 6 and 1011 Hz. 

One final result is required to analyse the weak field spectrum of CsF. We derived an expression for the matrix elements of the electric field perturbation earlier, without the inclusion of nuclear spin effects, in equation (8.278). We now repeat this derivation using the basis set employed above. Taking the direction of the electric field to define the p = 0 (Z) direction, the results are as follows ... 

Derivative Techniques 240 10.4 Lagrangian Techniques 242 10.5 Coupled Perturbed Hartree-Fock 244 10.6 Electric Field Perturbation 247 10.7 Magnetic Field Perturbation 248 10.7.1 External Magnetic Field 248 13.1 Vibrational Normal Coordinates 312 13.2 Energy of a Slater Determinant 314 13.3 Energy of a Cl Wave Function 315 Reference 315 14 Optimization Techniques 316... 

Buckingham and Pople refer to the effect of the electric field as a paramagnetic term, and it has the dependence of the second term in equation (5), Although equation (5) has the virtue of attempting to describe the true electronic environment of the proton, it has the disadvantages of intractability. The electric field perturbation model is mathematically simple but an extreme approximation. Since these two treatments lead to the same functional dependence on p, perhaps the electric field model provides a useful approximation to the more complete description of equation (5), Whether this proves to be true or whether the characteristic arbitrariness of the electrostatic model will deprive the model of more than qualitative predictive value is not yet clear. In any event, the two treatments do concur in shifting attention from the p" term to the p term with its opposite sign. 

Thus Eq. (4.7) for the optical activity tensors of the molecule are again employed, but now the summation is over all occupied LMOs and the vectors Ri define the positions of the orbital centroids. Once the wavefunctions are known, the polarizability of the ith LMO and the position of its centroid R( can be determined. The derivatives of a, and Rj with respect to the normal coordinates are calculated using the electric field perturbation approach recently shown to be very effective for the calculation of conventional infrared and Raman intensities61) the required derivatives of R= and a. are determined from the first and second derivatives, respectively, of the gradient of the molecular potential energy with respect to a small applied electric field. One important aspect of this method is that both infrared CD and ROA can be determined from the same conceptual and calculational method, which will enhance the study of the relationship between these two forms of vibrational optical activity. So far, only one ROA calculation using LMO methods has been reported59), and since that was for the model compound NHDT there has been no comparison with experimental data. 

Crystal field theory A theory of bonding in transition metal complexes in which ligands and metal ions are treated as point charges a purely ionic model. Ligand point charges represent the crystal (electric) field perturbing the metal s d orbitals that contain nonbonding electrons. 

In early years of quantum chemistry, several theoretical papers were devoted to calculations of linear and nonlinear responses of molecules to the electric field perturbations using the Uncoupled Hartree-Fock (UCHF) method. In comparison with the CI ansatz, the UCHF is less accurate in the description of electronic structure of molecules. Since this method was of some interest in computations of NLO properties we present this method in Section 5. 

Hyperpolarizabilities can be calculated in a number of different ways. The quantum chemical calculations may be based on a perturbation approach that directly evaluates sum-over-states (SOS) expressions such as Eq. (14), or on differentiation of the energy or induced moments for which (electric field) perturbed wavefunctions and/or electron densities are explicitly calculated. These techniques may be implemented at different levels of approximation ranging from semi-empirical to density functional methods that account for electron correlation through approximations to the exact exchange-correlation functionals to high-level ab initio calculations which systematically include electron correlation effects. 

The self-atom and atom-atom polarizabilities (nAA,nAB) defined using perturbation theory have been also employed to describe chemical reactivity [44], These quantities represent the effect of an electric field perturbation at one atom on the electronic charge at the same (nAA) or another atom (nAB), respectively. 

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