Higher order nonlinear effects

We can summarize these considerations by saying that higher-order nonlinear effects generally favor network patterns in which each pi-bond site has balanced or... 

FIGURE 10.9 (a) Spatial conhnement of the excitation efficiency of higher-order nonlinear effects. The hrst-order fnnction represents the distrihntion of the tip-enhanced held intensity as shown in Figures 10.2 and 10.3. (h) Energy diagram of the CARS process. 

Molecular polarizabilities and hyperpolarizabilities are now routinely calculated in many computational packages and reported in publications that are not primarily concerned with these properties. Very often the calculated values are not likely to be of quantitative accuracy when compared with experimental data. One difficulty is that, except in the case of very small molecules, gas phase data is unobtainable and some allowance has to be made for the effect of the molecular environment in a condensed phase. Another is that the accurate determination of the nonlinear response functions requires that electron correlation should be treated accurately and this is not easy to achieve for the molecules that are of greatest interest. Very often the higher-level calculation is confined to zero frequency and the results scaled by using a less complete theory for the frequency dependence. Typically, ab initio studies use coupled-cluster methods for the static values scaled to frequencies where the effects are observable with time-dependent Hartree-Fock theory. Density functional methods require the introduction of specialized functions before they can cope with the hyperpolarizabilities and higher order magnetic effects. 

A quantitative compaiison of our theory with existing measurements of the energy loss of antiprotons [13] (which unlike protons carry no bound states) in a variety of target materials can be achieved by combining our first-principles calculations of the Zj (linear-response) stopping power with Zj corrections in a FEG. Nevertheless, a comparison with experiment still requires the inclusion of losses from the inner shells, xc effects, and higher-order nonlinear terms. Work in this direction is now in progress. 

Particular nonlinear optical phenomena arise also when static electric or magnetic fields are applied. The molecular states and selection rules are thereby modified, leading, for instance, to higher-order, nonlinear-optical variants of the linear (Pockels) and quadratic (Kerr) electro-optical effect, or of the linear (Faraday) and quadratic (Cotton-Mouton) magneto-optical effect. 

These expressions are only correct for wave functions that obey the Hellmann-Feynman theorem. Flowever, these expressions have been used for other methods, where they serve as a reasonable approximation. Methods that rigorously obey the Flellmann-Feynman theorem are SCF, MCSCF, and Full CF The change in energy from nonlinear effects is due to a change in the electron density, which creates an induced dipole moment and, to a lesser extent, induced higher-order multipoles. 

The work of the present section shows that shock-compression experiments provide an effective method for determination of higher-order elastic properties and that, by the same token, the effects of nonlinear elastic response should generally be taken into account in investigations of shock compression (see, e.g., Asay et al. [72A02]). Fourth-order contributions are readily apparent, but few coefficients have been accurately measured. 

High-power pulsed lasers offer the possibility of studying nonlinear phenomena such as stimulated Raman scattering, the inverse Raman effect and the hyper-Raman effect. These investigations have contributed much to our knowledge of the solid-state and liquid stucture of matter and its higher order constants. 

It is evident from Table 7.1 that the shorter the effective pulse duration, the higher the nonlinear signal intensity. This is even more the case for third-order processes... 

The above-mentioned nonlinear optical effects can be described by the perturbation of the electromagnetic held intensity under the electric dipole approximation. Actually, this approximation is broken in optical near-helds. Hence, a perturbation effect of multipole such as electric quadrupole or magnetic dipole should also be considered, although such a higher-order effect is normally negligible. Indeed, electric quadrupole contributions can be comparable with electric dipole contributions... 

In this section we address formation of concentration shocks in reactive ion-exchange as an asymptotic phenomenon. The prototypical case of local reaction equilibrium of Langmuir type will be treated in detail, following [1], [51], For a treatment of the effects of deviation from local equilibrium the reader is referred to [51]. The methodological point of this section consists of presentation of a somewhat unconventional asymptotic procedure well suited for singular perturbation problems with a nonlinear degeneration at higher-order derivatives. The essence of the method proposed is the use of Newton iterates for the construction of an asymptotic sequence. 

In this section we examine the higher orders beyond the linear noise approximation. They add terms to the fluctuations that are of order relative to the macroscopic quantities, i.e., of the order of a single particle. They also modify the macroscopic equation by terms of that same order, as has been anticipated in (V.8.12) and (4.8). These effects are obviously unimportant for most practical noise problems, but cannot be ignored in two cases. First, they tell us how equilibrium fluctuations are affected by the presence of nonlinear terms in the macroscopic equation, in particular how... 

The discontinuity of the interface leads to two contributions to the second order nonlinear polarizability, the electric dipole effect due to the structural discontinuity and the quadrupole type contribution arising from the large electric field gradient at the surface. Under the electric dipole approximation, the nonlinear susceptibility of the centrosymmetric bulk medium 2 is zero. If the higher order magnetic dipole... 

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