Second-order moments

Here, f, Fx and are local field factors, a is a molecular parameter including the dipole moment, second order and third order hyperpolarizabilities. Under condition F = Fx = the formula does not agree with the experiment for 8CB. Indeed, it follows that rji should increase at the transition from the nematic to smectic A phase according to the increase in (F2) However, in the experiment, Fn markedly decreases. The similar temperature behavior was earlier observed for dielectric permittivity of 8CB. In the latter case the decrease in is due to the antiparallel correlation of molecular dipoles in the smectic A phase, which results in a decrease in the effective dipole moment /x Thus, the decrease... 

Assume, for a moment, second-order kinetics. Expressed in terms of S2, the Elovich equation (54) reads... 

In order to describe the second-order nonlinear response from the interface of two centrosynnnetric media, the material system may be divided into tlnee regions the interface and the two bulk media. The interface is defined to be the transitional zone where the material properties—such as the electronic structure or molecular orientation of adsorbates—or the electromagnetic fields differ appreciably from the two bulk media. For most systems, this region occurs over a length scale of only a few Angstroms. With respect to the optical radiation, we can thus treat the nonlinearity of the interface as localized to a sheet of polarization. Fonnally, we can describe this sheet by a nonlinear dipole moment per unit area, -P ", which is related to a second-order bulk polarization by hy P - lx, y,r) = y. Flere z is the surface nonnal direction, and the... 

A very useful quantity for the determination of a critical point which is directly based on order parameter moments is the fourth-order cumulant [179-181] or the second-order cumulant [182,183] t/. ... 

The second-order terms give the magnetizability. The first term is known as the diamagnetic part and it is particularly easy to calculate since it is just the expectation value of the second moment operators. The second term is called the paramagnetic part. 

The poles con espond to excitation energies, and the residues (numerator at the poles) to transition moments between the reference and excited states. In the limit where cj —> 0 (i.e. where the perturbation is time independent), the propagator is identical to the second-order perturbation formula for a constant electric field (eq. (10.57)), i.e. the ((r r))Q propagator determines the static polarizability. 

The second order central moment is used so frequently that it is very often designated by a special symbol o2 or square root of the variance, o, is usually called the standard deviation of the distribution and taken as a measure of the extent to which the distribution is spread about the mean. Calculation of the variance can often be facilitated by use of the following formula, which relates the mean, the variance, and the second moment of a distribution... 

The physical interpretation of these joint moments is similar in every respect to the interpretation already given for moments of the form ak = E[k]. Thus, a . .. provides a measure of the center of mass of the joint probability density function p 1,...,second order central moments provide a measure of the spread of this density function about its center of mass.30... 

Perhaps the most widely studied joint moments are the second-order moments—those for which kx k2 + + kn = 2. These moments areallof the form k = 0,1,2 i,j = 1 The correspond-... 

In the case of weak collisions, the moment changes in small steps AJ (1 — y)J < J, and the process is considered as diffusion in J-space. Formally, this means that the function /(z) of width [(1 — y2)d]i is narrow relative to P(J,J, x). At t To the latter may be expanded at the point J up to terms of second-order with respect to (/ — /). Then at the limit y -> 1, to — 0 with tj finite, the Feller equations turn into a Fokker-Planck equation... 

Comparison of formulae (2.51) and (2.64) allows one to understand the limits and advantages of the impact approximation in the theory of orientational relaxation. The results agree solely in second order with respect to time. Everything else is different. In the impact theory the expansion involves odd powers of time, though, strictly speaking, the latter should not appear. Furthermore the coefficient /4/Tj defined in (2.61) differs from the fourth spectral moment I4 both in value and in sign. Moreover, in the impact approximation all spectral moments higher than the second one are infinite. This is due to the non-analytical nature of Kj and Kf in the impact approximation. In reality, of course, all of them exist and the lowest two are usually utilized to find from Eq. (2.66) either the dispersion of the torque (M2) or related Rq defined in Eq. (1.82) ... 

The second procedure, several aspects of which are reviewed in this paper, consists of directly computing the asymptotic value by employing newly-developed polymeric techniques which take advantage of the one-dimensional periodicity of these systems. Since the polarizability is either the linear response of the dipole moment to the field or the negative of the second-order term in the perturbation expansion of the energy as a power series in the field, several schemes can be proposed for its evaluation. Section 3 points out that several of these schemes are inconsistent with band theory summarized in Section 2. In Section 4, we present the main points of the polymeric polarization propagator approaches we have developed, and in Section 5, we describe some of their characteristics in applications to prototype systems. 

Linear response function approaches were introduced into the chemistry literature about thirty years ago Ref. [1,2]. At that time they were referred to as Green functions or propagator approaches. Soon after the introduction it became apparent that they offered a viable and attractive alternative to the state specific approaches for obtaining molecular properties as excitation energies, transition moments and second order molecular properties. 

In this regard, it seems clear that the operator is approximated to second order by the difference operator, the structure of which involves the coefficients k p taken for all a and / either at one and the same moment t = fj + i/2 01 nl 3-ny another moment t 6 With these members,... 

G2, to G3, and to G4, the effective enhancement was 10%, 36%, and 35% larger than the value estimated by the simple addition of monomeric values. The enhancement included the local field effect due to the screening electric field generated by neighboring molecules. Assuming the chromophore-solvent effect on the second-order susceptibility is independent of the number of chro-mophore units in the dendrimers, p enhancement can be attributed to the inter-molecular dipole-dipole interaction of the chromophore units. Hence, such an intermolecular coupling for the p enhancement should be more effective with the dendrimers composed of the NLO chromophore, whose dipole moment and the charge transfer are unidirectional parallel to the molecular axis. 

Without loss of generality y = y can be assumed. If the dipole moment can be assumed to be a linear function of coordinate within the spread of the frozen Gaussian wave packet, the matrix element (gy,q,p, Pjt(r) Y,q, p ) can be evaluated analytically. Since the integrand in Eq. (201) has distinct maxima usually, we can introduce the linearization approximation around these maxima. Namely, the Taylor expansion with respect to bqp = Qq — Qo and 8po = Po — Po is made, where qj, and pj, represent the maximum positions. The classical action >5qj, p , ( is expanded up to the second order, the final phase-space point (q, p,) to the first order, and the Herman-Kluk preexponential factor Cy pj to the zeroth order. This approximation is the same as the ceUularization procedure used in Ref. [18]. Under the above assumptions, various integrations in U/i(y, q, p ) can be carried out analytically and we have... 

Applying a chromatographic method it is sometimes possible to separate copolymer molecules according to their size Z and composition [5]. The SCD found in such a way can be compared with that calculated within the framework of the chosen kinetic model. The first- and second-order statistical moments of SCD are of special importance. 

Indeed, both expressions predict quadratic dependence of AA on the dipole moment of the solute. As in the previous example, it is of interest to test whether this prediction is correct. Such a test was carried out by calculating AA for a series of model solutes immersed in water at different distances from the water-hexane interface [11]. The solutes were constructed by scaling the atomic charges and, consequently, the dipole moment of a nearly spherical molecule, CH3F, by a parameter A, which varied between 0 and 1.2. The results at two positions - deep in the water phase and at the interface - are shown in Fig. 2.3. As can be seen from the linear dependence of A A on p2, the accuracy of the second-order perturbation theory... 

From a study of the microwave spectrum of 2-methylselenophene, the second-order Stark effect in the ground state was determined.11 The technique used was double radiofrequency-microwave resonance. For the identification by the double resonance method transitions of chiefly the A-state were chosen. From these observations the components of the dipole moment of 2-methylselenophene and the total dipole moment were determined. 

The first attempts to rationalize the magnetic properties of rare earth compounds date back to Hund [10], who analysed the magnetic moment observed at room temperature in the framework of the old quantum theory, finding a remarkable agreement with predictions, except for Eu3+ and Sm3+ compounds. The inclusion by Laporte [11] of the contribution of excited multiplets for these ions did not provide the correct estimate of the magnetic properties at room temperature, and it was not until Van Vleck [12] introduced second-order effects that agreement could be obtained also for these two ions. 

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