Polarizabilities second-moment

Van Hook, W. A. and Wolfsberg, M., Comments on H/D isotope effects on polarizabilities. Correlation with virial coefficient, molar volume and electronic second moment isotope effects. Z Naturforsch. 49A, 563 (1994)... 

The resulting equations show that the first-order polarizabilities atJ depend on the second moments of the distribution, while the second-order polarizabilities pijk are functions of both the second and third moments of the polarizable body, as in... 

The fact that the parameter ai is connected to spectroscopic moments for absorption and emission transitions opens an interesting opportunity to derive the polarizability change of optical chromophores from spectroscopic first and second moments. The equation for the polarizability change is as follows ... 

Laplace s equation, V V = 0, means that the number of unique elements needed to evaluate an interaction energy can be reduced. For the second moment this amounts to a transformation into a traceless tensor form, a form usually referred to as the quadrupole moment [5]. Transformations for higher moments can be accomplished with the conditions that develop from further differentiation of Laplace s equation. With modern computation machinery, such reduction tends to be of less benefit, and on vector machines, it may be less efficient in certain steps. We shall not make that transformation and instead will use traced Cartesian moments. It is still appropriate, however, to refer to quadrupoles or octupoles rather than to second or third moments since for interaction energies there is no difference. Logan has pointed out the convenience and utility of a Cartesian form of the multipole polarizabilities [6], and in most cases, that is how the properties are expressed here. 

The second-moment polarizability evaluated with the origin shifted to x = h is obtained as... 

One simple but very effective logic procedure developed for DHF keeps track of the derivatives. An integer list is constructed for each particular derivative where there is one integer in the list for each parameter. For first-and second-multipole polarizabilities, there will be nine parameters, three for the first moment and six for the second moment, after ignoring equivalent elements in V. One way of ordering them is V, V, V, Vj,y, Vyy, Vy, and K22, and then nine integers, ordered the same, are associated with each a derivative ... 

The discussion here applies to any second-order property, indcidentally. Force constants are simply used as an example, but the troublesome second term affects polarizabilities, dipole moment derivatives, and so on, as well. 

Certain intrinsic molecular properties describe how and in what manner a molecule responds to an applied field. These properties offer a good illustration of what is at stake in a direct calculation of a property. For uniform electric fields, the kind developed between two oppositely charged parallel plates, the first-order energy response is the dipole or first moment. The first-order response to a field gradient is the second moment, the quadrupole, and so on. Polarizabilities correspond to the second-order response, and hyperpolarizabilities to third-order and higher order changes. So, each of these is obtained as a derivative. 

Cl2 and CO2 and used the DID model for the induced polarizability. They showed that for CO2 the collision induced contribution to the depolarised Rayleigh intensity was 25% of the total intensity and that the second moment was increased by about 50% by induced effects. The timescale separation was examined in N2 and C02 In their terminology is the collision induced contribution it was found to relax in a very similar way to the orientational function <°M(t).°M> and the spectra of the two terms were indistinguishable for practical purposes. Furthermore the cross-term was quite large. The net effect of the non-orientational terms was to reduce the amplitude of the spectrum at low frequencies and to increase it in the wings. Frenkel and McTague s results on nitrogen have been carefully compared with experiment by Sampoli de Santis and co-workers(54). 

It should be emphasized again, that the multipole polarizabilities and moments in Eqs. (5.1.10) and (5.1.15) are written there in the laboratory system of coordinates and depend on the mutual orientation of the interacting molecules. As a result, the first dipole hyperpolarizability of two interacting molecules is a function (surface) of several variables Euler angles (rotation of the first and the second molecule), the intermolecular separation R and the internal coordinates when the molecules are considered as nonrigid ones. 

Equations (6.5) and (6.12) contain terms in x to the second and higher powers. If the expressions for the dipole moment /i and the polarizability a were linear in x, then /i and ot would be said to vary harmonically with x. The effect of higher terms is known as anharmonicity and, because this particular kind of anharmonicity is concerned with electrical properties of a molecule, it is referred to as electrical anharmonicity. One effect of it is to cause the vibrational selection mle Au = 1 in infrared and Raman spectroscopy to be modified to Au = 1, 2, 3,. However, since electrical anharmonicity is usually small, the effect is to make only a very small contribution to the intensities of Av = 2, 3,. .. transitions, which are known as vibrational overtones. 

In the second type of interaction contributing to van der Waals forces, a molecule with a permanent dipole moment polarizes a neighboring non-polar molecule. The two molecules then align with each other. To calculate the van der Waals interaction between the two molecules, let us first assume that the first molecule has a permanent dipole with a moment u and is separated from a polarizable molecule (dielectric constant ) by a distance r and oriented at some angle 0 to the axis of separation. The dipole is also oriented at some angle from the axis defining the separation between the two molecules. Overall, the picture would be very similar to Fig. 6 used for dipole-dipole interaction except that the interaction is induced as opposed to permanent. 

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 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. 

Experimental and theoretical results are presented for four nonlinear electrooptic and dielectric effects, as they pertain to flexible polymers. They are the Kerr effect, electric field induced light scattering, dielectric saturation and electric field induced second harmonic generation. We show the relationship between the dipole moment, polarizability, hyperpolarizability, the conformation of the polymer and these electrooptic and dielectric effects. We find that these effects are very sensitive to the details of polymer structure such as the rotational isomeric states, tacticity, and in the case of a copolymer, the comonomer composition. 

We have shown in this paper the relationships between the fundamental electrical parameters, such as the dipole moment, polarizability and hyperpolarizability, and the conformations of flexible polymers which are manifested in a number of their electrooptic and dielectric properties. These include the Kerr effect, dielectric polarization and saturation, electric field induced light scattering and second harmonic generation. Our experimental and theoretical studies of the Kerr effect show that it is very useful for the characterization of polymer microstructure. Our theoretical studies of the NLDE, EFLS and EFSHG also show that these effects are potentially useful, but there are very few experimental results reported in the literature with which to test the calculations. More experimental studies are needed to further our understanding of the nonlinear electrooptic and dielectric properties of flexible polymers. 

In order to describe second-order nonlinear optical effects, it is not sufficient to treat (> and x<2) as a scalar quantity. Instead the second-order polarizability and susceptibility must be treated as a third-rank tensors 3p and Xp with 27 components and the dipole moment, polarization, and electric field as vectors. As such, the relations between the dipole moment (polarization) vector and the electric field vector can be defined as ... 

Values for the Zero-Field Energies, Dipole Moments (p), Polarizabilities (a), HyperpolarizabiUties (fi), and Second Hyperpolarizabilities (y) for the Non-BO H2 Isotopomer Series"... 

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