Polarizabilities, frequency-independent

That the frequency dependent values can be obtained as readily as the frequency independent ones is an attractive feature of response theory as experimental measurements of (hyper)polarizabilities most often are carried out at non-zero frequencies. 

The polarizability of the individnal molecules is also frequency dependent, but the characteristic values are of the order of lO Vs and lO Vs for the rotational and electronic polarization, respectively. " Therefore, in the typical frequency domain for investigation of dispersions (1/s < co < 10 /s) the polarizability, e, of the material building up the particles is frequency independent. On the other hand, the disperse medium (which is usually an electrolyte solution) has a dielectric permittivity, Ej, for which the freqnency dependence can be described by the Debye-Falkenhagen theory. Besides, the characteristic relaxation time of the bulk electrolyte solutions is also given by Eqnation 5.385. ... 

This expression is known as the Clausius Mossotti relation. To simplify, the polarizability of the crystal can be taken as the sum of the electronic and atomic contributions. The electronic polarizability, aeiec, corresponds to the coupling of the electronic cloud of the otherwise immobile atoms with the electromagnetic wave, and it is a high-frequency process, whose contribution can be considered more or less frequency-independent below Es. The atomic... 

Determine the frequency-independent polarizability and the fre-qucncy-depcndent polariziibility at a frequency E = 0.1 a.u. for HeH. ... 

Carry out a coupled Hartree-Fock (CHF) calculation of the frequency-independent polarizability tensor for the closed-shell HeH system. To achieve this goal, follow the eps given below. 

Carry out a CMCHF calculation of the frequency-independent polarizability tensor. 

Comparing this expression to that of the CHF approach [Eq. (5.16)] shows that these two ways of writing the frequency-independent polarizability are indeed identical. 

By next inserting the unit matrix of Eq, (6.124) before and after the above inverse matrix and then using Eq. (6.127), we can write the frequency-independent polarizability in the form... 

In eq. 2 E51 is added to realize the quasi static contribution of permittivity due to polarizability of the lattice plus electronic part. Additional dipole contribution with relaxations between frequencies as realized here and the far infrared could contribute. A separation of the frequency independent and frequency dependent contributions could be obtained using first derivatives of eq. 1 and 2, namely... 

The dielectric constant of a polymer (K) (which we also refer to as relative electric permittivity or electric inductive capacity) is a measure of its interaction with an electrical field in which it is placed. It is inversely related to volume resistivity. The dielectric constant depends strongly on the polarizability of molecules tvithin the polymer. In polymers with negligible dipole moments, the dielectric constant is low and it is essentially independent of temperature and the frequency of an alternating electric field. Polymers with polar constituents have higher dielectric constants. When we place such polymers in an electrical field, their dipoles attempt... 

It is interesting to note that these conclusions seem to be inapplicable to model protonated Schiff bases in solution. Thus Brith-Linder et al, (246) have shown that the C=C stretching frequency in a series of PRSB in a variety of solvents is almost independent of the spectral shift (vc=c = 1588 + 6 cm" in the range 435 nm < Amax < 542 ran). This implies that the excited state, rather than the ground state, is sensitive to the electrostatic effects of the counterion and to the solvent polarizability. This behavior which is in variance with the substantial slope of vs. Amax for the pigments as shown in Fig. 7, raises... 

Table 5 shows the experimental specific refractivities, K X) = n(l) l]/ p, and the average polarizability as calculated from equation (1) at a number of frequencies for liquid and vapour phases. The values of the specific refractivity of the vapour have been obtained from the Cauchy dispersion formula of Zeiss and Meath.39 In this paper the authors assess the results of a number of experimental determinations of the refractive index of water vapour and its variation with frequency. Even after some normalization of the data to harmonize the absolute values from different determinations there is a one or two percent spread of results at any one wavelength. Extrapolation of the renormalized data for five independent sets of data leads to zero frequency values of K(7.) within the range (2.985-3.013) x 10-4 m3 kg 1, giving, via equation (1), LL — 9.63 0.10 au. Extrapolation of the earlier refractive index data of Cuthbertson and Cuthbertson40 by Russell and Spackman41 from 8 values of frequency between 0.068 and 0.095 au, leads to a zero frequency value, of y.i, 1,(0) = 9.83 au. While the considerable variation between the raw experimental data reported in different determinations is cause for some uncertainty, it appears that the most convincing analysis to date is that of... 

We saw in Section III that the polarization propagator is the linear response function. The linear response of a system to an external time-independent perturbation can also be obtained from the coupled Hartree-Fock (CHF) approximation provided the unperturbed state is the Hartree-Fock state of the system. Thus, RPA and CHF are the same approximation for time-independent perturbing fields, that is for properties such as spin-spin coupling constants and static polarizabilities. That we indeed obtain exactly the same set of equations in the two methods is demonstrated by Jorgensen and Simons (1981, Chapter 5.B). Frequency-dependent response properties in the... 

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