Zero frequency polarizability

In linear, spherical and synnnetric tops the components of a along and perpendicular to the principal axis of synnnetry are often denoted by a and respectively. In such cases, the anisotropy is simply Aa = tty -If the applied field is oscillating at a frequency w, then the dipole polarizability is frequency dependent as well a(co). The zero frequency limit of the dynamic polarizability a(oi) is the static polarizability described above. 

The normalization condition ((0 i) = 0) imposes to move the origin to the center of electronic charge ((u) = 0,u = x,y,z) thus, the polarizability may be written very simply in the limit of zero frequency ... 

To use Equation (32) it is necessary to extrapolate to infinite wavelength (or zero frequency) to obtain the unperturbed polarizability since the electric field of the light also alters the molecule. Failure to carry out such an extrapolation introduces far less error, however, than is introduced by an approximation such as Equation (27). 

Debye," for the coupling between the permanent dipole and the molecular polarizability, at zero frequency [see Table S.8.b and Eq. (L2.178)] ... 

This equation is accurate in the gas phase but the refractive index should first be extrapolated to infinite wavelength (or zero frequency) to obtain the static polarizability ... 

Before presenting the results which have been obtained for sodium chloride it is necessary to indicate the sources of the data used for this salt. Van der Waals coefficients were taken from a tabulation by Mayer (11) and electronic polarizabilities of the ions at zero frequency from the work of Tessman, Kahn, and Shockley (13). A value of 4.802 X 10-10 e.s.u. was used for the charge of the electron and the repulsive parameters by and p were taken from recent publications by Cubicciotti (5, 6). A value of a — 2.794 A. was computed for the nearest... 

Having obtained the zero frequency limit of the dynamic polarizability i.e., a = Iin, o7 (—wja ), we use a simplified approach to evaluate the screened dynamic response. This is necessary, since the expression given above, Eq. (40), for the polarizability neglects the induced collective effects essentially due to direct and exchange terms of the Coulomb interaction. To treat this screening approximately, we have used the simplified approach of Bertsch et al. [96] to include the induced electron interaction in the Ceo molecule, by a simple RPA type correction [92,95]... 

The system comprising the resistor Re and capacitor C in series provides an example of a class of systems for which, at the zero-frequency or dc limit, current cannot pass. Such systems are considered to have a blocking or ideally polarizable electrode. Depending on the specific conditions, batteries, liquid mercury electrodes, semiconductor devices, passive electrodes, and electroactive polymers provide examples of systems that exhibit such blocking behavior. 

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. 

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... 

Zeiss and Meath. However a number of recent theoretical papers take the Zeiss-Meath value at 514.5 nm as the basis for the experimental value of the polarizability anisotropy and the Cuthbertson value for the limiting zero frequency polarizability. Examination of Table 5 and Figure 1 shows that this would imply that the dispersion at the longer wavelength end of the optical range is very small. 

The calculations give the electronic part of the polarizability. It is thought that the ZPYA correction is the major part of the vibrational contribution and has a value of about 0.29 au at zero frequency. In Table 6 this correction has been added to the electronic part to give the values in the final column, which can be compared with the two possible experimental values in the first two rows. The most accurate of the CCSD type work should be that of Christiansen... 

The role of the medium, in which contacting and pull-off are performed, has been mentioned but not considered so far. However, the surroundings obviously influence surface forces, e.g., via effective polarizability effects (essentially multibody interactions e.g., by the presence of a third atom and its influence via instantaneous polarizability effects). These effects can become noticeable in condensed media (liquids) when the pairwise additivity of forces can essentially break down. One solution to this problem is given by the quantum field theory of Lifshitz, which has been simplified by Israelachvili [6]. The interaction is expressed by the (frequency-dependent) dielectric constants and refractive indices of the contacting macroscopic bodies (labeled by 1 and 2) and the medium (labeled by 3). The value of the Hamaker constant Atota 1 is considered as the sum of a term at zero frequency (v =0, dipole-dipole and dipole-induced dipole forces) and London dispersion forces (at positive frequencies, v >0). 

Let us return to the problem of solving the response of the quantum mechanical system to an external electric field. The zeroth-order wave function of the quantum mechanical system is obtained by use of any of the standard approximate methods in quantum chemistry and the coupling to the field is described by the electtic dipole operator. There exist a number of ways to determine the response functions, some of which differ in formulation only, whereas others will be inherently different. We will give a short review of the characteristics of tire most common formulations used for the calculation of molecular polarizabilities and hyperpolarizabilities. The survey begins with the assumption that the external perturbing fields arc non-oscillatory, in which case we may determine molecular properties at zero frequencies, and then continues with the general situation of time-dependent fields and dynamic properties. 

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. 

In recent years, experimental investigation of the depolarized Rayleigh scattering of several liquids composed of optically anisotropic molecules has confirmed the existence of a doublet-symmetric about zero frequency change and with a splitting of approximately 0.5 GHz (see Fig. 12.1.1). The existence of this doublet had been predicted on the basis of a hydrodynamic theory several years previously by Leontovich (1941). This theory assumes that local strains set up by a transverse shear wave are relieved by collective reorientation of individual molecules. Later, Rytov (1957) formulated a more general hydrodynamic theory for viscoelastic fluids that reduces to the Leontovich theory in the appropriate limit. The theories of Rytov and Leontovitch are different from the present two-variable theory, in that the primary variable is the stress tensor and not the polarizability. 

There are a number of assumptions implicit in this equation, such as that the refractive index of the solvent in the visible-region wavelengths is the same as at zero frequency. This is reasonable for solvents other than water, but for water there is quite a large difference between the values at zero and visible-region frequencies. The ground-state isotropic polarizability of the solute is assumed to be a jl. 

The values of a and for a group can be obtained from experimental refractive index vs. wavelength data. The value of a is essentially the electronic polarizability at zero frequency it can be approximated by a (v) for the Na D line. We can also obtain values of a and Vq corresponding to transitions with frequencies greater than v. 

Polarizability is a general concept that quantifies the response of an electron cloud of an ion to the apphcation of a time-dependent electromagnetic field resulting in a frequency-dependent polarizability. Our strict concern is with static, or zero-frequency polarizability as variations of an electric field induced by thermal fluctuations of an electrolyte operate at timescales much larger than the timescales of inner dynamics of an electron cloud. Frequency-dependent polarizability leads to other interesting effects, such as the London forces [32], when spontaneous fluctuations of electronic structure of two molecules become correlated at close spacial separations. These interactions, however, play secondary role when compared to induced interactions that arise from static polarizability [33, 34]. 

In the previous sections it was shown that frequency-dependent linear response prop>-erties, such as frequency-dependent polarizabilities, can be obtained as the value of the polarization propagator for the appropriate operators. Furthermore, all static second-order properties discussed in Chapters 4 and 5 can be calculated as the value of a polarization propagator for zero frequency. 

Polarizability(a) 4.9 x 10-23esu Dipole Moment(p) 7.3 x 10-18 Hyperpolarizability, 1367nm(pi367) -44x 10-30 Hyperpolarizability, extrapolated to zero frequency(po) -12x 10-30... 

The first calculations of ROA using ab initio molecular orbital methods have been carried out recently. This has been achieved by starting with expressions for the polarizability and Rayleigh optical activity tensors in the zero-frequency limit of the FFR approximation as... 

At zero frequency, the TDLDA equations have been solved to yield excellent results for closed shell atomic polarizabilities. At finite frequencies, calculations are facilitated by exploiting the identityll ... 

Choosing a non-zero value for uj corresponds to a time-dependent field with a frequency u, i.e. the ((r r)) propagator determines the frequency-dependent polarizability corresponding to an electric field described by the perturbation operator QW = r cos (cut). Propagator methods are therefore well suited for calculating dynamical properties, and by suitable choices for the P and Q operators, a whole variety of properties may be calculated. " ... 

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