Frequency-dependent polarizabilities

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. 

To give a simple classical model for frequency-dependent polarizabilities, let me return to Figure 17.1 and now consider the positive charge as a point nucleus and the negative sphere as an electron cloud. In the static case, the restoring force on the displaced nucleus is d)/ AtteQO ) which corresponds to a simple harmonic oscillator with force constant... 

Frequency-dependent polarizability a and second hyperpolarizability y corresponding to various third-order nonlinear optical processes have been... 

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. " ... 

Saue, T. and Jensen, H.J.Aa. (2003) Linear response at the 4-component relativistic level Application to the frequency-dependent dipole polarizabilities of the coinage metal dimers. Journal of Chemical Physics, 118, 522-536. 

The coefficients Cn may be derived from the (imaginary) frequency dependent polarizabilities summed over the entire frequency range [46]. If one employs only dipole polarizabilities the dispersion expansion is truncated at the leading term, with n = 6. In the current EFP2 code, an estimate is used for the n = 8 term, in addition to the explicitly derived n = 6 term. Rather than express a molecular C as a sum over atomic interaction terms, the EFP2 dispersion is expressed in terms of LMO-LMO... 

To obtain Raman spectra one needs the trajectories of the pq tensor elements of the chromophore s transition polarizability. Actually, for the isotropic Raman spectrum one needs only the average transition polarizability. This depends weakly on bath coordinates and this, together with the weak frequency dependence of the position matrix element, was included in our previous calculations [13, 98, 121]. For the VV and VH spectra, others have implemented... 

In the equation s is the measured dielectric constant and e0 the permittivity of the vacuum, M is the molar mass and p the molecular density, while Aa and A (po2) are the isotope effects on the polarizability and the square of the permanent dipole moment respectively. Unfortunately, because the isotope effects under discussion are small, and high precision in measurements of bulk phase polarization is difficult to achieve, this approach has fallen into disfavor and now is only rarely used. Polarizability isotope effects, Aa, are better determined by measuring the frequency dependence of the refractive index (see below), and isotope effects on permanent dipole moments with spectroscopic experiments. 

The final frequency-dependent polarizability is given by the sum of the plus and minus components... 

In the response function terminology [47] the i, j component of the frequency-dependent dipole polarizability tensor — w) (or the ij, kl component of the traceless quadrupole polarizability tensor is defined through... 

The last method used in this study is CCSD linear response theory [37]. The frequency-dependent polarizabilities are again identified from the time evolution of the corresponding moments. However, in CCSD response theory the moments are calculated as transition expectation values between the coupled cluster state l cc(O) and a dual state... 

Figure 6.25 Frequency dependence of (a) total polarizability and (b) power loss. From K. M. RaUs, T. FI. Conrtney, and J. Wulff, Introduction to Materials Science and Engineering. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission John Whey Sons, Inc.

In the previous section several equations were described that can be used to calculate MCD spectra. If the spectra are to be calculated using the transition-based approach described in Sections II.A.1-II.A.4, a number of quantities must be evaluated. These include the perturbed and unperturbed excitation energies, the perturbed and unperturbed transition moments between the ground and excited states, and/or the magnetic moment of the ground state. If an MCD spectrum is to be calculated with the imaginary Verdet approach described in Section II.A.6, then the first-order correction to the frequency-dependent polarizability due to a magnetic field is required. 

If the frequency-dependent perturbation is an electric field then pn) can be used to calculate the frequency-dependent polarizability that must be perturbed to evaluate the imaginary Verdet constant. 

The frequency-dependent polarizability, and therefore p(1> diverges when is equal to an excitation energy. Casida (54) made use of this property to derive a TDDFT equation for the excitation energy to an excited state J that is essentially the DFT form of the random phase approximation (RPA) equation ... 

The second approach to calculating MCD starts from its definition in terms of the real part of first-order correction to the frequency-dependent polarizability in the presence of a magnetic field (Section II.A.6). This definition can be used to consider all types of MCD linear in the magnetic field (9). Our current implementation is restricted to systems with a closed-shell ground state. We shall therefore only consider the calculation of A and terms by this method. 

The frequency-dependent polarizability is given by Eq. (38). Differentiation of this equation with respect to a static magnetic field provides the quantity needed to calculate MCD (or MOR). 

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