Dispersion response functions

Figure 2 shows the spectral response functions (5,(r), Eq. 1) derived firom the spectra of Fig. 1. In order to adequately display data for these varied solvents, whose dynamics occur on very different time scales, we employ a logarithmic time axis. Such a representation is also useful because a number of solvents, especially the alcohols, show highly dispersive response functions. For example, one observes in methanol significant relaxation taking place over 3-4 decades in time. (Mdtiexponential fits to the methanol data yield roughly equal contributions from components with time constants of 0.2, 2, and 12 ps). Even in sinqrle, non-associated solvents such as acetonitrile, one seldom observes 5,(r) functions that decay exponentially in time. Most often, biexponential fits are required to describe the observed relaxation. This biexponential behavior does not reflect any clear separation between fast inertial dynamics and slower diffusive dynamics in most solvents. Rather, the observed spectral shift usually appears to sirrply be a continuous non-exponential process. That is not to say that ultrafast inertial relaxation does not occur in many solvents, just that there is no clear time scale separation observed. Of the 18 polar solvents observed to date, a number of them do show prominent fast components that are probably inertial in origin. For example, in the solvents water [16], formamide, acetoniuile, acetone, dimethylformamide, dimethylsulfoxide, and nitromethane [8], we find that more than half of the solvation response involves components with time constants of 00 fs.

Now, as already mentioned in Section 1.3, the h of that section s Eq. (6) is of just the same form as the well-known Cole-Cole dielectric dispersion response function (Cole and Cole [1941]). In its normalized form, the same / function can thus apply at either the impedance or the complex dielectric constant level. We may generalize this result (J. R Macdonald [1985a,c,d]) by asserting that any IS response... 

In a recent publication [22] we reported the implementation of dispersion coefficients for first hyperpolarizabiiities based on the coupled cluster quadratic response approach. In the present publication we extend the work of Refs. [22-24] to the analytic calculation of dispersion coefficients for cubic response properties, i.e. second hyperpolarizabiiities. We define the dispersion coefficients by a Taylor expansion of the cubic response function in its frequency arguments. Hence, this approach is... 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

In the normal dispersion region below the first pole, response functions can be expanded in power series in their frequency arguments. The four frequencies, associated with the operator arguments of the cubic response function are related by the matching condition a -fwfl +wc -t-U ) — 0. Thus second hyperpoiarizabiiities or in general cubic response properties are functions of only three independent frequency variables, which may be chosen as u>b, ljc and U > ... 

In Chapter 2, Jansson describes the determination of the system response function for a dispersive spectrometer system. We have made a number of such determinations using very-low-pressure samples of, for example, CO in the 5-fim region. As discussed by Jansson, one records the data and then removes the Doppler profile using deconvolution, yielding the system response function. 

The effects of coupling of the DTO and RB units in not only one- but also three-dimensional arrays are discussed below and molecular weight trends illustrated. A fundamental connection between relaxation times and normal mode frequencies, shown to hold in all dimensions, allows the rapid derivation of the common viscoelastic and dielectric response functions from a knowledge of the appropriate lattice vibration spectra. It is found that the time and frequency dispersion behavior is much sharper when the oscillator elements are established in three-dimensional quasi-lattices as in the case of organic glasses. 

Fig. 4.23 also indicates a slight decrease of the signal plateau which, at a first glance, was unexpected. In the following, a reactive dispersion model given in ref. [37] is applied to deduce rate constants for different reaction temperatures. A trapezoidal response function will be used. The temperature-dependent diffusion coefficient was calculated according to a prescription by Hirschfelder (e.g., [80], p. 68 or [79], p. 104] derived from the Chapman-Enskog theory. For the dimensionless formulation, the equation is divided by M/A (with M the injected mass and A the cross-section area). This analytical function is compared in Fig. 4.24 with the experimental values for three different temperatures. The qualitative behavior of the measured pulses is well met especially the observed decrease of the plateau is reproduced. The overall fit is less accurate than for the non-reactive case but is sufficient to now evaluate the rate constant. 

The response function for a reacting gas (the dispersion coefficients fcr and the coefficient fcv were given in [38]) ... 

The induction-dispersion contribution, in turn, can be interpreted as the energy of the (second-order) dispersion interaction of the monomer X with the monomer Y deformed by the electrostatic field of the monomer Z (note that we have six such contributions). In particular, when X=A, Y=B, and Z=C the corresponding induction-dispersion contribution in terms of response functions is given by,... 

Experimentally, plasmonic engineering of SEF substrates requires then the consideration of the following variable The response function of the metal to polarization (dispersion of the dielectric function) plasmon resonances can be tuned using different shapes (such as triangles, squares, spheroids, rods), nanowires, shells, rings or holes. 

Figure 4 shows as examples of complete frequency coverage the dispersion and absorption curves of pure propylene carbonate (PC) and acetonitrile (AN). The maximum of e"( ) and the inflexion point of e (i/), situated at the same frequency, indicate the relsixation times r (PC) = 43 ps smd (AN) = 3.2 ps charsuiterizing the response function Fp of orientational polarization F" = > / V (/T,-, dipole moment of particle i V, volume... 

Figure 10.7. Spectra of cyclohexane obtained on FT-Raman and dispersive/CCD spectrometers, without correcting for instrument response function.

Waals envelope of the solvent. This projection can be expressed in terms of a response function, whose kernel contains a damping factor (the dielectric constant ) very near to the optical dielectric constant of water, eopt, when the water molecules are held fixed, or rapidly increasing towards the static dielectric constant, when water molecular motions are allowed and their number in the cluster increases. This is the origin of our PCM model (more details can be found in Tomasi, 1982). Surely, similar considerations spurred Rivail and coworkers to elaborate their SCRF method (Rivail and Rinaldi, 1976). An additional contribution to the formulation of today continuum models came from the nice analysis given by Kolos (Kolos, 1979 dementi et al., 1980) of the importance of dispersion contributions. 

TD-DFT can also be used to get better approximations to the ground-state exchange-correlation energy based on frequency dependent response functions. For example, the long-range dispersion term between two well-separated systems can be obtained from the frequency dependent susceptibility of the two systems [228,229]. 

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