Response function spectral representation

Polarization propagators or linear response functions are normally not calculated from their spectral representation but from an alternative matrix representation [25,46-48] which avoids the explicit calculation of the excited states. 

Using the spectral representation, it is possible to obtain an informative expression for the linear response function where we denote the energy difference between the state given by n> and the reference state 0 > as con = En — E0. 

In the case of the cubic response function we can, using the spectral representation, write the cubic response function as... 

For the quadratic response function, we have that the energy difference between the excited state q) and the ground state 0) is co o = q-Eo and for the external electric field we have the frequencies co and > . We are able to write the spectral representation of the quadratic response function as... 

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.

In addition to the linear response function r r we have also introduced the quadratic r r r , and the cubic r r r r , j, 3 response functions. The relation (9) only ensures that the real part of the second term is correct. In fact it turns out (Zubarev, 1974, Chap. 15 Oddershede et al., 1984, Chaps 2.1 and 2.2) that r r is the spectral representation of the retarded polarization propagator... 

In time domain measurements, the electrochemical system is subjected to a potential variation that is the resultant of many frequencies, like a pulse or white noise signal, and the time-dependent current from the cell is recorded. The stimulus and the response can be converted via Fourier transform methods to spectral representations of amplitude and phase angle frequency, from which the desired impedance can be computed as a function of frequency. 

Dynamic Response Functions. - The perturbation series formula or spectral representation of the response functions can be used only in connection with theories that incorporate experimental information relating to the excited states. Semi-empirical quantum chemical methods adapted for calculations of electronic excitation energies provide the basis for attempts at direct implementation of the sum over states (SOS) approach. There are numerous variants using the PPP,50,51 CNDO(S),52-55 INDO(S)56,57 and ZINDO58 levels of approximation. Extensive lists of publications will be found, for example, in references 5 and 34. The method has been much used in surveying the first hyperpolarizabilities of prospective optoelectronically applicable molecules, but is not a realistic starting point for quantitative calculation in un-parametrized calculations. 

Independently of the approach used, in the exact case one obtains the well-known spectral representation [also known as the sum-over-states (SOS) expression] of the linear response function (LRF)... 

The spectral representations above are not computationally efficient, as they would require knowledge of all intermediate excited states. Computationally tractable formulas for the response functions within the various approximate methods are obtained instead through the following steps (1) choose a time-independent reference wavefunction (2) choose a parametrization of its time-development, for instance an exponential parametrization (3) set up the appropriate equations for the time development of the chosen wavefunction parameters (4) solve these equations in orders of the perturbation to obtain the wavefunction (parameters) (5) insert the solutions of these equations into the expectation value expression and obtain the RTFs and (6) identify the excited-state properties from the poles and residues. The computationally tractable formulas for the response functions therefore differ depending on the electronic structure method at hand, and the true spectral representations given above are only valid in the limit of a frill-configuration interaction (FCI) wavefunction. For approximate methods (i.e., where electron correlation is only partially included), matrix equations appear instead of the SOS expressions, for example. 

After the introduction of the various interrelated response functions and basic concepts like the Debye-process and the derived spectral representations we come now in the second part of this chapter to the description and discussion of actual polymer behavior. In fact, relaxation processes play a dominant role and result in a complex pattern of temperature and frequency dependent properties. 

However, Eq. (3.108) is only the definition of the Fourier transform, which then has to be applied to the expression for the time-dependent polarization propagator in Eq. (3.107) with t —t replaced by t. This leads us [see Exercise 3.8] to the spectral representation of the polarization propagator or hnear response function... 

The off-diagonal terms in Eq. 76 are determined imposing a coherence function. The simulation of m-variate response-spectrum-compatible accelerograms is then performed extending the spectral-representation method described in Eqs. 76, 77, 78, 79, and 80 and 83, 84, 85, 86,87,88, and 89 to a wavelet simulation-based procedure. 

The response of the system concerned to an external electromagnetic field is conveniently described in terms of double-time Green s function (GF) which can be introduced in a variety of representations.144,218 221 In what follows we will involve the representation in Matsubara s frequency space218 which is accepted in the theory of anharmonic crystals197 and provides a number of exact solutions in the case of a single adsorbed molecule.I50,1 2 In this approach, the spectral line shape for high-frequency vibrations can be determined as follows 184... 

In the late fifties, Eringen and his co-workers [1-3] have analyzed the responses of beams and plates to random loads. Since these pioneering works, response analysis of structures subjected to random excitations has attracted considerable attention in the past thirty years. An extensive review of the recent developments have been provided by Crandall and Zhu [4]. Most of the earlier studies on nonstationary random vibrations were concerned with the analysis of mean-square response statistics [5,6]. Recently, evaluation of the time-dependent power spectra of structural response has attracted considerable interest. Priestley [7] introduced the orthogonal representation of a random function. Hammond [5], Corotis and Vanmarcke [8] and To [9] have studied the time-dependent spectral content of responses of single- and multi-degree-of-freedom structures. 

The presented results show that the method of normal mode and orthogonal representations of random functions provides a useful technique for analyzing random responses of shear beam structures. The method provides, in addition to the mean-square response statistics, information concerning the time-dependent spectral content of the response. The presented examples also... 

Fig. 18. Schematic representation of the spectral density (or intensity function) for spin couplings in the frame of the three-component analysis of molecular motions in polymers. The dipolar broadening region represented as the hatched section at low fluctuation rates is predominantly responsible for the transverse relaxation rate (compare Eq. 40 and Ref. [2]). Variation of the temperature shifts the components across the fluctuation rate defined by the motional-averaging condition, so that the influence of the individual components changes one by one. Variation of the molecular weight or the polymer concentration likewise shifts the molecular weight or concentration dependent components across the motional averaging fluctuation rate...

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