Generalized linear response theory

The above theory is usually called the generalized linear response theory because the linear optical absorption initiates from the nonstationary states prepared by the pumping process [85-87]. This method is valid when pumping pulse and probing pulse do not overlap. When they overlap, third-order or X 3 (co) should be used. In other words, Eq. (6.4) should be solved perturbatively to the third-order approximation. From Eqs. (6.19)-(6.22) we can see that in the time-resolved spectra described by x"( ), the dynamics information of the system is contained in p(Af), which can be obtained by solving the reduced Liouville equations. Application of Eq. (6.19) to stimulated emission monitoring vibrational relaxation is given in Appendix III. 

The paper [50] was written, when our general linear response-theory (ACF method) was still in progress, so the derivation of the spectral function described in Ref. 50 is more specialized than that given, for example, in VIG and GT. 

Validity of our formulas for the resonance lines, which express the complex susceptibility through the spectral function, could be confirmed as follows. We have obtained an exact coincidence of the equations (353), (370), (371), which were (i) directly calculated here in terms of the harmonic oscillator model and (ii) derived in GT and VIG (see also Section II, A.6) by using a general linear-response theory. 

The next two chapters are devoted to ultrafast radiationless transitions. In Chapter 5, the generalized linear response theory is used to treat the non-equilibrium dynamics of molecular systems. This method, based on the density matrix method, can also be used to calculate the transient spectroscopic signals that are often monitored experimentally. As an application of the method, the authors present the study of the interfadal photo-induced electron transfer in dye-sensitized solar cell as observed by transient absorption spectroscopy. Chapter 6 uses the density matrix method to discuss important processes that occur in the bacterial photosynthetic reaction center, which has congested electronic structure within 200-1500cm 1 and weak interactions between these electronic states. Therefore, this biological system is an ideal system to examine theoretical models (memory effect, coherence effect, vibrational relaxation, etc.) and techniques (generalized linear response theory, Forster-Dexter theory, Marcus theory, internal conversion theory, etc.) for treating ultrafast radiationless transition phenomena. 

In this chapter, the ultrafast radiationless transition processes are treated theoretically. The method employed is based on the density matrix method, and specifically, a generalized linear response theory is developed by applying the projection operator technique on the Liouville equation so that non-equilibrium cases can be handled properly. The ultrafast molecular... 

In this section we develop a general linear response theory for which the coupling of the system to the field in Eqs. [91] is small [i.e., fjt) 1]. Linear response theory should, in general, take into account the possibility of phase space compression coming from either the coupling to the external field or from the extended phase space variables. In traditional formulations of linear response theory, phase space compression has not been carefully considered. The present formulation treats the first level of compressibility exactly, for example, the compressibility due to the presence of an extended system (such as ther-... 

As shown from Eq. (4.26), the dynamics of both population p(At) and coherence p(At) /(n 7 n )is involved in the time-resolved experiment (the probe experiment here), and Eq. (4.26) can be applied to optical absorption and stimulated emission. Furthermore, we recover the ordinary linear response theory where p / = 0 and pnn represents the Boltzmann distribution. In other words, Eq. (4.26) denotes the generalized linear response theory (GLRP). Pumping experiments can be treated similarly by using Eq. (4.21). With a short-pulse pumping laser, both population... 

As mentioned, this equivalence is a consequence of the fluctuation-dissipation theorem (the general basis of linear response theory [51]). In (12.68), we have dropped nonlinear terms and we have not indicated for which state Variance (rj) is computed (because the reactant and product state results only differ by nonlinear terms). We see that A A, AAstat, and AAr x are all linked and are all sensitive to the model parameters, with different computational routes giving a different sensitivity for AArtx. 

The role of quadmpole polarizabilities is less pronounced. Jens Oddershede, e.g., has studied the quadmpole polarizability of N2 [10]. Furthermore, there are studies which point out the need for calculations of quadmpole polarizabilities, e.g., for the interpretation of spectra obtained by surface-enhanced Raman spectroscopy [42,43]. Generally the interest in multipole polarizabilities increases due to new experimental data. We decided, therefore, to also study how different linear response theory methods perform in the calculation of quadmpole polarizabilities. 

The important step of identifying the explicit dynamical motivation for employing centroid variables has thus been accomplished. It has proven possible to formally define their time evolution ( trajectories ) and to establish that the time correlations ofthese trajectories are exactly related to the Kubo-transformed time correlation function in the case that the operator 6 is a linear function of position and momentum. (Note that A may be a general operator.) The generalization of this concept to the case of nonlinear operators B has also recently been accomplished, but this topic is more complicated so the reader is left to study that work if so desired. Furthermore, by a generalization of linear response theory it is also possible to extract certain observables such as rate constants even if the operator 6 is linear. 

In another paper, R. Kuho (Kcio University, Japan) illustrates in a rather technical and mathematical fashion tire relationship between Brownian motion and non-equilibrium statistical mechanics, in this paper, the author describes the linear response theory, Einstein s theory of Brownian motion, course-graining and stochastization, and the Langevin equations and their generalizations. 

Recall that in the slow modulation limit, the Robertson and Yarwood ACF reduces in turn to that used by Bratos, in his pioneering work dealing with the SD of H-bonded species within the linear response theory. Also recall that there are two generalizations of the semiclassical model of Robertson and Yarwood, one by Sakun [78] and the other by Abramczyk [79]. The first is incorporating memory functions and the second rotational structure. 

The next section (Sect. 2) is devoted to a lengthy discussion of the molecular hypothesis from the point of view of quantum field theory, and this provides the basis for the subsequent discussion of optical activity. Having used linear response theory to establish the equations for optical activity (Sect. 3), we pause to discuss the properties of the wavefunctions of optically active isomers in relation to the space inversion operator (Sect. 4), before indicating how the general optical activity equations can be related to the usual Rosenfeld equation for the optical rotation in a chiral molecule. Finally (Sect. 5), there are critical remarks about what can currently be said in the microscopic quantum-mechanical theory of optical activity based on some approximate models of the field theory. 

In linear response theory the optical activity is obtained from the part of the generalized susceptibility involving the temporal correlations of the electric and magnetic polarization fields46,47. For a system such as a normal fluid, described by a statistical operator that is invariant under space and time translations, the appropriate retarded Green function is,... 

Linear response theory, applied to the particle velocity, considered as a dynamic variable of the isolated particle-plus-bath system, allows to express the mobility in terms of the equilibrium velocity correlation function. Since the mobility p(co) is simply the generalized susceptibility %vx(o ), one has the Kubo formula... 

Equations (161) and (162) are two equivalent formulations of the second FDT [30,31]. The Kubo formula (162) for the generalized friction coefficient can also be established directly by applying linear response theory to the force exerted by the bath on the particle, this force being considered as a dynamical variable of the isolated particle-plus-bath system. We will come back to this point in Section VI.B. 

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