Spectrum semiclassical

Figure 41 shows the absorption spectrum for the 24-mode model of pyrazine. As was done by Raab et al. [277], we have included a phenomenological dephasing time of T2 = 150 fs to model the experimental broadening due to hnite resolution and rotational motion. It can be seen that the inclusion of all 24 normal modes of the pyrazine molecule leads to a shape of the spectrum which is in good agreement with the experimental result (Fig. 38b). The semiclassical result is seen to be in fairly good agreement with the quantum result. The spurious structure in the semiclassical spectrum is presumably due to the statistical error.

Quantized chaos, or quantum chaology (see Section 4.1), is about understanding the quantum spectra and wave functions of classically chaotic systems. The semiclassical method is one of the sharpest tools of quantum chaology. As discussed in Section 4.1.3 the central problem of computing the semiclassical spectrum of a classically chaotic system was solved by Gutzwiller more than 20 years ago. His trace formula (4.1.72) is the basis for all semiclassical work on the quantization of chaotic systems. 

This method generates line positions as well as intensities, and the similarity found between this semiclassical spectrum and the quantal one is another example of the applicability of classical trajectories to intramolecule dynamics. 

It is emphasized that the two kinds of quantum relaxation mechanisms lead, in agreement with experiment, to a shift of the first moment of the spectrum toward low frequencies, whereas the RY semiclassical indirect relaxation mechanism does not produce this effect. 

Muller and Stock [227] used the vibronic coupling model Hamiltonian, Section III.D, to compare surface hopping and Ehrenfest dynamics with exact calculations for a number of model cases. The results again show that the semiclassical methods are able to provide a qualitative, if not quantitative, description of the dynamics. A large-scale comparison of mixed method and quantum dynamics has been made in a study of the pyrazine absorption spectrum, including all 24 degrees of freedom [228]. Here a method related to Ehrenfest dynamics was used with reasonable success, showing that these methods are indeed able to reproduce the main features of the dynamics of non-adiabatic molecular systems. 

Rather than looking at the spectrum obtained from the secular determinant (5), we will here consider the spectrum SG for fixed wavenumber k and than average over k. One can write the spectrum in terms of a periodic orbit trace formula reminiscent to the celebrate Gutzwiller trace formula being a semiclassical approximation of the trace of the Green function (Gutzwiller 1990). We write the density of states in terms of the traces of SG, that is,... 

When studying the border of universality, we always need to consider the limit of large graphs, that is, ue —> oo. This limit is in general not well defined, but may often be obvious from the examples considered. We will thus define the semiclassical limit loosely via a family of unitary-stochastic transition matrices Tn and associate USE s and take ue —> oo. The leading term in (13) then gives a condition for a family to show deviations from RMT statistics in terms of the spectrum of T the diagonal term must obey... 

Describing complex wave-packet motion on the two coupled potential energy surfaces, this quantity is also of interest since it can be monitored in femtosecond pump-probe experiments [163]. In fact, it has been shown in Ref. 126 employing again the quasi-classical approximation (104) that the time-and frequency-resolved stimulated emission spectrum is nicely reproduced by the PO calculation. Hence vibronic POs may provide a clear and physically appealing interpretation of femtosecond experiments reflecting coherent electron transfer. We note that POs have also been used in semiclassical trace formulas to calculate spectral response functions [3]. 

Figure 38. Absorption spectrum of pyrazine in the energy region of the S1-S2 conical intersection. Shown are (a) quantum mechanical (full line) and semiclassical (dotted line) results for the four-mode model (including a phenomenological dephasing constant of T2 = 30 fs) and (b) the experimental data [271],...

Figure 41. Absorption spectrum for the 24-mode pyrazine model. The full line is the quantum result [277], and the dotted line is the semiclassical result. In both spectra a phenomenological dephasing constant of = 150 fs was used.

To conclude, the results presented in this section demonstrate that the semiclassical implementation of the mapping approach is able to describe rather well the ultrafast dynamics of the nonadiabatic systems considered. In particular, it is capable of describing the correct relaxation dynamics of the autocorrelation function as well as the structures of the absorption spectrum of... 

Now, the eigenenergies of the Hamiltonian can be detected directly if the time dependence of the above average exhibits quantum beats. This will be the case if the spectrum is not too dense and the linewidths are smaller than the level spacings. From a Fourier transform of the autocorrelation function, we then obtain an expression of the form (2.26)-(2.27), which can be evaluated semiclassically in terms of periodic orbits and their quantum phases. 

B. A. Hess Dr. Gaspard has introduced the vibrogram as a tool to extract periodic orbits from the spectrum by means of a windowed Fourier transform. This raises the question whether other recent techniques of signal analysis like multiresolution analysis or wavelet transforms of the spectrum could be used to separate the time scales and thus to disentangle the information pertinent to the quasiclassical, semiclassical, and long-time regimes. Has this ever been tried ... 

A linear approximation of the potential is certainly too sweeping a simplification. In reality, Vr varies with the internuclear separation and usually rises considerably at short distances. This disturbs the perfect (mirror) reflection in such a way that the blue side of the spectrum (E > Ve) is amplified at the expense of the red side (E < 14).t For a general, nonlinear potential one should use Equations (6.3) and (6.4) instead of (6.6) for an accurate calculation of the spectrum. The reflection principle is well known in spectroscopy (Herzberg 1950 ch.VII Tellinghuisen 1987) the review article of Tellinghuisen (1985) provides a comprehensive list of references. For a semiclassical analysis of bound-free transition matrix elements see Child (1980, 1991 ch.5), for example. 

This ultrasimple classical theory is, of course, too crude for practical applications, especially for highly excited states of the parent molecule. Its usefulness gradually diminishes as the degree of vibrational excitation increases, i.e., as the initial wavefunction becomes more and more oscillatory. If both wavefunctions oscillate rapidly, they can be approximated by semiclassical WKB wavefunctions and the radial overlap integral of the bound and the continuum wavefunctions can subsequently be evaluated by the method of steepest descent. This leads to analytical expressions for the spectrum (Child 1980, 1991 ch.5 Tellinghuisen 1985, 1987). In particular, relation (13.2), which relates the coordinate R to the energy E, is replaced by... 

Comparisons of the correlation functions calculated quantum mechanically and semiclassically, like those presented in Fig. 6.2, show that the correction due to the dipole moment gradient included in (6.34) sometimes improves the accuracy especially for short propagation times. This correction affects not only the amplitude of the correlation function oscillation, but also its frequency and distortions due to the presence of high harmonics in the spectrum. An analysis of the spectrum of the correlation function indicates that including this correction in the formula enables additional quantum effects to be taken into account. 

The low temperature refractive properties of the He gas have not been studied extensively. However, the second virial Kerr coefficient can be related to the zeroth moment of the polarized Raman spectrum, and thus deduced from the Raman experiment. For the helium gas at the liquid nitrogen temperature the experiment gives 1.46 a.u.416, the full quantum calculation 1.45328, while the classical result computed according to Eq. (1-260) gives 1.63 328. This shows that also for the Kerr effect the quantum corrections are important. A systematic study of these corrections and of the convergence of the semiclassical expansion has not been reported thus far, even though all necessary expressions are derived328. 

So far as the classification of the type of spectroscopy performed is concerned, the characterisation of the dynamical motions of the nuclei and electrons within a molecule is more important than the region of the electromagnetic spectrum in which the corresponding transitions occur. However, before we come to this in more detail, a brief discussion of the nature of electromagnetic radiation is necessary. This is actually a huge subject which, if tackled properly, takes us deeply into the details of classical and semiclassical electromagnetism, and even further into quantum electrodynamics. The basic foundations of the subject are Maxwell s equations, which we describe in appendix 1.1. We will make use of the results of these equations in the next section, referring the reader to the appendix if more detail is required. 

On the applied side of quantum chaology we find serious efforts to forge the semiclassical method into a handy tool for easy use in connection with arbitrary classically chaotic systems. Quite frankly, the current status of semiclassical methods is such that they are immensely helpful in the interpretation of quantum spectra and wave functions, but are only of limited power when it comes to accurately predicting the quantum spectrum of a classically chaotic system. In this case numerical methods geared toward a direct numerical solution of the Schrodinger equation are easier to handle, more transparent, more accurate and cheaper than any known semiclassical method. It should be the declared aim of applied semiclassics to provide methods as handy and universal as the currently employed numerical schemes to solve the spectral problem of classically chaotic quantum systems. 

The first improved theory addressing the weakly coupled, or non-adiabatic electronic excitation transfer was the semiclassical vector model proposed by Forster [15]. It was further developed and refined by Levinson [16], Kasha [17], and others [18], who sometimes referred to it as the molecular exciton theory . Notably, this was the first successful attempt to link the rate of electronic excitation transfer with readily available experimental parameters, such as the absorption spectrum of the... 

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