Absorption correlation function

The absorption correlation function a(t) at temperature T, is given by58... 

Figure B3.4.7. Schematic example of potential energy curves for photo-absorption for a ID problem (i.e. for diatomics). On the lower surface the nuclear wavepacket is in the ground state. Once this wavepacket has been excited to the upper surface, which has a different shape, it will propagate. The photoabsorption cross section is obtained by the Fourier transfonn of the correlation function of the initial wavefimction on tlie excited surface with the propagated wavepacket.

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

To complete the description and get the connection with the solute emission and absorption spectra, there is need of the correlation functions of the dipole operator pj= (a(t)+af(t))j and, consequently, the differential equation for the one solute mode has to be solved. The reader is referred to [133] for detailed analysis of this point as well as the equations controlling the relaxation to equilibrium population. The energy absorption and emission properties of the above model are determined by the two-time correlation functions ... 

The Fourier transform of the correlation functions Cem(x) and Cabs(x) are the emission (Sem) and absorption (Sabs) spectra, respectively Sem = jdxCem(x)exp(-icox) the integration limits are (-00,-1-00),... 

Both RRGM and SLP have been used to compute various transition amplitudes with high efficiency and accuracy. Their applications, which have been reviewed in the literature,56 57 59 include laser-molecule interaction,43 44 99 correlation functions,45 104 absorption and emission spectra,100 103 105-107 intramolecular energy transfer,108-115 vibrational assignment,103 116 117 and reaction dynamics. ... 

Evidently, correlation functions for different spherical harmonic functions of two different vectors in the same molecule are also orthogonal under equilibrium averaging for an isotropic fluid. Thus, if the excitation process photoselects particular Im components of the (solid) angular distribution of absorption dipoles, then only those same Im components of the (solid) angular distribution of emission dipoles will contribute to observed signal, regardless of the other Im components that may in principle be detected, and vice versa. The result in this case is likewise independent of the index n = N. Equation (4.7) is just the special case of Eq. (4.9) when the two dipoles coincide. 

The Time Dependent Processes Section uses time-dependent perturbation theory, combined with the classical electric and magnetic fields that arise due to the interaction of photons with the nuclei and electrons of a molecule, to derive expressions for the rates of transitions among atomic or molecular electronic, vibrational, and rotational states induced by photon absorption or emission. Sources of line broadening and time correlation function treatments of absorption lineshapes are briefly introduced. Finally, transitions induced by collisions rather than by electromagnetic fields are briefly treated to provide an introduction to the subject of theoretical chemical dynamics. 

One of the most direct methods of examining reorientational motion of molecules is by far infrared absorption spectroscopy or dielectric absorption. In the absence of vibrational relaxation, the relaxation times obtained by IR and dielectric methods are equivalent. In both these techniques we obtain the correlation function, [Pg.209]

To model the experimental data we used a global-fit procedure to simulate EPS, integrated TG, heterodyne-detected TG, and the linear absorption spectrum simultaneously. The pulse shape and phase were explicitly taken into account, which is of paramount importance for the adequate description of the experimental data. We applied a stochastic modulation model with a bi-exponential frequency fluctuation correlation function of the following form ... 

Figure 2 IR absorption in the O-H/D stretching region for the 7D model of SA (H black, D grey). The small broadening results from an artificial damping of the correlation function with a time constant of 0.5 ps. The spectra of the laser pulses used in Fig. 3 are shown as dashed lines.

Time-dependent correlation functions are now widely used to provide concise statements of the miscroscopic meaning of a variety of experimental results. These connections between microscopically defined time-dependent correlation functions and macroscopic experiments are usually expressed through spectral densities, which are the Fourier transforms of correlation functions. For example, transport coefficients1 of electrical conductivity, diffusion, viscosity, and heat conductivity can be written as spectral densities of appropriate correlation functions. Likewise, spectral line shapes in absorption, Raman light scattering, neutron scattering, and nuclear jmagnetic resonance are related to appropriate microscopic spectral densities.2... 

It was recently shown that a formal density expansion of space-time correlation functions of quantum mechanical many-body systems is possible in very general terms [297]. The formalism may be applied to collision-induced absorption to obtain the virial expansions of the dipole... 

Constant acceleration approximation. An approximation introduced to the time-dependent intermolecular correlation function G, which was commonly referred to as the constant acceleration approximation (CAA), was used to compute the line shapes of collision-induced absorption spectra of rare gas mixtures, but the computed profiles were found to be unsatisfactory [286], It does not give the correct first spectral moment. 

The shape of the vibration-rotation bands in infrared absorption and Raman scattering experiments on diatomic molecules dissolved in a host fluid have been used to determine2,15 the autocorrelation functions unit vector pointing along the molecular axis and P2(x) is the Legendre polynomial of index 2. These correlation functions measure the rate of rotational reorientation of the molecule in the host fluid. The observed temperature- and density-dependence of these functions yields a great deal of information about reorientation in solids, liquids, and gases. These correlation functions have been successfully evaluated on the basis of molecular models.15... 

To examine in detail some of the time-correlation functions that enter into the theories of transport, light absorption, and light scattering and neutron scattering. 

Using the standard expression of the dipole-dipole correlation function the CC absorption cross section can be derived as [9,42] ... 

Interestingly enough, one sees differences between the various variants of Markovian and non-Markovian theories already in static linear absorption spectra. In the regime of second-order perturbation theory in the coupling to the electromagnetic field the linear absorption line-shape / (ui) can be calculated from the Fourier transform of the dipole-dipole correlation function as... 

---