Excited autocorrelation function

Figure 7. Decay of an excited autocorrelation function relative to the distribution pi in Section III (—) and the corresponding equilibrium autocorrelation function (—). p - 0.5,... 

Figure 11. Theoretical calculations via the CFP for a nonlinear itinerant oscillator with eflective potential harder than the harmonic one. The equilibrium autocorrelation function (—) and the excited autocorrelation function corresponding to an excited distribution P2 (—) are shown in the case of (F -1, 0 0.01, F - 0.03, and r — 3. For comparison with the linear case, we have also plotted the equilibrium autocorrelation function at (F - 0 (- -).

Arcsine distribution, 105, 111 Assumption of molecular chaos, 17 Asymptotic theory, 384 of relaxation oscillations, 388 Asynchronous excitation, 373 Asynchronous quenching, 373 Autocorrelation function, 146,174 Autocovariance function, 174 Autonomous problems, 340 nonresonance oscillations, 350 resonance oscillations, 350 Autonomous systems, 356 problems of, 323 Autoperiodic oscillation, 372 Averages, 100... 

The autocorrelation function G(t) corresponds to the correlation of a time-shifted replica of itself at various time-shifts (t) (Equation (7)).58,65 This autocorrelation defines the probability of the detection of a photon from the same molecule at time zero and at time x. Loss of this correlation indicates that this one molecule is not available for excitation, either because it diffused out of the detection volume or it is in a dark state different from its ground state. Two photons originating from uncorrelated background emission, such as Raman scattering, or emission from two different molecules do not have a time correlation and for this reason appear as a time-independent constant offset for G(r).58... 

Tj-hem TD = to2/4Dt the chemical relaxation time is much larger than the characteristic diffusion time so that there is no chemical exchange during diffusion through the excitation volume. The autocorrelation function is then given by... 

Triplet state kinetics can also be studied by FCS (Widengren et al., 1995). In fact, with dyes such as fluoresceins and rhodamines, additional fluctuations in fluorescence are observed when increasing excitation intensities as the molecules enter and leave their triplet states. The time-dependent part of the autocorrelation function is given by... 

When the excitation light is polarized and/or if the emitted fluorescence is detected through a polarizer, rotational motion of a fluorophore causes fluctuations in fluorescence intensity. We will consider only the case where the fluorescence decay, the rotational motion and the translational diffusion are well separated in time. In other words, the relevant parameters are such that tc rp, where is the lifetime of the singlet excited state, zc is the rotational correlation time (defined as l/6Dr where Dr is the rotational diffusion coefficient see Chapter 5, Section 5.6.1), and td is the diffusion time defined above. Then, the normalized autocorrelation function can be written as (Rigler et al., 1993)... 

A semiclassical description is well established when both the Hamilton operator of the system and the quantity to be calculated have a well-defined classical analog. For example, there exist several semiclassical methods for calculating the vibrational autocorrelation function on a single excited electronic surface, the Fourier transform of which yields the Franck-Condon spectmm [108, 109, 150, 244]. In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator [104-111, 245-248], which circumvent the cumbersome root-search problem in boundary-value-based semiclassical methods, have been successfully applied to a variety of systems (see, for example, Refs. 110, 111, 161, and 249 and references therein). The mapping procedure introduced in Section VI results in a quantum-mechanical Hamiltonian with a well-defined classical limit, and therefore it... 

The first relevant quantity required to obtain the rates is the autocorrelation function which are shown in Fig.8 for the ground vibrational level of the two excited electronic states. The two cases present a very similar behavior. Simply, for the A case its decay seems much faster. What is notorious is the large difference between the EP halfwidths as a function of the energy for the two electronic states, of approximately 2-3 orders of magnitude, as shown in Fig.9. This is explain by the norm of the initial wavepackets, which is much smaller for the B state, because its well is at larger R and shorter r, where the non-adiabatic couplings are much smaller. 

In the previous section, the phenomenological description of Brownian motion was presented. The Langevin analysis leads to a velocity autocorrelation function which decays exponentially with time. This is characteristic of a Markovian process, as Doobs has shown (see ref. 490). Since it is known heyond question that the velocity autocorrelation function is far from such an exponential function, the effect that the solvent structure has on the progress of a chemical reaction cannot be assessed very reliably by means of phenomenological Langevin description. Since the velocity of a solute is correlated with its velocity a while before, a description which fails to consider solute and solvent velocities can hardly be satisfactory. Necessarily, the analysis requires a modification of the Langevin or Fokker—Plank description. In this section, some comments are made on this new and exciting area of research. 

Here <( t ) f(t")> is the autocorrelation function of the electromagnetic field. For the case of excitation by a conventional light source, where the amplitudes and the phases of the field are subject to random fluctuations, the field autocorrelation function differs from zero for time intervals shorter than the reciprocal width of the exciting source. In the limit 8v A, that is when the spectral width, 8v, of the source exceeds the inhomogenously broadened line width, the field autocorrelation function can be represented as a delta function... 

Although the preparation of the excited state has been described in terms of a delta function excitation, the same results should be obtained for the case of excitation by a broad-band, random, conventional light source. We have pointed out, in Section VI, that in the case of the non-radiative decay of an excited state, the same behavior is predicted to follow excitation by a light source characterized by a second-order autocorrelation function which describes random phases and excitation by a delta function pulse. A similar situation prevails when the radiative decay channel is also taken into account. 

A time response function of the apparatus can be measured by upconversion of the excitation beam. The width of such measured instrument response function is 280fs (FWHM). Comparing this result with the width of the autocorrelation function of the dye laser 110fs we observe 170fs broadening of the instrument response function due to group velocity... 

In this section it will be outlined how the different molar masses contribute to the TDFRS signal. Of especial interest is the possibility of selective excitation and the preparation of different nonequilibrium states, which allows for a tuning of the relative statistical weights in the way a TDFRS experiment is conducted. Especially when compared to PCS, whose electric field autocorrelation function g t) strongly overestimates high molar mass contributions, a much more uniform contribution of the different molar masses to the heterodyne TDFRS diffraction efficiency t) is found. This will allow for the measurement of small... 

Expression (4.2) relates the energy dependence of the spectrum to the evolution of the molecular system in the excited electronic state as it is reflected by the autocorrelation function (Lax 1952 Gordon 1968 Cederbaum and Domcke 1977 Kulander and Heller 1978 Heller 1981a,b Koppel, Domcke, and Cederbaum 1984 Zare 1988 ch.3 Weissbluth 1989 ch.V). Structures in the spectrum can thus be explained in terms of features of S(t) which in turn can be traced back to the evolution of the molecular system. 

The autocorrelation function S(t) provides the link between the spectrum 0tot(w) on one hand and the (classical or quantum) dynamics in the excited electronic state on the other hand. 

If we switch on the coupling to the continuum at t = 0 the excited bound states begin to decay with the consequence that the wavepacket and therefore the autocorrelation function decay too. In order to account for this we multiply, according to Equation (7.14), each term in (7.18) by... 

The term diffuse structures is not well defined in the literature. It is used whenever structures in the spectrum cannot be unambiguously assigned. In the context of this chapter we identify diffuse structures with very broad resonances. The excited complex is so short-lived that the corresponding autocorrelation function exhibits one or at most two recurrences. 

The excited complex breaks apart very rapidly and only a minor fraction performs, on the average, one single internal vibration. Therefore, the total stationary wavefunction does not exhibit a clear change of its nodal structure when the energy is tuned from one peak to another (Weide and Schinke 1989). In the light of Section 7.4.1 we can argue that the direct part of the total wavefunction, S dir-, dominates and therefore obscures the more interesting indirect part, Sind- The superposition of the direct and the indirect parts makes it difficult to analyze diffuse structures in the time-independent approach. In contrast, the time-dependent theory allows, by means of the autocorrelation function, the separation of the direct and resonant contributions and it is therefore much better suited to examine diffuse structures. 

To understand the development or the absence of reflection structures one must imagine — in two dimensions — how the continuum wavefunction for a particular energy E overlaps the various ground-state wave-functions and how the overlap changes with E. This is not an easy task Figure 9.9 shows two examples of continuum wavefunctions for H2O. Alternatively, one must imagine how the time-dependent wavepacket, starting from an excited vibrational state, evolves on the upper-state PES and what kind of structures the autocorrelation function develops as the wavepacket slides down the potential slope. 

Fig. 15.8. Schematic one-dimensional illustration of electronic predissociation. The photon is assumed to excite simultaneously both excited states, leading to a structureless absorption spectrum for state 1 and a discrete spectrum for state 2, provided there is no coupling between these states. The resultant is a broad spectrum with sharp superimposed spikes. However, if state 2 is coupled to the dissociative state, the discrete absorption lines turn into resonances with lineshapes that depend on the strength of the coupling between the two excited electronic states. Two examples are schematically drawn on the right-hand side (weak and strong coupling). Due to interference between the non-resonant and the resonant contributions to the spectrum the resonance lineshapes can have a more complicated appearance than shown here (Lefebvre-Brion and Field 1986 ch.6). In the first case, the autocorrelation function S(t) shows a long sequence of recurrences, while in the second case only a single recurrence with small amplitude is developed. The diffuseness of the resonances or vibrational structures is a direct measure of the electronic coupling strength.

MD simulation is advantageous for obtaining dynamic properties directly, since the MD technique provides not only particle positions but also particle velocities that enable us to utilize the response theory (e.g., the Kubo formula [175,176]) to calculate the transport coefficients from time-dependent correlation functions. For example, we will examine the self-diffusion process of a tagged PFPE molecular center of mass (Fig. 1.49) from the simulation to gain insight into the excitation of translational motion, specifically, spreading and replenishment. The squared displacement of the center mass of a molecule or a bead is used as a measure of translational movement. The self-diffusion coefficient D can be represented as a velocity autocorrelation function... 

E>vib and Our also show up in the theory of spontaneous Raman spectroscopy describing fluctuations of the molecular system. The functions enter the CARS interaction involving vibrational excitation with subsequent dissipation as a consequence of the dissipation-fluctuation theorem and further approximations (21). Equations (2)-(5) refer to a simplified picture a collective, delocalized character of the vibrational mode is not included in the theoretical treatment. It is also assumed that vibrational and reori-entational relaxation are statistically independent. On the other hand, any specific assumption as to the time evolution of vib (or or), e.g., if exponential or nonexponential, is made unnecessary by the present approach. Homogeneous or inhomogeneous dephasing are included as special cases. It is the primary goal of time-domain CARS to determine the autocorrelation functions directly from experimental data. 

---