Autocorrelation function excited, decay

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. 

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)... 

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. 

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. 

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 nature of the excitation has a profound influence on the subsequent relaxation of molecular Uquid systems, as the molecular dynamics simulations show. This influence can be exerted at field-on equiUbrium and in decay transients (the deexdtation effect). GrigoUni has shown that the effect of high-intensity excitation is to slow the time decay of the envelope of such oscillatory functions as the angular velocity autocorrelation function. The effect of high-intensity pulses is the same as that of ultrafast (subpicosecond laser) pulses. The computer simulation by Abbot and Oxtoby shows that... 

The autocorrelation function Q(t) is evaluated at equilibrium, whereas A (r) is a transient property requiring preliminary excitation of the variable of interest A. Evans investigated this problem by monitoring via computer simulations the time behavior of a liquid sample after the instantaneous removal of a strong external held of force E. He found that at the point liE/kT = 12, A (t) decays considerably faster than Here is the dipole of the tagged molecule and is the energy associated with the held of force E. In that case A is the component of the dipole along the Z axis. 

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,... 

Thus, as mentioned earlier, time-resolved depolarization measurements afford a means of recording the time profile of the rotational autocorrelation function. The steady state technique, with continuous sample excitation, produces merely the time average of the emission anisotropy, F. For a rotating chromc hore with a sin e fluorescence decay time Tf, F is related to r(t) by the following expression... 

Time-resolved optical experiments rely on a short pulse of polarized light from a laser, synchrotron, or flash lamp to photoselect chromophores which have their transition dipoles oriented in the same direction as the polarization of the exciting light. This non-random orientational distribution of excited state transition dipoles will randomize in time due to motions of the polymer chains to which the chromophores are attached. The precise manner in which the oriented distribution randomizes depends upon the detailed character of the molecular motions taking place and is described by the orientation autocorrelation function. This randomization of the orientational distribution can be observed either through time-resolved polarized fluorescence (as in fluorescence anisotropy decay experiments) or through time-resolved polarized absorption. 

Fig. 11. Rise and decay curve of the excimer fluorescence of (A) 1Py(3)1Py adsorbed on Si-C- 8 (3 x 10 6 mol/g) and (B) Py adsorbed on Si-C- 8 (5-5 10 5 mol/g), fitted to three exponentials. The excitation pulse (337 nm) is also depicted in each case. The values for the decay parameters A 1 (in ns) and their amplitudes are given (see text). The weighted deviations in units of a (expected de- viation), the autocorrelation function (A-C), and the value for x " are also indicated. (Reprinted with permission from the Journal of Physical Chemistry, 86 (1985) 3521, our ref. (38), Copyright (1985) American Chemical Society).

Figure 9. Fluorescence decay of the donor, MNA, in MNA-BPTI in 50% glycerol in 0.05M Bicine buffer, pH 7.5 at -30°C. The experimental curve was fitted to a monoexponential decay function. The shortest pulse,L(t), is the trace of the excitation pulse. The second pulse, F(t), shows the experimental and calculated donor fluorescence decay curves- The lower inset shows the deviation between the two curves, and the upper right inset shows the autocorrelation of the residuals-These show that the decay rate is raonoexponential (RMS of fit 0-0030)-...

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