Autocorrelation function coupling

When translational diffusion and chemical reactions are coupled, information can be obtained on the kinetic rate constants. Expressions for the autocorrelation function in the case of unimolecular and bimolecular reactions between states of different quantum yields have been obtained. In a general form, these expressions contain a large number of terms that reflect different combinations of diffusion and reaction mechanisms. 

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. 

The velocity autocorrelation function (VAF) may be used to investigate the possibility of coupling between translational and rotational motions of the sorbed molecules. The VAF is obtained by taking the dot product of the initial velocity with that at time t. It thus contains information about periodic fluctuations in the sorbate s velocity. The Fourier transform of the VAF yields a frequency spectrum for sorbate motion. By decomposing the total velocity of a sorbate molecule into translational and rotational terms, the coupling of rotational and translational motion can be investigated. This procedure illustrates one of the main strengths of theoretical simulations, namely to predict what is difficult or impossible to determine experimentally. 

In more complex molecular systems, increased coupling between the translational motion and both rotational and vibrational modes occurs. It is difficult to separate these effects completely. Nevertheless, the velocity autocorrelation functions of the Lennard—Jones spheres [519] (Fig. 52) and the numerical simulation of the carbon tetrachloride (Fig. 39) are quite similar [452a]. 

If F is an operator working on the system S alone this autocorrelation function describes the fluctuations in S under influence of the bath. We compute it to second order in the coupling constant a. 

The friction on a tagged particle is expressed in terms of the time-dependent force-force autocorrelation function. Although the bare, short-time part of the friction that arises from binary collisions can be calculated from kinetic theory, the long-time part needs the knowledge of the solvent and the solute dynamics and the coupling between them. The solvent dynamic quantities... 

It is well known that the velocity autocorrelation function decays as f3/2 in the asymptotic limit due to the coupling between the tagged particle motion and the transverse current mode of the solvent [23, 56, 57]. The asymptotic limit of the Rn term can be calculated by assuming that Fs(q, t) and Ctt(q, t) have simple diffusive behavior. Thus the expression for Rn in this limit takes the following form ... 

To obtain an approximate expression for the density autocorrelation function, first we consider that the density fluctuation is coupled only to the longitudinal current fluctuation, and its coupling to the temperature fluctuation and other higher-order components are neglected. 

With this consideration die relaxation equation will give rise to a set of coupled equations involving the time autocorrelation function of the density and the longitudinal current fluctuation, and also there will be cross terms that involve the correlation between the density fluctuation and the longitudinal current fluctuation. This set of coupled equations can be written in matrix notation, which becomes identical to that derived by Gotze from the Liouvillian resolvent matrix [3]. 

There have been various approaches in the mode coupling theory [9, 37, 57, 176]. All these theories have exhibited the presence of t 3/2 of the velocity autocorrelation function in the asymptotic limit in three dimensions. Extending each of these theories for studies in two dimensions we can show that the velocity autocorrelation function has r1 tail in the asymptotic limit. Since the diffusion coefficient is related to Cv(t) through Eq. (337), it can be shown that D diverges in the long time due to the presence of this t l tail in the VACF. 

In the specific cases of the spectra of the metal compounds discussed in this chapter, a third mode is observed with no evidence of coupling to the Qx and Qy coordinates. In the framework of the time-dependent theory, the total autocorrelation function is the product of < 4> 4>(t) > of the coupled coordinates discussed above and the <4> 4>(t)> from a separate calculation for the third mode. 

The decrease in < 0 < (t) > depends on the slope of the potential surface at the point at which the wavepacket is initially placed. The slope of the potential in the Qy direction is steeper on the positive Qx side of the surface than on the negative Qx side. When the slope in the Qy dimension is large, the Qy part of the two-dimensional wavepacket will rapidly change its shape and < 010 (t) > will decrease rapidly. Therefore, for a positive Qx displacement in the coupled potential, the autocorrelation function will decrease more rapidly than it would for a negative displacement. 

The magnitudes of < (j>(t)> versus time are shown in Fig. 4. The autocorrelation function for the positive displacement along Qx in the coupled potential (lowest curve in Fig. 4), starts out at 1 and drops to 0 over a shorter period of time than in the uncoupled potential (middle line). The Fourier transforms of these < (t) > give the spectra shown in Fig. 3. This reasoning explains why a positive displacement results in a broader progression and a negative displacement results in a narrower progression. 

Fig. 4. Autocorrelation functions plotted versus time for the spectra described in Fig. 3. The autocorrelation function shown by the middle curve corresponds to the reference spectrum (Fig. 3a), k.j, = Ocm-1, the autocorrelation function shown by the lowest curve corresponds to a positive displacement in the quadratically coupled potential surface, and the autocorrelation function shown by the top curve corresponds to a negative displacement in the quadratically coupled potential surface...

Each rotational state is coupled to all other states through the potential matrix V defined in (3.22). Initial conditions Xj(I 0) are obtained by expanding — in analogy to (3.26) — the ground-state wavefunction multiplied by the transition dipole function in terms of the Yjo- The total of all one-dimensional wavepackets Xj (R t) forms an R- and i-dependent vector x whose propagation in space and time follows as described before for the two-dimensional wavepacket, with the exception that multiplication by the potential is replaced by a matrix multiplication Vx-The close-coupling equations become computationally more convenient if one makes an additional transformation to the so-called discrete variable representation (Bacic and Light 1986). The autocorrelation function is simply calculated from... 

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... 

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.

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