Slow mode potential

Examination of Eqs. (54) shows that, within the quantum representation III, the excitation of the fast mode displaces the origin of the slow mode wave functions toward shorter lengths. That may be viewed as a translation of the slow mode potential, that is induced by the excitation of the fast mode. In order to visualize this potential displacement, it is suitable to consider the potential as Morse-like. That is depicted on the right-hand side of Fig. 2. Here, the left-hand side is devoted to the quantum representation //, where there is no potential... 

As a consequence of the translation of the origin of the slow mode potential induced by the excitation of the fast mode, there are in representation ///, overlaps between the wave functions of the H-bond bridge corresponding, respectively, to the ground state of the fast mode and to its first excited state, that is,... 

Figure 1. Main anharmonicities of the fast mode. F, fast mode S, slow mode (intrinsic anharmonic Morse potential) B, bending mode.

If we want to remove the driven term in the potential of the slow mode, when k = 0 (ground state of the fast mode), it is suitable to perform the following phase transformation ... 

Several studies of Fermi resonances in the absence of H bond have been made [76-80]. We shall account for this situation by simply ignoring the anharmonic coupling between the fast and slow modes (a = 0). The theory then describes the coupling between the fast mode and a bending mode through the potential Htf, with both of these modes being damped in the same way. Because aG = 0, the slow mode does not play any role, so that the total Hamiltonian does not refer to it ... 

Equation (21) gives the potential energy of the system as a function of nuclear positions only. These are slow modes on the time scale of the electronic transitions. However, we must also realize that the system contains fast modes, such as the solvent electronic polarizability. Since we are not interested here in the dynamics of these modes, we will assume that ensemble averages calculated with the potential energy in equation (21) have all the fast modes equilibrated to the temperature T. Thus the free energy difference between the two states is given by... 

We have already encountered this type of potential in eq. (4.6) where it governed Q, intermonomer modes. It tends towards the harmonic potential of eq. (6.1) when the parameter 8, characteristic of this ID anharmonicity in q, tends towards 0. The vibrational potential experienced by the slow mode when the rapid mode q is in its nth excited state, is then equal to... 

Table III contains correlation times and dominant eigenvalues for a second rank observable in a first rank potential. There are still roughly two ranges of decay rates when the solvent body is slow (D, 1). The slower range is mostly due to the FRD of the solvent body, while the faster one is described by motions of the solute body and/or dynamic interactions. This faster decay is hardly described by a single frequency, unlike the case of a first rank correlation function. Rather, it is controlled by a few eigenvalues of the same order of magnitude. Thus for Dj = 0.01 and Uj = 2 the slow mode, entry 3a in Table III, is largely described by a 7, = 0, 72 = 2 term the fast mode 3b is mostly due to dynamic interactions (7j = 1, 72 = 1 and 7, = 1, 72 = 3 are the important terms) and the fast...

Finally, results on second rank correlation functions for a second rank potential are collected in Table V. The situation is now very similar to the corresponding set of data for a first rank potential (Table III), since even rank correlation functions are not sensitive, for symmetry reasons, to jump motions. The slow mode is then again mostly due to the FRD of the larger solvent body while the fast modes are mainly dominated by motions of the first body (cf. the entries for D2 = 0.01, Uj = 2 and 4 in Table V). Note that other faster eigenvalues are present, with smaller weights, whose nature is mostly mixed, but are not listed in the table. Their individual contribution to the overall decay of the correlation function is small, but their cumulative weights may be around 0.1-0.3 or even more. 

One of them is much slower than the others, and the fastest one becomes Hbrational (i.e., it acquires a detectable imaginary part) in the inertial regime (wi = 0.5). Note, however, that since the second rank potential coupling provides two potential minima in which the solute can reorient (with the possibility of jump motions), the nature of the slow mode in... 

Nothing of this sort is observed for second rank correlation functions, since the dominant slow mode is simply a FRD of the solvent body. In both first and second rank correlation functions one notes that librational modes are slightly more important when the potential coupling is second rank than they were for a first rank potential coupling. This may be due to the increased curvature of the potential near the minima. 

The complex nature of the slow mode responsible for the long-time behavior of first rank correlation functions for a first rank interaction potential is illustrated by the composition of the eigenvector corresponding to the slow mode 11a in Table XI, for Uj = 3 and o) = 0.5. Note that n 1, tij, ii, J2 describe the magnitudes and the orientations of the momentum vectors Lj and L2 j is referred to the orientation of L, -t- Lj, 7, and J2 are related to the orientations of the two bodies, and the total orientational angular operator defines the quantum number J finally J, which is not included in this table, is the total angular momentum quantum number, and it is always equal to 1 for first rank orientational and momentum correlation functions, and to 2 for second rank correlation functions. In Fig. 11 we show the first rank correlation functions for different collision frequencies of body 1. The second rank correlation function decays are plotted in Fig. 12. The librational motions in the wells are more important than they were in the first rank potential case (since there is now a more accentuated curvature of the potential wells). 

The first question, which is raised by this simple analogy, concerns the very possibility of exciting slow mode wave packets in a hydrogen bond at all. Taking a different perspective it touches the very issue of interpretation of the notoriously complex IR spectra [9]. In the condensed phase much of this complexity is hidden under bands broadened by the solvent interaction. Hence it was only recently that coherent wave packet motion of a 100 cm i hydrogen bond mode could be observed after OH-stretch excitation, although in a system which has only a single minimum potential [10]. Meanwhile coherent low-frequency dynamics has also been observed in a double minimum system (acetic acid dimer) [11]. With this... 

Figure 15.1 Anharmonic coupling ofthe O-H stretching mode q and a low-frequency hydrogen bond (0...0) mode Q. (a) Potential energy diagram for the low-frequency mode in a single hydrogen bond. The potential energy surfaces as defined by the stretching mode and the quantum levels ofthe low-frequency mode are plotted for the Voh = 0 and 1 states as a function of the slow-mode coordinate Q.

One should keep in mind that the rate constant increases monotonically with coupling only if the relaxation time of the fast mode is infinitely small. If it is finite, as it should be, the rate versus couphng dependence starts as the Golden Rule prescribes, then continues according to the Sumi-Marcus theory, but finally saturates at a certain value determined by the relaxation time of the fast mode. This effect is illustrated in Figure 9.18(h) for the case where both fast and slow modes relax exponentially. In this case, the problem is reduced to solution of a system of 2D diffusion equations along parabolic potential surfaces corresponding to two coordinates and... 

To describe the effect of the above parameters on interdiflfusion, de Gennes [67] used the chemical potential gradient as the driving force for interdiffusion. Assuming that the fluxes of the two components were equal, but opposite, Brochard-Wyart et al. [68] derived the slow-mode theory for interdiffusion at polymer interfaces. 

It is inefficient and potentially not conclusive, especially when the process possesses fast and slow modes ... 

The low probability of mincycEN = 2 in slow mode, triple search is the reason why the length of the linear pathway in the event-related potentials of subjects using this variant is shorter than the reaction time data (with mincycEN = 2) suggest. In the reaction time distribution of these subjects the break between the linear and the cyclical pathway does not lie mincycEN = 2ET left of the first peak but for example 4ET left of the first peak (see above). 

Brown and collaborators interpret their spectra as showing that S q, t) in some systems has multiple slow modes, some being -independent while others scale as (43,48). Brown, et a/. s interpretation potentially explains all slow mode behaviors, namely in different circumstances the slow mode is dominated by a -dependent or by a -independent component. A spectrum whose modes have -dependent shapes might in some cases also be described as a mixture of and °-dependent relaxations whose relative amplitudes are not constant. The relationship between the spectral analyses of Brown and Stepanek(29), who interpret their spectra via a regularized Laplace transform method, and the work of Phillies and collaborators(87), who interpret S q,t) as a sum of stretched exponentials whose parameters depend on q and c, has not been completely analyzed. The latter interpretation has the virtue of supplying quantitative parameters for further analysis. 

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