Vibrational transitions double-well potential

The contour plot is given in fig. 43. As remarked by Miller [1983], the existence of more than one transition states and, therefore, the bifurcation of the reaction path, is a rather common event. This implies that at least one transverse vibration, q in the case at hand, turns into a double-well potential. The instanton analysis of this PES has been carried out by Benderskii et al. [1991b]. The... 

Figure 2. Double-well potential V(q) with corresponding vibrational levels Ev and wave functions v(q) for the model 2,6-dicyanoethylmethylsemibullvalene (SBV) (adapted from Ref. 26). The reaction coordinate q indicates the Cope rearrangement of the model SBV from the reactant (R) isomer versus the transition state 1 to the product (P) isomer. Vertical arrows indicate the laser control of the isomerization R - — P by two IR femtosecond/picosecond laser pulses cf. Fig. 6 and Table I.

The potential (6.37) corresponds with the previously discussed projection of the three-dimensional PES V(p,p2,p3) onto the proton coordinate plane (pi,p3), shown in Figure 6.20b. As pointed out by Miller [1983], the bifurcation of reaction path and resulting existence of more than one transition state is a rather common event. This implies that at least one transverse vibration, q in the case at hand, turns into a double-well potential. The instanton analysis of the PES (6.37) was carried out by Benderskii et al. [1991b], The existence of the onedimensional optimum trajectory with q = 0, corresponding to the concerted transfer, is evident. On the other hand, it is clear that in the classical regime, T > Tcl (Tc] is the crossover temperature for stepwise transfer), the transition should be stepwise and occur through one of the saddle points. Therefore, there may exist another characteristic temperature, Tc2, above which there exists two other two-dimensional tunneling paths with smaller action than that of the one-dimensional instanton. It is these trajectories that collapse to the saddle points at T = Tcl. The existence of the second crossover temperature Tc2 for two-proton transfer was noted by Dakhnovskii and Semenov [1989]. 

Figure 17. Two-dimensional double-well potential for reaction in the Sumi-Marcus model spanned by slow (diffusive) molecular-arrangement fluctuations in solvents on the abscissa for the coordinate X and fast (ballistic) intrasolute vibrational fluctuations on the ordinate. Also shown is a reactive trajectory surmounting the transition-state barrier on the line C with an A -dependent rate constant...

Figure 6.1 Harmonic (dash), Morse (dots, S = 0.31) and double-well potentials for the O-H stretch (coordinate q) motion in an 0-H---0 bond with a fixed 0---0 distance = 2.65 A. Eigenvalues corresponding to observable values of vibrational energies, and defined by indices = 0, 1, etc., are shown for each potential. All three potentials have the same curvature at g = 1 A, defined by the parameter (o which is such that w/2ttc = 3000cm (c velocity of light in cm sec ). With this value the 0 —> 1 transition of the harmonic potential, represented by a vertical arrow, falls at f = 3000 cm .

The electronic ground state of this non-polar species is still of interest because classical rotationally resolving methods are difficult to apply. Bermejo et al. [02Ber] report on high-resolution rovibrational Raman spectra of Cl2, Cl2, and Cl Cl, and their spectroscopic constants for the vibrational states v = 0,1,2. Wang et al. [98Wan] have measured rotationally resolved vacuum ultraviolet REMPI detected laser spectra of the Cl2 1 <— X transition, and obtained rotational constants of levels loealized in the double-well potential of... 

A lot of theoretical work on displacive phase transitions has focussed on a simple model in which atoms are connected by harmonic forces to their nearest neighbors, and each neighbor also sees the effect of the rest of the crystal by vibrating independently in a local potential energy well (Bruce and Cowley 1980). For a phase transition to occur, this double well must have two minima, and can be described by the following function ... 

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