Intramolecular multidimensional

The surface in Fig. 12 demonstrates that there is little coupling between the C—F translation coordinate and the bending coordinate of the complex. Stated another way, the time scale for intramolecular vibrational redistribution between these coordinates is slow compared to the time scale for breaking the C—F bond. These conclusions are not obvious upon examination of the minimum energy path shown in Fig. 11, and indeed such diagrams, while generally instructive, can lead to improper conclusions because they hide the multidimensional nature of the true PFS. A central assumption of statistical product distribution theories... 

If there were no intramolecular interactions (such as bonding or excluded volume), then V(R) = 0, and the next guess for the density profile can be obtained directly from Eq. (75). The presence of V(R) necessitates either a multidimensional integration or (more conveniently) a single-chain simulation. 

Water is an interesting and important liquid. As the combination of multidimensional vibrational spectroscopy and molecular dynamics helps us understand water better, more and more complex dynamics have been revealed [8,9]. We can briefly explain why a liquid of triatomic molecules turns out to be so immensely complicated Water s three atoms bestow all the complexity of multiple intramolecular vibrations, and in addition in water there are more hydrogen bonds (—3.57) than atoms ... 

J. Troe Professor Marcus, you were mentioning the 2D Sumi-Marcus model with two coordinates, an intra- and an intermolecu-lar coordinate, which can provide saddle-point avoidance. I would like to mention that we have proposed multidimensional intramolecular Kramers-Smoluchowski approaches that operate with highly nonparabolic saddles of potential-energy surface [Ch. Gehrke, J. Schroeder, D. Schwarzer, J. Troe, and F. Voss, J. Chem. Phys. 92, 4805 (1990)] these models also produce saddle-point avoidances, but of an intramolecular nature the consequence of this behavior is strongly non-Arrhenius temperature dependences of isomerization rates such as we have observed in the photoisomerization of diphenyl butadiene. 

The one-dimensional potential curves depicted in Figures 1.1-1.3 represent the dissociation of diatomic molecules for which the potential V(Rab) depends only on the internuclear distance between atoms A and B. However, if one or both constituents are molecules, V is a multidimensional object, a so-called potential energy surface which depends on several (at least three) nuclear coordinates denoted by the vector Q = (Ql,Q2,Q3,---) (Margenau and Kestner 1969 Balint-Kurti 1974 Kuntz 1976 Schaefer III 1979 Kuntz 1979 Truhlar 1981 Salem 1982 Murrell et al. 1984 Hirst 1985 Levine and Bernstein 1987 ch.4 Hirst 1990 ch.3). The intramolecular and intermolecular forces, defined by... 

The sin7o term reflects the fact that the intramolecular vector r is expressed in polar coordinates it is especially important for the dissociation of linear molecules and must not be forgotten The delta-function S(Hf — Ef) selects only those points (i.e., trajectories) in the multidimensional phase-space that have the correct energy Ef. It ensures that the quantum mechanical resonance condition Ef = Ei + Ephoton is fulfilled. 

Fig. 1. Potential diagram of the TICT forming reaction for DMABN and derivatives in a medium polar solvent [24]. The reaction coordinate comprises intramolecular twist, relaxation of the surrounding solvent molecules and a further relaxation coordinate, possibly connected with the pyramidalization at the amino group [22] and is therefore multidimensional...

All intermolecular interactions can be adequately described, at least in principle, by multidimensional scalar and vector fields representing the energetics of a molecular system as functions of both intermolecular distances and orientations as well as intramolecular structure data. The visualization of these fields, however, has to be based on a three-dimensional picture or a two-dimensional projection because human pattern recognition ability is strongly related to the two- and three-dimensional world. Consequently, the multidimensional field has to be reduced to a two- or three-dimensional representation. In molecular science this can be done in many different ways. 

The chemical reaction is assumed to proceed when the molecule passes irreversibly from a region of configuration space identified as the reactant to a region identified as the product through a saddle point in the multidimensional potential surface. It is assumed that this saddle point is characterized by a local maximum of the potential along one degree of freedom (the reaction coordinate), while the other intramolecular degrees of freedom maintain stable oscillations about their local minima. This saddle point constitutes the potential barrier to the reaction. 

There have been two general directions for classical trajectory studies of IVR (Gomez and Poliak, 1992). One involves the analysis of classical Hamiltonians for molecules and is concerned with the structure of the multidimensional phase space for excited molecules and the mechanism(s) for intramolecular energy transfer (Lichtenberg and Lieberman, 1991). The other is concerned with determining how to use classical mechanics to represent the initially excited zero-order state i and then propagate I (/). There is also an interest in the correspondence between classical and quantal descriptions of IVR (Brumer and Shapiro, 1988). 

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