Dynamic reaction coordinate

Ln = Pr, Nd, Pm, Sm, Dy, Ho, Er, and Tm X = F, Cl, Br, and I. Ground electronic states for all trihalides were established, assuming that the molecular symmetry was planar (Dsa) rather than pyramidal (Csv). Spin-orbit interaction was ignored. Comparison of calculated Ln-X bond lengths with experimental data showed that description of dynamic electron correlation was absolutely necessary for correct results. These studies on lanthanide systems were later extended to hydration models of trivalent rare-earth ions [253] for Y +, La +, Gd +, and Lu + geometry optimization was carried out at the MP2 level for hydrates containing from one to ten water molecules. In addition, ab initio molecular dynamics simulations (by following the dynamical reaction coordinate) for the systems with more water molecules, [M(H20)24] (M = Y, La) and [La(H20)64] ", were done and both radial distribution function and coordination number were obtained. 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

Haynes G R and Voth G A 1995 Reaction coordinate dependent friction in classical activated barrier crossing dynamics when it matters and when it doesn t J. Chem. Phys. 103 10 176... 

Carter E A, Ciccotti G, Hynes J T and Kapral R 1989 Constrained reaction coordinate dynamics for the simulation of rare events Chem. Phys. Lett. 156 472-7... 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

In order to define how the nuclei move as a reaction progresses from reactants to transition structure to products, one must choose a definition of how a reaction occurs. There are two such definitions in common use. One definition is the minimum energy path (MEP), which defines a reaction coordinate in which the absolute minimum amount of energy is necessary to reach each point on the coordinate. A second definition is a dynamical description of how molecules undergo intramolecular vibrational redistribution until the vibrational motion occurs in a direction that leads to a reaction. The MEP definition is an intuitive description of the reaction steps. The dynamical description more closely describes the true behavior molecules as seen with femtosecond spectroscopy. 

Both molecular dynamics studies and femtosecond laser spectroscopy results show that molecules with a sufficient amount of energy to react often vibrate until the nuclei follow a path that leads to the reaction coordinate. Dynamical calculations, called trajectory calculations, are an application of the molecular dynamics method that can be performed at semiempirical or ah initio levels of theory. See Chapter 19 for further details. 

The original microscopic rate theory is the transition state theory (TST) [10-12]. This theory is based on two fundamental assumptions about the system dynamics. (1) There is a transition state dividing surface that separates the short-time intrastate dynamics from the long-time interstate dynamics. (2) Once the reactant gains sufficient energy in its reaction coordinate and crosses the transition state the system will lose energy and become deactivated product. That is, the reaction dynamics is activated crossing of the barrier, and every activated state will successfully react to fonn product. 

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