The generalised Langevin equation and reactions in solution

Adelman [530] and Stillman and Freed [531] have discussed the reduction of the generalised Langevin equation to a generalised Fokker— Planck equation, which provides a description of the probability that a molecule has a velocity u at a position r at a time t, given certain initial conditions (see Sect. 3.2.). The generalised Fokker—Planck equation has important differences by comparison with the (Markovian) Fokker— Planck equation (287). However, it has not proved so convenient a vehicle for studies of chemical reactions in solution as the generalised Langevin equation (290). 

They solved the equation of motion for the two reactants and their respective nearest neighbour shells with a potential energy of interaction between the two iodine atoms of the Morse oscillator type. A Monte Carlo technique was used and reaction occurred immediately the iodine atoms came into contact. The results of these simulations ate shown in Fig. 55. As the frequency of oscillation coe, of the iodine atom in the 

There have been several studies of the iodine-atom recombination reaction which have used numerical techniques, normally based on the Langevin equation. Bunker and Jacobson [534] made a Monte Carlo trajectory study to two iodine atoms in a cubical box of dimension 1.6 nm containing 26 carbon tetrachloride molecules (approximated as spheres). The iodine atom and carbon tetrachloride molecules interact with a Lennard—Jones potential and the iodine atoms can recombine on a Morse potential energy surface. The trajectives were followed for several picoseconds. When the atoms were formed about 0.5—0.7 nm apart initially, they took only a few picoseconds to migrate together and react. They noted that the motion of both iodine atoms never had time to develop a characteristic diffusive form before reaction occurred. The dominance of the cage effect over such short times was considerable. 

Stace and Murrell [535] numerically simulated the reaction of iodine atoms in the inert gases, helium, argon and xenon using a method similar to that of Bunker and Jacobson above. Most of their interest was in the reaction process at low inert gas pressures. At high inert gas pressures, a cage effect was noted over times longer than 50 ps. 

A slightly different technique was used by Allen [536]. A large number of solvent molecules were allowed to move each according to a Langevin equation in a force field of all the other molecules (which interact by Leonard-Jones potentials). In this system, there are two reactants, AB and C, and the reaction is of the type 

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