Boltzmann distribution periodic potentials

Due to thermal movement of the solvent molecules in the solvation shell, the potential energy of the reduced and oxidized species may deviate from their equilibrium value. If we could probe the energy of an individual species over a sufficiently long time period, we would find a distribution in its energy around the minimum. The probability that the energy of the oxidized and reduced species deviates from the equilibrium value by E(p) - E is given by the Boltzmann distribution ... 

The setup of the initial configuration, the application of the interaction potential, and the periodic boundary conditions are identical in MD and MC methods. In the case of MD, a Boltzmann distribution of velocities appropriate to the temperature is also assigned to the atoms. The atoms move under the gradients (the negative of the gradients gives the force that is actually used in the MD simulation) of the potential for a time step (At) according to Newton s laws of motion (F = ma = -dVIdr) to obtain a new set of coordinates, and the process is repeated for N time steps to obtain a simulation time of NAt.42... 

The particular model used in the original simulation i- of this reaction was that of a Cl + CI2 like reaction as modeled by a LEPS potential energy surface. The barrier for this symmetric reaction was normally taken to be 20 kcal/mol (—33 kT at room temperature). Other simulations used 10 and 5 kcal/mol barriers. The reactants were placed in either a 50 or 100 atom solvent (Ar in the earliest simulations Ar, He, or Xe in the later work) with periodic truncated octahedron boundary conditions. To sample the rare reactive events, as described previously, this system was equilibrated with the Cl—Cl—Cl reaction coordinate constrained at its value at the transition state dividing surface (specifically, the value of the antisymmetric stretch coordinate was set equal to zero). From symmetry arguments, this constraint is the appropriate one (except in the rare case where the solvent stabilizes the transition state sufficiently such that a well is created at the top of the gas phase barrier). For each initial configuration, velocities were chosen for all coordinates from a Boltzmann distribution and molecular dynamics run for 1 ps both forward and backward in time. 

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