Microcanonical normal mode sampling

To form a microcanonical ensemble random values for the P( and Qt are chosen so that there is a uniform distribution in the classical phase space of ( , Q) [17]. Two ways are described here to accomplish this. For one method, called microcanonical normal-mode sampling [18], random values are chosen for the mode energies Et, which are then transformed to random values for Pt and Qt. In the second method, called orthant sampling [11], random values for P and Q are sampled directly from the phase space. 

The remaining steps for orthant sampling are the same as steps 2 and 3 described above for microcanonical normal-mode sampling. 

Cartesian coordinates and momenta for a polyatomic reactant are found from the energies of its normal modes [Eq. (2.33)] and the components of its angular momentum. The procedure is given by Eqn. (210) and steps 1-3 for microcanonical normal-mode sampling in Section II.A.3.a, and is applied to both reactants. Each reactant is randomly rotated through its Euler angles, as described by Eqs. (3.24) and (3.25), and the impact parameter, center-of-mass separation, and relative velocity added as described by Eqs. (3.26) and (3.27). 

Microcanonical normal mode sampling adds a constant total energy E to n normal modes of a molecule. The probability that normal mode I has energy E is proportional to the number of ways the energy E — E may be added to the remaining n — 1 oscillators. This is given by the classical density of states for these oscillators. Thus, the normalized probability that normal mode I contains energy E is... 

Microcanonical normal mode sampling. Microcanonical normal mode sampling may be used to prepare approximate... 

For many applications, it may be reasonable to assume that the system behaves classically, that is, the trajectories are real particle trajectories. It is then not necessary to use a quantum distribution, and the appropriate ensemble of classical thermodynamics can be taken. A typical approach is to use a microcanonical ensemble to distribute energy into the internal modes of the system. The normal-mode sampling algorithm [142-144], for example, assigns a desired energy to each normal mode, Qa as a harmonic amplitude... 

Trajectories were initiated by generating initial conditions with the efficient microcanonical sampling or quasiclassical normal-mode sampling procedures at 54.6 or 146.0 kcalmoC1 of vibrational energy for trimethylene. Trimethylene was then placed in the center of a box, with periodic boundary conditions, and surrounded by an argon bath with an equilibrium temperature and density. Initially, trimethylene was in a nonequilibrium state with respect to the bath, since its coordinates and momenta were held fixed while the bath was equilibrated, and the trajectories were propagated until either cyclopropane or propene was formed. 

Both microcanonical normal-mode and orthant sampling give random sampling of the phase space and prepare microcanonical ensembles. For each, the average energy in a normal-mode is E/n. 

Carlo sampling schemes are described for exciting A randomly with a microcanonical ensemble of states and nonrandomly with specific state selection. For pedagogical purposes, selecting a microcanonical ensemble for a normal mode Hamiltonian is described first. 

If bottlenecks restrict intramolecular vibrational energy redistribution," the unimolecular dissociation is not random and not in accord with equation (4). There is considerable interest in identifying which unimolecular reactions do not obey equation (4). In this section Monte Carlo sampling schemes are described for exciting A randomly with a micro-canonical ensemble of states and nonrandomly with mode selective excitation. For pedagogical purposes, selecting a microcanonical ensemble for a normal mode Hamiltonian is described first. 

This sampling, with the normal-mode/rigid-rotor Hamiltonian, provides an exact microcanonical ensemble for this Hamiltonian, but an approximate microcanonical ensemble for the actual anharmonic and reactive Hamiltonian with vibrational-rotational coupling. 

In the previous sections random sampling procedures were described for preparing a classical microcanonical ensemble of states. However, for comparison with experiment and to study the rate of intramolecular vibrational energy redistribution (IVR) [12,27] within a molecule, it is important to sample the states (i.e., phase space) of the molecule nonrandomly. Here, both normal and local mode [28] quasi-classical [2-4] sampling schemes are described for nonrandom excitation. 

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