Microcanonical trajectory

In deriving the RRKM rate constant in section A3.12.3.1. it is assumed that the rate at which reactant molecules cross the transition state, in the direction of products, is the same rate at which the reactants fonn products. Thus, if any of the trajectories which cross the transition state in the product direction return to the reactant phase space, i.e. recross the transition state, the actual unimolecular rate constant will be smaller than that predicted by RRKM theory. This one-way crossing of the transition state, witii no recrossmg, is a fiindamental assumption of transition state theory [21]. Because it is incorporated in RRKM theory, this theory is also known as microcanonical transition state theory. 

The first classical trajectory study of iinimoleciilar decomposition and intramolecular motion for realistic anhannonic molecular Hamiltonians was perfonned by Bunker [12,13], Both intrinsic RRKM and non-RRKM dynamics was observed in these studies. Since this pioneering work, there have been numerous additional studies [9,k7,30,M,M, ai d from which two distinct types of intramolecular motion, chaotic and quasiperiodic [14], have been identified. Both are depicted in figure A3,12,7. Chaotic vibrational motion is not regular as predicted by tire nonnal-mode model and, instead, there is energy transfer between the modes. If all the modes of the molecule participate in the chaotic motion and energy flow is sufficiently rapid, an initial microcanonical ensemble is maintained as the molecule dissociates and RRKM behaviour is observed [9], For non-random excitation initial apparent non-RRKM behaviour is observed, but at longer times a microcanonical ensemble of states is fonned and the probability of decomposition becomes that of RRKM theory. 

Apparent non-RRKM behaviour occurs when the molecule is excited non-randomly and there is an initial non-RRKM decomposition before IVR fomis a microcanonical ensemble (see section A3.12.2). Reaction patliways, which have non-competitive RRKM rates, may be promoted in this way. Classical trajectory simulations were used in early studies of apparent non-RRKM dynamics [113.114]. 

Several VTST techniques exist. Canonical variational theory (CVT), improved canonical variational theory (ICVT), and microcanonical variational theory (pVT) are the most frequently used. The microcanonical theory tends to be the most accurate, and canonical theory the least accurate. All these techniques tend to lose accuracy at higher temperatures. At higher temperatures, excited states, which are more difficult to compute accurately, play an increasingly important role, as do trajectories far from the transition structure. For very small molecules, errors at room temperature are often less than 10%. At high temperatures, computed reaction rates could be in error by an order of magnitude. 

If the coupling parameter (the Bath relaxation constant in HyperChem), t, is too tight (<0.1 ps), an isokinetic energy ensemble results rather than an isothermal (microcanonical) ensemble. The trajectory is then neither canonical or microcanon-ical. You cannot calculate true time-dependent properties or ensemble averages for this trajectory. You can use small values of T for these simulations ... 

Another problem with microcononical-based CA simulations, and one which was not entirely circumvented by Hermann, is the lack of ergodicity. Since microcanoriical ensemble averages require summations over a constant energy surface in phase space, correct results are assured only if the trajectory of the evolution is ergodic i.e. only if it covers the whole energy surface. Unfortunately, for low temperatures (T << Tc), microcanonical-based rules such as Q2R tend to induce states in which only the only spins that can flip their values are those that are located within small... 

H. Waalkens, A. Burbanks, and S. Wiggins, Efficient procedure to compute the microcanonical volume of initial conditions that lead to escape trajectories from a multidimensional potential well, Phys. Rev. Lett. 95, 084301 (2005). 

Because T -> V energy transfer does not lead to complex formation and complexes are only formed by unoriented collisions, the Cl" + CH3C1 -4 Cl"—CH3C1 association rate constant calculated from the trajectories is less than that given by an ion-molecule capture model. This is shown in Table 8, where the trajectory association rate constant is compared with the predictions of various capture models.9 The microcanonical variational transition state theory (pCVTST) rate constants calculated for PES1, with the transitional modes treated as harmonic oscillators (ho) are nearly the same as the statistical adiabatic channel model (SACM),13 pCVTST,40 and trajectory capture14 rate constants based on the ion-di-pole/ion-induced dipole potential,... 

Thus, every trial trajectory that connects. (/ and 28 is accepted. In the microcanonical case momentum displacements <5p must be selected such that the system does not change its total energy. How to do that and how to take other constraints such as conserved total linear and angular momentum into account is explained in detail in [8],... 

For Newtonian dynamics and a canonical distributions of initial conditions one can reject or accept the new path before even generating the trajectory. This can be done because Newtonian dynamics conserves the energy and the canonical phase-space distribution is a function of the energy only. Therefore, the ratio plz ]/p z at time 0 is equal to the ratio p[.tj,n ]/p z ° at the shooting time and the new trajectory needs to be calculated only if accepted. For a microcanonical distribution of initial conditions all phase-space points on the energy shell have the same weight and therefore all new pathways are accepted. The same is true for Langevin dynamics with a canonical distribution of initial conditions. 

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... 

This function is normahzed to take the unit value for 0 = 2n. For vanishing wavenumber, the cumulative function is equal to Fk Q) = 0/(2ti), which is the cumulative function of the microcanonical uniform distribution in phase space. For nonvanishing wavenumbers, the cumulative function becomes complex. These cumulative functions typically form fractal curves in the complex plane (ReF, ImF ). Their Hausdorff dimension Du can be calculated as follows. We can decompose the phase space into cells labeled by co and represent the trajectories by the sequence m = ( o i 2 n-i of cells visited at regular time interval 0, x, 2x,..., (n — l)x. The integral over the phase-space curve in Eq. (60) can be discretized into a sum over the paths a>. The weight of each path to is... 

Figure 2). The calculations were done in the microcanonical ensemble at a temperature of 300K 5K. Energy was well conserved throughout the trajectories, and no overall drifts in molecular temperature were observed. Small ensembles of trajectories (12 for SI and 6 each for the other minima) were calculated for the averaging of system properties. Each trajectory was equilibrated by velocity reassignments during an initial period of 20ps, followed by another 20ps of dynamics used for data collection.

To conclude our brief overview of ab initio MD, we note that the dynamics defined by Eq. (9.16) define a microcanonical ensemble. That is, trajectories defined by this Lagrangian will conserve the total energy of the system. Similar to the situation for classical MD simulations, it is often more useful to calculate trajectories associated with dynamics at a constant temperature. One common and effective way to do this is to add additional terms to the Lagrangian so that calculations can be done in the canonical ensemble (constant N, V, and T) using the Nose-Hoover thermostat introduced in Section 9.1.2. 

First let us assume that the system has been undisturbed for so long that it is in a macrostate of thermal equilibrium. Trajectories will then pass through the bottleneck region equally often from left to right and from right to left, and the probabilities of different microstates in the bottleneck region, as in any part of phase space, will be given by the formulas of equilibrium statistical mechanics (e.g. the equilibrium microcanonical density,... 

An efficient implementation of microcanonical classical variational transition state theory was applied to Si—H bond fission in SiFF and compared with trajectory calculations on the same potential surface235. 

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. 

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