Microcanonical sampling

In the full-quantum dynamics method, the distribution of nuclear positions is accounted for in nuclear wavepacket form, that is, by a function that defines the distribution of momenta of each atom and the distribution of the position in the space of each atom. In classical and semi-classical or quasi-classical dynamics methods, the wavepacket distribution is emulated by a swarm of trajectories. We now briefly discuss how sampling can generate this swarm. 

The starting conditions are generated by converting the zero-point vibrational energy into an initial geometry and velocity, where the partition is generated via a random number. 

The total energy is formally split into elementary individual Qi and P contributions that are related respectively to V and T. 

In the case of a photochemical reaction, the species is instantaneously promoted toward the excited surface by the photon absorption. The species on the excited-state surface (generally or 83) will initially have all the characteristics of the species on the Sq surface. The typical sampling for a photochemical dynamics calculation should therefore be carried out on the Sq surface. When studying a very rare photochemical event (e.g., one associated with a huge activation barrier), there is no guarantee that any of the resulting trajectories will ever follow the desired path. There are several approaches to address this issue. 

In the most favorable cases, advantage can be taken of the fact that the equations of motion are reversible by computing the product— reactant (P— R) rather than the R— P trajectory. In this case, the sampling is conducted on the product. Very recent techniques that explicitly connect the product (P) to the reactant (R) are therefore of general interest 

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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. 

We demonstrate here how to create a microcanonical sampling and concern ourselves with only the distribution of energy over the vibrational modes. In a microcanonical ensemble, aU molecules have the same total energy, but different position in phase space, meaning different coordinates and/or momentum. The total vibrational energy is given as... 

Fig. 15.1. Trajectory Al6 —AI5 + A1 lifetime distribution, following microcanonical sampling. The histogram plot represents the number of Al6 dissociations per time interval. The dashed line represents the random lifetime distribution of Eq. (3). The energy is 40 kcal/mol and the angular momentum is zero. Adapted from Ref. [50].

Fig. 15.2. Trajectory lifetime distributions for HCO, HNO, and HO2 dissociation, following microcanonical sampling. Extensive intrinsic non-RRKM dynamics are present in HCO and HNO decomposition. The smooth curves are obtained by fitting the data to an exponential function, with the initial fast decomposition times being excluded. Adapted from Ref. [51].

Let us now consider the phase space if(P,Q), defined as a space of dimension 6N (3N dimensions for the coordinates q and 3N dimensions for the momenta p, where N is the total number of degrees of freedom) of all possible distributions so that the sum of their potential and kinetic energies is a constant. Microcanonical sampling is a sampling procedure whose key idea is to generate a (classical) uniform distribution in this phase space. Practically, it is based on an analytical frequency calculation at the reference geometry and on the use of a random number R (see the right-hand side of Scheme... 

Two key questions for the assessment of dynamics calculations are (a) are the initial conditions realistic and (b) how many trajectories should be run before the swarm can be said to be of sufficient size To investigate these two points, we ran three series of trajectories in a 3-2IG basis on the potential surface for cis trans conversion of model chromophore 1 and Im (Scheme 2.9). The first series used a genuine (T + V) microcanonical sampling of the velocity and geometry simultaneously. In the second series (T sampling only), we used the same velocity as in the first series but did not allow any... 

G. Nyman, S. Nordholm, and H. W. Schranz, J. Chem. Phys., 93, 6767 (1990). Efficient Microcanonical Sampling for a Selected Total Angular Momentum. Applications to HO and HjO. H. W. Schranz, S. Nordholm, and G. Nyman,/. Chem. Phys., 94,1487 (1991). An Efficient Microcanonical Sampling Procedure for Molecular Systems. H. W. Schranz, /. Phys. Chem., 95, 4581 (1991). On the Microcanonical Weight Function. 

Decomposition of methyl nitrite by concerted elimination (CH2O and HNO formation) and O-N bond dissociation (CH3O and NO formation) has been investigated by classical trajectories and statistical variational efficient microcanonical sampling-transition state theory. The dissociation, which exhibits an inverse deuterium isotope effect, is markedly faster than the four-centre elimination process. The UHF/AMl MO method has been used to study thermolyses of aroyl nitrites. ... 

Other algorithms have been developed for speeding up the uniform sampling of phase space points. For example, the efficient microcanonical sampling series of schemes exploits the possibility of sampling independently the spatial coordinates and momenta, simply by weighting the sampled geometries by their associated momentum space density. 

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