Phase space averages

The last step consists of proving that ( (a5 F/d ))F = 0 For Hamiltonian systems, it is generally true that the phase-space average of the time derivative of any function of q and p is equal to zero. For an arbitrary function /(q, p) and a Hamiltonian function 34 ... 

With the IVR approach the quantum mechanical propagator can be semiclassically approximated by a phase-space average that involves all possible initial conditions for the classical trajectories. Consider a simple version of IVR— that is, the so-called coordinate space IVR. In this case the quantum propagator is given by... 

Equation (369) indicates that to obtain the semiclassical reaction rate constant k T) one needs to carry out the multidimensional phase-space average for a sufficiently long time. This is far from trivial, since the integrand in Eq. (369) is highly oscillatory due to quantum interference effects between the sampling classical trajectories. The use of some filtering methods to dampen the oscillations in the integrand may improve the accuracy of the semiclassical calculation. 

The details of the pair potential used in the simulations are given in Table I. This consists of an -trans model of the sec-butyl chloride molecule with six moieties. The intermolecular pair potential is then built up with 36 site-site terms per molecular pair. Each site-site term is compost of two parts Lennard-Jones and charge-charge. In this way, chiral discrimination is built in to the potential in a natural way. The phase-space average R-R (or S-S) potential is different from the equivalent in R-S interactions. The algorithm transforms this into dynamical time-correlation functions. 

It is clear from the derivation presented above that the phase space average, Equation (38), is exactly equal to the desired ensemble average. That is, all phase points with energy E are included with equal probability. Consider the phase space volume, f2 NVE), the number of states with energy E given physical volume, V, and N particles. As the phase space volume increases, obviously, the number of microstates increases and the entropy should increase. This suggest that we postulate that S NVE) = F f2 NVE)) where Q NVE) is now referred to as the microcanonical partion function and F must be a monotonically increasing, function to be determined. 

Prom our postulates these quantities MUST arise from phase space averages. Therefore, F must be a constant times the logarithm (A log) so that, for example. 

The classical phase-space averages for bound modes in Eq. (11) are replaced by quantum mechanical sums over states. If one assumes separable rotation and uses an independent normal mode approximation, the potential becomes decoupled, and onedimensional energy levels for the bound modes may be conveniently computed. In this case, the quantized partition function is given by the product of partition functions for each mode. Within the harmonic approximation the independent-mode partition functions are given by an analytical expression, and the vibrational generalized transition state partition function reduces to... 

The most naive approach to quantizing the TST expression for N(E), Eq. (15), is as follows the phase-space average becomes a quantum mechanical trace, ... 

Applying Eqs (5.9) and (5.11) to this phase-space average gives. 

If the system is at equilibrium, the term on the left becomes zero in the limit that T - 00, because the position vectors are bounded by the limits of the container, and the momenta will be extremely large only over very small time intervals. In this same limit, the time averages of the first two terms on the right side may be replaced by the phase-space averages, using the equilibnum distribution function. The first term (from (Eq. A.3)) is ... 

Nonadiabatic Electron Wavepacket Dynamics in Path-branching Representation 197 6.2.1.4 Phase-space averaging vs. force averaging... 

The method presented in this section, which is a combined use of phase-space averaging and natural branching, is referred to as Phase Space Averaging and Natural Branching (PSANB). 

Fig. 6.17 The upper panel Two adiabatic potential curves that couple with each other through the coupling element (the Gaussian-like curve). The lower panel magnifies the coupling function, which illustrates the starting point of the phase-space averaging and the exit from which for two branching paths to emerge. (Reprinted with permission from T. Yonehara et al, J. Chem. Phys. 130, 214113 (2009)).

One possible way to treat such a case is to use an approximated approach of the nonadiabatic electron wavepacket theory, the phase-space averaging and natural branching (PSANB) method [493], or the branching-path representation, in which the wavepackets propagate along non-Born-Oppenheimer branching paths. 

Many of the system properties are phase variables (B) which means that they are instantaneous functions only of F and t. There are two equivalent formulae to calculate the phase space averages of the phase variables. 

A formally exact (and direct ) expression for N(E) can be obtained by quantizing the dynamically exact classical expression, equation (12) (with equation 11). The classical phase space average becomes a quantum trace, and classical functions become operators ... 

Monte Carlo (MC). In Monte Carlo algorithms, we do not calculate time averages but phase space averages. For example, in the Ccmonical (NVT) ensemble, the average of a static property A is equal to... 

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