Phase-space integration

The bulk of the infomiation about anhannonicity has come from classical mechanical calculations. As described above, the aidiannonic RRKM rate constant for an analytic potential energy fiinction may be detemiined from either equation (A3.12.4) [13] or equation (A3.12.24) [46] by sampling a microcanonical ensemble. This rate constant and the one calculated from the hamionic frequencies for the analytic potential give the aidiannonic correctiony j ( , J) in equation (A3.12.41). The transition state s aidiannonic classical sum of states is found from the phase space integral... 

The canonical ensemble partition function is the phase space integral... 

If one is interested in spectroscopy involving only the ground Born Oppenheimer surface of the liquid (which would correspond to IR and far-IR spectra), the simplest approximation involves replacing the quantum TCF by its classical counterpart. Thus pp becomes a classical variable, the trace becomes a phase-space integral, and the density operator becomes the phase-space distribution function. For light frequency co with ho > kT, this classical approximation will lead to substantial errors, and so it is important to multiply the result by a quantum correction factor the usual choice for this application is the harmonic quantum correction factor [79 84]. Thus we have... 

Equation 4.117 makes complete sense. One of the first things one learns in dealing with phase space integrals is to be careful and not over-count the phase space volume as has already been repeatedly pointed out. In quantum mechanics equivalent particles are indistinguishable. The factor n ni is exactly the number of indistinguishable permutations, while A accounts for multiple minima in the BO surface. It is proper that this factor be included in the symmetry number. Since the BO potential energy surface is independent of isotopic substitution it follows that A is also independent of isotope substitution and cannot affect the isotopic partition function ratio. From Equation 4.116 it follows... 

In a mixed quantum-classical calculation the trace operation in the Heisenberg representation is replaced by a quantum-mechanical trace (tTq) over the quantum degrees of freedom and a classical trace (i.e., a phase-space integral over the initial positions xq and momenta Po) over the classical degrees of freedom. This yields... 

As a consequence, the semiclassical propagator is given as a phase-space integral over the initial conditions qo and Po, which is amenable to a Monte Carlo evaluation. For this reason, semiclassical initial-value representations are regarded as the key to the application of semiclassical methods to multidimensional systems. 

It is also useful to define the transformed operator L whose operation on a function f is L f = L[Peqf). This operator coincides with the time reversed backward operator, further details on these relationships may be found in Refs. 43,44. L operates in the Hilbert space of phase space functions which have finite second moments with respect to the equilibrium distribution. The scalar product of two functions in this space is defined as (f, g) = (fgi q. It is the phase space integrated product of the two functions, weighted by the equilibrium distribution P The operator L is not Hermitian, its spectrum is in principle complex, contained in the left half of the complex plane. 

QTST is predicated on this approach. The exact expression 50 is seen to be a quantum mechanical trace of a product of two operators. It is well known, that such a trace can be recast exactly as a phase space integration of the product of the Wigner representations of the two operators. The Wigner phase space representation of the projection operator limt-joo %) for the parabolic barrier potential is h(p + mwtq). Computing the Wigner phase space representation of the symmetrized thermal flux operator involves only imaginary time matrix elements. As shown by Poliak and Liao, the QTST expression for the rate is then ... 

The quantum version of the partition function is obtained by replacing the phase space integral and the classical Boltzmann distribution with the trace operation of the quantum Boltzmann operator, giving the usual expression... 

Again, the quantum mechanical expressions can be written in a form that is analogous to the classical expressions for the rate constant given in Section 5.1, remembering that a classical phase-space integral is equivalent to a quantum mechanical trace [9], and classical functions of coordinates and momenta are equivalent to the corresponding quantum mechanical operators. 

One view of this trace operation is that the usual phase space integral may be obtained by representing the thermal density matrix e in plane-wave momentum states, and performing the trace in that state space (Landau et al, 1980, Section 33. Expansion in powers of h ). Particle distinguishabihty restrictions are essential physical requirements for that calculation. In this book we will confine ourselves to the Boltzmann-Gibbs case so that e = Q n, V, T)/n, since the... 

The density of states is a central concept in the development and application of RRKM theory. The density of states of a classical system is the number of states of that system per unit energy, expressed as a function of energy. This quantity may be formulated as a phase space integral in several ways. 

It is easy to see that the definition above is essentially a differential. It is therefore entirely equivalent to find the total number of states up to and including by a suitable phase space integral and then differentiate with respect to E. 

In Section 4.3, example macroscale transport equations are derived for selected moments of the NDF. Having introduced the precise forms of the mesoscale advection models in Eq. (5.2), it is of interest to derive explicitly some example moment source terms resulting from these models. In order to do so, we will use the rules presented in Section 4.3.1 for phase-space integration. For simplicity, we consider only the advection term involving (Afp)i and assume that the only phase-space variables of interest are v and Vf, and that the model in Eq. (5.2) reduces to... 

By removing the phase-space integration, Eq. (B.46) can be rewritten as a ki-dependent advection term for the NDF ... 

The average in (3.48) denotes a microcanonical expectation value. The classical limit of (3.48) is a phase space integral ... 

The superposition method extends Stillinger and Weber s division of the PES into catchment basins for each minimum to calculate thermodynamic properties. In this approach the configurational part of the phase space integral in the definition of the density of states, Q E), or partition function, Z T), is divided into separate integrals for each minimum, giving... 

Because hindered rotors involve densely spaced energy levels, it is possible to treat the problem classically. This has been done for the case of a two-dimensional hindered rotor (Jordan et al., 1991) which is particularly important in the dissociation of loose transition states, a topic to be discussed in the following chapter. The classical phase space integral is solved using the Hamiltonian ... 

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