Phase Space Theory PST

The association rate data determined in this study can be used to make quite a precise binding energy estimate for the aluminum ion-benzene complex. The relation between the association rate constant and the binding energy was made with use of phase space theory (PST) to calculate as a function of E, with a convolution over the Boltzmann distribution of energies and angular momenta of the reactants (see Section VI). PST should be quite a reasonable approximation for... 

Phase space theory (PST) has been widely used for estimation of rates and energy partitioning for ion dissociations. It can be considered within the framework of transition-state theory as the limiting case of a loose transition state, in which all product degrees of freedom are statistically fully accessible at the transition state. As such, it is expected to give an upper limit for dissociation rates and to be best suited to barrierless dissociations involving reaction coordinates with simple bond cleavage character. 

It must be emphasized that such phenomena are to be expected for a statistical system only in the regime of low level densities. Theories like RRKM and phase space theory (PST) (Pechukas and Light 1965) are applicable when such quantum fluctuations are absent for example, due to a large density of states and/or averaging over experimental parameter such as parent rotational levels in the case of incomplete expansion-cooling and/or the laser linewidth in ultrafast experiments. However, in the present case, it is unlikely that such phenomena can be invoked to explain why different rates are obtained when using ultrafast pump-probe methods that differ only in experimental detail. 

Figure 32 Dissociation rates of rotationally cold NO2 as function of the mean excitation energy. Thick solid line phase space theory (PST). The two vertical dashed lines are reaction thresholds for NO2 starting in its lowest rotational level (i.e., Do) and its first excited rotational level. Reprinted, with permission of the American Institute of Physics, from Ref. 275.

We consider next phase-space theory (PST), which was designed principally to calculate energy distributions of the reaction products.12 In PST a loose transition state is assumed and, rather than using the approximation of l J,l is chosen so as to satisfy angular momentum conservation, namely, the triangle inequality,... 

Figure 5. Comparison of the observed product rotational state distributions (solid bars) and the results of two statistical models the statistical adiabatic channel model (SACM, open bars) and phase space theory (PST, hatched bars). The distributions are an average of those observed, or those calculated, at several excitation wavelengths in the region of (a) the 5tbu main band, (b) the 5voh combination band, (c) the 6vOH main band, and (d) the 6v0H combination band of HOOH. (Reproduced with permission from Ref. 41.)...

The statistical dissociation rate constant can be calculated from the point of view of the reverse reaction, namely the recombination of the products to form a complex. This approach, commonly referred to as phase space theory (PST) (Pechukas and Light, 1965 Pechukas et al., 1966 Nikitin, 1965 Klots, 1971, 1972) is limited to reactions with no reverse activation energy, that is, reactions with very loose transition states. PST assumes the decomposition of a molecule or collision complex is governed by the phase space available to each product under strict conservation of energy and angular momentum. The loose transition state limit assumes that the reaction potential energy surface is of no importance in determining the unimolecular rate constant. 

PHASE SPACE THEORY (PST) PRODUCT ENERGY DISTRIBUTIONS... 

For the abbreviations used see text. (-) or (4-) in connection with RRKM(—AM) or Phase-Space Theory (PST) concerns (dis)agreement with experiment. 

For the theoretician, clusters are also convenient model systems to evaluate the performance of dissociation rate theories. By comparing the results of numerically exact molecular dynamics (MD) trajectories to the predictions of rate theories, the various approximations inherent to these theories can be unambiguously tested and possibly improved upon. Previous authors have critically discussed how the Rice-Ramsperger-Kassel (RRK), ° Weisskopf, and Phase Space Theory of Light and Pechukas, Nikitin, Klots, Chesnavich and Bowers respectively compare for the thermal evaporation of atomic clusters. This work was subsequently extended by the present authors to rotating and molecular clusters. From these efforts it was concluded that phase space theory (PST), in its orbiting transition state version, was quantitatively able to describe statistical dissociation. This chapter is not devoted to a detailed presentation of phase space theory and the reader is encouraged to consult the cited work. 

Variations of the evaporation rate constant of the (H2O)50 cluster, as predicted by phase space theory (PST) in its orbiting transition state version, and values of the rate constant obtained from statistical molecular dynamics (MD) trajectories at high energies. The inset shows the decay of the number of clusters N(t) having resisted evaporation as a function of time, at three internal energies denoted next to the curves and in logarithmic scale. 

Figure 4.6 Variations of the average (a) size (b) angular momentum and (c) internal temperature of a 28-atom LJ cluster undergoing successive evaporation of monomers, as a function of time. The results of molecular dynamics (MD) simulations are compared with the predictions from phase space theory (PST) with calibrated rates and anharmonic state densities and also to approximate PST calculations in which the rates are either not calibrated or obtained from harmonic state densities.

Different rotor TS models have been advanced and, for each, the standard assumption is that H, 2 and H 2 are conserved as the fragments associate to form the variational TS, i.e. the vibrational frequencies of the fragments are constant and do not change (although these frequencies may be changed as a function of r if considered important). For the phase space theory (PST) model of Klotz, the variational TS is assumed to be in the product limit, with the two fragments freely rotating. As a result the and V terms in eqn (20.42)... 

While the results of trajectory calculations provide an accurate testing ground for more approximate theories, and, in the parameterised form developed by Su, Chesnavich and Bowers [25,26], a widely applied means of calculating capture rate coefficients for these more complex interactions, they provide less insight into reaction mechanisms and rate coefficient determinants than more analytic approaches. The simplest approach is provided by phase space theory (PST) which assumes an isotropic potential between the reactants [31]. The centrifugal term in the effective potential in (3.2) can be expressed in terms of the orbital angular momentum quantum number, , for the collision, so that the equation for Vejf (Rab) becomes ... 

Figure 5 The derived kinetic energy release from the energy-selected Ar2CO + ions as a function of the trimer ion internal energy. The solid line is a calculated kinetic energy release based on the statistical theory of dissociation phase space theory (PST) or the version of PST due to C.E. Klots. AP is the threshold energy for ArCO+ formation. The onset leads to a heat of formation of the trimer ion. Reproduced with permission from Mahnert J, Baumgartel H and Weltzel KM (1997) The formation of ArCO+ ions by dissociative Ionization of argon/carbon monoxide clusters. Journal of Physical Chemistry 07 6667-6676.

The limit with a = 00 in the adiabatic channel model corresponds to phase space theory " (PST, see Figure 1). PST can thus be considered to be a simplified limiting model of the SACM. From this simple fact one can conclude that if... 

---