Theory, phase space

To invoke microscopic reversibility and obtain a rate expression for a unimolecular reaction through considering the association of its products is an idea with a long history [706] and has been taken up in recent years in mass spectrometry [486, 489]. There are a number of treatments [165, 166, 486, 489], which are similar to each other and all of which can be considered to be, in essence, reformulations of QET. There is a tendency to refer to these reformulations, either individually of collectively, as phase space theory [165, 166, 452, 485] and this term is used in a collective sense here. 

Phase space theory considers the long-range polarisation between the associating ion and neutral and this interaction has generally been described on the basis of the Langevin model [808]. As has been realised for some time [482, 486, 547, 614], the phase space theory is equivalent to transition state theory in the extreme case of a loose transition state. The term loose as used here means that the incipient fragments can each freely rotate [287, 333] within the confines of conservation of angular momentum. 

Phase space theory can be thought of as, in effect, considering a loose or, as it is sometimes called, orbiting [333] transition state regardless of the nature of the reaction. The need to select transition state properties for each individual reaction considered is avoided and it has been argued that a virtue of the theory is that it gives definite predictions [452]. 

Phase space theory has been able to reproduce experimental distributions of translational energy releases closely for a number of decompositions which do not have energy barriers to the reverse reactions [165, 485] (see Sect. 8). Phase space theory does focus attention on a very late stage of reaction since the degrees of freedom of the loose transition state can be identified as vibrations, rotations and translations. 

Recent mechanistic discussions of unimolecular decompositions of organic ions have invoked ion—molecule complexes as reaction intermediates [102, 105, 361, 634]. The complexes are proposed to be bound by long-range ion—dipole forces and to be sufficiently long-lived to allow hydrogen rearrangements to occur. The question of lifetime aside, there is more than a close similarity between the proposed ion—dipole intermediate and the assumed loose or orbiting transition state of phase space theory. 

As has been noted, the discussion in the preceding two sections only concerns the orbiting cross-section or cross-section for complex formation. These theories cannot predict the relative importance of various exit channels, the energy distribution in products, nor the effect of internal 

The problem now is how to calculate the proper phase-space available. At present, this is only possible for the atom (ion)—diatom systems. For such three-particle systems, the phase-space element dF is 

Making use of many available data on atomic and molecular constants, computer calculations have been made for some three-body ion—molecule reactions [54, 55, 73]. The results give cross-sections for 10 possible reaction channels as functions of the relative kinetic energy of the colliding particles [73]. Results on the reactions He + Hj and He + Hj [55] + HD and X + HD (X is He, Ne, Ar, Kr) [73] and + Nj, N  

The theory was also extended to include reactions with activation energy 

The quantum mechanical phase space theory has also been developed [75—77] and applied to ion—molecule reactions, including some endothermic reactions [78, 79]. While results again seem to show that the theory is promising, the apparent future problem would be how to extend the treatment to systems involving more than three atoms. 

---

The intennolecular Hamiltonian of the product fragments is used to calculate the sum of states of the transitional modes, when they are treated as rotations. The resulting model [28] is nearly identical to phase space theory [29],... 

Since the observation made in study of the formation HeH+ indicated that this product was not formed by reaction of He + with H2, it had been assumed that the exothermic heat of reaction of He+ ions with H2 is probably deposited in the product HeH + as internal energy, decomposing the product into H+ and He. This idea was cited by Light (16) in his phase space theory of ion-molecule reactions to account for the failure to observe HeH+ from reactions with He+ ions. The experimental difficulty in the mass spectrometric investigation of this process is that H + formed by electron impact tends to obscure the ion-molecule-produced H+ so that a sensitive quantitative cross-section measurement is difficult. 

Rosenstock (55) pointed out that the initial formulation of the theory failed to consider the effect of angular momentum on the decomposition of the complex. The products of reaction must surmount a potential barrier in order to separate, which is exactly analogous to the potential barrier to complex formation. Such considerations are implicit in the phase space theory of Light and co-workers (34, 36, 37). These restrictions limit the population of a given output channel of the reaction com-... 

The phase space theory in its present form suffers from the usual computational difficulties and from the fact it has thus far been developed only for treating three-body processes and a limited number of output channels. Further, to treat dissociation as occurring only through excitation of rotational levels beyond a critical value for bound vibrational states is rather artificial. Nevertheless, it is a useful framework for discussing ion-molecule reaction rates and a powerful incentive for further work. 

In contrast to the results obtained for dehydrogenation reactions, kinetic energy release distributions for alkane elimination processes can usually be fit with phase space theory. Results for the loss of methane from reaction 9 of Co + with isobutane are shown in Figure 10b. In fitting the... 

The success of the phase space theory in fitting kinetic energy release distributions for exothermic reactions which involve no barrier for the reverse reaction have led to the use of this analysis as a tool for deriving invaluable thermochemical data from endothermic reactions. This is an important addition to the studies of endothermic reactions described above. As an example of these studies, consider the decarbonylation reaction 11 of Co+ with acetone which leads to the formation of the... 

To determine the optimal value of quantum correction y, several criteria have been proposed, all of which are based on the idea that an appropriate classical theory should correctly reproduce long-time hmits of the electronic populations. (Since the populations are proportional to the mean energy of the corresponding electronic oscillator, this condition also conserves the ZPE of this oscillator.) Employing phase-space theory, it has been shown that this requirement leads to the condition that the state-specihc level densities... 

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. 

Figure 3. Thermal rate constants for capture of HC1 by H3 (PST locked-dipole capture corresponding to phase-space theory, Eq. (16) SACM statistical adiabatic channel model, Eqs. (26)-(34) [15] SACMci classical SACM, Eqs. (28H31) [15] CT classical trajectories, Eqs. (26) and (27) [1]).

Figure 6. Number of open channels for the interaction between H3 and HC1 (SACM calculations from Ref. IS PST phase-space theory full curves permanent + induced dipole dashed smoothed curves permanent dipole J total angular momentum of H3-HCI complex).

Statistical phase-space theory Statistical phase-space theory Statistical phase-space theory... 

The latter process is not competitive with the former above the thermodynamic threshold. Furthermore, as long as the available energy was kept below the thermochemical threshold for the production of vibrationally excited CN radicals, it was possible to fit the observed rotational distributions with phase space theory. The upper electronic state that is involved in the two-photon dissociation was shown to originate below 22,000 cm l and is thought to be repulsive. It could be the same state that has its absorption maximum at 270 nm. 

If redissociation into reactants is faster than stabilization, equations (3.15) and (3.16) simplify into a product of k,/k, and either kr or kcoll. Under these conditions, to obtain a theory for a total association rate coefficient, one must calculate both k,/k i and kr or kco . Three levels of theory have been proposed to calculate k, /k, . In the simplest theory, one assumes (Herbst 1980 a) that k, /k 3 is given by its thermal equilibrium value. In the next most complicated theory, the thermal equilibrium value is modified to incorporate some of the details of the collision. This approach, which has been called the modified thermal or quasi-thermal treatment, is primarily associated with Bates (1979, 1983 see also Herbst 1980 b). Finally, a theory which takes conservation of angular momentum rigorously into account and is capable of treating reactants in specific quantum states has been proposed. This approach, called the phase space theory, is associated mainly with Bowers and co-workers... 

The modified thermal and phase space theories reproduce most three body association data equally well, including the inverse temperature dependence of the rate coefficient (Herbst 1981 Adams and Smith 1981), and are capable of reproducing experimental rate coefficients to within an order of magnitude (Bates 1983 Bass, Chesnavich, and Bowers 1979 Herbst 1985b). They should therefore be this accurate for radiative association rate coefficients if kr is treated correctly. 

---