W. Forst

Comparative Test of Approximations for Calculation of Energy‐Level Densities

Two molecular models, one representing a “small” molecule and the other a “large” molecule, both with several free rotors, are used to test, by comparison with exact count, seven approximation formulas for the calculation of density of harmonic vibration–rotation states, at energies between 1000 and 20 000 cm−1. The techniques of the classical approach, the Laplace transform, steepest descents, and symmetric functions are represented among the seven formulas. Also shown is the effect of the various approximations on the energy dependence of the computed unimolecular rate constant, and on ion breakdown curves calculated by the statistical theory of mass spectra. It is concluded that the techniques of the Laplace transform [J. Chem. Phys. 46, 3736 (1967)] and steepest descents, as formulated in an Appendix, yield the best all‐around approximation, while a semiclassical‐type formula [J. Chem. Phys. 41, 1883 (1964)] requires least computational effort and still gives excellent results in most cases. In an App...

Transfer of vibrational-rotational energy in thermal reactions: approximations, application to diatomics

Abstract A thermal system is considered, consiting of a reactant diluted in a heat bath of inert gas, undergoing decomposition at low pressure; thus collisional activation is the rate-determining process. If collissions are assumed to transfer both vibrational and rotational energy, the master equation becomes two-dimensional, ultimately involving an intractable four-fold integration. Two approximate solutions are developed that permit removing one of the dimensions from the integrations. On the basis of a simplified model, involving exponential vibrational and rotational transition probabilities, it is shown that the socalled fixed- v approximation is the most reasonable one. It represents essentially the rate of rotational dissociation at a given vibrational energy, averaged over an appropriate non-equilibrium vibrational distribution function. The model and the approximation are tested on the thermal dissociation of H 2 and shown to give excellent results. The treatment can be readily generalized to polyatomic reactants and other forms of the transition probabilities.

Exclusion of Disallowed States from State Density and Its Effect on Unimolecular Rate

Statistical rate theory requires the knowledge of the density of allowed states of the decomposing molecule. States may be disallowed if they are unbound, or if they are excluded by angular‐momentum conservation requirements. To investigate such exclusion, the generalized method of steepest descents, as recently formulated by the authors [J. Chem. Phys. 51, 3006 (1969)], is used to compute the density and sum of states for a molecular species represented by a system of independent quantum oscillators and independent classical free rotors under the restriction that (1) all oscillators (harmonic or anharmonic) are subject to vibrational cutoff; or (2) one anharmonic oscillator is coupled with a two‐dimensional rotor whose rotational energy is restricted by angular‐momentum conservation, while its vibrational energy is not allowed to exceed its effective dissociation energy, all other oscillators being subject to Restriction (1). The molecular model used is a “small molecule” where the two restrictions produ...

Non-adiabatic unimolecular dissociation of polyatomic molecules. II. The thermal dissociation of nitrous oxide

Abstract The theory presented in part I of this series is applied to the non-adiabatic spin-forbidden thermal dissociation N 2 O( 1 Σ + )→N 2 ( 1 Σ + g )+O( 3 P) as a test case. The molecular model is multidimensional and includes all vibrational modes of the molecule. Specifically considered is the fact that the initial singlet state of N 2 O is linear and the final triplet state is bent. The best available data are used for describing the intersection of singlet and triplet potential energy surfaces. Calculated microcanonical rate constants are averaged over Boltzmann distribution of energies and compared with k co , the high-pressure rate constant deduced from experiment. The agreement between theory and experiment is satisfactory. Analysis of the calculations shows that the driving force for the N 2 O dissociation is the flow of energy into the bending vibrations. This is because the bendings have very different equilibrium angles in the initial and final states.

Analytic solution of relaxation in a system with exponential transition probabilities. II. The initial transients

Using an exponential transition probability model normalized in the (0,∞) energy domain, we have obtained an analytical solution for the time‐dependent population density below threshold, c (x,t), in the form of the eigenfunction expansion where x is internal energy, t is time, A0 and Aj are constants that depend on initial conditions, S and Rj are solutions of a determinant of a matrix of coefficients and k0−1 and τj are the relaxation times, the lowest of which (subscript 0) represents the reciprocal of the steady‐state rate constant. From c (x,t) we then obtain all other time‐dependent properties such as the non‐steady‐state rate constant and average energy, as well as incubation times and dead times for both number density and vibrational energy. Calculations relative to shock tube decomposition of N2O, CO2, and O2 in inert gas are compared with experiment, with generally good results. For the triatomics, average energy transferred per collision, as calculated from the experimental relaxation time, co...

Adiabatic Rotations in Unimolecular Rate Theory

The term “adiabatic rotations” describes rotations that are constrained to remain in the same quantum state when the activated complex is formed from the active molecule because of restrictions imposed by angular momentum conservation. It is shown that these rotations decrease the activation energy for dissociation and therefore increase the rate; a priori, the effect is likely to be largest at the high‐pressure limit of a unimolecular decomposition, and least at the low‐pressure limit. The general case is considered of a polyatomic symmetric‐top molecule decomposing unimolecularly. Assuming a Boltzmann distribution of rotational energies of the molecules undergoing decomposition, an expression is derived for the factor (f) by which the rate constant is increased due to adiabatic rotations. Two approaches are used: one involving the average moment of inertia of complex (I‡), if known from other information, and the second involving the assumption of inverse sixth‐power potential between the two incipient ...

Transfer of vibrational-rotational energy in thermal reactions: application to polyatomics

Abstract A simple model is used to deal with vibrational-rotational energy transfer in a thermal dissociation of a polyatomic molecule under non-equilibrium conditions when collisional energy transfer is rate-determining. Transition probabilities are assumed to be factorizable into purely vibrational and purely rotational terms, both given by an exponential model; a gaussian model is also used for the vibrational term. The master equation leads to a two-dimensional integral equation which is solved by means of the so-called “fixed-υ” approximation developed previously, in which no vibrational energy transfer takes place while rotational energy is being transferred. Two sets of calculations are performed, one with rotational energy transfer much faster than vibrational energy transfer, and the other with both of equal importance. Relative to purely vibrational energy transfer, rotational energy transfer is shown to lead to increased rate of dissociation and increased depopulation of levels near threshold, especially in the first set of calculation, and also to a further decrease of activation energy below the bond-dissociation energy. The model system used is the dissociation of H2O2 and the calculated data are in good agreement with experiment where available.

Analytic solution of relaxation in a system with exponential transition probabilities. III. Macroscopic disequilibrium

The exponential transition probability, in the version that permits an analytical solution of the relaxation problem, is used to compute a number of macroscopic (bulk) observables for a model system based on multiphoton excitation of SF6 coupled to a rare‐gas heat bath. Two extreme cases are considered: Initial excitation as a delta function, or as a Poisson distribution. It turns out that regardless of initial conditions, all macroscopic observables are functions of time, including the relaxation time, so that the system does not undergo simple exponential decay. This is because the first moment of the exponential transition probability does not satisfy the linear ‘‘sum rule.’’ The exponential transition probability causes the overall (or bulk) average of energy transferred (〈〈ΔE〉〉) to be constrained to a maximum which is independent of the nature and level of initial excitation, thus producing a bottleneck in the macroscopic relaxation process when excitation is sufficiently high. The consequence is tha...

Relaxation of internal energy and collisional energy transfer

Abstract Using the exponential model for the collisional transition probability, it is shown that relaxation of average internal energy is a measure of bulk-average energy transfer ⪡Δ E ⪢. This is a macroscopic property which is a complicated function of both time and initial excitation and is only distantly related to average energy transferred per collision ⪡Δ E ⪢, a microscopic property.

Low-Pressure Non-Equilibrium Activation Energy of Unimolecular Reactions,

Abstract It is shown that regardless of the exact nature of the collisional transition probability (representing efficient, intermediate and inefficient energy transfer), the non-equilibrium activation energy of a unimolecular reaction at the low pressure limit is decreased by a most 2 kT relative to the equilibrium value. The important parameter is the average transferred in a deactivating collision and not the actual transition probability.