Theory Of Unimolecular Reactions Research Articles

The commutative property of strong-transition rate matrices

The (n − 1)-fold degenerate set of transition probabilities used in our recent reformulation of unimolecular reaction theory is operationally equivalent to the set of effective strong-transition probabilities introduced recently by Nordholm. We have found that the relaxation matrix corresponding to these transition probabilities commutes with any other relaxation matrix which will drive the same system to a Boltzmann distribution at infinite time. Some useful results stemming from this commutative property are presented; we expect that these results will help to clarify further the nature of the assumptions underlying strong- and effective strong-collision theories.

Simple bond rupture reactions in multiphoton dissociation of molecules

A molecular beam study was carried out of the infrared multiphoton dissociation of seven molecules dissociating by a simple bond rupture. Translational energy distributions for the dissociation products were obtained, and the results can be described very well by a statistical (RRKM) theory of unimolecular reactions. A general discussion of the main features of the multiphoton dissociation process is given.

Unimolecular decay paths of electronically excited species. II. The C2H+4 ion

The reaction paths of dissociation and the mechanisms of electronic relaxation of the ethylene cation have been calculated ab initio. Internal rotation is shown to bring about radiationless transition of states ? 2A and ? 2B2 to the ground state. Two competing channels are available for the first excited ? 2B3: it can either undergo internal conversion to the ground state ? 2B3 or lose a hydrogen atom by simple bond cleavage and give C2H+3+H fragments. The competition is governed by nonadiabatic interaction around a conical intersection. The coupling matrix element 〈?‖∂/∂q‖?〉 has been calculated and is shown to obey the linear model of nonadiabatic interaction quite well. The ground state ? 2B3 cannot eliminate directly a hydrogen molecule. It must first undergo rearrangement to a 2E state of a CH3CH+ structure via a bridged structure. Jahn–Teller interaction takes place at the doubly degenerate 2E state. Dissociation of CH3CH+ via 1,1-hydrogen elimination is then allowed and leads to C2H+2+H2. There is a small (possibly inexistent) energy barrier along this path. The complicated nature of this multistep reaction path brings about energy redistribution among many vibrational degrees of freedom and provides a plausible explanation for the success of statistical theories of unimolecular reactions (RRKM and QET).

Theory of unimolecular reactions induced by monochromatic infrared radiation

A theory of unimolecular reactions induced by coherent, monochromatic infrared radiation (URIMIR) in the absence of collisions is presented. It is shown that the set of first order linear differential equations for the amplitudes of molecular states (Schrödinger equation) can be reduced, under specified conditions, to a much smaller set of first order linear differential equations for the coarse grained populations of levels for polyatomic molecules (master equation). Four limiting situations are identified in which such linear rate equations provide a reasonable approximation. Rate coefficients are obtained as a function of spectroscopic parameters (energy levels and transition moments). Solutions for the master equations are given as a function of time and at steady-state. Simple limiting cases (Strong Field Limit, Weak Field Limit, Reaction Threshold Bottleneck, etc.) are identified and very simple rate expressions are obtained for these cases. A complete statistical mechanical theory of URIMIR is formulated and the computational approaches for the quantitative treatment for any molecule are summarized. Predictions are made concerning the dependence of the unimolecular rate constant and product (state) distributions upon radiation intensity. In particular at high intensities a less than proportional increase of the rate constant with intensity is predicted. The possibility of specific pumping of reaction paths with high energy thresholds is discarded. Comparison with thermal unimolecular reactions shows that collisionless URIMIR are quite different in all respects. Fundamentally these differences are traced to the fact that the underlying molecular distribution functions are different. They are Boltzmann distributions for thermal reactions and ’’microcanonical’’ [without exp (−E/kT)] in URIMIR. Although real distribution functions are still different in general, the underlying distribution functions often dominate the dynamical behavior.

Dynamical treatment of unimolecular decomposition reactions. II. Short-range interfragment coupling and incomplete randomization

We investigate the dynamics of an isolated polyatomic molecule undergoing unimolecular dissociation. The intramolecular vibrational energy transfer step is treated by applying a theory of vibrational relaxation previously developed for permanently bound molecules. In contrast to an earlier study, the decomposition step is treated here subject to the assumption that, when sufficient energy is concentrated in the reaction coordinate, transitions between internal states of the molecule occur only while the incipient fragments are near their minimum classically allowed separation. Under such circumstances, it is generally not possible to divide configuration space cleanly into a randomized and a nonrandomized region, as required by the RRKM theory of unimolecular reactions. In spite of this, it is shown that, under certain conditions, the RRKM specific rate constant expression may remain valid. More generally, it is shown that the accurate specific rate constant assumes a form predicted by the transition state treatment of unimolecular reactions. The explicit rate expression derived here reveals factors which determine the accuracy of the transition state approximation for unimolecular reactions—the basic, simplifying assumption that a transition state exists which coincides with a ’’configuration of no return’’ for both decomposition and association reactions. Estimates based on our rate formulas suggest that the transition state approximation may often be sufficiently accurate to justify use of the RRKM rate expression. As part of our analysis of randomization and decomposition dynamics, we examine the time-dependent behavior of a molecule which has fragment–fragment interactions as described above but which is constrained to remain bound by the presence of an artificial ’’wall’’ which prevents the fragments from separating completely. It is found that statistical equilibrium may be achieved in this system—a significant result because the system violates some rather stringent assumptions previously invoked to prove randomization in bound molecules. This motivates a future search for a more general theory of intramolecular relaxation.

A reformulation of the theory of unimolecular reactions

The theory of unimolecular reactions is reformulated in a way which makes no reference to the concepts of a transition state, a reaction coordinate, or of an activated complex: the present theory is based, instead, on a simple master-equation approach using the estimated positions and lifetimes of the rotation–vibration energy levels of the molecule. The fundamental basis of this reformulation is that the unimolecular rate is derived from the perturbation of the normal modes of internal relaxation of the reactant molecule by the microscopic reaction processes. The thermal decompositions of cyclobutane, cyclopropane, ethyl isocyanide, ethyl chloride, and methyl isocyanide are examined in detail, and it is shown that not only is the present theory much easier to use than is conventional RRKM theory, but also that it gives better results.An Appendix provides a set of algorithms required for these calculations.

On the theory of unimolecular reactions: Application of mean first passage time to reaction rates

We apply the theory of first passage times to the process of collision free unimolecular decomposition of large molecules at a given total energy. The theory is intermediate between the dynamical and statistical theories. We find that the rate of decomposition is given in terms of a series, the leading term of which has the form used in the RRKM theory. We analyze the conditions for which this term is dominant, and find that when the molecule is large or the total energy is low or the rate of decomposition is small, our expression coincides with the phase space approach. We show that the separation of time scales (energy redistribution vs decomposition) is large when the same conditions are met. Our analysis reduces to the Montroll–Shuler model for the decomposition of a diatomic molecule in a bath of inert atoms, in the appropriate limits.

An extension of the RRKM theory of unimolecular reactions - the effective strong collision frequency

It is proposed that the accuracy of the RRKM theory can be improved by replacing the usual collision frequency by an effective collision frequency defi

Theory of unimolecular reactions in one dimension

The widely used transition state theory including corrections for tunnelling and quantum zero point energy as applied to a one-dimensional reaction coo

Collision theory treatment of monomolecular and bimolecular gas reactions

AbstractAn exact collision theory of unimolecular and bimolecular gas phase reactions is derived from a general quantum‐mechanical formulation of reactions rates based on the assumption that the reactants are in thermal equilibrium. In this way the quantum corrections to the classical collision theory expressions are rigorously defined. Approximate formulas for these corrections make it possible to determine well the temperature ranges within which the classical and the semiclassical approximations are valid. A comparison is made between the collision and the transition state theory with emphasis on some conceptual difficulties of the latter in treating the simple decomposition and recombination reactions. It is shown that in the classical (high temperature) limit these theories are incompatible except when the reaction coordinate is entirely separable (i.e., when the transition state theory is no longer useful).

Semiclassical calculation of bound states in a multidimensional system for nearly 1:1 degenerate systems

The method is devised to calculate eigenvalues semiclassically for an anharmonic system whose two unperturbed modes are 1:1 degenerate, by introducing a curvilinear Poincaré surface of section. The results are in reasonable agreement with the quantum ones. The classical trajectories also frequently show a large energy exchange among the two unperturbed normal modes. Implications for Slater’s theory of unimolecular reactions, which neglects this effect, and for ’’quantum ergodicity’’ are described.

A spectral analysis method of obtaining molecular spectra from classical trajectories

Vibrational classical trajectories of anharmonic molecules are used here to obtain the classical vibrational autocorrelation function and, via a Fourier transform, the power (or infrared) spectrum of the dynamical variables. In the vibrationally quasiperiodic regime the spectrum consists of sharp lines, for any given initial amplitude. The initial amplitudes are chosen semiclassically. The spectral lines are compared with quantum mechanical calculations for systems with two and three coordinates, with excellent agreement. The method is also useful for obtaining a classical spectrum in the ergodic regime; the spectral lines are then ’’broad’’ rather than narrow. The method can be used in the analysis of trajectories for unimolecular reactions, infrared multiphoton dissociations, and for obtaining molecular spectra from force fields. The spectral analysis itself has implications for the theory of unimolecular reactions.

Theory of thermal unimolecular reactions at low pressures. I. Solutions of the master equation

The master equation for a thermal unimolecular reaction in gases at low pressures is formulated. Steady-state solutions are derived in analytical form with an exponential model of collisional transition probabilities, (i) for vibrational energy transfer in molecules with variable densities of states, and (ii) for combined rotational and vibrational energy transfer in molecules with variable heights of the centrifugal barriers. Other models of transition probabilities are treated numerically. The diffusion limit of energy transfer is discussed. In all cases, the nonequilibrium populations of excited states and the weak collision efficiency factors βc are calculated.

Collisionless Multiphoton Dissociation of SF6: A Statistical Thermodynamic Process

A calorimetric experiment, which combines picosecond C${\mathrm{O}}_{2}$-laser pulses with acoustic detection techniques, proves that the collisionless multiphoton dissociation of S${\mathrm{F}}_{6}$ is a statistical thermodynamic process. The standard Rice-Ramsberger-Kassel-Marcus theory of unimolecular reactions provides a quantitative explanation of the dissociation effect. The reaction rate is primarily limited by transitions in the quasicontinuum rather than by the anharmonicity of the first few discrete levels.

Theory of thermal unimolecular reactions. Calculation of strong collision rate constants in the low pressure limit

Strong collision rate constants for thermal unimolecular reactions in the low pressure limit are calculated with a new and more rigorous model. Various potential surfaces are considered in order to determine the sensitivity of the calculated rate constants to uncertainties in the centrifugal potential. Model calculations are performed for the FNO, CINO, INO, H 2O, NO 2 and O 3 systems.

Unimolecular decomposition of chemically activated methylallylether

AbstractThe unimolecular decomposition of chemically activated methylallylether (MAE) formed by the cross combination of methoxymethyl and vinyl radicals was studied in the gas phase. The experimentally determined rate constant was found to be 1.11 × 108 sec−1 at 9.6°C for the decomposition of MAE into propene and formaldehyde. The decomposition of MAE via the six‐center retro‐“ene” type transition state is analyzed by using the RRKM unimolecular reaction theory. For the molecular parameter assignments of energized MAE, a model which contains one internal rotational mode is supported, and MAE decomposition is characterized by a tight complex model. The best agreement between experimental and theoretical results was found when a critical energy of 40.1 kcal/mol was used.

Energy Distributions in Unimolecular Reactions

AbstractThe present paper describes results on modes of anharmonically coupled oscillators in a molecule, the role of energy exchange and neardegeneracy, and quasi‐periodic versus ergodic theory of unimolecular reactions. The quasi‐periodic one is a formal generalization of Slater's harmonic oscillator theory. Spectral decomposition of a classical trajectory for the molecule provides insight into the molecular spectrum, both in the low (quasi‐periodic) and high (ergodic) energy regimes. The role of quantization of transition states in unimolecular reactions and a relation to vibrational adiabaticity are analyzed. Results obtained on energy distribution of translational energy of molecular beam reactions involving, intermediate complexes and tight exit channel transition states are described together with the relation to intramolecular energy randomization.

An alternative view of unimolecular reaction theory

A primitive form of unimolecular reaction theory is constructed, based on the master equation. Application of the theory depends upon the way in which eigenvalues of large stochastic matrices behave as certain of the matrix elements are varied. Some of the required eigenvalue properties have been deduced by numerical experiment, and others are available from earlier work on diatomic dissociation reactions. Comparison of the results from this primitive approach with those of existing theories of unimolecular reactions suggests some reinterpretation of parameters used in those theories: most interestingly, the number of effective oscillators occurring in Kassel theory (or its Slater-theory equivalent) appears to be the ratio, at infinite pressure, of the total rate to the rate of the only reaction channel which survives at the low-pressure limit.

Translational energies from ionic fragmentation

Statistical theories of unimolecular reactions are applied to determine the translational energies of fragments of ionic decomposition. The results are found to be in conflict with experimental measurements of translational energies of ions formed by electron or photon impact. However, a full application of phase-space theory results in much better agreement with ion–molecule experiments than the theory of Safron et al. [S. A. Safron, N. D. Weinstein, D. R. Herschbach, and J. C. Tully, Chem. Phys. Lett. 12, 564 (1972)]. A classical model of a triatomic molecule is constructed, and the partitioning of energy in dissociating trajectories is examined and compared to statistical theories. Discrepancies between the trajectories and statistical theories are noted and possible explanations discussed.

Kinetic energy release distributions for the dissociation of internal energy selected CH3I+ and CD3I+ ions

Product translational energy distributions have been measured from an analysis of symmetrically broadened time-of-flight peaks of CH3+ and CD3+ ions in photoion–photoelectron coincidence spectra of CH3I and CD3I, respectively. The reactions were investigated at various precursor ion internal energies up to 1.7 eV above the dissociation threshold. We found that the experimentally obtained kinetic energy release distributions can be fitted accurately by simple exponential functions over the energy range considered. The experimental results are compared to predictions of the statistical theory of unimolecular reactions. An angular momentum conserving version of the quasiequilibrium theory, formulated in terms of the Langevin collision cross section model for the reverse association reaction, predicts kinetic energy release distributions which agree very well with the experimental results at precursor ion energies up to 0.65 eV above the dissociation onset. At approximately this energy, the average kinetic energy released begins to deviate sharply from the theoretically predicted average, possibly indicating the accessibility of a new reaction channel.