Energy-grained Master Equation Research Articles

On the accuracy of the chemically significant eigenvalue method.

We study the accuracy and convergence properties of the chemically significant eigenvalues method as proposed by Georgievskii et al. [J. Phys. Chem. A 117, 12146-12154 (2013)] and its close relative, dominant subspace truncation, for reduction of the energy-grained master equation. We formally derive the connection between both reduction techniques and provide hard error bounds for the accuracy of the latter which confirm the empirically excellent accuracy and convergence properties but also unveil practically relevant cases in which both methods are bound to fall short. We propose the use of balanced truncation as an effective alternative in these cases.

TUMME 2023: Tsinghua University Minnesota Master Equation program. New version announcement

This is an updated version of TUMME (Tsinghua University Minnesota Master Equation program), which is a computer program for setting up and solving master equations for chemical kinetics of unimolecular and bimolecular reactions. The master equation is discretized in terms of energy bins, and rate constants are calculated from chemically significant eigenmodes. TUMME has interfaces to Gaussian 16, Polyrate 2023, and MSTor 2023 output files that allow the master equation code to obtain the microcanonical flux coefficients needed for the kernel of the master equation as calculated by conventional transition state theory (TST), variational transition state theory (VTST) with various tunneling methods, or multi-structural or multi-path VTST (MS-VTST or MP-VTST) with various tunneling methods. The tunneling methods supported include zero-curvature tunneling (ZCT), small-curvature tunneling (SCT), large-curvature tunneling (LCT), and microcanonically optimized multidimensional tunneling (μOMT). For mechanisms involving only unimolecular isomerization (no bimolecular pairs), TUMME can set up and solve a conservative master equation for both rate constants and time-dependent energy-bin populations. For mechanisms involving bimolecular pairs, TUMME 2023 can set up and solve two kinds of master equation: (i) a nonconservative master equation for calculating rate constants of bimolecular reactions and (ii) a conservative master equation that includes bimolecular association in the transition matrix and that can be used for calculating the time evolution of the concentration of a pseudo-first-order bimolecular reactant. TUMME is written in double precision with Python 3; quadruple and octuple precision are also available for some subtasks in C++. The Python code can run in serial or parallel (MP or MPI), and C++ code can run on a single processor or on multiple processors with OpenMP. The program includes a manual and a tutorial. New version program summaryProgram Title: TUMME 2023CPC Library link to program files:https://doi.org/10.17632/whcnvm2mc9.2Developer's repository link:https://doi.org/10.5281/zenodo.7943283Licensing Provisions: Apache-2.0 for the program and CC-BY-4.0 for the manualProgramming languages: Python 3 and C++Does the new version supersede the previous version?: YesJournal reference of previous version: Comput. Phys. Commun. 270 (2022) 108140 [1]External libraries: Numpy, Scipy, Numba, mpi4py (optional), modified mpack (optional), qd (optional), omp (optional)Interfaces to other programs: Gaussian 16 [2], Polyrate 2023 [3], and/or MSTor 2023 [4]Nature of problem: Characterize and calculate rate constants for a temperature-dependent and pressure-dependent complex reaction systemSolution method: Solve the energy-grained master equation based on the chemically significant eigenmodes to get phenomenological rate constants and time evolutions of the populations [1,5–7].Reason for new version: The new version increases the functionality and accuracy, and it offers an easier way to run the program. In addition, some bugs have been corrected.Summary of revisions:1.Added an option (available in high precision) to treat pseudo-first-order bimolecular reactions to obtain the time evolution of the reactants and intermediates. [8]2.Added an inverse-Laplace-transform option in the REACTION block.3.Abandoned rounding the sizes of energy bins to an integer wavenumber (ESOT).4.Changed the way to run TUMME by providing an executable bash script.5.Slightly adjusted the strategy for eigenmode mergers to deal with some unexpected cases.6.Adjusted the output for the partition functions.7.Added options for large-curvature tunneling (LCT) and microcanonically optimized multidimensional tunneling (μOMT).8.Added the keyword pyrfile to support a Polyrate 2023 [3] file that ends with a suffix .fu100.9.Fixed some bugs.10.Updated the manual and added a tutorial to the program distribution, so the distribution now has both a manual and a tutorial.

Investigating the Association Reactions of HOCH2CO and HOCHCHO with O2: A Quantum Computational and Master Equation Study

Glycolaldehyde, HOCH2CHO, is an important multifunctional atmospheric trace gas formed in the oxidation of ethylene and isoprene and emitted directly from burning biomass. The initial step in the atmospheric photooxidation of HOCH2CHO yields HOCH2CO and HOCHCHO radicals; both of these radicals react rapidly with O2 in the troposphere. This study presents a comprehensive theoretical investigation of the HOCH2CO + O2 and HOCHCHO + O2 reactions using high-level quantum chemical calculations and energy-grained master equation simulations. The HOCH2CO + O2 reaction results in the formation of a HOCH2C(O)O2 radical, while the HOCHCHO + O2 reaction yields (HCO)2 + HO2. Density functional theory calculations have identified two open unimolecular pathways associated with the HOCH2C(O)O2 radical that yield HCOCOOH + OH or HCHO + CO2 + OH products; the former novel bimolecular product pathway has not been previously reported in the literature. Master equation simulations based on the potential energy surface calculated here for the HOCH2CO + O2 recombination reaction support experimental product yield data from the literature and indicate that, even at total pressures of 1 atm, the HOCH2CO + O2 reaction yields ∼11% OH at 298 K.

Quantitative Kinetics of HO2 Reactions with Aldehydes in the Atmosphere: High-Order Dynamic Correlation, Anharmonicity, and Falloff Effects Are All Important.

Kinetics provides the fundamental parameters for elucidating sources and sinks of key atmospheric species and for atmospheric modeling more generally. Obtaining quantitative kinetics in the laboratory for the full range of atmospheric temperatures and pressures is quite difficult. Here, we use computational chemistry to obtain quantitative rate constants for the reactions of HO2 with HCHO, CH3CHO, and CF3CHO. First, we calculate the high-pressure-limit rate constants by using a dual-level strategy that combines conventional transition state theory using a high level of electronic structure wave function theory with canonical variational transition state theory including small-curvature tunneling using density functional theory. The wave-function level is beyond-CCSD(T) for HCHO and CCSD(T)-F12a (Level-A) for XCHO (X = CH3, CF3), and the density functional (Level-B) is specifically validated for these reactions. Then, we calculate the pressure-dependent rate constants by using system-specific quantum RRK theory (SS-QRRK) and also by an energy-grained master equation. The two treatments of the pressure dependence agree well. We find that the Level-A//Level-B method gives good agreement with CCSDTQ(P)/CBS. We also find that anharmonicity is an important factor that increases the rate constants of all three reactions. We find that the HO2 + HCHO reaction has a significant dependence on pressure, but the HO2 + CF3CHO reaction is almost independent of pressure. Our findings show that the HO2 + HCHO reaction makes important contribution to the sink for HCHO, and the HO2 + CF3CHO reaction is the dominant sink for CF3CHO in the atmosphere.

Nonadiabatic Kinetics in the Intermediate Coupling Regime: Comparing Molecular Dynamics to an Energy-Grained Master Equation.

We propose and test an extension of the energy-grained master equation (EGME) for treating nonadiabatic (NA) hopping between different potential energy surfaces, which enables us to model the competition between stepwise collisional relaxation and kinetic processes which transfer population between different electronic states of the same spin symmetry. By incorporating Zhu-Nakamura theory into the EGME, we are able to treat NA passages beyond the simple Landau-Zener approximation, along with the corresponding treatments of zero-point energy and tunneling probability. To evaluate the performance of this NA-EGME approach, we carried out detailed studies of the UV photodynamics of the volatile organic compound C6-hydroperoxy aldehyde (C6-HPALD) using on-the-fly ab initio molecular dynamics and trajectory surface hopping. For this multichromophore molecule, we show that the EGME is able to capture important aspects of the dynamics, including kinetic timescales, and diabatic trapping. Such an approach provides a promising and efficient strategy for treating the long-time dynamics of photoexcited molecules in regimes which are difficult to capture using atomistic on-the-fly molecular dynamics.

Reaction and relaxation at surface hotspots: using molecular dynamics and the energy-grained master equation to describe diamond etching

The extent to which vibrational energy transfer dynamics can impact reaction outcomes beyond the gas phase remains an active research question. Molecular dynamics (MD) simulations are the method of choice for investigating such questions; however, they can be extremely expensive, and therefore it is worth developing cheaper models that are capable of furnishing reasonable results. This paper has two primary aims. First, we investigate the competition between energy relaxation and reaction at ‘hotspots’ that form on the surface of diamond during the chemical vapour deposition process. To explore this, we developed an efficient reactive potential energy surface by fitting an empirical valence bond model to higher-level ab initio electronic structure theory. We then ran 160 000 NVE trajectories on a large slab of diamond, and the results are in reasonable agreement with experiment: they suggest that energy dissipation from surface hotspots is complete within a few hundred femtoseconds, but that a small fraction of CH3 does in fact undergo dissociation prior to the onset of thermal equilibrium. Second, we developed and tested a general procedure to formulate and solve the energy-grained master equation (EGME) for surface chemistry problems. The procedure we outline splits the diamond slab into system and bath components, and then evaluates microcanonical transition-state theory rate coefficients in the configuration space of the system atoms. Energy transfer from the system to the bath is estimated using linear response theory from a single long MD trajectory, and used to parametrize an energy transfer function which can be input into the EGME. Despite the number of approximations involved, the surface EGME results are in reasonable agreement with the NVE MD simulations, but considerably cheaper. The results are encouraging, because they offer a computationally tractable strategy for investigating non-equilibrium reaction dynamics at surfaces for a broader range of systems.This article is part of the themed issue ‘Theoretical and computational studies of non-equilibrium and non-statistical dynamics in the gas phase, in the condensed phase and at interfaces’.

Atmospheric chemistry of enols: a theoretical study of the vinyl alcohol + OH + O(2) reaction mechanism.

Enols are emerging as trace atmospheric components that may play a significant role in the formation of organic acids in the atmosphere. We have investigated the hydroxyl radical ((•)OH) initiated oxidation chemistry of the simplest enol, vinyl alcohol (ethenol, CH2═CHOH), using quantum chemical calculations and energy-grained master equation simulations. A lifetime of around 4 h was determined for vinyl alcohol in the presence of tropospheric levels of (•)OH. The reaction proceeds by (•)OH addition at both the α (66%) and β (33%) carbons of the π-system, yielding the C-centered radicals (•)CH2CH(OH)2, and HOCH2C(•)HOH, respectively. Subsequent trapping by O2 leads to the respective peroxyl radicals. About 90% of the chemically activated population of the major peroxyl radical adduct (•)O2CH2CH(OH)2 is predicted to undergo fragmentation to produce formic acid and formaldehyde, with regeneration of (•)OH. The minor peroxyl radical HOCH2C(OO(•))HOH is even less stable and undergoes almost exclusive HO2(•) elimination to form glycolaldehyde (HOCH2CHO). Formation of the latter has not been proposed before in the oxidation of vinyl alcohol. A kinetic mechanism for use in atmospheric modeling is provided, featuring phenomenological rate coefficients for formation of the three main product channels ((•)O2CH2CH(OH)2 [8%]; HC(O)OH + HCHO + (•)OH [56%]; HOCH2CHO + HO2(•) [37%]). Our study supports previous findings that vinyl alcohol should be rapidly removed from the atmosphere by reaction with (•)OH and O2 with glycolaldehyde being identified as a previously unconsidered product. Most importantly, it is shown that direct chemically activated reactions can lead to (•)OH and HO2(•) (HOx) recycling.

Rate Rules, Branching Ratios, and Pressure Dependence of the HO2 + Olefin Addition Channels

In this work, we present high-pressure rate rules and branching ratios for the addition of HO2 to olefins through the concerted addition channel to form an alkyl peroxy radical (HO2 + olefin → RO2) and through the radical addition channel to form a β-hydroperoxy alkyl radical (HO2 + olefin → β-QOOH). These rate rules were developed by calculating rate constants for a series of addition reactions involving olefins with varying degrees of branching. The individual rate expressions were determined from electronic structure calculations performed at the CBS-QB3 level of theory combined with TST calculations. The calculated rate constants were found to be in good agreement with those reported in the literature. Next, we calculated apparent pressure- and temperature-dependent rate constants for HO2 addition to the terminal site of 1-butene using an energy-grained master equation (ME) approach and QRRK calculations with a modified strong collision (MSC) approximation. The two methods gave similar results for both reaction classes. We found that, for the radical addition reaction, the high-pressure limit for the stabilization channel is not reached until unusually high pressures (>1000 atm). Instead, this reaction leads to the direct formation of an oxirane + OH. In general, the results for the major channels are in reasonable agreement with prior theoretical and experimental data. Finally, to explicitly examine the effect of pressure, we compared concentration-time profiles for the reactions of HO2 plus butene in air that were obtained using both high-pressure and pressure-dependent mechanisms at 10 and 100 atm. These simulations showed that, contrary to general expectations, the manifestation of pressure falloff effects in kinetic modeling studies might be more prevalent at increasing pressures. This behavior is attributed to the reaction of β-QOOH with O2, the rate of which increases with increasing pressure of air. This bimolecular reaction competes with the unimolecular reactions of β-QOOH under conditions where falloff effects are important for that channel.

Time-Resolved Cavity Ringdown Measurements and Kinetic Modeling of the Pressure Dependences of the Recombination Reactions of SiH2 with the Alkenes C2H4, C3H6, and t-C4H8

Room-temperature rate constants for the pressure-dependent reactions SiH2 + ethene, propene, and t-butene have been determined at total pressures of 3.3 mbar < or = p < or = 300 mbar with Ar as buffer gas. SiH2 was detected by means of time-resolved cavity ringdown spectroscopy (CRDS), and the deconvolution of ringdown, kinetics, and laser bandwidth effect was accomplished with the extended simultaneous kinetics and ringdown model (eSKaR). In this way, pseudofirst-order rate constants could be extracted from nonexponential ringdown profiles. The recombination reactions, including the reaction SiH2 + i-butene, have been modeled based on the simplified statistical adiabatic channel model (SACM) and weak collision energy-grained master equation (ME) simulations. The influence of an interfering fast isomerization channel was investigated based on the Rice, Ramsperger, Kassel, Marcus theory (RRKM) and was found to be only important for the C2H4 reaction. Using ab initio energies (G3) and structures (MP2/6-311G(d,p)) as input parameters for the kinetic models, a consistent description of the pressure and temperature dependences of all four reactions was possible. In the temperature range 295 K < or = T < or = 600 K, the extrapolated limiting high-pressure rate constants, k(infinity)(C2H4)/(cm(3) x mol(-1) x s(-1)) = 1.9 x 10(14) (T/K)(-0.065), k(infinity)(C3H6)/(cm(3) x mol(-1) x s(-1)) = 1.3 x 10(14) (T/K)(0.075), k(infinity)(i-C4H8) = 1.8 x 10(14) cm(3) x mol(-1) x s(-1), and k(infinity)(t-C4H8)/(cm(3) x mol(-1) x s(-1)) = 4.6 x 10(13) (T/K)(0.21), are close to the collision number and are more or less temperature independent. In the case of ethene, probably due to the approximate treatment of rotational effects and/or the interfering isomerization process, the applied model slightly underestimates the falloff and thus yields too high extrapolated rate constants at p < 10 mbar.

HCO formation in the thermal unimolecular decomposition of glyoxal: rotational and weak collision effects

The multi-channel thermal unimolecular decomposition of glyoxal was experimentally investigated in the temperature range 1106 K < T < 2320 K and at total densities of 1.7 x 10(-6) mol cm(-3) < rho < 1.9 x 10(-5) mol cm(-3) by monitoring HCO (frequency modulation spectroscopy, FMS), (CHO)(2) (UV absorption), and H atom (atom resonance absorption spectroscopy, H-ARAS) concentration-time profiles behind shock waves. With a branching fraction of 48% at T = 2300 K and rho = 1.6 x 10(-5) mol cm(-3), the so-far-neglected, energetically unfavourable HCO-forming decomposition channel, (CHO)(2)--> 2HCO, was found to play a crucial role and in fact represents the major decomposition pathway at high temperatures and high total densities. A theoretical analysis of the experimental results in terms of Rice-Ramsperger-Kassel-Marcus theory (RRKM), the simplified statistical adiabatic channel model (SACM), and an energy-grained master equation (ME) was based on input parameters from ab initio calculations (G3 and MP2/6-311G(d,p)) and literature data on branching ratios from collision-free photolysis experiments. A consistent description of the temperature and density dependences was achieved, revealing that both rotational and weak collision effects are reflected in the measured branching ratios. Overall, a product channel switching occurs with the CH(2)O-forming channel, (CHO)(2)--> CH(2)O + CO, dominating at low temperatures/densities and the HCO-forming channel dominating at high temperatures/densities. Additionally, the so-called triple-whammy channel, (CHO)(2)--> 2CO + H(2), significantly contributes to the total decomposition rate at intermediate temperatures/densities whereas the HCOH-forming pathway, (CHO)(2)--> HCOH + CO, is predicted to be the least important one. The temperature and pressure dependences of the different decomposition channels are parametrized in terms of two-dimensional Chebyshev polynomials.

Fluence and pressure effects on IR multiphoton dissociation of ethylbromide

The IR multiphoton dissociation of C2H5Br was studied as a function of the laser fluence and substrate pressure using a pulsed TEA CO2 laser at the 10P(20) (10.59μm,944 cm−1) laser line. The effects of added H2, He, N2 and NO were also studied. The reaction products were found to be C2H2, C2H4, C2H6 and CH4. The increase in the partial pressure of the products with increasing sustrate pressure was interpreted in terms of the large contribution from a collision-induced between the reactant molecules excited below the reaction threshold. At a focal fluence of less than 100 J cm−2, C2H4 was the only reaction product detected, whereas C2H2 and CH4 also appeared at higher fluence. The formation of C2H2 and CH4 at higher fluence was attributed to additional laser photon absorption by vibrationally excited ethylene (C2H4*) primarily in the quasi-continuum state. The production of C2H6 was due to CBr bond fission. The observed dissociation data were compared with the theoretical predictions obtained using the energy-grained master equation.

Infrared multiphoton dissociation of CTF3: Computer simulation using an energy-grained master equation.

The TEA CO2 laser-induced infrared multiphoton dissociation (IRMPD) of CTF3 diluted in a large excess of a CHF3 buffer gas cooled at -78 °Cwas studied as a function of laser fluence and gas pressure. The dissociation rate for CTF3 increased with an increase in pressure in the region below 45 Torr owing to collisional relaxation of rotational bottle-necking, while it decreased above 45 Torr owing to collisional de-excitation. An energy-grained master equation (EGME) incorporating these pressure effects was used to simulate the experimental results on the CTF3 dissociation. The EGME was numerically solved from trajectory calculations based on a stochastic method. We found that the EGME successfully predicted the pressure dependence of the IRMPD of CTF3 over a wide range of experimental parameters.

The approximation of collisional energy relaxation by imbedding diffusion processes

It is demonstrated that a diffusion process may be constructed which imbeds an exponential-down collisional energy relaxation process. The imbedding diffusion is shown to provide a good description of relaxation even under conditions of jump size and frequency far removed from the diffusion limit. Its validity is demonstrated by comparison between solutions of the diffusion equation with numerical solutions of an energy-grained master equation for the same model system.

Infrared multiphoton dissociation of 1-chloro-1-fluoro ethylene: Competitive reactions and pressure dependence

We have examined the infrared multiphoton dissociation of CH2CFCl as a function of pressure and laser fluence. The pressure dependence could be interpreted in terms of a competition between rotation hole filling and vibrational relaxation. The fluence-dependent studies showed that the competitive hydrogen halide elimination reactions could be influenced by optical pumping rates near the dissociation threshold. Numerical modeling of the results using the energy-grained master equation allowed us to obtain numerical values for the vibrational relaxation efficiencies of a number of bath gases. The numerical modeling suggests that while our numerical approach is adequate to describe the population changes near the dissociation threshold, and under the influence of collisions, it is (not surprisingly) inadequate to describe potential coherent effects at low levels of internal excitation.

Infrared multiphoton decomposition and energy-dependent absorption cross sections of chloroethane-d, -2-d1, and -2,2,2-d3

The infrared photochemistry of chloroethane-d0, -2-d1, and -2, 2, 2-d3 have been investigated. In all cases, the multiple infrared photon induced dissociation is found to proceed by four-center elimination of HCl (or DCl). An energy-grained master equation is used to interpret the data; in this way, the energy-dependent photon absorption cross section σ(E) is determined for all three species. The functional form of σ(E) approaches that for a ‘‘large’’ (i.e., high density of states) molecule with increasing deuteration. Ensemble properties, such as vibrational energy distributions and average energy deposition, can then be determined; but these observables do not by themselves suffice to determine a unique form for σ(E). A form for σ(E) can, however, be specified within certain limits by applying constraints imposed by intramolecular vibrational coupling models.

Master equation description of the multiphoton decomposition of ethyl acetate

In experimental observations of the multiphoton decomposition of ethyl acetate by a CO 2 laser at 1045.0 cm −1 fluences up to 4 J cm −2 were employed to dissociate 2 Pa of ethyl acetate in up to 600 Pa of N 2, He, Ne, Ar, Kr, Xe, ethylene and acetone bath gases. The fraction dissociated was measured in a closed cell, either by rate of removal of reactant, monitored by a probe laser and spectrophone, or by rate of pressure rise, monitored by a capacitance manometer. Total absorption and transient absorption changes were also measured in separate experiments. Data were analysed by finite difference solution of the energy-grained master equation, incorporating collisional effects and changes in transitional temperature. The convergence of the solutions was checked with decrease in the size of the energy grain. Microscopic reaction rates were described by an RRKM formulation with parameters from thermal experiments. Radiation absorption was described by an energy-dependent cross section with one free parameter chosen to match independent data. The collisional energy transfer function was an exponential form with the mean down transfer energy <Δ E> as a parameter. Dissociation versus pressure and fluence was fitted by one value of <ΔE> for each bath gas: N 2-820, He-550, Ne-700, Ar-930, Kr-905, Xe-920, C 2H 4-3000, CO(CH 3) 2-5000 cm −1. The technique appears to be a reliable means of obtaining energy transfer data at low temperatures. In addition to the experiments with reactant diluted in bath gas, the decomposition of undiluted reactant was also observed, at pressures in the range 1 to 800 Pa; here, reaction was terminated by collisions of irradiated molecules with molecules outside the beam; an approximate theory which appears to confirm the hypotheses is presented.

Effect of intensity on fragment internal state distributions in the infrared multiphoton dissociation of vinyl cyanide

The technique of laser induced fluorescence has been used to observe the C2 and CN fragments produced in the infrared multiphoton dissociation of vinyl cyanide CH2 = CHCN in a low pressure flowing gas. The rotational distributions of the lowest vibrational level of the C2 fragment were found to be well characterized by temperatures. As was previously observed for the CN fragment of this parent molecule, a distinct decrease in the values of these temperatures is seen to occur during the photolysis laser pulse. This is interpreted as the result of an intensity effect in the later stages of the infrared pumping process. Additional information is also presented on the behavior of the CN radical. Modeling calculations within the framework of the energy-grained master equation support this interpretation.

IR photochemistry: A unified approach for single-channel reactions. I. Theory and computational examples

A model for infrared multiphoton decomposition of polyatomic molecules based on an energy-grained master equation (EGME) shows remarkably uniform behavior over a wide range of physically reasonable model parameters. These parameters encompass the size of the molecule, the reaction threshold energy, the variation of the decomposition rate constant above the threshold, and the functional dependence of the infrared absorption process on the internal energy of the molecule. The results are shown to be uniformly well represented by a cumulative log–normal distribution function for reaction yield versus time or fluence. A complementary approach based on the distribution of first passage times for a Markovian stochastic process in discrete state space and continuous time shows that the yield behavior of the EGME can be conveniently calculated without solution of the full set of equations. The uniform behavior of the model results yields a powerful method for presenting and analyzing experimental data.

Collisional energy transfer from excited polyatomic molecules produced by infrared multiple photon absorption

A simple energy-grained master equation is used to model data on the multiphoton dissociation of CF2HCl as a function of laser fluence and argon bath–gas pressure. Information on energy transfer from vibrationally excited CF2HCl is obtained from an analysis of the competition between laser pumping and collisional energy removal using suitable parametrizations of the laser intensity and the collision process.

Infrared multiphoton decomposition: A comparison of approximate models and exact solutions of the energy-grained master equation

The approximate ’’thermal’’ model and ’’continuum’’ model have been compared to exact calculations, and neither gives satisfactory results. In particular, the thermal model is based on an inaccurate and unphysical approximation to the correct molecular density of states, as well as a restrictive expression for the absorption cross sections. As an alternative to the approximate models, the exact stochastic method gives exact results and can be implemented using small computers or programmable pocket calculators. The results obtained from this method are exact and precision depends only upon the number of ’’trajectories’’ calculated. It was discovered that the yield versus fluence results from exact numerical integration of the master equation approximately conform to a cumulative log–normal distribution function in decomposition time or fluence. Thus, to a good degree of approximation, computed yield versus fluence results can be expressed simply in terms of a mean and a dispersion parameter. This suggests that the use of probability graph paper and the log–normal distribution function may prove to be a useful means for the presentation and analysis of experimental and computed data. Moreover, deviation from the log–normal distribution may indicate that the simple master equation is not a sufficient description of the experimental results.