J . Phys. Chem. 1990, 94, 6696-6106
6696
Are Classical Molecular Dynamics Calculations Accurate for State-testate Transition Probabilities in the H -4- D, Reaction? Meishan Zhao, Donald G . Truhlar,* Department of Chemistry, Chemical Physics Program, and Supercomputer Institute, University of Minnesota, Minneapolis. Minnesota 55455- 043 I
Normand C. Blais, Chemistry and Laser Sciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
David W. Schwenke.* N A S A Ames Research Center, Mail Stop 230-3, Moffett Field, California 94035
and Donald J . Kouri* Department of Chemistry and Department of Physics, University of Houston, Houston, Texas 77204-5641 (Received: January 25, 1990)
We present converged quantum dynamics for the H + D, reaction at a total energy high enough to produce H D in the L.' = 3,;' = 7 vibrational-rotational state and for total angular momenta J = 0, 1, and 2. We compare state-to-state partial H D (v' = 0-2, j ? + H and H + D, (o = 1, j = 6. J = 0-2) HD cross sections for H + D, (c = 0-2, j = 0 , J = 0-2) (L" = 0-2, j ? H as calculated from classical trajectory calculations with quantized initial conditions, Le., a "quasiclassical" trajectory (QCT) simulation, to the results of converged quantum dynamics calculations involving up to 654 coupled channels. Final states in the QCT calculations are assigned by the quadratic smooth sampling (QSS) method. Since the quasiclassical and quantal calculations are carried out with the same potential energy surface, the comparison provides a direct test of the accuracy of the quasiclassical simulations as a function of the initial vibrational-rotational state and the final vibrational-rotational state. The test of the QCT-QSS method against quantum mechanics for the four initial states considered yields the following conclusions (since the v' = 3 cross sections are all very small, we consider only c' = 0, I , and 2 for these comparisons). QCT-QSS partial cross sections summed over c'and j ' a r e accurate to 2 4 % . QCT-QSS partial cross sections summed overj'for specified c'are accurate to 5-1092 for c ' = 0, 8-l7% for e ' = I . and 7-29% for c ' = 2, with an average error of these I2 partially summed cross sections of 1310. The ratio of the partial cross section for producing o' = 0 to that for producing e' = I is accurate to 2-23%, and the (c' = I)/(c' = 2) ratio is accurate to 2-545'6; the average error in these eight ratios is 26%. The average value ofj'in a given c'manifold is accurate to 2-37% with average error of 17% for initial states with; = 0, but it is accurate to 1-4% with an average error of only 2% for the initial state with j = 6. The root-mean-square width of thej'distribution for a given c'is accurate to 1-294 with an average error of 12%, where we have included both J = 0 and; = 6 initial states in this comparison since the j = 6 results are not more accurate for this product attribute. The peak in the QCT-QSS /'distributions for a given c'agrees (within sampling statistics) with the accurate quantal peak position in all eight cases for e' = 0 and c' = 1 but in only two of four cases for e' = 2. Fully resolved state-to-state cross sections show larger errors than the partially summed ones in many cases, and in general the quantal product state distributions have more structure than the QCT-QSS ones, even though the quantal oscillations are partially damped by summing over three to five total angular momentum/parity blocks. We conclude that classical mechanics, although very accurate for the reaction cross section summed over all final states and qualitatively correct for product state distributions, shows significant quantitative errors on the order of 20% for partially summed cross sections and low moments of the product distribution, with larger errors in some cases for individual state-to-state cross sections.
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Introduction
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The H D2reaction is of special interest in chemical kinetics because we have more complete information about product vibrational-rotational energy distributions as a function of initial translational energy for this reaction than for any other.',* One surprising aspect of the interpretation of these results has been thc generally very good agreement between experiment's2 and quasiclassical trajectory simulations for product-state rotational ( I ) Gerrity, D. P.; Valentini, J. J . J . Chem. Phys. 1983, 79, 5102; 1984, 81, 1298; 1985,82, 1323; 1985, 83, 2207. Viers. D. K.; Rosenblatt, G . M.; Valentini, J. J . J . Chem. Phys. 1985. 83, 160s. Levene. H. B.; Phillips, D. L.;Nieh, J.-C.; Gerrity, D. P.; Valentini, J . J . J . Chem. Phys. Left. 1988, 143, 317. Valentini. J . J.: Phillips. D. L. Bimolecular Collisions; Ashford, M. N. R.. Baggott. J . E.. Eds.; Royal Society of Chemistry: London, 1989; p 1 ( 2 ) Marino. E. E.; Rettner. C. T.; Zare, R. N.J . Chem. Phps. 1984. 80. 4141. Rinnen, K.-D.: Kliner. D. A. V.; Blake, R. S.; Zare. R. N . Chem. Phys Letr. 1988. 153. 371. Rinner. K.-D.; Kliner, D. A . V.; Zare, R. N. J . Chent. Ph.rs. in press. and references therein
0022-3654/90/2094-6696$02.50/0
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d i s t r i b ~ t i o n s . ~This leads to increased confidence in the potential cnergy ~ u r f a c e sused ~ , ~ for these studies, even a t relatively high cncrgies, and i t also gives us hope that the easilj interpretcd classical calculations may be useful for understanding cvcn very detailedaspects of state-to-state dynamics for systems involving light atoms, Another set of comparisons6 has involved integral cross sections and opacity functions calculatedby t r a jectory simulations and classical and quantal rotational sudden approximations for the ground vibrational state of D, and high (3) Blais, N . C.; Truhlar, D. G. Chem. Phys. Lett. 1983, 102. 120: J . Chmr. Phps. 1985, 83. 2201; Chem. Phys. Lett. 1989, 162, 503. (4) Liu, B. J . Chem. Phys. 1973, 58, 1925. Siegbahn, P.: Liu, B. J . Chem. Phps. 1978. 68, 2457. Truhlar. D. G.;Horowitz. C. J. J . Chem. Phys. 1978.
68. 2466: 1979. 71. 1514(E). ( 5 ) Varandas. A. J. C.; Brown, F. B.; Mead, D. A,; Truhlar. D G.; Blais. \ C. J . Chem. P h j s . 1987, 86, 6258. (6) Baer, M. Chem. Phys. 1988, 123, 365. Last, 1.: Ron. S.; Baer. M. 1sr. J Chem 1989. 29. 451
C I990 American Chemical Society
H
+ D2 Reaction
The Journal oj”Physical Chemistry, Vol. 94, No. 17, 1990 6691
translational energies (0.8-2.7 eV). From the comparison of the latter set of results it was concluded that quantum integral cross sections may be about 10% higher than classical ones even at high cncrgies, with the approximate quantal reaction probabilities exceeding the approximate classical ones a t both low and high orbital angular momenta. Further exploration of this inferred adequacy of trajectory simulations for product distributions and of the inferred systematic errors in overall reaction probabilities are the main subjects of this paper. In particular we present the first accurate quantum dynamical transition probabilities for this isotopic combination, employing the most recent5 accurate potential energy surface for this system, and we present new trajectory calculations for the same conditions and the same potential energy surface. W e consider four different initial states of D,. The present quantal and trajectory results may be compared to each other without any ambiguities due to averaging over initial conditions, state-dependent detection efficiencies or calibration errors, or difference of the best available theoretical potential energy function from the (unknown) one involved in the experiments, and thus they provide a direct test of the trajectory simulation of the state-to-state dynamics. Until recently, such tests of classical dynamics against completely converged quantum dynamics were limited to collinear collisions7~* and to the unlabeled H H, reaction a t low energy.9 In other recent work we have carried out tests for three-dimenH2I0and D H211312 a t energies high enough for r‘ sional H = I states to be produced (c is the vibrational quantum number, and the prime denotes final state). In the present work we extend the accurate three-dimensional quantum dynamics for such comparisons to a system with two atoms heavier than protium and to an energy where I” = 3 states are open. We present results for three total angular momenta ( J = 0, 1, and 2 ) . Summing the state-to-state transition probabilities over a small range of total angular momenta averages quantum fluctuations that occur only at the experimentally unobservable single S matrix level, but it provides more resolution of the state-to-state dynamics than integral cross sections would. W e believe that this kind of quantitative assessment of classical simulation methods at a detailed partial cross section level enables future simulations to be interpreted with more realistic expectations.
where OK is the contribution to the reactance matrix due to H:, u hich induces only nonreactive scattering, BXis the distorted wave Born approximation reactance matrix for reactive and vibrationally inclastic scattering, and the b and C matrices are expressed as
(3) (4)
In these equations (@818=l, 2 , , . , M is the L2basis set used to ex and the reactive amplitude densities, .I/” is a distorted wave, G, -I? is a principal value Green’s function, U* is the coupling potential multiplied by ( - 2 p / h 2 ) ,p is the three-body reduced mas^,^^,^' (3 is the basis set index, n8 is the arrangement label associated with basis function fi, and n is a channel index, which includes the complete set of arrangement ( a ) ,vibrational (0).rotational (j), and orbital ( I ) quantum numbers. For H D2 we use the symmetry of the two identical product arrangements to block diagonalize C prior to the solution of (2).15 After obtaining the K matrix, we calculate the transition matrix (T matrix), whose elements are the state-to-state transition amplitudes with physical boundary conditions, by
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+
+
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T = 2i(l - iK)-IK
(5)
Transition Probabilities and Cross Sections. In all calculations we take account of the conserved quantum numbers J , the total angular momentum, and P, the parity, the latter of which is given by
p =
(-])I+‘ = (-1 )?.+I’
(6)
We treat thc D atoms as identical but distinguishable, which is consistent with the classical simulations we are testing and which should cause negligible error for the results to be Considered. The completely resolved transition probabilities are given by
(7) These quantities contain more detail than we usually wish to see. Thus we will tabulate three kinds of partially summed results: vibrationally resolved reaction probabilities
Quantum Mechanical Calculations
Theory. The quantum mechanical transition probabilities are obtained by L2 variational solutions of the coupled integral equations for reactive amplitude densities by the generalized Newton variational principal (GNVP).I3-” For each atomdiatom arrangement. the Hamiltonian operator is partitioned into two parts: H = H : + ~
+
(1)
+
+ bTC-]b
P$ = CFy$ c‘
(2)
(7) Truhlar, D. G. J . Phys. Chem. 1979, 83, 188. (8) Raff, L. M.; Thompson, D. L. Theory of Chemical Reaction Dynamics; Baer. M.. Ed.; CRC Press: Boca Raton, FL, 1985; Vol. 3, p I . (9) Schatz, G . C.; Kuppermann, A . J . Chem. Phys. 1976, 65, 4642. (IO) Zhao, M.; Mladenovif. M.; Truhlar, D. G.; Schwenke, D. W.; Sun, Y.; Kouri, D. J.; Blais, N. C. J . Am. Chem. SOC.1989, / I / . 852. ( I I ) Blais. N. C.; Zhao, M.; Mladenovif, M.; Truhlar, D. G.; Schwenke. D. W.; Sun, Y . ; Kouri, D.J. J . Chem. Phys. 1989, 91. 1038. (12) Blais, N. C.; Zhao, M.; Truhlar, D. G.;Schwenke. D. W.: Kouri. D. J . Chem. Phys. Lett. 1990. 166, 1 I . (13) Schwenke, D. W.; Haug, K.; Truhlar, D. G.; S u n , Y . ;Zhang, J. Z. H.; Kouri. D. J. J . Phys. Chem. 1987, 9 / , 6080. (14) Schwenke, D. W.; Haug, K.; Zhao. M.; Truhlar, D. G.; Sun, Y . ; Zhang, J. 2 . H.; Kouri, D.J. J . Phys. Chem. 1988, 92, 3202. (15) Schwenke. D. W.; Mladenovif. M.; Zhao. M.: Truhlar. D. G.; Sun. Y . ; Kouri. D. J . Supercomputer Algorithms f o r Reacticity. Dynamics. and Kinetics of Small Molecules; Laganl, A,, Ed.; Kluwer: Dordrecht, The Netherlands, 1989; p 13I
(9)
and partial integral cross sections for initial vibrational state c and initial rotational state j
+
where a = I denotes H D,, a = 2 and 3 denote H D D, H: is a rotationally coupled distorted Hamiltonian operator associated with nonreactive scattering for arrangement a, and contains the vibrational coupling in a given arrangement as well as the coupling between arrangements. T h e variationally correct reactance matrix is given by
K = OK
total reaction probabilities
Jm
d,V 4 6 F'inal r o t a t i o n a l s t a t e j '
t3
Figure 4. Final rotational-statedistribution of the state-to-state partial cross sections for initial state u = I , j = 6 : (a) final vibrational level v' = 0: (b) u' = 1: (c) u' = 2.
are determined and are compared to results calculated by quasiclassical trajectories. The quantal and the quasiclassical results on an accurate potential energy surface are generally in good agreement with each other for the cross sections summed over final quantum numbers for four initial states considered here and for the partial cross sections summed over the final j'or summed over final u'. The comparison of the state-to-state cross sections shows larger differences. The accuracy of the quasiclassical trajectory method is assessed from the results. For all four initial states studied the trends in the QCT-QSS predictions of the H D o' = 0 distributions are in good agreement with accurate quantum dynamics, the j'distributions in the HD c'= 1 level also show generally good agreement in the trends, and
J . Phys. Chem. 1990, 94, 6706-6718
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thc 1.' = 2 level, which has very small partial cross sections, shows a larger discrepancy between quantum mechanics and the QCTQSS method. For the higher final vibrational levels when reaction begins in a j = 0 state of the c = 0, 1 , or 2 vibrational level. the quantal calculations show much more structure than the QCTQSS ones, even though we have summed over three total angular momenta; some of the peaks in the quantum distributions are missing in the QCT-QSS ones. This effect is especially clear in c ' = I levels, and it also occurs for H D c ' = 1 produced by the reaction H D,(u= 1 j = 6 ) , where we have summed over five total angular momentum/parity blocks.
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Acknowledgment. We are grateful to Richard N . Zare, Dahv Klincr, and Klaus Rinnen for stimulating discussions and to David Chatficld for checking some of the calculations. This work was supportcd in part by the National Science Foundation and the Minnesota Supercomputer Institute. Supplementary Material Available: Four tables o f reaction probabilities and convergence checks for all initial states in each of four total angular momentumfparity blocks at total energy 1.49 eV (8 pages). Ordering information is given on any current masthead page.
Thermal Decomposition of Energetic Materials: Temporal Behaviors of the Rates of Formation of the Gaseous Pyrolysis Products from Condensed-Phase Decomposftion of Octahydro-l,3,5,7-tetranitro-l,3,5,7-tetrazocine Richard Behrens, J r . Combustion Research Facility, Sandia National Laboratories, Livermore, California 94551 (Received: January 22, 1990; In Final Form: April 19, 1990)
The temporal behaviors of the rates of formation of the gaseous products from the pyrolysis of HMX (octahydro-1,3,5,7tetranitro-1,3,5,7-tetrazocine)are determined by simultaneous thermogravimetric modulated beam mass spectrometry (STMBMS). The gaseous products formed from the pyrolysis of HMX, and its deuterium- and 'SN02-labeledanalogues, between 210 and 235 'C are H 2 0 , HCN, CO, C H 2 0 , NO, N 2 0 , CH,NHCHO, ( C H 3 ) 2 N N 0 , l-nitroso-3,5,7-trinitro1,3,5,7-tetrazocine (ONTNTA), and their isotopic analogues. In addition, a nonvolatile residue (NVR) is formed. The NVR is a polyamide that decomposes between 250 and 780 OC producing NH,, HCN, HNCO, H,NCHO, CH,NHCHO, and products that include long-chain hydrocarbons. The temporal behaviors of the rates of gas formation of the pyrolysis products along with the macroscopic and microscopic structure of the NVR indicate that complex physical processes and chemical mechanisms within the condensed phase of HMX control the decomposition. The temporal behaviors of the rates of gas formation show induction, acceleratory, and decay stages that are characteristicof either condensed-phaseor autocatalytic decomposition. The NVR is composed of broken ellipsoidal shells whose diameters range from 0.3 to 5 pm. The shells appear to be remnants of gas-filled bubbles that are formed within the HMX particles during the pyrolysis. The pressures within the gas bubbles may exceed 7 MPa. The major portion of the pyrolysis products observed in the experiments comes from gases that have been contained within the bubbles. Gases released earlier in the decomposition are contained in smaller bubbles and therefore are formed under higher pressure conditions. This results in the variation of the relative rates of formation of the gas products. For example, the ratio of the rates of formation of C H 2 0 to N 2 0 is between 0.7 and 0.8 during the induction stage, decreases to a minimum of 0.36 during the acceleratory stage, and increases to 1 at the end of the decay stage. Variations in the relative rates of formation of the other products are also observed. The changes in the rates of release of products from the initial to final stages of the decomposition indicate that different species are formed when the pyrolysis products are contained at different pressures. The effects of these different physical processes on the relation between the observed pyrolysis products and the underlying chemical decomposition mechanisms are discussed.
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Introduction The cyclic nitramines octahydro-1,3,5,7-tetranitro-1,3,5,7tetrazocine ( H M X , I) and hexahydro-l,3,5-trinitro-s-triazine ( R D X , 11) are energetic ingredients that are used in various
I
I1
propellants and explosives. Understanding the complex physicochemical processes that underlie the combustion of these materials can provide a link between the physical properties and molecular structure of these molecules and their combustive behavior which, in turn, may lead to methods for modifying propellant and explosive formulations in order to obtain better control of their ignition, combustion, or sensitivity. The overall goal of our work is to understand the relationship between the physical properties and molecular structures of
different nitramines and their combustive behavior. This requires understanding the reaction kinetics and transport processes in both the gas and condensed phases. T h e gas-phase reaction kinetics and transport properties of the various gaseous pyrolysis products of the nitramines are relatively well understood.'*2 On the other hand, the reaction kinetics and transport properties associated with the reactions occurring in the condensed phase are not well understood. The thrust of our work is to obtain a better understanding of the physical processes and reaction mechanisms that occur during the decomposition of nitramines in the condensed phase so that the identity and rate of release of the pyrolysis products can be predicted based on the physical properties and molecular structure of the material. This information can then be used along with the gas-phase reaction models to correlate (1) Melius, C. F.; Binkley, J. S. In Proceedings o f t h e 2 l s f Symposium (International) on Combusrion;The Combustion Institute: Pittsburgh, PA,
1986; p 1953. (2) Melius, C. F. Proceedings of the 25th JANNAF Combustion Meeting CPIA Publ. 498; Chemical Propulsion Information Agency. The John Hopkins University Applied Physics Laboratory, Laurel, MD, 1988; Vol. 2, p 155.
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