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Quantum mechanical H+02+OH+0 calculation of the rate constant for the reaction Claude Leforestie?) and William H. Miller Department of Chemistry, University of California, and Chemical SciencesDivision, Lawrence Berkeley Laboratory, Berkeley, California 94720 (Received 13 July 1993; accepted26 October 1993) Rigorous three-dimensional quantum mechanical calculations have been carried out for the cumulative reaction probability of the titled reaction for total angular momentum, J=O. Higher values of J are included approximately, and the resulting thermal rate constant agreeswell with experimental values for 8OO(T<3O@JK. The recent review by Miller, Kee, and Westbrook’has emphasizedthe central importance of the title reaction for modeling combustion kinetics (“...the single most important reaction in combustion...“). A very thorough summary of recent theoretical and experimental work is also given by Varandas, Brandao, and Pastrana2in their classical trajectory study of this reaction and its isotopic variants. The reaction is complicated becausethe potential energy surface (PES) has a deep well ( -2 eV) corresponding to the stable radical HO2 [cf. Fig. 1(a)] which leads to formation of a collision complex, and the small number of atoms and/or disparity of their masses seemsto invalidate the use of simple statistical models for a quantitative description of the reaction rate.’ The purpose of the present paper is to report the results of quantum mechanical calculations for the thermal rate constant of the reaction over the range T = 800-3000 K, using the DMBE IV potential energy surface of Pastrana et al3 Rigorous calculations-which should be essentially exact for the assumedpotential energy surfacehave been carried out for total angular momentum J=O of the three-body system, with the effectsof higher J included approximately (vide infra). The theoretical approach is that of Seideman and Miller,4 with specific further developmentsintroduced by Manthe and Miller.’ Thus the cumulative reaction probability (CRP) N(E) is calculated directly (i.e., without having to solve the complete state-to-statereactive scattering problem), and the thermal rate constant given in terms of it by a Boltzmann averageover total energy E, k(T)=[2tiQr(T)]-‘J:m dEe-“kriV(E), (1) where Q, is the reactant partition function (per unit volume). Manthe et aL5 expressionfor the CRP is ME) =TrE&E) I= ~PAE), (24 where the reaction probability operator @ is ii(E) =4Z;‘2&E)+ip&(E)6;‘2. (2b) Here 6(E) = (E+ir!-I?) -’ is the Green’s function, where l? is the Hamiltonian operator and c=&+Zp an absorbing potential which enforces outgoing wave boundary conditions, t?,(gpP,being the part of the absorbingpotential in the J. Chem. Phys. 100 (l), 1 January 1994 0021-9606/94/l reactant (product) region. Manthe et al. showed that a convenient way to evaluatethe trace of&E) in Eq. 2(a) is to determine its eigenvalues f&(E)}, the eigenreaction probabilities, whose values all lie between 0 and 1. Any convenient basis set can be used to represent the operators in Eq. (2); it is necessaryonly that the ( L2) basis span the interaction region between the absorbing potentials (and somewhat into them) and that the absorbing potentials be located outside the region where the reaction dynamics is actually determined.4The present calculation is challenging because of the deep well in the interaction region [cf. Fig. 1(a)] which necessitatesthe use of a large basis set. Jacobi coordinates of the reactants [cf. Fig. 1(b)] are used to specify a set of grid points for a discrete variable representation ( DVR)6 of the following size: r: 60 sine-DVR7 points in the interval (0,6); R: 213 sine-DVR points in the interval (2,lO) contracted to 87 points by meansof the HEG method,* using the eigenfunctions of { ?R + V,,( R ) }, where V,dR) =min,e VW,@; r: 32 symmetry-adapted DVR points built from the restricted Legendre basis set {P1(cos 0); I= 1,3,5,...,63;only odd functions are used because the ground state of the O-O-H system is odd upon the interchange of the two oxygen atoms This primitive basis is then contracted via the sequential adiabatic reduction scheme of Bacic and Lightg: at each grid point Rp, the (r,e) Hamiltonian is diagonalized and only the eigenfunctions q5,(r,t3;Rp) retained for which the corresponding eigenvalueE,,(R,) ~3.6 eV (relative to the bottom of the O-O-H well). The final basis ] (b,(R,) ) 1RP) consists of 4338 functions. The absorbing potentials were chosen to be quartic functions, e,(Q) =A (Q- Q0)4/ (Q,,,- Q0)4 for a=r and p. For the reactant absorbing potential the coordinate Q is r [cf. Fig. 1(b)], with rc=4.75 ao, rmaX=6.0ao, and A=3.2 eV, and for the product absorbing potential the coordinate Q is R [cf. Fig. 1(b)], with Ro=8.5 ao, R,,,= 10 ao, and A=2.2 eV. Extensive convergencetests were carried out, varying both the strength and location of the absorbing potentials, to ensure that the values obtained for N(E) are independentof them. It was 00(1)/733/3/$6.00 @ 1994 American institute of Physics 733 Downloaded 19 May 2005 to 169.229.129.16. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp Letters to the Editor 734 OH+0 0.58 H+02 0.8 0.6 Ei 2 0.4 (a) 0.0 0.6 0.8 E W) 0.9 1 FIG. 2. Eigenreaction probabilities--i.e., the eigenvalues of the matrix P(E) of Eq. (2)-as a function of total energy E. N(E) = 2 (W+ l)N(E,J), (3) J FIG. 1. (a) Energetics (in eV) of the H-O-O potential energy surface. (b) Jacobi coordinates used to describe the H+O,-OH+0 reaction. necessary,e.g., for R,, (the location of the absorbingpotential in the product O+OH valley) be as large as it is in order to get the reaction threshold correct. The*eigenreactionprobabilities bk} i.e., the eigenvalues of P were computed by the Lanczos procedure described in Ref. 5. Thus if there are n nonzero eigenvalues bk} which contribute to the sum in Eq. 2(a), then -n Lanczos iterations (plus maybe one or two) are required, and this in turn meansthat -2n operationsof the Green’s function onto a vector are required. The operation of the Green’s function is carried out by LU decomposition.” For each energy E, the LU decomposition needs to be carried out only once (requiring - 248 s on&theCray C90), and then every subsequentoperation of G onto a vector requires only back substitution (requiring - 0.2 s) . Therefore, the computation of N(E) requires - (248+0.4n) s for each energy, and n varies from two to ten for energies E from threshold up to - 1.1 eV. Figure 2 shows the individual eigenreactionprobabilities Cpk(E)} as a function of energy, and Fig. 3 their sum N(E) . Resonancestructure resulting from a collision complex causedby the deep potential well appearsin both the individual probabilities and in the CRP. The fact that the reaction probabilities do not all rise promptly to unity with increasing energy is also a manifestation of dynamics that violates the basic tenet of transition state theory; i.e., all trajectories-using the languageof classical mechanicswhich enter the interaction region do not proceedto products, but some turn around and becomenonreactive. One should calculate the CRP separately for each value of total angular momentum J,iV( E,J), and then sum over J, but for the present only calculations for J=O have been carried out. Higher J values are taken into account by the “J-shifting” approximation” (essentially a molecular version of the modified wave number approximation12 of atomic physics), N(E,J) zz 2 N(E+,O), (44 K= -J where I$,, is the rotational energy of the O-O-H complex. It is then easyto show that the integral over E in Eq. ( 1), with sum over J in Eq. (3)) simplifies to 5.0 EW FIG. 3. The cumulative reaction probability, given by Eq. (2), as a function of energy E for total angular momentum J=O. J. Chem. Phys., Vol. 100, No. 1, 1 January 1994 Downloaded 19 May 2005 to 169.229.129.16. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp Letters to the Editor 10” 7 2.x.X. 10” :Li Icy B %. . ; 10” IO’O 0.3 0.5 0.7 1000/T (OK) 0.9 1.1 FIG. 4. Arrhenius plot of the thermal rate constant, as given by Eq. (6)) over the range T= 750-3ooO K. The solid curve connecting the circles is the experimental result (Ref. 13), the dashed curve connecting the squares the present results with the G-G-H intermediate geometry used for o’, and the long-short dashed curve connecting the diamonds the present result with the transition state geometry used for &, . dE ewElkT $ (W+ l)N(E,J) =&ot,,( T) I--COdE e-E’kTN(E,J=O), (4b) where Q& is the rotational partition function for the O-O-H complex, i e-E::/kT. (&I K= --J Also, the reactant partition function in Eq. ( 1) is given by ef,,( T) = E (W+ 1) QA T) = QelQvibQrotQtrans 9 (5) where Qviband Q,,, are the standard vibrational and rotational partition functions for O,, Q,r=3 is the electronic degeneracy,and Q,, = (pkT/2n-# ) 3’2is the translational partition function (per unit volume) for relative motion of H and O2 (CLbeing the relevant reduced mass). Since this reaction is not dominated by a simple transition state, it is not obvious what geometry one should use in Eq. 4(c) to determine the rotation energy levels and thus gm, for the J-shifting approximation. Therefore Fig. 4 shows the rate constant given by Eq. ( 1) with Eq. (4)) i.e., k(T)~:[2~Q,(T)l-‘ejbt(T) m dE e-E’kTN(E,J=O), X I -02 (6) for the temperaturerange T r 800-3000 K, for two choices of the (O-O-H)t geometry: the geometry of the O-O-H stable intermediate [i.e., the bottom of the potential well in Fig. 1(a)], and the geometry of the transition state (which 735 lies on the product side of the intermediate). The two results differ by an approximately constant factor of -2.5 and are seen to bracket the latest experimental results of Du and Hessler’3 (and also the classical trajectory calculations of Ref. 2). Further analysis of these rigorous quantum calculations will shed more light on the qualitative featuresof the dynamics of this important reaction, and calculations for J>O will be useful to eliminate the uncertainty due to the J-shifting approximation. The very fact that the present calculations could be carried out, however, is evidenceof the progressbeing made in theoretical reaction dynamics. Note added in proo$ We call attention to the very recent paper14which has also treated this reaction, using the same potential energy surface. Specifically, they have carried out coupled channel scattering calculations for J=O, producing state-to-statereaction probabilities and thus also the cumulative reaction probability. Their results for the latter quantity are in reasonable agreement with our present values (Fig. 3). W.H.M. is pleasedto acknowledgesome helpful comments of Dr. Russell T Pack. This work has been supported by the Director, Office of Energy Research,Office of Basic Energy Sciences,Chemical SciencesDivision of the U.S. Department of Energy under Contract No. DEAC03-76SFOOO98. The calculations were carried out on the Cray C-90 at the National Energy ResearchSupercomputer Center (NERSC) at the Lawrence Livermore National Laboratory. CL has been supported in part by National ScienceFoundation Grant No. CHE89-20690. “)Permanent address: Laboratoire de Chimie ThCorique, Universid de Paris-Sud, Bltiment 490, 91405 Orsay Cedex, France. ‘J. A. Miller, R. J. Kee, and C. K. Westbrook, Annu. Rev. Phys. Chem. 41, 345 (1990). *A. C. Varandas, J. Brandao, and M. R. Pastrana, J. Chem. Phys. 96, 5137 (1992). ‘M. R. Pastrana, L. A. M. Quintales, J. BrandXo, and A. J. C. Varandas, J. Phys. Chem. 94, 8073 ( 1990). 4 (a) T. Seideman and W. H. Miller, J. Chem. Phys. 96,4412 (1992); 97, 2499 (1992); (b) W. H. Miller, Act. Chem. Res. 26, 174 (1993). ‘U. Manthe and W. H. Miller, J. Chem. Phys. 99, 3411 ( 1993). 6J. V. Lill, G. A. Parker, and J. C. Light, J. Chem. Phys. 85,900 (1986). ‘C. Leforestier, J. Chem. Phys. 94, 6388 (1991). ‘D. 0. Harris, G. G. Engerholm, and W. D. Gwinn, J. Chem. Phys. 43, 1515 (1965). 9Z. Bacic and J. C. Light, J. Chem. Phys. 85, 4594 (1986); 87, 4008 (1987). “W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge University, Cambridge, 1986). “J. M. Bowman, J. Phys. Chem. 95, 4960 ( 1991). I2 (a) K. Takayanagi, Prog. Theor. Phys. 8, 497 ( 1952); (b) see also, E. E. Nikitin, Elementary Theory of Atomic and Molecular Processes in Gases, translated by M. J. Kearsley (Oxford University, Oxford, 1974), pp. 58-60. 13H. Du and J. P. Hessler, J. Chem. Phys. 96, 1077 (1992). 14R T Pack, E. A. Butcher, and G. A. Parker, J. Chem. Phys. 99, 9310 (l.993). J. Chem. Phys., Vol. 100, No. 1, 1 January 1994 Downloaded 19 May 2005 to 169.229.129.16. 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