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J. Phys. Chem. 1986, 90, 4305-4311 A scenario which has not been excluded, and which this study suggests, is that the transient absorption they report as superposition of triplet-triplet absorption and radical absorption is really the 9-anthryloxy ion, which is formed with unit efficiency from the triplet state of 9-NA; i.e., the observed transient absorption would be from the 9-anthryloxy radical and photochemically derived species. Our data suggests this to be the case since the quantum yield is approximately 35 times smaller in the PVB film, where recombination may play a major role. It is noteworthy that when we initiated this investigation we did not know what type of hologram to expect, other than perhaps it would be easier to interpret in the long wavelength region (>450 nm) where photoproducts do not interfere. Clearly the results indicate that the photochemistry of 9-NA is much more complicated in the UV region. Further, our results suggest that the wavelength dependence of holographic behavior could provide useful mechanistic information. In cases where complicated
photochemical events and dark reactions prevail, such information could help to identify the primary photochemical event. In summary the holographic behavior of 9-NA in a PVB film at 466 and 458 nm indic &esa simple one-photon event leading to 9-anthrol via the 9-anthryloxy radical. The low quantum yield of (8.3 f 1.7) X measured at 458 nm suggests that recombination of the 9-anthryloxy radical with NO may be an important relaxation mode in the polymer film.
Acknowledgment. Support from the Swiss National Science Foundation is gratefully acknowledged. A. C . Testa is very grateful for the excellent facilities extended to him at the ETH during the academic year 1984-85, where this work was done. We thank Dr. Alois Renn for his valuable assistance in the early phases of this study. Registry No. 9-NA, 602-60-8; 9-anthrol, 529-86-2; 9-anthryloxy radical, 24690-75-3.
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Evaluation of Dynamical Approximations for Calculating the Effect of Vibrational H,(n =0,1) OH(n =0,1) H Excitatlon on Reaction Rates. 0
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Bruce C. Garrett,* Chemical Dynamics Corporation, Columbus, Ohio 43220
Donald G. Truhlar,* Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455
Joel M. Bowman? Department of Chemistry, Illinois Institute of Technology. Chicago, Illinois 6061 6
and Albert F. Wagner Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439 (Received: February 10, 1986)
We previously proposed a generalization of variational transition-state theory (VTST) to allow the calculation of excited-state rate constants, summed over final states, by assuming that the excited degree of freedom is adiabatic from the beginning of the collision until the free energy of activation bottleneck is reached. Here we propose a modification in which the reaction is assumed to be vibrationally adiabatic only up to the first Occurrence in proceeding from reactants to products of an appreciable local maximum in the reaction-path curvature; at that point the reaction is treated as if all flux is suddenly diverted to the ground vibrational channel. The new theory is applied to calculate both forward and reverse excited-state rate constants for collinear 0 + Hz OH H for three quite different potential energy surfaces, and it is tested against collinear exact quantal rate constants in each case. We draw conclusions about the conditions for the validity of variational transition-state theory for excited-state reaction rates as well as about the possible accuracy of the potential energy surfaces that have been proposed for this reaction.
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I. Introduction The reaction 0 H2 OH + H is of special interest as it is the simplest important reaction in oxygen combustion chemistry. In fact, after H O2 OH + 0, this almost thermoneutral reaction is the second most important chain branching step in the H2/02system governing high-temperature combustion chemistry.' As a consequence its thermal rate constant has been widely studied experimentally,'.2 and the vibrationally excited-state-selected reaction rate of 0 + H 2 ( n = l ) OH + H, where n is the vibrational quantum number, has also been m e a ~ u r e d . ~ The thermal and state-selected reactions have also been the subject of numerous detailed theoretical dynamical ~tudies."~ Although the state-selected reaction rates are probably not needed for modeling reactions of combustible mixtures under flame condi-
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Address after Sept 1986: Department of Chemistry, Emory University, Atlanta, GA.
0022-3654/86/2090-4305$01.50/0
tions,14 they are important for understanding laser-initiated reaction systems and other nonequilibrium situations, and they (1) (a) Dixon-Lewis, G.; Williams, D. J. In Comprehensive Chemical Kinetics; Bamford, C . H., Tipper, C. F. H. Eds.; Elsevier Scientific: Amsterdam, 1977, Vol. 17, p 1. (b) Warnatz, J. In Combustion Chemistry; Gardiner, W. C., Jr., Ed.; Springer-Verlag: New York, 1984; p 197. (2) (a) Baulch, D. L.; Drysdale, D. D.; Horne, D.; Lloyd, A. C. Evaluated Kinetic Data for High- Temperature Reactions; Butterworths; London, 1972; Vol. I, p 49 and references therein. (b) Cohen, N.; Westberg, K. J . Phys. Chem. Ref. Dara 1983, Z2, 531 and references therein. (c) Pamidimukkala, M.; Skinner, G. B. J . Chem. Phys. 1982,76,311. (d) Frank, P.; Just, T. Ber. Bunsen-Ges. Phys. Chem. 1985, 89, 181. (e) Presser, N.; Gordon, R. J. J . Chem. Phys. 1985, 82, 1291. (f) Sutherland, J. W.; Michael, J. V.; Klemm, R. B.; Pirragalia, A. N., to be published. (3) Light, G. C. J . Chem. Phys. 1978, 68, 2831. (4) Johnson, B. R.; Winter, N. W. J . Chem. Phys. 1977, 66, 4116. (5) Muckerman, J. T.; Faist, M. B. J . Phys. Chem. 1979, 83, 79. (6) Garrett, B. C.; Truhlar, D. G.; Grev, R. S.; Magnuson, A. W. J. Phys. Chem. 1980, 84, 1730; 1983,87, 4554(E).
0 1986 American Chemical Society
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The Journal of Physical Chemistry, Vole90, No. 18, 19'86
Garrett et al.
In this series they reported systematic trends in the reaction provide a stringent test of detailed dynamical theories. dynamics as a function of saddle-point location of the potential The first modern theoretical work on the thermal reaction was energy All of the surfaces except the DIM one apby Johnson and Winter! They and, later, Muckerman and Faist5 peared consistent with experiment for both thermal and n = 1 used the quasi-classical trajectory (QCT) method and the potential rate constants? In a previous paper Truhlar, Garrett, and Runge" energy surface of Johnson and Winter4 to calculate thermal rate have shown that the VTST/K method reproduces the dynamical constants over the temperature range 300-1000 K. Garrett et trends as a function of potential energy surface as found by aL6 used variational transition-state theory both without (VTST) collinear exact quantum calculations by Bowman, Wagner, and and with (VTST/K) a multidimensional semiclassical transmission c o - ~ o r k e r s ~for - ~the thermal reaction, which is dominated by n coefficient (K)to calculate the same rate constants for the same = 0 and in the present article we will compare the predictions of potential energy surface over the range 200-2400 K. At 1000 the VTST/K method, including new refinements in the semiK the three methods give rate constants that agree within a factor classical transmission coefficients" and the treatment of anharof 1.4 or better, but at 300 K the QCT rate constant exceeds the monicity,20 to collinear exact quantal results for three of these VTST one by a factor of 4.8. Since neither method includes O H + H and the reverse surfaces for both 0 H2(n=1) tunneling this may be interpreted as an inability of the QCT H 0 H2. These comparisons will reaction O H ( n = l ) method to predict the reaction probability in the threshold region enable us to better judge the validity both of the VTST/Kmethods because it does not adequately include the quantum mechanical for calculating excited-state rate constants and also of the various energy requirements in the dynamical bottleneck region of conpotential energy surfaces that have been considered for this system. figuration space.I5 Inclusion of tunneling by the VTST/K method In this regard we also discuss the results of previously published was calculated6 to increase the 300 K rate constant by a factor TST/CEQB/Gi0 and VTST/dZ calculations for three-dimensional of 8.2. Thus both quantum mechanical energy requirements a t rate constants on the 0 H2(n=0,1) system. dynamical bottlenecks and tunneling through potential barriers Section I1 reviews the theoretical dynamical methods for the have large effects on the thermal reaction rate. Furthermore it new calculations reported here. Section 111 presents the exact was shown6 that one-dimensional tunneling calculations based on quantal and VTST/K results for the collinear cases. Section IV the minimum-energy reaction path severely underestimate the discusses these collinear calculations as well as the previous tunneling contributions, and the rate constant obtained by such three-dimensional calculations in the context of the new undera one-dimensional tunneling calculation was a factor of 3.3 below standing gained by the present comparisons. Major findings are the best estimate at 300 K and a factor of 10.6 below the best summarized in section V. estimate at 200 K. As a consequence of these findings we will be concerned in this article only with methods that attempt to 11. Theory include at least two-dimensional quantal effects in the description of the reaction dynamics. Other methods, besides the VTST/K A . Variational Transition-State Theory and Semiclassical approach, for including such quantal effects in practical rate Tunneling Calculations. In earlier work an adiabatic meconstant calculations are to employ conventional transition-state thod,'1bv2"-23an adiabatic method with a localized classical twotheory with transmission coefficients based on two-dimensional state diabatic transition24and an adiabatic method with delocalized exact quantum calculations on the collinear reaction (TST/CEQ) quantal multistate nonadiabatic transitionsz5were used to genor on reduced-dimensional exact quantum calculations with an eralize transition-state theory to treat excited-state reactions. adiabatic incorporation of ground-state bending motion (TST/ These methods were applied successfully to several one-dimenCEQB/G). These approaches have been applied to the reactions siona12"-z3,25 and threedimensiona11'bp24 reactions, but they would 0 H2(n=0,1) O H H by Bowman, Wagner, and co-workers not be equally valid for 0 H2(n=l) on all the potential energy in a four-article series7-10presenting a comparative study of the surfaces studied here. The existence of two almost equal maxima reaction dynamics for five different potential energy surfaces. In in the excited-state vibrationally adiabatic potential curves for addition to the Johnson-Winter (JW) ~ u r f a c ethey , ~ considered this reaction on the JW potential energy surface plus the variation the fit of Schinke and Lester to the ab initio calculations of in saddle point locations for the various 0 H2 potential energy Howard, McLean, and Lester (HML-SL),I6 a rotated-Morsesurfaces require a more careful analysis of the adiabatic aposcillator-spline fit to the diatomics-in-molecules surface of proximation. In this section we provide this analysis and propose Whitlock et al. (DIM-RMOS),I7 and two new surfaces (called a more generally valid approach. This approach has already been ' ~ for a calculation presented at a conferenceI2 but without PolCI and ModPolCI) based on new ab initio c a l c u l a t i ~ n s . ~ * l ~ ~ used extensive discussion. Here we emphasize conceptual points and (7) Schatz, G. C.; Wagner, A. F.; Walch, S.P.; Bowman, J. M. J . Chem. the fundamental assumptions that provide a basis for the appliPhys. 1981, 74, 4984. cation of variational transition-state methods to a state-selected ( 8 ) Lee, K. T.; Bowman, J. M.; Wagner, A. F.; Schatz, G. C. J . Chem. reaction. At the end of the section we very briefly summarize Phvs. 1982. 76. 3563. the computational details. (9) Lee, K. T ; Bowman, J . M.; Wagner, A F ; Schatz, G. C. J Chem. Phys. 1982, 76, 3583. Our formulation of variational transition-state theory6,21begins (IO) Bowman, J. M.; Wagner, A. F ; Walch, S. P.; Dunning, T. H., Jr. J. with the definition of a reaction path and of the reaction coordinate Chem. Phys. 1984, 81, 173% as the signed distance s along this path, with s = -m at reactants, ( 1 1 ) Truhlar, D. G.; Runge, K.; Garrett, B. C. Twentieth Symposium s = 0 at the saddle point, and s = +m at products. We define (Internariond) on Combustion; Combustion Institute: Pittsburgh, 1984; (a) p 585, (b) p 594. the reaction path as the union of the two steepest descents paths (12) Garrett, B. C.; Truhlar, D. G. Inr. J . Quantum Chem., in press. through a mass-scaled Cartesian coordinate system, one beginning (13) Broida, M.; Persky, A. J. Chem. Phys. 1984, 80, 3687. at the saddle point and proceeding toward reactants and the other (14) Brown, R. C. Combust. Flume 1985, 62, 1 . beginning at the same point and proceeding toward products. For (15) (a) Truhlar, D. G. J. Phys. Chem. 1979,83, 188. (b) Truhlar, D. G.; Isaacson, A. D.; Skodje, R. T.; Garrett, B. C. J. Phys. Chem. 1982,86,2252; a three-dimensional, three-atom system, A + BC AB C, with l983,87,4554(E). nc) Bondi, D. K.; Clary, D. C.; Connor, J. N. L.; Garrett, a collinear minimum-energy path, the degrees of freedom orB. C.; Truhlar, D. G. J. Chem. Phys. 1982, 76, 4986. (d) Blais, N. C.; thogonal to the reaction path consist of a stretching vibration, a Truhlar, D. G.; Garrett, B. C. J . Chem. Phys. 1983, 78, 2363.
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(16) Schinke, R.; Lester, W. A., Jr. J . Chem. Phys. 1979, 70, 4893. (17) (a) Whitlock, P. .a, Ph.D. Thesis, Wayne State University, Detroit, MI, 1976. Whitlock, P. A.; Muckerman, J. T.; Fisher, E. R. Research Institute for Engineering Sciences Technical Report, Wayne State University, Detroit, MI, 1976. (b) Wagner, A. F.; Schatz, G. C.; Bowman, J. M. J . Chem. Phys. 1981, 74, 4960. (18) (a) Walch, S. P.; Dunning, T. H., Jr.; Bobrowicz, F. W.; Raffenetti, R. J . Chem. Phys. 1980, 72,406. (b) Walch, S. P.; Wagner, A. F.; Dunning, T. H., Jr.; Schatz, G. C. J . Chem. Phys. 1980,72,3894. (c) Dunning, T. H., Jr.; Walch, S. P.; Wagner, A. F. In Potential Energy Surfaces and Dynamics Calculations; Truhlar, D. G., Ed.; Plenum: New York, 1981; p 329.
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(19) Garrett, B. C.; Truhlar, D. G. J . Chem. Phys. 1983. 79, 4931. (20) Garrett, B. C.; Truhlar, D. G. J . Chem. Phys. 1984, 81, 309. (21) Garrett, B. C.; Truhlar, D. G. J . Phys. Chem. 1979, 83, 200, 1079, 3058(E); 1980, 84, 682(E); 1983, 87, 4553(E). (22) Garrett, B. C.; Truhlar, D. G. J . Phys. Chem. 1985, 89, 2204. (23) Steckler, R.;Truhlar, D. G.;Garrett, B. C. J . Chem. Phys., submitted. (24) Truhlar, D. G.; Isaacson, A. D. J . Chem. Phys. 1982, 77, 3516. (25) Garrett, B. C.; Abusalbi, N.; Kouri, D. J.; Truhlar, D. G. J . Chem. Phys. 1985, 83, 2252.
Calculation of Excited-State Rate Constants doubly degenerate bending vibration, two rotational degrees of freedom, and three translational degrees of freedom. We remove the latter by transforming to the center-of-mass coordinate system. Let n denote the quantum number for the stretching vibration and y the collection of quantum numbers for the bend and rotation. A generalized transition state is a system with fixed s but otherwise free to vibrate and rotate. The adiabatic energy of a generalized transition state a t s is denoted c(n,y,s), which is an adiabatic potential curve for motion along the reaction coordinate. In improved canonical variational transition-state theory we compute an improved standard-state free energy of activation AGIGTv0(T,s), where T i s the temperature, for the generalized transition state at s by averaging cGT(n,y,s)over a truncated canonical ensemble, where the truncation corresponds to omitting states at energies below by the microcanonical variational transition-state theory threshold energy. The improved canonical variational transition state is the generalized transition state with the maximum improved free energy of activation, and the thermal rate constant is
The Journal of Physical Chemistry, Vol. 90, No. 18, 1986 4307 considered that portion of the reaction coordinate that follows the last appreciable local maximum of the reaction-path curvature;29 if there is a local maximum in AGIGT*O(n,T=O,s)in that exitchannel region and if it exceeds the local maximum of AGIGTv0(T=O,s)that controls the non-state-selected reaction rate, it may be related to a product-state-specific threshold energy. In the present treatment we generalize this idea of adiabaticity along a portion of a reaction path. We will call this partial-reaction-path (PRP) adiabaticity. To implement the P R P adiabaticity approximation in greater generality we divide every reaction path for a given direction of reaction into two parts, an entrance region defined as --OD < s S s+ and an exit region defined as s+ < s C m, where s+ is the location of the first occurrence of an appreciable local maximum of the reaction-path curvature. Then, for both endoergic and exoergic reactions, we calculate the state-selected reaction rate constants for vibrational state n of reactants by
kICVT(T)= ( k B T / h ) K f oe x p [ - m a ~ A . G ' ~T~J~) ~] ( (1)
--03
Q s Q s+
S
AGIGT~a(n=O, T,s)
where kB is Boltzmann's constant, h is Planck's constant, and fl is the reciprocal of the standard-state concentration. To generalize this to state-selected reactions we simply fix n. Averaging over a truncated canonical ensemble for the other degrees of freedom yields AGIGT,o(n,T,~), and the state-selected rate constant is
The interpretation of this expression is that the vibration remains adiabatic until s = s+ and then an appreciable probability of vibrational nonadiabaticity occurs. After that the system can proceed to products in any vibrational state so if the reaction kICVT(n,T)= ( k B T / h ) K f oe ~ p ( - A G ' ~ ~ ~ ~ [ n , T , s ~ ' (~2 ~ ) ~ ( n probability ,T)]) is summed over final states only the adiabatic barriers for the ground state impede the total reaction probability. The where s.ICVT(n,T) is the location of the maximum of AGIGT2Osudden nonadiabatic transition at s = s+ is of course an ideali(n,T,s). Both kICVT(T ) and kICVT(n,T)correspond to classical zation. If an excited-state barrier beyond s+ does impede a sigmotion along the reaction coordinate and quantal motion in other nificant amount of flux, then kPRP-ICVT will likely overestimate degrees of freedom. If tunneling or other quantal effects on the state-selected rate constant, but kICVT(n,T)based on the full reaction-coordinate motion are important, they are included by reaction path (FRP) probably underestimates the state-selected a transmission coefficient (see below). rate constant because it requires all flux to pass the exit-region The above theory may be generalized to systems with nonvariational state with the originally excited vibrational quantum collinear reaction paths by adding an extra rotational degree of number n. The real situation lies between these extremes. Since, freedom to y2('and to systems with three or more atoms by adding for n 2 1, vibrational nonadiabaticity is almost always expected extra vibrational quantum numbers to y.27 It may also be speto be facile in the vicinity of s = s+, we expect that if there is a cialized to collinear atom-diatom reactions by deleting the bending large difference between kICVT(n,7')and kPRP-ICVT(n,T) the true and rotational degrees of freedom. hybrid rate constant lies closer to the latter, larger estimate. In The treatment of state-selected reaction rates just outlined is the case where the highest adiabatic barrier for state n occurs most valid if the stretching vibration characterized by quantum before s+, kPRP-IcVT(n,T)and kICVT(n,T)are the same. number n adjusts adiabatically to reaction-coordinate motion. If Tunneling contributions are added by the least action (LA) s = -m to s = +m, then this adiabatic approximation is valid from appro~imation.'~ If K ~ ~ ( denotes ~ , T ) the semiclassical least action kICVT(n,T)corresponds to a specified vibrational state of both transmission coefficient for the FRP and ~ ~ ~ 7') ~ denotes - ~ ~the( n , reactants and products. If the adiabatic approximation is valid semiclassical least action transmission coefficient for the PRP, only from s = --m to s = slicVT(n,T),then kICVT(n,T)corresponds then the corresponding state-selected rate constants are to a "state-selected rate constant," by which we mean a rate kICVT/LA(n T ) = kICVT(O-bLA(n,T) constant for a selected vibrational state of reactants but summed (4) over final vibrational states. The latter interpretation is the one and we have used in previous work.11bq20'24 kPRP-ICVT/LA(n,T ) = kPRP-ICVT(n,OKPRP-LA(n,T ) Regions of large reaction-path curvature, K ( s ) , ~promote vi(5) brational nonadiabaticity.** Usually reaction-path curvature is An important element in the calculation of K ~ ~ ( ~ is , the T ) largest in the region where the reaction path changes its primary state-selected adiabatic potential curve eg(n,s), which is e(n,y,s) character from A approaching BC to AB receding from C. If for the case that all degrees of freedom except the vibration in s*lCVT(n,T)precedes this region, or at least does not follow it, then state n are in their gound states and is closely related to kICVT(n,T)might provide a reasonable approximation to the @(n,s)plays the role of an effective potential AGIGT.o(n,T=O,~); state-selected reaction rate constant for state n. In previous work curve in regions where the semiclassical tunneling calculation is on excited-state rate constants, when we applied the adiabatic based on the adiabatic approximation. To calculate K ~ ~ ~ - ~ ~ ( ~ , T ) generalized transition-state theory to appreciably unsymmetrical eg(n,s) is replaced by eg(n=O,s) for s > s+, and all other n-dereactions we always calculated state-selected reaction rates in the pendent quantities (vibrational frequencies and classical turning exoergic direction so that s11C*(n,7') preceded the region of highest points) required for the LA calculation are also replaced by their reaction-path curvature. Alternatively, we considered state-testate n = 0 analogues in this region. This is a straightforward genreactions with very late dynamical bottlenecks, but we only 7'). Note that eralization of the physical model behind kPRP-ICVT(n, even in the case where the maximum of AGTGTs0(n,T,s)occurs (26) Rai, S . N.; Truhlar, D. G. J . Chem. Phys. 1983, 79, 6040. (27) (a) Isaacson, A. D.; Truhlar, D. G. J . Chem. Phys. 1982, 76, 1380. (b) Isaacson, A. D.; Sund, M. T.; Rai, S. N.; Truhlar, D. G. J . Chem. Phys. 1985.82, 1338. (28) Truhlar, D. G.; Dixon, D. A. In Atom-Molecule Collision Theory;
Bernstein, R. B., Ed.; Plenum: New York, 1979; p 595 and references therein.
(29) (a) Steckler, R.; Truhlar, D. G.; Garrett, B. C.; Blais, N. C.; Walker, R. B. J . Chem. Phys. 1984, 81, 5700. (b) Brown, F. B.; Steckler, R.; Schwenke, D. W.; Truhlar, D. G.; Garrett, B. C. J . Chem. Phys. 1985, 82, 188.
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The Journal of Physical Chemistry, Vol. 90, No. 18, 1986
before s+ so that kICVT(n,T)and kPRP-ICVT(n,T) are the same, K ~ ~ ( ~and I , KPRP-LA(n,T) T) may be different. For collinear reactions the quantum numbers y are omitted, and the superscripts g are unnecessary. The methods employed for the variational transition-state theory and tunneling calculations have all been explained elsewhere.6*19-2's23For the thermally averaged rate constant the transmission coefficient is based on the ground-state reaction, and we add a G to the abbreviation to denote this ground-state approximation, Le., ICVT/LAG. For all calculations reported in this paper the coordinates are scaled such that all motions correspond to the same reduced mass p , where
Garrett et al. 14 21
12
19
10
17
a
/-. -
E
,--.
1 '.
c
0
2
15
10 0
6
v
V
A
v,
A
P
= m0(2m~)/(m0+ 2 m ~ )
(6)
v)-
C
13
4
11
2
9
0
7
-2
Y
V
The stretching vibrations are treated by the WKB approximation, and the bending vibrations are treated by a harmonic-quartic approximation with Taylor series force constants. Rotation is treated by the quantum mechanical rigid-rotator approximation. For collinear reactions the electronic partition functions are set equal to unity. For three-dimensional reactions they are calculated by including only the most important electrcnic states, namely 3Pz, 3P,, and 3P0for 0, XIB,+ for H2, and X3A" for OHz. B. Accurate Quantal Calculations for Collinear Reactions. The accurate quantal reaction probabilities P(n,EIeI)for collinear reactions of molecules in state n at relative translational energy Erclwere calculated by methods presented elsewhere.30 The state-selected collinear rate constants are then given by3'
kCEQ(n, T ) = ( 2 V . k T)-'/' ~
mdEreIexp(-Erei/kB T)P(n,E,,J (7)
>"
Figure 1. Two lowest energy vibrationally adiabatic potential curves
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(solid) and reaction-pathcurvature (dashed) for the ModPolCI potential energy surface for 0 + H2(n=0,1) OH(n=O,l) + H, proceeding left to right. The scale for the vibrationally adiabatic potential curves is on the left and that for the reaction path curvature is on the right. The abscissa is distance along the minimum-energy path through mass-scaled coordinates scaled to the reduced mass of eq 6 .
The excited-state forward rate constant will be called simply k(n=l,T), and the excited-state reverse rate constant will be called krev(n=1 The thermally averaged forward rate constant, obtained by averaging kCEQ(n,T)over a Boltzmann distribution of n, will be called k(7').
,n.
111. Calculations and Results We will focus attention on three of the five potential energy surfaces considered previously, namely the best of the three surfaces based on ab initio calculations, which are the ModPolCI surface* and the two semiempirical surfaces, the J W surface4 and the DIM-RMOS surface.]' The J W surface is of the extended London-Eyring-Polanyi-Sato (LEPS) functional form,32and the DIM-RMOS and ModPolCI surfaces are both of rotatedMorse-oscillator-spline (RMOS) form.'7b An important qualitative difference between the a b initio surfaces and the semiempirical surfaces is the geometry of the saddle point. In all cases the saddle point is collinear, and we label the atoms O-H,-Hb, with H, in the central position. On all three a b initio surfaces the saddle point occurs relatively early with the saddle-point value of the new bond length RLH, = 2.27-2.30~~.In contrast RbH, = 2.1 lao for the J W surface and 2 . 0 1 for ~ ~ the DIM-RMOS surface. There is a similar pro ression in the saddle-point values of the breaking bond length RHaHa,which is 1 . 7 4 - 1 . 7 9 ~for ~ the a b initio surfaces, 1.80ao for the J W surface, and 2 . 0 7 ~for~ the DIM-RMOS surface. Thus the H2 bond is broken to a much greater extent a t the late DIM-RMOS saddle point than at any of the other saddle points. Quantum mechanical and quasiclassical trajectory calculations for n = 0 and n = 1, as reported in ref 8, also show quite different behavior for the DIM-RMOS surface as compared to the other surfaces. The choices of s+ values for the present cases are fortunately unambiguous. Figures 1-3 show the relevant vibrationally adiabatic potential curves and reaction-path curvatures for the three
f
(30) Schatz, G. C.; Bowman, J. M.; Kuppermann, A. J. Chem. Phys. 1975, 63, 614. (31) Truhlar, D. G.; Kuppermann, A. J . Chem. Phys. 1972, 56, 2232. (32) (a) Sato, S.J . Chem. Phys. 1955.23, 592, 3465. (b) Kuntz, P. J.; Nemeth, E. M.; Polanyi, J. C . ; Rosner, S . D.; Young, C. E. J . Chem. Phys. 1966, 44. 1168.
A
A
v,C
v)
13
4
11
2
9
0
V
k
v
>"
7
-
-7
-15 - 1 0
-05
00
0 5
1 0
15
s (a01 Figure 2. Same as Figure 1 except for the Johnson-Winter potential
energy surface.
potential energy surfaces studied here. For collinear reactions the vibrationally adiabatic potential curves are defined by6S2' v,(n,s) =
+ +v)
~ M E P ~ )
(8)
where VMEp(s)is the Born-Oppenheimer potential energy along the MEP. Figure 1 shows that for the ModPolCI surface the first appreciable maximum of the reaction-path curvature (which locates s+ for the forward reaction) is at s = -O.lao, and the last appreciable maximum (which locates s+ for the reverse reaction) ~ 2 and 3 show that for the J W surface is at s = 0 . 3 ~Figures these PRP boundaries are at s = f0.2a0 and for the DIM-RMOS surface they are at s = - 0 . 3 ~and ~ +O.lao. The primary question addressed in the present paper is whether variational transition-state theory with semiclassical tunneling
The Journal of Physical Chemistry, Vol. 90, No. 18, 1986 4309
Calculation of Excited-State Rate Constants
TABLE I 1 Collinear Exact Quantal Rate Coefficients (cm molecule-' s-l) for the Reverse Reaction
T. K 200 300 400 600 800 1000 1200
A
L
1 '. 0
ModPolCI
JW
DIM-RMOS
1.57 (4) 2.48 (4) 3.27 (4) 4.58 (4) 5.65 (4) 6.55 (4) 7.31 (4)
1.27 (2) 9.34 (2) 2.70 (3) 8.45 (3) 1.56 (4) 2.30 (4) 3.00 (4)
6.00 8.77 (1) 3.96 (2) 1.98 (3) 4.56 (3) 1.54 (3) 1.05 (4)
0
Y
v
TABLE IIk Ratios of ICVT/LAC Rate Constants to CEQ Ones for the Thermal Rate Constant
A
'" C
T. K 200 300 400 600 800 1000 1200 1600 2000
v
>"
ModPolCI
JW
DIM-RMOS
1.01 0.76 0.88 1.19 1.35 1.43 1.49 1.58 1.66
1.30 1.01 0.90 0.84 0.85 0.88 0.9 1 0.99 1.08
0.84 1.25 1.59 2.01 2.28 2.52 2.75 3.21 3.66
s (a,> Figure 3. Same as Figure 1 except for the DIM-RMOS potential energy
surface. TABLE I: Collinear Rate Constants (cm molecule-' 9-l) at 300 K
surface ModPolCI
method CEQ ICVT/LA PRP-ICVT/LA JW CEQ ICVT/LA PRP-ICVT/LA DIM-RMOS CEQ ICVT/LA PRP-ICVT/LA
k(T) 1.1 (-1)" 8.5 (-2) 8.5 (-2)b 3.0 (-2) 3.0 (-2) 3.0 (-2)b 9.5 (-3) 1.2 (-2) 1.2 (-2)b
k(n-1.T) 2.4 (1) 5.0 (1) 5.0 (1) 2.1 (2) 2.9 (2) 2.6 (2) 2.6 (4) 2.6 (1) 3.8 (4)
k,(n=l,T) 2.5 (4) 1.6 (2) 3.5 (4) 9.2 (2) 5.9 (2) 1.3 (3) 8.8 (1) 2.8 (2) 2.8 (2)
'In tables, numbers in parentheses are powers of 10. bNote: As a consequence of the definitions in section I1 the PRP-ICVT/LAG results for the thermally averaged rate constant are identical with the ICVT/LAG ones. corrections can adequately account for the reaction probabilities, for both n = 0 and n = 1, for all three surfaces, despite the quantitative and qualitative differences in the surfaces themselves as well as in their associated quantal and classical dynamics. This question is answered in Table I. Here we see that, as the saddle-point location becomes later, the collinear exact quantal thermal rate constant decreases monotonically by a factor of 12, the excited-state forward rate constant increases monotonically by a factor of 1.1 X 103, and the excited-state reverse rate constant decreases monotonically by a factor 2.8 X lo2. The PRPICVT/LA calculations correctly reproduce all three monotonic trends and are even semiquantitatively accurate for the three factors, yielding factors of 7 and 7.6 X lo2 for the increase in the thermal rate constant and the excited-state forward rate constant, respectively, and a factor of 1.3 X lo2 for the decrease in the excited-state reverse rate constant. Thus the question is answered affirmatively. Table I also shows that in the three cases where the highest excited-state barrier lies beyond the first appreciable local maximum of the reaction path curvature, namely the forward reaction on the DIM-RMOS surface and the reverse reaction for the ModPolCI and J W surfaces, kPRP-lCVT/LA(n,T) lies above, and kICVTILA(n,T)lies below, the true rate constant. This possibility was anticipated in section 1I.A. The three cases where the highest excited-state barrier lies beyond s+, the PRP and FRP results not only bracket the exact CEQ results for n = 1 but they differ very appreciably from one another by factors of 4.5-1500. In the other three cases for n = 1, the P R P and F R P results are identical or slightly different because of tunneling effects and are always larger
TABLE I V Ratios of PRP-ICVT/LA Rate Constants to CEQ Ones for the Excited-State Forward Rate Constant
T, K 200 300 400 600 800
ModPolCI
JW
DIM-RMOS
0.97 2.12 2.43 2.50 2.56
1.24 1.26 1.32 1.42 1.49
1.55 1.48 1.46 1.44 1.46
-
TABLE V Ratios of PRP-ICVT/LA Rate Constants to CEQ Ones for OH(n=l) H 0 H2
+
T, K 200 300 400 600 800 1000
+
ModPolCI
JW
DIM-RMOS
1.40 1.41 1.42 1.45 1.46 1.48
1.63 1.43 1.40 1.42 1.45 1.48
3.85 3.15 2.83 2.62 2.63 2.72
than the CEQ results. (The joint overestimation of the CEQ rate constants in these three cases as well as the overestimation of the CEQ results by the PFW ones in the other three cases is probably due to recrossing effects as discussed below.) Further results are shown in Tables 11-V. Table I1 presents the CEQ values for kw(n=l,T) as a function of temperature; the CEQ results for the forward reaction are tabulated in ref 9 for the temperature range 200-2000 K for n = 0 and for 200-800 K for n = 1. Tables 111-V present ratios of the final ICVT/LAG and PRP-ICVT/LA rate constants to the CEQ ones for k ( T ) , k(n= l,T), and krev(n=l,T), respectively.
IV. Discussion A . Thermal Collinear Rate Constants. First consider the thermal rate constants in Table 111. The table updates an earlier comparison presented in ref 1l a in which the less reliable Morse I approximation2' was used for stretching anharmonicity, whereas here we used the WKB a p p r o x i m a t i ~ n .(The ~ ~ ~WKB ~ ~ results for the ModPolCI surface at 300 and 600 K were presented previously in ref 20.) Table I11 shows that the ICVT/LAG results for the ModPolCI and J W surfaces are accurate within a factor of 1.7 or better over the whole factor-of-10 temperature range. The good agreement at low temperature is a striking confirmation of the accuracy of our tunneling calculations. For example, if we had used the hybrid ICVT result without a transmission coefficient, the ratios to the accurate quantal results at 200 K would have been 0.005 for the ModPolCI surface and 0.017 for the J W one. Quantal effects on reaction-coordinate motion remain very important at 300 K,
4310
The Journal of Physical Chemistry, Vol. 90, No. 18, 1986
where they increase the ModPolCI and J W rate constants by factors of 6.3 and 9.4, respectively, and even at 400 K, where the corresponding factors are 2.3 and 4.1. The values of AGtGTs0[ T,s.IcvT( T ) ] are not very different from AGIGT*O(T,s=O) in these cases, accounting for at most a factor of 1.5 in the hybrid rate constants, and the good overall agreement of the variational transition-state theory results with the accurate quantal ones is an indication that classical recrossing of the best dynamical bottleneck regions near the saddle point is not very extensive. For the DIM-RMOS surface, the rate constant is smaller and ICVT/LAG theory is not as successful at temperatures of 400-500 K and higher. Ideally we would like to relate this fact to the special features observeds in the previously reported trajectories on this surface. We will discuss two possibly relevant features of these trajectories. The two features observed in the quasi-classical trajectories are that even above threshold most quasi-classical trajectories do not reach the saddle-point region and also that a small number of trajectories react at low energies, well below the adiabatic energy requirement at the variational transition state and even at the saddle point. Those few trajectories that react classically at energies below the adiabatic bottleneck are a definite indication of a dynamical difference between the QCT and VTST methods, thus at least one of the methods is incorrect in this regard, but it is not clear which one. The QCT method does not enforce a quantal vibrational energy requirement in the direction orthogonal to the reaction path and hence it may allow too much vibrational nonadiabaticity as compared to a quantal treatment. The VTST method attempts to enforce a quantal energy requirement on the degrees of freedom orthogonal to the reaction path, but it does so by enforcing the full adiabatic quantized energy requirement as if generalized transition states were stable states. This treatment may be too adiabatic. VTST does, however, include some lower energy reactive events by the tunneling mechanism. The above-threshold classical reflection, prior to the saddle point, of most of the trajectories for 0 H2 on the DIMRMOS surface may have two causes. One cause is that the equilibrium flux through a phase-space-generalized transition-state dividing surface is much smaller than the equilibrium flux through dividing surfaces in the region of classical reflection; VTST correctly accounts for this kind of reflection. A second possibility is that there is a large amount of nonadiabatic reflection from the region of high reaction-path curvature that precedes the saddle point in this case (see Figure 3, where the saddle point is at s = 0). If this is occurring, then trajectories starting from the O H H asymptote may also reflect nonreactivity from this region. Since we find that the variational transition state for the thermal 0 H2 reaction on the DIM-RMOS surface is always later than for the saddle point (by 0.11-0.18ao for the temperature range of Table III), these trajectories would recross the variational transition state as well as the saddle point. This would cause VTST to overestimate the rate constant. Apparently this kind of recrossing trajectory is more common for 0 + H 2 on the DIMRMOS potential energy surface than for most other cases. In over 40 comparisons to exact quantal results,33the only other reaction for which VTST rate constants have been found to be too large by comparable amounts is F H2 FH H.34 In that case the barrier is even more asymmetric. It is fortunate for the validity of VTST in general that these overestimates do not occur very commonly. In particular we note that in all other cases of asymmetric barriers that have been studied VTST does not overestimate the rate constant by a factor of more than 1.9, even at high temperature. These cases include I H2 on two quite different potential energy surfaces (maximum overestimate of a factor of 1.03 for one surface and 1.2 for the other surface),6*21 H C12 (maximum overestimate of a factor of 1.1),35 F + D2
+
+
+
+
-
+
+
+
(33) For reviews, see: (a) Truhlar, D. G.; Hase, W. L.; Hynes, J. T. J . Phys. Chem. 1983, 87, 2664, 5523(E). (b) Truhlar, D. G.; Garrett, B. C. Annu. Rev. Phys. Chem. 1984, 35, 159. (34) Garrett, B. C.; Truhlar, D. G.; Grev, R. S.; Magnuson, A. W.; Connor, J. N. L. J . Chem. Phys. 1980, 73, 1721. (35) Garrett, B. C.; Truhlar, D. G.; Grev, R. S. J . Phys. Chem. 1980, 84, 1749.
Garrett et al.
+
(maximum overestimate of a factor of 1.6),34H F2, D + F2, and T F2 (maximum overestimate for any of these of a factor of 1.03),34and C1+ HBr (maximum overestimate of a factor of l.9).25All of the cases mentioned are more asymmetric than 0 H2 on the DIM-RMOS surface. We note that the ICVT/LAG theory satisfies detailed balance for thermal rate constants so that the ratios in Table 111 are the same for the reverse reaction as for the forward one. B. Excited-State Collinear Reactions. In general excited-state rate constants are much more sensitive to anharmonicity than are ground-state ones, and so it is harder to predict them satisfactorily. In this light the results in Tables IV and V are satisfactory in many respects. First, for both directions of reaction and a factor of 4 in temperature (200-800 K), 90% of the PRP-ICVT/LA rate constants are accurate to a factor of 2.6 or better. The average error in the forward direction is a factor of 1.65 and that in the backward direction is 1.97 over the 200-800 K range. Furthermore in almost all cases the PRP-ICVT/LA prediction is too high, as anticipated in section 11. For the forward excited-state reaction on the ModPolCI and JW surfaces the FRP and PRP results are identical within a few percent or better in all cases except the J W case at low temperature (where the difference is a factor of 1.35 at 200 K and a factor of 1.12 at 300 K). This indicates that the first state-selected improved free energy of activation barrier is higher than the second and the only effect of the second adiabatic barrier is on verylow-energy tunneling probabilities. For the DIM-RMOS surface the FRP and P R P results differ significantly at all temperatures, e.g., a factor of 3.6 X lo4 at 200 K, a factor of 26 at 700 K, and a factor of 2.6 at 2400 K. This, plus the good agreement of the P R P results with the CEQ ones, indicates that in this case the second state-selected improved free energy of activation maximum is larger than the first, but it does not limit the reactive flux because the system experiences significant vibrational nonadiabaticity before reaching the maximum. We note that such nonadiabaticity does not limit the accuracy of thermal reaction rates to any significant extent because it occurs at too high a total energy. Examination of the CEQ resultss bears this out in that n = 0 reaction has a significantly lower threshold the n = 1 energy than the n = 1 adiabatic reaction. When we consider the excited-state reverse reaction, the trends are quite understandably reversed. Here the ICVT/LA and PRP-ICVT/LA results differ significantly for the ModPolCI and JW surfaces but agree to 0.04% or better from 200 to 2400 K for the DIM-RMOS surface. We find appreciable vibrational nonadiabaticity in the reverse reactions in all cases, and the simplest interpretation is that this occurs prior to reaching the highest adiabatic barrier for the ModPolCI and JW surfaces but afterward for the DIM-RMOS surface. In three of the six cases in Tables IV and V, the highest adiabatic barrier occurs beyond the region of maximum curvature and the PRP-ICVT/LA rate constants are an overestimation (by a factor of 1.4-1.6) as expected. However, two of the remaining three cases (ModPolCI forward and DIM-RMOS reverse) contain the largest overestimation, generally >2. These overestimations are a feature of the ICVT, not the PRP, part of the theory. Note that the scale of overestimation is somewhat similar to that for the thermal rate constant on the DIM-RMOS surface (Table 111). Furthermore these two cases have the earliest location of the operative adiabatic barrier, relative to their respective reactants, of any of the six cases in Tables IV and V. From the previous discussion regarding thermal rate constants for 0 H2 on the DIM-RMOS surface and for F + H2, we know that surfaces with exceptionally early or late adiabatic barrier locations for either thermal or vibrationally excited rate constants sometimes exhibit exceptionally large recrossing of even the best dynamical bottleneck, and the excited-state forward reaction on the ModPolCI surface and the excited-state reverse reaction on the DIM-RMOS surface are apparently two more examples of this phenomenon. The JW forward rate constant is the other case where the adiabatic barrier occurs before the region of maximum curvature, and kPRP-lCVT(n,T) is generally in better agreement with the exact
+
+
-
+
4311
J. Phys. Chem. 1986, 90,431 1-4317 results than for any other case. The JW has the most centrally located potential barrier and hence its adiabatic barriers are more centrally located than corresponding barriers on the other surfaces. Thus the first excited adiabatic barrier is not exceptionally early and agreement with exact results is good. However, if recrossing becomes more important as barrier location becomes earlier, kPRP-ICVT(n,T) would be expected to become more and more an overestimation as n increases beyond 1, since on most surfaces with a potential barrier adiabatic barriers in the entrance channel become earlier as the initial excitation increases. It would be interesting to perform further tests of ICVT and PRP-ICVT calculations to study this possibility in general. C. Implications for Three-Dimensional Reactions. The results presented here indicate that the ICVT/LAG method should be reliable for the thermal 0 H2reaction, and the PRP-ICVT/LA method should be reliable for the 0 H2(n=l) reaction. In a separate paper, application of these two theories to the M2 potential energy surface, which reduces to the ModPolCI surface of collinear geometries but also includes a fit to a b initio bend potentials,1° yielded excellent agreement with experiment2v3for both k(T) and k(n=l,T). Similar good agreement was obtained by a reduced-dimensionality study employing the collinear ModPolCI potential energy surface modified by adiabatic
+
+
ground-state bending energy eigenvalues.1° For the DIM-RMOS surface, in contrast, TST rate constants with CEQ transmission coefficients predict a k(n=l,T) value that is 1.5 orders of magnitude larger than e ~ p e r i m e n t .The ~ present calculations provide a greater understanding of this result and lead to greater confidence in the conclusion that the M2 potential energy surface is consistent with experiment, whereas the DIM-RMOS one is not. V. Conclusions We conclude that variational transition-state theory with semiclassical tunneling corrections can be applied to calculate state-selected reaction rates of vibrationally excited systems provided one ignores adiabatic dynamical bottlenecks that occur after the first Occurrence in proceeding from reactants to products of an appreciable local maximum in the reaction-path curvature. Acknowledgment. This work at Chemical Dynamics Corp. was supported in part by the Army Research Office through Contract No. DAAG29-84-3-0011 and that at the University of Minnesota, Illinois Institute of Technology, and Argonne National Laboratory was supported in part by the U S . Department of Energy, Office of Basic Energy Sciences. Registry No. 0, 17778-80-2;H2,1333-74-0.
Time-Dependent Mass Spectra and Breakdown Graphs. 8. Dissociative Photoionization of Phenol Y. Malinovich and C. Lifshitz* Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (Received: February 26, 1986)
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Time-resolvedphotoionization mass spectrometry in the millisecond range has been employed to study the reaction C6HSOH’+ c-C5H,’+ + CO in phenol. Experimental photoionization efficiency curves and breakdown graphs at t = 6 ws and 2 ms were compared to those predicted by the statistical theory (RRKM/QET) and by previous photoelectron photoion coincidence spectrometry results. The experimental breakdown curves for 2 ms are the first to be obtained by photoionization for such a long reaction time in any system. A sensitivity analysis yielded the following activation parameters: critical energy of activation, E , = 67.6 0.9 kcal/mol, and entropy of activation, AS*(1000 K) = 2.2 1.2 eu.
*
Introduction Activation parameters (activation energy and activation entropy) are known for only a small number of unimolecular ionic dissociation reactions. A major problem in determining the critical energy (or activation energy) for several interesting reactions of some complexity has been the fact that the ions spend only a short time (several microseconds) in the flight tube of an ordinary mass spectrometer, while some of these reactions are very slow at near-threshold energies. As a result, an excess internal energy above the critical energy (the so-called “kinetic shift”) is required in order to observe them. The lowest energy dissociation channel of the phenol ion, reaction 1, is one such reaction. There are at present two experimental methods to circumvent the problem of kinetic shifts, which allow the determination of activation parameters and which have been applied to the phenol ion reaction. In the first method, the reactant ions are trapped for variable times and the appearance energy (AE) of the product ions is measured as a function of storage time.’ In the second method, the reaction dynamics of energy-selected reactant ions is studied by photoelectron photoion coincidence (1) Lifshitz, C.; Gefen, S.Org. Mass Spectrom. 1984, 19, 197.
0022-3654/86/2090-431 1$01.50/0
*
(PEPICO) spectrometry. The microcanonical rate coefficient,
k ( E ) , is determined experimentally as a function of energy over a range of energies, corresponding to ion lifetimes in the microsecond regime. An RRKM/QET model calculation is performed to fit these experimental data and this provides the activation energyS2 The measurements of time-resolved AE’s by electron impact (EI) demonstrated a large kinetic shift for reaction 1,’ indicating that the reaction is very slow at near-threshold energies. The appearance energy at the longest trapping time of 0.8 ms was 11.5 eV.’ The RRKM/QET calculation, which was in agreement with experimental dissociation rates between lo4 and 3 X lo6 s-l, predicts2 a 0 K onset of 11.59 eV and a 298 K onset of 1 1.46 eV, in excellent agreement with the E1 time-resolved AE at 0.8 ms. The agreement was considered* to be in part fortuitous due to the poor energy resolution of E1 experiments. Several additional intersting studies of the phenol ion system have appeared. These include multiphoton ionization mass spectrometry3 and multiphoton ionization photoelectron spectroscopy: the latter study resulting in vibrational frequencies for (2) Fraser-Monteiro, M. L.; Fraser-Monteiro, L.; de Wit, J.; Baer, T. J . Phys. Chem. 1984, 88, 3622. (3) Pandolfi, R. S.;Gobeli, D. A.; Lurie, J.; El-Sayed, M. A. Laser Chem. 1983, 3, 29. Yang, J. J.; El-Sayed, M. A,; Robentrost, F. Chem. Phys. 1985, 96. 1.
0 1986 American Chemical Society
