10678
J. Phys. Chem. 1994, 98, 10678-10688
FEATURE ARTICLE Four-Atom Reaction Dynamics David C. Clary Department of Chemistry, University of Cambridge, Cambridge CB2 IEW, UK Received: April 4, 1994; In Final Form: July 18, 1994@
It is now possible to do quantum reactive scattering calculations on the state-to-state dynamics of gas phase chemical reactions involving four atoms. This article gives an overview of this recent progress. A survey is also given of results obtained using the rotating bond approximation (RBA). This general method is stateselective in the stretching and bending vibrations of ABC, the vibrations of AB and CD, and the rotational AB CD. Results for the OH H2, OH CO, OH HC1, and states of AB in the reaction ABC D O H HBr reactions are presented and compared with experiment. Calculations of product rotational and vibrational distributions, differential cross sections, bond and mode-selective effects, lifetimes of collision complexes, and unusual temperature dependences in rate constants are discussed. The predictions and comparisons with experiment show that the RBA is a useful and versatile quantum theory for studying the state-to-state dynamics of reactions involving polyatomic molecules.
+
+
-
I. Introduction Chemical reactions in the gas phase that involve four atoms are currently of considerable interest. 1-6 These reactions raise many new features that are not met in the study of three-atom reactions, and new experimental and theoretical methods are required. Immediate questions that arise include the following: How do the initial vibrational and rotational states of AB and CD affect the reaction cross sections and product rovibrational distributions in AB CD ABC D? Are these effects independent of each other, are they cooperative, and are they mode- or bond-selective? Do the bending vibrations of ABC enhance the ABC D reaction rate as much as stretching vibrations? What is the branching ratio for producing the different product channels AB CD and AD BC from the ABC D reaction? All of these questions on how the reaction dynamics relates to the intemal quantum states of the molecules are also relevant to understanding the kinetics of these reactions such as the temperature dependence of rate constants. For example, many triatomic molecules have vibrational bending modes that have frequencies low enough for the fist excited state to be populated at temperatures of interest, and if the excitation of this bending mode enhances reactivity, the effect on the overall rate constant for the ABC D reaction might be significant. Furthermore, many reactions between diatomic free radicals have particularly large rate coefficients with unusual temperature dependences, and a fundamental understanding of this might be obtained from the more detailed reaction dynamics studies.’ It is only relatively recently that technological developments have enabled detailed experimental investigations to be carried out on the reaction dynamics of four-atom reactions. The H H20 H2 OH reaction, and its reverse reaction, is being intensively studied and is rapidly becoming the four-atom equivalent of the H H2 reaction. Furthermore, OH H2 is one of the key reactions involved in the hydrogen-oxygen explosion reaction,8 and the OH + H2 collision is thought to
-
+
+
+
+
+
+
+
-
+
+
+
@
+
Abstract published in Advance ACS Abstracts, September 1, 1994.
0022-365419412098-10678$04.50/0
+
+
+
+
be important in interstellar cherr~istry.~ There have been several reports of rate constants for the OH H2 reaction,1° and crossed molecular beam measurements of angular distributions for the OH D2 HOD D reaction have also been measured.” The ability to create “hot” hydrogen atoms with well-defined collision energy, together with laser methods for exciting particular vibrational modes of H20 and for detecting the vibration-rotation states of OH and Ha, has enabled the H H2O OH H2 reaction to be studied experimentally at the state-selected leve1.1,2J2-14 Other related four-atom reactions such as C1 HCN HC1 CN and C1-k H2O HC1+ OH are also starting to receive similar experimental attenti~n.’~ The OH HC1- H2O C1 and OH HBr H20 Br reactions are also very important in atmospheric chemistry as they produce halogen atoms that react with ozone. l6 However, the experimentally measured temperature dependence of the rate constants for these fast reactions is currently not well understood. l73l8 The OH CO COS H reaction is another very important four-atom reaction that, unlike the previously mentioned reactions, reacts via a long-lived c0mp1ex.l~It is also a key reaction in combustion and in flames.20 Vibrationally-selected rate constant measurements have been made on this reaction,21and the vibrational product distribution of the C02 molecules has been measured.22 Crossed molecular beam measurements of angular distributions have also been made on this reaction.23 Furthermore, hot H atom experiments have been done on the H C02 OH CO reaction.24 Particularly exciting experiments have been done on this reaction by both Zewail and c o - ~ o r k e r and s ~ ~Wittig and co-workers.26 They measure the lifetime of the HOC0 transition state by performing timeresolved measurements on the OH products produced by the photoinduced reaction of the van der Waals complex IHC02 hv- OH CO I. Given the intrinsic fundamental interest in four-atom reaction dynamics and the large number of experiments that are now being carried out on these reactions, and also the importance of these reactions in atmospheric, combustion, and interstellar
+
-
-
+
+
+
+
-
+
+
+
+
+
+
-
-
+
-
+
-
+
+
+
+
0 1994 American Chemical Society
+
Feature Article
J. Phys. Chem., Vol. 98, No. 42, 1994 10679
-
+
+
chemistry, it is no surprise that theoreticians have turned to these H and the reverse reaction H H20 OH H2 were studied, problems. Quantum chemistry calculations of quite good quality and several subsequent calculations on this reaction have been can be carried out of points on the potential energy surfaces reported with the same t h e 0 r y . 4 ~ - ~In~the fullest form of this for reactions such as OH H2 H2O H (ref 27) and H m e t h ~ d , ”which ~ has since been called the rotating bond approximation (RBA)?O all the vibrations in the H2O are treated C02 OH CO (ref 19), and it is possible to fit this ab initio explicitly by close-coupling calculations, as are the vibrations data to flexible potential energy surfaces.28 Quasiclassical of the OH and HZ and the rotation of OH. However, the trajectory calculations, including state selection, have been done on these reactions by Schatz and c o - w o r k e r ~ ~and ~ - 0~t~h e 1 - s . ~ ~ ~ rotational ~~ motions of H2 and H20 are not treated explicitly in These quasiclassical studies have illustrated many interesting this approach, although they can be treated approximately by features of the reaction dynamics such as mode and bond using adiabatic bend corrections. These calculations have also shown that the vibration of the OH reactant is largely a spectator selectivity in the H H20 reaction. However, these and other for the OH H2 reaction.49 classical studies have also highlighted the problems of performing classical trajectory calculations on reactions involving The RBA is a general and computationally inexpensive theory polyatomic molecules where neglect of zero-point energy effects that has the particular advantage that it can treat explicitly both can be more significant than in atom diatom reactions. the bending mode of triatomic ABC and rotational states of AB Furthermore, tunneling effects are not treated in the classical in the reaction ABC D AB CD. This special feature calculations. was exploited in RBA calculations on the reaction OH CO C02 H,50where lifetimes of the reaction complex HOC0 Quantum reactive scattering calculations are much harder to were also calculated and compared with e ~ p e r i m e n t . ~The ~.~~ perform than quasiclassical trajectory computations. They have method has also been applied to OH HCl H20 C1 and the distinct advantage that, when done accurately, they can give OH HBr H20 Br, and the RBA has given new insight the exact results for the given potential energy surface. Even into the temperature dependence of the rate constants for these when done approximately, the quantum reactive scattering reaction^.^^^^^ Also, the RBA can be extended to reactions of results give considerable insight into the reaction dynamics, and larger molecules and has recently been used in calculations on many comparisons of approximate and accurate calculations for the OH CHq H20 CH3 reaction.53 atom diatom molecule reactions have given invaluable indications of the validity of approximations to the quantum Echave and Clary have developed an exact quantum medynamics.35 The atom diatom molecule quantum reactive chanical hyperspherical coordinate theory for four-atom reaction scattering problem has almost become a solved problem in three dynamics with planar geometry.54 This method has been applied dimension^,^^^^^ and four-atom reactions now present a more to the OH H2 reaction, with the length of the OH spectator formidable theoretical and computational ~hallenge.43~ bond held fixed. This technique has the advantage over the RBA that the rotations of OH, H2, and HzO are treated Sophisticated forms of transition-state theory have also been accurately for planar geometry. The only degree of freedom applied to four-atom reaction^.^^-^^ These calculations have omitted from this theory is the out-of-plane torsional mode. The been very useful in examining the importance of mode coupling calculations of reaction probabilities obtained with this method and tunneling in these reactions. They also normally provide show that the RBA is quite an accurate technique in providing quite good rate constants from given potential energy surfaces reaction probabilities that are state-selected in the initial and are much cheaper in computer time than the quantum vibrational states of H2O for the H H20 reaction. Therefore, reactive scattering calculations. However, transition state theory the RBA is expected to be reliable for studying vibrational mode does not provide the state-selected results that are measured in selectivity for polyatomic reactions. many modem reaction dynamics experiments, and the range of accuracy of the approach is still uncertain for polyatomic Bowman and co-workers have also reported calculations on reactions. the OH H2 reaction55by applying essentially the same method that they used for the H HCN r e a c t i 0 n . 4 ~Their ~ ~ approach The fvst quantum reactive scattering calculations on a realistic involves treating explicitly both the stretching vibrations of H20 four-atom reaction were done independently by groups at and the vibrations of OH and H2. However, all other degrees Cambridge University”2 and Emory Uni~ersity?~who both of freedom, including the bending vibration of H20 and the calculated reaction probabilities for the H2 CN H HCN rotations of OH, are treated adiabatically in this approach. The reaction with linear geometry. The method of Brooks and Clary method, therefore, does not give cross sections that are stateused a new set of hyperspherical coordinates for this reaction selected explicitly in the bending mode of H20 and rotations that are essentially identical to spherical polar In of OH in the H H20 OH H2 reaction. principle, this set of rather general coordinates can treat not only an abstraction channel such as AB CD ABC D but Recently, other quantum scattering calculations have been also a double exchange channel such as AB CD AC reported for the H H20 reaction. Baer and co-workers have BD. This calculation showed that the CN bond in the H2 adapted their method of complex absorbing potentials to this CN reaction largely acts as a spectator bond and hardly reaction.56 Combined with a “breathing sphere” approximation, influences the reaction. Bowman and co-workers also reported this enables the vibrational state selectivity of the H20 molecule calculations of reaction probabilities for the H2 CN reacti0n.4~ to be examined. The accuracy of their approximation remains They used a set of coordinates that are the same as those used to be tested. for calculations on atom-diatom reactions for collinear geomAlso Zhang and Zhang have performed time-dependent wave etry with the addition of an oscillator basis for the CN spectator packet calculations that have given accurate 3D reaction bond. They also turned their reaction probabilities into rate probabilities for the OH H2 reaction summed over all product constantsu by making use of the reduced dimensionality rovibrational states of H20 but state selected in the initial approximation with adiabatic bend corrections that they have rotational states of OH and H2. In the first of these calculations previously used successfully for atom diatom reactions.45 the initial OH bond was held fixed in length,57but their method has recently been extended to the full-dimensional calculation The first quantum reactive scattering calculations on a stateof accurate reaction probabilities for rotationally selected OH selected four-atom reaction with nonlinear geometry were and H2 summed over all final quantum states of HzO.~* reported by Clary in 1991.46 The reaction OH H2 H20
-
+
+
-
+
+
+
+
+
-
+
-
+
+
-
+
+
+
+
+
+
-
+
+
+
-+ -+
+
-
+
+ +
+
+
+
+
+
-
+
+
+
+
+
-
+
+
-
+
10680 J. Phys. Chem., Vol. 98, No. 42, 1994
Clary
B
state selective in CD, and that is the main approximation of the RBA. Related “reduced-dimensionality’’ approaches have been applied successfully to atom diatom r e a c t i o n ~ ! ~ , ~ ~We -~~ emphasize, however, that the special attribute of the RBA, which is not a feature of most other methods that have been proposed to date for four-atom reactive scattering, is that the rotational states of AB and the bending vibration of ABC are treated explicitly. We have also found in calculations on the OH HZreaction that it is an excellent a p p r o ~ i m a t i o nto ~ ~freeze the OH “spectator” bond and a CN spectator bond was also found in calculations on the CN HZ HCN H reaction?2 However, although the length of AB has been held fixed in most of the calculations on the reactions discussed in this article, this is not a necessary requirement of the method, and in the theory described here, the AB vibration is included explicitly. The key coordinates in the chemical reaction that refer to the bonds being broken and formed are R, the distance between the centers of mass of AB and CD, and r, the length of the CD bond. It is convenient to transform R and r to polar coordi-
+
+
+
+
-
+
Figure 1. Coordinates for the AB CD ABC D reaction. The angle that describes the out-of-planetorsional motion of AB with respect to CD is 4.
Miller and c o - w ~ r k e r shave ~ ~ calculated exact cumulative reaction probabilities that refer to a sum over all initial and final rovibrational states of both reactant OH H2 and product Hz0 for J=O and a given total energy. By applying a “J and K-shifting” approximation, they have also calculated rate constants. Very recently, NeuhausefiO has also used a timedependent method to calculate accurate reaction probabilities for this reaction, summed over all product states of H20. As is described in section 111, these recent studies provide very useful benchmarks to test the accuracy of the more approximate theories. The RBA has, to date, given the most extensive set of stateto-state results for four-atom reactions. This versatile and computationally inexpensive technique has enabled detailed studies to be carried out on reactions with barriers in the potential (e.g., OH H2 H2O H), on a reaction involving the formation of a long-lived complex (OH CO C02 H), and on fast reactions with hardly any activation energy (OH HC1 H20 C1 and OH HBr H2O Br). It is appropriate to collect together all of these results, and that is the main emphasis of the remainder of this paper. The RBA results are also compared with those obtained using other theories and with experimental data. Included in the discussion are product rotational and vibrational distributions, differential cross sections, bond and mode-selective effects, lifetimes of collision complexes, and temperature dependences of rate constants. The comparisons with many different types of experiment show that the RBA is a versatile and computationally inexpensive theory for the reactions of state-selected polyatomic molecules.
+
+
+
-
+
-
+
+
-
-
+
+
+
11. The Rotating Bond Approximation Here we give a simple overview of how the rotating bond approximation is applied to the four-atom reaction AB(vj)
+ CD(v’) - ABC(l,m,n) + D
The quantum numbers v and v’ are the vibrational levels of AB and CD, respectively,j is the rotational level of AB, and (I,m,n) are the symmetric stretch, bend, and asymmetric stretch vibrational quantum numbers of ABC, respectively. More complete details of the method are described in refs 46-52. Figure 1 presents the coordinates for the three dimensional interaction of molecules AB and CD. The main idea in the full form of the RBA is to treat explicitly the coordinates s, R, r, and 8 in quantum scattering calculations while averaging over the coordinates y and 4, which are associated with the rotation of the CD molecule. Therefore, the theory is not rotationally
-
+
R = pl@ COS(&, r = h e sin(d) where p1 and p2 are mass factors46and e and Q are known as the hyperradius and hyperangle, respectively. The wave function for the problem is then expanded in the coupled-channel form
where the {?$k(d,8,s;@)}functions, which have the energies { E t (e)},are computed by diagonalizing the RBA Hamiltonian on a grid of fixed e values. The basis functions used for the diagonalizations are kept intentionally simple: spherical harmonics for 8, distributed Gaussians for d, and harmonic oscillators for s, the bond length of AB. Note that this approach enables both the rotations of AB and all the bending vibrations of ABC to be described accurately. The idea of using a simple spherical harmonic basis set to describe the rotations of one molecule taking part in a diatom diatom reaction, while treating the rotational motion of the other molecule in a more averaged way, goes back to our own calculations on fast diatom diatom reactions where various such approaches were considered and tested.65 In later theoretical work on four atom reaction^:^ this possibility was also discussed but not applied. Plots of the energies {E,(@)} against e are called hyperspherical adiabats. For large @. the {&(@)} tend in magnitude to the energies associated with either the reactant (vj,v’) or product (l,m,n) quantum states. If two of the hyperspherical adiabats correlating with different arrangement channels show an avoided crossing, then a reactive transition is likely to occur between the two quantum states that label these adiabats for large Scattering boundary conditions are applied to the solution of the coupled-channel equations for large values of e when the interaction between the reactant and product channel becomes in~ignificant.~~ This gives the reaction probabilities (E>, which are obtained for fixed values of the collision energy E, total angular momentum J , and projection K of the AB angular momentum j along the intermolecular axis. Reaction cross sections are calculated from
+
+
Feature Article
J. Phys. Chem., Vol. 98, No. 42, 1994 10681
Maxwell-Boltzmann averaging these cross sections over collision energy and surnfning over all final quantum states then gives the rate constant kv,M (r). It is assumed in the calculations that K is a good quantum number. This “coupled states” approximation is expected to be valid for reactions with strongly localized reaction paths,35and calculations by B a n g and B a n g also suggest that this should be appropriate for the OH H2 reacti0n.~8 The rotational motion of the CD molecule is not treated explicitly in the RBA scattering calculations. This can be accounted for by using a variety of “reduced dimensionality” approximations that have been developed by Bowman and coworkers, initially for atom-diatom reactions.45 The simplest energy shift harmonic (ESH) approximation that was used in some of our earlier calculations&-49 is to set y = 0 (see Figure 1) in the scattering computations. The rotational motion of CD is accounted for approximately by calculating the harmonic inplane { E;:} and out-of-plane { coutk} bending energies of the CD molecule at the transition state of the reaction. This makes it possible to calculate “cumulative” reaction probabilities that refer to a sum over all the CD initial rotational states for a given total energy E. This is done from
+
Rate constants can also be calculated with the ESH approximation from the formula66
and this enables the required harmonic bending zero-point energies Ein(s,g,6) and Eout(s,g,6)to be obtained, with corresponding harmonic quantum numbers ltin and nout. The potential used in the scattering calculations is then
v(q,6,S,8,ym,$,)
+ w, + 1woUt(S,g,~) +
min+ l)Ein(W,~)
where Vis the full six-dimensional potential energy surface and nout= ni, = 0 for reactions with a well-defined transition state. In the AB approximation, the rate constants for a reaction with a well-defined transition state are calculated from a formula analogous to eq 4 but without the
( ;?) €in
exp - -
+
+
+
In. Application to the OH where QCDis the rotational partition function and cin and cout are the harmonic zero-point energies of the in-plane and outof-plane bending modes of CD at the transition state. MaxwellBoltzmann averaging over the initial states (vjKv’) gives the final rate constant k(T). In practice, the K dependence of the rate constants can be obtained by finding the rotational constants of the ABCD transition state and using these to estimate the cumulative reaction probabilities for K t 0 from those for K = 0 with a “K-shifting” a p p r ~ x i m a t i o n .However, ~~ we do not do an analogous “J-shifting” approximation to carry out the partial wave summation over J to obtain cross sections from the reaction probabilities as we actually calculate the reaction probabilities for different J values. A more accurate way of accounting for the neglect of the rotations of CD which we have used in more recent RBA calculations is the adiabatic bend (AB) approximation in which the zero-point energy of the in-plane and out-of-plane bending modes of CD, E&,@& and Eout(s,g,6), are added to every point on the potential energy s u r f a ~ e . ~We , ~ ~implement .~~ this by f i s t of all choosing a fixed value of s, g, and 6 and then determining the angular coordinates Om, ym, and $m that minimize the potential. Then the harmonic energies of the three bending modes of ABCD are obtained by using the FG matrix method6’ about these minimum-energy points. The bendstretch coupling terms in the FG matrix are ignored so that the bending modes can be separated The bending mode with largest frequency is identified with the bending mode of ABC,
-
term in the numerator. For a reaction which has no barrier in the potential, such as OH HBr H2O Br, and coutare not clearly defined. In this case, cross sections are calculated for all possible values of ni, and noutthat allow reaction, these cross sections are then summed and Maxwell-Boltzmann averaged over collision energy, and the rate constant so obtained is divided by QCDto give the final kyjw (7‘). In practice, the energy shift and adiabatic bend approaches give very similar quantum state dependences of the cross sections, but the energy shift approximation underestimates the absolute values of the reaction probabilities at very low energies (see section 111.3). For reactions involving the OH molecule, such as OH Hz, it is also necessary to divide the rate constant by the electronic partition function of OH,38which is close to a factor of 2 at higher temperatures. Furthermore, the rate constant for a reaction such as OH -I- H2 calculated using the above procedures needs to be multiplied by a factor of 2 to account for the indistinguishability of the H atoms in H2.
-
+ H2
-
H20
+ H Reaction
+
In this section we describe RBA calculations on the OH H2 H20 H reaction and its reverse. Comparisons are made with other calculations and with experimental results. Several RBA calculations have been done recently on this r e a ~ t i o n , ~ - ~ ~ and we present some illustrative examples. The results demonstrate the versatility of the RBA in calculating rate constants, mode and vibrationally selected cross sections, OH rotational product distributions, and differential cross sections. The first dynamical computations on this reaction were done by Schatz and co-~orkers~*-~* using the quasiclassical trajectory method. Several comparisons of quantum and quasiclassical results for this reaction have been p ~ b l i s h e d . ~The , ~ potential ~,~~ energy surface used in the calculations is a slightly modified v e r ~ i o nof~ a* form ~ ~ due to Schatz and ElgersmaZ8that is a fit to ab initio data.27 This potential has a classical barrier of 0.26 eV, and the classical exoergicity of the reaction is 0.66 eV. It should be noted that a molecule such as OH has a 211electronic ground state, but the electronic orbital and spin angular momentum is not treated in the dynamics calculations reported here. 1. Spectator Bond in OH Hz. It is of interest to examine whether the vibration of the initial OH molecule acts like a “spectator bond”. Figure 2 shows plots of RBA cross sections for the OH(v=Oj=O) H*(v‘=O) H20 H reaction obtained from two separate calculations: (a) in which the length of the OH spectator bond is held fixed throughout the calculation and (b) in which this OH bond is allowed to vibrate so that all three vibrations of H20 are treated explicitly. The comparison for
+
+
+
-
+
10682 J. Phys. Chem., Vol. 98, No. 42, 1994 OH+H2
1.6
I
I
1.4
1.2
-a.-
2
1
2
0.8
a
0.6
e
0.4
0.2
0.05
0.1
0.15 0.2 0.25 0.3 Translational energy (eV)
+
0.35
-
+
Figure 2. Cross sections for the OH(v=Oj=O) Hz(v'=O) H20 H reaction as described in section 111.1.The curves with dashed lines refer to RBA computations in which the OH spectator bond is held fixed (calculation (a)) while the curves with unbroken lines refer to RBA calculations in which all three vibrations of H20 are treated explicitly (calculation (b))49The circles refer to cross sections summed over all product H20 vibrational states while the squares refer to cross sections into only the (100) (001) levels of H2O (calculation (b)) or the n = 1 local mode of HzO (calculation (a)).
+
cross sections obtained by methods (a) and (b) and summed over all final vibrational states of H20 is excellent. Also shown in the figure is a comparison of cross sections that refer to reaction into the local n = 1 OH stretching vibration of H2O for calculation (a) and to a summation over the product vibrations (100) and (001) of H20 for calculation (b). It can be seen that the agreement is once again good. This comparison provides strong evidence to suggest that, for AB CD ABC -I-D reactions in which the AB bond length in reactant, transition state, and products is almost the same, the length of the AB spectator bond can be held fixed in scattering calculations. In this case, cross sections for reaction into the local AB stretching vibration of ABC correspond to a sum of cross sections into the excited asymmetric and symmetric stretch vibrations of ABC. In many of the calculations reported in the rest of this article, the AB bond length is held fixed, and this gives a considerable saving in computer time with no significant loss in accuracy of the results. Several experimental studies also indicate the spectator nature of the bond not broken in some four-atom r e a ~ t i o n s . ' ~ ~Furthermore, J~J~ the accurate calculation of reaction probabilities for the OH H2 reaction by Zhang and Zhang also confirms this view.57%58 2. Comparison with Exact Planar Reaction Probabilities. The calculation by Echave and Clarys4 in which exact reaction probabilities were calculated for the OH H2 reaction with planar geometry and with the rotations of both OH and HZ treated explicitly is very useful in testing the accuracy of the RBA. Figure 3 presents exact and RBA reaction probabilities for the vibrationally selected reaction H HzO(m,n) OH H2 for planar geometry. In these calculations, the OH spectator bond length was held fixed, m labels the bending vibration, and n labels the local OH stretching vibration in H20. The calculations refer to a sum over all rotations of H20, OH, and HZfor a fixed collision energy. It can be seen that the agreement between the RBA and exact planar probabilities is good, and this suggests that the RBA should be quite an accurate method for calculating vibrationally selected reaction probabilities in four-atom reactions. 3. Comparison with Exact Cumulative Reaction Probabilities. Manthe et have calculated exact cumulative reaction probabilities for the OH H2 reaction. These are not state-selected results but refer to a sum over all initial rovibrational states of OH and H2 and all final rovibrational states of
+
0 0.2
0.3
0.5
0.4
0.1
0.6
0.8
1
0.9
Kinetic Energy(eV)
-
Figure 3. Exact54and RBA cumulative reaction probabilities for the HzO(m,n) H Hz OH reaction with planar geometry as a function of initial kinetic energy. Here m and n refer to the quantum numbers of the bending mode of HzO and local OH stretching mode of HzO, respectively. The probabilities refer to a sum over all rotational states of H20, OH, and Hz (see section III.2).
+
-S
I
,
l
+
l
,
l
,
1
"
,
l
,
l
l
l
l
/
l
j
j
l
,
I
2 '
c J .d
s0
n
1
2Q 0C
0
+ V
L W
-
+
+
+
+
-
+
0.1
0.2
"
1
"
"
1 " 0.3
"
1 0.4
0.5
8.4
0.5
T r a n s l a t i o n a l Energy(eV1
C
-I U
2
-2
0,
c .d
n
-1
8.1
0.2
0.3
T r a n r t at i o n o l Energy(e V 1
+
-
+
Figure 4. Cumulative reaction probabilities for OH HZ HzO H obtained in the RBA-AB (upper panel) and RBA-ESH (lower panel) calculations for J = 0. Also shown are the exact resultssg(see section III.3).
H20 for a given energy E and zero total angular momentum. Figure 4 presents a comparison of these exact probabilities with those obtained using two versions of the RBA: the energy shift (ESH) and adiabatic bend (AB) approximations described in section 11 that account in different ways for the rotation of the H2. It can be seen that the ESH probabilities agree well with the exact results for energies above 0.3 eV but fall below the exact probabilities for lower energies. The AB probabilities,
Feature Article
J. Phys. Chem., Vol. 98, No. 42, I994 10683
1
0
1
’
io
1
N
003
a
ETrans=O.leV OOlz 0-H stretch 0-D stretch 010= HOD bend 100=
-
+
1
1.5
2 2.5 1000 I T(K)
3.5
3
+
Figure 5. Arrhenius plot of the rate constants for the OH Hz reaction. See section III.4 for a description of the different calculations. Also shown in the diagram as error bars are the experimental results1° and the exact rate constants calculated by Manthe et aLS9
+
Figure 6. Cross sections for the H HOD(lmn) Hz OD reaction The for different excitations of the three vibrational modes in initial translational energy is 0.1 eV for each initial state (see section 111.5).
-
+
+
reaction H HOD(1mn) H2 OD for a thermal collision energy are shown for up to three quanta in any one vibrational mode in HOD. (Here I, m, and n refer to the local OD stretch, HOD bend, and OH stretching vibration, respectively.) This however, agree quite well with the exact results over the whole figure clearly shows the effect of exciting different vibrational energy range. This comparison suggests that the ESH method modes and HOD(003), with three quanta in the OH local mode, is appropriate to use for calculations at higher energies where gives by far the largest cross section. Reaction from HOD there is much interest in state-selective effects on the chemical vibrational states such as (102), (201), (012), and (030) do not reactions but should be used with caution for lower energies. give nearly such large cross sections. At these collision The reduced dimensionality approach of Bowman also gives energies, excitation of the HOD bending mode has a similar good agreement with the exact quantum reaction probabilities effect to excitation of the OD spectator vibration. Therefore, for this reacti0n.5~ it is vibrational excitation of the bond explicitly being broken 4. Rate Constants. Figure 5 presents rate constants k(T) in the reaction that has the most significant effect on the reaction for the OH H2 reaction obtained by several different methods cross sections. This is also the finding of quasiclassical of calculation and by experiment.’O The J = 0 reaction trajectory c a l c ~ l a t i o n s ~ *and - ~ ~ the reduced dimensionality probabilities of Manthe et al. were turned into rate constants59 computations of Bowman and c o - w o r k e r ~ .This ~ ~ is in agreeby using a “J- and K-shifting” approximation that is expected ment with the overall conclusion of the experiments performed to be accurate for this reaction, and these results are shown in by Crim and co-workers and by Zare and co-workers, although the diagram. Also shown are the RBA rate constants obtained the experimental results do not provide such state-selected detail with the AB approximation and averaged over all initial as is provided in the RBA calculations. rotational states of OH and H2. We also show the rate constants The RBA also predicts49 interesting product vibrational for the OHU=O) reaction obtained with the ESH and AB distributions for the H HOD(Zmn) reactions: HOD(003) gives approaches. As expected, the full RBA-AB rate constants mainly H2(v=l) products while HOD(001) gives OD(v=O) agree well with the “exact” ones while the RBA-AB rate Hz(v’=O) and HOD(100) gives OD(v=l) H~(v’=O). This constants for OHU=O) are about a factor of 2 lower than these remains to be confirmed in experiments. at 300 K. Both the RBA-ESH and experimental rate constants 6. OH Rotational Product Distributions. Several experifall below the “exact” values by a factor close to 4 at 300 K. ments on H H20 OH HZmeasure the OH product However, above 700 K all the methods do give similar rate rotational distributions and these are calculated explicitly in the constants that agree fairly well with experiment. RBA. Figure 7 compares such distributions, calculated by These results suggest that the barrier in the Schatz-Elgersma with the results of four Nyman and Clary with the potential28is slightly too low, but the reasonable agreement with different e x p e r i m e n t ~ . ’ , ~ J ~It- ~can ~,~ be~ seen that, in every experiment at higher temperatures suggests that the potential case, the agreement between the RBA and experimental should still be realistic for dynamics calculations at higher rotational distributions is quite good with a very low fraction collision energies. Not shown in this diagram are the rate of available energy going into rotational excitation in the product constants of Zhang and Zhang5*which should be exact and are OH. The product OH rotational distributions seem to be not computed with any shifting approximations. These rate relatively insensitive to the collision energy and even to initial constants refer to reaction out of OHU=O) H20’=0) and are excitation of the H20 stretching vibrations. This has been very similar to the RBA-AB rate constants for OHU=O) over explained by a simple Franck-Condon model4’ that projects the entire temperature range. the ground-state bending mode of H2O onto spherical harmonics 5. Mode and Bond Selectivity. Of particular interest in that describe the product OH. Since the bond angle of H20 in the H H2O reaction is the effect of different vibrational states the transition state of this reaction is close to the geometry for of HzO on the reaction cross sections. The experiments of Crim isolated H20, this approximation would be expected to work and co-workers pump quite high overtones of the OH stretching quite well. Other Franck-Condon models have also been applied vibrations in the H20 molecules, which then react with thermal to this r e a c t i ~ n . Excitation ~ ~ . ~ ~ of the initial bending mode in H atoms1J2. The experiment of Zare and co-workers excites the v = 1 stretching modes in H20 and uses “hot” H a t ~ m s . ~ J ~ H20 does lead to slightly more rotational excitation in the OH product, and there is some evidence for this in the experiments A detailed comparison between some of these experimental by Zare and co-workers on the H D20 rea~ti0n.l~ results on mode selectivity and those obtained from various methods of calculation has been presented in refs 48, 49, 55, 7. Differential Cross Sections. In a crossed molecular beam experiment, Casavecchia and co-workers have measured the and 68. differential cross section for the OH Figure 6 shows cross sections for conditions quite similar to D2 D HOD reaction.” Figure 8 shows a comparison between the RBA48 those of the Crim experiment.’,l2 RBA cross sections49for the
+
+
+
+
+
-
+
+
+
+
+
-
+
10684 J. Phys. Chem., Vol. 98, No. 42, 1994
I
0
2
4 6 8 OH r o t a t i o n a l quantum number
OH
+
+
D2-
I
10
ETIans=0.27 eV 1.0
c1
t
I
5 -
n
4,O
'
0
4 6 OH r o t a t i o n a l quantum number
2
5,0 0,2
10
8
~
HOD(bend,stretch) product vibration 35,
I
I
0
2
4 6 OH r o t a t i o n a l quantum number
-
8
Figure 7. (a) Top panel: Rotational product distributions of OHG) for the H H20(0,4) H2 OHU) reaction. RBA calculations4* (unbroken line) are compared with experimental results.I2Here (0,4) refers to four quanta in the OH local stretching mode of H20. The RBA calculation had an initial translational energy of 0.05 eV, while the experimental results refer to an initial temperature of 300K.(b) Middle panel: Rotational product distributions of OHU) for the H + HzO(O,O,O) H2 + OHQ) reaction at a collision energy of 1.5 eV. RBA calculation^^^ (unbroken line) are compared with experimental result^.*^^^' (c) Bottom panel: Rotational product distributions of OHU) for the H H~0(0,0,0) H2 + OHU) reaction at a collision energy of 1.0 eV. RBA calculation^^^ (unbroken line) are compared with experimental results.I4 and experimental differential cross section for this reaction. It can be seen that the agreement is very good, with strong backward peaking. Recent quasiclassical trajectory angular distributions for this reaction68 also agree well with the RBA results. The product vibrational distributions of HOD obtained from the RBA48for the OHG=O) DZreaction are shown in Figure
+
+
-
+
-
+
-
Figure 9. HOD(m,n) vibrational product distributions calculated4*for the OH(j=O) Dz HOD(m,n) D reaction, where m and n are bending and OD stretching vibrations of HOD, respectively.
+
+
9. Here it can be seen that the most likely product state of HOD has one quantum in the HOD bending mode, and there is hardly any excitation in the product state with two quanta in the OD stretching vibration. The proportion of available energy going into product vibration isfv = 29% in the RBA calculation. Assuming very low product this gives the percentage of energy going in to product translation as ft = 71%. This seems to be in disagreement with the results of the molecular beam experiment which gives ft = 32%." Quasiclassical trajectory calculations also differ from experiment giving fv = 46% with very little energy going into product rotation (fi = 9%).68 This suggests again, in line with the conclusion from the comparison of the calculated rate constant with experiment, that the Schatz-Elgersma potential will need improvement for comparison with the more detailed experimental data that is becoming available.
+ CO - H + COZReaction Unlike the OH + H2 reaction, the OH + CO - C02 + H
IV. The OH
reaction goes via a long-lived complex, and it is of interest to see whether the RBA can be applied to it successfully. Detailed comparisons of the RBA calculations on this reaction with several different types of experiment and with the results of quasiclassical trajectories are made in ref 50. Here, we concentrate on the lifetimes of the HOC0 complex formed in the reaction and the differential cross sections. The potential energy surface for this reaction is due to Schatz and co-workers and is obtained as a fit to extensive ab initio computation^.^^.^^ This potential has a very small barrier in the entrance OH CO channel, a deep well of about 1.6 eV
+
Feature Article
J. Phys. Chem., Vol. 98, No. 42, 1994 10685 1
.
0 -8
1 l1
~
,
, 0.4
,
/
,
~
,
,
.
0.5
l
,
,
,
,
0.6
,
t
,
0.7
Energy ( e V )
-
+
Figure 10. Logarithm of reaction probabilities for OH COg’=O) C02 H calculated with the M A S 0for J=O and plotted against total energy. The probabilities are summed over all the product vibrational states of CO?.
+
100 I
1
U
45
0
90
.
135
~
180
O R / degrees
-
+
+
Figure 12. Differential cross sections for the OH CO C02 H reaction obtained by the RBASOand experiment23for a translational energy of 0.6 eV. The distributions are normalized to the values at f3 = 0”.
2D to 3D. These lifetimes compare quite well with those computed in trajectory computations in 2D, 3D, and 6D.32,50,71 All of the computations give lifetimes in the range 0.1 -Ips for higher energies, and this agrees well with the experimental results of Wittig et ~ 1(although . ~ the ~ calculated values have shorter lifetimes than those of Zewail et ~ 1 . As ~ ~can ) be seen from Figure 11, the calculated 2D and 3D quantum lifetimes do show a non-monotonic progression as a function of collision energy. Recent 2D calculation^^^ suggest that this is because resonances in which the HOCO complex has OH(v=2) excitation have much longer lifetimes than those with OH(v=3) or OH(v=l). This prediction awaits experimental verification.
2. Differential Cross Sections and Other Results for OH Casavecchia and co-workersZ3 have measured the differential cross sections for the OH CO reaction and find significant peaking in both the backward and forward directions, with a slight preference for forward scattering over backward scattering. The RBA differential cross section50does also show peaks in both the forward and backward direction with a preference for forward scattering, but the probability for sideways scattering is much lower than in the experimental case (see Figure 12). This is because the RBA differential cross section is dominated by scattering resonances from only a small number of partial waves with large J. Calculations treating more degrees of freedom will give more resonances, and more partial waves will then contribute significantly to the differential cross section. The contributions from these partial waves can then interfere to give less dominance from the 0” and 180” scattering angles. Although the RBA does give the correct qualitative features of the differential cross section, it will clearly be necessary to treat explicitly more degrees of freedom in quantum scattering calculations for a quantitative description of a property such as this for this reaction. The rate constants calculated from the RBA for the OH CO C02 H reaction are smaller than those calculated from quasiclassical calculations for temperatures below 500 K32and are smaller than experiment.21 This is also the conclusion from RRKM calculations32. It is likely that the effective barrier in the potential energy surface for the exit H + COz channel is too large. Furthermore, the quasiclassical trajectory calculations do not account for zero-point energy effects, and “leaking” through the effective potential barrier is then possible so that a reaction cross section can be obtained that is too large for the given potential energy s ~ r f a c e . ~ ~ . ~ ~ The vibrational product distributions of the C02 molecule obtained from the RBA computations on the OH CO C02 H reaction show only limited e x ~ i t a t i o n ,and ~ ~ this is in agreement with e ~ p e r i m e n t and ~ ~ .trajectory ~~ computation^.^^
+ CO.
-
+
+
+
+
+
-
10686 J. Phys. Chem., Vol. 98, No. 42, 1994 -9.5
1'
35.0
' ' ' " " ' ~ ' ' ' ' ~ ' ' ' ' ~ " ' ' ~ ' " ' '
I Exp
J
Calc A
I
4
So
E 3,
25.0
"
.-
-11.5
OH+HCI
-12.0
CI
1
Y
I
I
I
1
I
30*01 1
.
N
I
i=o
20*o! 15.0
1
\
6
- 13.0 0
50
100
200
150
250
300
T IK
Figure 13. RBA rate constants51for the OH -tHC1 and OH reactions compared with the experimentall7J8results.
+ HBr
Sixty-five percent of the energy available to products goes into translation in the RBA calculations, which is in good agreement with the measuredz3 value of 64%.
Final H,O (bend, stretch) state
-
Figure 14. RBA cross sections5!for the HCl(v=O) -t OH(j,K=O) C1 HlO(m,n) reaction plotted against HzO product vibrational state ( m n ) with j = 0 and 1. The initial translational energy is 0.36 eV. Here m is a vibrational bending quantum number and n is a local OH stretching quantum number of H20.
+
V. Reactions of OH with HC1 and HBr
-
--
+
The RBA has also been applied to the reactions OH HC1 HzO C1 and OH HBr H20 Br for thermal collision energiess1 and to C1 HOD HC1 OD and C1 HzO HCl OH for higher collision energies.52 The form of potential energy surface used was simple and contained a LondonEyring-Polanyi-Sat0 (LEPS) function to describe the bonds breaking and forming in these reactions and also included a very accurate potential for ~ ~The potential 0 . function ~ was ~ parametrized to give quite good agreement with the transition state geometry for the OH HCl reaction calculated at the unrestricted Hartree-Fock (UHF) level. The transition state vibrational frequencies of the potential also agree quite well with the UHF calculation. The OH HBr potential was based on the OH HC1 one and is expected to be much more approximate. Both of these reactions are quite fast: OH HCl appears to have a small classical barrier to reaction17 while OH HBr has no activation energy at a11.18 However, the ab initio calculations suggest that no long-lived complex is formed during these reaction^.^^ Figure 13 shows the -AS1 and e ~ p e r i m e n t a l ' ~rate J~ constants for the OH HC1 and OH HBr reactions. It can be seen that the agreement for OH HC1 is quite good, and the flat temperature dependence of the rate constants is obtained. In the case of the faster OH HBr reaction, the calculated rate constants are slightly below the experimental values, but the strong negative temperature dependence of the rate constants is obtained. The rovibrationally selected dynamics of the OH -I- HX (X = Br, C1) reactions is particularly interesting. Figure 14 shows RBA cross sections for the OH(j,K=O) HCl HZO(m,n) C1 reaction in which m is a quantum number for the bending mode of H20 and n is a local OH stretching mode quantum number of H Z O . ~It ~can be seen that the total reactive cross sections forj=O is much larger than that for j.0. This is true over the whole thermal energy range. It is also seen that the ground vibrational state (0,O) of HzO is the only state with appreciable population out of OH(j=O) while the state (1,O) with one quantum of energy in the H20 bending mode is the most likely state to be formed from OH(j=l,K=O). This can be explained by a plot of the hyperspherical adiabats {Ek(e)} discussed in section 11. The OH HX reactions are "heavylight-heavy" (HLH) reactions involving the transfer of a H atom, and it is known from calculations on atom diatom reactions that these plots are particularly useful in explaining reaction propensities for HLH reactions.63
+
+ +
+
+ +
+
-
+
+
+
+
+
+
+ +
+
+
-
+
+
+
IS
I8
28
15
'9 Cl+H,O(m,n)
p/bohr
-
Figure 15. Plot of the hyperspherical adiabats5' {E&)} for the HC1(v) OH(j,K=O) C1+ HzO(m,n) reaction. Note the avoided crossing between the OH(j=O) and H20(0,0) adiabats in the region of the broken line (the "ridge en erg^''^*.^^).
+
Figure 15 presents a plot of the hyperspherical adiabats { E k -
(e)} for the OH + HC1 reacti~n.~'It can be seen that there is a strongly avoided crossing between the adiabat correlating with HCl OH(j=O) and that correlating with HzO(0,O) C1, and the transition probability between these states is therefore large. Also, the adiabat correlating with HCl OH(j=l,K=O) has an avoided crossing with that labeled by HzO(1,O) C1. As H20(1,O) has one quantum of energy in the H20 bending mode, the (j=1, K=O) state would seem to have an effective activation energy larger than that for j=O by the frequency of the HzO bending vibration at the transition state of the reaction. This "rule" is also found to apply in calculations for other values of j and K the cross sections for j = K are similar to those for j = 0, while those for j > IKl are very small.51 A consequence of this is that the degeneracy-averaged cross sections and rate constants depend approximately on (2j l)-I, and this results in the approximate form for the rate constant51
+
+
+
+
+
where b ( T ) is the rate constant for reaction out of OH(j=O), B is the rotor constant of OH, and k6 is the Boltzmann constant. In OH HC1, h ( T ) has a slightly positive temperature dependence which is matched by the T1I2 term to produce an overall k(T) that is almost independent of t e m ~ e r a t u r e . ~ ~ Furthermore, the ratio for the rate constants for the OH HC1 and OD HC1 reactions is predicted by this formula to be approximately d2, and this is in good agreement with experiment.7s
+
+
+
J. Phys. Chem., Vol. 98, No. 42, 1994 10687
Feature Article
I
10"
I
~ , , , , , , , , , 1 , , , , 1 , , , , 1 , , , , , , , , , 1 ~
50
0
100
150
200
250
300
T IK Figure 16. Fit (unbroken line) of the expression k(T) = A/{&1 exp(-AE/T)]} to the experimental rate constants18(error bars) for the OH HBr reaction. The AE value is 181 K.
+ OH r o t a t i o n a l quantum number
+
+
In the case of OH HBr, the main dependence of kdT) on temperature is due to the electronic partition function of OH, and eq 5 then suggests the simple form for the rate constant of k(T) =
P2[1
+
A exp(-~~/~)l
(6)
where AE is the difference in energy between the 2113/2 and 2111/2 states of OH and A is a constant. Figure 16 presents a fit of this functional form to the experimental rate constants for the OH HBr reaction, and it is seen that it provides rather a good explanation of the strong negative temperature dependence observed in the experimental rate constants.18 The OH HX reactions are excellent examples of where detailed insight into the rovibrational dynamics of a chemical reaction has enabled an important and more macroscopic quantity such as the temperature dependence of a rate constant to be explained. The RBA would seem to be an ideal theoretical tool for enabling this connection to be made. One experiment which is related to this finding on rotational effects in the OH HC1 reaction is the measurement by Crim and co-workers of the rotational distributions for the C1 H20(0,4) HCl OH(j,K) reaction," where (0,4) refers to 4 quanta in the local OH stretching vibration. This experiment found rotational excitation in the OH product which is very low and dies away quickly as the OHV) quantum number increases. This is expected from the above analysis, and the experimental rotational distributions are in quite good agreement with the RBA prediction^^^ (see Figure 17a). Another interesting prediction from the RBA calculations are some small oscillations in the OH rotational product distributions when plotted as a function of j . These are explained by Figure 17b, which shows the rotational distributions of OH for each populated vibrational state v of HC1 product. It can be seen that the most likely OH product states are j = 1 and 4 for v = 3 and 2, respectively, and this produces the small peaks in the v-summed OH rotational distributions at the same values of j . The vibrational state HCl(v=3) is also predicted to be the most likely product state for this reaction, and this is in agreement with e~periment.'~
+
+
-
+
+
+
VI. Conclusions The aim of this article is to show that quantum scattering calculations can now be applied to a variety of important stateselected four-atom reactions. The rotating bond approximation that we have developed has the special feature that it is stateselective explicitly in the vibrational bending and stretching modes of ABC and rotational states of AB in the reaction ABC D AB CD. This enables many detailed comparisons to be carried out with experimental results on the dynamics of
+
-
+
OH r o t a t i o n a l quantum number
-
Figure 17. (a) RBA (straight lines)52and experimentallS (rhombus) OH rotational product distributions for the C1 H20(0,4) HCl OHV) reaction at an initial translational energy of 0.16 eV. To model the experimental conditions,the calculations are for rotational projection quantum number K = 1 and, therefore, j = 0 is not populated in the RBA computations. Here, (0.4) refers to four quanta in the OH local stretching mode of H20.(b) As for (a) but the cross sections for reaction into OHV) for each of the HCI(v) states are shown.
+
+
four-atom reactions. We have demonstrated this through applications to several important examples including H H20, OH CO, OH HC1, and OH HBr. In all cases, RBA calculations on the rovibrational dynamics have given new insight into these systems that range from slow to complexforming to fast reactions. Furthermore, the RBA is not very expensive computationally. However, the method does have some limitations, which have been discussed. One of the other useful contributions from these types of calculations is to demonstrate that quantum scattering calculations, albeit approximate, are now possible for reactions more complicated than atom diatom. This has stimulated the development of new quantum theories that are now capable of giving exact rate constants from a given potential energy surface for a reaction such as OH H2. In tum,these accurate quantum theories have been very useful in establishing the accuracy of the RBA. It seems very likely that exact quantum scattering calculations on the reaction OH H2 H2O H will soon be possible for selected rovibrational states of both reactants and products. This suggests there is now an urgent need for the best quantum chemistry methods to be applied to the calculation of very accurate potential energy surfaces for fouratom reactions such as this. It will then be possible to make predictions that will compete with experiment in their accuracy. Furthermore, the RBA can also be applied to reactions containing more than four atoms (e.g., OH C& H2O CH3).53 Clearly, quantum reactive scattering calculations have now gone
+
+
+
+
+
+
-
+
+
+
-
+
10688 J. Phys. Chem., Vol. 98, No. 42, 1994
+
well beyond atom diatom problems, and the future for this field is very promising.
Acknowledgment. This work has been supported by the Science and Engineering Research Council and the European Community. The author is very grateful to several colleagues who have made major contributions to the work described in this article including Julian Echave, Gunnar Nyman, Marta Hernandez, and Ramon Hernandez. References and Notes (1) Crim, F. F.; Hsiao, M. C.; Scott, J. L.; Sinha, A.; Vander Wal, R. L. Philos. Trans. R. Soc. London, Ser. A 1990, 332, 259. (2) Bronikowski, M. I.; Simpson, W. R.; Girard, B.; Zare, R. N. J. Chem. Phys. 1991, 95, 8647. Adelman, D. E.; Filseth, S. V.; Zare, R. N. J. Chem. Phys. 1993, 98, 4636. (3) Schatz, G. C. J. Chem. Phys. 1979, 71, 542. (4) Clary, D. C.; Echave, J. In Advances in Molecular Vibrations and Collision Dynamics: Bowman, J. M., Ed.; JAI: Greenwich, 1994; Vol2A, p 203. (5) Wang, D.; Bowman, J. M. In Advances in Molecular Vibrations and Collision Dynamics, Bowman, J. M., Ed.; JAI: Greenwich, 1994: Vol. 2B, p 187. (6) Casavecchia, P.; Balucani, N.; Volpi, G. G. Research in Chemical Kinetics, Compton, R. G.; Hancock, G., Eds.; Elsevier: Amsterdam, 1994; Vol 1, p 1. (7) Smith, I. W. M. J. Chem. Soc, Faraday Trans. 1991, 87, 2271. (8) Creighton, J. R. J. Phys. Chem. 1977, 81, 2520. (9) Millar, T. J.; Williams, D. A,, Eds. Rate Coefjcients in Astrochemistry; Kluwer; Dordrecht, 1988. (10) Tully, F. P.; Ravishankara, A. R. J. Phys. Chem. 1980, 84, 3126. Ravishankara, A. R.; Nicovich, J. M.; Thompson, R. L.; Tully, F. P. J. Phys. Chem. 1981, 85, 2498. Smith, I. W. M.; Zellner, R. J. Chem. Soc., Faraday Trans. 2 1974, 70, 1045. (1 1) Alagia, M.; Balucani, N.; Casavecchia, P.; Stranges, D.; Volpi, G. G. J. Chem. Phys. 1993, 98, 2459. (12) Sinha, A.; Hsiao, M. C.; Crim, F. F. J. Chem. Phys. 1990,92,6333. Sinha, A.; Hsiao, M. C.; Crim, F. F. J. Chem. Phys. 1991, 94,4928. Hsiao, M. C.; Sinha, A.; Crim, F. F. J. Phys. Chem. 1991, 95, 8263. (13) Bronikowski, M. J.; Simpson, W. R.; Zare, R. N. J. Phys. Chem. 1993, 97, 2194, 2204. (14) Jacobs, A,; Volpp, H.-R.; Wolfrum, J. Chem. Phys. Lett. 1992,196, 249. Jacobs, A,; Volpp, H.-R.; Wolfrum, J. J. Chem. Phys. 1994,100, 1936. (15) Sinha, A.; Thoemke, J. D.; Crim, F. F. J. Chem. Phys. 1992, 96, 372. Crim, F. F. Private communication. Metz, R. B.; Pfeiffer, J. M.; Thoemke, J. D.; Crim, F. F. Chem. Phys. Lett. 1994, 221, 347. (16) Wayne, R. P. Chemistry of the Atmospheres, Oxford University Press: London, 1991. (17) Sharkey, P.; Smith, I. W. M. J. Chem. Soc., Faraday Trans. 1993, 89, 631. (18) Sims, I. R.; Smith, I. W. M.; Clary, D. C.; Bocherel, P.; Rowe, B. R. J. Chem Phys. 1994, 101, 1748. (19) Schatz, G. C.; Fitzcharles, M. S.; Harding, L. B. Faraday Discuss. Chem. Soc. 1987, 84, 359. (20) Warnatz, J. Combustion Chemistry; Gardiner, W. C., Jr., Ed.; Springer-Verlag: New York, 1984; p 197. (21) Smith, I. W. M.; Zellner, R. J. Chem. Soc., Faraday Trans. 2 1973, 69, 1617. Gardiner, W. C., Jr. Acc. Chem. Res. 1977,10,326. Ravishankara, R. R.; Thompson, R. L. Chem. Phys. Lett. 1983, 99, 377. Mozurkewich, M.; Lamb, J. J.; Benson, S. W. J. Phys. Chem. 1984,88,6429,6435. Jonah, C. D.; Mulac, W. A.; Zeglinski, P. J. Phys. Chem. 1984, 88,4100. Baulch, D. L.; Cox, R. A.; Hampson, R. F., Jr.; Kerr, J. A,; Troe, J.; Watson, R. T. J. Phvs. Chem. Ref: Data 1984. 13. 1359. (2.2) Frost, M. j . ; Sharkey, P:; S h t h , I. W. M. Faraday Discuss. Chem. Soc. 1991, 91, 305. (23) Alagia, M.; Balucani, N.; Casavecchia, P.; Stranges, D.; Volpi, G. G. J. Chem. Phys. 1993, 98, 8341. (24) Rice, J. K.: Baronavsky, A. P. J. Chem. Phys. 1991, 94, 1006. Nickolaisen, S. L.; Cartland, H. E.; Wittig, C. J. Chem. Phys. 1992, 96, 4378. Quick, C. R.; Tiee, J. J. Chem. Phys. Lett. 1983, 100, 223. (25) Scherer, N. F.; Khundkar, L. R.; Bernstein, R. B.; Zewail, A. H. J. Chem. Phys. 1987, 87, 1451. Scherer, N. F.; Sipes, C.; Bemstein, R. B.; Zewail, A. H. J. Chem. Phys. 1990, 92, 5239. (26) Ionov, S. I.; Brucker, G. A.; Jaques, C.; Valachovic, L.; Wittig, C. J. Chem. Phys. 1992, 97, 9486. Jaques, C.; Valachovic, L.; Ionov, S.;
Bohmer, E.; Wen, Y.; Segall, J.; Wittig, C. J. Chem. Sor. Faraday Trans. 1993, 89, 1419. (27) Walch, S. P.; Dunning, T. H. J . Chem. Phys. 1980, 72, 1303. (28) Schatz, G. C.; Elgersma, H. Chem. Phys. Lett. 1980, 73, 21. (29) Schatz, G. C. J. Chem. Phys. 1981, 74, 1133. (30) Schatz, G. C.; Colton, M. C.; Grant, J. L. J. Phys. Chem. 1984,88, 297 1. (31) Kudla, K.; Schatz, G. C. ChemPhys. Lett. 1992, 193, 507. Kudla, K.; Schatz, G . C. J. Chem. Phys. 1993,98,4644. Kudla, K.; Schatz, G. C. Chem. Phys. 1993, 175, 71. (32) Kudla, K.; Schatz, G. C.; Wagner, A. F. J. Chem. Phys. 1991, 95, 1635. (33) Rashed, 0.;Brown, N. J. J. Chem.Phys. 1985, 82, 5506. (34) Hanison, J. A.; Mayne, H. R. J. Chem. Phys. 1988, 88, 7424. (35) Clary, D. C., Ed. The Theory of Chemical Reaction Dynamics; Reidel: Dordrecht, 1986. (36) Miller, W. H. Annu. Rev. Phys. Chem. 1990, 41, 245. (37) Manolopoulos, D. E.; Clary, D. C. Annu. Rep. C. R. SOC. Chem. 1989, 86, 95. (38) Truhlar, D. G.; Isaacson, A. D. J. Chem. Phys. 1982, 77, 3516. (39) Isaacson, A. D. J. Phys. Chem. 1992, 96, 531. (40) Schatz, G. C.; Walch, S. P. J. Chem. Phys. 1980, 72, 776. (41) Cohen, M. J.; Willetts, A.; Handy, N. C. J. Chem. Phys. 1993, 99, 5885. (42) Brooks, A. N.; Clary, D. C. J. Chem. Phys. 1990,92,4178. Brooks, A. N. Ph.D. Thesis, University of Cambridge, 1989. (43) Sun, Q.;Bowman, J. M. lnt. J. Quantum Chem. Symp. 1989, 23, 115. Sun, Q.; Bowman, J. M. J. Chem. Phys. 1990, 92, 5201. (44) Sun, Q.; Yang, D. L.; Wang, N. S.; Bowman, J. M.; Lin, M. C. J. Chem. Phys. 1990, 93, 4730. (45) Bowman, J. M.; Wagner, A. F. The Theory of Chemical Reaction Dynamics, Clary, D. C., Ed.; Reidel: Dordrecht, 1986; p 47. Bowman, J. M. J. Phys. Chem. 1991, 95, 4960. (46) Clary, D. C. J. Chem. Phys. 1991, 95, 7298. (47) Clary, D. C. J. Chem. Phys. 1992, 96, 3656. (48) Nyman, G.; Clary, D. C. J. Chem. Phys. 1993, 99, 7774. (49) Clary, D. C. Chem. Phys. Lett. 1992, 192, 34. (50) Clary, D. C.; Schatz, G. C. J. Chem. Phys. 1993, 99, 4578. (51) Clary, D. C.; Nyman, G.; Hernandez, R. J. Chem. Phys. 1994,101, 3704. (52) Nyman, G.; Clary, D. C. J. Chem. Phys. 1994, 100, 3556. (53) Nyman, G.; Clary, D. C. J. Chem. Phys., in press. (54) Echave, J.; Clary, D. C. J. Chem. Phys. 1994, 100, 402. (55) Wang, D.; Bowman, J. M. J. Chem. Phys. 1992,96,8906.Bowman, J. M.: Wane. D. J. Chem. Phvs. 1992.96.7852. Wane. D.: Bowman. J. M. J. Chem. Piys. 1993, 98, 6235. (56) Szichman, H.: Last, I.: Baer, M. J. Phvs. Chem. 1994, 98, 828. Szichman, H.; Baer, M. J. Chem. Phys. 1994, i O l , 2081. (57) Zhang, D. H.; Zhang, J. Z. H. J. Chem. Phys. 1993, 99, 5615. (58) Zhang, D. H.; Zhang, J. Z. H. J. Chem. Phys. 1994, 100, 2697. (59) Manthe, U.;Seideman, T.; Miller, W. H. J. Chem. Phys. 1993,99, 10078. (60) Neuhauser, D. J. Chem. Phys. 1994, 100, 9272. (61) Child, M. S. Mol. Phys. 1967, 12,401. Connor, J. N. L.; Child, M. S. Mol. Phys. 1970, 18, 653. Wyatt, R. E. J. Chem. Phys. 1969, 51, 3489. (62) Walker, R. B.; Hayes, E. F. The Theory of Chemical Reaction Dynamics; Clary, D. C., Ed.; Reidel: Dordrecht, 1986; p 105. (63) Ohsaki, A.; Nakamura, H.Phys. Rep. 1990, 187, 1. (64) Kuppermann, A. Chem. Phys. Lett. 1975, 32, 374. Kuppemann, A.; Kaye, J. A,; Dwyer, J. P. Chem. Phys. Lett. 1980, 74, 257. Romelt, J. Chem. Phys. Lett. 1980, 74, 263. Romelt, J. The Theory of Chemical Reaction Dynamics; Clary, D. C., Ed.; Reidel: Dordrecht, 1986; p 77. (65) Clary, D. C. Mol. Phys. 1984, 53, 3. (66) The formula in eq 25 of ref 46 is incorrect. As a result, the calculated rate constants of Table IV and Figure 9 of that paper are in error. Equation 4 and Figure 5 of the present paper give the corrected results. (67) Wilson, E. B.; Decius, I. C.; Cross, P. C. Molecular Vibrations: McGraw-Hill: New York, 1955. (68) Bradley, K. S.; Schatz, G. C. J. Phys. Chem. 1994, 98, 3788. (69) Kessler, K.; Kleinemanns, K. Chem. Phys. Lett. 1992, 190, 145. (70) Wang, D.; Bowman, J. M. Chem. Phys. Lett. 1993, 207, 227. (71) Schatz, G. C.; Dyck, J. Chem. Phys. Lett. 1992, 188, 11. (72) Hernindez, M. I.; Clary, D. C. J. Chem. Phys. 1994, 101, 2779. (73) Mandelshtam, V. A,; Ravuri, T. R.; Taylor, H. S. Phys. Rev. Lett. 1993, 70, 1932. (74) Murrell, J. N.; Carter, S. J. Phys. Chem. 1984, 88, 4887. (75) Smith, I. W. M.; Williams, M. D. J. Chem. Soc., Faraday Trans. 2 1986, 82, 1043.
-
