The Journal of
Physical Chemistry ~~
~~
VOLUME 99, NUMBER 30, JULY 27,1995
0 Copyright 1995 by the American Chemical Society
ARTICLES Simulation of Effective Vibrational-Translational Energy Exchange in Collisions of Vibrationally Excited OH with 0 2 on the Model Potential Energy Surface. Can the Relaxation of OH@)Be One-Quantum for Low and Multiquantum for High v? Dmitrii V. Shalashilin*Jy*and Alexandre V. Michtchenko Institute of Chemistry, National University of Mexico, Coyoachn, 04510, Mexico D.F., Mexico
Stanislav Ya. Umanskii and Yulii M. Gershenzon Institute of Chemical Physics, Russian Academy of Science, I1 7977, Kosygina 4, Moscow, Russian Federation Received: September 28, 1994; In Final Form: May 23, 1995@
-
The collision of OH with the oxygen molecule is studied by the trajectory simulation technique on the model potential energy surface of OH 0 2 03 H chemical reaction. Although the reaction channel is closed, we aim to demonstrate that the L-shape of the OH 0 2 valley leads to the effective coupling of OH(v) vibration with the relative motion of collisional partners and therefore explains the high value of the vibrational relaxation rate constant observed experimentally. The characteristic feature of the mechanism considered is the predominance of one-quantum relaxation for low and multiquantum transitions for high OH vibrational levels. To estimate state-to-state vibrational relaxation rate constants, the method of dynamical corrections of transition state theory is used. The expression for the rate constant consists of a transition state term and a correction factor, determined in two-dimensional classical trajectory calculations. We also demonstrate the instability of motion on the potential energy surface with the L-shape valley, resulting from the scattering of the trajectory on the “comer” of the potential energy surface and the presence of regular and chaotic motions.
+
+
1. Introduction
The vibrational relaxation of OH in collisions with various collisional partners was the subject of a number of recent experimental and theoretical ~tudies.I-~The importance of this problem is related to the role of OH for the processes occurring in the meso~phere,~-~ in chemical lasers,I0 and in liquids and crystals. Moreover, an understanding of the mechanisms of relaxation is of serious interest for the theory of elementary proces~es.’~,’~ The physics of OH vibrational relaxation is very Permanent address: Institute of Chemical Physics, Russian Academy of Science, 117977, Kosygina 4, Moscow, Russian Federation. Present address: Department of Chemistry, Oklahoma State University, Stillwater, OK 74078-0447, U.S. Abstract published in Advance ACS Abstracts, July 1, 1995.
*
@
0022-365419512099-11627$09.00/0
+
rich. To reconcile with the experimental data, a number of mechanisms should be involved The vibrational energy transfer in collisions of OH with Ar and N2 has been analyzed p r e v i ~ u s l y . ~ -It~ was shown that the competition of vibrational-to-rotational (VR)energy transfer and electronically nonadiabatic mechanism (NA) must be taken into account when considering these systems. Both mechanisms are well known. For example, VR exchange determines relaxation of HC1 and HF.I4 Electronically nonadiabatic mechanism has been DroDosed . . to describe relaxation of N o i 5 This paper focuses on the theory of the dynamics of collision of OH with 0 2
OWvJ
-
+ 02(v2)
O W v , - AvJ
0 1995 American Chemical Society
+ 02(v;)
(1)
11628 J. Phys. Chem., Vol. 99, No. 30, 1995
Shalashilin et al.
The experimental information available about the OH-02 system demonstrates a significant difference between the physics of vibrational energy transfer in OH-02 and that considered previously for the OH-Nz colli~ion.~ OH(v, = 1-9)
+ N2(u2= 0) -.
2.5
OH(v; = u1 - 1)
+ N2(v2= 1) (2)
For process 2 the rate constant of the VR mechanism was concluded to exceed the rate constant of the nonadiabatic one in ref 7. Recent reexaminationI6of the VR and NA competition for a new potential energy surface1’ confirms this result. The predominance of VR relaxation in the OH-Nz system is the consequence of a high activation energy of the curve-crossing point, responsible for the NA mechanism, and a small transition probability in this point. Finally, the rate constant of process 2 decreases sharply with an increase of energy mismatch, hAw, being proportional to the factor
k
-
where I is the moment inertia of OH and yo is an interaction parameter. Such an expression of the rate constant is a characteristic feature of the VR For Aul = 1, v2 = 0 and u; = 1 the energy mismatch of process 1 is 779 cm-’ greater, but the experimental rate constant of OH-02 vibrational energy transfer is several orders of magnitude higher than that of (2). Thus, the VR energy transfer theory applied successfully to OH-N2 cannot reproduce the rate constant order of magnitude for the OH-02 system. The arguments for predominance of the VR to NA mechanism remain valid for OH-02, and the high efficiency of process 1 observed experimentally in refs 1-4 cannot be explained by the electronically nonadiabatic mechanism. In addition, the relaxation of a highly excited OH has qualitative preculiarities. The analysis of the ground-based observations demonstrates that the OH Meinel emission in the upper atmosphere can be described only if multiquantum relaxation of OH(v = 7-9) in collisions with 0 2 is i n v ~ l v e d . ~ In ref 1 the relaxation of OH(u = 1-6) was investigated experimentally. Possible multiquantum transitions were taken into account in the kinetic analyses of the experiment, and the following rate constant dependence on Aul was introduced.
-
I/(AvJ
With this assumption the experimental kinetic curves’ where shown to be described by n >> 1. This fact was interpreted as a predominance of one-quantum transitions. The relaxation of the levels with u = 1-6 was supposed to be by one-quantum transitions in the experiments as well as in the kinetic scheme for gound-based observation^.^ In order to explain the above mentioned features of the vibrational energy transfer in the OH-02 system, the following mechanism is proposed. Both OH and 0 2 particles have unpaired electrons, and the chemical reaction 0 2
+ OH(U)
0,
+H
4
I
0.5 VOH
=0
0.0
-1.0
q=X**Z-y**Z
(A**Z)
-1.5
Figure 1. Reaction path profiles used for the construction of PES for reaction 3. Curves 1 and 2 are according to ref 24, and curve 3 is the averaged profile.
e~p(-3(ho~Iy~/2kT)‘/~)
k
2.0
1
(3)
is possible. This reaction is highly endothermic for v < 9, and its rate is very small. But in the OH-02 collisions the OH bond is elongated. This leads to the strong coupling of vibrational and translational motions even if the reaction channel is closed. Such a mechanism was previously used for the description of vibrational relaxation of a molecule colliding with potentially reactive open shell atoms and in radical-radical
collisions.I2 In contrast to the OH-N2 system, the mechanism that we suppose is responsible for relaxation in OH-02 is specific for chemically active particles, in particular for molecular radicals with open electronic shells. It can be named as the mechanism of aborted chemical reaction (ACR). The ACR mechanism proposed is closely connected with the general properties of chemical reaction potential energy surfaces (PES). According to the Polanyi rule, known also as a Hammond p o s t ~ l a t e ,the ’ ~ saddle ~ ~ ~ point on the PES is shifted into the shallow valley. As a result, the deep valley (the OH0 2 valley in the system considered) has a L-shape. According to refs 19 and 20, such an asymmetric structure of PES leads to vibrational excitation of exothennic reaction products, in particular, to OH(v) vibrational excitation in the exothermic reaction
(4) observed in the The ACR mechanism considered reproduces the high efficiency of process 1 and a predominance of one-quantum transitions for low vibrational levels of OH but multiquantum transitions for high levels. We have also analyzed a possible correlation of the vibrational distributions for the chemical reaction and for the relaxation. This correlation could be expected because the same PES determines processes 1 and 4. Moreover, the energy of OH vibrational levels v = 8, 9 is close to the energy of the ground state H 03 system (see Figure 1). It will be shown later that this correlation is only partial due to the appearance of regular and chaotic motion on the PES. The relaxation mechanisms connected with the shape of the valley were analyzed previously for Br HC1.I2 For this system the relaxation was less effective than in our case because the bend of the valley was more smooth. Recently, the mechanism of “transient chemical bond formation” has been proposed in ref 13 to explain the results of their experiments on the self-relaxation of highly vibrationally excited NO and 0 2 . In particular, the role of the shift of the transition state into the 0 0 3 valley and “late barrier” of the reaction 0 2 0 2 0 0 3 has been assumed to explain the sharp increase of self-relaxation and multiquantum transitions for high u in this system. In section 2, on the basis of ab initio calculation^,^^ we construct the two-dimensional model PES with a L-shape OH-
+
+
-
+ +
+
Energy Exchange between OH and 0
J. Pkys. Chem., Vol. 99, No. 30, 1995 11629
2
valley and describe classical trajectory calculations on it. In section 3 the method of dynamical corrections to the transition state theory, developed in refs 25-28, is applied to the rate constant evaluations. This approach incorporates the onedimensional quantum or two-dimensional classical calculations to the transition state theory. The modeling approach cannot provide a quantitative description of the process. Nevertheless, (1) it reproduces the rate constant order of magnitude, which cannot be done by other theories of vibrational relaxation and (2) it explains the predominance of one-quantum relaxation for low and of multiquantum relaxation for high v. Unfortunately, the details of PES for HO3 are not known, but recent consideration of the OH CO H C02 chemical reaction29shows the “late barrier” and sharp bend of the OH CO valley. The well-known structure of the HO CO PES confirms indirectly the idea of the present work. The information available about the OH-02 PES is not sufficient to obtain the quantitative agreement with the experimental rate constants. But here, we have used a flexible analytical expression for the PES. We subsequently vary the PES parameters and reproduce experimental results for processes 1 and 4. Nevertheless, realization of this methodology appears to be very difficult for the system considered. The main reason is the instability of motion on the potential energy surface with a L-shape valley, resulting from the scattering of the trajectory on the “comer” of PES and the presence of regular and chaotic motions. The demonstration of this phenomenon is also the goal of the present study. 0 2
+
-
+
+
+
2. Potential Energy Surface and Collision Dynamics The starting point in the PES construction is the paper24which reports the quantum chemical investigation of the reaction path for processes 3 and 4. It was shown that in reaction 4 the H atom attacks the ozone molecule perpendicular to the 0 3 plane, H
I
o-o+o-o-o-o
I
H
-0-0-0
q
(5)
H
TABLE 1: Coordmates and Energies of Points on the Reaction Path Calculated in Reference 24“
~~~~
~
xo= r - re [24]
0.889 -0.038 0.0130 (A) (1.5) (0.889) (0.0081) (0.0012) (0.0009) y = R - [24] 0.002 0.1131 1.0931 (4 (0.0009) (0.0016) (0.1731) (1.0931) (1.5) U , profile 1, [24] (kcaymol) 0 20 15 19.1 71 U , profile 2, [24] (kcaymol) 0 2.6 -13.8 19.7 17 U , profile 3 (kcaUmol) 0 0 0 19.1 71
The corrected values of x and y used in this work are in brackets. R and r are shown in ( 5 ) . Re and re are their equilibrium values. describes the reaction path. The hyperbolic coordinate grid (x - u)O, - u ) = ~0 was introduced. The value of u characterizes the shift of the hyperbole with respect to the reaction path xy = m. After that we calculated new coordinates u(x,y) and q(x,y) from the following equations (x (x
- u)o, - u ) = uo
(6)
- u)2 - o, - u)2 = q
--
The value of q plays a role of the reaction coordinate. In the HO 0 2 and H 0 3 valleys q tends to and respectively. That is why it is mathematically convenient to use it for PES fitting.30-32 Parameter ~0 determines the curvature of the reaction path in the region of its bend. It was taken in accordance with ref 24 as
+
u,, = (x,y,
+
+ xcyc + x9y,)/3
= 1.41
+-,
A2
(7)
Coordinate u characterizes the shift of the point (x,y) = (q,u) with respect to the reaction path xy = m. In 02-OH and 03-H valleys it corresponds to the vibrational coordinates of OH and 0 3 . We use the following expression for the PES30-32
W,Y) = F(q) + D(q){exp(-2(a(q))u) - 2 exp(-a(q>u)l
R
The 0-0 bond in the direct and inverse reactions can be considered as frozen (vi = v2 in (l)),and two coordinates, R = RQ-O and r = rOH, describe the collision if the angle y of the O-H bond with respect to the 0 3 plane is fixed ( y = d 2 ) . The coordinates of the saddle and minima points along the reaction path calculated in ref 24 are presented in Table 1, as well as two sets of energies at these points taken from the ref 24. Two potential energy profiles, curves 1 and 2, reported in ref 24 are shown in Figure 1 as a function of the reaction coordinate (see below). Due to the great difference between curves 1 and 2, we used an “averaged” model profile 3 without the H03 complex and a saddle point between HO 0 2 and HO3. For all three profiles the energy of H 0 3 and H-O3 potential energy barrier were taken from ref 24. When fitting the PES of a reactive system, we used the idea in refs 30 and 31 to introduce a new set of coordinates convenient for PES representation, instead of the Cartesian coordinates x and y, and used an analytical expression for PES with these coordinates. The m e t h ~ d ~is~effective ,~’ if only a few points on the reaction path along with vibrational frequencies of reagents and products are known. Within the accuracy of quantum chemistry calculation the coordinates of the saddle and minima points were slightly corrected in order to satisfy the equation ( r - re)(R - Re) = xy = uo, where re and Re are equilibrium bond lengths in OH and ozone. This equation
+
+
(8) Function F(q) describes the reaction path profile (see Figure 1). It was fitted by cubic spline. The expressions for D(q) and a(q)are as follows
(9) Thus, our PES reproduces parameters of OH and 0 3 vibrations in HO-02 and H-O3 valleys taken from refs 33 and 34. B is a changing-over constant. The PES also fits the energies at points B = (HO-02), C = (HO3), and D = (H-03) on the reaction path (see Table 1 and Figures 1 and 2a). Its equipotential lines are shown in Figure 2. The energies of these lines correspond to the energies of OH (v = 3, 6, 9, 12) levels. The characteristic features of the model PES obtained are the sharp bend of the reaction path in the HO 0 2 valley and its L-shape. This follows from the structure of the H-03 transition state with a strongly elongated O-H bond, calculated in ref 24. To describe the dynamics of collision on the PES the classical trajectory simulation was used. Due to the low mass of the H atom, the OH center mass is close to the mass of the 0 atom.
+
11630 J. Phys. Chem., Vol. 99, No. 30, 1995
a
3.50
1Y ( Q
trajectory .
0 2
+
12
OH(v) ,v=3
2.50
Shalashilin et al.
Px=-,
au
au
1
b y = - - ?Y x = - Px
H
1.50
PO-H
y=-
PY P02-OH
0.50
-
03+H xtil ~~
0.50
\--,
2.50
1.50
+
+
J
12
OH(v)
6 I I
v=3
2.50
These equations describe the collisions of nonrotating OH with 0 2 . The OH bond is supposed to be perpendicular to 0 3 plane. According to ref 24 this geometry is optimal for chemical reaction 4 and for the ACR mechanism of relaxation. The integration of (10) was started at y, = 2.5 A and was stopped after the collision (yf = 2.5 A). The initial vibrational energy E, = p:/2p0-, V(x,,y,)was taken to be equal to the energy of the quantum state OH(v).
I
E,, = p0+Xf/2
0
+ U(x,,y,)= h[wOH(v, + 112) + X,@OH(V, + 1/2)2] (11)
For any initial vibrational and translational energy E, = 400 trajectories with different initial vibrational pX/2po2-,, 2 phases were calculated. The final vibrational quantum number vf was prescribed to the trajectory with final vibrational energy Ef within the interval
H
{E,f - (E”,- E,f-,)/21 < Ef < {E,f + (E,,+, - E ” p l (12) 0 3+ -0.5OL
-0.50
. . . . . . .
3
.
7
0.50
H
u t8) --,--,
.............................. I
.so
2.50
-
The relative number of these trajectories determines the probability of the v, vf transition
P”,,r 0.70
Vf
Here, Ea is the activation energy of a transition state. Finally, for the state-to-state rate constant we have
0.60 n
0.50
x
t
0.40
6
where is the partition function of the transition state from the free particles 0 2 OH to the H03 collisional complex and Fo2and FOHare those for free 0 2 and OH. The transition state is represented by the point B in Figure 2a and is marked as HO-02 in Figure 1. The parameters of HO3 vibrations are not known. Nevertheless, we can estimate its partition function reliably and with understandable accuracy on the basis of the experimental rate constanP
+
A = 1.4 x
kreact= A exp(-E,,JT),
cm3/sec, Eeff= 480 K (17)
of the chemical reaction
0,
+ H - 0 2 + OH(v)
In Appendix 1, we obtain the following expression for the rate constant kv,vf= A e-n(T/T)na/T1‘2(PV,v) Here, a = 1.879 K’’2, = 300 K, and -1/2 < n < 312. Variation of n results from uncertainty of 03-H transition state parameters, and formula 18 provides an estimation of the rate constant by the order of magnitude. Poor information about PES is the main source of error. In such a situation use of the model PES and simplified approach to dynamics appear justified. A more accurate approach will require threedimensional dynamical calculations and more detailed information about the PES that is still not available. If the PES is not known, the gain in accuracy of a three-dimensional calculation will be compensated by the uncertainty of necessarily introduced additional parameters of PES.
4. Results and Discussion 4.1. Rate Constant Estimations. Typical trajectories on model PES are presented in Figure 2. These demonstrate an effective vibrational-translational energy exchange. For all the trajectories the initial vibrational quantum number vi = 6. The averaged transition probabilities (PvIvf) for vi = 2, 3,4, 6, 9 are shown in Figure 3. Figure 4 represents relaxation rate constants VI
kv, = c k v , v- S , V , v=o
as well as experimental rate constants and the rate constants of
0
- v=2 - v=3
-
v=4 v=6 v=9
0.30 0.20 0.10 V’
0.00
0 Figure 3. Vibrational distributions (P,,,,,,) of products of the relaxation (1).
VR exchange in process 1 for AVI= 1, v2 = 0, vi = 1 and AVI = 1, v2 = 0, vi = 2, estimated by theory7-I4(see Appendix 2). The distance between curves 1 and 2, obtained by eqs 18 and 19 for n = - 1/2 and n = 312, characterizes the error of model dynamics due to poor infomation about vibrational modes of the transition state. The discrepancy with experiment for low levels is a consequence of poor information about the two-dimensional PES. In contrast to the VR relaxation (curves 4 and 5 in Figure 4) the mechanism proposed explains qualitatively the high efficiency of vibrational relaxation of OH in collision with the oxygen molecule. Table 2 shows the ratio kv,v,-l/kv,which characterizes the contribution of one-quantum relaxation. Figure 3 and Table 2 demonstrate one-quantum transitions to dominate for the levels vi = 2, 3. For the higher levels probabilities of multiquantum transitions are comparable or are greater. Thus, within this mechanism the tendency for one-quantum relaxation to dominate for the lower levels and multiquantum transitions for higher OH vibrational levels is reproduced as well as the rate constant order of magnitude. According to ref 9, the multiquantum transitions are effective for the higher levels (v = 7-9) than those in our calculations (v = 4-9). The probable reason for quantitative discrepancies is discussed below. It is not surprising that multiquantum vibrational relaxation is important if strong coupling of a vibrational mode with the rest of the degrees of freedom takes place. The challenge for the theory is to describe the physical reason for coupling. For example, the attractive interaction has been found to be responsible for multiquantum vibrational transitions in HF colliding with C0.36 There is no dipole-dipole interaction in the OH-02 system, and it is not the potential well but rather the bend of the valley that provides strong coupling. The “late barrier” on the PES of chemical reactions 0 2 0 2 0 3 0 and NO NO NO2 N has been assumed to explain the multiquantum transitions observed in the experiment^.'^ For
+
-
+
+
-
+
11632 J. Phys. Chem., Vol. 99, No. 30, 1995
Shalashilin et al.
k( cm**3/sec) 10
-'Dl
0.80
0.60
4
w w
0.40
\
Q 0.20 0.00
Figure 4. Theoretical and experimental rate constants. Curves 1 and 2 are our calculations by formulas 18 and 19 with n = -1/2 and n =
-0.20
3/2. Position 3 represents the experimental rate constants.],* Curves 4 and 5 are rate constants of VR energy exchange for processes (A17) and (A18).
-0.40
8
0.00
*ooi
,
0.20
0.40
0.60
0.80
1 .oo
'P/m Figure 6. Dependence of vibrational energy transferred to the translational degree of freedom on the phase of OH vibration. A& is
180
the region of chaotic motion.
called Sinai's billiard, see, e.g., refs 40 and 41 where trajectories are unstable and chaos appears. The value of A& is estimated as
-
0 0.00
,
'
1 0.20
0.40
0.60
0.80
1 .oo
P/2n Figure 5. Dependence of the collisional time on the initial phase of OH vibration. A& is the region of chaotic motion.
+
these systems and for OH 0 2 , considered in this work, the reason for strong coupling appears to be the same. Previously, the relaxation mechanisms connected with the shape of the valley were analyzed for H2 H (see, for example, a review in ref 39) and for Br HC1.I2 For these systems the relaxation was less effective than in our case because the bend of the valley was more smooth. 4.2. Characteristic Features of Motion on the PES with L-Shape Valley and Test of the PES. The motion on the model PES with a tum has many interesting peculiarities. Figures 5 and 6 demonstrate a dependence of the collisional time and energy transfer on the initial phase of OH vibration. We can see two different types of motion, which can be interpreted as regular and chaotic. A typical trajectory of the former motion is depicted in Figure 2a. The initial phase of the latter is in the interval A& in Figures 5 and 6. The trajectory of chaotic motion is shown in Figure 2b. Figure 2c demonstrates the origin of instability and chaotic properties of motion. After running across the region of instability, marked in Figure 2c, the trajectories start to diverge. All chaotic trajectories have a turning point in this region where the equipotential line has a bend. The motion here resembles the reflection from the surface with a negative curvature (billiard with convex walls or the so-
+
+
-=A& a
