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Comparison of reduced dimensionality and accurate cumulative reaction probabilities for O(3P)+H2(ν=0, 1)
Volume I4 1,number 6
CHEMICAL PHYSICS LETTERS
27 November 1987
COMPARISON OF REDUCED DIMENSIONALITY AND ACCURATE CUMULATIVE REACTION PROBABILITIES FOR 0 (3P) +H,( v= 0,l) * Joel M. BOWMAN Department of Chemistry, Emory University,Atlanta, GA 30322, USA Received 3 August 1987; in final form 2 September 1987
Reduced dimensionality (CEQBIG) cumulative reaction probabilities are compared with exact quantum ones for the O(3P) +H2(v=0, 1) reaction for zero total angular momentum. The semi-empirical LEPS surface of Johnson and Winter as modified by Schatt is used in these calculations.
1. Introduction We have developed an approximate three-dimensional theory of atom plus diatom reactive scattering based on an adiabatic treatment of the bending motion in the vicinity of the transition state [ 1,2]. This approximation effectively decouples the internal angular motion from the motion of the two bond displacements, which are fully coupled. Thus, the dimensionality of the space where explicit coupling is considered is reduced to two. For zero total angular momentum, J=O, the Schriidinger equation describing the reactive scattering looks very much like the one for collinear scattering with, however, an effective potential given by the reference potential (in the two stretch coordinates) plus the adiabatic bending eigenvalue, which depends on the two stretch coordinates. In a particularly useful version of the theory developed for collinear reaction paths, the reaction probability for excited bending states is related to the one for the ground bend state by a simple energy shift: the acronym for this theory is CEQB/G. The total reaction probability summed over all bending states for a given initial and final vibrational state is the cumulative reaction probability [ 11, i.e. the sum of state-to-state probabilities, as shown below. The CEQB/G theory has been tested against * Work supported in part by the Department of Energy (DEFGOS-86ER13568).
accurate three-dimensional quantum calculations for H+H2( v=O, u= 1) and found to be quite accurate [ 11. This and a simpler version of it have been applied to numerous reactions, most recently to 0( 3P) +Hz, DZ and HD 13-61. Among the potential energy surfaces employed in these studies was the semiempirical surface due to Johnson and Winter [ 71. That surface, with a slight modification, was recently used in coupled-states distorted wave [ 81 and exact quantum [ 91 calculations of the reaction probability for J= 0. In those calculations as well as the present ones it is assumed that the reaction occurs electronically adiabatically on a single electronic surface. In this Letter we compare reduced dimensionality cumulative reaction probabilities with the exact ones. Before doing that a brief review of the reduced dimensionality theory is given, followed by the results and a discussion of them.
2. Theory The vibrational state-to-state ( u-+ u’) cumulative reaction probability for an atom-diatom system for a given value of the total angular momentum J and total energy E is given by
where P$,n_y.,.12 is the detailed state-to-state reac-
0 009-2614/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
545
CHEMICALPHYSICSLETTERS
Volume 141,number6
2.0
tion probability between the initial and final vibration/rotation states (u, j, 8) and (v’, j’, s2’) respectively. .(2 (a’) is the projection of j 0”) on the body-fixed z axis, and its range is -min(j (j’), J) to min(j (j’), J) [ lo]. In the reduced dimensionality theory Sz is assumed to be a conserved quantum number (as in the centrifugal sudden approximation [lo]) as is the bending quantum number &, and the probability Pi_U. (E) is given by [ 1,2]
&v’(E)
= &%.‘(&
nb,
n)
,
=P::9B'G(E)
+ C P$:‘,EgB’G(E-AE;,a,o), ltb>O
(3)
where pz:$“’ (E) is the reduced dimensionality reaction probability for J= 0 and the ground bend state. The probabilities for excited bending states are obtained from the probability for the ground bend state by a simple energy shift 6El,,n,o
=E&R=o -Ef,,=on=o
,
(4)
where l&, o is the bend energy at the transition state for &+?=O. In the harmonic approximation
for a linear transition state 8 varies from - & to r$, in steps of two. Thus, for G?=Oonly even values of & contribute to the sum in eq. (3).
3. Results and discussion Eq. (3) was used to obtain cumulative reaction for v=O and v= 1 summed over v’, for the 0( 3P) + Hz( v) reaction using the LEPS surface of Johnson and Winter as modified slightly by Schatz [ 81. The adiabatic ground-state bending eigenvalue was calculated using the approximate harmonic/ quartic theory of Garrett and Truhlar [ 111. At the saddle point, which is the transition state for the u= 0 reaction, the energy obtained using that method, 0.093 eV, was within 0.4W of the exact bending energy. The harmonic approximation gave an energy of 0.090 eV, which is very good agreement with the harmonic/quartic result. Thus, for simplicity we used
probabilities
546
CEqB/G
(SOLID CWRVE)
-6.0
(2)
where & and r(2 are the quantum numbers of the bending state of the three-atom system and Q ranges from -J to J. In the CEQB/G theory
P:::,(E)
27 November I987
0.4
1.0
0.6
E (e; Fig. 1. Semi-logplot of exact quantum (EQ) (ref. [ 91) and reduced dimensionality (CEQB/G) cumulative reaction probabilities summedover final vibrational statesand the two arrangement channels versus the total energy E for the O(‘P) +H2( u=O, 1) reaction.
harmonic energies to calculate the shifts in energy for excited bending states, even though the groundstate bending energy surface was calculated with the harmonic/quartic theory, This should be adequate for the graphical comparisons given below. For the reaction with ZJ=1 the transition state is not at the saddle point [ 12,131. We calculated the ground-state bending energy at the v= 1 transition state and over the energy range we consider, excited bending states make a negligible contribution to the cumulative reaction probability for v= 1, according to eq. (4). In fig. 1 the present CEQB/G and the exact quantum (EQ) three-dimensional cumulative probabilities are compared in a semi-log plot, where good agreement is seen. (The probabilities are summed over the two identical product channels.) The semilog plot emphasizes very small values of the probability and it is satisfying to see good agreement for these small values. A comparison of the EQ and CEQB/G cumulative reaction probabilities on a linear scale is given in fig. 2, where good agreement is seen. The CEQBIG probability for v=O shows a marked step-like structure. This is due to the contribution of excited bending states to the cumulative probability. These steps are separated by the energy difference of consecutive, even excited bending states. In the harmonic ap-
Volume 141, number 6
CHEMICAL PHYSICS LETTERS
mlI
/I
CEQB/G (SOLIDCURVE)
t EP (OPEN CIRCLES)
27 November 1987
as the CSDW, LA and CEQB/G are the only viable ones to obtain thermal rate constants for the 0( 3P) +H,, Dz, and HD reactions.
Acknowledgement I thank Professor Donald Truhlar for supplying the accurate quantum cumulative probabilities.
0.4
0.6
0.6
1.0
References
E (4
Fig. 2. Same as fig. 1 except on a linear scale.
proximation these energy differences are equal and given by 2fiwi, where o$, is the bend frequency of the transition state. In the present case Zfiw’, equals 0.18 eV, which, by inspection, is roughly the difference in energy between the step-like features. That the EQ cumulative probability for v=O displays a similar step-like feature is quite significant. It demonstrates that the exact bending motion does display adiabaticity, even for excited states, as assumed in the reduced dimensionality theory. The present comparisons between the CEQB/G and EQ cumulative reaction probabilities complement comparisons already reported [ 9,131 with other approximate methods, i.e. the coupled-state distorted wave (CSDW) [8] and least-action semiclassical (LA) approximations [ 141. The comparison with the CSDW calculations [ 181 was for the cumulative reaction probability for H2( v= 0) and for Hz(v=O, j=O) and the comparison with the LA method was for the cumulative reaction probability for Hz( u= 0, 1). The accuracy of these approximate methods and the CEQB/G method, shown here, are about the same and all quite good. This is encouraging because the LA and CEQBIG methods have been used in calculations of thermal rate constants using a (presumably) more accurate ab initio potential energy surface. And these rate constants have subsequently been compared with experiment [ 5,6,15-l 71. At present approximate methods such
[ 11 J.M. Bowman, Advan. Chem. Phys. 61 (1985) 115. [2] J.M. Bowman and A.F. Wagner, in: The theory of chemical reaction dynamics, ed. D.C. Clary (Reidel, Dordrecht, 1986) p. 47. [3] K.-T. Lee, J.M. Bowman, A.F. Wagner and G.C. Schatz, J. Cbem. Phys. 76 (1982)3563. [ 41 J.M. Bowman, A.F. Wagner, S.P. Walch and T.H. Dunning Jr., J. Chem. Phys. 81 (1984) 1739. [S] J.M. Bowman and A.F. Wagner, J. Chem. Phys. 86 (1987) 1967. [ 61 A.F. Wagner and J.M. Bowman, J. Chem. Pbys. 86 (1987) 1976. [ 71 B.R. Johnson and N.W. Winter, J. Chem. Phys. 66 (1977) 4116. [ 81 G.C. Schatz, J. Chem. Phys. 83 (1985) 5677. [ 91 K. Haug, D.W. Schwenke, D.G. Truhlar, Y. Zhang, J.Z.H. Zhang and D.J. Komi, J. Chem. Phys. 87 (1987) 1892. [ IO] G.C. Schatz, in: The theory of chemical reaction dynamics, ed. D.C. Clary (Reidel, Dordrecht, 1986) p. I. [ 111 B.C. Garrett and D.G. Truhlar, J. Phys. Chem. 83 (1979) 1915. [ 121 K.-T. Lee, J.M. Bowman, A.F. Wagner and G.C. Schatz, J. Chem. Phys. 76 (1982) 3563. [ 131J.Z.H. Zhang, Y. Zhang, D.J. Kouri, B.C. Garrett, K. Haug, D.W. Schwenke and D.G. Truhlar, Lz Calculations of Accurate Quanta] Dynamical Reactive Scattering Transition Probabilities and Their Use to Test Semiclassical Applications, Faraday Discussions Chem. Sot. 84 (Dynamics of Elementary Gas Phase Reactions), to be published. [ 141 B.C. Garrett and D.G. Truhlar, J. Chem. Phys. 79 (1983) 4931; 81 (1984) 309. [ 151 B.C. Garrett, D.G. Truhlar, J.M. Bowman, A.F. Wagner, D. Robie, S. Arepalli, N. Presser and R.J. Gordon, J. Am. Chem. Sot. 108 (1986) 3515. [ 161 D. Robie, S. Arepalli, N. Presser, T. Kitsopoulos and R.J. Gordon, Chem. Phys. Letters 134 (1987) 579. [ 17] B.C. Garrett and D.G. Truhlar, Intern. J. Quantum Chem. 29 (1986) 1463; 31 (1987) 81.
547
CHEMICAL PHYSICS LETTERS
27 November 1987
COMPARISON OF REDUCED DIMENSIONALITY AND ACCURATE CUMULATIVE REACTION PROBABILITIES FOR 0 (3P) +H,( v= 0,l) * Joel M. BOWMAN Department of Chemistry, Emory University,Atlanta, GA 30322, USA Received 3 August 1987; in final form 2 September 1987
Reduced dimensionality (CEQBIG) cumulative reaction probabilities are compared with exact quantum ones for the O(3P) +H2(v=0, 1) reaction for zero total angular momentum. The semi-empirical LEPS surface of Johnson and Winter as modified by Schatt is used in these calculations.
1. Introduction We have developed an approximate three-dimensional theory of atom plus diatom reactive scattering based on an adiabatic treatment of the bending motion in the vicinity of the transition state [ 1,2]. This approximation effectively decouples the internal angular motion from the motion of the two bond displacements, which are fully coupled. Thus, the dimensionality of the space where explicit coupling is considered is reduced to two. For zero total angular momentum, J=O, the Schriidinger equation describing the reactive scattering looks very much like the one for collinear scattering with, however, an effective potential given by the reference potential (in the two stretch coordinates) plus the adiabatic bending eigenvalue, which depends on the two stretch coordinates. In a particularly useful version of the theory developed for collinear reaction paths, the reaction probability for excited bending states is related to the one for the ground bend state by a simple energy shift: the acronym for this theory is CEQB/G. The total reaction probability summed over all bending states for a given initial and final vibrational state is the cumulative reaction probability [ 11, i.e. the sum of state-to-state probabilities, as shown below. The CEQB/G theory has been tested against * Work supported in part by the Department of Energy (DEFGOS-86ER13568).
accurate three-dimensional quantum calculations for H+H2( v=O, u= 1) and found to be quite accurate [ 11. This and a simpler version of it have been applied to numerous reactions, most recently to 0( 3P) +Hz, DZ and HD 13-61. Among the potential energy surfaces employed in these studies was the semiempirical surface due to Johnson and Winter [ 71. That surface, with a slight modification, was recently used in coupled-states distorted wave [ 81 and exact quantum [ 91 calculations of the reaction probability for J= 0. In those calculations as well as the present ones it is assumed that the reaction occurs electronically adiabatically on a single electronic surface. In this Letter we compare reduced dimensionality cumulative reaction probabilities with the exact ones. Before doing that a brief review of the reduced dimensionality theory is given, followed by the results and a discussion of them.
2. Theory The vibrational state-to-state ( u-+ u’) cumulative reaction probability for an atom-diatom system for a given value of the total angular momentum J and total energy E is given by
where P$,n_y.,.12 is the detailed state-to-state reac-
0 009-2614/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
545
CHEMICALPHYSICSLETTERS
Volume 141,number6
2.0
tion probability between the initial and final vibration/rotation states (u, j, 8) and (v’, j’, s2’) respectively. .(2 (a’) is the projection of j 0”) on the body-fixed z axis, and its range is -min(j (j’), J) to min(j (j’), J) [ lo]. In the reduced dimensionality theory Sz is assumed to be a conserved quantum number (as in the centrifugal sudden approximation [lo]) as is the bending quantum number &, and the probability Pi_U. (E) is given by [ 1,2]
&v’(E)
= &%.‘(&
nb,
n)
,
=P::9B'G(E)
+ C P$:‘,EgB’G(E-AE;,a,o), ltb>O
(3)
where pz:$“’ (E) is the reduced dimensionality reaction probability for J= 0 and the ground bend state. The probabilities for excited bending states are obtained from the probability for the ground bend state by a simple energy shift 6El,,n,o
=E&R=o -Ef,,=on=o
,
(4)
where l&, o is the bend energy at the transition state for &+?=O. In the harmonic approximation
for a linear transition state 8 varies from - & to r$, in steps of two. Thus, for G?=Oonly even values of & contribute to the sum in eq. (3).
3. Results and discussion Eq. (3) was used to obtain cumulative reaction for v=O and v= 1 summed over v’, for the 0( 3P) + Hz( v) reaction using the LEPS surface of Johnson and Winter as modified slightly by Schatz [ 81. The adiabatic ground-state bending eigenvalue was calculated using the approximate harmonic/ quartic theory of Garrett and Truhlar [ 111. At the saddle point, which is the transition state for the u= 0 reaction, the energy obtained using that method, 0.093 eV, was within 0.4W of the exact bending energy. The harmonic approximation gave an energy of 0.090 eV, which is very good agreement with the harmonic/quartic result. Thus, for simplicity we used
probabilities
546
CEqB/G
(SOLID CWRVE)
-6.0
(2)
where & and r(2 are the quantum numbers of the bending state of the three-atom system and Q ranges from -J to J. In the CEQB/G theory
P:::,(E)
27 November I987
0.4
1.0
0.6
E (e; Fig. 1. Semi-logplot of exact quantum (EQ) (ref. [ 91) and reduced dimensionality (CEQB/G) cumulative reaction probabilities summedover final vibrational statesand the two arrangement channels versus the total energy E for the O(‘P) +H2( u=O, 1) reaction.
harmonic energies to calculate the shifts in energy for excited bending states, even though the groundstate bending energy surface was calculated with the harmonic/quartic theory, This should be adequate for the graphical comparisons given below. For the reaction with ZJ=1 the transition state is not at the saddle point [ 12,131. We calculated the ground-state bending energy at the v= 1 transition state and over the energy range we consider, excited bending states make a negligible contribution to the cumulative reaction probability for v= 1, according to eq. (4). In fig. 1 the present CEQB/G and the exact quantum (EQ) three-dimensional cumulative probabilities are compared in a semi-log plot, where good agreement is seen. (The probabilities are summed over the two identical product channels.) The semilog plot emphasizes very small values of the probability and it is satisfying to see good agreement for these small values. A comparison of the EQ and CEQB/G cumulative reaction probabilities on a linear scale is given in fig. 2, where good agreement is seen. The CEQBIG probability for v=O shows a marked step-like structure. This is due to the contribution of excited bending states to the cumulative probability. These steps are separated by the energy difference of consecutive, even excited bending states. In the harmonic ap-
Volume 141, number 6
CHEMICAL PHYSICS LETTERS
mlI
/I
CEQB/G (SOLIDCURVE)
t EP (OPEN CIRCLES)
27 November 1987
as the CSDW, LA and CEQB/G are the only viable ones to obtain thermal rate constants for the 0( 3P) +H,, Dz, and HD reactions.
Acknowledgement I thank Professor Donald Truhlar for supplying the accurate quantum cumulative probabilities.
0.4
0.6
0.6
1.0
References
E (4
Fig. 2. Same as fig. 1 except on a linear scale.
proximation these energy differences are equal and given by 2fiwi, where o$, is the bend frequency of the transition state. In the present case Zfiw’, equals 0.18 eV, which, by inspection, is roughly the difference in energy between the step-like features. That the EQ cumulative probability for v=O displays a similar step-like feature is quite significant. It demonstrates that the exact bending motion does display adiabaticity, even for excited states, as assumed in the reduced dimensionality theory. The present comparisons between the CEQB/G and EQ cumulative reaction probabilities complement comparisons already reported [ 9,131 with other approximate methods, i.e. the coupled-state distorted wave (CSDW) [8] and least-action semiclassical (LA) approximations [ 141. The comparison with the CSDW calculations [ 181 was for the cumulative reaction probability for H2( v= 0) and for Hz(v=O, j=O) and the comparison with the LA method was for the cumulative reaction probability for Hz( u= 0, 1). The accuracy of these approximate methods and the CEQB/G method, shown here, are about the same and all quite good. This is encouraging because the LA and CEQBIG methods have been used in calculations of thermal rate constants using a (presumably) more accurate ab initio potential energy surface. And these rate constants have subsequently been compared with experiment [ 5,6,15-l 71. At present approximate methods such
[ 11 J.M. Bowman, Advan. Chem. Phys. 61 (1985) 115. [2] J.M. Bowman and A.F. Wagner, in: The theory of chemical reaction dynamics, ed. D.C. Clary (Reidel, Dordrecht, 1986) p. 47. [3] K.-T. Lee, J.M. Bowman, A.F. Wagner and G.C. Schatz, J. Cbem. Phys. 76 (1982)3563. [ 41 J.M. Bowman, A.F. Wagner, S.P. Walch and T.H. Dunning Jr., J. Chem. Phys. 81 (1984) 1739. [S] J.M. Bowman and A.F. Wagner, J. Chem. Phys. 86 (1987) 1967. [ 61 A.F. Wagner and J.M. Bowman, J. Chem. Pbys. 86 (1987) 1976. [ 71 B.R. Johnson and N.W. Winter, J. Chem. Phys. 66 (1977) 4116. [ 81 G.C. Schatz, J. Chem. Phys. 83 (1985) 5677. [ 91 K. Haug, D.W. Schwenke, D.G. Truhlar, Y. Zhang, J.Z.H. Zhang and D.J. Komi, J. Chem. Phys. 87 (1987) 1892. [ IO] G.C. Schatz, in: The theory of chemical reaction dynamics, ed. D.C. Clary (Reidel, Dordrecht, 1986) p. I. [ 111 B.C. Garrett and D.G. Truhlar, J. Phys. Chem. 83 (1979) 1915. [ 121 K.-T. Lee, J.M. Bowman, A.F. Wagner and G.C. Schatz, J. Chem. Phys. 76 (1982) 3563. [ 131J.Z.H. Zhang, Y. Zhang, D.J. Kouri, B.C. Garrett, K. Haug, D.W. Schwenke and D.G. Truhlar, Lz Calculations of Accurate Quanta] Dynamical Reactive Scattering Transition Probabilities and Their Use to Test Semiclassical Applications, Faraday Discussions Chem. Sot. 84 (Dynamics of Elementary Gas Phase Reactions), to be published. [ 141 B.C. Garrett and D.G. Truhlar, J. Chem. Phys. 79 (1983) 4931; 81 (1984) 309. [ 151 B.C. Garrett, D.G. Truhlar, J.M. Bowman, A.F. Wagner, D. Robie, S. Arepalli, N. Presser and R.J. Gordon, J. Am. Chem. Sot. 108 (1986) 3515. [ 161 D. Robie, S. Arepalli, N. Presser, T. Kitsopoulos and R.J. Gordon, Chem. Phys. Letters 134 (1987) 579. [ 17] B.C. Garrett and D.G. Truhlar, Intern. J. Quantum Chem. 29 (1986) 1463; 31 (1987) 81.
547