J . Phys. Chem. 1990, 94, 3231-3236
3231
Rapid Convergence of Basis Set Expansions for Quantum Mechanical Reactive Amplitude Densities: Channel-Dependent Expansion Lengths Philippe Halvick, Donald G. Truhlar,* Department of Chemistry and Supercomputer Institute, University of Minnesota. Minneapolis, Minnesota 55455-0431
David W. Schwenke, NASA Ames Research Center, Mail Stop 230-3, Moffett Field, California 94035
Yan Sun, and Donald J. Kouri Department of Chemistry and Department of Physics, University of Houston, Houston, Texas 77204-5641 (Received: September 29, 1989)
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We report that very well converged quantum mechanical initial-state-selected and state-to-state reaction probabilities for the three-dimensional reaction 0 + H2 OH + H can be obtained with an average of 21/2-31/2 square-integrable (L2) basis functions per channel. This yields results accurate to 1-296, even for very small transition probabilities.
Introduction The accurate calculation of quantum mechanical reactive scattering probabilities presents a great computational challenge to modern theoretical c h e m i ~ t r y . I - ~In~ a continuing series of ( I ) Clary, D. C., Ed. The Theory o/Chemical Reaction Dynamics; Reidel: Dordrecht, 1986. Schatz, G. C. Annu. Rev. Phys. Chem. 1988, 39, 317. (2) H a w , K.; Schwenke, D. W.; Shima, Y.; Truhlar, D. G.; Zhang, J.; Kouri, D. J. J . Phys. Chem. 1986, 90,6757. Zhang, J. Z. H.; Kouri, D. J.; Haug, K.; Schwenke, D. W.; Shima, Y.; Truhlar, D. G. J . Chem. Phys. 1988, 88. 2492. (3) Haug, K.; Schwenke, D. W.; Truhlar, D. G.; Zhang, Y.; Zhang, J. Z. H.; Kouri, D. J. J . Chem. Phys. 1987,87, 1892. Zhang. J. Z. H.; Zhang, Y.; Kouri, D. J.; Garrett, 8. C.; Haug, K.; Schwenke, D. W.; Truhlar, D. G. Faraday Discuss. Chem. Soc. 1987,84, 371. Haug. K.: Schwenke, D. W.; Shima, Y.; Truhlar, D. G.; Zhang, J. Z. H.; Zhang, Y.; Sun, Y.; Kouri, D. J.; Garrett, B. C. In Science and Engineering on Cray Supercomputers; Cray Research: Minneapolis, MN, 1987; p 427. (4) Schwenke, D. W.; Haug, K.; Truhlar, D. G.; Schweitzer, R. H.; Zhang, J . Z. H.; Sun, Y.: Kouri, D. J. Theor. Chim. Acta 1987, 72, 237. Kouri, D. J.; Sun, Y.; Mowrey, R. C.; Zhang, J. Z. H.; Truhlar, D. G.; Haug, K.; Schwenke, D. W. I n Mathematical Frontiers in Computational Chemical Physics; Truhlar, D. G., Ed.; Springer-Verlag: New York, 1988; p 207. ( 5 ) Zhang, Y. C.; Zhang, J. Z. H.; Kouri, D. J.; Haug, K.; Schwenke, D. W.; Truhlar, D. G. Phys. Reo. Lett. 1988, 60, 2367. (6) Schwenke, D. W.; Haug, K.; Truhlar, D. G.; Sun, Y.; Zhang, J. Z. H.; Kouri. D. J. J . Phvs. Chem. 1987. 91. 6080. Schwenke. D. W.: Haue. K.: Zhao, M.: Truhla;, D. G.: Sun, Y.; Zhang, J. Z. H.; Kouri, D. J. J . phys. Chem. 1988, 92, 3202. (7) Mladenovic. M.; Zhao. M.: Truhlar, D. G.; Schwenke, D. W.; Sun, Y.; Kouri, D. J. Chem. Phys. Lett. 1988, 146, 358. Zhao, M.; Mladenovic, M.; Truhlar. D. G.: Schwenke. D. W.; Sun, Y.: Kouri, D. J.; Blais. N. C. J. Am. Chem. Soc. 1989, 1 1 1 , 852. (8) Schwenke, D. W.; Mladenovic, M.; Zhao, M.; Truhlar, D. G.; Sun, Y.; Kouri. D. J. In Supercomputer Algorithms /or Reactivity, Dynamics, and Kinetics o/Sma11 Molecules; Lagani, A., Ed.; Kluwer: Dordrecht, Holland, 1989; p 131. (9) Mladenovic. M.; Zhao, M.; Truhlar, D. G.; Schwenke, D. W.; Sun, Y.; Kouri. D. J. J . Phys. Chem. 1988,92,7035. Zhao, M.; Truhlar, D. G.; Kouri, D. J.; Sun, Y . ; Schwenke. D. W. Chem. Phys. Lett. 1989, 156, 281. Blais, N. C.; Zhao. M.; Mladenovic, M.; Truhlar, D. G.; Schwenke, D. W.; Sun, Y.; Kouri, D. J. J . Chem. Phys. 1989, 91, 1038. Zhao, M.; Mladenovic. M.: Truhlar, D. G.; Schwenke, D. W.; Sharafeddin, 0.;Sun, Y.; Kouri, D. J. J . Chem. Phys. 1989, 91, 5302. (IO) Lynch, G. C.; Halvick, P.; Truhlar. D. G.; Garrett, B. C.; Schwenke, D. W.; Kouri, D. J. Z . Natur/orsch. 1989, 44a, 427. ( 1 I ) Sun. Y.:Yu. C.-h.: Kouri, D. J.; Schwenke, D. W.; Halvick, P.: Mladenovic, M.;Truhlar, D. G. J . Chem. Phys. 1989, 91, 1643. Sun, Y.; Yu. C.-h.; Kouri, D. J.; Schwenke, D. W.: Zhao, M.; Mladenovic, M.;Truhlar, D. G., unpublished work. (12) Yu, C.-h.; Sun, Y.; Kouri, D. J.; Halvick, P.; Truhlar. D. G.; Schwenke, D. W. J . Chem. Phys. 1989, 90, 7608. Yu, C-h.; Kouri. D. J.: Zhao. M.; Truhlar, D. G.:Schwenke, D. W. Chem. Phys. Lett. 1989. 157,491; Inr. J. Quantum Chem. Symp. 1989. 23, 45. (13) Sharafeddin, 0.;Sun, Y.; Kouri, D. J.; Schwenke, D. W.; Zhao, M.: Truhlar, D. G. unpublished work. (14) Zhang, Y.; S u n . Y.; Kouri, D. J.; Zhao. M.; Truhlar, D. G.; Schwenke. D. W.. unpublished work.
0022-3654/90/2094-323 1 $02.50/0
studies we have demonstrated the efficiency of L2expansions of the reactive amplitude density for carrying out such calculations, and we have made several applications to problems of current interesta2-I4In all applications published so far, we have used basis sets in each chemical arrangement that are direct products of a
( I 5 ) Duneczky, C.; Wyatt, R. E.; Chatfield, D.; Haug, K.; Schwenke, D. W.; Truhlar, D. G.; Sun, Y.; Kouri, D. J. Comput. Phys. Commun. 1989,53, 357. (16) Garrett, B. C.; Truhlar, D. G.; Schatz, G. C. J . Am. Chem. Soc. 1986, 108, 2876. (17) Schwenke, D. W.; Truhlar, D. G.; Kouri, D. J. J. Chem. Phys. 1987, 86, 2772. (18) Neuhauser, D.; Baer, M.; Judson, R. S.; Kouri, D. J. J . Chem. Phys. 1989, 90, 5882. (19) Colton, M. C.; Schatz, G. C. Chem. Phys. Lett. 1986, 124, 256. (20) Schatz, G. C. Chem. Phys. Lett. 1988, 150, 92; 1988, 151, 409; J . Chem. Phys. 1989, 90, 3582. (21) Kuppermann, A.; Hipes, P. G. J. Chem. Phys. 1986,84,5962. H i p , P. G.; Kuppermann, A. Chem. Phys. Lett. 1987, 133, I . Cuccaro, S. A.; Hipes, P. G.; Kuppermann, A. Chem. Phys. Lett. 1989,154, 155; 1989,157, 440. (22) Webster, F.; Light, J. C. J . Chem. Phys. 1986, 85, 4744; 1989, 90, 265, 300. (23) Hermann. M. R.; Miller, W. H. Chem. Phys. 1986, 109, 163. (24) Miller, W. H.; Jansen op de Haar, B. M. D. D. J . Chem. Phys. 1987, 86,6213. Zhang, J. Z. H.; Miller, W. H. Chem. Phys. Lett. 1987, 140, 329. Zhang. J. Z. H.; Chu, S.-I.; Miller, W. H. J . Chem. Phys. 1988, 88, 6233. Zhang, J . Z. H.; Miller, W. H. J. Chem. Phys. 1988,89,4454; Chem. Phys. Lett. 1988, 153, 465; 1989, 159, 123. (25) Shima, Y.; Baer, M. Chem. Phys. Lett. 1982, 91, 43. Shima, Y.; Baer, M. J . Phys. B 1983, 16, 2169. Shima. Y.; Baer, M.; Kouri, D. J. J . Chem. Phys. 1983, 78,6666. (26) Baer, M.; Shima. Y. Phys. Rev. A 1987, 35, 5252. Baer, M. J . Phys. Chem. 1987, 91, 5846. (27) Neuhauser, D.; Baer, M. J . Chem: Phys. 1988,88, 2856. (28) Baer, M. J . Chem. Phys. 1989, 90, 3043. (29) Baer, M. Phys. Rep. 1989, 178, 99. (30) Linderberg, J. Int. J . Quantum Chem. Symp. 1986, 19, 467. Linderberg, J.; Vessal, B. Int. J. Quantum Chem. 1987, 31, 65; 1989, 35, 801. Linderberg, J. In Supercomputer Algorithms/or Reactivity, Dynamics, and Kinetics ofSma11 Molecules; Lagani, A., Ed.;.,Kluwer: Dordrecht, Holland, 1989; P 215. Linderberg, J.; Padjaer, S. B.; Ohm, Y.; Vessal. B. J . Chem. Phys. 1989, 90, 6254. (31) Parker, G. A.; Pack, R. T; Archer, B. J.; Walker, R. B. Chem. Phys. Lett. 1987. 137. 564. Pack. R. T: Parker. G. A. J. Chem. Phvs. 1987.87, 3888. Parker, G. A.; Pack, R. T;Lagan& A,; Archer, B. J.; K r k . J. B.; BaEiC, 2.In Supercomputer Algorithms for Reactivity, Dynamics, and Kinetics of Small Molecules; Lagani, A,, Ed.: Kluwer: Dordrecht, Holland, 1989; p 105. Kress, J. D.; Bacic, 2.;Parker, G. A.; Pack, R. T Chem. Phys. Lett. 1989, 157, 484. (32) Launay, J. M.; Lepetit, B. Chem. Phys. Lett. 1988, 144, 346. Lepetit, B.; Launay, J. M. Chem. Phys. Lett. 1988, 151, 287. (33) Manolopoulos, D. E.; Wyatt, R. E. Chem. Phys. Lett. 1988,152,23. Manolopoulos, D. E.; Wyatt, R. E. Chem. Phys. Lett. 1989, ISY, 123.
0 I990 American Chemical Society
3232 The Journal of Physical Chemistry, Vol. 94, No. 8, 1990 radial translational basis and a vibrational-rotational-orbital basis. Furthermore, because we have been solving new kinds of problems for which no previous benchmarks were available, we have been very careful in terms of convergence, often using overkill as far as the basis set size is concerned. For example, although we can obtain very good convergence for the H + H2 reaction with a rotationally coupled distortion potential and 4-5 radial translational basis functions per channel6 most of our published results for this system and the related D H2 system were obtained with 8-10 basis functions per channe1.7-9*11 We also have found that some systems containing heavier atoms, such as 0 HD,'oq'l H + HBr,Ss'3 F H2,12and CI + H2I4may require more translational basis functions per channel than H H2 and D + H2. Since the computational steps in a basis-set quantum scattering calculation scale as f i 2and where tii is the average number of radial translational basis functions per channel, it is important to learn how efficiently such problems can be solved if we change our emphasis from obtaining converged results for qualitatively new kinds of systems (firstgeneration strategy) to obtaining converged results as efficiently as possible (second-generation strategy). In this paper we report significant progress in this direction. In particular we retain the basis of distributed Gaussian functions for the radial translational degree of freedom of the amplitude density that we have employed for most of our previous work, but we drop the restriction of using only direct products of radial translational and channel bases. I n a first step, we allow different parameters for the DGFs and a different number of DGFs in each vibrational manifold. In a second step, we go further by using different Gaussian basis sets even for some channels with the same vibrational quantum number. In a related paper35we will examine another strategy, namely contracted basis functions.6,36 In that paper we will again use non-direct-product basis functions, and we will again examine the restriction of using the same basis set for all channels within a given vibrational manifold. This allows for computational efficiencies in memory management and storage of intcrmcdiatc quantities in the calculation.8 Thus we are very interested in learning the minimum basis sizes required to solve state-of-the-art reactive scattering problems with this restriction. Our computer program has, however, been generalized to allow arbitrary non-direct-product primitive basis sets, and, as stated above, in this paper we also examine convergence in the more general case. The non-direct-product basis set approaches presented here and in the contracted basis function paper, in both cases for the L2 expansion of the Fock-coupled reactive amplitude d e n ~ i t y , ~ - I ~ should also bc uscful for other basis-set treatments of reactive scattering employing superpositions of basis functions defined in Jacobi coordinatcs in various arrangements, e.g., the Kohn method,24937*38 the Schwinger Fock-scheme hyperspherical propagation method^,'^.^^ arrangement-channel coupling array L2 m ~ t h o d s . ~the ~ - basis-set ~~ log derivative method,33 and the scattcrcd wavc variational p r i n ~ i p l e . ~These ~ , ~ ~basis set efficicncies should also be useful for calculations on more complicated problcms such as clcctronically nonadiabatic scattering.
+
+
+
+
f i 3 , 4 9 6 3 ' 5
Calculations
The calculations reported here were carried out with the generalized Newton variational principle.6 We use a rotationally (34) Hamilton. I . P.; Light, J. C. J . Chem. Phys. 1986. 84, 306. Schwenke, D. W.; Yu, C.-h.; Kouri, D. J . (35) Zhao, M.;Truhlar. D. G.;
J . Phys. Chem.. to be published. (36) Abdallah, Jr.. J.; Truhlar, D. G. J . Chem. Phys. 1974. 60, 4670. (37) Mortcnscn, E. A,; Gucwa, L. D. J . Chem. Phys. 1969, 51. 5695. (38) Truhlar. D. G.; Abdallah, Jr.. J.; Smith, R. L. I n Aduances in C'henticul Physics; Prigogine, 1.. Rice, S.A,, Eds.; Wiley: New York, 1974; Vol. 25. p 21 I . Truhlar, D. G.;Abdallah, Jr., J. Phys. Rec. A 1974, 9, 1188. (39) Schlcssingcr. L. Phys. Rec. 1968, 168. 141I . Nuttall, J.; Cohen. H. L. P h ~ vRe. . 1969. 188. 1542. Pieper. S.C.; Wright, J.; Schlessinger, L. Phys. Rei.. D 1971. 3. 2419. (40) Sun, Y.; Kouri. D. J.; Truhlar. D. G.; Schwenke. D. W. Phys. Rea. A. 10 be published.
Halvick et al. coupled distortion potential and real-valued Green's functions. Details of the theory and its implementation are presented else-
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+
We consider the reaction 0 + H2 O H H on a single potential energy surface; we use the surface of Johnson, Winter, and Schatz4' and we consider a total angular momentum of zero. Converged results for this problem have been presented elseThe first set of results3 were obtained with the method of moments for the amplitude density and a basis set of 41-51 DGFs in 0 + H2 channel and 33-47 DGFs in each OH + H channel. The second set of results" was obtained with the generalized Newton variational principle and 4-5 Gaussians per channel. Here we use the generalized Newton variational principle with nile DGFs in 0 + H2 channels with vibrational quantum number L' and m2cDGFs in OH + H channels with vibrational quantum number c. We will attempt to find the minimal values for the parameters {maulthat will still yield converged results. We consider four sets of parameters, given in Table I . The meanings of the parameters are the following. arrangement ( I , 0 + H,; 2, 3, OH + H) number of channels in arrangement a number of DGFs in each channel in arrangement a and vibrational manifold u maximum vibrational quantum number maximum rotational quantum number for vibrational quantum number u number of harmonic oscillator functions used to expand the radial parts of vibrational-rotational eigenstates number of finite difference nodes for calculation of regular solutions and half-integrated Green's functions for distortion Hamiltonian mass-scaled radial coordinate of first node in arrangement a
mass-scaled radial coordinate of last node number of quadrature points for single-arrangementvibrational quadratures number of quadrature points for single-arrangement angular quadratures number of points in each Gauss-Legendre quadrature for radial exchange integrals number of quadrature segments, Le., number of repetitions of the Gauss-Legendre quadrature number of points in angular exchange integrals -log (exchange integral screening parameter) mass-scaled radial coordinate of center of first DGF in arrangement a and vibrational level u mass-scaled radial coordinate of center of last DGF spacing of DGFs in arrangement a and vibrational level u In all cases the final DGF in a channel is located at and the overlap parameter c of the DGFs was 1.4. Parameters without an explicit a dependence or for which values are labeled (Y in Table I were taken to be the same in all arrangements for the present calculations. Further details on all parameters are given e l s e ~ h e r e . ~ . ~ Results All calculations reported here for a total energy of E = 15 kcal/mol (=0.6505eV) and total angular momentum J of zero. There are 3 I open channels at this energy: u = 3, j = 0-6 for the 0 + Hz arrangement and u = 0 , j = 0-1 I for each O H + H arrangement. We report results for 14 runs, using full A + B2 symmetry and parameters sets 1-14, respectively. We also made a run with parameter set I but with only homonuclear diatomic symmetry, without arrangement-channel symmetry. This run will be called run IA. Table 11 gives reaction probabilities defined as follows p"P' = p Ja- 0~ , d j ~ I a,cj/(E)12 = "J'
olt,'
where Sitnis an element of the scattering matrix and I is the orbital (41) Johnson, B. R.; Winter, N. W. J . Chem. Phys. 1977, 66, 4116. Schatz, G. C. J . Chem. Phys. 1985.83, 5677.
Reactive Scattering Probabilities
TABLE I:
Panmeten for Runs 1 4 (a)," 5-8 (b), 9-11 (e), 12-14
Ne h
The Journal of Physical Chemistry, Vol. 94, No. 8, 1990 3233
a
j,, (0 jmx (0 j,, (o j,, (v j,, (0
0) = 1)
= 2) = 3)
= 4) NHO) N ( F) RLl(ao) Rpo)
set I
set 2
set 3
43 3 12 11 10 6
52 4 13 12 II 7 4
a
a a
60 956 2.2 1.1 19.0 12 34 16 13 56 14
50 680 2.4 I .3
%;(F)(aO)
N?: rvQGL
l w
N?:
€ RIe,,(ao),u = 0-2
a
a
18.0
a
10 30 14 12 50 I2 3.7
a a a a a a a
a a
a a a
a
a
(a)
(a) Runs 1-4 set 4 Rl,u,l(ao), u = 3-4 a a Adao), u = 0-2 a Al,(ao), u = 3-4 R2,,,l(ao),u = 1-2 a a a a
R3,3,1(a0)
R3,,,l(ao),u = 4-5 Au(ao), o = 1-2 &,(ao),u = 3-5
a
a a a a
mlo
a
m14
a a a
m20 m21
a
m23
a
4
3.6
m
MI1
m12 m13
set I 4.05 0.35 0.40 3.6 4.0 4.35 0.375 0.5 7 6 6 3
a a a
set 3
set 4
a
3.95 0.30 0.35 3.5 3.9 4.15 0.33 0.45 9 8
a
a a a a
a
a
a
a
a
a a a
a
4.3
7 6 6 3 3 5 4 3 1 1 4.0
set 5
set 6
set 7
set 8
0 0 0
6 4 4.7
5 3 3.7
6 2 2.7
5 4 3 I
m22
set 2
4
8
a a
4
a
7 5 4 2
a
a a a a
5.8
a
(b) Runs 5-8
set 5 Rl,,l(ao). u = 0-3 A,,"(ao), 0 = 0-3 Rz,,,,(ao).u = 0-3 AZe(ao).u = 0-3
Rl,,l(ao)+u = 0-2 RIA1b o ) Al,(ao). L' = 0-2 A13(a0)
R2,0.1(ao)
R2.1.1b o ) R2.2.1b o ) R2,3,1(ao) Au(ao),
Rlj,,lb AIJc ntlJc
R2Jy.l
11
= 0, 1
set 6
set 7
set 8
3.7 0.35 3.6 0.375
3.75 0.38 3.7 0.4
3.8 0.42 3.8 0.5
set 9
set 10
3.7 4.05 0.38 0.5 3.6 3.6 4.0 4.35 0.4
3.75 4.2 0.42 0.5 3.65 3.65 4.2 4.35 0.5
jo= 0-12, jl = 0 - 1 I , j2= 0.7 J 2 = 8-10? j3= 0-6 j o = 0-12. jl = 0-1 1, j2= 0-2 j2= 8-10, j3= 0-6 jo= 0-8 jo= 9-12, jl = 0-11, j2= 0-7 J2 = 8-10. j3= 0-6 10 = 0 - 1 2 jl = 0-8 jl = 9-11 j2= 0-8 j2= 9-10 j3= 0-6
ml,u,u = 0-3 m2,u,o = 0-3
m
(c) Runs 9-1 I set 11 3.8 4.2 0.48 0.5 3.7 3.85 4.2 4.35 0.55
42630) MI0
mlu.o = 1-2 m13
m20 m21 4
2
m23
m
set 12
set 13
(d) Runs 12-1 4 set 14
3.7 4.05 0.38 0.5 6 5 2 3.6 3.6 3.7 4.0 4.2 4.35
3.75 4.2 0.42 0.5 5 4 1 3.65 3.65 3.65 4.2 4.2 4.35
3.8 4.2 0.48 0.5 4 3 1 3.7 3.85 3.85 4.2 4.2 4.35
AzjO
mZj,
til
set 9
set 10
set I 1
0.55 6 5 2 4 3 2 1 3.4
0.55 5 4
0.55 4 3 1 2 1 1
jo= 0-12, jl = 0-8 jl = 9-11 j2= 0-8 j2= 9-10
j, = 0-6 jo= 0-8 jo= 9-12 jl = 0-8 jl = 9-11 j2= 0-8 j2= 9-10, j, = 0-6
I 3 2 1 I 2.5
1
1.9
set 12
set 13
set 14
0.4 0.5 0.55 0.63 0.63 4 3 3 2 2 1 3.2
0.5 0.5 0.55 0.55 0.55 3 3 2 2
0.55 0.55 0.55 0.55 0.55 2 2 1
I
1 1 1.8
1
2.4
1
a In parts b, c, and d, all parameters not mentioned have the same values as in run I . Also, parameters whose value is given as a in part a are the same as in run I . For R, A, and m, we use the indices n and ju, instead of a and u as in parts a-c of this table. This is because, in part d, the paramctcrs of the DGB and the number of DGBs depend now on both vibrational and rotational quantum numbers.
angular momentum quantum number; notice that one must multiply by 2 to obtain the sum over both OH H arrangement channels. For all probabilities in Table 11, runs I and I A agreed to seven or more significant figures. Run 2 differs from run 1 in having a larger number of rotational and vibrational channels, and run 3 has all numerical parameters refined by about 10%. Table I I shows then that the results are converged to much better than I % with respect to the channel basis and bcttcr than four significant figures with respect to the numcrical paramctcrs. Thus this channel basis and set of numcrical parameters are more than adequate for testing convergence
+
with respect to number of DGFs used to expand the radial translational part of the reactive amplitude density. In run 4 the number of DGFs is increased in every channel and the locations of all DGFs are varied from run I . Table I1 shows that the results are stable to better than 1%. This shows that run 1 is well converged. (Many additional runs were made, and they verify this conclusion.) Thus, having a set of well converged probabilities as a reference, we now try to find a basis set with less translational basis functions such that all the reaction probabilities larger than IV5still agree with the reference run to better than 1%. In all these runs we freeze the number of channels at
3234 The Journal of Physical Chemistry, Vol. 94, No. 8, 1990 TABLE 11: Reaction Probabilities( v = 0, j = I ) run 1 run 2 run 3 I 2 0 1.296(-2)0 1.300(-2) 1.296(-2) 1 4.012(-2) 4.022(-2) 4.01 1(-2) 2 4.869(-2) 4.883(-2) 4.869(-2) 3 2.71 I(-2) 2.720(-2) 2.71 I(-2) 4 5.832(-3) 5.851(-3) 5.831(-3) 5 2.958(-4) 2.973(-4) 2.958(-4) 6 7.658(-7) 7.693(-7) 7.661(-7) 2 1 0 3.594(-3) 3.604(-3) 3.594(-3) 1 1.069(-2) 1.072(-2) 1.069(-2) 2 1.732(-2) 1.737(-2) 1.732(-2) 3 2.256(-2) 2.262(-2) 2.256(-2) 4 2.499(-2) 2.506(-2) 2.499(-2) 5 2.335(-2) 2.342(-2) 2.335(-2) 6 1.772(-2) 1.770(-2) 1.772(-2) 7 1.019(-2) 1.022(-2) 1.019(-2) 8 3.890(-3) 3.905(-3) 3.890(-3) 9 6.821(-4) 6.851(-4) 6.820(-4) IO 9.033(-6) 9.109(-6) 9.034(-6) II b b b a a' j run 5 run 6 run 7 1 2 0 1.223(-3) 1.299(-2) 1.297(-2) I 3.784(-3) 4.021(-2) 4.014(-2) 2 4.618(-3) 4.880(-2) 4.874(-2) 3 2.587(-3) 2.717(-2) 2.71 5(-2) 4 5.662(-4) 5.843(-3) 5.839(-3) 5 2.987(-5) 2.962(-4) 2.949(-4) 6 8.216(-8) 7.635(-7) 7.515(-7) 2 1 0 3.393(-4) 3.602(-3) 3.596(-3) 1 I .009(-3) 1.072(-2) 1.070(-2) 2 1.636(-3) 1.736(-2) 1.734(-2) 3 2.1 32(-3) 2.262(-2) 2.258(-2) 4 2.364(-3) 2.505(-2) 2.501(-2) 5 2.214(-3) 2.340(-2) 2.337(-2) 6 I .688(-3) 1.776(-2) 1.774(-2) 7 9.780(-4) 1.021(-2) 1.020(-2) 8 3.780(-4) 3.896(-3) 3.834(-3) 9 6.761(-5) 6.813(-4) 6.818(-4) IO 9.233(-7) 8.994(-6) 9.055(-6) I1 b b b a a' i run 9 run IO 1 2 0 1.293(-2) 1.281(-2) I 4.002(-2) 3.97 I(-2) 2 4.861 (-2) 4.826(-2) 2.683(-2) 3 2.708(-2) 4 5.776(-3) 5.828(-3) 5 2.948(-4) 2.921(-4) 7.508(-7) 7.536(-7) 6 2 1 0 3.533(-3) 3.586(-3) 1.067(-2) I I .052(-2) 1.729(-2) 2 1.706(-2) 3 2.252(-2) 2.225(-2) 4 2.495(-2) 2.469(-2) 2.331(-2) 5 2.315(-2) 1.770(-2) 6 1.765(-2) 7 I .018(-2) 1.022(-2) 8 3.886(-3) 3.943(-3) 9 6.8 I 8(-4) 7.007(-4) IO 3.108(-6) 9.405(-6) II b b a a' i run 12 run 13 1 2 0 1.293(-2) 1.281(-2) I 4.003(-2) 3.971(-2) 2 4.862(-2) 4.826(-2) 3 2.709(-2) 2.689(-2) 4 5.829(-3) 5.773(-3) 5 2.948(-4) 2.920(-4) 6 7.5 I9(-7) 7.455(-7) 2 1 0 3.587(-3) 3.533(-3) I I .067(-2) 1.052(-2) 2 1.729(-2) I .706(-2) 3 2.252(-2) 2.225(-2) 4 2.495(-2) 2.470(-2) 5 2.332(-2) 2.315(-2) 6 1.770(-2) 1.765(-2) 7 1.018(-2) 1.023(-2) 8 3.888(-3) 3.943(-3) 9 6.826(-4) 7.006(-4) IO 9.1 1 l(-6) 9.403(-6) II b b OThis denotes 1.296 X *Less than a
a'
run 4 1.299(-2) 4.021(-2) 4.88 I(-2) 2.717(-2) 5.842(-3) 2.962(-4) 7.653(-7) 3.603(-3) 1.072(-2) 1.737(-2) 2.262(-2)
j
Halvick et al. TABLE 111: Deviations from Run 1 of the Transition Probabilities run m AP(rms) N(devl%)
2.505(-2) 1.021(-2)
2.150(-2)
1.618(-2) 9.196(-3) 3.455(-3) 5.905(-4) 7.607 (-6) b
5.8
2.56(-5)"
340
I56
5
0
1.02(-2)
16
480
6 7 8
4.7 3.7 2.7
2.33(-5) 2.84(-5) 1.60(-4)
338 315 172
158 181
324
9 II
3.4 2.5 1.9
4.18(-5) 1.73(-4) l.l2(-3)
307 176 55
189 320 44 I
12 13 14
3.2 2.4 1.8
4.77(-5) 1.74(-4) l.l3(-3)
297 170 53
199 326 443
IO
2.341(-2) 1.776(-2) 3.897(-3) 6.818(-4) 9.026(-6) b run 8 1.277(-2) 3.957(-2) 4.817(-2) 2.683(-2) 5.757(-3) 2.899(-4) 7.412(-7) 3.538(-3) 1.053(-2) 1.707(-2) 2.225(-2) 2.467(-2) 2.307(-2) 1.755(-2) 1.013(-2) 3.891(-3) 6.915(-4) 9.385(-6) b run I I 1.213(-2) 3.7 19(-2) 4.478(-2) 2.477(-2) 5.378(-3) 2.8 58 (-4) 8.239(-7) 3.355(-3) 9.975(-3) 1.614(-2) 2.097(-2) 2.314(-2) 2.1 5 1(-2) 1.619(-2) 9.201(-3) 3.458(-3) 5.912(-4) 7.620(-6) b run 14 1.213(-2) 3.718(-2) 4.476(-2) 2.476(-2) 5.371(-3) 2.8 54(-4) 8.1 I I(-7) 3.354(-3) 9.973(-3) 1.613(-2) 2.097(-2) 2.314(-2)
4
(1
Denotes 2.56
X 1 O-5.
129, and in fact all numerical and other parameters, except the Gaussian basis set parameters in Table Ib-d, are frozen at their values in run I . First, however, in run 5, we try a calculation without any Gaussian functions; Le., we compute the probabilities by the distorted-wave Born approximation. It can be seen that these probabilities are 1 order of magnitude different from the reference; this shows explicitly that the case chosen for study is indeed a difficult one, and it shows how much work the variational principle has to do to get the correct probabilities. (Interestingly, the dependence of the reaction cross section o n j is reasonably accurate; Le., the error is about the same for all initial states. The fact that the relative cross sections are more accurate than the absolute ones reminds us of earlier distorted-wave c a l c ~ l a t i o n on s ~the ~~~~ H + H2, and H+ D2reactions in which the product-state relative distributions were found to be more accurate than the absolute cross sections. Further examination of this question, however, is extraneous to the goal set for the present research.) In runs 6, 7, and 8, we use the same number of distributed Gaussian functions for all channels in a given arrangement, and we progressively decrease these numbers in going from run 6 to run 8. This test will show what is the best we can do with a direct product basis in each arrangement. Table I1 shows that runs 6 and 7 both give well converged probabilities (deviations smaller than I%), and the minimum average number of Gaussians per channel for a coverged run is 3.7. In runs 9, 10, and 1 I , we use a different number of Gaussian functions in each vibrational level, which is the first step mentioned in the Introduction. In addition we changed the placement and spacing of the Gaussians such that for higher u the Gaussians start farther out and are more widely spaced. Table I1 shows that reaction probabilities converged to 1% can be obtained with run 9, which has an average of only 3.4 Gaussian functions per channel. Finally, in runs 12, 13, and 14, we use a different number of Gaussian functions in individual channels or small subsets of channels. For the 0 + H2 arrangements, the parameters used for a given u , j channel with energy higher than the u + I , j = 0 channel were the same as the ones for the latter channel. For the OH H arrangements, we also moved the first Gaussians out and used less Gaussians for some higher j . Table I1 shows that reaction probabilities converged to 1% are obtained with only 3.2 Gaussians per channel in run 12. I f we temporarily increase our convergence criterion to 2.3%, we can actually achieve this with an average of only 2.5 L2basis functions per channel with run 13! Several other runs with 1.8-2.7 Gaussians per channel, in particular those for parameter sets 8, IO, I I , and 14, also show maximum deviations of around 2% from the reaction probabilities larger than in the reference set; this shows thc good stability of the generalized Newton variational principle, even with real-valued Green's functions and Gaussian basis sets smaller than 3 functions per channel.
+
(42) Clary, D. C.; Connor, .I.N. L. Mol. Phys. 1981, 43, 621. (43) Connor. J. N . L.: Southall, W.J. E. Chem. Phys. Lett. 1986,123, 139.
The Journal of Physical Chemistry, Vol. 94, No. 8, 1990 3235
Reactive Scattering Probabilities
TABLE IV: Average Deviation from Run 1 of the Probabilities for Each Order of Magnitude of the Transition Probabilities' run 1 2 3 4 5 6 7 8 9 0.15 0.15 0.24 0.27 0.15 0.05 0.20 4 0.02 0.08 78.41 73.56 74.86 89.68 89.00 90.89 21.90 84.11 6.58 5 0.17 0.25 0.13 0.26 0.46 0.31 0.05 0.18 6 0.02 7 0.18 0.5 1 0.46 I .93 2.08 3.17 0.04 0.12 0.03 2.01 4.16 4.51 6.44 1 .oo 3.51 0.32 1.19 8 0.12 0.59 0.78 1.68 1.22 1.21 9 0.03 0.13 0.22 0.20 2.31 2.70 0.14 0.31 1.21 I .48 1.71 3.93 3.26 IO 8.53 3.00 6.85 7.60 12.44 5.61 12.48 9.87 0.97 II 0.60 0.89 1.91 2.03 1.16 0.04 0.13 0.21 0.20 12 0.14 0.31 1.21 1.51 3.21 1.76 3.98 3.70 2.89 13 7.16 11.51 8.91 5.74 12.59 14 0.97 3.01 6.88 8.60
no.
59
52
83
39
27
17
13
IO
2
IO 0.65 83.80 I .05
2.65 6.75 0.46 3.70 5.13 5.43 7.00 9.48 L
"The column headed i shows the average deviation in percent for the probabilities larger than IO-' and smaller than '-OI TABLE V Deviations from Run 4 of the Transition Probabilities run A AP(rms) N(devl%) I 4.3 2.56(-5)' 340 I56 2 4.0 I .45(-5) 322 174 194 3 4.0 2.59(-5) 302 16 I .02(-2) 480 5 0 6 4.7 151 8.I5(-6) 345 7 3.7 3.73( - 5 ) 321 175 332 I .85(-4) I64 8 2.7 9 3.4 6.23(-5) 309 187 1.91(-4) IO 2.5 335 161 11 1.9 55 441 l.l4(-3) 6.56(-5) 293 203 12 3.2 I .92(-4) 13 2.4 335 161 14 1.8 I . I5(-3) 53 443 .
I
'Denotes 2.56 X Tables Ill-VI show convergence statistics for the state-to-state probabilities. There are 31 open channels and hence 31(32)/2 = 496 transition probabilities. Table 111 gives the root-meansquare of the deviation of run 1 from each of runs 4-14 for the entire set of transition probabilities and also the number of probabilities converged to within 1% and not within I%, respectively. Table IV shows the average deviation in percent of the transition probabilities larger than classed by order of magnitude. The last row of Table IV tells how many of the transition probabilities were in the given range. For the runs where the reaction probabilities were well converged (i.e., runs 6 , 9, and 12), we observe that the average deviation of the state-to-state probabilities larger than IO" is also under 1%. Run 13, with an average of 2.4 Gaussians per channel, is converged to an average of 1.79%)for all state-to-state transition probabilities greater than
which would be quite adequate for many purposes. Table V and VI are just like Tables 111 and IV except they use run 4 rather than run 1 as the reference. This confirms that the conclusions are not strongly affected by the choice of reference run.
Discussion The results obtained above are quite encouraging. Depending on the accuracy required, we can get converged results with 2'/2-31/2 L2basis functions per channel, even with no non-L2 functions in the basis. This shows the good convergence properties of the variational expansion of the reactive amplitude density with a multichannel distortion potential and the Fock coupling scheme, even with real-valued Green's functions. There are only a small number of converged calculations in the literature for systems with an atom heavier than and the other methods used in such c a l c ~ l a t i o n s ,which ~ ~ ~ ~are ' ~hyperspherical ~~ propagation methods, are sufficiently different from the basis-set expansions of the amplitude density that we have e m p l ~ y e d ~that ~ ~it*is~ ~ ' ~ hard to make a comparison of efficiencies. But the number of basis functions per channel in the present calculations is sufficiently small in an absolute sense that basis-set techniques are clearly confirmed as powerful methods for calculating accurate quantum mechanical reaction probabilities. In addition the present method appears to be the most rapidly convergent basis-set approach. It is also interesting to comment on the straightforward way in which efficient non-direct-product basis sets were created because this has an important bearing on the convenience of the method for new applications. The placement of the DGFs for run 1 was optimized in earlier runs by trial and error. The placement of basis functions in these runs was guided by the principle that low-u manifolds have channels with high local translational energies and hence closer-in classical turning points, requiring smaller Rz,o,lvalues, and they have smaller deBroglie
TABLE VI: Averaee Deviation from Run 4 of the Probabilities for Each Order of run 1 2 3 4 5 I 0.02 0.05 0.20 0.15 0.15 2 0.01 0.03 0.06 0.17 0.36 3 0.02 0.05 0.20 0.15 0.16 5 6.59 21.95 84.16 78.42 73.54 6 0.01 0.02 0.02 0.05 0.12 7 0.03 0.07 0.18 0.19 0.63 8 0.14 0.37 1.37 1.13 2.12 9 0.05 0.17 0.72 0.42 0.33 IO 0.16 0.35 1.37 1.53 1.74 11 0.98 3.04 7.03 8.63 5.63 12 0.05 0.17 0.4I 0.33 0.74 13 0.16 0.35 1.37 1.55 1.79 14 0.99 3.06 7.06 8.69 5.76 no. 59 52 83 39 27
D,3959'b12920*3'932
Magnitude of the Transition Probabilities' 6 7 8 9 0.24 0.27 0.15 0.08 0.75 0.77 0.43 0.19 0.20 0.21 0.07 0.09 74.83 89.67 89.01 90.90 0.33 0.34 0.32 0.23 3.25 0.43 1.77 2.01 6.52 4.10 4.38 3.46 1.63 1.33 1.29 0.79 2.78 3.97 3.34 2.41 12.35 12.32 9.93 7.66 2.13 1.25 0.94 1.98 3.31 2.97 4.01 3.78 1 1.42 8.96 7.23 12.44 17 13 IO 2
uThe column headed i shows the average deviation in percent for the probabilitites larger t h a n IO" and smaller than IO-'".
IO 0.65 2.04 0.95 83.91 0.40 1.99 6.05 0.19 4.32 5.75 6.04 7.60 10.07 2
J . Phys. Chem. 1990, 94, 3236-3241
3236
wavelengths, r uiring smaller Auavalues. The general trend in the optimized % RUy,]and Auo values confirms these expectations. Similar considerations were used to construct the basis sets for rotational submanifolds that performed so efficiently in runs 12 and 13. Thus these principles may be anticipated to be valid for other systems as well, and this provides some guidance for basis set optimizations for other systems in the future.
only 3.4 distributed Gaussian functions per channel. Releasing this restriction allows us to obtain well converged results with only 3.2 basis functions per channel. In both cases we use only L2 basis functions, and the convergence criterion is that all reaction probabilities greater than be converged to 1% or better. This demonstrates the excellent basis-set efficiency of variational L2 expansions of the reactive amplitude density.
Conclusion Even with the restriction of using the same basis set for all channels in a given vibrational level, we Can obtain well converged results for the reaction 0 + Hz O H + H with an average of
Acknowledgment. This work was supported in part by the National Science Foundation, the National Aeronautics and Space Administration, and Minnesota Supercomputer Institute. Registry No. 0, 7782-44-7; H2,1333-74-0.
-
Ab Initio Theoretlcal Studies of the CH, 4- H
CH:,”
CH
+ H2 Reactions
Mutsumi Aoyagi, Ron Shepard, Albert F. Wagner,* Thomas H. Dunning, Jr., Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439
and Franklin B. Brown Supercomputer Computations Research Institute. Florida State University, Tallahassee, Florida 32306 (Received: October 17, 1989; In Final Form: January 30, 1990)
Ab initio electronic structure characterizations of the reactants and parts of the addition reaction path for the title reaction are described. The wave function allows extensive correlation of all seven of the valence electrons via a CASSCF plus singleand double-excitation configuration interaction expansion. A large correlation-consistent basis set (triple valence plus polarization) is used. The calculated properties of the reactants and of CHI compare favorably to experiment. Variational transition state theory calculations suggest little temperature dependenceto the high-pressure limiting rate constant for CH2 + H addition.
I. Introduction The reaction CH2(’B,)
+H
CH(211) + H2
CH3*
(I)
is of considerable interest for three reasons. The thermal rate constants for addition in both directionsk-” and the thermal dissociation rate constant12 have been measured. In the direction as written, there is controversy’-5 over the temperature dependence of the addition rate constant. The isotopic variations of the reverse reaction have also been measuredP*’ Second, recently, molecular beam m e a s ~ r e m e n t s have ~ ~ * ~characterized ~ the state-resolved reactive and inelastic cross sections for the reverse reaction and its isotopic variants. Finally, the spectroscopy of CH, has been measuredI5J6with special attention to the out-of-plane umbrella ~~
~
~~~~~~~~
~~
( I ) Bohland, T.; Temps, F. Ber. Bunsen-Ges. Phys. Chem. 1984,88,459. (2) Grebe, J.: Homann, K. H. Ber. Bunsen-Ges. Phys. Chem. 1982, 86, 581. (3) (4) (5) (6)
Frank, P.; Bhaskaran, K.A.; Just, Th. J. Phys. Chem. 1986.90.2226. Lohr, R.; Roth, P. Ber. Bunsen-Ges. Phys. Chem. 1981, 85, 153. Peeters, J.; Vinckier, C. Symp. (Int.) Combust., [Proc.] 1974, 15, 969. (a) Berman, M. R.; Lin, M. C. J. Chem. Phys. 1984,81, 5743. (b) Zabarnick, S.; Fleming, J. W.; Lin, M. C . J. Chem. Phys. 1986, 85, 4373. (7) Becker, K. H.; Engelhardt, B.; Wiesen, P.; Bayes, K. D. Chem. Phys. Lett. 1989, 154, 342.
(8) Braun, W.; McNesby, J. R.; Bass, A. M. J. Chem. Phys. 1967, 46, 2071. (9) Butler, J. E.; Goss. L. P.; Lin, M. C.; Hudgens, J. W. Chem. Phys. Lett. 1979, 63. 104. (IO) Anderson, S. M.; Freedman, A.; Kolb. C. E. J. Phys. Chem. 1987, 91. 6272. ( I 1 ) Bosnali. M. W.; Perner, D. Z . Nuturforsch. 1971. M A , 1768. ( 12) Roth. P.; Barner, U.; Lohr, R. Ber. Bunsen-Ges. Phys. Chem. 1979, 83, 929. (13) Liu, K.; Macdonald, G.J . Chem. Phys. 1988, 89, 4443. (14) Liu, K.; Macdonald, G. J. Chem. Phys., in press. (15) (a) Snelson, A . J. Phys. Chem. 1970, 74, 537. (b) Pacansky, J.; Bargon, J. J. Am. Chem. Soc. 1975, 97,6896. (c) Milligan, D. E.; Jacox, M. E. J. Chem. Phys. 1967.47, 5146. (d) Jacox, M. E. J. Mol. Specrrosc. 1977, 66. 272.
0022-3654/90/2094-3236$02.50/0
motion which has quartic characteristics. Studies of reaction 1 have a bearing on a variety of other reactions, such as H 2 C 0 dissociationk7(which involves a similar competition between atomic and molecular dissociation channels) and non-carbon analogues, e.g., BH + H2I8 and SiH + Hz.l9 Reaction -I is also one of the simplest reactions with a nonleast-motion pathway;20Le., C H does not insert into H2 along a C, path. For this and other reasons, it has been the subject of several ab initio electronic structure studies*= which have mapped out the general nature of either the C H H220-21 or the CH2 + H2* reaction path. However, a relatively small basis set and modestly correlated wave function were used in these Calculations, resulting in limited accuracy in the reaction path characterization. This paper is the first of a series of theoretical studies of reaction 1. This report will focus on the theoretical description of the fragments CH, CH2, CH3, and H2 and the CHI H addition reaction path at the harmonic level, as well as an estimate of the high-pressure limiting rate constant of reaction I . All electronic structure calculations in this work were performed with the COLUMBUS program system2, and are of significantly higher quality
+
+
(16) (a) Riveros, J. J . Chem. Phys. 1969,51, 1269. (b) Tan, L. Y.; Winer, A. M.;Pimentel, G.C. Ibid. 1972, 57, 4028. (c) Yamada, C.; Hirota, E.; Kawaguchi, K. Ibid. 1981, 75, 5256. (d) Ellison, G. B.; Engelking, P. C.; Lineberger, W. C. J. Am. Chem. Soc. 1978,100,2556. (e) Hermann, H. W.; Leone. S.R. J . Chem. Phys. 1982,76,4766. (0 Amano, T.; &math, P. F.; Yamada, C.; Endo, Y . ; Hirota, E. Ibid. 1982, 77, 5284. (g) Holt, P. L.; McCurdy, K.E.; Weisman, R. B.; Adams, J. S.; Engel, P. S. Ibid. 1984,8I, 3349. (17) Moore. C. B.; Weisshaar, J . C. Annu. Rev. Phys. Chem. 1983, 34, 525. ( I 8) Caldwcll, N. J.; Rice, J. K.; Nelson, H. H.; Adams, G. F.; Page, M.
J. Phys. Chem., this issue. (19) Bcgemann, M. H.; Dreyfus, R. W.: Jacsinski, J. M. Chem. Phys. Leu. 1989, f55, 351. (20) Brooks, B. R.; Schaefer, 111, H. F. J. Chem. Phys. 1977, 67, 5146. (21) Dunning, Jr.. T. H.; Harding, L. B.; Bair, R. A.; Eades, R. A.; Shepard, R . L. J. Phys. Chem. 1986, 90, 344. (22) Merkel, A.; Zulicke, L. Mol. Phys. 1987, 60, 1379.
0 I990 American Chemical Society
