The Journal of
Physical Chemistry ~~
0 Copyright, 1988, by the American Chemical Society
VOLUME 92, NUMBER 25 DECEMBER 15,1988
LETTERS Converged Quantum Mechanical Caicuiatlon of the Product Vibration-Rotation State Distribution of the H p-HP Reaction
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Mirjana Mladenovic, Meishan Zhao, Donald G. Truhlar,* Department of Chemistry, Chemical Physics Program, and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455
David W. Schwenke,+ Eloret Institute, Sunnyvale, California 94087
Yan Sun, and Donald J. Kouri Department of Chemistry and Department of Physics, University of Houston, Houston, Texas 77004 (Received: September 1. 1988)
We have calculated converged quantum mechanical state-testate cross sections for the reaction of H with H2at a total energy of 1.0687 eV. The calculations involve 41 total-angular-momentum/parityblocks with up to 1005 channels per block for production runs and up to 1035 channels per block for convergence checks. The converged vibrational branching ratio is in good agreement with recent experiments but the rotational distribution is not.
Introduction The H H2reaction is a prototype chemical reaction for both theoretical and experimental study.' The first accurate quantum mechanical calculations of state-to-state cross sections for this reaction were reported by Schatz and Kuppermann 12 years ago: but these were restricted to energies below the vibrational excitation threshold. Although there have been many approximate cross-section calculations at higher energies, no completely converged quantum mechanical calculation of the state-to-state cross sections at an energy permitting vibrational excitation has been reported. The present Letter reports such a calculation.
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'Mailing address: NASA Ames Research Center, Mail Stop 230-3, Moffett Field. CA 94035.
0022-3654/88/2092-7035$01.50/0
Computation Methods The calculations were carried out using the 1: generalized Newton variational principle (GNVP).3-5 In particular, the reactive amplitude density in chemical arrangement a is expanded (1) Truhlar, D. G.; Wyatt, R. E.Annu. Rev. Phys. Chem. 1976, 27, 1. (2) Schatz, G. C.; Kuppermann, A. J . Chem. Phys. 1976,65, 4668. ( 3 ) Schwenke, D. W.; Haug, K.;Truhlar, D. G.; Sun, Y.; Zhang, J. Z. H.; Kouri, D. J. J. Phys. Chem. 1987,91,6080. (4) Schwenke, D. W.; Haug, K.; Zhao, M.; Truhlar, D. G.; Sun, Y.; Zhang, J. 2.H.; Kouri, D. J. J. Phys. Chem. 1988.92, 3202. ( 5 ) Schwenke, D. W.; Mladenovic, M.; Zhao, M.; Truhlar, D. G.; Sun, Y.; Kouri, D. J., to be published in Proceedings of the NATO Advanced Research Workshop on Supercomputer Algorithms f o r Reactivity, Dynamics, and Kinetics of Small Molecules; NATO AS1 Series; Reidel: Dordrecht.
0 1988 American Chemical Society
7036 The Journal of Physical Chemistry, Vol. 92, No. 25, 1988 in a square-integrable product basis set formed from asymptotic vibrational eigenvectors,6 Arthurs-Dalgarno’ laboratory-frame rotational-orbital functions, and distributed Gaussians* in the radial relative translational coordinate. The theory and the details of its implementation for the present calculations are completely described in two full papers:ss and two other applications have also been r e p ~ r t e d . ~The . ~ ~present calculations are the first in which we utilize several new computational improvement^,^ including more efficient quadratures for the exchange integrals and use of identical-particle symmetry to partially decouple the equations for the coefficients in the variational expansion of the amplitude density. The problem is block diagonal in total angular momentum (J) and parity (P). The calculations for each JP block involve three
Letters TABLE I: Sets of Parameter Values for Convergence Checks’ parameter set A set B
Calculations and Convergence Tests All large calculations were carried out on the Cray-2/4-256 supercomputer at the Minnesota Supercomputer Center, with simple preparation and analysis tasks carried out on the University of Minnesota Chemistry Department VAX minicomputer. The calculations were carried out using the double-manybody-expansion (DMBE) potential energy surface13 at a total energy of 1.0687 eV, measured with respect to H + H2(re). The basis set for a given JP block is defined by the maximum vibrational quantum number urnax, the maximum rotational quantum number in each vibrational level brnax(u)Iu= 0, 1, ..., urnax),and the number m and parameters of the distributed Gaussian basis (DGB). In convergence checks for J = 0, for both parities of J = 1, and for the smaller JP blocks with J = 2 , 3, 6, 10, and 15, all calculations were carefully converged with respect to these parameters and with respect to all FDBVM and quadrature parameters. Finally the cross sections were converged = 20. Since there are two with respect to J , which requires JmaX parity blocks for each J 2 1 and one for J = 0, this involved 41 JP blocks. The largest jrnaX(u)used for production runs was 13 so the size of the basis set increases with J for J = 0-13 and then remains the same for J = 13-20. For each J in the latter range, the larger parity block has 1005 channels, and the smaller one has 843 channels. The production runs for most J (the exception is J = 1, see below) involve m = 9 so the number of basis functions per
540. (8) Hamilton, I. P.; Light, J. C. J. Chem. Phys. 1986, 84, 306.
(9) Mladenovic, M.; Zhao, M.; Truhlar, D. G.;Schwenke, D. W.; Sun, Y.; Kouri, D. J. Chem. Phys. Lett. 1988, 146, 358. (10) Zhao. M.; Mladenovic, M.; Truhlar, D. G.; Schwenke, D. W.; Sun, Y.; Kouri, D. J. J. Am. Chem. Soc., in press. ( I 1) Sun, Y.; Kouri, D. J.; Schwenke, D. W.; Zhao, M.; Halvick, P.; Truhlar, D. G., to be submitted for publication. (12) Dongarra, J. J.; Moler, C. B.; Bunch, J. R.; Stewart, G. W. LINPACK User’s Guide; SIAM: Philadelphia, 1979; Chapter 5. (13) Varandas, A. J. C.; Brown, F. B.; Mead, C. A,; Truhlar, D. G.; Blais, N. C. J. Chem. Phys. 1987, 86, 6258.
m RIG,au
9 1.871 0.354 4.700 1.4 50 524 0.336 12.909 10 30 50 12 14 4.312
C
NWO) N( F) R,,, au %v(F)$ au
nah”
d:
W L NQR
R,, au
14 13 11 9 7 4 10 1.694 0.354 4.876 1.3 70 845 0.364 12.165 12 40 60 14 15 4.554
‘Notation defined in ref 4.
TABLE I 1 Selected Reaction Probabilities J P v j v ’ j ’ production run convergence test
o
+
1
+
1
-
2
-
3
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6
1
1 (6) Zhang, J. 2. H.; Kouri, D. J.; Haug, K.; Schwenke, D. W.; Shima, Y.; Truhlar, D. G. J. Chem. Phys. 1988, 88, 2492. (7) Arthurs, A. M.; Dalgarno, A. Proc. R. SOC.London Ser. A 1960, 256,
13 12 10 8 6
A, au RmG,au
step^:^-^ (i) We define a distortion potential that includes all intraarrangement rotational coupling but is diagonal in vibrational and arrangement quantum numbers. The half-integrated Green’s functions for this problem are obtained on Gauss-Legendre grids in the relative translational radial coordinates by the finite difference boundary value method (FDBVM)$v6 Two FDBVM calculations are carried out for each unique vibrational level, one for p-H2 and one for o-H,. (ii) Integrals over the half-integrated Green’s functions and basis functions are carried out in terms of angular momentum coupling coefficients and repeated Gauss-Legendre quadratures, which are evaluated by using fast matrix multiplication routines. (iii) The variational equations for the basis set coefficients, decoupled to the maximum extent p o ~ s i b l e by ~ , ~symmetry, ~ are solved by Gaussian elimination, using a vectorized L I N P A C K ~ * library subprogram for symmetric matrices.
jmax(u=O) jmax(u=1) jmaSu=2) jmax(u=3) jmax(v=4) jmax(v=5)
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0
5
o
o
o
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 2 1 2 1 2 0 2 0 2 0 0 0 0 0 0 0 0 0 0 1 2 0 2 1 2 0 2 1 2 0 2 0 2 0 2 0 2 1 2 0 2 0 2 1 2 1 2 1 0 2 0 0 2 0 0 2 0 0 2 0 1 0 2 1 0 2 1 0 2 1 + 0 2 0
1 2 7 4 4 5 1 5 8 0 6 8 9 4 7 5 9 5 1 7 8 9 5 2 7 1 4 5 3 7 9 0 1 4 5 2
2.033 X 8.768 X 7.880 X 3.094 X 8.303 X 2.576 X 2.804 X 3.480 X 3.172 X 1.070 X 2.484 X 1.174 X 3.483 X 2.953 X 2.631 X 5.180 X 1.692 X 2.969 X 2.249 X 4.194 X 6.608 X 2.202 x 2.890 X 3.045 X 6.210 X 1.384 X 3.271 X 1.489 X 1.907 X 8.364 X 8.570 X 2.170 X 1.122 x 3.044 X 2.277 X 1.676 X
0
2
0
9
1.362 X
0
2
1
1
1.049 X
‘Integration parameter check.
lo-’ lo-)
lo4 10“
lo4 lo-’
lo4 10” 10“
lo4 10-5 10“
lo4 10”
10-5 10”
2.034 X 8.766 X 7.911 x 3.091 X 8.285 X 2.604 X 2.808 X 3.465 X 3.174 X 1.071 X 2.478 X 1.162 X 3.470 X 2.956.X 2.639 X 5.211 X 1.709 X 2.991 X 2.250 X 4.198 x 6.612 X 2.216 X 2.898 X 3.041 X 6.212 X 1.392 X 3.286 X 1.502 X 1.909 X 8.366 X 8.598 X 2.209 X 1.124 X 3.035 X 2.453 X 1.676 X 1.677 X 1.364 X 1.365 X 1.050 X 1.050 X
lo-’ 10-3
lo4 IO”
lo-*
lo4 10-I
lo-’ lo4 IO-’ 10“ IOw5 10”
lo-* 10-3
lo4 10“
lo-’ IO4 10”
10“
lo-’
Channel check.
block in the calculations with J 2 13 is 9045 or 7587. In one of the J = 15 convergence checks the small parity block was increased to 1035 channels, which was the largest single run in terms of number of channels. Table I gives the numerical parameters and Table I1 gives selected state-to-state probabilities for convergence tests for H
The Journal of Physical Chemistry, Vol. 92, No. 25, 1988 7037
Letters
0.21
TABLE III: Root Mean Square Errors in Completely Selected Transition Probabilities no. of J P range probabilities rms dev 0 110-4 181 2.5 X lod ~
~~~
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210-7
1
+
1
-
2
-
3
+
6
-
10
-
15
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all 110-4 2 10-7
all 210-4 210-7
all 210-4 2 10-7
all 210-4 2 10-7
all 2104 2 10-7
all 210-4 2 10-7
all 2104 10-7 all
259 324 136 191 256 578 884 1156 398 658 900 612 1221 1764 90 1 2361 4356 606 2145 5716 255 1007 5176
1.8 X 1.4 x 3.3 x 2.3 X 1.7 X 4.8 X 3.2 x 2.4 X 7.3 x 4.4 x 3.2 X 2.8 X 1.4 X 9.7 x 2.6 X 1.0 x 5.5 x 1.9 x 5.4 x 2.5 X 5.2 X 1.3 X 2.3 X
to” 10-6 lo“ 10“ 10” 10-7 10-7 10-7 10-7 10-7 10-7 10-7 10-7 10-8 10-7 10-7 10-8 10-7 10-8 lo-* 10-*/4.h X 10-8/1.1 X 10-9/1.9 X 10”“
Deviation for changing the integration parameters is followed by deviation for changing the channel basis.
+ p-H,(uj)
-
0.14
J
Figure 1. Convergence of the state-to-statereaction cross sections, eq 2, with respect to the maximum value of the total angular momentum J included in the sum. v = j = v’ = 0. TABLE I V Converged State-to-State Reaction Cross Sections (AZ) V‘ i‘ v=O,j=O v=O,j=2 0.109 0.103 0 1 0.205 0.199 0 3 0.199 0.187 0 5 0.0623 0.0587 0 7 0.0009 0.0009 0 9 sum(u‘=O) 0.577 0.549 1 1 t
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H o-H2(u’jl), where u and j are vibrational rotational quantum numbers, and primes denote final values. Table I shows two sets of parameters, sets A and B, that were used for the production and convergence runs, respectively, for all J except J = 1. For both parities of J = 1 we used an improved parameter set for the production run; this set involved the incorporation of the DGB basis from the set B into the set A; the convergence test run shown for J = 1 had m = 11, RIG= 1.694 au, A = 0.315 au, RmG= 4.876 au, and the other parameters were unchanged from set B. The convergence test for J = 15 was done in two steps; first the channel basis set was tested by increasing the set of maximum rotational quantum numbers (jmax(u)Iu = 0, ..., 5) to the set B values, and then the other parameters were increased with the channel basis set from set A. Since the convergence tests involve changing all numerical and channel parameters in the calculation, the deviations between transition probabilities from the production runs and the convergence tests should be a reasonable way to estimate how reliably we have solved the Schrodinger equation. Table 111summarizes the convergence defined by tests for the reactive state-to-state probabilities
Le., as absolute squares of off-diagonal transition matrix elements averaged over C when there is more than one C for a given Jj pair and summed over C’ and both identical arrangements for a fixed parity P. The results presented in Table I1 are for the initial states u = 0 , j = 0,2 for J = 0 and 1 and for the initial state u = 0, j = 2 for the other five test JP blocks. As seen from Table 11, only the transition probabilities into the most rotationally excited products, which are (much) smaller than show deviations greater than 1%. Although the present calculations yield all the state-to-state reactive cross sections for the total energy studied, we first consider results for just two initial states, namely u = 0, j = 0, for which the relative translational energy is Elel= 0.800 eV, and u = 0, j = 2, for which it is 0.756 eV. For the JP blocks represented in Table I1 there are 126 values of reactive P$l, probabilities with and only one of u = 0 and j = 0 or 2 that are greater than these (Le., the transition u = j = 0 u’ = 0,j’ = 8 from the 1block, see Table 11) shows a deviation between the production run and the test run greater than 1%. Furthermore, of 3 2 probabilities
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0.0197 0.0032 a 0.0229
1 3 5
sum(v’= 1)
“Less than 3
X 10”.
“Less than 2
0.01 11 0.003 1 b 0.0142
X
in the range 10-7-10-4, only seven show deviations greater than 1% and only one shows a deviation greater than 2.3%. Table 111 gives more global measures of convergence for the JP blocks represented in Table 11. All completely specified aujC a’u’j’C‘ transition probabilities with a = 1 and a’ = 2 (Le., squares of individuals transition matrix elements) were sorted according to their magnitude, and the rms error was calculated either for all probabilities greater than a selected minimum or for all probabilities. These results indicate that the rms deviation depending on the JP block. is between 2.4 X 10“ and 2 X Further checks on these JP blocks showed that only eight out of the 2450 completely selected transition probabilities that are greater than have deviations greater than 1%, with a maximum deviation of 1.7%, and only 43 out of 5727 probabilities greater than lo4 have deviations greater than 1%. The transitions with deviations greater than 1% are invariably those involving highly rotationally excited diatomic states with j j’ L 8. Finally we note that the average absolute percentage deviation of all transition probabilities greater than lo4 varies from 0.04% to G.3%, depending on which JP block is considered. For all transition probabilities greater than lo-’ this increases only to 0.1-0.8%. We conclude that the results are very well converged.
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Results and Discussion The convergence with respect to J,,, reaction cross sections, i.e.
of quantal state-to-state
(where k , is the initial wave number and J,,, is the maximum value of total angular momentum included in the sum) from initial states u = j = 0 and u = 0, j = 2 into the product levels u‘ = 0 and u’ = 1 are presented in Figures 1-4 and in Table IV. In Figures 1-4, the quantal results are shown as circles connected together to guide the eye. Both initial states show similar behavior with respect to convergence of the sum over J, Le., in both cases the cross sections for u’ = 0 are converged by J KZ 18, and those
7038 The Journal of Physical Chemistry, Vol. 92, No. 25, 1988 0.024
j
Letters
VSj.O,V’.l
0 o.oooo
5
10
15
L
O.OzO
5
10
15
20
J
20
Figure 5. Convergence of the vibrational branching ratio, eq 3, as a function of the maximum value of the total angular momentum J included in the various cross sections.
J
Figure 2. Same as Figure 1 except u’ = 1.
TABLE V Comparison of Present Results to Experiment quantity exptI4 theory R 0.055 0.033 u(u’=lj’=l) u(u’= lj’=3)
5
15
10
20
J
Figure 3. Same as Figure 1 except j = 2.
P t
-8 ID g
‘E
0.007
b$
0.000
0
5
10
15
20
J
Figure 4. Same as Figure 3 except u’ = 1.
for u‘ = 1 by J g 10. Table IV summarizes the converged values for selected para-ortho transitions. The vibrational branching ratio (
Xgojlf )
R = - i’ (
c
0.015 0.009
0.016 0.0031
been measured e~perimentally’~ (as have some of the state-testate cross sections as discussed below). Although the cross sections computed here correspond to fixed total energy rather than fixed initial relative translational energy, the difference in the latter is less than the experimental spreadi4 in Ere,.Thus the present results were averaged over j = 0 and 2, with the experimental 5248 weighting and compared with the corresponding experimental values. The results are shown in Table V. (A similar calculation of the branching ratio R at Ed = 0.815 eV including contributions from J = 0, 1, and 26 showed that the difference between averaging over j = 0 and 2 states with a given total energy or a given Ercl is only 2%.) Figure 5 shows that the vibrational branching ratio calculated here is well converged with respect to increasing J. When the experimental error bard4 are considered, the theoretical branching ratio in Table V is in good agreement with the experimental values. Figure 5 shows that the theoretical ratio would be much higher if only low-J contributions were included, as in previous9J0 comparisons. Table V also gives the two absolute cross sections that have been reported by the experimentalists. Again the theoretical results are averaged over a 52:48 mixture of initial j states. The cross section for u’ = 1, j ’ = 1 is in excellent agreement with experiment. The smaller cross section for u’ = 1, j ’ = 3, however, for which the experimental error bar is larger,Is is not in good agreement with experiment. Further work will be required, therefore, to assess the reliability of the potential energy surface and the experimental j ’ distribution. Acknowledgment. This work was supported in par‘ by the National Science Foundation and the Minnesota Supercomputer Institute.
gojoy )
j’
where the averages are over a 52:48 mixture o f j at fixed ,EreI, has
(14) Nieh, J.-C.; Valentini, J. J. Phys. Reo. Lett. 1988, 60,519.
(1 5 ) Valentini, J. J., personal communication.
