Effect of Rotational Excitation on State-testate Differential Cross Sections

7074

J . Phys. Chem. 1990, 94, 7074-7090

heterogeneous free-radical reactions as a function of t e m p e r a t u r e and o t h e r experimental parameters.

valuable discussions and for making some of his e x p e r i m e n t a l results available t o us.

Acknowledgment. We thank Amoco Research and Develop-

Registry No. CH,, 74-82-8; C2H,, 74-84-0; CaO, 1305-78-8; Sr, 7440-24-6; La2O,, 1312-81-8; CH,, 2229-07-4.

m e n t c e n t e r for providing financial s u p p o r t and Mark Barr for

Effect of Rotational Excitation on State-testate Differential Cross Sections: D HD -I- H

+ H,

-

Meishan Zhao, Donald G.Truhlar,* Department of Chemistry, Chemical Physics Program, and Supercomputer Institute. University of Minnesota, Minneapolis, Minnesota 55455

David W. Schwenke, NASA Ames Research Center, Mail Stop 230-3, Moffett Field, California 94035

and Donald J. Kouri Department of Chemistry and Department of Physics, University of Houston, Houston, Texas 77204 (Received: December 5, 1989: In Final Form: March 7 , 1990)

-

+

+

The differential cross sections for D H2(v = 0 , j = 0 or 1) HD(u',j? H, where u and j are vibrational and rotational q u a n t u m numbers (without primes for precollision values a n d with primes for postcollision values), are calculated a t five total energies in the range 0.82-1.35 eV by variational quantum dynamics with the most accurate available potential energy surface. Results a r e compared to previous calculations on a similar potential energy surface for the ground initial state, and the effect of rotational excitation on converged differential cross sections is illustrated for the first time. T h e effect of rotational excitation on the angular distribution is substantial, and it is much larger than the effect of rotational excitation on integral cross sections or than the difference between results obtained for the two most accurate available potential energy surfaces.

Introduction

The reaction of D with H2 has been the subject of several recent experiments'-6 that should provide a stringent test of state-oft h e - a r t dynamical theory. Converged quantum dynamical calculations have been reported for the two most accurate potential energy surfaces a t b o t h low9 and high"'-I5 energies, (1) Gotting, R.; Mayne, H. R.; Toennies, J. P. J . Chem. Phys. 1986,85, 6396. Gotting, R.;Herrero, V.; Toennies, J. P.; Vodegel, M. Chem. Phys. Lett. 1987, 137, 524. Buchenau, H.; Herrero, V. J.; Toennies, J. P.; Vodegel, M. MOLEC VII: Abstracts of Invited Talks and Contributed Papers, Assissi (Perugia), Italy, Sept 5-9 1988; pp 191-192. (2) Buntin, S. A.; Giese, C. F.; Gentry, W. R. J . Chem. Phys. 1987, 87, 1443. Buntin, S. A. Ph.D. Thesis, University of Minnesota, Minneapolis, 1987. (3) Phillips, D. L.; Levene, H. B.; Valentini, J. J. J . Chem. Phys. 1989, 90, 1600. (4) Continetti, R. E.; Balko, B. A.; Lee, Y . T. Paper presented at the International Sympsoium on Near-Future Chemistry in Nuclear Energy Field, Ibarako-Ken, Japan, Feb 15-16, 1989, and to be published in proceedings [Lawrence Berkeley Laboratory Technical Report, University of California: Berkeley: February 1989, LBL-267621. ( 5 ) Michael, M. V.; Fisher, J. R. J . Phys. Chem. 1990, 94, 3318. (6) (a) Kliner, D. A. V.; Zare, R. N. J . Chem. Phys., in press. (b) Kliner. D. A . V.; Rinnen, K.-D.; Zare, R. N. To be published. (7) Liu,B. J . Chem. Phys. 1984,80, 581. Siegbahn, P.; Liu, B. J . Chem. Phys. 1978, 68. 2457. Truhlar, D. G.; Horowitz. C. J. J . Chem. Phys. 1978. 68. 2466; errata: 1979, 71, 1514. (8) Varandas, A. J . C.; Brown, F. B.; Mead, C. A,; Truhlar, D. G.; Blais, N . C. J . Chem. Phys. 1987, 86, 6258. (9) Schatz, G. C. In The Theory of Chemical Reaction Dynamics, Clary, D. C., Ed.; Reidel: Dordrecht, 1986; p 1. Garrett, B. C.; Truhlar, D. G.: Schatz, G . C. J . Am. Chem. SOC.1986, 108, 2876.

0022-3654/90/2094-1014.$02.5O f 0

although the calculations are still limited in terms of numbers of total angular momenta or initial rotational angular momenta that have been studied. Nevertheless, it already appears that theory and experiments show surprisingly large differences, and differences between theory16-22 and experiment have also been found (IO) Haug, K.; Schwenke, D. W.; Shima, Y . ;Truhlar, D. G.; Zhang, J. Z. H.; Kouri, D. J. J . Phys. Chem. 1986, 90,6757. ( I I ) Zhang, J. 2.H.; Kouri, D. J.; Haug, K.; Schwenke, D. W.; Shima, Y . ;Truhlar, D. G. J . Chem. Phys. 1988,88, 2492. (12) Schwenke, D. W.; Mladenovic, M.; Zhao, M.; Truhlar, D. G.; Sun, Y . ;Kouri, D. J. In Supercomputer Algorithms for Reactivity, Dynamics, and Kinetics of Small Molecules; Lagan& A., Ed.; Kluwer: Dordrecth, 1989; p 131. .. .

(13) Zhao, M.; Truhlar, D. G.; Kouri, D. J.; Sun, Y . ;Schwenke, D. W. Chem. Phys. Lett. 1989, 156, 281. (14) Blais, N. C.; Zhao, M.; Mladenovic, M.; Truhlar, D. G.; Schwenke, D. W.; Sun, Y . ;Kouri, D. J. J . Chem. Phys. 1989, 91, 1038. ( 1 5 ) Zhang, J. Z. H.; Miller, W. H. J . Chem. Phys. 1989, 91, 1528. (16) Schatz, G. C.; Kuppermann, A. J . Chem. Phys. 1976, 65, 4668. (17) Schatz, G. C. Chem. Phys. Lett. 1983, 94, 183. Colton, M. C.; Schatz, G. C. Chem. Phys. Lett. 1986, 124, 256. Schatz, G. C. Annu. Rev. Phys. Chem. 1988, 39, 317. (18) (a) Mladenovic, M.; Zhao, M.; Truhlar, D. G.; Schwenke. D. W.; Sun, Y . ; Kouri, D. J. Chem. Phys. Lett. 1988, 146, 358. (b) Zhao, M.; Mladenovic, M.; Truhlar, D. G.; Schwenke, D. W.; Sun,Y . ; Kouri, D. J.; Blais, N . C. J . Am. Chem. Sot. 1989, 1 1 1 , 852. (19) Mladenovic, M.; Zhao, M.; Truhlar, D. G . ;Schwenke, D. W.; Sun. Y . ;Kouri, D. J. J . Phys. Chem. 1988, 92, 7035. (20) Zhang, J. Z. H.; Miller, W. H. Chem. Phys. Lett. 1988, 153, 465. (21) (a) Manolopoulos, D. E.; Wyatt, R. E. Chem. Phys. Lett 1989, 159, 123. (b) Manolopoulos, D. E.; Wyatt, R. E. J . Chem. Phys. 1990. 92, 810.

0 1990 American C h e m i c a l Society

Differential Cross Sections for D

-

+

+ Hz

-+

HD

+H

The Journal of Physical Chemistry, Vol. 94, No. 18, 1990 1015

+

for H para-Hz ortho-Hz H, both in full cross-section calculations and in qualitative features observed for low total angular momenta. Since the quantum dynamics is converged for a given potential energy surface, the discrepancies are presumably due to inaccuracies in the potential energy surfaces, incomplete theoretical simulation of the conditions in the experiments, or experimental error. The present contribution addresses two of these issues, namely, the choice of potential energy surface and effect of changing the initial rotational state. Additional interest in this system is also stimulated by recent and accomparisons of trajectory c a l c u l a t i ~ n s ,experiment^,^^ ~~ curate quantum dynamicszs for the H Dz H D D reaction.

TABLE I: Initial Relative Translational Energies

Methods

where k , is the wavenumber in the initial state. Summing over and summing this over u’yields the total reaction j’yields u~,~,, cross section. The state-testate differential cross sections are calculated from the distinguishable-atom reactive scattering amplitude by3’

+

-

+

The methods used for the present calculations are the same as described in detail p r e v i ~ u s l y . ’ ~In~particular ~ ~ ~ ~ we use the generalized Newton variational principle (GNVP) with complex Green’s functions for a rotationally coupled distortion potential, and the reactive amplitude density is expanded in an Lzbasis of Arthurs-Dalgarno rotational-orbital functions,29 asymptotic vibrational eigenstates expanded in harmonic oscillator functions,i1 and distributed G a u s ~ i a n in s ~the ~ radial translational coordinate. Computational efficiencies such as the use of symmetry, the direct calculation of half-integrated Green’s functions on Gauss-Legendre quadrature grids, and the use of real arithmetic at all stages up to the last step are described elsewhere.lz For a given total angular momentum J and total energy E, the state-to-state distinguishable-atom reaction “probabilities” from the initial arrangement D + H2to the two equivalent final arrangements H H D are defined in this paper by

+

where u is the initial diatomic vibrational quantum number,j is the initial rotational angular momentum quantum number, I is the orbital angular momentum quantum number associated with the relative translational motion, primes denote quantum numbers in the final state, P is the parity which is defined by

p = (-I)j+/ (2) and S~!opu’j7,rru I is a scattering matrix element with final arrangement label a’ and initial arrangement label a. The factor of 2 in ( I ) is from the two final equivalent arrangements, Le., the scattering amplitude for a’ = 3 is the same as for a’ = 2. For a specific final vibrational quantum number v’, the reaction probabilities summed over all final rotational states j ’ are given by (3)

and this in turn is further summed over u’in some of the tables. The state-to-state integral cross sections are

(4) (22) Zhao, M.; Mladenovic, M.;Truhlar, D. G.; Schwenke, D. W.; Sharafeddin, 0.;Sun, Y.; Kouri, D. J. J . Chem. Phys. 1989, 91, 5302. (23) Blais. N.C.; Truhlar, D. G. J . Chem. Phys. 1985, 83, 2201. Blais, N.C.; Truhlar. D. G. Chem. Phys. Lett. 1989, 162, 503,and references therein. (24)Marinero, E. E.;Rettner, C. T.; Zare, R. N. J . Chem. Phys. 1984, 80,4141.Gerrity, D. J.; Valentini, J. J. J . Chem. Phys. 1984,81, 1298. For a review see: Valentini, J. J.; Phillips, D. L. In Bimolecular Collisions; Ashford, M. N. R., Baggott, J. E., Eds.; Royal Society of Chemistry: London, 1989;p I . (25)Blais, N.C.; Zhao, M.;Truhlar, D. G.; Schwenke, D. W.; Kouri, D. J. To be published. (26)Schwenke, D. W.;Haug, K.; Truhlar, D. G.; Sun, Y.; Zhang, J. Z. H.; Kouri, D. J. J . Phys. Chem. 1987, 91, 6080. (27)Schwenke, D. W.;Haug, K.; Zhao, M.; Truhlar, D. G.; Sun, Y.; Zhang, J. Z. H.; Kouri, D. J. J. Phys. Chem. 1988, 92, 3202. (28)Sun. Y.;Yu, C.-h.; Kouri, D. J.; Schwenke, D. W.; Halvick, P.; Mladenovic, M.; Truhlar, D. G. J . Chem. Phys. 1989, 9 / , 1643. (29)Arthurs, A. M.;Dalgarno, A. Proc. R . SOC.London, A 1960, 254, 540. (30) Hamilton, I. P.; Light, J. C. J . Chem. Phys. 1986, 84, 306.

Ere19

E, eV 0.8200 0.9300 1.0860 1.2500 1.3500

eV

u = 0, j = 00 0.5513 0.6613 0.8173 0.9813 1.0813

Internal energy 0.2687eV (9.874X 0.2834eV (1.041X hartree).

v=O,j= Ib 0.5366 0.6466 0.8026 0.9666 1.0666 hartree).

Internal energy

where the reactive amplitude is defined by

where (...I...) is a Clebsch-Gordan ~ o e f f i c i e n t Ylm , ~ ~is a spherical harmonic function, and O,, and &, are the center-of-mass scattering angles. The angle between the incident velocity vector of H and the final velocity vector of H D in the center-of-mass coordinate system is the conventional scattering angle” and is given by 0 = K - O,, and the final result of ( 5 ) is independent of &.,

Calculations Calculations are presented for five total energies, E, and two initial states. The initial vibrational quantum number is v = 0, and we consider initial rotational quantum number j equal to both 0 and 1. The initial relative translational energies, E,,,, for the 10 sets of initial conditions considered are given in Table I. All calculations are based on the double-many-body-expansion (DMBE) potential energy surface of ref 8. As before we performed checks to illustrate the degree of convergence. We used smaller basis sets than in some of our previous calculation^,^^-^^ which was possible for two reasons: (1) Our previous calculations were the first completely converged calculations at energies above a vibrational threshold for this system on each of the two potential energy surfaces, and it always requires more care in obtaining converged results for the first time. ( 2 ) Previously we converged the whole S matrix whereas in the present work we are concerned only with experimental accuracy for reactive transitions from the two lowest energy initial states and not with convergence for higher states. As examples of the degree of convergence we will compare state-testate reaction probabilities for pairs of parameter sets for total angular momenta J = 10 and even parity at three energies. For the lower three energies the parameter sets typical of those used for J 3 5 is called 1, and for the higher two energies it is called 3. The parameter sets compared are completely different for all the numerical and basis set parameters involved in the calculation (except for the inner-quadrature vibrational-wavefunction cutoff parameter,lZ which is set equal to the very safe value in the range 10-9-10-20in all runs), and they are specified in Table 11. Typically we used larger basis sets, called 1L and 3L, for J = 0-4, as detailed in the supplementary material. The convergence checks for J = 10 are given in Tables 111 and IV, (31)Blatt, J. M.; Biedenharn, L. C. Rev. Mod. Phys. 1952, 24, 258. (32)Condon, E. U.;Shortley, G. H. Theory of Atomic Specrro; Cambridge University Press: Cambridge, England, 1935. (33)Herschbach, D. R. Ado. Chem. Phys. 1966, 10, 319.

7076 The Journal of Physical Chemistry, Vol. 94, No. 18. 1990

Zhao et al. m 3

0 (0

:

,

I

/ ' " " " ' " ' " " " " ' " ' ' ' "

-

Figure 2. Integral cross section for D + Hz(u = 0, j = 0 or 1 ) HD(u' = 0 or 1, j ' ) as a function of final rotational quantum number j', calculated by (4) at E = 0.93 eV. The open circles are the present results for j = 0, the filled circles are the results of Zhang and MillerI5for j = 0, and the open triangles are the present results for j = 1. (a, top) u'= 0. (b, bottom) u' = 1.

'>

T

'5

CL

previous calculations, the convergence is acceptable for comparison to experiment. We carried out calculations with both parities for all J from 0 through J,,,, where J,,, = 31-33, depending on the energy. These values are large enough to converge the reactive differential cross sections for all u'and j'for all scattering angles 6 at all five energies and for both initial j . Complete sets of results demonstrating this convergence and tabulating all state-to-state differential cross sections at 5' intervals for two values of J,,, are given in Tables S549 (in the supplementary material). Note that calculations for j = 0 require only one parity per J , but the results for j = 1 involve one parity block for J = 0 and two for J L 1. Thus the results with J,,, = 33 are based on 67 totalangular-momentum/parity blocks. The number of channels per total-angular-momentum/parity block ranges from 12 1 for parameter set 1 L at J = 0 to 604 for parameter set 3 for the larger parity blocks for J > 12 and 1005 for parameter set 4 for the larger parity block with J = 20.

1.35 eV v'=l

0 i T .-

a"

Results 0

1.25 eV v'=l

E? 0

b

o

i=O

A

]=I

T

Some converged reaction probabilities for J = 0 and 20, summed over j'as in (3), are given in Table V. Integral cross sections are given in Table VI. For the ground state 0' = 0), we can compare our results to those of Zhang and Millerfs for the LiuSiegbahn-Truhlar-Horowitz (LSTH) potential energy surface are compared in Figure I , which also of ref 7 . Values of shows our values for j = I , and values of are compared in Figures 2-5, which also show are values for j = I . State-to-state differential cross sections were calculated at intervals of l o , and they are plotted for the most populated final states in Figures 6-10. (As mentioned in the Calculations section, all differential cross sections are tabulated at intervals of 5' in the supplementary tables.) Differential cross sections summed over j ' for each u'are given in Figures 11-14. The differential cross sections for two backward scattering angles, 0 = 165' and 180°, are shown for both initial states for those final states with significant cross sections (larger than IO6 A2/sr) in Table VII. Our differential cross sections for j = 0 in Figure 9a may be compared to those in Figure 22 of ref 15, and the differential cross sections for j = 0 in Figures 1 1 and 13 may be compared to those in Figures I7 and I8 of ref 15. There are no previous accurate results f o r j # 0. The variations in the final rotational state distributions with initial rotational state and scattering angle are illustrated in Figures

ejd

a

a

0 9

O O

0

4

8

12

16

J

20

24

-

28

Figure 1. Reaction probability for D + H2(u = 0, j = 0 or I ) HD(u' = 0 or I , summed over j ' ) + H as a function of total angular momentum J , including both parities as in ( I ) . The open circles are the present results for j = 0, the open triangles are the present results for j = 1. and the filled-in circles are the results of Zhang and Millerls for j = 0. (a, top) E = 1.086 eV, u' = 0. (b) E = 1.25 eV, u' = 0. (c) E = 1.086 and 1.35 eV, u' = I . (d, bottom) E = 1.25 eV, u'= I .

and additional checks for J = 0 and 20 are given in Tables SI-S4. (Tables prefixed with an S are in the supplementary material, which is included in the microfiche edition of the Journal or may be ordered separately from the ACS. See the note at the end of article.) Although the basis sets are smaller than we used in some

15-19.


-

+ H2

HD + H c

I

"

?

I

'

1.35 eV v'= 1

0

0

2

6

4

8

10

o

i=O

A

]=1

12

j' Figure 3. Same as Figure 2 except E = 1.086 eV. 0

9

O O

2

7

6

4

8

10

12

14

-

j'

Figure 5. Same as Figure 2 except E = 1.35 eV. O B , , , , . , , , , , , , , , i-

,

,

/

,

'

1

-__-

t

a

05L

t

E=OEZeV

-,'=1

...... 1'=2 ]'=3

___ .... 1'=5 ]'=4

B n

B

03

t

9 O O

2

6

4

8

10

12

80

'

"

'

100

'

'

'

"

"

'

"

"

~

0

20

40

BO

120

140

160

180

0

20

40

60 80 100 I20 Scattering angle (des)

140

160

180

Scattering angle (deg)

14

j' Figure 4. Same as Figure 2 except E = 1.25 eV.

Moments of the final vibrational and rotational quantum

numbers are given in Table VI11. They are computed from the integral cross sections by (u') =

EU'U"jd/&jd VI

v'

(7)

and (j')(o3

Cj'auj~y/E~ujuy i' i'

(8)

We also computed (J')(u',O) analogously to (8) but using the differential cross sections at scattering angle 0.

-

Figure 6. Differential cross sections for the reaction D + H2(u = 0.j) HD(u'= 0, j ' ) + H for j ' = 1-5 and total collision energy E = 0.820 eV. (a, top) j = 0. (b, bottom) j = I .

Discussion The differential cross sections in this paper are calculated (within the convergence tolerances documented in Tables 111, IV, and Sl-S9)under the assumption that reaction proceeds on a

Zhao et al.

7078 The Journal of Physical Chemistry, Vol. 94, No. 18, 1990

__ c

/

-

" " I

"i."

" I " ' I " '

I

("-0. j=O. v'=O, j')

20

0

60 60 100 I20 Scsltering angle ( d e l )

40

140

160

0

160

20

40

n

(v=O, j = l , v'=O, j')

140

80 100 120 Scallering angle (deg)

60

160

I80

/ ,J'

/

/ ,'

03

0

-

+

Figure 7. Differential cross sections for the reaction D H2(u = 0, j ) HD(u'= 0, j ? + H for j ' = 2-6 and total collision energy E = 0.930 eV. (a, top) j = 0. (b, bottom) j = I . " '

C"' .05

1

E

1

" ' l " ' ~ " ~ l " ' l " ' l " ' I ~ "

,

eV

1

#

/

E = 135

05

0 5 k

e

-

60

60 LOO 120 Scattering angle (del)

160

140

160

+

Figure 9. Differential cross sections for the reaction E H2(o = 0, j) HD(o' = 0, j ' ) + H for j' = 5-9 and total collision energy E = 1.250 eV. (a, top) j = 0. (b, bottom) j = I .

1.066 V :

E = 1.086

" ~ " " " ' ~ " ' ~ " " " ' ~ " '

40

-

0

05

20

20

1

1

1

,

1

1

1

,

1

I I I / I I l

#

,

I

#

,

#

l 7 r

eV

40

E = 1 3 5 e V

60

80 100 120 Scattering angle (deg)

140

180

180

3

e

+

Figure 8. Differential cross sections for the reaction D H2(u = 0, j ) HD(o'= 0, j ' ) H for j ' = 3-7 and total collision energy E = 1.086 cV. (a, top) j = 0. (b, bottom) j = I .

+

single potential energy surface given by the DMBE function of ref 8. The assumption of a single potential energy surface is most

-

Figure 10. Differential cross sections for the reaction D + H2(u = 0, j) HD(o' = 0 , j ' )+ H for j' = 6-10 and total collision energy E = 1.350 eV. (a, top) j = 0. (b, bottom) j = I .

questionable at the highest energy, but even that energy, 1.35 eV, which is 1.40 eV below the lowest energy conical

Differential

Cross Sections for D + H2

-

+H

HD


TABLE 11: Parameter Setsa

j,,,.Su=

H+HD

IO

j,,(u=O)

JmX(u=2) j,,,.Ju=3) jmaa(u=4)

H+HD

12 8 1 2 7 2.00 0.38 4.28 1.3 55 0.60 0.0267 20.5 60 35 13 13 14

5 2.20 0.40 3.80 1.4 50 0.40 0.0335 15.0 50 30 12 12 12

e,,, 2m

D+H2

14 12 9 1 2 7 2.00 0.38 4.28 1.3 55 0.30 0.0203 15.5 60 35 13 13 14

IO

5

5

mQ

D+H2

11 9 7

8 6 5

1)

set 3

set 2

set 1 D+H2

set 4

D+Hz 13

H+HD

H+HD

I5

11 9 6 4

IO

II

13

7 5

9 8 2

8 2

5 2.20 0.40 3.80 1.4 50 0.80 0.0441 20.0 50 30 12 12 12

5 2.20 0.40 3.80 1.4 50 0.40 0.0335 15.0 50 30 12 12 12

1

1

2.00 0.38 4.28 1.3 55 0.60 0.0267 20.5 60 35 13 13 14

2.00 0.38 4.28 1.3 55 0.30 0.0203 15.5 60 35 13 13 14

12

IO

Oj,,,.,(u) is maximum rotational uantum number in the channel basis for vibrational level u, ma is number of distributed Gaussians per channel in A(G), and are the center of the first Gaussian, the spacing between Gaussians, and the center of the final Gaussian in each arrangement a. is the number of harmonic oscillator functions used to expand the vibrational channel in arrangement a,c is the Gaussian overlap parameter, eigenstates, RE,o and RE,N(v+l are the boundaries of the finite difference grid for the regular solutions of the distortion problem and the half-integrated Green's function, Aa is the average grid spacing on the part of the finite difference grid that is based on Gauss-Legendre nodes and interdispersed points, @$ is the number of points in the Gauss-Legendre quadrature for angular exchange integrals, and the other parameters are quadrature parameters defined in refs 12 and 18b. --rr-r25

I

C

"

eo

"'

l " ' l '

"

04

Lecend

-d820 eV _._ 0930 ev ...... 1080 eV

1

I "

l " ' l " ' l " ' / " ' I " ' / " '

I " Legend

-0 820

(v=O,~=Ov=l,rummed over

j )

_-

eV

-__

1250 eV .... 1350 eV

,'

0

20

0

,

,

,,'

i

1

/

I I ~ I l i ~ I I I ~ ~ l I ~ l l l l l l I , I BO 80 100 120 140 160 180

40

Scattering angle (deg)

-

Figure 11. Differential cross sections for the reaction D + H2(u = 0, j = 0) HD(u' = 0, summed over j') + H at various total collision energies.

f

-

Figure 13. Differential cross sections for the reaction D + H2(o = 0, j = 0) HD(u' = I , summed over j ' ) H at various total collision energies.

+

"

I " " '

' l " " " ' " " ~

Legend 0 8 2 ev 0.93 eV -..--1.080 eV 1.25 eV ..._135 ev

1t

I

180

"'T'

1 $

d

___

15

L

.^

0

L

0

-

~

20

~

40

I ' ' ' " I 60 UO 100 120 Scattering angle (dep)

'

'1

' "

140

'

150

--

180

-

Figure 12. Differential cross sections for the reaction D + H2(u = 0, j = 1) HD(o' = 0, summed over j ' ) + H at various total collision energies.

Figure 14. Differential cross sections for the reaction D H2(u = 0, j = 1) HD(u' = I , summed over j') + H a t various total collision energies.

is a t 2.75 eV. T h e validity of t h e second part of t h e assumption, t h a t t h e analytical form of t h e surface is given by t h e DMBE

function, is h a r d e r t o judge. W e note t h a t t h e DMBE function was especially designed t o be valid even a t higher energies,* b u t

+

7080 The Journal of Physical Chemistry, Vol. 94, No. 18, 1990 TABLE 111: Reaction Probabilities for D + H2(v = 0, j ) E i j' set 1 or 3 0 2.02E-02O 0.82 0 0 I 5.24E-02 0 0 2 6.76E-02 0 0 3 6.83842 0 0 4 5.98E-02 0 0 5 4.48E-02 0 0 6 2.7 3E-02 0 0 7 1.21E-02 0 0 8 2.89E-03 0 0 9 1.75E-04 0 0 IO I .42E-07 0 0 sum over u' = 0 3 S5E-0 I 0 2.21 E-06 0 1 1 1 5 . I6E-06 0 2 5.30E-06 0 I 0 1 3 2.84E-06 4 2.24E-07 0 1 1.57E-05 sum over o ' = 1 3.558-0 1 sum over u' = 0 and I I 0 0 1.40E-02 I 0 1 3.7 1 E-02 2 4.90E-02 I 0 I 0 3 4.998-02 4 4.3 2 E-02 I 0 5 3.15E-02 I O 6 1.87E-02 1 0 7 8.16E-03 I 0 I 0 8 2.03E-03 I 0 9 1.3 1 E-04 IO 1.13E-07 I 0 sum over u' = 0 2.53 E-0 1 I I 0 1.74E-06 1 3.876-06 I 1 I 1 2 3.56E-06 I I 3 1.82E-06 4 1.52E-07 I I 1. I I E-05 sum over u' = I sum over u' = 0 and 1 2.80E-01 0 3.17E-03 1.25 0 0 1 6.08 E-0 3 0 0 2 1.26E-02 0 0 3 4.36E-02 0 0 4 9.83E-02 0 0 5 1.35E-01 0 0 6 1. I8E-01 0 0 7 6.22E-02 0 0 8 I .9 1 E-02 0 0 9 1.13E-02 0 0 IO 1.19E-02 0 0 I 1 6.09E-03 0 0 12 1.04E-03 0 0 5.30E-0 1 sum over u t = 0

-

DH( v', j 3

Zhao et al.

+ H at Total Energy E (in eV) and at Total Angular Momentum J = 10

set 2 or 4

E

j

2.02E-02 5.248-02 6.76E-02 6.8 1E-02 5.97 E-02 4.508-02 2.74 E-02 1.228-02 3.02E-03 1.88E-04 1.70E-07 3.56E-0 I 2.41 E-06 5.55E-06 5.62E-06 2.96E-06 2.4 I E-07 1.688-05 3.568-01 I .40E-02 3.70E-02 4.8 9 E-02 4.99E-02 4.3 I E-02 3.17E-02 I .89E-02 8.39E-03 2.14E-03 1.43E-04 1.39847 2.54E-0 1 1.90E-06 4.17E-06 3.78 E-06 1.90E-06 I .64E-07 1.19E-05 2.548-0 1 3.62E-03 6.708-03 1.238-02 4.27E-02 9.69E-02 I .35E-01 1.18E-01 6.09E-02 1.888-02 1 . 1 OE-02 1.21E-02 6.1 7E-03 1.13E-03 5.23E-0 1

1.25

0 0 0 0 0 0 0 0 0 0 0

u'

1

I I 1 1 1 1

0

I

1

1 1

2 3 4 5 6 7 8 9

I 1 1 1 1 1 1

sum over u ' = 1 0 2 0 2 0 2 0 2 0 2 sum over o f = 2 sum over u' = 0, I , and 2 1 0 1 1 1 1 1 1

j'

0 0 0 0 0 0 0 0 0 0 0 0

IO 0 1

2 3 4

0 1

2 3 4 5 6 7 8 9

IO 11

12

sum over u' = 0 1

I I I I 1 1

1 1 1 1

1 1 1 1 1 1 1 1 1 1

I sum over u ' = 1 I 2 I 2 I 2 1 2 1 2 sum over u' = 2 sum over u' = 0, I , and 2

0 I 2 3 4 5 6 7 8 9 IO 0 1 2 3 4

set 1 or 3 7.65E-04 6.04E-03 1.66E-02 2.61E-02 2.528-02 I S9E-02 6.23E-03 1.4 1 E-03 3.04E-04 1.548-05 2.94E-08 9.86E-02 9.32 E-05 I .83E-04 1.248-04 3.60E-05 2.7 3E-06 4.39E-04 6.29E-0 1 1.96E-03 1.03E-02 2.67 E-02 4.5 1 E-02 5.61E-02 5.968-02 6.058-02 5.41E-02 3.56E-02 1.54E-02 5.36E-03 2.868-03 8.66E-04 3.74E-0 1 1.33E-03 7.13E-03 1.648-02 2.20E-0 2 1.968-02 1.14E-02 4.95E-03 1.78E-03 6.12E-04 5.23E-05 1.50E-07 8 S4E-02 6.628-05 1.38E-04 9.66E-05 3.1OE-05 2.42E-06 3.35 E-04 4.60E-01

set 2 or 4 9.08E-04 6.33E-03 1.71E-02 2.61 E-02 2.55 E-02 1.62842 6.42E-03 1.50E-03 2.28E-04 1.938-05 6.748-08 1 .OOE-01 I .06E-04 2.07E-04 1.36E-04 3.85E-05 2.888-06 4.92E-04 6.25E-0 I 1.81E-03 9.7 7E-03 2.58E-02 4.39E-02 5.48842 5.808-02 5.86E-02 5.338-02 3.59842 1.52E-02 5.36E-03 3.78 E-03 1.23E-03 3.67E-0 1 1.35E-03 7.14843 1.658-02 2.21 E-02 1.91E-02 I . 13E-02 5.08E-03 2.18E-03 8.36E-04 I .06E-04 3.02E-07 8.5 7E-02 7.948-05 1 S9E-04 1.12E-04 3.42845 2.78 E-06 3.87 E-04 4.54E-0 1

' 2 . 0 2 8 4 2 means 2.02 X

the effects of the inevitable inaccuracies in any analytical potential are hard to gauge. Indeed one of the main motivations of the present study is to provide state-testate differential cross sections that can be compared to experimental differential cross s e c t i ~ n s l ~ , ~ to test the potential surface. Simulations of the experimental signals of ref 2 using the present backward differential cross sections, as well as those of ref 15, are already under way.)* In the present paper, while awaiting those comparisons, and presumably other theoretical-experimental comparisons in the future, we concentrate on the trends that can be discerned from the theoretical results on their own as well as on the comparison of integral cross sections to the recent experiments of Kliner and Zare.6b Table V and Figure 1 show generally good agreement of the present reaction probabilities for j = 0 and j = I . [The much larger difference for J = 0, j = 1 is simply a consequence of the way the (2j 1 ) factor enters the formal definition ( 1 ) and the

+

(34) Porter. R. N.; Stevens, R. M.; Karplus, M.J . Chem. Phys. 1968.49, 5163. (35) Buntin, S. A.; Gentry, W.R. Personal communication.

fact that there is only a single parity block for J = 0.1 The difference between the present results for j = 0 on the DMBE surface and the j = 0 results of ref 15 for the LSTH surface is slightly larger, on the average, than the difference between the two sets of results for j = 0 and 1 on the DMBE surface. On an absolute scale through, the dependence of the j = 0 results on potential energy surface is small, indicating-as in previous comparisons1*b~2'b~36-that although the functional forms of the two surfaces are entirely different, they predict very similar dynamics. The "bumps" in the Poi-, curves of ref 15 at 1.35 eV (see Figure I C in this paper) do not show up in the present results; they may, however, be a convergence artifact in the results of ref 15 rather than a true consequence of the difference in the surfaces. Table VI and Figures 2-5 show that the state-to-state integral cross sections are more sensitive than the probabilities discussed above, which are summed over j', to differences in j and in the potential energy surface. The differences are mainly in the magnitudes of the cross sections and not in the shape of their (36) Garrett, B. C.; Truhlar, D. G.; Varandas, A. J . C.; Blais, N. C. In(. J . Chem. Kinef. 1986, 18, 1065.


+ H2

-

HD

+H

-

TABLE IV: Reaction Probabilities for D + H,( v = 0,j ) DH( v’, j ? + H at Total Energy E = 1.35 eV and at Total Angular Momentum I = 10 i U’ if set 3 set 4 0 0 0 4.98E-03O 4.388-03 9.048-03 7.99843 0 0 1 0 0 9.54E-03 9.05E-03 2 0 0 3.12E-02 3.048-02 3 4 0 0 8.368-02 8.19E-02 0 0 1.31E-01 1.28E-01 5 1.18E-01 0 0 6 1. I 8E-0 1 7 0 0 6.24842 6.14E-02 0 0 2.348-02 8 2.38E-02 9 0 0 2.208-02 2.22E-02 0 0 2.078-02 1.98E-02 IO 8.298-03 0 0 II 7.49843 0 0 1.29E-03 1.388-03 12 5.258-01 sum over u’ = 0 5.15E-0 1 1.91E-04 0 1 0 3.738-05 0 1 3.78E-03 1 3.16E-03 1.41E-02 0 1 2 1.37E-02 0 1 2.75E-02 2.768-02 3 0 1 4 3.33E-02 3.35E-02 0 1 2.28E-02 5 2.48E-02 0 1 1.08E-02 1.08E-02 6 7 0 1 3.20E-03 2.408-03 0 1 8.63E-04 8 1.99E-03 9 0 1 9.85E-04 2.68E-04 0 1 IO 3.228-05 1 . I 2E-04 1.16E-01 sum over u ’ = 1 1.19E-01 0 2 2.198-03 0 1.99843 0 2 4.49E-03 1 4.51E-03 0 2 4.478-03 4.51E-03 2 0 2 2.448-03 3 2.838-03 4 0 2 7.98E-04 I .08E-03 0 2 1.10E-04 5 1.598-04 7.41E-07 0 2 6 2.5 5E-06 1.45E-02 sum over u’ = 2 1.5 1 E-02 sum over o f = 0, 1 and 2 6.568-01 6.50E-0 1 I 0 1.568-03 0 1.38E-03 I 0 8.62E-03 1 8.608-03 I 0 2.408-02 2 2.428-02 I 0 4.168-02 3 4.14E-02 4 I 0 5.08E-02 4.95E-02 I 0 5.238-02 5 5.088-02 I 0 5.60E-02 5.41 E-02 6 7 I 0 5.59E-02 5.43842 I 0 8 4.00E-02 3.96E-02 1 0 1.86E-02 1.82E-02 9 1 0 8.20E-03 IO 8.62E-03 I 0 6.20E-03 6.828-03 11 I 0 3.338-03 12 2.958-03 sum over u‘ = 0 3.67E-0 1 3.61E-01 I 1 5.61 E-04 0 6.27E-04 I 1 1 4.268-03 4.62E-03 I 1 1.32842 2 1.41E-02 I 1 3 2.26E-02 2.258-02 I 1 4 2.378-02 2.28E-02 I 1 1.65E-02 1.56E-02 5 I I 8.25E-03 6 8.17E-03 1 1 7 4.1 1E-03 3.69E-03 I 1 2.18E-03 8 2.36E-03 1 1 4.56E-04 9 7.79E-04 I 1 2.76E-05 IO 7.35E-05 sum over u ‘ = 1 9.50E-02 9.62E-02 I 2 1.90E-03 0 1.83E-03 I 2 4.358-03 1 4.208-03 I 2 4.08E-03 2 4.15E-03 I 2 2.38E-03 3 2.528-03 I 2 4 7.25E-04 9.04E-04 I 2 1.30E-04 5 1 .O1E-04 1 2 8.0 1E-07 6 2.12E-06 sum over v’ = 2 1.35E-02 1.378-02 sum over u’ = 0, 1 , and 2 4.7 1 E-0 1 4.768-01 “ 4 . 6 3 8 4 3 means 4.63

X

dependence on j’. A very significant result of this study, since theory and experiment have often been compared with differential initial conditions for j , is that raising j by one quantum does not increase the j’value at which the final distribution peaks. The effect of changing the potential energy surface on the j’distributions, although small, is not entirely insignificant. In particular the LSTH surface leads to a (very slightly) hotter j’distribution


-

TABLE V Reaction Probabilities for D + H2(v = 0,j ) HD( v’, H for Selected E and J J E U‘ j = O j = l S.l6(-ly 2.85(-1) 0 0.82 0 1 1.02(-4) 2 .OO (-4) 2.85(-1) sum 5.16(-1) 4.70(-1) 2.23(-1) 0.93 0 7.49(-2) 1 1.27(-1) 5.97(-1) 2.98(-1) sum 5.21(-1) 2.09(-1) ,086 0 7.76(-2) 1 1.l2(-1) 2.87(-1) sum 6.33(-1) 5.21 (-1) 1.78(-1) .25 0 9.38(-2) 1 1.41(-1) 1.14(-2) 8.89(-3) 2.81 (-1) sum 6.73(-1) 0 1.50(-1) 5.02(-1) .35 I . I 2(-1) 1 1.86(-1) 2 2.1 O(-2) 2. I3(-2) 2.83(-1) 7.09(-1) sum 1 0.82 0 5.11(-1) S.IO(-l) 1 9.52(-5) 2.23(-4) 5.10(-1) sum 5.11(-1) 4.92(-1) 0.93 0 4.79(-1) 1 7.01(-2) 1.18(-1) sum 5.62(-1) 5.97(-1) 1.086 5.14(-1) 5.21(-1) 0 1.17(-1) 1 1.10(-1) 6.3l(-l) sum 6.3 I (-I) 5.22(-1) I .25 0 5.24(-1) 1 1.43(-1) 1.49(-1) 2 1.09(-2) 8.76(-3) 6.80(-1) sum 6.78(-1) 1.35 0 5.14(-1) 5.25(-1) l.79(-1) l.68(-1) 1 2 2.14(-2) 2.51 (-2) 7.14(-1) 7.18(-1) sum 1.61(-4) 20 0.82 0 1.36(-4) 1 5.7(-12) 6.7(-12) 1.61(-4) sum 1.36(-4) 1. I I(-2) 0.93 0 9.41 (-3) 1 2.51(-7) 2.13(-7) 1.11(-2) sum 9.41 (-3) 1.086 0 1.19(-1) 1.17(-1) 2.02(-4) 1 2.29(-4) sum I . I 7(-1) 1.19(-1) I .25 0 3.00(-1) 2.75(-1) 1 6.01(-3) 7.09(-3) 1.59(-9) 1.4 1 (-9) 2 sum 3.06(-1) 2.82(-1) 3.59(-1) 3.35(-1) 1.35 0 1 2.58(-2) 2.24(-2) 8.12(-7) 9.88(-7) 2 sum 3.85(-1) 3.67(-1)

Summed over j’)

+

“5.16(-1) means 5.16

X

IO-’.

than the DMBE one. The effect is smaller than the effect produced in a previous model study3’ of two surfaces that differ primarily in the bending potential for asymmetric geometries. Thej’distribution for u ’ = 1 in Figure 5 can be compared to the recent measurement of Kliner et a1.,6bwhich was carried out under similar initial conditions (room temperature j distribution, relative translational energy of about 1.05 eV). The experimental distribution peaks at j’ = 4 with a value 90% as high as j ’ = 3 and has a full width at half-maximum (fwhm) of SI/., quanta. This distribution is in very good agreement with all three curves in the bottom of Figure 5 , which peak at j ’ = 3 or 4 and have fwhm’s of between 5 and 5 ’ / , quanta. In contrast the quasiclassical trajectory distribution of j’ for u’ = 1 at this energy, as calculated3*using the LSTH surface, peaks at j’ = 5 and has a fwhm of 5 1 / 2 quanta. Table VI also shows the cross sections summed over j’for a (37) Blais, N. C.; Truhlar, D. G.;Garrett, B. C. J. Chem. Phys. 1985.82, 2300. (38) Blais, N. C.; Truhlar, D. G . J. Chem. Phys. 1988, 88, 5457.

The Journal of Physical Chemistry, Vol. 94, No. 18, 1990

7082

TABLE VI: Integral Cross Section (A') for D E

0.82

j'

u'

0 0 0 0 0 0 0 0 0 0 0 sum over u' = 0 I I

0 I 2 3 4 5 6 7 8

9 IO 0 1 2 3 4

1

I I sum over u ' = 1 sum over u' = 0 and 1 0.93 0 0 0 0 0 0 0 0 0 0 0 0 sum over u' = 0

0 1

2 3 4 5 6 7 8 9 IO 11

0 1 2 3 4 5 6

1

I I I I 1

I sum over u t = 1 sum over u' = 0 and 1 1.086 0 0 0 0 0 0 0 0 0 0 0 0 0 sum over u' = 0 I

0 1

2 3 4 5 6 7 8

9 IO 11

12 0

I

1

I

2 3 4 5 6 7

I I I I I I sum over u ' = 1 sum over u' = 0 and 1 1.25 0 0

O4.71(-2) means 4.71

X

8

0 I

+ H2( = 0, j ) Y

-

HD( v', j 3 E

Zhao et al.

+ H at Various Total Energies E (in eV)

j = o

j = 1

u'

j'

j-0

j = \

4.71(-2)' 1.29(-1) IXO(-l~ 1.93(-1) l.69(-1) 1.19(-1) 6.51(-2) 2.54(-2) 5.56(-3) 3.09(-4) 2.25(-7) 9.35(-1) 8.75(-6) 1.82(-5) l.61(-5) 8.11(-6) 7.31(-7) 5.19(-5) 9.35(-1) 3.63(-2) 1.04(-1) l.59(-1) l.97(-1) 2,13(-l) l,97(-l) 1.53(-1) 9.15(-2) 3.88(-2) 8.87(-3) 4.57(-4) 2.83(-7) I .20 6.19(-3) l.21(-2) 8.52(-3) 3.02(-3) 4.68(-4) 1.46(-5) 8.02(-7) 3.03(-2) 1.23 2.36(-2) 6.99(-2) 1.14(-1) l.57(-1) 1.96(-1) 2.21(-1) 2.24(-1) 1.92(-1) l.36(-1) 6.71(-2) 1.95(-2) 1.94(-3) 3.27(-9) 1.42 1.06(-2) 2.61(-2) 2.94(-2) 2.15(-2) 1.06(-2) 3.17(-3) 4.86(-4) 5.95(-5) 8.71(-7) 1.02(-1) 1.52 l.65(-2) 4.82(-2)

3.83(-2) I .08(-1) I.%(-]) I .74(- I ) I .55(-l) I . 10(-1) 5.96(-2) 2.28(-2) 4.85(-3) 2.67(-4) 1.93(-7) 8,3I(-l) I .85(-5) 3.55(-5) 2.47(-5) 8.94(-6) 6.99(-7) 8.82(-5) 8.32(-1) 3.25(-2) 9.44(-2) 1.47(-1) l.84(-1) 1.99(-1) 1.84(-1) 1.44(-1) 8.67(-2) 3.7 1(-2) 8.55(-3) 4.42(-4) 2.69(-7) 1.12 3.95(-3) 7.87(-3) 5.76(-3) 2.23(-3) 4.39(-4) 2.96(-5) 7.34(-7) 2.03(-2) 1.14 2.29(-2) 6.87(-2) 1.13(-1) l.56(-1) 1.91(-1 ) 2.12(-1) 2.1 2(-I) 1.82(-I) l.29(-1) 6.64(-2) 1.97(-2) 1.99(-3) 6.78(-9) 1.38 8.19(-3) 2. I7(-2) 2.7 I(-2) 2. 18(-2) 1.18(-2) 3.99(-3) 6.9 I(-4) 6.40(-5) 8.14(-7) 9.54(-2) 1.47 1.54(-2) 4.66(-2)

0 0 0 0 0 0 0 0 0 0 0 sum over u' = 0

2 3 4 5 6 7 8 9 10

1

0 I 2 3 4 5 6 7 8 9 IO

7.84(-2) 1.09(-1 j l.42(-1) 1 .SO(-I) 2.20(-I) 2.40(-1) 2.22(-I) l.69(-1) 9.14(-2) 3.13(-2) 4.93(-3) I .55 9.38(-3) 2.63(-2) 3.76(-2) 4.02(-2) 3.45(-2) 2.24(-2) 1.13(-2) 3.74(-3) 8.74(-4) 5.20(-5) 1.23(-7) 1.86(-1) 6.14(-4) 1 . I I(-3) 6.69(-4) 1.74(-4) 1.3I(-5) 2.58(-3) 1.75 1.37(-2) 3.9 7(-2) 6.36(-2) 8.67(-2) 1.13(-1) l.47(-1) I .89(-1) 2.27(-1) 2.47(-1) 2.20(-1) I .55(-1) 7.45(-2) 2.05(-2) 2.27(-3) 1.60 7.69(-3) 2.29(-2) 3.68(-2) 4.53(-2) 4.53(-2) 3.55(-2) 2.40(-2) 1.23(-2) 4.99(-3) 1.23(-3) 7.59(-5) I .03(-7) 2.36(-1) 2.23(-3) 5.31(-3) 5.65(-3) 3.76(-3) 1.35(-3) 2.10(-4) I .61(-6) 1.85(-2)

7.86(-2) I . I 2(-I j l.48(-1) l.84(-1) 2.16(-1) 2.30(-1) 2.1 2(-1) 1.65(-1) 9.20(-2) 3.26(-2) 5.23(-3) 1.54 7.90(-3) 2.3 5(-2) 3.55(-2) 3.87(-2) 3.32(-2) 2.14(-2) 1.1 I(-2) 4.06(-3) 1.05(-3) 6.62(-5) 1.47(-7) l.77(-1) 4.79(-4) 8.67(-4) 5.28(-4) 1.39(-4) 1.07(-5) 2.02(-3) 1.72 1.31(-2) 3.89(-2) 6.42(-2) 9.08(-2) 1.21(-1) l.55(-1) l.93(-1) 2.23(-1) 2.37(-I) 2.1 1(-1) l.53(-1) 7.78(-2) 2.27(-2) 2.49(-3) 1.60 6.79(-3) 2.10(-2) 3.49(-2) 4.32(-2) 4.3 I(-2) 3.41 (-2) 2.31(-2) 1.19(-2) 4.84(-3) 1.25(-3) 7.97(-5) 1.09(-7) 2.24(-I) 2.32(-3) 5.54(-3) 5.79(-3) 3.67(-3) 1.23(-3) 1.80(-4) 1.45(-6) 1.87(-2)

1.85

1.85

I 1 1 1 1 1

1 1

1

I sum over 2 2 2 2

11

12

L.'= 1

0 1

2 3 4

sum over u" = 2 sum over u'= 0, 1 and 2 I .35 0 0 0 0 0 0 0 0 0 0 0 0 0 0 sum over u' = 0

0 I 2 3 4 5 6 7 8 9 IO 11

12 13

0 1 2 3 4 5 6 7

I 1 1

I

8

1 1 1 1

9 IO 11

sum over 2 2

c'= I

L

2 L

2 2 sum over u' = 2 sum over c' = 0, 1 and 2 ~

0 1 2 3 4 5 6

IO-*.

given u', and from these results we can compute the vibrational branching ratio Rj =

nojl/

nojo

These ratios are listed in Table IX. Exceot at the lowest enerev where the amount of vibrational excitation is very small, rotational

excitation slightly decreases the vibrational branching ratio. For both initial rotational states, the vibrational branching ratio is a monotonically increasing function of energy, contrary to the experimental3 finding. The controversy3~'3~'5~18-H)~22~39 about whether

I .

(39) Nieh, J.-C.; Valentini, J . J. Phys. Rev. Lett. 1988, 60, 519


-

+ H2

HD

+H

The Journal of Physical Chemistry, Vol. 94, No. 18, 1990 7083 E = 0.82 eV; scattering angle =135 degrees

E = 0.82 eV; scattering angle = 90 degrees 0

v'-O,J=O A v ' = O , J = l

0.036

0.012

-.-62 c

c .-m v

m

'0

e,

6

L

VI L

0.009

0.027

0 Q

Q

E

VI

c

I

VI

VI

m

050

5

? t

5

0.006

x

v

v

0

.-0

U

U

VI al

VI al

C

C

c

VI

-

-

0.018

3

VI cr

VI

e -

0.003

0.009

U

m C 2 0,

miC L al

c v

.-

u

0

0 0

1

2

3

4

5

6

7

8

9

1

1

2

9

10

A V'-D,]-l

-B -6 c

VI

0.042

z0

Q)

Q

-$

-8

VI

0.04

VI

m

m

05

05 2 40-

0.028

cr v c VI

VI

v

-

c .-0

.-0 c

u

4 Eu -

8

0.06

c

9

7

0 V'-D,j-O

B -6 ??

6

E 0.82 eV; scattering angle =180 degrees

A V'-O,J-1

0.056

L

5

-

E = 0.82 eV; scattering angle =165 degrees

VI

4

final rotational state j'


0 v'.O,j.O

3

U

0.02

VI

VI

VI

0

0.014

L U

m

m

.-c c

c

C

?? G e,

2

.-

v

-0

0

0

0

1

2

3

4

5

6

7

8

9

0

10

-

1

2

3

4

5

6

7

8

9

10


final rotational state j' Figure 15. Distribution of final rotational states for D + H2(u = 0, j = 0 or 1 ) HD(o' = 0, j'j at various scattering angles for E = 0.82 eV. denotes the results fori = 0, and A denotes the results for j = 1. (a) 0 = 90'. (b) 0 = 135'. (c) 0 = 165'. (d) 0 = 180'. there are resonance peaks in the vibrational branching ratio as a function of energy remains unsettled, even though we now include the effect of rotational excitation of H2. To try to understand the discrepancy better, we will compare the individual state-to-state cross sections. The cross sections at 0.93 and I .086 eV can be compared to the experimental results of Phillips et al.,3 who measured HD

II

product distributions at relative translational energies, E,,, of 0.67 and 0.79 eV. The difference of these translational energies from ours (see Table I ) is smaller than the experimental spread, so the comparison should be informative. Since the experiment was carried out with a Boltzmann distribution of H, rotational states at room temperature, and since the population of t h e j = 1 state is 1.68 times larger t h a n j = 0 for H2 at 293 K, we weighted the

7084

The Journal of Physical Chemistry, Vol. 94, No. 18, 1990

Zhao et al.

E = 0.93 eV; scatteringangle = 90 degrees

0.044

0.014

C

fI

v

-f in L

0.01 05

0.033

Q,

n v)

c v)

131

2e, L

4 W

0.007

0.022

v)

v

C

e

e

U v) 0

I

0.0035

v) v)

2U m

-

0.01 1

c

C

0)

5 ! t .v

0

0 0

1

2

3

4

5

6

7

8

9

1

0

1

0

1

1

2

3

4

5

6

7

8

9

10

I



E = 0.93eV; scattering angle = 180 degrees

E E 0.93eV; scattering angle = 165 degrees

/.C % v

e

Q,

e v)

0.0435

L.

Q,

Q v)

c

v)

D

s

0,

L

0.029

3 m

v) W

c

e

c U

in 0, v) v)

2

0.0145

U

m

c

c

0

lc3+lL0

Figure 16.

1

2

3

4

5

6

7

8

9

1

0

e,

L

e,

G -0

1

1

final rotational state j' Same as Figure 15 except for E = 0.93 eV.

= 1 states in the ratio 1.00:1.68 for this comparison, which is given in Table X . First of all we see that the theoretical cross sections are systematically lower than the experimental' ones. Nevertheless they agree with experiment within a factor of 2 except for the following: (2) The u' = 1 cross sections at the lower energy are too low by factors of 8-30; (ii) the u'= 0, j ' = 1 and 2 cross sections are lower than experiment by more than a factor of 2 , and (iii) the high-j'

j = 0 and j

0

1

2

3

4

5

6

7

8

9 1 0 1 1

final rotational state j' tails (u' = 0, j ' = 8 and 9; u' = 1, j ' = 4-6) at the higher energy fall off more rapidly than experiment. The reason for i is that the u ' = 1 cross sections are much larger than expected at the lower energy; e.g., they are much larger than the higher energy. This was interpreted3 as a resonance, but we pointed out12on the basis of earlier calculations on the DMBE surface that theory does not agree with experiment for the occurrence or magnitude of this resonance. Zhang et aI.l5 subsequently found a similar discrepancy


+ H2

-

HD

+H


-

E = 1 .OMeV; scattering angle = 135 degrees

E E 1.086 eV; scattering angle 90 degrees

l o v ~ o .i.0

A v'.~.

v*-1.

i-1

i-0 0 v ' - 1 , 1 - 1

0.024

0.048

-

c

C

m

d

5

s

$

w

e

v)

v)

L Q)

i

0.016

c

z5

c v)

v)

0,

4

Q)

z

Q)

2

0.024

5:

v) 0-

v

v

-

-

C

C

.-0

.-0

U

4

0.036

n

Q

U

x -g

0.008

v)

v)

2 U

.-m

L 0

e,

-2

E D

t '0

0.012

.-m

C

C

Q)

0 0

1

2

3

4

5

6

7

8

9

1 0 1 1 1 2

0

1

2

3


v'-OJ-0

A v'-O,j-l

5

6

7

8

9

101112

final rotational state j' E = 1.086 eV; scattering angle = 180 degrees

E = 1 . O B eV; scattering angle = 165 degrees 0

4

N V'-l,j-0

0

V'-1,j-1

0

v'.O,j.O

A v'-0,j-1

3

5

X

V'm1,j-0

P

V'.l,j-1

0.052

c

B

-s

al n

0.039

-i

0.04

z v)

$

0.026

5:

v

-

c .-0

U

0.02

v) e, v)

0

b .-m

0.013

c

C

2

0

0 0

1

2

3

4

5

6

7

8

9 1 0 1 1 1 2


0

-

1

2

4

6

7

8

9

101112


Figure 17. Distribution of final rotational states for D + H2(u = 0, j = 0 or 1 ) HD(u'= 0 or 1 , j') at various scattering angles for E = 1.086 eV. 0 denotes the results f o r j = 0, u ' = 0; X forj = 0,u ' = 1; A for j = 1, u ' = 0; and v f o r j = 1, u t = 1 . (a) B = 90'. (b) B = 135'. (c) B = 165'. (d) 0 = 180'.

for the LSTH surface. This is one of the main unresolved problems in our understanding of this reaction, unless the experiment is wrong. Discrepancy ii was also noted by Zhang and Miller,Is who stressed that the difference could be due to the limitation of their calculations t o j = 0. However, the inclusion o f j = 1 in the present calculations does not remove the problem, nor does the change

in potential energy surface. Discrepancy iii is also troubling, especially in light of the good agreement of theory23and experiment" for j'distributions in H + D2 and in D H2 a t E = 1.35 eV, where the latter comparison is discussed above. There does not appear to be any sign of a resonance at E = 1.086 eV, so this discrepancy is apparently not clouded by that problem. We note,

+

7086 The Journal of Physical Chemistry, Vol. 94, No. 18, 1990

Zhao et at. E = 1.25eV; scattering angle = 135 degrees


0.048

0'033

I C

.a -0

-6 ln L Q)

Q

-

0.032

ln [5,

05

Q)

L

9 lCT n Y

c 0 .-e V

4

0.016

v) ln

V m

.-c K

e Q)

s -0 0 0

1

2

3

4

5

6

7

8

9 loll'



-


E 1.25 eV; scettering angle 165 degrees I

0.044

0.036 h

C

43

s

c v)

0.033

Bn 0.024 c in

i?

5

0.022

CT

ln

Y

C

.-0

I

V

4

0.012

ln v)

g

0.011

zm C 0

L

0

0

1

2

3

4

5

6

7

8

9 101112


0

1

2

3

4

5

6

7

8

9 1011'


Figure 18. Same as Figure 17 except E = 1.25 eV.

however, a similar disagreement of converged cross sections at a nonresonant energy for the H + para-H2 reaction, where the theoreti~al~~j'distribution for u' = 1 product molecules decreases much more rapidly than the e ~ p e r i m e n t a one. l~~ Figures 6a, 7a, 8a, 9a, and 10a show, as found previously by Zhang and MillerIs for the LSTH surface, that the reactive scattering shifts forward asj'increases. (We have also found this effect for H H,.18b*40)The present differential cross sections

+

are somewhat smoother than those in ref 15; our experience indicates that this is probably a consequence of better convergence. It is not clear whether the remaining small-scale oscillations in our own results, for E 1 1.086 eV near 0' and for E = 1.25 eV (40)Dong, H.;Mladenovic, M.; Zhao, M.; Truhlar, D. G.; Schwenke, D. W. Unpublished results.

Differential Cross Sections for

- -

D + H2 HD + H


-

-

E 1.35 eV; scattering angle 135 degrees

E = 1.35 eV; scattering angle 90 degrees

0 v'.O.j.O

A V'.O.jml

X

V'm1,j-0

0

~'-1,j-l

0.04

0.04

I

0.03

0.03

0.02

0.02

0.01

0.01

0

0



E = 1.35 eV; scattering angle = 165 degrees 0

0'03

v'-O.j-0

A v'm0,j-1

X

v'-l,j-O

1

v'-l,j-1

I 0.03

0.02

0.02

0.01 0.01

0

0 0

final rotational state j' Figure 19. Same as Figure 17 except E = 1.35 eV.

near 180°, are a quantitative artifact of incomplete convergence (e.g.,jmaxvalues), but since these features are very small and the general good convergence of our calculations is well documented, we did not study these features in detail. The comparison of part a to part b in each of Figures 6-10 is the central result of this study. We see that, quantitatively, the dependence of the differential cross sections on initial j is very large. This is somewhat easier to appreciate in Table VII. where

1

2

3

4

5

6

7

8

9 10111213


we compare differential cross sections for the backward scattering directions studied in ref 2. At 0 = 165O,the ratio of the stateto-state differential cross sections to a particular final state from j = 1 to that for j = 0 varies from 0.72 to 1.83 at 0.82 eV, from 0.63to 4.4at 0.93 eV, from 0.56to 3.0 at 1.086 eV, from 0.60 to 2.7 for the larger cross sections at 1.25 eV and to 3.9 for the smallest cross section at that energy, and from 0.61 to 1.6 for the larger cross sections at 1.35 eV with a ratio of 0.25for the smallest

7088 The Journal of Physical Chemistry, Vol. 94, No. 18, 1990 TABLE VII: State-testate Differential Cross Sections (A*/sr) for D 6 = 165' e = 1800 E o ' j ' j = O j = I j = O j = l 0.82

0.93

1.086

1.25

0

0 0 0 0 0 0 0 0 0 0 0 I

IO O

1

1

I 1 I 0 0 0 0 0 0 0 0 0 0 0 0 I I I 1 1

2 3 4 0 1 2 3 4 5 6 7 8 9 IO II O 1 2 3 4

1 2 3 4 5 6 7 8 9

1

5

1 0 0 0 0 0 0 0 0 0 0 0 0 0 I

6 0 1 2 3 4 5 6 7 8 9 10 11

I2 O

1

1

1 1 I 1 1 I 1 0 0 0

2 3 4 5 6 7 8 0 1 2

1.588-02' 4.14E-02 5.34842 5.12E-02 3.94E-02 2.39E-02 1.13E-02 4.0 I E-03 8.29E-04 4.15E-05 2.84E-08 7.01 E-06 1.22E-05 6.87 E-06 I .87E-06 1.298-07 1.85E-02 4.70E-02 5.61 E-02 4.8 1 E-02 3.23E-02 1.73E-02 7.48 E-03 2.62E-03 7.858-04 1.40E-04 6.44E-06 6. I9E-09 3.04E-03 6.01 E-03 4.34E-03 1.61E-03 2.65E-04 6.51 E-06 2.10E-07 I .76E-02 4.43E-02 5.1 5E-02 4.1 5E-02 2.5 5E-02 I .24E-02 7.01 E-03 6.19E-03 5.2OE-03 3.34E-03 9.55844 9.45 E-05 I .02E-09 3.7OE-03 8.34E-03 7.97E-03 4.80E-03 1.89E-03 4.5 1 E-04 5.14E-05 6.55E-06 I .22E-07 1.22E-02 3.04E-02 3.488-02

"1.58E-02 means 1.58

1.13E-02 3.07E-02 4.19E-02 4.3 1 E-02 3.58 E-02 2.35E-02 1.21E-02 4.48E-03 9.12E-04 4.46E-05 2.60E-08 1.16E-05 2.10E-05 1.26E-05 3.3OE-06 1.80E-07 1.16E-02 3.11E-02 4.17E-02 4.278-02 3.67E-02 2.6 9 E-02 1.688-02 8.37E-03 3.1 3E-03 6. I2E-04 2.83E-05 1.548-08 2.048-03 4.09E-03 3.04E-03 I .20E-03 2.27 E-04 9.678-06 1.33E-07 1.04E-02 2.86E-02 3.94E-02 4.09842 3.47 E-02 2.46E-02 1.50E-02 8.3 1 E-03 4.5 3 E-03 2.24E-03 5.87E-04 5.26E-05 1.93E-09 3.22E-03 7.21 E-03 6.90E-03 4.268-03 1.84E-03 6.078-04 1.528-04 1.8 1 E-05 1.54E-07 7.27843 2.028-02 2.86E-02

1.838-02 4.7OE-02 5.84E-02 5.348-02 3.89E-02 2.21 E-02 9.77E-03 3.328-03 6.66E-04 3.20E-05 2.58E-08 8.89 E-06 1.52E-05 8.078-06 I .90E-06 1.2 1E-07 2.27E-02 5.56E-02 6.2OE-02 4.76842 2.688-02 1.06E-02 2.69E-03 4.20844 8.16E-05 2.54E-05 1.27E-06 3.89E-09 3.368-03 6.64E-03 4.79E-03 1.788-03 2.938-04 6.86E-06 2.29 E-07 2.37E-02 5.65 E-02 5.71 E-02 3.5 5E-02 1.32E-02 2.40E-03 3.738-03 7.948-03 7.52 E-03 4.49E-03 I . 17E-03 1.03E-04 1.09E-09 4.35E-03 9.43E-03 8.36E-03 4.4 2E-03 1.37843 2.05E-04 3.43E-06 2.61 E-06 7.368-08 1.92842 4.25E-02 3.65E-02

+ H,(

1.23E-02 3.32E-02 4.46E-02 4.498-02 3.638-02 2.33E-02 1 . I 7E-02 4.258-03 8.43E-04 3.96E-05 2.468-08 1.37E-05 2.468-05 1.46E-05 3.63 E-06 1.878-07 1.30E-02 3.43E-02 4.478-02 4.42842 3.628-02 2.518-02 1.48E-02 6.90E-03 2.448-03 4.48E-04 2. OOE-0 5 1.25E-08 2.24E-03 4.508-03 3.34E-03 1.328-03 2.45E-04 9.64E-06 1.35E-07 1.17E-02 3.17E-02 4.28E-02 4.22E-02 3.32E-02 2.14E-02 I . I5E-02 6.15E-03 3.80E-03 2.3 1E-03 6.6 7E-04 6.06 E-0 5 2.07 E-09 3.65E-03 7.88E-03 7.078-03 3.99E-03 I .5 I E-03 5.01E-04 I S5E-04 2.07E-05 1.63E-07 7.9 1 E-03 2.32E-02 3.47E-02

Y

= 0,j )

E

-

HD( v', j ' )

+ H at Various Total Energies E (in eV) e = 1650

0'

j'

j = O

0 0 0 0 0 0 0 0 0

3 4 5 6 7 8 9 IO I1 12 O

2.81E-02 1.958-02 1.57E-02 1.66E-02 1.72E-02 1.34E-02 7.62E-03 2.92E-03 7.89E-04 8.14E-05 4.33E-03 9.72E-03 8.89E-03 4.80E-03 1.998-03 1.25843 I .47E-03 8.73E-04 2.13E-04 1.03E-05 3.98E-09 2.23E-04 3.98E-04 2.38E-04 6.028-05 4.20E-06 9. I6E-03 2.27E-02 2.62E-02 2.20E-02 1.72E-02 I .68E-02 1.96E-02 2.06E-02 1.66E-02 9.628-03 4.42E-03 1.87E-03 4.47E-04 3.62E-05 3.75E-03 8.42843 7.66E-03 4.47E-03 2.91E-03 2.998-03 3.45 E-03 2.47E-03 1.05E-03 I .80E-04 6.71E-06 8.07E-09 3.90E-04 9.1 7E-04 9.93E 4 4 7.09E-04 3.01 E-04 5.64E-05 4.84 E-07

0

I 1

1

1 1 1 1 1 1 1 1 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 3 4 5 6 7 8 9 IO 0 1 2 3 4 0 1 2 3 4 5 6 7 8 9 IO II 12 13

1 1

0 1

1 1 1 1 1 I 1 1 I

2 3 4 5 6 7 8 9 IO II 0 1 2 3 4 5 6

1

1.35

Zhao et al.

1

2 2 2 2 2 2 2

j-1

3.10E-02 2.87E-02 2.4OE-02 1.958-02 1.58842 1.19E-02 8.05843 4.478-03 1.55E-03 2.19E-04 3.69E-03 8.46E-03 8.228-03 5.18E-03 2.87E-03 I .84E-03 1.588-03 8.07 E-04 2.1 IE-04 I . 19E-05 I .57E-08 I .84E-04 3.29E-04 1.698-04 4.89E-05 3.35E-06 5.65E-03 3.57E-02 2.22E-02 2.448-02 2.32E-02 2.098-02 1.90E-02 1.798-02 1.60E-02 1.19E-02 7.1 8E-03 3.02E-03 7.1OE-04 4.86E-05 3.01E-03 6.98E-03 7.00E-03 4.95E-03 3.80E-03 3.25E-03 2.97E-03 1.798-03 6.99844 1.1OE-04 4.14E-06 1.99849 4.60 E-04 1.05E-03 1.09E-03 7.048-04 2.63E-04 4.5 2E-05 4.06E-07

6 = 180° j = O

j = ]

1.638-02 5.57E-03 I . I2E-02 2.20E-02 2.41 E-02 1.498-02 5.63E-03 9.33 E-04 6.60E-05 6.6 I E-07 5.19E-03 I . 1OE-02 8.908-03 3.708-03 8.7 5E-04 1.01E-03 1.888-03 I .20E-03 2.97E-04 1.49E-05 5.92E-09 2.3 5E-04 4.19E-04 2.50E-04 6.308-05 4.4OE-06 I .58E-02 3.33E-02 2.52E-02 9.128-03 5.8 1E-03 1.78E-02 2.92E-02 2.7 1E-02 1.4I E-02 4.03E-03 1.88E-03 2.17E-03 7.47E-04 6.5 3E-05 4.44E-03 9.3 I E-03 7.40E-03 3.088-03 1.97E-03 3.308-03 4.33E-03 2.998-03 1.14E-03 1.58E-04 5.19E-06 8.38 E-09 4.17E-04 9.87E-04 9.84E-04 6.8 I E-04 2.968-04 5.78E-05 4.99E-07

3.70E-02 2.99E-02 1.99E-02 1.38E-02 1.40E-02 1.38E-02 1.02E-02 4.98E-03 1.24E-03 1.238-04 4.19E-03 9.36E-03 8.408-03 4.748-03 2.35E-03 1.74E-03 1.75E-03 8.99E-04 2.07 E-04 I .04E-05 1.28E-08 I .94E-01 3.45 E-04 2.05E-04 5.13E-05 3.60E-06 5.73E-03 1.678-02 2.48E-02 2.628-02 2.218-02 1.728-02 1.61 E-02 1.82E-02 1.79E-02 I .3 1 E-02 6.90E-03 2.23E-03 4.27E-04 2.53E-05 3.318-03 7.39E-03 6.7 2 E-03 4.1 5E-03 3.39E-03 3.70E-03 3.45E-03 1.83E-03 6.078-04 8.50E-05 3.34E-06 1.27E-09 4.84E-04 1.06E-03 1.07E-03 6.84E-04 2.608-04 4.65 E-05 4.26E-07

X

cross section at the highest energy. These ratios fluctuate even more at 180'; e.g., at 0.93 eV they vary from 0.57to 30, and at 1.086 eV they vary from 0.49 to 45! Clearly, differential cross sections for backward scattering with rotational resolution are very sensitive to the initial rotational state. The dramatic nature of the dependence of the differential cross sections on initial rotational excitation is even more striking if we compare it to the very small effect of rotational excitation on the state-testate integral cross sections, as seen in Table VI. The differential cross sections summed over j', as seen in Figures 11-14, all move forward as the incident energy increases, a well-known trend.4'

Another measure of the sensitivity of the final state distributions is the average value of j' for a given u, j , and u'. These values are presented in Table VIII. At a more detailed level, Figures 15-1 9 show j'"spectran at fixed scattering angles. The trends in the evolution of the distributions for u ' = 0 are fascinating. At 0.82 eV (Figure 1 3 , the distributions are very similar in shape for both sideways and backward scattering. At 0.93 eV we see

(41) Eight references are given by: Truhlar, D. G.; Dixon, D.A.In Atom-Molecule Collision Theory: Bernstein, R. B.,Ed.;Plenum: New York, 1979; p 595, where such trends are reviewed.


-

+ H2

HD + H

-

TABLE VIII: Average Values of j' for D + HI( Y = 0,j ) HD( v', j ' ) H at Various Total Energies E (in eV) E i u' from u,,,,, t9 = 90' e = 1350 e = 180° 3.17 2.78 3.45 0.82 0 0 2.69 I 1S O 1.62 1.74 1.10 I O 3.22 3.04 3.35 2.96 1 I .30 1.53 1.33 1.15 4.01 4.53 4.12 0.93 0 0 2.58 I 1.32 1.37 1.32 1.35 4.38 I O 4.03 4.07 3.10 I 1.37 1.38 1.49 1.39 6.35 1.086 0 0 5.10 4.78 2.50 1 2.07 1.68 2.44 1.63 6.13 I O 5.08 4.75 3.15 1 2.23 2.16 2.38 1.76 6.19 7.72 5.45 1.25 0 0 3.64 1 3.08 3.81 3.03 2.07 2 1.17 1.21 1.17 1.16 1 0 6.17 7.46 5.48 4.13 1 3.14 3.82 3.02 2.29 1.25 2 1.18 1.18 1.16 1.35 0 0 6.84 8.31 5.67 4.51 I 4.65 3.70 3.30 3.02 2 I .86 1.61 2.01 1.89 8.08 6.80 I O 5.71 4.82 1 3.73 4.59 3.37 3.02 2 1.66 1.92 1.81 1.81

+


04

It

4

4

-

relative t r a n s l a t i o n a l e n e r g y ( e v )

Figure 20. Mean value of ( u ' ) for D + H2(q = 0, j = 1) HD(u') + H as computed from both quantum mechanics (QM) and trajectories (QCT-QSS). l

7

'

l

l

..

,

l

I

l

l

I

A A A Q M

t 6t-

QCT-QSS

0

ll

>

TABLE IX: Vibrational Branching Ratios at Various Total Energies E (in eV)

E

RO

R,

0.82 0.93 1.086 I .25 I .35

5.6 x 10-5 0.025 0.072 0.120 0.148

1.1 x 10-4 0.018

0.069 0.1 15 0.140

TABLE X: Absolute Cross Sections (ao') for Production of Specific Final States E,,, = 0.65-0.67 eV E,,, = 0.79-0.82 eV u' i' exDt" theoryb,C exDta theorvb" 0 0 0.03 0.02 1 0.34 f 0.12 0.10 0.23 f 0.08 0.07 2 0.36 f 0.12 0.15 0.30 f 0.10 0.1 1 3 0.36 f 0.12 0.19 0.32 f 0.1 1 0.16 4 0.40 f 0.14 0.20 0.35 f 0.12 0.19 5 0.31 f 0.10 0.19 0.32 f 0.1 1 0.22 6 0.28 f 0.10 0.15 0.35 f 0.12 0.22 7 0.18 f 0.07 0.09 0.26 f 0.09 0.19 8 0.04 0.27 0.09 0.13 9 0.009 0.18 f 0.06 0.07 1 0 0.005 0.009 1 0.07 f 0.02 0.009 0.03 f 0.01 0.02 2 0.09 f 0.03 0.007 0.03 f 0.01 0.03 3 0.09 f 0.03 0.003 0.04 f 0.01 0.02 4 0.08 f 0.03 0.0004 0.06 f 0.02 0.01 5 0.00002 0.03 f 0.01 0.004 0.000001 0.025 f 0.007 6 0.006

A

-

L

3

A

,

I

6 8 1 relative t r a n s l a t i o n a l e n e r g y ( e \ )

4

2

-

...

QCT-QSS

.

A

--

*

a Phillips et al., ref 3. Present, weighted average over j = 0 and 1 as discussed in text. = 0.93 eV. d E = 1.086 eV.

similar patterns for 0 = 90' and 135O, but for j = 1 the rotational distribution for backscattering begins to pick up a more extended high-j'tail. At 1.086 eV, the enhanced tail is noticeable as far forward as 135"; at 165' it has become a shoulder; and at 180" it has become a secondary local maximum. The results for 0 = 90" and 135" at 1.25 eV are similar to the previous energy, but now at 165" and 180" we see pronounced secondary local maxima for j = 0 and the structure is disappearing for j = 1. The results at 1.35 eV are similar to those at 1.25 eV except at 0 = 180". Here we see pronounced structure in the j'distributions for both initial rotational states. With so much detail in the results, the comparison to experimental state-testate differential cross should

A

I

I

>

ti

9

1

1

I

l

1

l

,

11

relative t r a n s l a t i o n a l e n e r g y ( e v )

-.

Figure 21. Mean value of (j') for D + H2(u = 0, j = 1) HD(u',j') + H as computed from both quantum mechanis (Q.M.) and trajectories (QCT-QSS). (a, top) u' = 0. (b, middle) u' = 1. (c, bottom) u' = 2.

7090

J . Phys. Chem. 1990, 94. 7090-7096

be quite a challenge for theory. The two energies for which Continetti et al. have reported results so far4 correspond to the first and fourth total energies on our list (see Table I ) . Their velocity-angle contour map4 at 0.51-eV relative translational energy does appear to show rotationally hotter products at 135" deg than 180°, in qualitative agreement with parts b and d of Figure 15. (The effect was actually predicted by trajectory calculations3* prior to the experiment.) The experimental velocity-angle contour map for a relative translational energy of 1.25 eV also shows a shift toward higher rotational energies at 0 shifts from 180" to 135' to 90°, in qualitative agreement with parts d, b, and a of Figure 18. However, there is no sign in that contour4 of the added structure in Figure 18d. This confirms that a more detailed comparison of theory and experiment may be quite informative. The moments in Table VI11 provide a more averaged picture of the trends, but they again show how the differential cross sections are more sensitive to rotational excitation of H2 than are integral cross sections. Consider, for example, the effect of raising j from 0 to 1 on (j')(vf=O). At all five energies the effect is 10.05 for the integral cross sections, a surprisingly small effect. The effect is also small at 135". However, at 90' it is typically 0.2 of a rotational quantum, and at 180" it ranges from 0.31 to 0.65 of a rotational quantum. We conclude the discussion by comparing our results to the quasiclassical trajectory calculations of Blais and one of the aut h o r ~for~ D~ + H2(u = 0, j = 1) on a similar' potential energy surface. First consider Figure 20, which shows very good agreement for the average final vibrational quantum number; the differences are only about 0.02-0.04 quantum at any energy. The generally good accuracy afforded by the trajectory calculations is even more evident in Figure 21a,b for the mean final rotational

quantum number. The trajectory calculations do, however, yield slightly 'hotter" final rotational distributions.

Concluding Remarks We have found, by converged quantum dynamics calculations on a realistic potential energy surface, that the state-to-state differential cross sections for the D + H2 reaction are very sensitive to putting 1 quantum of rotational excitation into the H2 reactant, even though this has only a small effect on the integral cross sections. For example, for u = 0, u f = I , and j' = 6 at 1.086 eV the backscattering reaction cross section increases by a factor of 45 when j is increased from 0 to 1. The complete tables of converged state-to-state cross sections for two initial states and five total energies, presented in the supplementary material, should be useful not only for comparisons to future experiments but also for testing approximate dynamical methods. Acknowledgment. We are grateful to Steven Mielke for additional convergence checks, David Chatfield and Haozhe Dong for assistance, the authors of refs 4-6 for preprints, and Ron Gentry for stimulating discussions of the data. This work was supported in part by the National Science Foundation, the Minnesota Supercomputer Institute, and the Cray Research University Grant Program. Registry No. D, 16873-17-9; H2,1333-74-0

Supplementary Material Available: Tables S 1-S9 of convergence checks, including tabulations of the state-to-state differential cross sections at 5" intervals for all IO sets of initial conditions (98 pages). Ordering information is given on any current masthead page.

Influence of Radiative Transport on Energy Transfer from Fluorene to Pyrene in n-Hexane J . M. R. d'oliveira, V. R. Pereira, J. M. C.Martinho,* Centro de Qulmica- F h c a Molecular, Instituto Superior TZvnico, 1096 Lisboa Codex, Portugal

and M. A. Winnik Lash Miller Laboratories, Department of Chemistry and Erindale College, University of Toronto, Toronto, Canada MSS I A I (Received: December I I , 1989; In Final Form: March I S , 1990)

A model describing the influence of radiative transport over donors on the decay curves and fluorescence spectra of a donor

able to transfer its electronic energy to an acceptor is applied to fluorene-pyrene in n-hexane. Radiative transport is important a t high concentrations of fluorene (donor) and low concentrations of pyrene (acceptor) and depends upon both excitation and emission wavelengths. Stern-Volmer plots of lifetimes corrected from the influence of the radiative processes yielded an energy-transfer rate constant of kQ = (6.9 f 0.4) X 10l0M-' s-' independent of fluorene concentration and both excitation and emission wavelengths in close agreement with the theoretical value kQ = 7.1 X 1OIo M-' s-l. The nondependence of the transfer rate constant on fluorene concentration indicates that nonradiative transport is insignificant, in accordance with the small calculated value (A = 3.1 X IO-* cm2 s-I) of the nonradiative energy migration coefficient.

Introduction The transfer of electronic energy between organic molecules in solution has received considerable attentionl since Forster2 and Galanin3 obtained the donor decay curve for the case of donors and acceptors randomly distributed in a rigid solution. The energy can be transferred by direct absorption of the donor emission by the acceptor (trivial mechanism) or nonradiatively due to a Coulombic or an exchange interaction mechanism. The nonradiative energy-transfer rate for a donoracceptor pair was obtained 'To whom correspondence should be addressed. 0022-3654/90/2094-7090$02.50/0

by Forster4 for a dipole-dipole interaction and by DexterS who extended the theory to cover the multipole-multipole and exchange-type interactions. In rigid solution the transfer occurs between donor-acceptor pairs at several distances, the transfer (1) (a) Birks, J. B. Photophysics of Aromatic Molecules; Wiley: London, 1970. (b) Gosele, U. M. Prog. Reaer. Kiner. 1984, 13, 63. (c) Rice, S.A. In Comprehensive Chemical Kinetics; Bamford, C . H., Tipper, C. F. H., Compton, R. G.,Eds.; Elsevier: New York, 1985; Vol. 2 5 . ( 2 ) Forster, Th.Z. Naturforsch. 1949, 40, 321. (3) Galanin, M. D. Sou. Phys. JETP 1955, 1 , 317. (4) Forster, Th. Ann. Phys. 1948, 2, 5 5 . ( 5 ) Dexter, L. D. J. Chem. Phys. 1953, 21, 836.

0 1990 American Chemical Society

Effect of Rotational Excitation on State-testate Differential Cross Sections

Recommend Documents