12822
J. Phys. Chem. 1994, 98, 12822-12823
Exchange Processes via Electronic Nonadiabatic Transitions: An Accurate Three-Dimensional Quantum Mechanical Study of the F(2P1/2,2P3/2) H2 Reactive Systems
+
Miquel Gilibertt and Michael Baer* Department of Physics and Applied Mathematics, Soreq NRC, Yavne 81800, Israel Received: August 15, 1994; In Final Form: October 27, 1994@
This Letter presents the first study yielding formally accurate three-dimensional quantum mechanical reactive probabilities for a reactive (exchange) process taking place via an electronic nonadiabatic transition. It uses a method based on the application of negative imaginary potentials (NIPs) that decouple all the asymptotes. Reactive and electronic nonadiabatic J = 0 transition probabilities were calculated for the F ( ' P I / ~ , ~ P ~HZ /~) system. /
+
All quantum-mechanical three-dimensional (3-D) formally accurate treatments performed so far have been for chemical exchange processes taking place on one single potential energy surface (PES) and therefore not involving any electronic nonadiabatic transition. In this Letter we present the first fully 3-D quantum mechanical probabilities for a reactive (exchange) atom-diatom process taking place via an electronic nonadiabatic transition. This calculation is canied out by solving the 3-D Schrodinger equation, employing negative imaginary potentials (NIPs) to decouple the various exchange arrangement channels (AC).1,2 This way of performing the numerical treatment yields the total wave function W,cz)for the whole configuration space, excluding the various asymptotes. Here ;1stands for the initial (reagents) AC, and i designates an asymptotic electronic state. To solve YA('), we present it as an N-component vector (for the sake of brevity N is assumed to be equal to 2), where each component refers to an electronic state:
Here i , j = 1 , 2 ( i f j), yjn(') is the unperturbed part of the wave function describing the incoming flux and xu('(k ) = i , j ) are the perturbed parts of YY,(').In other words, yj~(')is the part of the wave function responsible for the construction of the source term of the inhomogeneous Schrodinger equation and therefore is related to the electronic state of the reagents; (k = i, j) is mainly responsible for the close interaction part of this function and correlates with the kth electronic asymptotic component of the total wave function. The two functions are derived from the following equation:
x
part of the potential belonging to the reagents' electronic state. The introduction of the NIPs1s2permits the application of L2 basis sets3 to solve the xu(')Thus, . in this sense eq 2 is solved as in the single-surface case. The reagents' translational coordinate is divided into M sectors, and a single translational localized basis function (Gaussian) and a whole set of adiabatic internal functions are attached to each s e c t ~ r . ~The . ~ number of these functions is controlled by a single parametes (more details will be given elsewhere). It is important to mention that for each translational grid point a single vib-rotationalelectronic basis set is calculated. Once the components of Yi(') are derived, it can be shown that the various S-matrix elements follow from the expression
where U,V) and yjaU) are the perturbation potential and the unperturbed (elastic) wave function, respectively, related to the jth electronic state in the a (=A, Y) AC, Y ~ Ais( ~the ) jth component of !If,V,cij) $'is)the , previously defined electronic diabatic coupling term, and d,J is the corresponding (full) elastic phase shift. This expression must be somewhat modified for the elastic case. This new approach is applied to the following processes:
F('P,,,)
+ H2(v=Oj=O) - F(2P,,2) + H2(v'=0, u)(Ia) - HF('&'~') + H F('P312)
where HA(@( k = i,j) is the heavy-particle (namely the nuclear) Hamiltonian related to the kth electronic potential energy surface also contains the required NIPs), V~('11is the potential (diabatic) coupling term, and UA(')is the interaction (perturbed) On leave from the Department of Physical Chemistry, University of Barcelona, Barcelona, Spain. * To whom correspondence should be addressed. +
+ H,(v=Oj=O)
-
HF(Cv'j')
+H
(11)
To carry out these calculations the M5 potential energy surface6 is used as the ground electronic state, the Blais-Truhlar surface7 is taken as the upper electronic state, and the two are assumed to be coupled by the spin-orbit interaction.* (More details can be found in refs 9 and 10.) Figure 1 shows the J = 0 probabilities for reactions la and Ib as a function of total energy. For comparison we also include the collinear results? and as can be seen, they fit the 3-D ones very nicely. In Figure 2 we show the results for reaction I1 as calculated first with a single surface2 and secondly employing the two coupled surfaces. It is noticed that the electronic
0022-365419412098-12822$04.50/0 0 1994 American Chemical Society
Letters
0.03
J. Phys. Chem., Vol. 98, No. 49, 1994 12823 phenomenon was observed in a collinear treatment.9 There, like here, the resonance position was shifted toward higher energies, and in general larger reactive probabilities (for the same energy) were encountered. The similarities between collinear and 3-D J = 0 probabilities may not necessarily be indicative of the reliability of the present treatment, but such similarities were frequently observed in single-surface calculations and among other cases also for the F H2 system.* The fact that the spin-orbit coupling may significantly affect the reactive probabilities of this system should also be taken into account when comparing theoretical results with spectroscopic measurements, as was recently done for the FH2system.’(’
1
+
0.60
Et, (eV)
--
Figure 1. Electronic nonadiabatic J = 0 transition probabilities as a
+
+
F(’P1/2) Hz(j=O,v=O) F(2P3/2) function of total energy: (0,O) H2 (inelastic process); (0, B) F(2P~/z) Hz(j=O,v=O) HF H2 (exchange process); (-) results due to full 3-D (J = 0) calculation; (- - -) results from a collinear treatment (ref 9).
+
0.0 L-0 0.3
’
I
0.4
I
I
0.5
+
I
0.6
,E ( e W Figure 2. Reactive J = 0 transition probabilities for the process F(2P3/z) iHz(j=O,v=O) HF H : (- - -) a two-surface calculation; (-) a single-surface calculation (ref 2).
-
+
nonadiabatic coupling, although weak (10.05 eV), has a relatively large effect on the dynamics taking place on the ground electronic state. It is important to mention that a similar
Acknowledgment. We are indebted to the “Centre de Supercomputacio de Catalunya” (CESCA) at the University of Barcelona, Spain, for computer time and resources made available for this study. This work was supported by the German-Israel Foundation (GIF) and by a grant (BE92/285) from the regional Catalonian Govemment, CIRIT), Spain. References and Notes (1) Neuhauser, D.; Baer, M. J. Chem. Phys. 1989, 91, 4651. Baer, M.; Neuhauser, D.; Oreg, Y. J. Chem. Soc., Faraday Trans. 1990,86, 1721. Baer, M.; Nakamura, H. J. Chem. Phys. 1992,96,6565. Last, I.; Baer, M. J. Chem. Phys. 1992, 96, 2017. Last, I.; Baer, M. Chem. Phys. Lett. 1992, 189, 84. Last, I.; Baram, A.; Szichman, H.; Baer, M. J. Phys. Chem. 1993, 97,7040. Baram, A.; Last, I.; Baer, M. Chem. Phys. Lett. 1993,212,645. (2) Gilibert, M.; Last, I.; Baram, A,; Baer, M. Chem. Phys. Lett. 1994, 221, 327. (3) Nuttal, J.; Cohen, H. L. Phys. Rev. 1969,188, 1542. Rescigno, T. N.; Reinhardt, W. P. Phys. Rev. A 1973,8, 2828. Heller, E. J.; Yamani, H. A. Phys. Rev. A 1974, 9, 1201, 1209. Miller, W. H.; Jansen-Op-de-haar, B. M. D. D. J. Chem. Phys. 1986, 86, 6213. Sun, Y.; Kouri, D. J. Chem. Phys. Lett. 1991, 179, 142. Gilibert, M.; Baram, A.; Last, I.; Szichman, H.; Baer, M. J . Chem. Phys. 1993, 99, 3503. Zhang, J. Z. H.; Chu, S. I.; Miller, W. H. J . Chem. Phys. 1988, 88, 6233. (4) Bacic, Z.; Light, J. C. J . Chem. Phys. 1986, 85, 4595; 1980, 86, 3065. ( 5 ) Zhang, J. Z. H.; Miller, W. H. J. Phys. Chem. 1980, 94, 7785. (6) Muckerman, J. T. In Theoretical Chemistry: Advances and Prospects; Eyring, H., Henderson, D., Eds.; Academic Press: New York, 1981; Vol. 6A, p 1. (7) Blais, N. C.; Truhlar, D. G. J . Chem. Phys. 1978, 69, 846. (8) Zimmerman, I. H.; Baer, M.; George, T. F. J . Chem. Phys. 1979, 71, 4132. Komomiki, A.; Morokuma, K.; George, T. F. J . Chem. Phys. 1977, 67, 5012. (9) Last, I.; Baer, M. J . Chem. Phys. 1985,82,4954 (see also: Lepetit, B.; Launay, J. M.; Le Doumeuf, M. Chem. Phys. 1986, 106, 111). (10) Manolopoulos, D. E.; Stark, K.; Werner, H.-J.; h o l d , D. W.; Bradforth, S. E.; Neumark, D. M. Science 1993, 262, 1852.
