2872
J . Phys. Chem. 1989, 93, 2872-2873
Wave-Packet Approach To Treat Low-Energy Reactive Systems: Accurate ProbaMmtes for H 4- H, Daniel Neuhauser and Michael Baer* Department of Physics and Applied Mathematics, Soreq Nuclear Research Center, 70600 Yavne, Israel (Received: November 7, 1988)
A new method to study reactive collisions employing the quantum mechanical wave-packet approach is presented. The method makes use of the projection operator formalism to form a coupled system of time-dependent SchrGdinger equations and of absorbing potentials to circumvent the necessity of employing product arrangement channel coordinates. As an example, the method is applied to the collinear (reactive) H + H, system.
The time-dependent wave-packet approach to treat molecular collisions has recently been getting a lot of attention. Following the pioneering work of McCullough and Wyatt,' Heller,* Kulander,3 and Kosloff and co-workers4 revived the subject by injecting beautiful new ideas and making it a workable method for most collisional processes, as has since been d e m ~ n s t r a t e d . ~ , ~ However, reactive scattering, and in particular low-energy reactive scattering, belongs to a field in which the wave-packet approach did not "scoren many successes. In the present Letter, we shall show what kind of modification must be made in the existing approaches in order to make them relevant for reactive scattering processes as well. Although we consider a collinear system, the improvements we suggest are general and therefore applicable to any collisional process, including the reactive three-dimensional process. As an example, we show a reproduction of the low-energy reactive transition probability curve for H + H2 including the resonance region which is, as will be shown, well reproduced. The collinear reactive system employing scaled coordinates is shown in Figure 1 (insert). The whole configuration space is divided into three regions: the reagent arrangement (RE), the product arrangement (PR),and the interaction region (IN). Attention is devoted mainly to R E and IN, whereas PR can be ignored, because we employed a short-range absorbing potentia1*7-8 along the line r = r p (This subject is extensively discussed by us elsewhere7 and will only briefly be mentioned here.) Since only the regions R E and I N are explicitly considered, it suffices to treat the whole process employing reagent coordinates only. The time-dependent Schrodinger equation i ( W / a t ) = HJ/ ( h = I ) is solved; H stands for the ordinary collinear H a m i l t ~ n i a n . ~ If $,,(r) are the (asymptotic) eigenfunctions of the diatomic reagent molecule, then J/(R,r,t)in the asymptotic region is assumed to be N
lim J/(R,rJ) = C v n ( R J ) M r ) R-m
n= I
(1)
McCullough, Jr., E. A.; Wyatt, R. E. J. Chem. Phys. 1971,54,3592. Heller, E. J. J. Chem. Phys. 1975, 62, 1544. Kulander, K. C. J . Chem. Phys. 1978, 69, 5064. (a) Kosloff, D.; Kosloff, R. J . Comput. Phys. 1983, 52, 35. (b) TalEzer, H.; Kosloff, R. J. Chem. Phys. 1984, 81, 3967. (c) Kosloff, R.; Kasloff, D. J. Comput. Phys. 1986, 63, 363. (5) Mowrey. R. C.; Bowen, H.F.; Kouri, D. J. J. Chem. Phys. 1987,86, 244 I . (6) Mohan, V.; Sathyamurthy, N. Compuf. Phys. Rep. 1988, 7, 213. (7) Neuhauser, D.; Baer, M. J . Chem. Phys., in press. ( 8 ) Leforestier, C.; Wyatt, R. E. J. Chem. Phys. 1983, 78, 2334. (9) Baer, M. In The Theory of Chemical Reaction Dynamics; Baer, M., Ed.; CRC: Boca Raton, FL, 1985; Vol. I, Chapter 3. (1) (2) (3) (4)
0022-3654/89/2093-2872$01.50/0
where N is equal to (or perhaps greater than) the number of open states and vn(R,t)are the translational functions to be solved (at t = 0 all of them are zero, except the one with n = no). Accordingly, we introduce the following projection operators:
which operate on the r coordinate only. Operating with P,,, n= 1, ..., N , and Q on the time-dependent Schriidinger yields the following system of coupled differential equations:
a
i -(PnJ/) = P,HJ/ = at
which can be shown to become a set of coupled equations for qn(R,t),n = 1, ..., N , and for X(R,r,t) defined as It is important to mention that the introduction of projection operators guarantees that x(R,r,t)becomes zero when R is large enough, Le., R > R , (see insert, Figure 1). The introduction of the projection operators enables dividing the (RE + IN) region with two kinds of grids. In the R E region a one-dimensional grid is employed to propagate the one-dimensional vn(R,r)functions, whereas in the I N region, two different grids are encountered: the above-mentioned one-dimensional grid, which is continued from the R E region, for the calculation of v,,(R,t) and the ordinary two-dimensional grid to solve for the X(R,r,t) function. It is important to indicate that all current methods utilize two-dimensional grid functions for all regions, including the asymptotic. This limits the possibility of extending the initial wave packet to the required width in case of low energies or resonances, due to the immense increase in computer time consumption. The fact that, in our approach, in the asymptotic reagent region, we have to calculate only one-dimensional functions enables the use of wave packets as wide as required, without significantly increasing the numerical effort (for instance, in one case a wave packet 36 8, long was used). Moreover, the number of the two-dimensional grid points is therefore reduced significantly (in the present calculation 32 X 32 grid points were considered). The resulting coupled system of equations is solved by applying the Chebyshev polynomials to expand the propagation operator4a and the fast Fourier transformation to evaluate the spatial der i v a t i v e ~ .The ~ ~ outgoing flux is calculated along the line r = r, (see insert, Figure l ) , which is chosen to be outside the main interaction region. An imaginary short-range linear negative potential7 is assumed for r > rI (see shaded area in Figure I), and this significantly reduces the amount of computation for r > r I . All details of the calculations, as well as the explicit form of the coupled equations to be solved, will be given elsewhere. Still, we
0 1989 American Chemical Society
(a).
J. Phys. Chem. 1 9 8 9 , 9 3 , 2 8 7 3 - 2 8 7 5
I /
I
I 0.6
0.7
Tom,'"qy
0.8
0.9
1.0
(0)
Collinear reactive transition probability for H + H,: -, time-independent calculations (ref IO): 0,present results. Insert: The collinear mass-scaled potential energy surface. The imaginary negative potential is located along the line I = q (shaded area). Figure 1.
would like to emphasize that by employing this new method the savings in computer time is significant, in particular in the case of resnnanm or very low energies, where a reduction of more than 1 order of magnitude is achieved.
I
2873
As an exampkof an application of the above-described approach, we give the low-energy reactive transition probability for the collinear H + H,. In Figure 1 we show the curve of Walker et al.lo and the results of the present treatment (designated as full circles). As for the energy value of the resonance, two results are shown: one is obtained employing a Gaussian of initial width 2 8, and the other (the more accurate one) of initial width 6 A. It should be mentioned that with this method we also managed to obtain rate constants for temperatures as low as 200 K." To our knowldege, no other time-dependent approach was able to produce rate constants for temperatures lower than 500 K (see also ref 6). As for the prospects of extending the method of three dimensions, we would like to indicate that, together with Judson and Kouri, we recently achieved this extension.l' The first 3-D timedependent reactive probabilities for H H, (J = 0) were found to be identical with those obtained by the well-established time-independent methods.
+
Registry No. H, 12385.13-6; H ,1333-74-0
(10) Walker, R. B.;Stechcl.
,om
E.B.; Light, J. C. 3. Chem.Phys. 1978,69,
.I__.
(II) Neuhaucr. D.; h e r . M..submitted for publication.
(12) Ncuhauxr. D.; Bacr. M.; Judson. R
S;Koun. D. J. I n preparation
Dynamic Burstein-Moss Shift in Semiconductor Colloids' Prashant V. Kamat,' Nada M. Dimitrijevic, Notre Dame Radiation Laboratory, Notre Dame, Indiana 46556 and A. J. Nozik* Solar Energy Research Instituie, Golden, Colorado 80401 (Receioed: November 28, 1988)
Photoinduced blue shifts in CdS colloids have been time resolved by picosecond pump-probe measurements. The blue shift appears within the time domain of the pump pulse (18 ps) and is found to increase with increased pump light intensity. Calculations of the predicted blue shift from a dynamic Buntein-Moss shift agree with the experimental results at the lower laser intensities, but the predicted shifts are greater than observed values at the higher light intensities.
In recent years, considerable interest has been shown in the photophysical and photochemical properties of colloidal semiconduct0rs.l Of particular interest is the observation by several authors of photoinduced blue shifts in the absorption spectra of semiconductor colloids with very small particle sizes (
