J . Phys. Chem. 1994,98, 1868-1874
1868
Distributed Approximating Function Approach to Atom-Diatom Reactive Scattering: Time-Dependent and Time-Independent Wavepacket Treatments Youhong Hung,+ Wei Zhu,+and Donald J. Kouri’vt Department of Chemistry and Department of Physics, University of Houston, Houston, Texas 77204-5641
David K. Hoffman$ Department of Chemistry and Ames Luboratory,I Iowa State University, Ames, Iowa 5001 I Received: August 27, 1993’
The recently developed distributed approximating function (DAF) method for evaluating the action of the kinetic energy evolution operator, and the kinetic energy portion of the Hamiltonian, is applied to treat collinear reactive scattering. The DAF approach yields highly banded representations of these operators, while permitting the relevant matrix-vector multiplications to be done by fast convolution. Both time-dependent and timeindependent wavepacket propagation schemes are employed, along with the DAFs, and accurate results obtained for the standard H Hz collinear reactive scattering system.
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I. Introduction Recent years have seen enormous progress made in the theoretical treatment of reactive scattering dynamics.182 The initial advances were achieved mainly with time-independent approaches.’ More recently, the largest calculations have employed time-dependent (wavepacket) approachesa2 A key ingredient in these advances has been the development of supercomputers, up to now predominantly based on vector processing. The introduction of negative imaginary absorbing potentials to decouple arrangements has also provided a powerful tool for increasing the complexity of the systems which can now be treated.2 Other powerful methods for evaluating the action of the kinetic energy operator have included the fast Fourier transform3 (FFT) and the discrete variable representation4 (DVR). More recently, we have introduced a different method for evaluating the action of differential operators (including infinite-order ones like the free evolution operator), called the distributed approximating function(a1)W or DAF method. It appears to offer an extremely promising tool for quantum dynamicsbecause it yields a highly banded representation for the kinetic energy or free evolution operators; and because it also yields Toeplitz matrices, its action on a vector can be evaluated by fast convolution? We have been carryingout systematicstudies of various quantum dynamics systems to understand and demonstrate the method. In this paper, we report the calculation of H Hz collinear reactive scattering using the DAF method both in a timedependent and a time-independent framework. The DAF yields an integral representation of differential operators, and the DAF representation of the action of the operator on an (M 1)-degree polynomial is exact. The DAF is a localized functional approximation to the Dirac 6 function and gives a banded matrix representation. Physically, the DAF can be considered as a filter in momentum space and results in a filtered propagated wave function. Because of the Toeplitz structure of the DAF, we have
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under R. A. Welch Foundation Grant E-0608. Supported in part under National Science Foundation Grant CHE8907429. 1 Supported in part under National Science Foundation Grant CHE9201967. * The Ames Laboratory is operated for the Department of Energy by Iowa State University under Contract No. 2-7405-ENG82. * Abstract published in Advance ACS Abstracts, January 1, 1994. t Supported 2
0022-3654/94/2098-1868$04.50/0
been able to use fast convolution to carry out the time-dependent wavepacket propagation in our previous papers? and we have demonstrated that the basic DAF/fast convolution method outperforms the FFT method with respect to both storage and CPU time requirements. In this paper, we not only use fast convolution, we also truncate the grid. This leads to a substantial increase in efficiency of the method. We also carry out part of our calculations using the recently developed time-independent wavepacket Schrodinger equation (TIWS) and the time-independent wavepacket Lippmann-Schwinger quation (TIWLS).1”12 For the initial wavepacket we employ, solutions of TIWS and TIWLS are proportional to the standard outgoing boundary condition wave function. The TIWS and TIWLS equations combine desirable features of the time-dependent wavepacket and the time-independent approaches. In this paper, we use our recently developed Chebychev series representation of the causal full Green’s function to solve TIWLS for the reactive scattering problem.11 This paper is organized as follows. In section 11, we briefly review the DAF method and the TIWS and TIWLS formalism. Section I11 briefly describesthe Chebychev series solution of the TIWLS equation, the fast convolution and grid truncation used, and the final state analysis. Numerical results for the case of H Hzcollinear reactive scattering calculation are presented in section IV. Our conclusions are given in section V.
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11. Theoretical Background
A. Distributed Approximating Functions (DAFs). In a coordinate representation, the DAF can be written as-
where Hzn((x- x ’ ) / f i a ( O ) ) is the usual Hermite polynomial. A function 4 ( x ) is approximated by the DAF according to-
It has been shown that r$M(x)is exactly equal to $(x) provided 0 1994 American Chemical Society
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The Journal of Physical Chemistry, Vol. 98, No. 7, 1994 1869
that 4(x) is a polynomial of degree up to M + 1. On an evenly spaced grid, eq 2 may be approximated by quadrature ass8
and
m
(3) respectively, where where the DAF, W(j- I ) , is defined by g(M(j - 1)
=
(4)
Here, A is the grid spacing and a(0) the width parameter for the DAF, which, along with the highest degree, M,of the Hermite polynomial, determines the bandwidth of 6(m. Because of the exponential function in eq 4, the infinite sum in eq 3 can be replaced accurately by a finite sum. The operation of differential operators (dl/dx') on the DAF function (1) can be evaluated analytically and yields the expression
This relation makes the DAF method very useful for solving dynamical equations, which typically involve differential operators. Two fundamental operators in quantum mechanics, the kinetic energy operator and the free evolution operator, are differential operators. The action of the kinetic energy operator and the free evolution operator on a wave function are given bylo
and
in continuous coordinates, respectively, where
and
= ~'(0)
+ ih.r/m
(12) The finite sum in eqs 9 and 10 again is the result of the real exponential function appearingin the DAF function. Thequantity w is defined as the half-bandwidth which in general is broader for the DAF propagator than for the DAF. Equations 9 and 10 are also discrete convolutionswhich can be performed numerically with the same order of operations as an FFT. Furthermore, due to the bandedness of the DAF, a fast convolution, with overlap save technique, can be used to carry out the matrix-vector multiplications involved in these two equations. In the next section, this method is briefly discussed. B. TieIndependent Wavepacket Schrodjnger (TIWS)Equation and Time-Independent Wavepacket Lippma&hwinger (TIWLS) Equation. Quantum wavepacket dynamics, similar to classical dynamics, describes events occurring in finite space and during a finite time interval while energy eigenstate dynamics in general involves infinite space and infinite time. Since physical processes carried out in a laboratory apparatus can be described locally, one can utilize a variety of methods to make possible descriptions of scattering using L2functions. This includes the use of imaginary absorbing potentials as a means to localize the system, as well as a sequence of L2functions which, in a finite region, become closer and closer to the exact, non-L2 definite energy scattering state. The TIWS and TIWLS formalisms represent a rigorous framework to achieve this.10J1 They are derived as follows. Starting from the solution to the time-dependent Schrodinger equation, ~ ' ( 7 )
Ix(t))
= exp(-rT;(t H - tO))IX(tO))
where I X ( t o ) ) is the initial wavepacket and H = HO+ V, a timeto-energy transform of I x ( t ) ) in eq 13 is carried out, yielding
I\k(E)) = -LJIdt eiEf/heAW-tO)/hIx(t0))
When discretized by quadrature, eqs 6 and I become
(13)
(14)
In a chemical scattering problem, the interaction potential Vis usually local, and an absorbing potential Vab can be introduced into Hin eq 14 in order to decrease the region required to describe the dynamics. In the time-dependent description, the detailed scattering information is not altered, provided the absorbing potential does not overlapthe target Vand absorbs all the incident wavepacket without reflection. Neuhauser and Baer have given an excellent discussion of this method.I3 Thus, we can add Vab into the Hamiltonian H as
where Vabmust be positive definite, and can carry out the integral in eq 14. This results in the time-independent wavepacket
1870 The Journal of Physical Chemistry, Vol. 98, No. 7, 1994
Huang et al.
LippmannSchwinger (TIWLS) equation
and
or the equivalent time-independent wavepacket Schrodinger (TIWS) equation
Here, H- is the maximum energy and H h is the minimum energy significantly contained in the initial wavepacket. Equation 21 can now be expanded into the Chebychev series
i h iE/hlo IX(t0))
( E - H + i~,,)lWEl+)) = z;;"
(17)
where the plus sign indicates the outgoing wave boundary condition. This condition is imposed naturally by the momentum distribution in x(to) and by the limits in the integral (14). It can be shown that if the initial 1-D wavepacket IX(to))consists of positive definite linear momenta,
where
and T, is the usual Chebychev polynomial of degree n. The spectrumofH-rangesfrom-1 to+l. Using theorthonormality relation of the Chebychev polynomials,
andiscenteredon theprecollisionsideofthepotential,thesolution [@(El+))of (16) and (17) is proportional to the outgoing wave boundary condition LippmannSchwinger solution the coefficient un(a*) is expressed as (19)
in the region on the "potential" side of the initial wavepacket Ix(to))and is proportional to the scattered wave in the region behind the wavepacket,12
l\k(El+)) = -(I*+(E))
-eikx)
h2k It should be noted that if Vp, is taken to be constant and small, it plays the same role as the B parameter in the usual causal Green's function. Thus one may solve eq 16 or 17 by explicit construction of the action of (E - H k)-1 on the initial wavepacket. In the following section, we discuss how the TIWLS equation is solved by a Chebychev expansion of the Green's function in eq 16.11 We also discuss several other aspects of the computational methods we employ in our time-dependent and time-independent wavepacket calculations.
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111. Computational Methods and Final State Analysis
The units used in this and the following sections are atomic units; i.e. h = 1, me = 1, and c = 1. A. Chebychev Expansion of Green's Function. The TIWLS equation, eq 16, can be solved by various methods. Here, we use a method involving a Chebychev expansion of the full causal Green's function acting on the initial wavepacket.I1 The energy dependence of the Green's function is isolated in the coefficients of the Chebychev polynomials, which contain all dependence on the Hamiltonian. The application of the Hamiltonian is independent of the energy. Including the final state analysis, the storage requirement of the method includes three vectors for use in the three-term Chebychev recursion, and the values of each Chebychev term at several grid points. Explicitly, we write eq 16 in the alternative form i
E - H f iV,, where
Then, eq 26 can be written as
-- i - 1 AHE - Hnomf iq
(21)
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where the replacement z = s i( 1 - s2)1/2 is made. The contour integral, evaluated on the unit circle, can be performed using the residue theorem. There are poles a t z=o
z -,
- a- * i
d 1 -az-
(28)
and z:z? = 1. The two pairs of poles, (0'2:)) and (0,z;), are inside the contour. They correspond to the causal and anticausal Green's function, respectively. The result of the integration in eq 27 is then ( 2 -),6 an(d =
[ ( E - II) r i d ( m z - ( E - mZln
(mnd(m2 - ( E - m2
(29)
The solution of the TIWLS equation can thus be obtained by the Chebychev expansion as
where an(aa)is given by eq 29 and Tn(Hnm)IX(fo))is obtained by the recursion relation Tn(Hnom)IX(to))= 2Hnom Tp+1(Hnom)lx(to)) -Tp+2(Hnom)lx(to)) (31) For a finite number of Chebychev terms, the L2 result is written as J\kN(El+)), which becomes an increasingly good representation of the exact (non-U) l@(El+)) in a given finite region as the number of Chebychevs included increases. An adequate number of Chebychev terms, N, can be estimated by
N-MT
(32)
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The Journal of Physical Chemistry, Vol. 98, No. 7, 1994 1871
where T i s the time required classically for the major portions of the wavepacket to travel past the point xo at which one is doing the final state analysis and AH is given in (22). The estimation in eq 32 can be justified by noticing that the Chebychev expansion, eq 23, can also be done at xo using the time-dependentapproach,14 i.e. lim ( x o v.bi-a 'E - H
1
* ivab'~ ( 0 )=)i ( x o l K d t eiE'e-'H'IX(0)) = i K d t elE' x(x0lt)
= i c d t elE' x(x0lt)
(33)
since the wavepacket passes through the point xo after a finite time T (i.e. ~ ( x o l tN ) , 0, t > 7'). Equation 33 shows that the action of Green's function on the initial wavepacket, evaluated in a finite region of space, is equivalent to a time integration over a finite time, i.e.
wherei4
bn(t) = ( 2 - ad)(-i)" exp(-iRt)J,,(AHt)
(35)
H,Hnom, and M a r e defined in eq 26, and Jn(AHt)is the cylinder Bessel function of order n. Equation 35 is obtained by expanding e-iHf into a Chebychev series. It is readily seen that, for n 1 N = AHT, J,,(AHT) becomes vanishingly small and does not contribute to (xoll/(E- H + iO+)lx(O)). Thus, the infinite sum in eq 34 can be replaced by a finite sum, and the integral and the sum in this equation can be interchanged. The final result is
can be found in our previous paper^.^ In this paper, we use this technique to partition the 2-D collinear reactive scattering into four segments of the Jacobi coordinates, R and R'. Here, Rand R'are the distancesbetween the H and H2 in reactant and product arrangements, respectively, and the four segments are as follows: large R, small R'; large R', small R; small R', small R; and large R, large R'. The physically negligible segment involving both large R and large R'is eliminated or truncated in the DAF/fast convolutioncalculation. This truncation speeds up thecalculation by a factor of 4/3, compared to the untruncated calculation. C. Final State Analysis. In a reactive scattering problem, the natural description of rearrangement configurations requires a transformation of the coordinates in the asymptotic regions. To make thecalculation efficient, we use the following method, which only requires informationat several points in the asymptoticregion of interest. The number of the points needed is equal to the number of internal states allowed by energy conservation. We note that the method of final state analysis based on a time-toenergy Fourier transform is closely related to those in ref 2b and 2c. In the time-dependentapproach, the time-to-energytransform oftheevolvingwavepacketis takenat thespecificpointsofinterest,
In the time-independent treatment, \k(x~,y,lE)is the solution of eq 16 or eq 17. Since the initial wavepacket is constructed in such a way that only the positive semidefinitelinear momenta are included (as shown in eq 18), \k(x,,ydE) is proportional to the scattered wave portion of the LS-equation solution in the region behind the initial wavepacket (nonreactive elastic and inelastic scattering) and in the product arrangement region (no incoming wave). Thus, we have
for the elastic and inelastic components in the region behind the initial wavepacket, Ix(O)), and N
where a,(E) = is,'dt
eiE'b,(t)
(37)
Note that the causal solution is used here as an example and to is set to zero without loss of generality. Another feature of our computationalapproach is that it permits the use of fast convolutions and grid truncation. We now briefly discuss these techniques. B. Fast Convolutionand Truncation. Fast convolution is wellknown in the fields of information science and image processing. In contrast to the FlT, whose application involves the complete data stream, fast convolution can be performed on segments of the complete data stream and outperforms FFT by an order of -log2(Ns) operations, where N8 is the number of segments. It also leads to reduction in the required working space memory. The DAFs used for fitting, free propagation, and differentiation are all convolutions. The key for fast convolution to provide an advantage over FFT is that the width, w, of the DAF must be small compared to the total number of grid points. In a fast convolution, the data stream is partitioned into segments, with padding of the data according to the bandwidth of the DAF. Then the convolution is performed by means of FFTs in each segment. We omit the description of the DAF/fast convolution implementation here, since a detailed discussion of this subject
for the reactive or product components in the rearrangement region of configurationspace. The number of points ( X I ,y J (equal also to the number of equations and unknowns) is n k 1 for nonreactive scattering and niax+ 1 for reactive. The reduced masses, pI and p2, are
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and
where mg and mc are the masses of the reactant diatom, mA and mg are the masses of the product diatom, I$,,and cp, are the internal states of reactant and product, having energies c, and in, respectively. The wavenumbers k and k,are related by the energy conservation
The transformation between the reactant coordinates ( x ,y) and the product coordinates (n, 9) is
Huang et al.
1872 The Journal of Physical Chemistry, Vol. 98, No. 7, 1994
x=-
We now proceed to describethe results of the numerical study.
x + mB(mA+ mg + mc) Ymc + mB ( m A + mB)(mB + mc)
IV. Numerical Study Collinear H+H2 reactive scattering has been well studied, and it is for this reason that we choose it as the example for testing the DAF formulation of the evolution operator and the Hamiltonian, and of the Chebychev expansion of G+(E).lOJl The LSTH potential15 is used with the various numerical parameters given in Table 1. To reduce the size of the grid, two absorbingpotentials are placed in the asymptotic reactant and product regions for the time-dependent approach. The symmetricsplit operatormethodI6
or
mC
J=XmC
+mB
lx(t
Y
(43)
Solving the set of linear equations for the coefficients r(noln)and t(noln), one obtains the correspondingreflection and transmission probabilities (44) and (45)
respectively. In the calculation for H+H2 collinear reactive scattering, we use the reactant coordinates (x, y) for the evaluation of 9(+IE). We use DAF fitting, eq 3, to evaluate the product internal states p&) at the specified reactant coordinate grid points (x, y), for use in eq 40. Thus,
where the reactant grid point (x)! yd) is obtained from ( Z j , 91) by eq 43. Here, ( ~ ~ , jis+a) grid point in the product coordinates. The internal states &(y) and pn@)are obtained by solving the eigenvalue problems
+ 7 ) ) = e-iH71X(t)) = e-i/2J'Te-fKTe-i/2vT IxW) (50)
is used to propagate the wavepacket. We could as easily have used the Chebychevexpansionmethod*4for theevolutionoperator. We chose not to simply illustrate the use of the DAF evolution operator. Use of the Tal-Ezer and Kosloff formalismwould have involved using the DAF Hamiltonian, but since that is already being illustrated in the TIW treatment, with the Chebychev expansion of G+(E),we use instead the split operator form of exp(-iHt/h). Thus, the free propagation in eq 50 is carried out using the DAF representation, as given in eq 10. In both timedependent wavepacket and time-independent wavepacket approaches, we have been able to calculatethe transition probabilities for a wide energy range, from 0.5 eV (elastic scattering) to 1.4 eV (involving three open vibrational states of H2) in a single wavepacket propagation. The time-dependent wavepacket prop agation was performed on a Cray-YMP, and the time-independent wavepacket approach was performed on an IBM AIX 560 workstation. The initial wavepacket is taken to be a Gaussian times an initial Hz vibrational state
where
The amplitude, A ( k ) , in momentum space is A(k) = and
G$r) exp(-'/,x;(k -'l4 - k,,)') exp(i(k - k,,)xo) (53)
respectively, where the masses are defined as
m1 =
mBmC m0
+ "c
and m2 =
mAmO
m~ + m~
For the H+H2 system, since ml = m2, V(x,y)l,, = V(X,j&and t, = ;", and we use eq 47 to calculate q5n and (p., In terms of the DAFs, eq 47 is written by
The initial wavepacket travels toward the target from right to left. The same DAF parameters are used in calculating the internal states, the wavepacket propagation, the action of the kinetic energy, and the final state interpolation. The DAF/fast convolution method is used in the time-dependent wavepacket calculation. The 2-D grid is partitioned into the four segments, as discussed earlier, and the grid is truncated such that the segment with both large x and 2 is omitted. The results of several stateresolved transition probabilities are given in Table 2 and compared with numerical results calculated using the timeindependent S-matrix version of the Kohn variational principle and the DVR.17 Results obtained using the time-independent wavepacket treatment are presented in Table 3. We see that satisfactory results are obtained by both wavepacket approaches, demonstrating both the ability of the DAFs to treat reactive scattering and the validity of the TIWS and TIWLS equations for reactive scattering.
V. Conclusion
where V(l) = V(x,y,)l,-.
We have presented successful applications of the DAF formalism and the TIWS/TIWLS equations for calculating collinear H+H2 reactive scattering probabilities. These applications have shown that DAFs can be used to accurately and
DAF Approach to Reactive Scattering
The Journal of Physical Chemistry, Vol. 98, No. 7, 1994 1873
TABLE 1: Parameters of Grid, Wavepacket, DAF, Absorbing Potential ( V&), Time Step ( T ) , Total Time (II), H-, Number of Chebychev Iterations (N)in Atomic Units (au)
Hdnand the
grid time-dependent X&
time-independent X& = -1.0 x, = 43.0 y& = 4 - 0 y , = 40.0 Nx = 482 Ny= 482
= -1 .o
x, = 29.0 x. = 19.w x, = 17.0b y& = 4 . 0 y , = 26.0 y , = 16.W y , = 12.w Nx = 264 N, = 264
wavepacket xo = 14.0 X I = 0.6 k,, = 7.0
DAF ux(0)= 2.85Ax
vor= 7.0 x 1
= 2.85Ay MJ2 = 40 w=30
= 7.5 T = 2.25 X l(r H , = 1.0 H- = 0.0 N = 2500
u,.(O)
~ 3
7
x,, is where the absorbing potential in the reactant region starts. It ends at x,. x, is where the analysis of the reflection is done. e y,, is where the absorbing potential in the product region starts. It ends at ymx. d y sis where the analysis of the reaction is done.
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TABLE 2 Transition Probabilities for H H2 Collinear Reactive Scattering by the Time-Dependent Wavepacket Approach transition
energy (ev)
0.5 0.8 1-1
1+2 1-3
1.1 1.4
0.8 1.1 1.4 1.4
inelastic TDWA.
SMKVb
reactive TDWA
8.32 X l P z 0.923 0.917 6.23 X 1P2 6.22 X 1P2 0.936 0.173 0.172 0.298 0.300 6.02 X l P z 0.302 2.08 x i t 5 2.62 x 1 ~ 2.20 5 x1 ~ 0.153 0.153 0.384 0.221 0.220 0.225 9.05 X 1P2 8.91 X 1P2 0.104
SMKV 8.30 X 1P2 0.938 0.296 5.96 X 1P2 3.74 5 x1 ~ 0.380 0.224 0.107
a TDWA: time-dependentwavepacket approach. SMKV: S-matrix Kohn variational principle.
TABLE 3: Transition Probabilities for H + H2 Collinear Reactive Scattering by the Time-Independent Wavepacket ADDroach with Chebrchev ExDansion of the Green's Function transition
energy (ev) . ,
0.5 0.8 1-1
1+2 1-3
1.1 1.4
0.8 1.1 1.4 1.4
inelastic TIWLS'
SMKV
reactive TIWLS
SMKV
0.924 0.917 8.35 X 1P2 8.30 X 1P2 0.938 6.20 X 1P2 6.22 X 1 t 2 0.939 0.295 0.296 0.171 0.172 0.300 5.97 X l e z 5.96 X 1 t 2 0.296 2.95 X 1P5 2.62 X l P 5 4.81 X l P 5 3.74 X l P 5 0.382 0.380 0.152 0.153 0.224 0.224 0.220 0.220 9.36 X 1P2 8.91 X 1P2 0.108 0.107
0 TIWLS: time-independentwavepacket LippmannSchwinger equation.
conveniently evaluate the action of the kinetic energy operator and the free evolution operator for reactions, and also for multidimensional fitting of a numerically determined function. In the time-dependent approach, the numerical computation was facilitated by the use of the overlap-savefast convolution method. Furthermore, the fast convolution method with truncation of the grid has been implemented in this paper, yielding results in good agreement with those of other methods. A Chebychev representation of the full, causal Green's function was used to solve the TIWLS equation (16). This procedure has a very convenient dependence on energy, so that very little work has t o be done to obtain results at many energies. Thus, a single wavepacket propagation yields results for a range of energies. During the numerical testing, we have found two weaknesses of the methods used in this paper. First, the method used for the final state analysiscan be sensitiveto the points chosen to generate the simultaneous equations, due to the phase of the evolving wavepacket. Second,the Chebychev expansion of G+(E)requires a larger grid size in order to obtain converged results. This is due to the fact that the absorbing potential causes the expansion to
5
diverge, making its use in the Chebychev expansion impossible. The first problem can be overcome by means of a newly developed wavepacket form of various variational principles1*or by the use of "test" functions which average over a range of "observation points". The second one also can be overcome by using other polynomial representations of G+(E). In addition, no attempt to optimize the absorbing potential parameter was employed, and it is well-known that much shorter ranged absorbers can be implemented.&,*bJ3 In fact, little attention was paid to parameter optimization in general, since our goal here was to show that the DAF and the time-independent wavepacket methods can be applied to obtain correct results for reactive scattering. Furthermore, no attempt was made to eliminate grid points (other than in the fully dissociative region) on the basis of energetics. This can be done easily, yielding additional improvementin efficiency. Thus, there is substantial reason to believe that fully optimized versions of these new methods can make a significant contribution to the array of tools available for treating quantum scattering. In conclusion, we have shown through applicationto a standard test problem that DAFs yield a general, accurate, and potentially convenient numerical method for treating reactive quantum dynamics. The method is ideally suited to both vector and massively parallel computers. We will be improving efficiency and testing the approach on larger scattering problems in the near future. We also have shown that the TIWS and TIWLS equations provide attractive alternatives to the standard Schrodinger and LS equations, for reactive scattering, because of the appearance of a universalsource of scattered waves. This greatly decreases the energy dependence of the calculation, thereby transfering to time-independentreactive scattering the principal advantage of time-dependent quantum dynamics.
VI. Acknowledgement The authors have benefited from many discussions with Dr. Daniel Colbert. We also gratefully acknowledge that Dr. Colbert supplied us with his results obtained with the S-matrix version of the Kohn variational principle, using the DVR method for evaluating the action of the Hamiltonian.
References and Notes (1) See, e&: (a) Zhang, J. 2.H.; Chu, S I . ; Miller, W. H. J . Chcm. Phys. 1980,88,6233. (b) 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. (c) Manolopoulos, D.; Wyatt, R. E. Chem. Phys. Lett. 1988, 152, 23. (d) Launay, J. M.; Lepetit, B. Chem. Phys. Lett. 1988, I l l , 346. (e) Neuhauser, D.; Baer, M. J . Chem. Phys. 1988,88,2858. (f) Bear, M. J. Chem. Phys. 1989,90,3043. (8) Schatz, G. C. Chem. Phys. Lett. 1988,150,92. (h) Pack, R. T.; Parker, G. A. J. Chem. Phys. 1988,87,3888. (i) Webster, F.; Light, J. C.J. Chem. Phys. 1989,90,300. 6)Cuccaro,I.;Hipes,P.G.;Kuppermann, A. Chem. Phys. Left. 1989, 154, 155. (k) Kress, J. D.; Bacic, Z.; Parker, G. A.; Pack, R. T. Chem. Phys. Lett. 1989,157,484. (2) (a) Neuhauser, D.; Judson, R. S.;Kouri, D. J.; Adelman, D. E.; Shafer, N. E.; Kliner, D. A. V.; Zare, R. N. Science 1992, 257, 519. (b)
Huang et al.
1874 The Journal of Physical Chemistry, Vol. 98, No. 7, 1994 Neuhauser, D.; Judson, R.S.;Baer, M.; Kouri, D. J. In Advances in Molecular Vibrations and CollisionDynamics: Quantum Reactive Scattering, Bowman, J. M.,Ed.; JAI: Greenwich, CT, 1973;Vol. IIB. (c) The procedure of using a time-to-energy transform for extracting scattering information was also independently developed by: Hoffman, D. K.; Sharafeddin, 0. A.; Judson, R.S.; Kouri, D. J. J. Chem. Phys. 1990,.92,4167. (3) Feit, M.D.; Fleck, J. A. J. Chem. Phys. 1982,78,301.Kosloff, D.; Kosloff, R. Ibid. 1983,79,1823. (4).Lill, J. V.;Parker, G. A.; Light, J. C. Chem. Phys. Lett. 1982,89, 483. Light, J. C.; Hamilton, J. P.; Lill, J. V. Chem. Phys. Lett. 1985,82,
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