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Reactant–product decoupling approach to state-resolved reactive scattering Time-independent wavepacket formulation Stuart C. Althorpe,a¤ Donald J. Kouria”, David K. Ho†manb and John Z. H. Zhangc° a Department of Chemistry and Department of Physics, University of Houston, Houston, T X 77204-5641, USA b Department of Chemistry and Ames L aboratoryÒ, Iowa State University, Ames, IA 50011, USA c Department of Chemistry, New Y ork University, New Y ork, NY 10003, USA

Recently a formalism has been derived that rigorously partitions the time-dependent wavepacket for a reactive scattering system into mutually disjoint pieces satisfying uncoupled equations. Equations are set up for the time-evolution of the various arrangement pieces of the wavepacket by using absorbing potentials to decouple the initial arrangement dynamics from the other arrangements, and emitting potentials to provide sources for the scattering in each product arrangement. Here, we present the time-independent wavepacket equations that result from this reactantÈproduct decoupling and illustrate their solution with an example calculation on collinear H ] H . 2

1 Introduction In recent years, time-dependent wavepacket (TDW) methods have proven to be powerful tools for carrying out numerically exact simulations of 3D gas-phase reactive scattering of atomÈ diatom1h7 and diatomÈdiatom8h12 systems. As was stressed early by Kouri and co-workers,5,13,14 the power of the TDW method derives from (a) the fact that the computational e†ort, for molecular scattering in the body frame, centre-of-mass coordinate system, scales as a combination of N3@2 and M log M types of labour, with N denoting the number of rota2 tional states in a body-frame basis set expansion of the wavepacket dependence on internal angle(s) and M being the number of grid points in a discretization of radial coordinates and (b) the fact that a single wavepacket propagation yields results over a whole range of energies without requiring any additional propagations. In addition, the approach does not su†er from any stability problems associated with the fact that at any particular energy, there may be many “ closed channels Ï that may be needed to converge the results. Thus, in a closecoupled wavepacket study of molecule-corrugated surface scattering, at the lowest energy calculated, there were ca. 13 071 closed channels and, at the highest energy calculated, (in the same wavepacket evolution), there were 4457 closed channels.15 A major factor in making it possible to apply TDW methods to many extremely difficult problems has been the use of absorbing potentials to eliminate the need to propagate the wavepacket into arrangements which are of no ¤ Supported under National Science Foundation Grant CHE9403416. ” Supported in part under R. A. Welch Foundation Grant E-0608. Partial support from the Petroleum Research Fund, administered by the American Chemical Society, is acknowledged. ° Alfred P. Sloan, Fellow, and Camille Dreyfus, Teacher-Scholar. Supported in part by the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, U.S. Dept. of Energy under Grant No. DE-FG02-94ER14453. Partial support from the Petroleum Research Fund, administered by the American Chemical Society, is also acknowledged. Ò The Ames Laboratory is operated for the Department of Energy by Iowa State University under Contract No 2-7405-ENG82.

interest,16 see also ref. 1, 2 and 17. Neuhauser et al. showed how one could obtain quantitatively correct deÐnite energy total reaction probabilities,1,2,6 vibrationally resolved reaction probabilities,3 and completely state-resolved reaction probabilities4 by appropriate placement of the absorbing potential. Neuhauser et al. pointed out that such absorbing potentials provide the basis of an exact quantum mechanical transition state theory.17 Subsequently, Seideman and Miller derived elegant expressions for the cumulative reaction probability, as well as state-to-state S-matrix elements using the ÑuxÈÑux correlation function formalism combined with appropriately placed absorbing potentials in the incident and product arrangements.18 Most recently, Peng and Zhang19 have shown that one may use the absorbing potential, together with its adjoint (an emitting potential), rigorously to partition the total TDW for a reactive collision into disjoint pieces, which exactly describe the dynamics in each arrangement. The beautiful feature of the formalism is that one Ðrst solves an equation for the initial reactant piece of the wavepacket, t , placing absorbing potenr tials, [iV , just past the “ point-of-no-return Ï in each product ja arrangement j. However, one now stores V t (t), and then ja r can solve the equation for the p-product piece of the wavepacket, t (t), whenever one desires. If one wants only the total p reactive probability, this can be gained by a previously tested method,1,6 and one does not have to solve any of the t (t) p equations. Thus, the sizes of grids are enormously reduced and this leads to greatly reduced storage requirements and computational e†ort (since the matrixÈvector products all involve an H-matrix and either a t or t vector on a grid in p r a single arrangement). The speciÐc equations Ðrst given by Peng and Zhang19 are s(t) \ s (t) ] ; s (t) (1) r p p where s(t) is the total wavefunction, and the reactant and product arrangement pieces of the wavefunction satisfy i+

Ls (t) r \ Hs (t) [ i ; V s (t) r pr r Lt p

J. Chem. Soc., Faraday T rans., 1997, 93(5), 703È708

(2) 703

i+

Ls (t) p \ Hs (t) ] iV s (t) p pr r Lt

(3)

It is trivial to show that the sum of eqn. (2) and (3) (for each p) yields exactly the usual time-dependent SchroŽdinger equation, L s \ Hs i+ Lt

(4)

Peng and Zhang proposed a direct symmetric split operator method for solving eqn (3) for any p [after one solved eqn. (2) by any of the existing methods19]. However, in a recent series of papers, Kouri and coworkers20h27 have proposed a new form of time-independent SchroŽdinger equation [called the time-independent wavepacket SchroŽdinger equation (TIWSE)], which results from a halfFourier time-to-energy transform of the basic time-dependent (TD) SchroŽdinger equation : i s(t \ 0) (E [ H)m(E) \ 2n

(5)

It should be noted that the state m(E) has a very nice physical interpretation which is much more easily related to how, e.g., crossed molecular beams experiments are carried out. This is because the initial packet is a Ðnite distance from the scattering region, and this initial condition is explicitly present in the TIWSE. The major attraction, from a computational standpoint, of this new form of SchroŽdinger equation is that its solutions, m(E) \

i G(E)s(t \ 0) 2n

t : p1 i tr1 tr1 Ls 1 dt exp(iEt/+) r \ H dt exp(iEt/+)s (t) r 2n Lt 2n+ tr0 tr0 1 tr1 [i ; V dt exp(iEt/+)s (t) pr 2n+ r p tr0 and

P

P

P

i 2n

P

tp1

tp0

i ; g (E )F (H )s(t \ 0) (8) n norm n norm 2n n i 4 ; g (E )g (9) n norm n 2n n all the energy dependence is contained in the analytically known coefficients g and the Mg N are an energy independent n n set of vectors for representing m(E) at any energy E (for which the basis set used is accurate). Several closely related versions of this approach are being used for quantum dynamics calculations.20h30 The object of the present paper is to show how one may use Peng and ZhangÏs arrangement-partitioned TDW equations to derive analogous arrangement-partitioned TIW equations. We derive the TIW partitioned equations in Section 2 and propose a method for solving them in Section 3. We report the application of this method to collinear H ] H reactive 2 scattering in Section 4 and discuss the results in Section 5. We conclude by summarizing the di†erences between the TIW and TDW arrangement-partitioned equations in Section 6.

2 Time-independent wavepacket equations with arrangement partitioning We now multiply eqn. (2) and (3) by dt exp(iEt/+)/2n+ and integrate the Ðrst from t to t and the second from t to r0 r1 p0 704

J. Chem. Soc., Faraday T rans., 1997, V ol. 93

P

tp1

tp0

P

dt exp(iEt/+)s (t) p

tp1

dt exp(iEt/+)s (t) (11) r tp0 The left-hand sides of the above two equations are integrated by parts in the same way, leading to i [exp(iEt /+)s (t ) [ exp(iEt /+)s (t )] r1 r r1 r0 r r0 2n ] \H ]

P

tr1

tr0 and

E 2n+ 1 2n+

P P

tr1

tr0 tr1

tr0

dt exp(iEt/+)s (t) r 1 dt exp(iEt/+)s (t) [ i ; V r pr 2n+ p

dt exp(iEt/+)s (t) r

(12)

i [exp(iEt /+)s (t ) [ exp(iEt /+)s (t )] p1 p p1 p0 p p0 2n

where s(t \ 0) is a wavepacket (a superposition of many energies), allow an extremely useful separation of the Edependence from that of the Hamiltonian, H. Thus, one may develop polynomial expansions of G(E),24

m(E) \

Ls 1 p\H Lt 2n+

1 ] iV pr 2n+

(6)

G(E) \ ; g (E )F (H ) (7) n norm n norm n where E and H are the suitably normalized energy and norm norm Hamiltonian. Then

dt exp(iEt/+)

(10)

]

E 2n+

\H

P

tp1

tp0

1 2n+

P

dt exp(iEt/+)s (t) p tp1

tp0 1 ] iV pr 2n+

dt exp(iEt/+)s (t) p

P

tp1

dt exp(iEt/+)s (t) (13) r tp0 Now suppose we project both equations onto a Ðnite distance conÐguration of the atoms of the system and examine the lefthand sides of these equations in the case where t , t ] [O r0 p0 and t , t ] ]O. At any Ðnite distance conÐguration of the r1 p1 atoms, both s and s must tend to zero in these limits. [This r p is provided s(t \ 0) has zero overlap with any bound states of the full Hamiltonian. This condition is met by placing the initial packet outside the scattering potential region.] We then obtain the equations Et (E) \ Ht (E) [ i ; V t (E) r r pr r p

(14)

Et (E) \ Et (E) ] iV t (E) p p pr r

(15)

and

where 1 t (E) 4 r 2n+ and 1 t (E) 4 p 2n+

P

=

~=

P

=

dt exp(iEt/+)s (t) r

(16)

dt exp(iEt/+)s (t) (17) p ~= The fact that the above deÐnitions of t (E) and t (E) involve r p complete Fourier transforms shows that they are partitions of the standard time-independent wavefunction. In fact, eqn. (14) and (15) were also given in the original paper of Peng and

Zhang ;19 the above constitutes a direct derivation of these equations starting with the more fundamental time-dependent partitioned equations. Next, we consider the situation where t \ 0 and t \ t r0 p0 0 and t \ t \ O, again evaluating eqn. (12) and (13) at any r1 p1 Ðnite conÐguration of the atoms in the system. It is still true that s (O) and s (O) vanish for any such Ðnite-distance conr p Ðguration, but s (0) and s (t ) are not necessarily zero. r p p0 However, s (t) only begins to accumulate non-zero amplitude p when the Ðrst product Ñux emerges past the point-of-noreturn. Thus, in general (measuring both times from a common origin, so that t is the last instant of time before p0 amplitude begins building up in the product arrangement channel) we obtain the new partitioned TIW equations i s (0) [ i ; V m (E) (E [ H)m (E) \ pr r r 2n r p

(18)

(E [ H)m (E) \ iV m (E) p pr r

(19)

and

with 1 m (E) \ r 2n+ 1 m (E) \ p 2n+

P P

=

0 =

dt exp(iEt/+)s (t) r

(20)

dt exp(iEt/+)s (t) (21) p tp0 Thus, eqn. (19) has only the source due to the adjoint of [iV m (E) since s (t \ t ) 4 0. Note that to obtain the rightpr r p p0 hand side of eqn. (18), we make use of the fact that, for all times t \ t , s (t) has zero overlap with the V , for all p. We p0 r pr therefore have extended the lower limit of the integral in the last term on the right-hand side of eqn. (12) from t to zero. p0 We may solve eqn. (18) and (19) by the usual deÐnite energy GreenÏs functions noting that the full Hamiltonian in eqn. (18) is non-Hermitian due to the absorbing potentials. [iV m (E) in pr r eqn. (19) is strictly a source term and does not a†ect the GreenÏs function for this equation.] Thus, we obtain solutions m (E) \ r

A

i

2n E [ H ] i ; V pr p

B

s (0) r

(22)

and

polynomial expansion which has recently been developed by Mandelshtam and Taylor.28,29 In this expansion one indirectly includes (complex, energy dependent) absorbing potentials, C (E), by expanding in terms of damped generalizations of pr the Chebychev polynomials of H. Thus we actually solve modiÐed versions of eqn. (22) and (23) given by, m \ r

i

A

B

2n E [ H [ ; C (E) pr p

s (0) r

(24)

and [1 m (E) \ C (E)m (E) p r (E [ H) pr

(25)

We note that these equations may be derived by replacing each [iV in eqn. (2) and (3) and (10)È(13) by the complex pr potential C (E). For each value of E, there are then a set of pr TDW functions, sE(t) and sE(t), which may be half-Fourier r p transformed to yield a set of TIW functions, mE(E@) and mE(E@). r p When one chooses E@ \ E the latter are the m (E) and m (E) of r p eqn. (24) and (25). We solve eqn. (24) by direct application of Mandelshtam and TaylorÏs expansion,29 thereby obtaining m (E) in the form, r Nr 1 ; (2 [ d )exp([in/)g (26) m (E) \ n0 rn r 2n*H sin / n/0 where the g functions are generated by, rn g \ s (0) (27) r0 r g \< d H g r1 pr norm r0 p

(28)

g [ < d2 g (29) g \2 < d H pr norm rn~1 pr rn~2 rn p p The (coordinate-dependent) functions, d , are damping pr factors, each of which is related to C (E) by, pr C (E) \ *H[cos /V R [ i sin /V I ] (30) pr pr pr where (1 [ d )2 pr (31) 2d pr 1 [ d2 pr (32) VI \ pr 2d pr Note that V R and V I are functions of position only, and the pr pr energy dependence of C (E) arises solely from the /pr dependence. Each d is equal to one, except in the region of pr C (E), where it decays smoothly to a value that is between pr one and zero. Tests20,28,29 have shown that d can be chosen pr so that C (E) satisÐes the usual requirements of a good pr absorbing potential.5 Note that eqn. (26)È(32) employ the normalizations, H \ (H [ H3 )/*H, and cos / \ (E [ H1 )/*H, norm where H1 and *H are chosen so that the spectrum of H is norm contained within or on the interval [[1, 1]. To solve for m (E) in eqn. (25), we substitute for m (E) and p r C (E) using eqn. (26) and (30), thus yielding the expression, pr Nr [1 ; exp([in/) m (E) \ p 2n sin / n/N0 ] [cos /mR (E) [ i sin /mI (E)] (33) pn pn in which VR \ pr

i V m (E) m (E) \ p (E [ H) pr r

(23)

where is(0)/2n is the source for the m (E) waves and iV m (E) is r pr r the source for the m (E) waves. p We observe that eqn. (22) has the same form as the TIW equation given in eqn. (6), so that m (E), the wavefunction in r the reactant arrangement, can be calculated using the polynomial expansion method as described in eqn. (7)È(9). Eqn. (23), however, cannot be solved in this way since, unlike eqn. (6) and (22), it contains the reactant wavefunction, m (E), as a r factor, thereby resulting in an energy-dependent source. We shall discuss the calculation of m (E) in the next Section. r

3 Method for solving the arrangement partitioned time-independent wavepacket equations We have developed a method to solve the arrangementpartitioned TIW equations in which the GreenÏs operators in eqn. (22) and (23) are expanded in terms of polynomials of H. This method contains many features that were developed in previous work on the TIW equation,20h27 so we shall concentrate here on what is particular to eqn. (22) and (23). Rather than include the absorbing potentials, [iV , directly pr in the polynomial expansion, we use the damped Chebychev

1 VR g mR (E) \ pn E [ H pr rn J. Chem. Soc., Faraday T rans., 1997, V ol. 93

(34) 705

1 mI (E) \ VI g pn E [ H pr rn

(35)

The sum over n in eqn. (33) starts at N , the smallest value of 0 n for which g overlaps C (E). It is known20 that there is a rn pr correspondence between the index n and time, so that N cor0 responds to how long it would take a time-dependent wavepacket to reach C (E). The source terms (the g ) in eqn. (34) pr rn and (35) are energy independent, so that each mR (E) and mI (E) pn pn can now be calculated by expanding 1/(E [ H) in terms of Chebyshev polynomials of H.20 Hence, each mR (E) function is pn expanded as, Np 1 ; (2 [ d )exp([in@/)gRn{ mR (E) \ n{0 rn pn i*H sin / n{/0

[M Nj ; exp([in /) m (E) B j p 2n sin / j (38) ] [cos /mR (E) [ i sin /mI (E)] pnj pnj Here, M is an integer, and the n are all values of n such that j N O n O N and n \ jM. The results of testing this approx0 r j imation are reported in the next Section.

4 Computational illustration for collinear H + H 2 reactive scattering We have solved the arrangement-partitioned TIW equations, using the method described above, for collinear H ] H reac2 tive scattering on the LSTH31 potential surface. We thus solved two independent scattering problems, one in the reactant arrangement, using eqn. (24), and the other in the product arrangement, using eqn. (25). In each arrangement, we used a discrete grid of equally spaced points in the appropriate Jacobi coordinates, placing four points per de Broglie wavelength at an energy of 1.4 eV. The grid dimensions used are given in Table 1. We shall refer to the coordinates as (x , y ) in r r the reactant arrangement, and (x , y ) in the product arrangep p ment, where x is the distance between the H atom and the H 2 centre of mass, and y is the H bondlength. The Hamiltonian 2 was represented on the grid using the DAF representation,32h35 in which the potential is diagonal, and the kinetic energy operator is represented by a sparse, Toeplitz matrix. The action of the latter was evaluated using a discrete fast DAF convolution algorithm.36 The (energy dependent) partitioning potential, C (E), was pr placed along the top of the (x , y ) reactant grid by choosing r r the damping factor,37

B

706

for o y o [ y r damp

(39)

otherwise

(40)

J. Chem. Soc., Faraday T rans., 1997, V ol. 93

x min x max y min y max x damp y damp

test functions

reactant

product

1.0 11.0 0.0 6.5 8.0 3.5

3.5 14.25 0.0 3.5 11.25

reactant x 0 x l k av

product

6.5 0.5 7.0

9.75 0.5 7.0

initial wavepacket x 0 x l k av

6.5 0.5 [7.0

Atomic units are used throughout. All parameters are deÐned in the text.

gRn{ \ T (H)V R g (37) rn n{ pr rn A similar expression holds for mI (E). pn We note that the exact solution of eqn. (33) evidently requires the propagation of (N [ N ) of the functions, mR (E) r 0 pn and mI (E), and is therefore likely to be very expensive, since pn typical values of N are of the order of 1000. We therefore r consider a simple approximate way to solve eqn. (33), in which we regard the sum over n as the evaluation of an integral by numerical quadrature. (This is similar to the standard method of evaluating, e.g. the partition function for the translational or rotational motion in quantum statistical mechanics.) We then (further) approximate the integral by choosing a coarser spacing of quadrature points, so that m (E) is now approxp imated by

oy o[y 3 r damp d(x , y ) \ 1 [ 0.4 r r y [y max damp \1

grid parameters

(36)

where

A

Table 1 Grid and wavepacket parameters used in the reactant and product arrangement calculations

We also included damping of this form along the right-hand edge of both the (x , y ) and (x , y ) grids, in order to prevent r r p p reÑection. The values used for all the damping parameters are given in Table 1. We chose an initial wavepacket, s (0), of the form, r (x [ x )2 1 0 exp [ r sl(x , y o 0) \ r r r 2x2 J(x n1@2) l l ] exp(ik x )/ ( y ) (41) av r l r where / ( y ) is the lth vibrational wavefunction of H . We l r 2 also used wavepackets of this form in the Ðnal state analysis as “ test functions Ï, s@(0) and s@ (0) ; that is, we evaluated the r p S-matrix elements using the TIW Kohn-variational formula,23

A

B

i+2J(k k ) l l{ Ss@ (0) o G`(E) o s (0)T (42) Sll{ \ r rj k(2n)2A (k )A*(k ) j r l j l{ where k is the reduced mass for motion along the x coordinate, and j denotes r or p ; A (k ) and A (k ) are the Fourierr l j l{ momentum components of s (0) and s@ (0), evaluated at the r j appropriate momenta, k and k . Note that the action of l l{ G`(E) on s (0) is, here, a symbolic representation of the calcur lation of either m (E), or m (E). The values taken by the paramr p eters, x , x and k , in s (0), s@(0) and s@ (0) are given in Table 0 l av r r p 1. We calculated the reactant arrangement wavefunction, m (E), with N [the largest value of n in eqn. (26)] set to N \ r r r 1000, and *H and H1 (the normalizing factors) both set to 0.3 a.u. In Table 2, we present the inelastic transition probabilities that were obtained from the Ðnal state analysis of the calculated m (E). These results are compared with the S-matrix r Kohn Variational (SKV) results of Colbert and Miller.38 Having calculating m (E), we retained all the g which were r rn non-zero in the region of C (E), and transformed them (in the pr region where C (E) is non-zero) from the (x , y ) to the (x , y ) pr r r p p coordinates by using a DAF Ðtting algorithm.26 We found that N , the smallest value of n for which g overlapped 0 rn Table 2 Inelastic transition probabilities obtained from the TIW calculation in the reactant arrangement and the SKV results of Colbert and Miller energy/ eV

TIW

SKV

0]0

0.5 0.8 1.1 1.4

0.934 6.06 ] 10~2 0.170 0.303

0.917 6.22 ] 10~2 0.172 0.300

0]1

0.8 1.1 1.4

6.71 ] 10~4 0.152 0.219

2.62 ] 10~5 0.153 0.220

0]2

1.4

8.66 ] 10~2

8.91 ] 10~2

transition

C (E), took the value N \ 120. We then used the transpr 0 formed g functions to form the source terms required to calrn culate m (E). Rather than propagate each mR (E) and mI (E) as p pn pn in eqn. (34) and (35) (which would have required 1780 propagations), we solved eqn. (25) directly, propagating m (E) p at four values of E. The results are given in Table 3 in the column labelled as N \ 880. j We next tested the quadrature approximation in eqn. (38) by calculating m (E) as a sum over the N g functions, and r j rnj repeating the calculation of m (E). This was done using values p of M [in eqn. (38)] of 24, 20 and 16 ; these values corresponded to N \ 37, 44 and 55. In all the calculations, N (the j p number of propagation steps for the product arrangement channel) was set to N \ 500. H1 and *H were both set to 0.3 p a.u. The reactive transition probabilities that were obtained from the calculated m (E) functions are also presented in Table p 3, where they are compared with the SKV results.

5 Discussion We Ðnd in Table 3 that using only a small fraction of the g nr that overlap C (E) in eqn. (38) leads to results which are suffipr ciently accurate (better than 5% error, except for the smallest probabilities). This is an unexpected result and it has extremely interesting implications as to the behaviour of the g . Basically, it implies that the g are almost linearly depennr nr dent, a result that was also observed independently in the different context of the spectral density approach to bound state energies and eigenfunctions.39 As a consequence, one need not include every g -source term in generating the m (E) [see eqn. nr p (38)]. In fact, this feature is crucial to the practicality of the arrangement partitioned TIW approach. An interesting question in relation to these results is whether a similar behaviour holds in the case of the PengÈ Zhang TD SchroŽdinger equations, and leads to equally accurate results. That is, one can solve for s (t) also using the r Chebychev-type expansion (including damping). The g one nr calculates in that case are identical to the ones computed here in the TIW version. The only change is that the expansion coefficients multiplying the g are now cylinder Bessel funcnr tions depending on time rather than energy. If the basic property that enables our approach to work for the TIW equations is the near-linear dependence of the g , for nr lM \ n \ (l ] 1)M, l \ 0, 1, . . . , N , then we expect it to work j also in the time-dependent case. This is currently under study. The results of our example application to collinear H ] H 2 reactive scattering show clearly that the arrangement decoupling, achieved via the use of appropriate partitioning potentials, works very well. The analogous TD version has been tested previously40 and also yielded quantitatively accurate results. Thus, this “ divide and conquer Ï strategy is now established both in the time and the energy domain. This then

opens up a variety of possible avenues of study. For example, can one gain any advantage by using a “ mixed Ï approach (e.g. using the energy domain to generate the g , but combining nr them to produce the source for scattering in the product arrangement in the time domain) ? The point is that in Peng and ZhangÏs approach,19 one must store s (t) in the region of r V for a large number of time steps, and also transform them pr in that region from reactant to product coordinates. But once a “ complete set Ï of g values are calculated, one can superpose nr them to obtain s (t) at any time during which it is being r absorbed. Furthermore, if one needs to include only a small subset of the g values to construct the time dependence, this nr will greatly reduce the amount of computational e†ort involved. Another important point which should be emphasized is that the present results demonstrate that one may use the damped Chebychev recursion,28,29 and thereby expand in terms of a Hermitian Hamiltonian. That is, one can take advantage of the simple relationship between the energydependent absorbing potential underlying this damping (eqn. (30)] to separate the energy dependence of the m (E) from the r spatial dependence. This enables one again to obtain results as a continuous function of energy, E, from propagating a single s (t \ 0) packet. The alternative could involve, e.g. expanding r a GreenÏs operator containing a non-Hermitian Hamiltonian using Faber polynomials.24 While these avoid instabilities in the ordinary Chebychev expansion, they do so at the expense of a larger spectral range, due to the non-Hermitian Hamiltonian. This, in turn, requires inclusion of more terms in the Faber expansion. Finally, the fact that the arrangement partitioned TIW equations work implies that the partitioned time-independent SchroŽdinger equations given by Peng and Zhang,19 and derived rigorously herein, should also work. However, because they do not contain the initial source of waves, s (0), one must r work with the more usual LippmannÈSchwinger type equation. In this case, the source term will essentially involve V t0(E), where t0(E) is not a wavepacket state, but rather an r r eigenstate of an unperturbed Hamiltonian, e.g., K (so V \ H [ K). This implies that V t0(E) changes with E in a r non-trivial fashion, and it will not be possible in general to isolate the energy dependence in analytically known coefficients. Thus, the g will be energy dependent in this case. nr

6 Conclusions In this paper we have derived the time-independent wavepacket (TIW) form of the arrangement-partitioned (AP) equations of Peng and Zhang.19 We have developed a method for solving the equations, in which the (damped) Chebyshev polynomial expansion is used in a way which permits separate applications of the GreenÏs function in the reactant and the product arrangements. In developing the method, we discovered that the Chebyshev expansion can be treated as a

Table 3 Reactive transition probabilities obtained from the TIW calculation in the product arrangement and the SKV results of Colbert and Miller TIW transition

energy/ eV

N \ 37 j

N \ 44 j

N \ 55 j

N \ 880 j

SKV

0]0

0.5 0.8 1.1 1.4

8.77 ] 10~2 0.934 0.300 5.62 ] 10~2

8.27 ] 10~2 0.934 0.300 5.61 ] 10~2

8.21 ] 10~2 0.934 0.301 5.66 ] 10~2

8.21 ] 10~2 0.934 0.301 5.65 ] 10~2

8.30 ] 10~2 0.938 0.296 5.96 ] 10~2

0]1

0.8 1.1 1.4

9.35 ] 10~6 0.381 0.234

7.03 ] 10~6 0.381 0.230

8.27 ] 10~6 0.381 0.230

8.26 ] 10~6 0.381 0.230

3.74 ] 10~5 0.380 0.224

0]2

1.4

0.108

0.107

0.107

0.107

0.107

N denotes the number of g functions used in the expansion of the energy-dependent source term (see text). j rn

J. Chem. Soc., Faraday T rans., 1997, V ol. 93

707

quadrature approximation to an integral, and thereby further approximated by retaining only a fraction of the terms. This approximation greatly reduces the amount of work required to Ðt the reactant wavepacket to the product coordinates, and reduces the number of propagations in the product arrangement. We applied the TIW AP method to collinear H ] H reac2 tive scattering and obtained results which were in very good agreement with the S-matrix Kohn variational results of Colbert and Miller.38 We have therefore demonstrated that the AP equations can be solved in the TIW form using the Chebyshev expansion method, as well as in the TDW form using the split-operator method (that was originally proposed by Peng and Zhang19). In future work we plan to investigate ways of simplifying the structure of the TIW AP equations, so that only one propagation is necessary in the product arrangement. When this is done, the TIW and TDW methods of solving the AP equations will di†er mostly in that they employ di†erent propagators. From previous work,26 it is known that the Chebyshev propagator typically requires fewer propagation steps than the split-operator propagator. This property, together with the quadrature approximation mentioned above, suggests that the TIW AP method will require fewer operations of the Hamiltonian than the TDW AP method. The split-operator propagator, however, has the advantage that it can include absorbing potentials directly ; to date this has proved to be more efficient than the indirect inclusion of absorbing potentials that is used in the damped Chebyshev propagator.28,29 This suggests that the TDW AP method will require smaller partitioning regions than the TIW AP method. Numerical tests on di†erent systems will probably be required to determine which of the two methods is the more efficient. References 1 2 3 4 5 6 7 8

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Paper 6/06300D ; Received 12th September, 1996