J. Phys. Chem. 1992,96, 1179-1184
1179
Distributed Approximating Functlon Theory: A General, Fully Quantal Approach to Wave Propagation David K. Hoffman Department of Chemistry and Ames Laboratory.’ Iowa State University, Ames, Iowa 5001 1 and Donald J. Kauri' Department of Chemistry and Department of Physics, University of Houston, Houston. Texas 77204-5641 (Received: August 15, 1991)
A new procedure for developing a coarse-grained representation of the free particle propagator in Cartesian, cylindrical, and spherical polar coordinates is presented. The approach departs from a standard basis representation of the propagator and the state function to which it is applied. Instead, distributed approximating functions (DAFs), developed recently in the context of propagating wave packets in 1-D on an infinite line, are used to create a coarse-grained, highly banded matrix which produces arbitrarily accurate results for the free propagation of wave packets. The new DAF formalism can be used with nonuniform grid spacings. The banded, discretized matrix DAF representation of (xIexp(-iKT/h)lx’) can be employed in any wave packet propagation scheme which makes use of the free propagator. A major feature of the DAF expression for the effective free propagator is that the modulus of the x,,xI: element is proportional to the Gaussian exp(-d(0)(xj X ~ ) ~ / ~ ( U ~+ ( Oh2rz/mz)). ) The Occurrence of a 7-dependent width is a manifestation of the fundamental spreading of a wave packet as it evolves through time, and it is the minimum possible because the DAF representation of the free propagator is based on evolving the Gaussian generator of the Hermite polynomials. This suggests that the DAFs yield the most highly banded effective free propagator possible. The second major feature of the DAF representation of the free propagator is that it can be used for real time dynamics based on Feynman path integrals. This holds the possibility that the real time dynamics for multidimensional systems could be done by Monte Carlo methods with a Gaussian as the importance sampling function.
I. Introduction Basis set expansion methods are the primary tool used for solving partial differential equations, and the development of new basis functions especially suited for solving particular classes of problems is extremely important. In the field of quantum mechanics, one is faced with solving either the time-independent SchrMinger equation (TISE), subject to appropriate boundary conditions,’ or the timedependent Schrijdinger equation (TDSE), In the case of timeindependent subject to an initial quantum scattering, the boundary conditions are such that one must solve large numbers of simultaneous linear algebraic equations in order to determine the appropriate expansion coefficients for expressing the physical solution in terms of the basis set employed. By contrast, the TDSE yields the scattering information without requiring the solution of a large, linear algebraic system of equations, and this leads to a slower scaling of the necesSary computational effort with basis set size. For this reason, we have been focusing substantial efforts on developing methods for solving the TDSE.49~s1~s3~ssJ8”0~67~70 However, in either approach many of the same basis set considerations hold. Thus,one desires a rapid rate of convergence, ease of construction of the matrix representation of the Hamiltonian in the basis, a simple structure of the relevant operators in the basis (e.g., block diagonal, banded, or sparse structures), good behavior of the basis functions at boundaries (e.g., no spurious reflections and uniform fitting of the wave function), the ability to employ nonuniformly spaced grid points for evaluating matrix elements, the ability to treat infinite or semiinfinitedomains, and the ability to place few grid points in regions where the potential prevents significant penetration of the wave packet. (Notethat we consider the use of grid methods to be a particular type of basis set method.) Some popular choices for bases are the Fourier,12-13discrete variable representation (DVR),” and eigenstates for various bound degrees of freedom.2s In fact, most TDSE problems aredealt with using combinations of the above types of basis functions (Fourier methods are typically used for unbound or Cartesian variables, DVR basis sets can be used for both bounded and unbounded Amea Laboratory is operated for the US.Department of Energy by Iowa State University under Contract No. 2-7405-ENG82.
0022-365419212096-1179$03.00/0
degrees of freedom, and energy eigenstate bases are typically used for internal degrees of freedom). (1) See,e.g., the discussions of a variety of modern methods for time-independent scattering theory (and references therein): (a) Kuppermann, A.; Hipes, P. B. J . Chem. Phys. 1986, 84, 5962. (b) Linderberg, J. Int. J. Quantum Chem., Quantum Chem. Symp. 1986, 19, 467. (c) Webster, F.; Light, J. C. J . Chem. Phys. 1986,85,4744. (d) Linderberg, J.; Vessel, B. Int. J. Quanfum Chem. 1987,31,65. (e) Zhang, J. Z. H.; Kouri, D. J.; Haug, K.; Schwenke, D. W.; Shima, Y.; Truhlar, D. G. J . Chem. Phys. 1988,88, 2492. (f) Pack, R. T; Parker, G. A. J. Chem. Phys. 1987, 85, 5252. (g) 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. (h) Baer, M.; Shima, Y. Phys. Rev. A 1987,35,5252. (i) Baer, M. J. Phys. Chem. 1987,91,5846. (j) Zhang, J. Z. H.; Miller, W. H. J. Chem. Phys. 1988,88,449. (k) Launay, J. M.; kpetit, B. Chem. Phys. Lett. 1988, 144, 346. (1) Neuhauser, D.; Baer, M. J. Chem. Phys. 1988,88,2858. (m) Schatz, G.C. Chem. Phys. Leff. 1988, 150, 92. (n) Manolopoulos, D. E.; Wyatt, R. E. J. Chem. 3hys. 1989, 91, 6096. ( 0 ) Baer, M. J. Chem. Phys. 1989,90,3043. (p) Halvick, P.; Truhlar, D. G.;Schwenke, D. W.; Sun, Y.; Kouri, D. J. J. Phys. Chem. 1990,94,3231. (9)Zhao, M.; Truhlar, D. G.;Schwenke, D. W.; Yu, C.-H.; Kouri, D. J. J. Phys. Chem. 1990,94, 7602. (r) Manolopoulos, D. E.; D’Mello, M.; Wyatt, R. E. J . Chem. Phys. 1990, 93, 403. (s) Zhang, J. Z. H.; Yeager, D. L.; Miller, W. H. Chem. Phys. Lett. 1990,173,489. (t) Zhang, J. Z. H. J. Chem. Phys. 1991,94,6047. (u) Sun, Y.; Kouri, D. J.; Truhlar, D. G. Nucl. Phys. 1990, A508,41c. (v) Sun, Y.; Kouri, D. J.; Truhiar, D. G.; Schwenke, D. W. Phys. Rev. A. 1990,41,4857. For some earlier developmental papers on basis functions in timaindependent scattering, see: (w) Abdallah, J., Jr.; Truhlar, D. G.J . Chem. Phys. 1974,61,30. (x) Light, J. C.; Walker, R. B. J. Chem. Phys. 1976,65,4272. (y) Baer, M.; Drolshagen, G.; Toennies, J. P. J. Chem. Phys. 1980,73,1690. (z) Kuppermann, A.; Schatz, G. C.; Baer, M. J. Chem. Phys. 1976, 65,4596. (aa) Mullaney, N. A.; Truhlar, D. G. Chem. Phys. 1979,39,91. (ab) Pack, R. T; Parker, G. A. J. Chem. Phys. 1987,82,3888. (ac) Staszewska, G.;Truhlar, D. G. J. Chem. Phys. 1987,86, 1646. (2) Some recent reviews include: (a) For molecule-surface scattering, Gerber, R. B.; Kosloff, R.; Bennan, M. Compuf.Phys. Rep. 1986,5,59. (b) For collinear gas phase reactive scattering, Mohan, V.; Sathymurthy, N. Ibid. 1988,7,213. (c) For a general review, Kosloff, R. J. Phys. Chem. 1988.92, 2087. (3) Mazur, J.; Rubin, R. J. J . Chem. Phys. 1959, 31, 1395. (4) McCullough, E. A,; Wyatt, R. E. J. Chem. Phys. 1971,543578,3592. ( 5 ) Zubert, Ch.; Kamal, T.; Zulike, L. Chem. Phys. Leff. 1975, 36, 396. (6) (a) Heller, E. J. J . Chem. Phys. 1975,62, 1544,1976,65,4979. (b) Kulander, K. C.; Heller, E. J. Ibid. 1978,69,2439. (c) Drolshagen, G.; HeUer, E. J. Ibid. 1983, 79, 2072. (7) Askar, A.; Cakmak, A. S. J . Chem. Phys. 1978,68, 2194. ( 8 ) (a) Kulander, K. C. J. Chcm. Phys. 1978,69,5064. (b) Sray. J. C.; Fraser, G. A.; Truhlar, D. G.;Kulander, K. C. Ibid. 1980, 73, 5726. (c) Orel, A. E.; Kulander, K. C. Chem. Phys. Lett. 1988, 146,428.
0 1992 American Chemical Society
1180 The Journal of Physical Chemistry, Vol. 96, No. 3, 1992
In this paper, we continue the development of a new approach in which we abandon the idea of using a basis to construct an
(9) LeForestier, C.; Bergerson, G.; Hiberty, P. C. Chem. Phys. Lett. 1981, 84, 385. (See also LeForestier, C. Chem. Phys. 1974.87, 241.)
(10)Agrawal, P. M.; Raff, L. M. J . Chem. Phys. 1982, 77,3946. (11)Heller, E. J.; Sundberg, R. L.; Tannor, D. J. J. Phys. Chem. 1982, 86, 1822. (12)(a) Feit, M. D.; Fleck, J. A. J . Chem. Phys. 1982,78, 301;1983, 79, 301;1984,80, 2578. (b) DeVries, P. in NATO ASI Ser. 1988,BI71, 113. (13)Kosloff, D.; Kosloff, R. J. Comput. Phys. 1983,52,35. Kosloff, D.; Kosloff, R. J . Chem. Phys. 1983, 79, 1823. (14) (a) LeForestier, C. Chem. Phys. 1984,87,241. LeForestier, C. In Theory of Chemical Reaction Dynamics; Clary, D. C., Ed.; Reidel: Dordrecht, The Netherlands, 1986. (b) le Quere, F.; LeForestier, C. J . Chem. Phys. 1990, 92,247. (15)Drolshagen, G.;Heller, E. J. J. Chem. Phys. 1983,79, 2072. (16)Tal-Ezer, H.;Kosloff, R. J . Chem. Phys. 1984,81,3967. (17) Kosloff, R.; Cerjan, C. J . Chem. Phys. 1984,81,3722. (18) (a) Yinnon, A. T.; Kosloff, R.; Gerber, R. B. Surf.Sci. 1984, 148, 148. (b) Gerber, R. 8.; Yinnon, A. T.; Kosloff, R. Chem. Phys. Lett. 1984, 105,523. (19)Imre, D.; Kinsey, J.; Sinha, A,; Krenos, J. J . Phys. Chem. 1984,88, 3956. (20)Skodje, R. T. Chem. Phys. Lett. 1984,109,227. (21)Mowrey, R. C.; Kouri, D. J. Chem. Phys. Lett. 1985, 119, 285. (22)(a) Jackson, B.; Metiu, H. J . Chem. Phys. 1985, 83, 1952. (b) Sawada, S.; Heather, R.; Jackson, B.; Metiu, H. Ibid. 1985,83, 3009. (23)Zhang, Z. H.; Kouri, D. J. Phys. Reu. 1986,A34, 2687. (24)(a) Jackson, B.; Metiu, H. J . Chem. Phys. 1986, 84, 3535. (b) Jackson, B.; Metiu, H. Ibid. 1986,85,4192.(c) Jackson, B.; Metiu, H. Ibid. 1987,86,1026. (25)Mowrey, R. C.; Kouri, D. J. J . Chem. Phys. 1986,84, 6466. (26)Park, T. J.; Light, J. C. J . Chem. Phys. 1986,85,5870. (27)(a) Mohan, V.; Sathymurthy, N. Curr. Sci. 1986, 55, 115. (b) Thareja, S.; Sathymurthy, N. J . Phys. Chem. 1987,91,1970. (28)Mowrey, R. C.; Kouri, D. J. J . Chem. Phys. 1987,86,6140. (29)Heather, R.; Metiu, H. J . Chem. Phys. 1987,86, 5009;1989,90, 6116;1989,91, 1596. (30)Kouri, D. J.; Mowrey, R. C. J. Chem. Phys. 1987,87,2087. (31)Sun, Y.; Mowrey, R. C.; Kouri, D. J. J . Chem. Phys. 1987,87,339. (32) (a) Mowrey, R. C.; Bowen, H. F.; Kouri, D. J. J . Chem. Phys. 1987, 86,2441.(b) Mowrey, R.C.; Bowen, H. F.; Kouri, D. J.; Yinnon, T.; Gerber, R. B. Ibid. 1988,89,3925;manuscript in preparation. (33) (a) Huber, D.;Heller, E. J. J . Chem. Phys. 1987,87,5302;1988,89, 4752. (b) Huber, D.; Heller, E. J.; Littejohn, R. G. J . Chem. Phys. 1987.87, 5302. (c) Huber, D.; Ling, S.; Imre, D. G.; Heller, E. J. Ibid. 1989,90, 7317. (34)Sun,Y.;Kouri, D. J. J . Chem. Phys. 1988.89,2958. (35)Jackson, B. J. Chem. Phys. 1988.88, 1383; 1988,89,2473;in press. (36)Chaseman, D.; Tannor, D. J.; Imre, D. G. J . Chem. Phys. 1988,89, 6667. (37)Williams, S.0.; Imre, D. J. J . Phys. Chem. 1988, 92,6b18. (38)Tannor, D. J.; Rice, S.A. Adu. Chem. Phys. 1988,70,441. (39)Sun, Y.;Judson, R. S.; Kouri, D. J. J . Chem. Phys. 1989,90,241. (40)Neuhauser, D.; Baer, M. J . Chem. Phys. 1989,90,4351.Neuhauser, D.; Baer, M. J . Phys. Chem. 1989,93,2872. Neuhauser, D.; Baer, M. J. Chem. Phys. 1989,91,4651. (41)Neuhauser, D.; Baer, M.; Judson, R. S.; Kouri, D. J. J. Chem. Phys. 1989,90,5882;1990,93,312. Neuhauser, D.; Baer, M.; Judson, R. S.; Kouri, D. Compur. Phys. Commun. 1991, 63,460. (42)Jiang, X.-P.; Heather, R.;Metiu, H. J. Chem. Phys. 1989,90,2555. (43)Jacon, M.; Atabek, 0.; LcForestier, C. J. Chem. Phys. 1989,91,1585. (44) Zhang, J.; Imre, D. G. J . Chem. Phys. 1989,90, 1666. (45)Gray, S.K.; Wozny, C. E. J. Chem. Phys. 1989,91,7671. (46)Dixon, R. N. Mol. Phys. 1989,68,263. (47)Judson, R. S.; Kouri, D. J.; Neuhauser, D.; Baer, M. Phys. Rev. 1990, A42, 351. (48) Zhang, J. Z. H. Chem. Phys. Lett. 1989,160,417.Zhang, J. Z. H. J . Chem. Phys. 1990, 92,324. (49)Hoffman, D. K.; Sharafeddin, 0.;Judson, R. S.;Kouri,D. J. J . Chem. Phys. 1990,92,4167. (50)Das, S.; Tannor, D. J. J . Chem. Phys. 1990, 92,3403. (51)Sharafeddin, 0.Judson, ; R. S.;Kouri, D. J.; Hoffman, D. K. J. Chem. Phys. 1990, 93,5580.
Hoffman and Kouri “exact” matrix representation of the relevant operators and the state vector. We focus instead on developing an approach based on the accurate description of the result of the action of relevant operators on the appropriate class of state vectors. In a sense, our approach is akin to pursuing an “analog” versus a “digital” strategy. It will be applicable to any TDSE solution method that involves the free propagator, exp(-iKT/h) (e.g., our kinetic referenced modified Cayley m e t h ~ d or ~ ~the. ~split operator methodI2). Our strategy is to work in the discretized coordinate representation rather than to utilize, for example, the discretized momentum representation followed by a transformation to the discretized coordinate representation. Thus, the two discretized Fourier transforms required to apply exp(-iKT/ h ) to a discretized packet are replaced by a single matrix multiplication of a banded, coarse-grained, “effective” free propagator matrix times an approximation to the discretized packet.6o@Our procedure (although independently developed) bears some similarities to the new “wavelets” approach to problems in computational fluid dynami c ~ . ’ ~The specific procedure68we employ to coarse grain exp(-iKr/ h) derives from the effect of this operator when applied to an arbitrary packet expressed in terms of a set of distributed approximating functions (DAFs), or simply “fitting” functions, denoted by a,(%)E %(x - x,,). The function q,(x - x,,) is not equal to Si, at a grid point xi; that is, we are not using an interpolation scheme, which is required to yield the packet of interest exactly on the discrete set of grid points. The DAFs, however, are required to fit exactly all polynomials of order n or less. This condition is converted to an infinite set of simultaneous, linear algebraic equations by expanding the DAFs in Hermite polynomials (times the usual Gaussian weight function). The Hermites are chosen because, by means of their generating function, the action of the free particle propagator on them can be expressed analytical1y.68.73-74 This infinite set of linear algebraic equations is solved approximately by replacing discrete sums by integrals and making use of orthogonality of the Hermite functions. This leads to analytic expressions for the DAFs in terms of flnite sums of (52)Judson, R. S.; McGarrah, D. B.; Sharafeddin, 0.; Kouri, D. J.; Hoffman, D. K. J. Chem. Phys. 1991, 94,3577. (53) Neuhauser, D.; Judson, R. S. J . Chem. Phys., submitted for publication. (54)Neuhauser, D.; Judson, R. S.; Baer, M.; Kouri, D. J. Chem. Phys. Lett. 1990, 169,372. (55) Sharafeddin, 0.A.; Bowen, H. F.; Kouri. D. J.; Hoffman, D. K. J . Comput. Phys., in press. (56) Viswanathan, R.; Shi, S.; Villalonga, E.; Rabitz, H. J. Chem. Phys. 1989,91,2333. (57)Neuhauser, D. J . Chem. Phys., in press. (58)Sharafeddin, 0. A.;Kouri, D. J.; Nayar, N.; Hoffman, D. K. J. Chem. Phys., in press. (59) Hoffman, D. K.; Sharafeddin, 0.A.; Kouri, D. J.; Carter, M.; Nayar, N.: Gustafson. J. Theor. Chim. Acta 1991. 79,297. (60)Sharafeddin, 0.A.;Kouri, D. J.;.Nayar, N.; Hoffman, D. K. J . Chem. -Phvs.. in rmess. ,- --- --(61) Heller, E. J. J . Chem. S h p . 1991, 94. 2723. (62)Gray, S. K.; Wozney, C. E. J . Chem. Phys. 1991, 94, 2817. (631 Bandrauk. A. D.; Shen, H. Chem. Phys. Lett. 1991, 176,428. (64) Williams, C. J.; Qian, J.; Tnnnor, D. J. J. Chem. Phys., submitted for publication. (65)(a) Dixon, R. N.; Marston, C. C.; Baht-Kurti, G. G. J. Chem. Phys. 1990,93, 6520. (b) Baht-Kurti, G. G.; Dixon, R. N.; Marsten, C. C. J. Chem. Soc., Faraday Trans. 1990,86,1741. (66)Founargiotakis, M.; Light, J. C. J . Chem. Phys. 1990. 93, 633. (67) Carter, M.; Nayar, N.; Gustafson, J.; Hoffman, D. K.; Sharafeddin, 0. A.; Kouri, D. J. To be published. (681 Hoffman, D. K.; Nayar, N.; Sharafeddin,0.A.; Kouri, D. J. J. Phys. Chem.. 1991, 95,8299. 1691 Bala. P.; McCammon. A. J.; Truong, T.; Tanner, J.: Kouri, D. J.; Hoffman, D: K. Manuscript in preparation: (70)Parker, G. A.; Hoffman, D. K.; Kouri, D. J. Manuscript in preparation. (71)(a) Lill, J. V.; Parker, G. A.; Light, J. C. Chem. Phys. Lett. 1982, 89,483. (b) Light, J. C.; Hamilton, I. P.; Lill, J. V. J. Chem. Phys. 1985, 82, 1400. (72) (a) Grossman, A.; Morlet, J. SIAM J. Math. Anal. 1984, IS, 723. (b) Albert, B. K. Sparse Representation of Smooth Linear Operators. Ph.D. Dissertation, Yale University, Dec 1990. (73)Hoffman, D. K.; Kouri, D. J., Glowinski, R. Research in progress. (74) Feynmann, R. P.; Hibbs, A. R. Quantum Mechanics and Path Integrals; McGraw-Hill: New York, 1965.
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The Journal of Physical Chemistry, Vol. 96, No.3, 1992 1181
Distributed Approximating Function Theory Hermite functions. The wave packet at point x is thereby expressed as a sum of the fitting functions at x multiplied by the values of the wave packet on the corresponding grid points. The wave packet at time t T is evaluated by first analytically propagating the Hermite functions, thereby obtaining analytic expressions for the time-propagated DAFs. These then are multiplied by the packet values at time t on the grid to determine the propagated function. It follows that the free propagator is approximated by a discrete matrix whose elements in a column are the values of the propagated DAF (the DAF index being the same as the column index) evaluated on the grid. The time-evolved (t T ) packet is obtained by multiplying this matrix times the discrete vector of previous (at time t ) wave packet values on the grid. By appropriately choosing the width of the fitting function, one can affect the degree to which this approximating matrix is banded. (However, bandednw is also influenced by the spreading in time of the propagated DAF. Spreading is governed by the uncertainty principle; its rate increases as the initialfitting function is narrowed.) We note that although the propagated DAFs spread as time goes on the elements of the approximating matrix representation of the free propagator, for a single time step T , are constant. Thus the degree of bandedness of the approximating matrix depends on the time step, T , but not on the absolute time,
Here P ( T is ) the 1-D (indicated by the superscript (1)) free propagator, which is defined by
+
+
t.
A different way to view the present approach is as a generalized quadrature evaluation of the integral resulting from the action of the coordinate representation free propagator (continuous) matrix on a general packet. It then represents an alternative to the trapezoidal rule quadrature coarse-graining procedure we have discussed earlier.60 Desirable features of the DAFs, which we explore in this paper, are as follows: (1) arbitrary spacing of grid points is possible; and (2) coordinates other than Cartesian can be employed. (3) In addition, the DAFs result in a highly banded propagator matrix structure. (4) All calculations are done in the discretized coordinate representation. ( 5 ) DAFs can be conveniently used for other operator functions of the kinetic energy such as the free Green's function. This paper is organized as follows. In section I1 we summarize the theory of the distributed approximating function for the case of a uniform, 1-D Cartesian grid. Next, in section I11 we discuss the 1-D Cartesian grid case with nonuniform spacing. In section IV we deal with the generalization of the DAFs to multidimensional, Cartesian variables and use them to treat the case of non-Cartesian variables (cylindrical and spherical polar coordinates). Finally, section V contains our conclusions and indicates our future research directions.
P)(T) = exp(-iTKc')/h)
(5)
where Kc') is the kinetic energy operator in 1-D. Thus, the functions hn(x,Tlu) are timetranslated basis functions which incorporate, analytically, the free particle propagation. These functions at T = 0 can be used as a basis to construct a set of "fitting" functions, aj')(x,O) which we call distributed approximating functions (DAFs). They are indexed on the integers, and the jth function is symmetrically peaked about x = j , according to ajl)(x,O) = a&l)(x-j,O)
(6)
and L
ab')(x,o) =
C b2$b(x,Olu) n=O
(7)
where the integer j ranges from --m to m. Again, the superscript indicates dimensionality. The upper limit on the sum, designated here by L,is not infinite but instead is roughly fixed by u (as described in ref 68). Let d(x,t) be a 1-D wave packet which is presumed to be known on the grid of points, xi = j . (For the present we are considering an evenly spaced grid on the whole line. The grid points can be conveniently made to be the integers by appropriately scaling the problem.) The DAFs can be used tc approximate d(x,t) at any time, t , according to the equatiod8 m
d(x,t) =
C aJl)(x,o)d(j,r) j=-m
(8)
It is very important to note that the a)')(x,O) are not Kronecker deltas on the grid, and therefore, eq 8 yelds only an approximation to d(x,t) on the grid points. That is, eq 8 is not an interpolation formula. This fact has implications for the discretized free propagator which we will shortly explore. In general, the a p proximation becomes more accurate with increasing u. However, large values of u require correspondingly large values of L and also cause the function aj')(x,O) to become less localized about x = j . Making use of eq 4, the effect of the P')(T) on the packet is easily computed. Thus,the value of the packet at point x after it is freely propagated for a time T is given by ",
II. Introduction to Theory of Distributed Approximating Functiom for the 1-D Cartesian Case Recently a new way to treat the effect of the free propagator in 1-D has been put forward68and used as the numerical basis of scattering calculations employing the modified Cayley propagation method. The approach, however, is applicable for any procedure which involves a free propagator; in particular, it also will be useful for the split operator method.I2 It has the advantage that one can work entirely in the coordinate representation without using finite difference methods. The essence of the approach is as follows. We begin by defining a set of Hermite functions where Ifnis the nth Hermite polynomial and y and u are defined bY
and &(T)
= $(O)
+ ihr/m
(3)
It is shown in ref 68 that ~ , ( X , T ~ has U ) the property that 14 O (35a)
sdr) = ab''(r,o)/y/,o(~=O), sn(r) = [ @ ( ~ t t , O ) + (-l)'~b')(i+t~,O)] /Y/,o(e=O)
n > 0 (35b)
and thus
a&l)(p-n,O)
(29) It is obvious that eq 29 is simply a direct product extension of cq 11 to the 3-D case. However, it is often the case that for reasons of symmetry one wants to use non-Cartesian coordinates. Polar coordinrtes. The angular momentum operator commutes with the free propagator and hence the angular momentum state of a 3-D packet is unchanged as it freely evolves in time. This fact can be used to simplify the free propagation of a packet in a pure angular momentum state. The choice of coordinate system, of course, depends ultimately on the scattering system one wishes to consider. Thus, in cylindrical polar coordinates, we wish to deal with packets of the form f , ( p , ~ , t + ~ ) e " + = .H3)(~)&(p,z,t)dms]m = integer (30a) q&?,,jk(T)
(33a)
= aj1)(i',T)"')0'',T)a~'')(k',T)
where p, 4, and z are the usual variables in this coordinate system, and for spherical polar coordinates
and
Finally, by combining eqs 31, 32, and 34, we have that m
fm(p,zJ)e'm+
=
C cn(p)@+fm(nqzvt) n=O
(364
m
g/(rit)Y/,m(e,4) e
n=O
Sn(r)g/(nJ)Y/,m(e,4)
(36b)
Equations 36a and 36b provide us, respectively, with the cylindrical polar and spherical polar analogues of eq 8. We now turn our attention to determining the effect of the free propagator acting on these functions. Making use of eq 30a,b, we find that
Hoffman and Kouri not a particularly important consideration. Computational studies in progress indicate that a = 0.5 (Le., having the grid spacing) seems to be satisfa~tory.~~ V. Conclusion In this paper, we have succeeded in generalizing the DAF procedure, developed earlier?* to treat the cases of a 1-D, nonuniform grid and 3-D, non-Cartesian coordinates. Included are radial variables occurring both in cylindrical polar and spherical polar coordinates. These generalizations should enable the DAF formalism to be applied to a wide range of scattering problems. It is expected, however, that applications to systems involving more than three particles will normally be done using Cartesian coordinates. Some specific systems of interest include electron (37b) tunneling in 3-D quantum nanostructures, proton tunneling in fully 3-D models of genetically important molecules, gas-phase atomwhere, in analogy to eq 9, we have introduced the functions molecule scattering, and semiclassical dynamics in which some degrees of freedom are treated by fully quantal wave packet ai2)(p,7)= ai2)(p,0,O,~)= (fi2)(~)c,,eim4}(p,4=0)(38a) propagation with other degrees of freedom being treated classically. and Included in this class of problems are the interaction of radiation with quantum systems, with the radiation treated as a classical ai3)(r,r)= Bi3)(r,0,0,7)= (fi3)(r)snY~,o(6)/Y~,o(6=O)J(r,6=O) field, and molecular dynamics simulations in which a few quantum (38b) degrees of freedom are treated with wave packets, and all other particle dynamics treated classically, including Ehrenfest forces Here the a functions with tildes are the Cartesian form of the due to the quantum degrees of freedom. functions without tildes. In writing eq 38a, we have made explicit In addition, we have pointed out the fundamental connection use of the fact that the z evolution is uncoupled from that in the between the banding of the propagator and the basic nature of p plane. wave packets to spread as they evolve in time. Because the The quantity a i 2 ) ( p = k , ~is) the k,nth matrix element of the DAF-effective free propagator results from evolving the Hermite discretized propagator for a function of the radial variable in functions expressed in terms of a Gaussian generating function, cylindrical polar coordinates (cf., eq 11). Similarly, ai3)(r=k,7) it is expected that the present results are the best that can be is the k,nth matrix element of the radial propagator in spherical achieved." This is reflected in the observation that the matrix polar coordinates. Both matrix elements, of course, implicitly elements of the DAF-effective free propagator decay with (xi depend on the appropriate angular momentum eigenstate of the xi.) according to the Gaussian envelope exp(-02(0)(xj system. (d(0) h2T2/m2)).The possibility of obtaining such a Gaussian The time evolutions in eq 38a,b now can be completed easily behavior for the effective free propagator is a direct result of using the method of choice. For example, the time evolutions could relaxing the requirement that the a.(xf,O) equals a Kronnecker be carried out by Fourier transform. (The calculation need only delta, ajP That is, it is made p s i h e because we do not insist be done once for a fmed time step 7.) The time evolution can also on a standard interpolation but rather on an (accurate) approxbe done using DAF theory. Thus, we find that imation to the function on the grid points. A second fundamental m m result following from the Gaussian banding of the DAF-effective u i 2 ) ( p , ~= ) al')(p,7/a2)~jl)(O,~/a2)iSi2)(~i,OIj,0,.I) j*-m j = - m free propagator is that it opens the possibility for Monte Carlo evaluation of the real-time, many-body propagator with Gaussian (394 importance sampling. This would allow real-time, many-body and dynamics to exploit the same powerful methods that have been m m m used so effectively in equilibrium statistical mechanics where one Ui3)(f,T) = Ul''(f,T/a2)Uj''(o,T/oa1;')(0,T/a2) x uses purely imaginary times?' We are currently pursuing both j--m ji-m ki-m the above aspects of our if3(rai,-aj,-ak,7)(39b) Finally, we have noted that the DAF method is related to the so-called "wavelet" approach used in CFD.72 We are currently Here the parameter a defines a new grid spacing for the purpose carrying out studies to elucidate precisely this relation~hip.'~We of evaluating the h functions. From eq 5 and the fact that Kc') expect that the DAF procedure can, in fact, be used for a wide is a second derivative with respect to position, it is seen that if variety of problems describable in terms of wave propagation. x scales by a then T scales by 1/a2.It is clear that this new spacing must in fact be somewhat finer than the original spacing, Le., 0 Acknowledgment. This work was supported in part under < a < 1. There is no limit to how small a can be, however, as National Science Foundation Grant CHE89-07429 and in part a decreases the effective time step is rapidly increased and one under R. A. Welch Foundation Grant E-608. must include more terms in the sums for convergence (because of the spreading of the fitting functions with time). Since these (89) Naresh, N.; Hoffman, D. K.; Kouri, D. J. Research in progress. sums are done only at the start of a scattering calculation to (90) Hoffman, D. K.; Kouri, D. J. Research in progress. determine the propagator weights, economy in doing the sums is (91) Kouri, D. J.; Hoffman, D. K. Research in progress.
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