J. Phys. Chem. 1993,97, 4984-4988
4984
Distributed Approximating Function Theory for an Arbitrary Number of Particles in a Coordinate System-Independent Formalism David K. Hoffman* Department of Chemistry and Ames Laboratory, f Iowa State University, Ames, Iowa 5001 I
Donald J. Kouri Department of Chemistry and Department of Physics, University of Houston, Houston, Texas 77204-5641 Received: January 5, 1993
A general, multidimensional distributed approximating function theory is developed that applies to any system which has one possible configurational representation in Cartesian variables. That is, the configuration of the system can be expressed as a generalized N-dimensional vector which has the usual transformation properties under multidimensional rotations. In particular, the theory is applicable to a scattering system with an arbitrary number, A , of scattering centers and projectiles (atoms), for which N = 3A. The approach makes possible the realization of distributed approximating functions (DAFs) in any orthogonal, curvilinear coordinates including spherical polar, cylindrical polar, and hyperspherical, as well as elliptic and parabolic coordinates.
1. Introduction
The distributed approximating function (DAF) approach to the time evolution of wave packets is a powerful, new method aimed a t taking advantage of the local nature of the potential in the coordinate representation for ordinary chemical collisions, and the “almost-local” nature of the kinetic energy in this same representation.’-6 These features of the Hamiltonian lead to a highly banded expression for the propagator, specific to a restricted class of wave packets (the “DAF class”). With control of the parameters which determine it, the DAF class can be made to include the wave packets of interest in any particular timedependent quantum problem. Because the propagator is banded, its application to the propagation of wave packets on a grid (1) scales like the number of grid points, N , in any dimension (which is the ultimate for any grid method; the scaling constant depends on the band width); (2) requires reduced communication time when implemented on massively parallel computers; and (3) minimizes the storage requirements for the propagator. Despite these advantages, progress on systems with many degrees of freedom has been somewhat hampered because there has been only limited development of DAF theory for non-Cartesian variables. Although some results for radial variables have been reported,2 a general procedure for obtaining DAFs independent of the choice of coordinate system has not been available. It is this problem which is addressed in the present paper. In previous formulations of the theory, Cartesian variables play a special role in that they are required to obtain an analytic expression for the DAF free propagator (since use is made of the fact that the free particle evolution operator commuteswith the derivative operators which produce the Hermite polynomials by acting on the Gaussian generating function).lJ We now present a general derivation of the DAFs which is clearly valid for any system of orthogonal (curvilinear or Cartesian) coordinates. We also present the formalism in a way applicable to any number of particles. Before presenting these developments, we first briefly review crucial elements of DAF theory and the progress which has been made in its application to date. Any physical, 152 wave packet has a limited bandwidth in momentum space which it accesses over the course of a timedependent quantum calculation. This fact can be used to develop Ames Laboratory is operated for the Department of Energy by Iowa State University under Contract No. 2-7405-ENG82. +
0022-3654/93/2097-4984%04.00/0
a restricted coordinate-representation identity operator (the DAF), which is a truncated expansion of the Hermite polynomial representation of the &function. The DAF class of wave packets is defined to consist of all Lz wave packets which, to the desired accuracy, can be approximated as Mth degree polynomials under the DAF envelope (or, in more general formulations, such polynomials times a phase factor).Id The terms in the expansion provide a “Hermite function” window in momentum space. Hermite functions are products of Hermite polynomials times their Gaussian generating functions.’ Because of the Gaussian factor in the Hermite functions, the DAF is highly localized. Taking advantageof this property, it is possible to obtain a highly banded representation of the full evolution operator which is appropriate for propagating any wave packet, which, to the desired accuracy, is reproduced by this “filtered” identity. The DAF is an exact identity for polynomials of a given degree,l*3,5but such functions are not L2and, in fact, there are no normalizable wave packets for which the DAF provides a rigorous identity. This produces some interesting aspects of the theory. A crucial feature of the DAF (and the one motivating the use of Hermite functions to construct the filtered DAF identity) is that it can be analytically propagated by the free propagator.Is3 This fact implies that the DAF approach can be employed in any formulation of the full propagator which uses the free propagator combined with other operators to incorporate the effects of the potential. This includes various versions of the split operator approach,’ as well as the recently developed modified Cayley approach.* Other ways of implementing the effects of the potential into propagation with DAFs have also been derived and applied, including the interacting-DAF (IDAF)6 and the generalized quadratic-DAF (GQDAF) .9 In addition to DAF approaches to wave packet propagation, new methods for treating time-independent quantum dynamics have also been developed.’O Indeed, promising time-independent wave packetSchrSdingerand wave packet-LippmannSchwinger equations have been derived and applied to some simple collision systems. Thesemethods makeuseof a “DAFHamiltonian” which is also highly banded. By now, a number of computational implementations of DAFs has been carried out. Included are electrons tunneling through double barriers’ I (including extremely long-lived resonances and, therefore, long propagation times), atom-collinear diatom inelastic scattering,12 and atom-corrugated surface scattering in 3D.13 Calculations are underway for atom-collinear diatom 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 19, 1993 4905
Distributed Approximating Function Theory reactive scattering. A key feature in several of the above studies is the explicit demonstration that the highly banded nature of the DAF operators enables efficient computer codes to be written. With regard to efficiency, DAF-based methods compete and in some cases outperform the powerful, and widely used, fast Fourier transform (FFT) methods. The DAF proves to be most efficient, compared to FFTs, when high accuracy is required, but our experienceindicates that some combinationof methods produces the most efficient ~ o d e s . ~However, ~ J ~ the DAF requires much less memory, which is an extremely important considerationsince memory can be as costly (or more so) than CPU time. Also, the DAF requires less communication time on massively parallel computers. These matters will be fully explored in subsequent papers. Another extremely promising and attractive feature of the DAFs is the fact that they lead to expressions for real-time Feynman path integrals which automatically contain Gaussians for use in random sampling.’ This has enabled US to carry out Monte Carlo-Gaussian sampling of real.time path integral expressions for a 1D autocorrelation function (the survival time).’ Indeed, combining the DAFs with the modified Cayley wave packet propagation expressions shows substantial improvement of such Monte Carlo-Gaussian sampling calculations of path integrals. l4 This paper is organized as follows. In the next section, we present the derivation of the DAFs independent of the choice of coordinates and show how the Hermite functions become generalized for arbitrary numbers of degrees of freedom and choice of orthogonal coordinates. In section 3 we discuss the propertiesof the generalized “DAF polynomials” associatedwith different coordinates and numbers of degrees of freedom, and in section 4 we show how angular momentum conservationleads to a radial propagator for spherical waves. Finally, in section 5 we give our conclusions.
Hermite polynomials are given in terms of their generator by
where V, is the x component of the gradient operator. This equation provides two equivalent and useful ways in which eq 4 can be expressed. That is,
O))-1’2x m
6(x-x ’) = (2 m 2 (
(-1 /4)“( n!)-’[2a2(O)] nV$ X
n=O
exp[-(x - ~ ’ ) ~ / 2 a ~ ( O(6a) )] or m
b(x-x ’) = (2 ~ a ~ ( O ) ) - ”(-1~ /4)”(n!)-’ x [2a2(0)]”V,,2” X n=O
exp[-(x - x’)’/2a2(0)] (6b) where in the second equation Vx2nhas been replaced by Vx/2n. The two are equivalentbecause the derivativeacts on an even function of x - x’. If eq 6a or 6b is substituted into eq 3, the general term in the resulting multidimensional sum contains the factor [nl!!t2!...nN!]-I, where the integer n, is the sum index for the 6 function for the ith cbmponent. Making use of the fact that n!/[nl!n2! ...n ~ ! ]for n = Czln, is the general multinomial expansion coefficient for an N-sum, we can perform the multidimensional sum by first summing over the set of indices (n,) which add to some fixed n (by invoking the multinomialexpansion theorem) and then summing over all values of n. The result is m
6(R-R’) = ( 2 ~ ~ ~ ( 0 ) ) - ~ ~ ~ ~ ( - 1 / 4[2a2(o)]”vR2” )”(n!)-~X n=O
exp[-(R - R’)2/2a2(0)] (7) 2. Theory
We consider a system which can be represented in a general configuration space by a vector R with an arbitrary number N of Cartesian components,X I , x2, ...,xN. By “Cartesian* we mean that the volume element for the space is given by N
dR = n d x j j= I
An arbitrary function, \k(R) in Hilbert space can be represented by q ( R ) = JdR’ 6(R-R’) q(R’)
(2)
where 6(R-R’) is the 6 function in N dimensions which can be written in the form N
6(R-R’) = nS(x-x’)
(3)
j= 1
The theory of distributed approximating functions is based on the Hermite function representation of the 1D 6 function which is
where V Ris ~the N - D Laplacian operator. Here, for simplicity, it is assumed that the parameter u(0) is the same for all degrees of freedom. This need not be the case; one may simply introduce scaled coordinates, (x, - x’J/u,(O). The resulting expressions no longer contain u(0) explicitly.’ The advantage of this expression is that it is independent of the particular choiceofcoordinatesystemand wecanusewhatever coordinates are convenient for a particular problem (e.g., Cartesian, cylindrical polar, spherical polar, hyperspherical, elliptical, etc., as the case might be). If R is conveniently divided into two or more subvectors, Le., R = Rl, R2, ..., R,, for the purposes of a particular calculation, then we can write 6(R-R’) = 6(R1-R’,) 6(R,-R’2)
... 6(Rm-R’,)
(8) where the &function for each subspace is given by eq 7 and we can represent eachR, in thecoordinate systemof choice. Equation 3, the original starting place, is a special case when m = N. Let R be an N - D coordinate vector (or a subvector of the coordinate vector as in eq 8) which we want to treat in some specified coordinate system. The simplest version of distributed approximating function (DAF) theory results from truncating the sum in eq 7 at some maximum value M / 2 to obtain
6 [R,R’;M,u(O)] = M /2
~ ( x - x ?= ( ~ T u ~ ( o ) ) -exp[-(x ’/~
- x ’ ) ~ / ~ ~ ~ x( o ) I
n=O
m
C(-1
(~ T U ~ ( O ) ) -(-~1//4)”( ~ C n!)-’[2a2(O)]n V c X
- X ? / (2) 1 / 2 d 0 ) /n! I (4)
exp[-(R - R’)2/2a2(0)] (9)
Here Hzn is a Hermite polynomial and u(0) is the width of the Gaussian generator of the Hermite polynomials in eq 4. The
(This particular truncation gives rise to the so-called “stationary” DAF (SDAF). Following the procedures outlined in ref 5 , a more general “traveling” DAF (TDAF) truncation can be given.
/4)”%[(X
n=O
4986
Hoffman and Kouri
The Journal of Physical Chemistry, Vol. 97, No. 19, 1993
The result, however, is a straightforward extension of what we present here and consequently will not be pursued further.) As indicated by the notation, the distributed approximatingfunction, G[R,R’;M,u(O)], depends on M a n d u(0) (or the set {ui(O))),but it does not depend on the coordinate system per se. However, the functional form of its realization, of course, does depend on the coordinate system, and it is this that dictates the system of choice. Almost always it is most convenient to use a “spherical”coordinate system in the N - D space, and we now confine our attention to such cases (i-e., we exclude elliptical or parabolic systems, etc.). By spherical, we mean that one coordinate is Ri, the magnitude of Ri, and the other N - 1 coordinates are angles. The space of the subvector, Ri, could be 1D (a special case), 2D, 3D, or hyperspherical. If we define a set of “scalar” polynomials Li by
L : ( R ~exp(-R2) ) = vR2“ exp(-R2)
(10) then we can write G[R,R’;M,a(O)] in the more explicit form MI2
b [R,R’;M,u(O)] = ( 2 7 r ~ ~ ( O ) ) - ~ /(-1 ’ E / 4)”(n!)-’Li [ (R n=O
R’)’/(2~~(0))’/~] exp[-(R - R’)2/2u2(0)] (1 1) The polynomial L:(R2) is scalar (as indicated by the superscript), in that it depends only on the square magnitude of the vector R, and is thus invariant to rotations in the N - D space. In the next section we show that the set of functions Li, n = 0-m, forms a complete set of orthogonal polynomials in R2. We also demonstrate that if b[R,R’;M,u(O)] replaces 6(R-R’) in eq 3, the result is exact to the extent that *(R’) can be represented as a (tensor) polynomial of degree M in R’ in a neighborhood about R. The size of the neighborhood is controlled by u(O), through the Gaussian function exp[-(R - R’)2/2u2(0)] and by M. The functions which can be so represented are said to comprise the DAF class.’ (Again, one can also associate different neighborhoods with different degrees of freedom by introducing separate width parameters, (uix(0),uiy(0),u,=(O)),for the components of Ri).3 Of course, any analytic function which is strictly a polynomial in some region is a polynomial everywhere, and thus normalizable wave packets (and indeed all nonpolynomic functions) never strictly satisfy this condition. Thus, the definition of a DAF class for wave packets inherently involves specification of the error one is willing to tolerate when 6(R-R’) is replaced by G[R,R’;M,u(O)] in eq 2. This can be made arbitrarily small, but is always nonzero. If we choose our initial coordinates to be mass-weighted, then the kinetic energy operator k is given by
h = -(h2/2h)V,Z
(12)
where p is a generalized “reduced“ mass.15 The effect of the free propagation operator, exp(-i&/h), acting on the DAF is then given by exp(-ihT/ h ) 6[R,R’;M,u(O)] = MI2
(2Xu2(o))-~’2C(-i/4)n(n!)-1[2u2(o)lnv,Z” x n=O
exp(-ikr/h) exp[-(R - R’)2/2u2(0)] (13) Here we have made use of the fact that the free propagator operator commutes with the gradient. Making use of the wellknown result thatL6 exp(-ikT/h) exp[-(R - R’)2/2u2(0)] = exp[-(R - R’)2/2u2(~)] (14)
+
where ~ ~ ( =7 u2(0) ) i h r / p and performing the indicated differentiation with the aid of eq 10, we find that F[R,R’;M,u(O)Ir], the DAF-class free propagator, is given by
F[R,R’;M,u(O)lr]
exp(-ihr/ h) S[R,R’;M,u(O)l
MI2
= (~ ~ U ~ ( O ) )(-1 - ~/ 4)”( / ~ n!)-’ E [u(0)/ u( 7 ) ]2nL; [(R n=O
R’)2/(2u2(~))’/2]exp[-(R - R’)2/2u2(~)] (15) The polynomial
R’)’/ (2u2(T))1’2] is a multidimensionalgeneralizationof the previously introduced “shape p~lynomial”.~ Although the DAF-class free propagator is not a local operator, it is localized by virtue of the Gaussian factor, exp[-(R - R’)2/2u2(~)],which makes its matrix elements effectively nonzero only in a neighborhood for which (R- R’l is small. Within specified accuracy limits, the domain of this neighborhood is determined by U ( T ) and M . The free propagation of any wave packet in the DAF class is awmplished by *(R,f+T) = JdR’ F[R,R’;M,u(O)lr] *(R’,t)
(16)
and the integral can be calculated on any discretized grid for which the spacing of the points is sufficiently close. By using a split operator formalism,’ modified Cayley formalism: or some similar approaches where the effects of free proagation are separated from the propagation by the potential, the action of the full propagator can be obtained. 3. Properties of the Polynomials
Equation 10 defines a set of polynomials, which we call generalized Hermite polynomials, the salient properties of which we now explore. We will also have need for a similar set of vector functions defined by L:(R) exp(-R2) = -
v ~ exp(-R2) v ~
(17)
Under rotations in the N dimensional space Li transforms according to the scalar representation of the full rotation group and L: according to the vector representation. Following the convention for the 3D rotation group, we denote the former by “s” and the latter by “p”. In 1D these functions are simply the even and odd Hermite polynomials, respectively. We first wish to establish that these are orthogonal functional sets. Consider first the integral JdR L y mexp(-R2) and assume for the case of argument that n Im. Then from eq 11 we have that JdR L r mexp(-R2) = JdR L;V,Zm exp(-R2) = JdR (VR2”L;) exp(-R2) (1 8) where the latter equality is obtained by successive integrations by parts. From this result we immediately establish that the integral vanishes unless n = m. Furthermore, it is easily shown by direct differentiation using the defining relation of eq 10 that coefficient of ( R ~ )in” L; = 4”
(19) Knowing the term of highest degree in R2, it is then a simple matter to evaluate the integral of eq 18 for n = m. A nearly identical argument shows that JdR L:Lk exp(-R2) also vanishes unless n = m, and using the fact that the lead term of L: is -2R(4RZ)” the value of the integral for n = m is also readily established. In this manner the orthogonality relations
Distributed Approximating Function Theory
The Journal of Physical Chemistry, Vol. 97,No. 19, 1993 4987
JdR LzLk exp(-R2) =
P+’(m!){r(m
+ 1 + ~ / 2 ) / r ( ~ / 2 ) } # / ~ 6(21) ~,~
are obtained. Another useful set of relations result from lowering operations. To obtain these, we first examine V;Li. It obviously is a polynomial of degree n - 1 in R2 and thus can be expressed as
[email protected],where ak is an expansion coefficient. Now from the orthogonality relation of eq 20 we have that akJdR LF; exp(-R2) must equal JdR (VR2L,“,)Li exp(-R2), and if we integrate the latter by parts, we obtain JdR (V:Li)Li
exp(-R2) = JdR LiV:{LS,
exp(-R2)) =
JdR L z i , , exp(-R2) (22) where we have made use of eq 10. From this result it is clear that the only nonzero coefficient is an-l, and hence VcL: is proportional to Li-l. A similar argument establishes that V;L,P is proportional to LR,. The proportionality constants can be obtained by doing the indicated integrals to obtain the lowering relations VR2L: = 4(2n)(2n
+ N - 2)LL1
(23)
+ N)Lp,
(24)
VR2L,P= 4(2n)(2n
Another lowering relationship established in much the same way is VRL; = 4nLE-,
(25)
The above results are useful for deriving the polynomial recursion relations. We start with the fact that R’L,”, is a polynomial of n + 1 degree in R2 and can thus be expressed as
are also complete for any function of R2 (Le., any function that transforms like a scalar in the N-space). We now wish to establish that G[R,R‘;M,u(O)] is an acceptable approximation to 6(R-R’) for all functions which can be adequately represented by a polynomial of degree M under the DAF envelope (Le., for functions in the DAF class). To do this we consider a general polynomic function of the form Q(R’) = Qo(R)
+ (R’-R)Q,(R) + (1/2)(R’-R)(R’-R):QO2(R)
+ ... (32)
where the highest term is of degree M in R’ - R, and examine the integral JdR’ G[R,R’;M,u(O)] 9(R’). Now from eq 11 we have that 6 [R,R’;M,u(O)] transforms like a scalar under a general rotation about R and thus from group theoretical considerations only scalarlike terms in 9(R’) survive the integration. Since the Li for k = 0 to M/2 are complete for such terms, we need only consider integrals of the form JdR’ G[R,R’;M,u(O)] L;[(R’ R)2/(2?ru2(0))1/2] to analyze the original integral. From eq 20 we find that JdR’ G[R,R’;M,a(O)]Li[(R’ - R)2/(2~a2(0))’/2] = (-4)kr(k+N/2)/r(N/2) (33) but the right hand side is just the constant term of Li as can be readily verified from the recursion relation of eq 3 1. Therefore we have JdR’ G[R,RM,u(O)]Li[(R’ - R)2/(2?ra2(0))’/2]= JdR’ G(R-R’)Li[(R’ - R)2/(2?ra2(0))”2] (34) and, more generally, JdR’ G[R,R’;M,u(O)] Q(R’) = JdR’ 6(R-R’) Q(R’)
(35)
which establishes the original contention. 4. Free Propagator for a Spherical Wave
It is often convenient to reduce the dimensionality of a scattering problem by invoking angular momentum conservation. Consider, for example, a particle described by a wave packet of the form2 where the coefficients bk are to be determined. Now VR2exp(-R2) = [4R2- 2Nl exp(-R2)
Q W ) = Q/(R,t) (27)
and, hence, from eq 26 and the orthogonality of the polynomials, we have bkJdR LiLi exp(-R2) = 1/4JdR Lz:[V:
+ 2Nl exp(-R2)
(28)
Integrating the right hand side by parts and rearranging, we have [bk- (N/2)6k,n]JdR LiLi exp(-R2) = 1/4JdR x [(v:Li)Li
+ Li(v~’Li)+ 2(v&i)(v&i)]
exp(-R2) (29)
+ 2n)6,,,
Q1(R,t+7)
y;”(k) = M/2
- ( 2 W n + N - 2)6k,n-l- 1/46k,n-l = 0
(30) which, when substituted into eq 26, leads to the recursion relation [R2-2n-N/2]LS, = (2n)(2n+d-2)L;-,
+ 1/4Li+l
(36)
freely propagating in 3D. Here r;“ is a spherical harmonic with total angular momentum quantum number 1 and azimuthal quantum number m. Because angular momentum is conserved under free motion, the form of eq 36 is retained for all times under free propagation. Thus, describing the free motion of the wave packet is reduced to the 1D problem of determining the time evolution of 9(R,t), which is turn is governed by an effective 1D Hamiltonian which is 1 dependent. Instead of immediately eliminating the angular variables, it is convenient to make use of eq 16. This equation together with eq 36 yields
and by using eqs 20, 21, 23, 24, and 25, we obtain b, - (N/2
r(R
(31)
Noting from eq 19 that the coefficient of the (R2)* does not vanish, we immediately have that the set of orthogonalpolynomials
[ 2 ~ a ~ ( O ) ]~(l/n!)[-a2(0)/4u2(7)]nJomdR’ -~/~ X n=O
R”QI(R’,t)JdR‘exp[-(R2
+ (R’)2 - 2RR’cos e)/
+
2u2(7>]L,”,[(R2 (R’)’ - 2RR’cos O)/2u2(~)]Y;“(&J(37) where cos 9 = kk‘,and the approximate equality holds to the extent that the original packet of eq 36 is in the DAF class. We now multiply both sides of this equation by (21 + l)-’r*(j?),
4988
The Journal of Physical Chemistry, Vol. 97, No. 19, 1993
sum over m making use of the addition theorem
and integrate over k to obtain
q I ( R , f + 7= ) SomdR’(R’)2F’[R,R’;M,u(0)lr] */(R,t) (39) HereF/[R,R’;M,u(o)l~]is the free propagatorfor the radialmotion and is given explicitly by
P[R,R’;M,u(0)17] =
1 ; d x exp[-(R2
+ (I?’)* - 2RR4)/2u2(7)] X
Li[(R2+ (R’)2- 2RR’x)/2u2(7)]PI(x) (40) Since both L: and PI are polynomials the integral over x can be done in closed form, but the result is algebraically complicated. Terms of two kinds result from the integration; one is proportional to exp[-(R - R’,V/2u2(r)] and one to exp[-(R + R’)2/2u2(~)]. The latter is important only when both R and R’are near the origin. 5. Conclusions
In this paper we have shown how the DAF formalism can be developed in a way which is independent of any specific choice of coordinate system. This allows one to derive a DAF and DAFclass propagator appropriate to any choice of orthogonal coordinates, including spherical polar, cylindrical polar, hyperspherical, parabolic, elliptic, etc. In addition, the DAF formalism has been developed for any number of particles, which are free to move in ordinary 3D space, or in reduced dimensions (e&, atomdiatom collisions restricted to collinear,12 coplanar, or full 3D, electron transport in 1D, 2D, or 3D nanostructures,’I etc.). Special attention has been focused on generalized spherical coordinate systems. As part of the new formalism, the concept of the Hermite function and shape polynomial has been extended: and we have discussed properties of the resulting associated, generalized Hermite polynomials. Of the possible orthogonal coordinates, one of special interest for the generalized DAFs is spherical polar coordinates, since it
Hoffman and Kouri is generally computationally efficient to take advantage of conservation of total angular momentum in carrying out quantum dynamics calculations by wave packet propagation. Although some progress in developing the DAF propagator for radial variables had been reported previously,2 the present approach is much more powerful and general. The present analysiswill form the basis of a complete DAF treatment of molecular collision processes. Because of the banded nature of the DAF-close propagator, the ability to utilize this approach in any curvilinear coordinate system should make wave packet propagation extremely efficient. We are currently initiating computational implementations of the approach for some example collision systems.
Acknowledgment. The work of D.K.H. was supported in part under National Science Foundation Grant CHE92-01967, and that of D.J.K. was supported in part under National Science Foundation Grants CHE89-07429 and CHE92-005 18.
References and Notes (1) Hoffman,D. K.;Nayar,N.;Sharafeddin,O. A.;Kouri, D. J.J. Phys. Chem. 1991.95,8299. (2) Hoffman, D. K.;Kouri, D. J. J . Phys. Chem. 1992, 96, 1179. (3) Kouri, D. J.; Zhu, W.; Ma, X.;Pettitt, B. M.; Hoffman, D. K. J . Phys. Chem. 1992,96,9622. (4) Kouri, D. J.; Hoffman, D. K. J. Phys. Chem. 1992,96,9631. (5) Hoffman, D. K.; Arnold, M.; Kouri, D. J. J . Phys. Chem. 1992,96, 6539;1993,97, 1 1 10. (6) Hoffman, D. K.; Arnold, M.; Zhu, W.; Kouri, D. J. J. Chem. Phys., in press. (7) (a) Feit, M. D.; Fleck, J. A., Jr. J . Chem. Phys. 1983,79,301;1984, 80,2578. (b) DeVries, P. NATOASISer. 1988,8171,113. (c) Bandrauk, A. D.; Shen, H. Chem. Phys. Lett. 1991, 176,428. (8) (a) Judson, R.S.;McGarrah, D. B.; Sharafeddin, 0. A.; Kouri, D. J.; Hoffman, D. K. J . Chem. Phys. 1991,94,3577.(b) Sharafeddin, 0.A.; Kouri, D. J.; Hoffman, D. K. Can.J . Chem. 1992,70,686.(c) Sharafeddin, 0.A.; Kouri, D. J.; Judson, R.S.; Hoffman, D. K. J . Chem. Phys. 1992,96, 5039. (9) Marchioro, T.L.; Hoffman, D. K.; Huang, Y.; Kouri, D. J. To be published. (IO) Kouri, D. J.; Arnold, M.; Hoffman, D. J. Chem. Phys. Lett. 1993, 203, 166. (11) Nayar, N.;Hoffman, D. K.; Ma, X.;Kouri, D. J. J . Phys. Chem. 1992,96,9637. (12) Huang, Y.;Kouri, D. J.; Arnold, M.; Marchioro, T. L.;Hoffman, D. K. J . Chem. Phys., in press. (13) Huang,Y.;Kouri, D. J.;Arnold, M.; Marchioro,T. L.;Hoffman, D. K. Compur. Phys. Commun., in press. (14) Ma, X.;Kouri, D. J.; Hoffman, D. K. Chem. Phys. Lett., in press. (15)See,e.g.: Delves, L. M. Nucl.Phys. 1960,20, 275. (16) Powell, J. L.; Crasemann, B. Quantum Mechanics;Addison-Wesley: Reading, MA, 1961;pp 77-81.
