J. Phys. Chem. 1992, 96, 6539-6545
6539
Properties of the Optimum Distributed Approximating Function Class Propagator for Discretized and Continuous Wave Packet Propagations David K. Hoffman, Mark Amold, Department of Chemistry and Ames Laboratory,? Iowa State University, Ames. Iowa 5001 1
and Donald J. Kouri*l* Department of Physics and Department of Chemistry, University of Houston, Houston, Texas 77204-5641 (Received: December 2, 1991)
A new, more concise derivation of the continuous and discretized distributed approximating function (CDAF and DDAF) class free propagators and a detailed explication of their properties are presented. The DAF class propagators are characterized by three factors: namely, a real Gaussian, a unimodulator oscillatory factor, and a ‘shape polynomial” of degree M (with M even) which has complex coefficients. For the continuous version of the theory, these factors are solely a function of the ~], x)’/4h~], and g,,,(x - x I u ( O ) = ( ~ T / ~ ) I / ~respectively, ,T), time step T and take the forms exp[-m(x’- ~ ) ~ / 4 hexp[im(x’where o(0) = (hr/m)’I2is the width of the Gaussian envelope of the CDAF. The discrete version of the theory requires slightly different forms for these factors, because the choice of a(0) depends also on the grid spacing. The relationship ~(0) = (hT/m)ll2,which gives the optimum choice of the Gaussian envelope of the CDAF for minimizing its spread in time T , is derived. The relationship between the Gaussian width o(0) and the degree of the shape polynomial is given, and it is shown that the specification of the time step T is sufficient to fix all other parameters. The time step T is detennined by the characteristics of the propagator algorithm being used.
I. Introduction Recently, we have begun a study of wave packet propagation in which the objective is to take advantage of the local (diagonal) character of the coordinate representation of the potentials normally encountered in chemical dynamics and the almost local (banded) character of the coordinate representation of the kinetic This necessitates an abandonment of the highly energy popular (and successful) fast Fourier transform (FFT) method for calculating the action of the kinetic energy ~perator.~,’ The success of such an approach is clearly dependent on the degree to which the free particle evolution operator can be banded when put in the coordinate representation. By making use of the fact that the action of the free particle evolution operator on Hermite functions (the product of a Hermite polynomial times its Gaussian generator) can be evaluated analytically, it has been shown that the discretized effective free propagator appropriate for propagating a particular class of wave functions has a bandwidth determined by a Gaussian envelope.’ Although the initial derivation involved the use of Cartesian coordinates and equispaced grids, it was subsequently shown that the procedure could be extended to radial variables and to nonequally spaced discretizations.2 Next, the extension of the DAFs to yield a continuous effective free propagator was given, making possible a general quantum formalism for Monte Carlo integration of real time Feynman path integrals with Gaussian importance ampl ling.^ In the companion paper,4 we examined properties of the exact free and full propagators in the Feynman path integral formalism with emphasis on the consequences of the purely mathematical necessity of including infinitely large momenta (or energies), with all momenta contributing equally, in calculating the matrix element (xlexp(-iKr/h)lx) or (xlexp(-iHr/h)lx). Such general matrix elements are required if one writes expressions capable of treating all possible physical problems, including systems of indefinitely high momenta or energies. Any particular physical wave packet propagation requiresinclusion of energies or momenta with relative amplitudes that decrease rapidly with increasing magnitude of momentum. Earlier, Makri*assumed that for a given initial wave packet and potential, all contributionsto the propagator, associated with higher energies or momenta than the cutoff, are irrelevant ~~~
‘Am- Laboratory is operated for the US.Department of Energy by Iowa State University under Contract No. 2-7405-ENG82. *Supported in part under National Science Foundation Grant CHE8907429 and R. A. Welch Foundation Grant E-608. 0022-3654/92/2096-6539%03.00/0
for its propagation, and the result is elimination of some of the problematic properties of the exact, coordinate representation, free propagator matrix elements. However, Makri’s effective propagator still has sufficient oscillations as to make long time path integration unfeasible by Monte Carlo methods? The CDAF class propagator, with its bare Gaussian envelope, provides a promising new tool for banding the discretized propagator and for enabling Monte Carlo evaluation of path integrals, without introducing a sharp momentum cutoff.’-s However, these earlier studies did not consider the details of how, e.g., the optimum values of the relevant parameters of the DAF class free propagator were to be chosen. In addition, the roles of the three factors comprising the respective CDAF and DDAF propagators in determining their behavior have not been fully investigated. These are extremely important issues in order to provide a general, powerful scheme for mast efficiently carrying out either discretized wave packet propagations or Monte Carlo evaluation of path integrals with Gaussian importance sampling. In the third paper of this series? it was noted that the CDAF class propagator shows promise of overcoming problems encountered in Makri’s8 Fourier transform based effective propagator procedure. In this paper, we discuss in detail the way in which the various factors of the DAF effective propagator combine to yield the desired behavior. We also analyze the parameters in the CDAF and obtain their optimum values. This paper is organized as follows. In section I1 we present a new derivation of the DDAF and CDAF class zero-time propagators. This enables us to analyze how the various parameters in the DAF class propagators are interrelated and how sufficiently accurate fits of the initial packet can be constructed. In section I11 we extend the derivation to treat the nonzero-time DAF class free propagators. Finally, our conclusions are given in section IV. 11. Derivation of the DAF Function as the “Identity” for a
Class of Function
The general philosophy of our approach has been explicated in some detail in earlier papers.14 Briefly, the idea is that, whereas the formal operators in the theory must apply to every function in the Hilbert space, for any physical problem conservation principles, and, in particular, conservation of energy, require that only a certain class of functions of the total space will be sampled. From the beginning we set out to establish a theory which is only valid in the class of functions of the problem,’ a process we call 0 1992 American Chemical Society
Hoffman et al.
6540 The Journal of Physical Chemistry, Vol. 96, No. 16, 1992
“prefiltering”. By taking this approach, we avoid the problematic properties associated with operators that are used in the whole Hilbert space: constant modulus of the amplitude for the x x’ transition and violent oscillations whose frequency increases with the distance Ix’- XI. To begin we consider a continuous function,Ax), on the infinite line. It can be formally represented as
-
It should be remembered, of course, that although ~(x-x’) can be formally manipulated in many applications as though it were a well-behaved function, it is not a function at all in the true sense. Because eq 1 must fit all functions, the &function can only have a value at x = x’ and must be normalized to unity. By contrast, if we have some information about the function Ax), the &function in eq 1 can be replaced by a function which is not infinitely spiked. That is, the theory of the &function can be developed by considering it to be the limit of a sequence of analytic functions, S(w(x-x’), which become increasingly spiked in the h i g h 4 limit. That is 6(x-x’) = M-lim tW(x-x’)
(2)
Shape Polynomial with M = 8 and p-(O)/dx = .98 4
3-
2-
-5
-4
-3
-2
-1
0
1
2
3
4
5
Figure 1. Shape polynomial at T = 0, for u(O)/Ax = 0.98 and with degree M = 8. For this choice of u(O)/Ax and M,the zeros lie approximately on the integers.
where is the free propagator, and we will want to approximate the h c t i o n at time t in termsof the values at an earlier time t - 7 . In other cases we will want to consider approximations on a grid. In light of these considerations, the CDAFs in eq 4 are for a continuous DAF distribution with a time step of 7 = 1 M/2 0. exp[-(~-x’)~/2u*(O)] (-1 / 4 ) n ( n ! m ) - lX 4 0 ) n=O We now turn our attention to a discretized problem in which H2n([~- x ! I / [ ~ ~ ~ ( O ) I(3) - ~ / ~ ) we require an approximation formula on a grid. Obviously, it is nonsense to attempt to discretize the integration in eq 1. However, = exp [-(x - x ’)2/ 2u2(O)]gM(x-x1u(0),7=0) this can be done easily for eq 4, providing that a(0) in the CDAF is sufficiently large compared to the grid spacing. Thus, eq 4 where gdx-xla(O),r=O) (defined by this equation) is a polybecomes nomial of degree M which we call the (zero-time) “shape polynomial”. We later discuss a generalized ( 7 # 0) version of this polynomial. If the &function in eq 1 is replaced by b(w(x-x’), Ax) = ,=-m (Ax)W(x-x,)f(xj) (6) then thisequation is valid for the class of functions discussed below. As M increases, the shape polynomial serves to make 6(m(x-x’) where Ax is the grid spacing (which, for the moment, we assume correspondingly increasingly spiked. to be uniform). It is apparent that, in order to develop a satisThe function b(m(x-x’) is dominated by the Gaussian envelope factory approximation, it will be necessary for the Gaussian enexp[-(x - ~’)~/(2~(0)2)], which serves to define the effective extent velope to capture a number of grid points. The degree of accuracy of the function. Thus, we expect ~(“(x-x’) will be a suitable of the approximation depends on how smooth the function is and approximation for the &function for any function which can be what is the grid spacing. These questions cannot be answered in adequately expressed as a polynomial of degree M under the the abstract but rather require some knowledge of the physical Gaussian envelope. It is obvious that for any analytic function, problem to be solved so that the appropriate “prefiltering” can Ax), we can make eq 1 to be valid to arbitrary accuracy by be done. Some simple considerations of this type are explored choosing M to be sufficiently large. As a general rule, the more in ref 1. As is to be expected, it is a general rule that accuracy oscillatory isdx), the higher order the polynomial that will be is improved as u(O)/Ax is increased, but for the minimum comneeded to represent it. However, the oscillatory nature of a wave putational effort one, of course, wants this quantity to be as small packet is governed by the momentum distribution of which it is as possible. We have found that this ratio must be approximately composed, and for any physical system this is controlled by energy in the range 1-5 for the problems we have examined. conservation. Thus, at the outset in considering a specific dyAs remarked in ref 1, the approximation is asymptotic in M. namical problem, we can “prefilter” by choosing an M which is That is, increasing the degree of the shape polynomial increases valid for a given a(0). the accuracy of the approximation for a range of M,but beyond The quantity 6(w(x-x’) is an example of what we call a cona certain M value the fit deteriorates. Furthermore, this happens tinuous distributed approximating function or CDAF? The name in a very systematic way.’ To understand this behavior, it is implies that we view the equation necessary to examine the behavior of 6(“(x). The zeros of this function-are determined by the zeros of the shape polynomial, Ax) = 1:dx’ 6(m(x-x’) Ax’) (4) which for fiied M a r e invariant when exwessed in units of u(0) as is evident from eq 3. They are, of’course, symmetrically as an approximating equation where a continuous distribution of distributed about the origin (because the shape polynomial is even) CDAF functions, each indexed by the x’value around which it and are approximately evenly spaced, with the exception that there is centered, is placed on the line. The CDAF is not a projection is a maximum instead of a node at the origin. This is illustrated operator, but loosely speaking, it is the identity for the class of in Figure 1 for u(O)/Ax = 0.98 and M = 8. In general, there functions for which eq 4 holds. To approximate the function, each are M real zeros. As M is increased, the zeros, of course, increase CDAF is multiplied byflx’) (the value of the function at the point in number on the fringe and move toward the origin so that the at which it is centered), and the approximate result is then obtained peak at the origin gets steeper and narrows, as is to be expected. by adding together the contributions from all DAFs. The most accurate approximation occurs when the first zeros of In more general cases we will consider the function we want the DAF (symmetrically placed about the origin) are about equal to propagate as being a function of time according to to the grid spacing, as indicated in Figure 2 for a(O)/Ax = 2.36 f(x,t) = e-iK‘/hA x ,t=0) and M = 54 in conjunction with Figure 3 (which is taken from (5) There are many ways to do this. For our purposes the GaussHermite representation is most convenient. Thus, we write W(x-x’) =
-c
2
The Journal of Physical Chemistry, Vol. 96, No. 16, 1992 6541
Optimum Distributed Approximating Function
@ h dM
Modulus of OAF 1.2
'Modulus'
-
'-1
VS. qo)/AX '
~
/
I
901
80
-
70 -
0.8
/
60-
40 -
0.6
50 I
0.4
30 -
L
0.2
-10
-5 U-101 / A
X
0 = 2.36;
5
M
/ ,
54
Figure 2. Modulus of the DDAF at T = 0, for a(O)/Ax = 2.36 and with degree M = 54.
0.3
0
/ 1
1.5
2
2.5
3
3.5
Figure 4. Shown is the value of M that yields the best fit as a function of a(O)/Ax. This value causes the zeros of the DDAF to fall at or near
the grid points.
Goodness of fit as a fcn of ~-(O)/AX and M 16 14
*O/ 10
10
0
i
z"
One of the advantages of working in the coordinate representation is that it is a relatively easy matter to introduce a nonuniform grid of pointsa2 This is not possible (at least in a straightforward manner) using Fourier transform methods. The reason one might wish to employ a nonuniform grid is that it allows one to p i t i o n sampling points strategically relative to the potential in real scattering problems, so as to capture the important features of the evolving wave packet with a minimum number of points. (For a continuous problem there is obviously an analogous concept of a nonuniform weighting or scale function in the configuration space.) The most obvious way to incorporate a nonuniform grid is to write eq 6 in the form
5
Ax) * je-, (W+(%-xJ M = degree of Shape Polynomial
Figure 3. This plot is reproduced from ref 1; it gives the number of correct digits in a fit of the arbitrarily chosen function [exp(-x2/32) cos (0.7x)I at x = 0.8 against the degree of the shape polynomial. The four curves represent different values of a(O)/Ax, and each is peaked precisely at the optimal M value.
ref 1). This is reasonable because it implies that the functional value at the origin contributes strongly to the approximate function evaluated at the origin but contributes very little to the approximate value of the function at other grid points. This behavior is to be compared, for example, with the Fourier interpolation f i t i o n sin ( r x ) / ( r x ) ,which is exactly equal to unity at the origin and zero at all other grid points.' As M is increased beyond the optimum value, so that the first zero of the DAF appears between the first grid point and the origin, the accuracy of the approximation gradually decreases, giving rise to the asymptotic character of the approximation in M. The larger the value of u(O)/Ax, the larger the degree of the polynomial that is needed to make the spacing of zerm the same as the grid spacing, as is illustrated by comparing Figures 1 and 2. In Figure 4 a plot of the optimal value of M as a function of u(O)/Ax is given. (Note that the peaks in Figure 3 coincide with the optimal M values in Figure 4.) The accuracy of the fit with increasing M increases much less steeply for large values of u(O)/Ax than for small values, because it takes a higher degree of polynomial to represent the function under a broader Gaussian. However, the best fit improves as u(O)/Ax increases because the structure of the DAF is broader. (See Figure 3.) It is, in fact, apparent that any analytic function can be fitted to arbitrary accuracy by increasing the value of u(O)/Ax and choosing M optimally, since in this limit the discretized integral becomes equivalent to eq 4. As has been argued,' this gives rise to an arbitrarily accurate approximation for analytic functions.
AX,)
(7)
where
= (xj+l - xj-l)/2 (8) is the grid spacing for the jth grid point. For a nonuniform grid it is clearly no longer possible to bring all of the zeros of the shape polynomial into register with the grid spacing by appropriate choice of M,and as a consequence, the accuracy of the approximation provided by eq 7 suffers. A more sophisticated method of introducing a nonuniform grid has been presented in ref 2. From the point of view of the present paper, we can reformulate the discussion there by writing eq 1 in the form ( b ) j
where x(y) is the desired mapping of the grid points onto the space. The mapping is chosen so that x(y=j) = x, is the desired jth grid point for integer j . In they space the grid points are evenly spaced, and so the DAF approximation formula can be used effectively in the manner
The function b(m(y(x)-y) does not propagate analytically under the action of the free propagator, but it can be easily and efficiently expanded in Hermite functions in x . As is fully discussed in ref 2, this gives rise to modified DAF functions, which involve odd as well as even Hermite functions in x, which can be propagated analytically. For applications using the CDAF theory the weight function dx/dy in eq 9 can be incorporated into the bias sampling. X One might want to use the WKB-like damping, e~p[(2mE)'/~ h-'.f:, dx' (V(x?/E - 1)'/2], developed in another paper, to provide a bias against sampling points in the highly nonclassical regions for example? Here xl is a classical turning point and x I x,.
6542 The Journal of Physical Chemistry, Vol. 96, No.16, 1992
III. "be DAF Class Free Propagator The free propagator operator is given by F(7) = e-'KT/h
(11)
where h2 2m is the kinetic energy operator and T is the time step for the propagation. Applying F(T) to eq 1, we obtain F(X,XIT) = F(T) 6(X-X') (12) as the exact coordinate representation matrix element of the free propagator. (Here x is considered to be the variable and x'an index.) The analytic expression for this exact matrix element is well-known and is easily obtained by first writing the &function in the momentum representation
K = - -(d2/dx2)
and then making use of the fact that the kinetic energy, and hence the free propagator, is diagonal in this representation. The result is9 F(x,xIr) = ( m / 2 ~ i h r ) ' /exp[(im/2h~)(x ~ - x ' ) ~ ](14) The probIematic behavior of thii coordinate representation matrix element has been explored in the companion paper.4 By applying F(T) to eq 4, we obtain the CDAF class free propagator F""(X,XlT) = F(T)6(""(X-X') (15) The action of the free propagator on ~(""(x-x') can be obtained analytically.' An outline of the procedure for doing this is as follows. First, we note that HAY) exP(-Y2) = (-l)"(d"dY") exP(-Y2) (16) which is the expression for the generator of the Hermite polynomials where y = (x - x')/[2%(0)] (17) Thus 6(""(x-x') =
But, from the defining relations of eqs 11 and 1l', it is seen that the commutator relation [F(r),d"/dy"] = 0 (19) holds, and thus, by making use of the well-known resultlo F(4bP[-(X
- X?2/(2.2(o))l)
= [ ~ ( O ) / ~ ( ~ ) l ( e x P [- (x12/(2u2(7))1) x (20)
we have that
1
= -{exp[-(x 47)
- x ' ) ~ / ( ~ u ~ ( T x) ) ] )
Here
= u2(0) + ih.r/m
(22) is the complex variance of the Gaussian. a,the sequence It is important to realize that, in the limit M of the CDAF class free propagators converges to F(x,x~T)(the &(T)
-
Hoffman et al. exact coordinate representation matrix element of the free propagator), regardless of the choice of u(0). In the limit u(0) 0, eq 14 is clearly recovered, but this is also the case for any finite u(0) in the M limit. That is, there is no natural or preferred value of u(0). This result is simply a manifestation of the fact that the exact expression for the free propagator contains contributions from all momenta such that the variance in the momentum is infinite. To pursue this point further, we note from the theory of Fourier transforms that if u(0) is the variance of the Gaussian in coordinate space, then h/u(O) is the variance in momentum space. Clearly, it is not desirable to choose a value of M which is larger than that necessary to represent the wave packet by a polynomial of degree M across the spread of the Gaussian in all regions and at all times. This is related to the distribution of momenta required to express the wave packet at all times. Consequently, if we choose a large value of u(O), with a narrow momentum spread, h/u(O), we must then correspondingly choose a large enough M value to build a CDAF function that is sufficiently oscillatory to capture only the dominant momentum components necessary to accurately approximatethe wave 0 limit, packet. (As a special case we note that, in the u(0) all momenta are included equally in the momentum distribution of the Gaussian envelope, and only the n = 0 term is required in the sum of eq 21.) Making use of the fact that
-
-
-
[U2(T)]-' = [u2(0) + i h ~ / m ] - I = [u2(0) -ihr/m]/[u4(0)
+ ( h ~ / m ) ~(23) ]
the free DAF class propagator of eq 21 can be decomposed as a product of three factors according to fiw(X,XIT) exp[-(x - ~ ? ~ 2 ( 0 ) / ( 2 [ & + 9 ( h ~ / m ) ~ 1 )x1 exp[i(x - x ')2 h r / (2m [ u4(0) + (h r / m)2])]g&x-x Iu(O),r) (24) where
This polynomial is of degree M and has complex coefficients; it is called the "shape polynomial" and is the generalization of the zero-time polynomial introduced in eq 3. For r = 0 its structure has been previously discussed. The first term in eq 24 provides the bare Gaussian envelope for the CDAF propagator; it ultimately controls the asymptotic nature of the propagator matrix element as a function of x - x'. The second factor is oscillatory in form and is very similar to the exponential term in the exact free propagator of eq 14. Although this term also becomes highly oscillatory as x - x'increases, this behavior is ultimately damped out by the Gaussian. The shape polynomial has the effect of lifting the oscillatory term in the wings. The result is that the CDAF propagator spreads out from the a, Gaussian envelope with increasing M. In the limit as M the result of eq 14 is obtained, although for any finite M the matrix element in the wings is damped by the Gaussian. In Figure 5 it is shown how, for T * 0, the Gaussian peak is sharpened by the shape polynomial to give the zero-time CDAF. This obviously occurs because all of the zeros of the shape polynomial lie on the real axis for T = 0. For nonzero r , the zeros of the shape polynomial move off of the real axis, and as a result the CDAF modulus is broadened as is illustrated in Figure 6. It should be noted in this figure that the CDAF at T = 0 is narrower than its Gaussian envelope, whereas the CDAF propagated for a time T extends beyond its envelope. This is consistent with the uncertainty principle and nicely illustrates that a Gaussian packet undergoes minimum uncertainty spreading. Even though the shape polynomial causes the propagated CDAF to extend beyond the wings of the Gaussian, the CDAF is ultimately dominated by the Gaussian. Advantage was taken of this behavior in ref 3 to
-
Optimum Distributed Approximating Function
The Journal of Physical Chemistry, Vol. 96, No. 16, 1992 6543
DAF, Bare Gaussian Envelope and Scaled Shape Polynomial
Envelope and Oscillation Factors
>I
I
I
i
2.5 2-
1.5 -
"",,
-1.5l' -2
-1.5
-1
-0.5
0
1.5
1
0.5
1 12 Figure 7. The Gaussian envelope and the real part of the oscillatory factor of the CDAF as shown.
~ ( O ) / A X = 0.5;?= 0.0;M = 20
Figure 5. This plot shows, for T = 0, the Gaussian envelope, the shape polynomial, and their product, the CDAF. The shape polynomial has been scaled by a factor of l / l Q only 10 of its 20 zeros are shown. DAF and Bare Gaussian EnvcloDe lor 7= 0 and 0.25
which determines the optimal relationship between the initial Gaussian and the time step. Substituting eq 27 into eq 24 finally yields
[
exp i-(x m 4hT 1.5
I
This expression has some very interesting features. First, only ~ T ) ; are no other free the time step T determines ~ ; < M ( X , X there parameters. From eqs 20 and 27 it is seen that, through uncertainty spreading, the Gaussian envelope width increases by a factor of dj during the time step. Second, this fixed relationship between u(0) = (h7/m)'I2and 7 means that the Gaussian and the oscillatory phase factor stand in fmed relationship to each other. The wavelength of the first oscillation about the origin is determined by
.. ...
....
0 -3
- ~ ' ) ~ ] g n r ( x - x ~ ( h ~ / m ) "(28) ',r)
-2
-1
0 T(O)/dx
1
2
3
= 0.5; M = 10
Figure 6. Two pairs of curves are shown: the Gaussian envelope at times T = 0 and T = 0.25 and the modulus of the D A F at times T = 0 and T = 0.25.
introduce an "expanded" Gaussian envelope, under which the overall CDAF envelope lies. This yields an improved bias sampling function which ensures proper sampling in the wings of the CDAF envelope, while also decreasing the magnitude of the part of the integrand not included in the Gaussian bias. We now turn our attention to the optimum choice of ~(0).Two quite different cam present themselves. For applications requiring CDAFs (as in, for example, the evaluation of Feynman path integrals), it is important to minimize the number of oscillations in the CDAF propagator to reduce the statistical error that results from the Monte Carlo evaluation of integrals. This requires a minimum bandwidth for the propagated CDAF. For fucd u(0) the accuracy of the approximation increases with increasing M,for fued M the accuracy increases with decreasing u(0). The width of the CDAF propagator is controlled by these considerations plus the uncertainty spreading of the packet, which in turn is governed by the complex variance C ( T ) of eq 22. From eq 24 we see that minimum spreading of the Gaussian envelope for time T is obtained when [u4(0) + ( h ~ / m ) ~ ] / u ~=([Re O ) (1/uz(~))]-'
(26) is a minimum. Minimizing this quantity with respect to u(0) at fixed T leads to u(0) = (hT/nl)'/Z
(27)
( m / 4 h ~ ) ( X / 2 )= ~r
(29)
where X is the wavelength. Substituting Ix - X I = X/2 into the Gaussian envelope factor we have that
[
exp - z4hT ( A / 2 ) z ] = e- = 0.0432 Thus, the first wave of the oscillation basically fills the Gaussian envelope, as is illustrated in Figure 7. The wavelength falls off linearly as a function of Ix - X I , and subsequent oscillations are of shorter wavelength but are ultimately damped by the Gaussian. The effect of the shape polynomial is to lift these outlying oscillations in the wings. Of course, as M is increased, the oscillations in the wings are elevated to resemble the free propagator as illustrated in Figures 8-10 for the real part of the CDAF. (The imaginary part of the CDAF exhibits similar behavior.) Finally, it is interesting to note that the oscillatory factor in P ) ( x , x ~ T ) is of exactly the same form as the exact propagator of eq 14 except that T is replaced by 27. For the discretized problem the considerationsare somewhat different. The nature of the oscillations in the DDAF class propagator is not an important consideration. The critical quation is one of accuracy of the initial fitting by DDAF expansion. As previously explained, this is a question of grid spacing and the choice of u(0) and M (which is determined by u(0)). The propagation preserves the accuracy of the initial fitting. The places where the DDAF class propagator must be evaluated are completely prescribed by the grid, and thus the oscillations of the DDAF propagator are not an issue. To achieve satisfactory accuracy with the optimum choice of u(0) given by eq 27, it is necessary to choose a grid spacing which is much smaller than
6544 The Journal of Physical Chemistry, Vol. 96, No. 16, 1992 Real Part of Free Propogator and DAF 1 'Free'
-4
- 3.
-2
-1 c(O)/d x
-
0 0.5;
5 -
1 0.25; M
-
2
-
3
4
6
F i i 8. The real part of the exact (analytically known) free propagator and the real part of the CDAF with M = 6 as shown. Real P a r t of Free Propogator and DAF
1 'Free'
-
0.8 0 6 0.4
0 2
0 -0.2
-0.4 -0.6 -0.8 -4
-3
-2
-
-1
r ( o ) / ox
1
0
0.5:
t- 0.25;
n
-
2
3
4
iz
Figure 9. Same as Figure 8, but for M = 12. Compare with Figure 8
and note the improved fit and broader effective support. Real P a r t of Free Pronoa.3tor and DAF 1 0.8
0.6 0.4
0.2 0
-0.2 -0.4
-0.6 -0.8 -4
-3
-2
-
-1 U I O i l ~ x
0 0.5;
r
-
1 0.25;
M
-
2
3
4
18
Figure 10. Same as Figures 8 and 9, but for M = 18.
would otherwise be necessary. In our experience with certain model problems, a(0) which is 3 or more times greater than the optimum value is required for the grid spacing we have employed. Thus, in summary, (h7/m)1/2is a lower bound for the choice of a(O), and the actual value is determined by the grid spacing and the accuracy required of the DAF propagator. IV. conclusions In this paper we have given a new derivation of the CDAF and DDAF class free propagators which greatly clarifies their prop
Hoffman et al. erties. At time t = 0, the CDAF functions are shown, loosely speaking, to be a resolution (in terms of a sum of M Hermite polynomials times their Gaussian generator) of the 'identity operator" for the class of functions that can be accurately expressed in terms of polynomials of degree M o r less, within the envelope of the CDAF. (Although, it should be emphasized that the zero-time CDAF is nor a projection operator.) The DDAF is obtained from the CDAF fitting formula by quadrature. The relationship between the initial width, o(O),of the CDAF and the number M of Hermites is examined, and it is shown that the two parameters change in concert with one another. Thus, decreasing the variance a(0) allows one to decrease M,and increasing a(0) requires increasing M so as to obtain an accurate fit of the class of functions of interest. It is noted also that decreasing a(0) implies an increasing uncertainty or variance in momentum space. This, in turn, leads to spreading of the CDAF during the time 7 . Thus, it is not desirable to make u(0) so small that the CDAF at time 7 has spread too much (since this results in a less banded propagator). Application of the free propagator, on the CDAF yields the time-evolved "identity" on the class of functions of interest. Any function belonging to the appropriate class of functions at initial time r is propagated to the same class at time t + 7 because momentum is considered in free propagation. The action of the free propagator on the class of functions identity 6(M(x - x') yields the CDAF class free propagator within the class of functions. The new width of the Gaussian part of the CDAF is u(7) and is chosen by minimizing the spread. This implies that, for a given time step T , the choice of a(0) to produce minimal spreading yields the optimum CDAF since it will be banded as much as is possible (due to its minimal spread during the propagation time 7 ) . This analysis of the various CDAF parameters showed that (for the minimal spreading choice of a(0)) all are determined by specifying the time step 7. This time step, in turn, is determined by the specific algorithm used to integrate the timedependent Schrainger equation (i.e., the kinetic referenced modified Cayley, the kinetic or potential referenced symmetric split operator, etc.). In contrast to our previous work, wg have developed DAF grid methods by discretizing the continuum problem. It is shown that if one defines a grid in x by a Ax, then the zero-time DDAF that fits the packet will achieve its highest accuracy when the fmt mros of the zero-time shape polynomial fall on the grid points. This occurs as M is increased since this systematically moves the smallest zeros of the shape polynomial toward the origin. The M zeros for the polynomial of degree M are interspersed between the zeros of the polynomial for M + 2. (Recall that the degree of the polynomial is even because the Dirac delta function is even.) We conclude that the CDAF and DDAF procedures for obtaining prefitered free propagators should provide very powerful tools for carrying out both path integral calculations (by capitalizing on the rapidly decaying, approximately Gaussian envelope of the CDAF) and matrix propagation of wave packets (by capitalizing on the highly banded structure of the DDAF). We are currently carrying out calculations utilizing both forms of DAFs as well as extending the DAF theory to eliminate violent oscillations that can arise in highly nonclassical regions of space." Acknowledgment. D.K.H. and D.J.K. gratefully acknowledge the hospitality of the Institute for Theoretical Physics, University of California-Santa Barbara, where part of this work was carried out. Stimulating conversations with Horia Metiu and Chris B6ttcher are gratefully acknowledged.
References and Notes (1) Hoffman, D. K.; Nayar, N.; Sharafeddin, 0. 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, K. J.; Hoffman, D. K. J . Phys. Chem., submitted for publication. (4) Kouri, D. J.; Hoffman, D. K. J . Phys. Chem., submitted for publica-
tinn
( 5 ) See also: Sharafeddin,0. A.; Kouri, D. J.; Nayar, N.; Hoffman, D. K. J . Chem. Phys. 1983, 79, 1823. (6) Feit, M.D.; Fleck, J. A. J . Chem. Phys. 1982, 78, 301.
J. Phys. Chem. 1992,96,6545-6549 (7)Kosloff, D.;Kosloff, R. J. Compur. Phys. 1983,52,35; J. Chem. Phys.
1983. 79. 1923.
( 8 ) (a) Makri, N . Chem. Phys. Lerr. 1989, 159, 489. (b) Makri, N . Comput. Phys. Commun. 1991,63, 389. (9) (a) Feynman, R. P. Rev. Mod. Phys. 1948,20, 367. (b) Feynman, R.
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P.;Hibbs, A. R. Quantum Mechanics and Parh Integrals; McGraw-Hill: New York. 1965. (IO) Powell, J. L.; Crasemann, B. Quantum Mechunics; Addison-Wesley: Reading, MA, 1961. (1 1) Hoffman, D.K.; Kouri, D.J. Manuscript in preparation.
Tuning of Photoinduced Energy Transfer in a Bichromophoric Coumarin Supermolecule by Cation Binding B. Valeur,* J. Pouget, J. Bouson, Laboratoire de Chimie GenPrale (CNRS URA llO3), Conservatoire National des Arts et Metiers. 292 rue Saint-Martin, 75003 Paris, France
M.Kaschke,f and N. P. Ernsting Max-Planck-Institut fiir biophysikalische Chemie, Abteilung Luserphysik, 0-3400 Gbttingen, Germany (Received: February 12, 1992; In Final Form: April 20, 1992)
A bichromophoric molecule consisting of two coumarins linked by a pentakis(ethy1ene oxide) spacer can efficiently bind Pb2+ions in acetonitrile and in propylene carbonate. The resulting changes in photophysical properties are reported with special attention to photoinduced electronic energy transfer. Steady-state data allow one to determine the transfer efficiency and stoichiometry of the complex, whereas subpicosecond time-resolved experiments by the excite-and-probe technique provide information on the transfer kinetics. The stoichiometry of the complex is 1:l in acetonitrile and 1:3 (ligandmetal) in propylene carbonate. A significant increase in efficiency and rate of energy transfer for the complex is observed in acetonitrile, whereas there is almost no change in propylene carbonate. The results are discussed in terms of a possible structure of the complex which is different in the two solvents.
Introduction Photoinduced energy transfer is a photoprocess of paramount importance involved in many fields: molecular and supramolecular photophysics, polymer physics, laser physics, biology, and molecular devices. Systems of particular interest are bichromophoric molecules containing a donor able to transfer its excitation energy to an acceptor linked to it by a spacer. Such supermolecules are well suited for the study of fundamental aspects of electronic energy transfer and for applications as laser dyes, frequency converters of light, and more generally, as molecular devices.’ An interesting basic question arises as to whether it is possible to control and tune electronic energy transfer. Since this process is distance and orientation dependent, the rate of transfer can be modified by inducing changes in the mutual distance and orientation of the donor and acceptor moieties by means of an extemal perturbation acting on the spacer. Figure 1 illustrates two possibilities: (i) the spacer is a short chain able to form a complex with metal ions (e.g., polyether chains, or more generally chains containing 0, N, or S atoms), and the external perturbation is the addition of a cation; (ii) the spacer is photoisomerizable, and the extemal perturbation is light. The supermolecule DXA described in this paper offers an example of the first case.
0
a
- -
(0 CHz CH+
DONOR
NH ACCEPTOR
DXA This supermoleculeconsists of two coumarins linked by a short flexible chain (pentakis(ethy1ene oxide)) capable of complexation with cations; the spacer is indeed an open-chain ligand of the crown type The emission spectrum of the coumarin donor
’
Resent address: Carl Zeiss, Zentralbereich Forschung, D-7082Oberkochen, Germany.
strongly overlaps the absorption spectrum of the coumarin acceptor so that effkient electronic energy transfer is possible after optical excitation. In previous papers, we reported studies of the distribution of interchromophoric distances by means of conformational calculations,’ and steady-state’ and time-resolved‘ energy-transfer experiments. Here we report the photophysical changes induced by cation binding with special attention to electronic energy transfer. A preliminary account was published r e c e n t l ~ . ~
Materials and Methods Materials. The synthesis of DXA was previously described.’ As model compounds, 7-ethoxycoumarin and coumarin 500 were used for the donor and acceptor moieties, respectively. Ltad perchlorate (Pb(C1O4),-3H2O)from Aldrich was kept in vacuum. Acetonitrile (spectroscopic grade) was purchased from Janssen. Propylene carbonate was from Aldrich and doubly distilled prior to use. Preparation of Solutions. A stock solution of DXA (1-2.5 X M) in acetonitrile or propylene carbonate containing a supporting electrolyte (tetraethylammonium perchlorate, 0.1 M) was prepared at the appropriate optical density (0.154.45).Part of this solution was taken, and lead perchlorate (Pb(C1O4),-3H2O) was added to a concentration of 1.2 X le2M in acetonitrile and 6.2 X M in propylene carbonate. This solution of Pb2+was gradually added by means of micropipets to 2 mL of the stock solution of DXA directly into the cuvette. In this way, the concentrations in DXA and supporting electrolyte were kept constant. Instruments. The absorption spectra were recorded on a d Kontron Uvikon 940 spectrophotometer. The c ~ ~ e c t eemission and excitation spectra were measured on a SLM 8000C spectrofluorometer. Transient absorption spectra were recorded with a laser system combining an excimer laser with a colliding-pulse mode-locked (CPM) dye laser. It produces pump pulses at 308 nm with pulse durations of less than 250 fs (fwhm), and pulses at 616 nm with less than 100-fs duration. The red pulses are used for the generation of the spectral continuum, which probes the transient absorption of the DXA after donor excitation by the short UV
0022-365419212096-6545$03.00/0 0 1992 American Chemical Society
