A computational demonstration of the distributed approximating

J. Phys. Chem. 1992, 96,9637-9643

9637

A Computational Demonstration of the Distributed Approximating Function Approach to Real Time Quantum Dynamics Naresh Nayar, Department of Computer Science and Ames Laboratory,f Iowa State University, Ames, Iowa SO01 I

David K. Hoffman,$ Department of Chemistry and Ames Laboratory,+Iowa State University, Ames, Iowa 50010

Xin Ma,$ and Donald J. Kouri**S Department of Chemistry and Department of Physics, University of Houston, Houston, Texas 77204-5641 (Received: March 27, 1992)

We report computational applications of the newly developed distributed approximating function (DAF) approach to real time quantal wavepacket propagation for several one-dimensionalmodel problems. The DAF is constructed to fit all wavepackets accurately which can be represented, to the same accuracy, by a polynomial of degree M,or less, within the envelope of the DAF. (This defines the "DAF class" of functions.) By expressing the DAF (and thus the wavepacket to be propagated) in terms of Hermite functions (Each a product of a Hermite polynomial and its Gaussian generating function), the DAF approximation to the wavepacket is propagated freely and exactly for a short time 7 . The Hermite functions are the natural basis states for describing the free evolution of a localized particle and yield a highly banded representation for the free particle propagator. Combining the DAF class free propagation scheme with any of several short time approximations to the full propagator enables one to propagate the wavepacket through a potential. The DAF results for the propagated wavepacket and various scattering amplitudes are shown to be in good agreement with those obtained by more standard methods.

scheme ideally suited to modern supercomputers and especially 1. Introduction to massively parallel computers.8 The Gaussian generator also Recently it has been shown that the action of the free particle plays a key role in the path integral formulation, since it provides propagator on a well defined class of wavepackets can be evaluated, an approach to the evaluation of the real time path integral by analytically and exactly, by expressing the wavepackets in terms Gaussian biased sampling Monte Carlo method^.^ In a paper4 of "distributed approximating functions", or DAFs. The DAFs which considered general properties of the exact free and full are constructed in such a way that (a) they accurately approximate propagators, it was noted that the fitting of the wavepacket must any wavepacket that can be represented, to the same level of be accurate enough so that repeated application of the potential accuracy, as an Mth degree polynomial within the DAF envelope portions of the full propagator will not remove the wavepacket and (b) The action of the free particle evolution operator on the from the DAF class. This can be done by appropriately choosing DAF can be obtained exactly and analytically. the width of the Gaussian generator of the Hermite polynomials, This class of wavepackets is termed the DAF classe6 These and the highest degree polynomial contained in the DAF.'q5 properties are guaranteed by expressing the DAF in terms of a Our previous papers have focused on the formal development new basis comprised of "Hermite functions", each of these being of the DAF approach, and the basic structure of the DAFs, and the product of a Hermite polynomial and its Gaussian generator. included calculations of survival probabilities, but did not contain These Hermite functions possess the fundamental significance that any applications of the method to scattering problems. In this they are the natural functions for describing the freely evolved paper, we provide several computational examples showing that wavepacket of localized particles. Thus, they provide a compact representation of the freely evolved DAF class of wa~epackets.**~-~ the DAF approach can provide accurate results for real time quantum dynamics, even for systems requiring extremely long This procedure of developing an accurate representation of a class propagation times. For the examples studied, we have concenof wavepackets in terms of exactly and analytically propagatible trated on the discretized or matrix version of the DAFs. We fitting functions leads to expressions for freely evolving DAF class illustrate the ability of the DAF formulation to propagate wawavepackets. The DAF fits only a ratricted class of wavepackets, vepackets accurately for long times by applying it to the following and the freely evolved DAF fits the freely evolved restricted class problems: (a) an electron scattering (in 1-D) off a double square of wavepackets. Effectively, the dynamics has been 'prefiltered", barrier, (b) an electron scattering (in I-D) off a double square so that one is no longer depending on interference to eliminate barrier, plus a linear potential, (c) an electron scattering (in 1-D) nonphysical high-frequency oscillations of the exact free propaoff a double square barrier, plus a sinusoidal time dependent g a t ~ r . ~ .Both ' discretized and continuous versions of the DAF potential representing an external radiation field, and (d) an have been developed; the former yields a discrete matrix for fitting electron scattering (in 1-D) off a double square barrier, plus a the discretized DAF class wavepackets and for propagating term varying sinusoidally with position and a linear (with position) them,l+*while the latter is used in the real time Feynman path potential term. integral form of wavepacket propagati~n.~ The Gaussian generator This paper is organized as follows. In the next section, we factor produces a highly banded matrix structure for the DAF briefly summarize the discretized DAF expressions (DDAF) for class free propagator. This, along with the "translation" property the propagated wavepacket, and in section 3 we present the results of the DAF (that enables the DAF class free propagator matrix for the model problems mentioned above. Finally, we present our to be generated from a single row), leads to a matrix propagation conclusions in section 4. 'Am= Laboratory is operated for the Department of Energy by Iowa State University under Contract No. 2-7405-ENG82. 'Supported in part under National Science Foundation Grant CHE-9201 967. $Supported in part under National Science Foundation Grants CHE8907429 and CHE-92-00518 and R. A. Welch Foundation Grant E-608.

2. Theory of Wavepacket hopagation Using Discretized Distributed Approximating Functions The propagation schemes which we use in this paper are the kinetic referenced modified Cayley93Io(KRMC) and the kinetic

0022-3654/92/2096-9637$03.00/00 1992 American Chemical Society

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The Journal of Physical Chemistry, Vol. 96, No. 24, 1992

referenced symmetric split operator"*12(-0) methods. Both involve the short time free particle evolution operator, exp(i K ~ / h )but , they differ in how the potential enters. Explicitly, they yield the wavepacket at time t in terms of its values at time t - I , according to9J0

Nayar et al.

TABLE I: Potential P a ~ t e for n Short and Long Time proprortion

(a) Double Barrier 0

otherwise

400 time steps

a = 35 A, c = 75 A,

I q ( t ) ) = e-iTv/zhe-iKT/he-'Tv/2~IW - 7 ) )

V, = 6.147641 4 x 78413, 100OO0, 120000 time steps a = 35 A, c = 70 A, V, = 6.1476414 x

(2)

The strategy for calculating the time evolution by either a p proach is that of utilizing the coordinate representation in order to take advantage of the diagonality of the potential in this r e p resentation. As has been discussed in earlier work, the DAF procedure provides a computationally tractable and accurate way of obtaining the result of the free particle evolution operator on a specific class of wavepackets (the DAF class, consisting of all L2 packets that can be accurately represented as Mth degree polynomials within the envelope of the DAF). The coordinate representation of the DAF version of eqs 1 and 2 can be expressed \k(xlt) = Q(x) S m -m F M ( x , x l T ) Q * ( x ? - l ~ ( x l t- ~ ) d x ' ( 3 )

V ( x ) = VD(x)

VL(x) = eEx

eE/m, =

lozaA/s2

Double Barrier + Time Dependent Cosine V(x,r) = -VIcos[ ( ~ ( x ct)/a)l + V ( x ) VI = 8.1985806X J, a = 200 c = speed of light

x,

(d) Double Barrier + Linear + Cosine V ( x ) = V,(x) V&) - V'COS (.x/a) VI = 8.1985806 X lo-'' J, (I = 200 A

+

TABLE II: Initial Packet Propagation Parameters for Short and Long Time Propagations Initial Gaussian Packet Short Time Propagations

+ &V(x))'

(4)

for eq 1 and z

e-irv(*)Ph

(5)

for eq 2, provided V ( x ) is real. P ( x , x ~ T is) the continuous DAF class free propagator given byS

with ~ ~ ( =7 d(0) )

+ ih7/m

(7)

and where Hznis a Hermite polynomial. The width parameter, a(O), is chosen so that the DAF, Le., FM(x,xlO), can be used to approximate any polynomial of degree M accurately within the DAF envelope. Then for any wavepacket representable as such a polynomial within the DAF envelope, P ( X , ~ ~provides T) an equally accurate time evolved wavepacket. For an initial Gaussian wavepacket, very accurate results, with a reasonably low M,can be obtained using a a(0) which is about 10-20 times smaller than the initial width of the Gaussian wavepacket. The discretized version of eq 3 is easily obtained by introducing a trapezoidal quadrature for the integration over x', yielding' co

!P(xjlt) = Q(xj)Ax

(b) Double Barrier + Linear + VL(x)

potential (a)

Xmin=. -1500 A, X,,, = 1500 A, NX potentials (b)-(d) X,jn = -2800 A, X,,,

Q(x)

C FM(X,,X,,IT)Q*(Xi,)-'~(Xflt jk-co

- T)

(8)

In the present calculation we choose Ax so that o(O)/A(x) L 3

J

(c)

where Q(x) is Q ( x ) = (1

J

(9)

Le., we take at least three quadrature points under each DAF. It is important to note that, due to the Gaussian factor, exp[-(xj - X ~ ) ~ / ~ U ~the( TDAF ) ] , class propagator matrix, F M ( x j , x f l ~ ) , is very highly banded. In addition, it depends only on the difference (xi - xi,), so that the full matrix can be generated from one row, and it also depends on only even powers of this difference. These are important features for the development of highly efficient wavepacket propagation schemes. (However, our purpose

405,AX = 3000 A1404

2000 A, NX = 809,AX

4800 A1808

Long Time Propagation potential (a) X,, = -12465 A,&,, = 12465 A, NX = 214,hx = 1.52174 DAF and FFT Propagation Parameters Short Time Propagations potential (a) T = 1.6 X s, no. of time steps = 400 DAF Gaussian width Z(0) = 3.5355 A maximum Hermite M = 40 potentials (b)-(d) T = 4 x s, no. of time steps = 1200 DAF Gaussian width Z(0) = 3.5355 A maximum Hermite M = 40 Long Time Propagation potential (a) 7 = 5 x 10-16 s no. of time steps = 78413; 100ooO; 120000 DAF Gaussian width 2(0) = 3.5355 8, maximum Hermite M = 40

in the present paper is to demonstrate that the DAF approach can be used to obtain accurate scattering information, and so we do not concern ourselves with developing the mast highly optimized codes.) For the purposes of comparing results, we have also implemented eqs 1 and 2 by standard methods,llJzusing the momentum representation for evaluating exp(-KT/ h), the coordinate representation for evaluating the Q and (Q*)-I, and the fast fourier transform to go between these representations."J3J4We now turn to give the results for the four scattering systems. 3. Computational Results Calculations have been camed out for the four model problems in order to verify that the DAF approach is capable of yielding accurate results for real time quantum dynamics requiring very long propagation times. The potentials and their parameters are summarized in Table I, and the initial packet, grid, and propagation parameters are in Table 11. The double square barrier potential presents a significant challenge because it requires an

The Journal of Physical Chemistry, Vol. 96, No. 24, 1992 9639

Applications of the Distributed Approximating Function 0.05

I

I

Linear + Double B a r r i e r P o t e n t i a l 1

I

I

I

I

I

I

( 1 2 0 0 Time s t e p s ) I

I

I

1

0.04 0.03

0.02 0.01

U

k

a

a

0

4

a a,

-0.01 -0.02

-0.03

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'

-0.05 -2000

I

I

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I

-1800

-1600

-1400

-1200

-1000 -800 Grid ( i n angstroms)

I

I

I

I

-600

-400

-200

0

Figure 1. TABLE 111: Comparison of the Real and Imaginary Parts of the Wavepacket After 400 Time Steps for the 1-D Double Barrier Potential Re\k(z,t) Zm\k(z,t) z (A) DAF FFT DAF FFT 0.415E-4 -0.204E-4 0.419E-4 -0.204E-4" -1321.782 0.517E-4 -0.8187E-3 -0.81878-3 -1002.475 0.517E-4 0.358 17E-2 0.27734E-2 0.27734E-2 -750.000 0.358 17E-2 -400.990 -0.12040E-1 -0.120402E-1 0.2441 17E-1 0.2441 16E-1 -200.495 0.414835E-1 0.414838E-1 -0.99686E-2 -0.99683E-2 0.00000 0.186421E-1 0.186436E-1 0.144016E-1 0.144016E-1 304.455 -0.20575E-1 -0.2057538-1 0.99862E-2 0.99863E-2 504.950 0.102290E-1 0.1022918-1 0.552438-2 0.55244E-2 750.000 -0.30300E-2 -0.30301 E-2 0.86610E-3 0.8662E-3 980.198 -0.910E-4 -0.910E-4 -0.5789E-3 -0.5789E-3 1240.01 0.510E-4 0.51 1E-4 0.99E-5 0.100E-4 a

E-4 denotes 1O-4.

extremely long propagation time and it is discontinuous. Calculations were first carried out for problem (a) using the DAF and FFT methods, combined with KRMC equations, to evaluate the action of exp(-iKT/h), and propagating a total of 400 time steps. A quantitative comparison of the DAF and FFT results for the real and imaginary parts of the wavepacket at randomly selected points is given in Table 111. The agreement is seen to be excellent both with regard to the magnitude and phase of the wavepacket. However, it is also found that the wavepacket remains substantially trapped in the interbarrier region. Therefore, it is important to carry out the propagation for a sufficiently long time, so that transmission and reflection probability amplitudes can be calculated. The transmission and reflection amplitudes are sensitive to the phase of the wavepacket and are typical of the type of quantity of interest in scattering problems. It is found that this double barrier problem is extremely challenging because it takes a very long time for the wavepacket to completely decay out of the double barrier region! The collision is substantially over after about 40 ps, but calculations were continued to 60 ps to complete the scattering and test the stability of the method. It is very encouraging that the DAF approach is able to evolve the wavepacket correctly, including phase information, for such long times. The resulting transmission probabilities obtained by the DAF and FFT methods are given in Table IV and agree with time

TABLE IV: Converged Reflection and Transmission Probabilities for Electron Scattering Off the Double Barrier Potential no. of time stem 78413 100 000 120 000 Reflection DAF FFT analytical"

0.510 0.5 10 0.509

DAF FFT analytical"

0.479 0.490 0.49 1

0.506 0.509

0.505 0.505 0.509

Transmission 0.486 0.49 1

0.487 0.485 0.49 1

" Results obtained by standard, time independent boundary matching of the wave function and derivative at potential discontinuities. independent results to within a percent. Problem (b) has a linear potential (constant classical force) added to the previous double barrier potential. This system was propagated by the DAF and FFT methods for a total of 1600 time steps, but with a time step which was 1/4 of that used in the study of the double barrier problem. In Figures 1 and 2, we present the comparison of the DAF and FFT results for the modulus and real parts of the wavepacket. It is seen that they agree very well. The constant acceleration due to the linear potential leads to ever increasing oscillations in the real and imaginary parts of the wavepacket, and these high-frequency oscillations are accurately reproduced by the DAF class free propagator. However, continued propagation by either the same DAF class free propagator or FFT grid eventually will be unable to accurately follow the wavepacket because, from the DAF point of view, the wavepacket will become so oscillatory as to leave the DAF class, while the FFT spatial grid implies a maximum momentum which can be correctly propagated, so that it also will be unable to accurately propagate the packet. Model problem (c) adds to the double barrier a time dependent, oscillating potential of the form given in Table I. The modulus and real part of the wavepacket are shown in Figures 3 and 4, where we again see excellent agreement between the DAF and FFT results. The time dependence of the potential makes this an interesting problem for the KRMC propagation equations

9640

Nayar et al.

The Journal of Physical Chemistry, Vol. 96, No. 24, 1992 L i n e a r + D o u b l e B a r r i e r P o t e n t i a l (1200 Time steps) 0.05

1

1

1

1

1

1

1

1

1

1

FFT m e t h o d +--

0.045 0.04

0.035 0.03

0.025 0.02 0.015 0.01 0.005

-2000

-1800

-1600

-1400

-1200

-1000

-800

-400

-600

0

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Grid ( i n a n g s t r o m s )

Figure 2. C o s i n e Time-Dependent

0.05

1

1

Double Barrier P o t e n t i a l ( 4 0 0 Time steps)

t 1

1

1

1

FFT m e t h o d -eDAF m e t h o d -+-.

0.041,

1

0.04 0.035 0.03 0.025

0.02 0.015 0.01 0.005 0 -400

-300

-200

-100 0 100 Grid ( i n angstroms)

200

300

400

Figure 3.

because they are valid both for time independent and time dependent potential^.^^.'^ The oscillating field is seen to have a substantial effect on the overall probability distribution of where the electron is likely to be observed. A portion of the electron’s wavepacket moves a significant distance, and a portion of the wavepacket is significantly delayed by the influence of the oscillating field. Finally, the most challenging model incorporates the double barrier plus linear potential and adds a spatially sinusoidally varying term. In order to follow the wavepacket accurately, the

time steps are reduced to 4.0 X s, and the real part of the wavepacket is shown in Figures 5-7 for a sequence of 400,800, and 1200 time steps. As per previous calculations, the imaginary part is reproduced with similar accuracy, and the absolute value is much smoother than either the real or imaginary part. The dynamics of the wavepacket are seen to be quite complicated with the linear term producing ever increasing oscillations, and the sinusoidal potential causing a delay in part of the wavepacket (compared to Figure 1 and 2) and increased acceleration in part of the packet. The DAF procedure again is seen to yield results

The Journal of Physical Chemistry, Vol. 96, No. 24, 1992 9641

Applications of the Distributed Approximating Function 0 0 0 0

-0

-0 -0

-0

-0 Grid ( i n angstroms)

Figure 4.

0.06

Linear + Cos + Double B a r r i e r P o t e n t i a l ( 4 0 0 Time s t e p s ) I

1

I

0

500

I

1

0.04

0.02 u

Lc

K!

a d

0

K! Q)

-0.02

-0.04

-0.016 -1000

-500

1000

1500

Grid ( i n angstroms) Figure 5.

in quantitative agreement with those obtained using FFTs. 4. Conclusions

The objective of this paper has been to provide computational demonstrations of the ability of the DAF approach to treat correctly real time quantum dynamics. The model potentials chosen for this purpose were taken from a group of double barrier systems which are useful in studying electron dynamics in quantum hetero~tructures.'~The double square barrier by itself actually poses a substantial challenge because (a) it involves infinite forces (but finite impulses), (b) it can cause considerable distortion of

the wavepacket, and (c) it can drastically delay the wavepacket in the region between the barriers, thereby requiring very long propagation times before the collision of the electron with the potential is over. Additional difficulties are created by adding to the double barrier potential a linear potential, a time varying sinusoidal potential, or a linear plus spatially oscillating potential. The constant force due to the linear term creates ever increasing oscillations in the wavepacket, while the additional oscillating field compounds the high frequency oscillations with considerable distortion in the spatial distribution of the electron's probability distribution. We consider the fact that the DAF effective free

9642

Nayar et al.

The Journal of Physical Chemistry, Vol. 96, No. 24, 1992 0.06

I

+

Linear + Cos I

I

Double B a r r i e r P o t e n t i a l (800 T i m e stems) I

I

I

I

I

FFT method 4-

DAF method

-+-.

0.04

0.02

0

.....

......

(0

al Lc

-0.02

-0.04

I I -0.06 -1600 -1400 -1200

I

I

I

I

-600

I

-200 Grid ( i n angstroms)

-800

-1000

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I

1

I

0

200

400

600

Figure 6. 0.04

+

Linear FFT method

Cos

+

+ Double B a r r i e r P o t e n t i a l I

I

( 1 2 0 0 Time s t e p s ) 1

I

0.03

0.02

0.01

0

-0.01

-0.02

-0.03

-0.04 -2500

-2000

-1000

-1500

-500

0

500

Grid ( i n angstroms)

Figure 7

propagator, used both in the KRMC and the KRSSO methods, was able to produce highly accurate results for all four of these models and provides convincing evidence of the validity of the DAF formalism. It is also of interest to compare the efficiency of the DAF approach with that of using FFTs to apply the free particle propagator portion of the KRMC and KRSSO methods. Since we have designed the DAF approach with implementation on massively parallel supercomputers as the primary objective, we focus on the key aspects most relevant to efficiency on such machines. The first comparison is of the number of operations

for the DAF and complex FIT, the former being 8nb and the latter being 2 X 5n log, n (the factor of 2 arising because one must do 2 FFTs per time step). Here, n is the grid size and 6 is the band width of the DAF class free propagator matrix. Therefore, dividing out the common factor n, one compares 86 to 10 log, n. We have considered l-D problems which involved 10 Ib 5 20 and 5 12 In I32 768. For these extremes, the DAF factor ranges from 80 to 160, and the FFT factor ranges from 80 to 170. This indicates that the two methods do not involve large differences in the operations count. However, the DAF under consideration has not taken any advantage of being able to use nonuniform grids

9643

J. Phys. Chem. 1992, 96,9643-9650 nor any of several possible procedures for contracting the total grid extent required for the propagation. Although grid shortening methods also exist for FFT-based procedures, they cannot be used simply with nonuniform grids. The second important aspect for implementation on massively parallel supercomputers is that of communications. The communication time for an FFT on the hypercube is 2(ts tmmmn/p)log, p , and that for the DAF algorithm is 2(t, t,,,b/2). Here, t, is the start-up time, t,,, is the communication time for a word, n is the length of the input vector (I grid size), p is the number of prwessors, and b is the band width of the DAF class matrix propagator. For given tsand t,,,, the FFT has the term 24 log, p compared to the DAF 24, the difference being (log, p - 1)2t,. For 128 processors, this would be 124 longer for the FFT. For the second term, one compares (2t,,,n/p) log, p for the FFT, with t,,,b for the DAF, so the extremes (with 10 I 6 I20 and 512 In I32768) are 56t,,, to 3584t,,, for the FFT and lOt,,, to 20t” for the DAF. It is clear that the DAF is extremely well suited to implementation on massively parallel supercomputers. On the basis of these results, we are optimistic that other versions of the DAF formalism will also be successful and provide useful new tools for real time quantum dynamics.’+ We are carrying out many such computational studies now, with particular emphasis on the Gaussian biased sampling-Monte Carlo evaluation of the DAF path integral scattering amplitude6 and the quadrature (DDAF) and Monte Carlo (CDAF) evaluation of real time dynamics.

+

+

Acknowledgment. The authors gratefully acknowledge helpful

suggestions and comments on this work by Dr. J. Gustafson.

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, D. J.; Zhu, W.; Ma, X.;Pettitt, B. M.; Hoffman, D. K. J. Phys. Chem., companion paper in this issue. (4) Kouri, D. J.; Hoffman, D. K. J. Phys. Chem., companion paper in this issue. (5) Hoffman, D. K.; Arnold, M.; Kouri, D. J. J . Phys. Chem. 1992, 96, 6539. (6) Hoffman, D. K.; Arnold, M.; Zhu, W.; Kouri, D. J. J. Chem. Phys.,

submitted for publication. (7) Makri, N. Chem. Phys. Lerr. 1989, 159, 489. Makri’s effective prop agator is based on introducing a sharp cutoff, Pmax,into the Fourier integral evaluation of the free particle propagator. The resulting effective free prop agator processes a structure which decays with the distance propagated, (x - x’), as sin [Pmnr(x- x’)]/(x - x’), which is much weaker than the Gaussian decay of the DAF class free propagator. (8) Hoffman, D. K.; Sharafeddin, 0. A.; Kouri, D. J.; Carter, M.;Nayar, N.; Gustafson, J. Theor. Chem. Acru 1991, 79, 297. (9) Judson, R. S.;McGarrah, D. B.; Sharafeddin, 0. A.; Kouri, D. J.; Hoffman. D. K. J. Chem. Phvs. 1990. 94. 3577. (10) Sharafeddin, 0. A.; Kouri, D.’J.; Judson, R. S.;Hoffman, D. K. J . Chem. Phys. 1992, 96, 5039. (1 1) Feit, M. D.; Fleck, J. A. J . Chem. Phys. 1982, 78, 301; 1983, 79, 302; 1984,80, 2578. (12) Devries, P. NATO AS1 Series. 1988 8171, 113. (13) Kosloff, D.; Kosloff, R. J . Comput. Phys. 1983, 52, 35. (14) Kosloff, R.;Kosloff, D. J . Chem. Phys. 1983, 79, 1823. (15) Kouri, D. J.; Hoffman, D. K. Chem. Phys. Lorr. 1991, 186, 91. (16) Truong, T. N.; McCammon, J. A.; Kouri, D. J.; Hoffman, D. K.J . Chem. Phys. 1992, 96, 8136. (1 7) Luscomb, J. H.; Frensley, W. R. Nanorech. 1990, 1, 13 1.

Photophysics of Donor-Acceptor Substituted Stilbenes. A Time-Resolved Fluorescence Study Using Selectively Bridged Dimethylamino Cyano Model Compounds R e d Lapouyade,*.t Konstantin Czeschka; Wilfried Majenz,*Wolfgang Rettig,*gt Eric Gilabert,s and Claude Rullibe*v* Photophysique et Photochimie Moleculaire, URA du CNRS No. 348, Univ. de Bordeaux I, 351, cours de la LibPration, F-33405 Talence, France, I. N. Stranski-Institute, Techn. Univ. Berlin, Strasse des 17. Juni 112, 0-IO00 Berlin 12, FRG, and Centre de Physique MolCculaire Optique et Hertrienne, U.A. du CNRS No. 283, Univ. de Bordeaux I, 351, cours de la LibCration, F-33405 Talence, France (Received: December 16, 1991; In Final Form: August 17, 1992) Several selectively bridged 4-(dimethylamino)-4’-cyanostilbenesare investigated. All of them show strongly red-shifted fluorescence spectra connected with a considerable dynamical Stokes shift in polar solvents. Bridging, however, affects the photophysics (fluorescence quantum yields and lifetimes) dramatically. Blocking of the double bond twist by a sufficiently rigid bridge (