Potential Energy Surface from Spectroscopic Data in the Photodissociation of P

Articles

Potential Energy Surface from Spectroscopic Data in the Photodissociation of P

yatomic M

ecules

Hwa Joong KimandYoungSik Kim#

Department of ChemicalEngineering, ‘Departmentof Basic Science, Hong-ikUniversity,Seoul121-791, Korea Received October 27, 2000

The time-dependent tracking inversion methodis studied toextract the potential energy surface of theelectronic excitedstate inthephotodissociation of triatomic molecules.Basedon the relay of the regularized inversion procedure and time-dependent wave packet propagation, thealgorithm extractsthe underlyingpotential energy surface piecebypiecebytracking thetime-dependent data, which can be synthesized from Raman excitation profiles. Wehave demonstrated thealgorithmtoextract the potential energy surface of electronicexcited state forNO2 moleculewherethewavepacketsplitonasaddle-shapedsurface. Finally,we describe themerits of thetime-dependent tracking inversionmethod comparedwith thetime-independent inversion method and dis­ cussedseveralextensions of thealgorithm.

Keywords: Potential energy surface,Wave packet propagation, Inversionmethod,Photodissociationdyna­ mics, Raman spectra.

Introduction

The dissociation of molecules through the absorption of UV photonsis the most fundamental problem ofphotochem­

istry and chemical processes. This phenomenon is well suited for the study ofintra- and intermoleculardynamics, coupling among different bonds, energy transfer from one modetothe others, thebreaking of bonds and theformation ofnew ones. During the last two decades, the improving state of the artofcrossed-beam techniques incombination with ultrashort laser pulses and high-resolution spectro­ scopictechniqueshas provided details ofphotodissociation processes in polyatomicmolecules. Complete identification ofthe initial state in the ground state and dynamicscalcula­ tions for the excited state surfaces enable researchers to obtain a sequenceofsnapshots of amolecularsystem from its preparation all theway through tothe product.

Usinghighly successful modern experimental techniques, theoreticians have greatlyadvancedcomputational methods to treat photodissociation processes. The powerofmodern computers allowsaccess to the dissociationdynamicsoftri- atomic molecules by exact quantum mechanical methods without anyapproximation. In the field of photodissociation, thecloseinterplay between experiment and theoryhas con­

tributedgreatly to the understanding of photochemical pro­

cesses.

However, the potential energy surface (PES) oftriatomic molecules has complicated topology, especially for excited electronic states. Avoiding crossingsof the PES of interest with thePES’s ofotherelectronicstatescan lead to potential barriers or potential wells. Also, in addition to the unique dissociationchannel,there are otherchannels,includingthe

breakupintothree atoms.Thus, the exactinformation of the PES is important to the studyof dissociation dynamics.1 The determination of PES has been studied for many years by two different approaches.2-4 One approach isto perform ab initio calculation's within the Born-Oppenheimer approxi- mation,5 which is a forward method, and theother,whichis an inverse method,6-8 istoperforminversionofexperimental data.

In recent years, ab initio calculations have been very advanced but still limited to relatively small systems and lacking the accuracy demanded by quantitative dynamical calculation. Most of existing numerical procedures to extract PES fromlaboratory data use a traditionalleast-squares fit­ ting methods,9 butthese methodshavethedrawback of con­

straining PES’s to follow a fixed function form, making goodfits difficulttoobtain. Without the assumption of any PESform, a directinversion method is available for the sim­ plecase of diatomicmolecules. It is known as thesemi-clas­ sical Rydberg-Klein-Rees (RKR) method.10Although the RKR method has been used successfully to determine of one­

dimensional PES, itcannot beapplied to replusive PES's and to a PES with morethanone minimum.11

Recently, a regularized inversion method was developed withinthe framework of the Tikhonov regularizationproce- dure.12 The method is related to the iterative inversionalgo­ rithm, which extractstheunderlyingPES, based on the func­

tionalderivatives of theexperimental data withrespect to the PES. One important aspect of this method is thatitallows thevalueof thePESat every point inspacetovary indepen­

dently compared with theparameter-fitting methods. Because therearemorediscretizationpoints for PESthan data points (i.e.,ill-posednessofthe inversion12-16), reasonable regulari­

zationconstraintshaveto be included for the stability ofthe

solution, e.g., smoothnessof thepotential.

In general, experimentaldata can be obtained from either stationary (frequency-domain)7-8 or time-dependent experi­

ments. The latter justrecently became available as exciting real-time dynamicalexperiments,suchasfemtosecondtran­ sition-state spectroscopy (FTS).17-22Bothfrequency-domain and time-dependent measurements can reflect the same basic information aboutthe molecules involved.20Gruebele et al.21 have reported on the use ofFTS to yield accurate spectralinformation(vibrational and rotational)via Fourier transformation and to directly invert to the PES, usingthe semiclassical RKRmethod.(i.e., frequency-domain inversion).23 In the present study,weexpand our earliertime-dependent tracking inversion method to systems with more than one internal degree of freedom. This method applies to the tri- atomic system toextract PESoftheelectronic excitedstate from trackingthe time-dependent data24 (i.e.thecorrelation function)which can be synthesizedfromfrequency-domain measurements, such as theRaman excitation profile (REP) in photodissociation processes. The merits of the time­

dependent tracking inversion method24 are as following;

First, there is noneedto assignenergylevellabels. Inpoly­ atomicmolecules,this task is notoriously difficult in thefre­ quency-domaininversionapproach. However,the assignment problem is completely avoided in the tracking algorithm, sincethe time ordering of data doesthe assigning. Second, compared withthe iterative inversion method in frequency domain,theproposedinversionalgorithm is direct and com­ putationallyefficient because it solvestheinversionproblem ateachtime stepextracting thePESpiece by piece.

In the following section, we outline the time-dependent inversion method basedontheTikhonovregularizationpro­ cedure to extract the PES from the correlation function.

Next, we simulate the inversion of the PES of the excited electronicstateofNO2 wherethewave packet spliton a sad­ dle-shaped surface fromthe REP in photodissociationpro­ cesses. Thisis followedbydiscussions onpossibleapplica­ tions of the inversion method to problems involving the dipolefunction and thePES, simultaneously.

Theory

Figure 1 showsthe photodissociationof a triatomic mole­

cule ABC into products A andBC. Before the absorption process, the molecule is assumed to beina particular initial vibrational state|@> in the ground electronicstate (S0). Absorp­ tion of a photon promotes a moleculeto the excited elec­ tronic state (Si) on which the initialwave packet|^(0)> is propagated, according to the time-dependent schr dinger’s equation. The corresponding PES’s will be denoted by Vg and Ve, respectively. If the upper-state potential is purely repulsive along the dissociation coordinateR, themolecule dissociatesimmediately intothefragments A+BC. If, onthe other hand, Ve has a barrier that blocks direct dissociation, thebonddissociation willbedelayed.

When molecules are excited to aphotodissociative conti­

nuum,theabsorption spectrum Gabs(w)measures the proba-

disso이ation coodinate R

Figure 1. The UV photodissociation of triatomic molecule ABC into products A+BC. The electronic excitation occurs from the molecule in a particular initial state |0i > in the ground electronic state (S0) to the excited singlet electronic state (S1). The corresponding PES are Vg and V?, respectively. Cabs(w') and %(企)denote the absorption cross section and the Raman scattering cross section, respectively.31

bilityofabsorbinga photon with specific frequency irrespective of thefate of the excitedmolecule. It dependsprimarily on the shape of thePESinthe Franck-Condon (FC)region.The vast majority of excited molecules dissociate without any emittedradiation. A very small fraction willexchangeone of theincidentphotons a) for one ofadifferentwavelength 国^, thereby returning to a different vibrational level of the groundelectronic state 物 >.Becausethe lifetime ofthe dis­

sociatingmoleculeisusually lessthanthe order ofpicosec­ onds and much shorter than the lifetime for spontaneous emission,which is of theorder of nanoseconds,theintensity ofthe emittedlight is extremely low. Nevertheless, it can be measured with sensitive detectors, andits intensity is given bytheRamanscatteringcross-sectionfw).

^fi(a) = aa"| afi(m)\2, (1) where aand a arethe incidentfrequency and thescattered frequency, respectively.

The dependence ofthe cross-section for a giventransition upon the excitationfrequency awill be referred to as the Ramanexcitationprofile (REP). Itreflectsthe motion of the dissociating complex in the excited electronic state and is used to extract information about Ve. Because the REP involvesthe overlap of the wave function intheupperelec­ tronic statewiththeexcitedvibrational states in the ground electronicstate, it issensitiveto a wider regionofthe upper PES thanthe absorptionspectrum Gabs (w). fa^》inEq. (1) is the polarizability appropriate to this transition and expressedas

广8 i(ffl + E; h +ir)t

a^fi(a) =

J

whereEi is theenergy of theinitialvibrationalstate and r is the usual phenomenological damping parameter (i.e., the inverse of the lifetime). Sfi(t) is the correlation function defined as

S^fi(t) = 시卩e^H决"Vw》, (3)

where He is the nuclear Hamiltonian in theexcitedelectronic state, and 卩representsthetransitiondipolemoment.

For simplicity, weassume thatthe groundelectronic state is understood so well that its vibrational eigenfunctionsand energies are available. Then, with the knowledge of the excited electronic state’s PES, Veand the transition dipole moment 仏 it is straightforward to calculate the REP for every possibletransition giveninEqs (2)and (3). However, is it possible to get the information about Ve from theREP’s.

With the known initial wave packet and the correlation function, which can be obtained from the REP, we wantto extractthePESof the excitedelectronic stateon which the wave packet is propagated. In the limit of the Condon approximation (to ignore dependence of the transition moments on internal nuclear coordinates), the initial non- stationary wavepacket onthe excitedPES in the FCregion can be prepared as 0>= |必(。)>,and the initial wave packet is propagated on the excited PES according to the time-dependent Schrodinger's equation. The solutioncan be expressedas e~iHet/h ©(0)>= | v(t) >. Then, Eq. (3)can be rewritten as following

Sft( t)=挪f V (t)〉. (4) Further differentiating Sfi(t) with respectto t, the PES of theexcitedstatecan be relatedto it as

ih요d- =〈 wfihdv(^t)〉

dt dt

=^〈W^fT V (t)〉+ 3^fK| v( t)〉, (5) wheretheHamiltonian of theexcited electronicstate is parti­

tioned into kinetic and potential operators He= T+Ve. The time derivative of the correlation function and the kinetic energy operatorT isknownand the potential Ve is what we want to extract. Rearranging Eq. (5) gives the Fredholm integral equation ofthefirst kind as

匚 dRKf(R, t)f(R)R ^2 = g^f(t), (6)

wherethe known term g(t) and the kernel K(R, t) inEq.(6) are

gf、t) = ih 쓰絳〉-〈珈 T v( t)〉; f( R) = V^e (7)

"金 *之 ■土 金2 小、

Kf、R, t)=如(R)Wⁱ(R，t)/R . (8) If we put t=0 in Eq. (5), the solution of theintegral equa­

tion gives the local excited potential Ve0, where the initial non-stationary states 妙(0) are localized in the FC region.

Then, this piece of potential Ve0 is used to propagate the wave function仍(0) to the nexttime 仍(t), accordingtothe time-dependent schr dinger equation by the split operator method ofFeit and Fleck.27 The wave function atthe next timestepcan be expressed as

|V( t)〉=。「血"|叫(0 )〉

—iTt/2h ~iVet/ h -iTt/ 2hi ^"\、\

-e e e |v (0 )〉. (9)

With the known Wf and 切(t) [i.e., using the correlation function &(t)], Eq. (6) gives thenextpiece of the potential

V?. With V}, the wavefunction atthe nexttime step W(2t) canbe obtained by propagation, and wave functions Wf and

"2f) allowobtaining V fromEq. (8),etc. Therelayofthe inversion process and thewavepacketpropagationcontinue until the wave functions have sampled all dynamically accessible regions of the PES. Finally, the potential is the sumof these pieces, Vei.

Ingeneral, with the known PES’s, thecorrelation function can be calculated from Eq. (4), because this forward prob­ lem is usually well-posedproblem and it has aunique solu­ tion. However, the inverseproblemofextracting PES from experimental data is ill-posed problem because it corre­

sponds to the determination of the continuous PES from a givensetofdiscretedata(observablemeasurements).A rea­ sonableregularizationconstraint, e.g. smoothnessofPES, is needed to solve Eq. (7). For this, Eq. (7) can betransformed, viaintegration bypartsas

g^f =

J

R'), and

Here wehave taken into account the fact thatthe potential, as wellas its derivative, Vn)(R),usuallyapproach zero quickly as the dissociation coordinate becomes infinite. Then, the regularized solution in Eq. (10) is achieved by minimizing thefollowing functional7

Jfn, g^f]=习

J

+ aJ； dRfⁿ)( R )]². (11) The regularization parameter a denotes the tradeoff between reproducing the dataand obtaining a smoothsolu­ tion. If we choose too small an avalue, PES is overfitted with the data and lose smoothness. On the other hand, the resolutionof PES may be verypoor ifwetry tooversuppress the smoothness of PES by choosing too large an avalue.

Consequently, itis necessary toproperlychooseana value for Eq. (11) to yield anadequate PES. The solutionof Eq.

(11) can be obtained by a singular value decomposition method as

/ T 、

严=

z

where u方 Vj( R) are the singularfunction for the kernel Kj and K with singular values Wj

Z

j

J

Fora moredetailedderivation, see Ref.12.

Calculations

In our previous numerical study, a simple diatomic-like model wasused to demonstratetheapplicabilityof the time­

dependent tracking inversion method to absorption spectra.

In the present paper, we expand this method tosystemswith more thanoneinternal degreeoffreedom. Ourtargetmole­ cule is NO².28-36 The coordinates ri, r and 0 define the dynamics for NO2,where ri and rdenotethedistancesfrom N to eachof the O atoms. The coordinate 0isthe bending angle betweenthetworadial NO directions. In these coordi­ nates, the two-dimensional Hamiltonian for NO2isdefinedby

cos0 자 m^N dr 1dr2

V(七，r2, (15)

wherep^--=mo-1 + mN^-and mo and mNare themasses of theoxygen and nitrogen,respectively.

The time-dependent study of NO2 photodissociation is definedby two PES, S^oand S¹, corresponding to the elec­ tronic ground and excitedstates, respectively.29 TheSo sur­ face typicallyhas a pronouncedwellleadingtothepresence of bound states. The S1 surface may have a shallow well, which may not be ableto supportboundstates. The S1 sur­ face is based on the ab initio data. The original expression given byHirsch andBuenker37as

V(r^】,r2, 0) = 1*1) + V^no2(r2) + [ 1-Q(r^】)]

X [ 1-Qa( r2)]旗 bklmQ^a(尸1^0( h^)Q O, (16)

k, l, m

where

Q^a(r) = 1 - exp[-ft( r -1%)] (17a) Q^b(0) = «1(0-0^eq)+ a2(0-0^eq)2+ «3(0- 0^eq)3 (17b)

V^no (r) = a2Qa(r )²+a3 Qa(r )³ +a4 Qa( r )⁴. (17c) All parametersare givenin Ref.31.

The ground PES is used for determiningthe groundstate wave functionto be used astheinitialwavepacketon theS1

surface. For simplicity, the bending angle 0is frozen at a selectedequilibrium value 0eq. Using the equilibrium value re = 2.27 a.u., for r1, r2,the groundMorsePES is givenas

V(r¹, r2) = De[ 1-e「a(，1-"叩 + De[1-e"1-'^r)]2 (18) and the correspondinggroundstate gaussianwavefunctions are given as

机 n2(r¹, r2^{) =} Pⁿ¹(r1 )Pn2( r ) (19a)

Pⁿ(r) ^{= [씨쏘2느쯔广D] X}X^{'-n"玲-2n-1/ 、 一x}/2 (x )e , (19b) whereLn(x) is a Laguerrepolynomial,

—a(- re)

x ⁼ 2te (19c)

t = (2pD)',2/ha . (19d) These parameters in atomic unit (D, re, a) =(o.2953, 2.27, 1.3164) can be found in Ref. 36.

To perform the wave packet dynamics calculation, the groundstatewavefunction wasfirstcomputedon the S^o sur­ face. This ground state wavefunctionvert|山> was vertically shiftedto the S1 surface to provide the initial wave packet propagatedontheexcited PES. Starting at an N-O distance of 1.6 a.u., 256 gridpoints were placed every o.o5 a.u. in each coordinate. It should be pointedoutthat inallcalcula­

tions presented here the vibronic coupling between the So

and the S¹ surface is neglected. However, the dynamics on the isolated diabatic S1 surface of NO² gives reasonable spectroscopicdata andprovides a significant testfor thefea­ sibility of the time-dependent trackinginversion method.

Figure 2 showsthewavepacketvert | i/(t)> as a functionof the symmetric stretchcoordinate r¹ +r2 and theasymmetric stretch coordinate r¹ -r² at o, 12, 24, 48 and12ofs. The cor­

responding PES V(r1, r², 0^o) of the S¹ state is given for a fixed value 0eq =133.89o, which is the equilibrium bending angle inthe ground state. It has been found that the wave packet spreads along the dissociation coordinate during the propagation. Initially, thecompact wavepacketmovesin the direction of the symmetric stretch coordinate r¹ + r². Since the motion occurson thesaddle of the PES, strongspreading of thewavepacketin thedirection of theasymmetric stretch coordinate r¹ -r² wasfound. After about 24 fs,the turning point of thesymmetric stretch motion is reached. The wave packet becomes extremely broad and starts moving toward the potential valleys.

Theamplitude of thecorrelationfunction is showninFig­ ure 3. Two recurrences are apparent, one at approximately 48 fsand the otherat12o fs. Thefirst recurrencecorrespondsto the symmetric stretch resonance that occurs because some portion of the wavepacket is reflectedfromthe symmetric stretchbarrier and returns to the FCregion. Another portion is reflected from the barriertowardthe repulsivewall to the dissociation valleys. Because of the geometry of the PES, some portion is reflectedback toward the barrier and then redirected back to theFCregionatapproximately 12o fs.

The REP fE) is obtained from Eq. (1) and (2) as the Fouriertransform of thecorrelationfunction. This is shown

Figure 2. the wave packetvert g) > as a function of the symmetric stretch coordinate ri + r and asymmetric stretch coordinate ri-r2 at (a) 0, (b) 12, (c) 24, (d) 48 and (e) 120 fs (处 133.89o).

in Figure 4 as the solid line for f = (0,0). When the final vibrationalstate fis thesameastheinitial state, the correla­

tionfunction becomes theauto-correlationfunction and the REP becomes the absorption spectrum. The overall enve­ lopeoftheabsorptionspectrum is controlled bythewidth of the correlationfunctions andsmall spikes can be explained bythe oscillation period ofthe symmetric stretchresonance and thehyperspherical resonance.

From differentiating Sfi(t) withrespectto t and initial con­

ditions, wehave performedtherelay of theinversion proce­

dureandthe wave packet propagation to extractthe underlying PES piece by piece. Comparisonsofthe invertedpotentials with the exact one, according to the time evolution of the wavepacket, areshownin Figure 5. InFigure5(a),thecom­ parison is shownalong ri at the fixed equilibrium position (r2 = 2.4125 a.u.) of S1 surface. Figure 5(b) displays along the symmetric stretch coordinate (r1 + r2) at a fixedasym­

metric stretch coordinate (i.e. r1 = r2). We found that the inversion algorithm extracts the potential beyond the FC region function, and that the inverted potential closely resembles

Figure 3. The amplitude of the correlation function (6= 133.89°).

theexactone until thewave packetreaches theturning point inthesymmetriccoordinate. Beyond theturning point,there is a sizable difference fromthe exactones at the large dis­

tancewherethewavepacketcannot reach.

Table 1 presents the FCregion and invertedPES region.

The data show that thetime-dependentinversion algorithm canextracttheexcitedelectronic PESbeyondthe FC region until thewavepacket reachestheturning point.

The FC region of the S1 surface of the NO2 molecule is inverted exactly because it has two kinds ofresonances and is trappedtemporarily in theresonancebefore the dissocia­

tion. However,the S⁰ surfacehas a pronounced well and the groundstatewavefunctionsarelocalized aroundthe equilib­

rium position, the amplitude of the correlation function decays quickly in exit channels. ExtractionofthePESalong theentiredissociationchannelmaybeaccomplished by using thePES of anotherknownelectronic state for the finalstate with shallow well. Anotherway toextract thePESalongthe dissociation coordinate is to usethe measured mean position of atoms in themolecules by ultrafast x-ray or electron dif­ fraction.

Discussion

Thetime-dependentinversion method hasbeenappliedto extractthe PES ofthe excitedelectronicstate from tracking the time-dependent data (i.e.thecorrelation function),which canbesynthesized fromthe Ramanexcitationprofile (REP) in the photodissociation process. To demonstrate the feasi­

bility of the time-dependent inversion algorithm, we have simulated the inversionof the PES of the NO2 molecule in the UV photodissociation process. Using the inversion algorithm and the wave packet propagation by turns, we have demonstrated that this algorithm extracts underlying PESpieceby piece.

Theresultsshow that we canobtainthe excitedelectronic PES alongthe dissociation path fromthe FC region to the exit channel as far as the propagated wave packet on the excited PES overlaps with excitedvibrational states in the ground electronic state. This numerical experiment should be regarded as extremely encouraging for extracting the detailed structure of non-FC regions from frequency mea­ surement. Because the inverse algorithm included the asymptotic condition oftheexitchannelsin Eq. (10),itcan

Figure 4. Raman Excitiation Profile. of(E). f = (0,0), (0,1), (1,0), (1,1) (6= 133.89o).

Figure 5. The comparison of the inverted potentials at a certain evolution time (0, 10, 20, 30, 40, 50 fs) with the exact one according to the wave packet propagation. (a) for a fixed value r = 2.4125 a.u.

at the equilibrium position of S1 surface (。= 133.89o). (b) for a fixed value symmetric coordinate r^】一r² = 0.

Table 1. FC region and Inverted PES region

FC region Inverted PES region

Symmetric coordinate 4.2-5 ^^^■4.-5.4

r1 (fixed『2 = 2.4125 a.u.) 2.3-2.65 2.1-3.1

extract the PES smoothly without any assumption. Also, even in the case of splitting dissociation, such as NO231, FNO31,39 etc, the inversion algorithm extracts the potential beyond the FC region and the inverted potential closely resembles the exact one until the wave packet reaches the turn­ ing points in the symmetric and the asymmetric coordinates.

Compared with the time-independent inversionmethods, thetime-dependent tracking inversion approach has several advantages. In the time-independentmethod, itisnecessary toassignquantumnumberstocomparetheexperimentaldata with the corresponding datagenerated from thetrial poten­ tial. The assignmentprocess is verydifficult in many situa­

tions, for example, whenthe potential is multidimensional and when the spectra are non-separable. However, in the tracking approach the assignment is avoided completely,

sincethetimecarries out theassigning.

Also, the proposed tracking inversion methodis a quite different approach from the fitting methods. Usually, the deviation ofthe fitted from the exact potentials becomes largerasthe dissociation coordinatesmove further fromthe equilibrium position. But, the trackingalgorithm automati­ cally satisfies the boundary condition, avoiding the discre­

pancyfar fromtheequilibriumposition.

Moreover, we note that the proposed approach can be extended to extractthe underlying PES and dipolefunctions simultaneously from tracking the mean position of wave packets.Mostof thePES's canbe obtained bythecontrol of thewave packet trajectories.

Acknowledgment. This workwassupportedby grant No.

981-0306-030-2fromBasic Research program ofthe Korea Science&EngineeringFoundation.

References

1. Levine, R. D.; Bernstein, R. B. Molecular Reaction Dyna­

mics and Chemical Reactivity; Oxford University Press:

New York, 1987.

2. Buck, U. Rev. Mod. Phys. 1974, 46, 369; Comput. Phys.

Rep. 1986, 5, 1.

3. Gerber, R. B. Comments At. Mol. Phys. 1985, 17, 65.

4. Buckingham, A. D.; Fowler, P. W.; Hutson, J. M. Chem.

Rev. 1988, 88, 963.

5. Friesner, R. A. Annu. Rev. Phys. Chem. 1991, 42, 341.

6. Ho, T.-S.; Rabitz, H. J. Chem. Phys. 1988, 89, 5614; 1991, 94, 2305; 1992, 96, 7092; J. Phys. Chem. 1993, 97, 13447.

7. Kim, H. J.; Kim, Y. S. J. Korean Chem. Soc. 1996, 40, 20.

8. Kim, H. J.; Kim, Y. S. Bull. Korean Chem. Soc. 1997, 18, 1281.

9. Hutson, J. M. Annu. Rev. Phys. Chem. 1990, 41, 123.

10. Rydberg, R, Z. Phys. 1931, 73, 376; 1993, 80, 514.; Rees, A. L. G. Proc. Phys. Soc. 1947, 59, 998.

11. Roth, R. M.; Ratner, M. A.; Gerber, R. B. Phys. Rev. Lett.

1984, 52, 1288.

12. Tikhonov, A. N.; Arsenin, V. Solution of Ill-posed Prob­

lem; Wiley: Washington, DC, 1977.

13. Snieder, R. Inverse Probl. 1991, 7, 409.

14. Bates, H. Inverse Scattering Problems in Optics; Springer Verlag: Berlin, 1980.

15. Bertero, M.; Del Mol, C.; Pike, E. R. Inverse Probl. 1985, 1, 301; 1988, 4, 573.

16. Hansen, P. C. Inverse Probl. 1992, 8, 849; SIAM Rev.

1992, 34, 561.

17. Zewail, A. H. Science 1988, 242, 1645.

18. Bernstein, R. B.; Zewail, A. H. J. Phys. Chem. 1990, 170, 321.

19. Roberts, G.; Zewail, A. H. J. Phys. Chem. 1991, 95, 7973.

20. Zewail, A. H. J. Phys. Chem. 1993, 97, 12427.

21. Grueble, M.; Roberts, G.; Dantus, M.; Bowman, R. M.;

Zewail, A. H. Chem. Phys. Lett. 1990, 166, 459.

22. Kosloff, R. J. Phys. Chem. 1988, 92, 2087.

23. Baer, R.; Kosloff, R. J. Phys. Chem. 1995, 99, 2534.

24. Lu, Z.; Rabitz, H. Phys. Rev. 1995, A52, 1961.

25. Scoles, G.; Atomic and Molecules Beam Methods; Oxford University Press: New York, 1988; Vol. 1.

26. Stewart, O. W.; Imre, D. G. J. Phys. Chem. 1988, 92,

3363; 1988, 92, 6636.

27. Feit, M. D.; Fleck, J. A. Jr.; Steiger, A. J. Compl. Phys.

1982, 47, 412.

28. Vasan, V S.; Cross, R. J. J. Chem. Phys. 1982, 3871.

29. Mayor, F. S.; Askar, A.; Rabitz, H. A. J. Chem. Phys.

1999,111, 2423.

30. Untch, A.; Weide, K.; Schinke, R. J. Chem. Phys. 1991, 9, 6496.

31. Huber, J. R.; Schinke, R. J. Phys. Chem. 1993, 97, 3463.

32. Manthe, U.; Meyer, H.-D.; Cederbaum, L. S. J. Chem.

Phys. 1992, 97, 3199.

33. Johnson, B. R.; Kinsey, J. L. Femtosecond Chemistry 1994,

354.

34. Mohan, V; Sathyamurthy, N. Quantal Wavepacket Calcu­

lations of Reactive Scattering 1988, 214.

35. McDonald, J. K.; Merritt, J. A.; Kalasinsky, V. F.; Heusel, H. L.; During, J. R. J. Mol. Spectrosc. 1986, 117, 69.

36. Schinke, R.; Novella, M.; Suter, H. O.; Huber, J. R. J.

Chem. Phys. 1990, 93, 1098.

37. Hirsch, B.; Buenker, R. J. Mol. Phys. 1993, 80, 145.

38. Bruce, R. J.; William, P. R. J. Chem. Phys. 1986, 85, 4538.

39. Ogai, A.; Brandon, J.; Reisler, H.; Suter, U. H.; Huber, J.

R.; Dirke, M. V; Schinke, R. J. Chem. Phys. 1992, 96,6643.