Roi Baer and Ronnie Kosloff- Inversion of Ultrafast Pump-Probe Spectroscopic Data

8/3/2019 Roi Baer and Ronnie Kosloff- Inversion of Ultrafast Pump-Probe Spectroscopic Data

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2534 J . Phys. Chem. 1995,99, 2534-2545

Inversion of Ultrafast Pump-Probe Spectroscopic DataRoi Baer and Ronnie KoslofPDepartment of Physical Chemistry and the Fritz Haber Research Center, The Hebrew University,Jerusalem 91904, IsraelReceived: May 26, 1994; In Final Form: August 9, 1994@

Spectroscopic observables aregoverned by the dynam ics on the ground and excited potential energy surfaces.An inversion schem e is presented to iteratively construct the potential surface which reproduces experimentaldata. Special attention is draw n to the nonlinear character of the inversion problem and, in particular, to therole of ultrafast pum p-pro be spectroscopy for dealin g with it. Th e regions of inversion, Le., the nuclearconfigurations for which the potential is to be determined, are identified by calculating the observable-potential sensitivity function. A method is introduced for calculating these sensitivity functions in a numericallyconverged time-dependent quantum mechanical fashion. These functions are the basic building blocks ofthe inver ted potential. Tw o demonstrations of the procedure are presented, both use simulated pump-probespectroscopic data. The first, applied to the IC N molecule, reconstructs the medium- and long-range parts ofthe dissociative excited surface. The second attempts to reconstruct the bound excited potential surface ofNCO.

1. IntroductionIn recent years ultrafast pump-probe experiments havematured considerably. From an esoteric discipline criticizedfor reproducing experimental data already known in thefrequency domain, the method is now stand ing on its own right.Initially, the main asset of time domain methods was insight,derived from the natural tendency to think about molecularencounters in causal terms where a dynamical process is initiatedby the pump pulse and followed through time by the probe pulse.Recently experimental ultrafast techniques have been able tocompete with frequency domain methods for bound systems insupplying quantitative information on molecular dynamics.' For

unbound systems with intermediates and in condensed phasesultrafast techniques supersede frequency domain methods. Inthis paper it is shown that the unique features of time domainspectroscopy can be exploited to give new quantitative resultsconcerning the molecular structure and properties, by enablinginversion of potential energy surfaces.Molecular processes can be elucidated using the adiabatictheorem, which exploits the fact that nuclear motion is slowcompared to the stir of the underlying electrons. The resultingpicture is one in which the electron energy eigenvalues formpotential energy surfaces, on which the nuclear motion isexecuted.2 Thus, in principle, the molecular energy surfacescan be calculated from first principles just by solving thiseigenvalue equation. Ab initio methods for performing this task

are today very advanced but still lack, for many systems, theaccuracy demanded by q uantitative dynamical calculations. T h einversion methods attempt to overcome this limitation anddesign techniques w hich correct the ab initio predictions usingexperimental data.The weak ness of the inversion process is that frequently morethan one possible potential can faithfully reproduce the labora-tory measurements, so that additional assumptions have to bemade. In the theory of mathematical inversion these assump-tions usually impose analytical properties as well as oth er knownfeatures on the potential su rface, e.g., the asymptotic vanish ingof the potential and its derivatives for large internuclear@Abstractpublished in Advance ACS Absrracts, February 1, 1995.

0022-3654/95/2099-2534$09.0~/0

distances. Even under such stringent conditions, mathematicalinversion procedures are effectively limited to one dimension,because of the large degree of underdetermination in higherdimensions.Historically, the model problem of the field has been theinversion of central potentials using elastic scattering crosssections and phase shift^.^,^ The nonuniqueness of the inversionwas lucidly demo nstrated by New ton3 as he constructed anonzero central potential to yield zero phase shifts for all orbitalangular momentum 1at a given scattering energy. Despite these,Shapiro and Gerber5 have show n it possible to use the Bornseries expansion in order to invert molecular potential energysurfaces from elastic amplitudes. A good summary of scatteringdata inversion can be found in the book by N e ~ t o n . ~For inversions using rovibrational spectroscopic measure-ments, the semiclassical Rydberg-Klein-Rees (RKR) schemeis most widely used to determine diatomic potential curve^.^-^The method is based on locating the distance between the innerand outer classical turning points of the potential. As in otherinversions, here too nonuniqueness exists, as it stems from thefact that mappings of the potential, which preserve the actionbetween tuming point pairs, produce the same observedspectrum.'O The shortcom ing of the RKR method is that it isinherently one dimensional since the multidimensional analogueof "tuming points" is undefineable. To overcome this limitation,Gerber and Ratner' ' . I 2 utilized the self-consistent field (SCF)approximation in order to break down the multidimensionalspace into a coupled set of one-dimension al systems. Theseare then inverted by the RKR method sequentially, and aniterative procedure is used to solve the self-consistent set ofcoupled equations. A different approach to rovibrationalspectroscopy inversion is the perturbation based scheme workedo ut by K osm an and H i n ~ e . ' ~ . ' ~his method is based onminimizing the sum of the differences squared between meas-ured term values and calculated energy levels by varying theunderlying potential surface using an inverse perturbationapproach.The diffi culty in direct inve rsion has led to the use of heuristicindirect procedures. These are based on hypothesizing aparametrized functional form for the potential and use theempirical data to fit the parameters. This approach can be used

0 1995 American Chemical Society


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Inversion of Pump-Probe Spectroscopic Datain conjunction with the known functional form of the potentialfor diatomic fragments. Th e main drawback of the heuristicapproach is that it is arbitrarily dependent on the functional formof the potential chosen for the mo deling; thus good fits are hardto obtain.

Another technique of inversion, especially suited for boundpolyatomic systems, is based on the expansion of the potentialsurface in a Taylor series around its minimum and it systemati-cally alters the coefficients in order to obtain a fit to experi-mental data.I5 Recently, McCoy et al.16 improved these methodsby exploiting fourth-order Van Vleck perturbation theory.Direct inversion procedures have been revived lately, espe-c ia lly by Ra bitz and c o - w ~ r k e r s '~ - ~ ~s well as by the presentauthors.2' A comprehensive review of methods and numerousapplications for inverting scattering and rovibrational spectro-scopic data by Rabitz et al. has recently been published.22 TheRabitz approach consists of iteratively improving upon an initialbest-guess potential. The method is based on functionalsensitivity analysis and uses the Tikonov regularization methodto ov ercome the inherent instabilities caused by the ill-posednessof the inversion problem.In this paper the pump-probe ultrashort pulse spectroscopyis used for direct potential surface inversion. The goal is to

use explicitly the special characteristics of pump-probe experi-ments, which have made them so appealing to intuition. Th ebasic motivation for the work23 as been the classical mechanicalinversion procedure of Berns te in and Z e ~ a i l ~ ~or the ICNphotodissociation on the excited-state potential. Although theclassical inversion scheme has been criticized by Krause et al.,25this study show s that a full quantum inversion of pump-probeexperiments is possible and that the optimal inversion schemefollows closely the causal classical picture. For this purpose anew direct inversion method is develop ed, applicable for excited-state potential surfaces, as well as for the ground-state case.Following Rabitz and co-workers,22 functional sensitivityanalysis is used as the basic tool for inversion, thus linearizingthe problem. Solving the linearized equation employing all

experimental data simultaneou sly limited the inversion to caseswhere the initial guess potential was very close to the final form.It should be stressed that this effect is caused by the highlynonlinear nature of the pump-probe inversion problem. Thus,the main point of this paper is how to cope with the nonlinearityof the inversion. It is found that by starting from areas in whicha good knowledge of the potential exists, one can graduallybuild into the ambiguous regions. This makes explicit use oflocal aspects of the sensitivity functions. It was found that thepump-prob e experiments have extreme sensitivity to theFranck-Condon region of the potential, therefore this sectionshould be determined from other experiments. For this reasonthe inversion scheme from absorption spectra of photodissoci-ating molecules was developed.21 The pump-probe inversionenables us to go beyond the Franck-Condon area, and thedesired region of inversion, the sensitivity region, can becontrolled by the experimentalist.

The sco pe of the inversion can be extended beyond the simpleZewail type pump-probe experiments. The procedure canemploy phase-locked ultrashort pulses similar to the type usedby S cherer et a1.26 but us ing s trong field pulses , of which thepump is a complex pulse, specially designed to create highvibrational excitation on the excited potential surface. As ademonstration for inverting this type of spectroscopic data, wepresent a two-dimensional inversion of the bound ex cited-statepotential of NCO.The arbitrariness of the procedure is minimized, by usin g allavailable information on the form of the target potential surface,

ab initio data and intuition included. All this is incorporated

J . Phys. Chem., Vol. 99, No . 9, 1995 2535during the construction of an initial guess, called the referencepotential. The iterative procedure developed in the followingsections than alters this potential surface until the experimentalmeasurements are reconstructed. The first step is to map outthe regions of the potential to which the measurements aresensitive, using high-quality quantum mechanical methods forsimulating the experiment. After the sensitivity functions areobtained, their structure is scrutinized and the appropriate formof inversion is decided upon.The simulation of the experiment and the calculation ofsensitivity functions is performed using the combination of theFourier grid representation of the wave function with theChebychev polynomial expansion of the evolution operator.These mehods have been shown to provide extremely accurateand stable r e s ~ l t s . ~ l , ~ ~ - ~ ~he method was used to calculatedirectly the absorption29 or R aman spectra27 of w eak fieldexcitation. In a previous paper, we used these methods toefficiently invert frequency domain abso rption spectra into theexcited potential of ICN molecule at the Franck-Condonregion.21 The paper demo nstrates that by exploiting the specialfeatures of time domain spectroscopy , inversion of the medium -and long-range parts of the potential energy surface can alsobe performed.The section structure of this paper is as follows. In section2 the theory and the calculation algorithms for performing thefully quantum mechanical inversion is presented. This sectionis divided into four subsections. Section 2A presents theinversion theory, together with the basic algorithm proposedfor solving it. The basic concept of this scheme is the sensitivityfunction, about which theory and calculation methods arepresented in subsection 2B, for Hamiltonian dynamics and insubsection 2C for the Liouvillian dynamics. A new algorithmfor performing the time integral of two wave functions, neededfor the calculation of sensitivity functions but useful fo r otherapplications in molecular dynamics, is given in su bsection 2D.The inversion theory is applied to two case studies, wheresimulated experimetnal data were used to reconstruct theunderlying potential surface. The first case study, presented insection 3 is that of the Zewail ultrashort pump-probe experi-ment for inverting an excited-state surface of the ICN molecule.The second case study, shown in section 4, econstructs boundexcited-state potential of NCO, using phase-locked strong fieldpulses.2. Inversion Scheme

A. Inversion Equation. The inversion problem is intimatelyrelated to the postulates of q uantum mechanics since it addressesthe central issue of measu rement. Considering the measurementof an observable A , quantum mechanics tifferentiates betweenthe state of the system and the operator A that corresponds tothe observable. Only a combination of the two entities yieldsthe observed measurement result A. This statement is sum-marized by Von Neumann in the following equation:31

where the state of the system is represented by the densityoperator &. When this state is a pure state, it may then bedescribed by the wave function Y(t), eading to the formulation(2.2)( t )= (A)= (Y(t)[A[Y(t))

Equations 2.1 and 2.2 represent the static postulate of quantummechanics. The objective of the inversion procedure is todeduce the exact forces governing the nuclear motion from theobserved values of various time dependent experimentallymeasurable entities. For a nondissipative system this motion


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2536 J . Phys. Chem., Vol. 99, No. 9, 1995is generated by the Hamiltonian

Baer and Kosloff

H(t) = T + 9 + W ( t ) (2.3)where T is the nuclear-kinetic energy operator,? s the potentialenergy operator, and W(t) is a known time-dependent perturba-tion, caused by an ex ternal, experimentally controllable, source.Since both the kinetic energy and the external perturbations areknown, the target for inversion is the potential energy operator,compo sed of the Bom -Oppenhe imer potential energy surfaces,and possibly some nonadiabatic coupling terms.For the present theoretical analysis it is more convenient touse the Heisenberg picture, where it is the operator which istime dependent, while the state of +e system is fixed. In thispicture the motion of the operator A in time is determined bythe following similarity transformation:

A(t) = U+( f ,O) AU(t,O) (2.4)where the evolution operator U(t ,O) is a unitary transformationgoverned by the Schrodinger equation and boundary condition:

ihdUldt = H(t) U( t ,O) ; U(0,O)= i (2.5)The-theoretical prediction of the experimentally observed valueof A at time t , as given by eq 2.2, is thus determined by theoperatorAV. To conc lude, the time dependenc? of the observab leA ( t )= (A(t)) is a functional of the potential V. This functionalis nonlinear and in general has a very complicated form.The Schrodinger equation is a recipe for predicting experi-mental measurem ents, once the potential is given. Is it possibleto invert this procedure? Can the potential operator bedetermined from experime ntal data? It is the purpose of thispaper to give a partially positive answer to these questions.Furthermore, a detailed method for carrying out this inversionprocedure is now described.Assume that M experimentally measured results a, (m= 0 ,...,M - 1 ) are obtained by M different expFrimenta1 setups,corresponding respectively to M observables A,,,. The theoreti-cal prediction for the measured values, the quantum mechanicalexpectation values A,, are M functionals of the potential andthe inversion problem is now a problem of solving the followingM functional equations:

A,(+) = a, (m= 0, ...,M - 1 ) (2.6)Since these equations are nonlinear, they are solved iteratively,starting from an initial best estimate potential, called thereference potential Vref, hat yields, when inserted into thefunctional A,, values which dev iate by damfrom those observed:

(2.7)If the refereqce p?tenti?l is close to the true potential q, hedifference 6V = VI- V,f is everywhere small, then a linearequation relating 6V to 6a m s obtained:

Am(Vref) a, - dam ( m= 0, ...,M - )

~ ~ d 9da m ( m = 0, ...,M - 1 ) (2.8)where J , is the sensitivity functional of the mth observable tothe potential: it operates on the 6V operators to give a number.Symbolically

J , = dAm/d9 (2.9)Equation 2.8, in the limit of d? - is in fact the de finition ofthe operator derivative in eq 2.9.The linearized inversion eq 2.8 must now be solved. Thesuccessful linearization of the inversion does not remove theill-posedness of the problem. Bertero3* gives an excellent

review of linear inversion problems and methods of theirso!ution. In this study the equation s were solyed by expan ding6V as a linear combination of M operators B,:= x b , B , (2.10)

m

where the B operators are called targeting operators since theywill be used to localize the sensitivity functional, and b, areexpansion coefficients, solutions of the following M x M linearequation:

C(J,B,,)b,. = damm

(2.11)Here, the highly nonunique nature of thejnv ersion problem isevident: for different choices of the B operator, differentsolutions are obtained. It should be stressed that although theill-posedness is removed by choosing a certain B operator, theresulting equations can still be ill-conditioned. This issue willbe addressed in the case-study sections.For the present inversion schemes the potential is diagonalin the R representation therefore the sensitivity functional, whenapplied to a functionf(R), acts to multiply it by the sensitivityfunction Jm (R)= dA,/dV(R) and integra te the result over R:

j , J = j J , ( R ) f ( R ) dR = j ( d A J d V ( R ) ) f ( R ) R (2.12)Just as for the potential, the B operator is assumed diagonal inR representation:

B m - m(R) (2.13)To calculate these entities, a realization by discretizing theR space on a mesh of N grid points R, with spacing AR, isconstructed. Thus transforming the potential difference to a Nvector 6V, = 6V(R,), and the sensitivity functional J to a M xN matrix with J,, = Jm(R,)AR,. In matrix notation eq 2.8 thenbecomes

J d V = da (2.14)This is the discretized inversion equation. Typically, the numberof equations M is much smaller than N , the totality of unknow ndV,s, because these represent a continuous R space, while theformer are a finite count of measurem ents made. Since thereare more unknowns than equations, the linear system (2.14)admits at least N - M independen t solutions. As imm ediatelyverifiable by inserti on into eq 2.14,a general solution is obtainedby the following expression:

d V = BT(JBT)-da (2.15)where BT s an N x M matrix, constructed so that the square Mx M matrix, JBT is nonsingular. This equation is the discretizedanalogue of eq 2.11, and the columns of the BT matrix areactually the Bm(R) f eq 2.13.Since dV is not determined uniquely by eq 2.15, here is roomleft to permit the incorporation of a priori know ledge concerningthe inve rted potential. This can be a chieved by a suitable choiceof the Bm (R) argetting functions which linearly com bine to form6V.It is important to conside r the nonlinea r nature of the inversionproblem (see eq 2.6). Since the equations are solved bylinearization, the reference potential must be very close to thetrue potential. It is found that good results are obtainedtypically for deviations of 0.01 eV. But this limitation can berelaxed and deviations of even more than 0.1 eV can be dealtwith if the special features of pump-prob e observables are


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Inversion of Pump-Probe Spectroscopic Dataexploited. The pertient feature of the pump-probe observableis the fact that its sensitivity function is usually well localizedin the nuclear configuration space. Furthermore, by alteringthe pump and probe pulse parameters, it is possible for theexperimentalist to control and determine the domain of highsensitivity. Therefore, by choosing different pump-probeparameters, the experimentalist can see different sections ofthe potential surface.Since the sought-for potential is usually known to highaccuracy in certain regions (e.g., pump Franck-Condo n andasymptotic regions), the inversion process can start from oneof these and gradually enter the locations in wh ich it is vague.Thus the first iterations are based on the experimental setupsthat are sensitive to sections close to the known part of thepotential. After inverting the potential in this region, additionalexperimental results are added, extending a bit further into theunknow n parts. Thus, each iteration alters the potentialsurface in a new section, adjacent and overlapping the regionmodified in the previous one, paving the way for its successor.This way, the specific section of the potential surface beingcurrently under construction is always close to the truepotential. An algorithm for implementing this idea is outlinedforthwith:

(1) Select a sequence of experimental setups (observables)that gradually build sensitivity from nuclear configurations inwhich the potential surface is accurately known into areas inwhich it is inexact. One way to do this, which works well inthe cases studied, is to use probe-resonance pulses. Practi-cally, this means that for each time delay in an ascending seriesof times t, (m = 0, ..., M - 1) the probe wavelength I , whichyields maximum probe absorption (or induced emission, de-pending on the experiment) is located, the series a(Im,tm)husobtained, is used as the sequence of observables for the inversionprocess.(2) Set M = 1. S et V-the target-o f-invers ion poten tial tothe above-mentioned best-estimate potential surface, which ishighly accurate at asymptotic regions and at the pump Franck-Condon region.(3) Set Vref- .(4) Calculate the VIef based expectation values a$ andcompare with the results measured in the laboratory to determinedam= a, a, .(5) Calcu late the sensitivity functions J,(R,)on the gridrepresenting the potential.(6) Calculate the required potential correction 6V using eq2.15. The BT columns should be built out of the sensitivityfunctions themselves, so that 6V obtained is zero at nuclearconfigurations for which the observable sensitivity functionsare zero. The most natural choice is to take B,(R,) = Jm(Rfl).This howev er is not always a good choice, since the sensitivityfunction J,(R) can have undesirable features, such as high-

frequency oscillations, that shou ld not be built into the potential.It was found that these high-frequency oscillations once formedin the potential tend to be amplified in successive iterationsyielding unphysical potential energy surfaces and damaging thestability of the inversion process. One way to overcome this isto choose smooth BT columns, which nevertheless resemble thesensitivity functions essential structure (smooth peaks forinstance). This point is application dependent and is discussedin the specific case studies in sections 3 and 4.

calc - empi

(7) Correct the potential surface: V- ref + 6V.(8) Add an experiment from the sequence prepared in step(1): M -M -t 1. And redo from step 3, unless the experimentsequence is exhausted.To apply the algorithm, two crucial quantities have to becalculated. The first is the expectation valueA (t) = (A(t)) (step

J. Phys. Chem., Vol. 99 , No . 9, 1995 25374), and the second is the sensitivity function J(R) = dA/dv(R)(step 5 ) . Equations 2.1 and 2.2 give working expressions forexpectation values; the method of calculating the sensitivityfunctions will be described in the following subsections. Theresulting expressions, as shown in section 2B are readilyavailable from efficient numerical calculations.B. Sensitivity Function y d e r Hamiltonian Dynamic?The time-dependent operator A(t) depends on the potential Vtk ou gh the evolution equation (2.4). An infinitesimal change6V in the poten tial will, in the Heisenbe:g pictu re, cause acorresponding variation in the operator A(t). This inducedchange is given by

&t) = Cf(t,O) RdU(t,O) + hc (2.16)The change in U can be expressed, using first-order time-dependent perturbation theory, as

ih&J(t,O)= Je(t,z) Sir6(z,O) dz (2.17)Plugging this result jnto eq 2.16, yields the causality relationbentween the cause 6V and the induced change in the observable6A(t):

ih&) = [6(t), S i r ( t ) dz] (2.18)where &(z) is defined by

s i r@)= U+(z,o) irU(z,O) (2.19)We now assum: that nonadiabatic couplings are kept constant,and therefore 6V is a diago nal operator in the IR) representation,and

sir = jdR IR)GV(R)(RI (2.20)(Note that the integration can also include summations ondiscrete indexes: R can represent both continuous variables anddiscrete quantum numbers.) We can now retum to the wave-packet Schrodinger picture, and the expectation values, at themeasurement time t , of the two equated entities in eq 2.18 areeasily cast into the following form:d A ( t )= ( 2 h ) s d R dV(R ) I m(J ds x * ( R , t )Y(R,z)) (2.21)where

W t , R ) = (RlU(z,O)IY,) (2.22)is the coordinate representation of the state of the system, [YO)is the initial state, and

x(z ,R)= ( R I U f ( t d W ) (2.23)Note that x is a wave packet canying the measurementinformation: it depends on the form of the operator whichreprfsents the observable. Now, the sensitivity function of A ( t )= (A@))with respect to the changes dV(R) s deduced from eq2.21 and the operator derivative definition eq 2.9 becomes inthe R representation, a functional derivative:

6A(t ) /dV(R)= (2 h) Im(Jdt x*(R,z)Y (R , t ) ) (2.24)Notice that the product x*(R,z) Y ( R , t )= &(t)IR)(RIY(z))sthe probability amplitude for a transition from state IY(z)) ostate Ix(z)) at the point 6. The state ~ ( z ) , haracterized by thefinal condition x(t) = AY(t), is evolved backward in time.Consequen tly, the sensitivity function becom es proportional to


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2538 J . Phys. Chem., Vol. 99, No . 9, 1995the imaginary part of the sum over all times of the transitionamplitudes at the position R between the forward moving wavefunction Y(t) and the backward moving information carryingwave function ~ ( t ) .It is interesting to examine the expression for the sensitivityfunction (eq 2.24) in a simple case for which the inducedvariation in the expectation value is immediately apparent.Adding a constant to the potential should not change thedynamics and therefore the expectation values. Consider aconstant change in the potential A Y ( R )= constant. Using thedefinite form for the previously arbitrary basis IR), it iscomposed of continuous R s (e.g., nuclear configurations) anddiscrete j s (e.g., quantum numbers of electronic eigenstates).The change in the observable (A( t ) ) s computed by multiplyingthe sensitivity function by this constant and then integratingover allR and summing over allj. To be consistent,this processmust yield zero chang e in (A ( t ) ) . Consequently, he total integralof the sensitivity function must be zero, which sho uld of coursebe the case for the above form of the sensitivity function. Thiscan indeed be proved as follows. First, note the fact that theIRj) basis is complete and therefore

Baer and Kosloff

(2.25)Next, for any intermediate time t:

This quantity is real because a is Hermitian; therefore, the totalintegral of the sensitivity function, proportional to its imaginarypart, must be zero, QED.C. Sensitivity Function under LiouvillianDynamics. Realexperiments are described by mixed initial states and usuallyby dissipative evolution. The formalism of the sensitivityfunction has to be extended to include mixed states and non-Hermitian Liouville operators.A brief sk etch of the formalism to be used here is presented.The inner product between two operators is defined by(Ai81= r{A+ii} (2.27)

The definition of the inner or scalar product is used to constructa Hilbert space of operators also known as the Hilbert-SchmidtThis means that any operator can be expanded as alinear combination of a complete set of operators.The linear operators which map the operator space into itselfare called superoperators. A definite example is the Liouvil-lian3, efined byz$ -(i/h)[9,A] = -(i/h)(?A - A+) (2.28)The normalized state of the system is represented by a densityoperator 4 which is a Hermitian unit-traced linear ~p er at or ,~ and the dynamics of the system is governed by the Liouvillevon Neumann equation:

dldt @ ( t )=Y& = -(i/h)[H(t), @ ( t ) ] (2.29)This equation is the starting point for many generalizedtreatments of quan tum systems coupled to heat baths or systemsundergoing random external perturbation^.^^ Dissipative terms& added to +Y$ yield the systems Liouville superoperator J f t )= +Y$ +&, and the generalized Liouville equation is then

dldt @ ( t )=zt) ( t ) (2.30)The time ev olution operator which corresponds to the Liouvilleequation is the superoperator q t , O ) , which contrary to the pure-state, isolated system, case is nonunitary. This superoperatorevolves in time according to eq 2.30, and the solution for Q ( t )is formally written as

@ ( t )= 4 t , o >@(O>= @(O>k(t,O) (2.3 1)The expectation value of an operator a at time t is defined by(2.32)(cf. eq 2.1). Equation 2.32 paves the way to the Heisenbergpicture, where the system state is stationary, while the dynamicaloperators are time-dependent, evolving according to

A ( t )= $( t ,O) A(0) (2.33)The quan tity of iqterest is the variation of the ex pectation valueof the operator A(t) when a small change in the molecularpotential operator is made. From eq 2.33 it can be seen thatthis may be obtained o nce the change in the evolution operatorhas been figured out. It is explicitly assumed that the changein the evolution operator due to a perturbation &(t) is

A ( t )= &f ) ) = r{@(t)A} r(e(0) @ ( t , O ) A }

sqqt,O)= L&r,s)dx t ) qz,O) d t ( 2.34)This equation is analog ous to eq 2.17 of the pure-state case. Itcan be proved that a necessary condition for eq 2.34 is thatq t , O ) is invertible. Even for dissipative dynamics the super-operator of evolution is invertible for finite propagation time.The treatment here is then as general as the assumption that&(t,O) is perturbed according to this equation. Using eqs 2.32,2.33, and 2.34, the induced change in A(?)= ( A ( t ) ) s given by

&(t) = Ltr{@(O) ( t , O ) &&(t,t) A} dz (2.35)Taking the trace with respect to the R basis in which d ? isassumed diagonal, one obtains

&t ) = !dRL(Rle(r) -&&(t,z)AIR) d t (2.36)Defining the operator a( t , z ) s

A ( t , t )= $(t,t)A (2.37)and using the cyclic permutation symmetry of the trace, thefinal expression for the sensitivity function becomes

a A ( f ) / d C ( R ) (i/h)L(RI[@(t), (t,t)]R) dz (2.38)As a relatively simple application of this general equation,consider a system initially in a pure state Yo governed by adissipative evolution described by a non-Hermitian Hamiltonian.In this case, the time evolution superoperator takes a specialform, and eqs 2.31 and 2.37 are written as

Remember that since the evolution operator U(t,z) is no longerunitarian, this is neither a similarity nor a uct ar y transformation(note, however, that A(t,t) is Hermitian if A is). Equation 2.39may be used to calculate the commutation relation in eq 2.38.The crucial step in this calculation shows that


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Inversion of Pump-Probe Spectroscopic Data J. Phys. Chem., Vol. 99, No . 9, 1995 2539

Upon insertion of this expression into eq 2.38, eq 2.24 isrecovered. This result generalizes that derived in section 2Bfor purely Hermitian Hamiltonians. There is an importantconceptual difference between the two cases. In the case of aHermitian Hamiltonian eq 2.23 defines the state ~ ( z ) s thebackward evolution (from time t to time T)pf state Y(t).Now,however, since the evolution operator U(t,O) is no longerunitarian, Ut(t,O) is not a usual "backward evolution", indeed,it is a very special backward evolution, one in wh ich thedamping caused by the non-Hermitian terms in H(t) is notreconstructed, therefore leading to the loss of informationcontained in the experimental result from the final time to thepast.D. Numerical Evaluation of Time-Overlap Integrals. Anumerical scheme for calculating the time integration ofcorrelation functions constructed from a product of two distinctwave packets is presented in this section. The integration whichappears in the right-hand side of eq 2.24 is an example. Thistype of integral is extremely useful for other applications aswell, especially for flux calculations in scattering and reactivescattering problems. The integration scheme will enable anumerically exact integration, without having to propagate thewave packets using extremely small time steps. It will be shownthat the integral can be replaced analytically by an infinite series,which in a numerical calculation are truncated.The integral of interest is given by the following equation:Z(t,R)= h ' x*( t ,R)Y(@) dz (2.41)

where both Y(t) and x(t) are two wave functions evolved bythe same time-independent Hamiltonian H:W O )(z) e-(i/Wfir

x(T> = e-('/h)'tX(O) (2.42)In a previous paper;' a method for calculating the time-correlation function of two wave packets was presented. Theproblem addressed here does not resemble a time-correlationintegral but can be represented as one. Instead of looking at

X(T) as the forward evolution of x(O), t can be thought of asthe backward evolution of ~ ( t ) .n this way the time integral isdressed as a time correlation of two wave packets:-(i/h$r~ ( t , R ) ] * e Y(0,R)dz (2.43)- ( i /h ) @t - t )W , R )=

The method described in ref 21 may now be gainfullyemployed and in the following is briefly outlined. First, thetwo wave packets are propagated using the truncated Chebychevexpansion o p e r a t ~ r : ~ * , ~ ~

N

N

In the series a, is proportional to the nth Bessel function J,,more specifically a&) = (2 - dno)Jn(x). The symbols &@)andl &(R) denote the functions obtained by operating withT,(HN) respectively on Y(0)Aandon ~ ( t ) ,here Tn(x)are theChebychev polynomials and HN s the normalized Hamiltonian.These and other symb ols are defined in the following equations:

do = W(0)1 i H & O

HN= ( 2H - ) /v

@ = -= - 2 4 v # % + 6 - 1 (2.45)

v = (E,,,,, - Emin)/2hCi,= v 4- E,,,& (2.46)

With this expansion, the time integral (2.43) becomesZ(t,R)= e - ' " 'C [ h r a m ( v( t - ) ) U,(VZ)dt]Bm(R)*@JR )

(2.47).mAt this point a valuable property of the Bessel functions can beused to replace their correlation integral by an infinite series:35

m

Rearranging terms and using eq 2.48 finally enable the timeintegral (2.41) to be represented by the following expression:z ( ~ , R ) (2/v)ei"Z(- 1)"(2 - n0)(2-

n,mdm,)Am+,(vt)@,*(R> (2.49)

there defining the coefficients A,+,(?):

For a given value of t , the Bessel series (Jn(vt)}SP>yrxhibitsexponentially rapid decay to zero as n is increased. Therefore,the sums in eqs 2.49 and 2.50 both display exceedingly stablenumerical convergence.3. Case Study 1: Inversion of ICN Excited-StatePotential

Fo llo win g B em stein a nd Z e ~ a i l , ~ ~he pump-probe experi-ments on the photod issociation of ICN constitute an appropriateexample for inversion of the excited-state potential. The originalmethod h as been criticized by K rause et al.25 arguing thatbecause of the quantum nature of the process the classicallybased inversion is inappropriate. Nevertheless in this sectionit is shown that using a full quantum description, the Bemstein-Zewail idea is fruitful and can lead to a viable inversionprocedure.The experimental setup of the pump-probe experiments andtheir interpretation have been described in detail by Rosker etal.36 The experiment consists of exposing an ICN molecule,initially in the ground electronic state, to a sequence of twoultrashort laser pulses (see Figure 1). The fr s t pulse, the pump,is designed to shift a portion of the molecular wave function toan excited state of repulsive nature. The second pulse, the probe,carefully timed relative to the pump, interacts with the evo lvingexcited wave packet and creates an amplitude on a third excitedstate correlated to an excited CN fragment (denoted CN*).Measuring the intensity of fluorescence originating from theradiative decay, the yield of CN* product is determined as afunction of the time delay and wavelength of the two pulses.In carrying out the inversion process, it is assumed that theground electronic potential is known to a fair accuracy and thesecond excited state is practically structureless with respect to


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2540 J . Phys. Chem., Vol. 99, No. 9, 1995 Baer and Kosloff

I4.7 6.7 8.7 10.7 12.7R I-CN] (Bohr)Figure 1. The three relevant potential energy surfaces for the ICNpump-p robe experiment. Th e molecule is initially in the vibrationalground state of the E+urface. The pump pulse transfers someamplitude to the excited 31T~+tate which starts accelerating towarddissociation in response to the repelling force. At a time delay At theprobe pulse interacts with the molecule and raises some of the am plitudeto the CN*-correlated surface, measured by recording the CN *- Nfluorescence. The probe-induced excitation is registered as a functionof the time delay At and probe wavelength w . Notice that the excitedstate possesses a potential well with a minimum at R 6. 3 eV(indicated by a small arrow).the I-CN internuclear coordinate. Thus the target of inversionis reduced to the first excited repulsive electronic surface.The sim plest quantum mechanical description of this systemis obtained by considering three electronic surfaces and only o nedynamical degree of freedom, the R 1 - c ~ separation. Otherpossible electronic states, nonadiabatic couplings, as well asother ICN degrees of freedom (bending, CN vibrations) areneglected. Also, for simplicity, the electronic transition dipoleoperators are taken R independent. These approximations, crudeas they are, drastically reduce the complexity of the calculations,while retaining the essential physics of the ICN ph otodissocia-tion and m aking it possible to explore the inversion possibilitiesand performance.The Hamiltonian of the system is denoted3' by

where the two pulses are described by the radiative couplingterms:W( t )= W+( t )= - P*( q ( t ) cos(w,t) + E&) cos(o, t))

(3.2)where ,b are the dipole operators and e n ( t ) ,n = 1, 2 , are theelectric field envelopes of the two laser pulses, both varyingslowly relative to the central frequency w,-l time scale. Sincethe delayed CN* - N fluorescence is proportional to thepopulation on surface f, he corresponding observable is thefollowing projection operator:

(3.3)The target of inversion is the first excited-state potential 9,.Using the formalism developed in section 2 , in the R representa-tion, the sensitivity function of the observable (3.3) to thispotential becomes

SPjSV(R) = ( 2 h ) I m { h i , * ( t ,R ) Y e ( r , R ) t} (3 .4)

Pump-probe delay (fs)Figure 2. Two normalized transients measured by the CN* fluores-cence for the ICN pump-probe setup. The circled curve correspondsto 6 fs pulses, and the diamond curve to 25 fs ones. This figure clearlyshows the effect of leveling off the important details of the potentialsurface.where t is a time after the laser pulses die out. The ICNmolecule is prepared initially on its ground electronic surfaceY(0) = Yo,. The following definitions for Ye(t)and ~ , ( r )are used:

Ye(t)= PeY(t) P,U(t,O)Y(0)x , ( s )= P&) = P,U(t,t) PfY(t)

(3.5)(3.6)

The three relevant potential surfaces are shown in Figure 1the ground-state surface is cast into the Morse functional form,designed to fit the know n dissociation and vibrational excitationenergies of ICN. The excited surface is taken as the 3110+diabatic surface calculated by Yabushita et al.38 This surfaceexhibits a potential well at RI-CNM 6.3 bohr. The secondexcited surface is independent of R. The two other electronicsurfaces, I l l , and 3111,re ignored as are the nonadiabaticcouplings between Ill, and 3110+.In the original experiments 100fs (fwhm) pulses were used.These broad pulses have been found to eliminate importantdetails of the potential surface. For example as can be seen inFigure 2 , the broad pulse response is not influenced by thepotential well of the 311~+urface while the shorter pulse is.These considerations naturally lead to the use short pulses of250 au ( ~ 6s) for the inversion demonstration. Examples ofvarious transients for these pulses are shown in Figure 3.To use the inversion algorithm of section 2A, special attentionmust be paid to the sensitivity functions structure and behavior,since they determine the correct choice of the columns of thetargeting matrix BT.The most striking feature of the sensitivity functions is thedominance of sensitivity in the Franck-Condon region of the

pump pulse (R1-c~ 5 . 2 bohr), as shown in Figure 4. Thisdominance means that, un attended, any inversion procedure willmake chan ges almost entirely in this region. To overcome thisproblem, it is assum ed that the potential in this region is knownfrom other sources and therefore is excluded from the inversiontarget. This region can be determin ed by an inversion procedurebased on the absorption spectra.21 Consequently the BT olumnsare chosen to ignore the sensitivity at the Franck-Condonregion.An important feature of the sensitivity functions is demon-strated in Figure 5. Here the probe pulse catches most of theexcited wave-packet am plitude as it is located on the right wallof the excited potential well. Howev er, since there is still somewave-packet amplitude on the opposite side, it also becomes asensitivity-determining egion, thus yielding a nonlocal sensitiv-


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2542 J . Phys. Chem., Vol. 99 , No . 9, 1995 Baer and Kosloff

-; .3.4g 3.98 3.7Ss

3.5 ,3.33.9

-0.150.03

-0.05 ,v0 -0.10

1.0 , II 1

-0.15--0.15-.7 6.7 8.7 10.73.3-.5 7.5 9.5 11.5R X-CNI (Bohr) R I-CN (Bohr)Figure 7. ICN inversion progress. Left panel: The dotted line is thetrue potential, the dashed line is the reference potential and the solidline is the inverted potential, shown in various stages of the inversion.The potential differences are shown on the right panel. Th e true toreference ( a Gaussian) is depicted by a dotted line: the true to inverteddifference is traced out by the solid line. The inversion stages are (a)after 15 experiments: (b ) after 25 experiments; (c ) after 30 experiments;(d) after 50 experiments.

the largest eigenvalue to the smallest one). When a matrix issingular or rank deficient, this ratio is infinite, since at leastone eigenvalue is zero. On testing for the optimal maximumcondition number which yields a good inversion, a value oflo4 was found. The SVD ensures that the condition numberdoes not exceed a given value by changing the matrix in sucha way that effectively eliminates troublesome sensitivity func-tions. The maximum condition number should be determinedby the magnitude of the experimental error. It should be notedthat other methods for dealing w ith the ill-conditioned equationscan also be employed, especially the use of Tikonov regulariza-t i o n ~ , ~ ~hich systematically approximates the ill-conditionedequations by well-conditioned ones.Figure 7 demonstrates how the inverted potential is graduallyreconstructed. The reference potential, identical with the truepotential near the Franck-Condon region, deviates from it by0.1 eV in larger distances, over a wide internuclear region ofapproximately 3 bohr. The transients calculated using thereference potential are significantly different from the em piri-cal ones (those calculated using the true potential), as is seenin Figure 8.The calculation was performed using the Chebychev expan-sion of the time-independent evolution operator.40 Since thepulse induced evolution is time dependent, use was made ofthe rotating wave approximation eliminating high frequencyoscillations enabling, to substantially decrease the number oftime steps needed for the propagation. Various parametersconcerning the calculations are given in Table 1.

4. Case Study 2: Inversion of NCO Excited-StatePotentialTo extend the experience gained on the inversion procedure,a two-dimensional problem is undertaken. The experimentalsetup studied uses ideas from laser control employing a sequenceof strong pulses, thus extending the region of inversion. In thedynamical treatment of this molecule we assume that NCO

molecule i s linear throughout the process. This approximation

0.00.0 25.0 50.0 75.0 100.0Pump-Probe delay (fs)

Figure 8. Comparison of a reference potentials transient (opentriangles), with the experimental one (dark circles). The probewavelength for these transients is 3.37 eV .TABLE 1: Parameters of ICN Calculations (au)

grid AR = 0.017pulse parameters E y = 0.0001 fwhm = 250no . of expts 60time delay difference between At = 40expts

facilitates the numberic calculations but otherw ise is somewhatarbitrary. Having a colinear configuration, we use the the twoJacobi coordinates r = RN-c, R = R ~ c - 0 o describe thedynamics where the bending motion is frozen. The Hamiltonian2. 3 is given by

1( t ) = i ,,(t) T +9,T + tgW,,(f) (4.1)where T is the kinetic energy operator in Jacobi coordinates,and V, and V, are respectively the ground-state (X%) andexcited-state (A2C)potential surfaces calculated by Li et ashown in Figure 9. The ground-state wave-packet mean positionis at ( r , R) = (2.314, 3.498) bohr, its width being- (or, .1 =(0.070, 0.065) bohr. The nondiagonal operators W,, = W,,in the Hamiltonian matrix represent the time-dep endent couplingcaused by the electromagnetic field of the pulses:

(4.2)Here peg s the dipole operator, and ~ ( t )s the electric field ofthe pulse. This field consists of a pump pulse followed by aprobe pulse. Different from the usual pump-prob e experi-mental setups, these fields are strong and the pump pu lse is nota simple pulse but a series of three Gau ssian pulses, for reasonsthat are now discussed. The facts that the excited-state potentialis strongly bound and that its minimu m configuration is locatedclose to the groundstate one m ean that it is difficult to obtain asignificant vibrational excitation on th e excited state employinga weak-field pump pulse. This can be seen in Figu re 1 0, wherepulse 1 excites the ground-state wave packet to point a. Theextent of vibrational oscillations produced on the excited surfaceis between points a and points b, approximately 0.2 bohr. Toincrease the inversion range, two extra strong pulses are usedwhich dump and repump the wave packet.2 This causes a muchwider range of excit ation, resulting in the wave packet oscillatingbetween points c and d. This compoun d pump sequence is ableto attain sensitivities in a much wider range of the excitedpotential surface, thereby enabling meaningful inversions inextensive domains. The actual parameters concerning the three-Gaussian pump and the Gaussian probe are depicted in Table2.


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Inversion of Pump-Probe Spectroscopic Data

x 'n ---

I I I I I I

J. Phys. Chem., Vol. 99, No . 9, 1995 25432.62.52.4 7z,o2.32.22.1

B

R [NC-0] (Bohr)

A Z Z - 2.5- 2.4

z9- 2.3- 2.2

2.1

2

3

3.2 3.3 3.4 3.5 3.6 3.7 3.0R [NCO] (8ohr)

Figure 9. NCO potentials surfaces of the ground-state XT I (top) andthe excited-state AZZ+(bottom) state. The potential difference betweentwo contour lines is 0.068 eV.The observable of this pump-probe experiment is the residualpopulation of the excited state. Thus the operator representingthe observable is

(4.3)Pump-probe transients are depicted in Figure 11. The gridparameters used for this (and the sensitivity function) calculationare shown in Table 3. The periodic motion of the wave packeton the excited surface is clearly observed by these transients.The nature of the wave packets motion on the excited-statepotential can be traced by em ploying a classical trajectory. Thistrajectory is shown in Figure 12. The dots on the trajectorycurve represent the classical predictions of the wave-packetcenter at the dealy times used in the inversion. It can be seenthat due to the nonharmonic coupling in the potential surface,the motion corresponds to a combination of NC and COvibrations.The target for inversion was chosen as the excited potential,under the assumption that the ground-state potential is knownto good accuracy. Thus the sensitivity functions are given by

SP,/SV(R) = (2/h) Im(h'x*,(r,R) Y,(z,R) dz } (4.4)where we assume that the NCO molecule was in its vibrationalground state prior to the interaction with the laser fields, thus"(0) = Yo, and the definitions of Yu,(t)and x e ( t ) are

Ye(z)= P,Y(z) = P,U(z,O) Y(0) (4.5)(4.6), ( t ) = PJ(z ) = P J ( z ) = P,U(z,t) PeY(t)

-14.0 I

-18.0 I-0.4 -0.2 0.0 0.2 0.4Reaction Coordinate (Bohr)Figure 10. Approx imate illustration of the three-pulse pump, designedto achieve wider wave-packet oscillations on the excited potentialsurface. The first pulse raises 92% of the ground-state wave-pac ketamplitude to the excited state at point a . A second pulse carefullytimed relative to the first catches the excited wave packet at point band dumps most of its amplitude back to the ground surface, where athird delayed pulse pumps it back to the excited potential at point c.This compound pump achieves 84% excitation of the excited-statepotential. For this scheme to succeed, the pulses must be strong-fieldx-pulses.

I I0.0 10.0 20.0 30.0 40.0 0.0 10.0 20.0 30.0 40.0Pump Probe Time Delay (fsec)0.01 ' ' ' ' ' '

Pump Probe Time Delay (fsec)Figure 11. Typical NCO transients measured by the fluoresc ence fromth e AZ surface. For selected probe waveleng ths.TABLE 2: Parameters of the Three-Pulse Pump and Probefor NCO Inversion (au)

parameter pulse 1 pulse 2 pulse 3 probefwhm 250 250 250 250w 0.009 0.009 0.009 0.009photon energy (freq) 0.1060 0.0988 0.1 131 0.100max am plitude time 0 52 0 1000 0-1900locked-phase difference 0 0 0 0

TABLE 3: Parameters of NC O Calculations (au)grid Ar[N-C] = 0.028 AR[NC-0] = 0.038time delay differencebetween exptsno. of expts 80At = 20

Notice the analogy of the form alism here to that developed forthe ICN molecule (see eq 3.1 to 3.6).The sensitivity functions structures are scrutinized for thepurpose of characterizing the B,(R) functions, required forinversion. Figure 1 3 displays sensitivity functions for variouspump-probe time delays. It is clear that the sensitivityfunctions are relatively smooth; this is the result of the largenumber of pulses and the periodic motion of the wave packet


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2544 J . Phys. Chem., Vol. 99, No. 9, 1995 Baer and Kosloff

12.5start

i 2 6

I 12.03. 2 3.3 3. 4 3.5 3.6 3 . 7 3.8R F C - 0 1 Bohr)

Figure 12. Classical trajectory on the excited AX surface starting fromthe center of the excited wave function after the third pulse of the pump.The points on the trajectory correspond to time delays used for theinversion.-.2 3.3 3.4 3.5 3.6 3.7 3.8

R 1NC.q (Bohr)

r 1 2.6

I 1 2.8exprlment 6 4 i 5-.03.2 3.3 3.4 3.5 3.6 3.7 3.0R [NC-01 (Bohr)

Figure 13. Selection of sensitivity functions fo r the NCO pump-probe transient. Although smooth, these exhibit a complex structure.on the excited surface washing out structural peculiarities. Thisfact suggests the use of the sensitivity functions Jm(R) hem-selves for the targeting Bm(R) functions. Indeed, this choiceenabled significant inversions with initial differences betweenthe true and the reference potentials extending to more than0.1 eV, covering a large portion of the excited potential wellbottom. Figure 14demonstrates the inversion process at work.5. Discussion

In this work it is demonstrated that an inversion procedurefor ultrafast pump-probe experiments is feasible and advanta-geous, where the inversion is based on a full quantum mechani-cal description of the molecular dynamics.Experience gained in many inversion attempts, most of themnot described in the paper, has shown that it is common thatthe inversion process becomes numerically singular. Thissingularity is the result of overlapping sensitivity functions. Toovercome this problem, numerical schemes have been employedsuch as the singular value decomposition. Another problemarises when the sensitivity functions have high-frequencycomponen ts. This phenomena can introduce high frequenciesinto the inverted potential which are amplified in further

I i .6

2 5

2 42 3 r[NCl(sOhrl222 17

2 52 42 3 r[NC] M r )2 22 12

I

3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.2 3.3 3.4 3.5 3.6 3.7 3.8R I N C q Wr) R [NC.W (Bohr)

Figure 14. Progress of NCO inversion, shown on equipotential contourplots. The potential difference between two contours is 0.068 eV . Thetop left shows the reference potential superimposed on top of the trueexcited potential surface. The differences between the tw o reach almost0.15 eV close to the bottom of the well. Typical evolutionary stagesof the inversion are shown in subsequent plots. The time delay betweentwo successive experiments is 0.48 fs (20au). The probe wavelengthsfor each time delay is chosen as resonance pulse, following thecriterion of maximum dumping to ground state.iterations and result in unphy sical inverted potentials and severenumerical instabilities. Dealing with these cases requires specialmeasures. In this paper we suggest the use of smooth targetingB functions, centered on the sensitivity functions peak. Bycontrast, Ho and Rabitz** use a Tikonov regularization, whichimposes smoothness by minimizing the norm of the nth-orderderivative of the potential.It should be stressed that although the targeting functionsapproach removes the ill-posedness of the problem, the resultinglinear equations can still be ill-conditioned (Le., because ofoverdetenninance). Solving the ill-conditioned equations canbe done in several ways. We have chosen the SVD approach,which essentially locates the minimum (L2norm) of the least-squares solutions to the problem. A different approach is touse a Tikonov regularization also in this case.Another severe problem of the inversion procedure is itsnonlinearity, which demand s that the initial reference potentialbe very close to the true potential. This is partially remediedby using features pertinent to pump-probe observables, Le.,the locality of the sensitivity functions in coordinate space andthe fact that the time delay between the two pulses is a clockingdevice which controls the extent of sensitivity. We use thesefeatures to construct the potential in an incremental manner,starting from regions in which it is accurately known, graduallyentering the unknown ones.Considering the pump-probe inversion of Bemstein andZewail, which used 100 fs pulses, the sensitivity functions arespread in space, smearing important details of the potentialsurface. We found that by decreasing the pulse widths, thesensitivity functions become narrow, more localized and togetherwith the empirical data have higher information content


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Inversion of Pump-Probe Spectroscopic Dataconcerning the potential surface peculiarities. These charac-teristics of course enhance the inversion capabilities. It is foundthat the step-by-step method follows closely the intuition andgeneral behav ior predicted by classical mechanics-features usedby Bernstein an d Zewail in their classical mechanical inversion.In the NCO example a simple sequence of pump-probepulses created sensitivity functions in a very limited region ofinternuclear distances. The suggested solution was to requirethat a more active role be played by the experimentalist andthat a strong field three-pulse pump should be used to excitethe molecule, thereby extending the sensitivity region. Weshould note here that we also tried to perform inversion (on adifferent system) using weak-field phase-locked pulses ofidentical frequencies, as sug gested by S cherer et al.26 We foundthat the sensitivity functions of the resulting observables wereoscillatory, spread out over the entire inversion region, andshowed a very high sensitivity to the potential. This prohibitedsignificant inversions, since the algorithms we devised couldalways recover the experimental data by minute corrections tothe reference potential.It is seen in this report more than once that successfulinversion demands an active role to be played by the experi-mentalist. This active intervention has the flavor of coherentcontrol of molecular m o t i ~ n . ' ~ , ~ ~ , ~he need for localizationof the sensitivity function makes it natural to think in terms ofoptimal control theory, the target of which is the electromagneticfield to drive and localize the sensitivity functions in regionswhere the potential is vaguely For example, it ispossible by chirping the pump pulse to control the width of thesensitivity. In pioneering calculations we performed on thismatter, we found that this also results in diminishing theoversensitivity of the observables to the Franck-Condon region.There is another close relation between the inversion schemeand coherent control of molecular motion. In the theory ofo ptim al c o n t r 0 1 ' ~ ~ ~ ~he field-dependent potential is used to steerthe system to a final target state. This can also be viewed asan inversion procedure w here the target of inversion is the fieldand the target state plays the role of the measurement. Thissimilarity can be traced by com paring the first order perturbationcontrol equ ation of Yan et al.46 and eq 2.38.

J . Phys. Chem., Vol. 99 , No. 9, I995 2545

Acknowledgment. It is a pleasure to thank David TannorSandy Ruhman and Stuart Rice for active encouragement anddiscussions. This research was supported by the BinationalUnited States-Israel Science Foundation. The Fritz HaberResearch Center is supported by the Minerva Gesellschaft fiirdie Forschung, GmbH Munchen, FRG.References and Notes

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New York, 1979.

8, 905; 1968, 9, 1241.Springer-Verlag: Berlin, 1989.

(7) Rydberg, R. Z . Phys. 1931, 73, 376; 1933, 80, 514. Klein, 0. 2.Phys. 1932, 76, 226. Rees, A. L. G. Proc. Phys. SOC.1947, 59 , 998.(8) Schutte, C. J. The Theory of Molecular Spectroscopy; North-Holland: Amsterdam, 1976. Rees, A. L. G. Proc. Phys. SOC., ondon 1947,59, 998.(9) Child, M. S. Molecular Collision Theory; Academic Press: NewYork, 1974.(10) Katriel, J. ; Rosenhouse, A. Phys. Rev. D. 985, 32, 884.(1 1) Gerber, R. B. Comments At. Mol. Phys. 1985, 17, 65.(12) Gerber, R. B.; Ratner, M. A. J . Phys. Chem. 1988, 92, 3252.(13) Kosman, W. M.; Hinze, J. J . Mol. Spectrosc. 56, 93.(14) Vidal, C. R. Comments At. Mol. Spectrosc. 1986, 17, 173.(15) Carter, S . ; Mills, I. M.; Murrell, J. N.; Varandas, A. J. C. Mol.(16) McCoy, A. B. ; Sibert, E. L. J . Chem. Phys. 1992, 97, 2938.(17) Shi, S . ; Woody, A.; Rabitz, H. J . Chem. Phys. 1988, 88, 6870.(18) Shi, S.; Rabitz, H. Comput. Phys. Rep. 1989, 10 , 1.(19) Ho, T.-S.; Rabitz, H .J . Chem. Phys. 1988,89,561; 988,89,5614;

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Shi, S.;Rabitz, H. Comput. Phys. Commun. 1991, 63, 71.1989, 90, 1519: 1989. 91. 7590: 1991, 94, 2305.(20) Heo, H.; Ho, T.-S.; Lehmann, K. K.; Rabitz, H. J . Chem. Phys.1992, 97, 852.(21) Baer, R.; Kosloff, R. Chem. Phys. Lett. 1992,200, 183.(22) Ho, T. S.; Rabitz, H. J . Phys. Chem. 1993, 97, 13447.(23) Kosloff, R.; Baer, R. Mode Selective Chemistry; Jerusalem Sym-posia on Quantum Chemistry and Biochemistry; Jortner, J. , Levine, R.,Pullman, B., Eds.; Kluwer Academic Publishers: Dordrecht, 1991; Vol.24, p 347.(24) Bemstein, R. B.; Zewail, A. H. J . Chem. Phys. 1990, 90, 829.(25) Krause, J. L.; Shapiro, M.; Bersohn, R. J . Chem. Phys. 1991, 94,5499.(26) Scherer, N. F.; Carlson, R. J. ; Matro, A ,; Du, M.; Ruggiero, A. J.;Romero-Rochin, V.; China, J. A,; Fleming, G. R.; Rice, S. A. J . Chem.Phys. 1991, 95, 1487.(27) Kosloff, R. J . Phys. Chem. 1988, 92, 2087.(28) Kosloff, R.; Tal-Ezer, H. Chem. Phys. Lett. 1986, 127, 223.(29) Hartke, B.; Kosloff, R.; Ruhman, S. Chem. Phys. Lett. 1989, 158,

(30) Huang, Y.; Zhu, W.; Kouri, D. J.; Hoffman, D. K. Chem. Phys.Lett. 1993, 206, 96.(31) von Neumann, J. Mathematical Foundations of Quantum Mechan-ics; Princeton University Press: Princeton, NJ, 1955.(32) Bertero, M. In Inverse Problems; Talenti, G., Ed.; Lecture Notesin Mathematics; Springer-Verlag: Berlin, 1986; Vol. 1225.(33) Davis, E. B. Quantum Theory of Open Systems; Academic Press:London, 1976.(34) Alicki, R.; Landi, K. Quantum Dynamical Semigroups and Ap-plications; Springer-Verlag: Berlin, 1987.(35) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Func-tions; Dover: London, 1972.(36) Rosker, M. J. ; Dantus, M.; Zewail, A. H. J . Chem. Phys. 1988,89,61 13. Dantus, M.; Rosker, M. J. ; Zewail, A . H. J . Chem. Phys. 1988,89,6128.(37) Williams, S. 0.; mre, D. G. J . Phys. Chem. 1988, 92, 6636; J .Phys. Chem. 1988, 92, 6654.(38) Yabushita, S . ; Morokuma, K. Chem. Phys. Lett. 1990, 175, 518.(39) Belward, J. L. In Numerical Algorithms; Mohamed, J. L., Walsh,

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Roi Baer and Ronnie Kosloff- Inversion of Ultrafast Pump-Probe Spectroscopic Data

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