State-selective excitation of molecules by means of optimized

J. Phys. Chem. 1993, 97, 12609-12619

12609

State-Selective Excitation of Molecules by Means of Optimized Ultrashort Infrared Laser Pulses Werner Jakubetz' Institut f i r Theoretische Chemie und Strahlenchemie, Uniuersitht Wien, Wahringerstrasse 17, A - 1090 Vienna, Austria

Elmar Kades Physikalisch-Chemisches Institut, Uniuersitat Zurich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland

Jorn Manz Institut f i r Physikalische und Theoretische Chemie, Freie Uniuersitat Berlin, Takustrasse 3, 0-14195 Berlin 33, FRG Received: June 16, 1993; In Final Form: August 12, 1993"

Optimal control theory is used to design ultrashort (subpicosecond) infrared laser pulses inducing state-selective vibrational excitation processes. For a Thiele-Wilson model Hamiltonian with parameters adapted for the H D O molecule, complete control of vibrational excitation by such pulses is demonstrated, including the selective excitation of O H or O D local vibrations (mode or bond selectivity). It is also shown how the constraint of fluence minimization can be used to achieve the additional objective of keeping the laser intensity as low as possible. We find that the approach works well from a computational point of view, and no numerical difficulties are encountered in a conjugate-gradient implementation of the pulse optimization. The spectral composition of the resulting minimum-fluence pulses follows a simple pattern. These pulses can be understood as superpositions of few components, each one inducing resonant transition between two levels in a series forming a ladder from the initial state to the target state. Thus in this ultrafast regime stepwise excitation by overlapping, phaseadjusted subpulses of low photonicity is seen to be more efficient than mechanisms related to or derived from direct multiphoton excitation. Fluence minimization is found to be an essential prerequisite for keeping the laser intensities below the range where molecular ionization and dissociation become nonnegligible processes, but even so the intensity requirements are formidable and limit the application of this technique to moderate degrees of vibrational excitation. The suitability of this approach as a tool in mode- or bond-selective chemistry is discussed.

1. Introduction

State-selective multiphoton excitation of molecular vibration by means of infrared (IR) laser pulses has recently attracted considerable attention.'-3 The principles governing the resonant excitation dynamics of multiphoton pumping are well understood: Basically, the Hamiltonian describing the interactions between an electromagnetic field and a system of (asymptotically) bound molecular eigenstates can be recast in the form of an effective two-level system expressed in a basis of Floquet states4vs Altogether, the usual features of two-level systems are displayed, with modificationsreflecting the system's true multistate nature. Thus resonant level switching may be achieved by application of suitable "generalized" ~-pulses,6,~ which represent a fairly straightforward variant of the r-pulses familiar, e.g, from twostate spin systems interacting with weak field lasers.* The time scale on which this behavior can be observed is limited (from below) by the onset of power-broadening effects? which for toohigh field strengths and accordingly for too-short pulse lengths lead to the coupling of more than two molecular eigenstates and thus destroy both the resonance characteristics and the desired selectivity of the spectroscopictransition."J The onset of powerbroadening in turn is linked to the shape of the laser pulse. Pulses with smoothly evolving instantanteous Floquet states, i.e., those with envelopes displaying an "adiabatic" switch-on and switchoff behavior,sJlJ*are more efficientthan those with a more abrupt threshold behavior, in the sense that full selectivity is retained for Abstract published in Aduance ACS Abstrucfs. November 1, 1993.

higher laser intensities,and hence shorter pulse lengths. Important examples are sine-square-shaped pulses or pulses with Gaussian shape. A second important factor limiting the applicability of such pulses is the maximum field strength to which a molecule can be subjected without being destroyed by either dissociation or ionization. At laser intensities in the range beyond about 20 TW cm-2, ionizationof the molecule is beoming a fast and unavoidable (but undesirable) process6J3 (for an early discussion of the ionization limit in ir photochemistry see ref 14). Similarly at high laser intensities molecular dissociation, which invariably accompanies vibrational excitation at any field strength, can no longer be neglected. Assuming model Hamiltonians with parameters characteristic of stretch vibrations in small molecules or chromophors, using sine-square-shaped intense IR pulses, it is found that virtually complete level switching from the ground state to highly excited states, including local overtone modes corresponding to the vibrational excitation of individual bonds, may be achieved with pulse lengths in the picosecond range.'OJ1 If shorter excitationtimes are an important factor of consideration, e.g., in order to compete with collisional or intramolecular relaxation processes or to cope with any of the challenges in the realm of femtochemistryor mode-specificchemistry,Is selectivity, and intensity limitations prevail, and one has to switch to different techniques of laser control. One might either attempt to modify the shape, the frequency characteristics or the phase relations within the IR range,3,6J6-1* or one might apply IR pulse trains or overlapping pulse sequences corresponding to sequential vibrational excitation.'qJO Alternatively a number of different control schemes are available which mainly operate in the

0022-3654/93/2097-12609$04.00/00 1993 American Chemical Society

12610 The Journal of Physical Chemistry, Vol. 97, No. 48, 1993

Jakubetz et al.

address transitions between bound molecular eigenstates of the ultraviolet regime and which involve electronically excited electronic ground state induced by the pulse. In semiclassical intermediate states.21-24 dipole approximation this interaction is expressed by the total In the present paper we pursue the first of these two strategies. Hamiltonian Extending a previous investigation,I7 we use optimal control the0ry2s2~(OCT) in order toobtain a laser pulseof givenduration, suitably designed soas to guarantee maximum attainabletransfer of population to a specified target state. Since physical considthe laser pulse being specified by the field strength vector B(t) erations require us to avoid destructively high laser intensities, of an electromagnetic field suitably polarized with respect to a we also need a means of of limiting the intensity of the optimized molecular axis, which hence should be viewed as fixed in space. pulse, and this objective is achieved through a constraint demanding that the pulse fluence be as small as p o ~ s i b l e .By ~ ~ , ~ ~Any of a number of experimental setups will be compatible with this assumption, like embedding the molecule in a sclid matrix. choosing the strategy of fluence minimization, our method differs For the laser pulse, the scalar field strength E ( t ) = llE(r)ll is zero from other implementations of OCT,21,22where the fluence is a t times r smaller than 0 or larger than rp. fixed in advance and remains unchanged throughout the optimization. Throughout our investigations, we shall assume that neither In our present application we are addressing a two-dimensional excited electronic states nor continuum states of the molecule problem of coupled Morse oscillators. In particular, the Thieleneed to be considered. In this way, we restrict the validity of our Wilson-type model Hamiltonian31 we are employing is paramcalculations to the range where dissociation and ionization are etrized in a way adequate for describing stretching vibrations of unimportant, but this is entirely in accord with the physical the HDO molecule. Due to the existence of pronounced situation we want to describe. Thus I*), the (time-dependent) progressions of local O D and OH vibrations,32 choosing our pulse wave function of the system, may be expanded in a finite basis lengths to fall into the femtosecond regime, this Hamiltonian set of the bound eigenstates Ik) of Ho: allows us to study laser control of ultrafast bond-selective vibrational excitation. Such processes play an important part as possible intermediate steps in the bond- (or isotope-) selective photodissociation of HDO,33 and more generally they may be considered as simple and manageable modes for the ultrafast The time-dependent Schriidinger equation corresponding to the preparation of suitably excited reactants in mode-selective Hamiltonian H chemistry.lJ5 We shall also see that the constraint of fluence minimization (3) can be used as a powerful tool identifying efficient excitation pathways. The optimal pulses so obtained will generally appear can then be rewritten in algebraic form as as a superposition of components attributable to low-order (notably single-photon) transitions between various vibrational levels of ih8 = Hi. (4) the molecule forming a suitably spaced ladder from the initial where? is thevector of expansion coefficientsCk(t), and the matrix state to the target state. They show an oscillatory pattern, with elements of the Hamiltonian matrix H are overall shapes characterized by the presence of several superimposed beat patterns with periods corresponding to the differences in the "resonance" frequencies of the single components of the pulse. In eq 5, & denotes the energy eigenvalue of Ho corresponding We organize this paper in the following way. In the next three to Ik) and the pk/ are the dipole matrix elements (kkll), into sections, we will present some theoretical and computational which we have ais? absorbed the geometrical factor describing details of our investigation. First in section 2 we give an account the projection of p onto &). We shall use the symbol M to of the OCT implementation, stressing especially the augmendenote the full dipole matrix. tations to the basic equations which arise from fluence minimiInitially, a t time t = 0, we assume the molecule to be prepared zation. Section 3 is devoted to the molecular properties. We in a molecular eigenstate, I\k(t=O)) = ti), and throughout we briefly describe the molecular Hamiltonian and the molecular shall adopt the choice li) = IO), i.e., the molecular ground state. dipole moment function used to model HDO, as well as the Our objective is to choose E(r) so as to obtain, a t time t = rp, resulting vibrational level structure. In section 4, we collect the maximum population of a specified final target state v): information on simple monochromatic pulse forms with analytical shapes, e.g., the sine-square pulses mentioned before, which we use as input to the iterative OCT algorithm. This chapter will also illustrate how far laser control of selective vibrational excitation can be driven using multiphoton pumping with single OCT provides a framework to achieve such an optimization, and simple laser pulses. The subsequent sections are then devoted to the practical applicability of this scheme for laser excitation of the main body of our results. The OCT-optimized pulses effecting molecular systems has been demonstrated for several implemensubpicosecond population transfer to selected excited vibrational tations.'7.21,22.2"27 In particular, Zhao and Rice28 have shown levels are presented in section 5 along with an analysis of their that within the subset of bound states of a quantum system structure in terms of power spectra. This analysis will also lead possessing also continuum states, optimal control is feasible. Here us to some interesting findings concerning the mechanism of we modify30 the implementation of ref 17, incorporating also the ultrafast selective vibrational excitation. Finally, our conclusions additional condition that a t the same time as maximizing PA$), and an outlook on the possible use of OCT pulse shaping as a tool the pulse fluence I,: in mode-selective chemistry are found in section 6. 2. Computational Techniques

We consider the interaction of an oriented molecule, described by the m_olecular Hamiltonian HO and the molecular dipole operator p , with a laser pulse of length r, and in particular we

Ip = E , C C E ( dt ~)~

(7)

i.e., the time-integrated laser intensity or, equivalently, the total energy carried by the pulse, should be kept as small as possible. Introducing a "tunable" weight parameter wf, we perform a

State-Selective Excitation of Molecules

The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 12611

constrained minimization of the cost-functional J

with respect to arbitrary variations 6E(t) of the pulse, where the constraints are imposed by demanding that each of the expansion coefficients c&) be solution of eq. (4). Note also that wr has to carry suitable (atomic) units. As in earlier work,17 in order to satisfy eq. (4) we adopt a Lagrangean formalism,minimizing the (real) Lagrangean target functional L,

L = J- 2llXe[X+(ihE - HE)]dt,

x+

(9)

x.

where denotes the Hermitian conjugate of In the case of perfect level switching to the target state D, Ic&p)12 =,,S and with a negligible fluence term, wrZp- 0, the Lagrangean (9) will assume its absolute minimum, L = -1. In computational praxis we will be concerned with local minima of L,for which in general we have to 4xpect L > -1. To determine such a local minimum, the vector A, whose elements are the N Lagrangcan functions A&), is advantageously ~ h o s e n 1 ~ J ~toobey . ~ z ~the ~ .SchrMinger~~ type equation ihX =

HX

(10)

with "final" boundary conditions

If the whole set of equations is discretized in time, e.g., at the Nt 1 integration points used in a fixed step size numerical integration of the Schrainger eqs 4 and 10, then it is straightforward to determine the gradient of L and to set up any gradientbased algorithqminimizing Lin eq 9. Replacing the timevariable t by the vector 7 = (TO, TI, ...,t ~ , )wheF , TO = 0 and r ~=,tp, the amplitude function E ( t ) by the vector e = (eo, (1, ...,e ~ , ) , where ef = E(rl),and finally the functional k=L(E(t)) by the modified Lagrangean target function t = E(€), the n-umerical problem turns into the search of stationary points of L. Hence we have

+

grad

t= 0

or, invoking eqs 6-1 I

at/&, = w,E(T,)- X+(T,)

ME(7,) - E+(T,) MX(7,) = 0; i = O , l , ...,Nt (13)

With y!a specified electric field ,; the final populations Pn(tp)as well as Land its gradient are thus obtained from two integrations of a Schrtdinger equation: a standard integration of eq 6 and an inLegration of q 11 backward in time. Starting with a trial field Q, an optimization technique like the conjugafe-gradieflt algorithmu will iteratively produce improved fields cb (2, ...,e&, hopefully converging toward a local minimum of L with an associated 'locally" optimal laser pulse. Hence in general the solution to the-OCT optimization will depend on the choice of the trial field Q. Disregarding the fluence term wIZp in q 8, some of the local minima of L will correspond to various multiphoton excitation processes of different photonicity or to pathways with sequential or overlapping stepwiseexcitation mechanisms. Among the locally optimal fields, our actual, 'physically motivated" interest is to single out pulses leading to values of Pj(tP)reasonably close to unity, which at the same time are characterized by reasonably low fluence and perhaps by reasonably low complexity, and we certainly should be prepared to accept some degree of trading-in among these possibly conflicting requirements. While the formal implementation of the third of these objectives issomewhat elusive, the second one is taken care of by the fluence term wJp Clearly the ambiguity caused by the presence of the tunable parameter

w~to some extent reflects the vagueness in the physical objectives as formulated above. Given an arbitrary trial field, convergence toward a 'desirable" minimum will require a well-judged choice of wr: choosing wr too small may fail to eliminate high-fluence minima in the vicinity of the trial field, while choosing it too large may result in a vanishing field, E ( t ) = 0 for all t. Finally the first objective might be handled by a nonlinear constraint of the form P,(tp)> constant (0.98, say), but we did not attempt to incorporate this constraint, preferring to monitor the target populations by inspection. More formally, we are interested in a minimization of Zp subject to constraints on P,(tp) and Z, but this problem is not readily implemented. In the model calculations reported in the following section, we generally used pulse lengths tp = 500 fs = 20 671 atomic units (au; 1 atomic time unit = 0.002 42 fs), which we divided into N, = 214 - 1 = 16 383 intervals. This large value of N, was chosen in order to enhance the resolution of the power spectra which are computed in order to analyze the optimized pulses; a smaller number of integration points would have sufficed for convergence of the Runge-Kutta integration. Still, even with a total of 16 384 free optimization variables em, no numerical difficulties were encountered in any of our numerical applications,and convergence to a maximum value of P,(tp) reasonably close to unity could always be achieved with a comparatively small number of conjugate gradient iterations; Le., no more than about 300 with fluence minimization turned on, and less than 40 with w~ = 0, each iteration requiring four to seven solutions of eq 6 and one solution of eq 11. Finally we note that the laser intensity of 20 TW cm-2 = 200 PW m-2, which marks the onset of molecular ionization, corresponds to a field strength of about 12.3 GV m-1 = 0.024 au (1 au of field strength = 1 hartree qe-l ao-I = 5.142 25 GV cm-I), and that the fluencecarried by rectangular pulses with thiscritical field strength and with a pulse length of 500 fs amounts to 100 kJ m-2 = 64.2 au. Hence, to be on the safe side, in our discussion later in this paper we shall consider pulses with a fluence significantly in excess of 60 au as potentially damaging for the molecule.

3. Molecular hoperties

For the simulationsreported below, all models and parameters were chosen so as to provide a good descriptionof the stretching vibrations of the HDO molecule. The molecular Hamiltonian HOis of theThiele+Wi1son3l form and represents twokinematically coupled Morse oscillators describing bond vibrations, with the bond angle fixed at its experimental equilibrium value. The vibrational bending mode is neglected, as is molecular rotation and potential coupling. A bond dipole-model is assumed for the molecular dipole moment operator p. The individual bond moments, which are each represented by Mecke's function?S are added vectorially to yield the overall molecular dipole moment as a function-of the two bond distances rOH and roD. This model for HOand p had been used before in a study of multiphoton vibrationalexcitationin H2010apart from the obvious adaptation of the atomic masses (mo = 16.000 u, mH = 1.008 u, mD = 2.014 u ; l u = 1.660 53X 10-27kg),theentiresetofdatahasbeentaken directly from ref 10,where a complete list of formulas, parameters, and units is given. This simple model Hamiltonian is adequately providing a manageable test case for our application with realistic parameters. Of course any treatment aiming at a quantitative description of selective excitations in HDO would have to include the neglected degrees of freedom, notably bending vibrations and molecular rotation, which are well-known to play an important role in IR multiphotonexcitation.2~36We note that more involved and more accurate Hamiltonians for the HDO molecule are a~ailable.3~ The molecular eigenfunctionsIk) and eigenvalues E&for the Thiele-Wilson Hamiltoniandescribed above have been obtained)*

Jakubetz et al.

12612 The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 -2.5

-2.5 local

OH [S.OI -

non-local _ _ (2.41

local

OD (0.71

(1.51 (4.11

13.21

-3.0

(2.31

l4,Ol-

-( 0 . 6 1

-3.0

(1.41 (3.11

-( 2 . 2 1

-

-3.5

%

-( 0 . 5 1 -3.5

11.31

13,Ol-

12,lI

-10.4

I

(1.21

m h

L-4.0 al

(2.01

C

(0.31

-

-4.0

r1.11

we have considered various molecular orient2tions with respect to the directions of the electric field vector E(t). Since we did not encounter any marked qualitative effects in the orientation dependence, we again restrict our attention to one particular choice: all results reported below are for an x-polarized field, with the HDO molecule placed in the xz plane, and the z axis bisecting the DOH angle. To conclude this section, we note that the molecular Hamiltonian described above, as well as the basis sets employed in our computations, have also been used by Manz and Paramonov. In a complementary investigation?O similarly addressing the problem of mode-selective vibrational excitation in HDO, these authors investigated the technique of using sequences and superpositionsof simple unmodulated sine-squareshaped IR pulses. 4. Simple Pulse Forms

(0.21

-4.5

-4.5

(1.01 -

The input trial fields, which we use for starting the iterative OCT-pulse shaping algorithm, are of the form

10.11

-5.0

lO.01

[O,O

-5.0

Figure 1. Energy level diagram of the lowest 25 vibrational eigenstates of HDO as resulting from the Thiele-Wilson Hamiltonian described in the text. The OH and OD local progressions (m,O) and (0,n)are set out at the left and right of the figure, respectively. Exited states identified by larger labels arc those used as target states in the present investigation.

by diagonalizing HOin a basis of products of Morse functions li(r0H)) and V(r0D)). In general, for the bound eigenfunctions there is a strongly dominant leading product term Im(roH))In(roD)) in the expansion, and it is appropriate to label the eigenfunctions Ik) using the corresponding pair of pseudo-quantum numbers (m,n). Hence (0,O)denotes the vibrational ground state, and (m,n)may loosely be interpreted as a state with m and nvibrational quanta respectively, in the OH and OD bonds. The dominance of the leading term is particularly pronounced in the series (m,O) and (O,n), defining two sets of local modes corresponding to vibrational excitation of the OD and OH bonds. Forming pseudodiatomic anharmonic progressions, the local modes thus can serve as ladders for IR multiphoton excitation in the frequency range near the fundamental transitions. The lowest 25 energy levels obtained for the present Hamiltonian are plotted in Figure 1. The levels constituting the local progressions are shown at the outer left and right of the figure, while the remaining levels correspond to nonlocal vibrational modes. The two basis sets actually used in the expansion (2)were composed respectively of the lowest 60 and the lowest 32 eigenstates of Ha. In our presentation below, we will focus on the results obtained for selective excitation of four specific vibrational states of our HDO model: the local modes (3,0),(4,0), and (0,4), and the nonlocal level (2,l). With their very pronounced local nature corresponding to OH and OD vibrations, and with their energetic near-degeneracy, (3,O) and (0,4) are well-suited for a discussion of the virtues and shortcomings of the present approach. In addition, we will use level (4,O) to discuss the requirements of populating more highly excited target states. Finally, to explore the possibly different conditions arising in the excitation of nonlocal modes, we will also give some attention to the state labeled (2,1), this level being energetically close to (3,O) and (0,4). Actually we also performed calculations for several others of the levels shown in Figure 1, but it is unnecessary to give any details of these results, since in qualitative terms, there is not much to be learned from the additional calculations. Similarly,

with carrier frequency w, and with simple shapes described by analytical amplitude functions A ( t ) . Our main choice are pulses with square-sinusoidal amplitude functions:

A ( t ) = A, sin’ r t / t p

(14a) here denoted “sine-square pulses”. It is well documented that sine-square pulses are suitable for inducing resonant multiphoton transitions between molecular vibrational states. In particular, it has beendemonstrated that for systems with’typical” molecular parameters, vibrational selectivity is maintained down to the picosecond range,i1J2J9and hence the degree of extrapolation required in order to adapt the pulses for use in the subpicosecond regime may be expected to be moderate in many cases. Both tocheck thesensitivityof the conjugategradient algorithm to variations in the input fields and to investigate nature and distribution of local minima of the target functional L,we also performed a restricted number of OCT optimizations starting with rectangular pulses:

A ( t ) = A, (1 4b) Such pulses may provide a useful starringpoint for an optimization due to their relation to continuous wave (cw) lasers “on” for a period tp,for which a number of scaling relations between A0 and tp are available. However, since the semiclassical dipole approximation (1) is not valid for pulse forms such as (14b), there is no other physical significance to calculations using rectangular pulses than that of marking a point of reference. At low field intensities, population switching between a specified pair of levels, with transition probabilities closely approaching unity, can be induced by generalized r - p u l s ~ s .With ~ ~ ~suitable values of tp, Ao, and a,,pulses of the form (14a) will fulfill the requirements, as will pulses of any other nonpathologic shape (again, as a point of reference suitable parameter values can also be obtained for rectangular pulses). Usually, to a given pulse length there will be more than one matching pair Ao, a,, corresponding to resonant multiphoton transitions at various photonicities. To achieve such transitions with successivelyshorter pulse lengths, the field strength has to be increased,’, and the effects of power-broadening accompanying this increase limit the extent to which selectivity can be maintained in the ultrashort regime. Table I lists properties of a number of pulses with rp = 500 fs, which have been obtained by adjusting the parameters A0 and ai so as to obtain maximum target state population. Before discussing these results, it is necessary to comment on the convergence of the calculations. Our results have been obtained



TABLE I: Optimized Simple Pulses for Selected Vibrational Multiphoton Transitions in H W b target pulse state photonicity shapeC Aod we irf PJtfPP (0~4) 4 s.sq. 0.0675 0.012 046 2643.7 0.646 2 s.sq. 0.4060 0.023 820 5227.9 0.475 4 rect. 0.0318 0.012 062 2647.3 0.248 (390) 3 ssq. 0.0460 0.016 451 3610.6 0.985 3 rect. 0.0256 0.016 431 3606.2 0.494 (4m 4 ssq. 0.0680 0.016 035 3519.3 0.961 (291) 4 ssq. 0.3400 0.011 610 2548.0 0.013 3 ssq. 0.2070 0.015 41 1 3382.4 0.038 a Pulse length f p = 500 fs; initial state is (0,O). For convergence properties and physicalsignificancesee text. ssq. = sine square, eq 14a; rect. =rectangular, eq 14b. Peak field strength in au (1 au = 1 Eh4e-lCzO-l = 514.2 GV m-I). CCarrier (angular) frequency in au (1 au = 1 Ehh-’ = 4.13413 X 10-16s-I). f Wavenumber incm-I. g Target statepopulation at f = f p . with the 32-state basis set, and convergence has been checked against results for the 60-state basis. Only the calculations for pulses with peak field strengths below about 0.05 au are converged with respect to the basis set expansion; the remaining results should be be seen as numerical results for a system of 32 bound states, not properly describing the physical situation in HDO. They have a significance as reference data illustrating the numerical aspects of our calculations, but most importantly they serve the purpose of defining startup pulses suitable for a subsequent OCT treatment. At high field strengths “converged” results would require the inclusion of the dissociative continuum. Within the OCT formalism, the next step in accounting for the continuum characteristics of the system such an approach might involve the inclusion of resonance states with complex eigenvalues to the bound-state basis set? yet in view of the neglect of ionization even results of such an implementation would lack physical significance. With these limitations in mind, the main conclusion to bedrawn from the data in Table I is the following item: (i) For our specific problem of selective vibrational excitation in HDO in the subpicosecond regime, the energetic requirements are formidable, with peak field strengths above the limit imposed by the onset of field-induced ionization even in the most favorable case of 3-photon pumping to the local mode (0,3). Also to give some room for the underlying principles of multiphoton pumping, for the next three items, we shall temporarily ignore the limitations set by the ionization and dissociation limits. For other systems with different level spacings and dipole characteristics, items ii-iv may become relevant: (ii) It is clear that for all cases we witness the breakdown of selective vibrational excitation, although for the different levels included this is true to vastly different extents. For the OH local vibrations, the manifestation of this breakdown is just barely noticeably: the maximum target state populations P(3,0)(tp)= 0.985 and P(4,0)(tp)= 0.961 arequiteclose tounity,andppulation switching may be considered “quantitative” for any practical purpose. For the O D local state (0,4), with P(0,4)(rp)= 0.646, coupling effects to other levels are sizeable, although the specified target state is still clearly the prefered level of excitation. In marked contrast, for the nonlocal state (2,1), the breakdown of the resonance conditions is complete. (iii) It will be much more difficult to design pulses for multiphoton excitation of nonlocal modes than of local ones. Note that efficient vibrational multiphoton excitation is greatly assisted by the existence of a ”ladder” of dipole-coupled levels with regular spacing, like the-slightly-anharmonic local OH and OD progressions or, e.g., the hyperspherical progression in H20.10,39 Because of the lack of a suitable regular ladder, the significantly different behavior of local and nonlocal target states with respect to selective multiphoton excitation apparent from the data in Table I is well expected. In Table I we include results for two

TABLE II: Properties of Optimized Pulses for Selected Vibrational Transitions in H W * target state startup field optimizationc max(lE(f)l]d Ipe PJtf,,)f (0,4) 4-photon,sine-square none 0.068 193 0.646 WI =0 0.068 154 0.998 WI >0 0.030 47 0.994 4-photon rectangular none 0.032 115 0.248 160 0.999 WJ 0 0.050 wi >0 0.032 47 0.992 2-photon sine-square none 0.406 6970 0.475 WI = 0 0.400 7055 0.992 WI> 0 0.052 51 0.991 (3,O) 3-photon sine-square none 0.046 89 0.985 WI = 0 0.046 91 0.999 WI> 0 0.024 21 0.993 3-photon rectangular none 0.026 74 0.494 WI = 0 0.034 a9 0.999 Wl> 0 0.024 20 0.993 (4,O) 4-photon sine-square none 0.068 195 0.961 WI = 0 0.084 210 0.997 w1>0 0.032 34 0.994 (2,l) 4-photon sine-square none 0.340 4885 0.013 WI = 0 0.890 18540 0.995 w1> 0 0.027 24 0.995 3-photon sine-square none 0.207 1810 0.038 WI = 0 0.240 1908 0.994 WI> 0 0.025 23 0.996 a Pulse length, f p = 20671 au = 500 fs; initial state is (0,O).b For convergenceproperties and physical significancesee text. None: startup field. W I = 0: OCT pulse optimizationwithout constraint on the fluence. W J > 0: OCT pulseoptimizationwith simultaneousfluence-minimization. Maximum field strength in au (1 au = 1 Ehqe-Iao-l = 514.2 GV m-l). e Pulse fluence in au (1 au = 1 &Uf2 = 1.557 kJ m-2).fTarget state population at t = t,. pulses in the frequency ranges nominally corresponding to 4- and 3-photon processes as obtained from a crude scan of the parameter space. At the very high laser intensities required, coupling is so strong that population is efficiently channeled into the more easily excitable local modes, and (2,l) remains virtually unpopulated. (iv) For selective multiphoton excitation of local states, processes of high photonicity are prefered. Generally in situations where an efficient multiphoton ladder exists, scaling laws show*JO that as the pulse length is more and more decreased, the highest order multiphoton process will ultimately be the one requiring the least energy expenditure (or laser fluence). Hence 3-photon transition to (3,O) and 4-photon transition to (0,4) and (4,O) in the ir frequency range near the fundamental (0,O) (1,O) and (0,O) (0,l) transitions are the obvious choices to be used as starting points for a further OCT treatment of ultrashort pulses. As an example demonstrating this behavior, sine-square pulses corresponding both to the 4-photon and the 2-photon excitation pathways of (0,4) are included in Table I, and the much higher laser intensity required for the two-photon mechanism is clearly evident. Note however once more, that the two photon-process, aswellas the(0,O)-(2,l) transitionsinitem(iii),arehypothetical processes. In a real system, the extremely intense laser pulses involved would instantly destroy the HDO molecule. Item (i) makes clear that for the ultrafast selective excitation even of moderately highly excited states, direct multiphoton pumping using single simple IR laser pulses is not a viable route. To be useful devices in laser control, sine-square pulses (or other simple pulses inducing multiphoton pumping) will have to be modified in some way. In the following section we will describe OCT pulses obtained by reshaping the simple pulses presented in Table I, i.e., by using the simple pulses as startup fields in the iterative conjugate gradient minimization algorithm. An alternative approach using optimized sequences of simple pulses with lower photonicity is explored in a paper by Manz and Paramonov published in the same issue,20 and it will be seen in the following section that the two approaches in some way converge.

-

-

Jakubetz et al.


OPTIMIZED FIELDS 1 0 , O l-.(m,nl

0

200

POWER SPECTRA LINEAR

w*>o

400

0

5000

FUNDAMENTAL RANGE

LOGARITHMIC

0

2500

5000

3000

3500

4000

m 1.0

.a2 0.

.5

-. 02

II

13.0111

I

1.0

.5

1.0

.5

1.0

.5

0

200

time ( f s l

400

0

5000

0

5000

wave number

(

cm-‘ I

Figure 2. Electric fields and power spectra for 500-fs OCT pulses inducing level switching from the vibrational ground state (0,O)to selected excited states (m,n), as obtained with simultaneous fluence minimization. The one-sided power spectra shown in both linear and logarithmic scales are normalized to unity for the largest component; the ticks in the logarithmic plots represent successively falling powers of 10. All pulses, which are derived using the optimal multiphoton sine-square pulses of Table I as startup fields, induce virtually complete population transfer with PAtp) > 0.99. Results shown in each row are for target levels (m,n) as identified in the outer left panels. The outer right panels show a magnified view of the power spectra in the frequency range of the fundamental transitions. At the top of each of these panels, long tick marks correspond to energy differences between the states forming a ladder for successive excitation from (0,O) to (m,n)and demonstrate that the OCT pulses induce selective excitation by a stepwise mechanism involving overlapping and phase-adjusted sequences of lower-order transitions (see text). For the identification of the individual tick marks; see Figures 3a,d.

5. Control of Vibrational Excitation by Means of OCT Laser Pulses 5.1. Results and Numerical Considerations. We will now present OCT-designed laser pulses suitable for the control of state-selectivevibrational excitation of HDO. Starting from the simple mu1tiphoton pulses discussed in the previous section, we have obtained optimized pulses both without and with fluence minimization. Collectively these two situations will be labeled “w1= 0” for the case of no fluence minimization and “WI > 0” for fluence minimizationwith suitably chosen weight parameters wl; cf. eq 8. An overview of the results is provided by the data in Table 11,while more details of the pulses themselves,an analysis of their spectral frequency composition in terms of (one-sided) power spectra, and information about the population dynamics of the target state are shown in Figures 2 and 3. In particular, Figure 2 presents the minimum fluence pulses, which are the main result of our investigations,whereas Figure 3 demonstrates the relations between the startup pulses and the OCT pulses and illustrates the important effects of fluence minimization. Concentrating first on the numerical side of the optimizations, threedifferent aspects of convergence have to be considered.Two of them correspondto standard numerical concepts: convergence of the OCT conjugate gradient iterations toward a locally or globally optimal electric field, and convergence of the solution of the Schriidinger eq 4 with respect to the basis set expansion. Finally, in the case of fluence minimization we should also like

to know whether our choice of WI has “converged”, i.e., among all pulses leading to reasonably complete population transfer, did we really find the one with lowest fluence? Nominally, for all cases listed in Table I1 convergence of the OCT iterations toward a pulse inducing complete population transfer with Pxt,) > 0.99 could be obtained, although for the runs with WI > 0 this required not only careful adjustment of w1, but in fact frequent redefinition of its value as the iteration! proceeded. The iterative minimization of the target function L is stopped if one of the specified convergencecriteria on the amount of change of its value during an iteration or 03!he magnitude of its gradient are met. Sincearound its minima, L(c)usually appears to be flat, there may be considerable flexibility in the variation parameters q, Le., the detailed form and shape of the electric field, and the pulses as given here may visually deviate from the formal optimum pulses, without however conceding any sizeable deviation of Ip or PA$) from their respective optimum values. This variability can be understood as flexibility in the phase relations between the spectral components, since at the same time the gross spectral composition as well as the total fluence of the pulses are found to be stable and hence the converged quantities (and thus are also well suited for the characterization of the “optimized” pulses). As to convergence with respect to the basis set expansion, the discussion given in the previous section again applies. However, since it is mainly the electric field we are interested in, not so

The JournaI of Physical Chemistry, VoI. 97, No. 48, 1993 12615


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Figure 3. Electric fields, population dynamics of initial and target states and parts of the power spectra for 500-fs pulses variously optimized for maximum population transfer from (0,O)to (m,n). In a-d, top rows show best multiphoton sine-square pulse, middle and bottom rows show OCT pulse without and with fluence minimization, respectively, starting from the best sine-square pulse. "Populations" are l~{(t)1~, with i = (0,O)(initial state; dashed lines) and i = (m,n) (target state; full lines). The one-sided power spectra are normalized to unity for the largest component. Along with the power spectra long tick marks at the top of the panels correspond to the nominal energy differences between the states forming a ladder for successive excitation from (0,O)to (m,n). (a) Target state (0,4). From left to right, the tick marks accompanying the power spectra correspond to the transitions (0,3) (0,4), (0,2) (0,3), (0,l) (0,2)and (0,O) (0,l). (b) Target state (3,O). From left to right, the tick marks accompanying the power spectra correspond to the transitions (2,O) (3,0), (1,O) (2,O) and (0,O) (1,O). (c) Target state (4,O). From left to right, the tick marks accompanying the power spectra correspond to the transitions (3,O) (4,0), (2,O) (3,0), (1,O) (2,O) and (0,O) (1,O). (a) Target state (2,l). From left to right, the tick marks accompanying the power spectra correspond to the energy differences for three goups of near degenerate (2,l)near 2800 cm-'; OH transitions (unresolved at the scale of the figure): OD fundamental transition (0,O) (O,l), (1,O) (l,l), and (2,O) transition (1,O) (2,O) and (1,l) (2,l) near 36OOcm-'; OH fundamental transition (0,O) (1,O) and (0,l) (1,l) near 3800 cm-l. Thecalculations shown for the startup pulse (top row) and the wj = 0 pulse (middle row) are unconverged. They are included as reference for the fluence-minimized pulse shown in the bottom row.

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much the set of individual level populations, the conditions on convergence may be relaxed to some extent. Among the present examples, the calculations for pulses with a maximum field strength below 0.07 au, which in particular include all minimum fluence pulses, are converged in the following sense: If the optimum fields resulting from calculationswith the 32-state basis are taken as startup fields for a reoptimization using the @-state basis, then we obtain renewed convergence within one or two

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conjugate-gradient iterations, with only slight modifications to the final field. In view of the high field strengths associated with the remaining examples, it is not surprising that the results for the 32-state basis do not constitute properly converged calculations. However, in view of the extremely high laser intensities involved and the associated dominance of molecular ionization or dissociation, any attempt to achieve convergence for these exampleswould just amount to unwarranted computationaleffort.

Jakubetz et al.

12616 The Journal of Physical Chemistry, Vol, 97, No. 48, 1993

No definitiveconclusionscan be reached for the third question, namely, by a different choice of w1, could we have done better, or did we manage to find the lowest-fluence pulse fulfilling our objective of virtually complete population transfer to the target state? Generally, in a high-dimensional optimization problem there is no simple way to identify a global minimum, and on top of this we have to deal with the additional complication that our real interest concerns the global minimum of Ip subject to suitable constraints (a problem we have not in fact treated!). In any case, the optimized pulses with wI> 0 do conform to all of our objectives, combininghigh selectivity,drastically reduced fluencecompared with the startup pulses or with those obtained for WI = 0, laser intensitiesdown to the range below the limit of 20 TW cm-2, and even fairly simplestructures as judged from the relative simplicity of the power spectra and from the fairly regular beat patterns, Another very important feature of the minimum-fluence OCT pulses is the independence of the final pulse from the startup pulse, provided the weight parameters wI are suitably adjusted, as is the case in our examples. Remembering that the convergence of the laser fields should be judged by the convergence of their power spectra and of their fluences, the three minimum fluence pulses ( w ~> 0) associated with the (0,O) (0,4) transition, which are listed in Table 11,represent the “same” optimized pulse, even if the table suggests some differences in E(?). Analogous statements apply to the remaining transitions. In other words, we may not have reached the globallyoptimal pulses, but certainly our pulses have useful properties and are acceptably “optimal” ones in view of our objectives. In section 2 we have estimated a fluence of 60 au to mark the upper limit of the physically meaningful intensity range, above which laser pulses must be considered to be destructive for a molecule. Table I1shows that under theconditionschosen, fluence minimization is an indispensableprerequisite for obtaining pulses of sufficiently low intensity. None of the OCT pulses obtained without fluence minimization meet the physical requirements. Hence for this set of results, we will only give a very brief discussion of those aspects which help to illustrate the performance of the OCT technique. For each different combination of target state and of shape and pbotonicity of the startup pulse, a close lying local minimum of L is readily found within few iterations. In each case the optimal pulse determined with WI = 0 is strongly reminescent of the respective startup pulse. lXX7;*h the very important qualification that we are strictly_tal bout pulses obtained within a certain limited basis set, L ha- - --y rich local minimum structure, the situation resembling the weak field case, where generalized?r-pulsesof many different shape and frequency characteristics may induce population switching. It appears that on this level (that is, in a system composed of a finite number of bound states) the loss of selectivity by power broadening, which accompanies the transition to the ultrashort regime, can be compensated for by fairly unconspicuous modifications of the simple startup pulses. Most of the qualitative properties of the input pulses are retained, notably the approximate value of the fluence and the gross spectral composition. Examples of OCT pulses for the (0,O) (0,4) transition obtained with W I = 0, which are shown in Figure 4, clearly demonstrate this behavior. In marked contrast, apart from the decrease in the fluence, there is also a complete change in the compositionof the power spectra when fluence minimization is operating. This is best seen from Figure 3 by inspecting the sequences of power spectra: startup pulse-OCT pulse, W I = 0-OCT pulse, wI > 0. 5.2. Minimum Fluence Pulses and the Mechanism of Ultrafast Selective VibrationalExcitation. In the first place, the final result of an OCT optimization is an instruction about how to shape a pulse in order to achieve a specified goal. However, far beyond this technical information, the structure and properties of the optimized pulses in some way reflect the principles and requirements underlying ultrafast molecular excitation processes. There-

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Figure 4. Electric fields and power spectra for 500-fs OCT pulses inducing level switching from the vibrational ground state (0,O) to the OD local mode (0,4), as obtained without fluence minimization. The one-sided power spectra shown both in linear and logarithmic scales are normalized to unity for the largest component. From top to bottom, the startup fields correspond in turn to optimal 4-photon rectangular, 4-photon sine-square, and 2-photon sine-square pulses with parameters shown in Table I. Within the limited basis set of 32 ’bound vibrational eigenfunctions, all pulses induce virtually complete population transfer with Pj(tp) > 0.99 and represent different local optima dependent on the startup field. See however the text for a discussion of convergence properties and physical significance of the results.

fore, the properties of the optimized pulses contain important information about the mechanisms of the processes they induce. In particular, the minimum fluence pulses may reveal which excitation mechanismis the most efficientone for a given transition and pulse length. The composition of the power spectra shows that there is a dramatic trend away from high-order multiphoton transitions. Throughout, the power spectra split into several components with similar weights, which all fall into the frequency range of the fundamental OD and OH local excitationsand which all have similar weights. The frequencies of these components can be correlated with transitions among various vibrationallevels falling on a ladder from the initial to the target state. The overall excitation of the target state is thus achieved in a stepwise mechanism, and to a first approximation the laser pulse can be interpreted as a quite uncomplicated Superposition of subpulses separately inducing the individual subtransitions. It requires inspection of the power spectra on a logarithmic scale to discern the components correcting this first-order interpretation. The additional spectral components have weights clearly below 1% of the main contributions and fall into both the fundamental and the first overtone ranges of the spectra. To be sure, the concept of pulse sequences is an ubiquitous one, being also at the heart of all pumppump or pumpdump strategies. The important difference in the OCT realization of this concept is the perfect organisation of the subpulses, and the nontrivial aspect of the present analysis is the very sharp breakup of the multiqauntum transition into a set of almost undisturbed transitions of very low order. Since the effects are most pronounced, or most clearly resolved, for the case of the (0,O) (2,l) transition, it is appropriate to choose this particular example for a detailed discussion. In the outer right panel of Figure 2, part of the power spectrum of the optimized pulse is shown in the form of a histogram, the stepsize

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corresponding to the resolution of the power spectra obtained from the discrete representation of E(r). The tick marks on the top axis of the graph correspond to energy differences for various subtransitionsoftheform(m,n)-(m,n+l)or (m,n)-(m+l,n). In our particular example there are three relevant groups of neardegenerate transitions of that general form, which are unresolved at the scale of the figure. In detail, we are concerned with the three 0 1 transitions in the OD bond near 2800 cm-l, (m,O) (m,l), with m = 0, 1, or 2, the two 1 2 transitions in the O H bond near 3600 cm-I, (1,n) (2,n),with n = 0 or 1, and the two 0 1 transitions in the OH bond near 3800 cm-l, (0,n) (l,n), with n = 0 or 1 . The close correspondence between these zero-order resonance energies and the principal components of the power spectrum suggests consecutive vibrational excitation by one or more the three pathways

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although in view of the low resolution of the power spectra, the somewhat lower weight of the 1 2 transitions in the O H bond and the uncertainties in the frequency matching due to resonance defects, parallel excitation via a pathway involving two-photon excitation in the O H bond: (090) (290) (291) (154 cannot be excluded. For the same reasons a full investigation of the relative importance of each of the pathways (15a-d) cannot be attempted on the basis of the gross power spectra. However, additional information can be obtained from an analysis of the interference patterns in the electric field and from an analysis of the system’s wavefunction, I*). We shall return to this point below. The breakup of the overall process into subexcitations of low photonicity and the excellent matching of the spectral components with well-defined transitions along a ladder of states is also observed for the remaining transitions shown in Figure 2. Some apparent imperfections in the spectral composition of the optimized pulses, like the diffuseness of two of the three principal components making up the pulse exciting the OH local mode (3,0), could,be interpreted as signature of the participation of both single- and two-photon processes along the local progression, corresponding to the pathways

or even to pathways proceeding via level (4,O). However it is tempting to speculate that these imperfections may disappear for a more judicious choice of WI, giving way to one preferred mechanism of consecutive low-order transitions by pulses made up of few simple components. The optimal time delays and phase relations between the components making up the overall pulse are automatically built into the OCT pulses and their values are accessibleonly by indirect methods. In favorable cases, some semiquantitative estimates can be made by analyzing the beat patterns of the pulses. Thus for the pulse exciting (2,1), a low-frequency beat from the interference of the two closely spaced OH components near 3600 and 3800 cm-l and a higher frequency beat due to the interaction of the OH and OD components can be clearly resolved. The onset of the low frequency beat is discernible near t = 80 fs,

0

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time [fsl Figure 5. Time evolution of the level populations, Ict(t)12,for all molecular levels Ik) acquiring significant intermediate population in the interaction of HDO with a minimum-fluence OCT pulse. Left panels: (0,O) (2,l) transition. Right panels: (0,O) (3,O) transition.

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which suggests that 1 2 pumping of the OH bond vibration sets in very early in the pulse, while 0 1 pumping must still be proceeding strongly. The optimization is able to create a very compact and intertwined sequence or superposition of subpulses characterized by very short time delays and a considerable amount of overlapping between consecutive steps. An analogous analysis of the shorter frequency beat shows that both an OH and an OD component must be present in about equal amounts right away from the beginning of the pulse. This seems to imply that mechanisms starting with a (0,O) (0,l) transition and those starting with a (0,O) (1,O) transition are present in parallel. This interpretation is further corroborated by an analysis of the time dependence of the expansion coefficients ck(r) of the total wavefunction I*), eq 2, as shown in Figure 5 for two of the minimum-fluence OCT pulses. An interpretation of the Ick(t)I* as level dynamics during the excitation is entirely consistent with the mechanism deduced above, and can in fact help to resolve some of the remaining ambiguities. Accordingly, for the (0,O) (2,l) transition, the most efficient excitation pathway, involving the intermediate levels ( 1 , O ) and ( 1 , 1 ) , is seen to be pathway ( 1 5b), and the sequential nature of the overall transition is clearly displayed. Furthermore it is seen that, to a weaker extent, both the pathways ( 1 5a) and (15c) are present as parallel mechanisms. Similarly, the dominance of the sequential one-photon ladder mechanism (16a) for the (0,O) (3,O) transition is vindicated by the Jck(t)l*vs t plots of Figure 5. Some intermittent dumping of population into the (4,O) state is also apparent, but it is not clear whether this “imperfection” is essential for achieving perfect selectivity. Altogether, interference effects between competing pathways may play a role in ultrafast selective vibrational excitation, but a conclusive answer to this question requires additional research. One of the striking contrasts between the minimum-fluence

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pulses and the pulses related to multiphoton excitation is the fact that for the former there is no difference between selective excitation of a nonlocal state like (2,l) and a local state like (3,O). Both processes require pulses of about the same fluence and the same spectral complexity, and both apparently occur through successive steps which can loosely be interpreted as Av = 1 excitations of bond vibrations. The local or nonlocal nature of the target mode is of little influence, the condition for ready excitation being that the target state and all intermediate states are reasonably pure states in a direct-product expansion of bond functions. As noted in section 3, for the present Hamiltonian this is the case for most states, and at least for the modes with up to four vibrational quanta the excitation dynamics with minimum-fluence pulses is similar for all target states, the main computational challenge being the right choice of W I . Certainly the mechanism of excitation may change as the number of vibrational quanta is increased, but a direct investigation of this point, which would require calculations with larger basis sets, has yet to be carried out. Before closing this discussion,we note that the breakup of the overall transition into a sequence of lower order transitions mediated by simply shaped subpulses, as suggested by the analysis of the OCT pulses, closely corresponds to the kind of approach heuristicallyintroduced by Manz, Paramonov, and wworkers.l9J0 The present findings suggest that their approach may in some sense be an optimal one. 6. Outlook and Conclusions

In this investigation, we have demonstrated the ready applicability of OCT techniques to the problem of ultrafast stateselective vibrational excitation in a two-dimensional system. In particular, we have introduced a fluence-minimizing strategy, which in addition to its use as a pulse-shaping technique is also providing insight into the mechanism behind the dynamics of excitation. Using the minimum-fluence OCT strategy, we have succeeded in designing IR laser pulses with a pulse length of 500 fs, which in theory meet the conditions necessary for the stateselective population of moderately highly excited local OD or OH modes of HDO (admittedly represented by a simplistic model). This process is an important step in certain schemes for the bond selective dissociation of HDO,33 but more generally, similar local excitation processes may be used in mode-selective chemistrylJs to prepare intermediateswith the energy temporarily stored in a selected mode, available for eventually triggering a specific followup process. Our application to a model system, of course, represents just a very limited aspect of what is required to make the technique a useful tool in mode-selective chemistry. Although many extensions appear to be feasible, some critical comments are in order. The vibrational transitions we have been considering in this paper populate fairly low-lying states of HDO and correspond to only moderately strong vibrational excitation. Yet despite successful fluence reduction, the required laser intensities are still found to be formidable even in the most favorable cases and come close to the intensity limit imposed by the onset of molecular i ~ n i z a t i o n . ~Even J ~ assuming that the fluences for optimized pulses scale only linearly with the total number of vibrational quanta to be excited, the limitations quoted above would allow subpicosecond excitation of at most the sixth overtone levels in HDO. There are little prospects for accessing more highly excited states in this time domain, let alone for attempting ‘real” femtochemistry like the direct, mode-selective dissociation of HDO via selective vibrational excitation into local resonancestates. Certainly this limitation to some extent depends on the level spacing and dipole properties of the molecule in question, and more highly excited states may be accessiblein other molecular systems. Nevertheless, the intensity problems have to be taken seriously, and it may well be that it is the picosecond rather than

Jakubetz et al. the femtosecond domain where shaped IR laser pulses exert their full power in controlling selective intramolecular excitation processes. It should alsobenoted that the possibilityof efficient vibrational excitation may to some extent be an artifical property of our model Hamiltonian. The stepwisemechanism of excitation which is connected with the present minimum fluence pulses is tied to the fact that all states involved, including the intermediate ones, are reasonably pure states in a product basis of bond vibrations. However, almost by construction this property is imposed on the eigenstates of a multidimensional system consisting of individual one dimensionaloscillatorswithout any potential-coupling terms, as is the case for the present Thiele-Wilson Hamiltonian. It remains to be seen if and how OCT could also yield pulses of comparable simplicity and efficiency for a system with strong potential coupling, and indeed it might be instructive to analyze the excitation mechanisms in such a system by methods similar to the ones used in the present investigation. Extensions of the technique, which might enhance its use in mode-selective chemistry, include the obvious, but computationally demanding addition of bending vibrations and rotation, but also the addition of a weakly coupled background modes@ representing a larger molecular or intermolecular environment. Other extensions are suggested by the relative ease of implementation of the minimum-fluence constraint and its striking success. Several other constraints on the properties of the laser field can be envisaged,notably one producing a smoother transition at the beginning and the end of the pulse than is obtained for the present pulses. An interesting byproduct of the analysis of the minimumfluence OCT pulses is the strong footing it gives to methods breaking up single simple multiphoton pulses like sine-square pulses into sequences or superpositionsof such p u l s e ~ . I ~Indeed .~~ the present investigations would have shown the way to such methods, had they not already been introduced before. A combination of the two approaches, e.g., using OCT to identify efficient pathways and other parameters, followed by the construction of an overall pulse from few simple subpulses may be a viable strategy for obtaining pulse forms which can be realized experimentally. Such an approach will workonly if the optimized pulses are robust and can endure some amount of surgery without losing the property of selectivity. We did not explicitlyinvestigate the robustness of our OCT pulses, but we note that Tersigni et al. have reported22 robustnessof OCT pulses for a five-level model system. In addition, the ease with which the present, extremely high-dimensional optimization problem converges towards selective pulses may be taken as a hint that selectivity is upheld over fairly large stretches of configuration space of the variation parameters q, and hence of E(t). On the other hand, the wl = 0 pulses seem to demonstrate that the minor components of the power spectra, might have a considerable role to play for establishing selectivity. After all, Figure 4 implies that the main difference in the power spectra between the nonselective simple startup pulses and the selective OCT pulses is in the existence of the minor components associated with the latter. In any case, this point needs further investigation,and its analysis may benefit from recently introducedapproaches expresslydevised for dealing with imperfections and uncertainties in the Hamiltonians and fields.2.29 Summing up and considering all the aspects discussed, we conclude that the technique of designing minimum fluence IR laser pulses by OCT may prove to be a valuable tool in modeselective chemistry, not least because of the insights it might provide into the mechanisms and dynamics of selective excitation and the capability to identify efficient excitation pathways. Acknowledgment. Partial financial support of this work by the Deutsche Forschungsgemeinschaft under Project No. MA 515/9 is gratefully acknowledged by J.M. and E.K. The initial

State-Selective Excitation of Molecules stage of the research was conducted while E.K.and J.M. were at the Institute for Physical Chemistry at the University of Wiirzburg (FRG),where also some of the preparatory computations were carried out. Most of the computations were performed at the Computer Center of the University of Vienna under the auspices of IBM’s European Academic Supercomputer Initiative (EASI). We thank all institutions involved for their generous support.

References and Notes (1) Brumer, P.; Shapiro, M. Annu. Rev. Phys. Chem. 1992, 43, 257. Combariza, J. E.; Daniel, C.; Just, B.; Kades, E.; Manz, J.; Malisch, W.; Paramonov, G. K.; Warmth, B. In Isotope Effects in Gas-Phase Chemistry; ACS Symposium Series Vol. 502; Kaye, J. A,, Ed.; American Chemical Society: Washington, DC, 1992; p 310. Bandrauk, A. D., Ed. Atomic and Molecular Processes with Short Intense Luser Pulses; Plenum: New York, 1988 and references therein. (2) Lupo, D. W.; Quack, M. Chem. Reu. 1987.87, 181. (3) Yan, Y. J.; Gillian, R. E.; Whitnell, R. M.; Wilson, K. R.; Mukamel, S.J . Phys. Chem. 1993, 97, 2320 and references therein. (4) Chu, S.-I.Adu. Chem. Phys. 1989, 73,733. Leasure, S.;Wyatt, R. E. Opt. Eng. 1980, 19, 46. (5) Breuer, H. P.; Dietz, K.; Holthaus, M. Z . Phys. D 1988, 8, 349. Breuer, H. P.; Holthaus, M. Z . Phys. D 1989, 11, 1. Breuer, H. P.; Dietz, K.; Holthaus, M. J. Phys. B: At. Mol. Opt. Phys. 1991, 24, 1343. (6) Chelkowski, S.; Bandrauk, A. D.; Corkum, P. B. Phys. Rev. Lett. 1990,65, 2355. (7) Holthaus, M.; Just, B. Preprint. (8) Allen, J.; Eberly, J. H. Optical Resonance and Two-Leuel Atoms; Wiley: New York, 1975. (9) Shirley, J. H. Phys. Reu. 1965, 138, 8979. (10) Jakubetz, W.; Manz, J.; Mohan, V. J . Chem. Phys. 1989.90.3686. (11) Paramonov, G. K.; Sawa, V. A. Phys. Lett. A 1983, 97, 340. Paramonov,G. K.; Sawa, V. A.;Samson,A. M. InfraredPhys. 1985,25,201.

Dolya, Z. E.; Nazarova, N. B.; Paramonov, G. K.; Sawa, V. A. Chem. Phys.

Lett. 1988,145,499. Paramonov, G. K. Chem. Phys. Lett. 1990, 169,573. Paramonov, G. K. Phys. Lett. A 1991, 169, 573. (12) Jakubetz, W.; Just, B.; Manz, J.; Schreier, H.-J. J . Phys. Chem. 1990,94, 2294. (13) Dietrich, P.; Corkum, P. B. J . Chem. Phys. 1992, 97, 3187. Chelkowski, S.; Zuo, T.; Bandrauk, A. D. Phys. Rev. A 1992, 46, 5342. (14) Quack, M. Ber. Bunsen-Ges. Phys. Chem. 1979,83, 757. (15) Bloembergen, N.; Zewail, A. H. J . Phys. Chem. 1984, 88, 5459. Zewail, A. H. Science 1988,242,1645. Brumer, P.; Shapiro, M. Acc. Chem. Res. 1989, 22, 407. Crim, F. F. Science 1990, 249, 1387. (16) Chelkowski, S.;Bandrauk, A. D. Chem. Phys. Lett. 1991,186,264. Just, B.; Manz, J.; Trisca, I. Chem. Phys. Lett. 1992, 193, 423. (17) Jakubetz, W.; Manz, J.,Schreier, H.-J. Chem. Phys. Lett. 1990,165, 100.

The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 12619 (18) (19) 429. (20) (21)

Shi, S.;Rabitz, H. J. Chem. Phys. 1990, 92, 364. Just, B.; Manz, J.; Paramonov, G.K. Chem. Phys. Lett. 1992,193,

Manz, J., Paramonov, G. K. J . Phys. Chem., this issue. Tannor, D. J.; Rice, S.A. Adu. Chem. Phys. 1989,70,441. Kosloff, R.; Rice, S. A.; Gaspard, P.; Tersigni, S.H.; Tannor, D. J. Chem. Phys. 1989, 139, 201. (22) Tersigni, S.H.; Gaspard, P.; Rice, S.A. J . Chem. Phys. 1990, 93, 1670. (23) Gaubatz, U.; Rudecki, P.; Schiemann, S.;Bergmann, K. J . Chem. Phys. 1990,92,5363. Coulston, G.W.; Bergmann, K. J. Chem. Phys. 1992, 96, 3467. (24) Kosloff, R.; Hammerich, A. D.; Tannor, D. J. Phys. Reu. Lett. 1992, 69, 2172. (25) Huang, G. M.; Tarn, T. J.; Clark, J. W. J . Math. Phys. 1983, 24, 2608. Barnett, S.; Cameron, R. G. Introduction to Mathematical Control; Oxford University: New York, 1985. (26) Peirce, A. P.; Dahleh, M. A,; Rabitz, H. Phys. Reu. A 1988,37,4950. Shi, S.;Rabitz, H. Chem. Phys. 1989,139, 185. Shi, S., Rabitz, H. J . Chem. Phys. 1990, 92, 2927. Dahleh, M. A,; Peirce, A. P.; Rabitz, H. Phys. Reo. A 1990,42, 1065. (27) Shi, S.;Woody, A.; Rabitz, H. J . Chem. Phys. 1988, 88, 6870. (28) Zhao, M.; Rice, S.A. J . Chem. Phys. 1991, 95, 4265. (29) Gross, P.; Neuhauser, D.; Rabitz, H. J . Chem. Phys. 1993,98,4557. (30) Schreier, H.-J. Dissertation, University of Wiirzburg, FRG, 1990.

Kades, E., Diplomarbeit (Diploma Thesis), University of Wiirzburg, FRG, 1990. (31) Thiele, E.; Wilson, D. J. J . Chem. Phys. 1961, 35, 1256. (32) Child, M. S.; Halonen, L. Adu. Chem. Phys. 1984, 57, 1. (33) Segev, E.; Shapiro, M. J. Chem. Phys. 1982, 77, 5604. Engel, V.; Schinke, R. J . Chem. Phys. 1988,88,6831. Imre, D. G.;Zhang, J. Chem. Phys. 1989, 139, 89. Amstrup, B.; Henriksen, N. E. J . Chem. Phys. 1992, 97,8285. Successful application of an alternative approach to isotopeselective

HDO photodissociation, using the concept of coherent radiative control, has recently been reported in: Brumer, P.; Shapiro, M. J . Chem. Phys. 1993,98, 201. For accounts of the coherent radiative control technique, see the articles by Brumer and Shapiro in refs 1 and 15. (34) Press, W. H.; Flannery, B. P.; Teukolsky, S.A.; Vetterling, W. T. Numerical Recipes; Cambridge University: Cambridge, 1986. (35) Mecke, R. Z. Elektrochem. 1950, 54, 38. (36) Chang, J.; Wyatt, R. E.J. Chem. Phys. 1986,85,1840. Quack, M.; Sutcliffe, E. Chem. Phys. Lett. 1984, 105, 147. (37) Jensen, P. J. Mol. Spectrosc. 1989,133,438. Gabriel, W.; Rosmus, P. J . Phys. Chem., this issue. See also the related work on the H20 Hamiltonian: Bacic, Z.; Watt, D.; Light, J. C. J. Chem. Phys. 1988,89,947. Choi, S.E.; Light, J. C. J. Chem. Phys. 1992, 97, 7031. (38) Hartke, B.; Manz, J.; Mathis, J. Chem. Phys. 1989, 139, 123. (39) Manz, J.; Schor, H. H. R. Chem. Phys. Lett. 1984, 107, 542. (40) Wyatt, R. E. Adu. Chem. Phys. 1988,72,231. Guldberg, A,; Billing, G.D. Chem. Phys. Lett. 1992, 191, 455.

State-selective excitation of molecules by means of optimized

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