20 August 1999

Ž .Chemical Physics Letters 309 1999 495–501www.elsevier.nlrlocatercplett

A classical phase r-centroid approach to molecular wave packetdynamics – illustrating the danger of using an incomplete set of

initial states for thermal averaging

Tony Hansson a,b,)

a Department of Physics, Section of Atomic and Molecular Physics, Royal Institute of Technology, KTH, SE-100 44 Stockholm, Swedenb Department of Chemistry, Royal Institute of Technology, KTH, SE-100 44 Stockholm, Sweden

Received 5 May 1999; in final form 24 June 1999

Abstract

An inexpensive semiclassical method to simulate time-resolved pump–probe spectroscopy on molecular wave packets isapplied to NaK molecules at high temperature. The method builds on the introduction of classical phase factors related to ther-centroids for vibronic transitions and assumes instantaneous laser–molecule interaction. All observed quantum mechanicalfeatures are reproduced – for short times where experimental data are available even quantitatively. Furthermore, it is shownthat fully quantum dynamical molecular wave packet calculations on molecules at elevated temperatures, which do notinclude all rovibrational states, must be regarded with caution, as they easily might yield even qualitatively incorrect results.q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

Despite considerable recent effort spent on theproblem of the full quantum dynamical treatment ofmolecular wave packets, tractable systems remainlimited in the number of degrees of freedom. Evenfree diatomic molecules become challenging systemsat high temperatures if rotational and vibrationalmotion should be included completely and on anequal footing. It can be done, however, as in the

w xelucidating work by Gruebele and Zewail 1 . In thisstudy, many of the qualitative features due to cou-pling of vibrational and rotational motion at elevatedtemperatures were discussed and illustrated for I2

molecules, such as the washing out of vibrational

) Fax: q46-8-200430; e-mail: tony.hansson@atom.kth.se

beat structures arising from rotational averaging. Theauthors also made brief comparisons of quantumdynamical calculations to classical and simple semi-classical methods.

Due to the complexity of the full quantum me-chanical treatment, methods capable of treating accu-rately, although not necessarily exactly, wave packetdynamics at elevated temperatures, i.e. room temper-ature or higher, are needed to extend the range oftractable systems at elevated temperatures to includemore degrees of freedom. One such approach was

w xmade by Andersson et al. 2 , who studied the wavepacket dynamics of NaK molecules at 700 K. Theytreated the quantum dynamics exactly for both rota-tional and vibrational degrees of freedom and alsoincluded the time-dependent classical laser field–molecule interaction. To reduce the calculational ef-

0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0009-2614 99 00709-5

( )T. HanssonrChemical Physics Letters 309 1999 495–501496

forts, however, the averaging over rotational stateswas treated approximately, by selecting a representa-tional set of J numbers. They could in this mannershow that the inclusion of rotational averaging isessential in order to get a quantitative match betweenexperimental and calculated short-time dynamics ofmolecular wave packet dynamics. As will be shownbelow, though, this approach does not give reliableresults for the long-term wave packet dynamics.

w xA recent paper by Ermoshin et al. 3 treats theelevated temperature molecular wave packet dynam-ics in a mixed quantumrclassical fashion. Thus, thevibrational degree of freedom of I molecules in a2

bath of perturbers is treated exactly, whereas therotational and all other degrees of freedom of thesystem are considered classical. As expected, therotation–vibrational coupling is found to lead towashing out of most of the observable vibrationalcoherence structures, as do interactions with thebath. The price paid in choosing this approach is togive up the quantum mechanical nature of the rota-tional degree of freedom.

We demonstrate here the application of an inex-pensive semiclassical method to include all rovi-bronic states in the calculation of molecular wavepacket transients at elevated temperatures. The modelassigns a classical phase to each coherent pair ofeigenstates in the wave packet and relates the pumpand probe transitions to the most probable internu-clear separations in the form of vibronic r-centroids.It also includes coherence overtones. The agreementboth with experiments and full quantum dynamicalsimulations at elevated temperature is found be verysatisfactory – within experimental noise limits in thestudy case of the NaK molecule in the A state. Thus,the semiclassical simulations reproduce correctly vi-brational beat splitting seen in quantum dynamicalcalculations, as well as partial revivals with correctphases. In addition, it is shown that the approach

w xchosen in Ref. 2 to make a representative J selec-tion is inadequate and leads to incorrect long-timewave packet dynamics.

2. Theory

The signal in a time-resolved pump–probe experi-ment on a thermally excited molecule is best de-

w xscribed 4 in terms of the evolution of the molecularŽ .density matrix, r t , where t is the time delay

between pump and probe pulses. The final signal,Ž . w xI t , is then given by 1

I t sTr O q r t , 1Ž . Ž . Ž . Ž .�q4

� 4where Tr is the trace operator, q indicates that thetrace should be taken over all quantum numbersdefining the molecules state after the probe pulse,

Ž .and O q is a factor weighting the q states accord-ing to their detection probability.

Ž .The expansion of Eq. 1 in terms of Hilbert spacew xrovibrational wave functions is performed in Ref. 1 ,

and there also some very useful approximations areobtained. Of special interest here is when ‘magic

w xangle’ 5 configuration of the experiment is as-sumed, in which rotational anisotropy effects in thesignals are absent. The signal for two well-separatedpulses then becomes,

I t s g exp yhcE J rk TŽ . Ž .ÝÕ n 0R B RJ

= yA f J cos 2pctŽ . �Ý i ji , j-i

= E Õ , Jy1 E Õ , Jy1 qDwŽ . Ž . 4Õ i Õ j i j

qqf J cos 2pct E Õ , Jq1Ž . Ž .� Õ i

=E Õ , Jq1 qDw . 2Ž .Ž . 4Õ j i j

The summations run over all rotational states J inthe electronic ground state and pairs of vibrationalstates Õ in the excited electronic state. The nucleari j

spin degeneracy is denoted g and E and E aren 0R Õ

Ž .the rotational electronic ground state and vibra-Ž .tional term values excited electronic state , respec-

"Ž .tively. The f J factors correspond to the statisti-cal weights of the rotational transitions and dependon J and laser field polarisation. All other informa-tion on initial state population, transition probabili-ties, laser field frequency and intensity, detectionprobabilities, and so forth are contained in the Afactors. Of the remaining symbols, the Dw repre-i j

senting phase factors introduced by the laser–mole-cule interaction is the only one needing explanation.In fact, these phase factors contain all informationwhich characterises the shape and position of thewave packet after excitation, as well as the location

( )T. HanssonrChemical Physics Letters 309 1999 495–501 497

of the probe transition and weighting of differentstates in the probe process. Compare for instancewith the doorway–window formalism developed by

w xMukamel et al. 4 .Ž .In order to evaluate Eq. 2 we make the semi-

classical approximation that the Dw factors can bei j

obtained as

probe probeDw ™ Dw J s"w r JŽ . Ž .i j i j i j c , i j

pump pump"w r J . 3Ž . Ž .i j c , i j

X ? X Ž .@Here w r J is the classical phase of a Morsei j c, i jw x X Ž .oscillator 6 corresponding to the position r J ,c, i j

and there is in all four possible combinations of thesigns of the phases leading to distinct Dw ’s. Thei j

dependence of the Morse oscillator phase on therotational state was found to be very well reproducedby simply shifting the equilibrium distance of the

w xoscillator, r , according to 7e

a"J Jq1Ž .r J sr 0 . 4Ž . Ž . Ž .e e 3 2m r 0 vŽ .e e

Ž .In Eq. 4 , m is the reduced mass of the Morseoscillator, v the vibrational constant of it, and a ise

X Ž .a constant. The position r J is taken to be thec, i j

most probable internuclear separation for the X tran-w xsition, as given by its r-centroid 8,9 . That is,

² X X < < Y Y :Õ J r Õ JŽ . Ž .X XXr J s , 5Ž . Ž .X X Y Yc , i j ² < :Õ J Õ JŽ . Ž .X X

where J is representing the rotational state of themolecule in the excited state. hence, J can either beJ X or JY depending on if X denotes a pump or aprobe transition. To accurately reflect the spread inthe r-centroids and thus in phase shifts, every coher-

Ž .ence pair in Eq. 2 was assigned two different phaseshifts; corresponding to the phases for the populationand detection of each of the levels in the pair. In thecase of an overtone coherence, i.e. when DÕ)1 in

Ž .Eq. 2 , we note that as the potential and totalenergies are still the same as for DÕs1 this corre-sponds to a smaller reduced mass of the Morse

Ž .2oscillator. We thus in this case let m™mr DÕ ,which is a good approximation for small DÕ’s, acondition which certainly applies here.

Finally, focussing on the long-time dynamics ofthe wave packet, we treat the laser–molecule interac-tion as instantaneous, but take into account the finitespectral width of light field. This is done by weight-ing every transition according to the intensity of thefield at resonance and neglecting all non-resonantcontributions. The non-resonant contributions defi-nitely influence the phase of the wave packet, but donot have a simple classical correspondence. In fact, itis the neglect of these non-resonant terms whichnecessitates the introduction of a classical phase in

Ž Ž . w x.the calculations cf. Eq. 13.17 in Ref. 4 .

3. Calculations and results

We apply the semiclassical model described inSection 2 to the wave packet dynamics of the NaKmolecule in the A 1

Sq state, which has been the

w xsubject of several recent experimental 10–12 andw xtheoretical 2 studies. There has been some confu-

sion regarding the nature of the probe state in thew xexperiments, but recent work 2,12,13 seems to have

settled this issue. Thus, we adopt the conclusionfrom the latter studies that the 3 1

P state is the majorcontributor to the signal. The potential energy curveswe used were the adiabatic ones by Magnier and

w xMillie 14 , which have been verified experimentally´w xto be very accurate 2,12,13,15–18 . The initial rovi-

brational levels in the electronic ground state takeninto account were Õ s0–15 and J s0–150. In the0 0

excited and final states all Õ’s 0–25 were included,with the corresponding J ’s given by the electricdipole selection rules.

Except for the rovibrational matrix elements allcalculations were done on a rather modest PC,equipped with a 233 MHz AMD-K6 microprocessorand 32 Mbyte RAM, and the program code waswritten in Cqq for Windows 95. A trace extending

Ž 8.over 100 ps at 700 K including all ;10 initialrovibrational transitions took of the order of 2 h tocalculate on this machine. The rovibrational matrixelements, r-centroids, and transition energies were

w xcalculated using the Level 6.1 software 19 . Thisonce-and-for-all calculation was for practical reasonsdone on a DEC 3000 workstation and took another;3 h.

( )T. HanssonrChemical Physics Letters 309 1999 495–501498

Fig. 1. Calculated relative pump–probe intensity for l sl s766 nm, DÕ s190 cmy1 , T s50 K, and T s10 K. The˜pump probe laser vib rot

arrows mark the approximate times for the occurrence of the mrn partial revivals, T .mr n

Two different experiments were considered in thequantum dynamical calculations by Andersson et al.w x2 , and we address both of them here and comparethe results to both the quantum dynamical and theexperimental ones. Consider first the molecular beam

w xexperiment by Heufelder et al. 11 . In this experi-ment the pump and probe pulses were identical witha wavelength of 766 nm and the level populations ofthe molecules were described by rotational and vi-brational temperatures of 10 and 50 K, respectively.With 3 1

P as the final state the quantum dynamicalcalculations exhibit a pronounced initial splitting ofthe wave packet signal, as well as clear T , T ,1r6 1r4

T , and T partial revivals. All these features are1r3 1r2

also obtained by our semiclassical method, as shownin Fig. 1. Furthermore, the times at which thesepartial revivals occur are very close to those obtainedin the quantum dynamical calculations.

The second set of calculations by Andersson et al.concerned an experiment where the two pulses againwere of the same colour, 790 nm, but now with themolecules at thermal equilibrium at 700 K. Oursimulated signal for this case up to 10 ps is found inFig. 2 where it is also compared to the experimentaltrace. Obviously, the phase shift of p of the oscilla-tions obtained at ts0 in the experiment is wellreproduced, as is also the decay of the beat ampli-tude. A dephasing between the experimental andcalculated traces seems to occur for t)6 ps, whichis also taking place for the quantum dynamical trace.

The same calculated trace as in Fig. 2 but ex-tended up to 100 ps is displayed in the lowermost

panel of Fig. 3. Note in particular that no partialrevivals are observed. This is in agreement with

w xexperimental findings 2,10 but stands in contrast tow xthe quantum dynamical results in Ref. 2 , which

Ždisplay a pronounced half-revival ;30% of the.initial amplitude of the wave packet at ;95 ps.

w xHowever, Andersson et al. 2 made in their calcula-tion a selection of a set of representative rotationalquantum states over which they averaged the signal.In the topmost panel of Fig. 3 we show the traceobtained by our simulation when making the same J

w xsampling as in Ref. 2 . The trace now exhibitsstrong revival at ;95 ps. To further clarify theeffect of J sampling we show in Fig. 3 the traces

Ž . Ž .Fig. 2. Calculated thick line and experimental thin relativepump–probe intensity for l s l s790 nm, DÕ s173˜pump probe laser

cmy1 , and T sT s700 K.vib rot

( )T. HanssonrChemical Physics Letters 309 1999 495–501 499

Fig. 3. The effect of incomplete sampling of J levels on thermally averaged pump–probe signals. All conditions in the calculations are0

identical to those in Fig. 2, except for the J selection. In each panel all J ’s in the range 0–150 have been sampled with the indicated0 0

frequency.

obtained for some intermediate choices of the Jsampling frequency.

4. Discussion

Turning our attention first to the cold moleculecalculations in Fig. 1 we note that all quantummechanical features displayed in the pump–probe

signal are reflected in our semiclassical trace. More-over, a thorough comparison of the quantum dynami-cal long-time trace to our semiclassical one in Fig. 3reveals that even the intrinsically very non-classical

w xphase of the T -revival 20 is correct.1r2

The semiclassical calculation also reproduces veryclosely the splitting of the signal at short timesobserved in the quantum dynamical calculation un-der the conditions pertaining to Fig. 1. As explained

( )T. HanssonrChemical Physics Letters 309 1999 495–501500

w xin Ref. 2 , this splitting is due to the fact that themost likely probe position is located not at theclassical outer turning point of the A state butsomewhat short of it. From this it is clear why weintroduce the most probable transition point in theform of the r-centroid, instead of simply applyingthe classical turning point. In fact, the r-centroid isclosely related to the difference potential used bymany authors and discussed in detail by Mullikenw x w x21 . It can namely be shown 8,9 , that

EXÕ

X yEYy

Y fV X r X yV Y r X , 6Ž . Ž . Ž .Ž . Ž .X X X X X c , i j X c , i j

Ž .where V r is a molecular potential energy. Thus,the r-centroid value approximately corresponds tothe internuclear separation at which the differencepotential matches the transition frequency for they

X§y

Y vibronic transition, which is not necessarilyat a classical turning point.

So far, we have shown that the semiclassicalapproach used here qualitatively reproduces all quan-tum dynamical features of pump–probe spectroscopyon molecular wave packets. The value of the pre-sented semiclassical model is not limited by this,though. Comparing the simulated trace to the experi-mental one in Fig. 2 it becomes apparent that at leastthe short-time decay of the beat amplitude in theexperiment is quantitatively reproduced. The phaseshift between the theoretical traces, semiclassical aswell as quantum dynamical, and the experimentalone for t)6 ps is most likely due to a perturbation

Ž 1 q. w xof the NaK A S state 11 .It was mentioned above that there is a discrepancy

between the experimental and quantum dynamicalw xresults in Ref. 2 at high temperatures and long

times – the quantum dynamical calculations display-ing a pronounced T -revival, which was never1r2

observed experimentally. The semiclassical calcula-tions, however, predict an almost complete washingout of the revival in accordance with the experimen-tal finding. It is from Fig. 3 clear that the reason forthis long-time failure of the quantum dynamical cal-culations is related to how the elevated temperaturewas taken into account. In that figure we illustratethe general effect of the J sampling approach Ander-

w xsson et al. 2 used. Up to D J s5 it seems to work0

reasonably well, but for D J s10 the T -revival0 1r2

structure blows up and yields a qualitatively erro-

neous result. Hence, the J sampling approach has tobe exercised with great caution, or the long-termdynamics might be not even qualitatively correct, as

w xin Ref. 2 where a D J of 15 was used. Thus, we0

conclude that the strong T -revival obtained in Ref.1r2w x2 is an artefact due to incomplete thermal averagingover rotational states.

To sum up, we have introduced an inexpensivesemiclassical method to simulate time-resolvedpump–probe spectroscopy on molecular wave pack-ets. The method builds on the introduction of classi-cal phase factors in the form of the r-centroids forvibronic transitions and assumes instantaneouslaser–molecule interaction. Even though we in thisapproach throw away all information on the wavefunction in the actual propagation of the wave packetall observed quantum mechanical features are repro-duced – for short times where experimental data areavailable even quantitatively. Furthermore, we showthat fully quantum dynamical molecular wave packetcalculations at elevated temperatures, which do notinclude all rovibrational states, must be regardedwith caution, as they easily might yield even qualita-tively incorrect results.

Acknowledgements

I thank Peter van der Meulen for valuable discus-sions. Sylvie Magnier and Phillippe Millie most´kindly made their unpublished adiabatic potential

w xcurves from the calculations in Ref. 14 available tome. This work was supported by the Swedish Natu-

Ž .ral Science Research Council NFR .

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