Theory for the ultrafast structural response of optically excited small clusters:Time dependence of the ionization potential

H. O. Jeschke, M. E. Garcia, and K. H. BennemannInstitut fur Theoretische Physik der Freien Universita¨t Berlin, Arnimallee 14, 14195 Berlin, Germany

~Received 22 February 1996; revised manuscript received 25 April 1996!

Combining an electronic theory with molecular-dynamics simulations we present results for the ultrafaststructural changes in small clusters. We determine the time scale for the change from the linear to a triangularstructure after the photodetachment process Ag3

2→Ag3. We show that the time-dependent change of theionization potential reflects in detail the internal degrees of freedom, in particular coherent and incoherentmotion, and that it is sensitive to the initial temperature. We compare with experiment and point out the generalsignificance of our results.@S1050-2947~96!50512-3#

PACS number~s!: 31.70.Hq, 36.40.Mr, 33.80.Eh

The excitation of a cluster by a laser pulse induces time-dependent changes of its electronic and atomic structure.These changes involve bond formation and bond breaking.The understanding of the relaxation mechanisms induced bythe excitation is of general interest. In particular, it is offundamental importance to study how the system approachesequilibrium in order to determine how the time scales of therelaxation processes can be controlled by varying the experi-mental conditions.

Recently, a pump and probe experiment has been per-formed on mass selected Ag3

2 clusters, which serves as anexample for the investigation of the structural relaxationtimes@1#. The initially negatively charged clusters were neu-tralized through photodetachment by the pump pulse and af-ter a delay timeDt ionized by the probe pulse in order to bedetected. Due to the remarkable differences in the equilib-rium geometries of the ground states of Ag3

2 , which islinear @2#, and Ag3, which consists of an obtuse isoscelestriangle @3#, the ultrashort photodetachment process puts theneutralized trimer in an extreme nonequilibrium situation. Asa consequence of that, a structural relaxation process occurs.The experimental signal, consisting of the yield of Ag3

1,was measured as a function ofDt and the frequency of theprobe laser pulse. For a frequency slightly above the ioniza-tion potential~Vi) of Ag3 a sharp rise of the signal is ob-served atDt.750 fs. After a maximum is reached, there is

a saturation of the signal, which then remains constant for atleast 100 ps, which is the longest time delay used in theexperiment. New features appear for higher frequencies.Again the signal increases sharply, but after reaching a maxi-mum it decreases to a constant value.

A preliminary interpretation of these results uses theFranck-Condon principle@1#. The first laser pulse creates aneutral linear silver trimer that bends and comes to a turningpoint near the equilateral equilibrium geometry of the posi-tive ion @3#. After rebounding, the neutral trimer starts pseu-dorotating through its three equivalent obtuse isosceles equi-librium geometries. This would explain the saturationbehavior of the signal. However, this would mean that thepseudorotations have an extremely long mean life, whichseems improbable. Furthermore, this model does not explainwhy the signal changes as a function of the frequency of thelaser pulse.

In this paper we perform a theoretical analysis ofthe physics underlying the ultrafast dynamics of Ag3 clus-ters produced by photodetachment. In particular, we analyzethe time evolution of the ionization potential and the depen-dence of the dynamics on the initial temperature of the clus-ters. We show that the experimental results can be explainedusing a physical picture which can be generally applied toother ultrashort-time processes. In our calculations, we com-bine molecular-dynamics~MD! simulations in the Born-

PHYSICAL REVIEW AATOMIC, MOLECULAR, AND OPTICAL PHYSICS

THIRD SERIES, VOLUME 54, NUMBER 6 DECEMBER 1996

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Oppenheimer approximation with a microscopic theory todescribe the time-dependent electronic structure of the clus-ters.

In order to determine the potential-energy surface~PES!needed for the MD simulations, we start from a Hamiltonianof the formH5HTB1 1

2( iÞ jf(r i j ), where the tight-bindingpartHTB is given by

HTB5 (i ,a,s

« iacias† cias1 (

iÞ j ,sa,b

Via jbcias† cibs . ~1!

Here, the operatorcias† (cias) creates ~annihilates! an

electron with spin s at the site i and orbital a ~a55s, 5px , 5py , 5pz). « ia stands for the on-site energy, andVia jb for the hopping matrix elements. For simplicity, andsince the 5s electrons are expected to be rather delocalized,we neglect the intra-atomic Coulomb matrix elements.f(r i j ) refers to the repulsive potential between the atomiccores i and j . For the distance dependence of the hoppingelements and the repulsive potential we use the functionalform proposed in Ref.@4#. By diagonalizingHTB ~takinginto account the angular dependence of the hopping elements@5#!, and summing over the occupied states, we calculate as afunction of the atomic coordinates the attractive parts of theelectronic ground-state energiesEattr

2 andEattr0 of Ag3

2 andAg3

0 , respectively. Then, by adding the repulsive part ofH we obtain the PES, which we need to perform the MDsimulations. In order to determine the forces acting on theatoms we make use of the Hellman-Feynman theorem. Thus,the a component of the force acting on atomiFia52]E/]r ia is given by

Fia52(k

occ.

K kU ]HTB

]r iaUkL 2

1

2 (iÞ j

]f~r i j !

]r ia. ~2!

Here theuk& ’s are the eigenstates ofHTB . The parameters ofH are determined in the following way. The on-site energieswere obtained from atomic data@8#. We fit the parametersVab and the potentialf(r i j ) in order to reproduce the equi-librium bond lengths of the silver dimers Ag2

2 , Ag2 , andAg2

1 obtained by effective core-potential–configuration-interaction calculations@2,3#. We assumed the hopping ele-mentsVab to fulfill Harrison’s relations@9#. The best fit wasobtained by the following parameters:Vsp50.954 eV,r c54.33 Å, nc52, m55.965, andA50.605 eV, wherer cand nc refer to the cutoff radius and exponent, whereasmandA stand for the exponent and strength of the repulsivepotentialf(r i j ) @4#. Using these parameters we have calcu-lated the vibrational frequencies of the dimers, which com-pare reasonably well with the experimental values@10# andquantum-chemical calculations@2,3#. Then, we determinedthe equilibrium geometries of the ground states of the silvertrimers Ag3

2 , Ag3 , and Ag31 which again yielded excel-

lent agreement with the all-valence-electron calculations ofRefs.@2# and @3#. We have also determined a linear equilib-rium geometry for the excited state2Sg

1 of Ag3, which canbe reached by photodetachment of Ag3

2 @6#. However, inour calculations this state lies 1.2 eV higher than the groundstate~in relatively good agreement with photodetachment ex-periments@6#! and therefore cannot be excited by the laser

frequencies used in Ref.@1#. Therefore we consider in thefollowing only the dynamics of the ground state of Ag3.Note, our hopping parameters are comparable to those ofsilver bulk@7#. This gives another justification for the neglectof Coulomb interactions.

The MD simulations are performed applying the Verletalgorithm in its velocity form. We used a time step ofDt50.05 fs. This ensures an energy conservation up to1026 eV after 105 time steps. The equilibrium structureswere obtained by performing simulated annealing. Startingwith the equilibrium geometry of Ag3

2 , we generate anensemble of approximately 1000 clusters characterized bythe ensemble temperatureT, defined as the time average ofthe kinetic energy for a long trajectory (;106 time steps!@11#.

In Fig. 1~a! we show the time dependence of the fractionof Ag3 clusters p(hn,t) with ionization potentialsVi(t)<hn. We scalehn with Vi

0 , which is the minimal ion-ization potential of Ag3 in the equilateral equilibrium geom-etry of Ag3

1 . The quantityp(hn,t) can be interpreted as thetime-dependent probability for ionization with a laser pulseof frequencyhn ~i.e., half of this energy, if the ionization isachieved in a two-photon process as in Ref.@1#!. p(hn,t) isproportional to the signal detected in the experiment. Note,slightly above the ionization threshold (hn51.02Vi

0), clus-

FIG. 1. ~a! Time dependence of the fraction of clustersp(hn,t) with Vi(t) smaller thanhn. The sharp increase and theoverall time dependence ofp(hn,t) for increasinghn should becompared with the experimental results of Ref.@1#. The initial tem-perature wasT5317 K. ~b! Time dependence of the fraction ofclustersptriang(t) having triangular structures for different initialtemperatures.

R4602 54H. O. JESCHKE, M. E. GARCIA, AND K. H. BENNEMANN

ters can be ionized only after a delay timet0(hn).750 fs, atwhich p(hn,t) begins to increase.p(hn,t) displays the fea-tures observed in the experiment, namely, first a sharp in-crease, then a maximum, and finally a decrease to a smallervalue, which remains constant. There is, however, a narrowrange of frequencieshn, where the enhancement of the sig-nal after the sharp increase is largest. For higher laser fre-quencies the maximum becomes broadened and the enhance-ment factor is smaller.

These results can be understood physically as follows.Upon photodetachment of a binding electron vibrational ex-citations occur, in particular, those of the central atom alongthe chain direction. Due to the shape of the PES the motionof the central atom dominates the ultrashort-time responseover the first few hundred femtoseconds. Then, the slowerthermally activated bending motion comes into play andyields triangular bonded Ag3. The resultant bond formationis exothermic. The excess energy can in turn cause bondbreaking or, in case of uniform energy distribution, also aregular vibrational mode such as pseudorotations.

In order to demonstrate the temperature dependence of the

bond-formation and bond-breaking processes, we present inFig. 1~b! results for the time-dependent probabilityptriang(t) of finding triangular clusters for different initialtemperatures.ptriang(t) is defined as the fraction of clustershaving all angles between 45° and 90°. Comparison of thebehavior ofptriang(t) for T5317 K with the curves of Fig.1~a! clearly shows that only these particular cluster geom-etries contribute to the signal observed in the experiment@1#.Note that the onset of the abrupt increase ofptriang(t) showsa strong temperature dependence. For increasing initial tem-perature the clusters bend faster. Notice also the remarkablefact that at low temperature the maximum inptriang(t) dis-appears.

Figure 2 shows contour plots for the time development ofthe distribution of ionization potentials of neutral silver tri-mers after photodetachment att50. The most prominentfeature of these pictures is the coherent oscillation ofVi ofall clusters, which continues for about 700 fs in the case ofclusters starting atT529 K and which is destroyed muchmore quickly at the higher temperatures. These coherent os-cillations ofVi are due to the internal vibrations of the linearchain before and during bending. High values ofVi corre-spond to a rather symmetric geometry, while the lower val-ues stand for the turning points of the oscillation. The second

FIG. 2. Distribution of photodetached clusters as a function oftime andV i at different temperatures. Black regions indicate largenumber of clusters, whereas no clusters are present in the whiteparts. Note the progressive destruction of the coherent motion afterphotodetachment for increasing initial temperatures.

FIG. 3. ~a! Average distribution of neutral silver clusters overvalues ofVi for 2 ps<Dt<20 ps after photodetachment.~b! Cal-culatedp(hn,t) for a large value ofhn. Note that also the internalvibration of the chain before and during bending can be detected.~c! Spectral analysis of the pseudorotations. Note that the lifetime issmaller than one pseudorotation period.

54 R4603THEORY FOR THE ULTRAFAST STRUCTURAL . . .

feature to be noted is the areas in the plot with values ofVi below approximately 1.07Vi

0 , which become populatedonly after time delays of 1500, 700, and 300 fs, respectively,and which represent those clusters that have bent to formgeometries close to the equilibrium geometry of Ag3. ForT51048 K we observed fragmentation of 31% of the trim-ers ~after 20 ps), and in Fig. 2~c! we show only the distri-bution corresponding to the nonfragmented clusters.

With the help of Fig. 2 it is also possible to explain themaximum ofVi as a result of the coherent vibrational motionof the clusters: while a large fraction of the clusters bendcollectively, they still continue their vibrational motion,which results in a pronounced minimum ofVi in the sameway as for the unbent trimers. This effect accounts for thedark areas in Figs. 2~b! and 2~c! for Vi(t)/Vi

0 below 1.07approximately and is quantified by integration over energiesin the calculation ofp(hn,t) @Fig. 1~a!#. Since the clustersresponsible for this effect have gained an average energy of900 K while descending the PES towards the equilibriumstructure of Ag3, their coherent motion is quickly destroyedand this explains why no further maxima occur. This is sup-ported by the fact that the cluster ensembles starting out withtemperatures of 29 and 104 K do not show this maximum.Here, the coherent motion has already disappeared before theclusters bend toward the equilibrium geometry of Ag3.

Figure 2 reveals that for longer times (t.2000 fs) theclusters approach a time-independent distribution over thevalues of Vi or, equivalently, over the Ag3 geometries.These asymptotic distributions are plotted in Fig. 3~a!.

In Fig. 3~b! we have plotted the quantityp(hn,t) as inFig. 1~a!, but for an initial temperature ofT529 K and for ahigher energyhn51.17Vi

0 A new feature appears whichconsists of sharp oscillations of the probability between 0and 1 within the time range 0<t<700 fs. These oscillationsresult from the dominant vibrational mode of the trimers forthe first 700 fs approximately. This interesting effect wasnot observed in the experiment@1#, because the laser ener-gies used were not high enough. In order to demonstrate thatthe excited state2Sg

1 does not contribute to the signal ob-served in the experiment of Ref.@1#, we have performed MDsimulations on the corresponding PES and confirmed its one-dimensional dynamics. Its signalp(hn,t) consists of oscilla-

tions with no sharp increase such as that observed in Ref.@1#.However, it is important to point out that this structurelesssignal could interfere with the bending or stretching dynam-ics of the ground state if the state2Sg

1 were excited by thepump pulse.

The last important point to be investigated is the role ofpseudorotations. In order to study these vibrational excita-tions we determine the autocorrelation function

G~t!5(kqx~ tk!qx~ tk1t!

(kqx~ tk!qx~ tk!, ~3!

whereqx is one of the normal coordinates of the triangularAg3 molecule@12#. If the pseudorotations had a long life-time,G(t) would show an oscillatory behavior. However, asshown in the inset of Fig. 3~c!, the correlations are stronglydamped. This becomes clear by analyzing the Fourier trans-form of G(t). The lifetime of the pseudorotations, deter-mined from the width of the peak at half maximum, is onlyslightly lower than the duration of one cycle. This shows thatpseudorotations do not play an important role in the dynam-ics of the trimers. The high kinetic energy of the atomsseems to prevent the occurrence of a regular mode such aspseudorotations. The saturation of the signal is rather a sta-tistical effect induced by the temperature.

In summary, by employing an electronic theory and MDsimulations we have analyzed the femtosecond dynamics ofAg3 upon photodetachment. Our results show that an un-complicated theory that can easily be extended to deal withlarger systems is able to account for all experimental obser-vations. Most importantly, we find that the time scales forbond formation and breaking are affected by temperature.Our results might also be of general interest regarding lasercontrol of chemical reactions. Figure 3~a! suggests that tem-perature can be used to design special distributions for theionization potentials or for the cluster geometries present inthe initial ensemble. Figure 3~b! immediately suggests newexperiments with a probe laser pulse energy ofhn51.17Vi

0 . Thus a direct measurement of the dominantvibration of the trimers seems possible.

This work was supported by the Deutsche Forschungsge-meinschaft.

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R4604 54H. O. JESCHKE, M. E. GARCIA, AND K. H. BENNEMANN