Dynamics of structural relaxation upon Rydberg excitation of an impurity in an Ar crystal

Ž .Chemical Physics 233 1998 343–352

Dynamics of structural relaxation upon Rydberg excitation of animpurity in an Ar crystal

Sonia Jimenez a,b, Alfredo Pasquarello b,c, Roberto Car b,c, Majed Chergui a

a Institut de Physique Experimentale, UniÕersite de Lausanne, CH-1015 Lausanne, Switzerland´ ´b ( )Institut Romand de Recherche Numerique en Physique des Materiaux IRRMA , Ecublens, CH-1015 Lausanne, Switzerland´ ´

c Departement de Physique de la Matiere Condensee, UniÕersite de GeneÕe, CH-1211 GeneÕe, Switzerland´ ` ´ ´ ` `

Received 12 December 1997

Abstract

We study the ultrafast dynamics of structural relaxation induced by the Rydberg excitation of an NO molecule in an Arcrystal. We used classical molecular dynamics simulations and normal mode analysis to describe the dynamics of the cage ofAr atoms surrounding the NO molecule. The results show a behaviour characterized by an impulsive expansion of the cage

˚Ž .radius at short times F 250 fs , followed by multimodal oscillations over several picoseconds around a radius of ;4 A.This corresponds to a dilatation of the ground state cage radius of ;9%. The dynamics show a high degree of nuclearvibrational coherence. The relaxation process is described by the damping of mainly five vibrational modes. Theirfrequencies range from 20 to 75 cmy1 and correspond to resonant modes of the crystal. The associated lifetimes range from0.5 to 16 ps. The mode of highest frequency being the most anharmonic. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction

Extensive configurational rearrangement follow-ing absorption of light are operative in a large classof many-body systems such as biological molecules,condensed phase chemical media and in solids such

w xas insulators and semiconductors 1 . Pure and dopedrare gas media have long been recognized as modelsystems for describing and understanding the basicprinciples behind configurational rearrangementsw x2,3 . Indeed, their simple structural properties andthe good knowledge of their physical and thermody-namic properties make them more easily amenable tomodelization. Over the ten past years or so, molecu-

lar dynamics simulations have been used to describeŽsimple photochemical reactions e.g. photo-dissocia-

. w xtion in rare gas liquids and fluids 4–7 , solidsw x w x6,8–11 and clusters 8,9,12,13 . In such photochem-ical events, it is the photoinduced intramolecularmotion which induces nuclear dynamics of the sur-rounding cage. Another way by which nuclear dy-namics of the surrounding species is induced in-volves for instance the charge redistribution of anexcited center in a given medium. This leads tosevere configurational changes as in the case of color

w xcenter in alkali halide solids 14 or in the case ofatoms and molecules excited to their Rydberg states

w x w xin rare gas 1,2,15,16 or van der Waals solids 17 . It

0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0301-0104 98 00154-2

( )S. Jimenez et al.rChemical Physics 233 1998 343–352344

is known that the excitation of low-n Rydberg statesof impurities in rare gas solids leads to a large bluespectral shift in absorption, as compared to the gasphase. This is due to the strong short-range repulsionbetween the Rydberg electron and the closed shell of

w xthe rare gas atoms 2,16 . The strong repulsion in-duces a relaxation of the cage surrounding the ex-cited center to a new equilibrium configuration. Fromthis new configuration, fluorescence takes place. Thelarge observed absorption-emission Stokes shifts aresignatures of extensive lattice rearrangements aroundthe excited species. The basic mechanism is consid-

Žered to be a radial expansion of the cage the so-called. w xelectronic ‘‘bubble’’ formation 1,2,15,16 and is

w xalso operative in rare gas liquids and clusters 1,2,18 .Molecular dynamics simulations of ‘‘bubble’’ for-

mation in Xe-doped Ar clusters upon photoexcita-n

tion of the lowest dipole-allowed Rydberg state ofXe have been carried out by Jortner and co-workersw x19–21 . Both specific size and site effects wereidentified and analyzed. In the case of large clustersŽ .ns146 , they established two general configura-tion relaxation phenomena: a ‘‘bubble’’ formation

Žfor interior sites and a ‘‘spring’’ formation stretch-.ing of the excited Xe atom outside the cluster for

the surface site. Jortner and co-workers identify fourinequivalent interior sites, among which the so-calledcentral site is the most relevant to the solid phase.Vertical excitation of such sites results in an impul-sive expansion of the surrounding cage atoms in;170–280 fs. Thi is followed by a complex oscilla-tory behaviour of the cage in the 0.2–2 ps range anda further cage expansion for times larger than 2 ps.This later expansion is related to a center-of-massdiffusion of the Xe atom in the cluster. These simula-tions were used to interpret the absorption and fluo-rescence spectra of the Xe-doped Ar clusters ob-

w xtained by Moller and co-workers 18 .¨The cage relaxation upon Rydberg state excitation

of impurities trapped in rare gas and van der Waalssolids, has been intensively studied over the past few

w xyears 2,15–17,22,24,25 . Chergui and co-workershave mainly investigated the case of NO-doped rare

w x w xgas solids 16,23,24 and H solids 17,25 . In the2

electronic ground state X 2P, the NO molecule pos-

sesses one unpaired electron in the 2 pp orbital,which has a non spherical geometry. Promotion ofthis electron to the 3ss orbital yields the first Ryd-

berg state A2S

q. In the Rydberg state, the electronreaches an extension of the order of the nearestneighbour distance. By virtue of the Pauli exclusionprinciple, a strong repulsion occurs between the elec-tronic closed shell of the matrix atoms and the NO3ss electronic orbital. In the case of Ar matrices,this leads to a blue shift of ;0.86 eV for the

2 2 q w xX P™A S transition 23 and to an absorption-w xemission Stokes shift of 0.58 eV 16 . Indeed, the

Ž .NO 3ss –Ar distances increase dramatically until anew equilibrium distance is reached. This process isachieved by the creation of phonons in the crystaland gives rise to the formation of a microcavityaround the excited molecule, thus the strong Stokesshift.

Chergui et al. performed a moment analysis of theabsorption and emission lineshape in the configura-tion coordinate model and in the harmonic approxi-

w xmation 16 . It indicates that the ‘‘bubble’’ formationin the excited state corresponds to an increase by;10% of the ground state NO–Ar distance and that;330 meV are dissipated in the form of phonons.The analysis also showed that effective phonons of5.8 meV energy are involved in the dissipation ofenergy. This leads to Huang–Rhys factorsŽ .electron–phonon coupling strengths of ;60 inexcitation and ;40 in emission. Such factors are thelargest so far reported in the literature and, as already

w xpointed out 16 , they probably overshadow the con-tribution of anharmonic effects which are not consid-ered in the model. Since the moment analysis isbased on the effective mode and the harmonic ap-proximations, it can only give an approximate de-scription of the mechanisms of lattice relaxation andenergy dissipation. By combining molecular dynam-ics simulations and a normal mode analysis, whichare the object of the present contribution, moreinsight can be obtained. This contribution presents astudy of the dynamics of structural relaxation arounda Rydberg excited NO molecule embedded in an Arcrystal. Our aim is to gain insight into the fundamen-tal processes by considering an idealized situation.We are interested in describing the immediate re-sponse of the crystal upon photoexcitation. We alsodescribe the lattice modes therein involved and thesubsequent exchange of energy between the excitedcenter and the lattice. This work is motivated by theabove-mentioned experimental studies by Chergui

( )S. Jimenez et al.rChemical Physics 233 1998 343–352 345

and co-workers using steady state spectroscopyw x16,17,23 and femtosecond pump-probe spec-

w xtroscopy 24,25 .

2. Methodology

2.1. Intermolecular potentials

All intermolecular interactions were modeled byŽ .Lennard-Jones LJ pair potentials:

12 6s si j i j

V R s4e y ,Ž .i j i j i j ž / ž /R Ri j i j

5 5R s r yr , 1Ž .i j j i

where e and s are the LJ parameters and R isi j i j i j

the intermolecular distance between particles i andj.

The potential parameters for the Ar–Ar,Ž 2 . Ž 2 q.NO X P –Ar and NO A S –Ar interactions are

given in Table 1. In the ground state, the nonspherical orbital of the 2 pp electron implies ananisotropic interaction between the NO molecule andthe Ar atom, as shown by the crossed beam studies

w xof Thuis et al. 26 . Even though the angular part isŽnot negligible in the ground state it amounts to

w x.about 20% of the isotropic part 26 , it was ne-glected in this work. Indeed, we treat the NO

Ž .molecule as an atom on the grounds that: a thedramatic effects induced by the electronic excitationon the energy and the structure are assumed to belittle affected by the anisotropy of the ground state

Ž .intermolecular potential, b the intramolecular NOvibrations need not be considered because at low

Ž .temperature ;4 K , the vibrational levels of theŽ .NO molecule are not populated and, c the

Ž 2 q.NO A S –Ar interaction in the first Rydberg statecan be approximated as isotropic because the Ryd-

Table 1Lennard-Jones pair potential parameters.

˚Ž . Ž .e eV s Ai j i j

w xAr–Ar 30 0.0104 3.402Ž . w xNO–Ar X P 26 0.0119 3.23

2 qŽ . w xNO–Ar A S 27 0.0073 4.09

berg electron is in a nearly spherical 3ss orbital.Ž 2 .The NO X P –Ar potential parameters for the iso-

w xtopic part were taken from Thuis et al. 26 . TheŽ 2 q.NO A S –Ar pair potential was modeled by Tsuji

w xet al. 27 in terms of an isotropic LJ potential fittedto spectroscopic data of bound-bound and bound-freetransitions between the X 2

P and the A2S

q state ofNO complexed with one Ar atom.

2.2. Molecular dynamics

In order to study the dynamical response of the Arcrystal upon electronic excitation of the NO impu-

Ž .rity, we used classical molecular dynamics MD .At the experimental temperature of 4 K, the nu-

clear motion of the Ar atoms is dominated by zero-point motion. Bergsma et al. showed how thesequantum effects can approximately be included inclassical MD by using a higher effective temperaturew x29 . We follow the same approach here. This ap-proach was successfully used by Zadoyan et al. tomodelize the dissociation and caging dynamics of I2

w xin solid Ar and Kr 10 .Our MD simulations were performed at constant

w xenergy using the velocity-Verlet algorithm 28 . Theclassical trajectories were integrated with a time-stepof 10.18 fs in the X 2

P state and 5.09 fs in theexcited A2

Sq state of the NO molecule. In order to

speed up the calculations, the interactions were trun-cated at a cutoff distance which corresponds to an

˚interatomic distance of 12 A. A smooth function wasintroduced in order to preserve a continuous dynam-ics at the cutoff distance.

The simulated system consists of a f.c.c. supercellstructure with 499 Ar atoms and one NO moleculeplaced at a substitutional site. Periodic boundaryconditions were used to simulate an infinite crystal.The size of the supercell contains 10 shells of inde-pendent neighbours corresponding to 200 particles.

The procedure was based on the following steps:1. We first altered the atomic positions by random

shifts of the atoms and introduced initial veloci-ties. The velocities were then adjusted in order toachieve zero total initial momentum.

2. The next step was the thermalization of the sys-tem in the ground state. In order to model theexperimental system at Ts4 K, we scaled the


velocities to an effective temperature T X given,w xaccording to Bergsma et al. 29 , by

y1"v "v

XT s tanh . 2Ž .ž /ž /2k 2k TB B

Ž y1 .Using the Debye frequency of solid Ar 75 cm ,an effective temperature of T X s49 K was re-quired for the thermalization of the system. This

Žgave an initial configuration positions and veloci-.ties for the integration of trajectories in the mi-

crocanonical ensemble. Thus, the temperature ofthe resulting MD was 49 " 3 K.

3. The vertical transition was induced by instanta-neous switching from the ground state potentialenergy surface to the excited one.The temporal evolution is given by the numerical

integration of the equations of motion for a systemof Ns500 interacting particles described by theHamiltonian:

N 2piH r , p s qV r , . . . ,r , 3Ž . Ž . Ž .Ý 1 N2miis1

whereN N

1V r , . . . ,r s V R . 4Ž . Ž .Ž .Ý Ý1 N i j i j2is1 js1, j/i

Ž . Ž .V R is given by Eq. 1 .i j i j

The nuclear motion in response to the excitationis visualized by the temporal evolution of the cageradius:

121 5 5

)R t s r t yr t , 5Ž . Ž . Ž . Ž .Ý j NO2js1

which corresponds to the mean distance betweenŽ 2 q. ŽNO A S and the first shell of Ar atoms 12.atoms .

2.3. Normal mode analysis

To complement the MD simulations, we alsostudied the vibrational properties of the system per-forming a normal mode analysis. We calculated the

w x3N=3N dynamical matrix 30 :

1 E 2UD s , 6Ž .i ja b ž /E u E um m( i ji j a b u s0;u s0i ja b

U is the total potential energy and u is the dis-ia

Ž .placement of the i-th particle i, js1, . . . , N alongŽone of the three Cartesian coordinates a ,bs

.1, . . . ,3 . Diagonalization of D gives the vibra-i ja b

Ž .tional eigenfrequencies v ns1, . . . ,3N and theirn

corresponding normalized eigenmodes E n.The structural change following the electronic

excitation can be described in terms of a distributionof vibrational modes in the Rydberg excited state. Tothis end, we defined the vector:

D rsr yr s D r , . . . ,D r , 7Ž . Ž .ground excited 1 N

corresponding to the displacement of the particles’positions in the relaxed ground state with respect tothose in the relaxed excited state. In order to distin-guish the modes generated by the photoexcitation,we projected D r on the eigenmodes of the excitedsystem:

3N nEia

D r s A , 8Ž .Ýi na m( ins1

where the A coefficients are dimensionless. ThenŽ .density of excited modes in the Rydberg state r v

is then given by

< < 2r v s A d vyv . 9Ž . Ž . Ž .Ý n nn

The anharmonic contribution for a given eigen-mode was estimated by comparing the potential en-ergy obtained in the harmonic approximation with

Ž .the exact potential energy given by Eq. 4 . WeŽ .introduced the displacement D r n,l of the parti-ia

cles as a function of a parameter l such that:3N

D r s D r n ,ls1 , 10Ž . Ž .Ýi ia a

ns1

where

Enia

D r n ,l sl A and 0FlF1. 11Ž . Ž .i na m( i

Ž .D r n,ls1 corresponds to the contribution of theia

Ž .mode v to the displacement D r given in Eq. 7 .n

Every vibrational mode v can be considered sepa-n

rately as a function of l. The total potential energyin the harmonic approximation, U harm, is given byn

1harm 2 2 2U l sU q v l A , 12Ž . Ž .n 0 n n2

LJŽ .and can be compared to the potential energy U ln


Ž . 1as derived from Eq. 4 . The anharmonic contribu-tion in the vibrational mode v is then characterizedn

by the coefficient:

U harm ls1 yU LJ ls1Ž . Ž .n nb s , 13Ž .n harmU ls1 yUŽ .n 0

U being the cohesive energy of the crystal.0

3. Results and discussion

The large absorption gas-to-matrix shift character-izes the strong short-range repulsion between theexcited NO molecule and its environment. The con-sequence of this repulsion is the structural relaxationwhose signature is the Stokes shift. It is thereforeimportant to see how the simulation reproduces theseexperimental quantities. We estimated the verticaltransition energies for the X 2

P–A2S

q transition ofNO in solid Ar by using the classical Franck princi-ple. A long trajectory was integrated in the thermal-ized ground state. At each time-step, we calculatedthe total potential energy V . For the same configura-g

tion, we also calculated the total potential energy Ve

in the first Rydberg state. We obtained a verticalabsorption energy of 6.3 eV by taking the timeaverage of the energy difference V yV . This im-e g

plies a blue gas-to-matrix shift of 0.8 eV for theX 2

P™A2S

q transition, quite close to the experi-w xmental value of 0.86 eV 23 . The same method was

used to evaluate the vertical emission energy, yield-ing 5.6 eV. The corresponding Stokes shift is ;0.7eV, in fair agreement with the experimental value of

w x;0.58 eV 16 . The Stokes shift gives the totalenergy released to the crystal in one absorption-re-laxationremission-relaxation cycle. We found thatthe calculated Stokes shift results from an energydissipation of 430 meV in the excited state afterabsorption and of 270 meV in the ground state afteremission. The approximate agreement between ex-perimental and simulated shifts is an indication thatthe potentials used in this work, albeit simple, pro-vide a sufficiently correct description of the system

1 LJŽ . wŽ Ž ..12 Ž Ž ..6 xU l sÝ 4e s rR n,l y s rR n,l ,i j i/ j i j i j i j i j i jŽ . 5Ž Ž .. Ž Ž ..5

) )R n,l s r yD r n,l y r yD r n,l .i j NO i NO j

to justify a detailed study of the dynamics of cageformation.

3.1. Structural relaxation

Ž .The mean cage radius, R t is obtained by aver-Ž .aging the expression given in Eq. 5 over 100

atomic trajectories. This is plotted in Fig. 1 as afunction of time. The oscillatory multimode evolu-

Ž .tion of R t implies a high degree of nuclear vibra-w Ž .xtional coherence. In the short time regime Fig. 1 b ,

˚an impulsive increase of the cage radius from 3.69 AŽwhich corresponds to the equilibrium NO–Ar dis-

˚.tance in the ground state to 4.23 A takes place inabout 250 fs. This expansion of the cage radiusresults from the strong short-range repulsion betweenthe Rydberg excited molecule and the surroundingAr atoms. The perturbation induced in the crystal bythe excited NO molecule displaces the Ar atoms far

Ž . Ž .Fig. 1. a Time evolution of the mean cage radius R t following2 2 q Ž . Ž .the X P™A S transition of NO in Ar matrices. b R t

Ž . Ž . Ž .thick line and fit with five damped oscillators R t thin linefitŽ .given in Eq. 14 .


from equilibrium. After 250 fs, the motion of the Aratoms in the first shell is reversed due to theirinteractions with the next shells of atoms and the

˚cage compresses back to a radius of ;4 A. Thisfirst response of the cage takes place in ;500 fs.

The above ultrafast configurational rearrangementis followed by a complex oscillatory pattern around

˚the average NO–Ar distance of ; 4 A, which is theequilibrium cage radius in the Rydberg state asconfirmed by the MD evolution of the system in theexcited state. The amplitude of the oscillations isdamped in the course of time, as the system isundergoing relaxation. The size of the cage radius of

˚;4 A in the Rydberg state corresponds to an in-crease of ;9% of the initial cage radius. Thisincrease is accompanied in our finite system by arise in temperature of ;4 K.

Ž .The power spectrum of the time evolution of R tis shown in Fig. 2. The vibrational modes involvedin the relaxation process have frequencies of up to80 cmy1. They are characterized by bands withfrequencies peaking around 20 cmy1, 32 cmy1, 43cmy1, 50 cmy1 and ;80 cmy1. The narrow band-width of the lowest frequency mode at 20 cmy1

suggests that it is longer lived than the higher fre-Ž .quency modes. This also appears in Fig. 1 a : the

high frequency oscillations are damped out rapidly,while the low frequencies modes survive longer.

The behaviour of the mean cage radius and theform of its power spectrum suggest a structuralrelaxation in terms of damped oscillators:

Nyt rt jR t s A cos v qa e , 14Ž . Ž . Ž .Ýfit j j j

js1

where the jth mode is characterized by the amplitudeA , the frequency v , the phase a and the relaxationj j j

Ž .time t . Indeed the temporal evolution of R t canjŽ .be very well represented by Eq. 14 . A fit using thew xLevenberg–Marquardt algorithm 31 gives very good

results with Ns5. The frequencies of these fiveoscillators correspond to the five main bands of theŽ .R t spectrum in Fig. 2. The parameters of the fit are

reported in Table 2. In the fit, we considered onlyŽ .the first 30 ps of the R t evolution, since beyond

this time the oscillations are essentially due to ther-Ž . Ž .mal motion. In Fig. 1 b , the resulting function R tfit

Ž .is compared to the calculated R t for the first 10 ps,illustrating the quality of the fit. The power spectrum

Ž .of R t is given in Fig. 2 and agrees very closelyfitŽ .with the power spectrum of R t . The long tailŽextending at high frequencies beyond the Debye

Ž . Ž . Ž . Ž .Fig. 2. Power spectrum of the time evolution over 50 ps of the mean cage radius R t thin line and of the fit R t thick line .fit


Table 2w Ž .xParameters for the fit Eq. 14 of the mean cage radius with five

damped oscillators.

y1˚Ž . Ž . Ž . Ž .j A A v cm a rad t psj j j j

y21 4.1=10 21 3.54 15.6y22 8.7=10 34 3.48 2.6y23 6.2=10 43 4.15 2.8y24 2.6=10 49 3.94 6.2y25 19.8=10 74 3.15 0.5

.cutoff of the crystal results from the very shortlifetime of the mode at 74 cmy1. Because of its largeamplitude, the mode at 74 cmy1 plays an importantrole in the relaxation process. Its very short relax-ation time of ;500 fs indicates that after the firstcage expansion and contraction, it has transferredmost of its energy to the rest of the crystal. Themodes at 32 cmy1 and 43 cmy1 still have sizeableamplitudes but have longer relaxation times of 2.6and 2.8 ps, respectively. The remaining modes at 21

Fig. 3. Density of vibrational states of the NO-doped Ar crystalŽ . Ž .a in the ground state and b in the excited state. A Gaussianbroadening of FWHMs8 cmy1 is used for plotting the fre-quency lines. The dotted lines correspond to the contribution ofthe NO motion.

Ž . Ž . Ž .Fig. 4. a Density of excited vibrational states r v thick line .Ž .The contribution of the first shell of Ar atoms thin line together

with the cumulative contribution of the first ten shells of Ar atomsŽ . Ž . Ž .thick dashed line are also shown. b Radial thick line and

Ž .tangential thin line contributions to the excited vibrational modesŽ .in the first 10 shells of Ar atoms thick dashed line .

cmy1 and 49 cmy1 have smaller amplitudes andmuch longer relaxation times. Hence, we can distin-

Ž . Žguish two regimes: a an ultrafast relaxation F3.ps resulting from the fast damping of the modes of

Ž .largest amplitude and, b a slower relaxation result-ing from the damping of the modes of smallestamplitudes.

When comparing our results with those of Gold-w xberg and Jortner 21 for Xe-doped Ar clusters at146

Ž .10 K, we may notice the following: a Contrary tothe cluster case we do not see diffusion of theexcited center in the present case. This is due to the

w xfact that bulk solids are more rigid than clusters 32 .Ž .b The first bubble expansion of the cage of Aratoms takes place in about 250 fs both in the crystaland in the cluster. In the crystal, this bubble expan-sion can be described as a damped oscillatory modewith a lifetime of 500 fs. However, in the cluster,such an oscillatory behaviour is not observed and thetime scale only refers to the expansion of the cage


Ž .radius. c In qualitative agreement with our findingsfor the crystal, the cage relaxation in the cluster isdominated by two modes at 28 cmy1 and 72 cmy1

with damping constants of 6.1 ps and 2.1 ps, respec-tively.

3.2. Normal mode analysis

We now clarify the origin of the vibrational modesinvolved in the relaxation process on the basis of thenormal mode analysis described in Section 2.3. Thecalculated vibrational density of states of the dopedAr crystal is reported in Fig. 3, where we used aGaussian broadening with FWHMs8 cmy1. In Fig.Ž .3 a , we show the density of states obtained when

the NO impurity is in the ground state, while weŽ .report in Fig. 3 b the density of states corresponding

to the NO impurity in the excited state. The contribu-tion of NO motion to the vibrational modes is alsoshown in the figure as a dotted line. When NO is in

w Ž .xits ground state Fig. 3 a , the vibrational motion ofNO is characterized by a broad distribution centered

y1 w Ž .xat ;35 cm . In the Rydberg state Fig. 3 b , theNO vibrations are characterized by a localized modeat frequency ;105 cmy1 and by a broad distribu-tion of modes within the vibrational spectrum of thecrystal.

Ž . w Ž .xIn Fig. 4, we present the density r v Eq. 9 ofthe excited vibrational modes following the X 2

P™

A2S

q electronic transition. Again, a Gaussian

Fig. 5. Potential energy of the vibrational modes with frequencies: v s20 cmy1 , v s32 cmy1 , v s46 cmy1 , v s52 cmy1 and1 2 3 4y1 Ž .v s75 cm as a function of the displacement parameter l, within the harmonic approximation solid line and using the exact LJ5

Ž .potential dashed line . The left and top scales refer to the mode v , while the right and bottom scales to the modes v , v , v and v .5 1 2 3 4


broadening with FWHMs8 cmy1 was used. In Fig.Ž .4 a , the thick line is obtained by projecting the

w Ž .xdisplacement vector D r Eq. 7 onto the normalw Ž .xmodes of the excited system Eq. 8 . It can be seen

Ž .that the main vibrational modes in r v have fre-quencies: v s20 cmy1, v s32 cmy1, v s461 2 3

cmy1, v s52 cmy1 and v s75 cmy1. There is a4 5

good correspondence between these frequency modesand the main frequencies in the power spectrum ofŽ . Ž .R t Fig. 2 . Some vibrational modes of the impu-

Ž .rity-doped crystal do not appear in Fig. 4 a , mostnotably the localized mode at 105 cmy1. This is aconsequence of the high symmetry of the initial

Ž .configuration used to define D r in Eq. 7 , whichcorresponds to the ground state of the system atTs0 K. Indeed if we give a small random displace-ment to the initial configuration, a weak contributioncorresponding to the vibrational modes missing in

Ž .Fig. 4 a appears.We can also examine more closely the vibrational

motions of the different shells of Ar atoms aroundŽ .the NO impurity. Fig. 4 b shows the decomposition

of the motion of Ar atoms into radial and tangentialcontributions. It appears that radial contributionsŽ .thick line largely dominate the density of excited

Ž .modes r v . This is particularly the case at thefrequency of 75 cmy1, which corresponds to a

Ž .breathing mode. In Fig. 4 a , the thin line shows theŽ .contribution of the first shell of Ar atoms to r v . It

results that the first shell largely contributes to thebreathing mode at 75 cmy1. Furthermore, our analy-sis reveals that the 4th and the 9th shells also give animportant contribution to the mode at 75 cmy1. Thiscan be understood as a consequence of the f.c.c.symmetry of the Ar crystal: these shells containatoms that are situated on the axis of the NO moleculeand the first shell of Ar atoms. Therefore the motionof the 4th and the 9th shells is directly connected tothe motion of the first shell.

We now consider the anharmonic contributions tothe main vibrational modes. We compare the total

harm Ž . w Ž .x LJŽ . 2potential energies U l Eq. 12 and U ln n

for the following frequencies: v s20 cmy1, v s1 2

32 cmy1, v s46 cmy1, v s52 cmy1 and v s753 4 5

2 See footnote 1.

cmy1. The energies in Fig. 5 are given with respectto the cohesive energy of the crystal. In order toquantify the anharmonic contribution, we computed

Ž .the coefficient b defined in Eq. 13 . We found:n

b s1.7%, b s0.2%, b s4.8%, b s2.4% and1 2 3 4

b s8.6% for the five frequencies given above. In5

particular this indicates that the highest frequencymode v s75 cmy1 is the most anharmonic mode.5

This also relates to the fact that it has the shortestlifetime.

4. Conclusion

Molecular dynamics simulations and normal modeanalysis of electronic bubble formation have beencarried out on a model system consisting of solid Ardoped with an NO impurity. The time evolution ofthe average cage radius reveals an impulsive cageexpansion followed by a cage contraction in about500 fs. The system then undergoes oscillations aroundan average cage radius 9% larger than the groundstate cage radius. The bubble formation is dominatedby five resonant modes whose lifetimes range be-tween 0.5 and 16 ps. The fastest relaxation processcorresponds to a breathing mode motion of the cage.An important finding of this work is that the cageevolution is characterized by a high degree of nu-clear coherence. Indeed, after performing these simu-lations, coherent cage oscillations have been ob-served in femtosecond pump-probe experiments on

w xNO-doped Ar crystals 33 .

Acknowledgements

We wish to thank Prof. J.-F. Loude for providingfitting programs and the referee for valuable sugges-tions. This work was supported by the Swiss Na-tional Science Foundation under grant N821-47100.96. Most of the calculations were performed

Ž .at the Swiss Center for Scientific Computing CSCSin Manno.

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Dynamics of structural relaxation upon Rydberg excitation of an impurity in an Ar crystal