Surrogate Hamiltonian study of electronic relaxation in the...

JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 3 15 JULY 2003

Surrogate Hamiltonian study of electronic relaxation in the femtosecondlaser induced desorption of NO ÕNiO„100…

Christiane P. Koch,a) Thorsten Kluner,b) and Hans-Joachim FreundFritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany

Ronnie KosloffDepartment of Physical Chemistry and The Fritz Haber Research Center, The Hebrew University,Jerusalem 91904, Israel

~Received 11 February 2003; accepted 2 April 2003!

A microscopic model for electronic quenching in the photodesorption of NO from NiO~100! isdeveloped. The quenching is caused by the interaction of the excited adsorbate–substrate complexwith electron hole pairs (O 2p→Ni 3d states! in the surface. The electron hole pairs are describedas a bath of two level systems which are characterized by an excitation energy and a dipole charge.The parameters are connected to estimates from photoemission spectroscopy and configurationinteraction calculations. Due to the localized electronic structure of NiO a direct optical excitationmechanism can be assumed, and a reliable potential energy surface for the excited state is available.Thus a treatment ofall steps in the photodesorption event from first principles becomes possible forthe first time. The surrogate Hamiltonian method, which allows one to monitor convergence, isemployed to calculate the desorption dynamics. Desorption probabilities of the right order ofmagnitude and velocities in the experimentally observed range are obtained. ©2003 AmericanInstitute of Physics.@DOI: 10.1063/1.1577533#

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I. INTRODUCTION

A molecule which is adsorbed on a surface can betached by thermal or photoexcitation. For chemisorbed mecules, desorption involves the cleavage of a chemical band it represents the simplest case of a chemical reacSince molecule and surface degrees of freedom are couenergy is transferred from the surface to the moleculevice versa. The nature of the excitation process reveals iin the observables of the desorption experiment: Therdesorption results in the distribution of energy onto all dgrees of freedom while for laser induced desorption thecitation of specific quantum states resulting in a nontherdistribution of energy can be observed. The theoreticalscription of laser induced desorption must therefore be qutum mechanical and start from first principles. In particulall aspects of the problem should be treated on the sameof rigor.

The photodesorption process consists of three stepexcitation of the molecule due to the interaction with tlaser pulse, relaxation of the excitation energy into the sface, and subsequent desorption. In previous treatments,1 theemphasis has been placed on the molecular dynamics.the goal of this paper to go one step further and also treadynamical interaction between molecule and surface onab initio level. To this end, the surrogate Hamiltoniamethod2,3 to model the dissipative quantum dynamicscombined withab initio potential energy surfaces obtainedprevious work.4,5

Two mechanisms for laser induced desorption based

a!Electronic mail: koch–[email protected]!Electronic mail: [email protected]

1750021-9606/2003/119(3)/1750/16/$20.00

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short-lived electronically excited states have besuggested.1 The Menzel–Gomer–Redhead~MGR! model6,7

assumes the electronically excited state to be repulsive, win a variation of the MGR model going back to Antoniewic8

the excited state is bound. In both models, the excitatfrom the electronic ground to an excited state and the reation back to the ground state are modeled as vertical tsitions of a classical particle or a quantum mechanical wpacket. If an ensemble of wave packets is considered9,10

each wave packet spends a certain residence time onexcited state. Expectation values can then be computestochastic averages of the ensemble where the resontime, i.e., the lifetime of the excited state, enters as a weiThis procedure is usually termed Gadzuk’s wave pacjumping. A quantitative theoretical description has tobased on anab initio treatment of the participating potentiaenergy surfaces and excitation and deexcitation mechani

The calculation of reliable potential energy surfacesgeneral, and for excited states in particular, is still an opproblem. However, for NO/NiO~100!4,5 excited state potentials were obtained on anab initio level. The topology of therepresentative excited state potential for NO/NiO~100! whichwas used in the calculations will be discussed in Sec. II B

Irradiation by nanosecond pulses can well be describy a Franck–Condon transition. The theoretical descriptof femtosecond experiments, however, requires an impromodel since excitation, excited state dynamics, and reation all occur on the same time scale. For metals, the ttemperature model1 has been introduced to describe femtsecond excitation of the surface. The excitation mechanis assumed to be substrate-mediated, i.e., the pulse genea cloud of hot electrons which can attach to or scatter fr

0 © 2003 American Institute of Physics

license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

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1751J. Chem. Phys., Vol. 119, No. 3, 15 July 2003 Electronic relaxation in laser induced desorption

the adsorbate. On oxides, the substrate is also involved inexcitation process. This can be seen in the linear dependof the desorption cross section on the laser energy oncelaser energy is larger than the band gap.11 The excitationmechanism can, however, be thought of as directwithin thesubstrate–adsorbate complex as will be discussed inII C. Then the full time-dependence of the pulse enterstheoretical model.

The electronic excitation of the adsorbate is dissipainto the surface due to interaction with surface electronsholes or phonons. The lifetime of the electronic excitationextremely short.12 For metals it is estimated to bet&1 fswhile for oxides it is assumed to be somewhat largert'20– 30 fs. If the interaction with charge carriers in the sface is seen as the primary cause of relaxation, this difence corresponds to the different density of states in meand insulators. In both cases the short lifetime leads toconclusion that the interaction with the surface muststrong.1 If a fully quantum mechanical description of thproblem is desired, an open quantum system approshould be used to describe the relaxation process. Theface electron–hole pairs and phonons are then modeleenvironment or bath.

There are two standard approaches to the problemopen quantum systems, perturbation theory and the dyncal semigroup formalism. In both cases, the total Hamtonian is separated into parts describing the~primary! systemand the~secondary! bath,HS and HB ,

Htot5HS1HB1HSB. ~1!

In a perturbation theory treatment,13,14 the coupling betweensystem and bath,HSB, is assumed to be weak. Equationsmotion for the reduced density operator, i.e., the densityerator of the systemrS , can then be derived which depenupon system operators only. The derivation in a most gensense is done by the projection operator technique.14,15 Thedrawback of this reduction is the occurrence of memoryfects describing the correlations between system and bais well known, however, that the dynamics of the reducdensity operator does not obey complete positivity.16 Thisobscures the interpretation of diagonal matrix elements orS

as probabilities. The condition of complete positivity tgether with the Markov assumption is the starting pointthe second approach. These two conditions are fulfilled ifLiouville superoperator generating the dynamics ofrS is ofso-called Lindblad form.17,18 Dissipation is modeled by system operators which have to be chossemi-phenomenologically.19

Both perturbation theory and semigroup formalism leto an equation of motion for the density operator of the stem which needs to be solved. A description based onsystem wave function with a more favorable scaling innumerical solution is also possible. The influence of the bon the system is then treated as a stochastic force andmethod is hence termed the stochastic wave packet or MCarlo wave function~MCWF! method.20–23 The MCWFmethod was shown to be equivalent to the semigroformalism.20–22 For laser induced desorption, the MCW


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method is furthermore equivalent to Gadzuk’s wave pacjumping if a coordinate-independent relaxation rateassumed.24,25

While the Markov approximation is intrinsic for thsemigroup approach, the equation of motion obtained iperturbation theory approach in general is non-MarkoviSince the solution of integro-differential equations is ffrom being straightforward, the Markov approximation usally needs to be made. It is, however, not necessarily justiin the case of moderate or strong coupling between sysand bath which can be expected for laser induced desorpWithin second-order perturbation theory different approacexist to treat the memory kernel in the reduced equationmotion. One possibility consists in transforming the integdifferential equation into an algebraic equation by expandthe reduced density operator in a suitable polynomial bae.g., Laguerre polynomials.26 The method is limited, how-ever, to weak field excitation. If the interaction with stronexternal fields is to be considered, the non-Markovian eqtion of motion for the system density matrix can be tranformed into a set of coupled Markovian equations for treduced density matrix and auxiliary density matrices whincorporate the memory effects.27 This method is best suitedfor high temperature calculations. No further assumptioother than the weak coupling approximation need tomade.27

The path integral formalism supplies a different aproach to open system quantum mechanics.28–30 For a lin-early coupled bath of harmonic oscillators, the bath variabcan analytically be integrated. The effect of the bath is thcaptured in an influence functional.28 Non-Markovian effectsresult in an influence functional containing correlationstime between different paths,31 i.e., they can in principle beaccounted for. However, the numerical evaluation of a mtidimensional integral remains computationally challengidespite recent progress.31 This limits the applications of pathintegrals to relatively simple model systems. Furthermothe treatment of time-dependent Hamiltonians is not psible.

The theoretical description of femtosecond laser indudesorption thus poses a dilemma: Strong interaction withenvironment excludes perturbation theory. The impossseparation of excitation and relaxation time scales manon-Markovian effects likely to be important. An anhamonic environment~the electron–hole pairs in the surface!,comparatively low temperature, and an explicit time depdence of the Hamiltonian are not well suited for a paintegral approach.

The number of existing theoretical studies is therefosmall. The above-mentioned two-temperature model wused in a semigroup treatment for NO/Pt~111!.32,33The indi-rect treatment of the pulse in the two-temperature model plonged the excitation time scale justifying the Markov asumption inherent in the semigroup approach. Anotindirect treatment of the excitation process has been appto CO/Cu~111!.34,35The surface was treated as stochasticvironment leading to an optical potential in the systeHamiltonian. Memory effects were considered to be neggible due to the internal energy transfer in the surface. Dir


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1752 J. Chem. Phys., Vol. 119, No. 3, 15 July 2003 Koch et al.

optical excitation without further justification has been asumed for NH3 /Cu(111)36 although the excitation is knownto be substrate-mediated.37 The deexcitation was also modeled by an optical potential, i.e., assuming ad-correlatedenvironment. However, the field is known to affect tdissipation.38 Since excitation and deexcitation are bocaused by electrons in the copper surface, and the time sof excitation and relaxation are comparable, correlationstween excitation and dissipation have to be expectedwere not accounted for.

The present paper presents a theoretical study of femsecond laser induced desorption of NO/NiO~100!. This sys-tem has been the subject of numerous epxerimental11,39–41aswell as theoretical studies.4,42,43The most prominent featurof the experimental results is a bimodality of the desorptvelocity distributions and a coupling of rotational and tranlational energy for the fast desorption channel.39 These find-ings could be explained in terms of the topology of the ecited state potential and the excited state dynamics withtwo-dimensional jumping wave packet treatment.42 How-ever, the lifetime of the excited state had to be chosemiempirically. Furthermore, such a stochastic approachglects correlations between excitation and relaxation whbecome important in femtosecond experiments as curreperformed. The introduction of femtosecond laser technogy requires a theoretical treatment of excitation and decitation mechanisms on the same level of rigor. In particua microscopic description of the dissipation has not battempted so far. We therefore combine the information frprevious studies4,11,39,40,42,43with the surrogate Hamiltonianmethod2,3 to develop a model which treats the nuclear dnamics as well as the excitation and relaxation mechanon the same level of rigor.

The surrogate Hamiltonian is complementary to othapproaches to dissipative quantum dynamics which are bon a reduced description of the system. This is particulainteresting in light of the rigorous proof that a reduced dnamics in general does not exist for quantum systems.44 Forthe surrogate Hamiltonian the starting point is a descriptof the total system and bath. This description is appromated in a controlled way yielding a model whose treatmis numerically feasible but whose validity is limited in timNo weak coupling assumption needs to be made, andcoupling constants can be derived from first principles. Tenvironmental modes are described as two level syst~TLS!. Such an excitonic bath is particularly well-suitedmodel the electron–hole pairs in the surface. The interacwith phonons is not likely to play a role: It requires lifetimeat least on the picosecond to nanosecond time scale.temperature dependence of desorption observables couldthermore be explained purely by initial population of grouvibrational states.45 The interaction of the excited adsorbatesubstrate complex with phonons is therefore neglectedprinciple, however, it could be incorporated in the modelemploying a second bath.

The treatment of time-dependent fields can be inclunaturally into the description. Furthermore, the strengththe surrogate Hamiltonian lies in the low temperaturegime. Since the focus is on developing a microscopic und


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standing of the interaction between substrate, adsorbate,laser pulse, the model will, however, be restricted to onuclear dimension—the desorption coordinate. Observaof interest are therefore desorption yield and desorptionlocities. Once a fully quantum mechanical description of tdesorption event including electronic states, excitation,relaxation mechanisms has been obtained, a generalizatimore degrees of freedom is possible. This would allowcalculating furthermore rotational and vibrational distribtions. Such a generalization is beyond the scope ofpresent work which serves as one more step toward a cplete quantum description of laser induced desorption.

II. THE SURROGATE HAMILTONIAN METHODAPPLIED TO LASER INDUCED DESORPTION OF NOFROM NIO„100…

When the total system is separated into primary sysand secondary bath parts, its Hamiltonian is given by Eq.~1!.If in addition the interaction with a laser field is explicitlconsidered, the Hamiltonian can be written as

Htot5HS1HSF~ t !1HSB1HBF~ t !1HB , ~2!

where the interaction of the primary system and of the enronment with the field is described byHSF(t) and HBF(t),respectively. In the present study, the system HamiltonianHS

describes the adsorbed NO molecule on a finite part ofNiO surface~cf. Sec. II B!, while the remaining part of thesurface is modeled as environment or bathHB ~cf. Sec. II D!.The effect of this environment on the system is capturedthe interaction termHSB ~cf. Sec. II E!. Before proceedingwith the application of the surrogate Hamiltonian to lasinduced desorption of NO/NiO~100!, the general idea of themethod2,3 is recalled.

A. Brief review of the surrogate Hamiltonian method

In quantum mechanics the effort to solve a problescales exponentially with the number of degrees of freedExcept for a few special, analytically solvable cases, Eqs.~1!and~2! therefore state an extremely complicated problemwhich approximations are unavoidable. Since the bathgrees of freedom themselves are not interesting, and otheir influence on the system is important, the first stepapproximation consists in an implicit description of the baby abstract, representative modes. The core idea of therogate Hamiltonian2 is the truncation of the infinite numbeof representative bath modes in a well-defined way. Thispossible if the modes are chosen such that the ones winteract strongest with the system are always included indescription. This leads to a new, ‘‘surrogate’’ Hamiltonian fthe total system which generates the time evolution o‘‘surrogate’’ wave function,C(Z,a,s1 ,s2 ,...,sN). Z repre-sents the nuclear coordinate of the system,a the electroniclevel, ands i the bath degrees of freedom. Observablesassociated with operators of the primary system. They cadetermined from the reduced density operator, i.e., the dsity operator of the primary system:


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rS~Z,Z8,a,a8!5trB$uC~Z,a,$s i%!&

3^C* ~Z8,a8,$s i8%!u%, ~3!

where trB$ % denotes a partial trace over the bath degreefreedom. The system density operator is thus construfrom the total system–bath wave function while only thwave function is propagated. Thisa posteriori constructionof the density operator is different from most other aproaches to dissipative quantum dynamics14,46,47 where thetrace over the bath is performed before time propagatpossibly neglecting correlations between system and batha consequence of the surrogate model, all correlationstween system and bath which the Hamiltonian allows forincluded. Furthermore, since the Schro¨dinger equation for awave function is solved, the treatment of time-dependentternal fields poses no additional problems.

In the limit of an infinite number of bath modes, thsurrogate Hamiltonian is completely equivalent to the orinal, ‘‘true’’ Hamiltonian. Since, at least in principle, the number of modesN can be increased, it is possible to checonvergence. The truncation leading to the surrogate Hatonian relies on a time-energy uncertainty argument: Infinite time, t!`, the system can only resolve a finite number, N!`, of bath states and not the full density of statThe sampling density in energy of the finite set of bath stais determined by the inverse of the time interval. This argment leads to two observations—the surrogate Hamiltonis well-suited for the description of ultrashort events, andnumber of needed modes increases with the interacstrength between system and bath. Strong and intermecoupling strengths might therefore pose a computatiochallenge. From the above-mentioned derivation, it is clehowever, that no weak coupling assumption was neededaddition, the surrogate Hamiltonian method yields acontrol-lable approximation.

The bath is composed of TLS and described byHamiltonian

HB51S^ (i

« isi†si ~4!

with n i5si†si the occupation number operator and« i the

energy of thei th bath mode. ForN bath modes the HilberspaceHB of the bath has dimension 2N. This results from asingle TLS or spin-12 being defined on a two-dimensionHilbert space and the possibility to combine each of the tbasis states for allN modes. The dimension of the total Hibert spaceHS^ HB is then given by the product of the dmensions ofHS and 2N. Obviously, this dimension quicklygets very large when the number of bath modesN is in-creased. However, considering all 2N possibilities of combin-ing the bath modes corresponds to consideringall possiblesystem–bath correlations which might not be necessary.number of simultaneously allowed excitations can thenrestricted. In an extreme case, only single excitationsconsidered. This reduces the dimension of the total Hilbspace from 2N to N11. The approximation made is agacontrollable since the number of simultaneously allowedcitations can be increased. In the present study, the eigen


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ues« i and eigenstatesni of the bath were obtained withinmicroscopic model~cf. Secs. II D and II E!. The TLS bathcan also be viewed as a low temperature approximationharmonic oscillator bath with the possibility of connectinthe parameters of the two models.48–52

B. The primary system

The Hamiltonian of the primary system,HS , describestwo electronic states and one nuclear degree of freedomZ,which is the distance of the molecule from the surface,

HS5S T1Vg~ Z! 0

0 T1Ve~ Z!D . ~5!

T denotes the kinetic energy operator which is appliedmomentum space. Potential energy surfacesVg/e(Z) havebeen constructed by Klu¨ner and co-workers4,5 in twodimensions—distance from the surfaceZ and polar angleubetween the NO molecular axis and the surface normalthe present study the angle is kept fixed at the equilibrivalueu545°. The excited state potential has been calculain a valence configuration interaction~CI! approach for aNO/NiO5

82 cluster embedded in a point charge field.4,5 Theexcited state is a charge-transfer state which is characterby a deep potential well due to Coulomb interaction betweNO2 and the positively charged cluster and by a potenminimum at a distance of about 1.5 a.u. smaller thanelectronic ground state minimum. The wave packet wtherefore be accelerated toward the surface upon excitaand desorption will occur according to the Antoniewimechanism.8

CI has so far been the only method to obtain excistates for adsorbates on transition metal oxides.4,53 However,within such an approach based on a finite cluster only retive energies and the topology of the potential energy surfare expected to be reliable. Vertical excitation energiesonly be estimated due to orbital relaxation within the clusand due to extra cluster polarization.54 Since many excitedstates are located in the energy range probed by the lpulse, the field is likely to cause a resonant transition. Hoever, since the topology of these states is very similar,5 it ispossible to select one representative state whose excitaenergy coincides with the laser energy.

C. The interaction with the laser field

The Hamiltonian, Eq.~2!, includes both direct@(HSF(t)#

and substrate-mediated@HBF(t)# excitation of the primarysystem. However, in the following only direct optical excittion will be considered. For metal surfaces direct optical ecitation can be excluded due to the strong quenching of etrons in the conduction band. This situation is different foxide surfaces which have a considerable band gap. Msurements with different polarizations of the laser pufound a dependence of the desorption yield on the polartion while the desorption velocities were not influenced.55 Ifthe excitation is mediated by the substrate,1 the excitationenergy is dissipated into the surface via electron–elecand ~secondary! electron–phonon scattering. These multip


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scattering events rule out a symmetry dependence of thecitation. In contrast, a direct excitation is determined bysymmetry of the states involved. The polarization depdence of the desorption yield for NO/NiO~100! favorings-polarized light55 is compatible with calculated oscillatostrengths.4,5 It therefore supports an electronic excitatiomechanism which is determined by optical selection rui.e., a direct optical excitation within the adsorbate–substcomplex.

If a direct optical excitation of the adsorbate–substrcomplex is assumed, the primary system interacts withelectric fieldE(t) of the laser pulse which causes an eletronic transition,

HSF~ t !5S 0 E~ t !mtr~ Z!

E* ~ t !mtr~ Z! 0D . ~6!

mtr(Z) is the transition dipole operator depending on tnuclear coordinate. The fieldE(t) is treated semiclassicallyand its spatial dependence is neglected. The shape ofield is assumed to be Gaussian,

E~ t !5E0 expS 2(t2tmax)

2

2sP2 DeivLt. ~7!

Since the excitation is taken to be resonant, the laserquencyvL coincides with the difference ofVg andVe at theminimum of the ground state potential. The parameters cacterizing the pulse are its frequencyvL , the intensityE0 orthe pulse fluence which is related toE0 , and the full widthhalf maximum~FWHM! tP which is related to the standardeviation sP by tP52sPA2 ln 2. The transition dipolemtr(Z) can be obtained from the oscillator strengthf ,

f 5 23 Ef i um f i u2 ~8!

~in atomic units!. f was found to be approximatelf 5exp(2Z) in the ab initio calculations,4,5 and Ef i'4 eV50.15 a.u.

D. The electron hole pair bath

The lifetime of the excited state has been estimatedabout 15–25 fs, i.e., the charge transfer state is extremshort-lived,4 but the lifetime is still considerably larger thathose estimated for desorption from metal surfaces.1 Opticaldeexcitation and interaction with phonons require lifetimesleast on the picosecond to nanosecond time scale andtherefore be excluded as possible relaxation channels.interaction with phonons is furthermore not likely to playrole since the temperature dependence of desorption obables could be explained purely by initial populationground state vibrational states.45 The remaining possible relaxation channel is electronic quenching caused by the inaction with electron hole pairs, i.e., O 2p→Ni 3d chargetransfer states in the surface.

The electron hole pairs are described as a TLS bath

HB5«(i

si1si1

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creation and annihilation operators for thei th TLS, respec-tively. Equation~9! implies that one electron hole pair located at sitei is modeled by thei th TLS. The sites are Ni–Opairs in the lattice between which charge transfer can ocThe first term in Eq.~9! describes the excitation of localizeTLS at the sitesi . This is motivated by the Ni 3d statesbeing in general localized.56 Delocalization is brought abouby the O 2p states and introduced into the model by tsecond term in Eq.~9!. This term describes the transportexcitation from one electron hole pair to its nearest neibors. The bath is hence characterized by two paramete«andh. All electron hole pairs are assumed to have identiexcitation energy«. In a molecular orbital picture this is thtransition energy from the highest occupied molecular orbto the lowest unoccupied molecular orbital.h is the interac-tion strength between nearest-neighbor TLS and leadsfinite width of excitation energy, i.e., an energy band of tbath: If the bath Hamiltonian, Eq.~9!, is diagonalized, andNis the number of modes,N energies around« correspond tosingle excitations,N energies around 2« correspond todouble excitations, etc. The spread of these eigenvaaround« is determined byh. The scaling 1/log(N) of thesecond term in Eq.~9! needs to be introduced to make thprocedure convergent, i.e., to have the spread of eneraround« independent of the number of bath modesN. Itresults from the topology of the problem, i.e., from the maping of two dimensions of the bath onto one~cf. the Appen-dix!. The interaction itself does not scale withN since thebath modes are localized.48 To summarize, the parameter«can be viewed as the center of the bath energy band whihdetermines its width.

Equation~9! represents an abstraction from the compcated electronic structure of actual O 2p→Ni 3d chargetransfer states in the surface. Therefore, it should be possto estimate reasonable values of« andh from either electronspectroscopy or electronic structure theory. Froexperiment57,58 as well as theory,54,59 the band gap of NiO isknown to be about 4 eV with some surface states correspoing to d→d excitations of nickel at lower energies. Thwidth of the energy band of single electronic excitations wfound to be about 10 eV. However, laser energies betw3.2 and 6.4 eV41 do not probe the whole energy band. If«and h are chosen in direct correspondence to the elecenergy loss spectroscopy data and CI calculations, mowith energy higher than those probed by the laser arecluded. However, while these modes exist, they do not ctribute to the quenching dynamics. An opimal choice wtherefore put as many modes as possible into the physicrelevant range, i.e., the energy range set by the laser enA thorough discussion of the role of« andh will be given inSecs. III B–III D.

E. A microscopic model for the system–bathinteraction

The interaction of the electron hole pairs with the NO2

like intermediate leads to quenching of electronic excitatof the primary system. The electron hole pairs can be viewas dipoles, and the laser excitation creates a nonzero tra


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tion dipole in the system. The interaction is therefore moeled as dipole–dipole interaction which assumes thatelectric field can be described classically and that the sysdipole is in the far field of the bath dipoles. However, copared to the simplification of the O 2p→Ni 3d charge trans-fer states to TLS, these additional approximations arepected to be negligible. TheVi in the interactionHamiltonian,

HSB5S 0 1

1 0D ^ (i

Vi~si†1si !, ~10!

are then given by the scalar product of the system’s transidipole, m¢ S , and the electric field of the bath dipoles,EW i :

Vi5m¢ S•EW i5m¢ S•m¢ i

ur¢i u323

~m¢ S•r¢i !~m¢ i•r¢i !

ur¢i u5. ~11!

ur¢i u is the distance of thei th bath dipole from the systemdipole. Note that theVi are operators in the Hilbert spacethe system. Taking into account the expectation value oftransition dipole instead of the operatorm¢ S in Eq. ~11! cor-responds to a time-dependent self-consistent field~TD-SCF!approach.34 It introduces the fast time dependence of ttransition dipole into the Hamiltonian. Since evaluationthe operator expressions poses no difficulty when usinggrid representation, Eq.~11! was implemented using the operators and not expectation values. The bath dipoles aresumed to be located at the center of charge in betweenickel and an oxygen atom. The system dipole is locatedbetween the nickel atom and the NO molecule. Evaluatthe scalar products then leads to

Vi~ Z!56qa0m tr~ Z!

S S 1

2~ Z1a0!1ma0D 2

1n2a02D 3/2

73qa0m tr~ Z!Z2

S S 1

2~ Z1a0!1ma0D 2

1n2a02D 5/2, ~12!

wherea0 is the distance between the Ni and O atoms, ihalf the lattice constant (2a053.93 Å), andn, mPN. nlabels the sites within the surface~horizontal distance! whilem labels the layers~vertical distance, cf. the Appendix!. If aone-dimensional primary system is considered, i.e., theangle of NO versus the surface normal is neglected, onlybath dipoles parallel to the surface normal contribute tointeraction. The only parameter in Eq.~12! and therefore inthe interaction HamiltonianHSB is the dipole chargeq char-acterizing the completeness of charge transfer betweenickel and an oxygen atom. An estimate ofq is known fromab initio calculations.60 The role ofq will be discussed inSec. III E.

F. Dynamics and observables

The initial state is taken to be the vibrational groustate of the electronic ground state potential. This co


-em

-

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e

fe

as-a

ing

.,

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a

-

sponds to a factorizing initial state at zero temperaturedensity matrix formalism, i.e., no initial correlations betwesystem and bath are considered. Due to the large band gaabout 4 eV in NiO, no electron hole pairs are thermally ecited at timet50. Hence it is justified to neglect initial correlations between system and bath.

Since a direct excitation mechanism is assumed, theser pulse transfers population from the electronic groundthe electronically excited state around the Franck-Conpoint. The partial wave packets on the electronically excistate start to travel toward smaller distancesZ. Due to theinteraction with the bath, population is continuously tranferred to the electronic ground state. The process is enhawhen the energy of a bath mode matches the potentialferenceDV(Z). In principle, while both electronic quenching and electronic excitation of the system are possible,latter is much less probable. This becomes obvious wusing the rotating wave approximation~RWA! and moving tothe rotating frame. Then the bath creation operators couonly to the electronic annihilator of the system, and the bannihilators couple only to the electronic creation operatothe system. Electronic excitation of the system can thereonly occur after electronic quenching with the associated cation of bath excitations. The validity of the RWA has bethoroughly checked. The population which has been traferred to the electronic ground state will either be trappedthe potential well though vibrationally excited, or it hagained enough kinetic energy to leave the potential wellthe ground state and to desorb. The wave packet is progated on the electronic ground state until the trapped anddesorbing parts are well separated and the observables iasymptotic region are converged. The grid switching methproposed by Heather and Metiu is employed.61

The convergence of the surrogate Hamiltonian withspect to the propagation time is limited due to recurrencethe bath. Since the bath Hamiltonian is finite, energy traferred from the system to the bath will eventually be rflected and transferred back to the system. At this po~calledtS), the simulation should be stopped and the numof bath modes needs to be increased. In the simulationstS

can be determined by two criteria, the population backfland the bath distance criteria. For the first, the populationthe zeroth mode corresponding to all TLS being deexcitemonitored. If it increases, population and energy is traferred from the bath back into the system. For the latter,average distance of the excitation in the bath is calculaThis is possible since every bath mode is connected witNiO lattice position. A reflection of the excitation at thboundary due to the finite size of the bath leads to a decrein the bath distance.

Within the convergence time of the surrogate Hamtonian it is not possible to obtain converged expectation vues in the asymptotic region@Vg(Z)'0# which can be com-pared to observables of laser induced desorptexperiments. However, the surrogate Hamiltonian is neeonly to describe electronic quenching, it is not necessarydescribe the nuclear motion on the electronic ground sleading to a separation of the wave packet into a trappeda desorbing part: If the decay of the electronic excitation


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fast, the quenching happens on a much shorter time scale~fs!than the nuclear motion in the ground state~ps!, and the twophenomena can be separated.

After the interaction has been switched off at timetS the2N ground state wave packets can be propagated untilobservables in the asymptotic region are converged. Ifwave packet is still comparatively localized, it is, howevmore efficient to construct the ground state density matrixrS

of the system by tracing over the bath. Since no further dsipative processes are included in the description, the tevolution of this reduced density matrix is unitary. Therefoif rS is diagonalized,

rS~Z,Z8;t !5U1PU5(k

pkuck~Z;t !&^ck~Z8;t !u, ~13!

no further mixing of the wave functionsuck(Z)& will occurduring the time propagation. More than one eigenvaluepk

will be nonzero since the wave packet created by electroquenching and nuclear dynamics is a mixed state. Butnumber ofpk which contribute significantly in the sum in Eq~13! is small ~usually between 15 and 20 in the examppresented in the following!. Expectation values can be constructed asA(t)5(k pk^ck(t)uAuck(t)&. The computationalsavings depend on the number of modesN and can reachseveral orders of magnitude for largeN.

The observables in laser desorption experiments of NNiO~100! have been the desorption cross section whichrelated to the desorption probability and the quantum sresolved velocity of the desorbing molecules.11,39,40The de-sorption probability is obtained by weighting the populatiin the asymptotic region by the excitation probability. Inone-dimensional model, only average velocity distributiocan be observed. The velocity distribution corresponds toprobability density of the wave packet in the asymptoticgion in momentum representation.

III. RESULTS AND DISCUSSION

A. Convergence behavior

The time interval for which propagation with the surrgate Hamiltonian gives converged results depends onnumber of bath modesN. This interval can be prolonged bincreasing the number of bath modes. On the other handconvergence of observables with respect toN can bechecked. Figure 1 shows the population~left!, coordinate~top right!, and momentum~bottom right! expectation valueson the electronically excited state forN535,45,55. Further-more, forN535 and forN545, one and two simultaneouslallowed excitations~dotted and solid curves! are compared.The curves are indistinguishable, i.e., it is possible to ressimultaneously allowed excitations to one. This is not sprising, since the energy of double excitations is much higthan the laser energy. Only single excitations have enerin the right range to accept excitation energy from the sysand thus can be effective in the quenching. Exponentialcay of population can be observed after excitation bylaser pulse~Fig. 1 left!, while the wave packet is acceleratetoward the surface~Fig. 1 top right!. The observables can bconsidered converged up to about 27 fs forN535, 40 fs for


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/iste

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N545, and 60 fs forN555. The convergence timetc issomewhat smaller than the total lengthtS of the curves inFig. 1 since the interaction with the bath is switchedwhen recurrences reach the zeroth spinor component. Bytime the energy reflected at the boundary of the finite syshas already passed through the bath modes. The exponedecay of excited state population within the convergenceterval can be fitted to obtain decay rates or lifetimes. Derates versus the number of bath modes are plotted in Figtop panel, while the quality of exponential fit characterizby the correlation coefficient is shown in the bottom panTwo different values of dipole strengthq characterizing thestrength of interaction between system and bath~cf. Sec.III E ! have been used. The decay rates~lifetimes! saturate atabout 0.04 fs21 ~25 fs! for q50.10 and 0.075 fs21 ~13 fs! forq50.14. The correlation coefficient fluctuates betwe0.9980 and 0.9995 showing a good agreement betweenand exponential fit.

FIG. 1. The excited state population vs time is shown on the left, on a lin~top! and logarithmic scale~bottom!. The expectation value of distance~top!and momentum~bottom! of the wave packet on the excited state vs time ashown on the right. Increasing the number of modesN prolongs the conver-gence time. The number of simultaneously allowed excitations can bestricted to 1 since the black and gray curves are identical. The paramare«54.0 eV, h52.0 eV, q50.1, vL53.7 eV.

FIG. 2. The excited state decay rate obtained from exponential fit~top! andthe correlation coefficient of exponential fit~bottom! are plotted vs the num-ber of bath modesN. The decay rate reaches saturation when increasingN,while the correlation fluctuates in a range close to one.


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B. Role of center of bath energies for the quenchingdynamics

The TLS bath describes electron hole pairs in the surfwhich cause the quenching of electronic excitation. It is chacterized by the two parameters« andh in Eq. ~9!, the TLSenergy and the nearest-neighbor interaction strength. Tparameters are related to the center and width of the en‘‘band’’ of the bath. Figure 3 shows the range of baeigenenergies for different values of these parameters onright. Only single excitations are considered, i.e., the numof simultaneously allowed excitations of the bath is restricto one. Double excitations would lead to a second bathergy band at larger energies. The left of Fig. 3 displaysexcited state potential and the difference potential,DV(Z)5Ve(Z)2Vg(Z). The Franck–Condon point indicatedFig. 3 determines the classical turning points for the wapacket motion on the excited state~light gray arrow!. Thevalues of the difference potentialDV in between the classicaturning points specify the range of bath energies relevantquenching~bold dark gray!. Bath modes with energies withithis range can accept energy from or give energy to thetem causing a transition between electronic ground andcited state. The bath parameters should therefore be chto obtain the best possible convergence of observablesrespect to the number of modes («'3.7– 4.0 eV andh'0.7– 1.0 eV, cf. Figs. 4 and 5!. Many electron hole pairswith energies much higher than the laser energy exist58 butthey are not needed. This explains why double, triple, eexcitations can be neglected in the dynamics.

Figure 4 shows the influence of the TLS energy« on theexcited state dynamics. If« is considerably larger than thlaser energy, the range of bath eigenenergies does not mthe values of the difference potential between the classturning points. The TLS therefore cannot accept energy frthe system, and hence no decay of excited state populatiobserved~dotted curve in Fig. 4!. For « considerably smallerthan the laser energy, there are no matching bath modesto the Franck–Condon point. However, as the wave pactravels toward smaller distances, the value of the differe

FIG. 3. The range of bath eigenenergies needs to match the differpotentialDV for quenching to be efficient~left!. It is determined by the bathparameters« and h ~right!. The difference potential is fixed by assuminresonant excitation at the Franck–Condon point with a laser energy oeV.


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.,

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potential is decreased~cf. Fig. 3!, bath energies in the righrange are found, and decay of excited state population mbe observed with some delay~solid curve in Fig. 4!. Forintermediate values of«, the excited state population decaexponentially. Exponential decay corresponds to a consrelaxation rate and allows for a comparison of the surrogHamiltonian method with the jumping wave packet approaaccording to Gadzuk.5,42 The specific value of« determinesconvergence as can be seen from comparison of the band gray lines~101 and 91 bath modes, respectively! andfrom Table I. An optimal choice places« close to the laserenergy. This is reasonable also from a physical point of vi

ce

.7

FIG. 4. Dependence of excited state population vs time on the TLS en«: If « and therefore the center of the bath energy band is close to the syresonance fixed by the laser energyvL ~the dashed curves! the decay ofexcited state population is exponential, and the choice of« determines con-vergence. If the bath energies are larger than the system resonance~dottedcurve!, the system cannot give energy to the bath and no decay is obseFor bath energies smaller than the system resonance~solid curve!, the wavepacket needs to travel to a region where the bath energies match the ptial difference before decay can occur~see the text for further explanation!.N indicates the number of bath modes.

FIG. 5. Dependence of excited state population vs time on nearest-neiginteraction strengthh: For smallh ~0.05 and 0.3 eV! there is no transport ofrelaxed population out of the interaction region, the convergence timvery short and cannot be improved by increasing the number of modeN.Increasingh ~0.5 eV! leads to transport, however on a time scale larger ththe interaction with the system. Forh>0.7 eV, the transport is efficient, anexponential decay of excited state population is observed~curves in thebottom panel!. Largeh causes less quenching of excitation during the intaction of the system with the pulse, and hence a larger maximum populaof the excited state.


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Since the laser pulse induces an electronic transition innickel oxide surface, its energy needs to be equal to or lathan the band gap to which« is related.

C. Role of the nearest-neighbor interaction for thequenching dynamics

The dependence of excited state population onnearest-neighbor interaction strengthh of the TLS is shownin Fig. 5. Since the dipole–dipole interaction exhibits a 1/Z3

dependence, the system interacts only with electron hpairs which are very close to the NO molecule. To transpthe excitation away from this interaction region, the neareneighbor coupling between electron hole pairs is needed.his very small ~0.05 and 0.3 eV in Fig. 5!, the excitationcannot be given to TLS outside the interaction region, athe convergence time is determined by saturation of theTLS close to the primary system. In this case, increasingnumber of modes cannot prolong the convergence tiWhen h is increased, transport sets in leading to a lonconvergence time, to slower decay, and to a dependencthe number of modes. Furthermore, the quenching of exction during interaction of the system with the pulse is leefficient, hence the maximum population of the excited stis increased for largerh ~cf. Fig. 5, lower panel!. The excitedstate population decays exponentially—independent ofvalue ofh with the exception ofh50.5 eV. The decay rateobtained from an exponential fit of the excited state popution versus time~Fig. 5! are plotted versus the numbermodes in the upper panel of Fig. 6, while the lower pashows the goodness of the exponential fit characterizedthe correlation coefficient. The decay rate is decreasedincreasing nearest-neighbor interaction. This can be unstood from the following considerations:h determines howquickly relaxed population is transported away from the pmary system, but also from TLS close to the primary systeThe interaction energy, i.e., the expectation value of theteraction Hamiltonian, Eq.~10!, depends on the populatioof the primary system and of the bath modes close to itpopulation is removed from these bath modes, the interacenergy is decreased and the decay becomes slower. If melectron hole pairs are excited close to the NO molecule,decay becomes faster.

The same argument explains the increase of the mmum excited state population, i.e., decrease of quencduring the interaction with the laser pulse. Since the tiscale of this interaction is shorter than that of relaxation~thelaser FWHM was chosen as 5 fs!, the effect becomes visibleonly for large h ~cf. Fig. 5, right!. The correlation coeffi-

TABLE I. The quality of exponential fit for the decaying part of the excitstate population with TLS energy« varied is given by the correlation coefficient. The best fit is obtained for«5vL53.7 eV. In this case, the quenching is also most efficient~largest decay rate!.

«Decay rate~1/fs!

(N5101)Decay rate~1/fs!

(N591)Correlation of fit

(N5101)Correlation of

(N591)

3.5 0.050 0.046 0.988 0.9913.7 0.096 0.095 0.997 0.9984.0 0.011 0.010 0.977 0.967


eer

e

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cients of the exponential fit lie within the [email protected],1.0#with the exception ofh50.5 eV~Fig. 6, bottom!. If the datafor h50.5 eV is fitted only up to 25 fs, its correlation coeficient also lies [email protected],1.0#. A similar behavior is foundfor h50.6 eV, while the data forh50.4 eV is similar to theone with h50.3 eV. Thus, variation ofh does not changethe decay of excited state population qualitatively. Thisdifferent from the role of the TLS energy« ~cf. Fig. 4!.However, since« shifts the position of bath eigenenergiewhile h changes only their width~cf. Fig. 3!, this is notsurprising.

The considerations have so far only been numericalsuggest an optimal choice of about 0.7–1.0 eV forh. How-ever, an upper limit to the nearest-neighbor interactstrength is also set by the physics of the NO/NiO~100! sys-tem: The lowest states are surface states in the optical bgap at about 2.7 eV.57 These are not charge transfer statbut Ni d→d excitations. The charge transfer states lie engetically above the band gap. Therefore no bath modes wenergies much below 3.5 eV need to be considered.course, if the TLS energy« is shifted,h should be adjustedto result in a reasonable smallest bath eigenenergy. Accingly, the optimal choice of bath parameters leading to bpossible convergence of expectation values with respecthe number of bath modes is a combination of TLS energ«and nearest-neighbor interaction strengthh. In the following,«53.7 eV andh50.7 eV were used.

D. Asymptotic observables

So far it has been shown that converged observarelated to the excited state dynamics can be obtained. Wthe excited state dynamics are crucial for the outcome olaser desorption experiment, they are not directly accessin a single pulse experiment. Instead, the desorption yand the state resolved velocity of desorbing moleculesmeasured, i.e., observables in the asymptotic region. Duthe Antoniewicz-type mechanism of desorption, t

FIG. 6. The decay rates of excited state population which were obtafrom an exponential fit of the data shown in Fig. 5 are plotted vs nearneighbor interaction strengthh ~upper panel!. It decreases for larger valueof h. The goodness of exponential fit is more or less independent oh~lower panel!. Two values for the decay rate and the correlation coefficiare plotted forh50.5 eV, once the data of the whole range shown in Figand once only values of excited state population up to 25 fs were usefitting.


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asymptotic observables are detemined by partial wave pets which stay on the excited state for a comparatively lotime. In this case, the partial wave packets end up highthe repulsive part of the ground state potential after the etronic quenching. They can then gain enough kinetic eneto leave the potential well and reach the asymptotic regThe convergence of the surrogate Hamiltonian with respto the number of modes is limited in time. It is therefocomparatively easy to obtain converged excited state dynics, while it turned out to be more difficult to obtain converged asymptotic observables.

Two strategies can be employed to reach convergencthe asymptotic observables. Either the number of bmodes,N, or the number of layers in the surface,NL ~cf. theAppendix!, can be increased. An increase in the numberbath modes,N, enlarges the size of the bath horizontally, amore layers in the surface allow for vertical transport, i.transport into the surface. While both processes are equlikely for nickel oxide, they are not treated on the same foing in the model. This is discussed in more detail in tAppendix.

Considering several surface layers indeed leads tlonger convergence time~cf. Fig. 7!. The treatment of morethan one layer of dipoles introduces a new parameter intomodel, the coupling between layers,hL . Its influence on theexcited state dynamics is shown in Fig. 8. A small valuethe interlayer coupling~0.0 and 0.05 eV in Fig. 8! does notchange the lifetime of the excited state. Increasing the vaof hL leads to slower decay of the electronic excitation~0.2and 0.3 eV in Fig. 8!. This can be understood by an argumesimilar to the one explaining the dependence of the lifetion the nearest-neighbor interactionh. The strength of thesystem–bath interaction depends on the excited state plation of the system and on the population of bath moclose to the system. An increase ofhL results in quickertransport of excitation from bath modes close to the systo bath modes further away. Thus the system–bath intetion becomes weaker and the lifetime longer.

As a consequence of the slower decay, increasinginterlayer couplinghL leads to a larger desorption probabity ~cf. Fig. 9!. While the nearest-neighbor interactionh, i.e.,

FIG. 7. The population of the excited state vs time. The coupling betwelectronic states is switched off when recurrences occur, therefore the glengths indicate the convergence time.


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the intralayer coupling, has been directly related to the etronic structure of the system, such a connection has sonot been given for the interlayer coupling,hL . However,since in the NiO surface the transport of excitation is equalikely horizontally as well as vertically, it is reasonableassume thathL should be of the same order of magnitudeh. On the other hand,hL is related to the convergence proerties of the model. This means that there exists an optivalue for which the largest convergence time for a givnumber of modes and a given number of layers is obtainIf hL is small, an increase in the number of layers, i.e.,increase of the vertical bath size, will not result in a largconvergence time. In this case, the excitation hits the hzontal boundary of the bath before reaching the vertilimit. For large values ofhL the opposite is true: The verticais reached before the horizontal boundary. The best congence is achieved when at the same time both boundariereached. The optimal choice ofhL therefore depends on thnumber of bath modes and the number of layers.

For the parameters investigated, up to 21 layers withN<51 and up to 13 layers withN<101 bath modes in eaclayer were considered. The maximum convergence timeabout 90 fs. This was enough to obtain converged desorp

nph

FIG. 8. Increasing the interlayer coupling leads to a slower decay dufaster transport of relaxed population away from the interaction region~pa-rameters as in Fig. 7, butNL55).

FIG. 9. Increasing the number of layers leads to converged desorption pabilities ~left! and velocity distributions~right!. The interlayer couplinghL

influences the lifetime of the excited state and therefore also the desorprobability ~parameters as in Figs. 7 and 8!.


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probabilities and velocities~cf. Fig. 9!. Desorption probabili-ties between 1% and 20% were obtained. This is compatwith estimates from experiment.9,11,62 Furthermore, the ve-locities are found to be in the experimentally observed rabetween 0 and 2000 m/s.39

The population of the excited state was decreased fits maximum value of about 0.11 to about 0.005~at best0.0034 shown in Fig. 7!. This means that about 0.5% of thdensity was left out when the bath was switched off. Whthis number is small in absolute value, it might be a conserable portion of the desorbing part. The desorption probaity is obtained by weighting the norm in the asymptotic rgion by the excitation probability. One may assume thator a substantial part of the neglected density desorbs sthe coordinate expectation value approaches the value oclassical turning point. In case of a sudden electronic tration much of the kinetic energy would be gained by the wapacket. Weighting the value of 0.5% with the excitatiprobability results in a possible increase of the desorpprobability by 5%. While this is on the same order of manitude as the desorption probability itself, it is still wewithin the uncertainty of the experimental estimate.

The second strategy to increase the convergenceconsists in increasing the number of bath modes,N, withinone layer. A peculiarity is then observed: Above a certnumber,N* , of bath modes a further increase does not rein a prolongation of the convergence time. The exact valuN* depends on the parameters, in particular on the dipstrength,q, which determines the system–bath interactstrength and hence the convergence. For numbers of mlarger thanN* , the two criteria to monitor recurrences in thbath lead to different convergence timestS , while for smallN the tS are more or less equivalent. This means thatN.N* , the finite boundary does not seem to be reacwhen population backflow is observed. A possible interptation of this phenomenon consists in a polarization ofbath dipoles which interact with the system leading tobackflow of population. Both criteria rely on expectation vaues, i.e., averages. They can therefore both only give antimate of the time at which recurrences occur. In additionthe bath distance, also its variance has been examineswitch-off criterion, but no differences could be observed

A comparison of the two criteria is shown in Fig. 1Due to the structure of the interaction operator in the RWthe population backflow can be observed directly by ancrease in the excited state population~cf. the upper left paneof Fig. 10!. In spite of the backflow of population into thsystem, Fig. 10 shows that the switch-off criterion employthe distance of the bath is reasonable: The curves of excstate population overlap for an increasing number of bmodes for a time considerably longer than the convergetime given by the population backflow criterion~indicated bythe black arrow in Fig. 10!. The two switch-off criteria leadto different desorption probabilities~cf. the lower left panelof Fig. 10! owing to the different times the wave packspent on the excited state. The different times spent onexcited state furthermore result in different velocity distribtions ~cf. the right panel of Fig. 10!. In the case of the bathdistance switch-off criterion~gray curves in Fig. 10!, the


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propagation on the excited state continued sufficiently loto pass the classical turning point. Therefore an interferecan be observed in the velocity distribution. The interferenresults from different pathways of partial wave packewhich have reached the asymptotic region. It is acoherentphenomenon which cannot be observed within a odimensional stochastic wave function approach.

This point is clarified by Fig. 11, which shows the ecited state population versus time and velocity distributiofor a different number of bath modesN. For all three casesin Fig. 11, the switch-off criterion employing the bath ditance has been used. ForN5161 bath modes~gray curves inFig. 11!, the propagation with both electronic states had toswitched off before the classical turning point was reachThe corresponding velocity distribution therefore shows o

FIG. 10. If the distance of the bath excitation~gray! is used as convergenccriterion instead of the population of the system~black!, longer convergencetimes tS can be obtained~indicated by the black arrow!. The time the wavepacket spent on the excited state is crucial for both the desorption probity ~bottom left! and the shape of the velocity distributions~right, scaled forcomparison!. The second peak in the velocity distribution can be relatedthe passage of the classical turning point of the excited state potentia~in-dicated by gray arrow! which has only occurred for the gray curves witN5181 andN5201. The top left shows the population of the excited stwith the pulse~not to scale! indicated~parameters as in the previous figurebut NL51).

FIG. 11. The gray curves from Fig. 10 are plotted here for increasing nber of bath modesN. The appearance of the high velocity peak is relatedthe passage of the classical turning point, but is independent of the incrin excited state population, i.e., it is not caused by the recurrences inbath.


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a single peak. ForN5181 andN5201 modes, the excitedstate propagation continued beyond the classical turnpoint. Consequently, the velocity distributions exhibit anterference pattern. In particular, it can be concluded frFig. 11 that the interferences in the velocity distributioappear independent from the population backflow sincelatter is observed for all three cases presented in Fig.Such interference can also be observed with the populabackflow criterion—given that the propagation proceedlong enough on the electronically electronic state to passclassical turning point. This was the case, for example, ifunphysically small TLS energy« was chosen~cf. the solidcurves in Fig. 4!. The conclusion is that the interference patern is not caused by recurrences in the bath, but it canequivocally be related to the excited state dynamics.

E. Dependence on the dipole charge

There is only one parameter which enters the interacHamiltonian, Eq.~10!, and the interaction constants, E~11!—the dipole chargeq. It characterizes the completeneof charge transfer between a nickel and an oxygen atom<q<1, and it determines the system–bath interactstrength. The dynamics can therefore be expected to decrucially onq. Its value is related to the electronic structuof the substrate.

Figure 12 shows that an increase inq leads to a strongeinteraction between system and bath and therefore tsmaller lifetime of the excited state. But the excited stdynamics is influenced in a twofold way: Besides the exnential decay which can be observed after the pulse hasapplied, the maximum population of the excited state iscreased. The two phenomena are, of course, related. Thter, however, givesq the meaning of a parameter charactizing a metal to insulator transition, albeit in a vesimplified way. For largeq, no significant population of theexcited state is observed at all. This corresponds to theof metals where a direct optical excitation is immediatequenched due to the strong interaction with the substrate

The exponentially decaying part of the excited stpopulation versus time can be fitted to obtain decay ratelifetimes. The fit is indicated for three examples in the lowleft panel of Fig. 12~solid lines!. The obtained decay rateversusq are plotted in the right panel of Fig. 12. Forq

FIG. 12. The lifetime decreases with increasing dipole strengthq (NL

51).


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>0.1, a linear dependence is observed. This correspondthe coupling constants, Eq.~11!, being linear inq.

The lifetime of the electronic state was estimatedabout 25 fs.5,42 With the surrogate Hamiltonian, such a lifetime is obtained for a comparatively small value ofq'0.1. Avalue of this order of magnitude seems justified, however,the following consideration: The O 2p states are quite delocalized. One nickel atom therefore receives the electron frall its five or six neighboring oxygen atoms. But only the oor two charge transfer states with dipole moment parallethe surface normal contribute to the dipole–dipole intertion. This gives a rough estimate of 0.15&q&0.2. A similarnumber has been obtained independently in a populaanalysis of the O 2p→Ni 3d charge transfer states obtainein electronic structure calculations.63

The lifetime of the electronically excited state and thefore q determine the desorption probability. While this is nconfirmed by Fig. 13~lower left panel! due to the conver-gence problem explained in the previous section, it canobserved in Fig. 14. Figures 13 and 14 have been obtawith the population backflow versus the bath distance crrion for switching off the bath. The desorption probabilityPdes'0.005– 0.01 forq50.1 ~gray line in Fig. 13! is too

FIG. 13. The dipole strengthq determines the lifetime~top left! and it leadsto a very slight shift in mean velocity~right!. The desorption probability wasconverged well enough to give an order of magnitude estimation~bottomleft!.

FIG. 14. The same as Fig. 13 (q varied!, but with bath distance switch-offcriterion. The velocity distributions have been scaled for comparison.


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small because a comparatively large amount of populationeglected when switching off the bath. If the amount of nglected population is added to the desorbed norm, an ulimit for the desorption probability of aboutPdes50.2 is ob-tained. Forq50.14 ~black line in Fig. 13!, the observed desorption probability of about 4% is in the correct ordermagnitude. In this case, the amount of neglected populawas much smaller than forq50.1. While the bath distanccriterion leads to a higher desorption probability for smalq as expected, it leads to a trapping of population onexcited state which does not seem to be physical. One cargue that the excited state population does not have to din an overdamped way, i.e., purely exponentially. This wolead to additional, decaying oscillations in the excited stpopulation due to multiple electronic transitions. It is, hoever, beyond the current feasibility of the method to obtconvergence times long enough to test this hypothesis.

The desorption velocities~right panel of Figs. 13 and 14!depend only slightly onq. This is, however, subject to thconvergence behavior. The shape of the velocity distributimight be changed considerably by the density whichbeen neglected when switching off the bath. While it seereasonable to assume that all or most of the neglected plation reaches the asymptotic region and desorbs, it is impsible to estimate with which velocities the desorption occuIt should be pointed out, however, that in all cases the veity distributions show intensity in the experimentally oserved velocity range, and the desorption probability is ofexpected order of magnitude forq50.14, i.e., for a value ofq close to the estimate from electronic structure calculatio

F. Dependence on the pulse parameters

A characteristic result of femtosecond photodesorptexperiments has been the observation of a nonlinear dedence of the desorption yield or probability on the laser flence. This indicates a DIMET mechanism and a fluencependent transition from DIET to DIMET regimes.64 Figure15 therefore shows the dependence of the excitation queing and the excited state decay on the laser fluence,*2`

` E(t)dt. The arrow in Fig. 15 indicates the pulse fluenwhich has been used in the remaining calculations. Tvalue is still larger than the fluence of experimentally eployed pulses~about 200mJ in Ref. 41!. The comparativelylarge value can be justified, however, to compensate forsimplification of just a single excited state which is acounted for in the theoretical model. This excited state irepresentative of many, closely lying states which are inclose to resonance with the laser pulse in the experimThe population transfer will therefore be higher than pdicted by the model. This argument is supported by thedependence of experimentally observed state resolved veity distributions from the laser energy,\vL , which indicatesa manifold of excited states with a very similar topologytheir potential energy surfaces.40 A similar conclusion hasbeen reached by CI calculations.4,5

The excited state decay rate does not depend on theintensity ~cf. Fig. 15!. This is reasonable since the decaycaused by the substrate. For weak to moderate pulses


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excitation quenching~left panel of Fig. 15! shows a lineardependence on the fluence. For strong pulses, Rabi cycbetween the two electronic states becomes significant leato a nonlinear dependence. These intensities are very hand Rabi cycling is probably insignificant. It was furthemore shown in a simulation without bath, that Rabi cyclihas no influence on the desorption. In that case the poption transfer is solely caused by the coupling to the lapulse. The time spent on the excited state, however, turout to be insufficient for desorption—independent of puintensity and length. Since Rabi cycling is the only mechnism in the present model, which can lead to a nonlindependence of the desorption probability on the fluence,not surprising that DIMET cannot be observed. DIMET cinevitably only be modeled by taking into account substramediated excitation described byHBF(t).

The dependence of observables on the pulse duratioshown in Figs. 16 and 17. The lifetime of the excitationalso independent of pulse duration. This is expected andbe explained by the same argument as above: The dec

FIG. 15. Dependence of excitation quenching on pulse fluence: eneleading to about 10% excited state population as employed in the ocalculations are in the linear regime~indicated by an arrow!. The bath pa-rameters are«54 eV, h52 eV with resonant system excitation atvL

53.7 eV and a pulse duration oftFWHM55 fs. The excited state lifetime isnot affected by increasing the pulse energy. The quality of the exponentiis denoted by the correlation coefficientR.

FIG. 16. The lifetime of the excitation is independent of pulse duration. Tbath parameters are«54 eV, h52 eV and the excitation is resonant avL53.7 eV.


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caused by the substrate, and its rate should not be alterethe pulse. The independence of the decay rate from the pparameters points to a consistent treatment of the excitaprocess in the model. The excitation quenching, howeveinfluenced by the pulse FWHM—a longer pulse leads toincreased quenching~cf. Table II and Fig. 16!. The systeminteracts simultaneously with the field and the bath. Thefore, in case of a larger FWHM and hence a longer intertion with the field, more population can be quenched.

The time in which the system simultaneously interawith the field and the bath should furthermore influenceasymptotic observables. This is shown in Fig. 17. The resare, however, only preliminary due to the convergence prlem discussed earlier. When the pulse duration becocomparable with the vibrational period of the wave packetthe excited state potential, an interplay between pulsenuclear dynamics can be observed. A longer pulse duraexcites partial wave packets at times further away from eother. This leads to more different pathways which occurinterference pattern in the velocity distributions~Fig. 17,right!. While this is consistent within the model, some cation is advisable when drawing conclusions with respecexperiment. In the present treatment, electronic dephahas been completely neglected. Electronic dephasingcertainly wash out some of the observed quantum coences. This effect should become more pronounced aspulse duration is increased. The quantum coherencesfurthermore be attenuated in a higher dimensional desction.

IV. CONCLUSIONS

This study was aimed at a theoretical description of lainduced desorption where all aspects of the problemtreated on the same level of rigor. Potential energy surfa

FIG. 17. Desorption probability and velocity distributions for varied puduration. Interference phenomena can be observed when the time scathe pulse and of the nuclear dynamics become comparable.

TABLE II. The quenching of excitation is increased for longer pulses.

Pexcmax(bath)/Pexc

max(no bath) tFWHM55 fs tFWHM510 fs tFWHM525 fs

N559 0.8407 0.7053 0.4823N561 0.8403 0.7045 0.4822N563 0.8399 0.7038 0.4818


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obtained from first principles were combined with a micrscopic model of the interaction between the excitadsorbate–substrate complex and substrate electronpairs causing the finite lifetime. The picture is based onsimplified description of the electron hole pairs as a bathdipoles, and a dipole–dipole interaction between systembath. All parameters were connected to results from etronic structure calculations. This direct derivation of tcoupling constants from first principles is different from thmore common treatments based on a reduced descriptionharmonic baths.

In this first attempt to study laser induced desorptiwith the surrogate Hamiltonian method, converged excistate dynamics could be obtained. The convergenceasymptotic observables proved to be more difficult. The cvergence behavior with respect to the parameters ofmethod was characterized. As a fully quantum mechaniand therefore coherent method, the surrogate Hamiltonsuggests that experimentally observed bimodality of velocdistributions can be caused by quantum interferences knas Stu¨ckelberg oscillations. The surrogate Hamiltonian trement represents the first attempt to microscopically mothe relaxation which subsequently leads to desorption.only parameter entering the system–bath interaction wasdipole chargeq. This parameter was estimated by considing the geometry and electronic structure of the substrThe value obtained agrees very well with an estimate frCI calculations. The model leads to desorption probabilitin the same order of magnitude as the experiment and tovelocity distributions in the experimentally observed rang

The results obtained with the surrogate Hamiltonimethod, in particular the shape of the velocity distributioof desorbing molecules, should be considered as preliminsince the convergence requires improvement. They sugghowever, that the experimentally observed bimodality canexplained by quantum interferences due to different paways. A similar interpretation had been given in Ref. 6albeit with a simplified treatment of the relaxation. An altenative reason was suggested within a two-dimensionalchastic wave packet treatment.5,42 There, the experimentallyobserved bimodality was connected to a bifurcation ofwave packet on the excited state caused by the topologthe excited state potential energy surface. These two hypeses could be tested by an experiment as well as theorestudies which change the vibrational frequencies of thetential while leaving the chemistry invariant. This could baccomplished, for example, by using different isotopes ofNO molecule.

While in a MCWF approach the lifetime of the excitestate is empirically chosen and adjusted to give the cordesorption probability, the surrogate Hamiltonian approayielded expectation values in the right range for both expmental observables which can be captured within a odimensional treatment, the desorption yield and the destion velocities, without any adjustable parameters. The exshape of the velocity distributions could, however, notreproduced. The restriction of the present model to onemension is certainly a flaw. A two-dimensional stochaswave packet treatment showed a better compatibility w

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the experimental results, i.e., bimodal velocity distributioA combination of two-dimensionalab initio potential energysurfaces with the microscopic treatment of the dissipationthe surrogate Hamiltonian therefore paves the way towacomplete quantum mechanical description of the experim

In future work, vibrational relaxation for the grounstate dynamics should be included. This could be accplished by employing a second bath modeling the surfphonons. Vibrational relaxation for the excited state dynaics is insignificant due to the difference in time scales.contrast the ground state wave packet leaves the potewell on the time scale of picoseconds meaning that vibtional relaxation plays a role. Furthermore, the dynamisimulations of the surrogate Hamiltonian should be suppmented by a detailedab initio calculation of the dipolecharge of nickel oxide. While an estimate based on CI cculations has been given in Ref. 63, a more accurate intigation would further reduce the uncertainty of this paraeter in the surrogate Hamiltonian approach. Additionallyshould be investigated in more detail whether a substrmediated excitation is feasible within the surrogate Hamtonian. This would allow for a direct comparison of the curently applied direct and a purely substrate-mediaexcitation mechanism, and it would permit a theoreticalvestigation of the DIET to DIMET transition.64 A substrate-mediated, i.e., indirect excitation mechanism can be modby exciting the TLS by the laser pulse,

HBF~ t !5E~ t !(i

m i~si†1si !. ~14!

Indirect excitation of the adsorbate has so far been tresemi-phenomenologically by the two-temperature modelglecting, for example, the nonthermal nature of excielectrons.32,33 Another approach took into account the nolinear optical response of the substrate treating, howeverinteraction between substrate and adsorbate in a TD-framework.34,35 In contrast to these approaches, a surrogHamiltonian treatment would allow for a microscopic dscription of the interaction between laser pulse and surfelectrons. It needs to be investigated, however, whethcomparatively small number of bath modes would be sucient to model excitationand deexcitation of the system duto the bath, i.e., whether such an approach would be numcally feasible. While modeling substrate-mediated excitatwill require some effort, the description of a two-pulse eperiment with the direct excitation model as employed in tpaper is comparatively straightforward within the surrogHamiltonian. It requires, however, the characterization othird electronic state describing the NO molecule, anbound electron and the positively charged surface.

ACKNOWLEDGMENTS

We would like to thank Roi Baer for fruitful discussionand Stephan Thiel for reading the manuscript. Financial sport from the German–Israeli Foundation for Scientific Rsearch and Development~GIF! and from the Deutsche Forschungsgemeinschaft~SPP 1093! is gratefully acknowledged


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The Fritz Haber Center is supported by the Minerva Gesschaft fur die Forschung GmbH Mu¨nchen, Germany.

APPENDIX: MAPPING THE TWO-DIMENSIONS OF THEBATH ONTO ONE DIMENSION

The electron hole pairs which make up the bath aresumed to be localized on single Ni–O pairs. Then the battwo dimensional~2D! considering the uppermost layer of thNiO surface or three dimensional~3D! in case several layerare considered. However, only the distance of each elechole pair from the NO molecule and the direction of its dpole moment are important. In a one-dimensional treatmof the primary system only electron hole pairs with dipomoments parallel or antiparallel to the surface normal ctribute to the dipole–dipole interaction. Therefore the batheffectively one dimensional.

The fact that NiO has cubic lattice structure can be uto develop an algorithm to map a 2D or 3D bath onto odimension~the distance! and a sign~the direction of the di-pole!. In 2D, each Ni–O pair is located at a point of a qudratic lattice. The lattice points correspond to numbersn, 0<n<NBPN, for which n5 i 21 j 2 holds with i , j 50,...PN. This means that all square numbers and sums ofsquare numbers need to be found to determine the lapoints. A theorem from number theory can be employEvery integer can be factorized into prime numbersp52,p54m11, andp54m13. If and only if all prime factorsp54m13 of n occur an even number of times in the fatorization,n is a sum of two square numbers. The distancethis lattice point to the origin~which is the site below the NOmolecule! is then given by

distance of TLS5Ana0 ~A1!

with a0 half the lattice constant. The sign of the dipole mment is given by

i 1 j even→1,~A2!

i 1 j odd→2.

If n can be factorized into different pairs (i , j ), the num-ber of possible factorizations corresponds to the numbetimes this distance occurs,

occurrence5Number of different ~ i , j !•4, ~A3!

where the factor 4 accounts for fourfold symmetry. Sinpoints which are connected by a 90° rotation are identifithe surface slab is mapped onto a sphere. The usual asstion of periodic boundary conditions corresponds to a mping onto a torus and does not make use of four-fold symetry.

The outlined algorithm allows one to map a 2D banamely the dipoles in the uppermost layer of Ni–O paionto one dimension. If additional Ni–O layers shall btreated to account for transport into the surface, the simpapproach describes every layer as a separate bath~cf. Fig.18!. This means that all correlations between layers areglected. For the NO/NiO~100! system this should not poseserious restriction. From physical considerations, there isready one restriction on the correlations between layers:


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up and down dipoles marked in bold in Fig. 18 get excitedan electron transfer from the same oxygen atom, therefois very unfavorable that they are excited simultaneouAnalogously, it is unlikely that a nickel atom gets an electrfrom both the oxygen above and below. Therefore, thiscitation can be excluded. So what really is neglected arecorrelations between the down dipole marked in bold anddipoles in the first layer and the correlations between thedipole marked in bold and all dipoles in the second layKeeping in mind that so far only two to three simultaneoexcitations within one layer needed to be allowed, thisproximation should not be severe. It should be kept in mthat the approximation relies on the electronic structureNiO, in particular on the localizedd-orbitals. Thus this algo-rithm is not general, and the validity of the approximatimight break down for other oxides, for example. Howevthen the whole ansatz of Eq.~9! might become questionablefor example, more than nearest-neighbor interaction shobe included for a more delocalized electronic structure.

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