Time-Resolved Femtosecond Photoelectron Spectroscopy by Field

ARTICLE pubs.acs.org/JPCA

Time-Resolved Femtosecond Photoelectron Spectroscopy by Field-Induced Surface Hopping Roland Mitric,*,† Jens Petersen,‡ Matthias Wohlgemuth,‡ Ute Werner,‡ Vlasta Bonacic-Koutecky ,*,‡,§ Ludger W€oste,† and Joshua Jortner|| †

Fachbereich Physik, Freie Universit€at Berlin, Arnimallee 14, D-14195 Berlin, Germany, Institut f€ur Chemie, Humboldt-Universit€at zu Berlin, Brook-Taylor-Str. 2, D-12489 Berlin, Germany § Interdisciplinary Center for Advanced Science and Technology, University of Split, Mestrovicevo Setaliste bb., HR-21000 Split, Croatia School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel

)

‡

ABSTRACT: We present the extension of our field-induced surface hopping method for the description of the photoionization process and the simulation of time-resolved photoelectron spectra (TRPES). This is based on the combination of nonadiabatic molecular dynamics “on the fly” in the framework of TDDFT generalized for open shell systems under the influence of laser fields with the approximate quantum mechanical description of the photoionization process. Since arbitrary pulse shapes can be employed, this method can be also combined with the optimal control theory in order to steer the photoionization or to shape the outgoing electronic wavepackets. We illustrate our method for the simulation of TRPES on the prototype system of Ag3, which involves excitation from the equilibrium triangular geometry, as well as excitation from the linear transition state, where in both cases nonadiabatic relaxation takes place in a complex manifold of electronic states. Our approach represents a generally applicable method for the prediction of time-resolved photoelectron spectra and their analysis in systems with complex electronic structure as well as many nuclear degrees freedom. This theoretical development should serve to stimulate new ultrafast experiments.

1. INTRODUCTION The time-resolved photoelectron spectroscopy is one of the most powerful experimental techniques for the interrogation of ultrafast nonadiabatic processes since it is sensitive both to the nuclear dynamics as well as to the character of the electronic wave function.1-5 In this ultrafast pump-probe technique, a femtosecond pump pulse creates a wavepacket in the electronically excited state of the system whose time evolution is probed by photoionization induced by a time-delayed probe pulse. Both the energy distribution as well as the angular distribution of the photoelectrons can be measured and contain information about the coupled electronnuclear dynamics. In recent years, time-resolved photoelectron spectroscopy has been used to probe excited state dynamics in a variety of systems ranging from organic and biochromophores,6-10 metal clusters11-15 to molecular switches,16 etc. Parallel to the development of experimental techniques, theoretical approaches for simulation of TRPES using both the full quantum mechanical description of nuclear motion17-27 as well as the semiclassical description based on the Wigner distribution approach8,15,28-30 have been devised. However, the theoretical interpretation of TRPES is still challenging since, in addition to multistate nonadiabatic dynamics, the description of the photoionization process also has to be taken into account. Recently, Mitric et al. have introduced the field-induced surface hopping method (FISH) that allows both to treat nonadiabatic r 2010 American Chemical Society

effects as well as to include the laser fields directly into the excited state dynamics simulations.31 This method combines quantum electronic state population dynamics, in which both the nonadiabatic coupling between the states as well as the coupling induced by the laser field is included, with classical nuclear dynamics in a manifold of electronic states. Concerning the description of the photoionization, Mitric et al. have recently introduced the transition dipole moments between excited states in the framework of TDDFT and have extended this to the simulation of photoionization probabilities using the discretized continuum and the Stieltjes imaging procedures. This has been applied to the investigation of the ultrafast photodynamics of organic chromophores, providing the interpretation for the experimental TRPES.29,30 In this paper we present a formulation of the FISH method augmented by the approximate description of the ionization continuum and implemented in the framework of TDDFT. We also extend the formulation of TDDFT nonadiabatic dynamics as well as the calculation of transition dipole moments between Special Issue: Graham R. Fleming Festschrift Received: July 9, 2010 Revised: September 2, 2010 Published: October 13, 2010 3755

dx.doi.org/10.1021/jp106355n | J. Phys. Chem. A 2011, 115, 3755–3765

The Journal of Physical Chemistry A excited states to open shell systems. This method allows to simulate laser-driven photoionization processes using laser pulses with arbitrary shapes and therefore can be used both to simulate the spectroscopic observables as well as to control the photoionization process and to shape the electronic wavepackets. We illustrate the scope of our method on the prototype example of the small metal cluster Ag3, which involves electronic excitation from the equilibrium triangular ground state geometry, as well as from the linear transition state. This system already exhibits the complexity of excited states characteristic for metal nanoclusters, which is reflected in a high density of states as well as the presence of strong nonadiabatic relaxation effects connected with degeneracies of the electronic states at particular highly symmetrical geometries. The investigation of ultrafast processes in metal clusters is of fundamental importance for establishing the relation between structure, size, and optical properties and the time scales of nonradiative processes that determine their photoemission. This is particularly significant for the development of new photonic and biosensing materials in which small clusters are used as emitters. As we have previously shown on the example of anionic gold clusters,15 the lifetimes of excited electronic states and the mechanisms of nonradiative relaxation are strongly size dependent and can be changed by orders of magnitude just by adding or removing a single atom. In this sense the investigation of photochemical and photophysical properties of clusters is of fundamental interest and has been only scarcely explored so far. The dynamics of Ag3 has been studied in the framework of the negative ion-to neutral-to positive ion (NeNePo) spectroscopy allowing to identify the time scales of geometrical relaxation in the ground state as well as the fundamental features of internal vibrational energy redistribution and its nature such as resonant and dissipative.32 In contrast to the ground state dynamics of Ag3, which involves only adiabatic geometric relaxation, the excited state dynamics is expected to exhibit also strong nonadiabatic effects that can open new photochemical reaction channels not available in the ground state. Thus, in order to establish the connection between the TRPES signal and the underlying dynamical processes we will present the theoretical simulations of the timeresolved photoelectron spectra of Ag3 using our FISH method. The approximations underlying our approach and their influence on the appearance of the TRPES signal will be critically examined. The paper is structured as follows: first, the theoretical formulation of the FISH method extended by the discretized continuum approximation for the description of the photoionization and the simulation of TRPES signals will be presented. Subsequently, the method will be illustrated on the simulation of the TRPES for Ag3. The influence of Franck-Condon factors on the appearance of the spectra will also be discussed.

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2.1. Theory for Time-Resolved Photoelectron Spectra within the FISH Method. To describe laser-induced photoioni-

zation processes and to simulate TRPES, we present the extension of our FISH method31 by the discretized continuum approximation allowing the use of fields of moderate intensity and with arbitrary shape. Briefly, our FISH method has been formulated in the framework of the Wigner phase space representation of quantum mechanics33,34 and is based on the semiclassical limit of the quantum Liouville-von Neumann (LvN) equation ^0 - ! ^ ip^ F ¼ ½H μ 3! ε ðtÞ, F

ð1Þ

^ 0 represents the field-free nuclear for the density operator . H Hamiltonian for a molecular system with several electronic states in the Born-Oppenheimer approximation, and μB is the dipole moment operator determining the interaction with the laser field εB(t). In the lowest order in p, the commutators in the quantum LvN equation reduce to the classical Poisson brackets34 and the semiclassical time evolution equations for the diagonal (Fii) and off-diagonal (Fij) density matrix elements become F_ii ðq, p, tÞ ¼ fHi ðq, pÞFii g 2X Imð! μ ij ðqÞ 3 ! ε ðtÞFji Þ p j

ð2Þ

i μ ðqÞ 3 ! ε ðtÞðFjj - Fii Þ F_ij ðq, p, tÞ ¼ - iωij ðqÞFij þ ! p ij X i þ ð! μ ðqÞ 3 ! ε ðtÞFkj - ! μ kj ðqÞ 3 ! ε ðtÞFik Þ ð3Þ p k6¼ i, j ik where q and p represent the coordinates and momenta, respectively. The diagonal density matrix elements determine the quantum mechanical electronic state populations and the offdiagonal elements describe the electronic coherence. The curly braces denote the Poisson brackets, Hi is the Hamiltonian function for the electronic state i, ωij(q) is the energy gap, and μBij(q) the transition dipole moment between the electronic states i and j. As has been demonstrated previously15,29,31,32,35 this set of equations can be used to develop analytic expressions for the simulation of a variety of time-resolved observables such as pump-probe spectra and time-resolved photoelectron spectra by utilizing the timedependent perturbation theory and by restricting to Gaussian pulse shapes. The diagonal phase space functions Fii(q,p,t) corresponding to the electronic state populations can be represented by a swarm of independent trajectories propagated in the ground and excited electronic states, Fii ðq, p, tÞ

2. THEORETICAL FORMULATION In this section we present first the FISH method extended by the discretized continuum approximation for the description of photoionization processes in order to simulate TRPES. In addition, the influence of vibrational Franck-Condon factors on the appearance of the TRPES signals will be also accounted for. In the second part, we briefly outline the specific procedure for the calculation of nonadiabatic couplings and transition dipole moments in the frame of TDDFT generalized for openshell systems which are needed to carry out nonadiabatic molecular dynamics (MD) “on the fly”. In the third part, the computational aspects are presented.

¼

1 X k θ ðtÞδðq - qik ðt; q0 , p0 ÞÞδðp - pik ðt; q0 , p0 ÞÞ ð4Þ Ntraj k i

where each δ function represents a trajectory qik(t;q0,p0), pik(t;q0, p0) in the electronic state i with initial conditions q0 and p0, and Ntraj is the number of trajectories. The switching parameter θki (t) determines in which state the trajectory resides at the time t and has a value of one or zero if the trajectory k resides in the state i or not.35 The population transfer between the electronic states induced by the coupling with the laser field is simulated by allowing the trajectories to switch between the states in analogy with the Tully’s surface hopping method,36 which is commonly used in order to describe field free nonadiabatic transitions in 3756

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molecular systems.37-43 The simulation of the laser-induced dynamics in the framework of our previously derived FISH method31 is performed in the following three steps: (i) The initial conditions for an ensemble of trajectories are generated by sampling, for example, the canonical Wigner distribution function or a sufficiently long classical trajectory in the electronic ground state, allowing for the consideration of temperature effects. (ii) For each trajectory which is propagated in the framework of MD “on the fly” based on the chosen quantum chemical method, the density matrix elements Fij are calculated by numerical integration. If the initial electronic state is a pure state and decoherence effects can be neglected, as it is in our case, the set of eqs 2-3 can be reduced to the time-dependent Schr€odinger equation in the manifold of adiabatic electronic states coupled by the laser field εB(t) and by the nonadiabatic coupling terms Dij

simulation of spectroscopic observables in a manifold of several bound (non-ionized) electronic states.31 Discretized Continuum Approximation for Photoionization. For extending the applicability of the FISH method to the simulation of time-dependent ionization processes, the transition to the ionization continuum has to be included in addition to the description of the bound electronic state dynamics. For this purpose, we add to the bound electronic states of the molecular system in which the trajectories are propagated a manifold of singly ionized continuum states and allow the system to switch from the bound electronic states to the ionized states. In the framework of the FISH method, this can be achieved by augmenting the time-dependent Schr€odinger eq 5 by a set of equations describing the ionization continuum. The electronnuclear wave function of a combined system with both bound as well as continuum states can be represented as Ψðr, R, tÞ ¼

ip_ci ðtÞ ¼ Ei ðRðtÞÞci ðtÞ X jð! μ ij ðRðtÞÞ 3 ! ε ðtÞ þ ipDij ðRðtÞÞÞcj ðtÞ

i

þ ð5Þ

where ci(t) are the expansion coefficients of the electronic wave function from which the density matrix elements can be calculated as Fij = ci*cj. It should be noticed that the adiabatic state energies as well as the transition dipole matrix elements are parametrically dependent on the nuclear trajectory R(t). The inclusion of the nonadiabatic couplings d Dij ¼ ÆΨi ðRðtÞÞj Ψj ðRðtÞÞæ dt

X

ð6Þ

is mandatory since the potential energy surfaces in complex molecular systems commonly exhibit regions of strong nonadiabatic coupling, as for example in the vicinity of conical intersections, which have strong influence on the course of the dynamics of photoexcited molecules. In particular, these effects can be substantial after the field has ceased. The nuclear trajectories R(t) are propagated by solving Newton’s equations of motion, for example, by using the Verlet algorithm:44 X :: M RðtÞ ¼ θi ðtÞrR Vi ðRðtÞÞ ð7Þ i

In eq 7 the parameter θi(t) has a value of unity for the state in which the trajectory is propagated at the given time and zero for all other states and Vi(R(t)) is the adiabatic potential energy of the electronic state i. The forces rRVi(R(t)) acting on the nuclei are calculated “on the fly”. The solution of eq 7 provides continuous nuclear trajectories that reside in different electronic states according to the quantum mechanical occupation probabilities given by Fii. (iii) To determine in which electronic state the trajectory is propagated, the hopping probabilities are calculated in each nuclear time step from the change of the electronic state populations and are used in a stochastic process to decide if a state switch occurs. Although the individual trajectory is allowed to jump, the total number of trajectories in a given state is a continuous function of time. The phase of the electronic wave function is preserved, and our procedure gives rise to the full quantum-mechanical coherent state population. The above presented FISH method is suitable for the simulation and control of laser-induced processes as well as for the

ðNÞ

ðNÞ

χi ðR, tÞΨi ðr; RÞ

XZ

ðN - 1Þ

dE χj

ðN - 1Þ

ðR, E, tÞA ½Ψj

ðr; RÞφj ðEÞ ð8Þ

j

(N-1) where χ(N) (R,E,t) represent the nuclear wavei (R,t) and χj packet in the bound and continuum states, respectively. Ψ(N) i (r;R) are the eigenfunctions of the N-electron Hamiltonian, (r;R)φj(E)] rewhile the antisymmetrized product A [Ψ(N-1) j presents the continuum eigenfunctions of the combined ion-free electron Hamiltonian in which the electron-core interaction has (r;R) is the jth cationic been neglected. In this product, Ψ(N-1) j state, and φj(E) is a free electron scattering state. The summation extends over the whole manifold of singly ionized states. The wave function ansatz in eq 8 can be inserted in the full electronnuclear time-dependent Schr€odinger equation including the coupling to the electric field, and a set of equations for the time evolution of the continuum portion of the nuclear wavepacket of (R,E,t) can be derived as45 the ionized system χ(N-1) j

ðN - 1Þ

ipχ_ j

- 1Þ ðN - 1Þ ^ þ EðN ðR, E, tÞ ¼ ðT þ EÞχj ðR, E, tÞ j X ðNÞ ! ε ðtÞ 3 ! μ ij ðR, EÞχi ðR, tÞ ð9Þ i

In order to connect eq 9 with our trajectory-based FISH simulations, we have introduced the semiclassical approximation related to Tully’s original procedure51 as outlined in the Appendix. The derivation yields a set of equations for the time evolution (E,t) of a continuum state [Ψ(N-1) of the amplitudes c(N-1) j j þ E: (r;R)φj(E)] with the energy E(N-1) j ðN - 1Þ

ip_cj

ðN - 1Þ

ðN - 1Þ

ðE, tÞ ¼ ðEj ðRðtÞÞ þ EÞcj ðE, tÞ X ðNÞ ! ε ðtÞ 3 ! μ ij ðRðtÞ, EÞci ðtÞ

ð10Þ

i

where E is the kinetic energy of the released photoelectron. Equation 10 can serve as a basis for the approximate description of photoionization in the framework of the FISH method. Thus, in order to include the photoionization process, the time-dependent Schr€odinger eq 5 describing the bound electronic state dynamics along the classical trajectories can be augmented by eq 10. The system of eqs 5 and 10 provides a basis for the simulation of laserdriven photoionization processes and can be used to calculate the 3757



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TRPES. Notice that for Gaussian laser pulses and in the perturbation theory limit this procedure of calculating TRPES is equivalent to the analytical formulation derived in ref 32. Furthermore, since arbitrary pulse shapes can be employed, our FISH method extended for photoionization can be also combined with the optimal control theory in order to steer the photoionization process or to shape the outgoing electronic wavepackets. It should be emphasized that our approach represents a general framework for simulating time-dependent photoionization dynamics based on classical trajectories. Depending on the availability of transition dipole moments to the ionization continuum either the exact approach can be used or additional approximations can be made (e.g., the Condon approximation). In this contribution, we chose a pair of Gaussian pulses with the time delay t d to simulate time-resolved photoelectron spectra using eqs 5 and 10, representing the FISH approach augmented by discretized continuum states. The intensity of the TRPES signal at a particular value of the photoelectron kinetic energy E is obtained by averaging over the whole ensemble of individually propagated trajectories according to: 1 X X ðN - 1Þ jcj ðE, t f ¥Þj2 Sðtd , EÞ ¼ Ntraj j Ntraj

ð11Þ

However, due to the classical description of the nuclear motion, the vibrational degrees of freedom that can also influence the form of the TRPES signal were not taken into account. This approximation implies the assumption that the vibrational energy of the wavepacket does not change upon ionization. Role of Franck-Condon Factors. To investigate the qualitative influence of the Franck-Condon (FC) factors on the TRPES, we use the first order perturbation theory approach outlined in ref 32. Briefly, for this purpose eq 9 is projected onto the vibrational eigenstates |νjæ of the jth cationic electronic state, leading to

employed to derive an analytic expression for the continuum state population. After integrating over the duration of the probe pulse, the population of a specific cationic vibrational state |νj æ corresponding to the kinetic energy E of the photoelectron is given by XZ ¥ 2 ðN - 1Þ ∼ dτj! μ ij j2 jFi, νj j2 jcνj ðE, t f ¥Þj exp -

- 1Þ þ EðN νj

Sðtd , EÞ ¼ Z

is the energy of the minimum of the jth cationic where E(N-1) j0 state, F i,νj is a classical approximation for the time-dependent FC factor, and μB ij (R(t),E) is the transition dipole moment evaluated at the position of the nuclear trajectory R(t). This allows to study the dependence of the TRPES signal on the FC factors. For laser pulses of Gaussian form and low intensity, the first-order perturbation theory can be

ðNÞ - Ei

σ2pr p2

ðN - 1Þ

ðEpr - ðEj0 !

ðNÞ - Ei, vib Þ - EÞ2

¥

ðNÞ

σ2pr p

2

ðN - 1Þ

ðEpr - ðEj ðNÞ

- Ei ðRðτÞÞ - Ei, vib Þ - EÞ2

(N-1)

i

exp -

ð14Þ

Z Ntraj 1 X X X ¥ ðN - 1Þ dEj, vib Ntraj 0 j i

exp -

i

ðN - 1Þ

-¥

i

ðτ - td Þ ðN - 1Þ dτj! μ ij ðRðτÞ, EÞj2 jFij ðEj, vib Þj2 exp σ2pr -¥

ðN - 1Þ

- 1Þ - 1Þ - 1Þ ðE, tÞ ¼ ðEj0 þ EðN þ EÞcðN ðE, tÞ ip_cðN νj νj νj X ðNÞ ci ðtÞ! ε ðtÞ 3 ! μ ij ðRðtÞ, EÞFi, νj ð13Þ

!

where σpr is the temporal width of the probe pulse, E pr is its energy, while E (N) is the electronic energy and E (N) i i,vib the vibrational energy of the neutral wavepacket. For a classical description based on localized trajectories the quantum energy E (N-1) þ Eν j(N-1) consisting of the electronic and j0 the vibrational part can be approximated by the total energy of the classical system E (N-1) (R(t)) þ E(N-1) j j,vib . The FC factor then becomes a function of the continuous vibrational energy of the cation as F ij (E (N-1) j,vib ). Upon ionization the vibrational part of the energy can be partly transformed into the photoelectron kinetic energy. The total population for a given photoelectron kinetic energy E can thus be obtained by integration over the vibrational energy E (N-1) j,vib . Therefore, in the classical approximation the photoelectron signal can be calculated according to

- 1Þ - 1Þ - 1Þ ðE, tÞ ¼ ðEj þ EðN þ EÞcðN ðE, tÞ ip_cðN νj νj νj X ! ε ðtÞ 3 Æνj j! μ ij ðR, EÞjχðNÞ ðR, tÞæ ð12Þ -

, the with the vibrational expansion coefficients c νj of the cationic state j, and the electronic energy E(N-1) j vibrational energy E νj (N-1) of the corresponding vibrational state. The classical approximation for the neutral state (N) wavepacket χ(N) i (R,t) by a localized δ-function χ i (R,t) ≈ (N) c i (t)δ(R - R(t)) leads to

ðτ - td Þ σ 2pr

2

2

!

ðN - 1Þ

ðRðτÞÞ þ Ej, vib

ð15Þ

Assuming a Gaussian energy dependence of the FC factors around the initial vibrational energy E(N) i,vib of the neutral wave2 (N) (N-1) 2 ) = exp (-[1/(2σ )](E packet as Fij(E(N-1) j,vib i,vib - Ej,vib ) ) the (N-1) integration of eq 15 over Ej,vib can be carried out explicitly and the influence of the FC factors can be studied. 2.2. Nonadiabatic Couplings and Transition Dipole Moments in Time-Dependent Density Functional Theory for Open Shell Systems. For the simulation of time-resolved photoelectron spectra in the frame of the extended FISH method presented in Sec. 2.1 the nonadiabatic coupling terms DKI as well as the transition dipole moments μBKI between excited electronic states K and I are needed. In this section we present a general procedure for the calculation of these quantities in the frame of linear response TDDFT, which is an extension of the previously introduced approaches8,28-30,38 and is applicable both to openand closed-shell systems. For the calculation of these coupling 3758



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terms we use the CIS-like representation of the excited state wave function given by: X jΨK ðr; RðtÞÞæ ¼ cKR, i, a jΦR, i, a ðr; RðtÞÞæ i, a X cKβ, i, a jΦβ, i, a ðr; RðtÞÞæ ð16Þ þ i, a where |ΦR,i,a(r;R(t))æ and |Φβ,i,a(r;R(t))æ represent singly excited configurations in the R and β spin subspaces in which one electron has been promoted from the occupied Kohn-Sham orbital i to the virtual orbital a with spin R or β, respectively. The K and cKβ,i,a in eq 16 are determined such expansion coefficients cR,i,a that the wave function in eq 16 gives rise to the same density response as obtained by the linear response TDDFT procedure. As it has been shown in ref 28, the expansion coefficients are then connected to the solutions X and Y of the TDDFT eigenvalue problem according to cKR, i, a ¼ ðεR, a - εR, i Þ-1=2 ðXRK, ia þ YRK, ia Þ

ð17Þ

cKβ, i, a ¼ ðεβ, a - εβ, i Þ-1=2 ðXβK, ia þ YβK, ia Þ

ð18Þ

This allows us to define the excited state wave function |ΨK(r; R(t))æ, which is employed to calculate the nonadiabatic couplings as well as the transition dipole moments. Notice that in the case of a closed-shell system this procedure reduces to the expressions that have been previously published in refs 8, 28, and 29. The nonadiabatic coupling DKI defined in eq 6 can be approximated by using the finite difference for the time derivative, Δt 1 DKI R t þ ðÆΨK ðr; RðtÞÞjΨI ðr; Rðt þ ΔtÞÞæ 2 2Δt - ÆΨK ðr; Rðt þ ΔtÞÞjΨI ðr; RðtÞÞæÞ ð19Þ

In this way, the nonadiabatic coupling is obtained as the overlap of two electronic wave functions for the states K and I at subsequent timesteps t and t þ Δt and can be further reduced to the overlap of spatial Kohn-Sham orbitals as described in refs 8 and 28. The transition dipole matrix elements μBKI between two electronic states K and I can be calculated according to ! μ jΨ ðr; RðtÞÞæ ð20Þ μ ¼ ÆΨ ðr; RðtÞÞj^ K

KI

¼

I

PP i;a j;b

þ

I ^ jΦR;j;b ðr; RðtÞÞæ cK R;i;a cR;j;b ÆΦR;i;a ðr; RðtÞÞj μ

XX i;a

I ^ jΦβ;j;b ðr; RðtÞÞæ cK β;i;a cβ;j;b ÆΦβ;i;a ðr; RðtÞÞj μ

j;b

ð21Þ where a and b indicate virtual and i and j occupied orbitals, respectively. The dipole matrix elements on the right-hand side of eq 21 can be reduced to the time-dependent transition dipole moments between Kohn-Sham orbitals.29 2.3. Computational Procedures. The field-induced nonadiabatic dynamics of Ag3 in the frame of the FISH method has been performed using the spin-unrestricted linear response

TDDFT with the PBE exchange-correlation functional46 combined with the triple-ζ valence plus polarization atomic basis set (TZVP)47 and the relativistic 19 electron effective core potential of the Stuttgart group.48 The initial coordinates and momenta q0 and p0 have been sampled (i) from a long ground state trajectory at 300 K for the triangular start geometry and (ii) from a ground state trajectory at 300 K of the anionic species for the linear start geometry. A total number of 48 trajectories has been used for the further simulations. The FISH dynamics with direct inclusion of the pump laser pulse has been carried out in the ground state and (i) six excited states for the triangular geometry and (ii) 8 excited states for the linear geometry. The Newtonian equations of motion have been integrated using the velocity-Verlet algorithm with a time step of 1.0 fs. The electronic equations of motion (eqs 5 and 10) have been integrated with a time step of 10-4 fs. Notice that, due to the singular behavior of the nonadiabatic coupling in the vicinity of conical intersections, special precaution is needed for the accurate integration of the electronic part. The nonadiabatic couplings and transition dipole moments along the trajectories, needed for the FISH simulations, have been calculated in the frame of TDDFT according to eqs 19 and 21 using the procedure presented in Section 2.2. For each dynamics step, the hopping probabilities were calculated, and the decision if a switch between states occurs was made in a stochastic process. For times after 90 fs, when the duration of the pump pulse was over, the nuclear velocities were uniformly rescaled after hopping in order to ensure energy conservation. The photoionization due to the probe pulse has been simulated using the FISH procedure augmented by the discretized contiuum approximation outlined in Section 2.1. For this purpose, the electronic continuum corresponding to the cationic ground state was approximated by 110 discrete energy levels above the ionization limit with an equal spacing of 0.027 eV. The transition dipole moments for the ionization from the bound neutral electronic state in which the trajectory is propagated to the ionized states (μBij(R(t),E) in eq 10) have been all set to a constant value. Notice, that Takatsuka and McKoy have shown that the variation of transition dipole moments to the continuum with the nuclear geometry can be very important for the simulation of time-resolved photoelectron spectra.24 However, since the main purpose of our contribution is to extend our fieldinduced surface-hopping method to the simulation of photoelectron signals, and the transition dipole moments to the continuum in the frame of TDDFT are not available at the present stage, we employ here the Condon approximation for illustrative purposes. The implementation of transition dipole moments for ionization is under development and is out of scope of this contribution. The ionized population has been calculated along each trajectory for each given time delay td of the probe pulse by numerically propagating the time-dependent Schr€odinger eq 10 in the manifold of the current neutral excited electronic state and the states above the ionization limit under the influence of the probe pulse. The propagation of eq 10 was performed over the finite width of the Gaussian probe pulse in the time interval -3σ to þ3σ, where σ is connected to the full width at half-maximum (FWHM) of the pulse by σ = 1/2 FWHM (2 ln 2)-1/2. The final populations of the ionized states were obtained for the time after the probe pulse has ceased and were used to calculate the TRPES signal using eq 11. For studying the influence of FC factors the TRPES signal was also calculated according to eq 15 for Gaussian FC factor distributions of different widths. 3759



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Figure 2. Section of the potential energy surfaces for the ground and 8 excited states of Ag3 along the bending coordinate connecting the triangular with the linear geometry.

Figure 1. (a) Electronic absorption spectrum of triangular Ag3. Vertical bars represent the electronic transition, each being broadened by a Lorentzian distribution. The dominant excitations corresponding to the transition to the D4 state (marked by an arrow) are shown. (b) Electronic absorption spectrum of the transient linear geometry of Ag3. The dominant excitations corresponding to the transition to the D8 state (marked by an arrow) are shown.

3. RESULTS AND DISCUSSION We present here the simulated time-resolved photoelectron spectra of Ag3 based on multistate nonadiabatic molecular dynamics in the frame of TDDFT using the FISH method as outlined in Section 2. We wish to demonstrate the capability of our approach to simulate ultrafast spectroscopic observables of small metallic particles, such as Ag3, which are characterized by a manifold of excited states on which the nonadiabatic dynamics can take place. For this purpose the pump laser pulse is explicitly included and drives the bound state dynamics, while the probe pulse only serves to ionize the system without propagation of trajectories in the ionized state. We have carried out theoretical calculations of dynamics under two different sets of initial conditions: (i) the triangular equilibrium geometry of the Ag3 cluster in the ground 2B2 state; and (ii) the linear geometry of Ag3 in the ground electronic state. The linear geometry corresponds to the equilibrium configuration of the Ag3anionic species and constitutes a transition state for the neutral Ag3 cluster. Whereas the first simulation represents the excitation of an ensemble of neutral triangular Ag3 clusters, the second simulation reflects on the situation where the neutral, linear Ag3 clusters are produced by ultrafast photodetachment from the Ag3- anionic species, as conducted, for example, in the framework of NeNePo spectroscopy.32 The electronic absorption spectra for both triangular and linear geometries are shown in Figure 1a and 1b, respectively. The excited states that are populated by the pump pulse and

hence are the starting points for the nonadiabatic relaxation (D4 for the triangular structure, D8 for the linear one) are marked by arrows. In addition, the character of the corresponding transitions is indicated by the dominant excitations. For illustrative purposes we present in Figure 2 sections of the potential energy surfaces of the electronic states D0 - D8 along the bending coordinate connecting the triangular with the linear structure. It can be seen that the vicinity of the excited states is strongly structure dependent. The ground state, as already mentioned, has a minimum for an obtuse triangular structure. For the D4 state, which is mainly populated in the simulation starting in the triangular geometry, there is a minimum at a bending angle of about 80 corresponding to an opening of the triangle. Since at this geometry the energies of the states D2, D3, and D4 are nearly degenerate, nonadiabatic transitions from D4 to the lower states are expected to occur. The D8 state, which is mainly populated in the simulation starting at the linear geometry, has a shallow minimum at 150, where also near-degeneracy with lower-lying states occurs. In the following Sections 3.1 and 3.2 we wish to present both the excited state population dynamics and the simulated TRPES for the two sets of initial conditions corresponding to (i) the triangular structure (the equilibrium ground state electronic configuration) and (ii) the linear structure (the transition state in the ground electronic configuration). 3.1. Initial Conditions for Triangular Geometry. For initial excitation we have employed a Gaussian pump pulse with 50 fs FWHM and an energy which is resonant with the lowest-lying intense transition at 2.59 eV corresponding to the D4 state. The pump pulse intensity was 4.2 1011 W/cm2, which is sufficient to achieve considerable population in the D4 state. A higher intensity would excite Rabi oscillations, thus influencing the efficiency of the excitation. To illustrate the electronic state population dynamics we present in Figure 3a the quantum electronic state populations along one selected nuclear trajectory. The initial decay of the D0 population is caused by the interaction with the pump laser field, leading to the highest population of the D4 state. After the pump field has ceased the change of the electronic state population is induced by nonadiabatic coupling that is particularly strong in the vicinity of conical intersections. It should be noticed that although the electronic state populations change largely, the total population remains normalized, proving also the accuracy of the 3760


The Journal of Physical Chemistry A numerical integration. In addition, we present in Figure 3b selected transition dipole moments along the same nuclear trajectory, demonstrating their strong dependence on the nuclear geometry. The ensemble population dynamics induced by the pump pulse is shown in Figure 4a. In spite of the resonant pulse energy, the D4 state is only populated to about 50% even though the applied pulse intensity is relatively high. This is due to very low transition dipole moments to that state in a significant part of the initial ensemble (cf. also the large variation of the transition dipole moment μ04 during the dynamics shown in Figure 3b). After the pump pulse has ceased, the D4 state population decays with a time constant of 196 fs. The population is mainly transferred via the third to the second excited state, and the depopulation of the fourth state is essentially complete after 500 fs. The population of the D2 state decays within 2000 fs almost completely via the D1 to the ground state. In Figure 4b we present the time-dependent populations of the D0 and D4 states calculated using an ensemble

Figure 3. (a) Quantum electronic state populations along one selected nuclear trajectory of Ag3 started in the triangular geometry. (b) Absolute values of selected transition dipole moments along the same nuclear trajectory as in panel a.

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of 24 (dashed lines) as well as of 48 trajectories (full lines). Doubling the number of trajectories does not lead to a significant change of the electronic state populations, proving the convergence of our results with respect to the number of trajectories. The dynamical processes induced by the laser excitation are characterized by large geometrical fluctuations affecting all normal modes, which are caused by the gain of nuclear kinetic energy due to the nonradiative relaxation after the pump pulse. This leads to fragmentation of the cluster, mainly to Ag þ Ag2, for about 25% of the propagated trajectories. We present the simulated TRPES employing our approach outlined in Section 2.1 in Figure 5a. For the probe pulse, a 50 fs Gaussian with an energy of 5.71 eV and a maximum field intensity of 2.6 1010 W/cm2 was used in order to achieve efficient ionization. The probe pulse energy was chosen slightly below the ionization potential of the ground state (6.1 eV) in order to selectively probe the excited states population. It can be seen that the initial excitation to the D4 state is reflected in the appearance of a strong photoelectron kinetic energy (PKE) signal in the energy range from 1.7 to 2.7 eV. This signal decays and shifts to lower PKE mainly within 1000 fs, indicating the depopulation of the D4 state. For later times, only low intensity remains for PKE higher than 0.2 eV, corresponding to the part of trajectories that remains in excited states until the end of the simulation. The signal intensity near zero kinetic energy can be attributed to the trajectories in the ground state. To obtain a more quantitative picture of the PKE distribution we show in Figure 5b the time-dependent photoelectron signal intensities for different kinetic energy intervals. After initial excitation, a strong photoelectron signal intensity is obtained for the kinetic energies between 1.6 and 2.4 eV, which then decreases to around one tenth at 1000 fs. Beginning around 750 fs, the highest intensity is obtained for photoelectrons with kinetic energies between 0.0 and 0.4 eV. This strong decay of intensity for higher PKE values is also reflected in the photoelectron spectra for fixed probe pulse delays shown in Figure 5c. While for a short time delay of 200 fs there is a strong signal for PKEs between 1.5 and 2.5 eV, for later times the relative intensities for small PKEs around zero increase (from 0.4 at 200 fs to 1.0 at 1750 fs), whereas the maximum PKE decreases from 2.5 eV at 200 fs to 1.3 eV at 1750 fs. For investigating the influence of vibrational FC factors on the appearance of the photoelectron spectra, we have also calculated the TRPES for the triangular start geometry according to the eq 15 as outlined in the last part of Section 2.1. The TRPES for a model Franck-Condon factor distribution given by

Figure 4. (a) Population of the ground and 4 excited electronic states of Ag3 during the FISH simulation starting in the triangular geometry for an ensemble of 48 trajectories. The pump excitation with an energy of 2.59 eV mainly populates the D4 state. (b) Population of the ground and the D4 states for ensembles of 24 (dashed lines) and 48 (full lines) trajectories showing the convergence of the populations with respect to the ensemble size. 3761



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Figure 6. Time-resolved photoelectron spectra of Ag3 calculated using the first order perturbation theory expression of eq 15 for the triangular initial geometry. The probe pulse energy is 5.71 eV. An approximate Franck-Condon (FC) factor distribution was assumed to be (N) (N-1) 2 2 Fij(E(N-1) j,vib ) = exp (-(1/2σ )(Ei,vib - Ej,vib ) ). The signal has been calculated for two different widths of the FC factor distribution: (a) σ = 0.5 eV, (b) σ = 1.5 eV.

Figure 5. (a) Time-resolved photoelectron spectrum of Ag3 obtained from the FISH dynamics starting in the triangular geometry for a probe pulse of 5.71 eV. (b) Time-dependent photoelectron signal intensities for different photoelectron kinetic energy (PKE) intervals. (c) Photoelectron spectra for different time delays of the probe pulse. The signals in panels b and c have been normalized with respect to the highest peak. 2 (N) (N-1) 2 Fij(E(N-1) j,vib ) = exp (-(1/2σ )(Ei,vib - Ej,vib ) ) with an energy width σ of 0.5 and 1.5 eV are shown in Figure 6a and 6b, respectively. As can be seen, the main effect of expanding the energy range in which the Franck-Condon factors are large (i.e., increasing σ) consists in filling the photoelectron kinetic energy distribution that was previoulsly peaked at the maximum kinetic energies (cf. Figure 5a) into the range of lower energies. This corresponds to the transfer of a part of the energy of the probe pulse into the vibrational degrees of freedom and gives rise to photoelectrons with lower kinetic energy. 3.2. Initial Conditions for Linear Geometry. The population dynamics induced by a pump pulse resonant with the intense transition to the D8 state at 2.72 eV is shown in Figure 7. In contrast to the triangular geometry, the excitation of the transient linear geometry of Ag3 is far more efficient due to high transition dipole moments to the D8 state for the entire initial ensemble. Therefore, the application of a pump pulse of smaller intensity (7.6 1010 W/cm2) than in the case of the triangular structure

Figure 7. Population of the ground and 8 excited electronic states of Ag3 during the FISH simulation starting in the linear geometry. The pump excitation with an energy of 2.72 eV mainly populates the D8 state.

significantly populates the D8 state, after Rabi oscillations reaching a population of about 80%. The nonradiative relaxation of that state occurs with a time constant of 268 fs and is essentially completed after 750 fs. The population is transferred partially into the manifold of the states D4 - D7, but mainly in the second excited state whose population reaches ≈30% after 500 fs. For later times, the D2 population decays almost equally into the D1 and D3 excited states. However, the D3 state is also depopulated for times after 1000 fs. After the simulation time of 2000 fs, both the populations of the ground and the first excited state are about 30%, whereas 20% remain in the second and 10% in the higher excited states. In this simulation the dynamics is characterized by large geometric fluctuations similar to the previous case starting from the triangular geometry. The amount of cluster fragmentation of about 45% is even higher due to the more efficient excitation that leads to a stronger energy transfer from the laser field to the cluster. 3762



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100 fs) and lower intensities at longer delay times (about 0.2 after 1800 fs). For the higher energy photoelectrons up to 2.4 eV, there is only a low intensity during the whole simulation time, whereas for PKEs lower than 1.2 eV the intensity increases as a function of the delay time from 0 to 0.5. However, at the end of the simulation, there is still some intensity of photoelectrons up to energies of 1.6 eV. This behavior is also reflected in the photoelectron spectra for fixed probe pulse delay shown in Figure 8c. For the early delay times of 200 fs, the maximum PKE is centered at 1.7 eV with a relative intensity of 1.0. Then this maximum decreases to 0.15 at a delay time of 1750 fs. In addition, a new peak is evolving at a PKE of 0.4 eV starting with an intensity of 0.3 at 500 fs to almost 0.6 at 1750 fs which reflects the remaining population of the lower excited states D1 and D2. In summary, the comparison of simulated TRPES for the different choices of starting geometries (triangular vs linear) presented in Sections 3.1 and 3.2 illustrates the structural influence on the excited state dynamics and the corresponding time scales. In the triangular case, there is almost complete decay of the excited state population within 1 ps. For the linear starting geometry, which is of interest in the context of transition state spectroscopy and excited state dynamics, the depopulation of the highest excited state involved is almost equally fast. However, a significant part of the population is not transferred to the ground state but to lower-lying excited states within a time scale of about 2 ps. Thus, the decay of the total excited states population is much slower for the linear than in the triangular starting geometry. Our theoretical predictions can be confronted with the results of two types of experimental studies of TRPES. For the equilibrium starting geometry of Ag3 two-pulse driven TRPES, which involves excitation followed by ionization, will be desirable. For the linear starting geometry of the transition state of Ag3 a threepulse driven NeNePo TRPES, involving ionization of Ag3followed by a sequence of excitation and ionization of the resulting Ag3, will be of considerable interest. Figure 8. (a) Time-resolved photoelectron spectrum of Ag3 obtained from the FISH dynamics starting in the linear geometry for a probe pulse of 6.0 eV. (b) Time-dependent photoelectron signal intensities for different photoelectron kinetic energy intervals. (c) Photoelectron spectra for different time delays of the probe pulse. The signals in panels b and c have been normalized with respect to the highest peak.

The simulated TRPES is shown in Figure 8a. In this case, we used a 50 fs Gaussian probe pulse with an energy of 6.0 eV, and again a maximum field intensity of 2.6 1010 W/cm2. Due to the higher excitation efficiency of the pump pulse (see also Figure 7), the signal intensity after excitation is in general stronger than for the triangular case. After a Rabi oscillation the signal is centered around a PKE of 1.7 eV, indicating the population of the D8 state. During the dynamics, the intensity of this signal decreases and a new feature evolves around 0.4 eV, reflecting the population transfer from the D8 state to the lower-lying excited states. Furthermore, also the total signal intensity decreases as a result of population transfer to the ground state, which cannot be ionized by the applied probe pulse since the ionization potential for the linear structure is 7.1 eV. Overall, the decay of TRPES intensity is slower than in the triangular case. The time-dependent photoelectron signal intensities for selected PKE intervals shown in Figure 8b clearly confirm this observation. After excitation, the photoelectrons in the energy range between 1.2 and 2.0 eV have the highest intensity at short delay times (0.85 at

4. CONCLUSIONS We have presented the formulation of the FISH method augmented by the discretized continuum approximation for the description of the photoionization. This approach represents a general framework both for simulation of spectroscopic observables such as TRPES as well as for control of photoionization processes using laser fields with arbitrary shapes. The nonadiabatic FISH dynamics has been carried out using TDDFT extended for open-shell systems. In order to illustrate the method, we have chosen the prototype example of the Ag3 cluster that is characterized by a high density of excited electronic states and strong nonadiabatic effects. We have determined the mechanism and the time scale of the nonradiative excited state relaxation and have shown how their fingerprint is reflected in the TRPES. Moreover, the influence of the Franck-Condon factor distribution on the appearance of TRPES signals has been examined, which is important for the interpretation of experimental results. The scope of our approach is broad since it can be combined with the whole spectrum of the quantum chemistry methods, allowing for the simulation of the photoionization in complex systems such as metallic nanoclusters or biochromophores interacting with the environment and solvated systems. 3763



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’ APPENDIX

thus eq 26 becomes ðN - 1Þ

DAj ip

Starting from the set of equations of motion for the cationic (R, E, t), nuclear wavepackets χ(N-1) j ðN - 1Þ ðR, E, tÞ ipχ_ j

ðN - 1Þ þ Ej

ip 1 ðN - 1Þ 2 ðN - 1Þ ðN - 1Þ ðN - 1Þ rAj rSj þ Aj r Sj m 2 ! Z X i t ðN - 1Þ ðNÞ ðNÞ ! ! ε 3 μ ij Ai exp ðE þ E - Ei Þ dτ p 0 j i ¼ -

ðN - 1Þ þ EÞχj ðR, E, tÞ

^ ¼ ðT X ðNÞ ! ε 3! μ ij ðR, EÞχi ðR, tÞ

ð22Þ

i

we outline the introduction of the semiclassical approximation. First, the nuclear wavepackets are separated into amplitude and phase terms in the spirit of the Bohm-Madelung representation of quantum mechanics49,50 ðN - 1Þ

χj

ðN - 1Þ

ðR, E, tÞ ¼ Aj

ðR, E, tÞ exp

i ðN - 1Þ Sj ðR, E, tÞ p ð23Þ

ðNÞ χi ðR, tÞ

ðNÞ Ai ðR, tÞ

¼

i ðNÞ exp Si ðR, tÞ p

Dt

ð30Þ Comparison of eq 30 with the equation for the time evolution of the electronic state coefficients a(N-1) (E, t) of a multilevel j system in the interaction picture, ðN - 1Þ

ip

ð24Þ

daj

dt ¼ -

X i

ðNÞ ! ε 3! μ ij ai

! Z i t ðN - 1Þ ðNÞ exp ðE þ E - Ei Þ dτ p 0 j

Inserting these expressions into eq 22 and appropriately separating terms into two equations yields51 ðN - 1Þ

DSj

Dt

ðN - 1Þ

1 p2 r2 Aj ðN - 1Þ 2 ðrSj þ Þ þ Ej ¼ - 1Þ 2m 2m AðN j

ðN - 1Þ

ip

DAj

Dt

¼-

-

X i

ð25Þ

ip 1 ðN - 1Þ 2 ðN - 1Þ ðN - 1Þ ðN - 1Þ rAj rSj þ Aj r Sj m 2

i ðNÞ ðNÞ ðN - 1Þ ! ! ðS - Sj ε 3 μ ij Ai exp Þ ð26Þ p i

The first equation can be interpreted as a quantum mechanical analog to the Hamilton-Jacobi equation describing the nuclear motion in the electronic state j. The second equation describes (R, E, t). It the time evolution of the amplitude functions A(N-1) j can be seen that the temporal change of the amplitude function for a specific electronic state j depends both on the spatial change of amplitude and phase in that state, represented by the terms ðN - 1Þ

rAj

ðN - 1Þ

rSj

1 ðN - 1Þ 2 ðN - 1Þ þ Aj r Sj 2

ð27Þ

as well as on the amplitude functions of the other electronic states, represented by the terms X i ðNÞ ðNÞ ðN - 1Þ ! ε 3! μ ij Ai exp ðSi - Sj Þ ð28Þ p i Hence, the terms (28) are responsible for population transfer between different electronic states. Interpreting the phase terms and S(N-1) as classical actions leads to S(N) i j ðNÞ Si

ðN - 1Þ - Sj

Z ¼

t

ðN - 1Þ ðEj

ðNÞ þ E - Ei Þ dτ

ð31Þ shows the analogy between the terms describing the population transfer between different electronic states in both equations. (R, E, t) Thus, it is possible to relate the amplitude functions A(N-1) j (E, t) and to use the of the wavepacket to the coefficients a(N-1) j eq 31 for describing the change in the electronic state populations. In order to be consistent with our derivation of the FISH method outlined in Section 2.1, the eq 31 is transformed R to the Schr€odinger picture by using the definition aj = cj exp i/p t0(Ej þ E) d, leading to ðN - 1Þ

ip_cj

ðN - 1Þ

ðN - 1Þ

ðE, tÞ ¼ ðEj ðRðtÞÞ þ EÞcj ðE, tÞ X ðNÞ ! ε 3! μ ij ðRðtÞ, EÞci ðtÞ

ð32Þ

i

This equation describes the time evolution of the coefficients (E, t) representing a continuum state of energy E(N-1) þE c(N-1) j j corresponding to the jth electronic state of the cation and can be used for the approximate description of photoionization in the frame of the FISH method.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected] (R.M.); [email protected] (V.B.-K.).

’ ACKNOWLEDGMENT R. M. acknowledges the financial support in the framework of the DFG Emmy-Noether Program (MI-1236). J. P. thanks the Fonds der Chemischen Industrie for financial support. This work has been supported by the DFG in the framework of the SFB 450. ’ REFERENCES

ð29Þ

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Time-Resolved Femtosecond Photoelectron Spectroscopy by Field

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