Atomistic Modeling of Vibrational Action Spectra in Polyatomic

Article pubs.acs.org/JPCA

Atomistic Modeling of Vibrational Action Spectra in Polyatomic Molecules: Nuclear Quantum Effects F. Calvo,*,† C. Falvo,‡ and P. Parneix‡ †

ILM, Université Lyon I and CNRS UMR 5306, 43 Bd du 11 Novembre 1918, F69622 Villeurbanne Cedex, France Institut des Sciences Moléculaires d’Orsay, UMR CNRS 8214, Université Paris Sud 11, Bât. 210, F91405 Orsay Cedex, France

‡

S Supporting Information *

ABSTRACT: The response of a polyatomic molecule to an infrared (IR) laser pulse of varying frequency has been simulated by classical molecular dynamics simulations and by quantum methods based on the path-integral framework (PIMD), as well as quantum thermal baths (QTBs). The outcome of the trajectories was subsequently processed to predict a dissociation spectrum, from the precalculated rate constant. Naphthalene described by a tight-binding potential energy surface was chosen as a testing ground for the present problem, possibly emitting an hydrogen atom after a 12 ps long pulse. At low field intensities, the heating efficiency of the pulse is found to vary similarly as the IR absorption spectrum for all methods considered, reflecting the validity of linear response in this regime. At fields that are sufficiently high to induce statistical dissociation over mass spectrometry timescales, marked differences appear with the spectral features exhibiting additional broadenings and redshift, especially for quantum mechanical descriptions of nuclear motion. Those excessive broadenings are mostly caused by anharmonicities but also convey the inherent approximations of the semiclassical QTB method and point at limitations of the PIMD simulations when used in such strong out-of-equilibrium conditions.

1. INTRODUCTION

One difficulty in assigning structures by comparing a measured depletion spectrum to a calculated absorption spectrum is that the intensity in photodepletion measurements results from additional mechanisms than mere photon absorption, including energy relaxation, possible internal conversion, vibrational redistribution, and eventually dissociation. In some cases, such as two-color IR/UV spectroscopy, the times of exposure to laser excitations are short enough and the action spectrum is expected to mimic rather faithfully the absorption spectrum. In contrast, multiphoton excitations in the infrared range, as typically produced by free-electron lasers, involve long exposure times so that vibrational ladder climbing proceeds up to a dissociation threshold. The simultaneous occurrence of laser heating and vibrational redistribution gives rise to spectral variations due to anharmonicities, mostly additional redshifts and broadenings, which were already noted14−16 and analyzed before.17,18 In recent years we have attempted to rationalize those nonlinear effects by proposing two analyses both built on pure theory (i.e., without phenomenological considerations). A fully classical approach was inspired by earlier efforts in which a significant influence of an external electric field was found on the behavior of water19−21 or biomolecules22,23 simulated by

The detailed structure of complex molecules can often be characterized by spectroscopic methods probing the electronic (optical-UV range) or rovibrational (IR range) degrees of freedom. For isolated molecules produced in the gas phase, experiments rely on mass spectrometry to measure the propensity to dissociate, emit electrons, or ionize upon a specific laser excitation. For structural assignment such action spectra necessitate extensive comparison with theoretical spectra, and the standard procedure consists of evaluating the linear absorption line shape using quantum chemistry, typically at the density functional theory level. This approach often considers a harmonic description of vibrational motion, which is the most straightforward but suffers also from systematic errors that can be corrected by empirical scaling procedures.1,2 A more robust though computationally demanding framework to correct for anharmonicities consists of vibrational perturbation theory.3,4 The limitations of those static approaches, besides the accuracy of the underlying theory, mostly reside in the need for dedicated strategies extending past chemical intuition to produce likely candidate structures. Explicitly dynamical situations at finite temperature,5,6 or requiring a quantum mechanical treatment of nuclear motion to account for strong vibrational delocalization,7−13 can be used to tackle those issues and naturally account for nonperturbative anharmonicities. © 2014 American Chemical Society

Received: April 24, 2014 Revised: June 23, 2014 Published: June 23, 2014 5427

dx.doi.org/10.1021/jp5040147 | J. Phys. Chem. A 2014, 118, 5427−5436

The Journal of Physical Chemistry A

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reported before,37 artifacts on the results are not excluded. The comparison between equilibrium (absorption spectrum) and nonequilibrium (dissociation spectrum) properties will thus also serve as a quantitative test of validity of PIMD methods for explicit time-dependent perturbations in the weak and stronger regimes. As will be shown below, our simulations of vibrational action spectra show remarkable similarities with the absorption spectra, indicating that the most salient quantum effects on the spectral features are indeed correctly captured, including anharmonicities. However, those effects also appear sometimes exacerbated, due to various possible contributions but also pointing out some limitations of the present modeling, most likely in the use of PIMD methods under strong out-ofequilibrium perturbations. The article is organized as follows. In the next section, we lay out the general computational protocol used to evaluate the fragmentation spectrum in response to a single laser pulse and briefly describe the types of classical and quantum simulations implemented to monitor this response. The calculation of the dissociation rate constant, adapted from an earlier work on cationic species,38 is also detailed in this section. Section 3 presents and discusses our results in the limits of low and high laser fields. Some concluding remarks are finally given in Section 4.

molecular dynamics. This strategy was extended to predict a fragmentation probability upon exposure to a laser field, and its application to a small rocksalt cluster as a function of laser field frequency yields spectroscopic observables that show similarities, but also discrepancies, with respect to the absorption spectrum at equilibrium.24 As a complementary effort, a purely statistical, event-driven model based on ab initio ingredients and including photon emission and absorption, vibrational relaxation, and dissociation processes could be all treated quantum mechanically but assuming a simplified quartic force field.25 These two models confirmed the specific dynamical effects of bandshift and broadening on the vibrational action spectrum. Unfortunately, because they were applied to rather different systems, the predictions of the two models cannot be easily compared to each other, and it is unclear whether the observed effects are ultimately caused by thermal or quantum fluctuations. In particular, the models differ not only in the quantum description of vibrations (classical dynamics vs discrete vibrational levels) but also in the way the nuclei interact with the field (continuous work exchanged with the laser vs stepwise absorption and emission jumps). The goal of the present contribution is to bridge the gap between these two approaches, by modeling the vibrational response of a polyatomic molecule to an external IR laser pulse as a function of its frequency, treating the dynamics at the atomistic level of detail, but attempting to include quantum mechanical aspects in the description of nuclear motion. Nuclear quantum effects are already known to be potentially important in absorption spectra,10−13 especially when light atoms and anharmonicities are involved. Hence they should be at least as important in nonlinear vibrational action spectra. In order to address a similar compound as in ref 25, we have chosen the neutral naphthalene molecule as a benchmark system and the tight-binding (TB) model of Van-Oanh and coworkers26 to model its energy and dipole moment surfaces. The TB force field was initially derived by Wang and Mak27 and was reparametrized in the context of IR spectroscopy.26 It predicts harmonic line positions that are, on average, within 3% of quantum chemical calculations,4 hence of comparable accuracy of DFT itself (a more systematic comparison is given as Supporting Information). While the TB force field does not claim the same accuracy as density functional theory, it provides a suitable description compatible with explicit molecular dynamics on a statistical ground. Previous simulations using this model28,29 have validated the path-integral (PI) approaches of partially adiabatic centroid (PA-C) MD30,31 and ringpolymer (RP) MD,32 as well as the more approximate but computationally efficient quantum thermal bath,33,34 all in the context of thermal equilibrium and time-dependent observables (correlation functions) within linear response theory. The use of the RPMD and PA-CMD frameworks is justified in this respect, in the limit of low excitation fields for the response not to perturb the equilibrium state. These PIMD methods notably have been shown to hold in the limit of short times and for harmonic potentials.43,44 Those conditions are not met if the external field is strong enough so as to make the system depart from its initial equilibrium (e.g., when its temperature increases). While thermostated PIMD methods have been extended to the out-of-equilibrium problem of free-energy calculations,35,36 no such extension exists for the approximate RPMD and PA-CMD methods of quantum dynamics, and although their use beyond the linear response regime has been

2. METHODS The theoretical strategy followed to determine dissociation spectra relies on the time multiscale approach39 combining explicit MD simulations to cover the short picosecondnanosecond timescale together with a kinetic theory spanning the experimentally relevant microseconds−milliseconds timescale. In this section, we emphasize the main stages of those calculations for the present problem and, in particular, our tentative inclusion of nuclear quantum effects using semiclassical and path-integral frameworks. 2.1. General Protocol. All calculations are initialized at time t = 0 from an equilibrium sample at the desired temperature, fixed here at T = 300 K, and trajectories under the presence of the laser pulse are carried out until a fixed time tpulse. The final states of the trajectories are eventually used to predict the propensity for dissociation based on their internal energy, after a measurement time tobs. Those three steps, sketched in Figure 1, require a number of ingredients for their practical implementations, starting with a given potential energy surface V(R), where R denotes the set of atomic coordinates that is appropriate for the system under study. The dipole moment surface μ⃗ (R), also required for the absorption spectrum, is not directly needed for the interaction with the field in the TB model (see below). 2.1.1. Thermalization. Molecular dynamics simulations coupled with Nosé−Hoover thermostats are performed to generate a sample of 1000 phase space configurations separated consecutively by 1 ps. Nuclear quantum effects are included in the path-integral representation by replicating the classical degrees of freedom into P = 32 beads and propagating them according to either the partially adiabatic centroid MD30,31 or the ring-polymer MD 32 rules of evolution. Although fundamentally different, the PA-CMD and RPMD methods are very similar in their practical implementations, as they mostly differ in the choice of masses and in the way observables are calculated.41 Further details about those PIMD methods in the context of vibrational spectroscopy are given in recent 5428



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spreading of molecular orientations in experiments. For the phase space configurations, some distinction is in order in the case of the quantum systems. For the path-integral descriptions, the possible global translation and rotation arising from the nonconservation of linear and angular momenta in massive thermostatting are removed from the centroids, and the PIMD equations of motion are propagated normally, without thermostat but in the presence of the field. The same procedure to remove the finite linear and angular momenta is applied to the samples generated by the QTB method, but the propagation under field irradiation, performed without the thermostat, is thus entirely classical. This difference reflects the semiclassical nature of the QTB method, in Wigner’s sense, and in comparison with the more rigorous PIMD methods, it provides an interesting intermediate approach for separating the respective roles of quantum effects and laser heating on the final spectrum. As mentioned in the introduction, the use of the RPMD and PA-CMD approaches to simulate the quantum dynamics under the laser field is not strictly valid except at very low excitations. Our tentative application of those methods for relatively intense laser fields should thus be considered as a practical test for their possible use for modeling a complex question in which electronic and nuclear degrees of freedom all require a quantum treatment. As a final comment on this issue, it should be noted that we did not include the magnetic contribution of the laser field, as its magnitude is several orders of magnitude lower than the electric component. 2.1.3. Dissociation Kinetics. Once the trajectory in the presence of the field is stopped after 50 ps, its final state is recorded and the internal energy Ef is stored. In the case of path-integral methods, the virial estimate46,47 is used to determine Ef from the last 5 ps of the trajectory. The dissociation probability p(tobs) after some measurement time tobs is evaluated, assuming statistical relaxation of this internal energy among all vibrational degrees of freedom, the kinetics of dissociation being described by a first-order process with rate constant k(Ef). This assumption is valid as long as tobs is short enough with respect to the timescale of IR spontaneous emission, in which case the energy of the system is constant between the end of the pulse and its statistical dissociation. The probability pα that trajectory α will have dissociated at time tobs is simply given by pα = exp[−k(tobs − 2t0)] ≃ exp[−ktobs]. Averaging pα over the M trajectories yields the final dissociation probability p̅ = (∑αpα)/M. 2.2. Tight-Binding Model in External Electric Field. For the sake of brevity, the details of the potential energy and dipole moment surfaces of the tight-binding model specifically reparametrized for hydrocarbons can be found in the original reference of Van-Oanh et al.26 and will not be repeated here. In the presence of an external electric field ℰ⃗ , the electronic structure is modified and the atomic levels are shifted proportionally.45 The field thus alters the diagonal elements Hiα,iα of the electronic Hamiltonian according to

Figure 1. Sketched flowchart of the computational protocol used to evaluate the vibrational dissociation spectrum. The preparation of initial conditions at thermal equilibrium and the propagation under laser irradiation are both treated at the atomistic level by classical or quantum MD, whereas the fragmentation probability is obtained from a kinetic rate theory. See the text for more detail about the different quantum methods of quantum thermal bath (QTB), partially adiabatic centroid MD (PA-CMD), or ring-polymer MD (RPMD).

contributions from the Tuckerman40 and Marx41 groups. For those methods, massive thermostatting was used to equilibrate the system and generate a Boltzmann sample at 300 K. Alternatively to path-integral MD, nuclear quantum effects were considered in a more semiclassical fashion using a quantum thermal bath (QTB).34 Briefly, QTBs mimic vibrational delocalization using a colored noise Langevin thermostat designed in such a way as to populate oscillators according to their quantum mechanical harmonic thermal energy E(ω) = ℏω[1/2 + 1/(eβℏω − 1)]. Here, the portable version of QTB42 was used to create the colored noise and integrate the equations of motion. The colored noise is generated on-the-fly by direct and inverse Fourier transforms of the bare and filtered power spectrum in the frequency range of −2Ωmax ≤ ω ≤ 2Ωmax, and we chose a maximum frequency Ωmax of 4000 cm−1 and a discretization of this range into 200 frequency bins. 2.1.2. Interaction with the Laser Field. A statistically meaningful number M = 500 of independent trajectories were propagated in the presence of a laser pulse, taken as a uniform but time-dependent electric field ℰ⃗ = ℰ0ef⃗ (t), with e ⃗ a random unit vector, ℰ0 the maximum amplitude of the field, and f(t) its temporal profile given by ⎡ 1 ⎛ t − t ⎞2 ⎤ 0⎟ ⎥cos[ω(t − t0)] f (t ) = exp⎢ − ⎜ ⎣ 2⎝ σ ⎠ ⎦

(1)

With these notations, the pulse duration measured at mid height is τ = 2σ(2 ln 2)1/2 = 12 ps, the pulse being centered at time t0 = 25 ps for a total simulated time of 2t0 = 50 ps for the entire trajectory. The field amplitude ℰ0 is related to the laser intensity I through I = cε0ℰ20/2, where c and ε0 are the velocity of light and the dielectric permittivity of vacuum, respectively. The magnitude ℰ0 was given two sets of values depending on the frequency range, corresponding to a low excitation regime in which the molecule is assumed to respond linearly, or a high excitation needed to heat the molecules sufficiently so it can dissociate after some fixed macroscopic time. The initial configurations of those trajectories are taken from the pools of 1000 thermalized phase space structures, and a random orientation of the electric field is drawn to mimic the

⃗ Hiα , iα = Hi(0) α , iα − eri· ,

(2)

where ri denotes the position vector of atom i, H(0) iα,iα is the unperturbed diagonal term of the Hamiltonian for atom i and orbital index α, respectively. For the present TB model, s orbitals are considered for carbon and hydrogen atoms, with three additional p orbitals for carbon atoms.26 5429



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2.3. Dissociation Rate. Phase space theory (PST)48−50 was used to calculate the dissociation rate constant k as a function of internal energy E. Following earlier work on cationic naphthalene,38 the ingredients of this calculation were obtained by dedicated density-functional theory calculations of the binding energies Ei and harmonic frequencies {ωi,γ, γ = 1··· 3N − 6} for the naphthalene parent and the two 1-naphthyl and 2-naphthyl radicals that can be produced upon hydrogen emission. The choice of performing dedicated quantum chemistry calculations for those properties was dictated by the poor reliability of the TB model to describe these radicals and its inability to handle fragments. For the three systems, only the lowest spin multiplicity was considered, and the results of the calculations are given as Supporting Information. The overall rate k is the sum of the specific rates ki leading to the dissociation of each i-naphthyl product, and the PST expression for ki reads50 ki(E) =

g 2Jth hΩ(E)

∫0

E −ΔEi

Figure 2. Calculated dissociation rate constant of neutral naphthalene into C10H7 + H, for the two possible products of 1- and 2-naphthyl. The corresponding rate for the cationic molecule is also reported. The dashed lines highlight the energy where the rate equates 1 ms−1. The inset shows the variations of the number of rotational states with total (rotational + translational) kinetic energy released, for neutral and cationic naphthalene.

Γi(ε)Ωi(E − ΔEi − ε)dε (3)

where ΔEi denotes the dissociation energy of product i, Ωi its vibrational density of states, with Ω the density of vibrational states of the parent naphthalene and Γ the number of rotational states for a given translational and rotational kinetic energy released ε. In eq 3, Jth ∼ 65ℏ is a thermal angular momentum taken as in ref 38 from the rotational Boltzmann distribution at 300 K, g is a ratio of symmetry factors corresponding to the equilibrium structures of the parent and product (here g = 2/8 for D2h vs Cs groups for both products). The function Γ was calculated assuming a simple sphere + atom approximation for the products, leading to Γi(ε) = BjJrmax(Jrmax + 1) with Bi the rotational constant of product i and Jrmax(ε) the maximum orbital momentum compatible with mechanical laws of conservation. Note that we use here a semiclassical form for Γ, with Jrmax treated as an integer product of ℏ , in contrast with ref 38, where angular momenta were considered to be continuous classical variables. Contrary to the cationic case, where the products feel a −C4/r4 interaction at long distance, the two neutral products interact through a van der Waals potential −C6/r6. This impacts the maximum angular momentum Jrmax, which for a general potential in −Cp/rp satisfies the following equation in Jr:50 ε = BJr (Jr + 1) +

cationic naphthalene were also recalculated for comparison using the above semiclassical expression for Γ, rather than the classical form used in ref 38. The variations of k(E) for the cation are in very good agreement with this earlier work, indicating that the classical approximation for the rotational degrees of freedom is reliable. The much lower dissociation rate obtained for neutral naphthalene is thus mainly the result of a lower number of rotational states and, only marginally, of differences in the densities of states or dissociation energies. The dissociation rate obtained here is rather low, and for a significant fragmentation to take place at the typical time of tobs ∼ 1 ms, an energy of at least 8 eV is required in order that ktobs ∼ 1. Given the short duration of the pulse, this constraint will impose a lower limit on the field strength ℰ0. 2.4. Anharmonic Absorption Spectra. The anharmonic absorption spectra 0 (ω) were calculated for the four approaches describing nuclear motion from the usual method connecting 0 to the refraction coefficient n and to the Fourier transform of the dipole moment time autocorrelation function (ACF)53

Jr2p /(p − 2) Λp

0(ω) = (4)

2 Cp2/(p − 2)(μp) p /(p − 2) p−2

∞

∫−∞ ⟨μ⃗ (t )·μ⃗ (0)⟩e−iωt d t

(6)

where β = 1/kBT with kB the Boltzmann constant, and = the volume of the sample. It is reasonable to fix n(ω) as a constant for an isolated system. To calculate the dipole moment ACF, we perform classical or path-integral MD trajectories initiated from the corresponding thermal samples and propagate the equations of motion without thermostat for 100 ps per trajectory. For the PACMD and RPMD methods, the dipole moment vector (rather than operator) is obtained at the centroids or as the average over the P sets of beads, respectively.

with Λp =

2πω(1 − e−βℏω) 3=ℏcn(ω)

(5)

μ being the reduced mass of the dissociating system. Taking C6 from the known Lennard-Jones parameters of the OPLS database,51 the number of rotational states is found not to depend markedly on the isomer due to very similar rotational constants (see Table 1 in the Supporting Information). However, it is about 3 orders of magnitude lower than the value obtained for the cation, as depicted as in inset in Figure 2. After determining the densities of states Ω and Ωi by direct counting52 with the harmonic frequencies scaled by 0.97, the global rate constant k(E) is obtained by solving eq 3. The results are shown in Figure 2 as the main plot. The rates for

3. RESULTS AND DISCUSSION The influence of an infrared laser pulse on the naphthalene molecule has been studied in two spectral ranges close to IR active bands, namely 810−870 cm−1 and 2800−3150 cm−1. The former band corresponds to a very intense out-of-plane 5430



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bending mode classically located at 859.1 cm−1, which is nearly degenerate with a much less active mode at 858.5 cm−1. The other chosen band collects the eight C−H stretching modes spanning between 3112.7 and 3128.3 cm−1, from which only 5 are active, but for the present neutral molecule, with a much lower intensity than the out-of-plane bendings. MD simulations under laser excitation were conducted in these two ranges, varying the laser frequency by steps of 1 and 5 cm−1, respectively. In absence of selection rules for IR absorption in a (semi)classical description of nuclear motion, the efficiency of laser heating should directly reflect the IR activity, at least in the linear regime of low external fields. To illustrate the dependence of laser frequency on the heating efficiency, we first show in Figure 3a the variations of internal energy during

although both significantly interact with the molecule near the field peak. These fluctuations higher than in the classical case are also typical of the quantum preparation method that lead to a greater spreading of the momenta. The more efficient pulse at the lower frequency indicates a shift in resonance that is in keeping with the redshift of the quantum IR absorption spectrum,28 confirming that the QTB method is able to account for those intrinsically quantum spectral features for individual molecules.54 On a more quantitative footing, the average energy variations upon exposure to the laser pulse are shown in Figure 4 as a

Figure 4. Averaged energy ΔE transferred from a low intensity laser field to the system, as a function of frequency in the two spectral ranges ω = 810−870 and 2800−3150 cm−1 (connected dots). The results were obtained from the four methods of preparing and propagating the nuclear equations of motion described in the text. The anharmonic IR absorption spectrum is also shown in the background, and the vertical dashed lines locate the static IR active harmonic absorption lines.

Figure 3. Internal energy obtained from classical MD trajectories interacting with laser pulses of frequencies 840 and 855 cm−1, initiated from identical initial conditions obtained from (a) classical or (b) quantum thermostats. The internal energy is averaged over successive time intervals of 100 fs to filter out the thermal noise. The field envelope f(t) is superimposed in the background.

two typical classical MD trajectories performed at ω = 840 and 855 cm−1, for a field amplitude of ℰ0 = 0.01 V/nm. For the sake of a more stringent comparison, the initial conditions and field orientations are taken as identical for both trajectories, only differing in the IR laser frequency. Short-time averages over 100 fs were carried out to wash out most of the noise affecting the variations of internal energy during the interaction with the pulse. From the depicted variations, the resonant pulse at 855 cm−1 strongly interacts with the molecule and heats it up by about 0.05 eV. Conversely, the off-resonant pulse at 840 cm−1 has a poor net effect, although fluctuations in internal energy are clearly seen when the field is close to its maximum amplitude. Those results are consistent with the location of the main IR active band in the classical absorption spectrum. The same analysis can be attempted but using the sample of structures equilibrated by the quantum bath. Two trajectories starting from the same initial condition and field orientation but differing in the laser frequency are depicted in Figure 3b. For these semiclassically prepared simulations, the pulse at 840 cm−1 is much more efficient than the one at 855 cm−1,

function of the frequency ω, for a field amplitude corresponding to mild heating at maximum ℰ0 = 0.01 V/nm in the range of 810 ≤ ω ≤ 870 cm−1 and ℰ0 = 0.02 V/nm in the range of 2800 ≤ ω ≤ 3150 cm−1. At those magnitudes, the perturbation caused by the field is sufficiently weak not to heat the system too much and compromise the thermal equilibrium assumption underlying the validity of both RPMD and PACMD methods. The results are given for the four treatments of nuclear motion, and the figure also depicts the equilibrium IR absorption spectra corresponding to these different methods, scaled accordingly in each spectral range to facilitate comparison. The IR active modes at 0 K obtained from the harmonic frequencies are also superimposed as vertical lines. From a general point of view, the average energy variations mimic quite well the absorption spectrum and exhibit a main peak in each range, and an additional but smaller peak near 825−835 cm−1 becomes a shoulder in quantum descriptions of nuclear motion. This feature reflects some important 5431



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anharmonicity effects on both the position and IR activity of the 858.5 cm−1 harmonic mode. Quantum effects are clearly seen on the absorption spectra as significant redshifts (by ∼20 cm−1 for the band at 855 cm−1, ∼ 120 cm−1 for the C−H stretchings band) and noticeable broadenings. The band near 3000 cm−1 is less reliable (more noisy) with the RPMD method, as the consequence of interferences between physical and ring-polymer modes.28,40,41 The semiclassical spectrum obtained with the QTB preparation method resembles more the PA-CMD spectrum, except for a markedly broader C−H stretching band already noted earlier.54 The positions of the bands are remarkably similar, indicating that the response to such mild fields clearly follows the expected linear behavior. In particular, quantum effects, even if only present in the sample of initial conditions, are correctly reproduced in this response as the red-shifted and broadened peaks. The most apparent differences between the absorption spectra and the internal energy variations lie in the excessive broadening displayed by the latter in the 810−870 cm−1 range, an effect not systematically present in the higher frequency range. This broadening partly originates from the laser bandwidth, which is estimated as about 1 cm−1. The internal energy variations generally show comparable magnitude but higher fluctuations than the corresponding classical data. These fluctuations are particularly high in the case of the RPMD description in the C−H stretchings domain, consistently with the already mentioned interference problem affecting the absorption spectrum in this range.28,40,41 The averaged energies reached by the system after laser irradiation remain very limited in magnitude, even near the resonances where they barely reach 0.1 eV above the thermal energy Eth. Although Eth depends significantly on the classical or quantum description of nuclear motion through the possible presence of zero-point energy (about 4.2 eV in the present case of naphthalene), the final energies are not high enough to even dissociate over the millisecond timescale. The simulations have thus been repeated using much stronger fields of ℰ0 = 0.1 V/nm (810−870 cm−1 range) or 0.2 V/nm (2800−3150 cm−1) for the heating to be compatible with dissociation. Such high fields are not realistic as to mimic experimental IR lasers, and this should be kept in mind when comparing with related measurements.55,56 In addition and as aforementioned, they may bring the PIMD simulations outside of their validity range, at least because the thermal equilibration hypothesis could break down under efficient heatings. The average energy transferred by the laser to the system is represented in Figure 5 as a function of ω, in both spectral ranges and for the four types of simulations. The primary effect of the more intense field is to increase the internal energy transferred to the system, the approximate 2 orders of magnitude increase reflecting the factor ten in the maximum field amplitude E0. Besides this general amplification in the heating efficiency, the spectra exhibit some additional broadening and redshift with respect to both the IR absorption spectrum and the variations of internal energy at lower field. Those effects are mainly manifestations of stronger anharmonicities resulting from the higher heating.17,18,24,25 To further assess this statement, we have carried out a set of classical and quantum simulations at the same high fields but varying the length of the laser pulse around 12 ps and at the two fixed frequencies of 850 and 2950 cm−1. The average energies transferred to the system by the laser are shown in Figure 6 as a function of the pulse duration τ. At 850 wavenumbers, the laser

Figure 5. Averaged energy transferred from a higher intensity laser field to the system, as a function of frequency in the two spectral ranges ω = 810−870 and 2800−3150 cm−1 (connected dots). The results were obtained from the four methods of preparing and propagating the nuclear equations of motion described in the text. The harmonic and anharmonic IR absorption spectra are again shown as vertical dashed lines and in the background, respectively.

Figure 6. Average energy transferred to the molecule from a laser pulse at fixed frequency ω but with varying duration, in the classical and quantum (QTB) descriptions of nuclear motion. (a) ω = 850 cm−1 and ℰ0 = 0.1 V/nm; (b) ω = 2950 cm−1 and ℰ0 = 0.2 V/nm.

is on the red side of the classical resonance and on the blue side of the quantum resonance. The energy scales approximately linearly τ at short values; however, in the classical case, a maximum in ⟨ΔE⟩ is reached near τ = 14 ps. These nonmonotonic variations are a manifestation of dynamical anharmonicities and indicate that the resonance depends nontrivially on the laser exposure through the interplay between effective frequencies and continuously increasing internal energies: below 14 ps exposure, the resonance takes place at frequencies higher than the laser, but the energy 5432



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enough when off-resonant. Near resonance, the distributions obtained for classical and quantum initial conditions are relatively similar and span over 6−7 eV at mid height. They are centered with maxima close to each other near 7 eV above the equilibrium structure energy. The lower heating efficiency in the quantum case is thus ascribable to the zero-point energy shift, combined with a less intense response caused by greater delocalization. The dissociation probability after 1 ms resulting from the final internal energies reached by the molecule upon laser irradiation is represented in Figure 8 as a function of laser

transferred to the molecule is unsufficient to alter the resonance itself, so it coincides with the laser. Above 14 ps, the energy transferred is high enough for the peak to further shift to the red and cross the resonance. In the quantum case, the excitation is always off-resonance and on the blue side of the peak, with longer durations affecting the energy transferred owing to the rather broad peak, thus explaining the lower but smoothly increasing variations of ⟨ΔE⟩ in Figure 6a. In contrast, the excitation at 2950 cm−1 is always offresonance and on the red side of the classical peak, and even pulses as long as 16 ps are not sufficient to transfer significant amounts of energy. Quantum delocalization significantly redshifts the C−H stretching peak by more than 100 wavenumbers, and the pulse duration has a much larger, monotonic but roughly quadratic, influence on the average energy transferred. These nonlinearities are again a consequence of anharmonicities, and the progressive redshift of the peak as more energy is transferred into the molecule by the laser field. From Figure 5, and for the three approaches considered, nuclear quantum effects are found to generally exacerbate spectral shifts and broadenings with respect to the classical behavior, by small but noticeable extents. Even more strikingly, their main consequence is to attenuate the amount of heating by about half relative to the classical case and in both spectral ranges. To interpret those differences, it is instructive to consider the absolute internal energy distributions rather than the difference only, and we have done so in Figure 7 where the

Figure 8. Dissociation probability p̅ as a function of laser frequency in the two spectral ranges ω = 810−870 cm−1 and ω = 2800−3150 cm−1 (connected dots). The results were obtained from the four methods of preparing and propagating the nuclear equations of motion described in the text. The harmonic and anharmonic IR absorption spectra shown as vertical dashed lines and in the background, respectively.

frequency for the four descriptions of nuclear motion. The variations generally convey the broadened and red-shifted peaks of Figure 5, the wings of the action spectra being significantly more extended on both the red and blue sides in the quantum case. Those small but finite dissociation probabilities are caused by the events from the high-energy tail of the internal energy distribution. Although rare, these events significantly contribute to the average dissociation probability p̅ due to the exponential variations of the rate constant with internal energy. The classical action spectrum shows features that are in rather good agreement with the absorption spectrum, except for a slightly excessive broadening and a minor redshift caused by the laser bandwidth and dynamical anharmonicities.24,25 Quantum delocalization further shifts and broadens these features by comparable amounts for the three methods, and with the exception of the C−H stretching modes with the ringpolymer MD approach there is general consensus among them. Although there is no directly available experimental spectrum for this neutral molecule, a comparison can be made with the cation, for which IRMPD spectra have been recorded.55,56 The

Figure 7. Normalized internal energy distributions of the classical and quantum thermal populations obtained during the thermalizing stage and final distributions upon irradiation by laser pulses at frequencies of 840 and 855 cm−1.

internal energy distributions before and after the pulse are shown at 840 and 855 cm−1, as obtained in classical simulations initiated from classical or quantum (QTB) initial conditions. The two initial distributions represent the pure thermal populations and, as expected, the quantum distribution is shifted by a zero-point energy of several electronvolts, and it is broader due to vibrational delocalization. Those quantum fluctuations are responsible for the generally broader features in the data shown in Figures 4 and 5, starting with the IR absorption spectra. The final distributions are all significantly broader than the thermal distributions, especially for the laser frequencies closest to resonance (855 cm−1 in the classical case and 840 cm−1 for the quantum case) but also and significantly 5433



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the result of stronger anharmonicities reached at higher internal energies. Although the energy transferred by the laser to the system is markedly higher in the classical case, the propensity for dissociation is similar due to the compensating zero-point shift from which quantum systems start. However, the simulated dissociation spectrum remains too broad with respect to experiments56 or fully quantum model calculations,25 a feature we interpreted either as intrinsic to the approximate QTB method or, in the case of PIMD approaches, to the breakdown of validity of the thermal equilibrium assumption under stronger time-dependent perturbations. The relatively short duration of the pulse and its rather high intensity probably also contribute to those broadenings by preventing a more complete vibrational energy redistribution. As was observed for the IR absorption spectrum at equilibrium,54 the quantum thermal bath method used to prepare the initial conditions performs reasonably well in reproducing the specific quantum effects predicted by the pathintegral methods. Due to the relatively long times of irradiation and to the strength of the perturbation, the validity of those methods was clearly challenged. However, except for this excessive broadening, the relatively good agreement between equilibrium (absorption spectrum) and out-of-equilibrium (dissociation spectrum) properties indicates that even in the presence of a relatively strong perturbation, the PA-CMD and RPMD approaches could be suitable for large polyatomic systems. Future efforts toward a more realistic modeling of traditional laser pulses could target the systematic characterization of spectral features for a fixed energy deposit ℰ02τ. In addition, the mechanisms of laser heating and the energy redistribution accompanying and following the pulse should be devoted further attention by dedicated classical and quantum simulations, precisely trying to quantify the rates of those mechanisms. Of interest could also be the extension of the present work to other types of pulses to describe, for example, environments,57 chirping effects,58,59 or polarized light. Finally, a more ambitious investigation would reside in the determination of photodepletion spectra in the optical excitation range, using for instance ionized rare-gas clusters as a suitable testing ground. Those systems have received attention from both the experimental60 and computational61 points of view and could be convenient to get insight into the importance of the quantum nature of the field, treated either as an instantaneous vertical excitation on an electronic excited state, or from a continuous interaction with an electromagnetic field. Such a study could be useful already from the perspective of classical nuclei and would shed light onto complementary aspects of complex dynamics in photoexcited polyatomic molecules.

most significant differences appear to be the excessive broadenings in the quantum action spectra, whereas the redshift due to delocalization is, for its part, realistic. A fully quantum modeling of IR multiphoton dissociation spectra of cationic naphthalene based on a quartic force field also found narrower peaks and confirmed the additional redshift caused by dynamical laser heating.25 One significant difference between this kinetic model and the present atomistic approach is the quantum treatment of the laser field, which is fully classical here. This approximation notably precludes any spontaneous emission from occurring in the present treatment. However, at the rather low-field strengths considered here and for such lowenergy IR photons, it is unlikely that field quantization could explain alone such broadenings. Another cause could be the unrealistically high electric field amplitude needed to reach the dissociation threshold after only tens of picoseconds exposure. Such high fields may not be compatible with a complete vibrational energy redistribution, as occurring in experiments conducted under milder but longer excitations, and could artificially contribute to reaching strongly anharmonic regions of the potential energy surface on short timescales but still while under irradiation. While excessive broadenings naturally arise with the QTB method,54 the situation is not as clear for the PIMD methods. One immediate contribution to those fluctuations lies in the time averages used to calculate the internal energy from the virial estimate, full convergence requiring (in principle) infinitely long trajectories. This results in more noisy energy distributions than those obtained with the QTB method in Figure 7, although the corresponding action spectra in Figure 8 are comparably broad and noisy. On a more fundamental level, it seems safer to assume that those broadenings mark the limit of validity of the RPMD and PA-CMD approaches used here far away from equilibrium.

4. CONCLUSIONS Action spectroscopy stands as one of the few fingerprint methods to unravel the structure of gas phase compounds, through comparison with simulations. Depending on the type of experiment, measured depletion spectra may differ from absorption spectra that are usually considered for this comparison due to various dynamical effects reflecting the complex processes taking place from the excitation to the redistribution eventually leading to fragmentation. In the present contribution, we attempted to model such processes by combining explicit molecular dynamics simulations at the level of atomistic details with a statistical kinetic description of subsequent dissociation. As a complementary effort to related work on purely classical24 or fully quantum25 methods, the importance of nuclear quantum effects was under scrutiny here, either within the two path-integral frameworks of partially adiabatic centroid MD31 and ring-polymer MD32 or using the more approximate (and semiclassical) approach of quantum thermal baths.34 In the low-field regime, we found that the heating efficiency varies with laser frequency similarly as the IR absorption spectrum, except for a small excessive broadening due to the laser bandwidth for such short picosecond excitations. Quantum effects generally shift the spectra to lower frequencies and contribute to slightly broaden the peaks, two effects that are remarkably reproduced by these explicit simulations under laser exposure. At higher fields, the broadening effect and, to a lesser extent, the lineshifting effect both become enhanced as

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ASSOCIATED CONTENT

S Supporting Information *

Harmonic IR spectra for the present TB model and compared with density-functional theory calculations at the B97−1/TZVP and B3LYP/cc-pVDZ levels and quantum chemical data for neutral naphthalene and its two dehydrogenation products, 1and 2-naphthyl, as used to evaluate the dissociation rate constant from phase space theory. This material is available free of charge via the Internet at http://pubs.acs.org. 5434



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AUTHOR INFORMATION

Corresponding Author

*E-mail: fl[email protected]. Present Address

Laboratoire Interdisciplinaire de Physique, CNRS and Université Joseph Fourier Grenoble, France. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors wish to acknowledge generous computational resources from the regional Pôle Scientifique de Modélisation Numérique in Lyon.

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Atomistic Modeling of Vibrational Action Spectra in Polyatomic

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