On-the-fly ab initio semiclassical evaluation of electronic coherences in polyatomic molecules

Irradiation of a molecular system by an intense laser field can trigger dynamics of both electronic and nuclear subsystems. Being much lighter than the nuclei, the electrons usually move on much faster time scale that is reflected in the attosecond dynamics of the electron density along a molecular chain. Due to a strong correlation between electronic and nuclear motion, the slow nuclear rearrangement damps ultrafast electronic oscillations, leading to the decoherence of the electronic dynamics on the time scale of a few femtoseconds. Here we present an application of a simple single-trajectory semiclassical scheme for computing the electronic coherence time in polyatomic molecules. This technique employs direct on-the-fly evaluation of the electronic structure, avoiding thus the"curse of dimensionality"appearing in large systems. We argue that the proposed scheme can be used for fast preselection of molecules to be studied experimentally. An excellent agreement of the presented approach with the full-dimensional quantum calculations is demonstrated.

Irradiation of a molecular system by an intense laser field can trigger dynamics of both electronic and nuclear subsystems.Being much lighter than the nuclei, the electrons usually move on much faster time scale that is reflected in the attosecond dynamics of the electron density along a molecular chain.Due to a strong correlation between electronic and nuclear motion, the slow nuclear rearrangement damps ultrafast electronic oscillations, leading to the decoherence of the electronic dynamics on the time scale of a few femtoseconds.Here we present an application of a simple single-trajectory semiclassical scheme for computing the electronic coherence time in polyatomic molecules.This technique employs direct on-the-fly evaluation of the electronic structure, avoiding thus the "curse of dimensionality" appearing in large systems.We argue that the proposed scheme can be used for fast preselection of molecules to be studied experimentally.An excellent agreement of the presented approach with the full-dimensional quantum calculations is demonstrated.
Recent progress in laser technologies [1][2][3] has revolutionized the field of atomic and molecular physics.In particular, tremendous developments of coherent light sources enabled the creation of sub-femtosecond laser pulses with remarkably well controlled parameters [4].Using state-of-the-art lasers, one is able to initiate and probe processes that are driven solely by the electron correlation, i.e., to study and manipulate electron dynamics on its natural time scale [5].
While experimental measurements of the electron motion in isolated atoms were reported by many authors [6,7], the direct evidence of the ultrafast electron dynamics in molecules still remains a point of debate [8].In particular, there are contradictions between recent experimental studies [9,10] that have claimed observation of the ultrafast electronic processes in molecules and theoretical investigations [11,12] performed on systems of similar complexity.The disagreement is centered around the question on how strong is the influence of the slow nuclear motion on the dynamics of electronic density.It was demonstrated by extensive ab initio calculations performed for several molecules [11,12] that the nuclear dynamics leads to the decoherence of the electronic wave packet on the time scale of a few femtoseconds which can make experimental observations of the electronic motion impossible.At the same time, long-lasting electronic coherences were reported recently for the ionized propiolic acid [13] and iodoacetylene [14] molecules, suggesting that the influence of nuclear motion on the electronic dynamics is very case-specific and requires careful investigation of the system under consideration.
In order to explore the interplay between nuclear rearrangement and the ultrafast electronic motion, a concerted treatment of the electron-nuclear dynamics is required.The above-mentioned theoretical studies [11][12][13] are all based on the full quantum treatment of the electron-nuclear motion with the multi-configurational time-dependent Hartree (MCTDH) method [15,16].Although this rigorous technique, on the one hand, makes it possible to take into account all the quantum effects, such as tunneling and non-adiabatic transitions, on the other hand, it still suffers from an exponential scaling problem and also requires construction of global potential energy surfaces (PESs), which is a daunting task by itself.
An alternative strategy allowing the simulation of coupled electron-nuclear dynamics employs trajectory-based methods in combination with an "on-the-fly" evaluation of the electronic structure.These approaches are referred to as direct dynamics methods since the PESs are calculated along trajectories only, thus avoiding the precomputation of globally fitted surfaces, and sampling only the relevant regions of the configuration space.Direct dynamics techniques range from fully quantum methods, such as the variational multi-configurational Gaussians [17,18], ab initio multiple spawning [19], coupled coherent states [20], multi-configurational Ehrenfest [21], and Gaussian dephasing representation [22], to more approximate mixed quantum-classical and semiclassical approaches including, e.g. the surface hopping [23], Ehrenfest dynamics [24], and Herman-Kluk propagator [25], together with its extensions [26,27].
It was demonstrated in several papers [28][29][30][31] that the direct dynamics approaches can be successfully used to estimate the influence of the distribution of nuclear geometries on the decoherence of the electronic oscillations.In these studies, multiple trajectories representing an initially delocalized wave packet were propagated using Ehrenfest approximation.Although this technique allows one to take into account the cancellation of the electronic motion due to a superposition of coherent oscillations with different frequencies appearing at the respective nuclear geometries, it completely ignores the decoherence occurring due to the quantum motion of the arXiv:1911.03586v1[physics.chem-ph]9 Nov 2019 wave packet resulting in the accumulation of the phase along the propagating trajectory.The latter mechanism is referred to as the phase jitter [32] in general literature on quantum decoherence or more specifically as dephasing [12] in the case of electron-nuclear processes.
In this paper, we use one of the simplest semiclassical approaches for molecular dynamics, the so-called thawed Gaussian approximation (TGA) [33,34], to evaluate the influence of nuclear motion on the ultrafast electronic dynamics.Within this method, the nuclear wavefunction is described by a single Gaussian wave packet whose center follows Hamilton's equations of motion and whose time-dependent width and phase are propagated using the local harmonic approximation of the PES.Although the algorithm is very simple in nature, it allows us to approximately take into account quantum properties of the nuclear dynamics and thus to include the dephasing mechanism into consideration.It will be demonstrated that, in case of the ultrafast processes, this semiclassical method produces accurate results comparable with the state-of-the-art MCTDH calculations.Moreover, the presented scheme can be implemented within the on-thefly ab initio framework for calculating the required local properties of the PES.The method allows calculations of the electronic coherences under the influence of nuclear motion with minimal efforts for setting up the Hamiltonian of the system under study.The latter can be extremely useful for a fast preselection of molecules suitable for further experimental investigations.
We simulate coupled electron-nuclear dynamics taking place after outer-valence ionization of two polyatomic molecules: propiolic acid (HC 2 COOH) and its amide derivative propiolamide (HC 2 CONH 2 ).Propiolic acid provides us with a perfect system on which we can demonstrate correctness of the results obtained using the semiclassical TGA because the electronic coherences in this molecule were recently calculated using a full quantum MCTDH approach [13].The propiolamide molecule, in turn, is studied here for the first time.It will be shown that although both molecules have a very similar ionization spectrum, their electronic coherence times differ substantially.The latter observation illustrates that the interplay between electronic and nuclear motion is a rather specific and multifaceted phenomenon which depends strongly on the topology of the PESs involved in the dynamics.
The starting point of our investigations is a neutral molecule in its ground electronic state.The ionization of the system performed by the ultrashort laser pulse can bring the molecule to a non-stationary superposition of the ionic states, thus launching a coupled dynamics of electronic and nuclear wave packets.A coherent superposition of multiple electronic states triggers oscillations of the charge along a molecular chain.This purely electronic mechanism was termed charge migration [35,36] in order to distinguish it from a more common charge trans- FIG. 1. Electronic coherence measured by the time-dependent overlap χ13(t) of the nuclear wave packets propagated on the first and third cationic states of propiolic acid after removal of a HOMO electron.Dynamics was performed with the full quantum MCTDH method ("quantum") or with the semiclassical vertical-Hessian thawed Gaussian approximation ("semiclass."),and either with the vibronic coupling ("VC") or onthe-fly ("OTF") Hamiltonian, both obtained at the ab initio many-body Green's function ADC(3) level of theory.All simulations except the "quantum VC" calculation employed the diabatic approximation ("diab."),which neglects nonadiabatic couplings between the diabatic states, or the adiabatic approximation ("adiab."),which neglects nonadiabatic couplings between adiabatic states.
fer driven by nuclei [37].Even though the charge migration is governed by the electronic motion, it is strongly coupled with the nuclear dynamics and therefore can crucially affect the behavior of the whole molecule.Previous calculations of the ionization spectrum of propiolic acid showed [13,38] that, due to the electron correlation, the ground and the second excited ionic states of the molecule are a strong mixture of two one-hole configurations: an electron missing in the highest occupied molecular orbital (HOMO) and an electron missing in the HOMO-2.Therefore, if we suddenly remove an electron either from HOMO or from HOMO-2, we will create an electronic wave packet, which will initiate charge migration oscillations between the carbon triple bond and the carbonyl oxygen with a period of about 6.2 fs, determined by the energy gap between the first and the third cationic states [38].Importantly, due to the planar geometry of the propiolic acid and a large energy gap between remaining ionic states, the indicated superposition can be obtained in an experiment by an appropriate orientation of the molecule with respect to the laser polarization.
To describe the ionization process, we used the non-Dyson algebraic-diagrammatic-construction (ADC) scheme [39] to represent the one-particle Green's function.At the third-order ADC [ADC(3)] level, used in the present calculations, the Hamiltonian is consistent with the exact Green's function up to the third order of perturbation theory with respect to the Hartree-Fock reference Hamiltonian.Standard double-zeta plus polarization (DZP) basis sets [40] were employed to construct the noncorrelated reference states.Ground-state geometries of the neutral molecules were optimized using the density functional theory [41] with the B3LYP functional [42].The optimization was done with Gaussian 16 package [43].
Assuming that the system was in its nuclear and electronic ground states at the beginning, the dynamical simulations consist in the on-the-fly propagation of nuclear wave packets on the multitude of involved ionic states.The ionization is performed vertically from the neutral state (Franck-Condon approximation).A single trajectory description of the nuclear motion taking place on each involved electronic state is used.The trajectories propagating on different states are treated independently from each other (non-adiabatic effects are neglected).The center of a Gaussian wave packet is guided by classical Hamilton's equations of motion, while the width and the phase are propagated using the single-Hessian [44] variant of the TGA scheme.Although, due to a single Gaussian description of the dynamics, the quantum effects and anharmonicity of the PES are taken into account only partially, the TGA can be surprisingly accurate in capturing short-time effects.Indeed, the major goal of our study is not to describe a long-term nuclear motion as precisely as possible, but rather to evaluate the influence of slow nuclear rearrangement on the ultrafast dynamics taking place in the electronic subsystem.
Within the Born-Huang representation [45] of the molecular wavefunction, the expectation value of an electronic operator Ô(r, R) can be expressed as where O ij (R) = Φ * i (r, R) Ô(r, R)Φ j (r, R)dr denote the R-dependent matrix elements of the electronic operator taken between electronic states i and j, and the quantities χ i (R, t) are the time-dependent nuclear wave packets propagated on the corresponding PESs.
If both the operator Ô(r, R) and the electronic states {Φ i (r, R)} have a weak dependence on nuclear coordinates R, Eq. ( 1) can be further simplified as where the nuclear overlaps χ ij (t) = χ * i (R, t)χ j (R, t)dR represent the populations of electronic states when i = j and the electronic coherences [11][12][13] when i = j.Equation (2) shows that the factors χ ij (t) provide the only source of the time-dependence in the expectation value of the electronic operator and thus can serve as convenient properties to quantify the decoherence time.FIG. 2. First four cationic states of the propiolamide computed using the ab initio many-body Green's function ADC(3) method.The second and fourth states belong to the A symmetry, while the first and third states to A .The next ionic state is located at 14.5 eV.The spectral intensity is defined as the combined weight of all one-hole configurations in the configuration-interaction expansion of the ionic state.The orbitals involved in the hole-mixing are also depicted.
We computed the electronic coherences χ ij (t) created after sudden ionization of the electron from HOMO of the propiolic acid.The initial molecular wave packet is prepared from an equally weighted and phase-synchronized superposition of the first and third cationic states of the molecule.Figure 1 shows the real part of the electronic coherences evaluated by five different schemes.We adopted the vibronic-coupling (VC) Hamiltonian from Ref. [13] to perform MCTDH simulations taking into account all nuclear degrees of freedom of the molecule.The full quantum-mechanical calculations show (red solid line in Fig. 1) that the electronic oscillations are strongly influenced by the nuclear motion.The initially created coherences are completely suppressed within first 15 fs of the dynamics [13].
In order to validate the applicability of the TGA, we performed semiclassical calculations using adiabatic version of the VC Hamiltonian.In this case, PESs are ob-tained by diagonalization of the four-state VC model used in the MCTDH calculations.Our simulation shows (blue dashed line in Fig. 1) that on the short time scale the TGA gives results almost identical to the full quantum MCTDH calculations.The small deviations start to appear at longer times due to the nonadiabatic effects which TGA does not take into account.It is important to mention that TGA is exact within the VC model in the case when the diabatic PESs are not coupled to each other by nonadiabatic terms.It therefore provides results completely identical to the MCTDH simulations performed on such VC Hamiltonian (yellow solid line in Fig. 1).
Finally, we performed single-Hessian TGA calculations with on-the-fly evaluation of the electronic structure at the same ab initio level of theory as that used in the construction of the VC Hamilatonian (green dash-dotted line in Fig. 1).In this case, the dynamics of the system is not bound to the evolution within model representation of the PESs spanned on the precomputed normal modes.Instead, we use nuclear gradients generated on the fly in order to propagate the center of the Gaussian wave packet which can potentially go "between" normal modes.This makes it possible to take into account nuclear configurations which are not accessible within the VC model.Our semiclassical results predict the electronic motion with a similar oscillation period, although the nuclear dynamics in this case leads to a slightly faster decay of the electronic coherence.
Let us turn to the electron-nuclear dynamics driven by the ionization of the propiolamide molecule.Similar to the spectrum of the propiolic acid [38], in the energy range 10-14 eV only the four states shown in Fig. 2 are present.The strong electron correlation between valence orbitals in the neutral propiolamide leads to appearance of the almost equal in weights one-hole configurations in the ionic states.Therefore, an ultrashort (sudden) ionization of the molecule will inevitably create an electronic wave packet and trigger dynamics of the electron density between the carbon triple bond and amide moiety.The molecule is planar and thus belongs to the C s symmetry group which allows assignment of the ionic states to two irreducible representations: the second and the fourth states belong to the A , while the first and the third states to A .As in the case of propiolic acid, orientation of the molecule with respect to polarization of the ionizing laser field can be used to populate only the states of interest.
Figure 3 shows the evolution of the electronic coherence in the propiolamide.Taking advantage of the symmetry in the molecule, we simulate dynamics occurring after removal of the HOMO (populating the first and the third cationic states, A representation) and HOMO-2 (populating the second and the fourth cationic states, A representation) electrons.Starting from the equally weighted and phase-matched superposition of the electronic states, in both cases the nuclear motion perturbs oscillations of the electronic subsystem leading thereby to decoherence.A convenient quantity to compare the coherence times of different molecules is the purity function Tr(ρ(t) 2 ) [11], where the electronic density matrix ρ(t) is related to the matrix of nuclear overlaps from Eq. ( 2) by transposition: ρ ij (t) = χ ji (t).Due to decoherence, the purity decays from the value Tr(ρ(0) 2 ) = 1 for the initially pure state to the value 1/n for the equally weighted mixture of n states.
Our simulations demonstrate that, contrary to the propiolic acid molecule, for which long-lasting coherences were observed (see bottom panel of Fig. 3 and also Ref. [13]), the initially pure mixtures in the propiolamide evolve to a mixed state on the time scale of just a few femtoseconds.Importantly, the energy gaps between the involved electronic states of propiolamide are larger than those for the propiolic acid (see Fig. 2 and Ref. [38]), which leads to faster oscillations of electronic density along the molecular chain.Although the electronic coherence time in the propiolamide is relatively short, due to the faster charge migration, the electronic density has enough time to perform one clear oscillation.Moreover, the existence of the strong hole-mixing in both symmetries of the propiolamide can be used to induce oscillations of the charge along different directions in the molecule.The dependence of the charge migration on the orientation of the system provides an important advantage for experimental measurements utilizing the timeresolved high-harmonic generation (HHG) spectroscopy employed recently by Wörner and co-workers [5].Align-ment of the molecule with respect to the pump pulse should be reflected in the resulting HHG spectra and thus can be used as a direct evidence of the ultrafast electron dynamics.
Although trajectory-based direct dynamics methods were previously used to estimate the electronic coherences in various polyatomic molecules [28][29][30][31], the decoherence was attributed to the interference of electronic oscillations following from the distribution of nuclear geometries.The presented approach, in turn, allows one to take into account quantum properties of the propagated wave packet.In particular, using the TGA equations of motion one can approximately reconstruct the evolution of the quantum phase and thus include dephasing mechanism into consideration.The semiclassical vertical-Hessians TGA scheme used in this paper can be further improved by calculating Hessians along the propagated trajectory and thus take into account more complicated situations, e.g., dissociation of a molecule.Moreover, the improved versions of the TGA such as the extended thawed Gaussian approximation (ETGA) [46], which propagates a Gaussian wave packet multiplied by a general polynomial, or a so-called three thawed Gaussians approximation (3TGA) [47], benefiting from representation of the wave packet by multiple Gaussians, were recently reported which can make on-the-fly semiclassical simulations even more accurate.
In conclusion, we have presented an application of one of the simplest and most efficient semiclassical approaches to the study of the ultrafast electronic processes coupled with the slower nuclear dynamics.We demonstrated an excellent agreement of this simple technique with the state-of-the-art full-dimensional quantum calculations performed using MCTDH method.Although clearly limited to the description of situations when the nonadiabatic effects are small, the presented technique can help breaking the "curse of dimensionality" appearing in the quantum treatment of multidimensional systems and therefore can deal with much larger molecules.The latter can be crucial for full-dimensional simulations of ultrafast electronic processes taking place in biologically relevant systems [48].Being able to treat molecules with up to few hundred atoms, the presented technique can help to shed light on the continuing debates on the role of quantum coherence in biology [49].We argue that simple semiclassical schemes can be successfully used to support theoretically recent experimental observations of attosecond electron dynamics in realistic molecular systems.
The authors thank Alexander Kuleff for many valuable discussions.Financial support from the Swiss National Science Foundation through the NCCR MUST (Molecular Ultrafast Science and Technology) Network is gratefully acknowledged.N. V. G. thanks the Branco Weiss Fellowship-Society in Science, administered by the ETH Zürich, for financial support.

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FIG.3.Top panel: Time evolution of the electronic coherences χij(t) created by the equally weighted coherent superposition of the second and fourth cationic states (ionization from the HOMO-2, A symmetry), and the first and third cationic states (ionization from the HOMO, A symmetry).Bottom panel: Comparison of the electronic purity function Tr(ρ 2 ) for the propiolamide and propiolic acid molecules.