Coupled Electron-nuclear Dynamics Research Articles

Coupled Electron-Nuclear Dynamics Induced and Monitored with Femtosecond Soft X-ray Pulses in the Amino Acid Glycine.

The coupling of electronic and nuclear motion in polyatomic molecules is at the heart of attochemistry. The molecular properties, transient structures, and reaction mechanism of these many-body quantum objects are defined on the level of electrons and ions by molecular wave functions and their coherent superposition, respectively. In the present contribution, we monitor nonadiabatic quantum wave packet dynamics during molecular charge motion by reconstructing both the oscillatory charge density distribution and the characteristic time-dependent nuclear configuration coordinate from time-resolved Auger electron spectroscopic data recorded in previous studies on glycine molecules [Schwickert et al. Sci. Adv. 2022, 8, eabn6848]. The electronic and nuclear motion on the femtosecond time scale was induced and probed in kinematically complete soft X-ray experiments at the FLASH free-electron laser facility. The detailed analysis of amplitude, instantaneous phase, and instantaneous frequency of the propagating many-body wave packet during its lifecycle provides unprecedented insight into dynamical processes beyond the Born-Oppenheimer approximation. We are confident that the refined experimental data evaluation helps to develop new theoretical tools to describe time-dependent molecular wave functions in complicated but ubiquitous non-Born-Oppenheimer photochemical conditions.

Strong Laser Field-Driven Coupled Electron-Nuclear Dynamics: Quantum vs Classical Description.

We have performed a coupled electron-nuclear dynamics study of H2+ molecular ions under the influence of an intense few-cycle 4.5 fs laser pulse with an intensity of 4 × 1014 W/cm2 and a central wavelength of 750 nm. Both quantum and classical dynamical methods are employed in the exact similar initial conditions with the aim of head-to-head comparison of two methodologies. A competition between ionization and dissociation channel is explained under the framework of quantum and classical dynamics. The origin of the electron localization phenomena is elucidated by observing the molecular and electronic wave packet evolution pattern. By probing with different carrier envelope phase (CEP) values of the ultrashort pulse, the possibility of electron localization on either of the two nuclei is investigated. The effects of initial vibrational states on final dissociation and ionization probabilities for several CEP values are studied in detail. Finally, asymmetries in the dissociation probabilities are calculated and mutually compared for both quantum and classical dynamical methodologies, whereas Franck-Condon averaging over the initial vibrational states is carried out in order to mimic the existing experimental conditions.

Few-Femtosecond Isotope Effect in Polyatomic Molecules Ionized by Extreme Ultraviolet Attosecond Pulse Trains.

Following ionization by an extreme ultraviolet (XUV) attosecond pulse train, a polyatomic molecule can be promoted to more-than-one excited states of the residual ion. The ensuing relaxation dynamics is often facilitated by several reaction coordinates, making them difficult to disentangle by the usual spectroscopic means. Here, we show that in atto-chemistry isotope labeling can be an efficient tool for unraveling the relaxation pathways in highly excited photoionized molecules. Employing an XUV pump pulse and a near-infrared probe pulse, we found the nuclear as well as coupled electron-nuclear dynamics in ethylene to be almost 40% faster compared to that of its deuterated counterpart. The findings, which are supported by advanced nonadiabatic dynamics calculations, led to the identification of the relevant nuclear coordinates controlling the relaxation. Our experiment highlights the relevance of ultrashort XUV pulses to capture the isotopic effect in few-femtosecond molecular photodynamics.

Nuclear-Electron Correlation Effects and Their Photoelectron Imprint in Molecular XUV Ionisation.

The ionisation of molecules by attosecond XUV pulses is accompanied by complex correlated dynamics, such as the creation of coherent electron wave packets in the parent ion, their interplay with nuclear wave packets, and a correlated photoelectron moving in a multi-centred potential. Additionally, these processes are influenced by the dynamics prior to and during the ionisation. To fully understand and subsequently control the ionisation process on different time scales, a profound understanding of electron and nuclear correlation is needed. Here, we investigate the effect of nuclear–electron correlation in a correlated two-electron and one-nucleus quantum model system. Solving the time-dependent Schrödinger equation allows to monitor the correlation impact pre, during, and post-XUV ionisation. We show how an initial nuclear wave packet displaced from equilibrium influences the post-ionisation dynamics by means of momentum conservation between the target and parent ion, whilst the attosecond electron population remains largely unaffected. We calculate time-resolved photoelectron spectra and their asymmetries and demonstrate how the coupled electron–nuclear dynamics are imprinted on top of electron–electron correlation on the photoelectron properties. Finally, our findings give guidelines towards when correlation resulting effects have to be incorporated and in which instances the exact correlation treatment can be neglected.

A molecular clock for autoionization decay

The ultrafast decay of highly excited electronic states is resolved with a molecular clock technique, using the vibrational motion associated to the ionic bound states as a time-reference. We demonstrate the validity of the method in the context of autoionization of the hydrogen molecule, where nearly exact full dimensional ab-initio calculations are available. The vibrationally resolved photoionization spectrum provides a time–energy mapping of the autoionization process into the bound states that is used to fully reconstruct the decay in time. A resolution of a fraction of the vibrational period is achieved. Since no assumptions are made on the underlying coupled electron–nuclear dynamics, the reconstruction procedure can be applied to describe the general problem of the decay of highly excited states in other molecular targets.

Coupled Electron-Nuclear Dynamics on H2+ within Time-Dependent Born-Oppenheimer Approximation.

Quantum dynamical behavior of H2+ in the presence of a linearly polarized, ultrashort, intense, infrared laser pulse has been studied by numerically solving the time-dependent Schrödinger equation with nuclear motion restricted in one-dimension along the direction of laser polarization and electronic motion in three-dimensions. On the basis of the time-dependent Born-Oppenheimer approximation, we have constructed time-dependent potentials for the ground electronic state (1sσg) of H2+. Subsequent nuclear dynamics is then carried out on these field-dressed potential energy surfaces, and the dissociation dynamics is investigated. Our analyses reveal that although the electronic longitudinal degree of freedom plays the major role in governing the dissociation dynamics, contributions from the electronic transverse degree of freedom should also have to be taken into account to obtain accurate results. Also, modeling electron-nuclei Coulomb interactions in a one-dimensional calculation with an artificially chosen constant softening parameter leads to a discrepancy with the exact results. Comparing our results with other quantum and classical dynamical studies showed a good agreement with exact results.

Coupled electron-nuclear dynamics in resonant1σ→2πx-ray Raman scattering of CO molecules

We present a detailed experimental-theoretical analysis of O $K$-edge resonant $1\ensuremath{\sigma}\ensuremath{-}2\ensuremath{\pi}$ inelastic x-ray scattering (RIXS) from carbon monoxide with unprecedented energy resolution. We employ high-level ab initio calculations to compute the potential energy curves of the states involved in the RIXS process and simulate the measured RIXS spectra using the wave-packet-propagation formalism, including Coulomb coupling in the final-state manifold. The theoretical analysis allows us to explain all the key features of the experimental spectra, including some that were not seen before. First, we clearly show the interference effect between different RIXS channels corresponding to the transition via orthogonal $^{1}\mathrm{\ensuremath{\Pi}}_{x}$ and $^{1}\mathrm{\ensuremath{\Pi}}_{y}$ core-excited states of CO. Second, the RIXS region of 13 eV energy loss presents a triple structure, revealed only by the high-resolution measurement. In previous studies, this region was attributed solely to a valence state. Here we show a strong Coulomb mixing of the Rydberg and valence final states, which opens the forbidden RIXS channels to the ``dark'' final Rydberg states and drastically changes the RIXS profile. Third, using a combination of high-resolution experiment and high-level theory, we improve the $|4{\ensuremath{\sigma}}^{\ensuremath{-}1}2{\ensuremath{\pi}}^{1}\ensuremath{\rangle}$ final-state potential-energy curve by fitting its bottom part with the experiment. Also, the coupling constants between Rydberg and valence states were refined via comparison with the experiment. Our results illustrate the large potential of the RIXS technique for advanced studies of highly excited states of neutral molecules.

Sub-laser-cycle control of coupled electron–nuclear dynamics at a conical intersection

Nonadiabatic processes play a fundamental role in the understanding of photochemical processes in excited polyatomic molecules. A particularly important example is that of radiationless electronic relaxation at conical intersections (CIs). We discuss new opportunities for controlling coupled electron–nuclear dynamics at CIs, offered by the advent of nearly single-cycle, phase-stable, mid-infrared laser pulses. To illustrate the control mechanism, a two-dimensional model of the NO2 molecule is considered. The key idea of the control scheme is to match the time scale of the laser field oscillations to the characteristic time scale of the wave packet transit through the CI. The instantaneous laser field changes the shape and position of the CI as the wave packet passes through. As the CI moves in the laser field, it ‘slices’ through the wave packet, sculpting it in the coordinate and momentum space in a way that is sensitive to the carrier-envelope phase of the control pulse. We find that the electronic coherence imparted on the sub-laser-cycle time scale manifests during much longer nuclear dynamics that follow on the many tens of femtosecond time scale. Control efficiency as a function of molecular orientation is analyzed, showing that modest alignment is sufficient for showing the described effects.

The second-order Ehrenfest method

This article describes the Ehrenfest method and our second-order implementation (with approximate gradient and Hessian) within a CASSCF formalism. We demonstrate that the second-order implementation with the predictor–corrector integration method improves the accuracy of the simulation significantly in terms of energy conservation. Although the method is general and can be used to study any coupled electron–nuclear dynamics, we apply it to investigate charge migration upon ionization of small organic molecules, focusing on benzene cation. Using this approach, we can study the evolution of a non-stationary electronic wavefunction for fixed atomic nuclei, and where the nuclei are allowed to move, to investigate the interplay between them for the first time. Analysis methods for the interpretation of the electronic and nuclear dynamics are suggested: we monitor the electronic dynamics by calculating the spin density of the system as a function of time.

Communication: oscillating charge migration between lone pairs persists without significant interaction with nuclear motion in the glycine and Gly-Gly-NH-CH3 radical cations.

Coupled electron-nuclear dynamics has been studied, using the Ehrenfest method, for four conformations of the glycine molecule and a single conformation of Gly-Gly-NH-CH3. The initial electronic wavepacket was a superposition of eigenstates corresponding to ionization from the σ lone pairs associated with the carbonyl oxygens and the amine nitrogen. For glycine, oscillating charge migration (when the nuclei were frozen) was observed for the 4 conformers studied with periods ranging from 2 to 5 fs, depending on the energy gap between the lone pair cationic states. When coupled nuclear motion was allowed (which was mainly NH2 partial inversion), the oscillations hardly changed. For Gly-Gly-NH-CH3, charge migration between the carbonyl oxygens and the NH2 lone pair can be observed with a period similar to glycine itself, also without interaction with nuclear motion. These simulations suggest that charge migration between lone pairs can occur independently of the nuclear motion.

Coupled electron-nuclear dynamics: Charge migration and charge transfer initiated near a conical intersection

Coupled electron-nuclear dynamics, implemented using the Ehrenfest method, has been used to study charge migration with fixed nuclei, together with charge transfer when nuclei are allowed to move. Simulations were initiated at reference geometries of neutral benzene and 2-phenylethylamine (PEA), and at geometries close to potential energy surface crossings in the cations. Cationic eigenstates, and the so-called sudden approximation, involving removal of an electron from a correlated ground-state wavefunction for the neutral species, were used as initial conditions. Charge migration without coupled nuclear motion could be observed if the Ehrenfest simulation, using the sudden approximation, was started near a conical intersection where the states were both strongly coupled and quasi-degenerate. Further, the main features associated with charge migration were still recognizable when the nuclear motion was allowed to couple. In the benzene radical cation, starting from the reference neutral geometry with the sudden approximation, one could observe sub-femtosecond charge migration with a small amplitude, which results from weak interaction with higher electronic states. However, we were able to engineer large amplitude charge migration, with a period between 10 and 100 fs, corresponding to oscillation of the electronic structure between the quinoid and anti-quinoid cationic electronic configurations, by distorting the geometry along the derivative coupling vector from the D6h Jahn-Teller crossing to lower symmetry where the states are not degenerate. When the nuclear motion becomes coupled, the period changes only slightly. In PEA, in an Ehrenfest trajectory starting from the D2 eigenstate and reference geometry, a partial charge transfer occurs after about 12 fs near the first crossing between D1, D2 (N(+)-Phenyl, N-Phenyl(+)). If the Ehrenfest propagation is started near this point, using the sudden approximation without coupled nuclear motion, one observes an oscillation of the spin density--charge migration--between the N atom and the phenyl ring with a period of 4 fs. When the nuclear motion becomes coupled, this oscillation persists in a damped form, followed by an effective charge transfer after 30 fs.

Local control theory applied to coupled electronic and nuclear motion

Local control theory is applied to a model system consisting of a proton and electron which move in the field of two fixed ions in a single dimension. The simplicity of the model allows the field-induced coupled electron–nuclear dynamics to be described exactly by numerically solving the time-dependent Schrödinger equation. The results obtained are compared to those obtained from solutions of the Schrödinger equation for the nuclear motion in an adiabatic approach. In a first step, the field-induced transfer of the proton between potential wells in the isolated electronic ground state is investigated. In this case, the transfer can be obtained with a high efficiency. Employing the derived control field in an extended model including the first excited electronic state results in a loss of efficiency due to population transfer between the electronic states. The loss of controllability is even more pronounced when the coupled electron–nuclear dynamics is treated numerically exactly: it is found that the electron is completely detached from the system. Additional calculations show that the proton transfer in the present model cannot be performed with fields derived from local control theory if the correlated particle motion is treated computationally exactly. This hints at the difficulties to be encountered if the electronic degree of freedom is not taken properly into account in calculations on the laser control of molecular processes.

Geometric picture for coupled electron–nuclear dynamics

AbstractEquations of coupled electron–nuclear dynamics (CEND) that describe molecular dynamics within a fully classical approach have been derived. Utilization of the language of fiber bundles and connections (gauge fields) allows one to represent the resulting classical Hamilton equations in an invariant (coordinate‐covariant) form. Applications of the CEND equations to calculations of off‐resonant nonlinear molecular polarizabilities are outlined. The CEND equations are shown to resemble the END equations derived earlier once the Thouless representation of single Slater determinants is adopted. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002