Differential Shannon entropies and correlation measures for Born-Oppenheimer electron-nuclear dynamics: numerical results and their analytical interpretation.

We study the Born-Oppenheimer dynamics within a model for a coupled electron-nuclear motion. Differential Shannon entropies are calculated from the time-dependent probability densities of the combined system and, using single particle densities, entropies for the electronic and nuclear degrees of freedom are derived. These functions provide information on details of the wave packet motion. From the entropies, we determine the mutual information which characterizes particle correlations. This quantity is compared to other measures of electron-nuclear entanglement. Numerical results are interpreted within an analytically solvable approach, and it is documented how these functions depend on properties of the Born-Oppenheimer wave function and, in particular, how dynamical effects like wave packet focusing and dispersion influence the correlation between the particles.

Differential Shannon Entropies Characterizing Electron-Nuclear Dynamics and Correlation: Momentum-Space Versus Coordinate-Space Wave Packet Motion.

We calculate differential Shannon entropies derived from time-dependent coordinate-space and momentum-space probability densities. This is performed for a prototype system of a coupled electron-nuclear motion. Two situations are considered, where one is a Born-Oppenheimer adiabatic dynamics, and the other is a diabatic motion involving strong non-adiabatic transitions. The information about coordinate- and momentum-space dynamics derived from the total and single-particle entropies is discussed and interpreted with the help of analytical models. From the entropies, we derive mutual information, which is a measure for the electron-nuclear correlation. In the adiabatic case, it is found that such correlations are manifested differently in coordinate- and momentum space. For the diabatic dynamics, we show that it is possible to decompose the entropies into state-specific contributions.

Information Theoretical Approach to Coupled Electron-Nuclear Wave Packet Dynamics: Time-Dependent Differential Shannon Entropies.

We study differential Shannon entropies determined from position-space quantum probability densities in a coupled electron-nuclear system. In calculating electronic and nuclear entropies, one gains information about the localization of the respective particles and also about the correlation between them. For Born-Oppenheimer dynamics, the correlation decreases at times when the wave packet reaches the classical turning points of its motion. If a strong non-adiabtic coupling is present, leading to a large population transfer between different electronic states, the electronic entropy is approximately constant. Then the time dependence of the entropy reflects the information on the nucleus alone, and the correlation is absent. A decomposition of the entropy into contributions from the participating electronic states reveals insight into the state-specific population and nuclear wave packet localization.

Third-order pump-probe spectroscopy applied to molecular dimers: characterization of relaxation dynamics and exciton-exciton annihilation.

We investigate theoretically the excitonic dynamics in molecular dimers which is monitored by two time-delayed femtosecond laser pulses. A two-photon absorption leads to a wave packet dynamics in the manifold of second excited states. This opens up the channel for exciton-exciton annihilation (EEA) which involves non-radiative electronic transitions. It is shown that the time interval during which EEA takes place can be monitored by the detection of third-order signals which can be interpreted as originating from a pump-probe scheme. In the case of transient absorption, the spectra directly map intraband relaxation processes.

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.

Quantum flux densities for electronic-nuclear motion: exact versus Born-Oppenheimer dynamics.

We study the coupled electronic-nuclear dynamics in a model system to compare numerically exact calculations of electronic and nuclear flux densities with those obtained from the Born-Oppenheimer (BO) approximation. Within the adiabatic expansion of the total wave function, we identify the terms which contribute to the flux densities. It is found that only off-diagonal elements that involve the interaction between different electronic states contribute to the electronic flux whereas in the nuclear case the major contribution belongs to the BO electronic state. New flux densities are introduced where in both, the electronic and the nuclear case, the main contribution is contained in the component corresponding to the BO state. As a consequence, they can be determined within the BO approximation, and an excellent agreement with the exact results is found. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

Excitation localization in a trimeric perylenediimide macrocycle: Synthesis, theory, and single molecule spectroscopy.

A novel trimeric perylenediimide (PDI) macrocycle was synthesized, and its intramolecular electronic couplings were investigated by bulk and single-molecule optical spectroscopy and by various theoretical approaches. In polarization-resolved excitation spectroscopy at 1.2 K in a PMMA matrix, the appearance and disappearance of the three zero-phonon lines (ZPLs) of an individual trimer by changing the polarization in steps of 60° nicely reflect an approximate triangular geometry of the macrocycle and indicate localized excitations that are transferred by incoherent hopping processes at time scales of around 1 ps as inferred from the ZPL linewidths. The electronic coupling strength deduced from the low temperature data is found to be in good agreement with theoretical estimates. Bulk spectroscopy in toluene at room temperature indicates that the excitations are also localized under these conditions. Theory reveals that the reasons for the localized nature of the excitations at room and low temperatures are different. For a rigid macrocycle, the excitations are predicted to be delocalized, but molecular dynamics simulations point to considerable structural flexibility at ambient temperatures, which counteracts excitation delocalization. At 1.2 K in a PMMA matrix, this effect is too small to lead to localization. Yet, supported by simple model calculations, the disorder in the PMMA host induces sufficient differences between the PDI chromophores, which again result in localized excitations. By addressing crucial aspects of excitation energy transfer, our combined approach provides a detailed and quantitative account of the interchromophore communication in a trimeric macrocycle.

Wave packet dynamics in an harmonic potential disturbed by disorder: Entropy, uncertainty, and vibrational revivals.

We investigate the quantum and classical wave packet dynamics in an harmonic oscillator that is perturbed by a disorder potential. This perturbation causes the dispersion of a Gaussian wave packet, which is reflected in the coordinate-space and the momentum-space Shannon entropies, the latter being a measure for the amount of information available on a system. Regarding the sum of the two quantities, one arrives at an entropy that is related to the coordinate-momentum uncertainty. Whereas in the harmonic case, this entropy is strictly periodic and can be evaluated analytically, this behavior is lost if disorder is added. There, at selected times, the quantum mechanical probability density resembles that of a classical oscillator distribution function, and the entropy assumes larger values. However, at later times and dependent on the degree of disorder and the chosen initial conditions, quantum mechanical revivals occur. Then, the observed effects are reversed, and the entropy may decrease close to its initial value. This effect cannot be found classically.

Correlated three-dimensional electron-nuclear motion: Adiabatic dynamics vs passage of conical intersections.

We study the three-dimensional correlated motion of an electron and a proton. In one situation, the dynamics is restricted to the electronic ground state and is, thus, well described within the Born-Oppenheimer (BO) approximation. The probability and flux densities yield information about the coupled dynamics. Because the electronic flux density vanishes if determined from the BO wave function, another flux density is regarded, which provides insight into the directional motion of the electron. This flux density can be calculated within the BO approximation and agrees numerically well with the one derived from the full-dimensional calculation. Starting in the first excited electronic state at a similar geometry as chosen for the ground state dynamics results in a short-time dynamics that takes place in the same regions of the configuration space. Adopting the picture that evolves from the adiabatic expansion of the wave function, the nuclear wave packet motion in the two coupled adiabatic electronic states proceeds through a ring of conical intersections (CIs), which is accompanied by an effective population transfer. Nevertheless, the total nuclear probability and flux densities resemble very much those obtained for the ground state dynamics. While passing the CI, the electronic densities remain nearly constant, as expected for a diabatic dynamics. This confirms the conclusions obtained from our former two-dimensional study, namely, that also in three-dimensional space the wave packet dynamics does not exhibit features of the non-adiabatic dynamics.

Time-dependent momentum expectation values from different quantum probability and flux densities.

Based on the Ehrenfest theorem, the time-dependent expectation value of a momentum operator can be evaluated equivalently in two ways. The integrals appearing in the expressions are taken over two different functions. In one case, the integrand is the quantum mechanical flux density j̲, and in the other, a different quantity j̲̃ appears, which also has the units of a flux density. The quantum flux density j̲ is related to the probability density ρ via the continuity equation, and j̲̃ may as well be used to define a density ρ̃ that fulfills a continuity equation. Employing a model for the coupled dynamics of an electron and a proton, we document the properties of the densities and flux densities. It is shown that although the mean momentum derived from the two quantities is identical, the various functions exhibit a very different coordinate and time-dependence. In particular, it is found that the flux density j̲̃ directly monitors temporal changes in the probability density, and the density ρ̃ carries information about wave packet dispersion occurring in different spatial directions.