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.

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'.

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.

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.

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.

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.

Twisting versus Delocalization in CAAC- and NHC-Stabilized Boron-Based Biradicals: The Roles of Sterics and Electronics.

Invited for the cover of this issue is Bernd Engels, Holger Braunschweig, Volker Engel and their coworkers at University of Würzburg. The image depicts bridged boron compounds which possess fascinating relationships between their composition and their geometrical and electronic structures, the latter ranging from closed-shell to biradical triplet or singlet ground state. Read the full text of the article at 10.1002/chem.202004619.

Twisting versus Delocalization in CAAC- and NHC-Stabilized Boron-Based Biradicals: The Roles of Sterics and Electronics.

Twisted boron-based biradicals featuring unsaturated C2 R2 (R=Et, Me) bridges and stabilization by cyclic (alkyl)(amino)carbenes (CAACs) were recently prepared. These species show remarkable geometrical and electronic differences with respect to their unbridged counterparts. Herein, a thorough computational investigation on the origin of their distinct electrostructural properties is performed. It is shown that steric effects are mostly responsible for the preference for twisted over planar structures. The ground-state multiplicity of the twisted structure is modulated by the σ framework of the bridge, and different R groups lead to distinct multiplicities. In line with the experimental data, a planar structure driven by delocalization effects is observed as global minimum for R=H. The synthetic elusiveness of C2 R2 -bridged systems featuring N-heterocyclic carbenes (NHCs) was also investigated. These results could contribute to the engineering of novel main group biradicals.

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.

Born-Oppenheimer and non-Born-Oppenheimer contributions to time-dependent electron momenta.

Using a model system for a coupled electron-nuclear motion, we calculate time-dependent expectation values of the electronic momentum operator. Whereas, within the velocity form, this quantity vanishes if the Born-Oppenheimer (BO) approximation is applied, it differs from zero if the calculation employs the length form of the expectation value. Using the adiabatic expansion of the total wave function, it is analyzed which terms contribute to the mean electronic momentum. For an adiabatic motion, where the BO approximation holds, it is shown that in the length form, the BO wave function yields an excellent estimate of the momentum. On the other hand, in the velocity form, it is necessary to include non-BO terms to calculate its value. This illustrates the different convergence behavior of the matrix elements in the two formulations. In the diabatic limit where the electron density does only marginally change upon the nuclear motion, both approaches converge to a vanishing mean electronic momentum.

Naphthoquinones as Covalent Reversible Inhibitors of Cysteine Proteases-Studies on Inhibition Mechanism and Kinetics.

The facile synthesis and detailed investigation of a class of highly potent protease inhibitors based on 1,4-naphthoquinones with a dipeptidic recognition motif (HN-l-Phe-l-Leu-OR) in the 2-position and an electron-withdrawing group (EWG) in the 3-position is presented. One of the compound representatives, namely the acid with EWG = CN and with R = H proved to be a highly potent rhodesain inhibitor with nanomolar affinity. The respective benzyl ester (R = Bn) was found to be hydrolyzed by the target enzyme itself yielding the free acid. Detailed kinetic and mass spectrometry studies revealed a reversible covalent binding mode. Theoretical calculations with different density functionals (DFT) as well as wavefunction-based approaches were performed to elucidate the mode of action.

cAAC-Stabilized 9,10-diboraanthracenes-Acenes with Open-Shell Singlet Biradical Ground States.

Narrow HOMO-LUMO gaps and high charge-carrier mobilities make larger acenes potentially high-efficient materials for organic electronic applications. The performance of such molecules was shown to significantly increase with increasing number of fused benzene rings. Bulk quantities, however, can only be obtained reliably for acenes up to heptacene. Theoretically, (oligo)acenes and (poly)acenes are predicted to have open-shell singlet biradical and polyradical ground states, respectively, for which experimental evidence is still scarce. We have now been able to dramatically lower the HOMO-LUMO gap of acenes without the necessity of unfavorable elongation of their conjugated π system, by incorporating two boron atoms into the anthracene skeleton. Stabilizing the boron centers with cyclic (alkyl)(amino)carbenes gives neutral 9,10-diboraanthracenes, which are shown to feature disjointed, open-shell singlet biradical ground states.

Lewis-Base Stabilization of the Parent Al(I) Hydride under Ambient Conditions.

Aluminum(III) is inherently electron deficient and therefore acts as a prototypical Lewis acid. Conversely, Al(I) is a rare, nucleophilic variant of aluminum that is thermodynamically unstable under ambient conditions. While attempts to stabilize and isolate Al(I) species have become increasingly successful, the parent Al(I) (i.e, Al-H) remains accessible only under extreme temperatures/pressures or matrix conditions. Here, we report the isolation of the parent Al(I) hydride under ambient conditions via the reduction of a Lewis-base-stabilized alkyldihaloalane. Computational and spectroscopic analyses indicate that the ground-state electronic configuration of this monomeric aluminum species is best described as an Al(I) hydride with non-negligible open-shell Al(III) singlet diradical character. These findings are also supported by reactivity studies, which reveal both the p-centered lone pair donating ability and the hydridic nature of the parent aluminene.

Optically Induced Electron Transfer in Mixed-Valence States: A Model Study on Electronic Transitions, Relaxation Dynamics, and Transient Absorption Spectroscopy.

Quantum dynamical model calculations are performed on the optically induced electron transfer in a mixed-valence system interacting with different solvents. The simultaneously occurring processes of population transfer between electronic states and relaxation are studied in detail. Transient absorption traces, as recently recorded in our laboratory, are simulated, and the features of the spectra are related to the dynamics. The agreement with the experiment hints at the fact that the employed one-dimensional models catch the essentials of the photochemistry of the investigated systems and that they can be used for the interpretation of the transient absorption spectra. It is inferred that the ultrafast electron transfer processes take place on a sub-picosecond time scale and afterward relaxation occurs within several picoseconds.

Electronic and nuclear flux dynamics at a conical intersection.

A combined electronic-nuclear wave packet motion is accompanied by temporal changes of probability flux densities. Using a two dimensional model, we study such densities in the vicinity of a conical intersection (CI) between the potential energy surfaces of two electronically excited states. When the dynamics is accompanied by an efficient population transfer, the electronic flux density behaves nearly time-independent although the nuclear flux does not. The second case involves a nuclear motion where the CI is surrounded. There, the nuclear wave packet undergoes a bifurcation, and the electronic density shows characteristics of a rotation. The electronic flux, however, exhibits a constant directional dynamics during the nuclear motion. The geometrical phase which appears in comparing the nuclear dynamics derived from the coupled motion and the Born-Oppenheimer calculation is also seen in the nuclear flux dynamics.

A classical ride through a conical intersection.

Regarding the correlated electron-nuclear motion in a model system, we investigate the dynamics in the vicinity of a conical intersection (CoIn) between two excited state potential surfaces. It is documented that an ensemble of classical trajectories which move in the complete electronic-nuclear phase space tracks the quantum wave-packet motion through the CoIn which is accompanied by a non-adiabatic population transfer. On the contrary, for an adiabatic circular motion around the position of the CoIn, the quantum mechanical and classical densities deviate substantially. In the latter case, the Born-Oppenheimer classical nuclear motion on a single potential surface is able to track the quantum dynamics.

On the calculation of time-dependent electron momenta within the Born-Oppenheimer approximation.

In the case of an adiabatic motion in molecules, electrons adjust to the smoothly changing geometry of the nuclei. Although then the Born-Oppenheimer (BO) approximation is valid, it fails in predicting the time-dependence of electron momenta because, within its product ansatz for the wave function, the respective expectation values are zero. It is shown that this failure can be circumvented using the Ehrenfest theorem. Here we extend our former work [T. Schaupp et al., Eur. Phys. J. B 91, 97 (2018)] and regard models in higher dimensions and for more particles. We solve the time-dependent Schrödinger equation for the combined nuclear-electronic motion and compare the results to those derived from BO wave functions. For all situations, it is found that the time-dependent BO electronic momenta are in excellent agreement with the numerically exact results.

The dimer-approach to characterize opto-electronic properties of and exciton trapping and diffusion in organic semiconductor aggregates and crystals.

A fundamental understanding of photo-induced processes in opto-electronic thin film devices is a prerequisite for the rational design of improved organic semiconductor materials. Absorption and emission spectra provide important insights into the complicated electronic structure of and relaxation processes in organic semiconductor aggregates and crystals. They are of interest because they often limit the efficiencies of the devices. For an assignment of the spectra a close interplay between experiment and theory is essential because simulations are often necessary to entangle the various effects which determine the features of the spectra. In the present perspective we describe the so called dimer-approach and provide a few examples in which this approach could successfully deliver an atomistic picture of photo-induced relaxation effects in perylene-based materials and characterize their optical spectra. The model Hamiltonians of standard monomer-based approaches are also briefly discussed to reveal the differences between both methods and to shed some light on their strengths and shortcomings.

Mapping of exciton-exciton annihilation in MEH-PPV by time-resolved spectroscopy: experiment and microscopic theory.

Transient absorption traces taken on samples of the polymer MEH-PPV are measured as a function of the laser intensity. In increasing the laser power, different decay dynamics of the signal are obtained. This suggests that effective exciton-exciton annihilation takes place. The signals are interpreted using a microscopic quantum mechanical model. The analysis points at an ultrafast excitonic decay via interchain and intrachain annihilation, where the latter process is roughly thirty times slower. Afterwards, diffusion-induced annihilation and relaxation become effective and thus determine the long-time behavior of the excited-state decay.