Attosecond Photoionization Dynamics: from Molecules over Clusters to the Liquid Phase.

Photoionization is a process taking place on attosecond time scales. How its properties evolve from isolated particles to the condensed phase is an open question of both fundamental and practical relevance. Here, we review recent work that has advanced the study of photoionization dynamics from atoms to molecules, clusters and the liquid phase. The first measurements of molecular photoionization delays have revealed the attosecond dynamics of electron emission from a molecular shape resonance and their sensitivity to the molecular potential. Using electron-ion coincidence spectroscopy these measurements have been extended from isolated molecules to clusters. A continuous increase of the delays with the water-cluster size has been observed up to a size of 4-5 molecules, followed by a saturation towards larger clusters. Comparison with calculations has revealed a correlation of the time delay with the spatial extension of the created electron hole. Using cylindrical liquid-microjet techniques, these measurements have also been extended to liquid water, revealing a delay relative to isolated water molecules that was very similar to the largest water clusters studied. Detailed modeling based on Monte-Carlo simulations confirmed that these delays are dominated by the contributions of the first two solvation shells, which agrees with the results of the cluster measurements. These combined results open the perspective of experimentally characterizing the delocalization of electronic wave functions in complex systems and studying their evolution on attosecond time scales.

From metastable to coherent sets- Time-discretization schemes.

Given a time-dependent stochastic process with trajectories x(t) in a space Ω, there may be sets such that the corresponding trajectories only very rarely cross the boundaries of these sets. We can analyze such a process in terms of metastability or coherence. Metastable setsM are defined in space M⊂Ω, and coherent setsM(t)âŠ‚Ω are defined in space and time. Hence, if we extend the space Ω by the time-variable t, coherent sets are metastable sets in Ω×[0,∞) of an appropriate space-time process. This relation can be exploited, because there already exist spectral algorithms for the identification of metastable sets. In this article, we show that these well-established spectral algorithms (like PCCA+, Perron Cluster Cluster Analysis) also identify coherent sets of non-autonomous dynamical systems. For the identification of coherent sets, one has to compute a discretization (a matrix T) of the transfer operator of the process using a space-time-discretization scheme. The article gives an overview about different time-discretization schemes and shows their applicability in two different fields of application.

Exact Single-Electron Approach to the Dynamics of Molecules in Strong Laser Fields.

We present an exact single-electron picture that describes the correlated electron dynamics in strong laser fields. Our approach is based on the factorization of the electronic wave function as a product of a marginal and a conditional amplitude. The marginal amplitude, which depends only on one electronic coordinate and yields the exact one-electron density and current density, obeys a time-dependent Schrödinger equation with an effective time-dependent potential. The exact equations are used to derive an approximation that is a step towards general and feasible ab initio single-electron calculations for molecules. The derivation also sheds new light on the usual interpretation of the single-active electron approximation. From the study of model systems, we find that the exact and approximate single-electron potentials for processes with negligible two-electron ionization lead to qualitatively similar dynamics, but that the ionization barrier in the exact single-electron potential may be explicitly time dependent.

ORBKIT: A modular python toolbox for cross-platform postprocessing of quantum chemical wavefunction data.

ORBKIT is a toolbox for postprocessing electronic structure calculations based on a highly modular and portable Python architecture. The program allows computing a multitude of electronic properties of molecular systems on arbitrary spatial grids from the basis set representation of its electronic wavefunction, as well as several grid-independent properties. The required data can be extracted directly from the standard output of a large number of quantum chemistry programs. ORBKIT can be used as a standalone program to determine standard quantities, for example, the electron density, molecular orbitals, and derivatives thereof. The cornerstone of ORBKIT is its modular structure. The existing basic functions can be arranged in an individual way and can be easily extended by user-written modules to determine any other derived quantity. ORBKIT offers multiple output formats that can be processed by common visualization tools (VMD, Molden, etc.). Additionally, ORBKIT possesses routines to order molecular orbitals computed at different nuclear configurations according to their electronic character and to interpolate the wavefunction between these configurations. The program is open-source under GNU-LGPLv3 license and freely available at https://github.com/orbkit/orbkit/. This article provides an overview of ORBKIT with particular focus on its capabilities and applicability, and includes several example calculations. © 2016 Wiley Periodicals, Inc.

Electronic Flux Density beyond the Born-Oppenheimer Approximation.

In the Born-Oppenheimer approximation, the electronic wave function is typically real-valued and hence the electronic flux density (current density) seems to vanish. This is unfortunate for chemistry, because it precludes the possibility to monitor the electronic motion associated with the nuclear motion during chemical rearrangements from a Born-Oppenheimer simulation of the process. We study an electronic flux density obtained from a correction to the electronic wave function. This correction is derived via nuclear velocity perturbation theory applied in the framework of the exact factorization of electrons and nuclei. To compute the correction, only the ground state potential energy surface and the electronic wave function are needed. For a model system, we demonstrate that this electronic flux density approximates the true one very well, for coherent tunneling dynamics as well as for over-the-barrier scattering, and already for mass ratios between electrons and nuclei that are much larger than the true mass ratios.

Multireference calculations for ring inversion and double bond shifting in cyclooctatetraene.

We present multireference calculations for the characterization of ring inversion and double bond shifting in cyclooctatetraene. The results show that it is necessary to treat the dynamical correlation very accurately to obtain correct values for the barrier heights. This can be done, for example, with multireference configuration interaction or with perturbation theory of third order. However, detailed analysis also shows that already a complete active space self-consistent field treatment describes the processes surprisingly well. Thus, this method could be used as a computationally cheap method, for example, for dynamics simulations.

Electron density dynamics in the electronic ground state: motion along the Kekulé mode of benzene.

If the Born-Oppenheimer approximation is invoked for the description of chemical reactions, the electron density rearranges following the motion of the nuclei. Even though this approach is central to theoretical chemistry, the explicit time dependence of the electron density is rarely studied, especially if the nuclei are treated quantum mechanically. In this article, we model the motion of benzene along the Kekulé vibrational coordinate to simulate the nuclear dynamics and electron density dynamics in the electronic ground state. Details of the change of core, valence, and π electrons are determined and analyzed. We show how the pictures anticipated by drawing Lewis structures of the rearrangement correlate with the time-dependent quantum description of the process.

Low-energy electron distributions from the photoionization of liquid water: a sensitive test of electron mean free paths.

The availability of accurate mean free paths for slow electrons (<50 eV) in water is central to the understanding of many electron-driven processes in aqueous solutions, but their determination poses major challenges to experiment and theory alike. Here, we describe a joint experimental and theoretical study demonstrating a novel approach for testing, and, in the future, refining such mean free paths. We report the development of Monte-Carlo electron-trajectory simulations including elastic and inelastic electron scattering, as well as energy loss and secondary-electron production to predict complete photoelectron spectra of liquid water. These simulations are compared to a new set of photoelectron spectra of a liquid-water microjet recorded over a broad range of photon energies in the extreme ultraviolet (20-57 eV). Several previously published sets of scattering parameters are investigated, providing direct and intuitive insights on how they influence the shape of the low-energy electron spectra. A pronounced sensitivity to the escape barrier is also demonstrated. These simulations considerably advance our understanding of the origin of the prominent low-energy electron distributions in photoelectron spectra of liquid water and clarify the influence of scattering parameters and the escape barrier on their shape. They moreover describe the reshaping and displacement of low-energy photoelectron bands caused by vibrationally inelastic scattering. Our work provides a quantitative basis for the interpretation of the complete photoelectron spectra of liquids and opens the path to fully predictive simulations of low-energy scattering in liquid water.

Charge-Transfer Steps in Density Functional Theory from the Perspective of the Exact Electron Factorization.

When a molecule dissociates, the exact Kohn-Sham (KS) and Pauli potentials may form step structures. Reproducing these steps correctly is central for the description of dissociation and charge-transfer processes in density functional theory (DFT): The steps align the KS eigenvalues of the dissociating subsystems relative to each other and determine where electrons localize. While the step height can be calculated from the asymptotic behavior of the KS orbitals, this provides limited insight into what causes the steps. We give an explanation of the steps with an exact mapping of the many-electron problem to a one-electron problem, the exact electron factorization (EEF). The potentials appearing in the EEF have a clear physical meaning that translates to the DFT potentials by replacing the interacting many-electron system with the KS system. With a simple model of a diatomic, we illustrate that the steps are a consequence of spatial electron entanglement and are the result of a charge transfer. From this mechanism, the step height can immediately be deduced. Moreover, two methods to approximately reproduce the potentials during dissociation are proposed. One is based on the states of the dissociated system, while the other one is based on an analogy to the Born-Oppenheimer treatment of a molecule. The latter method also shows that the steps connect adiabatic potential energy surfaces. The view of DFT from the EEF thus provides a better understanding of how many-electron effects are encoded in a one-electron theory and how they can be modeled.

Alternative Approach for the Determination of Mean Free Paths of Electron Scattering in Liquid Water Based on Experimental Data.

Mean free paths of low-energy electrons in liquid water are of importance for modeling many physicochemical processes, but neither theoretical predictions nor experimental results have converged for these parameters. We therefore introduce an approach to determine elastic and inelastic mean free paths (EMFP, IMFP) based on experimental data. We show that ab initio calculations of electron scattering with water clusters converge with cluster size, thus providing access to condensed-phase scattering. The results are used in Monte Carlo simulations to extract EMFP and IMFP from recent liquid-microjet experiments that determined the effective attenuation length (EAL) and the photoelectron angular distribution (PAD) following oxygen 1s-ionization of liquid water. For electron kinetic energies from 10 to 300 eV, we find that the IMFP is noticeably larger than the EAL. The EMFP is longer than that of gas-phase water and the IMFP is longer compared to latest theoretical estimations, but both EMFP and IMFP are much shorter than suggested by experimental measurements of integral cross sections for amorphous ice.

Attosecond spectroscopy of liquid water.

Electronic dynamics in liquids are of fundamental importance, but time-resolved experiments have so far remained limited to the femtosecond time scale. We report the extension of attosecond spectroscopy to the liquid phase. We measured time delays of 50 to 70 attoseconds between the photoemission from liquid water and that from gaseous water at photon energies of 21.7 to 31.0 electron volts. These photoemission delays can be decomposed into a photoionization delay sensitive to the local environment and a delay originating from electron transport. In our experiments, the latter contribution is shown to be negligible. By referencing liquid water to gaseous water, we isolated the effect of solvation on the attosecond photoionization dynamics of water molecules. Our methods define an approach to separating bound and unbound electron dynamics from the structural response of the solvent.

On the Probability Density of the Nuclei in a Vibrationally Excited Molecule.

For localized and oriented vibrationally excited molecules, the qualitative features of the one-body probability density of the nuclei (one-nucleus density) are investigated. Like the familiar and widely used one-electron density that represents the probability of finding an electron at a given location in space, the one-nucleus density represents the probability of finding a nucleus at a given position in space independent of the location of the other nuclei and independent of their type. In contrast to the electrons, however, the nuclei are comparably localized. Due to this localization of the individual nuclei, the one-nucleus density provides a quantum-mechanical representation of the "chemical picture" of the molecule as an object that can largely be understood in a three-dimensional space, even though its full nuclear probability density is defined on the high-dimensional configuration space of all the nuclei. We study how the nodal structure of the wavefunctions of vibrationally excited states translates to the one-nucleus density. It is found that nodes do not necessarily lead to visible changes in the one-nucleus density: Already for relatively small molecules, only certain vibrational excitations change the one-nucleus density qualitatively compared to the ground state. It turns out that there are simple rules for predicting the shape of the one-nucleus density from the normal mode coordinates. A Python module for the computation of the one-nucleus density is provided at https://gitlab.com/axelschild/mQNMc.

How Interatomic Steps in the Exact Kohn-Sham Potential Relate to Derivative Discontinuities of the Energy.

Accurate density functional calculations hinge on reliable approximations to the unknown exchange-correlation (xc) potential. The most popular approximations usually lack features of the exact xc potential that are important for an accurate prediction of the fundamental gap and the distribution of charge in complex systems. Two principal features in this regard are the spatially uniform shift in the potential, as the number of electrons infinitesimally surpasses an integer, and the spatial steps that form, for example, between the atoms of stretched molecules. Although both aforementioned concepts are well known, the exact relationship between them remained unclear. Here we establish this relationship via an analytical derivation. We support our result by numerically solving the many-electron Schrödinger equation to extract the exact Kohn-Sham potential and directly observe its features. Spatial steps in the exact xc potential of a full configuration-interaction (FCI) calculation of a molecule are presented in three dimensions.