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We present a detailed study of the decoherence correction to surface hopping that was recently derived from the exact factorization approach. Ab initio multiple spawning calculations that use the same initial conditions and the same electronic structure method are used as a reference for three molecules: ethylene, the methaniminium cation, and fulvene, for which nonadiabatic dynamics follows a photoexcitation. A comparison with the Granucci−Persico energy-based decoherence correction and the augmented fewest-switches surface-hopping scheme shows that the three decoherence-corrected methods operate on individual trajectories in a qualitatively different way, but the results averaged over trajectories are similar for these systems.

By inverting the time-dependent Kohn-Sham equation for a numerically exact dynamics of the helium atom, we show that the dynamical step and peak features of the exact correlation potential found previously in one-dimensional models persist for real three-dimensional systems. We demonstrate that the Kohn-Sham and true current densities differ by a rotational component. The results have direct implications for approximate time-dependent density functional theory calculations of atoms and molecules in strong fields, emphasizing the need to go beyond the adiabatic approximation, and highlighting caution in the quantitative use of the Kohn-Sham current, Financial support from the National Science Foundation Award No. CHE-1940333 (DD) and from the Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences under Award No. DESC0020044 (NTM, LL) are gratefully acknowledged. J.F. acknowledges financial support from the European Research Council through Grant No. ERC-2016- StG-714870, and by the Spanish Ministry for Science, Innovation, and Universities: Agencia Estatal de Investigación through Grant No. RTI2018-099737-B-I00

Simulating electron-ion dynamics using time-dependent density functional theory within an Ehrenfest dynamics scheme can be done in two ways that are in principle exact and identical: propagating time-dependent electronic Kohn-Sham equations or propagating electronic coefficients on surfaces obtained from linear-response. We show here that using an approximate functional leads to qualitatively different dynamics in the two approaches. We argue that the latter is more accurate because the functionals are evaluated on domains close to the ground-state where current approximations perform better. We demonstrate this on an exactly-solvable model of charge-transfer, and discuss implications for time-resolved spectroscopy., Comment: 6 pages, 4 figures, 1 supplementary material, 3 movies

The standard description of cavity-modified molecular reactions typically involves a single (resonant) mode, while in reality, the quantum cavity supports a range of photon modes. Here, we demonstrate that as more photon modes are accounted for, physicochemical phenomena can dramatically change, as illustrated by the cavity-induced suppression of the important and ubiquitous process of proton-coupled electron-transfer. Using a multi-trajectory Ehrenfest treatment for the photon-modes, we find that self-polarization effects become essential, and we introduce the concept of self-polarization-modified Born–Oppenheimer surfaces as a new construct to analyze dynamics. As the number of cavity photon modes increases, the increasing deviation of these surfaces from the cavity-free Born–Oppenheimer surfaces, together with the interplay between photon emission and absorption inside the widening bands of these surfaces, leads to enhanced suppression. The present findings are general and will have implications for the description and control of cavity-driven physical processes of molecules, nanostructures, and solids embedded in cavities., Financial support from the US National Science Foundation under Grant No. CHE-1940333 (N.T.M.) and the Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences under Award No. DE-SC0020044 (L.L.) is gratefully acknowledged. N.M.H. gratefully acknowledges an IMPRS fellowship. This work was also supported by the European Research Council (Grant No. ERC-2015-AdG694097), the Cluster of Excellence (AIM), Grupos Consolidados (IT1249-19), and SFB925 Light induced dynamics and control of correlated quantum systems. The Flatiron Institute is a division of the Simons Foundation.

The exact time-dependent potential energy surface driving the nuclear dynamics was recently shown to be a useful tool to understand and interpret the coupling of nuclei, electrons, and photons in cavity settings. Here, we provide a detailed analysis of its structure for exactly solvable systems that model two phenomena: cavity-induced suppression of proton-coupled electron-transfer and its dependence on the initial state, and cavity-induced electronic excitation. We demonstrate the inadequacy of simply using a weighted average of polaritonic surfaces to determine the dynamics. Such a weighted average misses a crucial term that redistributes energy between the nuclear and the polaritonic systems, and this term can in fact become a predominant term in determining the nuclear dynamics when several polaritonic surfaces are involved. Evolving an ensemble of classical trajectories on the exact potential energy surface reproduces the nuclear wavepacket quite accurately, while evolving on the weighted polaritonic surface fails after a short period of time. The implications and prospects for application of mixed quantum-classical methods based on this surface are discussed.

We analyze a mixed quantum-classical algorithm recently derived from the exact factorization equations [Min, Agostini, Gross, PRL 115, 073001 (2015)] to show the role of the different terms in the algorithm in bringing about decoherence and wavepacket branching. The algorithm has the structure of Ehrenfest equations plus a "coupled-trajectory" term for both the electronic and nuclear equations, and we analyze the relative roles played by the different nonadiabatic terms in these equations, including how they are computed in practice. In particular, we show that while the coupled-trajectory term in the electronic equation is essential in yielding accurate dynamics, that in the nuclear equation has a much smaller effect. A decoherence time is extracted from the electronic equations and compared with that of augmented fewest-switches surface-hopping. We revisit a series of nonadiabatic Tully model systems to illustrate our analysis.

We find and analyze the exact time-dependent potential energy surface driving the proton motion for a model of cavity-induced suppression of proton-coupled electron transfer. We show how, in contrast to the polaritonic surfaces, its features directly correlate to the proton dynamics and we discuss cavity modifications of its structure responsible for the suppression. The results highlight the interplay between nonadiabatic effects from coupling to photons and coupling to electrons and suggest caution is needed when applying traditional dynamics methods based on polaritonic surfaces.

A decomposition of the exact exchange-correlation potential of time-dependent density functional theory into an interaction component and a kinetic component offers a new starting point for non- adiabatic approximations. The components are expressed in terms of the exchange-correlation hole and the difference between the one-body density matrix of the interacting and Kohn-Sham systems, which must be approximated in terms of quantities accessible from the Kohn-Sham evolution. We explore several preliminary approximations, evaluate their fulfillment of known exact conditions, and test their performance on simple model systems for which available exact solutions indicate the significance of going beyond the adiabatic approximation., 16 pages, 10 figures

It was recently shown [Y. Suzuki, L. Lacombe, K. Watanabe, and N. T. Maitra, Phys. Rev. Lett. 119, 263401 (2017)] that peak and valley structures in the exact exchange-correlation potential of time-dependent density functional theory are crucial for accurately capturing time-resolved dynamics of electron scattering in a model one-dimensional system. Approximate functionals used today miss these structures and con- sequently underestimate the scattering probability. The dynamics can vary significantly depending on the choice of the initial Kohn-Sham state, and, with a judicious choice, a recently-proposed non-adiabatic ap- proximation provides extremely accurate dynamics on approach to the target but this ultimately also fails to capture reflection accurately. Here we provide more details, using a model of electron-He + as illustration, in both the inelastic and elastic regimes. In the elastic case, the time-resolved picture is contrasted with the time-independent picture of scattering, where the linear response theory of TDDFT can be used to extract transmission and reflection coefficients. Although the exact functional yields identical scattering probabil- ities when used in this way as it does in the time-resolved picture, we show that the currently-available approximate functionals do not, even when they have the correct asymptotic behavior., 9 pages, 6 figures

We present a new class of non-adiabatic approximations in time-dependent density functional theory derived from an exact expression for the time-dependent exchange-correlation potential. The approximations reproduce dynamical step and peak features in the exact potential that are missing in adiabatic approximations. Central to this approach is an approximation for the one-body reduced density-matrix as a functional of the Kohn-Sham density-matrix, and we demonstrate two such examples., Comment: 22 pages, 4 figures

The asymmetric Hubbard dimer is used to study the density-dependence of the exact frequency-dependent kernel of linear-response time-dependent density functional theory. The exact form of the kernel is given, and the limitations of the adiabatic approximation utilizing the exact ground-state functional are shown. The oscillator strength sum rule is proven for lattice Hamiltonians, and relative oscillator strengths are defined appropriately. The method of Casida for extracting oscillator strengths from a frequency-dependent kernel is demonstrated to yield the exact result with this kernel. An unambiguous way of labelling the nature of excitations is given. The fluctuation-dissipation theorem is proven for the ground-state exchange-correlation energy. The distinction between weak and strong correlation is shown to depend on the ratio of interaction to asymmetry. A simple interpolation between carefully defined weak-correlation and strong-correlation regimes yields a density-functional approximation for the kernel that gives accurate transition frequencies for both the single and double excitations, including charge-transfer excitations. Many exact results, limits, and expansions about those limits are given in the Appendices., DC and JF wish to thank funding support from the Spanish Ministerio de Economía y Competitividad via grant FIS2012-34858. NTM thanks the US National Science Foundation CHE-1566197 for support. KB acknowledges DOE grant number DE-FG02-08ER46496.

We identify peak and valley structures in the exact exchange-correlation potential of time-dependent density functional theory that are crucial for time-resolved electron scattering in a model one-dimensional system. These structures are completely missed by adiabatic approximations which consequently significantly underestimate the scattering probability. A recently-proposed non-adiabatic approximation is shown to correctly capture the approach of the electron to the target when the initial Kohn-Sham state is chosen judiciously, and is more accurate than standard adiabatic functionals, but it ultimately fails to accurately capture reflection. These results may explain the underestimate of scattering probabilities in some recent studies on molecules and surfaces., 6 pages, 4 figures

It was recently shown that the exact potential driving the electron's dynamics in enhanced ionization of H can have large contributions arising from dynamic electron-nuclear correlation, going beyond what any Coulombic-based model can provide. This potential is defined via the exact factorization of the molecular wavefunction that allows the construction of a Schrödinger equation for the electronic system, in which the potential contains exactly the effect of coupling to the nuclear system and any external fields. Here we study enhanced ionization in isotopologues of H in order to investigate the nuclear-mass-dependence of these terms for this process. We decompose the exact potential into components that naturally arise from the conditional wavefunction, and also into components arising from the marginal electronic wavefunction, and compare the performance of propagation on these different components as well as approximate potentials based on the quasi-static or Hartree approximation with the exact propagation. A quasiclassical analysis is presented to help analyse the structure of different non-Coulombic components of the potential driving the ionizing electron., We acknowledge support from the European Research Council (ERC-2015-AdG-694097), Grupos Consolidados (IT578-13), and the European Union’s Horizon 2020 Research and Innovation programme under grant agreement no. 676580. A. K. and A. A. acknowledge funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreement no. 704218 and 702406, respectively. N. T. M. thanks the National Science Foundation, grant CHE1566197, for support. Open Access funding provided by the Max Planck Society.

We explore an asymmetric two-fermion Hubbard dimer to test the accuracy of the adiabatic approximation of time-dependent density functional theory in modelling time-resolved charge transfer. We show that the model shares essential features of a ground state long-range molecule in real-space, and by applying a resonant field we show that the model also reproduces essential traits of the CT dynamics. The simplicity of the model allows us to propagate with an "adiabatically-exact" approximation, i.e. one that uses the exact ground-state exchange-correlation functional, and compare with the exact propagation. This allows us to study the impact of the time-dependent charge-transfer step feature in the exact correlation potential of real molecules on the resulting dynamics. Tuning the parameters of the dimer allows a study both of charge-transfer between open-shell fragments and between closed-shell fragments. We find that the adiabatically-exact functional is unable to properly transfer charge, even in situations where the adiabatically-exact resonance frequency is remarkably close to the exact resonance, and we analyze why., Comment: 6 figures

Methods based on propagation of the one-body reduced density-matrix hold much promise for the simulation of correlated many-electron dynamics far from equilibrium, but difficulties with finding good approximations for the interaction term in its equation of motion have so far impeded their application. These difficulties include the violation of fundamental physical principles such as energy conservation, positivity conditions on the density, or unchanging natural orbital occupation numbers. We review some of the recent efforts to confront these problems, and explore a semiclassical approximation for electron correlation coupled to time-dependent Hartree-Fock propagation. We find that this approach captures changing occupation numbers, and excitations to doubly-excited states, improving over TDHF and adiabatic approximations in density-matrix propagation. However, it does not guarantee $N$-representability of the density-matrix, consequently resulting sometimes in violation of positivity conditions, even though a purely semiclassical treatment preserves these conditions., Submitted to Int. J. Quant. Chem

Adiabatic approximations in time-dependent density functional theory (TDDFT) will in general yield unphysical time-dependent shifts in the resonance positions of a system driven far from its ground-state. This spurious time-dependence is rationalized in [J. I. Fuks, K. Luo, E. D. Sandoval and N. T. Maitra, Phys. Rev. Lett. {\bf 114}, 183002 (2015)] in terms of the violation of an exact condition by the non-equilibrium exchange-correlation kernel of TDDFT. Here we give details on the derivation and discuss reformulations of the exact condition that apply in special cases. In its most general form, the condition states that when a system is left in an arbitrary state, in the absence of time-dependent external fields nor ionic motion, the TDDFT resonance position for a given transition is independent of the state. Special cases include the invariance of TDDFT resonances computed with respect to any reference interacting stationary state of a fixed potential, and with respect to any choice of appropriate stationary Kohn-Sham reference state. We then present several case studies, including one that utilizes the adiabatically-exact approximation, that illustrate the conditions and the impact of their violation on the accuracy of the ensuing dynamics. In particular, charge-transfer across a long-range molecule is hampered, and we show how adjusting the frequency of a driving field to match the time-dependent shift in the charge-transfer resonance frequency, results in a larger charge transfer over time., Comment: 16 pages, 10 figs

Under the terms of the Creative Commons Attribution License 3.0 (CC-BY)., It was recently shown that the exact factorization of the electron-nuclear wave function allows the construction of a Schrödinger equation for the electronic system, in which the potential contains exactly the effect of coupling to the nuclear degrees of freedom and any external fields. Here we study the exact potential acting on the electron in charge-resonance enhanced ionization in a model one-dimensional H2+ molecule. We show there can be significant differences between the exact potential and that used in the traditional quasistatic analyses, arising from nonadiabatic coupling to the nuclear system, and that these are crucial to include for accurate simulations of time-resolved ionization dynamics and predictions of the ionization yield., Financial support from the National Science Foundation CHE-1152784 (N. T. M), and Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences and Biosciences under Award DESC0008623, the European Research Council Advanced Grant DYNamo (ERC- 2010-AdG-267374) and Grupo Consolidado UPV/EHU del Gobierno Vasco (IT578-13) (E. K, A. A.) are gratefully acknowledged.

The adiabatic approximation in time-dependent density functional theory (TDDFT) yields reliable excitation spectra with great efficiency in many cases, but fundamentally fails for states of double-excitation character. We discuss how double-excitations are at the root of some of the most challenging problems for TDDFT today. We then present new results for (i) the calculation of autoionizing resonances in the helium atom, (ii) understanding the nature of the double excitations appearing in the quadratic response function, and (iii) retrieving double-excitations through a real-time semiclassical approach to correlation in a model quantum dot.

Exact time-dependent density functionals depend on both the entire history of the density and the initial wave function. Recently we have shown that the two effects are entangled by an exact condition that should be useful as a test in building accurate memory-dependent functionals. Here we consider this condition for small perturbations and derive an exact relation connecting the exchange-correlation kernel with initial-state derivatives. We discuss other exact conditions on the initial-state derivatives which shed some light on the elusive memory-dependence in time-dependent density functional theory. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Within the adiabatic approximation, time-dependent density functional theory yields only single excitations. Near states of double excitation character, the exact exchange-correlation kernel has a strong dependence on frequency. We derive the exact frequency-dependent kernel when a double excitation mixes with a single excitation, well separated from the other excitations, in the limit that the electron--electron interaction is weak. Building on this, we construct a nonempirical approximation for the general case, and illustrate our results on a simple model.

The exact time-dependent correlation energy in time-dependent density-functional theory is shown to sometimes become positive, which is impossible with most present TDDFT approximations. Both the correlation potential and energy can be strongly nonlocal in time. A new inequality is derived for the time-dependent exchange-correlation energy. The correlation energy appears to scale to a constant function of scaled time in the high-density limit. In the linear response regime, the correlation energy is shown to become purely adiabatic, but the correlation potential is generally nonadiabatic. The usefulness of the virial theorem as a test of numerical accuracy is demonstrated. All results are found or inspired by exact numerical solution of a simple model system (Hooke’s atom), and inversion of the corresponding Kohn–Sham equations.

The time-periodic density of a Floquet state of a time-periodic potential does not uniquely determine that potential. A simple example demonstrates this, and the implications are discussed.

We present a rigorous reformulation of the quantum mechanical equations of motion for the coupled system of electrons and nuclei that focuses on the dynamics of the electronic subsystem. Usually the description of electron dynamics involves an electronic Schr\"odinger equation where the nuclear degrees of freedom appear as parameters or as classical trajectories. Here we derive the exact Schr\"odinger equation for the subsystem of electrons, staying within a full quantum treatment of the nuclei. This exact Schr\"odinger equation features a time-dependent potential energy surface for electrons (e-TDPES). We demonstrate that this exact e-TDPES differs significantly from the electrostatic potential produced by classical or quantum nuclei.

The decomposition of electronic and nuclear motion presented in~[A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)] yields a time-dependent potential that drives the nuclear motion and fully accounts for the coupling to the electronic subsystem. Here we show that propagation of an ensemble of independent classical nuclear trajectories on this exact potential yields dynamics that are essentially indistinguishable from the exact quantum dynamics for a model non-adiabatic charge transfer problem. We point out the importance of step and bump features in the exact potential that are critical in obtaining the correct splitting of the quasiclassical nuclear wave packet in space after it passes through an avoided crossing between two Born-Oppenheimer surfaces, and analyze their structure. Lastly, an analysis of the exact potentials in the context of trajectory surface hopping procedure is presented, including preliminary investigations of velocity-adjustment, and the force-induced decoherence effect., Comment: 25 pages, 11 figures

We show that an asymmetric two-fermion two-site Hubbard model illustrates the essential features of long-range charge-transfer dynamics in a real-space molecule. We apply a resonant field that transfers one fermion from one site to the other. Via constrained search we find the exact ground-state exchange-correlation functional, and use it to propagate the Kohn-Sham system, giving the first "adiabatically-exact" calculation of time-resolved charge-transfer. This propagation fails to properly transfer charge. We analyze why by comparing the exact and adiabatically-exact potentials and discuss the role of the derivative discontinuity. The implication for real-space molecules is that even the best possible adiabatic approximation, despite capturing non-local step features relevant to dissociation and charge-transfer excitations, cannot capture fully time-resolved charge-transfer dynamics.

Representing the dynamics of a continuous time molecular system by a symplectic discrete time map can much reduce the computational time. The question then arises of whether semiclassical methods can be effectively applied to this reduced description: as in the classical case, the map should prove to be a much more computationally efficient description of the dynamics. Here we study the semiclassical propagation of the standard map, or kicked rotor, based on a Herman–Kluk propagator. This is a very interesting playground to test the feasibility of a semiclassical mapping approach, since it demonstrates a wealth of quantum and classical dynamical behavior: As the kick strength increases, the system goes from being very nearly integrable, through mixed phase space, to chaotic. The map displays phenomena that occur in generic molecular systems, so this study is also a test of how well semiclassics can describe such phenomena. In particular, we discuss (i) classically forbidden transport: the significance of branches of the semiclassical integrand in the complex phase plane must be understood in order for the semiclassics to be meaningful; (ii) sub-h structure: in the nearly integrable regime, the semiclassics can be poor due to the presence of islets of area less than Planck’s constant in phase space; (iii) dynamical localization: in the chaotic regime, the classical momentum diffuses, whereas the quantum localizes. Our results show that semiclassics also localizes, and we can confirm directly the theory that dynamical localization is due largely to phase interference.

The issue of quantum barrier crossing and reflection in the time domain is addressed. We find that (1) classically forbidden barrier tunneling and above-barrier reflection are well-defined and important processes in the time domain, (2) classically forbidden processes can overshadow allowed ones when both are present, and (3) classically allowed trajectories in the time domain are not, in general, sufficient to explain tunneling amplitudes in the energy domain. We also make clear the essential distinction of barriers which flatten out at large distance and those which do not. {copyright} {ital 1997} {ital The American Physical Society}

Recent work [Phys. Rev. Lett. 109, 266404 (2012)] showed that the exact exchange-correlation potential of time-dependent density-functional theory generically displays dynamical step structures. These have a spatially nonlocal and time-nonlocal dependence on the density in real-time dynamics. The steps are missing in the usual approximations, which consequently yield inaccurate dynamics. Yet these same approximations often yield good linear response spectra. Here we investigate whether the steps appear in the linear response regime, when the response is calculated from a real-time dynamics simulation, by examining the exact correlation potential of model two-electron systems at various times. We find there are no step structures in regions where the system response is linear. Step structures appear in the correlation potential only in regions of space where the density response is nonlinear; these regions, having exponentially small density, do not contribute to the observables measured in linear response.

arXiv:1211.2849, We show that whenever an electron transfers between closed-shell molecular fragments, the exact correlation potential of time-dependent density functional theory develops a step and peak structure in the bonding region. This structure has a density dependence that is nonlocal both in space and in time that even the exact adiabatic ground-state exchange–correlation functional fails to capture it. For charge-transfer between open-shell fragments, an initial step and peak vanish as the charge-transfer state is reached. The inability of usual approximations to develop these structures leads to inaccurate charge-transfer dynamics. This is illustrated by the complete lack of Rabi oscillations in the dipole moment under conditions of resonant charge transfer for an exactly solvable model system. The results transcend the model and are applicable to more realistic molecular complexes., We gratefully acknowledge financial support from the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences and Biosciences under Award DE-SC0008623 (N.T.M.) and a grant of computer time from the CUNY High Performance Computing Center under NSF Grant CNS-0855217. The European Research Council (ERC-2010-AdG-267374), Spanish: FIS2011-65702-C02-01), Grupos Consolidados (IT-319-07), and EC Projects CRONOS (280879-2) and CNS-0958379 are gratefully acknowledged. J.I.F. acknowledges support from an FPI fellowship (FIS2007-65702-C02-01).

The exact exchange-correlation (xc) potential of time-dependent density functional theory has been shown to have striking features. For example, step and peak features are generically found when the system is far from its ground-state, and these depend nonlocally on the density in space and time. We analyze the xc potential by decomposing it into kinetic and interaction potential components, and comparing each with their exact-adiabatic counterparts, for a range of dynamical situations in model one-dimensional (1D) two-electron systems. We find that often, but not always, the kinetic contri- bution is mostly responsible for these features, that are missed by the adiabatic approximation. The adiabatic approximation often makes a smaller error for the potential contribution, which we write in two parts, one being the Coulomb potential due to the time-dependent xc hole. These observations also held in non-equilibrium cases we studied where there are large features in the correlation po- tential although no step structure. In ground-state situations, step and peak structures arise in cases of static correlation, when more than one determinant is essential to describe the interacting state. We investigate the time-dependent natural orbital occupation numbers and find the corresponding relation between these and the dynamical step is more complex than for the ground-state case., Comment: 13 pages and 20 figures

In the thirty-two years since the birth of the foundational theorems, time-dependent density functional theory has had a tremendous impact on calculations of electronic spectra and dynamics in chemistry, biology, solid-state physics, and materials science. Alongside the wide-ranging applications, there has been much progress in understanding fundamental aspects of the functionals and the theory itself. This Perspective looks back to some of these developments, reports on some recent progress and current challenges for functionals, and speculates on future directions to improve the accuracy of approximations used in this relatively young theory.

Imagine being able to calculate the exact quantum mechanics of electrons in an atom, molecule, or solid by completely removing the Coulomb repulsion between the electrons and warping the attraction between the electrons and the nuclei. Doesn’t this sound crazy? Yet this is exactly the recipe for the remarkably successful, and rigorous, density-functional theory (DFT), which says that one can extract the correct observables of an interacting electronic system by instead treating the electrons as independent particles that interact with a fictitious potential.

Many recent applications of time-dependent density functional theory begin in an initially excited state and propagate it using an adiabatic approximation for the exchange-correlation potential. This, however, inserts the excited-state density into a ground-state approximation. By studying a series of model calculations, we highlight the relevance of initial-state dependence of the exact functional when starting in an excited state, discuss different valid choices of the initial Kohn-Sham state, and explore the errors inherent in the adiabatic approximation that neglects this dependence.

Correlated electron motion plays a significant role in the spectra described in the previous chapters. Further, placing an atom, molecule or solid in a strong laser field reveals fascinating non-perturbative phenomena, such as non-sequential multiple-ionization (see Chap. 18), whose origins lie in the subtle ways electrons interact with each other.

We show that the exact exchange-correlation potential of time-dependent density-functional theory displays dynamical step structures that have a spatially non-local and time non-local dependence on the density. Using one-dimensional two-electron model systems, we illustrate these steps for a range of non-equilibrium dynamical situations relevant for modeling of photo-chemical/physical processes: field-free evolution of a non-stationary state, resonant local excitation, resonant complete charge-transfer, and evolution under an arbitrary field. Lack of these steps in usual approximations yield inaccurate dynamics, for example predicting faster dynamics and incomplete charge transfer., Financial support from the National Science Foundation (CHE-1152784) and a grant of computer time from the CUNY High Performance Computing Center under NSF Grants No. CNS-0855217 and No. CNS-0958379 are gratefully acknowledged. J. I. F. acknowledges support from a FPI Fellowship (FIS2007-65702-C02-01). A.R. acknowledges financial support from the European Research Council Advanced Grant DYNamo (ERC-2010-AdG-Proposal No. 267374) Spanish Grants (No. FIS2011- 65702- C02-01 and No. PIB2010US-00652), Grupos Consolidados UPV/EHU del Gobierno Vasco (IT-319-07), and the European Commission Project CRONOS (280879-2).

In ground-state DFT, the fact that the xc potential is a functional of the density is a direct consequence of the one-to-one mapping between ground-state densities and potentials. In TDDFT, the one-to-one mapping is between densities and potentials for a given initial state. This means that the potentials, most generally, are functionals of the initial state of the system, as well as of the density; and, not just of the instantaneous density, but of its entire history.

We investigate the accuracy and efficiency of the semiclassical Frozen Gaussian method in describing electron dynamics in real time. Model systems of two soft-Coulomb-interacting electrons are used to study correlated dynamics under non-perturbative electric fields, as well as the excitation spectrum. The results show that a recently proposed method that combines exact-exchange with semiclassical correlation to propagate the one-body density-matrix holds promise for electron dynamics in many situations that either wavefunction or density-functional methods have difficulty describing. The results also however point out challenges in such a method that need to be addressed before it can become widely applicable.

Long-range charge-transfer excitations pose a major challenge for time-dependent density-functional approximations. We show that spin-symmetry breaking offers a simple solution for molecules composed of open-shell fragments, yielding accurate excitations at large separations when the acceptor effectively contains one active electron. Unrestricted exact-exchange and self-interaction-corrected functionals are performed on one-dimensional models and on the real LiH molecule within the pseudopotential approximation to demonstrate our results., We acknowledge support from MEC (Grant No. FIS2007-65702-C02-01), ACI promociona (Grant No. ACI2009-1036), Grupos Consolidados UPV/EHU del Gobierno Vasco (Grant No. IT-319-07), the e-I3 ETSF project (Contract No. 211956), the National Science Foundation (Grant No. CHE-0647913), and the Cottrell Scholar Program of the Research Corporation for Science Advancement, as well as a grant of computer time from the CUNY High Performance Computing Center under NSF Grants No. CNS-0855217 and No. CNS-0958379.

We present an exact decomposition of the complete wavefunction for a system of nuclei and electrons evolving in a time-dependent external potential. We derive formally exact equations for the nuclear and electronic wavefunctions that lead to rigorous definitions of a time-dependent potential energy surface (TDPES) and a time-dependent geometric phase. For the $H_2^+$ molecular ion exposed to a laser field, the TDPES proves to be a useful interpretive tool to identify different mechanisms of dissociation., 4 pages, 2 figures

Lack of memory (locality in time) is a major limitation of almost all present time-dependent density functional approximations. By using semiclassical dynamics to compute correlation effects within a density-matrix functional approach, we incorporate memory, including initial-state dependence, as well as changing occupation numbers, and predict more observables in strong-field applications., 4.5 pages, 1 figure

A recent paper [J. Schirmer and A. Dreuw, Phys. Rev A. 75, 022513 (2007)] challenges exact time-dependent density-functional theory (TDDFT) on several grounds. We explain why these criticisms are either irrelevant or incorrect, and that TDDFT is both formally exact and predictive.

Systems whose underlying classical dynamics are chaotic exhibit signatures of the chaos in their quantum mechanics. We investigate the possibility of using time-dependent density functional theory (TDDFT) to study the case when chaos is induced by electron-interaction alone. Nearest-neighbour level-spacing statistics are in principle exactly and directly accessible from TDDFT. We discuss how the TDDFT linear response procedure can reveal the mechanism of chaos induced by electron-interaction alone. A simple model of a two-electron quantum dot highlights the necessity to go beyond the adiabatic approximation in TDDFT., 8 pages, 4 figures

Linear response time-dependent density functional theory is used to study low-lying electronic continuum states of targets that can bind an extra electron. Exact formulas to extract scattering amplitudes from the susceptibility are derived in one dimension. A single-pole approximation for scattering phase shifts in three dimensions is shown to be more accurate than static exchange for singlet electron-He$^+$ scattering., 5 pages, 2 figures, J. Chem. Phys. accepted

Long-range charge transfer excited states are notoriously badly underestimated in time-dependent density functional theory (TDDFT). We resolve how {\it exact} TDDFT captures charge transfer between open-shell species: in particular the role of the step in the ground-state potential, and the severe frequency-dependence of the exchange-correlation kernel. An expression for the latter is derived, that becomes exact in the limit that the charge-transfer excitations are well-separated from other excitations. The exchange-correlation kernel has the task of undoing the static correlation in the ground state introduced by the step, in order to accurately recover the physical charge-transfer states., 2 figures

Adiabatic time-dependent density functional theory fails for excitations of a heteroatomic molecule composed of two open-shell fragments at large separation. Strong frequency-dependence of the exchange-correlation kernel is necessary for both local and charge-transfer excitations. The root of this is static correlation created by the step in the exact Kohn-Sham ground-state potential between the two fragments. An approximate non-empirical kernel is derived for excited molecular dissociation curves at large separation. Our result is also relevant for the usual local and semi-local approximations for the ground-state potential, as static correlation there arises from the coalescence of the highest occupied and lowest unoccupied orbital energies as the molecule dissociates., Comment: 7 pages, 2 figures

Despite the incorrect asymptotic behavior of its potential, the time-dependent local density approximation can yield accurate optical spectra. The oscillator strengths of Rydberg excitations appear in the calculated spectrum as continuum contributions with excellent optical intensity. We explain why, illustrate this for the neon and helium atoms, and also discuss when such calculations of the optical response will be inaccurate.