Johanna I. Fuks

Universal dynamical steps in the exact time-dependent exchange-correlation potential

We show that the exact exchange-correlation potential of time-dependent density-functional theory displays dynamical step structures that have a spatially nonlocal and time nonlocal dependence on the density. Using one-dimensional two-electron model systems, we illustrate these steps for a range of nonequilibrium dynamical situations relevant for modeling of photochemical or physical processes: field-free evolution of a nonstationary state, resonant local excitation, resonant complete charge transfer, and evolution under an arbitrary field. A lack of these steps in the usual approximations yields inaccurate dynamics, for example, predicting faster dynamics and incomplete charge transfer.

Dynamics of charge-transfer processes with time-dependent density functional theory

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.

Nonlinear phenomena in time-dependent density-functional theory: What Rabi oscillations can teach us

Through the exact solution of a two-electron singlet system interacting with a monochromatic laser we prove that all adiabatic density functionals within time-dependent density-functional theory are not able to discern between resonant and nonresonant (detuned) Rabi oscillations. This is rationalized in terms of a fictitious dynamical exchange-correlation (xc) detuning of the resonance while the laser is acting. The nonlinear dynamics of the Kohn-Sham system shows the characteristic features of detuned Rabi oscillations even if the exact resonant frequency is used. We identify the source of this error in a contribution from the xc functional to the set of nonlinear equations that describes the electron dynamics in an effective two-level system. The constraint of preventing the detuning introduces a new strong condition to be satisfied by approximate xc functionals.

Time-Resolved Spectroscopy in Time-Dependent Density Functional Theory: An Exact Condition

A fundamental property of a quantum system driven by an external field is that when the field is turned off the positions of its response frequencies are independent of the time at which the field is turned off. We show that this leads to an exact condition for the exchange-correlation potential of time-dependent density functional theory. The Kohn-Sham potential typically continues to evolve after the field is turned off, which leads to time dependence in the response frequencies of the Kohn-Sham response function. The exchange-correlation kernel must cancel out this time dependence. The condition is typically violated by approximations currently in use, as we demonstrate by several examples, which has severe consequences for their predictions of time-resolved spectroscopy.

Density functional theory beyond the linear regime: Validating an adiabatic local density approximation

We present a local density approximation (LDA) for one-dimensional (1D) systems interacting via the soft-Coulomb interaction based on quantum Monte Carlo calculations. Results for the ground-state energies and ionization potentials of finite 1D systems show excellent agreement with exact calculations obtained by exploiting the mapping of an N-electron system in d dimensions onto a single electron in Nxd dimensions, properly symmetrized by the Young diagrams. We conclude that 1D LDA is of the same quality as its three-dimensional (3D) counterpart, and we infer conclusions about 3D LDA. The linear and nonlinear time-dependent responses of 1D model systems using LDA, exact exchange, and the exact solution are investigated and show very good agreement in both cases, except for the well-known problem of missing double excitations. Consequently, the 3D LDA is expected to be of good quality beyond the linear response. In addition, the 1D LDA should prove useful in modeling the interaction of atoms with strong laser fields, where this specific 1D model is often used.

Time-dependent density-functional and reduced density-matrix methods for few electrons: Exact versus adiabatic approximations

To address the impact of electron correlations in the linear and non-linear response regimes of interacting many-electron systems exposed to time-dependent external fields, we study one-dimensional (1D) systems where the interacting problem is solved exactly by exploiting the mapping of the 1D N-electron problem onto an N-dimensional single electron problem. We analyze the performance of the recently derived 1D local density approximation as well as the exact-exchange orbital functional for those systems. We show that the interaction with an external resonant laser field shows Rabi oscillations which are detuned due to the lack of memory in adiabatic approximations. To investigate situations where static correlations play a role, we consider the time-evolution of the natural occupation numbers associated to the reduced one-body density matrix. Those studies shed light on the non-locality and time-dependence of the exchange and correlation functionals in time-dependent density and density-matrix functional theories.

Charge transfer in time-dependent density-functional theory via spin-symmetry breaking

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.

Kinetic and interaction components of the exact time-dependent correlation potential

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 components and comparing each with their exact-adiabatic counterparts, for a range of dynamical situations in model one-dimensional two-electron systems. We find that often, but not always, the kinetic contribution is largely responsible for these features that are missed by the adiabatic approximation. The adiabatic approximation often makes a smaller error for the interaction component, which we write in two parts, one being the Coulomb potential due to the time-dependent xc hole. Non-adiabatic features of the kinetic component were also larger than those of the interaction component in cases that we studied when there is negligible 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.

Time-dependent exchange-correlation functional for a Hubbard dimer: Quantifying nonadiabatic effects

We address and quantify the role of nonadiabaticity (“memory effects”) in the exchange-correlation (xc) functional of time-dependent density functional theory (TDDFT) for describing nonlinear dynamics of many-body systems. Time-dependent resonant processes are particularly challenging for available TDDFT approximations, due to their strong nonlinear and nonadiabatic character. None of the known approximate density functionals are able to cope with this class of problems in a satisfactory manner. In this work we look at the prototypical example of the resonant processes by considering Rabi oscillations within the exactly soluble two-site Hubbard model. We construct the exact adiabatic xc functional and show that (i) it does not reproduce correctly resonant Rabi dynamics, and (ii) there is a sizable nonadiabatic contribution to the exact xc potential, which turns out to be small only at the beginning and at the end of the Rabi cycle when the ground-state population is dominant. We then propose a “two-level” approximation for the time-dependent xc potential which can capture Rabi dynamics in the two-site problem. It works well both for resonant and for detuned Rabi oscillations and becomes essentially exact in the linear response regime.

Studies of spuriously shifting resonances in time-dependent density functional theory

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 explained in Fuks et al. [Phys. Rev. Lett. 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, the TDDFT resonance position for a given transition in the absence of time-dependent external fields and ionic motion 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.