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arXiv:2205.03691 · Chemical Physics Reviews

The exact exchange–correlation potential in time-dependent density functional theory: Choreographing electrons with steps and peaks

Davood B. Dar, Lionel Lacombe, Neepa T. Maitra

Published Sep 1, 2022• 8 citations
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Abstract

The time-dependent exchange–correlation potential has the unusual task of directing fictitious non-interacting electrons to move with exactly the same probability density as true interacting electrons. This has intriguing implications for its structure, especially in the non-perturbative regime, leading to step and peak features that cannot be captured by bootstrapping any ground-state functional approximation. We review what has been learned about these features in the exact exchange–correlation potential of time-dependent density functional theory in the past decade or so and implications for the performance of simulations when electrons are driven far from any ground state.

DOI
10.1063/5.0096627

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The time-dependent exchange–correlation potential has the unusual task of directing fictitious non-interacting electrons to move with exactly the same probability density as true interacting electrons. This has intriguing implications for its structure, especially in the non-perturbative regime, leading to step and peak features that cannot be captured by bootstrapping any ground-state functional approximation. We review what has been learned about these features in the exact exchange–correlation potential of time-dependent density functional theory in the past decade or so and implications for the performance of simulations when electrons are driven far from any ground state.
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.
In time-dependent density-functional theory, an important quantity is the exchange-correlation kernel whose knowledge allows for the calculations of the excitation spectrum of electronic systems. In most applications, the exchange-correlation kernel is approximated by the adiabatic local-density approximation, ignoring the frequency dependence and nonlocality. To gain insight into the nature of the exact exchange-correlation kernel, we have considered a two-dimensional one-band Hubbard model. The calculated exact exchange-correlation kernel reveals a number of striking features. It has a weak energy dependence up to the main excitation energy. On the other hand, the exact kernel shows a strong energy dependence in energy regions, where there are many-body excitations that are not contained in the local-density approximation, even though these excitations have small weight. It is found that a static approximation can well reproduce the main excitation peaks, but the satellite structures arising from many-body interaction are unlikely to be accounted for with a static approximation.
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.
Approximations for the ground-state exchange-correlation potential of density-functional theory have reached a high level of sophistication. By contrast, time- or frequency-dependent exchange-correlation potentials are still being treated in a local approximation. Here we propose a novel approximation scheme, which effectively brings the power of the generalized gradient approximation (GGA) and meta-GGA to time-dependent density-functional theory. The theory should allow a more accurate treatment of strongly inhomogeneous electronic systems (e.g. molecular junctions) while remaining essentially exact for slowly varying densities and slowly varying external potentials.
We illustrate the potential of Time Dependent Density Functional Theory (TDDFT) for describing the nonequilibrium behavior of strongly correlated (lattice) models. Starting from an exact time-evolution of the many-body wavefunction, we determine, via reverse engineering, the exact exchange correlation (xc) potential vxcv_{xc}vxc​ for small Hubbard chains of different lengths and electron fillings and exposed to time-dependent perturbations. We compare some of the exact results to those of adiabatic-like treatments, in order to extract some of the properties that approximate xc potentials should have. Finally, we provide details of work in progress and future directions.
We employ the density matrix renormalization group to construct the exact time-dependent exchange-correlation potential for an impurity model with an applied transport voltage. Even for short-ranged interaction we find an infinitely long-ranged exchange-correlation potential which is built up instantly after switching on the voltage. Our result demonstrates the fundamental difficulties of transport calculations based on time-dependent density functional theory. While formally the approach works, important information can be missing in the ground-state functionals and may be hidden in the usually unknown non-equilibrium functionals.