Derivative discontinuities in time-dependent density-functional theory

Double and Charge-Transfer Excitations in Time-Dependent Density Functional Theory

Time-dependent density functional theory has emerged as a method of choice for calculations of spectra and response properties in physics, chemistry, and biology, with its system-size scaling enabling computations on systems much larger than otherwise possible. While increasingly complex and interesting systems have been successfully tackled with relatively simple functional approximations, there has also been increasing awareness that these functionals tend to fail for certain classes of approximations. Here I review the fundamental challenges the approximate functionals have in describing double excitations and charge-transfer excitations, which are two of the most common impediments for the theory to be applied in a black-box way. At the same time, I describe the progress made in recent decades in developing functional approximations that give useful predictions for these excitations. Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

From Kohn-Sham to Many-Electron Energies via Step Structures in the Exchange-Correlation Potential

Accurately describing excited states within Kohn–Sham (KS) density functional theory (DFT), particularly those which induce ionization and charge transfer, remains a great challenge. Common exchange-correlation (xc) approximations are unreliable for excited states owing, in part, to the absence of a derivative discontinuity in the xc energy (Δ), which relates a many-electron energy difference to the corresponding KS energy difference. We demonstrate, analytically and numerically, how the relationship between KS and many-electron energies leads to the step structures observed in the exact xc potential in four scenarios: electron addition, molecular dissociation, excitation of a finite system, and charge transfer. We further show that steps in the potential can be obtained also with common xc approximations, as simple as the LDA, when addressed from the ensemble perspective. The article therefore highlights how capturing the relationship between KS and many-electron energies with advanced xc approximations is crucial for accurately calculating excitations, as well as the ground-state density and energy of systems which consist of distinct subsystems.

Piecewise linearity, freedom from self-interaction, and a Coulomb asymptotic potential: three related yet inequivalent properties of the exact density functional

Three properties of the exact energy functional of DFT are important in general and for spectroscopy in particular, but are not necessarily obeyed by approximate functionals. We explain what they are, why they are important, and how they are related yet inequivalent.

Simple correction to bandgap problems in IV and III-V semiconductors: an improved, local first-principles density functional theory

We report results from a fast, efficient, and first-principles full-potential Nth-order muffin-tin orbital (FP-NMTO) method combined with van Leeuwen-Baerends correction to local density exchange-correlation potential. We show that more complete and compact basis set is critical in improving the electronic and structural properties. We exemplify the self-consistent FP-NMTO calculations on group IV and III-V semiconductors. Notably, predicted bandgaps, lattice constants, and bulk moduli are in good agreement with experiments (e.g. we find for Ge 0.86 eV, 5.57 [Formula: see text], 75 GPa versus measured 0.74 eV, 5.66 [Formula: see text], 77.2 GPa). We also showcase its application to the electronic properties of 2-dimensional h-BN and h-SiC, again finding good agreement with experiments.

Nonadiabatic Dynamics in Single-Electron Tunneling Devices with Time-Dependent Density-Functional Theory

We simulate the dynamics of a single-electron source, modeled as a quantum dot with on-site Coulomb interaction and tunnel coupling to an adjacent lead in time-dependent density-functional theory. Based on this system, we develop a time-nonlocal exchange-correlation potential by exploiting analogies with quantum-transport theory. The time nonlocality manifests itself in a dynamical potential step. We explicitly link the time evolution of the dynamical step to physical relaxation timescales of the electron dynamics. Finally, we discuss prospects for simulations of larger mesoscopic systems.

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.

Charge transfer in time-dependent density functional theory

Charge transfer plays a crucial role in many processes of interest in physics, chemistry, and bio-chemistry. In many applications the size of the systems involved calls for time-dependent density functional theory (TDDFT) to be used in their computational modeling, due to its unprecedented balance between accuracy and efficiency. However, although exact in principle, in practise approximations must be made for the exchange-correlation functional in this theory, and the standard functional approximations perform poorly for excitations which have a long-range charge-transfer component. Intense progress has been made in developing more sophisticated functionals for this problem, which we review. We point out an essential difference between the properties of the exchange-correlation kernel needed for an accurate description of charge-transfer between open-shell fragments and between closed-shell fragments. We then turn to charge-transfer dynamics, which, in contrast to the excitation problem, is a highly non-equilibrium, non-perturbative, process involving a transfer of one full electron in space. This turns out to be a much more challenging problem for TDDFT functionals. We describe dynamical step and peak features in the exact functional evolving over time, that are missing in the functionals currently used. The latter underestimate the amount of charge transferred and manifest a spurious shift in the charge transfer resonance position. We discuss some explicit examples.

Time-Local Equation for the Exact Optimized Effective Potential in Time-Dependent Density Functional Theory

A long-standing challenge in the time-dependent density functional theory is to efficiently solve the exact time-dependent optimized effective potential (TDOEP) integral equation derived from orbital-dependent functionals, especially for the study of nonadiabatic dynamics in time-dependent external fields. In this Letter, we formulate a completely equivalent time-local TDOEP equation that admits a unique real-time solution in terms of time-dependent Kohn-Sham and effective memory orbitals. The time-local formulation is numerically implemented, with the incorporation of exponential memory loss to address the unaccounted for correlation component in the exact-exchange-only functional, to enable the study of the many-electron dynamics of a one-dimensional hydrogen chain. It is shown that the long time behavior of the electric dipole converges correctly and the zero-force theorem is fulfilled in the current implementation.

Tuned range separated hybrid functionals for solvated low bandgap oligomers

The description of charge transfer excitations has long been a challenge to time dependent density functional theory. The recently developed concept of "optimally tuned range separated hybrid (OT-RSH) functionals" has proven to describe charge transfer excitations accurately in many cases. However, describing solvated or embedded systems is yet a challenge. This challenge is not only computational but also conceptual, because the tuning requires identifying a specific orbital, typically the highest occupied one of the molecule under study. For solvated molecules, this orbital may be delocalized over the solvent. We here demonstrate that one way of overcoming this problem is to use a locally projected self-consistent field diagonalization on an absolutely localized molecular orbital expansion. We employ this approach to determine ionization energies and the optical gap of solvated oligothiophenes, i.e., paradigm low gap systems that are of relevance in organic electronics. Dioxane solvent molecules are explicitly represented in our calculations, and the ambiguities of straightforward parameter tuning in solution are elucidated. We show that a consistent estimate of the optimal range separated parameter (ω) at the limit of bulk solvation can be obtained by gradually extending the solvated system. In particular, ω is influenced by the solvent beyond the first coordination sphere. For determining ionization energies, a considerable number of solvent molecules on the first solvation shell must be taken into account. We demonstrate that accurately calculating optical gaps of solvated systems using OT-RSH can be done in three steps: (i) including the chemical environment when determining the range-separation parameter, (ii) taking into account the screening due to the solvent, and (iii) using realistic molecular geometries.

Restoration of the derivative discontinuity in Kohn-Sham density functional theory: an efficient scheme for energy gap correction

From the perspective of perturbation theory, we propose a systematic procedure for the evaluation of the derivative discontinuity (DD) of the exchange-correlation energy functional in Kohn-Sham (KS) density functional theory, wherein the exact DD can in principle be obtained by summing up all the perturbation corrections to infinite order. Truncation of the perturbation series at low order yields an efficient scheme for obtaining the approximate DD. While the zeroth-order theory yields a vanishing DD, the first-order correction to the DD can be expressed as an explicit universal functional of the ground-state density and the KS lowest unoccupied molecular orbital density, allowing the direct evaluation of the DD in the standard KS method without extra computational cost. The fundamental gap can be predicted by adding the estimated DD to the KS gap. This scheme is shown to be accurate in the prediction of the fundamental gaps for a wide variety of atoms and molecules.

Kohn-Sham self-interaction correction in real time

We present a solution scheme for the time-dependent Kohn-Sham self-interaction correction. Based on the generalized optimized effective potential approach, the multiplicative Kohn-Sham potential is constructed in real time and real space for the self-interaction corrected local density approximation. Excitations of different character, including charge-transfer excitations that had been regarded as prime examples for the failure of standard time-dependent density functionals, are described correctly by this approach. We analyze the time-dependent exchange-correlation potential and density, revealing features that are decisive for the correct description of the response.

Time-dependent density-functional theory simulation for electron-ion dynamics in molecules under intense laser pulses

We have developed a simulation method to describe three-dimensional dynamics of electrons and ions in a molecule based on the time-dependent density-functional theory. We solve the time-dependent Kohn-Sham equation for electrons employing the real-space and real-time method, while the ion dynamics are described in classical mechanics by the Ehrenfest method. For an efficient calculation in massively parallel computers, the code is parallelized dividing the spatial grid points. We apply the method to the Coulomb explosion of the H(2)S molecule under an intense and ultrashort laser pulse and investigate the mechanism of the process.

Adiabatic approximation in nonperturbative time-dependent density-functional theory

We construct the exact exchange-correlation potential of time-dependent density-functional theory and the approximation to it that is adiabatic but exact otherwise. For the strong-field double ionization of the Helium atom these two potentials are virtually identical. Thus, memory effects play a negligible role in this paradigm process of nonlinear, nonperturbative electron dynamics. We identify the regime of high-frequency excitations where the adiabatic approximation breaks down and explicitly calculate the nonadiabatic contribution to the exchange-correlation potential.

Generator coordinate method in time-dependent density-functional theory: Memory made simple

The generator coordinate (GC) method is a variational approach to the quantum many-body problem in which interacting many-body wave functions are constructed as superpositions of (generally nonorthogonal) eigenstates of auxiliary Hamiltonians containing a deformation parameter. This paper presents a time-dependent extension of the GC method as a new approach to improve existing approximations of the exchange-correlation (XC) potential in time-dependent density-functional theory (TDDFT). The time-dependent GC method is shown to be a conceptually and computationally simple tool to build memory effects into any existing adiabatic XC potential. As an illustration, the method is applied to driven parametric oscillations of two interacting electrons in a harmonic potential (Hooke's atom). It is demonstrated that a proper choice of time-dependent generator coordinates in conjunction with the adiabatic local-density approximation reproduces the exact linear and nonlinear two-electron dynamics quite accurately, including features associated with double excitations that cannot be captured by TDDFT in the adiabatic approximation.

Spurious fractional charge on dissociated atoms: Pervasive and resilient self-interaction error of common density functionals

Semilocal density functional approximations for the exchange-correlation energy can improperly dissociate a neutral molecule XY (Y not =X) to fractionally charged fragments X(+q)...Y(-q) with an energy significantly lower than X0...Y0. For example, NaCl can dissociate to Na(+0.4)...Cl(-0.4). Generally, q is positive when the lowest-unoccupied orbital energy of atom Y0 lies below the highest-occupied orbital energy of atom X0. The first 24 open sp-shell atoms of the Periodic Table can form 276 distinct unlike pairs XY, and in the local spin density approximation 174 of these display fractional-charge dissociation. Finding these lowest-energy solutions with standard quantum chemistry codes, however, requires special care. Self-interaction-corrected (SIC) semilocal approximations are exact for one-electron systems and also reduce the spurious fractional charge q. The original SIC of Perdew and Zunger typically reduces q to 0. A scaled-down SIC with better equilibrium properties sometimes fails to reduce q all the way to 0. The desideratum of "many-electron self-interaction freedom" is introduced as a generalization of the one-electron concept.

Adiabatic approximation of the correlation function in the density-functional treatment of ionization processes

The ionization of a one-dimensional model helium atom in short laser pulses using time-dependent density-functional theory is investigated. We calculate ionization probabilities as a function of laser intensity by approximating the correlation function of the system adiabatically with an explicit dependence on the fractional number of bound electrons. For the correlation potential we take the derivative discontinuity at integer numbers of bound electrons explicitly into account. This approach reproduces ionization probabilities from the solution of the time-dependent Schrödinger equation, in particular, the so-called knee due to nonsequential ionization.