Introduction to TDDFT

Time-Dependent Density Functional Theory for X-ray Absorption Spectra: Comparing the Real-Time Approach to Linear Response

We simulate X-ray absorption spectra at elemental K-edges using time-dependent density functional theory (TDDFT) in both its conventional linear-response implementation and its explicitly time-dependent or "real-time" formulation. Real-time TDDFT simulations enable broadband spectra calculations without the need to invoke frozen occupied orbitals ("core/valence separation"), but we find that these spectra are often contaminated by transitions to the continuum that originate from lower-energy core and semicore orbitals. This problem becomes acute in triple-ζ basis sets, although it is sometimes sidestepped in double-ζ basis sets. Transitions to the continuum acquire surprisingly large dipole oscillator strengths, leading to spectra that are difficult to interpret. Meaningful spectra can be recovered by means of a filtering technique that decomposes the spectrum into contributions from individual occupied orbitals, and the same procedure can be used to separate L- and K-edge spectra arising from different elements within a given molecule. In contrast, conventional linear-response TDDFT requires core/valence separation but is free of these artifacts. It is also significantly more efficient than the real-time approach, even when hundreds of individual states are needed to reproduce near-edge absorption features and even when Padé approximants are used to reduce the real-time simulations to just 2-4 fs of time propagation. Despite the cost, the real-time approach may be useful to examine the validity of the core/valence separation approximation.

A New Way to Predict the Efficiency of Optical Limiters without Providing an Experiment: Processing of TDDFT Calculations for the Case of Pure Two-Photon Absorption

Laser-power-limiting devices play a predominant role in photonics because of their potential for protecting human eyes and optical devices that are sensitive to intense laser beams. This paper describes a new methodology for predicting the efficiency of optical limiting based on electric-field-induced changes in absorption spectra calculated with the TDDFT quantum-chemical method. Analytical equations are derived to evaluate the optical thresholds and speed of switching on, the dynamic range, and the degree of nonlinear attenuation of the radiation fluxes for the case of two-photon absorption. Thus, the researcher does not need to conduct costly experiments to evaluate the suitability of nonlinear absorbers for the creation of optical limiters. The possibility of developing a forecasting model is demonstrated by an example of a series of stable slipped-cofacial phthalocyanine J-type dimers, which were synthesized and investigated previously.

High harmonic spectra computed using time-dependent Kohn-Sham theory with Gaussian orbitals and a complex absorbing potential

High harmonic spectra for H₂ and H₂ ⁺ are simulated by solving the time-dependent Kohn-Sham equation in the presence of a strong laser field using an atom-centered Gaussian representation of the density and a complex absorbing potential. The latter serves to mitigate artifacts associated with the finite extent of the basis functions, including spurious reflection of the outgoing electronic wave packet. Interference between the outgoing and reflected waves manifests as peak broadening in the spectrum as well as the appearance of spurious high-energy peaks after the harmonic progression has terminated. We demonstrate that well-resolved spectra can be obtained through the use of an atom-centered absorbing potential. As compared to grid-based algorithms, the present approach is more readily extensible to larger molecules.

Inelastic scattering of electrons in water from first principles: cross sections and inelastic mean free path for use in Monte Carlo track-structure simulations of biological damage

Modelling the inelastic scattering of electrons in water is fundamental, given their crucial role in biological damage. In Monte Carlo track-structure (MC-TS) codes used to assess biological damage, the energy loss function (ELF), from which cross sections are extracted, is derived from different semi-empirical optical models. Only recently have first ab initio results for the ELF and cross sections in water become available. For benchmarking purpose, in this work, we present ab initio linear-response time-dependent density functional theory calculations of the ELF of liquid water. We calculated the inelastic scattering cross sections, inelastic mean free paths, and electronic stopping power and compared our results with recent calculations and experimental data showing a good agreement. In addition, we provide an in-depth analysis of the contributions of different molecular orbitals, species and orbital angular momenta to the total ELF. Moreover, we present single-differential cross sections computed for each molecular orbital channel, which should prove useful for MC-TS simulations.

Tension between predicting accurate ground state correlation energies and excitation energies from adiabatic approximations in TDDFT

The connection between the adiabatic excitation energy of time-dependent density functional theory and the ground state correlation energy from the adiabatic connection fluctuation–dissipation theorem (ACFDT) is explored in the limiting case of one excited state. An exact expression is derived for any adiabatic Hartree-exchange–correlation kernel that connects the excitation energy and the potential contribution to correlation. The resulting formula is applied to the asymmetric Hubbard dimer, a system where this limit is exact. Results from a hierarchy of approximations to the kernel, including the random phase approximation (RPA) with and without exchange and the adiabatically exact (AE) approximation, are compared to the exact ones. At full coupling, the numerical results indicate a tension between predicting an accurate excitation energy and an accurate potential contribution to correlation. The AE approximation is capable of making accurate predictions of both quantities, but only in parts of the parameter space that classify as weakly correlated, while RPA tends to be unable to accurately predict these properties simultaneously anywhere. For a strongly correlated dimer, the AE approximation greatly overestimates the excitation energy yet continues to yield an accurate ground state correlation energy due to its accurate prediction of the adiabatic connection integrand. If similar trends hold for real systems, the development of correlation kernels will be important for applications of the ACFDT in systems with large potential contributions to correlation.

Data-driven modeling of S0 → S1 excitation energy in the BODIPY chemical space: High-throughput computation, quantum machine learning, and inverse design

Derivatives of BODIPY are popular fluorophores due to their synthetic feasibility, structural rigidity, high quantum yield, and tunable spectroscopic properties. While the characteristic absorption maximum of BODIPY is at 2.5 eV, combinations of functional groups and substitution sites can shift the peak position by ±1 eV. Time-dependent long-range corrected hybrid density functional methods can model the lowest excitation energies offering a semi-quantitative precision of ±0.3 eV. Alas, the chemical space of BODIPYs stemming from combinatorial introduction of-even a few dozen-substituents is too large for brute-force high-throughput modeling. To navigate this vast space, we select 77 412 molecules and train a kernel-based quantum machine learning model providing <2% hold-out error. Further reuse of the results presented here to navigate the entire BODIPY universe comprising over 253 giga (253 × 10⁹) molecules is demonstrated by inverse-designing candidates with desired target excitation energies.

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.

Density Functional Calculations Based on the Exponential Ansatz

This work explores the application of the singles-based exponential ansatz to density functional calculations. In contrast to the standard approach where Kohn-Sham (KS) orbitals are determined prior to computing molecular quantities of interest, we consider the single-reference Hartree-Fock wave function as a starting point. Applying the exponential ansatz to this single reference gives an auxiliary wave function that is employed to calculate the electronic properties of the system. This wave function is determined self-consistently through the standard KS Hamiltonian but evaluated over the Hartree-Fock molecular orbital basis. By using spin-symmetry breaking, we recover size-consistent results free of unphysical fractional charges in the dissociation limit. Our method shows consistency with standard KS density functional calculations when the system geometry is similar to the equilibrium one or in repulsive configurations. For moderately long distances between atoms, not at dissociation, because of self-interaction the exponential ansatz may give instabilities in the form of large cluster amplitudes. To avoid these, this work introduces a relatively simple regularization method that preserves size-consistency and penalizes high amplitudes of the cluster operator, whereas the results remain physically meaningful. We also present the time-dependent extension of our theory and show that it can feature quantum states where multiple electrons are excited.

Simulating Vibronic Spectra without Born-Oppenheimer Surfaces

We show how linear vibronic spectra in molecular systems can be simulated efficiently using first-principles approaches without relying on the explicit use of multiple Born-Oppenheimer potential energy surfaces. We demonstrate and analyze the performance of mean-field and beyond-mean-field dynamics techniques for the H₂ molecule in one dimension, in the later case capturing the vibronic structure quite accurately, including quantum Franck-Condon effects. In a practical application of this methodology we simulate the absorption spectrum of benzene in full dimensionality using time-dependent density functional theory at the multitrajectory Ehrenfest level, finding good qualitative agreement with experiment and significant spectral reweighting compared to commonly used single-trajectory Ehrenfest dynamics. These results form the foundation for nonlinear spectral calculations and show promise for future application in capturing phenomena associated with vibronic coupling in more complex molecular and potentially condensed phase systems.

What Is the Accuracy Limit of Adiabatic Linear-Response TDDFT Using Exact Exchange-Correlation Potentials and Approximate Kernels?

Calculation of vertical excitation energies by the adiabatic linear-response time-dependent density-functional theory (TDDFT) requires static Kohn-Sham potentials and exchange-correlation kernels. When these quantities are derived from standard density-functional approximations (DFA), mean absolute errors (MAE) of the method are known to range from 0.2 eV to over 1 eV, depending on the functional and type of excitation. We investigate how the performance of TDDFT varies when increasingly accurate exchange-correlation potentials derived from Hartree-Fock (HF) and post-HF wavefunctions are combined with different approximate kernels. The lowest MAEs obtained in this manner for valence excitations are about 0.15-0.2 eV, which appears to be the practical limit of the accuracy of TDDFT that can be achieved by improving the Kohn-Sham potentials alone. These findings are consistent with previous reports on the benefits of accurate exchange-correlation potentials in TDDFT, but provide new insights and afford more definitive conclusions.

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.

Double excitations from modified Hartree Fock subsequent minimization scheme

Doubly excited states have nowadays become important in technological applications, e.g., in increasing the efficiency of solar cells and therefore, their description using ab initio methods is a great theoretical challenge as double excitations cannot be described by linear response theories based on a single Slater determinant. In the present work we extend our recently developed Hartree-Fock (HF) approximation for calculating singly excited states [M. Tassi, I. Theophilou, and S. Thanos, Int. J. Quantum Chem. 113, 690 (2013)] in order to allow for the calculation of doubly excited states. We describe the double excitation as two holes in the subspace spanned from the occupied HF orbitals and two particles in the subspace of virtual HF orbitals. A subsequent minimization of the energy results to the determination of the spin orbitals of both the holes and the particles in the occupied and virtual subspaces, respectively. We test our method, for various atoms, H2 and polyene molecules which are known to have excitations presenting a significant double excitation character. Importantly, our approach is computationally inexpensive.