XC Kernel Research Articles

Viscous Current-Induced Forces.

We study the motion (translational, vibrational, and rotational) of a diatomic impurity immersed in an electron liquid and exposed to electronic current. An approach based on the linear response time-dependent density functional theory combined with the Ehrenfest dynamics leads to a system of linear algebraic equations, which account for the competing and counteracting effects of the current-induced force (electron wind) and the electronic friction. We find and emphasize the coupling between the center of mass motion and that of the nuclei relative to each other, the feature due to the mediation of the two-body interaction by the environment. The current-induced forces, by means of the dynamic exchange-correlation (xc) kernel f_{xc}(r,r^{'},ω), include the electronic viscosity contribution. Starting from the ground state at the equilibrium internuclear distance and applying a current pulse, we observe three phases of the motion: (i)acceleration due to the prevalence of the current-induced force, (ii)stabilization upon balancing of the two forces, and (iii)deceleration due to the friction after the end of the pulse. At lower, but still metallic, electron densities, the dynamic xc contribution to the force significantly affects the acceleration (deceleration) at the first (third) phase of the process. For the Cs density (r_{s}≈6 a.u.), this correction amounts up to 40% in the rotation regime.

Revealing quasi-excitations in the low-density homogeneous electron gas with model exchange–correlation kernels

Time-dependent density functional theory within the linear response regime provides a solid mathematical framework to capture excitations. The accuracy of the theory, however, largely depends on the approximations for the exchange–correlation (xc) kernels. Away from the long-wavelength (or q = 0 short wave-vector) and zero-frequency (ω = 0) limit, the correlation contribution to the kernel becomes more relevant and dominant over exchange. The dielectric function, in principle, can encompass xc effects relevant to describe low-density physics. Furthermore, besides collective plasmon excitations, the dielectric function can reveal collective electron–hole excitations, often dubbed “ghost excitons.” Besides collective excitons, the physics of the low-density regime is rich, as exemplified by a static charge-density wave that was recently found for rs > 69, and was shown to be associated with softening of the plasmon mode. These excitations are seen to be present in much higher density 2D homogeneous electron gases of rs ≳ 4. In this work, we perform a thorough analysis with xc model kernels for excitations of various nature. The uniform electron gas, as a useful model of real metallic systems, is used as a platform for our analysis. We highlight the relevance of exact constraints as we display and explain screening and excitations in the low-density region.

Averaging over atom snapshots in linear-response TDDFT of disordered systems: A case study of warm dense hydrogen.

Linear-response time-dependent density functional theory (LR-TDDFT) simulations of disordered extended systems require averaging over different snapshots of ion configurations to minimize finite size effects due to the snapshot-dependence of the electronic density response function and related properties. We present a consistent scheme for the computation of the macroscopic Kohn-Sham (KS) density response function connecting an average over snapshot values of charge density perturbations to the averaged values of KS potential variations. This allows us to formulate the LR-TDDFT within the adiabatic (static) approximation for the exchange-correlation (XC) kernel for disordered systems, where the static XC kernel is computed using the direct perturbation method [Moldabekov et al., J. Chem. Theory Comput. 19, 1286 (2023)]. The presented approach allows one to compute the macroscopic dynamic density response function as well as the dielectric function with a static XC kernel generated for any available XC functional. The application of the developed workflow is demonstrated for the example of warm dense hydrogen. The presented approach is applicable for various types of extended disordered systems, such as warm dense matter, liquid metals, and dense plasmas.

Linear-response time-dependent density functional theory approach to warm dense matter with adiabatic exchange-correlation kernels

We present a new methodology for the linear-response time-dependent density functional theory (LR-TDDFT) calculation of the dynamic density response function of warm dense matter in an adiabatic approximation that can be used with any available exchange-correlation (XC) functional across Jacob's Ladder and across temperature regimes. The main novelty of the presented approach is that it can go beyond the adiabatic local density approximation (ALDA) and generalized LDA (AGGA) while preserving the self-consistence between the Kohn-Sham (KS) response function and adiabatic XC kernel for extended systems. The key ingredient for the presented method is the combination of the adiabatic XC kernel from the direct perturbation approach with the macroscopic dynamic KS response from the standard LR-TDDFT method using KS orbitals. We demonstrate the application of the method for the example of warm dense hydrogen, for which we perform a detailed analysis of the KS density response function, the RPA result, the total density response function and of the adiabatic XC kernel. The analysis is performed using LDA, GGA, and meta-GGA level approximations for the XC effects. The presented method is directly applicable to disordered systems such as liquid metals, warm dense matter, and dense plasmas.

Excitonic effects in time-dependent density functional theory from zeros of the density response

We show that the analytic structure of the dynamical xc kernels of semiconductors and insulators can be sensed in terms of its poles which mark physically relevant frequencies of the system where the counter-phase motion of discrete collective excitations occurs: if excited, the collective modes counterbalance each other, making the system to exhibit none at all or extremely weak density response. This property can be employed to construct simple and practically relevant approximations of the dynamical xc kernel for time-dependent density functional theory (TDDFT). Such kernels have simple analytic structure, are able to reproduce dominant excitonic features of the absorption spectra of monolayer semiconductors and bulk solids, and promise high potential for future uses in efficient real-time calculations with TDDFT.

Assessing the accuracy of hybrid exchange-correlation functionals for the density response of warm dense electrons.

We assess the accuracy of common hybrid exchange-correlation (XC) functionals (PBE0, PBE0-1/3, HSE06, HSE03, and B3LYP) within the Kohn-Sham density functional theory for the harmonically perturbed electron gas at parameters relevant for the challenging conditions of the warm dense matter. Generated by laser-induced compression and heating in the laboratory, the warm dense matter is a state of matter that also occurs in white dwarfs and planetary interiors. We consider both weak and strong degrees of density inhomogeneity induced by the external field at various wavenumbers. We perform an error analysis by comparing with the exact quantum Monte Carlo results. In the case of a weak perturbation, we report the static linear density response function and the static XC kernel at a metallic density for both the degenerate ground-state limit and for partial degeneracy at the electronic Fermi temperature. Overall, we observe an improvement in the density response when the PBE0, PBE0-1/3, HSE06, and HSE03 functionals are used, compared with the previously reported results for the PBE, PBEsol, local-density approximation, and AM05 functionals; B3LYP, on the other hand, does not perform well for the considered system. Additionally, the PBE0, PBE0-1/3, HSE06, and HSE03 functionals are more accurate for the density response properties than SCAN in the regime of partial degeneracy.

Non-empirical Mixing Coefficient for Hybrid XC Functionals from Analysis of the XC Kernel.

We present an analysis of the static exchange-correlation (XC) kernel computed from hybrid functionals with a single mixing coefficient such as PBE0 and PBE0-1/3. We break down the hybrid XC kernels into the exchange and correlation parts using the Hartree-Fock functional, the exchange-only PBE, and the correlation-only PBE. This decomposition is combined with exact data for the static XC kernel of the uniform electron gas and an Airy gas model within a subsystem functional approach. This gives us a tool for the non-empirical choice of the mixing coefficient under ambient and extreme conditions. Our analysis provides physical insights into the effect of the variation of the mixing coefficient in hybrid functionals, which is of immense practical value. The presented approach is general and can be used for other types of functionals like screened hybrids.

Ab Initio Static Exchange-Correlation Kernel across Jacob's Ladder without Functional Derivatives.

The electronic exchange─correlation (XC) kernel constitutes a fundamental input for the estimation of a gamut of properties such as the dielectric characteristics, the thermal and electrical conductivity, or the response to an external perturbation. In this work, we present a formally exact methodology for the computation of the system specific static XC kernel exclusively within the framework of density functional theory (DFT) and without employing functional derivatives─no external input apart from the usual XC-functional is required. We compare our new results with exact quantum Monte Carlo (QMC) data for the archetypical uniform electron gas model under both ambient and warm dense matter conditions. This gives us unprecedented insights into the performance of different XC functionals, and it has important implications for the development of new functionals that are designed for the application at extreme temperatures. In addition, we obtain new DFT results for the XC kernel of warm dense hydrogen as it occurs in fusion applications and astrophysical objects. The observed excellent agreement to the QMC reference data demonstrates that presented framework is capable to capture nontrivial effects such as XC-induced isotropy breaking in the density response of hydrogen at large wave numbers.

First-principles wave-vector- and frequency-dependent exchange-correlation kernel for jellium at all densities

We propose a spatially and temporally nonlocal exchange-correlation (xc) kernel for the spin-unpolarized fluid phase of ground-state jellium, for use in time-dependent density functional and linear response calculations. The kernel is constructed to satisfy known properties of the exact xc kernel, to accurately describe the correlation energies of bulk jellium, and to satisfy frequency-moment sum rules at a wide range of bulk jellium densities, including those low densities that display strong correlation and symmetry breaking. These effects are easier to understand in the simple jellium model than in real systems. All exact constraints satisfied by the recent MCP07 kernel [A. Ruzsinszky, et al., Phys. Rev. B 101, 245135 (2020)] are maintained in the new revised MCP07 (rMCP07) kernel, while others are added. The revision $f_\mathrm{xc}^\mathrm{rMCP07}(q,\omega)$ differs from MCP07 only for non-zero frequencies $\omega$. Only at densities much lower than those of real bulk metals is the frequency dependence of the kernel important for the correlation energy of jellium. As the wavevector $q$ tends to zero, the kernel has a $-4\pi \alpha(\omega)/q^2$ divergence whose frequency-dependent ultranonlocality coefficient $\alpha(\omega)$ vanishes in jellium, and is predicted by rMCP07 to be extremely small for the real metals Al and Na.}

Progress towards understanding ultranonlocality through the wave-vector and frequency dependence of approximate exchange-correlation kernels

In the framework of time-dependent density functional theory (TDDFT), the exact exchange-correlation (xc) kernel $f_{xc}(n,q,\omega)$ determines the ground-state energy, excited-state energies, lifetimes, and the time-dependent linear density response of any many-electron system. The recently developed MCP07 xc kernel $f_{xc}(n,q,\omega)$ of A. Ruzsinszky et al. [Phys. Rev. B 101, 245135 (2020)] yields excellent uniform electron gas (UEG) ground-state energies and plausible plasmon lifetimes. As MCP07 is constructed to describe $f_{xc}$ of the UEG, it cannot capture optical properties of real materials. To verify this claim, we follow Nazarov et al. [Phys. Rev. Lett. 102, 113001 (2009)] to construct the long-range, dynamic xc kernel, $\lim_{q\to 0}f_{xc}(n,q,\omega) = -\alpha(\omega)e^2/q^2$, of a weakly inhomogeneous electron gas, using MCP07 and other common xc kernels. The strong wavevector and frequency dependence of the "ultranonlocality" coefficient $\alpha(\omega)$ is demonstrated for a variety of simple metals and semiconductors. We examine how imposing exact constraints on an approximate kernel shapes $\alpha(\omega)$. Comparisons to kernels derived from correlated-wavefunction calculations are drawn.

Real-Time Exciton Dynamics with Time-Dependent Density-Functional Theory.

Linear-response time-dependent density-functional theory (TDDFT) can describe excitonic features in the optical spectra of insulators and semiconductors, using exchange-correlation (xc) kernels behaving as -1/k^{2} to leading order. We show how excitons can be modeled in real-time TDDFT, using an xc vector potential constructed from approximate, long-range corrected xc kernels. We demonstrate, for various materials, that this real-time approach is consistent with frequency-dependent linear response, gives access to femtosecond exciton dynamics following short-pulse excitations, and can be extended with some caution into the nonlinear regime.

Insights from exact exchange-correlation kernels

The exact exchange-correlation (xc) kernel ${f}_{\text{xc}}(x,{x}^{\ensuremath{'}},\ensuremath{\omega})$ of linear response time-dependent density functional theory is computed over a wide range of frequencies for three canonical one-dimensional finite systems. Methods used to ensure the numerical robustness of ${f}_{\text{xc}}$ are set out. The frequency dependence of ${f}_{\text{xc}}$ is found to be largely due to its analytic structure, i.e., its singularities at certain frequencies, which are required in order to capture particular transitions, including those of double excitation character. However, within the frequency range of the first few interacting excitations, ${f}_{\text{xc}}$ is approximately $\ensuremath{\omega}$ independent, meaning the exact adiabatic approximation ${f}_{\text{xc}}(\ensuremath{\omega}=0)$ remedies the failings of the local density approximation and random phase approximation for these lowest transitions. The key differences between the exact ${f}_{\text{xc}}$ and its common approximations are analyzed, and cannot be eliminated by exploiting the limited gauge freedom in ${f}_{\text{xc}}$. The optical spectrum benefits from using as accurate as possible an ${f}_{\text{xc}}$ and ground-state xc potential, while maintaining exact compatibility between the two is of less importance.

Adiabatic generalized gradient approximation kernel in time-dependent density functional theory

A complete understanding of a material requires both knowledge of the excited states as well as of the ground state. In particular, the low energy excitations are of utmost importance while studying the electronic, magnetic, dynamical, and thermodynamical properties of the material. Time-Dependent Density Functional Theory (TDDFT), within the linear regime, is a successful \textit{ab-initio} method to access the electronic charge and spin excitations. However, it requires an approximation to the exchange-correlation (XC) kernel which encapsulates the effect of electron-electron interactions in the many-body system. In this work we derive and implement the spin-polarized XC kernel for semi-local approximations such as the adiabatic Generalized Gradient Approximation (AGGA). This kernel has a quadratic dependence on the wavevector, {\bf q}, of the perturbation, however the impact of this on the electron energy loss spectra (EELS) is small. Although the GGA functional is good in predicting structural properties, it generality overestimates the exchange spin-splitting. This leads to higher magnon energies, as compared to both ALDA and experiment. In addition, interaction with the Stoner spin-flip continuum is enhanced by AGGA, which strongly suppresses the intensity of spin-waves.

Local vertex corrections from exchange-correlation kernels with a discontinuity

The fundamental gap of an interacting many-electron system is given by the sum of the single-particle Kohn-Sham gap and the derivative discontinuity. The latter can be generated by advanced approximations to the exchange-correlation (XC) energy and is the key quantity to capture strong correlation with density functional theory (DFT). In this work we derive an expression for the derivative discontinuity in terms of the XC kernel of time-dependent density functional theory and demonstrate the crucial role of a discontinuity in the XC kernel itself. By relating approximate XC kernels to approximate local vertex corrections we then generate beyond-GW self-energies that include a discontinuity in the local vertex function. The quantitative importance of this result is illustrated with a numerical study of the local exchange vertex on model systems.

DFT-SAPT Intermolecular Interaction Energies Employing Exact-Exchange Kohn-Sham Response Methods.

Intermolecular interaction energies have been calculated by symmetry-adapted perturbation theory based on density functional theory monomer properties (DFT-SAPT) employing response functions from time-dependent exact-exchange (TDEXX) kernels. Combined with a new asymptotic correction scheme for the exchange-correlation (xc) potentials of the monomers, which is similar in its performance to standard asymptotic correction methods, it is shown that this DFT-SAPT[TDEXX] method delivers highly accurate intermolecular interaction energies for the S22, S66, and IonHB benchmark databases by Hobza et al. A corresponding DFT-SAPT approach employing the adiabatic TDEXX kernel in the response calculations has also been tested. This DFT-SAPT[ATDEXX] method performs almost as well as DFT-SAPT[TDEXX] for dispersion-dominated dimer systems but less accurately for hydrogen-bonded dimers. Compared to this, DFT-SAPT[TDEXX] yields a balanced description of the interaction energies for various interaction-type motifs, similar to the standard DFT-SAPT method that utilizes the ALDA xc kernel to compute the response functions.

Casimir-Polder Size Consistency: A Constraint Violated by Some Dispersion Theories.

A key goal in quantum chemistry methods, whether ab initio or otherwise, is to achieve size consistency. In this work we formulate the related idea of "Casimir-Polder size consistency" that manifests in long-range dispersion energetics. We show that local approximations in time-dependent density functional theory dispersion energy calculations violate the consistency condition because of incorrect treatment of highly nonlocal "xc kernel" physics, by up to 10% in our tests on closed-shell atoms.

Nonadiabatic exchange-correlation kernel for strongly correlated materials

We formulate a rigorous method for calculating a nonadiabatic (frequency-dependent) exchange-correlation (XC) kernel appropriate for accurate description of both equilibrium and nonequilibrium properties of strongly correlated systems within the time-dependent density functional theory (TDDFT) via the charge susceptibility, which is in turn obtained from dynamical mean field theory (DMFT) based on the effective multi-orbital Hubbard model. Application to the simple case of the one-orbital Hubbard model already shows the importance of the nonadiabatic kernel as it leads to significant modification of the excitation spectrum—shifting the (adiabatic) peak and disclosing another that is reminiscent of the solution from DMFT. The impact of dynamical effects, naturally included through the nonadiabaticity of the XC kernel, becomes even more transparent in our consideration of the nonequilibrium charge-density response of a multi-orbital perovskite, YTiO3, to a perturbation by a femtosecond (fs) laser pulse. These initial results indicate that electron–electron correlations and nonadiabatic features may significantly affect the spectrum and nonequilibrium properties of strongly correlated systems. We also propose an algorithm for extension of the approach to non-linear response. The transparency and computational efficiency of this non-adiabatic TDDFT+DMFT approach opens the door to examination of the spectra and response of multi-orbital systems with many nonequivalent atoms—bulk material, films and nanostructures.

Time-Dependent Density-Functional Theory and Excitons in Bulk and Two-Dimensional Semiconductors

In this work, we summarize the recent progress made in constructing time-dependent density-functional theory (TDDFT) exchange-correlation (XC) kernels capable to describe excitonic effects in semiconductors and apply these kernels in two important cases: a “classic” bulk semiconductor, GaAs, with weakly-bound excitons and a novel two-dimensional material, MoS2, with very strongly-bound excitonic states. Namely, after a brief review of the standard many-body semiconductor Bloch and Bethe-Salpether equation (SBE and BSE) and a combined TDDFT+BSE approaches, we proceed with details of the proposed pure TDDFT XC kernels for excitons. We analyze the reasons for successes and failures of these kernels in describing the excitons in bulk GaAs and monolayer MoS2, and conclude with a discussion of possible alternative kernels capable of accurately describing the bound electron-hole states in both bulk and two-dimensional materials.

Towards TDDFT for Strongly Correlated Materials

We present some details of our recently-proposed Time-Dependent Density-Functional Theory (TDDFT) for strongly-correlated materials in which the exchange-correlation (XC) kernel is derived from the charge susceptibility obtained using Dynamical Mean-Field Theory (the TDDFT + DMFT approach). We proceed with deriving the expression for the XC kernel for the one-band Hubbard model by solving DMFT equations via two approaches, the Hirsch–Fye Quantum Monte Carlo (HF-QMC) and an approximate low-cost perturbation theory approach, and demonstrate that the latter gives results that are comparable to the exact HF-QMC solution. Furthermore, through a variety of applications, we propose a simple analytical formula for the XC kernel. Additionally, we use the exact and approximate kernels to examine the nonhomogeneous ultrafast response of two systems: a one-band Hubbard model and a Mott insulator YTiO3. We show that the frequency dependence of the kernel, i.e., memory effects, is important for dynamics at the femtosecond timescale. We also conclude that strong correlations lead to the presence of beats in the time-dependent electric conductivity in YTiO3, a feature that could be tested experimentally and that could help validate the few approximations used in our formulation. We conclude by proposing an algorithm for the generalization of the theory to non-linear response.

Critical Assessment of Time-Dependent Density Functional Theory for Excited States of Open-Shell Systems: II. Doublet-Quartet Transitions

Compared with closed-shell systems, open-shell systems place three additional challenges to time-dependent density functional theory (TD-DFT) for electronically excited states: (a) the spin-contamination problem is a serious issue; (b) the exchange-correlation (XC) kernel may be numerically instable; and (c) the single-determinant description of open-shell ground states readily becomes energetically instable. Confined to flip-up single excitations, the spin-contamination problem can largely be avoided by using the spin-flip TD-DFT (SF-TD-DFT) formalism, provided that a noncollinear XC kernel is employed. As for the numerical instabilities associated with such a kernel, only an ad hoc scheme has been proposed so far, viz., the ALDA0 kernel, which amounts to setting the divergent components (arising from density gradients and kinetic energy density) simply to zero. The ground-state instability problem can effectively be avoided by introducing the Tamm-Dancoff approximation (TDA) to TD-DFT. Therefore, on a general basis, the SF-TDA/ALDA0 Ansatz is so far the only promising means within the TD-DFT framework for flip-up single excitations of open-shell systems. To assess systematically the performance of SF-TDA/ALDA0, in total 61 low-lying quartet excited states of the benchmark set of 11 small radicals [J. Chem. Theory Comput. 2016, 12, 238] are investigated with various XC functionals. Taking the MRCISD+Q (multireference configuration interaction with singles and doubles plus the Davidson correction) results as benchmark, it is found that the mean absolute errors of SF-TDA/ALDA0 with the SAOP (statistical averaging of model orbital potentials), global hybrid, and range-separated hybrid functionals are in the range of 0.2-0.4 eV. This is in line not only with the typical accuracy of TD-DFT for singlet and triplet excited states of closed-shell systems but also with the gross accuracy of spin-adapted TD-DFT for spin-conserving excited states of open-shell systems.