Time-dependent Density Functional Theory Formalism Research Articles

Optical properties of quasi-two-dimensional objects from time-dependent density functional theory: Longitudinal versus transverse dielectric functions

Comprehension of the electronic properties of nano-objects is a key to defining dedicated properties, which can be adjusted by changing their size. Beyond confinement effects, the presence of interfaces, i.e., places where there is an abrupt change of electronic density, should also play a role. Time-dependent density functional theory (TD-DFT) is a state-of-the-art ab initio formalism in which this effect is accounted for through the so-called local field effects. In an earlier paper [S. Mazzei and C. Giorgetti, Phys. Rev. B 106, 035431 (2022)], we showed that the framework inherited from three-dimensional crystals could not provide reliable absorption spectra. In the present paper, we propose to calculate the macroscopic average of the dielectric tensor of a quasi-two-dimensional (2D) object from the response function of the density to the total macroscopic potential in order to avoid use of the so-called Adler and Wiser formula. We evidence that the inclusion of interfaces in the thickness of the slab causes the response function for the out-of-plane component to move sharply from the bulk absorption resonance to the plasmon one. This shows that the longitudinal-longitudinal contraction of the dielectric tensor is no longer equal to the transverse-transverse one in a quasi-2D object for out-of-plane perturbation. Nevertheless, we also show that the macroscopic average of the dielectric tensor of an ultrathin slab calculated within the longitudinal formalism of TD-DFT depicts the properties of the transverse reflectance and transmittance spectra of a thin slab.

Orbital-free methods for plasmonics: Linear response.

Plasmonic systems, such as metal nanoparticles, are widely used in different areas of application, going from biology to photovoltaics. The modeling of the optical response of such systems is of fundamental importance to analyze their behavior and to design new systems with required properties. When the characteristic sizes/distances reach a few nanometers, nonlocal and spill-out effects become relevant and conventional classical electrodynamics models are no more appropriate. Methods based on the Time-Dependent Density Functional Theory (TD-DFT) represent the current reference for the description of quantum effects. However, TD-DFT is based on knowledge of all occupied orbitals, whose calculation is computationally prohibitive to model large plasmonic systems of interest for applications. On the other hand, methods based on the orbital-free (OF) formulation of TD-DFT can scale linearly with the system size. In this Review, OF methods ranging from semiclassical models to the Quantum Hydrodynamic Theory will be derived from the linear response TD-DFT, so that the key approximations and properties of each method can be clearly highlighted. The accuracy of the various approximations will then be validated for the linear optical properties of jellium nanoparticles, the most relevant model system in plasmonics. OF methods can describe the collective excitations in plasmonic systems with great accuracy and without system-tuned parameters. The accuracy of these methods depends only on the accuracy of the (universal) kinetic energy functional of the ground-state electronic density. Current approximations and future development directions will also be indicated.

Prediction of Effective Photoelectron and Hole Separation in Type-I MoS2/PtSe2 van der Waals Junction.

Understanding the dynamics of charge transfer at vertical heterostructures of transition metal dichalcogenide monolayers is fundamentally important for future technological applications, given the unique feature of van der Waals interactions at the interface. Here, we employ time-dependent density functional theory formalism combined with molecular dynamics to investigate photoexcited electrons and holes in the type-I MoS2/PtSe2 van der Waals heterobilayer. While type-I junctions have been traditionally viewed as being ineffective in photocarrier separation, we show that here a different mechanism from type-II is at play, which effectively separates photoelectrons from photoholes. The key is the phonon bottleneck, arising from the characteristically different dynamic band alignments in the valence and conduction bands, respectively, which only affects the transfer of holes but not electrons. The disparity between electron and hole transfer rates offers a new direction for effective control of charge separation at interfaces.

Plasmon Couplings from Subsystem Time-Dependent Density Functional Theory.

Many applications in plasmonics are related to the coupling between metallic nanoparticles (MNPs) or between an emitter and a MNP. The theoretical analysis of such a coupling is thus of fundamental importance to analyze the plasmonic behavior and to design new systems. While classical methods neglect quantum and spill-out effects, time-dependent density functional theory (TD-DFT) considers all of them and with Kohn-Sham orbitals delocalized over the whole system. Thus, within TD-DFT, no definite separation of the subsystems (the single MNP or the emitter) and their couplings is directly available. This important feature is obtained here using the subsystem formulation of TD-DFT, which has been originally developed in the context of weakly interacting organic molecules. In subsystem TD-DFT, interacting MNPs are treated independently, thus allowing us to compute the plasmon couplings directly from the subsystem TD-DFT transition densities. We show that subsystem TD-DFT, as well as a simplified version of it in which kinetic contributions are neglected, can reproduce the reference TD-DFT calculations for gap distances greater than about 6 Å or even smaller in the case of hybrid plasmonic systems (i.e., molecules interacting with MNPs). We also show that the subsystem TD-DFT can be also used as a tool to analyze the impact of charge-transfer effects.

Self-consistent predictor/corrector algorithms for stable and efficient integration of the time-dependent Kohn-Sham equation.

The "real time" formulation of time-dependent density functional theory (TDDFT) involves integration of the time-dependent Kohn-Sham (TDKS) equation in order to describe the time evolution of the electron density following a perturbation. This approach, which is complementary to the more traditional linear-response formulation of TDDFT, is more efficient for computation of broad-band spectra (including core-excited states) and for systems where the density of states is large. Integration of the TDKS equation is complicated by the time-dependent nature of the effective Hamiltonian, and we introduce several predictor/corrector algorithms to propagate the density matrix, one of which can be viewed as a self-consistent extension of the widely used modified-midpoint algorithm. The predictor/corrector algorithms facilitate larger time steps and are shown to be more efficient despite requiring more than one Fock build per time step, and furthermore can be used to detect a divergent simulation on-the-fly, which can then be halted or else the time step modified.

Kohn-Sham Decomposition in Real-Time Time-Dependent Density-Functional Theory: An Efficient Tool for Analyzing Plasmonic Excitations.

Electronic excitations can be efficiently analyzed in terms of the underlying Kohn-Sham (KS) electron-hole transitions. While such a decomposition is readily available in the linear-response time-dependent density-functional theory (TDDFT) approaches based on the Casida equations, a comparable analysis is less commonly conducted within the real-time-propagation TDDFT (RT-TDDFT). To improve this situation, we present here an implementation of a KS decomposition tool within the local-basis-set RT-TDDFT code in the free GPAW package. Our implementation is based on postprocessing of data that is readily available during time propagation, which is important for retaining the efficiency of the underlying RT-TDDFT to large systems. After benchmarking our implementation on small benzene derivatives by explicitly reconstructing the Casida eigenvectors from RT-TDDFT, we demonstrate the performance of the method by analyzing the plasmon resonances of icosahedral silver nanoparticles up to Ag561. The method provides a clear description of the splitting of the plasmon in small nanoparticles due to individual single-electron transitions as well as the formation of a distinct d-electron-screened plasmon resonance in larger nanoparticles.

Efficient and accurate modeling of electron photoemission in nanostructures with TDDFT

We review different computational methods for the calculation of photoelectron spectra and angular distributions of atoms and molecules when excited by laser pulses using time-dependent density-functional theory (TDDFT) that are suitable for the description of electron emission in compact spatial regions. We derive and extend the time-dependent surface-flux method introduced in Reference [Tao L and Scrinzi A 2012 New Journal of Physics 14 013021] within a TDDFT formalism and compare its performance to other existing methods. We illustrate the performance of the new method by simulating strong-field ionization of C$_{60}$ fullerene and discuss final state effects in the orbital reconstruction of planar organic molecules.

Two-component hybrid time-dependent density functional theory within the Tamm-Dancoff approximation.

We report the implementation of a two-component variant of time-dependent density functional theory (TDDFT) for hybrid functionals that accounts for spin-orbit effects within the Tamm-Dancoff approximation (TDA) for closed-shell systems. The influence of the admixture of Hartree-Fock exchange on excitation energies is investigated for several atoms and diatomic molecules by comparison to numbers for pure density functionals obtained previously [M. Kühn and F. Weigend, J. Chem. Theory Comput. 9, 5341 (2013)]. It is further related to changes upon switching to the local density approximation or using the full TDDFT formalism instead of TDA. Efficiency is demonstrated for a comparably large system, Ir(ppy)3 (61 atoms, 1501 basis functions, lowest 10 excited states), which is a prototype molecule for organic light-emitting diodes, due to its "spin-forbidden" triplet-singlet transition.

Properties of atoms in electronically excited molecules within the formalism of TDDFT.

The topological analysis of the electron density for electronic excited states under the formalism of the quantum theory of atoms in molecules using time-dependent density functional theory (TDDFT) is presented. Relaxed electron densities for electronic excited states are computed by solving a Z-vector equation which is obtained by means of the Sternheimer interchange method. This is in contrast to previous work in which the electron density for excited states is obtained using DFT instead of TDDFT, that is, through the imposition of molecular occupancies in accordance with the electron configuration of the excited state under consideration. Once the electron density of the excited state is computed, its topological characterization and the properties of the atoms in molecules are obtained in the same manner that for the ground state. The analysis of the low-lying π→π* singlet and triplet vertical excitations of CO and C6H6 are used as representative examples of the application of this methodology. Altogether, it is shown how this procedure provides insights on the changes of the electron density following photoexcitation and it is our hope that it will be useful in the study of different photophysical and photochemical processes.

Dynamic structure factors of Cu, Ag, and Au: Comparative study from first principles

We present a comparative theoretical and experimental study of dynamic structure factors (momentum-dependent loss functions) of the noble metals Cu, Ag, and Au in the energy range of 0$-$60 eV. The emphasis is on theoretical results that are compared with new as well as available experimental data. Dynamic structure factors are calculated within the linear-response formalism of time-dependent density-functional theory, using the full-potential linearized augmented plane-wave (FP-LAPW) method. For the studied energy range, local-field effects are found to be very important for Ag and Au and only marginally relevant for Cu. We present an explanation for this surprising behavior. Loss functions of all three metals possess a complex multi-peak structure. We classify the features in the loss function as being related to collective excitations, interband transitions, or mixed modes. The impact of short-range correlations on the dynamic response functions are evaluated by comparing the results of the random-phase approximation to those of the time-dependent local-density approximation. Exchange correlation effects are found to be weak for small momentum transfers, but increasingly important for larger momenta. The calculated structure factors agree well with experiments.

Action formalism of time-dependent density-functional theory

The Runge-Gross [E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984)] action functional of time-dependent density-functional theory leads to a well-known causality paradox; that is, a perturbation of the electronic density in the future affects the response of the system in the present. This paradox is known to be caused by an inconsistent application of the Dirac-Frenkel variational principle. In view of the recent solutions to this problem, the action functional employed by Runge and Gross in their formulation of time-dependent density-functional theory is analyzed in the context of the Keldysh contour technique. The time-dependent electronic density and the concept of causality are extended to the contour. We derive a variational equation that obeys causality and relates the exchange-correlation potential with its kernel and the functional derivative of the exchange-correlation action functional with respect to the density. It is shown that the adiabatic local-density approximation is a consistent solution of this equation and that the time-dependent optimized potential method can also be derived from it. The formalism presented here can be used to find new approximation methods for the exchange-correlation potential and to avoid the causality dilemma.

Permanent dipole moments and energies of excited states from density functional theory compared with coupled cluster predictions: Case of para-nitroaniline

Different ways to extract properties of excited states from time-dependent density functional theory (TD-DFT) calculations are compared to ab initio results obtained with the Equation of Motion Coupled Cluster approach. The recently proposed a posteriori Tamm–Dancoff approximation (ATDA) predicts the permanent dipole moments to be underestimated by 25% on average, close to the results of the relaxed density TD-DFT formalism, quadratic response formalism, and numerical energy derivatives, while the unrelaxed density approximation results are less accurate (40% overestimate). We also propose a correction for TD-DFT excitation energies, which are known to be problematic for charge transfer states. The static DFT energies evaluated on the relaxed densities of the excited states are found to be more accurate than TD-DFT excitation energies (RMSD is 0.7eV vs. 1.1eV, while maximum deviation is −1.0eV vs. −2.0eV). This validates ATDA for description of nonlinear optical properties of donor–acceptor molecules, exemplified by para-nitroaniline, and extends this method to improve the excitation energy predictions.

On the subsystem formulation of linear-response time-dependent DFT

A new and thorough derivation of linear-response subsystem time-dependent density functional theory (TD-DFT) is presented and analyzed in detail. Two equivalent derivations are presented and naturally yield self-consistent subsystem TD-DFT equations. One reduces to the subsystem TD-DFT formalism of Neugebauer [J. Chem. Phys. 126, 134116 (2007)]. The other yields Dyson type equations involving three types of subsystem response functions: coupled, uncoupled, and Kohn-Sham. The Dyson type equations for subsystem TD-DFT are derived here for the first time. The response function formalism reveals previously hidden qualities and complications of subsystem TD-DFT compared with the regular TD-DFT of the supersystem. For example, analysis of the pole structure of the subsystem response functions shows that each function contains information about the electronic spectrum of the entire supersystem. In addition, comparison of the subsystem and supersystem response functions shows that, while the correlated response is subsystem additive, the Kohn-Sham response is not. Comparison with the non-subjective partition DFT theory shows that this non-additivity is largely an artifact introduced by the subjective nature of the density partitioning in subsystem DFT.

Two-dimensional andπplasmon spectra in pristine and doped graphene

The spectra of electronic excitations in graphene are calculated using first principles time-dependent density functional theory formalism, and used to obtain $\ensuremath{\pi}$ and $\ensuremath{\pi}+\ensuremath{\sigma}$ plasmon dispersion curves. The spectra and dispersion are in excellent agreement with recent experimental results, and they are used to investigate the anisotropy and splitting of a $\ensuremath{\pi}$ plasmon, which has also been experimentally verified. The high accuracy of this calculation enabled the discovery of some different features in the spectra, especially the $\ensuremath{\Gamma}M$-$\ensuremath{\Gamma}K$ anisotropy of the two-dimensional (2D) plasmon dispersion curve, which qualitatively agrees with recent experimental results. Our ab initio 2D plasmon dispersion curves are compared with the ones obtained in some recently proposed 2D models. They show strong disagreement with the dispersion curve obtained using a simple one-band 2D theory, as well as some discrepancies with respect to the commonly used Das Sarma et al.'s dispersion curve, even in the isotropic region. Excellent agreement of the calculated spectrum in pristine graphene with the electron energy loss spectroscopy spectrum measured for lower momentum transfers is demonstrated.

Nonadiabatic couplings from time-dependent density functional theory: Formulation by the Kohn-Sham derivative matrix within density functional perturbation theory

We present an improved method for accurately calculating nonadiabatic couplings (NACs) between the ground and excited states of electronic systems, using time-dependent density functional theory (TDDFT) within the plane-wave pseudopotential framework. Based on our previous work that the TDDFT formulation of NACs by the Kohn-Sham derivative matrix elements can avoid the accuracy problem caused by the nonlocal pseudopotentials, we extend the evaluation of these matrix elements to the analytical scheme, achieved by density functional perturbation theory (DFPT). As DFPT has been a standard implementation in many ab initio codes, our scheme requires little effort in the coding. Extensive calculations of NACs near various Jahn-Teller or Renner-Teller intersections show the good accuracy of our results. A comparison of our present and previous work has also clarified an important fact in the TDDFT computation of NACs, regarding the evaluation of the nuclear derivative of the many-body Hamiltonian after mapping to the Kohn-Sham system.

Time-dependent density functional theory benchmarking for the calculations of atomic spectra: efficiency of exc-ETDZ basis set

We present a time-dependent density functional theory (TD-DFT) benchmarking of recently constructed basis set, namely exc-ETDZ (Guevara et al. in J Chem Phys 131: 064104, 2009) for predicting the atomic spectra of the first-row atoms. A systematic testing with 31 density functional methods has been performed to see whether convincing performance of this basis set carries over the TD-DFT formalism. The efficiency of exc-ETDZ basis set for reproducing atomic spectra has been compared with Pople- and Dunning-style basis sets. We focused on the atomic low-lying valence excited states with single excitation character for our benchmarking, and the calculated excitation energies were compared to experimental data. On average, the functionals providing the best match with exc-ETDZ basis are BMK, BH&HLYP and ωB97. Moreover, on the basis of comparison between the results of these superior functionals with CIS(D) estimates, it turned out that TD-DFT and CIS(D) errors are of the same order of magnitude, once the exc-ETDZ basis set is used. Finally, the results of present study indicate that different functionals show results that are highly dependent on the atomic configuration as well as the basis set.

Theoretical Study of New Ruthenium-Based Dyes for Dye-Sensitized Solar Cells

Two relevant, recently reported, ruthenium-based complexes to be used as sensitizers in Grätzel photovoltaic cells are theoretically studied. The UV/vis absorption spectra have been computed within the time-dependent density functional theory formalism. The obtained excitation energies are compared with the experimental results, and the nature of the transition is analyzed in terms of the electronic density. A preliminary study on the performance of different functionals against the equation of motion coupled cluster is performed on a smaller model system.

Excited-state spin-contamination in time-dependent density-functional theory for molecules with open-shell ground states

While most applications of the linear response formulation of time-dependent density-functional theory (TDDFT) have been to the calculation of the excited states of molecules with closed-shell ground states, Casida’s formulation of TDDFT opened the way to TDDFT calculations on molecules with open-shell ground states by allowing for different-orbitals-for-different-spin, Although a number of publications have now appeared applying TDDFT to molecules with open-shell ground states and give surprisingly good results for simple excitations, it is relatively easy to show that some excited states of open-shell molecules will have unphysically large amounts of spin contamination. There is thus a clear need for computational tools which can separate physical from unphysical excited spin states in TDDFT. We address this need by using analytic derivative techniques to develop formulae for the 1- and 2-electron reduced density difference matrices, in essential agreement with those obtained by Rowe in the equation-of-motion superoperator approach to Green’s functions in nuclear physics. The corresponding formula for excited-state spin contamination appears to be generally good enough for assigning excited-state spin symmetries, but does lead to a small overestimation of S ^ 2 in the cases considered here. This (apparently small) problem is eliminated when the Tamm–Dancoff approximation is used in TDDFT.

Exact-exchange kernel of time-dependent density functional theory: Frequency dependence and photoabsorption spectra of atoms

In this work we have calculated excitation energies and photoionization cross sections of Be and Ne in the exact-exchange (EXX) approximation of time-dependent density functional theory (TDDFT). The main focus has been on the frequency dependence of the EXX kernel and on how it affects the spectrum as compared to the corresponding adiabatic approximation. We show that for some discrete excitation energies the frequency dependence is essential to reproduce the results of time-dependent Hartree-Fock theory. Unfortunately, we have found that the EXX approximation breaks down completely at higher energies, producing a response function with the wrong analytic structure and making inner-shell excitations disappear from the calculated spectra. We have traced this failure to the existence of vanishing eigenvalues of the Kohn-Sham non-interacting response function. Based on the adiabatic TDDFT formalism we propose a new way of deriving the Fano parameters of autoionizing resonances.

Autoionizing resonances in time-dependent density functional theory

Autoionizing resonances that arise from the interaction of a bound single-excitation with the continuum can be accurately captured with the presently used approximations in time-dependent density functional theory (TDDFT), but those arising from a bound double excitation cannot. In the former case, we explain how an adiabatic kernel, which has no frequency-dependence, can yet generate the strongly frequency-dependent resonant structures in the interacting response function, not present in the Kohn-Sham response function. In the case of the bound double-excitation, we explain that a strongly frequency-dependent kernel is needed, and derive one as an a posteriori correction to the usual adiabatic approximations in TDDFT. Our approximation becomes exact for an isolated resonance in the limit of weak interaction, where one discrete state interacts with one continuum. We derive a "Fano TDDFT kernel" that reproduces the Fano lineshape within the TDDFT formalism, and also a dressed kernel, that operates on top of an adiabatic approximation. We illustrate our results on a simple model system.