Standard Time-dependent Density Functional Theory Research Articles

Minimal Auxiliary Basis Set Approach for the Electronic Excitation Spectra of Organic Molecules.

We report a minimal auxiliary basis model for time-dependent density functional theory (TDDFT) with hybrid density functionals that can accurately reproduce excitation energies and absorption spectra from TDDFT while reducing cost by about 2 orders of magnitude. Our method, dubbed TDDFT-ris, employs the resolution-of-the-identity technique with just one s-type auxiliary basis function per atom for the linear response operator, where the Gaussian exponents are parametrized across the periodic table using tabulated atomic radii with a single global scaling factor. By tuning on a small test set, we determine a single functional-independent scale factor that balances errors in excitation energies and absorption spectra. Benchmarked on organic molecules and compared to standard TDDFT, TDDFT-ris has an average energy error of only 0.06 eV and yields absorption spectra in close agreement with TDDFT. Thus, TDDFT-ris enables simulation of realistic absorption spectra in large molecules that would be inaccessible from standard TDDFT.

Efficient implementation of time-dependent auxiliary density functional theory.

The random phase approximation of time-dependent auxiliary density functional theory (TDADFT) is rederived from auxiliary density perturbation theory. Our exhaustive validation of TDADFT reveals an upshift of the excitation energies by ∼0.1eV with respect to standard time-dependent density functional theory. For the computationally efficient implementation of TDADFT, floating point operation optimized three-center electron repulsion integral recurrence relations and their double asymptotic expansions are implemented into the Davidson solver. The computational efficiency of TDADFT is benchmarked with four sets of molecules comprising alkanes, fullerenes, DNA fragments, and zeolites. The results show that TDADFT has a computational scaling between 1.3 and 1.9 with respect to the number of basis functions, which is lower than the scaling of standard time-dependent density functional theory. Due to its computational simplifications, TDADFT is particularly well suited for Born-Oppenheimer molecular dynamics simulations. As illustrative examples, we present the temperature effects on the gas-phase absorption spectra of benzene, naphthalene, and anthracene.

Spin-orbit couplings within spin-conserving and spin-flipping time-dependent density functional theory: Implementation and benchmark calculations.

We present a new implementation for computing spin-orbit couplings (SOCs) within a time-dependent density-functional theory (TD-DFT) framework in the standard spin-conserving formulation as well in the spin-flip variant (SF-TD-DFT). This approach employs the Breit-Pauli Hamiltonian and Wigner-Eckart's theorem applied to the reduced one-particle transition density matrices, together with the spin-orbit mean-field treatment of the two-electron contributions. We use a state-interaction procedure and compute the SOC matrix elements using zero-order non-relativistic states. Benchmark calculations using several closed-shell organic molecules, diradicals, and a single-molecule magnet illustrate the efficiency of the SOC protocol. The results for organic molecules (described by standard TD-DFT) show that SOCs are insensitive to the choice of the functional or basis sets, as long as the states of the same characters are compared. In contrast, the SF-TD-DFT results for small diradicals (CH2, NH2 +, SiH2, and PH2 +) show strong functional dependence. The spin-reversal energy barrier in a Fe(III) single-molecule magnet computed using non-collinear SF-TD-DFT (PBE0, ωPBEh/cc-pVDZ) agrees well with the experimental estimate.

Organic Emitters Showing Excited-States Energy Inversion: An Assessment of MC-PDFT and Correlation Energy Functionals Beyond TD-DFT

The lowest-energy singlet (S1) and triplet (T1) excited states of organic conjugated chromophores are known to be accurately calculated by modern wavefunction and Time-Dependent Density Functional Theory (TD-DFT) methods, with the accuracy of the latter heavily relying on the exchange-correlation functional employed. However, there are challenging cases for which this cannot be the case, due to the fact that those excited states are not exclusively formed by single excitations and/or are affected by marked correlation effects, and thus TD-DFT might fall short. We will tackle here a set of molecules belonging to the azaphenalene family, for which research recently documented an inversion of the relative energy of S1 and T1 excited states giving rise to a negative energy difference (ΔEST) between them and, thereby, contrary to most of the systems thus far treated by TD-DFT. Since methods going beyond standard TD-DFT are not extensively applied to excited-state calculations and considering how challenging this case is for the molecules investigated, we will prospectively employ here a set of non-standard methods (Multi-Configurational Pair Density Functional Theory or MC-PDFT) and correlation functionals (i.e., Lie–Clementi and Colle–Salvetti) relying not only on the electronic density but also on some modifications considering the intricate electronic structure of these systems.

An improved seminumerical Coulomb and exchange algorithm for properties and excited states in modern density functional theory.

A seminumerical algorithm capable of performing large-scale (time-dependent) density functional theory (TD-DFT) calculations to extract excitation energies and other ground-state and excited-state properties is outlined. The algorithm uses seminumerical integral techniques for evaluating Coulomb and exchange parts for a set of density matrices as occurring in standard TD-DFT or similar methods for the evaluation of vibrational frequencies. A suitable optimized de-aliasing procedure is introduced. The latter does not depend on further auxiliary quantities and retains the symmetry of a given density matrix. The algorithm is self-contained and applicable to any orbital basis set available without the need for further auxiliary basis sets or optimized de-aliasing grids. Relativistic two-component excited-state TD-DFT calculations are reported for the first time using the developed seminumerical algorithm for standard and local hybrid density functional approximations. Errors are compared with the widely used "resolution of the identity" (RI) approximations for Coulomb (RI-J) and exchange integrals (RI-K). The fully seminumerical algorithm does not exhibit an enlarged error for standard DFT functionals compared to the RI approximation. For the more involved local hybrid functionals and within strong external fields, accuracy is even considerably improved.

Ab Initio Many-Body Perturbation Theory Calculations of the Electronic and Optical Properties of Cyclometalated Ir(III) Complexes

Cyclometalated Ir(III) compounds are the preferred choice as organic emitters in organic light-emitting diodes. In practice, the presence of the transition metal surrounded by carefully designed ligands allows fine-tuning of the emission frequency as well as good efficiency of the device. To support the development of new compounds, experimental measurements are generally compared with absorption and emission spectra obtained from ab initio calculations. The standard approach for these calculations is time-dependent density functional theory (TDDFT) with a hybrid exchange-correlation functional like B3LYP. Because of the size of these compounds, the application of more complex quantum chemistry approaches can be challenging. In this work, we used many-body perturbation theory approaches, in particular the GW approximation with the Bethe-Salpeter equation (BSE) implemented in Gaussian basis sets, to calculate the quasiparticle properties and the absorption spectra of six cyclometalated Ir(III) complexes, going beyond TDDFT. In the presented results, we compared standard TDDFT simulations with BSE calculations performed on top of perturbative G0W0 and accounting for eigenvalue self-consistency. Moreover, in order to investigate in detail the effect of the DFT starting point, we concentrated on Ir(ppy)3 and performed GW-BSE simulations starting from different DFT exchange-correlation potentials.

Nonadiabatic electron dynamics in time-dependent density functional theory at the cost of adiabatic local density approximation

We propose a computationally efficient approach to the nonadiabatic time-dependent density functional theory (TDDFT) which is based on a representation of the frequency-dependent exchange correlation kernel as a response of a set of damped oscillators. The requirements to computational resources needed to implement our approach do not differ from those of the standard real-time TDDFT in the adiabatic local density approximation (ALDA). Thus, our result offers an exciting opportunity to take into account temporal nonlocality and memory effects in calculations with TDDFT in quantum chemistry and solid state physics for unprecedentedly low costs.

Direct Extraction of Excitation Energies from Ensemble Density-Functional Theory.

A very specific ensemble of ground and excited states is shown to yield an exact formula for any excitation energy as a simple correction to the energy difference between orbitals of the Kohn-Sham ground state. This alternative scheme avoids either the need to calculate many unoccupied levels as in time-dependent density functional theory (TDDFT) or the need for many self-consistent ensemble calculations. The symmetry-eigenstate Hartree-exchange (SEHX) approximation yields results comparable to standard TDDFT for atoms. With this formalism, SEHX yields approximate double excitations, which are missed by adiabatic TDDFT.

Large-Scale Quantum Many-Body Perturbation on Spin and Charge Separation in the Excited States of the Synthesized Donor-Acceptor Hybrid PBI-Macrocycle Complex.

The reliable calculation of the excited states of charge-transfer (CT) compounds poses a major challenge to the ab initio community because the frequently employed method, time-dependent density functional theory (TD-DFT), massively relies on the underlying density functional, resulting in heavily Hartree-Fock (HF) exchange-dependent excited-state energies. By applying the highly sophisticated many-body perturbation approach, we address the encountered unreliabilities and inconsistencies of not optimally tuned (standard) TD-DFT regarding photo-excited CT phenomena, and present results concerning accurate vertical transition energies and the correct energetic ordering of the CT and the first visible singlet state of a recently synthesized thermodynamically stable large hybrid perylene bisimide-macrocycle complex. This is a large-scale application of the quantum many-body perturbation approach to a chemically relevant CT system, demonstrating the system-size independence of the quality of the many-body-based excitation energies. Furthermore, an optimal tuning of the ωB97X hybrid functional can well reproduce the many-body results, making TD-DFT a suitable choice but at the expense of introducing a range-separation parameter, which needs to be optimally tuned.

Ultra-fast computation of electronic spectra for large systems by tight-binding based simplified Tamm-Dancoff approximation (sTDA-xTB).

The computational bottleneck of the extremely fast simplified Tamm-Dancoff approximated (sTDA) time-dependent density functional theory procedure [S. Grimme, J. Chem. Phys. 138, 244104 (2013)] for the computation of electronic spectra for large systems is the determination of the ground state Kohn-Sham orbitals and eigenvalues. This limits such treatments to single structures with a few hundred atoms and hence, e.g., sampling along molecular dynamics trajectories for flexible systems or the calculation of chromophore aggregates is often not possible. The aim of this work is to solve this problem by a specifically designed semi-empirical tight binding (TB) procedure similar to the well established self-consistent-charge density functional TB scheme. The new special purpose method provides orbitals and orbital energies of hybrid density functional character for a subsequent and basically unmodified sTDA procedure. Compared to many previous semi-empirical excited state methods, an advantage of the ansatz is that a general eigenvalue problem in a non-orthogonal, extended atomic orbital basis is solved and therefore correct occupied/virtual orbital energy splittings as well as Rydberg levels are obtained. A key idea for the success of the new model is that the determination of atomic charges (describing an effective electron-electron interaction) and the one-particle spectrum is decoupled and treated by two differently parametrized Hamiltonians/basis sets. The three-diagonalization-step composite procedure can routinely compute broad range electronic spectra (0-8 eV) within minutes of computation time for systems composed of 500-1000 atoms with an accuracy typical of standard time-dependent density functional theory (0.3-0.5 eV average error). An easily extendable parametrization based on coupled-cluster and density functional computed reference data for the elements H-Zn including transition metals is described. The accuracy of the method termed sTDA-xTB is first benchmarked for vertical excitation energies of open- and closed-shell systems in comparison to other semi-empirical methods and applied to exemplary problems in electronic spectroscopy. As side products of the development, a robust and efficient valence electron TB method for the accurate determination of atomic charges as well as a more accurate calculation scheme of dipole rotatory strengths within the Tamm-Dancoff approximation is proposed.

Rare reaction channels in real-time time-dependent density functional theory: the test case of electron attachment

We present an extension of standard time-dependent density functional theory (TDDFT) to include the evaluation of rare reaction channels, taking as an example of application the theoretical modelling of electron attachment to molecules. The latter process is of great importance in radiation-induced damage of biological tissue for which dissociative electron attachment plays a decisive role. As the attachment probability is very low, it cannot be extracted from the TDDFT propagation whose mean field provides an average over various reaction channels. To extract rare events, we augment TDDFT by a perturbative treatment to account for the occasional jumps, namely electron capture in our test case. We apply the modelling to electron attachment to H2O, H3O+, and (H2O)2. Dynamical calculations have been done at low energy (3−16 eV). We explore, in particular, how core-excited states of the targets show up as resonances in the attachment probability.

Testing time-dependent density functional theory with depopulated molecular orbitals for predicting electronic excitation energies of valence, Rydberg, and charge-transfer states and potential energies near a conical intersection.

Kohn-Sham (KS) time-dependent density functional theory (TDDFT) with most exchange-correlation functionals is well known to systematically underestimate the excitation energies of Rydberg and charge-transfer excited states of atomic and molecular systems. To improve the description of Rydberg states within the KS TDDFT framework, Gaiduk et al. [Phys. Rev. Lett. 108, 253005 (2012)] proposed a scheme that may be called HOMO depopulation. In this study, we tested this scheme on an extensive dataset of valence and Rydberg excitation energies of various atoms, ions, and molecules. It is also tested on a charge-transfer excitation of NH3-F2 and on the potential energy curves of NH3 near a conical intersection. We found that the method can indeed significantly improve the accuracy of predicted Rydberg excitation energies while preserving reasonable accuracy for valence excitation energies. However, it does not appear to improve the description of charge-transfer excitations that are severely underestimated by standard KS TDDFT with conventional exchange-correlation functionals, nor does it perform appreciably better than standard TDDFT for the calculation of potential energy surfaces.

Conductor‐like screening model for relaxed excited states: Implementation in the semiempirical method MSINDO

Two approaches to treat solvent polarization and reorientation effects for excited states of molecules and surfaces have been implemented in the recently developed MSINDO-sCIS method (Gadaczek, Krause, Hintze, Bredow, J. Chem. Theory Comput. 2011, 7, 3675). They allow for an efficient calculation of analytical energy gradients and hence open the opportunity to investigate fluorescence effects or photochemical reactions in solution for large molecules that are difficult to treat with high-level methods. Both approaches are based on the conductor-like screening model (COSMO) (Klamt and Schüürmann, J. Chem. Soc., Perkin Trans. 1993, 2, 799) in combination with the configuration interaction singles (CIS) method (Foresman, Head-Gordon, Pople, and Frisch, J. Phys. Chem. 1992, 96, 135). The paper gives a brief outline of the theoretical background. As a first application, solvent shifts of three well-studied, environment-sensitive fluorescent dyes (Kucherak, Didier, Mély, and Klymchenko, J. Phys. Chem. Lett. 2010, 1, 616) have been calculated and compared with experimental results and standard time-dependent density functional theory. A statistical evaluation of MSINDO-COSMO-sCIS is provided for a set of 39 molecules suggested recently by Jacquemin et al. (Jacquemin, Planchat, Adamo, and Mennucci, J. Chem. Theory Comput. 2012, 8, 2359). Calculated vertical and adiabatic excitation energies and fluorescence energies are compared to experimental data.

Assessment of charge-transfer excitations with time-dependent, range-separated density functional theory based on long-range MP2 and multiconfigurational self-consistent field wave functions

Charge transfer excitations can be described within Time-Dependent Density Functional Theory (TD-DFT), not only by means of the Coulomb Attenuated Method (CAM) but also with a combination of wave function theory and TD-DFT based on range separation. The latter approach enables a rigorous formulation of multi-determinantal TD-DFT schemes where excitation classes, which are absent in conventional TD-DFT spectra (like for example double excitations), can be addressed. This paper investigates the combination of both the long-range Multi-Configuration Self-Consistent Field (MCSCF) and Second Order Polarization Propagator Approximation (SOPPA) ansätze with a short-range DFT (srDFT) description. We find that the combinations of SOPPA or MCSCF with TD-DFT yield better results than could be expected from the pure wave function schemes. For the Time-Dependent MCSCF short-range DFT ansatz (TD-MC-srDFT) excitation energies calculated over a larger benchmark set of molecules with predominantly single reference character yield good agreement with their reference values, and are in general comparable to the CAM-B3LYP functional. The SOPPA-srDFT scheme is tested for a subset of molecules used for benchmarking TD-MC-srDFT and performs slightly better against the reference data for this small subset. Beyond the proof-of-principle calculations comprising the first part of this contribution, we additionally studied the low-lying singlet excited states (S1 and S2) of the retinal chromophore. The chromophore displays multireference character in the ground state and both excited states exhibit considerable double excitation character, which in turn cannot be described within standard TD-DFT, due to the adiabatic approximation. However, a TD-MC-srDFT approach can account for the multireference character, and excitation energies are obtained with accuracy comparable to CASPT2, although using a much smaller active space.

Assessment of range-separated time-dependent density-functional theory for calculating C6 dispersion coefficients

We assess a variant of linear-response range-separated time-dependent density-functional theory (TDDFT), combining a long-range Hartree-Fock (HF) exchange kernel with a short-range adiabatic exchange-correlation kernel in the local-density approximation (LDA) for calculating isotropic C6 dispersion coefficients of homodimers of a number of closed-shell atoms and small molecules. This range-separated TDDFT tends to give underestimated C6 coefficients of small molecules with a mean absolute percentage error of about 5%, a slight improvement over standard TDDFT in the adiabatic LDA which tends to overestimate them with a mean absolute percentage error of 8%, but close to time-dependent Hartree-Fock which has a mean absolute percentage error of about 6%. These results thus show that introduction of long-range HF exchange in TDDFT has a small but beneficial impact on the values of C6 coefficients. It also confirms that the present variant of range-separated TDDFT is a reasonably accurate method even using only a LDA-type density functional and without adding an explicit treatment of long-range correlation.

Calculating Excited States of Molecular Aggregates by the Renormalized Excitonic Method

In this paper, we apply the recently developed ab initio renormalized excitonic method (REM) to the excitation energy calculations of various molecular aggregates, through the extension of REM to the time-dependent density functional theory (TDDFT). Tested molecular aggregate systems include one-dimensional hydrogen-bonded water chains, ring crystals with π-π stacking or van der Waals interactions, two dimensional benzene aggregates and the general aqueous systems with polar and nonpolar solutes. The basis set factor as well as the effect of the exchange-correlation functionals are also investigated. The results indicate that the REM-TDDFT method with suitable basis set and exchange-correlation functionals can give good descriptions of excitation energies and excitation area for lowest electronic excitations in the molecular aggregate systems with economic computational costs. It is shown that the deviations of REM-TDDFT excitation energies from those by standard TDDFT are much less than 0.1 eV and the computational time can be reduced by one order.

Short-Range to Long-Range Charge-Transfer Excitations in the Zincbacteriochlorin-Bacteriochlorin Complex: A Bethe-Salpeter Study

We study, using the Bethe-Salpeter formalism, the singlet excitation energies of the zincbacteriochlorin-bacteriochlorin dyad, a paradigmatic photosynthetic complex. In great contrast with standard time-dependent density functional theory calculations with (semi)local kernels, charge transfer excitations are correctly located above the intramolecular Q-band transitions found to be in excellent agreement with experiment. Further, the asymptotic Coulomb behavior towards the true quasiparticle gap for charge transfer excitations at long distance is correctly reproduced, showing that the present scheme allows us to study with the same accuracy intramolecular and charge transfer excitations at various spatial ranges and screening environments without any adjustable parameter.

Time‐Dependent Density Functional Theory for Calculating Origin‐Independent Optical Rotation and Rotatory Strength Tensors

An approach to calculate origin-independent electronic chiroptical property tensors using time-dependent density functional theory (TDDFT) and gauge-including atomic orbital (GIAO) basis sets is evaluated. Computations of origin-dependent optical rotation tensors and of rotatory strengths needed to simulate circular dichroism spectra are presented. The optical rotation tensor computations employ solutions of coupled perturbed Kohn-Sham equations for a dynamic electric field and a static magnetic field. Because the magnetic field is time independent, the GIAO treatment is somewhat simplified compared to a previously reported method, at some added computational cost if hybrid functionals are employed. GIAO rotatory strengths are also calculated, using transition density matrices from a standard TDDFT excitation energy module. A new implementation in the NWChem quantum chemistry package is employed for representative computations of origin-invariant chiroptical response tensors for methyloxirane, norbornenone, and the ketosteroid androstadienone. For the steroid molecule the vibrational structure of the CD spectrum is modeled explicitly by using calculated Franck-Condon factors. The agreement with experiment is favorable.

Renormalization of Optical Excitations in Molecules near a Metal Surface

The lowest electronic excitations of benzene and a set of donor-acceptor molecular complexes are calculated for the gas phase and on the Al(111) surface using the many-body Bethe-Salpeter equation. The energy of the charge-transfer excitations obtained for the gas phase complexes are found to be around 10% lower than the experimental values. When the molecules are placed outside the surface, the enhanced screening from the metal reduces the exciton binding energies by several eVs and the transition energies by up to 1 eV depending on the size of the transition-generated dipole. As a striking consequence we find that close to the metal surface the optical gap of benzene can exceed its quasiparticle gap. A classical image charge model for the screened Coulomb interaction can account for all these effects which, on the other hand, are completely missed by standard time-dependent density functional theory.

Is charge transfer transitions really too difficult for standard density functionals or are they just a problem for time-dependent density functional theory based on a linear response approach

It is well known that standard time-dependent density functional theory (TD-DFT) affords both a quantitative and qualitative incorrect picture of charge transfer transitions between two spatially separated regions. It is shown here that the well-known failure can be traced back to the use of linear response theory. Further, it is demonstrated that the inclusion of higher order response terms readily affords a qualitatively correct picture even for simple functionals based on the local density approximation. By using the higher order response terms, we finally derive a correction that can be added as a perturbation to charge transfer excitation energies calculated by standard TD-DFT.