Adiabatic Time-dependent Density Functional Theory Research Articles

Reference Energies for Cyclobutadiene: Automerization and Excited States.

Cyclobutadiene is a well-known playground for theoretical chemists and is particularly suitable to test ground- and excited-state methods. Indeed, due to its high spatial symmetry, especially at the D4h square geometry but also in the D2h rectangular arrangement, the ground and excited states of cyclobutadiene exhibit multiconfigurational characters and single-reference methods, such as standard adiabatic time-dependent density-functional theory (TD-DFT) or standard equation-of-motion coupled cluster (EOM-CC), are notoriously known to struggle in such situations. In this work, using a large panel of methods and basis sets, we provide an extensive computational study of the automerization barrier (defined as the difference between the square and rectangular ground-state energies) and the vertical excitation energies at D2h and D4h equilibrium structures. In particular, selected configuration interaction (SCI), multireference perturbation theory (CASSCF, CASPT2, and NEVPT2), and coupled-cluster (CCSD, CC3, CCSDT, CC4, and CCSDTQ) calculations are performed. The spin-flip formalism, which is known to provide a qualitatively correct description of these diradical states, is also tested within TD-DFT (combined with numerous exchange-correlation functionals) and the algebraic diagrammatic construction [ADC(2)-s, ADC(2)-x, and ADC(3)]. A theoretical best estimate is defined for the automerization barrier and for each vertical transition energy.

Linear Response of Current-Dependent Density Functional Approximations in Magnetic Fields.

This Letter outlines the steps and derivations that are necessary to apply density functional approximations that depend on the current and kinetic energy density rigorously within the framework of linear-response methods, including adiabatic time-dependent current density functional theory. This includes systems with a non-zero current density in the ground state. The necessary exchange-correlation kernel for these density functional approximations is derived, and the matrix elements are given explicitly. Due to the gauge variance of the kinetic energy density in an external magnetic field, having access to the proper current-dependent exchange-correlation kernel is necessary to recover gauge invariance for excited states. As a proof of principle application, the excited states of two small molecules in strong external magnetic fields are calculated using linear-response time-dependent current density functional theory. Finally, the implications of the derived current density-dependent exchange-correlation kernel for systems with strong spin-orbit coupling are discussed.

Spin-Conserved and Spin-Flip Optical Excitations from the Bethe-Salpeter Equation Formalism.

Like adiabatic time-dependent density-functional theory (TD-DFT), the Bethe–Salpeter equation (BSE) formalism of many-body perturbation theory, in its static approximation, is “blind” to double (and higher) excitations, which are ubiquitous, for example, in conjugated molecules like polyenes. Here, we apply the spin-flip ansatz (which considers the lowest triplet state as the reference configuration instead of the singlet ground state) to the BSE formalism in order to access, in particular, double excitations. The present scheme is based on a spin-unrestricted version of the GW approximation employed to compute the charged excitations and screened Coulomb potential required for the BSE calculations. Dynamical corrections to the static BSE optical excitations are taken into account via an unrestricted generalization of our recently developed (renormalized) perturbative treatment. The performance of the present spin-flip BSE formalism is illustrated by computing excited-state energies of the beryllium atom, the hydrogen molecule at various bond lengths, and cyclobutadiene in its rectangular and square-planar geometries.

Refined standards for simulating UV–vis absorption spectra of acceptors in organic solar cells by TD-DFT

Photoexcitation property of acceptor plays a vital role to understand fundamental physical process of acceptor excitation in organic solar cells (OSCs), which should be described at a reliable level and characterized by ultraviolet-visible (UV–vis) absorption spectra. Generally, adiabatic linear-response time-dependent density functional theory (LR-TD-DFT) as a suitable framework, which requires to select exchange correlation functionals (XCFs) and basis sets to concentrate the approximation of models, is widely recognized. However, selecting reliable theoretical methods tends to consume computational cost and time, and optimal selections have not yet reached general standards. Herein, a systematic investigation including XCF, basis set, calculation density, solvation effect, and full width at half maximum (FWHM) was undertaken to quantify the photoexcitation properties of typical fullerene acceptors (FAs) and non-fullerene acceptors (NFAs). The results reveal that maximum absorption wavelength (λmax) of NFAs is strongly correlated with the fraction of “exact” Hartree-Fock exchange (EEX), thus establishing selection standards. Moreover, calculation density is the primary factor to ensure the integrity of spectra. This work provides reasonable schemes for characterization conditions, which further lays foundation for the reliability of theoretical calculations that will help to advance the theoretical characterization of optical properties of organic materials.

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.

Excited-State Vibrational Frequencies: Restricted Virtual Space Time-Dependent Density Functional Theory.

Calculating accurate vibrational frequencies for molecules with electronically excited states has an important function in many areas of photochemistry. However, calculations are often limited to smaller molecules due to the rapid growth in the degrees of freedom that must be taken into account to accurately describe larger systems. The applicability of the restricted virtual space (RVS) approximation has been studied within adiabatic linear response time-dependent density functional theory when calculating excited-state nuclear vibrational frequencies. Using the S1 and T1 electronic states of CO, CN-, HOF, H2CS, and C2H4 as representative examples, it is found that vibrational frequency calculations are particularly sensitive to this approximation, with no more than 10-20% of orbitals recommended for safe removal without a priori knowledge when using the 6-311+G(d,p) and aug-cc-pVTZ basis sets. Higher-frequency vibrations such as those with a high degree of CH bond stretching character are found to be less sensitive to the RVS than the lower-frequency vibrations, and several of the triplet states are also found to be less sensitive to this approximation than their equivalent singlet states. Occupied core orbitals and high-energy virtual orbitals with core character can also be removed without introducing significant error.

Reference Energies for Double Excitations

Excited states exhibiting double-excitation character are notoriously difficult to model using conventional single-reference methods, such as adiabatic time-dependent density functional theory (TD-DFT) or equation-of-motion coupled cluster (EOM-CC). In addition, these states are typical experimentally "dark", making their detection in photoabsorption spectra very challenging. Nonetheless, they play a key role in the faithful description of many physical, chemical, and biological processes. In the present work, we provide accurate reference excitation energies for transitions involving a substantial amount of double excitation using a series of increasingly large diffuse-containing atomic basis sets. Our set gathers 20 vertical transitions from 14 small- and medium-size molecules (acrolein, benzene, beryllium atom, butadiene, carbon dimer and trimer, ethylene, formaldehyde, glyoxal, hexatriene, nitrosomethane, nitroxyl, pyrazine, and tetrazine). Depending on the size of the molecule, selected configuration interaction (sCI) and/or multiconfigurational (CASSCF, CASPT2, (X)MS-CASPT2, and NEVPT2) calculations are performed in order to obtain reliable estimates of the vertical transition energies. In addition, coupled cluster approaches including at least contributions from iterative triples (such as CC3, CCSDT, CCSDTQ, and CCSDTQP) are assessed. Our results clearly evidence that the error in CC methods is intimately related to the amount of double-excitation character of the transition. For "pure" double excitations (i.e., for transitions which do not mix with single excitations), the error in CC3 can easily reach 1 eV, while it goes down to a few tenths of an electronvolt for more common transitions (such as in trans-butadiene) involving a significant amount of singles. As expected, CC approaches including quadruples yield highly accurate results for any type of transition. The quality of the excitation energies obtained with multiconfigurational methods is harder to predict. We have found that the overall accuracy of these methods is highly dependent on both the system and the selected active space. The inclusion of the σ and σ* orbitals in the active space, even for transitions involving mostly π and π* orbitals, is mandatory in order to reach high accuracy. A theoretical best estimate (TBE) is reported for each transition. We believe that these reference data will be valuable for future methodological developments aiming at accurately describing double excitations.

Bond-breaking excitations with diverging coupling matrix of response density functional theory from highest-level functionals

Bond-breaking excitations ωα are the problematic case of adiabatic time-dependent density functional theory (TDDFT). The calculated ωα erroneously vanishes with the bond elongation, since the Hartree-exchange-correlation kernel and the corresponding response coupling matrix K of standard approximations lack the characteristic divergence in the dissociation limit. In this paper an approximation for K is proposed constructed from the highest-level functionals, in which both occupied and virtual Kohn-Sham orbitals participate with the weights wp. The latter provide the correct divergence of K in the limit of dissociating two-electron bond. The present K brings a decisive contribution to the energy of the 1Σu+ in the prototype H2 molecule calculated for various H-H separations. At shorter separations it improves ωα compared to the zero-order TDDFT estimate, while at the largest separation it reproduces near-saturation of the reference excitation energy.

Quadratic Response Properties from TDDFT: Trials and Tribulations.

We report on the efficient turbomole implementation of quadratic response properties within the time-dependent density functional theory (TDDFT) context that includes the static and dynamic dipole hyperpolarizability, ground-to-excited-state two-photon absorption amplitudes (through a single residue) and state-to-state one-photon absorption amplitudes (through a double residue). Our implementation makes full use of arbitrary (including non-Abelian) point-group symmetry as well as permutational symmetry and enables the calculation of nonlinear properties with hybrid density functionals for molecules with hundreds of atoms and thousands of basis functions at a cost that is a fixed multiple of the cost of the corresponding linear properties. Using the PBE0 hybrid density functional, we show that excited-state absorption spectra computed within the pseudowavefunction approach contain the qualitative features of transient absorption spectra tracking excimer formation in perylene diimide dimers, two-photon absorption cross sections for a series of highly twisted fused porphyrin chains are semiquantitatively reproduced, and the computed dynamic hyperpolarizability of several calix[4]arene stereoisomers yield simulated hyper-Raleigh scattering signals consistent with experiment. In addition, we show that the incorrect pole structure of adiabatic TDDFT properties can cause incorrect excited-state absorption spectra and overly resonant hyperpolarizabilities, and discuss possible remedies.

Expectations of the Kohn-Sham operator in a natural-orbital functional environment: An adiabatic response theory with single and double excitations

A combined adiabatic approach is proposed in which orbital expectations of the Kohn-Sham operator (KSEs) are embedded into the matrix environment of time-dependent phase-including natural-orbital functional theory (TDPINOFT). Analytical diagonalization of a model molecular matrix eigenvalue problem shows that the resultant TDNOFT-KSE combines strong sides of both adiabatic time-dependent density functional theory (TDDFT) and TDPINOFT. Around the equilibrium, like in TDDFT, the energy of a single excitation of the model is evaluated with the KSEs. In its turn, the TDPINOFT environment provides an effective Hartree-exchange-correlation (Hxc) kernel, which properly diverges in the bond dissociation limit. This environment also generates a double excitation, the energy of which is related to the ratio of fractional occupation numbers of the natural orbitals.

Do fractionally incremented nuclear charges improve time-dependent density functional theory excitation energies as reliably as fractional orbital populations?

Gaiduk et al. (Phys Rev Lett 108:253005, 2012) showed that one can improve local, semilocal, and hybrid approximations to the Kohn–Sham effective potentials of atoms and molecules by removing a system-independent fraction of electron charge from the highest occupied molecular orbital (HOMO); if the corrected Kohn–Sham potential is used for adiabatic linear-response time-dependent density functional theory (TDDFT) calculations, accurate Rydberg excitation energies are obtained. One may ask whether the same effect could also be achieved by fractionally increasing the positive charges of the nuclei. We investigate this question and find that a small increase in nuclear charges can indeed substantially reduce errors in TDDFT Rydberg excitation energies. However, the optimal magnitude of the charge increase is system-dependent. In addition, the procedure is ambiguous for molecules, where one has to decide how to distribute the additional charge among individual nuclei. These two drawbacks of the fractional nuclear charge method make it disadvantageous compared to the HOMO depopulation technique.

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.

Unphysical divergences in response theory.

Transition densities between excited states are key for nonlinear theoretical spectroscopy and multi-state non-adiabatic molecular dynamics (NAMD) simulations. In the framework of response theory, these transition densities are accessible from poles of the quadratic response function. It was shown recently that the thus obtained transition densities within time-dependent Hartree-Fock (TDHF) and adiabatic time-dependent density functional theory (TDDFT) exhibit unphysical divergences when the difference in excitation energy of the two states of interest matches another excitation energy. This unphysical behavior is a consequence of spurious poles in the quadratic response function. We show that the incorrect pole structure of the quadratic response is not limited to TDHF and adiabatic TDDFT, but is also present in many other approximate many-electron response functions, including those from coupled cluster and multiconfigurational self-consistent field response theory. The divergences appear in regions of the potential energy surface where the ground state is perfectly well behaved, and they are frequently encountered in NAMD simulations of photochemical reactions. The origin of the divergences is traced to an incorrect instantaneous time-dependence of the effective Hamiltonian. The implications for computations of frequency-dependent response properties are considerable and call into question the validity of conventional approximate many-electron response theories beyond linear response.

Critical Assessment of TD-DFT for Excited States of Open-Shell Systems: I. Doublet-Doublet Transitions.

A benchmark set of 11 small radicals is set up to assess the performance of time-dependent density functional theory (TD-DFT) for the excited states of open-shell systems. Both the unrestricted (U-TD-DFT) and spin-adapted (X-TD-DFT) formulations of TD-DFT are considered. For comparison, the well-established EOM-CCSD (equation-of-motion coupled-cluster with singles and doubles) is also used. In total, 111 low-lying singly excited doublet states are accessed by all the three approaches. Taking the MRCISD+Q (multireference configuration interaction with singles and doubles plus the Davidson correction) results as the benchmark, it is found that both U-TD-DFT and EOM-CCSD perform well for those states dominated by singlet-coupled single excitations (SCSE) from closed-shell to open-shell, open-shell to vacant-shell, or closed-shell to vacant-shell orbitals. However, for those states dominated by triplet-coupled single excitations (TCSE) from closed-shell to vacant-shell orbitals, both U-TD-DFT and EOM-CCSD fail miserably due to severe spin contaminations. In contrast, X-TD-DFT provides balanced descriptions of both SCSE and TCSE. As far as the functional dependence is concerned, it is found that, when the Hartree-Fock ground state does not suffer from the instability problem, both global hybrid (GH) and range-separated hybrid (RSH) functionals perform grossly better than pure density functionals, especially for Rydberg and charge-transfer excitations. However, if the Hartree-Fock ground state is instable or nearly instable, GH and RSH tend to underestimate severely the excitation energies. The SAOP (statistically averaging of model orbital potentials) performs more uniformly than any other density functionals, although it generally overestimates the excitation energies of valence excitations. Not surprisingly, both EOM-CCSD and adiabatic TD-DFT are incapable of describing excited states with substantial double excitation characters.

First-order derivative couplings between excited states from adiabatic TDDFT response theory.

We present a complete derivation of derivative couplings between excited states in the framework of adiabatic time-dependent density functional response theory. Explicit working equations are given and the resulting derivative couplings are compared with derivative couplings from a pseudo-wavefunction ansatz. For degenerate excited states, i.e., close to a conical intersection (CI), the two approaches are identical apart from an antisymmetric overlap term. However, if the difference between two excitation energies equals another excitation energy, the couplings from response theory exhibit an unphysical divergence. This spurious behavior is a result of the adiabatic or static kernel approximation of time-dependent density functional theory leading to an incorrect analytical structure of the quadratic response function. Numerical examples for couplings close to a CI and for well-separated electronic states are given.

Dielectric response of laser-excited silicon at finite electron temperature

We calculate the dielectric response of excited crystalline silicon in electron thermal equilibrium by adiabatic time-dependent density functional theory (TDDFT) to model the response to irradiation by high-intensity laser pulses. The real part of the dielectric function is characterized by the strong negative behavior at low frequencies due to excited electron-hole pairs. The response agrees rather well with the numerical pump-probe calculations which simulate electronic excitations in nonequilibrium phase immediately after the laser pulse irradiation. The thermal response is also compared with the Drude model which includes electron effective mass and collision time as fitting parameters. We find that the extracted effective masses are in the range of 0.22-0.36 and lifetimes are in the range of 1-14 fs depending on the temperature. The short Drude lifetimes show that strong damping is possible in the adiabatic TDDFT, despite the absence of explicit electron-electron collisions.

Gas phase UV spectrum of a Cu(II)-bis(benzene) sandwich complex: experiment and theory.

Photofragmentation with tunable UV radiation has been used to generate a spectrum for the copper-bis(benzene) complex, [Cu(C6H6)2](2+), in the gas phase. The ions were held in an ion trap where their temperature was reduced to ∼150 K, whereby the spectrum revealed two broad features at ∼38,200 and ∼45,700 cm(-1). Detailed calculations using density functional theory (DFT) show the complex can occupy three minimum energy structures with C2v and C2 (staggered and eclipsed) symmetries. Adiabatic time-dependent DFT (TDDFT) has been used to identify electronic transitions in [Cu(benzene)2](2+), and the calculations show these to fall into two groups that are in excellent agreement with the experimental data. However, the open-shell electronic configuration of Cu(2+) (d(9)) may give rise to excited states with double-excitation character, and the single-excitation adiabatic TDDFT treatment leads to extensive spin contamination. By quantifying the extent of spin contamination and allowing for the inclusion of a small percentage (∼10%), the theory can provide quantitative agreement with the experimental data.

Applications of time-dependent and time-independent density functional theory to Rydberg transitions.

We have benchmarked the performance of time-independent density functional theory (ΔSCF and RSCF-CV-DFT) in studies on Rydberg transitions employing five different standard functionals and a diffuse basis. Our survey is based on 71 triplet or singlet Rydberg transitions distributed over nine different species: CO (7), CH2O (8), C2H2 (8), H2O (10), C2H4 (13), Be (6), Mg (6), and Zn (8). The best performance comes from the long-range corrected functional LCBP86 (ω = 0.4.) with an average root-mean-square deviation (RMSD) of 0.23 eV. Of similar accuracy are LDA and B3LYP, both with a RMSD of 0.24 eV. The largest RMSD of 0.32 eV comes from BP86 and LCBP86* (ω = 0.75). The performance of ΔSCF is considerably better than that of adiabatic time-dependent density functional theory (ATDDFT) and matches that of highly optimized long-range corrected functionals. However, it is not as accurate as ATDDFT based on highly tuned functionals. The reasonable success of ΔSCF is based on its well-documented ability to afford good estimates of ionization potentials (IP) and electron affinities (EA) even for simple local functionals after orbital relaxation has been taken into account. In ATDDFT based on semilocal functionals, both IP and -EA are poorly described, with errors of up to 5 eV. In the transition energy (ΔE = IP - EA), these errors are canceled to some degree. However, ΔE still carries an error exceeding 1 eV.

Applications of Time Dependent and Time Independent Density Functional Theory to the First π to π* Transition in Cyanine Dyes.

The first π → π* transition in a number of cyanine dyes was studied using both time dependent and time independent density functional methods using a coupled cluster (CC2) method as the benchmark scheme. On the basis of 10 different functionals, it was concluded that adiabatic time dependent density functional theory (ATDDFT) almost independently of the functional gives rise to a singlet-triplet separation that is too large by up to 1 eV, leading to too high singlet energies and too low triplet energies. This trend is even clearer when the Tamm-Dancoff (TD) approximation is introduced and can in ATDDFT/TD be traced back to the representation of the singlet-triplet separation by a HF-type exchange integral between π and π*. The time independent DFT methods (ΔSCF and RSCF-CV-DFT) afford triplet energies that are functional independent and close to those obtained by ATDDFT. However, both the singlet energies and the singlet-triplet separations increases with the fraction α of HF exchange. This trend can readily be explained in terms of the modest magnitude of a KS-exchange integral between π and π* in comparison to the much larger HF-exchange integral. It was shown that a fraction α of 0.5 affords good estimates of both the singlet energies and the singlet-triplet separations in comparison to several ab initio benchmarks.

Introducing constricted variational density functional theory in its relaxed self-consistent formulation (RSCF-CV-DFT) as an alternative to adiabatic time dependent density functional theory for studies of charge transfer transitions.

We have applied the relaxed and self-consistent extension of constricted variational density functional theory (RSCF-CV-DFT) for the calculation of the lowest charge transfer transitions in the molecular complex X-TCNE between X = benzene and TCNE = tetracyanoethylene. Use was made of functionals with a fixed fraction (α) of Hartree-Fock exchange ranging from α = 0 to α = 0.5 as well as functionals with a long range correction (LC) that introduces Hartree-Fock exchange for longer inter-electronic distances. A detailed comparison and analysis is given for each functional between the performance of RSCF-CV-DFT and adiabatic time-dependent density functional theory (TDDFT) within the Tamm-Dancoff approximation. It is shown that in this particular case, all functionals afford the same reasonable agreement with experiment for RSCF-CV-DFT whereas only the LC-functionals afford a fair agreement with experiment using TDDFT. We have in addition calculated the CT transition energy for X-TCNE with X = toluene, o-xylene, and naphthalene employing the same functionals as for X = benzene. It is shown that the calculated charge transfer excitation energies are in as good agreement with experiment as those obtained from highly optimized LC-functionals using adiabatic TDDFT. We finally discuss the relation between the optimization of length separation parameters and orbital relaxation in the RSCF-CV-DFT scheme.