A long-range-corrected time-dependent density functional theory

We apply the long-range correction (LC) scheme for exchange functionals of density functional theory to time-dependent density functional theory (TDDFT) and examine its efficiency in dealing with the serious problems of TDDFT, i.e., the underestimations of Rydberg excitation energies, oscillator strengths, and charge-transfer excitation energies. By calculating vertical excitation energies of typical molecules, it was found that LC-TDDFT gives accurate excitation energies, within an error of 0.5 eV, and reasonable oscillator strengths, while TDDFT employing a pure functional provides 1.5 eV lower excitation energies and two orders of magnitude lower oscillator strengths for the Rydberg excitations. It was also found that LC-TDDFT clearly reproduces the correct asymptotic behavior of the charge-transfer excitation energy of ethylene-tetrafluoroethylene dimer for the long intramolecular distance, unlike a conventional far-nucleus asymptotic correction scheme. It is, therefore, presumed that poor TDDFT results for pure functionals may be due to their lack of a long-range orbital-orbital interaction.     

Long-range excitations in time-dependent density functional theory

Adiabatic time-dependent density functional theory fails for excitations of a heteroatomic molecule composed of two open-shell fragments at large separation. Strong frequency dependence of the exchange-correlation kernel is necessary for both local and charge-transfer excitations. The root of this is the static correlation created by the step in the exact Kohn-Sham ground-state potential between the two fragments. An approximate nonempirical kernel is derived for excited molecular dissociation curves at large separation. Our result is also relevant when the usual local and semilocal approximations are used for the ground-state potential, as static correlation there arises from the coalescence of the highest occupied and lowest unoccupied orbital energies as the molecule dissociates.     

General excitations in time-dependent density functional theory

A general framework within time-dependent density functional theory is presented for the calculation of excitations to states of arbitrary multiplicity in molecular systems with a non-singlet ground state. The proposed approach combines generalized orbital excitation operators designed to generate excited states which have well-defined multiplicities and the noncollinear formulation of density functional theory and it can be straightforwardly implemented in currently existing density functional programs.     

Spin-multiplet energies from time-dependent density functional theory

Starting from a formally exact density-functional representation of the frequency-dependent linear density response and exploiting the fact that the latter has poles at the true excitation energies, we develop a density-functional method for the calculation of excitation energies. Simple additive corrections to the Kohn-Sham single-particle transition energies are derived whose actual computation only requires the ordinary static Kohn-Sham orbitals and the corresponding eigenvalues. Numerical results are presented for spin-singlet and triplet energies. © 1996 John Wiley & Sons, Inc.     

Continuum states from time-dependent density functional theory

Linear response time-dependent density functional theory is used to study low-lying electronic continuum states of targets that can bind an extra electron. Exact formulas to extract scattering amplitudes from the susceptibility are derived in one dimension. A single-pole approximation for scattering phase shifts in three dimensions is shown to be more accurate than static exchange for singlet electron-He(+) scattering.     

Intermolecular dispersion energies from time-dependent density functional theory

Non-expanded dispersion energies are calculated from time-dependent coupled-perturbed density functional theory (DFT) employing various non-hybrid and hybrid exchange-correlation potentials and suitable adiabatic local density approximations for the exchange-correlation kernel. Considering the dimer systems He₂, Ne₂, Ar₂, NeAr, NeHF, ArHF, (H₂)₂, (HF)₂, and (H₂O)₂ it is shown that the effects of intramonomer electron correlation on the dispersion energy are accurately reproduced with the PBE0AC exchange-correlation potential. In contrast, the uncoupled sum-over-states approximation yields inacceptable errors. These are mainly due to neglect of the Coulomb and exchange-correlation kernels and therefore, not substantially improved through an asymptotic correction of the exchange-correlation potential.     

Cubic response functions in time-dependent density functional theory

We present density-functional theory for time-dependent response functions up to and including cubic response. The working expressions are derived from an explicit exponential parametrization of the density operator and the Ehrenfest principle, alternatively, the quasienergy ansatz. While the theory retains the adiabatic approximation, implying that the time-dependency of the functional is obtained only implicitly-through the time dependence of the density itself rather than through the form of the exchange-correlation functionals-it generalizes previous time-dependent implementations in that arbitrary functionals can be chosen for the perturbed densities (energy derivatives or response functions). In particular, general density functionals beyond the local density approximation can be applied, such as hybrid functionals with exchange correlation at the generalized-gradient approximation level and fractional exact Hartree-Fock exchange. With our implementation the response of the density can always be obtained using the stated density functional, or optionally different functionals can be applied for the unperturbed and perturbed densities, even different functionals for different response order. As illustration we explore the use of various combinations of functionals for applications of nonlinear optical hyperpolarizabilities of a few centrosymmetric systems; molecular nitrogen, benzene, and the C(60) fullerene. Considering that vibrational, solvent, and local field factors effects are left out, we find in general that very good experimental agreement can be obtained for the second dynamic hyperpolarizability of these systems. It is shown that a treatment of the response of the density beyond the local density approximation gives a significant effect. The use of different functional combinations are motivated and discussed, and it is concluded that the choice of higher order kernels can be of similar importance as the choice of the potential which governs the Kohn-Sham orbitals.     

Excitonic effects in a time-dependent density functional theory

Excited state properties of one-dimensional molecular materials are dominated by many-body interactions resulting in strongly bound confined excitons. These effects cannot be neglected or treated as a small perturbation and should be appropriately accounted for by electronic structure methodologies. We use adiabatic time-dependent density functional theory to investigate the electronic structure of one-dimensional organic semiconductors, conjugated polymers. Various commonly used functionals are applied to calculate the lowest singlet and triplet state energies and oscillator strengths of the poly(phenylenevinylene) and ladder-type (poly)(para-phenylene) oligomers. Local density approximations and gradient-corrected functionals cannot describe bound excitonic states due to lack of an effective attractive Coulomb interaction between photoexcited electrons and holes. In contrast, hybrid density functionals, which include long-range nonlocal and nonadiabatic corrections in a form of a fraction of Hartree-Fock exchange, are able to reproduce the excitonic effects. The resulting finite exciton sizes are strongly dependent on the amount of the orbital exchange included in the functional.     

Simple preconditioning for time-dependent density functional perturbation theory

By far, the most common use of time-dependent density functional theory is in the linear-reponse regime, where it provides information about electronic excitations. Ideally, the linear-response equations should be solved by a method that avoids the use of the unoccupied Kohn-Sham states--such as the Sternheimer method--as this reduces the complexity and increases the precision of the calculation. However, the Sternheimer equation becomes ill-conditioned near and indefinite above the first resonant frequency, seriously hindering the use of efficient iterative solution methods. To overcome this serious limitation, and to improve the general convergence properties of the iterative techniques, we propose a simple preconditioning strategy. In our method, the Sternheimer equation is solved directly as a linear equation using an iterative Krylov subspace method, i.e., no self-consistent cycle is required. Furthermore, the preconditioner uses the information of just a few unoccupied states and requires simple and minimal modifications to existing implementations. In this way, convergence can be reached faster and in a considerably wider frequency range than the traditional approach.     

Formulation of magnetically perturbed time-dependent density functional theory

A formulation of time-dependent density functional theory (TDDFT) in the presence of a static imaginary perturbation is derived. A perturbational approach is applied leading to corrections to various orders in the quantities of interest, namely, the excitation energies and transition densities. The perturbed TDDFT equations are relatively straightforward to derive but the resulting expressions are rather cumbersome. Simplifications of these equations are suggested. Both the simplified and full expressions are used to obtain equations for first- and second-order corrections to the excitation energy, the first-order correction to the transition density, and the corrections for both quantities to first-order in two different perturbations. This formulation, called magnetically perturbed TDDFT, details how conventional TDDFT calculations can be corrected to allow for the inclusion of a static magnetic field and/or spin-orbit coupling.     

Excitation energies from range-separated time-dependent density and density matrix functional theory

Time-dependent density functional theory (TD-DFT) in the adiabatic formulation exhibits known failures when applied to predicting excitation energies. One of them is the lack of the doubly excited configurations. On the other hand, the time-dependent theory based on a one-electron reduced density matrix functional (time-dependent density matrix functional theory, TD-DMFT) has proven accurate in determining single and double excitations of H(2) molecule if the exact functional is employed in the adiabatic approximation. We propose a new approach for computing excited state energies that relies on functionals of electron density and one-electron reduced density matrix, where the latter is applied in the long-range region of electron-electron interactions. A similar approach has been recently successfully employed in predicting ground state potential energy curves of diatomic molecules even in the dissociation limit, where static correlation effects are dominating. In the paper, a time-dependent functional theory based on the range-separation of electronic interaction operator is rigorously formulated. To turn the approach into a practical scheme the adiabatic approximation is proposed for the short- and long-range components of the coupling matrix present in the linear response equations. In the end, the problem of finding excitation energies is turned into an eigenproblem for a symmetric matrix. Assignment of obtained excitations is discussed and it is shown how to identify double excitations from the analysis of approximate transition density matrix elements. The proposed method used with the short-range local density approximation (srLDA) and the long-range Buijse-Baerends density matrix functional (lrBB) is applied to H(2) molecule (at equilibrium geometry and in the dissociation limit) and to Be atom. The method accounts for double excitations in the investigated systems but, unfortunately, the accuracy of some of them is poor. The quality of the other excitations is in general much better than that offered by TD-DFT-LDA or TD-DMFT-BB approximations if the range-separation parameter is properly chosen. The latter remains an open problem.     

Average excitation energies from time-dependent density functional response theory

The authors present an occupation number averaging scheme for time-dependent density functional response theory (TD-DFRT) in frequency domain. The known problem that TD-DFRT within the local (spin) density approximation (LDA/LSDA) inaccurately predicts Rydberg and charge-transfer excitation energies has been reexamined from the methodology of linear response, without explicit correction of the exchange-correlation potential. The working equations of TD-DFRT are adapted to treat arbitrary difference of orbital occupation numbers, using the nonsymmetric matrix form of Casida's formulation of TD-DFRT [M. E. Casida, in Recent Advances in Density Functional Methods, edited by D. P. Chong (World Scientific, Singapore, 1995), Pt. I, p. 155]. The authors' scheme is applied to typical closed-shell and open-shell molecular systems by examining the dependence of excitation energies on the fraction of excited electron. Good performance of this modified linear response scheme is shown, and is consistent with the authors' previous examination by the real-time propagation approach, suggesting that the calculation of average excitation energies might be one of the ways to better decode excitation energies from LDA/LSDA. Different techniques for treating singlet, triplet, and doublet states are discussed.     

Adiabatic approximation of time-dependent density matrix functional response theory

Time-dependent density matrix functional theory can be formulated in terms of coupled-perturbed response equations, in which a coupling matrix K(omega) features, analogous to the well-known time-dependent density functional theory (TDDFT) case. An adiabatic approximation is needed to solve these equations, but the adiabatic approximation is much more critical since there is not a good "zero order" as in TDDFT, in which the virtual-occupied Kohn-Sham orbital energy differences serve this purpose. We discuss a simple approximation proposed earlier which uses only results from static calculations, called the static approximation (SA), and show that it is deficient, since it leads to zero response of the natural orbital occupation numbers. This leads to wrong behavior in the omega-->0 limit. An improved adiabatic approximation (AA) is formulated. The two-electron system affords a derivation of exact coupled-perturbed equations for the density matrix response, permitting analytical comparison of the adiabatic approximation with the exact equations. For the two-electron system also, the exact density matrix functional (2-matrix in terms of 1-matrix) is known, enabling testing of the static and adiabatic approximations unobscured by approximations in the functional. The two-electron HeH(+) molecule shows that at the equilibrium distance, SA consistently underestimates the frequency-dependent polarizability alpha(omega), the adiabatic TDDFT overestimates alpha(omega), while AA improves upon SA and, indeed, AA produces the correct alpha(0). For stretched HeH(+), adiabatic density matrix functional theory corrects the too low first excitation energy and overpolarization of adiabatic TDDFT methods and exhibits excellent agreement with high-quality CCSD ("exact") results over a large omega range.     

On correlated electron-nuclear dynamics using time-dependent density functional theory

We discuss possibilities and challenges for describing correlated electron and nuclear dynamics within a surface-hopping framework using time-dependent density functional theory (TDDFT) for the electron dynamics. We discuss the recent surface-hopping method proposed by Craig et al. [Phys. Rev. Lett. 95, 163001 (2005)] that is based on Kohn-Sham potential energy surfaces. Limitations of this approach arise due to the Kohn-Sham surfaces generally having different gradients than the true TDDFT-corrected ones. Two mechanisms of the linear response procedure cause this effect: we illustrate these with examples.     

Double excitations within time-dependent density functional theory linear response

Within the adiabatic approximation, time-dependent density functional theory yields only single excitations. Near states of double excitation character, the exact exchange-correlation kernel has a strong dependence on frequency. We derive the exact frequency-dependent kernel when a double excitation mixes with a single excitation, well separated from the other excitations, in the limit that the electron--electron interaction is weak. Building on this, we construct a nonempirical approximation for the general case, and illustrate our results on a simple model.     

Magnetic circular dichroism in real-time time-dependent density functional theory

We apply the adiabatic time-dependent density functional theory to magnetic circular dichroism (MCD) spectra using the real-space, real-time computational method. The standard formulas for the MCD response and its A and B terms are derived from the observables in the time-dependent wave function. We find real-time method is well suited for calculating the overall spectrum, particularly at higher excitation energies where individual excited states are numerous and overlapping. The MCD sum rules are derived and intepreted in the real-time formalism; we find that they are very useful for normalization purposes and assessing the accuracy of the theory. The method is applied to MCD spectrum of C(60) using the adiabatic energy functional from the local density approximation. The theory correctly predicts the signs of the A and B terms for the lowest allowed excitations. However, the magnitudes of the terms only show qualitative agreement with experiment.     

Theoretical and numerical assessments of spin-flip time-dependent density functional theory

Spin-flip time-dependent density functional theory (SF-TD-DFT) with the full noncollinear hybrid exchange-correlation kernel and its approximate variants are critically assessed, both formally and numerically. As demonstrated by the ethylene torsion and the C(2v) ring-opening of oxirane, SF-TD-DFT is very useful for describing nearly degenerate situations. However, it may occasionally yield unphysical results. This stems from the noncollinear form of the generalized gradient approximation, which becomes numerically instable in the presence of spin-flip excitations from the closed- to vacant-shell orbitals of an open-shell reference. To cure this defect, a simple modification, dubbed as ALDA0, is proposed in the spirit of adiabatic local density approximation (ALDA). It is applicable to all kinds of density functionals and yields stable results without too much loss of accuracy. In particular, the combination of ALDA0 with the Tamm-Dancoff approximation is a promising tool for studying global potential energy surfaces. In addition to the kernel problem, SF-TD-DFT is also rather sensitive to the choice of reference states, as demonstrated by the spin multiplet states of closed-shell molecules of H(2)O, CH(2)O, and C(2)H(4). Surprisingly, SF-TD-DFT with pure density functionals may also fail for valance excitations with large orbital overlaps, at variance with the spin-conserving counterpart (SC-TD-DFT). In this case, the inclusion of a large amount of Hartree-Fock exchange is mandatory for quantitative results. Nonetheless, for spatially degenerate cases such as CF, CH, and NH(+), SF-TD-DFT is more advantageous than SC-TD-DFT, unless the latter is also space adapted. These findings are very instructive for future development and applications of TD-DFT.     

Absorption spectra of blue-light-emitting oligoquinolines from time-dependent density functional theory

Recently, it has been discovered that a series of four conjugated oligomers, oligoquinolines, exhibits many desirable properties of organic materials for developing high-performance light-emitting diodes: good blue color purity, high brightness, high efficiency, and high glass-transition temperatures. In this work, we investigate the optical absorption of oligoquinolines in the gas phase and chloroform (CHCl3) solution, respectively, using time-dependent density functional theory with the adiabatic approximation for the dynamical exchange-correlation potential. Our calculations show that the first peak of optical absorption corresponds to the lowest singlet excited state, whereas several quasi-degenerate excited states contribute to the experimentally observed higher-frequency peak. We find that, compared with the gas phase, there is a moderate red shift in excitation energy in solution due to the solute-solvent interaction simulated using the polarizable continuum model. Our results show that the lowest singlet excitation energies of oligoquinolines in chloroform solution calculated with the adiabatic hybrid functional PBE0 are in a good agreement with experiments. Our simulated optical absorption agrees well with the experimental data. Finally, analysis of the natural transition orbitals corresponding to the excited states in question underscores the underlying electronic delocalization properties.     

Brief review related to the foundations of time-dependent density functional theory

The electron density n(r,t), which is the central tool of time-dependent density functional theory, is presently considered to be derivable from a one-body time-dependent potential V(r,t), via one-electron wave functions satisfying a time-dependent Schrödinger equation. This is here related via a generalized equation of motion to a Dirac density matrix now involving t. Linear response theory is then surveyed, with a special emphasis on the question of causality with respect to the density dependence of the potential. Extraction of V(r,t) for solvable models is also proposed.     

Molecular-orbital-free algorithm for the excited-state force in time-dependent density functional theory

Starting from the equation of motion in the density matrix formulation, we reformulate the analytical gradient of the excited-state energy at the time-dependent density functional theory level in the nonorthogonal Gaussian atom-centered orbital (AO) basis. Analogous to the analytical first derivative in molecular-orbital (MO) basis, a Z-vector equation has been derived with respect to the reduced one-electronic density matrix in AO basis, which provides a potential possibility to exploit quantum locality of the density matrix and avoids the matrix transformation between the AO and the MO basis. Numerical tests are finished for the excited-state geometry optimization and adiabatic excitation energy calculation of a series of small molecules. The results demonstrate the computational efficiency and accuracy of the current AO-based energy gradient expression in comparison with the MO-based scheme.