Molecular properties via a subsystem density functional theory formulation: a common framework for electronic embedding

In this article, we present a consistent derivation of a density functional theory (DFT) based embedding method which encompasses wave-function theory-in-DFT (WFT-in-DFT) and the DFT-based subsystem formulation of response theory (DFT-in-DFT) by Neugebauer [J. Neugebauer, J. Chem. Phys. 131, 084104 (2009)] as special cases. This formulation, which is based on the time-averaged quasi-energy formalism, makes use of the variation Lagrangian techniques to allow the use of non-variational (in particular: coupled cluster) wave-function-based methods. We show how, in the time-independent limit, we naturally obtain expressions for the ground-state DFT-in-DFT and WFT-in-DFT embedding via a local potential. We furthermore provide working equations for the special case in which coupled cluster theory is used to obtain the density and excitation energies of the active subsystem. A sample application is given to demonstrate the method.     

Pseudospectral time-dependent density functional theory

Time-dependent density functional theory (TDDFT) is implemented within the Tamm-Dancoff approximation (TDA) using a pseudospectral approach to evaluate two-electron repulsion integrals. The pseudospectral approximation uses a split representation with both spectral basis functions and a physical space grid to achieve a reduction in the scaling behavior of electronic structure methods. We demonstrate here that exceptionally sparse grids may be used in the excitation energy calculation, following earlier work employing the pseudospectral approximation for determining correlation energies in wavefunction-based methods with similar conclusions. The pseudospectral TDA-TDDFT method is shown to be up to ten times faster than a conventional algorithm for hybrid functionals without sacrificing chemical accuracy.     

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.     

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.     

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.     

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.     

Limits of the creation of electronic wave packets using time-dependent density functional theory

Explicitly time-dependent density functional theory (TDDFT) has often been suggested as the method of choice for controlling the correlated dynamics of many electron systems. However, it is not yet clear which control tasks can be achieved reliably and how this depends on the functionals used. In this article, we show that the control task of creating a simple wave packet, having a population of 50% in the excited state, can indeed be achieved if a certain condition is fulfilled. This result is in contrast to the observation that a full population inversion is extremely difficult to achieve. In addition, we identify a rule to predict when TDDFT produces the correct wave packet. To illustrate our findings, we study the molecules Li(2)C(2), Li(7)OH, and B(2)N(2)CO using two different functionals as well as time-dependent Hartree-Fock (TDHF). To assess the performance of TDDFT and TDHF, we compare with time-dependent configuration interaction calculations.     

Analytical Hessian of electronic excited states in time-dependent density functional theory with Tamm-Dancoff approximation

We present the analytical expression and computer implementation for the second-order energy derivatives of the electronic excited state with respect to the nuclear coordinates in the time-dependent density functional theory (TDDFT) with Gaussian atomic orbital basis sets. Here, the Tamm-Dancoff approximation to the full TDDFT is adopted, and therefore the formulation process of TDDFT excited-state Hessian is similar to that of configuration interaction singles (CIS) Hessian. However, due to the replacement of the Hartree-Fock exchange integrals in CIS with the exchange-correlation kernels in TDDFT, many quantitative changes in the derived equations are arisen. The replacement also causes additional technical difficulties associated with the calculation of a large number of multiple-order functional derivatives with respect to the density variables and the nuclear coordinates. Numerical tests on a set of test molecules are performed. The simulated excited-state vibrational frequencies by the analytical Hessian approach are compared with those computed by CIS and the finite-difference method. It is found that the analytical Hessian method is superior to the finite-difference method in terms of the computational accuracy and efficiency. The numerical differentiation can be difficult due to root flipping for excited states that are close in energy. TDDFT yields more exact excited-state vibrational frequencies than CIS, which usually overestimates the values.     

Spin-orbit relativistic long-range corrected time-dependent density functional theory for investigating spin-forbidden transitions in photochemical reactions

A long-range corrected (LC) time-dependent density functional theory (TDDFT) incorporating relativistic effects with spin-orbit couplings is presented. The relativistic effects are based on the two-component zeroth-order regular approximation Hamiltonian. Before calculating the electronic excitations, we calculated the ionization potentials (IPs) of alkaline metal, alkaline-earth metal, group 12 transition metal, and rare gas atoms as the minus orbital (spinor) energies on the basis of Koopmans' theorem. We found that both long-range exchange and spin-orbit coupling effects are required to obtain Koopmans' IPs, i.e., the orbital (spinor) energies, quantitatively in DFT calculations even for first-row transition metals and systems containing large short-range exchange effects. We then calculated the valence excitations of group 12 transition metal atoms and the Rydberg excitations of rare gas atoms using spin-orbit relativistic LC-TDDFT. We found that the long-range exchange and spin-orbit coupling effects significantly contribute to the electronic spectra of even light atoms if the atoms have low-lying excitations between orbital spinors of quite different electron distributions.     

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.     

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.     

Vacuum-ultraviolet electronic circular dichroism of L-alanine in aqueous solution investigated by time-dependent density functional theory

The electronic circular dichoism (ECD) of L-alanine in the vacuum-ultraviolet region was calculated for various optimized structures using time-dependent density functional theory (TDDFT) to assign the CD spectrum observed experimentally in aqueous solution down to 140 nm [Matsuo, et al. Chem. Lett. 2002, 826]. The structure of L-alanine in vacuo was optimized using density functional theory (DFT) at the B3LYP/6-31G* level. Its hydrated structure was optimized with nine water molecules (six and three around carboxyl and amino groups, respectively) using DFT and a continuum model (Onsager model). The dihedral angles of carboxyl and amino groups in the optimized hydrated structure differed greatly from those in the crystal and in nonhydrated structures optimized using a continuum model only. The ECD spectrum calculated for the hydrated structure had two successive positive peaks with molar ellipticities of about 2000 deg cm2 dmol(-1) at around 205 and 185 nm, which were close to those observed experimentally. These positive peaks were attributable to n pi* transitions of the carboxyl group, with the latter peak also influenced by the pi pi* transition of the carboxyl group that originates below 175 nm. A small negative peak observed at around 252 nm was also predicted from the hydrated structure. These results demonstrate that the hydrated water molecules around the zwitterions play a crucial role in stabilizing the conformation of L-alanine in aqueous solution and that TDDFT is useful for the ab initio assignment of ECD spectra down to the vacuum-ultraviolet region.     

New formulation of the analytical electronic energy and the regional density functional theory

A certain analytical model of E is proposed to satisfy the prerequisite exact density functional theory (DFT ) formulas such as μ = ∂E/∂N = −(I + A)/2 and 2η = ∂²E/∂N² = I −A. © 1996 John Wiley & Sons, Inc.     

Modeling ultrafast solvated electronic dynamics using time-dependent density functional theory and polarizable continuum model

A first-principles solvated electronic dynamics method is introduced. Solvent electronic degrees of freedom are coupled to the time-dependent electronic density of a solute molecule by means of the implicit reaction field method, and the entire electronic system is propagated in time. This real-time time-dependent approach, incorporating the polarizable continuum solvation model, is shown to be very effective in describing the dynamical solvation effect in the charge transfer process and yields a consistent absorption spectrum in comparison to the conventional linear response results in solution.     

Spin-adapted open-shell time-dependent density functional theory. III. An even better and simpler formulation

The recently proposed spin-adapted time-dependent density functional theory (S-TD-DFT) [Z. Li and W. Liu, J. Chem. Phys. 133, 064106 (2010)] resolves the spin-contamination problem in describing singly excited states of high spin open-shell systems. It is an extension of the standard restricted open-shell Kohn-Sham-based TD-DFT which can only access those excited states due to singlet-coupled single excitations. It is also far superior over the unrestricted Kohn-Sham-based TD-DFT (U-TD-DFT) which suffers from severe spin contamination for those excited states due to triplet-coupled single excitations. Nonetheless, the accuracy of S-TD-DFT for high spin open-shell systems is still inferior to TD-DFT for well-behaved closed-shell systems. The reason can be traced back to the violation of the spin degeneracy conditions (SDC) by approximate exchange-correlation (XC) functionals. Noticing that spin-adapted random phase approximation (S-RPA) can indeed maintain the SDC by virtue of the Wigner-Eckart theorem, a hybrid ansatz combining the good of S-TD-DFT and S-RPA can immediately be envisaged. The resulting formalism, dubbed as X-TD-DFT, is free of spin contamination and can also be viewed as a S-RPA correction to the XC kernel of U-TD-DFT. Compared with S-TD-DFT, X-TD-DFT leads to much improved results for the low-lying excited states of, e.g., N(2)(+), yet with much reduced computational cost. Therefore, X-TD-DFT can be recommended for routine calculations of excited states of high spin open-shell systems.     

Theoretical studies on electronic spectroscopy and dynamics with the real-time time-dependent density functional theory

Time-dependent density functional theory (TDDFT) has evolved into a general routine to extract the energies of low-lying excited states over the last decades. Driven by the remarkable progress of laser technology, the study of the interaction between matter and intense laser fields with ultrashort pulse duration develops rapidly. A great number of new strong field phenomena emerge. The requirement of a theoretical tool to study the intense field phenomena and dynamical processes of polyatomic systems is urgent. To extend the power of the TDDFT beyond the linear responses, an alternative scheme has been developed by numerically solving the time-dependent Kohn-Sham equations directly in real-time domain. In this article, we summarize the algorithms and capabilities of the real-time TDDFTon studying electron spectroscopy and dynamics of polyatomic systems. The failure of TDDFT with the adiabatic localdensity approximation on some dynamical processes and the possible solutions are synopsized as well. The numerical implementation of algorithms and applications of RT-TDDFT on the linear and nonlinear spectroscopies and electronic dynamics of nano-size nonmetal clusters are displayed.     

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.     

General formulation of spin-flip time-dependent density functional theory using non-collinear kernels: theory, implementation, and benchmarks

We report an implementation of the spin-flip (SF) variant of time-dependent density functional theory (TD-DFT) within the Tamm-Dancoff approximation and non-collinear (NC) formalism for local, generalized gradient approximation, hybrid, and range-separated functionals. The performance of different functionals is evaluated by extensive benchmark calculations of energy gaps in a variety of diradicals and open-shell atoms. The benchmark set consists of 41 energy gaps. A consistently good performance is observed for the Perdew-Burke-Ernzerhof (PBE) family, in particular PBE0 and PBE50, which yield mean average deviations of 0.126 and 0.090 eV, respectively. In most cases, the performance of original (collinear) SF-TDDFT with 50-50 functional is also satisfactory (as compared to non-collinear variants), except for the same-center diradicals where both collinear and non-collinear SF variants that use LYP or B97 exhibit large errors. The accuracy of NC-SF-TDDFT and collinear SF-TDDFT with 50-50 and BHHLYP is very similar. Using PBE50 within collinear formalism does not improve the accuracy.