Relaxation and dephasing in open quantum systems time-dependent density functional theory: Properties of exact functionals from an exactly-solvable model system

The dissipative dynamics of many-electron systems interacting with a thermal environment has remained a long-standing challenge within time-dependent density functional theory (TDDFT). Recently, the formal foundations of open quantum systems time-dependent density functional theory (OQS-TDDFT) within the master equation approach were established. It was proven that the exact time-dependent density of a many-electron open quantum system evolving under a master equation can be reproduced with a closed (unitarily evolving) and non-interacting Kohn-Sham system. This potentially offers a great advantage over previous approaches to OQS-TDDFT, since with suitable functionals one could obtain the dissipative open-systems dynamics by simply propagating a set of Kohn-Sham orbitals as in usual TDDFT. However, the properties and exact conditions of such open-systems functionals are largely unknown. In the present article, we examine a simple and exactly-solvable model open quantum system: one electron in a harmonic well evolving under the Lindblad master equation. We examine two different representitive limits of the Lindblad equation (relaxation and pure dephasing) and are able to deduce a number of properties of the exact OQS-TDDFT functional. Challenges associated with developing approximate functionals for many-electron open quantum systems are also discussed.     

Recent development of self-interaction-free time-dependent density-functional theory for nonperturbative treatment of atomic and molecular multiphoton processes in intense laser fields

In this paper, we present a short account of some recent developments of self-interaction-free density-functional theory (DFT) and time-dependent density-functional theory (TDDFT) for accurate and efficient treatment of the electronic structure, and time-dependent quantum dynamics of many-electron atomic and molecular systems. The conventional DFT calculations using approximate and explicit exchange-correlation energy functional contain spurious self-interaction energy and improper long-range asymptotic potential, preventing reliable treatment of the excited, resonance, and continuum states. We survey some recent developments of DFT/TDDFT with optimized effective potential (OEP) and self-interaction correction (SIC) for both atomic and molecular systems for overcoming some of the above mentioned difficulties. These DFT (TDDFT)/OEP-SIC approaches allow the use of orbital-independent single-particle local potential which is self-interaction free. In addition we discuss several numerical techniques recently developed for efficient and high-precision treatment of the self-interaction-free DFT/TDDFT equations. The usefulness of these procedures is illustrated by a few case studies of atomic, molecular, and condensed matter processes of current interests, including (a) autoionizing resonances, (b) relativistic OEP-SIC treatment of atomic structure (Z=2-106), (c) shell-filling electronic structure in quantum dots, (d) atomic and molecular processes in intense laser fields, including multiphoton ionization, and very-high-order harmonic generation, etc. For the time-dependent processes, an alternative Floquet formulation of TDDFT is introduced for time-independent treatment of multiphoton processes in intense periodic or quasiperiodic fields. We conclude this paper with some open questions and perspectives of TDDFT.     

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.     

Time-dependent density-functional theory for open electronic systems

Time-dependent density-functional theory(TDDFT)has been successfully applied to predict excited-state properties of isolated and periodic systems.However,it cannot address a system coupled to an environment or whose number of electrons is not conserved.To tackle these problems,TDDFT needs to be extended to accommodate open systems.This paper provides a comprehensive account of the recent developments of TDDFT for open systems(TDDFT-OS),including both theoretical and practical aspects.The practicality and accuracy of a latest TDDFT-OS method is demonstrated with two numerical examples:the time-dependent electron transport through a series of quasi-one-dimensional atomic chains,and the real-time electronic dynamics on a two-dimensional graphene surface.The advancement of TDDFT-OS may lead to promising applications in various fields of chemistry,including energy conversion and heterogeneous catalysis.     

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.     

Theoretical study of the electronic spectra of square-planar platinum (II) complexes based on the two-component relativistic time-dependent density-functional theory

In the present work the electronic spectra of [PtCl(4)](2-), [PtBr(4)](2-), and [Pt(CN)(4)](2-) are studied with a recently proposed relativistic time-dependent density-functional theory (TDDFT) based on the two-component zeroth-order regular approximation and a noncollinear exchange-correlation (XC) functional. The contribution to the double group excited states in terms of singlet and triplet single group excited states is estimated through the inner product of the transition density matrix obtained from two-component and scalar relativistic TDDFT calculations to better understand the double group excited states. Spin-orbital coupling effects are found to be very important in order to simulate the electronic spectra of these complexes. The results show that the two-component TDDFT formalism can afford excitation energies with high accuracy for the transition-metal systems studied here when use is made of a proper XC potential.     

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.     

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.     

Time-dependent transition density matrix

The transition density matrix (TDM) is a useful tool for analyzing and interpreting electronic excitation processes in molecular systems. For any transition between two eigenstates of a many-body system, the TDM provides a characteristic spatial map which indicates the distribution of the associated electron–hole pairs and allows one to identify their delocalization and coherence lengths. This is particularly useful for characterizing charge-transfer excitations in large molecular chains or light-harvesting molecules. We here extend these concepts into the real-time domain and define the time-dependent TDM and discuss it in the context of TDDFT. An approximation is proposed in terms of the Kohn–Sham Slater determinants. This provides a new tool for the real-time visualization of electronic excitation processes such as exciton formation, diffusion, recombination, or charge separation. We illustrate the time-dependent TDM for simple one-dimensional lattice systems with two spinless electrons which are either noninteracting of fully interacting.     

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.     

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.     

A unified approach to the density-potential mapping in a family of time-dependent density functional theories

It is shown that the density-potential mapping and the V-representability problems in the time-dependent current density functional theory (TDCDFT) are reduced to the solution of a certain many-body nonlinear Schrödinger equation (NLSE). The derived NLSE for TDCDFT links the earlier NLSE-based formulations of the time-dependent deformation functional theory (TDDefFT) and the time-dependent density functional theory (TDDFT). We establish a close relation between the nonlinear many-body problems which control the existence of TDCDFT, TDDFT, and TDDefFT, and thus develop a unified point of view on the whole family of the TDDFT-type theories.     

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.     

Electronic transitions involved in the absorption spectrum and dual luminescence of tetranuclear cubane [Cu4I4(pyridine)4] cluster: a density functional theory/time-dependent density functional theory investigation

We present a combined density functional theory (DFT)/time-dependent density functional theory (TDDFT) study of the geometry, electronic structure, and absorption and emission properties of the tetranuclear "cubane" Cu4I4py4 (py = pyridine) system. The geometry of the singlet ground state and of the two lowest triplet states of the title complex were optimized, followed by TDDFT excited-state calculations. This procedure allowed us to characterize the nature of the excited states involved in the absorption spectrum and those responsible for the dual emission bands observed for this complex. In agreement with earlier experimental proposals, we find that while in absorption the halide-to-pyridine charge-transfer excited state (XLCT*) has a lower energy than the cluster-centered excited state (CC*), a strong geometrical relaxation on the triplet cluster-centered state surface leads to a reverse order of the excited states in emission.     

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.     

Regarding the validity of the time-dependent Kohn-Sham approach for electron-nuclear dynamics via trajectory surface hopping

The implementation of fewest-switches surface-hopping (FSSH) within time-dependent Kohn-Sham (TDKS) theory [Phys. Rev. Lett. 95, 163001 (2005)] has allowed us to study successfully excited state dynamics involving many electronic states in a variety of molecular and nanoscale systems, including chromophore-semiconductor interfaces, semiconductor and metallic quantum dots, carbon nanotubes and graphene nanoribbons, etc. At the same time, a concern has been raised that the KS orbital basis used in the calculation provides only approximate potential energy surfaces [J. Chem. Phys. 125, 014110 (2006)]. While this approximation does exist in our method, we show here that FSSH-TDKS is a viable option for computationally efficient calculations in large systems with straightforward excited state dynamics. We demonstrate that the potential energy surfaces and nonadiabatic transition probabilities obtained within the TDKS and linear response (LR) time-dependent density functional theories (TDDFT) agree semiquantitatively for three different systems, including an organic chromophore ligating a transition metal, a quantum dot, and a small molecule. Further, in the latter case the FSSH-TDKS procedure generates results that are in line with FSSH implemented within LR-TDDFT. The FSSH-TDKS approach is successful for several reasons. First, single-particle KS excitations often give a good representation of LR excitations. In this regard, DFT compares favorably with the Hartree-Fock theory, for which LR excitations are typically combinations of multiple single-particle excitations. Second, the majority of the FSSH-TDKS applications have been performed with large systems involving simple excitations types. Excitation of a single electron in such systems creates a relatively small perturbation to the total electron density summed over all electrons, and it has a small effect on the nuclear dynamics compared, for instance, with thermal nuclear fluctuations. In such cases an additional, classical-path approximation can be made. Third, typical observables measured in time-resolved experiments involve averaging over many initial conditions. Such averaging tends to cancel out random errors that may be encountered in individual simulated trajectories. Finally, if the flow of energy between electronic and nuclear subsystems is insignificant, the ad hoc FSSH procedure is not required, and a straightforward mean-field, Ehrenfest approach is sufficient. Then, the KS representation provides rigorously a convenient and efficient basis for numerically solving the TDDFT equations of motion.     

Performance of the M11 and M11-L density functionals for calculations of electronic excitation energies by adiabatic time-dependent density functional theory

Adiabatic time-dependent density functional theory is a powerful method for calculating electronic excitation energies of complex systems, but the quality of the results depends on the choice of approximate density functional. In this article we test two promising new density functionals, M11 and M11-L, against databases of 214 diverse electronic excitation energies, and we compare the results to those for 16 other density functionals of various kinds and to time-dependent Hartree-Fock. Charge transfer excitations are well known to be the hardest challenge for TDDFT. M11 is a long-range-corrected hybrid meta-GGA, and it shows better performance for charge transfer excitations than any of the other functionals except M06-HF, which is a specialized functional that does not do well for valence excitations. Several other long-range-corrected hybrid functionals also do well, and we especially recommend M11, ωB97X, and M06-2X for general spectroscopic applications because they do exceptionally well on ground-state properties as well as excitation energies. Local functionals are preferred for many applications to extended systems because of their significant cost advantage for large systems. M11-L is a dual-range local functional and-unlike all previous local functionals-it has good performance for Rydberg states as well as for valence states. Thus it is highly recommended for excitation energy calculations on extended systems.     

Generator coordinate method in time-dependent density-functional theory: memory made simple

The generator coordinate (GC) method is a variational approach to the quantum many-body problem in which interacting many-body wave functions are constructed as superpositions of (generally nonorthogonal) eigenstates of auxiliary Hamiltonians containing a deformation parameter. This paper presents a time-dependent extension of the GC method as a new approach to improve existing approximations of the exchange-correlation (XC) potential in time-dependent density-functional theory (TDDFT). The time-dependent GC method is shown to be a conceptually and computationally simple tool to build memory effects into any existing adiabatic XC potential. As an illustration, the method is applied to driven parametric oscillations of two interacting electrons in a harmonic potential (Hooke's atom). It is demonstrated that a proper choice of time-dependent generator coordinates in conjunction with the adiabatic local-density approximation reproduces the exact linear and nonlinear two-electron dynamics quite accurately, including features associated with double excitations that cannot be captured by TDDFT in the adiabatic approximation.     

Electronic spectroscopy of UO2(2+), NUO(+) and NUN: an evaluation of time-dependent density functional theory for actinides

The performance of the time-dependent density functional theory (TDDFT) approach has been evaluated for the electronic spectrum of the UO(2)(2+), NUO(+) and NUN molecules. Different exchange-correlation functionals (LDA, PBE, BLYP, B3LYP, PBE0, M06, M06-L, M06-2X, CAM-B3LYP) and the SAOP model potential have been investigated, as has the relative importance of the adiabatic local density approximation (ALDA) to the exchange-correlation kernel. The vertical excitation energies have been compared with reference data obtained using accurate wave-function theory (WFT) methods.