Excitation energies from time-dependent density-functional theory beyond the adiabatic approximation

Size-dependent errors in real-time electron density propagation

Real-time (RT) electron density propagation with time-dependent density functional theory (TDDFT) or Hartree-Fock (TDHF) is one of the most popular methods to model the charge transfer in molecules and materials. However, both RT-TDHF and RT-TDDFT within the adiabatic approximation are known to produce inaccurate evolution of the electron density away from the ground state in model systems, leading to large errors in charge transfer and erroneous shifting of peaks in absorption spectra. Given the poor performance of these methods with small model systems and the widespread use of the methods with larger molecular and material systems, here we bridge the gap in our understanding of these methods and examine the size-dependence of errors in RT density propagation. We analyze the performance of RT density propagation for systems of increasing size during the application of a continuous resonant field to induce Rabi-like oscillations, during charge-transfer dynamics, and for peak shifting in simulated absorption spectra. We find that the errors in the electron dynamics are indeed size dependent for these phenomena, with the largest system producing the results most aligned with those expected from linear response theory. The results suggest that although the RT-TDHF and RT-TDDFT methods may produce severe errors for model systems, the errors in charge transfer and resonantly driven electron dynamics may be much less significant for more realistic, large-scale molecules and materials.

Relativistic time-dependent density functional theories

The foundations, formalisms, technicalities, and practicalities of relativistic time-dependent density functional theories (R-TD-DFT) for spinor excited states of molecular systems containing heavy elements are critically reviewed.

Performance of a nonempirical density functional on molecules and hydrogen-bonded complexes

Recently, Tao and Mo derived a meta-generalized gradient approximation functional based on a model exchange-correlation hole. In this work, the performance of this functional is assessed on standard test sets, using the 6-311++G(3df,3pd) basis set. These test sets include 223 G3/99 enthalpies of formation, 99 atomization energies, 76 barrier heights, 58 electron affinities, 8 proton affinities, 96 bond lengths, 82 harmonic vibrational frequencies, 10 hydrogen-bonded molecular complexes, and 22 atomic excitation energies. Our calculations show that the Tao-Mo functional can achieve high accuracy for most properties considered, relative to the local spin-density approximation, Perdew-Burke-Ernzerhof, and Tao-Perdew-Staroverov-Scuseria functionals. In particular, it yields the best accuracy for proton affinities, harmonic vibrational frequencies, hydrogen-bond dissociation energies and bond lengths, and atomic excitation energies.

Performance of a nonempirical exchange functional from density matrix expansion: comparative study with different correlations

Recently, Tao and Mo proposed a meta-generalized gradient approximation for the exchange–correlation energy with remarkable accuracy for molecules, solids, and surfaces.

Gauge-Invariant Formulation of Circular Dichroism

Standard formulations of magnetic response properties, such as circular dichroism spectra, are plagued by gauge dependencies, which can lead to unphysical results. In this work, we present a general gauge-invariant and numerically efficient approach for the calculation of circular dichroism spectra from the current density. First we show that in this formulation the optical rotation tensor, the response function from which circular dichroism spectra can be obtained, is independent of the origin of the coordinate system. We then demonstrate that its trace is independent of the gauge origin of the vector potential. We also show how gauge invariance can be retained in practical calculations with finite basis sets. As an example, we explain how our method can be applied to time-dependent current-density-functional theory. Finally, we report gauge-invariant circular dichroism spectra obtained using the adiabatic local-density approximation. The circular dichroism spectra we thus obtain are in good agreement with experiment.

Towards time-dependent current-density-functional theory in the non-linear regime

Time-Dependent Density-Functional Theory (TDDFT) is a well-established theoretical approach to describe and understand irradiation processes in clusters and molecules. However, within the so-called adiabatic local density approximation (ALDA) to the exchange-correlation (xc) potential, TDDFT can show insufficiencies, particularly in violently dynamical processes. This is because within ALDA the xc potential is instantaneous and is a local functional of the density, which means that this approximation neglects memory effects and long-range effects. A way to go beyond ALDA is to use Time-Dependent Current-Density-Functional Theory (TDCDFT), in which the basic quantity is the current density rather than the density as in TDDFT. This has been shown to offer an adequate account of dissipation in the linear domain when the Vignale-Kohn (VK) functional is used. Here, we go beyond the linear regime and we explore this formulation in the time domain. In this case, the equations become very involved making the computation out of reach; we hence propose an approximation to the VK functional which allows us to calculate the dynamics in real time and at the same time to keep most of the physics described by the VK functional. We apply this formulation to the calculation of the time-dependent dipole moment of Ca, Mg and Na2. Our results show trends similar to what was previously observed in model systems or within linear response. In the non-linear domain, our results show that relaxation times do not decrease with increasing deposited excitation energy, which sets some limitations to the practical use of TDCDFT in such a domain of excitations.

Gauge-invariant calculation of static and dynamical magnetic properties from the current density

In this work we solve two problems related to the calculation of static and dynamical magnetic properties with ab initio theories. First, we show that the dependence of the dynamical magnetic dipole moment on the reference point of the multipole expansion and on the gauge origin of the vector potential have a clear physical significance. They are due to a dynamical electric dipole moment and an electric field, respectively. Both are fully determined by the experimental setup and do not pose any fundamental problem, contrary to what is commonly assumed. Second, in the static case, any dependence on the gauge origin is an artifact of the computational method. We show that the artificial dependence on the gauge origin can be removed in an elegant way by the introduction of a sum rule that puts the diamagnetic and paramagnetic contributions on equal footing. Our approach can be applied to calculate any magnetic observable that can be derived from the current density, and can be used in combination with any ab initio theory from which it can be obtained. To illustrate our method we apply it here to time-dependent current-density-functional theory for the calculation of static and dynamical magnetizabilities of molecules.

Current density partitioning in time-dependent current density functional theory

We adapt time-dependent current density functional theory to allow for a fragment-based solution of the many-electron problem of molecules in the presence of time-dependent electric and magnetic fields. Regarding a molecule as a set of non-interacting subsystems that individually evolve under the influence of an auxiliary external electromagnetic vector-scalar potential pair, the partition 4-potential, we show that there are one-to-one mappings between this auxiliary potential, a sharply-defined set of fragment current densities, and the total current density of the system. The partition electromagnetic (EM) 4-potential is expressed in terms of the real EM 4-potential of the system and a gluing EM 4-potential that accounts for exchange-correlation effects and mutual interaction forces between fragments that are required to yield the correct electron dynamics. We prove the zero-force theorem for the fragmented system, establish a variational formulation in terms of action functionals, and provide a simple illustration for a charged particle in a ring.

Design of effective kernels for spectroscopy and molecular transport: time-dependent current-density-functional theory

Time-dependent current-density-functional theory (TDCDFT) provides an, in principle, exact scheme to calculate efficiently response functions for a very broad range of applications. However, the lack of approximations valid for a range of parameters met in experimental conditions has so far delayed its extensive use in inhomogeneous systems. On the other side, in many-body perturbation theory accurate approximations are available, but at a price of a higher computational cost. In the present work, the possibility of combining the advantages of both approaches is exploited. In this way, an exact equation for the exchange-correlation kernel of TDCDFT is obtained, which opens the way for a systematic improvement of the approximations adopted in practical applications. Finally, an approximate kernel for an efficient calculation of spectra of solids and molecular conductances is suggested and its validity is discussed.

Spatial distribution of electron densities during optical excitation of C60

We calculate the spatial distribution of the oscillating electron density for the low energy optical excitations in C(60), using ab initio time-dependent density functional theory. We contrast this with the unscreened free response of the Kohn-Sham eigenfunctions, in which the charge transfer is between hemispheres and the simple density oscillation patterns are very similar for electrons outside and inside the C(60) radius. In the screened case, the oscillation patterns involve charge transfer on the scale of interatomic distances. The exterior and interior densities play different roles in the oscillator strength and screening. Almost all of the dipole oscillation for optical absorption strength is from exterior electrons, a consequence of the superposition of π and σ electron oscillations. The lowest transition with appreciable strength is dominated by screening from π electrons.

Prediction of Excitation Energies for Conjugated Oligomers and Polymers from Time-Dependent Density Functional Theory

With technological advances, light-emitting conjugated oligomers and polymers have become competitive candidates in the commercial market of light-emitting diodes for display and other technologies, due to the ultralow cost, light weight, and flexibility. Prediction of excitation energies of these systems plays a crucial role in the understanding of their optical properties and device design. In this review article, we discuss the calculation of excitation energies with time-dependent density functional theory, which is one of the most successful methods in the investigation of the dynamical response of molecular systems to external perturbation, owing to its high computational efficiency.

Real-time electron dynamics with exact-exchange time-dependent density-functional theory

The exact exchange potential in time-dependent density-functional theory is defined as an orbital functional through the time-dependent optimized effective potential (TDOEP) method. We numerically solve the TDOEP integral equation for the real-time nonlinear intersubband electron dynamics in a semiconductor quantum well with two occupied subbands. It is found that memory effects become significant in the vicinity of intersubband resonances.

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.

Density functional theory for charge transfer: the nature of the N-bands of porphyrins and chlorophylls revealed through CAM-B3LYP, CASPT2, and SAC-CI calculations

While density functional theory (DFT) has been proven to be extremely useful for the prediction of thermodynamic and spectroscopic properties of molecules, to date most functionals used in common implementations of DFT display a systematic failure to predict the properties of charge-transfer processes. While this is explicitly manifest in Rydberg transitions of atoms and molecules and in molecular charge-transfer spectroscopy, it also becomes critical for systems containing extended conjugation such as polyenes and other conducting polymers, porphyrins, chlorophylls, etc. A new density functional, a Coulomb-attenuated hybrid exchange-correlation functional (CAM-B3LYP), has recently been developed specifically to overcome these limitations, and it has been shown to properly predict molecular charge-transfer spectra. Here, we demonstrate that it predicts qualitatively reasonable spectra for porphyrin, some oligoporphyrins, and chlorophyll. However, alternate density functionals developed to overcome the same limitations such as current-density functional theory are shown, in their present implementation, to remain inadequate. The CAM-B3LYP results are shown to be in excellent agreement with complete-active-space plus second-order Møller-Plesset perturbation theory and symmetry-adapted cluster configuration interaction calculations: These depict the N and higher bands of porphyrins and chlorophylls as being charge-transfer bands associated with localization of molecular orbitals on individual pyrrole rings. The validity of the basic Gouterman model for the spectra of porphyrins and chlorophylls is confirmed, rejecting modern suggestions that non-Gouterman transitions lie close in energy to the Q-bands of chlorophylls. As porphyrins and chlorophylls provide useful paradigms for problems involving extended conjugation, the results obtained suggest that many significant areas of nanotechnology and biotechnology may now be realistically treated by cost-effective density-functional-based computational methods.

Electric field effects on aromatic and aliphatic hydrocarbons: a density-functional study

The influence of a uniform static external electric field on some aliphatic and aromatic molecular species is studied within the density functional theory (DFT) employing the 6-311++G(2d,2p) basis set with B3LYP exchange-correlation prescription. The electric field perturbs the molecular geometry but drastically alters the dipole moments and engenders, to a varying degree, the molecular vibrational Stark effect, i.e., shifts in the infrared (IR) vibrational frequencies accompanied by spectral intensity redistribution. For polar molecules, significant negative ("red") and positive ("blue") frequency shifts are observed for field orientations both parallel and antiparallel to their permanent dipole moments. Further, a selective reordering of frontier orbitals is observed to be brought about by moderately intense fields. In particular, molecules having a lowest unoccupied molecular orbital (LUMO) with predominant pi character possess a threshold field beyond which energy gap between the highest occupied molecular orbital (HOMO) and LUMO diminishes rapidly. A time-dependent (TD) DFT analysis reveals that an increase in the applied field strength by and large increases the excitation energies corresponding to significant electronic transitions among frontier MOs with a concomitant decrease in their oscillator strengths.

Time-dependent density-functional theory beyond the adiabatic approximation: Insights from a two-electron model system

Most applications of time-dependent density-functional theory (TDDFT) use the adiabatic local-density approximation (ALDA) for the dynamical exchange-correlation potential V(xc)(r,t). An exact (i.e., nonadiabatic) extension of the ground-state LDA into the dynamical regime leads to a V(xc)(r,t) with a memory, which causes the electron dynamics to become dissipative. To illustrate and explain this nonadiabatic behavior, this paper studies the dynamics of two interacting electrons on a two-dimensional quantum strip of finite size, comparing TDDFT within and beyond the ALDA with numerical solutions of the two-electron time-dependent Schrodinger equation. It is shown explicitly how dissipation arises through multiple particle-hole excitations, and how the nonadiabatic extension of the ALDA fails for finite systems but becomes correct in the thermodynamic limit.

Time-Dependent Density-Functional Studies of the D2 Coulomb Explosion

Real-time first principle simulations are presented of the D(2) Coulomb explosion dynamics detonated by exposure to very intense few-cycle laser pulse. Three approximate functionals within the time-dependent density functional theory (TDDFT) functionals are examined for describing the electron dynamics, including time-dependent Hartree-Fock theory. Nuclei are treated classically with quantum corrections. The calculated results are sensitive to the underlying electronic structure theory, showing too narrow kinetic energy distribution peaked at too high kinetic energy when compared with recent experimental results (Phys. Rev. Lett. 2003, 91, 093002). Experiment also shows a low energy peak which is not seen in the present calculation. We conclude that while Ehrenfest-adiabatic-TDDFT can qualitatively account for the dynamics, it requires further development, probably beyond the adiabatic approximation, to be quantitative.

Time-dependent Kohn-Sham theory with memory

In time-dependent density-functional theory, exchange and correlation (xc) beyond the adiabatic approximation can be described by viscoelastic stresses in the electron liquid. In the time domain, the resulting velocity-dependent xc vector potential has a memory containing short- and long-range components, leading to decoherence and energy relaxation. We solve the associated time-dependent Kohn-Sham equations, including the dependence on densities and currents at previous times, for the case of charge-density oscillations in a quantum well. We illustrate xc memory effects, clarify the dissipation mechanism, and extract intersubband relaxation rates for weak and strong excitations.

Propagator corrections to adiabatic time-dependent density-functional theory linear response theory

It has long been known that only one-electron excitations are available from adiabatic time-dependent density functional theory (TDDFT). This is particularly clear in Casida's formulation of TDDFT linear response theory. Nevertheless the explicit inclusion of two- and higher-electron excitations is necessary for an adequate description of some excited states, notably the first excited singlet states of butadiene and quartet excited states of molecules with a doublet ground state. The equation-of-motion superoperator approach is used here to derive a Casida-like propagator equation which can be clearly separated into an adiabatic part and a nonadiabatic part. The adiabatic part is identified as corresponding to Casida's equation for adiabatic TDDFT linear response theory. This equivalence is confirmed by deriving a general formula which includes the result that Gonze and Scheffler derived to show the equivalence of TDDFT and Gorling-Levy adiabatic connection perturbation theory for the exchange-only optimized effective potential. The nonadiabatic part explicitly corrects adiabatic TDDFT for two- and higher-electron excitations. The "dressed TDDFT" of Maitra, Zhang, Cave, and Burke is obtained as a special case where the ground state is closed shell. The extension of dressed TDDFT to the case where the ground state is an open-shell doublet is presented, highlighting the importance of correctly accounting for symmetry in this theory. The extension to other ground state spin symmetries is a straightforward consequence of the present work.

Time-dependent four-component relativistic density functional theory for excitation energies

Time-dependent four-component relativistic density functional theory within the linear response regime is developed for calculating excitation energies of heavy element containing systems. Since spin is no longer a good quantum number in this context, we resort to time-reversal adapted Kramers basis when deriving the coupled Dirac-Kohn-Sham equation. The particular implementation of the formalism into the Beijing density functional program package utilizes the multipolar expansion of the induced density to facilitate the construction of the induced Coulomb potential. As the first application, pilot calculations on the valence excitation energies and fine structures of the rare gas (Ne to Rn) and Group 12 (Zn to Hg) atoms are reported. To the best of our knowledge, it is the first time to be able to account for spin-orbit coupling within time-dependent density functional theory for excitation energies.