Spin Currents via the Gauge Principle for Meta-Generalized Gradient Exchange-Correlation Functionals.

The prominence of density functional theory in the field of electronic structure computation stems from its ability to usefully balance accuracy and computational effort. At the base of this ability is a functional of the electron density: the exchange-correlation energy. This functional satisfies known exact conditions that guide the derivation of approximations. The strongly constrained and appropriately normed (SCAN) approximation stands out as a successful, modern, example. In this Letter, we demonstrate how the SU(2) gauge invariance of the exchange-correlation functional in spin current density functional theory allows us to add an explicit dependence on spin currents in the SCAN functional (here called JSCAN)-and similar meta-generalized-gradient functional approximations-solely invoking first principles. In passing, a spin-current dependent generalization of the electron localization function (here called JELF) is also derived. The extended forms are implemented in a developer's version of the crystal23 program. Applications on molecules and materials confirm the practical relevance of the extensions.

Electron Localization Function for Noncollinear Spins.

Understanding of bonding is key to modeling materials and predicting properties thereof. A widely adopted indicator of bonds and atomic shells is the electron localization function (ELF). The building blocks of the ELF are also used in the construction of modern density functional approximations. Here, we demonstrate that the ELF breaks down when applied beyond regular nonrelativistic quantum states. We show that for tackling general noncollinear open-shell solutions, it is essential to address both the U(1) gauge invariance, i.e., invariance under a multiplication by a position dependent phase factor, and SU(2) gauge invariance, i.e., invariance under local spin rotations, conjointly. Remarkably, we find that the extended ELF also improves the description of paradigmatic collinear states.

Electronic Excited States in Extreme Limits via Ensemble Density Functionals.

Density functional theory (DFT) has greatly expanded our ability to affordably compute and understand electronic ground states, by replacing intractable ab initio calculations by models based on paradigmatic physics from high- and low-density limits. But, a comparable treatment of excited states lags behind. Here, we solve this outstanding problem by employing a generalization of density functional theory to ensemble states (EDFT). We thus address important paradigmatic cases of all electronic systems in strongly (low-density) and weakly (high-density) correlated regimes. We show that the high-density limit connects to recent, exactly solvable EDFT results. The low-density limit reveals an unnoticed and most unexpected result-density functionals for strictly correlated ground states can be reused directly for excited states. Nontrivial dependence on excitation structure only shows up at third leading order. Overall, our results provide foundations for effective models of excited states that interpolate between exact low- and high-density limits, which we illustrate on the cases of singlet-singlet excitations in H_{2} and a ring of quantum wells.

Excitations of Quantum Many-Body Systems via Purified Ensembles: A Unitary-Coupled-Cluster-Based Approach.

State-average calculations based on a mixture of states are increasingly being exploited across chemistry and physics as versatile procedures for addressing excitations of quantum many-body systems. If not too many states should need to be addressed, calculations performed on individual states are also a common option. Here we show how the two approaches can be merged into one method, dealing with a generalized yet single pure state. Implications in electronic structure calculations are discussed and for quantum computations are pointed out.

Ensemble Reduced Density Matrix Functional Theory for Excited States and Hierarchical Generalization of Pauli's Exclusion Principle.

We propose and work out a reduced density matrix functional theory (RDMFT) for calculating energies of eigenstates of interacting many-electron systems beyond the ground state. Various obstacles which historically have doomed such an approach to be unfeasible are overcome. First, we resort to a generalization of the Ritz variational principle to ensemble states with fixed weights. This in combination with the constrained search formalism allows us to establish a universal functional of the one-particle reduced density matrix. Second, we employ tools from convex analysis to circumvent the too involved N-representability constraints. Remarkably, this identifies Valone's pioneering work on RDMFT as a special case of convex relaxation and reveals that crucial information about the excitation structure is contained in the functional's domain. Third, to determine the crucial latter object, a methodology is developed which eventually leads to a generalized exclusion principle. The corresponding linear constraints are calculated for systems of arbitrary size.

Nonlinear light absorption in many-electron systems excited by an instantaneous electric field: a non-perturbative approach.

Applications of low-cost non-perturbative approaches in real time, such as time-dependent density functional theory, for the study of nonlinear optical properties of large and complex systems are gaining increasing popularity. However, their assessment still requires the analysis and understanding of elementary dynamical processes in simple model systems. Motivated by the aim of simulating optical nonlinearities in molecules, here exemplified by the case of the quaterthiophene oligomer, we investigate light absorption in many-electron interacting systems beyond the linear regime by using a single broadband impulse of an electric field; i.e. an electrical impulse in the instantaneous limit. We determine non-pertubatively the absorption cross section from the Fourier transform of the time-dependent induced dipole moment, which can be obtained from the time evolution of the wavefunction. We discuss the dependence of the resulting cross section on the magnitude of the impulse and we highlight the advantages of this method in comparison with perturbation theory by working on a one-dimensional model system for which numerically exact solutions are accessible. Thus, we demonstrate that the considered non-pertubative approach provides us with an effective tool for investigating fluence-dependent nonlinear optical excitations.

Ensemble Density Functional Theory: Insight from the Fluctuation-Dissipation Theorem.

Density functional theory can be generalized to mixtures of ground and excited states, for the purpose of determining energies of excitations using low-cost density functional approximations. Adapting approximations originally developed for ground states to work in the new setting would fast-forward progress enormously. But, previous attempts have stumbled on daunting fundamental issues. Here we show that these issues can be prevented from the outset, by using a fluctuation dissipation theorem (FDT) to dictate key functionals. We thereby show that existing exchange energy approximations are readily adapted to excited states, when combined with a rigorous exact Hartree term that is different in form from its ground state counterpart, and counterparts based on ensemble Ansatzë. Applying the FDT to correlation energies also provides insights into ground statelike and ensemble-only correlations. We thus provide a comprehensive and versatile framework for ensemble density functional approximations.

Understanding real-time time-dependent density-functional theory simulations of ultrafast laser-induced dynamics in organic molecules.

Real-time time-dependent density functional theory, in conjunction with the Ehrenfest molecular dynamics scheme, is becoming a popular methodology to investigate ultrafast phenomena on the nanoscale. Thanks to recent developments, it is also possible to explicitly include in the simulations a time-dependent laser pulse, thereby accessing the transient excitation regime. However, the complexity entailed in these calculations calls for in-depth analysis of the accessible and yet approximate (either "dressed" or "bare") quantities in order to evaluate their ability to provide us with a realistic picture of the simulated processes. In this work, we analyze the ultrafast dynamics of three small molecules (ethylene, benzene, and thiophene) excited by a resonant laser pulse in the framework of the adiabatic local-density approximation. The electronic response to the laser perturbation in terms of induced dipole moment and excited-state population is compared to the results given by an exactly solvable two-level model. In this way, we can interpret the charge-carrier dynamics in terms of simple estimators, such as the number of excited electrons. From the computed transient absorption spectra, we unravel the appearance of nonlinear effects such as excited-state absorption and vibronic coupling. In this way, we observe that the laser excitation affects the vibrational spectrum by enhancing the anharmonicities therein, while the coherent vibrational motion contributes to stabilizing the electronic excitation already within a few tens of femtoseconds.

Density-Driven Correlations in Many-Electron Ensembles: Theory and Application for Excited States.

Density functional theory can be extended to excited states by means of a unified variational approach for passive state ensembles. This extension overcomes the restriction of the typical density functional approach to ground states, and offers useful formal and demonstrated practical benefits. The correlation energy functional in the generalized case acquires higher complexity than its ground state counterpart, however. Little is known about its internal structure nor how to effectively approximate it in general. Here we show that such a functional can be broken down into natural components, including what we call "state-" and "density-driven" correlations, with the former amenable to conventional approximations, and the latter being a unique feature of ensembles. Such a decomposition provides us with a pathway to general approximations that are able to routinely handle low-lying excited states. The importance of density-driven correlations is demonstrated, and an approximation for them is introduced and shown to be useful.

Asymptotic behavior of the Hartree-exchange and correlation potentials in ensemble density functional theory.

We report on previously unnoticed features of the exact Hartree-exchange and correlation potentials for atoms and ions treated via ensemble density functional theory, demonstrated on fractional ions of Li, C, and F. We show that these potentials, when treated separately, can reach non-vanishing asymptotic constant values in the outer region of spherical, spin unpolarized atoms. In the next leading order, the potentials resemble Coulomb potentials created by effective charges which have the peculiarity of not behaving as piecewise constants as a function of the electron number. We provide analytical derivations and complement them with numerical results using the inversion of the Kohn-Sham equations for interacting densities obtained by accurate quantum Monte Carlo calculations. The present results expand on the knowledge of crucial exact properties of Kohn-Sham systems, which can guide development of advanced exchange-correlation approximations.

Charge transfer excitations from exact and approximate ensemble Kohn-Sham theory.

By studying the lowest excitations of an exactly solvable one-dimensional soft-Coulomb molecular model, we show that components of Kohn-Sham ensembles can be used to describe charge transfer processes. Furthermore, we compute the approximate excitation energies obtained by using the exact ensemble densities in the recently formulated ensemble Hartree-exchange theory [T. Gould and S. Pittalis, Phys. Rev. Lett. 119, 243001 (2017)]. Remarkably, our results show that triplet excitations are accurately reproduced across a dissociation curve in all cases tested, even in systems where ground state energies are poor due to strong static correlations. Singlet excitations exhibit larger deviations from exact results but are still reproduced semi-quantitatively.

Hartree and Exchange in Ensemble Density Functional Theory: Avoiding the Nonuniqueness Disaster.

Ensemble density functional theory is a promising method for the efficient and accurate calculation of excitations of quantum systems, at least if useful functionals can be developed to broaden its domain of practical applicability. Here, we introduce a guaranteed single-valued "Hartree-exchange" ensemble density functional, E_{Hx}[n], in terms of the right derivative of the universal ensemble density functional with respect to the coupling constant at vanishing interaction. We show that E_{Hx}[n] is straightforwardly expressible using block eigenvalues of a simple matrix [Eq. (14)]. Specialized expressions for E_{Hx}[n] from the literature, including those involving superpositions of Slater determinants, can now be regarded as originating from the unifying picture presented here. We thus establish a clear and practical description for Hartree and exchange in ensemble systems.

Locality of correlation in density functional theory.

The Hohenberg-Kohn density functional was long ago shown to reduce to the Thomas-Fermi (TF) approximation in the non-relativistic semiclassical (or large-Z) limit for all matter, i.e., the kinetic energy becomes local. Exchange also becomes local in this limit. Numerical data on the correlation energy of atoms support the conjecture that this is also true for correlation, but much less relevant to atoms. We illustrate how expansions around a large particle number are equivalent to local density approximations and their strong relevance to density functional approximations. Analyzing highly accurate atomic correlation energies, we show that EC → -AC ZlnZ + BCZ as Z → ∞, where Z is the atomic number, AC is known, and we estimate BC to be about 37 mhartree. The local density approximation yields AC exactly, but a very incorrect value for BC, showing that the local approximation is less relevant for the correlation alone. This limit is a benchmark for the non-empirical construction of density functional approximations. We conjecture that, beyond atoms, the leading correction to the local density approximation in the large-Z limit generally takes this form, but with BC a functional of the TF density for the system. The implications for the construction of approximate density functionals are discussed.

Modeling solvation effects in real-space and real-time within density functional approaches.

The Polarizable Continuum Model (PCM) can be used in conjunction with Density Functional Theory (DFT) and its time-dependent extension (TDDFT) to simulate the electronic and optical properties of molecules and nanoparticles immersed in a dielectric environment, typically liquid solvents. In this contribution, we develop a methodology to account for solvation effects in real-space (and real-time) (TD)DFT calculations. The boundary elements method is used to calculate the solvent reaction potential in terms of the apparent charges that spread over the van der Waals solute surface. In a real-space representation, this potential may exhibit a Coulomb singularity at grid points that are close to the cavity surface. We propose a simple approach to regularize such singularity by using a set of spherical Gaussian functions to distribute the apparent charges. We have implemented the proposed method in the Octopus code and present results for the solvation free energies and solvatochromic shifts for a representative set of organic molecules in water.

Unveiling the Role of Spin Currents on the Giant Rashba Splitting in Single-Layer WSe2.

The Rashba spin splitting in uniaxial, inversion-asymmetric materials has attracted considerable interest for spintronic applications. The most widely used theoretical framework to model such states is Kohn-Sham density functional theory (DFT) in combination with standard (semi)local exchange-correlation density functional approximations (DFAs). However, in the presence of spin-orbit coupling, DFT misses contributions due to modification of the many-body interaction by spin currents J⃗. Inclusion of the latter effects requires a spin current DFT (SCDFT) formulation, which is seldom considered. We investigate the giant Rashba splitting in single-layer WSe2, and we quantify the effect of including spin currents in DFAs of the SCDFT. Crucially, we show that SCDFT allows fully capturing the giant Rashba band splitting in single-layer WSe2, otherwise previously systematically underestimated by standard (semi)local DFAs within the DFT framework. We find the inclusion of J⃗ on the DFA increases the Rashba splitting by about 20%.

Role of crystal orientation in attosecond photoinjection dynamics of germanium.

Understanding photoinjection in semiconductors-a fundamental physical process-represents the first step toward devising new opto-electronic devices, capable of operating on unprecedented time scales. Fostered by the development of few-femtosecond, intense infrared pulses, and attosecond spectroscopy techniques, ultrafast charge injection in solids has been the subject of intense theoretical and experimental investigation. Recent results have shown that while under certain conditions photoinjection can be ascribed to a single, well-defined phenomenon, in a realistic multi-band semiconductor like Ge, several competing mechanisms determine the sub-cycle interaction of an intense light field with the atomic and electronic structure of matter. In this latter case, it is yet unclear how the complex balance between the different physical mechanisms is altered by the chosen interaction geometry, dictated by the relative orientation between the crystal lattice and the laser electric field direction. In this work, we investigate ultrafast photoinjection in a Ge monocrystalline sample with attosecond temporal resolution under two distinct orientations. Our combined theoretical and experimental effort suggests that the physical mechanisms determining carrier excitation in Ge are largely robust against crystal rotation. Nevertheless, the different alignment between the laser field and the crystal unit cell causes non-negligible changes in the momentum distribution of the excited carriers and their injection yield. Further experiments are needed to clarify whether the crystal orientation can be used to tune the photoinjection of carriers in a semiconductor at these extreme time scales.

Intermolecular conical intersections in molecular aggregates.

Conical intersections (CoIns) of multidimensional potential energy surfaces are ubiquitous in nature and control pathways and yields of many photo-initiated intramolecular processes. Such topologies can be potentially involved in the energy transport in aggregated molecules or polymers but are yet to be uncovered. Here, using ultrafast two-dimensional electronic spectroscopy (2DES), we reveal the existence of intermolecular CoIns in molecular aggregates relevant for photovoltaics. Ultrafast, sub-10-fs 2DES tracks the coherent motion of a vibrational wave packet on an optically bright state and its abrupt transition into a dark state via a CoIn after only 40 fs. Non-adiabatic dynamics simulations identify an intermolecular CoIn as the source of these unusual dynamics. Our results indicate that intermolecular CoIns may effectively steer energy pathways in functional nanostructures for optoelectronics.