Double and Charge-Transfer Excitations in Time-Dependent Density Functional Theory

Extended N-centered ensemble density functional theory of double electronic excitations

A recent work (arXiv:2401.04685) has merged N-centered ensembles of neutral and charged electronic ground states with ensembles of neutral ground and excited states, thus providing a general and in-principle exact (so-called extended N-centered) ensemble density functional theory of neutral and charged electronic excitations. This formalism made it possible to revisit the concept of density-functional derivative discontinuity, in the particular case of single excitations from the highest occupied Kohn-Sham (KS) molecular orbital, without invoking the usual "asymptotic behavior of the density" argument. In this work, we address a broader class of excitations, with a particular focus on double excitations. An exact implementation of the theory is presented for the two-electron Hubbard dimer model. A thorough comparison of the true physical ground- and excited-state electronic structures with that of the fictitious ensemble density-functional KS system is also presented. Depending on the choice of the density-functional ensemble as well as the asymmetry of the dimer and the correlation strength, an inversion of states can be observed. In some other cases, the strong mixture of KS states within the true physical system makes the assignment "single excitation" or "double excitation" irrelevant.

ΔDFT Predicts Inverted Singlet-Triplet Gaps with Chemical Accuracy at a Fraction of the Cost of Wave Function-Based Approaches

Efficient OLEDs need to quickly convert singlet and triplet excitons into photons. Molecules with an inverted singlet-triplet energy gap (INVEST) are promising candidates for this task. However, typical INVEST molecules have drawbacks like too low oscillator strengths and excitation energies. High-throughput screening could identify suitable INVEST molecules, but existing methods are problematic: The workhorse method TD-DFT cannot reproduce gap inversion, while wave function-based methods are too slow. This study proposes a state-specific method based on unrestricted Kohn-Sham DFT with common hybrid functionals. Tuned on the new INVEST15 benchmark set, this method achieves an error of less than 1 kcal/mol, which is traced back to error cancellation between spin contamination and dynamic correlation. Applied to the larger and structurally diverse NAH159 set in a black-box fashion, the method maintains a small error (1.2 kcal/mol) and accurately predicts gap signs in 83% of cases, confirming its robustness and suitability for screening workflows.

Density functional theory from spherically symmetric densities: Ground and excited states of Coulomb systems

Recently, Theophilou [J. Chem. Phys. 149, 074104 (2018)] proposed a peculiar version of the density functional theory by showing that the set of spherical averages of the density around the nuclei determines uniquely the external potential in atoms, molecules, and solids. Here, this novel theory is extended to individual excited states. The generalization is based on the method developed in the series of papers by Ayers, Levy, and Nagy [Phys. Rev. A 85, 042518 (2012)]. Generalized Hohenberg-Kohn theorems are proved to the set of spherically symmetric densities using constrained search. A universal variational functional for the sum of the kinetic and electron-electron repulsion energies is constructed. The functional is appropriate for the ground state and all bound excited states. Euler equations and Kohn-Sham equations for the set are derived. The Euler equations can be rewritten as Schrödinger-like equations for the square root of the radial densities, and the effective potentials in them can be expressed in terms of wave function expectation values. The Hartree plus exchange-correlation potentials can be given by the difference of the interacting and the non-interacting effective potentials.

On the Topological Phase around Conical Intersections with Tamm-Dancoff Linear-Response Time-Dependent Density Functional Theory

Regions of nuclear-configuration space away from the Franck-Condon geometry can prove problematic for some electronic structure methods, given the propensity of such regions to possess conical intersections, i.e., (highly connected) points of degeneracy between potential energy surfaces. With the likelihood (perhaps even inevitability) for nonadiabatic dynamics simulations to explore molecular geometries in close proximity to conical intersections, it is vital that the performance of electronic structure methods is routinely examined in this context. In a recent paper [Taylor, J. T. J. Chem. Phys. 2023, 159, 214115.], the ability of linear-response time-dependent density functional theory within the adiabatic approximation (AA LR-TDDFT) to provide a proper description of conical intersections, in terms of their topology and topography, was investigated, with particular attention paid to conical intersections between two excited electronic states. For the same prototypical molecules, protonated formaldimine and pyrazine, we herein consider whether AA LR-TDDFT can correctly reproduce the topological phase accumulated by the adiabatic electronic wave function upon traversing a closed path around an excited-to-excited state conical intersection despite not using the appropriate quadratic-response nonadiabatic coupling vectors. Equally, we probe the ability of the ground-to-excited state intersection ring exhibited by AA LR-TDDFT in protonated formaldimine to give rise to a similar topological phase in spite of its incorrect dimensionality.

Excited-State Charge Transfer Coupling from Quasiparticle Energy Density Functional Theory

The recently developed Quasiparticle Energy (QE) scheme, based on a DFT calculation with one more (or less) electron, offers a good description of excitation energies, even with charge transfer characters. In this work, QE is further extended to calculate electron transfer (ET) couplings involving two excited states. We tested it with a donor-acceptor complex, consisting of a furan and a 1,1-dicyanoethylene (DCNE), in which two low lying charge transfer and local excitation states are involved. With generalized Mülliken-Hush and fragment charge-difference schemes, couplings from the QE approach generally agree well with those obtained from TDDFT, except that QE couplings exhibit better exponential distance dependence. Couplings from half-energy gaps with an external field are also calculated and reported. Our results show that the QE scheme is robust in calculating ET couplings with greatly reduced computational time.

WFOT: A Wave Function Overlap Tool between Single- and Multi-Reference Electronic Structure Methods for Spectroscopy Simulation

We report the development of a novel diagnostic tool, named wave function overlap tool (WFOT), designed to evaluate the overlap between wave functions computed at single-reference [i.e., time-dependent density functional theory or configuration interaction singles (CIS)] and multireference (i.e., CASSCF/CASPT2) electronic structure levels of theory. It relies on truncating the single- and multireference WFs to CIS-like expansions spanning the same configurational space and maximizing the molecular orbital overlap by means of a unitary transformation. To demonstrate the functionality of the tool, we calculate the transient spectrum of acetylacetone by evaluating excited state absorption signals with multireference quality on top of single-reference on-the-fly dynamics simulations. Semiautomatic spectra generation is facilitated by interfacing the tool with the COBRAMM package, which also allows one to use WFOT with several quantum chemistry codes such as Gaussian, NWChem, and OpenMolcas. Other exciting possibilities for the utilization of the code beyond the simulation of transient absorption spectroscopy are eventually discussed.

Theoretical insights into dopamine photochemistry adsorbed on graphene-type nanostructures

The equilibrium geometry structures and light absorption properties of the dopamine (DA) and dopamine-o-quinone (DAQ) adsorbed on the graphene surface have been investigated using the ground state and linear-response time-dependent density functional theories. Two types of graphene systems were considered, a rectangular form of hexagonal lattice with optimized C-C bond length as the model system for graphene nanoparticles (GrNP) and a similar system but with fixed C-C bond length (1.42 Å) as the model system for graphene 2D sheet (GrS). The analysis of the vertical excitations showed that three types of electronic transitions are possible, namely, localized on graphene, localized on the DA or DAQ, and charge transfer (CT). In the case of the graphene-DA complex, the charge transfer excitations were characterized by the molecule-to-surface (MSCT) character, whereas the graphene-DAQ was characterized by the reverse, i.e. surface-to-molecule (SMCT). The difference between the two cases is given by the presence of an energetically low-lying unoccupied orbital (LUMO+1) that allows charge transfer from the surface to the molecule in the case of DAQ. However, it was also shown that the fingerprints of excited electronic states associated with the adsorbed molecules cannot be seen in the spectrum, as they are mostly suppressed by the characteristic spectral shape of graphene.

Spectroscopy of End-On Copper(II) Superoxido Complexes: A Wave Function-Based Analysis

Wave function methods are employed to analyze the ground and low-lying excited states of bipyramid trigonal copper(II) superoxido complexes, up to their characteristic ligand to metal charge transfer band. Several multireference methods have been combined to provide new insights into the interpretation of their experimental absorption spectra. We show that the intraligand transition on the dioxygen leads to a dark state. Among the results, we shall highlight the finding of doubly excited states in the region of the d-d transitions and the subtle interplay between Cu(I) and Cu(II) in the ground and excited states. Some of these findings could be obtained only with multireference methods.

Tunable Tesla-Scale Magnetic Attosecond Pulses through Ring-Current Gating

Coherent control over electron dynamics in atoms and molecules using high-intensity circularly polarized laser pulses gives rise to current loops, resulting in the emission of magnetic fields. We propose, and demonstrate with ab initio calculations, "current-gating" schemes to generate direct or alternating-current magnetic pulses in the infrared spectral region, with highly tunable waveform and frequency, and showing femtosecond-to-attosecond pulse duration. In optimal conditions, the magnetic pulse can be highly isolated from the driving laser and exhibits a high flux density (∼1 T at a few hundred nanometers from the source, with a pulse duration of 787 attoseconds) for application in forefront experiments of ultrafast spectroscopy. Our work paves the way toward the generation of attosecond magnetic fields to probe ultrafast magnetization, chiral responses, and spin dynamics.

Oscillator strengths and excited-state couplings for double excitations in time-dependent density functional theory

Although useful to extract excitation energies of states of double-excitation character in time-dependent density functional theory that are missing in the adiabatic approximation, the frequency-dependent kernel derived earlier [Maitra et al., J. Chem. Phys. 120, 5932 (2004)] was not designed to yield oscillator strengths. These are required to fully determine linear absorption spectra, and they also impact excited-to-excited-state couplings that appear in dynamics simulations and other quadratic response properties. Here, we derive a modified non-adiabatic kernel that yields both accurate excitation energies and oscillator strengths for these states. We demonstrate its performance on a model two-electron system, the Be atom, and on excited-state transition dipoles in the LiH molecule at stretched bond-lengths, in all cases producing significant improvements over the traditional approximations.

Inverted Lowest Singlet and Triplet Excitation Energy Ordering of Graphitic Carbon Nitride Flakes

In organic light-emitting diodes (OLEDs), only 25% of electrically generated excitons are in a singlet state, S1, and the remaining 75% are in a triplet state, T1. In thermally activated delayed fluorescence (TADF) chromophores the transition from the nonradiative T1 state to the radiative S1 state can be thermally activated, which improves the efficiency of OLEDs. Chromophores with inverted energy ordering of S1 and T1 states, S1 < T1, are superior to TADF chromophores, thanks to the absence of an energy barrier for the transition from T1 to S1. We benchmark the performance of time-dependent density functional theory using different exchange-correlation functionals and find that scaled long-range corrected double-hybrid functionals correctly predict the inverted singlet-triplet gaps of N-substituted phenalene derivatives. We then show that the inverted energy ordering of S1 and T1 is an intrinsic property of graphitic carbon nitride flakes. A design strategy of new chromophores with inverted singlet-triplet gaps is proposed. The color of emitted light can be fine-tuned through flake size and amine substitution on flake vertices.

On the Prediction of Spectroscopic Fingerprints of Co2+ Complexes Relevant for the ZIF Nucleation Process

The nucleation process of zeolitic imidazolate frameworks (ZIFs) is to date not completely understood. Recently, it has been found that, during the formation of Co-ZIF-67, after mixing imidazole-type ligands with octahedral precursors containing oxygen-coordinated ligands, a metal-organic pool with a diversity of transition metal complexes (TMCs) is formed showing fingerprints of octahedral and tetrahedral Co2+ complexes with both types of ligands [Filez, M. Cell Rep. Phys. Sci. 2021, 2, 100680]. In order to further unravel this process, we aim to characterize the d-d transitions of the TMCs and focus on their number, intensity, and position, which change during the process and can thus serve as a fingerprint for the formed species. It was previously shown that the number of ligands and symmetry has a detrimental influence on the ground state properties of Co2+ TMCs. Herein, we investigate how far excited state properties of TMCs relevant during nanoporous formation processes can be predicted by time-dependent density functional theory (TDDFT) and ligand field density functional theory (LFDFT). As TMCs are known to be challenging systems with possibly degenerate ground states and double excitations, we first investigate the performance of both techniques on first-row octahedral aqua-complexes. With this knowledge, we then focus on tetrahedral Co2+ complexes with aqua and imidazole-type ligands in order to investigate in how far we can propose a spectroscopic fingerprint that allows us to follow the Co2+ complexes during the formation of Co-ZIF-67. The results of TDDFT and LFDFT are qualitatively in agreement and provide complementary information. We found that various features can be used to distinguish between the species. However, as LFDFT is not suited for TMCs possessing the extended imidazole-type ligands and double and spin-flip states are not included in TDDFT, both techniques need to be complemented with more advanced methods to obtain complete insight into the d-d excitations of TMCs with imidazole ligands. Therefore, we particularly explored ab initio ligand field theory, which is capable of describing double excitations and is, in contrast to LFDFT, suitable for TMCs with extended ligands.

Generalization of the Tensor Product Selected CI Method for Molecular Excited States

In a recent paper [JCTC, 2020, 16, 6098], we introduced a new approach for accurately approximating full CI ground states in large electronic active-spaces called Tensor Product Selected CI (TPSCI). In TPSCI, a large orbital active space is first partitioned into disjoint sets (clusters) for which the exact, local many-body eigenstates are obtained. Tensor products of these locally correlated many-body states are taken as the basis for the full, global Hilbert space. By folding correlation into the basis states themselves, the low-energy eigenstates become increasingly sparse, creating a more compact selected CI expansion. While we demonstrated that this approach can improve accuracy for a variety of systems, there is even greater potential for applications to excited states, particularly those which have some excited-state character. In this paper, we report on the accuracy of TPSCI for excited states, including a far more efficient implementation in the Julia programming language. In traditional SCI methods that use a Slater determinant basis, accurate excitation energies are obtained only after a linear extrapolation and at a large computational cost. We find that TPSCI with perturbative corrections provides accurate excitation energies for several excited states of various polycyclic aromatic hydrocarbons with respect to the extrapolated result (i.e., near exact result). Further, we use TPSCI to report highly accurate estimates of the lowest 31 eigenstates for a tetracene tetramer system with an active space of 40 electrons in 40 orbitals, giving direct access to the initial bright states and the resulting 18 doubly excited (biexcitonic) states.

Recent Advances in Real-Time Time-Dependent Density Functional Theory Simulations of Plasmonic Nanostructures and Plasmonic Photocatalysis

Plasmonic catalysis provides a possible means for driving chemical reactions under relatively mild conditions. Rational design of these systems is impeded by the difficulty in understanding the electron dynamics and their interplay with reactions. Real-time, time-dependent density functional theory (RT-TDDFT) can provide dynamic information on excited states in plasmonic systems, including those relevant to plasmonic catalysis, at time scales and length scales that are otherwise out of reach of many experimental techniques. Here, we discuss previous RT-TDDFT studies of plasmonic systems, focusing on recent work that gains insight into plasmonic catalysis. These studies provide insight into plasmon dynamics, including size effects and the role of specific electronic states. Further, these studies provide significant insight into mechanisms underlying plasmonic catalysis, showing the importance of charge transfer between metal and adsorbate states, as well as local field enhancement, in different systems.

Curing the Divergence in Time-Dependent Density Functional Quadratic Response Theory

The adiabatic approximation in time-dependent density functional theory is known to give an incorrect pole structure in the quadratic response function, leading to unphysical divergences in excited state-to-state transition probabilities and hyperpolarizabilties. We find the form of the exact quadratic response kernel and derive a practical and accurate approximation that cures the divergence. We demonstrate our results on excited state-to-state transition probabilities of a model system and of the LiH molecule.

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.

Using diketopyrrolopyrroles to stabilize double excitation and control internal conversion

Diketopyrrolopyrrole (DPP) is a pivotal functional group to tune the physicochemical properties of novel organic photoelectronic materials. Among multiple uses, DPP-thiophene derivatives forming a dimer through a vinyl linker were recently shown to quench the fluorescence observed in their isolated monomers. Here, we explain this fluorescence quenching using computational chemistry. The DPP-thiophene dimer has a low-lying doubly excited state that is not energetically accessible for the monomer. This state delays the fluorescence allowing internal conversion to occur first. We characterize the doubly excited state wavefunction by systematically changing the derivatives to tune the π-scaffold size and the acceptor and donor characters. The origin of this state's stabilization is related to the increase in the π-system and not to the charge-transfer features. This analysis delivers core conceptual information on the electronic properties of organic chromophores arranged symmetrically around a vinyl linker, opening new ways to control the balance between luminescence and internal conversion.

Accurate core excitation and ionization energies from a state-specific coupled-cluster singles and doubles approach

We investigate the use of orbital-optimized references in conjunction with single-reference coupled-cluster theory with single and double substitutions (CCSD) for the study of core excitations and ionizations of 18 small organic molecules, without the use of response theory or equation-of-motion (EOM) formalisms. Three schemes are employed to successfully address the convergence difficulties associated with the coupled-cluster equations, and the spin contamination resulting from the use of a spin symmetry-broken reference, in the case of excitations. In order to gauge the inherent potential of the methods studied, an effort is made to provide reasonable basis set limit estimates for the transition energies. Overall, we find that the two best-performing schemes studied here for ΔCCSD are capable of predicting excitation and ionization energies with errors comparable to experimental accuracies. The proposed ΔCCSD schemes reduces statistical errors against experimental excitation energies by more than a factor of two when compared to the frozen-core core-valence separated (FC-CVS) EOM-CCSD approach - a successful variant of EOM-CCSD tailored towards core excitations.

The intermediate state approach for doubly excited dark states in EOM-coupled-cluster theory

Solution of dark, doubly excited states using equation-of-motion coupled-cluster (EOM-CC) usually equires at least triple excitations or even quadruples beyond the standard singles and doubles (EOM-CCSD) for an appropriate treatment. A new route to obtain these doubly excited states using EOM-CCSD is demonstrated. Traditionally, EOM-CC is performed on a closed shell reference state that has a well-described single reference CC wavefunction. In this Communication, we attempt to use low spin open-shell states such as the MS = 0 triplet and open-shell singlet as a reference state. Using this intermediate excited state as a reference state provides us with the benefit of obtaining a doubly excited state, as a single excitation at the cost of EOM-CCSD.

Automated Generation of Optimized Auxiliary Basis Sets for Long-Range-Corrected TDDFT Using the Cholesky Decomposition

Range-separated hybrid functionals making use of a smooth separation of the Coulomb operator in terms of the error function and its complement have proven to be a valuable tool for improving Kohn-Sham density functional theory (DFT) calculations. This holds in particular for obtaining accurate excitation energies from linear-response time-dependent DFT. Evaluating the long-range exchange contributions represents one of the most time-consuming tasks in such calculations. Prefitted auxiliary basis sets can be employed to speed up this step. Here, we present a way to generate auxiliary basis sets optimized to fit the long-range exchange contributions only, contrary to the common optimization strategies on the basis of the full Coulomb operator. For this purpose, we use the atomic Cholesky decomposition technique. The basis sets are generated on-the-fly using the specific range-separation parameter defined in the exchange-correlation functional. We obtain excitation energies and oscillator strengths which are of similar or better accuracy than those obtained with conventional resolution-of-the-identity auxiliary basis sets while drastically reducing the number of auxiliary functions required. This is demonstrated for the QUESTDB#5 benchmark set. In addition, we outline the benefits of this approach in sequences of calculations employing varying range-separation parameters, as is the case in the optimally tuned range-separation strategy. Finally, we illustrate the efficiency of this approach for real-world examples, namely, a chlorophyll tetramer from photosystem II and a carotenoid-porphyrin-C₆₀ triad.