Variational Density Functional Calculations of Excited States: Conical Intersection and Avoided Crossing in Ethylene Bond Twisting

Benchmarking DFT-based excited-state methods for intermolecular charge-transfer excitations

Intermolecular charge-transfer is a highly important process in biology and energy-conversion applications where generated charges need to be transported over several moieties. However, its theoretical description is challenging since the high accuracy required to describe these excited states must be accessible for calculations on large molecular systems. In this benchmark study, we identify reliable low-scaling computational methods for this task. Our reference results were obtained from highly accurate wavefunction calculations that restrict the size of the benchmark systems. However, the density-functional theory based methods that we identify as accurate can be applied to much larger systems. Since targeting charge-transfer states requires the unambiguous classification of an excited state, we first analyze several charge-transfer descriptors for their reliability concerning intermolecular charge-transfer and single out the charge-transfer distance calculated based on the variation of electron density upon excitation (DCT) as an optimal choice for our purposes. In general, best results are obtained for orbital-optimized methods and among those, the maximum overlap method proved to be the most numerically stable variant when using the initial MOs as reference orbitals. Favorable error cancellation with optimally-tuned range-separated hybrid functionals and a rather small basis set can provide an economical yet reasonable wavefunction when using time-dependent density functional theory, which provides relevant information about the excited-state character to be used in the orbital-optimized methods. The qualitative agreement makes these fast calculations attractive for high-throughput screening applications.

Excited State-Specific CASSCF Theory for the Torsion of Ethylene

State-specific complete active space self-consistent field (SS-CASSCF) theory has emerged as a promising route to accurately predict electronically excited energy surfaces away from molecular equilibria. However, its accuracy and practicality for chemical systems of photochemical interest have yet to be fully determined. We investigate the performance of the SS-CASSCF theory for the low-lying ground and excited states in the double bond rotation of ethylene. We show that state-specific approximations with a minimal (2e,2o) active space provide comparable accuracy to state-averaged calculations with much larger active spaces, while optimizing the orbitals for each excited state significantly improves the spatial diffusivity of the wave function. However, the incorrect ordering of state-specific solutions causes excited state solutions to coalesce and disappear, creating unphysical discontinuities in the potential energy surface. Our findings highlight the theoretical challenges that must be overcome to realize practical applications of state-specific electronic structure theory for computational photochemistry.

State-Specific Coupled-Cluster Methods for Excited States

We reexamine ΔCCSD, a state-specific coupled-cluster (CC) with single and double excitations (CCSD) approach that targets excited states through the utilization of non-Aufbau determinants. This methodology is particularly efficient when dealing with doubly excited states, a domain in which the standard equation-of-motion CCSD (EOM-CCSD) formalism falls short. Our goal here to evaluate the effectiveness of ΔCCSD when applied to other types of excited states, comparing its consistency and accuracy with EOM-CCSD. To this end, we report a benchmark on excitation energies computed with the ΔCCSD and EOM-CCSD methods for a set of molecular excited-state energies that encompasses not only doubly excited states but also doublet-doublet transitions and (singlet and triplet) singly excited states of closed-shell systems. In the latter case, we rely on a minimalist version of multireference CC known as the two-determinant CCSD method to compute the excited states. Our data set, consisting of 276 excited states stemming from the quest database [Véril et al., WIREs Comput. Mol. Sci. 2021, 11, e1517], provides a significant base to draw general conclusions concerning the accuracy of ΔCCSD. Except for the doubly excited states, we found that ΔCCSD underperforms EOM-CCSD. For doublet-doublet transitions, the difference between the mean absolute errors (MAEs) of the two methodologies (of 0.10 and 0.07 eV) is less pronounced than that obtained for singly excited states of closed-shell systems (MAEs of 0.15 and 0.08 eV). This discrepancy is largely attributed to a greater number of excited states in the latter set exhibiting multiconfigurational characters, which are more challenging for ΔCCSD. We also found typically small improvements by employing state-specific optimized orbitals.

Orbital-Optimized Versus Time-Dependent Density Functional Calculations of Intramolecular Charge Transfer Excited States

The performance of time-independent, orbital-optimized calculations of excited states is assessed with respect to charge transfer excitations in organic molecules in comparison to the linear-response time-dependent density functional theory (TD-DFT) approach. A direct optimization method to converge on saddle points of the electronic energy surface is used to carry out calculations with the local density approximation (LDA) and the generalized gradient approximation (GGA) functionals PBE and BLYP for a set of 27 excitations in 15 molecules. The time-independent approach is fully variational and provides a relaxed excited state electron density from which the extent of charge transfer is quantified. The TD-DFT calculations are generally found to provide larger charge transfer distances compared to the orbital-optimized calculations, even when including orbital relaxation effects with the Z-vector method. While the error on the excitation energy relative to theoretical best estimates is found to increase with the extent of charge transfer up to ca. -2 eV for TD-DFT, no correlation is observed for the orbital-optimized approach. The orbital-optimized calculations with the LDA and the GGA functionals provide a mean absolute error of ∼0.7 eV, outperforming TD-DFT with both local and global hybrid functionals for excitations with a long-range charge transfer character. Orbital-optimized calculations with the global hybrid functional B3LYP and the range-separated hybrid functional CAM-B3LYP on a selection of states with short- and long-range charge transfer indicate that inclusion of exact exchange has a small effect on the charge transfer distance, while it significantly improves the excitation energy, with the best-performing functional CAM-B3LYP providing an absolute error typically around 0.15 eV.

GPAW: An open Python package for electronic structure calculations

We review the GPAW open-source Python package for electronic structure calculations. GPAW is based on the projector-augmented wave method and can solve the self-consistent density functional theory (DFT) equations using three different wave-function representations, namely real-space grids, plane waves, and numerical atomic orbitals. The three representations are complementary and mutually independent and can be connected by transformations via the real-space grid. This multi-basis feature renders GPAW highly versatile and unique among similar codes. By virtue of its modular structure, the GPAW code constitutes an ideal platform for the implementation of new features and methodologies. Moreover, it is well integrated with the Atomic Simulation Environment (ASE), providing a flexible and dynamic user interface. In addition to ground-state DFT calculations, GPAW supports many-body GW band structures, optical excitations from the Bethe-Salpeter Equation, variational calculations of excited states in molecules and solids via direct optimization, and real-time propagation of the Kohn-Sham equations within time-dependent DFT. A range of more advanced methods to describe magnetic excitations and non-collinear magnetism in solids are also now available. In addition, GPAW can calculate non-linear optical tensors of solids, charged crystal point defects, and much more. Recently, support for graphics processing unit (GPU) acceleration has been achieved with minor modifications to the GPAW code thanks to the CuPy library. We end the review with an outlook, describing some future plans for GPAW.

Orbital-optimized density functional calculations of molecular Rydberg excited states with real space grid representation and self-interaction correction

Density functional calculations of Rydberg excited states up to high energy are carried out for several molecules using an approach where the orbitals are variationally optimized by converging on saddle points on the electronic energy surface within a real space grid representation. Remarkably good agreement with experimental estimates of the excitation energy is obtained using the generalized gradient approximation (GGA) functional of Perdew, Burke, and Ernzerhof (PBE) when Perdew-Zunger self-interaction correction is applied in combination with complex-valued orbitals. Even without the correction, the PBE functional gives quite good results despite the fact that corresponding Rydberg virtual orbitals have positive energy in the ground state calculation. Results obtained using the Tao, Perdew, Staroverov, and Scuseria (TPSS) and r2SCAN meta-GGA functionals are also presented, but they do not provide a systematic improvement over the results from the uncorrected PBE functional. The grid representation combined with the projector augmented-wave approach gives a simpler and better representation of diffuse Rydberg orbitals than a linear combination of atomic orbitals with commonly used basis sets, the latter leading to an overestimation of the excitation energy due to confinement of the excited states.

Seniority and Hierarchy Configuration Interaction for Radicals and Excited States

Hierarchy configuration interaction (hCI) has recently been introduced as an alternative configuration interaction (CI) route combining excitation degree and seniority number and has been shown to efficiently recover both dynamic and static correlations for closed-shell molecular systems [ J. Phys. Chem. Lett. 2022, 13, 4342]. Here we generalize hCI for an arbitrary reference determinant, allowing calculations for radicals and excited states in a state-specific way. We gauge this route against excitation-based CI (eCI) and seniority-based CI (sCI) by evaluating how different ground-state properties of radicals converge to the full CI limit. We find that hCI outperforms or matches eCI, whereas sCI is far less accurate, in line with previous observations for closed-shell molecules. Employing second-order Epstein-Nesbet (EN2) perturbation theory as a correction significantly accelerates the convergence of hCI and eCI. We further explore various hCI and sCI models to calculate the excitation energies of closed- and open-shell systems. Our results underline that the choice of both the reference determinant and the set of orbitals drives the fine balance between correlation of ground and excited states. State-specific hCI2 and higher-order models perform similarly to their eCI counterparts, whereas lower orders of hCI deliver poor results unless supplemented by the EN2 correction, which substantially improves their accuracy. In turn, sCI1 produces decent excitation energies for radicals, encouraging the development of related seniority-based coupled-cluster methods.

Target State Optimized Density Functional Theory for Electronic Excited and Diabatic States

A flexible self-consistent field method, called target state optimization (TSO), is presented for exploring electronic excited configurations and localized diabatic states. The key idea is to partition molecular orbitals into different subspaces according to the excitation or localization pattern for a target state. Because of the orbital-subspace constraint, orbitals belonging to different subspaces do not mix. Furthermore, the determinant wave function for such excited or diabatic configurations can be variationally optimized as a ground state procedure, unlike conventional ΔSCF methods, without the possibility of collapsing back to the ground state or other lower-energy configurations. The TSO method can be applied both in Hartree-Fock theory and in Kohn-Sham density functional theory (DFT). The density projection procedure and the working equations for implementing the TSO method are described along with several illustrative applications. For valence excited states of organic compounds, it was found that the computed excitation energies from TSO-DFT and time-dependent density functional theory (TD-DFT) are of similar quality with average errors of 0.5 and 0.4 eV, respectively. For core excitation, doubly excited states and charge-transfer states, the performance of TSO-DFT is clearly superior to that from conventional TD-DFT calculations. It is shown that variationally optimized charge-localized diabatic states can be defined using TSO-DFT in energy decomposition analysis to gain both qualitative and quantitative insights on intermolecular interactions. Alternatively, the variational diabatic states may be used in molecular dynamics simulation of charge transfer processes. The TSO method can also be used to define basis states in multistate density functional theory for excited states through nonorthogonal state interaction calculations. The software implementing TSO-DFT can be accessed from the authors.

Atomic-scale observation of solvent reorganization influencing photoinduced structural dynamics in a copper complex photosensitizer

Photochemical reactions in solution are governed by a complex interplay between transient intramolecular electronic and nuclear structural changes and accompanying solvent rearrangements. State-of-the-art time-resolved X-ray solution scattering has emerged in the last decade as a powerful technique to observe solute and solvent motions in real time. However, disentangling solute and solvent dynamics and how they mutually influence each other remains challenging. Here, we simultaneously measure femtosecond X-ray emission and scattering to track both the intramolecular and solvation structural dynamics following photoexcitation of a solvated copper photosensitizer. Quantitative analysis assisted by molecular dynamics simulations reveals a two-step ligand flattening strongly coupled to the solvent reorganization, which conventional optical methods could not discern. First, a ballistic flattening triggers coherent motions of surrounding acetonitrile molecules. In turn, the approach of acetonitrile molecules to the copper atom mediates the decay of intramolecular coherent vibrations and induces a further ligand flattening. These direct structural insights reveal that photoinduced solute and solvent motions can be intimately intertwined, explaining how the key initial steps of light harvesting are affected by the solvent on the atomic time and length scale. Ultimately, this work takes a step forward in understanding the microscopic mechanisms of the bidirectional influence between transient solvent reorganization and photoinduced solute structural dynamics.