NAC-TDDFT: Time-Dependent Density Functional Theory for Nonadiabatic Couplings

Reverse intersystem crossing mechanisms in doped triangulenes

Thermally activated delayed fluorescence (TADF) has emerged as one of the most promising strategies in the quest for organic light emitting diodes with optimal performance. This computational study dissects the mechanistic intricacies of the central photophysical step, reverse intersystem crossing (rISC) in N and B doped triangulenes as potential multi-resonance TADF compounds. Optimal molecular patterns conducive to efficient rISC, encompassing dopant atom size, number, and distribution, are identified. Additionally, we assess various electronic structure methods for characterizing TADF-relevant molecular systems. The findings identify the distinct role of the direct and mediated mechanisms in rISC, and provide insights into the design of advanced TADF chromophores for next-generation OLED technology.

Estimating Nonradiative Excited-State Lifetimes in Photoactive Semiconducting Nanostructures

The time evolution of the exciton generated by light adsorption in a photocatalyst is an important feature that can be approached from full nonadiabatic molecular dynamics simulations. Here, a crucial parameter is the nonradiative recombination rate between the hole and the electron that form the exciton. In the present work, we explore the performance of a Fermi's golden rule-based approach on predicting the recombination rate in a set of photoactive titania nanostructures, relying solely on the coupling of the ground and first excited state. In this scheme the analysis of the first excited state is carried out by invoking Kasha's rule thus avoiding computationally expensive nonadiabatic molecular dynamics simulations and resulting in an affordable estimate of the recombination rate. Our results show that, compared to previous ones from nonadiabatic molecular dynamics simulations, semiquantitative recombination rates can be predicted for the smaller titania nanostructures, and qualitative values are obtained from the larger ones. The present scheme is expected to be useful in the field of computational heterogeneous photocatalysis whenever a complex and computationally expensive full nonadiabatic molecular dynamics cannot be carried out.

A Local Diabatisation Method for Two-State Adiabatic Conical Intersections

A methodology to locally characterize conical intersections (CIs) between two adiabatic electronic states for which no nonadiabatic coupling (NAC) vectors are available is presented. Based on the Hessian and gradient at the CI, the branching space coordinates are identified. The potential energy surface around the CI in the branching space is expressed in the diabatic representation, from which the NAC vectors can be calculated in a wave-function-free, energy-based approach. To demonstrate the universality of the developed methodology, the minimum-energy CI (MECI) between the first (S1) and second (S2) singlet excited states of formamide is investigated at the state-averaged complete active space self-consistent field (SA-CASSCF) and extended multistate complete active space second-order perturbation theory (XMS-CASPT2) levels of theory. In addition, the asymmetrical MECI between the ground state (S0) and S1 of cyclopropanone is evaluated using SA-CASSCF, as well as (ME)CIs between the S1 and S2 states of benzene using SA-CASSCF and time-dependent density functional theory (TDDFT). Finally, a CI between the S1 and S2 excited states of thiophene was analyzed using TDDFT.

Prediction of fluorescence quantum yields using the extended thawed Gaussian approximation

Spontaneous emission and internal conversion rates are calculated within harmonic approximations and compared to the results obtained within the semi-classical extended thawed Gaussian approximation (ETGA). This is the first application of the ETGA in the calculation of internal conversion and emission rates for real molecular systems, namely, formaldehyde, fluorobenzene, azulene, and a dicyano-squaraine dye. The viability of the models as black-box tools for prediction of spontaneous emission and internal conversion rates is assessed. All calculations were done using a consistent protocol in order to investigate how different methods perform without previous experimental knowledge using density functional theory (DFT) and time-dependent DFT (TD-DFT) with B3LYP, PBE0, ωB97XD, and CAM-B3LYP functionals. Contrasting the results with experimental data shows that there are further improvements required before theoretical predictions of emission and internal conversion rates can be used as reliable indicators for the photo-luminescence properties of molecules. We find that the ETGA performs rather similar to the vertical harmonical model. Including anharmonicities in the calculation of internal conversion rates has a moderate effect on the quantitative results in the studied systems. The emission rates are fairly stable with respect to computational parameters, but the internal conversion rate reveals itself to be highly dependent on the choice of the spectral line shape function, particularly the width of the Lorentzian function, associated with homogeneous broadening.

On the description of conical intersections between excited electronic states with LR-TDDFT and ADC(2)

Conical intersections constitute the conceptual bedrock of our working understanding of ultrafast, nonadiabatic processes within photochemistry (and photophysics). Accurate calculation of potential energy surfaces within the vicinity of conical intersections, however, still poses a serious challenge to many popular electronic structure methods. Multiple works have reported on the deficiency of methods like linear-response time-dependent density functional theory within the adiabatic approximation (AA LR-TDDFT) or algebraic diagrammatic construction to second-order [ADC(2)]-approaches often used in excited-state molecular dynamics simulations-to describe conical intersections between the ground and excited electronic states. In the present study, we focus our attention on conical intersections between excited electronic states and probe the ability of AA LR-TDDFT and ADC(2) to describe their topology and topography, using protonated formaldimine and pyrazine as two exemplar molecules. We also take the opportunity to revisit the performance of these methods in describing conical intersections involving the ground electronic state in protonated formaldimine-highlighting in particular how the intersection ring exhibited by AA LR-TDDFT can be perceived either as a (near-to-linear) seam of intersection or two interpenetrating cones, depending on the magnitude of molecular distortions within the branching space.

When do tripdoublet states fluoresce? A theoretical study of copper(II) porphyrin

Open-shell molecules rarely fluoresce, due to their typically faster non-radiative relaxation rates compared to closed-shell ones. Even rarer is the fluorescence from states that have two more unpaired electrons than the open-shell ground state, since they involve excitations from closed-shell orbitals to vacant-shell orbitals, which are typically higher in energy compared to excitations from or out of open-shell orbitals. States that are dominated by the former type of excitations are known as tripdoublet states when they can be described as a triplet excitation antiferromagnetically coupled to a doublet state, and their description by unrestricted single-reference methods (e.g., U-TDDFT) is notoriously inaccurate due to large spin contamination. In this work, we applied our spin-adapted TDDFT method, X-TDDFT, and the efficient and accurate static-dynamic-static second order perturbation theory (SDSPT2), to the study of the excited states as well as their relaxation pathways of copper(II) porphyrin; previous experimental works suggested that the photoluminescence of some substituted copper(II) porphyrins originate from a tripdoublet state, formed by a triplet ligand π → π* excitation antiferromagnetically coupled with the unpaired d electron. Our results demonstrated favorable agreement between the X-TDDFT, SDSPT2 and experimental excitation energies, and revealed noticeable improvements of X-TDDFT compared to U-TDDFT, not only for vertical excitation energies but also for adiabatic energy differences. These suggest that X-TDDFT is a reliable tool for the study of tripdoublet state fluorescence. Intriguingly, we showed that the aforementioned tripdoublet state is only slightly above the lowest doublet excited state and lies only slightly higher than the lowest quartet state, which suggests that the tripdoublet of copper(II) porphyrin is long-lived enough to fluoresce due to a lack of efficient non-radiative relaxation pathways; an explanation for this unusual state ordering is given. Indeed, thermal vibration correlation function (TVCF)-based calculations of internal conversion, intersystem crossing, and radiative transition rates confirm that copper(II) porphyrin emits thermally activated delayed fluorescence (TADF) and a small amount of phosphorescence at low temperature (83 K), in accordance with experiment. The present contribution is concluded by a few possible approaches of designing new molecules that fluoresce from tripdoublet states.

Nonadiabatic Derivative Couplings Calculated Using Information of Potential Energy Surfaces without Wavefunctions: Ab Initio and Machine Learning Implementations

In this work, we implemented an approximate algorithm for calculating nonadiabatic coupling matrix elements (NACMEs) of a polyatomic system with ab initio methods and machine learning (ML) models. Utilizing this algorithm, one can calculate NACMEs using only the information of potential energy surfaces (PESs), i.e., energies, and gradients as well as Hessian matrix elements. We used a realistic system, namely CH₂NH, to compare NACMEs calculated by this approximate PES-based algorithm and the accurate wavefunction-based algorithm. Our results show that this approximate PES-based algorithm can give very accurate results comparable to the wavefunction-based algorithm except at energetically degenerate points, i.e., conical intersections. We also tested a machine learning (ML)-trained model with this approximate PES-based algorithm, which also supplied similarly accurate NACMEs but more efficiently. The advantage of this PES-based algorithm is its significant potential to combine with electronic structure methods that do not implement wavefunction-based algorithms, low-scaling energy-based fragment methods, etc., and in particular efficient ML models, to compute NACMEs. The present work could encourage further research on nonadiabatic processes of large systems simulated by ab initio nonadiabatic dynamics simulation methods in which NACMEs are always required.

Analytic Gradient for Time-Dependent Density Functional Theory Combined with the Fragment Molecular Orbital Method

The analytic energy gradient of energy with respect to nuclear coordinates is derived for the fragment molecular orbital (FMO) method combined with time-dependent density functional theory (TDDFT). The response terms arising from the use of a polarizable embedding are derived. The obtained analytic FMO-TDDFT gradient is shown to be accurate in comparison to both numerical FMO-TDDFT and unfragmented TDDFT gradients, at the level of two- and three-body expansions. The gradients are used for geometry optimizations, molecular dynamics, vibrational calculations, and simulations of IR and Raman spectra of excited states. The developed method is used to optimize the geometry of the ground and excited electronic states of the photoactive yellow protein (PDB: 2PHY).

Analytical derivative couplings within the framework of time-dependent density functional theory coupled with conductor-like polarizable continuum model: Formalism, implementation, and applications

The nonadiabatic phenomena, which are characterized by a strong coupling between electronic and nuclear motions, are ubiquitous. The nonadiabatic effect of the studied system can be significantly affected by the surrounding environment, such as solvents, in which such nonadiabatic process takes place. It is essential to develop the theoretical models to simulate these processes while accurately modeling the solvent environment. The time-dependent density functional theory (TDDFT) is currently the most efficient approach to describe the electronic structures and dynamics of complex systems, while the polarizable continuum model (PCM) represents one of the most successful examples among continuum solvation models. Here, we formulate the first-order derivative couplings (DCs) between the ground and excited states as well as between two excited states by utilizing time-independent equation of motion formalism within the framework of both linear response and spin flip formulations of TDDFT/CPCM (the conductor-like PCM), and implement the analytical DCs into the Q-CHEM electronic structure software package. The analytic implementation is validated by the comparison of the analytical and finite-difference results, and reproducing geometric phase effect in the protonated formaldimine test case. Taking 4-(N,N-dimethylamino)benzonitrile and uracil in the gas phase and solution as an example, we demonstrate that the solvent effect is essential not only for the excitation energies of the low-lying excited-states but also for the DCs between these states. Finally, we calculate the internal conversion rate of benzophenone in a solvent with DC being used. The current implementation of analytical DCs together with the existing analytical gradient and Hessian of TDDFT/PCM excited states allows one to study the nonadiabatic effects of relatively large systems in solutions with low computational cost.

Analytical gradients and derivative couplings for the TDDFT-1D method

We derive and implement analytic gradients and derivative couplings for time-dependent density functional theory plus one double (TDDFT-1D) which is a semiempirical configuration interaction method whereby the Hamiltonian is diagonalized in a basis of all singly excited configurations and one doubly excited configuration as constructed from a set of reference Kohn-Sham orbitals. We validate the implementation by comparing against finite difference values. Furthermore, we show that our implementation can locate both optimized geometries and minimum-energy crossing points along conical seams of S₁/S₀ surfaces for a set of test cases.

A Review of Performance Prediction Based on Machine Learning in Materials Science

With increasing demand in many areas, materials are constantly evolving. However, they still have numerous practical constraints. The rational design and discovery of new materials can create a huge technological and social impact. However, such rational design and discovery require a holistic, multi-stage design process, including the design of the material composition, material structure, material properties as well as process design and engineering. Such a complex exploration using traditional scientific methods is not only blind but also a huge waste of time and resources. Machine learning (ML), which is used across data to find correlations in material properties and understand the chemical properties of materials, is being considered a new way to explore the materials field. This paper reviews some of the major recent advances and applications of ML in the field of properties prediction of materials and discusses the key challenges and opportunities in this cross-cutting area.

Transition orbital projection approach for excited state tracking

Quantitively comparing the features between different electronic excited states (ESs) is a crucial task in both potential energy surface (PES) studies and excited-state fragmentation approaches. However, it is still a challenging problem in regard to the comparison of complex and highly degenerate systems. Herein, we present a transition orbital projection (TOP) method to calculate the similarity of different ESs based on the configuration vectors of two types of transition densities. It fully considers four significant problems, including phase, hole-particle bijectivity, orbital permutation, and sign of configuration coefficients. TOP state-tracking-based excited-state optimization shows high robustness in several high-symmetric systems, which are difficult to describe with traditional state-tracking approaches. The TOP state-tracking method is expected to be widely applied to the PES of photochemical reactions, ES molecular dynamics to track the diabatic states, and fragmentation approaches for local excitation of large systems.

Spin-Orbit Couplings for Nonadiabatic Molecular Dynamics at the ΔSCF Level

A procedure for the calculation of spin-orbit coupling (SOC) at the delta self-consistent field (ΔSCF) level of theory is presented. Singlet and triplet excited electronic states obtained with the ΔSCF method are expanded into a linear combination of singly excited Slater determinants composed of ground electronic state Kohn-Sham orbitals. This alleviates the nonorthogonality between excited and ground electronic states and introduces a framework, similar to the auxiliary wave function at the time-dependent density functional theory (TD-DFT) level, for the calculation of observables. The ΔSCF observables of the formaldehyde system were compared to reference TD-DFT values. Our procedure gives all components (energies, gradients, nonadiabatic couplings, and SOC terms) at the ΔSCF level of theory for conducting efficient, full-atomistic nonadiabatic molecular dynamics with intersystem crossing, particularly in condensed phase systems.

Analytical Nonadiabatic Couplings and Gradients within the State-Averaged Orbital-Optimized Variational Quantum Eigensolver

We introduce several technical and analytical extensions to our recent state-averaged orbital-optimized variational quantum eigensolver (SA-OO-VQE) algorithm (see Yalouz et al. Quantum Sci. Technol. 2021, 6, 024004). Motivated by the limitations of current quantum computers, the first extension consists of an efficient state-resolution procedure to find the SA-OO-VQE eigenstates, and not just the subspace spanned by them, while remaining in the equi-ensemble framework. This approach avoids expensive intermediate resolutions of the eigenstates by postponing this problem to the very end of the full algorithm. The second extension allows for the estimation of analytical gradients and nonadiabatic couplings, which are crucial in many practical situations ranging from the search of conical intersections to the simulation of quantum dynamics, in, for example, photoisomerization reactions. The accuracy of our new implementations is demonstrated on the formaldimine molecule CH₂NH (a minimal Schiff base model relevant for the study of photoisomerization in larger biomolecules), for which we also perform a geometry optimization to locate a conical intersection between the ground and first-excited electronic states of the molecule.