Ab Initio Study of Low-Lying Excited States of Carotenoid-Derived Polyenes

Characterization of the Ultraviolet-B Absorption Band of Carotenoids Using Solvent-dependent Shifts in Steady-State and Transient Absorption Spectra

The versatile functions of carotenoids in biological systems are associated with the extended π-electron conjugation system. Strong visible absorption resulting from the optically allowed S2 (1Bu+) state and the low-lying optically forbidden S1 (2Ag-) state examined. Carotenoids also exhibit an absorption band in the ultraviolet-B region; however, the origin of this band (hereafter referred to as Suv state) is not well characterized. The Suv state is a candidate for the destination level of the well-known S1 → Sn transient absorption; however, an obvious energy mismatch has been observed. In this study, we examined the steady-state and picosecond transient absorption spectra of lycopene in various solvents. The Suv absorption of carotenoids with diverse conjugation lengths was also examined. The dependence of the energies on solvent polarizability and conjugation length revealed that both Suv and Sn are the "second" Bu+ state. The absorption spectrum for lycopene at 200 K revealed an additional vibrational band, which may be the vibrational origin of the S0 → Suv band. Considering the slow vibrational relaxation of the 2Ag- state, the S1 → Sn transition may represent the 2Ag- (v = 1) → 2Bu+ (v = 0) transition, and the energetic contradiction can be resolved.

Predicting Solvatochromism of Chromophores in Proteins through QM/MM and Machine Learning

Solvatochromism occurs in both homogeneous solvents and more complex biological environments, such as proteins. While in both cases the solvatochromic effects report on the surroundings of the chromophore, their interpretation in proteins becomes more complicated not only because of structural effects induced by the protein pocket but also because the protein environment is highly anisotropic. This is particularly evident for highly conjugated and flexible molecules such as carotenoids, whose excitation energy is strongly dependent on both the geometry and the electrostatics of the environment. Here, we introduce a machine learning (ML) strategy trained on quantum mechanics/molecular mechanics calculations of geometrical and electrochromic contributions to carotenoids' excitation energies. We employ this strategy to compare solvatochromism in protein and solvent environments. Despite the important specifities of the protein, ML models trained on solvents can faithfully predict excitation energies in the protein environment, demonstrating the robustness of the chosen descriptors.

Singlet Fission in Lycopene H-Aggregates

A theory of singlet fission (SF) in carotenoid dimers is applied to explain the SF in lycopene H-aggregates observed after high-energy photoexcitation. The explanation proposed here is that a high energy, delocalized bright 1Bu+ state first relaxes and localizes onto a single lycopene monomer. The high-energy intramonomer state then undergoes internal conversion to the 11Bu- state. Once populated, the 11Bu- state allows exothermic bimolecular singlet fission, while its internal conversion to the 21Ag- state is symmetry forbidden. The simulation of SF predicts that the intramonomer triplet-pair state undergoes almost complete population transfer to the intermonomer singlet-pair state within 100 ps. Simultaneously, ZFS interactions begin to partially populate the intermonomer quintet triplet-pair state up to ca. 2 ns, after which hyperfine interactions thermally equilibrate the triplet-pair states, thus forming free single triplets within 50 ns.

On the Fluorescence Properties and Nonradiative Transitions in Medium-Sized All-Trans Polyenes

The nonadiabatic photodynamics of all-trans linear polyenes with N = 4-8 conjugated double bonds is studied from an electronic structure perspective. Excitation energies and stationary points for the 1Bu and 2Ag singlet states have been computed by using the state-average complete active space (SA-CASSCF) method and its second-order perturbation theory variant (MS-CASPT2). The dependence of the two low-lying excited states on the "chain length" N has been elucidated. In addition, the 1Bu-2Ag crossing seam has been mapped out in a suitable two-dimensional coordinate space and its minimum within the subspace has been determined. This minimum is found to increase substantially and monotonously in energy with increasing N. This increase is discussed and interpreted in relation to the fluorescence properties of these systems. In particular, it allows to understand the crossover from S1(2Ag) fluorescence for smaller N to S2(1Bu) (or dual) fluorescence for larger N.

Understanding the Excited-State Relaxation Mechanisms of Xanthophyll Lutein by Multi-configurational Electronic Structure Calculations

The contradictory behaviors in light harvesting and non-photochemical quenching make xanthophyll lutein the most attractive functional molecule in photosynthesis. Despite several theoretical simulations on the spectral properties and excited-state dynamics, the atomic-level photophysical mechanisms need to be further studied and established, especially for an accurate description of geometric and electronic structures of conical intersections for the lowest several electronic states of lutein. In the present work, semiempirical OM2/MRCI and multi-configurational restricted active space self-consistent field methods were performed to optimize the minima and conical intersections in and between the 1Ag-, 2Ag-, 1Bu+, and 1Bu- states. Meanwhile, the relative energies were refined by MS-CASPT2(10,8)/6-31G*, which can reproduce correct electronic state properties as those in the spectroscopic experiments. Based on the above calculation results, we proposed a possible excited-state relaxation mechanism for lutein from its initially populated 1Bu+ state. Once excited to the optically bright 1Bu+ state, the system will propagate along the key reaction coordinate, i.e., the stretching vibration of the conjugated carbon chain. During this period of time, the 1Bu- state will participate in and forms a resonance state between the 1Bu- and 1Bu+ states. Later, the system will rapidly hop to the 2Ag- state via the 1Bu+/2Ag- conical intersection. Finally, the lutein molecule will survive in the 2Ag- state for a relatively long time before it internally converts to the ground state directly or via a twisted S1/S0 conical intersection. Notably, though the photophysical picture may be very different in solvents and proteins, the current theoretical study proposed a promising calculation protocol and also provided many valuable mechanistic insights for lutein and similar carotenoids.

Dynamical Simulations of Carotenoid Photoexcited States Using Density Matrix Renormalization Group Techniques

We present a dynamical simulation scheme to model the highly correlated excited state dynamics of linear polyenes. We apply it to investigate the internal conversion processes of carotenoids following their photoexcitation. We use the extended Hubbard-Peierls model, H^UVP, to describe the π-electronic system coupled to nuclear degrees of freedom. This is supplemented by a Hamiltonian, H^ϵ, that explicitly breaks both the particle-hole and two-fold rotation symmetries of idealized carotenoid structures. The electronic degrees of freedom are treated quantum mechanically by solving the time-dependent Schrödinger equation using the adaptive time-dependent DMRG (tDMRG) method, while nuclear dynamics are treated via the Ehrenfest equations of motion. By defining adiabatic excited states as the eigenstates of the full Hamiltonian, H^=H^UVP+H^ϵ, and diabatic excited states as eigenstates of H^UVP, we present a computational framework to monitor the internal conversion process from the initial photoexcited 1¹B_u⁺ state to the singlet triplet-pair states of carotenoids. We further incorporate Lanczos-DMRG to the tDMRG-Ehrenfest method to calculate transient absorption spectra from the evolving photoexcited state. We describe in detail the accuracy and convergence criteria for DMRG, and show that this method accurately describes the dynamical processes of carotenoid excited states. We also discuss the effect of the symmetry-breaking term, H^ϵ, on the internal conversion process, and show that its effect on the extent of internal conversion can be described by a Landau-Zener-type transition. This methodological paper is a companion to our more explanatory discussion of carotenoid excited state dynamics in Manawadu, D.; Georges, T. N.; Barford, W. Photoexcited State Dynamics and Singlet Fission in Carotenoids. J. Phys. Chem. A 2023, 127, 1342.

R2022: A DFT/MRCI Ansatz with Improved Performance for Double Excitations

A reformulation of the combined density functional theory and multireference configuration interaction method (DFT/MRCI) is presented. Expressions for ab initio matrix elements are used to derive correction terms for a new effective Hamiltonian. On the example of diatomic carbon, the correction terms are derived, focusing on the doubly excited ¹Δ_g state, which was problematic in previous formulations of the method, as were double excitations in general. The derivation shows that a splitting of the parameters for intra- and interorbital interactions is necessary for a concise description of the underlying physics. Results for ¹L_a and ¹L_b states in polyacenes and ¹A_u and ¹A_g states in mini-β-carotenoids suggest that the presented formulation is superior to former effective Hamiltonians. Furthermore, statistical analysis reveals that all the benefits of the previous DFT/MRCI Hamiltonians are retained. Consequently, the here presented formulation should be considered as the new standard for DFT/MRCI calculations.

Photoexcited State Dynamics and Singlet Fission in Carotenoids

We describe our simulations of the excited state dynamics of the carotenoid neurosporene, following its photoexcitation into the "bright" (nominally 1¹B_u⁺) state. To account for the experimental and theoretical uncertainty in the relative energetic ordering of the nominal 1¹B_u⁺ and 2¹A_g^- states at the Franck-Condon point, we consider two parameter sets. In both cases, there is ultrafast internal conversion from the "bright" state to a "dark" singlet triplet-pair state, i.e., to one member of the "2A_g" family of states. For one parameter set, internal conversion from the 1¹B_u⁺ to 2¹A_g^- states occurs via the dark, intermediate 1¹B_u^- state. In this case, there is a cross over of the 1¹B_u⁺ and 1¹B_u^- diabatic energies within 5 fs and an associated avoided crossing of the S₂ and S₃ adiabatic energies. After the adiabatic evolution of the S₂ state from predominately 1¹B_u⁺ character to predominately 1¹B_u^- character, there is a slower nonadiabatic transition from S₂ to S₁, accompanied by an increase in the population of the 2¹A_g^- state. For the other parameter set, the 2¹A_g^- energy lies higher than the 1¹B_u⁺ energy at the Franck-Condon point. In this case, there is cross over of the 2¹A_g^- and 1¹B_u⁺ energies and an avoided crossing of the S₁ and S₂ energies, as the S₁ state evolves adiabatically from being of 1¹B_u⁺ character to 2¹A_g^- character. We make a direct connection from our predictions to experimental observables by calculating the time-resolved excited state absorption. For the case of direct 1¹B_u⁺ to 2¹A_g^- internal conversion, we show that the dominant transition at ca. 2 eV, being close to but lower in energy than the T₁ to T₁^* transition, can be attributed to the 2¹A_g^- component of S₁. Moreover, we show that it is the charge-transfer exciton component of the 2¹A_g^- state that is responsible for this transition (to a higher-lying exciton state), and not its triplet-pair component. These simulations are performed using the adaptive tDMRG method on the extended Hubbard model of π-conjugated electrons. The Ehrenfest equations of motion are used to simulate the coupled nuclei dynamics. We next discuss the microscopic mechanism of "bright" to "dark" state internal conversion and emphasize that this occurs via the exciton components of both states. Finally, we describe a mechanism relying on torsional relaxation whereby the strongly bound intrachain triplet-pairs of the "dark" state may undergo interchain exothermic dissociation.

Low-Lying Excited States of Natural Carotenoids Viewed by Ab Initio Methods

Low-lying excited states of carotenoids (the optically dark 2A_g^- and bright 1B_u⁺) are deeply involved in energy transfer processes in photosynthetic antennas, such as light harvesting and non-photochemical quenching. Though any ab initio modeling of these phenomena has to rely on precise energies of the carotenoid electronic states, the accurate evaluation of these states remains a challenging problem due to their different natures. The paper aims to study the accuracy of the excitation energies of the low-lying excited states of certain open- and closed-chain carotenoids obtained by a state-of-the-art multireference approach for electronic structure calculation. Here, density matrix renormalization group SCF (DMRGSCF) and a perturbative approach based on driven similarity renormalization group second-order multireference perturbation theory (DSRG-MRPT2) were used to treat the static and dynamic correlation, respectively. Nuclear geometries of the electronic states were optimized with DFT-based approaches. It is demonstrated that spin-flip TD-DFT can replace multiconfigurational methods for the geometry optimization of the 2A_g^- state but not for the calculation of the excitation energy. Adiabatic excitation energies to the 1B_u⁺ state were shown to be within a margin of 1000 cm^-1 with an appropriate flow parameter value. Adiabatic excitation energies to the 2A_g^- state for the open-chain carotenoids lie within a range of experimental values (taking into account the broad range of experimental estimates); for the closed-chain ones, the error does not exceed 2000 cm^-1. Ab initio stationary (1A_g^- → 1B_u⁺) and transient (2A_g^- → 1B_u⁺) absorption spectra were modeled for violaxanthin and lycopene, and these spectra showed good agreement with the experimental ones both in terms of the vibronic structure and the transition energies.

Singlet Triplet-Pair Production and Possible Singlet-Fission in Carotenoids

Internal conversion from the photoexcited state to a correlated singlet triplet-pair state is believed to be the precursor of singlet fission in carotenoids. We present numerical simulations of this process using a π-electron model that fully accounts for electron-electron interactions and electron-nuclear coupling. The time-evolution of the electrons is determined rigorously using the time-dependent density matrix renormalization group method, while the nuclei are evolved via the Ehrenfest equations of motion. We apply this to zeaxanthin, a carotenoid chain with 18 fully conjugated carbon atoms. We show that the internal conversion of the primary photoexcited state, S₂, to the singlet triplet-pair state occurs adiabatically via an avoided crossing within ∼50 fs with a yield of ∼60%. We further discuss whether this singlet triplet-pair state will undergo exothermic versus endothermic intra- or interchain singlet fission.

Post-Density Matrix Renormalization Group Methods for Describing Dynamic Electron Correlation with Large Active Spaces

The ab initio density matrix renormalization group (DMRG) method has been well-established and has become one of the most accurate numerical methods for the precise electronic structure solution of large active spaces. In the past few years, to capture the missing dynamic correlation, various post-DMRG approaches have been proposed through the combination of DMRG and multireference quantum chemical methods or density functional theory. With this in mind, this work provides a brief overview of ab initio DMRG principles and the new developments within post-DMRG methods. For clarity, post-DMRG methods are classified into two main categories depending on whether high-order n-electron reduced density matrices are used, and their merits and disadvantages are properly discussed. Finally, we conclude by discussing unsolved bottlenecks and giving development perspectives of post-DMRG approaches, which are expected to yield quantitative descriptions of complex electronic structures in large strongly correlated molecules and materials.

Excited States of Xanthophylls Revisited: Toward the Simulation of Biologically Relevant Systems

Xanthophylls are a class of oxygen-containing carotenoids, which play a fundamental role in light-harvesting pigment-protein complexes and in many photoresponsive proteins. The complexity of the manifold of the electronic states and the large sensitivity to the environment still prevent a clear and coherent interpretation of their photophysics and photochemistry. In this Letter, we compare cutting-edge ab initio methods (CC3 and DMRG/NEVPT2) with time-dependent DFT and semiempirical CI (SECI) on model keto-carotenoids and show that SECI represents the right compromise between accuracy and computational cost to be applied to real xanthophylls in their biological environment. As an example, we investigate canthaxanthin in the orange carotenoid protein and show that the conical intersections between excited states and excited-ground states are mostly determined by the effective bond length alternation coordinate, which is significantly tuned by the protein through geometrical constraints and electrostatic effects.

Toward an Accurate Ab Initio Description of Low-Lying Singlet Excited States of Polyenes

The low-lying excited states of carotenoids play a crucial role in many important biophysical processes such as photosynthesis. Most of these excited states are strongly correlated, which makes them both challenging for a qualitative ab initio description and an engaging model system for trying out emerging multireference methods. Among these methods, driven similarity renormalization group (DSRG) and its perturbative version (DSRG-MRPT2) are especially attractive in terms of both accuracy and moderate numerical complexity. In this paper, we applied density matrix renormalization group (DMRG) followed by DSRG-MRPT2 for the calculation of vertical and adiabatic excitation energies into the 2A_g^-, 1B_u^-, and 1B_u⁺ electronic states of polyenes containing from 8 to 13 conjugating double bonds acting as a model for natural carotenoids. It was shown that the DSRG flow parameter should be adjusted to ensure both the energy convergence with respect to it and the agreement with the experimental data. With the increased flow parameter, the proposed combination of methods provides a reasonable agreement with the experiment. The deviations of the adiabatic excitation energies are less than 1000 cm^-1 for the 2A_g^- and less than 3000 cm^-1 for the excited states of the B_u symmetry, which in terms of accuracy significantly outperforms the N-electron valence state perturbation theory. At the same time, DSRG-MRPT2 is shown to be robust with respect to variation of quality of the DMRG reference wave function such as the orbital optimization or the number of electronic states in the averaging.