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

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.

Block2: A comprehensive open source framework to develop and apply state-of-the-art DMRG algorithms in electronic structure and beyond

block2 is an open source framework to implement and perform density matrix renormalization group and matrix product state algorithms. Out-of-the-box it supports the eigenstate, time-dependent, response, and finite-temperature algorithms. In addition, it carries special optimizations for ab initio electronic structure Hamiltonians and implements many quantum chemistry extensions to the density matrix renormalization group, such as dynamical correlation theories. The code is designed with an emphasis on flexibility, extensibility, and efficiency and to support integration with external numerical packages. Here, we explain the design principles and currently supported features and present numerical examples in a range of applications.

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.

Exploring electronic energy level structure and excited electronic states of β-carotene using DFT

Carotenoids are a class of natural pigments that play a fundamental role in photosynthesis and optoelectronics. However, the complexity of their energy level structure and electronic states has prevented a clear interpretation of their photophysics and photochemistry. The mediating nonradiative decay of the bright S₂ state to the dark S₁ state of carotenoids involves a population of bridging intermediate state. Herein, time-dependent DFT was used to study the energy level and electronic excitation process of β-carotene. A π-π* transition and π electron delocalization of electron excitation could be inferred based on the difference in the electron cloud distribution of the HOMO and LUMO orbitals. Through the electronic transition contribution in the UV-vis spectra and the electron density difference between the ground state and the excited state, the electronic energy level structure and possible dark state were analyzed. On this basis, the electronic excitation process of β-carotene was theoretically studied by combining electron-hole analysis and transition density matrix (TDM). There was a charge transfer from the β-ionone ring to the long-chain in the (S₀) → (S₂), (S₀) → (S₄) and (S₀) → (S₅).

Density Matrix Renormalization Group for Transcorrelated Hamiltonians: Ground and Excited States in Molecules

We present the theory of a density matrix renormalization group (DMRG) algorithm which can solve for both the ground and excited states of non-Hermitian transcorrelated Hamiltonians and show applications in molecular systems. Transcorrelation (TC) accelerates the basis set convergence rate by including known physics (such as, but not limited to, the electron-electron cusp) in the Jastrow factor used for the similarity transformation. It also improves the accuracy of approximate methods such as coupled cluster singles and doubles (CCSD) as shown by recent studies. However, the non-Hermiticity of the TC Hamiltonians poses challenges for variational methods like DMRG. Imaginary-time evolution on the matrix product state (MPS) in the DMRG framework has been proposed to circumvent this problem, but this is currently limited to treating the ground state and has lower efficiency than the time-independent DMRG (TI-DMRG) due to the need to eliminate Trotter errors. In this work, we show that with minimal changes to the existing TI-DMRG algorithm, namely, replacing the original Davidson solver with the general Davidson solver to solve the non-Hermitian effective Hamiltonians at each site for a few low-lying right eigenstates, and following the rest of the original DMRG recipe, one can find the ground and excited states with improved efficiency compared to the original DMRG when extrapolating to the infinite bond dimension limit in the same basis set. An accelerated basis set convergence rate is also observed, as expected, within the TC framework.

Ultrafast Excited-State Dynamics of Carotenoids and the Role of the SX State

Carotenoids are natural pigments with multiple roles in photosynthesis. They act as accessory pigments by absorbing light where chlorophyll absorption is low, and they quench the excitation energy of neighboring chlorophylls under high-light conditions. The function of carotenoids depends on their polyene-like structure, which controls their excited-state properties. After light absorption to their bright S2 state, carotenoids rapidly decay to the optically dark S1 state. However, ultrafast spectroscopy experiments have shown the signatures of another dark state, termed SX. Here we shed light on the ultrafast photophysics of lutein, a xanthophyll carotenoid, by explicitly simulating its nonadiabatic excited-state dynamics in solution. Our simulations confirm the involvement of SX in the relaxation toward S1 and reveal that it is formed through a change in the nature of the S2 state driven by the decrease in the bond length alternation coordinate of the carotenoid conjugated chain.

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.

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.

Analytic Energy Gradients for the Driven Similarity Renormalization Group Multireference Second-Order Perturbation Theory

We derive analytic energy gradients of the driven similarity renormalization group (DSRG) multireference second-order perturbation theory (MRPT2) using the method of Lagrange multipliers. In the Lagrangian, we impose constraints for a complete-active-space self-consistent-field reference wave function and the semicanonical orthonormal molecular orbitals. Solving for the associated Lagrange multipliers is found to share the same asymptotic scaling of a single DSRG-MRPT2 energy computation. A pilot implementation of the DSRG-MRPT2 analytic gradients is used to optimize the geometry of the singlet and triplet states of p-benzyne. The equilibrium bond lengths and angles are similar to those computed via other MRPT2s and Mukherjee's multireference coupled cluster theory. An approximate DSRG-MRPT2 method that neglects the contributions of the three-body density cumulant is found to introduce negligible errors in the geometry of p-benzyne, lending itself to a promising low-cost approach for molecular geometry optimizations using large active spaces.

The Static-Dynamic-Static Family of Methods for Strongly Correlated Electrons: Methodology and Benchmarking

A series of methods (SDSCI, SDSPT2, iCI, iCIPT2, iCISCF(2), iVI, and iCAS) is introduced to accurately describe strongly correlated systems of electrons. Born from the (restricted) static-dynamic-static (SDS) framework for designing many-electron wave functions, SDSCI is a minimal multireference (MR) configuration interaction (CI) approach that constructs and diagonalizes a [Formula: see text] matrix for [Formula: see text] states, regardless of the numbers of orbitals and electrons to be correlated. If the full molecular Hamiltonian H in the QHQ block (which describes couplings between functions of the first-order interaction space Q) of the SDSCI CI matrix is replaced with a zeroth-order Hamiltonian [Formula: see text] before the diagonalization is taken, we obtain SDSPT2, a CI-like second-order perturbation theory (PT2). Unlike most variants of MRPT2, SDSPT2 treats single and multiple states in the same way and is particularly advantageous in the presence of near degeneracy. On the other hand, if the SDSCI procedure is repeated until convergence, we will have iterative CI (iCI), which can converge quickly from the above to the exact solutions (full CI) even when starting with a poor guess. When further combined with the selection of important configurations followed by a PT2 treatment of dynamic correlation, iCI becomes iCIPT2, which is a near-exact theory for medium-sized systems. The microiterations of iCI for relaxing the coefficients of contracted many-electron functions can be generalized to an iterative vector interaction (iVI) approach for finding exterior or interior roots of a given matrix, in which the dimension of the search subspace is fixed by either the number of target roots or the user-specified energy window. Naturally, iCIPT2 can be employed as the active space solver of the complete active space (CAS) self-consistent field, leading to iCISCF(2), which can further be combined with iCAS for automated selection of active orbitals and assurance of the same CAS for all states and all geometries. The methods are calibrated by taking the Thiel set of benchmark systems as examples. Results for the corresponding cations, a new set of benchmark systems, are also reported.

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.