Excited-State Geometry Optimization with the Density Matrix Renormalization Group, as Applied to Polyenes

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.

Kylin 1.0: An ab-initio density matrix renormalization group quantum chemistry program

The accurate evaluation of electron correlations is highly necessary for the proper descriptions of the electronic structures in strongly correlated molecules, ranging from bond-dissociating molecules, polyradicals, to large conjugated molecules and transition metal complexes. For this purpose, in this paper, a new ab-initio quantum chemistry program Kylin 1.0 for electron correlation calculations at various quantum many-body levels, including configuration interaction (CI), perturbation theory (PT), and density matrix renormalization group (DMRG), is presented. Furthermore, fundamental quantum chemistry methods such as Hartree-Fock self-consistent field (HF-SCF) and the complete active space SCF (CASSCF) are also implemented. The Kylin 1.0 program possesses the following features: (1) a matrix product operator (MPO) formulation-based efficient DMRG implementation for describing static electron correlation within a large active space composed of more than 100 orbitals, supporting both U 1 n × U 1 S z $$ \mathrm{U}{(1)}_{\mathrm{n}}\times \mathrm{U}{(1)}_{{\mathrm{S}}_{\mathrm{z}}} $$ and U 1 n × SU 2 S $$ \mathrm{U}{(1)}_{\mathrm{n}}\times \mathrm{SU}{(2)}_{\mathrm{S}} $$ symmetries; (2) an efficient second-order DMRG-self-consistent field (SCF) implementation; (3) an externally contracted multi-reference CI (MRCI) and Epstein-Nesbet PT with DMRG reference wave functions for including the remaining dynamic electron correlation outside the large active spaces. In this paper, we introduce the capabilities and numerical benchmark examples of the Kylin 1.0 program.

Algorithm for analytic nuclear energy gradients of state averaged DMRG-CASSCF theory with newly derived coupled-perturbed equations

We present an algorithm for evaluating analytic nuclear energy gradients of the state-averaged density matrix renormalization group complete-active-space self-consistent field (SA-DMRG-CASSCF) theory based on the newly derived coupled-perturbed (CP) DMRG-CASSCF equations. The Lagrangian for the conventional SA-CASSCF analytic gradient theory is extended to the SA-DMRG-CASSCF variant that can fully consider a whole set of constraints on the parameters of multi-root canonical matrix product states formed at all the DMRG block configurations. An efficient algorithm to solve the CP-DMRG-CASSCF equations for determining the multipliers was developed. The complexity of the resultant analytic gradient algorithm is overall the same as that of the unperturbed SA-DMRG-CASSCF algorithm. In addition, a reduced-scaling approach was developed to directly compute the SA reduced density matrices (SA-RDMs) and their perturbed ones without calculating separate state-specific RDMs. As part of our implementation scheme, we neglect the term associated with the constraint on the active orbitals in terms of the active-active rotation in the Lagrangian. Thus, errors from the true analytic gradients may be caused in this scheme. The proposed gradient algorithm was tested with the spin-adapted implementation by checking how accurately the computed analytic energy gradients reproduce numerical gradients of the SA-DMRG-CASSCF energies using a common number of renormalized bases. The illustrative applications show that the errors are sufficiently small when using a typical number of the renormalized bases, which is required to attain adequate accuracy in DMRG's total energies.

Analytic Gradients for Selected Configuration Interaction

We develop analytic gradients for selected configuration interaction wave functions. Despite all pairs of molecular orbitals now potentially having to be considered for the coupled perturbed Hartree-Fock equations, we show that degenerate orbital pairs belonging to different irreducible representations in the largest abelian subgroup do not need to be included and instabilities due to degeneracies are avoided. We introduce seminumerical gradients and use them to validate the analytic approach even when near degeneracies are present due to high-symmetry geometries being slightly distorted to break symmetry. The method is applied to carbon monoxide, ammonia, square planar H₄, hexagonal planar H₆, and methane for a range of bond lengths where we demonstrate that analytic gradients for selected configuration interaction can approach the quality of full configuration interaction yet only use a very small fraction of its determinants.

Relativistic Kramers-Unrestricted Exact-Two-Component Density Matrix Renormalization Group

In this work we develop a variational relativistic density matrix renormalization group (DMRG) approach within the exact-two-component (X2C) framework (X2C-DMRG), using spinor orbitals optimized with the two-component relativistic complete active space self-consistent field. We investigate fine-structure splittings of p- (Ga, In, Tl) and d-block (Sc, Y, La) atoms and excitation energies of monohydride molecules (GeH, SnH, and TlH) with X2C-DMRG calculations using an all-electron relativistic Hamiltonian in a Kramers-unrestricted basis. We find that X2C-DMRG yields accurate ²P and ²D splittings compared to multireference configuration interaction with singles and doubles (MRCISD). We also investigated the degree of symmetry breaking in the atomic multiplets and convergence of electron correlation in the total energies. Symmetry breaking can be large in some cases (∼30 meV); however, increasing the number of renormalized block states m for the DMRG optimization recovers the symmetry breaking by several orders of magnitude. Encouragingly, we find the convergence of electron correlation to be close to MRCISDTQ5 quality. Relativistic X2C-DMRG approaches are important for cases where spin-orbit coupling is significant and the underlying reference wave function requires a large determinantal space. We are able to obtain quantitatively correct fine-structure splittings for systems up to 10¹⁹ number of determinants with traditional CI approaches, which are currently unfeasible to converge for the field.

Near-Exact Nuclear Gradients of Complete Active Space Self-Consistent Field Wave Functions

In this paper, we study the nuclear gradients of heat bath configuration interaction self-consistent field (HCISCF) wave functions and use them to optimize molecular geometries for various molecules.We show that the HCISCF nuclear gradients are fairly insensitive to the size of the "selected" variational space, which allows us to reduce the computational cost without introducing significant error.The ability of HCISCF to treat larger active spaces combined with the flexibility for users to control the computational cost makes the method very attractive for studying strongly correlated systems which require a larger active space than possible with complete active space self-consistent field (CASSCF).Finally, we study the realistic catalyst, Fe(PDI), and highlight some of the challenges this system poses for density functional theory (DFT).We demonstrate how HCISCF can clarify the energetic stability of geometries obtained from DFT when the results are strongly dependent on the functional.We also use the HCISCF gradients to optimize geometries for this species and study the adiabatic singlet-triplet gap. During geometry optimization, we find that multiple near-degenerate local minima exist on the triplet potential energy surface.

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.

Computational design of quantum defects in two-dimensional materials

Missing atoms or atom substitutions (point defects) in crystal lattices in two-dimensional (2D) materials are potential hosts for emerging quantum technologies, such as single-photon emitters and spin quantum bits (qubits). First-principles-guided design of quantum defects in 2D materials is paving the way for rational spin qubit discovery. Here we discuss the frontier of first-principles theory development and the challenges in predicting the critical physical properties of point defects in 2D materials for quantum information technology, in particular for optoelectronic and spin-optotronic properties. Strong many-body interactions at reduced dimensionality require advanced electronic structure methods beyond mean-field theory. The great challenges for developing theoretical methods that are appropriate for strongly correlated defect states, as well as general approaches for predicting spin relaxation and the decoherence time of spin defects, are yet to be addressed.

Theoretical Study of the Photoisomerization Mechanism of All-Trans-Retinyl Acetate

The compound 9-cis-retinyl acetate (9-cis-RAc) is a precursor to 9-cis-retinal, which has potential application in the treatment of some hereditary diseases of the retina. An attractive synthetic route to 9-cis-RAc is based on the photoisomerization reaction of the readily available all-trans-RAc. In the present study, we examine the mechanism of the photoisomerization reaction with the use of state-of-the-art electronic structure calculations for two polyenic model compounds: tEtEt-octatetraene and tEtEtEc-2,6-dimethyl-1,3,5,7,9-decapentaene. The occurrence of photoisomerization is attributed to a chain-kinking mechanism, whereby a series of S₁/S₀ conical intersections associated with kinking deformations at different positions along the polyenic chain mediate internal conversion to the S₀ state, and subsequent isomerization around one of the double bonds. Two other possible photoisomerization mechanisms are taken into account, but they are rejected as incompatible with simulation results and/or the available spectroscopic data.

Electronic Structure Methods for the Description of Nonadiabatic Effects and Conical Intersections

Nonadiabatic effects are ubiquitous in photophysics and photochemistry, and therefore, many theoretical developments have been made to properly describe them. Conical intersections are central in nonadiabatic processes, as they promote efficient and ultrafast nonadiabatic transitions between electronic states. A proper theoretical description requires developments in electronic structure and specifically in methods that describe conical intersections between states and nonadiabatic coupling terms. This review focuses on the electronic structure aspects of nonadiabatic processes. We discuss the requirements of electronic structure methods to describe conical intersections and nonadiabatic couplings, how the most common excited state methods perform in describing these effects, and what the recent developments are in expanding the methodology and implementing nonadiabatic couplings.

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.

Near-Exact CASSCF-Level Geometry Optimization with a Large Active Space using Adaptive Sampling Configuration Interaction Self-Consistent Field Corrected with Second-Order Perturbation Theory (ASCI-SCF-PT2)

An accurate description of electron correlation is one of the most challenging problems in quantum chemistry. The exact electron correlation can be obtained by means of full configuration interaction (FCI). A simple strategy for approximating FCI at a reduced computational cost is selected CI (SCI), which diagonalizes the Hamiltonian within only the chosen configuration space. Recovery of the contributions of the remaining configurations is possible with second-order perturbation theory. Here, we apply adaptive sampling configuration interaction (ASCI) combined with molecular orbital optimizations (ASCI-SCF) corrected with second-order perturbation theory (ASCI-SCF-PT2) for geometry optimization by implementing the analytical nuclear gradient algorithm for ASCI-PT2 with the Z-vector (Lagrangian) formalism. We demonstrate that for phenalenyl radicals and anthracene, optimized geometries and the number of unpaired electrons can be obtained at nearly the CASSCF accuracy by incorporating PT2 corrections and extrapolating them. We demonstrate the current algorithm's utility for optimizing the equilibrium geometries and electronic structures of six-ring-fused polycyclic aromatic hydrocarbons and 4-periacene.

DMRG on Top of Plane-Wave Kohn-Sham Orbitals: A Case Study of Defected Boron Nitride

In this paper, we analyze the numerical aspects of the inherent multireference density matrix renormalization group (DMRG) calculations on top of the periodic Kohn-Sham density functional theory using the complete active space approach. The potential of the framework is illustrated by studying hexagonal boron nitride nanoflakes embedding a charged single boron vacancy point defect by revealing a vertical energy spectrum with a prominent multireference character. We investigate the consistency of the DMRG energy spectrum from the perspective of sample size, basis size, and active space selection protocol. Results obtained from standard quantum chemical atom-centered basis calculations and plane-wave based counterparts show excellent agreement. Furthermore, we also discuss the spectrum of the periodic sheet which is in good agreement with extrapolated data of finite clusters. These results pave the way toward applying the DMRG method in extended correlated solid-state systems, such as point defect qubit in wide band gap semiconductors.

Molecular Photochemistry: Recent Developments in Theory

Photochemistry is a fascinating branch of chemistry that is concerned with molecules and light. However, the importance of simulating light-induced processes is reflected also in fields as diverse as biology, material science, and medicine. This Minireview highlights recent progress achieved in theoretical chemistry to calculate electronically excited states of molecules and simulate their photoinduced dynamics, with the aim of reaching experimental accuracy. We focus on emergent methods and give selected examples that illustrate the progress in recent years towards predicting complex electronic structures with strong correlation, calculations on large molecules, describing multichromophoric systems, and simulating non-adiabatic molecular dynamics over long time scales, for molecules in the gas phase or in complex biological environments.

Ultrafast Spectroscopy of Photoactive Molecular Systems from First Principles: Where We Stand Today and Where We Are Going

Computational spectroscopy is becoming a mandatory tool for the interpretation of the complex, and often congested, spectral maps delivered by modern non-linear multi-pulse techniques. The fields of Electronic Structure Methods, Non-Adiabatic Molecular Dynamics, and Theoretical Spectroscopy represent the three pillars of the virtual ultrafast optical spectrometer, able to deliver transient spectra in silico from first principles. A successful simulation strategy requires a synergistic approach that balances between the three fields, each one having its very own challenges and bottlenecks. The aim of this Perspective is to demonstrate that, despite these challenges, an impressive agreement between theory and experiment is achievable now regarding the modeling of ultrafast photoinduced processes in complex molecular architectures. Beyond that, some key recent developments in the three fields are presented that we believe will have major impacts on spectroscopic simulations in the very near future. Potential directions of development, pending challenges, and rising opportunities are illustrated.

A general automatic method for optimal construction of matrix product operators using bipartite graph theory

Constructing matrix product operators (MPOs) is at the core of the modern density matrix renormalization group (DMRG) and its time dependent formulation. For the DMRG to be conveniently used in different problems described by different Hamiltonians, in this work, we propose a new generic algorithm to construct the MPO of an arbitrary operator with a sum-of-products form based on the bipartite graph theory. We show that the method has the following advantages: (i) it is automatic in that only the definition of the operator is required; (ii) it is symbolic thus free of any numerical error; (iii) the complementary operator technique can be fully employed so that the resulting MPO is globally optimal for any given order of degrees of freedom; and (iv) the symmetry of the system could be fully employed to reduce the dimension of MPO. To demonstrate the effectiveness of the new algorithm, the MPOs of Hamiltonians ranging from the prototypical spin-boson model and the Holstein model to the more complicated ab initio electronic Hamiltonian and the anharmonic vibrational Hamiltonian with the sextic force field are constructed. It is found that for the former three cases, our automatic algorithm can reproduce exactly the same MPOs as the optimally hand-crafted ones already known in the literature.

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

Knowledge about excited states of carotenoids is essential for understanding photophysical processes underlying photosynthesis. However, due to the presence of a large number of optically dark states, experimental study of the excited-state manifold is limited to a significant extent. In this paper, we apply high-level ab initio quantum chemical methods to study the low-lying excited states of polyenes containing from 8 to 13 conjugated double bonds, which serve as a model for natural carotenoids. Vertical and adiabatic excitation energies from the ground 1A_g^- state to the excited 2A_g^-, 1B_u⁺, and 1B_u^- states were evaluated by means of density matrix renormalization group (DMRG) with NEVPT2 perturbative correction. The energies of all excited states are highly sensitive to nuclear geometry, especially the 2A_g^- state. Thus, the 2A_g^- and 1B_u⁺ states interchange their relative positions upon geometry relaxation, while the vertical excitation energy to the 2A_g^- state is rather high. At the same time, the 1B_u^- state energy is shown to be higher than other studied excited states at any geometry. With relaxed geometries of the excited states, absorption and transient absorption spectra were calculated within the Franck-Condon approximation bridging the gap between experimental spectroscopic data and computational results.

Multireference Electron Correlation Methods: Journeys along Potential Energy Surfaces

Multireference electron correlation methods describe static and dynamical electron correlation in a balanced way and, therefore, can yield accurate and predictive results even when single-reference methods or multiconfigurational self-consistent field theory fails. One of their most prominent applications in quantum chemistry is the exploration of potential energy surfaces. This includes the optimization of molecular geometries, such as equilibrium geometries and conical intersections and on-the-fly photodynamics simulations, both of which depend heavily on the ability of the method to properly explore the potential energy surface. Because such applications require nuclear gradients and derivative couplings, the availability of analytical nuclear gradients greatly enhances the scope of quantum chemical methods. This review focuses on the developments and advances made in the past two decades. A detailed account of the analytical nuclear gradient and derivative coupling theories is presented. Emphasis is given to the software infrastructure that allows one to make use of these methods. Notable applications of multireference electron correlation methods to chemistry, including geometry optimizations and on-the-fly dynamics, are summarized at the end followed by a discussion of future prospects.

Elucidating the multi-configurational character of the firefly dioxetanone anion and its prototypes in the biradical region using full valence active spaces

We analyzed the near-degenerate states of the firefly dioxetanone anion (FDO-) and its prototypes, especially in the biradical region, using multi-configurational approaches. The importance of utilizing full valence active spaces by means of density-matrix renormalization group self-consistent field (DMRG-SCF) calculations was described. Our results revealed that the neglect of some valence orbitals can affect the quantitative accuracy in later multi-reference calculations or the qualitative conclusion when optimizing conical intersections. Using all of the relevant valence orbitals of FDO-, we confirmed that there were two conical intersections, as reported in previous work, and that the intersecting states were changed when the active space was enlarged. Beyond these, we found that there were strong interactions between states in the biradical regions, in which the changes in entanglements can be used to visualize the interacting state evolution.

Approximate Analytical Gradients and Nonadiabatic Couplings for the State-Average Density Matrix Renormalization Group Self-Consistent-Field Method

We present an approximate scheme for analytical gradients and nonadiabatic couplings for calculating state-average density matrix renormalization group self-consistent-field wave function. Our formalism follows closely the state-average complete active space self-consistent-field (SA-CASSCF) ansatz, which employs a Lagrangian, and the corresponding Lagrange multipliers are obtained from a solution of the coupled-perturbed CASSCF (CP-CASSCF) equations. We introduce a definition of the matrix product state (MPS) Lagrange multipliers based on a single-site tensor in a mixed-canonical form of the MPS, such that a sweep procedure is avoided in the solution of the CP-CASSCF equations. We apply our implementation to the optimization of a conical intersection in 1,2-dioxetanone, where we are able to fully reproduce the SA-CASSCF result up to arbitrary accuracy.

Stimulated X-ray Raman and Absorption Spectroscopy of Iron-Sulfur Dimers

Iron-sulfur complexes play an important role in biological processes such as metabolic electron transport. A detailed understanding of the mechanism of long-range electron transfer requires knowledge of the electronic structure of the complexes, which has traditionally been challenging to obtain, either by theory or by experiment, but the situation has begun to change with advances in quantum chemical methods and intense free electron laser light sources. We compute the spectra for stimulated X-ray Raman spectroscopy (SXRS) and absorption spectroscopy of homovalent and mixed-valence [2Fe-2S] complexes, using the ab initio density matrix renormalization group algorithm. The simulated spectra show clear signatures of the theoretically predicted dense low-lying excited states within the d-d manifold. Furthermore, the difference in spectral intensity between the absorption-active and Raman-active states provides a potential mechanism to selectively excite states by a proper tuning of the excitation pump, to access the electronic dynamics within this manifold.

Analytic first-order derivatives of partially contracted n-electron valence state second-order perturbation theory (PC-NEVPT2)

A balanced treatment of dynamic and static electron correlation is important in computational chemistry, and multireference perturbation theory (MRPT) is able to do this at a reasonable computational cost. In this paper, analytic first-order derivatives, specifically gradients and dipole moments, are developed for a particular MRPT method, state-specific partially contracted n-electron valence state second-order perturbation theory (PC-NEVPT2). Only one linear equation needs to be solved for the derivative calculation if the Z-vector method is employed, which facilitates the practical application of this approach. A comparison of the calculated results with experimental geometrical parameters of O₃ indicates excellent agreement although the calculated results for O₃ ^- are slightly outside the experimental error bars. The 0-0 transition energies of various methylpyrimidines and trans-polyacetylene are calculated by performing geometry optimizations and seminumerical second-order geometrical derivative calculations. In particular, the deviations of 0-0 transition energies of trans-polyacetylene from experimental values are consistently less than 0.1 eV with PC-NEVPT2, indicating the reliability of the method. These results demonstrate the importance of adding dynamic electron correlation on top of methods dominated by static electron correlation and of developing analytic derivatives for highly accurate methods.

Quantum Trajectory Mean-Field Method for Nonadiabatic Dynamics in Photochemistry

The mixed quantum-classical dynamical approaches have been widely used to study nonadiabatic phenomena in photochemistry and photobiology, in which the time evolutions of the electronic and nuclear subsystems are treated based on quantum and classical mechanics, respectively. The key issue is how to deal with coherence and decoherence during the propagation of the two subsystems, which has been the subject of numerous investigations for a few decades. A brief description on Ehrenfest mean-field and surface-hopping (SH) methods is first provided, and then different algorithms for treatment of quantum decoherence are reviewed in the present paper. More attentions were paid to quantum trajectory mean-field (QTMF) method under the picture of quantum measurements, which is able to overcome the overcoherence problem. Furthermore, the combined QTMF and SH algorithm is proposed in the present work, which takes advantages of the QTMF and SH methods. The potential to extend the applicability of the QTMF method was briefly discussed, such as the generalization to other type of nonadiabatic transitions, the combination with multiscale computational models, and possible improvements on its accuracy and efficiency by using machine-learning techniques.

Triplet-Pair States in Organic Semiconductors

Entanglement of states is one of the most surprising and counterintuitive consequences of quantum mechanics, with potent applications in cryptography and computing. In organic semiconductor materials, one particularly significant manifestation is the spin-entangled triplet-pair state, which consists of a pair of localized triplet excitons coupled into an overall spin-0, -1, or -2 configuration. The most widely analyzed of these is the spin-0 pair, denoted ¹(TT), which was initially invoked in the 1960s to explain delayed fluorescence in acene films. It is considered an essential gateway state for triplet-triplet annihilation and the reverse process, singlet fission, enabling interconversion between one singlet and two triplet excitons without any change in overall spin. This state has returned to the forefront of organic materials research in recent years, thanks both to its central role in the resurgent field of singlet fission and to its implication in a host of exotic new photophysical behaviors. Here we review the properties of triplet-pair states, from first principles to recent experimental results.

Excited state tracking during the relaxation of coordination compounds

The ability to locate minima on electronic excited states (ESs) potential energy surfaces both in the case of bright and dark states is crucial for a full understanding of photochemical reactions. This task has become a standard practice for small- to medium-sized organic chromophores thanks to the constant developments in the field of computational photochemistry. However, this remains a very challenging effort when it comes to the optimization of ESs of transition metal complexes (TMCs), not only due to the presence of several electronic ESs close in energy, but also due to the complex nature of the ESs involved. In this article, we present a simple yet powerful method to follow an ES of interest during a structural optimization in the case of TMCs, based on the use of a compact hole-particle representation of the electronic transition, namely the natural transition orbitals (NTOs). State tracking using NTOs is unambiguously accomplished by computing the mono-electronic wave function overlap between consecutive steps of the optimization. Here, we demonstrate that this simple but robust procedure works not only in the case of the cytosine but also in the case of the ES optimization of a ruthenium nitrosyl complex which is very problematic with standard approaches. © 2019 The Authors. Journal of Computational Chemistry published by Wiley Periodicals, Inc.

Delocalization Errors in Density Functional Theory Are Essentially Quadratic in Fractional Occupation Number

Approximate functionals used in practical density functional theory (DFT) deviate from the piecewise linear behavior of the exact functional for fractional charges. This deviation causes excess charge delocalization, which leads to incorrect densities, molecular properties, barrier heights, band gaps, and excitation energies. We present a simple delocalization function for characterizing this error and find it to be almost perfectly linear vs the fractional electron number for systems spanning in size from the H atom to the C₁₂H₁₄ polyene. This causes the delocalization energy error to be a quadratic polynomial in the fractional electron number, which permits us to assess the comparative performance of 47 popular and recent functionals through the curvature. The quadratic form further suggests that information about a single fractional charge is sufficient to eliminate the principal source of delocalization error. Generalizing traditional two-point information like ionization potentials or electron affinities to account for a third, fractional charge-based data point could therefore permit fitting/tuning of functionals with lower delocalization error.

Role of the Dark 2Ag State in Donor-Acceptor Copolymers as a Pathway for Singlet Fission: A DMRG Study

The mechanism of intramolecular singlet fission in donor-acceptor-type copolymers, especially the role of the dark 2A_g state, is not so clear. In this Letter, the electronic structure of the benzodithiophene (B)-thiophene-1,1-dioxide (TDO) copolymer is calculated by density matrix renormalization group theory with the Pariser-Parr-Pople model. We find that the dark 2A_g state is the lowest singlet excited state and is nearly degenerate with the 1B_u state. So, a fast internal conversion from 1B_u to 2A_g state is highly possible. The 2A_g state has a strong triplet pair character, localized on two neighboring acceptor units, which indicates that it is an intermediate state for the intramolecular singlet fission process. With the increase of the donor-acceptor push-pull strength in our model, this triplet pair character of the 2A_g state becomes more prominent, and meanwhile the binding energy of this coupled triplet pair state decreases, which favors the separation into two uncoupled triplet states. We propose a model in which the competition between the singlet fission process and the nonradiative decay process from the 2A_g state would determine the final quantum yield.

Multiconfigurational Effects in Theoretical Resonance Raman Spectra

We analyze resonance Raman spectra of the nucleobase uracil in the short‐time approximation calculated with multiconfigurational methods. We discuss the importance of static electron correlation by means of density‐matrix renormalization group self‐consistent field (DMRG‐SCF) calculations. Our DMRG‐SCF results reveal that a minimal active orbital space that leads to a qualitatively correct description of the resonance Raman spectrum of uracil should encompass parts of the σ/σ* bonding/anti‐bonding orbitals of the pyrimidine ring. We trace these findings back to the considerable entanglement between the σ/σ* bonding/anti‐bonding as well as valence π/π* orbitals in the excited‐state electronic structure of uracil, which indicates non‐negligible non‐dynamical correlation effects that are less pronounced in the electronic ground state.