A Merger of the Spin-Flip ORMAS Approach and the MC-PDFT Method

The SF-ORMAS-PDFT (spin-flip occupation restricted multiple active space-pair density functional theory) approach combines the SF-ORMAS-CI method with the MC-PDFT method to treat both static and dynamic correlation in multiconfigurational systems. The static correlation description is generated via the spin-flip approach, which uses a high-spin single reference determinant to treat excited states with multiconfigurational characters. The on-top pair density functional theory uses a translation scheme applied to GGA density functionals. The SF-ORMAS-PDFT scheme has also been combined with virtual valence orbitals (VVO), a well-defined subspace of the virtual molecular orbitals, giving rise to significant speedups relative to the use of the full virtual space. The accuracy of the SF-ORMAS-PDFT method is tested by calculating 65 vertical excitation energies of 12 small- and medium-sized organic molecules. The SF-ORMAS-PDFT vertical excitation energies calculated with VVOs are comparable to those calculated with the full virtual space. The SF-ORMAS-PDFT/6-31G(d) level of theory predicts the rotational barrier of ethylene to be 65.5 and 65.9 kcal/mol, with full virtual space and VVOs, respectively. These predicted barrier heights compare well with the experimental value of 65 kcal/mol.

Tetrafluorenofulvalene as a sterically frustrated open-shell alkene

Electronic and steric effects are known to greatly influence the structure, characteristics and reactivity of organic compounds. A typical π bond is weakened by oxidation (corresponding to the removal of electrons from bonding orbitals), by reduction (through addition of electrons to antibonding orbitals) and by unpairing of the bonding electrons, such as in the triplet state. Here we describe tetrafluorenofulvalene (TFF), a twisted, open-shell alkene for which these general rules do not hold. Through the synthesis, experimental characterization and computational analysis of its charged species spanning seven redox states, the central alkene bond in TFF is shown to become substantially stronger in the tri- and tetraanion, generated by chemical reduction. Furthermore, although its triplet state contains a weaker alkene bond than the singlet, in the quintet state its bond order increases substantially, yielding a flatter structure. This behaviour originates from the doubly bifurcated topology of the underlying spin system and can be rationalized by the balancing effects of benzenoid aromaticity and spin pairing.

State-specific solvation for restricted active space spin-flip (RAS-SF) wave functions based on the polarizable continuum formalism

The restricted active space spin-flip (RAS-SF) formalism is a particular form of single-reference configuration interaction that can describe some forms of strong correlation at a relatively low cost and which has recently been formulated for the description of charge-transfer excited states. Here, we introduce both equilibrium and nonequilibrium versions of a state-specific solvation correction for vertical transition energies computed using RAS-SF wave functions, based on the framework of a polarizable continuum model (PCM). Ground-state polarization is described using the solvent's static dielectric constant and in the nonequilibrium solvation approach that polarization is modified upon vertical excitation using the solvent's optical dielectric constant. Benchmark calculations are reported for well-studied models of photo-induced charge transfer, including naphthalene dimer, C₂H₄⋯C₂F₄, pentacene dimer, and perylene diimide (PDI) dimer, several of which are important in organic photovoltaic applications. For the PDI dimer, we demonstrate that the charge-transfer character of the excited states is enhanced in the presence of a low-dielectric medium (static dielectric constant ɛ₀ = 3) as compared to a gas-phase calculation (ɛ₀ = 1). This stabilizes mechanistic traps for singlet fission and helps to explain experimental singlet fission rates. We also examine the effects of nonequilibrium solvation on charge-separated states in an intramolecular singlet fission chromophore, where we demonstrate that the energetic ordering of the states changes as a function of solvent polarity. The RAS-SF + PCM methodology that is reported here provides a framework to study charge-separated states in solution and in photovoltaic materials.

On the role of dynamic electron correlation in non-orthogonal configuration interaction with fragments

Two different approaches have been implemented to include the effect of dynamic electron correlation in the Non-Orthogonal Configuration Interaction for Fragments (NOCI-F) method. The first is based on shifting the diagonal matrix elements of the NOCI matrix, while the second incorporates the dynamic correlation explicitly in the fragment wave functions used to construct the many-electron basis functions of the NOCI. The two approaches are illustrated for the calculation of the electronic coupling relevant in singlet fission and the coupling of spin moments in organic radicals. Comparison of the calculated diabatic couplings, the NOCI energies and wave functions shows that dynamic electron correlation is not only efficiently but also effectively incorporated by the shifting approach and can largely affect the coupling between electronic states. Also, it brings the NOCI coupling of the spin moments in close agreement with benchmark calculations.

Heavy atom oriented orbital angular momentum manipulation in metal-free organic phosphors

Metal-free purely organic phosphors (POPs) are emerging materials for display technologies, solid-state lighting, and chemical sensors. However, due to limitations in contemporary design strategies, the intrinsic spin-orbit coupling (SOC) efficiency of POPs remains low and their emission lifetime is pinned in the millisecond regime. Here, we present a design concept for POPs where the two main factors that control SOC-the heavy atom effect and orbital angular momentum-are tightly coupled to maximize SOC. This strategy is bolstered by novel natural-transition-orbital-based computational methods to visualize and quantify angular momentum descriptors for molecular design. To demonstrate the effectiveness of this strategy, prototype POPs were created having efficient room-temperature phosphorescence with lifetimes pushed below the millisecond regime, which were enabled by boosted SOC efficiencies beyond 102 cm-1 and achieved record-high efficiencies in POPs. Electronic structure analysis shows how discrete tuning of heavy atom effects and orbital angular momentum is possible within the proposed design strategy, leading to a strong degree of control over the resulting POP properties.

Revealing the Contest between Triplet-Triplet Exchange and Triplet-Triplet Energy Transfer Coupling in Correlated Triplet Pair States in Singlet Fission

Understanding the separation of the correlated triplet pair state ¹(TT) intermediate is critical for leveraging singlet fission to improve solar cell efficiency. This separation mechanism is dominated by two key interactions: (i) the exchange interaction (K) between the triplets which leads to the spin splitting of the biexciton state into ¹(TT),³(TT) and ⁵(TT) states, and (ii) the triplet-triplet energy transfer integral (t) which enables the formation of the spatially separated (but still spin entangled) state ¹(T···T). We develop a simple ab initio technique to compute both the biexciton exchange (K) and biexciton transfer coupling. Our key findings reveal new conditions for successful correlated triplet pair state dissociation. The biexciton exchange interaction needs to be ferromagnetic or negligible to the triplet energy transfer for favorable dissociation. We also explore the effect of chromophore packing to reveal geometries where these conditions are achieved for tetracene.

An Open-Shell Coronoid with Hybrid Chichibabin-Schlenk Conjugation

A hexaradicaloid molecule with alternating Kekulé and non-Kekulé connectivities between adjacent spin centers was obtained by fusing two classic conjugation motifs, found respectively in the Chichibabin and Schlenk hydrocarbons, into a coronoid structure. 1 H NMR, ESR and SQUID experiments, combined with computational analyses reveal that the system has a singlet ground state, characterized by a significant hexaradicaloid character ( γ 0 = 0.826, γ 1 = γ 2 = 0.773). It possesses multiple thermally accessible high-spin states (up to the septet), with uniform energy gaps of ca 1.0 kcal/mol between consecutive multiplicities. In line with its open-shell character, the coronoid has a small electronic bandgap of ca. 0.8 eV and undergoes two consecutive one-electron oxidations at low potentials, yielding cationic forms with extended near-infrared absorption. The hexaradicaloid, which combines open-shell and macrocyclic contributions to its π conjugation, provides an example of a design strategy for multistate spin switches and redox-amphoteric NIR dyes.

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.

Spin-Conserved and Spin-Flip Optical Excitations from the Bethe-Salpeter Equation Formalism

Like adiabatic time-dependent density-functional theory (TD-DFT), the Bethe–Salpeter equation (BSE) formalism of many-body perturbation theory, in its static approximation, is “blind” to double (and higher) excitations, which are ubiquitous, for example, in conjugated molecules like polyenes. Here, we apply the spin-flip ansatz (which considers the lowest triplet state as the reference configuration instead of the singlet ground state) to the BSE formalism in order to access, in particular, double excitations. The present scheme is based on a spin-unrestricted version of the GW approximation employed to compute the charged excitations and screened Coulomb potential required for the BSE calculations. Dynamical corrections to the static BSE optical excitations are taken into account via an unrestricted generalization of our recently developed (renormalized) perturbative treatment. The performance of the present spin-flip BSE formalism is illustrated by computing excited-state energies of the beryllium atom, the hydrogen molecule at various bond lengths, and cyclobutadiene in its rectangular and square-planar geometries.

Spin-Flip Pair-Density Functional Theory: A Practical Approach To Treat Static and Dynamical Correlations in Large Molecules

We present a practical approach to treat static and dynamical correlation accurately in large multiconfigurational systems. The static correlation is taken into account by using the spin-flip approach, which is well-known for capturing static correlation accurately at low-computational expense. Unlike previous approaches to add dynamical correlation to spin-flip models which use perturbation theory or coupled-cluster theory, we explore the ability to use the on-top pair-density functional theory approaches recently developed by Gagliardi and co-workers (J. Comput. Theor. Chem., 2014, 10, 3669). External relaxations are performed in the spin-flip calculations through a restricted active space framework for which a truncation scheme for the orbitals used in the external excitation is presented. The performance of the approach is demonstrated by computing energy gaps between ground and excited states for diradicals, triradicals, and linear polyacene chains ranging from naphthalene to dodecacene. Accurate results are obtained using the new approach for these challenging open-shell molecular systems.

Spin-flip methods in quantum chemistry

This Perspective discusses salient features of the spin-flip approach to strong correlation and describes different methods that sprung from this idea. The spin-flip treatment exploits the different physics of low-spin and high-spin states and is based on the observation that correlation is small for same-spin electrons. By using a well-behaved high-spin state as a reference, one can access problematic low-spin states by deploying the same formal tools as in the excited-state treatments (i.e., linear response, propagator, or equation-of-motion theories). The Perspective reviews applications of this strategy within wave function and density functional theory frameworks as well as the extensions for molecular properties and spectroscopy. The utility of spin-flip methods is illustrated by examples. Limitations and proposed future directions are also discussed.

Analytic non-adiabatic couplings for the spin-flip ORMAS method

Analytic non-adiabatic coupling matrix elements (NACME) are derived and implemented for the spin-flip occupation restricted multiple active space configuration interaction (SF-ORMAS-CI) method.

Hydroxyl Radical-Coupled Electron-Transfer Mechanism of Flavin-Dependent Hydroxylases

Class A flavin-dependent hydroxylases (FdHs) catalyze the hydroxylation of organic compounds in a site- and stereoselective manner. In stark contrast, conventional synthetic routes require environmentally hazardous reagents and give modest yields. Thus, understanding the detailed mechanism of this class of enzymes is essential to their rational manipulation for applications in green chemistry and pharmaceutical production. Both electrophilic substitution and radical intermediate mechanisms have been proposed as interpretations of FdH hydroxylation rates and optical spectra. While radical mechanistic steps are often difficult to examine directly, modern quantum chemistry calculations combined with statistical mechanical approaches can yield detailed mechanistic models providing insights that can be used to differentiate reaction pathways. In the current work, we report quantum mechanical/molecular mechanical (QM/MM) calculations on the fungal TropB enzyme that shows an alternative reaction pathway in which hydroxylation through a hydroxyl radical-coupled electron-transfer mechanism is significantly favored over electrophilic substitution. Furthermore, QM/MM calculations on several modified flavins provide a more consistent interpretation of the experimental trends in the reaction rates seen experimentally for a related enzyme, para-hydroxybenzoate hydroxylase. These calculations should guide future enzyme and substrate design strategies and broaden the scope of biological spin chemistry.

Ab Initio Probing of the Ground State of Tetraradicals: Breakdown of Hund's Multiplicity Rule

The electronic structure of organic σ-type polyradical including 2,4,6-tridehydropyridine radical cation (246-TDHP) and three isomers of tetradehydrobenzene (TDHB) have been studied using a computationally robust and cost-effective second-order multireference perturbative model which provides a balanced treatment of nondynamic and dynamic contributions to the electron correlation problem in the ground or excited electronic states which are imperative for predicting structural properties (e.g., ground state multiplicity, energy gaps between high-spin and low-spin states, etc.) of polyradicals. Energy gaps are useful to capture insight into the degree of interaction between the radical sites. An important finding of this study is that the tetraradicals considered here possess singlet ground states, contrary to Hund's rule. Present findings are in close agreement with the available high-level ab initio estimates at attainable cost implying that a perturbative description of the systems is adequate. The impact of N⁺ on the nature of ground state for the 246-TDHP have also analyzed. The singlet-triplet energy gaps for 1245- and 1234-TDHB are smaller than for o-benzyne mainly due to the ring strain. 1235-TDHB is 14.42 and 11.05 kcal/mol lower in energy than 1245- and 1234-isomers, respectively. IVO-SSMRPT predicts ¹A₁-³B₂ and ¹A₁-⁵B₂ gaps of 25.84 and 105.15 kcal/mol, respectively for the 246-TDHP cation.

5,10-Dimesityldiindeno[1,2-a:2',1'-i]phenanthrene: a stable biradicaloid derived from Chichibabin's hydrocarbon

A diindenophenanthrene biradicaloid, formally derived from Chichibabin's hydrocarbon, is obtained in a short, scalable synthesis.

A diindenophenanthrene biradicaloid, formally derived from Chichibabin's hydrocarbon, is obtained in a short, scalable synthesis. The present system is electron-rich and devoid of conjugated substituents, and still exhibits very good stability under ambient conditions. The introduction of the diindeno[1,2-a:2′,1′-i] phenanthrene ring framework results in a singlet biradicaloid system with an easily accessible triplet state (ΔE_S–T = –1.30 kcal mol^–1) and a small electronic bandgap (1.39 V). The stability limits of the title hydrocarbon were explored systematically in the solid state, to reveal an unusual thermally initiated hydrogen-scrambling oligomerization process.

Fully Conjugated [4]Chrysaorene. Redox-Coupled Anion Binding in a Tetraradicaloid Macrocycle

[4]Chrysaorene, a fully conjugated carbocyclic coronoid, is shown to be a low-bandgap π-conjugated system with a distinct open-shell character. The system shows good chemical stability and can be oxidized to well-defined radical cation and dication states. The cavity of [4]chrysaorene acts as an anion receptor toward halide ions with a particular selectivity toward iodides ( K_a = 207 ± 6 M^-1). The interplay between anion binding and redox chemistry is demonstrated using a ¹H NMR analysis in solution. In particular, a well-resolved, paramagnetically shifted spectrum of the [4]chrysaorene radical cation is observed, providing evidence for the inner binding of the iodide. The radical cation-iodide adduct can be generated in thin solid films of [4] chrysaorene by simple exposure to diiodine vapor.

The double exciton state of conjugated chromophores with strong diradical character: insights from TDDFT calculations

We show that, for large diradical character and strong orbital localization, the double exciton state is captured by TDUB3LYP calculations.

A general spin-complete spin-flip configuration interaction method

A new spin-complete spin-flip configuration interaction (SF-CI) method was developed using the ORMAS-CI algorithm, along with a perturbative correction for capturing dynamic and non-dynamic correlation.

A simplified ab initio treatment of diradicaloid structures produced from stretching and breaking chemical bonds

With a proper choice of active spaces, the single root perturbation theory employing improved virtual orbitals can flawlessly describe the ground, excited, ionized, and dissociated states having varying degrees of degeneracy at the expense of low computational cost.

On couplings and excimers: lessons from studies of singlet fission in covalently linked tetracene dimers

Electronic factors controlling singlet fission (SF) rates are investigated in covalently linked dimers of tetracene. Using covalent linkers, relative orientation of the individual chromophores can be controlled, maximizing the rates of SF. Structures with coplanar and staggered arrangements of tetracene moieties are considered. The electronic structure calculations and three-state kinetic model for SF rates provide explanations for experimentally observed low SF yields in coplanar dimers and efficient SF in staggered dimers. The calculations illuminate the role of the excimer formation in SF process. The structural relaxation in the S1 state leads to the increased rate of the multi-exciton (ME) state formation, but impedes the second step, separation of the ME state into independent triplets. The slower second step reduces SF yield by allowing other processes, such as radiationless relaxation, to compete with triplet generation. The calculations of electronic couplings also suggest an increased rate of radiationless relaxation at the excimer geometries. Thus, the excimer serves as a trap of the ME state. The effect of covalent linkers on the electronic factors and SF rates is investigated. In all considered structures, the presence of the linker leads to larger couplings, however, the effect on the overall rate is less straightforward, since the linkers generally result in less favorable energetics. This complex behavior once again illustrates the importance of integrative approaches that evaluate the overall rate, rather than focusing on specific electronic factors such as energies or couplings.

Singlet Fission in a Covalently Linked Cofacial Alkynyltetracene Dimer

Singlet fission is a process in which a singlet exciton converts into two triplet excitons. To investigate this phenomenon, we synthesized two covalently linked 5-ethynyl-tetracene (ET) dimers with differing degrees of intertetracene overlap: BET-X, with large, cofacial overlap of tetracene π-orbitals, and BET-B, with twisted arrangement between tetracenes exhibits less overlap between the tetracene π-orbitals. The two compounds were crystallographically characterized and studied by absorption and emission spectroscopy in solution, in PMMA and neat thin films. The results show that singlet fission occurs within 1 ps in an amorphous thin film of BET-B with high efficiency (triplet yield: 154%). In solution and the PMMA matrix the S1 of BET-B relaxes to a correlated triplet pair (1)(T1T1) on a time scale of 2 ps, which decays to the ground state without forming separated triplets, suggesting that triplet energy transfer from (1)(T1T1) to a nearby chromophore is essential for producing free triplets. In support of this hypothesis, selective excitation of BET-B doped into a thin film of diphenyltetracene (DPT) leads to formation of the (1)(T1T1) state of BET-B, followed by generation of both DPT and BET-B triplets. For the structurally cofacial BET-X, an intermediate forms in <180 fs and returns to the ground state more rapidly than BET-B. First-principles calculations predict a 2 orders of magnitude faster rate of singlet fission to the (1)(T1T1) state in BET-B relative to that of crystalline tetracene, attributing the rate increase to greater coupling between the S1 and (1)(T1T1) states and favorable energetics for formation of the separated triplets.

Computational Quantum Chemistry for Multiple-Site Heisenberg Spin Couplings Made Simple: Still Only One Spin-Flip Required

We provide a simple procedure for using inexpensive ab initio calculations to compute exchange coupling constants, J(AB), for multiradical molecules containing both an arbitrary number of radical sites and an arbitrary number of unpaired electrons. For a system comprised of 2M unpaired electrons, one needs only to compute states having the Ŝ(z) quantum number M - 1. Conveniently, these are precisely the states that are accessed by the family of single spin-flip methods. Building an effective Hamiltonian with these states allows one to extract all of the J(AB) constants in the molecule. Unlike approaches based on density functional theory, this procedure relies on neither spin-contaminated states nor nonunique spin-projection formulas. A key benefit is that it is possible to obtain completely spin-pure exchange coupling constants with inexpensive ab initio calculations. A couple of examples are provided to illustrate the approach, including a 4-nickel cubane complex and a 6-chromium horseshoe complex with 18 entangled electrons.