Theory and practice of uncommon molecular electronic configurations

Observation of SOMO-HOMO Inversion in a Neutral Polycyclic Conjugated Hydrocarbon

We report the generation of a nonbenzenoid polycyclic conjugated hydrocarbon, which consists of a biphenyl moiety substituted by indenyl units at the 4,4' positions, on ultrathin sodium chloride films by tip-induced chemistry. Single-molecule characterization by scanning tunneling and atomic force microscopy reveals an open-shell biradical ground state with a peculiar electronic configuration wherein the singly occupied molecular orbitals (SOMOs) are lower in energy than the highest occupied molecular orbital (HOMO).

Bistability between π-diradical open-shell and closed-shell states in indeno[1,2-a]fluorene

Indenofluorenes are non-benzenoid conjugated hydrocarbons that have received great interest owing to their unusual electronic structure and potential applications in nonlinear optics and photovoltaics. Here we report the generation of unsubstituted indeno[1,2-a]fluorene on various surfaces by the cleavage of two C-H bonds in 7,12-dihydroindeno[1,2-a]fluorene through voltage pulses applied by the tip of a combined scanning tunnelling microscope and atomic force microscope. On bilayer NaCl on Au(111), indeno[1,2-a]fluorene is in the neutral charge state, but it exhibits charge bistability between neutral and anionic states on the lower-workfunction surfaces of bilayer NaCl on Ag(111) and Cu(111). In the neutral state, indeno[1,2-a]fluorene exhibits one of two ground states: an open-shell π-diradical state, predicted to be a triplet by density functional and multireference many-body perturbation theory calculations, or a closed-shell state with a para-quinodimethane moiety in the as-indacene core. We observe switching between open- and closed-shell states of a single molecule by changing its adsorption site on NaCl.

Chiral π-Conjugated Double Helical Aminyl Diradical with the Triplet Ground State

We report a neutral high-spin diradical of chiral C2-symmetric bis[5]diazahelicene with ΔEST ≈ 0.4 kcal mol-1, as determined by EPR spectroscopy/SQUID magnetometry. The diradical is the most persistent among all high-spin aminyl radicals reported to date by a factor of 20, with a half-life of up to 6 days in 2-MeTHF at room temperature. Its triplet ground state and excellent persistence may be associated with the unique spin density distribution within the dihydrophenazine moiety, which characterizes two effective 3-electron C-N bonds analogous to the N-O bond of a nitroxide radical. The enantiomerically enriched (ee ≥ 94%) (MM)- and (PP)-enantiomers of the precursors to the diradicals are obtained by either preparative chiral supercritical fluid chromatography or resolution via functionalization with the chiral auxiliary of the C2-symmetric racemic tetraamine. The barrier for the racemization of the solid tetraamine is ΔG‡ = 43 ± 0.01 kcal mol-1 in the 483-523 K range. The experimentally estimated lower limit of the barrier for the racemization of a diradical, ΔG‡ ≥ 26 kcal mol-1 in 2-MeTHF at 293 K, is comparable to the DFT-determined barrier of ΔG‡ = 31 kcal mol-1 in the gas phase at 298 K. While the enantiomerically pure tetraamine displays strong chiroptical properties, with anisotropy factor |g| = |Δε|/ε = 0.036 at 376 nm, |g| ≈ 0.005 at 548 nm of the high-spin diradical is comparable to that recently reported triplet ground-state diradical dication. Notably, the radical anion intermediate in the generation of diradical exhibits a large SOMO-HOMO inversion, SHI = 35 kcal mol-1.

Peculiar Differences between Two Copper Complexes Containing Similar Redox-Active Ligands: Density Functional and Multiconfigurational Calculations

Transition metal complexes featuring redox-active ligands often exhibit multiple redox states, influenced by the interplay between the metal center and the ligand. This study delves into the electronic structures of two mononuclear complexes of copper with two similar redox-active urea azine ligands. The ligands differ by the replacement of an NCH3 moiety by an S atom in the ligand backbone. Experimental analysis yields pronounced electronic structural disparities between these complexes, observable in both the solution and solid phases. Conventional quantum chemical methods, such as density functional theory using different functionals (B3LYP, TPSSh, and CAM-B3LYP), remain inadequate to rationalize the observed spectroscopic anomalies. However, a multiconfigurational approach elucidates the disparate behaviors of these complexes. Multireference perturbation theory, based on complete active space self-consistent field computations, identifies Cu(I) in the case of the complex with the NCH3 containing ligands and a state with substantial Cu(II) contributions in the case of the complex with the S atom containing ligands. In contrast, DFT indicates Cu(I) in both scenarios.

Electronic Properties of Graphene Nano-Parallelograms: A Thermally Assisted Occupation DFT Computational Study

In this computational study, we investigate the electronic properties of zigzag graphene nano-parallelograms (GNPs), which are parallelogram-shaped graphene nanoribbons of various widths and lengths, using thermally assisted occupation density functional theory (TAO-DFT). Our calculations revealed a monotonic decrease in the singlet-triplet energy gap as the GNP length increased. The GNPs possessed singlet ground states for all the cases examined. With the increase of GNP length, the vertical ionization potential and fundamental gap decreased monotonically, while the vertical electron affinity increased monotonically. Some of the GNPs studied were found to possess fundamental gaps in the range of 1-3 eV, lying in the ideal region relevant to solar energy applications. Besides, as the GNP length increased, the symmetrized von Neumann entropy increased monotonically, denoting an increase in the degree of the multi-reference character associated with the ground state GNPs. The occupation numbers and real-space representation of active orbitals indicated that there was a transition from the nonradical nature of the shorter GNPs to the increasing polyradical nature of the longer GNPs. In addition, the edge/corner localization of the active orbitals was found for the wider and longer GNPs.

From Stable Radicals to Thermally Robust High-Spin Diradicals and Triradicals

Stable radicals and thermally robust high-spin di- and triradicals have emerged as important organic materials due to their promising applications in diverse fields. New fundamental properties, such as SOMO/HOMO inversion of orbital energies, are explored for the design of new stable radicals, including highly luminescent ones with good photostability. A relation with the singlet-triplet energy gap in the corresponding diradicals is proposed. Thermally robust high-spin di- and triradicals, with energy gaps that are comparable to or greater than a thermal energy at room temperature, are more challenging to synthesize but more rewarding. We summarize a number of high-spin di- and triradicals, based on nitronyl nitroxides that provide a relation between the experimental pairwise exchange coupling constant J/k in the high-spin species vs experimental hyperfine coupling constants in the corresponding monoradicals. This relation allows us to identify outliers, which may correspond to radicals where J/k is not measured with sufficient accuracy. Double helical high-spin diradicals, in which spin density is delocalized over the chiral π-system, have been barely explored, with the sole example of such high-spin diradical possessing alternant π-system with Kekulé resonance form. Finally, we discuss a high-spin diradical with electrical conductivity and derivatives of triangulene diradicals.

Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory: Multireference Advantages with the Practicality of Linear Response Theory

The density functional theory (DFT) and linear response (LR) time-dependent (TD)-DFT are of the utmost importance for routine computations. However, the single reference formulation of DFT suffers in the description of open-shell singlet systems such as diradicals and bond-breaking. LR-TDDFT, on the other hand, finds difficulties in the modeling of conical intersections, doubly excited states, and core-level excitations. In this Perspective, we demonstrate that many of these limitations can be overcome by recently developed mixed-reference (MR) spin-flip (SF)-TDDFT, providing an alternative yet accurate route for such challenging situations. Empowered by the practicality of the LR formalism, it is anticipated that MRSF-TDDFT can become one of the major workhorses for general routine tasks.

Synergy of semiempirical models and machine learning in computational chemistry

Catalyzed by enormous success in the industrial sector, many research programs have been exploring data-driven, machine learning approaches. Performance can be poor when the model is extrapolated to new regions of chemical space, e.g., new bonding types, new many-body interactions. Another important limitation is the spatial locality assumption in model architecture, and this limitation cannot be overcome with larger or more diverse datasets. The outlined challenges are primarily associated with the lack of electronic structure information in surrogate models such as interatomic potentials. Given the fast development of machine learning and computational chemistry methods, we expect some limitations of surrogate models to be addressed in the near future; nevertheless spatial locality assumption will likely remain a limiting factor for their transferability. Here, we suggest focusing on an equally important effort-design of physics-informed models that leverage the domain knowledge and employ machine learning only as a corrective tool. In the context of material science, we will focus on semi-empirical quantum mechanics, using machine learning to predict corrections to the reduced-order Hamiltonian model parameters. The resulting models are broadly applicable, retain the speed of semiempirical chemistry, and frequently achieve accuracy on par with much more expensive ab initio calculations. These early results indicate that future work, in which machine learning and quantum chemistry methods are developed jointly, may provide the best of all worlds for chemistry applications that demand both high accuracy and high numerical efficiency.

Heteroatom-Based Diradical(oid)s

Heteroatom-centered diradical(oid)s have been in the focus of molecular main group chemistry for nearly 30 years. During this time, the diradical concept has evolved and the focus has shifted to the rational design of diradical(oid)s for specific applications. This review article begins with some important theoretical considerations of the diradical and tetraradical concept. Based on these theoretical considerations, the design of diradical(oid)s in terms of ligand choice, steric, symmetry, electronic situation, element choice, and reactivity is highlighted with examples. In particular, heteroatom-centered diradical reactions are discussed and compared with closed-shell reactions such as pericyclic additions. The comparison between closed-shell reactivity, which proceeds in a concerted manner, and open-shell reactivity, which proceeds in a stepwise fashion, along with considerations of diradical(oid) design, provides a rational understanding of this interesting and unusual class of compounds. The application of diradical(oid)s, for example in small molecule activation or as molecular switches, is also highlighted. The final part of this review begins with application-related details of the spectroscopy of diradical(oid)s, followed by an update of the heteroatom-centered diradical(oid)s and tetraradical(oid)s published in the last 10 years since 2013.

Structural, Thermodynamic, and Spectroscopic Evolution in the Hydration of Copper(II) Ions, Cu2+(H2O)2-8

Gas-phase clusters of the hydrated Cu(II) cation with 2-8 water molecules were investigated using ab initio quantum chemistry. Isomer structures, energies, and vibrational spectra were computed across this size range, yielding a qualitative picture of this ion as an intact Cu2+ hydrate that also partially oxidizes the surrounding water network at equilibrium. At sufficient cluster sizes, these ion hydrates also become thermodynamically preferred over competitive Cu(II) hydroxide hydrates. Competitive coordination environments were found to exist at some cluster sizes, due to both hydrogen-bonding and d-orbital chemical effects, and the dominant coordination number was found in some cases to be temperature-dependent. Clear spectral signatures of the ion's coordination environment were computed to exist at each cluster size, which should make experimental verification of these computational predictions straightforward. Through comparison to recent studies of hydrated CuOH+, the effective charge on the metal center was shown to converge to approximately +1.5 in both cases, despite qualitatively different behavior of their radical spin densities. Therefore, nominally Cu(II) ions exhibit considerable electronic, chemical, and structural flexibility. The electronic origins of this flexibility─including key roles played by the water network itself─are investigated in this work and should provide a conceptual foundation for future studies of copper-based, water-oxidation catalysts.

Rational Design of Persistent Phosphorus-Centered Singlet Tetraradicals and Their Use in Small-Molecule Activation

Biradicals are important intermediates in the process of bond formation and breaking. While main-group-element-centered biradicals have been thoroughly studied, much less is known about tetraradicals, as their very low stability has hampered their isolation and use in small-molecule activation. Herein, we describe the search for persistent phosphorus-centered tetraradicals. Starting from an s-hydrindacenyl skeleton, we investigated the introduction of four phosphorus-based radical sites linked by an N-R unit and bridged by a benzene moiety. By varying the size of the substituent R, we finally succeeded in isolating a persistent P-centered singlet tetraradical, 2,6-diaza-1,3,5,7-tetraphospha-s-hydrindacene-1,3,5,7-tetrayl (1), in good yields. Furthermore, it was demonstrated that tetraradical 1 can be utilized for the activation of small molecules such as molecular hydrogen or alkynes. In addition to the synthesis of P-centered tetraradicals, the comparison with other known tetraradicals as well as biradicals is described on the basis of quantum mechanical calculations with respect to its multireference character, coupling of radical electrons, and aromaticity. The strong coupling of radical electrons enables selective discrimination between the first and the second activations of small molecules, which is shown by the example of H₂ addition. The mechanism of hydrogen addition is investigated with parahydrogen-induced hyperpolarization NMR studies and DFT calculations.

TAO-DFT with the Polarizable Continuum Model

For the ground-state properties of gas-phase nanomolecules with multi-reference character, thermally assisted occupation (TAO) density functional theory (DFT) has recently been found to outperform the widely used Kohn-Sham DFT when traditional exchange-correlation energy functionals are employed. Aiming to explore solvation effects on the ground-state properties of nanomolecules with multi-reference character at a minimal computational cost, we combined TAO-DFT with the PCM (polarizable continuum model). In order to show its usefulness, TAO-DFT-based PCM (TAO-PCM) was used to predict the electronic properties of linear acenes in three different solvents (toluene, chlorobenzene, and water). According to TAO-PCM, in the presence of these solvents, the smaller acenes should have nonradical character, and the larger ones should have increasing polyradical character, revealing striking similarities to the past findings in the gas phase.

Redox-Addressable Single-Molecule Junctions Incorporating a Persistent Organic Radical

Integrating radical (open-shell) species into non-cryogenic nanodevices is key to unlocking the potential of molecular electronics. While many efforts have been devoted to this issue, in the absence of a chemical/electrochemical potential the open-shell character is generally lost in contact with the metallic electrodes. Herein, single-molecule devices incorporating a 6-oxo-verdazyl persistent radical have been fabricated using break-junction techniques. The open-shell character is retained at room temperature, and electrochemical gating permits in situ reduction to a closed-shell anionic state in a single-molecule transistor configuration. Furthermore, electronically driven rectification arises from bias-dependent alignment of the open-shell resonances. The integration of radical character, transistor-like switching, and rectification in a single molecular component paves the way to further studies of the electronic, magnetic, and thermoelectric properties of open-shell species.

TAO-DFT fictitious temperature made simple

Over the past few years, thermally-assisted-occupation density functional theory (TAO-DFT) [J.-D. Chai, J. Chem. Phys., 2012, 136, 154104] has been proved to be an efficient electronic structure method for investigating the ground-state properties of large electronic systems with strong static correlation effects. In TAO-DFT, the strength of static correlation in an electronic system at zero temperature is closely related to the so-called fictitious temperature (i.e., the temperature of the corresponding noninteracting reference system). In this work, we propose a simple model to define the optimal system-independent fictitious temperature of a given energy functional in TAO-DFT. Besides, we employ this model to determine the optimal system-independent fictitious temperature of a global hybrid functional in TAO-DFT as a function of the fraction of exact exchange. In addition, we adopt TAO-DFT with various global hybrid functionals and system-independent fictitious temperatures to explore the ground-state properties of several electronic systems with strong static correlation effects, such as the linear acenes and cyclic carbon chains. Furthermore, we discuss the role of exact exchange and an optimal system-independent fictitious temperature in TAO-DFT. Owing to the much reduced self-interaction error, TAO-DFT with exact exchange and an optimal system-independent fictitious temperature can accurately predict the radical character and bond length alternation of cyclic carbon chains (with even number of carbon atoms), which are challenging problems for traditional electronic structure methods.

Optimal system-independent fictitious temperature θ of TAO-GH as a function of the fraction of exact exchange a_x.

Carbazole Isomerism in Helical Radical Cations: Spin Delocalization and SOMO-HOMO Level Inversion in the Diradical State

We report a new molecular design to afford persistent chiral organic open-shell systems with configurational stability and an inversion in energy of the singly occupied molecular orbital (SOMO) and the highest doubly occupied molecular orbital (HOMO) for both mono- and diradical states. The unpaired electron delocalization within the designed extended helical π-conjugated systems is a crucial factor to reach chemical stabilities, which is not obtained using the classical steric protection approach. The unique features of the obtained helical monoradicals allow an exploration of the chiral intramolecular electron transfer (IET) process in solvents of different polarity by means of optical and chiroptical spectroscopies, resulting in an unprecedented electronic circular dichroism (ECD) sign inversion for the radical transitions. We also characterized the corresponding helical diradicals, which show near-infrared electronic circular dichroism at wavelengths up to 1100 nm and an antiferromagnetic coupling between the spins, with an estimated singlet-triplet gap (ΔE_ST) of about -1.2 kcal mol^-1. The study also revealed an intriguing double SOMO-HOMO inversion (SHI) electronic configuration for these diradicals, providing new insight regarding the peculiar energetic ordering of radical orbitals and the impact on the corresponding (chiral) optoelectronic properties.

Mechanistic Advantages of Organotin Molecular EUV Photoresists

Extreme ultraviolet (EUV)-induced radiation exposure chemistry in organotin-oxo systems, represented by the archetypal [(R-Sn)₁₂O₁₄(OH)₆](A)₂ cage, has been investigated with density functional theory. Upholding existing experimental evidence of Sn-C cleavage-dominant chemistry, computations have revealed that either electron attachment or ionization can single-handedly trigger tin-carbon bond cleavage, partially explaining the current EUV sensitivity advantage of metal oxide systems. We have revealed that tin atoms at different parts of the molecule react differently to ionization and electron attachment and have identified such selectivity as a result of local coordination chemistry instead of the macro geometry of the molecule. An ionization-deprotonation pathway has also been identified to explain the observed evolution of an anion conjugate acid upon exposure and anion mass dependence in resist sensitivity.

Defect-Induced π-Magnetism into Non-Benzenoid Nanographenes

The synthesis of nanographenes (NGs) with open-shell ground states have recently attained increasing attention in view of their interesting physicochemical properties and great prospects in manifold applications as suitable materials within the rising field of carbon-based magnetism. A potential route to induce magnetism in NGs is the introduction of structural defects, for instance non-benzenoid rings, in their honeycomb lattice. Here, we report the on-surface synthesis of three open-shell non-benzenoid NGs (A1, A2 and A3) on the Au(111) surface. A1 and A2 contain two five- and one seven-membered rings within their benzenoid backbone, while A3 incorporates one five-membered ring. Their structures and electronic properties have been investigated by means of scanning tunneling microscopy, noncontact atomic force microscopy and scanning tunneling spectroscopy complemented with theoretical calculations. Our results provide access to open-shell NGs with a combination of non-benzenoid topologies previously precluded by conventional synthetic procedures.

Electronic Properties of Carbon Nanobelts Predicted by Thermally-Assisted-Occupation DFT

Accurate prediction of properties of large-scale multi-reference (MR) electronic systems remains difficult for traditional computational methods (e.g., the Hartree-Fock theory and Kohn-Sham density functional theory (DFT)). Recently, thermally-assisted-occupation (TAO)-DFT has been demonstrated to offer reliable description of electronic properties of various large-scale MR electronic systems. Consequently, in this work, TAO-DFT is used to unlock the electronic properties associated with C-Belt[n] (i.e., the carbon nanobelts containing n fused 12-membered carbon rings). Our calculations show that for all the system sizes reported (n = 4-24), C-Belt[n] have singlet ground states. In general, the larger the size of C-Belt[n], the more pronounced the MR character of ground-state C-Belt[n], as evident from the symmetrized von Neumann entropy and the occupation numbers of active TAO-orbitals. Furthermore, the active TAO-orbitals are delocalized along the circumference of C-Belt[n], as evident from the visualization of active TAO-orbitals.

Femtosecond dynamics of diphenylpropynylidene in ethanol and dichloromethane

Carbon chains with an odd number of C atoms are reactive intermediates with a high biradical character. Here we report a joint experimental and computational investigation of the dynamics of diphenylpropynylidene, C₆H₅-C₃-C₆H₅, in dichloromethane and ethanol. The biradical is generated by ultraviolet light from 1,3-diphenyldiazopropyne. Electron paramagnetic resonance spectra are recorded to elucidate the spin multiplicity and geometry of the biradical. In both solvents a triplet ground state at 4 K is verified. Transient absorption spectra provide insight into the fate of the biradical. A study in deaerated dichloromethane permits us to follow the photophysics of diphenylpropynylidene and to extract time constants for its vibrational as well as electronic relaxation. In the presence of oxygen, a more complex photochemistry is observed that permits us to derive a model for the reaction of the biradical with O₂. In ethanol, the spectra recorded in the presence and absence of O₂ are very similar, which can be explained by the similarity of the chromophores of the reaction products.

Large magnetic exchange coupling in rhombus-shaped nanographenes with zigzag periphery

Nanographenes with zigzag edges are predicted to manifest non-trivial π-magnetism resulting from the interplay of concurrent electronic effects, such as hybridization of localized frontier states and Coulomb repulsion between valence electrons. This provides a chemically tunable platform to explore quantum magnetism at the nanoscale and opens avenues towards organic spintronics. The magnetic stability in nanographenes is thus far greatly limited by the weak magnetic exchange coupling, which remains below the room-temperature thermal energy. Here, we report the synthesis of large rhombus-shaped nanographenes with zigzag peripheries on gold and copper surfaces. Single-molecule scanning probe measurements show an emergent magnetic spin singlet ground state with increasing nanographene size. The magnetic exchange coupling in the largest nanographene (C₇₀H₂₂, containing five benzenoid rings along each edge), determined by inelastic electron tunnelling spectroscopy, exceeds 100 meV or 1,160 K, which outclasses most inorganic nanomaterials and survives on a metal electrode.

Short-range DFT energy correction to multiconfigurational wave functions for open-shell systems

Electronic structure methods emerging from the combination of multiconfigurational wave functions and density functional theory (DFT) aim to take advantage of the strengths of the two nearly antagonistic theories. One of the common strategies employed to merge wave function theory (WFT) with DFT relies on the range separation of the Coulomb operator in which DFT functionals take care of the short-distance part, while long-range inter-electronic interactions are evaluated by using the chosen wave function method (WFT-srDFT). In this work, we uncover the limitations of WFT-srDFT in the characterization of open-shell systems. We show that spin polarization effects have a major impact on the (short-range) DFT exchange energy and are of vital importance in order to provide a balanced description between closed and open-shell configurations. We introduce different strategies to account for spin polarization in the short range based on the definition of a spin polarized electron density and with the use of short-range exact exchange. We test the performance of these approaches in the dissociation of the hydrogen molecule, the calculation of energy gaps in spin-triplet atoms and molecular diradicals, and the characterization of low-lying states of the gallium dimer. Our results indicate that the use of short-range DFT correlation in combination with a (full-range) multiconfigurational wave function might be an excellent approach for the study of open-shell molecules and largely improves the performance of WFT and WFT-srDFT.

Energetics of Formation of Cyclacenes from 2,3-Didehydroacenes and Implications for Astrochemistry

The carriers of the diffuse interstellar bands (DIBs) are still largely unknown although polycyclic aromatic hydrocarbons, carbon chains, and fullerenes are likely candidates. A recent analysis of the properties of n-acenes of general formula C4n+2 H2n+4 suggested that these could be potential carriers of some DIBs. Dehydrogenation reactions of n-acenes after absorption of an interstellar UV photon may result in dehydroacenes. Here the reaction energies and barriers for formation of n-cyclacenes from 2,3-didehydroacenes (n-DDA) by intramolecular Diels-Alder reaction to dihydro-etheno-cyclacenes (n-DEC) followed by ejection of ethyne by retro-Diels-Alder reactions are analyzed using thermally assisted occupation density functional theory (TAO-DFT) for n=10-20. It is found that the barriers for each of the steps depend on the ring strain of the underlying n-cyclacene, and that the ring strain of n-DEC is about 75 % of that of the corresponding n-cyclacene. In each case, ethyne extrusion is the step with the highest energy barrier, but these barriers are smaller than CH bond dissociation energies, suggesting that formation of cyclacenes is an energetically conceivable fate of n-acenes after multiple absorption of UV photons.

Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory (MRSF-TDDFT) as a Simple yet Accurate Method for Diradicals and Diradicaloids

Due to their multiconfigurational nature featuring strong electron correlation, accurate description of diradicals and diradicaloids is a challenge for quantum chemical methods. The recently developed mixed-reference spin-flip (MRSF)-TDDFT method is capable of describing the multiconfigurational electronic states of these systems while avoiding the spin-contamination pitfalls of SF-TDDFT. Here, we apply MRSF-TDDFT to study the adiabatic singlet-triplet (ST) gaps in a series of well-known diradicals and diradicaloids. On average, MRSF displays a very high prediction accuracy of the adiabatic ST gaps with the mean absolute error (MAE) amounting to 0.14 eV. In addition, MRSF is capable of accurately describing the effect of the Jahn-Teller distortion occurring in the trimethylenemethane diradical, the violation of the Hund rule in a series of the didehydrotoluene diradicals, and the potential energy surfaces of the didehydrobenzene (benzyne) diradicals. A convenient criterion for distinguishing diradicals and diradicaloids is suggested on the basis of the easily obtainable quantities. In all of these cases, which are difficult for the conventional methods of density functional theory (DFT), MRSF shows results consistent with the experiment and the high-level ab initio computations. Hence, the present study documents the reliability and accuracy of MRSF and lays out the guidelines for its application to strongly correlated molecular systems.

Azadiphosphaindane-1,3-diyls: A Class of Resonance-Stabilized Biradicals

Conversion of 1,2-bis(dichlorophosphino)benzene with sterically demanding primary amines led to the formation of 1,3-dichloro-2-aza-1,3-diphosphaindanes of the type C6 H4 (μ-PCl)2 N-R. Reduction yielded the corresponding 2-aza-1,3-diphosphaindane-1,3-diyls (1), which can be described as phosphorus-centered singlet biradical(oid)s. Their stability depends on the size of the substituent R: While derivatives with R=Dmp (2,6-dimethylphenyl) or Ter (2,6-dimesitylphenyl) underwent oligomerization, the derivative with very bulky R=tBu Bhp (2,6-bis(benzhydryl)-4-tert-butylphenyl) was stable with respect to oligomerization in its monomeric form. Oligomerization involved activation of the fused benzene ring by a second equivalent of the monomeric biradical and can be regarded as formal [2+2] (poly)addition reaction. Calculations indicate that the biradical character in 1 is comparable with literature-known P-centered biradicals. Ring-current calculations show aromaticity within the entire ring system of 1.

Singly Occupied Molecular Orbital−Highest Occupied Molecular Orbital (SOMO−HOMO) Conversion

Singly occupied molecular orbital−highest occupied molecular orbital (SOMO−HOMO) conversion (inversion), SHC, is a phenomenon in which the SOMO is lower in energy than the doubly occupied molecular orbitals (DOMO, HOMO). A non-Aufbau electronic structure leads to unique properties such as a switch in bond dissociation energy and the generation of high-spin species on one-electron oxidation. In addition, the pronounced photostability of these species has been reported recently for application in organic light-emitting devices. In this review article, we summarise the chemistry of SOMO−HOMO converted (inverted) species reported to date.

Visible-Light-Mediated Heterocycle Functionalization via Geometrically Interrupted [2+2] Cycloaddition

The [2+2] photocycloaddition is the most valuable and intensively investigated photochemical process. Here we demonstrate that irradiation of N-acryloyl heterocycles with blue LED light (440 nm) in the presence of an IrIII complex leads to efficient and high yielding fused γ-lactam formation across a range of substituted heterocycles. Quantum calculations show that the reaction proceeds via cyclization in the triplet excited state to yield a 1,4-diradical; intersystem crossing leads preferentially to the closed shell singlet zwitterion. This is geometrically restricted from undergoing recombination to yield a cyclobutane by the planarity of the amide substituent. A prototropic shift leads to the observed bicyclic products in what can be viewed as an interrupted [2+2] cycloaddition.

In Pursuit of Panchromatic Absorption in Metal Coordination Complexes: Experimental Delineation of the HOMO Inversion Model Using Pseudo-Octahedral Complexes of Diarylamido Ligands

The ability of a compound to broadly absorb light across the incident solar spectrum is an important design target in the development of molecular photosensitizers. The 'HOMO inversion' model predicts that for [(tpy)₂Fe]²⁺ (tpy = 2,2':6',2″-terpyridine) compounds, adjusting the character of the highest occupied molecular orbital (HOMO) from metal-centered to ligand-centered can drastically improve photophysical properties by broadening absorption in the visible and increasing molar extinction coefficients. In an effort to experimentally realize strong, panchromatic absorption, a tridentate N^N^-^N diarylamido ligand bearing flanking benzannulated N-heterocyclic donors (^tBuL) was used to prepare deeply colored, pseudo-octahedral coordination complexes of a range of first-row transition and main-group metals [(^tBuL)₂M^0/+; M = Fe, Co, Ni, Zn, Ga]. While the Fe(II) congener exhibits the sought-after broad absorption, isostructural and isoelectronic complexes of other first-row transition and main-group metals show vastly different absorption and redox properties. Density functional theory (DFT) calculations point toward the relative energies of the metal d orbitals and ligand orbitals as the source of major changes in electronic structure, confirming aspects and limitations of the predictive 'HOMO inversion' model in experimentally realized systems with implications for the design of abundant transition-metal sensitizers with broad, panchromatic absorptive properties.

Heterocyclopentanediyls vs Heterocyclopentadienes: A Question of Silyl Group Migration

The reaction of the singlet biradical [P(μ-NHyp)]₂ (Hyp = hypersilyl, (Me₃Si)₃Si) with different isonitriles afforded a series of five-membered N₂P₂C heterocycles. Depending on the steric bulk of the substituent at the isonitrile, migration of a Hyp group was observed, resulting in two structurally similar but electronically very different isomers. As evidenced by comprehensive spectroscopic and theoretical studies, the heterocyclopentadiene isomer may be regarded as a rather unreactive closed-shell singlet species with one localized N═P and one C═P double bond, whereas the heterocyclopentanediyl isomer represents an open-shell singlet biradical with interesting photochemical properties, such as photoisomerization under irradiation with red light to a [2.1.0]-housane-type species.

Axially and Helically Chiral Cationic Radical Bicarbazoles: SOMO-HOMO Level Inversion and Chirality Impact on the Stability of Mono- and Diradical Cations

We report persistent chiral organic mono- and diradical cations based on bicarbazole molecular design with an unprecedented stability dependence on the type of chirality, namely, axial versus helical. An unusual chemical stability was observed for sterically unprotected axial bicarbazole radical in comparison with monocarbazole and helical bicarbazole ones. Such results were experimentally and theoretically investigated, revealing an inversion in energy of the singly occupied molecular orbital (SOMO) and the highest (doubly) occupied molecular orbital (HOMO) in both axial and helical bicarbazole monoradicals along with a subtle difference of electronic coupling between the two carbazole units, which is modulated by their relative dihedral angle and related to the type of chirality. Such findings allowed us to explore in depth the SOMO-HOMO inversion (SHI) in chiral radical molecular systems and provide new insights regarding its impact on the stability of organic radicals. Finally, these specific electronic properties allowed us to prepare a persistent, intrinsically chiral, diradical which notably displayed near-infrared electronic circular dichroism responses up to 1100 nm and almost degenerate singlet-triplet ground states with weak antiferromagnetic interactions evaluated by magnetometry experiments.

TAO-DFT-Based Ab Initio Molecular Dynamics

Recently, AIMD (ab initio molecular dynamics) has been extensively employed to explore the dynamical information of electronic systems. However, it remains extremely challenging to reliably predict the properties of nanosystems with a radical nature using conventional electronic structure methods (e.g., Kohn-Sham density functional theory) due to the presence of static correlation. To address this challenge, we combine the recently formulated TAO-DFT (thermally-assisted-occupation density functional theory) with AIMD. The resulting TAO-AIMD method is employed to investigate the instantaneous/average radical nature and infrared spectra of n-acenes containing n linearly fused benzene rings (n = 2-8) at 300 K. According to the TAO-AIMD simulations, on average, the smaller n-acenes (up to n = 5) possess a nonradical nature, and the larger n-acenes (n = 6-8) possess an increasing radical nature, showing remarkable similarities to the ground-state counterparts at 0 K. Besides, the infrared spectra of n-acenes obtained with the TAO-AIMD simulations are in qualitative agreement with the existing experimental data.

Ligand-Mediated Spin-State Changes in a Cobalt-Dipyrrin-Bisphenol Complex

The influence of a redox-active ligand on spin-changing events induced by the coordination of exogenous donors is investigated within the cobalt complex [CoII(DPP·2-)], bearing a redox-active DPP2- ligand (DPP = dipyrrin-bis(o,p-di-tert-butylphenolato) with a pentafluorophenyl moiety on the meso-position. This square-planar complex was subjected to the coordination of tetrahydrofuran (THF), pyridine, tBuNH2, and AdNH2 (Ad = 1-adamantyl), and the resulting complexes were analyzed with a variety of experimental (X-ray diffraction, NMR, UV-visible, high-resolution mass spectrometry, superconducting quantum interference device, Evans' method) and computational (density functional theory, NEVPT2-CASSCF) techniques to elucidate the respective structures, spin states, and orbital compositions of the corresponding octahedral bis-donor adducts, relative to [CoII(DPP·2-)]. This starting species is best described as an open-shell singlet complex containing a DPP·2- ligand radical that is antiferromagnetically coupled to a low-spin (S = 1/2) cobalt(II) center. The redox-active DPPn- ligand plays a crucial role in stabilizing this complex and in its facile conversion to the triplet THF adduct [CoII(DPP·2-)(THF)2] and closed-shell singlet pyridine and amine adducts [CoIII(DPP3-)(L)2] (L = py, tBuNH2, or AdNH2). Coordination of the weak donor THF to [CoII(DPP·2-)] changes the orbital overlap between the DPP·2- ligand radical π-orbitals and the cobalt(II) metalloradical d-orbitals, which results in a spin-flip to the triplet ground state without changing the oxidation states of the metal or DPP·2- ligand. In contrast, coordination of the stronger donors pyridine, tBuNH2, or AdNH2 induces metal-to-ligand single-electron transfer, resulting in the formation of low-spin (S = 0) cobalt(III) complexes [CoIII(DPP3-)(L)2] containing a fully reduced DPP3- ligand, thus explaining their closed-shell singlet electronic ground states.

TAO-DFT investigation of electronic properties of linear and cyclic carbon chains

It has been challenging to adequately investigate the properties of nanosystems with radical nature using conventional electronic structure methods. We address this challenge by calculating the electronic properties of linear carbon chains (l-CC[n]) and cyclic carbon chains (c-CC[n]) with n = 10-100 carbon atoms, using thermally-assisted-occupation density functional theory (TAO-DFT). For all the cases investigated, l-CC[n]/c-CC[n] are ground-state singlets, and c-CC[n] are energetically more stable than l-CC[n]. The electronic properties of l-CC[n]/c-CC[n] reveal certain oscillation patterns for smaller n, followed by monotonic changes for larger n. For the smaller carbon chains, odd-numbered l-CC[n] are more stable than the adjacent even-numbered ones; c-CC[[Formula: see text]]/c-CC[4m] are more/less stable than the adjacent odd-numbered ones, where m are positive integers. As n increases, l-CC[n]/c-CC[n] possess increasing polyradical nature in their ground states, where the active orbitals are delocalized over the entire length of l-CC[n] or the whole circumference of c-CC[n].

TAO-DFT Study on the Electronic Properties of Diamond-Shaped Graphene Nanoflakes

At the nanoscale, it has been rather troublesome to properly explore the properties associated with electronic systems exhibiting a radical nature using traditional electronic structure methods. Graphene nanoflakes, which are graphene nanostructures of different shapes and sizes, are typical examples. Recently, TAO-DFT (i.e., thermally-assisted-occupation density functional theory) has been formulated to tackle such challenging problems. As a result, we adopt TAO-DFT to explore the electronic properties associated with diamond-shaped graphene nanoflakes with n = 2-15 benzenoid rings fused together at each side, designated as n-pyrenes (as they could be expanded from pyrene). For all the n values considered, n-pyrenes are ground-state singlets. With increasing the size of n-pyrene, the singlet-triplet energy gap, vertical ionization potential, and fundamental gap monotonically decrease, while the vertical electron affinity and symmetrized von Neumann entropy (which is a quantitative measure of radical nature) monotonically increase. When n increases, there is a smooth transition from the nonradical character of the smaller n-pyrenes to the increasing polyradical nature of the larger n-pyrenes. Furthermore, the latter is shown to be related to the increasing concentration of active orbitals on the zigzag edges of the larger n-pyrenes.

One-electron oxidation of ds(5'-GGG-3') and ds(5'-G(8OG)G-3') and the nature of hole distribution: a density functional theory (DFT) study

Of particular interest in radiation-induced charge transfer processes in DNA is the extent of hole localization immediately after ionization and subsequent relaxation. To address this, we considered double stranded oligomers containing guanine (G) and 8-oxoguanine (8OG), i.e., ds(5'-GGG-3') and ds(5'-G8OGG-3') in B-DNA conformation. Using DFT, we calculated a variety of properties, viz., vertical and adiabatic ionization potentials, spin density distributions in oxidized stacks, solvent and solute reorganization energies and one-electron oxidation potential (E0) in the aqueous phase. Calculations for the vertical state of the -GGG- cation radical showed that the spin was found mainly (67%) on the middle G. However, upon relaxation to the adiabatic -GGG- cation radical, the spin localized (96%) on the 5'-G, as observed in experiments. Hole localizations on the middle G and 3'-G were higher in energy by 0.5 kcal mol-1 and 0.4 kcal mol-1, respectively, than that of 5'-G. In the -G8OGG- cation radical, the spin localized only on the 8OG in both vertical and adiabatic states. The calculated vertical ionization potentials of -GGG- and -G8OGG- stacks were found to be lower than that of the vertical ionization potential of a single G in DNA. The calculated E0 values of -GGG- and -G8OGG- stacks are 1.15 and 0.90 V, respectively, which owing to stacking effects are substantially lower than the corresponding experimental E0 values of their monomers (1.49 and 1.18 V, respectively). SOMO to HOMO level switching is observed in these oxidized stacks. Consequently, our calculations predict that local double oxidations in DNA will form triplet diradical states, which are especially significant for high LET radiations.

Probing the Partial Activation of Water by Open-Shell Interactions, Cl(H2O)1-4

The partial chemical activation of water by reactive radicals was examined computationally for small clusters of chlorine and water, Cl^•(H₂O)_n=1-4. Using an automated isomer-search procedure, dozens of unique, stable structures were computed. Among the resulting structural classes were intact, hydrated-chlorine isomers, as well as hydrogen-abstracted (HCl)(OH)(H₂O)_n-1 configurations. The latter showed increased stability as the degree of hydration increased, until n = 4, where a new class of structures was discovered with a chloride ion bound to an oxidized water network. The electronic structure of these three structural classes was investigated, and spectral signatures of this hydration-based evolution were connected to these electronic properties. An ancillary outcome of this detailed computational analysis, including coupled-cluster benchmarks, was the calibration of cost-effective quantum chemistry methods for future studies of these radical-water complexes.

Exchange Rules for Diradical π-Conjugated Hydrocarbons

A variety of planar π-conjugated hydrocarbons such as heptauthrene, Clar's goblet and, recently synthesized, triangulene have two electrons occupying two degenerate molecular orbitals. The resulting spin of the interacting ground state is often correctly anticipated as S = 1, extending the application of Hund's rules to these systems, but this is not correct in some instances. Here we provide a set of rules to correctly predict the existence of zero mode states as well as the spin multiplicity of both the ground state and the low-lying excited states, together with their open- or closed-shell nature. This is accomplished using a combination of analytical arguments and configuration interaction calculations with a Hubbard model, both backed by quantum chemistry methods with a larger Gaussian basis set. Our results go beyond the well established Lieb's theorem and Ovchinnikov's rule, as we address the multiplicity and the open-/closed-shell nature of both ground and excited states.

Electronic Properties of Triangle-Shaped Graphene Nanoflakes from TAO-DFT

Reliable prediction of the properties of nanosystems with radical nature has been tremendously challenging for common computational approaches. Aiming to overcome this, we employ thermally-assisted-occupation density functional theory (TAO-DFT) to investigate various electronic properties (e.g., singlet-triplet energy gaps, vertical ionization potentials, vertical electron affinities, fundamental gaps, symmetrized von Neumann entropy, active orbital occupation numbers, and visualization of active orbitals) associated with a series of triangle-shaped graphene nanoflakes with n fused benzene rings at each side (denoted as n-triangulenes), which can be extended from triangulene. According to our TAO-DFT results, the ground states of n-triangulenes are singlets for all the values of n studied (n = 3, 5, 7, 9, ..., and 21). Moreover, the larger the values of n, the more significant the polyradical nature of n-triangulenes. There are approximately (n - 1) unpaired electrons for the ground state of n-triangulene. The increasing polyradical nature of the larger n-triangulenes should be closely related to the fact that the active orbitals tend to be mainly concentrated at the periphery of n-triangulenes, apparently increasing with the molecular size.

High stability and luminescence efficiency in donor-acceptor neutral radicals not following the Aufbau principle

With their unusual electronic structures, organic radical molecules display luminescence properties potentially relevant to lighting applications; yet, their luminescence quantum yield and stability lag behind those of other organic emitters. Here, we designed donor-acceptor neutral radicals based on an electron-poor perchlorotriphenylmethyl or tris(2,4,6-trichlorophenyl)methyl radical moiety combined with different electron-rich groups. Experimental and quantum-chemical studies demonstrate that the molecules do not follow the Aufbau principle: the singly occupied molecular orbital is found to lie below the highest (doubly) occupied molecular orbital. These donor-acceptor radicals have a strong emission yield (up to 54%) and high photostability, with estimated half-lives reaching up to several months under pulsed ultraviolet laser irradiation. Organic light-emitting diodes based on such a radical emitter show deep-red/near-infrared emission with a maximal external quantum efficiency of 5.3%. Our results provide a simple molecular-design strategy for stable, highly luminescent radicals with non-Aufbau electronic structures.

Electronic Properties of Linear and Cyclic Boron Nanoribbons from Thermally-Assisted-Occupation Density Functional Theory

It remains rather difficult for traditional computational methods to reliably predict the properties of nanosystems, especially for those possessing pronounced radical character. Accordingly, in this work, we adopt the recently formulated thermally-assisted-occupation density functional theory (TAO-DFT) to study two-atom-wide linear boron nanoribbons l-BNR[2,n] and two-atom-wide cyclic boron nanoribbons c-BNR[2,n], which exhibit polyradical character when the n value (i.e., the number of boron atoms along the length of l-BNR[2,n] or the circumference of c-BNR[2,n]) is considerably large. We calculate various electronic properties associated with l-BNR[2,n] and c-BNR[2,n], with n ranging from 6 to 100. Our results show that l-BNR[2,n] and c-BNR[2,n] have singlet ground states for all the n values examined. The electronic properties of c-BNR[2,n] exhibit more pronounced oscillatory patterns than those of l-BNR[2,n] when n is small, and converge to the respective properties of l-BNR[2,n] when n is sufficiently large. The larger the n values, the stronger the static correlation effects that originate from the polyradical nature of these ribbons. Besides, the active orbitals are found to be delocalized along the length of l-BNR[2,n] or the circumference of c-BNR[2,n]. The analysis of the size-dependent electronic properties indicates that l-BNR[2,n] and c-BNR[2,n] can be promising for nanoelectronic devices.