Singlet-Triplet Inversion in Triangular Boron Carbon Nitrides

The discovery of singlet-triplet (ST) inversion in some π-conjugated triangle-shaped boron carbon nitrides is a remarkable breakthrough that defies Hund's first rule. Deeply rooted in strong electron-electron interactions, ST inversion has garnered significant interest due to its potential to revolutionize triplet harvesting in organic LEDs. Using the well-established Pariser-Parr-Pople model for correlated electrons in π-conjugated systems, we employ a combination of CISDT and restricted active space configuration interaction calculations to investigate the photophysics of several triangular boron carbon nitrides. Our findings reveal that ST inversion in these systems is primarily driven by a network of alternating electron-donor and electron-acceptor groups in the molecular rim, rather than by the triangular molecular structure itself.

Unlocking Ultrafast Spin Transfer in Single-Magnetic-Center-Decorated Triangulene Systems

Triangulene, as a typical open-shell graphene fragment, has attracted widespread attention for nanospintronics, promising to serve as building blocks in spin-logic units. Here, using ab initio calculations, we systematically study the laser-induced ultrafast spin-dynamic processes on triangulene nanoflakes, decorated with a transition-metal atom. The results reveal a competition between the induced magnetic center and the carbon edge of the triangulene, resulting in the coexistence of dual spin-density-distribution patterns on such single-magnetic-center systems, thus opening up possibilities of complex spin-dynamic scenarios beyond the spin flip. Interestingly, no matter what direction the spin points to, it is possible to achieve reversible spin-transfer processes using the same laser pulse. Increasing the pool of elementary processes to contain not only spin-direction-dependent but also spin-direction-independent scenarios allows for more versatile spin-logic operations, including classical handling of information and quantum computing. In the present work, we suggest downscaling nanospintronic devices by integrating triangulene-based nanostructures.

Topological Design and Synthesis of High-Spin Aza-triangulenes without Jahn-Teller Distortions

The atomic doping of open-shell nanographenes enables precise tuning of their electronic and magnetic states, which is crucial for their promising potential applications in optoelectronics and spintronics. Among this intriguing class of molecules, triangulenes stand out with their size-dependent electronic properties and spin states, which can also be influenced by the presence of dopant atoms and functional groups. However, the occurrence of Jahn-Teller distortions in such systems can have a crucial impact on their total spin and requires further theoretical and experimental investigation. In this study, we examine the nitrogen-doped aza-triangulene series via a combination of density functional theory and on-surface synthesis. We identify a general trend in the calculated spin states of aza-[n]triangulenes of various sizes, separating them into two symmetry classes, one of which features molecules that are predicted to undergo Jahn-Teller distortions that reduce their symmetry and thus their total spin. We link this behavior to the location of the central nitrogen atom relative to the two underlying carbon sublattices of the molecules. Consequently, our findings reveal that neutral centrally doped aza-triangulenes have one less radical than their undoped counterparts, irrespective of their predicted symmetry. We follow this by demonstrating the on-surface synthesis of π-extended aza-[5]triangulene, a large member of the higher symmetry class without Jahn-Teller distortions, via a simple one-step annealing process on Cu(111) and Au(111). Using scanning probe microscopy and spectroscopy combined with theoretical calculations, we prove that the molecule is positively charged on the Au(111) substrate, with a high-spin quintet state of S = 2, the same total spin as undoped neutral [5]triangulene. Our study uncovers the correlation between the dopant position and the radical nature of high-spin nanographenes, providing a strategy for the design and development of these nanographenes for various applications.

Polyradical character assessment using multireference calculations and comparison with density-functional derived fractional occupation number weighted density analysis

The biradicaloid character of different types of polycyclic aromatic hydrocarbons (PAHs) based on small band gaps is an important descriptor to assess their opto-electronic properties. In this work, the unpaired electron densities and numbers of unpaired electrons (NU values) calculated at the high-level multireference averaged quadratic coupled-cluster (MR-AQCC) method are used to develop a test set to assess the capabilities of different biradical descriptors based on density functional theory. A benchmark collection of 29 different compounds has been selected. The DFT descriptors contain primarily the fractional occupation number weighted electron density (FOD) based on simplified thermally-assisted-occupation density functional theory (TAO-DFT) calculations, but the singlet-triplet energy difference and other descriptors denoted as y0 and nLUNO have been considered as well. After adjustment of the literature-recommended finite temperatures, a very good, detailed agreement between unpaired density and FOD analysis is observed which is also manifested in excellent statistical correlations. The other two descriptors also show good correlations even though the absolute scaling is not satisfactory. A new linear fit of FOD data to the MR-AQCC reference values leads to an improved regression relation for determining the recommended finite temperature value in dependence of the Hartree-Fock exchange. This provides the basis for fast and reliable assessment of the biradical character of many classes of PAHs without the need for performing computationally extended MR calculations.

Correlation vs. exchange competition drives the singlet-triplet excited-state inversion in non-alternant hydrocarbons

In this work, we focus on the understanding of the driving force behind the S1-T1 excited-state energy inversion (which would thus violate Hund's rule, making the S1 state lower in energy than the T1 state) of two non-benzenoid non-alternant hydrocarbons, composed of odd-membered rings. The molecules considered here have identical chemical composition but different atomic configuration in space. The delicate interplay between structural and electronic factors that might induce inversion and its energy extension, only by a few meV, is systematically investigated here by state-of-the-art calculations. Qualitative and quantitative accurate predictions are obtained employing post-HF methods, thanks to the balanced and careful inclusion of electron correlation effects. The obtained results might guide and rationalize new searches for molecules violating Hund's rule, concomitantly demonstrating the importance of key contributions from the theoretical method of choice.

On-Surface Synthesis and Characterization of a High-Spin Aza-[5]-Triangulene

Triangulenes are a class of open-shell triangular graphene flakes with total spin increasing with their size. In the last years, on-surface-synthesis strategies have permitted fabricating and engineering triangulenes of various sizes and structures with atomic precision. However, direct proof of the increasing total spin with their size remains elusive. In this work, we report the combined in-solution and on-surface synthesis of a large nitrogen-doped triangulene (aza-[5]-triangulene) on a Au(111) surface, and the detection of its high-spin ground state. Bond-resolved scanning tunneling microscopy images uncovered radical states distributed along the zigzag edges, which were detected as weak zero-bias resonances in scanning tunneling spectra. These spectral features reveal the partial Kondo screening of a high-spin state. Through a combination of several simulation tools, we find that the observed distribution of radical states is explained by a quintet ground state (S=2), instead of the quartet state (S=3/2) expected for the neutral species. This confirms that electron transfer to the metal substrate raises the spin of the ground state. We further provide a qualitative description of the change of (anti)aromaticity introduced by N-substitution, and its role in the charge stabilization on a surface, resulting in an S=2 aza-triangulene on Au(111).

Magnetic Coupling Control in Triangulene Dimers

Metal-free magnetism remains an enigmatic field, offering prospects for unconventional magnetic and electronic devices. In the pursuit of such magnetism, triangulenes, endowed with inherent spin polarization, are promising candidates to serve as monomers to construct extended structures. However, controlling and enhancing the magnetic interactions between the monomers persist as a significant challenge in molecular spintronics, as so far only weak antiferromagnetic coupling through the linkage has been realized, hindering their room temperature utilization. Herein, we investigate 24 triangulene dimers using first-principles calculations and demonstrate their tunable magnetic coupling (J), achieving unprecedented strong J values of up to -144 meV in a non-Kekulé dimer. We further establish a positive correlation between bandgap, electronic coupling, and antiferromagnetic interaction, thereby providing molecular-level insights into enhancing magnetic interactions. By twisting the molecular fragments, we demonstrate an effective and feasible approach to control both the sign and strength of J by tuning the balance between potential and kinetic exchanges. We discover that J can be substantially boosted at planar configurations up to -198 meV. We realize ferromagnetic coupling in nitrogen-doped triangulene dimers at both planar and largely twisted configurations, representing the first example of ferromagnetic triangulene dimers that cannot be predicted by the Ovchinnikov rule. This work thus provides a practical strategy for augmenting magnetic coupling and open up new avenues for metal-free ferromagnetism.

Spin-Orbit Couplings of Open-Shell Systems with Restricted Active Space Configuration Interaction

In this work we perform electronic structure calculations to unravel the origin of spin-orbit couplings (SOCs) in open-shell molecules. For that, we select systems displaying di or polyradical character, e.g., trimethylene, and analyze the changes in the magnitude of SOC constants along molecular distortions of ethylene and in the presence of intermolecular interactions between open and closed-shell moieties in the O₂-C₂H₄ system. Calculations were performed by using nonrelativistic wave functions obtained with the restricted active space configuration interaction (RASCI) method, in conjunction with a recent implementation for the calculation of SOC based on the spin-orbit mean field approximation. Our results demonstrate the suitability of RASCI in the calculation of SOCs of open-shell systems, while providing a deep understanding of the relationship between couplings and the nature of the electronic states. Moreover, we introduce a new definition of the SOC constant for the study of molecular aggregates.

Carbon-based nanostructures as a versatile platform for tunableπ-magnetism

Emergence ofπ-magnetism in open-shell nanographenes has been theoretically predicted decades ago but their experimental characterization was elusive due to the strong chemical reactivity that makes their synthesis and stabilization difficult. In recent years, on-surface synthesis under vacuum conditions has provided unprecedented opportunities for atomically precise engineering of nanographenes, which in combination with scanning probe techniques have led to a substantial progress in our capabilities to realize localized electron spin states and to control electron spin interactions at the atomic scale. Here we review the essential concepts and the remarkable advances in the last few years, and outline the versatility of carbon-basedπ-magnetic materials as an interesting platform for applications in spintronics and quantum technologies.

Tuning the Diradical Character of Indolocarbazoles: Impact of Structural Isomerism and Substitution Position

In this study, a set of 10 positional indolocarbazole (ICz) isomers substituted with dicyanomethylene groups connected via para or meta positions are computationally investigated with the aim of exploring the efficiency of structural isomerism and substitution position in controlling their optical and electronic properties. Unrestricted density functional theory (DFT), a spin-flip time-dependent DFT approach, and the multireference CASSCF/NEVPT2 method have been applied to correlate the diradical character with the energetic trends (i.e., singlet-triplet energy gaps). In addition, the nucleus-independent chemical shift together with ACID plots and Raman intensity calculations were used to strengthen the relationship between the diradical character and (anti)aromaticity. Our study reveals that the substitution pattern and structural isomerism represent a very effective way to tune the diradical properties in ICz-based systems with meta-substituted systems with a V-shaped structure displaying the largest diradical character. Thus, this work contributes to the elucidation of the challenging chemical reactivity and physical properties of diradicaloid systems, guiding experimental chemists to produce new molecules with desirable properties.

Aza-Triangulene: On-Surface Synthesis and Electronic and Magnetic Properties

Nitrogen heteroatom doping into a triangulene molecule allows tuning its magnetic state. However, the synthesis of the nitrogen-doped triangulene (aza-triangulene) has been challenging. Herein, we report the successful synthesis of aza-triangulene on the Au(111) and Ag(111) surfaces, along with their characterizations by scanning tunneling microscopy and spectroscopy in combination with density functional theory (DFT) calculations. Aza-triangulenes were obtained by reducing ketone-substituted precursors. Exposure to atomic hydrogen followed by thermal annealing and, when necessary, manipulations with the scanning probe afforded the target product. We demonstrate that on Au(111), aza-triangulene donates an electron to the substrate and exhibits an open-shell triplet ground state. This is derived from the different Kondo resonances of the final aza-triangulene product and a series of intermediates on Au(111). Experimentally mapped molecular orbitals match with DFT-calculated counterparts for a positively charged aza-triangulene. In contrast, aza-triangulene on Ag(111) receives an extra electron from the substrate and displays a closed-shell character. Our study reveals the electronic properties of aza-triangulene on different metal surfaces and offers an approach for the fabrication of new hydrocarbon structures, including reactive open-shell molecules.

Organic Emitters Showing Excited-States Energy Inversion: An Assessment of MC-PDFT and Correlation Energy Functionals Beyond TD-DFT

The lowest-energy singlet (S1) and triplet (T1) excited states of organic conjugated chromophores are known to be accurately calculated by modern wavefunction and Time-Dependent Density Functional Theory (TD-DFT) methods, with the accuracy of the latter heavily relying on the exchange-correlation functional employed. However, there are challenging cases for which this cannot be the case, due to the fact that those excited states are not exclusively formed by single excitations and/or are affected by marked correlation effects, and thus TD-DFT might fall short. We will tackle here a set of molecules belonging to the azaphenalene family, for which research recently documented an inversion of the relative energy of S1 and T1 excited states giving rise to a negative energy difference (ΔEST) between them and, thereby, contrary to most of the systems thus far treated by TD-DFT. Since methods going beyond standard TD-DFT are not extensively applied to excited-state calculations and considering how challenging this case is for the molecules investigated, we will prospectively employ here a set of non-standard methods (Multi-Configurational Pair Density Functional Theory or MC-PDFT) and correlation functionals (i.e., Lie–Clementi and Colle–Salvetti) relying not only on the electronic density but also on some modifications considering the intricate electronic structure of these systems.

Narrowband Deep-Blue Multi-Resonance Induced Thermally Activated Delayed Fluorescence: Insights from the Theoretical Molecular Design

Multi-resonance thermal activated delayed fluorescence (MR-TADF) has been promising with large oscillator strength and narrow full width at half maxima of luminescence, overcoming the compromise of emission intensity and energy criteria of traditional charge transfer TADF frameworks. However, there are still limited theoretical investigations on the excitation mechanism and systematic molecular manipulation of MR-TADF structures. We systematically study the highly localized excitation (LE) characteristics based on typical blue boron-nitrogen (BN) MR-TADF emitters and prove the potential triangular core with theoretical approaches. A design strategy by extending the planar π-conjugate core structure is proposed to enhance the multiple resonance effects. Moreover, several substituted groups are introduced to the designed core, achieving color-tunable functions with relatively small energy split and strong oscillator strength simultaneously. This work provides a theoretical direction for molecular design strategy and a series of potential candidates for highly efficient BN MR-TADF emitters.

Influence of the Radicaloid Character of Polyaromatic Hydrocarbon Couplers on Magnetic Exchange Interactions

The molecular properties of the conjugated spacers, such as the π-conjugation, aromaticity, length of the couplers, etc., that couple two localized spin-centers influence the intramolecular magnetic exchange interactions (2J) mediated...

peri-Acenoacene molecules: tuning of the singlet and triplet excitation energies by modifying their radical character

The energy difference between singlet and triplet excitons, or ΔE_ST, is a key parameter for novel light-emission mechanisms (i.e., TADF or thermally activated delayed fluorescence) or other photoactivated processes. We have studied a set of conjugated molecules (peri-acenoacenes and their heteroatom-doped analogues) to observe the evolution of their excited-state properties upon increasing the system size with and without substitution with a pair of N atoms. Since these molecules exhibit a (ground-state) diradicaloid character, together with marked correlation effects influencing the excited-states formed, we have applied a variety of theoretical methods (FT-DFT, TD-DFT, SF-TD-DFT, CIS, CIS(D), SCS-CC2, SA-CASSCF, and SC-NEVPT2) to bracket the accuracy of the results while concomitantly providing insights into electronic structure. The results show how this chemical strategy (N-doping) largely modifies not only the excited-state energies but also the oscillator strengths and the ΔE_ST values, constituting versatile platforms for fine-tuned photophysical applications.

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.

Clar Goblet and Aromaticity Driven Multiradical Nanographenes

The Clar Goblet, the first radical bowtie nanographene proposed by Erich Clar nearly 50 years ago, was recently synthesized. Bowtie nanographenes present quasi-degenerate magnetic ground states, which make them so elusive as unique. A thorough analysis is presented of the spin-state energetics of Clar Goblet and bowtie nanographenes by a battery of existing and novel ab initio procedures ranging from density functional theory to complete active space Hamiltonians. With this, it was proven that π radicals of bowtie nanographenes sit on BP (Benzo[cd]Pyrene) moieties driven by their local aromaticity, a purely chemical concept, which confers global stability to the whole structure. Besides, a novel Pauli energy densities analysis provided a visual intuitive explanation for this preference. These findings allow envisioning that analogous bowtie nanographenes with arbitrary polyradical character are not only feasible at the molecular scale but will share Clar Goblet's peculiar properties.

Uncovering the Triplet Ground State of Triangular Graphene Nanoflakes Engineered with Atomic Precision on a Metal Surface

Graphene can develop large magnetic moments in custom-crafted open-shell nanostructures such as triangulene, a triangular piece of graphene with zigzag edges. Current methods of engineering graphene nanosystems on surfaces succeeded in producing atomically precise open-shell structures, but demonstration of their net spin remains elusive to date. Here, we fabricate triangulenelike graphene systems and demonstrate that they possess a spin S=1 ground state. Scanning tunneling spectroscopy identifies the fingerprint of an underscreened S=1 Kondo state on these flakes at low temperatures, signaling the dominant ferromagnetic interactions between two spins. Combined with simulations based on the meanfield Hubbard model, we show that this S=1 π paramagnetism is robust and can be turned into an S=1/2 state by additional H atoms attached to the radical sites. Our results demonstrate that π paramagnetism of high-spin graphene flakes can survive on surfaces, opening the door to study the quantum behavior of interacting π spins in graphene 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.

Benzo[cd]triangulene: A Spin 1/2 Graphene Fragment

How does edge modification affect spin distribution in open-shell graphene fragments? We investigated this effect by analyzing spin-delocalization in benzo[cd]-triangulene, a spin 1/2 graphene fragment composed of seven benzenoid rings fused in a hybrid zigzag/armchair fashion. Six rings of this system form the core of Clar's hydrocarbon triangulene, to which an additional ring is annulated in the zigzag region. The singly occupied molecular orbital (SOMO) of this hydrocarbon radical resembles both SOMOs of triangulene, but the spin density is distributed over the core in a nonuniform fashion. The uneven spin distribution is reflected in the reactivity-reaction with oxygen occurs selectively at a position with the highest spin density-and correlates nicely with relative stabilities of the corresponding Clar resonance structures. The spin distribution is different from that of a topologically similar compound composed of the same number of sp² carbon atoms but featuring six rings only, illustrating the impact of subtle structural changes on spin-density distribution. This compound was characterized by means of UV-vis and electron paramagnetic resonance spectroscopy, cyclic voltammetry, mass spectrometry, and X-ray crystallography. The experimental results are supported by density functional theory calculations.

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.

Optical band gap of cross-linked, curved, and radical polyaromatic hydrocarbons

The impact of cross-linking, curvature, and radical character on the optical band gap of polyaromatic hydrocarbons has been investigated.