Polyradical Character Research Articles

Leap from Diradicals to Tetraradicals by Topological Control of π-Conjugation.

In this work, we explore the series of diradical(oid)s based on 2,2'-(5,11-dihydroindolo[3,2-b]carbazole-3,9-diyl)dimalononitrile (further referred to as PH). Hydrogen atoms in the central benzenoid (CB) ring of PH are substituted by the series of substituents with various lengths of π-conjugated chain and electron-donating or electron-withdrawing properties to study how they modulate the diradical character of the parent compound. The diradical character of molecules increases up to 88-89% by two groups doubly bonded to both sides of the CB ring of PH in para relative positions. This breaks the direct π-conjugation between unpaired electrons that gives rise to two radical centers and restricts the minimal polyradical identity of the compound to diradical. We show that diradicals and tetraradicals can be designed, and their polyradical character can be modulated by controlling the topology of π-conjugation as long as there is sufficient aromatic stabilization. Henceforth, the bridge between diradicals and tetraradicals is established, leading to the tetraradical(oid) molecule, which has been predicted to have narrow low-spin to high-spin energy gaps in our recent Letter.

On the Unusual Global Aromaticity of Two Cyclopenta-Ring-Fused Oligo(m-Phenylenes).

Some years ago, Jishan Wu reported the synthesis of 8MC and 10MC, two homologues of the cyclopenta-ring-fused oligo(m-phenylene) macrocycles mMC, each behaving as an annulene-within-an-annulene (AWA). This was a surprising result as the AWA behavior is rare. Both molecules have a partial polyradical character, enforced by the quest for restoring some aromatic character of benzene rings. However, that restoration brings back some coupling between the two annulenes. Indeed, we found that the geometry and the magnetically induced currents indicate that, while 8MC does have an AWA character, this is not the case of the larger 10MC. Limitations of the design strategy of AWA molecules should be taken into account in future attempts to prepare novel large coronenes.

Size matters: Computational insight into magnetic properties of extended acenes

A DFT study of a series of extended acenes comprising from 7 to 35 conjugated six-membered rings (n) predicted their polyradical character and ascertained correlation between the length and magnetic properties of these compounds. A regularity was revealed indicating that increasing the chain length by every seven rings bring into the formation of two additional paramagnetic centers, leading to the presence of ten unpaired electrons in the acene, for which n = 35. The computational results obtained allow one to expect increasing the number of unpaired electrons with a further (n > 35) expanding the acene length.

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.

Impact of Di- and Poly-Radical Characters on the Relative 
 Energy of the Doubly Excited and La States of Linear Acenes and Cyclacenes

Linear and cyclic acenes are polycyclic aromatic hydrocarbons that can be viewed as building blocks of graphene nanoribbons and carbon nanotubes, respectively. While short linear acenes demonstrated remarkable efficiency in several optoelectronic applications, the longer members are unstable and difficult to synthesize as their cyclic counterparts. Recent progress in on-surface synthesis, a powerful tool to prepare highly reactive species, opens promising perspectives and motivates the computational investigations of these potentially functional molecules. Owing to their di- and poly-radical character, low-lying excited states dominated by doubly excited configurations are expected to become more important for longer members of both linear and cyclic molecules. In this work, we investigate the lowest-lying La and the doubly excited (DE) state of linear acenes and cyclacenes, with different computational approaches, to assess the influence of the di-/poly-radical characters (increasing with the molecular dimensions) on their relative order. We show that DFT/MRCI calculations correctly reproduce the crossing of the two states for longer linear acenes, while TDUDFT calculations fail to predict the correct excitation energy trend of the DE state. The study suggests a similarity in the excited electronic state pattern of long linear and cyclic acenes leading ultimately to a lowest lying dark DE state for both.

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.

Designing molecules with a high-spin (quintet,S= 2) ground state for magnetic and spintronic applications

High-spin quintet polyradicals are reported by embedding the electron-deficient 6,6-dicyanofulvene group. We present a design strategy delineating control of the ground state from closed-shell to open-shell polyradical character.

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.

Libwfa: Wavefunction analysis tools for excited and open‐shell electronic states

AbstractAn open‐source software library for wavefunction analysis, libwfa, provides a comprehensive and flexible toolbox for post‐processing excited‐state calculations, featuring a hierarchy of interconnected visual and quantitative analysis methods. These tools afford compact graphical representations of various excited‐state processes, provide detailed insight into electronic structure, and are suitable for automated processing of large data sets. The analysis is based on reduced quantities, such as state and transition density matrices (DMs), and allows one to distill simple molecular orbital pictures of physical phenomena from intricate correlated wavefunctions. The implemented descriptors provide a rigorous link between many‐body wavefunctions and intuitive physical and chemical models, for example, exciton binding, double excitations, orbital relaxation, and polyradical character. A broad range of quantum‐chemical methods is interfaced with libwfa via a uniform interface layer in the form of DMs. This contribution reviews the structure of libwfa and highlights its capabilities by several representative use cases.This article is categorized under: Software > Quantum Chemistry Theoretical and Physical Chemistry > Spectroscopy

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.

Donor-Acceptor Conjugated Macrocycles with Polyradical Character and Global Aromaticity.

SummaryPolyradical character and global aromaticity are fundamental concepts that govern the rational design of cyclic conjugated macromolecules for optoelectronic applications. Here, we report donor-acceptor (D−A) conjugated macromolecules with and without π-spacer derivatives to tune the antiferromagnetic couplings between the unpaired electrons. The macromolecules without π-spacer have a closed-shell electronic configuration and show global nonaromatic character in the singlet and lowest triplet states. However, the derivatives with π-spacer develop a nearly pure open-shell diradical and a very high polyradical character, not reported for D−A type macromolecules. Furthermore, the π-spacer derivatives display global nonaromaticity in the singlet ground state, but global aromaticity in the lowest triplet state, according to Baird's rule. The absorption spectra of the open-shell macromolecules calculated with time-dependent density functional theory indicate intensive light absorption in the near-infrared region and broadening to 2,500 nm, making these materials suitable for numerous optoelectronic applications.

Möbius and Hückel Cyclacenes with Dewar and Ladenburg Defects.

Cyclacene nanobelts have not been synthesized in over 60 years and remain one of the last unsynthesized building blocks of carbon nanotubes. Recent work has predicted that Hückel-cyclacenes containing Dewar benzenoid ring isomers are the most stable isomeric forms for several of the smaller sizes of cyclacene belts. Here, we give a more complete picture of the isomers that are possible within these nanobelt systems by simulating embedded Ladenburg (prismane) benzenoid rings in Hückel-[n]cyclacenes (n = 5-14) and embedded Dewar benzenoid rings in twisted Möbius-[n]cyclacenes (n = 9-14). The Möbius-[9]cyclacene isomer containing one Dewar benzenoid defect and the Hückel-[5]cyclacene isomer containing two maximally spaced Ladenburg benzenoid defects are found to be more stable than their conventional Kekulé benzenoid ring counterparts. The isomers that contain Dewar and Ladenburg benzenoid rings have larger electronic singlet-triplet energy gaps and lower polyradical character when compared with the conventional isomers.

Designing Donor-Acceptor Conjugated Macrocycles with Polyradical Character and Global (Anti)Aromaticity

Polyradical character and global (anti)aromaticity are fundamental concepts that govern rational design of cyclic conjugated macromolecules for optoelectronic applications. Here, we have designed donor-acceptor (D-A) macromolecules with ring topology along with their pi-spacer derivatives, and analyzed their unique electronic properties with density functional theory (DFT). The macromolecules with n = 8 and 16 repeat D-A units display similar singlet-triplet energy gap and have closed-shell electronic configuration with no diradical character (y0 = 0). These macromolecules also display global nonaromatic character, which is supported by NICS(0), AICD, and 2D-ICSS analysis. However, the derivatives with pi-spacer (n = 8-pi) develop near pure open-shell character (y0=1) with increased singlet-triplet gap. Furthermore, these derivatives display global antiaromaticity in the singlet ground-state according to Huckel's rule, and global aromaticity in the lowest triplet state following Baird's rule, and have a very high polyradical character, which is not reported for D-A type macromolecules. The calculated absorption spectra with time-dependent DFT (TDDFT) indicates intensive light absorption in the near-infrared (NIR) region broadening to 2500 nm. These macrocycles also display a size-dependent absorption maxima when the number of repeat units increased from n = 8 to 16. Furthermore, due to the development of open-shell character in the pi-spacer derivatives, a significant red-shift in the absorbance wavelength is observed.

Dewar Benzenoids Discovered In Carbon Nanobelts.

The synthesis of cyclacene nanobelts remains an elusive goal dating back over 60 years. These molecules represent the last unsynthesized building block of carbon nanotubes and may be useful both as seed molecules for the preparation of structurally well-defined carbon nanotubes and for understanding the behavior and formation of zigzag nanotubes more broadly. Here we report the discovery that isomers containing two Dewar benzenoid rings are the preferred form for several sizes of cyclacene. The predicted lower polyradical character and higher singlet-triplet stability that these isomers possess compared with their pure benzenoid counterparts suggest that they may be more stable synthetic targets than the structures that have previously been identified. Our findings should facilitate the exploration of new routes to cyclacene synthesis through Dewar benzene chemistry.

CASSCF with Extremely Large Active Spaces Using the Adaptive Sampling Configuration Interaction Method.

The complete active space self-consistent field (CASSCF) method is the principal approach employed for studying strongly correlated systems. However, exact CASSCF can only be performed on small active spaces of ∼20 electrons in ∼20 orbitals due to exponential growth in the computational cost. We show that employing the Adaptive Sampling Configuration Interaction (ASCI) method as an approximate Full CI solver in the active space allows CASSCF-like calculations within chemical accuracy (<1 kcal/mol for relative energies) in active spaces with more than ∼50 active electrons in ∼50 active orbitals, significantly increasing the sizes of systems amenable to accurate multiconfigurational treatment. The main challenge with using any selected CI-based approximate CASSCF is the orbital optimization problem; they tend to exhibit large numbers of local minima in orbital space due to their lack of invariance to active-active rotations (in addition to the local minima that exist in exact CASSCF). We highlight methods that can avoid spurious local extrema as a practical solution to the orbital optimization problem. We employ ASCI-SCF to demonstrate a lack of polyradical character in moderately sized periacenes with up to 52 correlated electrons and compare against heat-bath CI on an iron porphyrin system with more than 40 correlated electrons.

Heterogeneous CPU + GPU Algorithm for Variational Two-Electron Reduced-Density Matrix-Driven Complete Active-Space Self-Consistent Field Theory.

We present a heterogeneous central processing unit (CPU) + graphical processing unit (GPU) algorithm for the direct variational optimization of the two-electron reduced-density matrix (2RDM) under two-particle N-representability conditions. This variational 2RDM (v2RDM) approach is the driver for a polynomially scaling approximation to configuration-interaction-driven complete active-space self-consistent field (CASSCF) theory. For v2RDM-based CASSCF computations involving an active space consisting of 50 electrons in 50 orbitals, we observe a speedup of a factor of 3.7 when the code is executed on a combination of an NVIDIA TITAN V GPU and an Intel Core i7-6850k CPU, relative to the case when the code is executed on the CPU alone. We use this GPU-accelerated v2RDM-CASSCF algorithm to explore the electronic structure of the 3,k-circumacene and 3,k-periacene series (k = 2-7) and compare indicators of polyradical character in the lowest-energy singlet states to those observed for oligoacene molecules. The singlet states in larger circumacene and periacene molecules display the same polyradical characteristics observed in oligoacenes, with the onset of this behavior occurring at smallest k for periacenes, followed by the circumacenes and then the oligoacenes. However, the unpaired electron density that accumulates along the zigzag edge of the circumacenes is slightly less than that which accumulates in the oligoacenes, while periacenes clearly exhibit the greatest buildup of unpaired electron density in this region.

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.

Electronic Properties of M\xf6bius Cyclacenes Studied by Thermally-Assisted-Occupation Density Functional Theory

It has been extremely difficult for traditional theoretical methods to adequately predict the properties of systems possessing radical character (i.e., multi-reference systems), especially for multi-reference systems at the nanoscale. To circumvent this, we employ thermally-assisted-occupation density functional theory (TAO-DFT) to predict the electronic properties of Möbius cyclacenes, with the number of fused benzene rings (n) ranging from 8 to 100. In addition, to investigate the significance of Möbius topology, we also compare these properties with the respective properties of cyclacenes and acenes, containing the same number of fused benzene rings. From our TAO-DFT results, Möbius cyclacenes, cyclacenes, and acenes have singlet ground states for all the cases examined. However, unlike acenes, the electronic properties of Möbius cyclacenes and cyclacenes display clear oscillation patterns when n is small (e.g., n ≤ 10 for Möbius cyclacenes and n ≤ 23 for cyclacenes), and converge to the respective properties of acenes when n greatly exceeds 30. The polyradical character of the ground states of Möbius cyclacenes should increase with the molecular size, intimately correlated with the localization of active orbitals at the edges of molecules.

N]Cyclo-para-biphenylmethine Polyradicaloids: [n]Annulene Analogs and Unusual Valence Tautomerization

Summary Annulenes and their charged analogs play an important role in validating Huckel's aromaticity rule. Whereas annulenes with [4n + 2] π electrons tend to delocalize π electron density, annulenes with [4n] π electrons prefer to form local double bonds (bond alternation) resulting in two interconvertible valence isomers. However, it is very challenging to experimentally detect such a valence tautomerization process as a result of the very fast conversion. In this work, we synthesized a series of [n]cyclo-para-biphenylmethine macrocycles ([n]CPBM-An, n = 3–8) as analogs of [n]annulenes and observed the unusual valence tautomerization process by variable-temperature NMR measurements for [4]CPBM-An, which showed a moderate interconversion barrier (11.22 kcal/mol at coalescence temperature) via a C4 tetraradicaloid transition state. These macrocycles showed [n]annulene-like geometries but with unique open-shell polyradical character. Their optical, electrochemical, and magnetic properties were systematically investigated and compared with those of the recently studied [n]cyclo-para-phenylenes.

Qualitative Views on the Polyradical Character of Long Acenes.

Spin-symmetry breaking appears in the DFT treatment of polyacenes, beyond a certain length, the critical length depending on the exchange-correlation potential. This phenomenon may be attributed to an instability with respect to HOMO-LUMO mixing, which suggests a diradical character of long acenes. However, the increase of the S2 operator with acene length questions this simple view. It is shown that this increase cannot be attributed to spin polarization of the inner MOs, and that a second symmetry breaking takes place for the pentadecacene, with four unpaired electrons centered at the first and third quarters of the chain. The spin density distributions of broken-symmetry solutions support a qualitative picture in terms of tetra-methylene hexacenes separated by Clar sextets. A strategy is proposed to identify local symmetry breaking in polyradical systems.