Role of Kekulé and Non-Kekulé Structures in the Radical Character of Alternant Polycyclic Aromatic Hydrocarbons: A TAO-DFT Study

Effect of Oriented External Electric Fields on the Electronic Properties of Linear Acenes: A Thermally Assisted Occupation DFT Study

Recently, oriented external electric fields (OEEFs) have earned much attention due to the possibility of tuning the properties of electronic systems. From a theoretical perspective, one can resort to electronic structure calculations to understand how the direction and strength of OEEFs affect the properties of electronic systems. However, for multi-reference (MR) systems, calculations employing the popular Kohn-Sham density functional theory with the traditional semilocal and hybrid exchange-correlation energy functionals can yield erroneous results. Owing to its decent compromise between accuracy and efficiency for MR systems at the nanoscale (i.e., MR nanosystems), in this study, thermally assisted occupation density functional theory (TAO-DFT) is adopted to explore the electronic properties of n-acenes (n = 2-10), containing n linearly fused benzene rings, in OEEFs, where the OEEFs of various electric field strengths are applied along the long axes of n-acenes. According to our TAO-DFT calculations, the ground states of n-acenes in OEEFs are singlets for all the cases examined. The effect of OEEFs is shown to be significant on the vertical ionization potentials and vertical electron affinities of ground-state n-acenes with odd-number fused benzene rings. Moreover, the MR character of ground-state n-acenes in OEEFs increases with the increase in the acene length and/or the electric field strength.

[19]Starphene: Combined In-Solution and On-Surface Synthesis Towards the Largest Starphene

Starphenes are structurally appealing three-fold symmetric polycyclic aromatic compounds with potential interesting applications in molecular electronics and nanotechnology. This family of star-shaped polyarenes can be regarded as three acenes that are connected through a single benzene ring. In fact, just like acenes, unsubstituted large starphenes are poorly soluble and highly reactive molecules under ambient conditions making their synthesis difficult to achieve. Herein, we report two different synthetic strategies to obtain a starphene formed by 19 cata-fused benzene rings distributed within three hexacene branches. This molecule, which is the largest starphene that has been obtained to date, was prepared by combining solution-phase and on-surface synthesis. [19]Starphene was characterized by high-resolution scanning tunneling microscopy (STM) and spectroscopy (STS) showing a remarkable small HOMO-LUMO transport gap (0.9 eV).

Energy Decomposition Scheme for Rectangular Graphene Flakes

We show-to our own surprise-that total electronic energies for a family of m × n rectangular graphene flakes can be very accurately represented by a simple function of the structural parameters m and n with errors not exceeding 1 kcal/mol. The energies of these flakes, usually referred to as multiple zigzag chains Z(m,n), are computed for m, n < 21 at their optimized geometries using the DFTB3 methodology. We have discovered that the structural parameters m and n (and their simple algebraic functions) provide a much better basis for the energy decomposition scheme than the various topological invariants usually used in this context. Most terms appearing in our energy decomposition scheme seem to have simple chemical interpretations. Our observation goes against the well-established knowledge stating that many-body energies are complicated functions of molecular parameters. Our observations might have far-reaching consequences for building accurate machine learning models.

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.

Real-Time Extension of TAO-DFT

Thermally assisted occupation density functional theory (TAO-DFT) has been an efficient electronic structure method for studying the ground-state properties of large electronic systems with multi-reference character over the past few years. To explore the time-dependent (TD) properties of electronic systems (e.g., subject to an intense laser pulse), in this work, we propose a real-time (RT) extension of TAO-DFT, denoted as RT-TAO-DFT. Moreover, we employ RT-TAO-DFT to study the high-order harmonic generation (HHG) spectra and related TD properties of molecular hydrogen H2 at the equilibrium and stretched geometries, aligned along the polarization of an intense linearly polarized laser pulse. The TD properties obtained with RT-TAO-DFT are compared with those obtained with the widely used time-dependent Kohn-Sham (TDKS) method. In addition, issues related to the possible spin-symmetry breaking effects in the TD properties are discussed.

Intruder-free cumulant-truncated driven similarity renormalization group second-order multireference perturbation theory

Accurate multireference electronic structure calculations are important for constructing potential energy surfaces. Still, even in the case of low-scaling methods, their routine use is limited by the steep growth of the computational and storage costs as the active space grows. This is primarily due to the occurrence of three- and higher-body density matrices or, equivalently, their cumulants. This work examines the effect of various cumulant truncation schemes on the accuracy of the driven similarity renormalization group second-order multireference perturbation theory. We test four different levels of three-body reduced density cumulant truncations that set different classes of cumulant elements to zero. Our test cases include the singlet-triplet gap of CH2, the potential energy curves of the XΣg+1 and AΣu+3 states of N2, and the singlet-triplet splittings of oligoacenes. Our results show that both relative and absolute errors introduced by these cumulant truncations can be as small as 0.5 kcal mol-1 or less. At the same time, the amount of memory required is reduced from O(NA6) to O(NA5), where NA is the number of active orbitals. No additional regularization is needed to prevent the intruder state problem in the cumulant-truncated second-order driven similarity renormalization group multireference perturbation theory methods.

Orbital-symmetry effects on magnetic exchange in open-shell nanographenes

Open-shell nanographenes appear as promising candidates for future applications in spintronics and quantum technologies. A critical aspect to realize this potential is to design and control the magnetic exchange. Here, we reveal the effects of frontier orbital symmetries on the magnetic coupling in diradical nanographenes through scanning probe microscope measurements and different levels of theoretical calculations. In these open-shell nanographenes, the exchange energy exhibits a remarkable variation between 20 and 160 meV. Theoretical calculations reveal that frontier orbital symmetries play a key role in affecting the magnetic coupling on such a large scale. Moreover, a triradical nanographene is demonstrated for investigating the magnetic interaction among three unpaired electrons with unequal magnetic exchange, in agreement with Heisenberg spin model calculations. Our results provide insights into both theoretical design and experimental realization of nanographene materials with different exchange interactions through tuning the orbital symmetry, potentially useful for realizing magnetically operable graphene-based nanomaterials.

Carbon-[n]Triangulenes and Sila-[n]Triangulenes: Which Are Planar?

Using our recently suggested concept of a quasi-molecule ("tile") and, in the case of the planarity here at stake, its generalization to larger than tetratomics, we explain why carbon [n]triangulenes tend to be planar, while hybrids, where just a few or even all a- or b-type carbon atoms are silicon-substituted (sila-[n]triangulenes), tend to be planar/nonplanar when compared with the unsubstituted carbon-[n]triangulenes. Because other spin states of the parent carbon- and sila-[n]triangulenes tend to correlate with the same tiles, it is conjectured that no structural changes are expected to depend on their spin state. Other polycyclic and sila-compounds are also discussed.

Tuning the Diradical Character of Pentacene Derivatives via Non-Benzenoid Coupling Motifs

The development of functional organic molecules requires structures of increasing size and complexity, which are typically obtained by the covalent coupling of smaller building blocks. Herein, with the aid of high-resolution scanning tunneling microscopy/spectroscopy and density functional theory, the coupling of a sterically demanded pentacene derivative on Au(111) into fused dimers connected by non-benzenoid rings was studied. The diradical character of the products was tuned according to the coupling section. In particular, the antiaromaticity of cyclobutadiene as the coupling motif and its position within the structure play a decisive role in shifting the natural orbital occupancies toward a stronger diradical electronic character. Understanding these structure-property relations is desirable not only for fundamental reasons but also for designing new complex and functional molecular structures.

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.

Unveiling a new aspect of oxocarbons: open-shell character of 4- and 5-membered oxocarbon derivatives showing near-infrared absorption

Oxocarbon derivatives consisting of 4- and 5-membered rings, referred to as croconaine and squaraine dyes and regarded as closed-shell molecules, are found to have an intermediate open-shell character from the experimental results of ¹H-NMR, ESR spectroscopy, SQUID magnetometric analysis, and X-ray crystallography. We employed two chalcogenopyrylium moieties with O and S chalcogen atoms as substitutions on oxocarbons. The singlet-triplet energy gaps (ΔE _S-T) associated with the degree of diradical nature are smaller for croconaines than for squaraines and smaller for thiopyrylium than for pyrylium groups. The diradical nature impacts the electronic transition energy that decreased with a decreasing degree of diradical contribution. They exhibit substantial two-photon absorption in the region over 1000 nm. The diradical character y of the dye was determined experimentally from the observed one- and two-photon absorption peaks and the triplet energy level. The present finding provides new insight into diradicaloids with the contribution of non-Kekulé oxocarbon and also showcases the correlation between the electronic transition energy and their diradical character.

Electronic Properties of Hexagonal Graphene Quantum Rings from TAO-DFT

The reliable prediction of electronic properties associated with graphene nanosystems can be challenging for conventional electronic structure methods, such as Kohn-Sham (KS) density functional theory (DFT), due to the presence of strong static correlation effects in these systems. To address this challenge, TAO (thermally assisted occupation) DFT has been recently proposed. In the present study, we employ TAO-DFT to predict the electronic properties of n-HGQRs (i.e., the hexagonal graphene quantum rings consisting of n aromatic rings fused together at each side). From TAO-DFT, the ground states of n-HGQRs are singlets for all the cases investigated (n = 3-15). As the system size increases, there should be a transition from the nonradical to polyradical nature of ground-state n-HGQR. The latter should be intimately related to the localization of active TAO-orbitals at the inner and outer edges of n-HGQR, which increases with increasing system size.

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.

Phenalenyl Radical: Smallest Polycyclic Odd Alternant Hydrocarbon Present in the Graphene Sheet

Phenalenyl, a zigzag-edged odd alternant hydrocarbon unit can be found in the graphene nanosheet. Hückel molecular orbital calculations indicate the presence of a nonbonding molecular orbital (NBMO), which originates from the linear combination of atomic orbitals (LCAO) arising from 13 carbon atoms of the phenalenyl molecule. Three redox states (cationic, neutral radical, and anionic) of the phenalenyl-based molecules were attributed to the presence of this NBMO. The cationic state can undergo two consecutive reductions to result in neutral radical and anionic states, stepwise, respectively. The phenalenyl-based radicals were found as crucial building blocks and attracted the attention of various research fields such as organic synthesis, material science, computation, and device physics. From 2012 onward, a strategy was devised using the cationic state of phenalenyl-based molecules and in situ generated phenalenyl radicals, which created a new domain of catalysis. The in situ generated phenalenyl radicals were utilized for the single electron transfer (SET) process resulting in redox catalysis. This emerging range of applications rejuvenates the more than six decades-old phenalenyl chemistry. This review captures such developments ranging from fundamental understanding to multidirectional applications of phenalenyl-based radicals.

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.

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.

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.

Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange-correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear-electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an "open teamware" model and an increasingly modular design.

Theoretical Understanding of Structure-Property Relationships in Luminescence of Carbon Dots

Carbon dots (CDs) have excellent luminescence characteristics, such as good light stability, high quantum yield (QY), long phosphorescence lifetime, and a wide emission wavelength range, resulting in CDs' great success in optical applications. Understanding the structure-property relationships in CDs is essential for their use in optoelectronic applications. However, because of the complex nature of CD structures and synthesis processes, understanding the luminescence mechanism and structure-property relationships of CDs is a big challenge. This Perspective reviews the theoretical efforts toward the understanding of structure-property relationships and discusses the challenges that need to be overcome in future development of CDs.

Cluster many-body expansion: A many-body expansion of the electron correlation energy about a cluster mean field reference

The many-body expansion (MBE) is an efficient tool that has a long history of use for calculating interaction energies, binding energies, lattice energies, and so on. In the past, applications of MBE to correlation energy have been unfeasible for large systems, but recent improvements to computing resources have sparked renewed interest in capturing the correlation energy using the generalized nth order Bethe-Goldstone equation. In this work, we extend this approach, originally proposed for a Slater determinant, to a tensor product state (TPS) based wavefunction. By partitioning the active space into smaller orbital clusters, our approach starts from a cluster mean field reference TPS configuration and includes the correlation contribution of the excited TPSs using the MBE. This method, named cluster MBE (cMBE), improves the convergence of MBE at lower orders compared to directly doing a block-based MBE from a RHF reference. We present numerical results for strongly correlated systems, such as the one- and two-dimensional Hubbard models and the chromium dimer. The performance of the cMBE method is also tested by partitioning the extended π space of several large π-conjugated systems, including a graphene nano-sheet with a very large active space of 114 electrons in 114 orbitals, which would require 10⁶⁶ determinants for the exact FCI solution.

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.

Unravelling the Open-Shell Character of Peripentacene on Au(111)

Polycyclic aromatic hydrocarbons (PAHs) are a family of organic compounds comprising two or more fused aromatic rings which feature manifold applications in modern technology. Among these species, those presenting an open-shell magnetic ground state are of particular interest for organic electronic, spintronic, and non-linear optics and energy storage devices. Within PAHs, special attention has been devoted in recent years to the synthesis and study of the acene and fused acene (periacene) families, steered by their decreasing HOMO-LUMO gap with length and predicted open-shell character above some size. However, an experimental fingerprint of such magnetic ground state has remained elusive. Here, we report on the in-depth electronic characterization of isolated peripentacene molecules on a Au(111) surface. Scanning tunnelling spectroscopy, complemented by computational investigations, reveals an antiferromagnetic singlet ground state, characterized by singlet-triplet inelastic excitations with an experimental effective exchange coupling (J_eff) of 40.5 meV. Our results deepen the fundamental understanding of organic compounds with magnetic ground states, featuring perspectives in carbon-based spintronic devices.

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.

ZZ Polynomials for Isomers of (5,6)-Fullerenes Cn with n = 20–50

A compilation of ZZ polynomials (aka Zhang–Zhang polynomials or Clar covering polynomials) for all isomers of small (5,6)-fullerenes Cn with n = 20–50 is presented. The ZZ polynomials concisely summarize the most important topological invariants of the fullerene isomers: the number of Kekulé structures K, the Clar number Cl, the first Herndon number h1, the total number of Clar covers C, and the number of Clar structures. The presented results should be useful as benchmark data for designing algorithms and computer programs aiming at topological analysis of fullerenes and at generation of resonance structures for valence-bond quantum-chemical calculations.

Total non-lewis structures: An application to predict the stability and reactivity of linear and angular polyacenes

In the present work, the total non-Lewis structure (TNLS) is introduced for describing the stability and reactivity of linear and angular polyacenes. TNLS of a molecule is derived from the natural bond orbital theory representing the antibonding orbitals and/or electron delocalization of π^∗ orbitals. To verify this application, we obtained the TNLS values of thirteen linear and angular polyacenes and evaluated with other quantitative aromaticity probes based on reactivities, energetics, geometrics, and magnetic properties. Our results show that there is a remarkable second-order polynomial correlation between TNLS and seven popular global measures of aromaticity including hardness, resonance energy, aromatic stabilization energy, harmonic oscillator measure of aromaticity, magnetic susceptibility exaltation, global magnetic characteristics, and mean polarizability. By increasing the TNLS values of the systems the reactivity increases and stability decreases. It is worth mentioning that the angular polyacenes produce less TNLS values than the linear systems with the same number of benzene rings. Therefore, it can be stated that TNLS values confirmed with Clar's rule which in general prove the angular polyacenes are more stable than linear ones.

Excitation energies from thermally assisted-occupation density functional theory: Theory and computational implementation

The time-dependent density functional theory (TDDFT) has been broadly used to investigate the excited-state properties of various molecular systems. However, the current TDDFT heavily relies on outcomes from the corresponding ground-state DFT calculations, which may be prone to errors due to the lack of proper treatment in the non-dynamical correlation effects. Recently, thermally assisted-occupation DFT (TAO-DFT) [J.-D. Chai, J. Chem. Phys. 136, 154104 (2012)], a DFT with fractional orbital occupations, was proposed, explicitly incorporating the non-dynamical correlation effects in the ground-state calculations with low computational complexity. In this work, we develop TDTAO-DFT, which is a TD, linear-response theory for excited states within the framework of TAO-DFT. With tests on the excited states of H₂, the first triplet excited state (1³Σ_u ⁺) was described well, with non-imaginary excitation energies. TDTAO-DFT also yields zero singlet-triplet gap in the dissociation limit for the ground singlet (1¹Σ_g ⁺) and the first triplet state (1³Σ_u ⁺). In addition, as compared to traditional TDDFT, the overall excited-state potential energy surfaces obtained from TDTAO-DFT are generally improved and better agree with results from the equation-of-motion coupled-cluster singles and doubles.

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.

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.

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.

Band Gap of Atomically Precise Graphene Nanoribbons as a Function of Ribbon Length and Termination

We study the band gap of finite N A = 7 armchair graphene nanoribbons (7-AGNRs) on Au(111) through scanning tunneling microscopy/spectroscopy combined with density functional theory calculations. The band gap of 7-AGNRs with lengths of 8 nm and more is converged to within 50 meV of its bulk value of ≈ 2 . 3 eV , while the band gap opens by several hundred meV in very short 7-AGNRs. We demonstrate that even an atomic defect, such as the addition of one hydrogen atom at the termini, has a significant effect - in this case, lowering the band gap. The effect can be captured in terms of a simple analytical model by introducing an effective "electronic length".

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.

Local Density Approximation for the Short-Range Exchange Free Energy Functional

Analytical expressions for the exchange free energy per particle of the uniform electron gas (UEG) associated with the short-range (SR) interelectronic interaction at the low- and high-temperature limits are examined, yielding an accurate analytical parametrization for the SR exchange free energy per particle of the UEG as a function of the uniform electron density, temperature, and range-separation parameter. This parametrization constitutes the local density approximation for the SR exchange free energy functional, which can be the first step toward finding generally accurate range-separated hybrid functionals in both finite-temperature density functional theory and thermally assisted-occupation density functional theory.

Electronic Properties of Möbius 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.

On-surface light-induced generation of higher acenes and elucidation of their open-shell character

Acenes are an important class of polycyclic aromatic hydrocarbons which have recently gained exceptional attention due to their potential as functional organic semiconductors. Fundamentally, they are important systems to study the convergence of physico-chemical properties of all-carbon sp2-frameworks in the one-dimensional limit; and by virtue of having a zigzag edge topology they also provide a fertile playground to explore magnetism in graphenic nanostructures. The study of larger acenes is thus imperative from both a fundamental and applied perspective, but their synthesis via traditional solution-chemistry route is hindered by their poor solubility and high reactivity. Here, we demonstrate the on-surface formation of heptacene and nonacene, via visible-light-induced photo-dissociation of α-bisdiketone precursors on an Au(111) substrate under ultra-high vacuum conditions. Through combined scanning tunneling microscopy/spectroscopy and non-contact atomic force microscopy investigations, together with state-of-the-art first principles calculations, we provide insight into the chemical and electronic structure of these elusive compounds.

From cyclic nanorings to single-walled carbon nanotubes: disclosing the evolution of their electronic structure with the help of theoretical methods

We systematically investigate the relationships between structural and electronic effects of finite size zigzag or armchair carbon nanotubes of various diameters and lengths, starting from a molecular template of varying shape and diameter, i.e. cyclic oligoacene or oligophenacene molecules, and disclosing how adding layers and/or end-caps (i.e. hemifullerenes) can modify their (poly)radicaloid nature. We mostly used tight-binding and finite-temperature density-based methods, the former providing a simple but intuitive picture about their electronic structure, and the latter dealing effectively with strong correlation effects by relying on a fractional occupation number weighted electron density (ρFOD), with additional RAS-SF calculations backing up the latter results. We also explore how minor structural modifications of nanotube end-caps might influence the results, showing that topology, together with the chemical nature of the systems, is pivotal for the understanding of the electronic properties of these and other related systems.

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.

Electronic properties of the coronene series from thermally-assisted-occupation density functional theory

To fully utilize the great potential of graphene in electronics, a comprehensive understanding of the electronic properties of finite-size graphene flakes is essential. While the coronene series with n fused benzene rings at each side (designated as n-coronenes) are possible structures for opening a band gap in graphene, their electronic properties are not yet fully understood. Nevertheless, because of their radical character, it remains very difficult to reliably predict the electronic properties of the larger n-coronenes with conventional computational approaches. In order to circumvent this, the various electronic properties of n-coronenes (n = 2-11) are investigated using thermally-assisted-occupation density functional theory (TAO-DFT) [J.-D. Chai, J. Chem. Phys., 2012, 136, 154104], a very efficient electronic structure method for studying nanoscale systems with strong static correlation effects. The ground states of the larger n-coronenes are shown to be polyradical singlets, where the active orbitals are mainly localized at the zigzag edges.

Electronic and Hydrogen Storage Properties of Li-Terminated Linear Boron Chains Studied by TAO-DFT

It has been extremely difficult for conventional computational approaches to reliably predict the properties of multi-reference systems (i.e., systems possessing radical character) at the nanoscale. To resolve this, we employ thermally-assisted-occupation density functional theory (TAO-DFT) to predict the electronic and hydrogen storage properties of Li-terminated linear boron chains (Li₂B_n), with n boron atoms (n = 6, 8, …, and 16). From our TAO-DFT results, Li₂B_n, which possess radical character, can bind up to 4 H₂ molecules per Li, with the binding energies in the desirable regime (between 20 and 40 kJ/mol per H₂). The hydrogen gravimetric storage capacities of Li₂B_n range from 7.9 to 17.0 wt%, achieving the ultimate goal of the United States Department of Energy. Accordingly, Li₂B_n could be promising media for storing and releasing H₂ at temperatures much higher than the boiling point of liquid nitrogen.

Quantitative Structure-Property Relationships for the Electronic Properties of Polycyclic Aromatic Hydrocarbons

This study presents a development in quantitative structure-property relationships (QSPRs) for research in organic semiconductor materials by introducing a new structural descriptor called "degree of π-orbital overlap" based on two-dimensional structure information of aromatic molecules. Application of this method to predict the electronic properties of polycyclic aromatic hydrocarbon (PAH) molecules, which are known to be the core component of many organic semiconductor materials, is presented. Results demonstrated that QSPRs based on the new descriptor can predict rather accurate band gaps, ionization potentials and electron affinities for a large number of PAHs compared to those explicitly calculated by density functional theory method. This research opens new possibilities for developing QSPRs for other organic semiconductor classes in future.

Effect of Li Termination on the Electronic and Hydrogen Storage Properties of Linear Carbon Chains: A TAO-DFT Study

Accurate prediction of the electronic and hydrogen storage properties of linear carbon chains (C n ) and Li-terminated linear carbon chains (Li2C n ), with n carbon atoms (n = 5-10), has been very challenging for traditional electronic structure methods, due to the presence of strong static correlation effects. To meet the challenge, we study these properties using our newly developed thermally-assisted-occupation density functional theory (TAO-DFT), a very efficient electronic structure method for the study of large systems with strong static correlation effects. Owing to the alteration of the reactivity of C n and Li2C n with n, odd-even oscillations in their electronic properties are found. In contrast to C n , the binding energies of H2 molecules on Li2C n are in (or close to) the ideal binding energy range (about 20 to 40 kJ/mol per H2). In addition, the H2 gravimetric storage capacities of Li2C n are in the range of 10.7 to 17.9 wt%, satisfying the United States Department of Energy (USDOE) ultimate target of 7.5 wt%. On the basis of our results, Li2C n can be high-capacity hydrogen storage materials that can uptake and release hydrogen at temperatures well above the easily achieved temperature of liquid nitrogen.

Role of exact exchange in thermally-assisted-occupation density functional theory: A proposal of new hybrid schemes

We propose hybrid schemes incorporating exact exchange into thermally assisted-occupation-density functional theory (TAO-DFT) [J.-D. Chai, J. Chem. Phys. 136, 154104 (2012)] for an improved description of nonlocal exchange effects. With a few simple modifications, global and range-separated hybrid functionals in Kohn-Sham density functional theory (KS-DFT) can be combined seamlessly with TAO-DFT. In comparison with global hybrid functionals in KS-DFT, the resulting global hybrid functionals in TAO-DFT yield promising performance for systems with strong static correlation effects (e.g., the dissociation of H₂ and N₂, twisted ethylene, and electronic properties of linear acenes), while maintaining similar performance for systems without strong static correlation effects. Besides, a reasonably accurate description of noncovalent interactions can be efficiently achieved through the inclusion of dispersion corrections in hybrid TAO-DFT. Relative to semilocal density functionals in TAO-DFT, global hybrid functionals in TAO-DFT are generally superior in performance for a wide range of applications, such as thermochemistry, kinetics, reaction energies, and optimized geometries.

Self-consistent determination of the fictitious temperature in thermally-assisted-occupation density functional theory

We propose a self-consistent scheme for the determination of the fictitious temperature in thermally-assisted-occupation density functional theory (TAO-DFT) [J.-D. Chai, J. Chem. Phys., 2012, 136, 154104].