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Carbon atoms have the potential to produce a variety of fascinating all-carbon structures with amazing electronic and mechanical properties. Over the last few decades, several efforts have been made using experimental and computational techniques to functionalize graphene, carbon nanotubes and fullerenes for potential use in modern hi-tech electronic, medicinal, optical and nonlinear optical (NLO) applications. Since photons replaced electrons as a carrier of information, the field of NLO material design has drawn immense interest in contemporary materials science. There have been several reports of bridging the gap between the exciting fields of carbon nanomaterials and NLO materials by functionalizing carbon nanomaterials for potential NLO applications. In contrast to previous reports of the design of third-order NLO materials using conventional closed-shell materials, a novel strategy using open-shell diradical molecular systems has recently been proposed. Quantum chemically, diradical character is explained in terms of the instability of the chemical bonds in open-shell molecular systems. Interestingly, several carbon nanomaterials, which naturally possess open-shell singlet configurations, have recently gained momentum in the design of these classes of open-shell NLO materials with excellent NLO properties, stability and diversity. The present review establishes a systematic sequence of different studies (spanning over two decades of intense research efforts), starting from the simplest theoretical two-site diradical model, continuing to its experimental and theoretical realization in actual chemical systems, and finally applying the abovementioned model/rule to novel carbon nanomaterials to tune their NLO properties, particularly their second hyperpolarizability (γ). In the present report, we highlight several recent efforts to functionalize carbon nanomaterials by exploiting their open-shell diradical character to achieve efficient third-order NLO properties. Several issues and opportunities are discussed, including the inherited disadvantages of both experimental (the high reactivity and short life of diradical compounds) and quantum (need for multi-reference methodology) techniques when dealing with carbon nanomaterials. A comparative analysis of several quantum chemical investigations, along with contemporary experimental results, will be performed to emphasize the core issues and opportunities related to carbon nanomaterials with singlet open-shell diradical character. Thus, the present review will highlight carbon nanomaterials with diradical/biradical character for their prospective applications in the NLO field.

The buckyferrocenes, synthesized through face-to-face fusion of ferrocene and fullerene fragments (C60Me10), are expected to enjoy the rich scientific heritage of ferrocene and fullerene with an extensively large π-conjugation network between the two Fe atoms [Y. Matsuo, K. Tahara and E. Nakamura, J. Am. Chem. Soc., 2006, 128, 7154]. However, the addition of pentamethyl groups at each end of the fullerene fragment breaks the π-conjugation path as well as metal-metal spin correlation between the two-ferrocene faces in a buckyferrocene. We found that the unblocking of π-conjugation from different positions in fullerene fragments have substantial effects on their topologies, spin densities, diradical characters as well as nonlinear optical (NLO) properties of these buckyferrocenes. We study the topological dependence of open-shell diradical character and second hyperpolarizability (γ), the third-order NLO properties at the molecular scale, in several buckyferrocenes. On the basis of their different diradical characters (yi), which are defined by the occupation number of the lowest unoccupied natural orbital (LUNO) + i (i = 0, 1,…), these buckyferrocenes are categorized into three groups, i.e., closed-shell (yi = 0), intermediate open-shell singlet (0 < yi < 1), and almost pure open-shell singlet (yi = 1) compounds. For example, we found that buckyferrocenes including (CpFe)2η(5)C60Me10 and (CpFe)2η(5)C70Me10 have closed-shell configurations. The buckyferrocenes (CpFe)2η(5)C60, (CpFe)2η(5)C70, (CpFe)2η(5)C70Me8, (CpFe)2η(5)C70Me4 and (CpFe)2η(5)C30 are intermediate open-shell singlet, while (CpFe)2η(5)C60Me4 and (CpFe)2η(5)C60Me8 are pure open-shell singlet complexes. Interestingly, the γzzzz amplitude of (CpFe)2η(5)C60, an open-shell intermediate diradical complex, is about 41 times and 13 times as large as those of its closed-shell ((CpFe)2η(5)C60Me10) and pure diradical ((CpFe)2η(5)C60Me8) counterparts, respectively. Similarly, the γzzzz amplitudes of (CpFe)2η(5)C70, (CpFe)2η(5)C70Me4, and (CpFe)2η(5)C70Me8 with intermediate diradical character are about 36, 17, and 9 times as large as that of their closed-shell (CpFe)2η(5)C70Me10 counterpart. The fact that larger γzzzz values are obtained for buckyferrocenes with intermediate diradical characters is in line with the "y-γ correlation" obtained from the valence configuration interaction (VCI) results for a two-site diradical model [M. Nakano, et al., Phys. Rev. Lett., 2007, 99, 033001] as well as for fullerene and graphene systems. The γzzzz density analysis shows that the large positive contributions originate from the large γzzzz density distributions on the upper- and lower-extended edges of the buckyferrocenes, between which significant spin polarizations appear within the spin-unrestricted DFT level of theory. These results demonstrate that such buckyferrocenes are potential candidates for a novel class of open-shell singlet NLO systems, where γzzzz values are modulated by tuning their diradical character through the use of suitably modified fullerene fragments.

The impact of intermolecular interactions on the third-order nonlinear optical (NLO) properties of open-shell molecular aggregates has been elucidated by considering one-dimensional aggregates of π-π stacked phenalenyl radicals with different intermolecular distances and the long-range corrected spin-unrestricted density functional theory method. In the phenalenyl dimer, which can be considered as a diradicaloid system, the diradical character strongly depends on the intermolecular distance, and the larger the intermolecular distance is, the larger the diradical character becomes. Then, around the equilibrium stacking distance that corresponds to an intermediate diradical character, its second hyperpolarizability (γ) is maximized and its value per monomer exhibits about a 30-fold enhancement with respect to the isolated phenalenyl monomer. This suggests that equilibrium is an optimal compromise between localization and delocalization of the radical electron pairs in such pancake bonding. No such effect was observed for the closed-shell coronene dimer. Moreover, when going from the dimer (diradical) to the tetramer (tetraradical), the γ-enhancement ratio increases nonlinearly with the aggregate size, whereas switching from the singlet to the highest spin (quintet) state causes a significant reduction of γ. Finally, for the tetramer, another one-order enhancement of γ is achieved for the dicationic singlet relative to its singlet neutral state. These results demonstrate the key role of intermolecular π-π stacking interactions and charge in open-shell (supra)molecular systems to achieve enhanced third-order NLO properties.

From the viewpoint of ‘‘diradical character,’’ referred to as ‘‘diradicalology,’’ we investigate the second hyperpolarizability c—the molecular third-order nonlinear optical (NLO) property—of one-dimensional supermolecular systems composed of acetylene-linked phenalenyl/pyrene rings using long-range corrected spin-unrestricted density functional theory. It turns out that the pyrene-based superpolyenes (Py-n) behave like closed-shell systems, whereas phenalenyl-based superpolyenes (Ph1-n and Ph2-n) have different diradical characters depending on the linked form, that is, Ph1-n and Ph2-n have intermediate and pure diradical characters, respectively. In comparison with Py-n and Ph2-n, the longitudinal c of Ph1-n is significantly larger, and it displays larger enhancement as a function of system size. Substitutions to the terminal rings by donor (NH2) and acceptor (NO2) groups further enhance c in Ph1-n, more than in Ph2-n and Py-n. These results are in agreement with the structure‐ property relationships derived for open-shell NLO systems with symmetric and asymmetric charge distributions (Nakano et al., J. Chem. Phys. 2010, 133, 154302). V C 2012 Wiley Periodicals, Inc.

Using the long-range corrected spin-unrestricted density functional theory method, the impact of antidot structure on the open-shell character and the second hyperpolarizability (c) of graphene nanoflakes (GNFs) has been investigated for rectangular GNFs with and without antidot structure, referred to as antidot and perfect GNFs, respectively. It is found that the two GNFs exhibit different multiradical characters; the antidot GNF shows intermediate open-shell character in contrast to the perfect GNF presenting nearly pure tetraradical character. This antidot structure dependence of open-shell character can be explained in terms of their natural orbital distributions. Such difference in open-shell characters affects their c values; the c of the antidot GNF is about 1.7 times larger than that of the perfect GNF, the feature of which is in agreement with our previous results on the diradical character dependence of c. V C 2012 Wiley Periodicals, Inc.

We study the (hyper)polarizabilities of a singlet diradical molecule (2-borapropane-1,3-diyl) which contains a boron atom. For the purpose to control the (hyper)polarizabilities by tuning the diradical character of the boron compound, we employ water solvent as an electron donor to the boron atom. A series of quantum chemical calculations reveal that the electron donation from the lone pair of electrons in a water molecule to the vacant p orbital of boron atom gives rise to the increase of the diradical character, leading to the large variation of the polarizability of the molecule. We also investigate the solvation effects of the bulk water by utilizing the polarizable continuum model (PCM). It is shown that the PCM environment significantly increases the degree of enhancement of the second hyperpolarizability in the intermediate diradical regime of the compound. These results suggest that a singlet diradical boron compound immersed in an electron donor solvent can exhibit efficient third-order nonlinear optical properties, which can be tuned by designing the solute-solvent interactions. We examine the reliability of the PCM results by performing the additional QM/MM simulations. It is found that the QM/MM solvation effects slightly decrease the polarizability in contrast to the PCM calculation, which implies the possibility that PCM tends to overestimate the polarizability due to the lack of the ability to represent the solvation structures.

We have developed a method for analyzing the (hyper)polarizabilities of open-shell molecular systems. This method employs the (hyper)polarizability densities based on the natural orbitals and occupation numbers, which enables us to analyze the contributions of odd electrons having various open-shell (diradical) characters. Within broken-symmetry, i. e., spin-unrestricted, single-determinant molecular orbital and density functional theory approaches, we can also remove the spin contamination effects on these quantities through spin projection. To do that, an approximate spin projected method has been elaborated and applied to the analysis of the (hyper)polarizability of multi-radical systems. As examples, typical open-shell singlet systems, 1,3-dipoles and rectangular graphene nanoflakes, are examined.

Switching on an external electric field (F) along the electron correlation direction produces a giant enhancement of the second hyperpolarizability γ in a polyaromatic diradicaloid having intermediate diradical character. This has been evidenced by carrying out spin-unrestricted density functional theory calculations with the LC-UBLYP long-range corrected exchange-correlation functional for the s-indaceno[1,2,3-cd;5,6,7-c′d′]diphenalene (IDPL) diradical compound in comparison to a closed-shell analogue of similar size composed of two pyrene moieties (PY2). For IDPL, the field-induced enhancement ratio is estimated to reach 4 orders of magnitude for an electric field of 0.0077 a.u., whereas it is less than a factor of 2 for PY2. Moreover, an enhancement is also observed by substituting both-end phenalenyl rings of IDPL with donor (NH2)/acceptor (NO2) groups, but this enhancement is limited to about 2 orders of magnitude. These enhancements are associated with a reduction of the diradical character (and therefore an improved thermal stability) as well as with the appearance of substantial type-I contributions to γ.

To demonstrate the performance of a novel approximate spin-projected (ASP) spin-unrestricted density functional theory (UDFT) method, we examine the diradical character dependences of the static polarizability (α) and second hyperpolarizability (γ) of a model open-shell singlet molecule, p-quinodimethane (PQM), using several hybrid and long-range corrected (LC) exchange–correlation functionals. The ASP-LC-UBLYP method with a range-separating parameter μ = 0.47 is found to reproduce semi-quantitatively the diradical character dependences of α and γ of the PQM model calculated using the strongly-correlated UCCSD(T) method.

We investigate a singlet diradical molecule (2-borapropane-1,3-diyl) involving a B atom in an electron donor solvent, i.e., water, from the viewpoint of controlling the (hyper)polarizability through tuning the diradical character. The quantum chemical calculations for a model system reveal that a donation from the O lone pair electrons of a water molecule to the vacant p orbital of the B atom increases the diradical character, leading to a large variation in the (hyper)polarizability of the B-containing compound. The effect of the bulk solvent determined within the PCM scheme is also shown to increase the degree of enhancement of the second hyperpolarizability in the intermediate diradical regime of the B-containing compound.

We develop a novel method for the calculation and the analysis of the one-electron reduced densities in open-shell molecular systems using the natural orbitals and approximate spin projected occupation numbers obtained from broken symmetry (BS), i.e., spin-unrestricted (U), density functional theory (DFT) calculations. The performance of this approximate spin projection (ASP) scheme is examined for the diradical character dependence of the second hyperpolarizability (γ) using several exchange-correlation functionals, i.e., hybrid and long-range corrected UDFT schemes. It is found that the ASP-LC-UBLYP method with a range separating parameter μ = 0.47 reproduces semi-quantitatively the strongly-correlated [UCCSD(T)] result for p-quinodimethane, i.e., the γ variation as a function of the diradical character.

Quantum molecular engineering has been performed to determine the second-order nonlinear optical (NLO) properties in different halo complexes of decaborane (B(10)H(14)) and their derivatives using the density functional theory (DFT) method. These decaborane halo complexes of X(-)@B(10)H(14) (X = F, Cl, Br, and I) are found to possess noncovalent charge transfer interactions. The static polarizability (α(0)) and first hyperpolarizability (β(0)) among these complexes increase by moving down the group from F to I, partly due to the increase in size of their anionic radii and the decrease in their electron affinities. A two-level approximation has been employed to investigate the origin of β(0) values in these halo complexes, which show very consistent results with those by the finite-field method. Furthermore, in the same line, two experimentally existing complexes, I(-)@B(10)H(14) and I(-)@2,4-I(2)B(10)H(12), are found to have considerably large β(0) values of 2859 and 3092 a.u., respectively, which are about three times larger than a prototypical second-order NLO molecule of p-nitroaniline, as reported by Soscun et al. [Int. J. Quantum Chem.2006, 106, 1130-1137]. Besides this, the special effects of solvent, counterion, and bottom substitutions have also been investigated. Interestingly, 2,4-alkali metal-substituted decaborane iodide complexes show remarkably large second-order NLO response with β(0) amplitude as large as 62436 a.u. for I(-)@2,4-K(2)B(10)H(12) complex, which are explained in terms of their transition energies, frontier molecular orbitals and electron density difference plots. Thus, the present investigation provides several new comparative insights into the second-order NLO properties of halo complexes of decaborane, which possess not only large first hyperpolarizabilities, but also high tunability to get a robustly large second-order NLO response by alkali metal substitution effects.

One- and two-photon absorption cross sections of symmetric open-shell singlet molecular systems are investigated using a valence configuration interaction scheme. It is found for one-photon absorption (OPA) that the peak intensity decreases and the peak position moves to the low energy region as increasing the diradical character of the system. In contrast, a significant enhancement of two-photon absorption (TPA) peak is predicted for open-shell singlet systems with intermediate diradical character as compared to the closed-shell and pure diradical systems. It is revealed that the largest TPA peak intensities occur for open-shell singlet diradicals having a ferromagnetically-coupled ground state and strongly depends on the ratio between damping factors of the excited states.

Within the spin-unrestricted density functional theory (DFT) the long-range correction (LC) scheme combined with the Becke-Lee-Yang-Parr exchange-correlation functional, referred to as LC-UBLYP method, has been applied to the calculation of the second hyperpolarizability (gamma) of open-shell singlet diradical systems of increasing complexity and has demonstrated good performance: (i) for the simplest H(2) dissociation model, the gamma values calculated by the LC-UBLYP method significantly overshoot the full configuration interaction result but reproduce qualitatively the evolution of gamma as a function of the diradical character, (ii) for small singlet diradical 1,3-dipole systems, the diradical character dependence of gamma determined by the UCCSD and UCCSD(T) reference methods is reproduced semiquantitatively by the LC-UBLYP method except in the small diradical character region, where the spin-unrestricted solutions coincide with spin-restricted solutions, (iii) the LC-UBLYP method also closely reproduces the UCCSD(T) results on the diradical character dependence of gamma of the p-quinodimethane model system, particularly in the intermediate and large diradical character regions, whereas it shows an abrupt change for a diradical character (y) close to 0.2 originating from the triplet instability, (iv) the reliability of LC-UBLYP to reproduce reference coupled cluster results on open-shell singlet systems with intermediate and large diradical characters has also been substantiated in the case of gamma of 1,4-bis-(imidazol-2-ylidene)-cyclohexa-2,5-diene (BI2Y), then (v), for real systems built from a pair of phenalenyl radicals separated by a conjugated linker, the LC-UBLYP results have been found to closely match the UBHandHLYP values-which, for small systems are in good agreement with those obtained using correlated molecular orbital methods-whereas the UB3LYP results can be much different. These results are not only important from the viewpoint of an efficient determination of the nonlinear optical properties of open-shell singlet systems, but also from the viewpoint of defining new challenges for elaborating improved exchange-correlation functionals.

Using long-range corrected density functional theory, the relationships between the electronic, magnetic, and nonlinear optical properties are drawn for two families of organic compounds, the dicyclopenta-fused acenes (DPAs) and the polyacenes (PAs), containing up to N = 12 fused rings. First, the longitudinal second hyperpolarizability (γ) of singlet DPAs is significantly enhanced with increasing system size, in comparison to PAs. This behavior is associated with an increase in the longitudinal spin polarization between the terminal five-membered rings of DPAs and is consistent with previous studies where γ is maximized for intermediate diradical character. The size dependence of the diradical character is also found to cause a hump in the γ/N evolution for singlet DPAs around N = 8. In fact, in the case of singlet PAs, the diradical characters y(0) and y(1), the various magnetic properties and the γ/N values vary monotonically with N, whereas for singlet DPAs, the shielding, the magnetizability, and the γ/N values exhibit extrema near N = 8 due to the appearance of transversal spin polarization in the middle six-membered rings in addition to the longitudinal spin polarization between the terminal five-membered rings. Moreover, it is shown that for singlet DPAs the longitudinal spin polarization (characterized by y(0)) is associated with the antiaromaticity (N ≤ 3) and the slight- or non-aromaticity (N ≥ 4) of the terminal five-membered rings, whereas the appearance of transversal spin-polarization (characterized by y(1)) is associated with the decrease in the aromaticity in the inner six-membered rings as shown for large PAs. Therefore, the exceptional behaviors in singlet DPAs for small N (N9) are caused by the increase in diradical character y(0) correlated with the anti-aromaticity or the slight-/non-aromaticity of terminal rings and the corresponding emergence of a global aromatic character. Such a relationship between the aromaticity/antiaromaticity and the diradical character is useful for designing real open-shell NLO molecules through the control of their diradical characters.