Second hyperpolarizability of phenalenyl radical system involving acetylene π-conjugated bridge

The static second hyperpolarizability (γ) of a singlet diradical system involving phenalenyl radicals linked with acetylene π-conjugated bridge, BPLE, is investigated as well as reference closed-shell systems, BPRY1 and BPRY2, by the hybrid density functional theory method. The γ value of BPLE with intermediate diradical character is shown to be about four times as large as that of BPRY1 with similar π-conjugation length and to be about twice as large as that of BPRY2 with longer π-conjugation length. This feature is in agreement with our prediction that the molecules with intermediate diradical characters enhance γ values compared to the closed-shell molecules.     

Second hyperpolarizability (gamma) of singlet diradical system: dependence of gamma on the diradical character

The dependence of the second hyperpolarizability (gamma) on the diradical character (y) for singlet diradical systems is investigated using a model compound, the p-quinodimethane (PQM) molecule with different both-end carbon-carbon (C-C) bond lengths, by several ab initio molecular orbital and density functional theory methods. The diradical character based on UHF calculations indicates that at equilibrium geometry PQM is in a singlet ground state and primarily exhibits a quinoid structure, whereas the diradical character increases when increasing both-end C-C bond lengths. At the highest level of approximation, that is, using the UCCSD(T) method with the 6-31G+diffuse p (zeta = 0.0523) basis set, the longitudinal static gamma of PQM presents a maximum value for intermediate diradical character (y approximately 0.5) while the gamma values are larger for intermediate and large diradical character (y approximately 0.5-0.7) than for small diradical character (y < 0.2). This feature suggests that the gamma values of singlet diradical systems in the intermediate and somewhat strong correlation regimes are significantly enhanced as compared to those in the weak correlation regime. These results are substantiated by a complementary study of the variation in gamma upon twisted ethylene.     

Third-order nonlinear optical properties of open-shell supermolecular systems composed of acetylene linked phenalenyl radicals

The third-order nonlinear optical (NLO) properties, at the molecular level, the static second hyperpolarizabilities, γ, of supermolecular systems composed of phenalenyl and pyrene rings linked by acetylene units are investigated by employing the long-range corrected spin-unrestricted density functional theory, LC-UBLYP, method. The phenalenyl based superethylene, superallyl, and superbutadiene in their lowest spin states have intermediate diradical characters and exhibit larger γ values than the closed-shell pyrene based superpolyene systems. The introduction of a positive charge into the phenalenyl based superallyl radical changes the sign of γ and enhances its amplitude by a factor of 35. Although such sign inversion is also observed in the allyl radical and cation systems in their ground state equilibrium geometries, the relative amplitude of γ is much different, that is, |γ(regular allyl cation)/γ(regular allyl radical)| = 0.61 versus |γ(phenalenyl based superallyl cation)/γ(phenalenyl based superallyl radical)| = 35. In contrast, the model ethylene, allyl radical/cation, and butadiene systems with stretched carbon-carbon bond lengths (2.0 ?), having intermediate diradical characters, exhibit similar γ features to those of the phenalenyl based superpolyene systems. This exemplifies that the size dependence of γ as well as its sign change by introducing a positive charge on the phenalenyl based superpolyene systems originate from their intermediate diradical characters. In addition, the change from the lowest to the highest π-electron spin states significantly reduces the γ amplitudes of the neutral phenalenyl based superpolyene systems. For phenalenyl based superallyl cation, the sign inversion of γ (from negative to positive) is observed upon switching between the singlet and triplet states, which is predicted to be associated with a modification of the balance between the positive and negative contributions to γ. The present study paves the way toward designing a variety of open-shell NLO supermolecular systems composed of phenalenyl radical building blocks.     

Origin of the enhancement of the second hyperpolarizability of singlet diradical systems with intermediate diradical character

The origin of the diradical character dependence of the second hyperpolarizability (gamma) of neutral singlet diradical systems is clarified based on the perturbation formula of gamma using the simplest diradical molecular model with different diradical characters, i.e., H2 under bond dissociation. The enhancement of gamma in the intermediate diradical character region turns out to originate from the increasing magnitude of the transition moment between the first and second excited states and the decrease of that between the ground and first excited states, respectively, with the increase in diradical character. This feature confirms that open-shell singlet conjugated molecules with intermediate diradical characters constitute a new class of third-order nonlinear optical systems, whose gamma values can be controlled by the diradical character in addition to the conjugation length.     

Second hyperpolarizabilities (gamma) of bisimidazole and bistriazole benzenes: diradical character, charged state, and spin state dependences

The second hyperpolarizabilities of bisimidazole- and bistriazole-benzene compounds have been calculated at different levels of approximation to unravel the effects of diradical character as well as of charge and spin multiplicity. The largest second hyperpolarizabilities are associated with intermediate diradical character, provided positive charging does not compensate for this effect. For the neutral diradical bisimidazole compound, the singlet diradical species possesses a second hyperpolarizability two to three times larger than the corresponding triplet, demonstrating the possibility of spin state control of the third-order NLO responses for diradical species.     

Enhancement of second hyperpolarizabilities in open-shell singlet slipped-stack dimers composed of square planar nickel complexes involving o-semiquinonato type ligands

Using the spin-unrestricted hybrid density functional theory method, we have investigated the intermolecular interaction effects on the longitudinal static second hyperpolarizability (γ) of open-shell singlet slipped-stack dimers composed of singlet diradical square planar nickel complexes involving o-semiquinonato type ligands, Ni(o-C(6)H(4)X(2))(2) (where X = O, NH, S, Se, PH). For comparison, we have also examined the γ values of a closed-shell singlet slipped-stack dimer composed of closed-shell monomers Ni[o-C(6)H(4)S(NH(2))](2). It is found that for interplanar distance ranging between 3.0 and 5.0 ? the slipped-stack dimers with intermediate monomer diradical characters exhibit larger γ values per monomer (γ(dimer)/2) than those with large monomer diradical characters or than the closed-shell dimer. These results extend the domain of validity of the relationship found between γ and the diradical character for individual molecules. It also turns out that the ratio R = (γ(dimer)/2)/γ(monomer) increases upon decreasing the interplanar distance and that this increase is larger for intermediate diradical character than for the other cases. These phenomena have been analyzed by considering the γ density distributions of the dimers, demonstrating a significant field-induced third-order charge transfer between the monomers in the case of intermediate diradical character. The present results indicate that open-shell singlet slipped-stack aggregates composed of monomers with intermediate diradical characters constitute another mean for achieving highly efficient and tunable third-order nonlinear optical materials.     

Second hyperpolarizabilities of singlet polycyclic diphenalenyl radicals: effects of the nature of the central heterocyclic ring and substitution to diphenalenyl rings

Adopting density functional theory and a hybrid exchange-correlation functional, the relationship between the second hyperpolarizability (gamma) and the diradical character has been investigated for diphenalenyl-based compounds containing different heterocyclic five-membered central rings (C(4)H(4)X, where X = NH, PH, O, S, CH(2), SiH(2), BH, GaH, C=O, C=S, and C=Se) or substituted by donor (NH(2))/acceptor(NO(2)) groups. It turns out that these structural modifications can tune the diradical character from 0.0 to 0.968 and lead to variations of gamma over more than 1 order of magnitude, demonstrating the controllability of gamma in this family of compounds. In particular, when the central ring is strongly aromatic, the diradical character is larger than 0.7, which is associated with pretty large gamma values except for almost the pure diradical case (y approximately 1). On the other hand, when the aromaticity decreases--or the antiaromaticity increases--the diradical character and the second hyperpolarizability get smaller. These relationships are correlated to structural (bond length alternation) and charge distribution (charge transfer between the phenalenyl rings and the central ring) properties, which account for the relative importance of the resonance diradical, zwitterionic, and quinoid forms. Therefore, the diradical character and the second hyperpolarizability can be controlled by the aromaticity of the ring while the paradigm of the enhancement of gamma for intermediate diradical character is globally verified. Then, upon introducing donor groups, the zwitterionic character increases, leading to closed-shell species and small second hyperpolarizabilities. In the case of substitution by acceptor groups, the charge transfer is reduced but the diradical character and the second hyperpolarizability hardly changes.     

Theoretical study on second hyperpolarizabilities of singlet diradical square planar nickel complexes involving o-semiquinonato type ligands

Hybrid density functional theory method is applied for investigating the diradical character dependence of the second hyperpolarizability (gamma) of square planar nickel complexes involving several types of bidentate ligands [o-C6H4XY, where X = Y = O, NH, S, Se, and PH as well as (X, Y) = (NH, NH2) and (S, NH2)]. It is found that, as a function of the donor atoms, the diradical character of these complexes varies from 0.0 to 0.884 and is associated with substantial variations of gamma ranging from 14 x 10(3) to 819 x 10(3) au. In particular, the largest gamma values are associated with intermediate diradical characters in good agreement with the structure-property relationship obtained for pure hydrocarbon systems. Increasing the electronegativity of the X and Y donor groups of the ligands leads to larger diradical characters as a result of the enhancement of the double bond nature of the C=X(Y) bonds, which further stabilizes the diradicals on both-end benzene rings. This demonstrates that the electronegativities of the donor atoms of the ligands become a tuning parameter of the diradical character and then of the gamma values of these complexes.     

Oligoacenes: theoretical prediction of open-shell singlet diradical ground states

A series of oligoacenes from benzene to decacene were studied computationally with DFT and CASSCF methods. In contrast to the common view that acenes are closed-shell systems or may have a triplet ground state, these results offer the first theoretical predictions for the singlet ground state and diradical character for oligoacenes. The nature of the ground states of these molecules arises from the disjoint nature of the NBMOs that are singly occupied in the diradical.     

Size dependences of the diradical character and the second hyperpolarizabilities in dicyclopenta-fused acenes: relationships with their aromaticity/antiaromaticity

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 (N < 9) 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.     

Giant electric field effect on the second hyperpolarizability of symmetric singlet diradical molecules

This contribution reveals the effects of a static electric field on the static second hyperpolarizability γ of symmetric singlet diradical molecules using the valence configuration interaction scheme. It is found that under the effect of a field, the component of γ along the axis joining the two radical sites can be gigantically (approximately two to three orders) enhanced for symmetric diradicals having intermediate diradical characters with respect to those of closed-shell and pure diradical molecules in the absence of a field. Moreover, this electric field enhancement of γ increases as a function of the diradical character. These results and their analysis propose therefore a new strategy to design materials with exceptional nonlinear optical responses.     

Theoretical Study on Structural Stability and Nonlinear Optical Properties of Non-conjugated Carbon and Nitrogen Diradical Molecules

The geometrical structures and stability of non-conjugated C5H10 and C3H8N2 singlet and triplet diradical molecules have been investigated at the UCCSD/6-311g** level. The effects of molecular structure, radical position, amount of Hartree Hork (HF) exchange and spin multiplicity on the nonlinear optical (NLO) coefficients have been also investigated. The reliable UCCSD results show that the triplets of all diradical molecules are more stable compared to their singlet analogues. In addition, the α s and β tot values of C5H10 and C3H8N2 triplet diradical mo-lecules have been investigated by the UBHandHLYP, UB3LYP, UBLYP, UHF and UCCSD methods. The investigation shows that the variations in α s and β tot values are closely connected to the amount of HF exchange. The increasing amount of HF exchange results in monotonic decreases in α s and β tot values, while the α s and β totvalues of singlet diradical molecules and the γ s of C5H10 and C3H8N2 singlet and triplet diradical molecules have been studied by the UBHandHLYP method. The results illustrate that the NLO coefficients for our studied non-conjugated carbon and nitrogen diradical species can be tuned by molecular structure, radical position and spin multiplicity, which are very significant for designing NLO materials.     

Singlet-triplet energy gaps for diradicals from fractional-spin density-functional theory

Open-shell singlet diradicals are difficult to model accurately within conventional Kohn-Sham (KS) density-functional theory (DFT). These methods are hampered by spin contamination because the KS determinant wave function is neither a pure spin state nor an eigenfunction of the S(2) operator. Here we present a theoretical foray for using single-reference closed-shell ground states to describe diradicals by fractional-spin DFT (FS-DFT). This approach allows direct, self-consistent calculation of electronic properties using the electron density corresponding to the proper spin eigenfunction. The resulting FS-DFT approach is benchmarked against diradical singlet-triplet gaps for atoms and small molecules. We have also applied FS-DFT to the singlet-triplet gaps of hydrocarbon polyacenes.     

Local aromaticity of the lowest-lying singlet States of [N]acenes (N = 6-9)

The local aromaticities of the six-membered rings in the two lowest-lying singlet states of [n]acenes (n = 6-9) have been assessed by means of three probes of local aromaticity based on structural, magnetic, and electron delocalization properties. Important differences between the local aromaticities of the closed-shell and diradical singlet electronic states are found. Thus, while the inner rings have the largest aromatic character in the closed-shell singlet states, the outer rings become the most aromatic for the diradical singlet states.     

Theoretical evidence for the singlet diradical character of square planar nickel complexes containing two o-semiquinonato type ligands

Density functional theory and complete active space self-consistent field computations are applied to elucidate the singlet diradical character of square planar, diamagnetic nickel complexes that contain two bidentate ligands derived from o-catecholates, o-phenylenediamines, o-benzodithiolates, o-aminophenolates, and o-aminothiophenolates. In the density functional framework, the singlet diradical character is discussed within the broken symmetry formalism. The singlet-triplet energy gaps, the energy gained from symmetry breaking, the spin distribution in the lowest triplet state, and the form of the magnetic orbitals are applied as indicators for the singlet diradical character. Moreover, a new index for the diradical character is proposed that is based on symmetry breaking. All symmetry breaking criteria show that the complexes obtained from o-catecholates and o-benzodithiolates have the largest and the smallest singlet diradical character, respectively. The singlet diradical character should be intermediate for the complexes derived from o-phenylenediamines, o-aminophenolates, and o-aminothiophenolates. The diradical character of all complexes suggests the presence of Ni(II) central atoms. This is also indicated by the d-populations computed by means of the natural population analysis.     

Open‐Shell Characters and Second Hyperpolarizabilities of One‐Dimensional Graphene Nanoflakes Composed of Trigonal Graphene Units

The impact of topology on the open‐shell characters and the second hyperpolarizabilities (γ) has been addressed for one‐dimensional graphene nanoflakes (GNFs) composed of the smallest trigonal graphene (phenalenyl) units. The main results are: 1) These GNFs show not only diradical but also multiradical characters when increasing the number of linked units. 2) GNFs composed of an equivalent number of units can exhibit a wide range of open‐shell characters—from nearly closed‐shell to pure multiradical characters—depending on the linking pattern of the trigonal units. 3) This wide variation in open‐shell characters is explained by their resonance structures and/or by their (HOMO?i)?(LUMO+i) gaps deduced from the orbital correlations. 4) The change in the linking structure of the units can effectively control their open‐shell characters as well as their γ values, of which the longitudinal components are significantly enhanced for the singlet GNFs having intermediate open‐shell characters. 5) Singlet alternately linked (AL) systems present intermediate multiradical characters even in the case of a large number of units, which creates a significant enhancement of γ with increasing the size, whereas nonalternately linked (NAL) systems, which present pure multiradical characters, possess much smaller γ values. Finally 6) by switching from the singlet to the highest spin states, the γ values of NAL systems hardly change, whereas those of AL systems exhibit large reductions. These fascinating structure–property relationships between the topology of the GNFs, their open‐shell characters, and their γ values not only deepen the understanding of open‐shell characters of GNFs but aim also at stimulating further design studies to achieve giant NLO responses based on open‐shell graphene‐like materials.     

Enhancement of the third-order nonlinear optical properties in open-shell singlet transition-metal dinuclear systems: effects of the group, of the period, and of the charge of the metal atom

Metal-metal multiply bonded complexes in their singlet state have been predicted to form a novel class of "σ-dominant" third-order nonlinear optical compounds based on the results of dichromium(II) and dimolybdenum(II) systems (H. Fukui et al. J. Phys. Chem. Lett.2011, 2, 2063) whose second hyperpolarizabilities (γ) are enhanced by the contribution of the dσ electrons with an intermediate diradical character. In this study, using the spin-unrestricted coupled-cluster method with singles and doubles as well as with perturbative triples, we investigate the dependences of γ on the group and on the period of the transition metals as well as on their atomic charges in several open-shell singlet dimetallic systems. A significant enhancement of γ is observed in those dimetallic systems composed of (i) transition metals with a small group number, (ii) transition metals with a large periodic number, and (iii) transition metals with a small positive charge. From the decomposition of the γ values into the contributions of dσ, dπ, and dδ electrons, the γ enhancements are shown to originate from the dσ contribution, because it corresponds to the intermediate diradical character region. Furthermore, the amplitude of dσ contribution turns out to be related to the size of the d(z(2)) atomic orbital of the transition metal, which accounts for the dependence of γ on the group, on the period, and on the charge of the metal atoms. These dependences provide a guideline for an effective molecular design of highly efficient third-order nonlinear optical (NLO) systems based on the metal-metal bonded systems.     

Theoretical study on the relationship between spin multiplicity effects and nonlinear optical properties of the pyrrole radical (C4H4N.)

The geometrical structure and stability of neutral pi-conjugated C4H4N. with three spin states were investigated by using ab initio and density functional theory (DFT) methods. In addition, the linear and nonlinear optical properties were studied at the same level combined with the finite field approach. The calculated results show that conjugation and stability decreased with increasing spin multiplicity. These reliable UCCSD results show that the polarizability (alpha) values of C4H4N. with the quartet state are maximal, while those of C4H4N. with the doublet state are minimal. The order of betatot values is betasextet > betadoublet > betaquartet. The second hyperpolarizability (gamma) values exhibit positive values. The variation trends of gamma are consistent with alpha.     

Diradical Character Tuning for the Third‐Order Nonlinear Optical Properties of Quinoidal Oligothiophenes by Introducing Thiophene‐S,S‐dioxide Rings

To create a design guideline for efficient third‐order nonlinear optical (NLO) molecules, the chain‐length (n) dependences of the diradical character y and the longitudinal second hyperpolarizability γ of quinoidal oligothiophenes (QTs), from monomers to octamers, involving thiophene‐S,S‐dioxide rings are investigated by using the density functional theory method. It turns out that the diradical character of the modified QTs is reduced as compared to those of the pristine QTs. By introducing an appropriate number of oxidized rings into the QT framework, intermediate y values can be achieved even in the systems with large values of n, in which the pristine QTs are predicted to have pure diradical character. Such intermediate diradical oligomers are shown to exhibit enhanced γ values as compared to the pristine QTs with the same value for n. From the calculation results, the introduction of the optimal number of thiophene‐S,S‐dioxide rings is predicted to be an efficient chemical modification for optimizing the third‐order NLO properties of open‐shell QTs through tuning the diradical characters.     

A Highly Stable Organic Luminescent Diradical

It is very challenging to obtain stable room-temperature luminescent open-shell singlet diradicals. Herein we report the first stable Müller's hydrocarbon TTM-PhTTM with luminescent properties. Variable-temperature electron paramagnetic resonance spectroscopy measurements and theoretical calculations show that TTM-PhTTM has an open-shell singlet ground state with a diradical character of 90 %. Because of a small singlet-triplet energy gap, the open-shell singlet ground state can be thermally excited to a triplet state. TTM-PhTTM shows room-temperature deep-red emission in various solutions. Unusually high stability of TTM-PhTTM was also observed owing to effective steric hindrance and spin delocalization. Our results are beneficial to the rational design and discovery of more stable luminescent diradical materials.