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WIREs Comput Mol Sci
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Nonlinear optical properties in open‐shell molecular systems

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For more than 30 years, nonlinear optical (NLO) properties of molecular systems have been actively studied both theoretically and experimentally due to their potential applications in photonics and optoelectronics. Most of the NLO molecular systems are closed‐shell species, while recently open‐shell molecular species have been theoretically proposed as a new class of NLO systems, which exhibit larger NLO properties than the traditional closed‐shell NLO systems. In particular, the third‐order NLO property, the second hyperpolarizability γ, was found to be strongly correlated to the diradical character y, which is a quantum‐chemically defined index of effective bond weakness or of electron correlation: the γ values are enhanced in the intermediate y region as compared to the closed‐shell (y = 0) and pure open‐shell (y = 1) domains. This principle has been exemplified by accurate quantum‐chemical calculations for polycyclic hydrocarbons including graphene nanoflakes, multinuclear transition‐metal complexes, main group compounds, and so on. Subsequently, some of these predictions have been substantiated by experiments, including two‐photon absorption. The fundamental mechanism of the y–γ correlation has been explained by using a simple two‐site model and the valence configuration interaction method. On the basis of this y–γ principle, several molecular design guidelines for controlling γ have been proposed. They consist in tuning the diradical characters through chemical modifications of realistic open‐shell singlet molecules. These results open a new path toward understanding the structure—NLO property relationships and toward realizing a new class of highly efficient NLO materials. WIREs Comput Mol Sci 2016, 6:198–210. doi: 10.1002/wcms.1242

Singlet three states {g, k, f} and a triplet sate {T} together with excitation energies and transition moments of the two‐site diradical model with two electrons in two orbitals.
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(a) Dependence of longitudinal γ/n as a function of yav of regular H2n chains (n = 1–5) calculated by the UCCSD(T)/(6)‐31(+)+G(*)* method, where the dotted line represents the displacement of the maximum γ/n value; (b) Dependences of diradical character y0 and γ/monomer (in the stacking direction) on the intermolecular distance d for the π‐π stacked phenalenyl dimer model.
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Resonance structures for open‐shell singlet molecules including diphenalenyl compounds (1(n), 3), oligoacenes (4), rectangular graphene nanoflake (GNF) (5), and hexagonal GNF (6), as well as a closed‐shell analogue 2 for 1(1). The γ (γzzzz) values are calculated using the UBHandHLYP/6‐31G (d) method.
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Resonance structures [quinoid (closed‐shell) and benzenoid (open‐shell)] of p‐quinodimethane model (a) and dependence of the diradical character (y) as a function of the length of the exo‐cyclic carbon–carbon bonds (R1) from 1.350 to 1.700 Å (b) under the bond‐length constraint of R2 = R3 = 1.4 Å. Note that the equilibrium geometry (R1 = 1.351, R2 = 1.460, R3 = 1.346 Å, optimized by the RB3LYP/6‐311G(d)) gives the lowest diradical character (y = 0.146).
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γ values of model compounds as determined at different levels of approximation. (a) stretched H2 molecule (aug‐cc‐pVDZ basis set); (b, c) p‐quinodimethane with different diradical characters (6‐31G(d)+p basis set).
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Variations of second hyperpolarizabilities, γDL, γIIDL, and γIII2DL, as a function of diradical character (y) in the case of rK = 0 (a), y dependences of γIII2DL, μgkDL2μkfDL2, and EkgDL2EfgDL in the case of rK = 0 (b), and contours of γDL (−5.0 ≤ γDL < 5.0, interval = 0.1) on the plane (y, rK), where solid, dotted, and dashed black lines indicate positive, negative, and zero lines of γDL, respectively (c). In (c), the red line represents the ridge line connecting the (y, rK) points giving maximum γDL values. The blue curve indicates rJ = 0 and the lower and upper regions of this curve represent the singlet (antiferromagnetic) and triplet (ferromagnetic) ground states, respectively. Region A and B–D represent closed‐shell (traditional NLO systems) and open‐shell (theoretically predicted systems) regions, respectively.
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Diradical character dependences of dimensionless excitation energies (EDLkg and EDLfg), squared transition moments ((μDLgk)2 and (μDLkf)2).
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Diradical character y versus |U/tab|. Effective bond order q = 1–y is also shown at delocalization (y = 0) and localization (y = 1) limits.
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