# Electronic Structure of Open-Shell Singlet Molecules: Diradical Character Viewpoint

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## Abstract

This chapter theoretically explains the electronic structures of open-shell singlet systems with a wide range of open-shell (diradical) characters. The definition of diradical character and its correlation to the excitation energies, transition properties, and dipole moment differences are described based on the valence configuration interaction scheme using a two-site model with two electrons in two active orbitals. The linear and nonlinear optical properties for various polycyclic aromatic hydrocarbons with open-shell character are also discussed as a function of diradical character.

## Keywords

Diradical character Open-shell singlet Excitation energy and property Valence configuration interaction Nonlinear optical property## 1 Introduction

*π*–

*π*stack open-shell aggregates [10]. These features are known to originate in the open-shell character in the ground electronic states of those open-shell singlet systems [11, 12, 13, 14, 15, 16, 17, 18]. The open-shell nature of PAHs is qualitatively understood by resonance structures. For example, benzenoid and quinoid forms of the resonance structures of zethrene species and diphenalenyl compounds correspond to the closed-shell and open-shell (diradcial) states, respectively (Fig. 1a, b). Also, for acenes, considering Clar’s aromatic

*π*-sextet rule [19], which states that the resonance forms with the largest number of disjoint aromatic

*π*-sextets (benzenoid forms) contribute most to the electronic ground states of PAHs, it is found that the acenes tend to have radical distributions on the zigzag edges as increasing the size (Fig. 1c). Indeed, recent highly accurate quantum chemical calculations including density matrix renormalization group (DMRG) method clarify that the electronic ground states of long acenes and several graphene nanoflakes (GNFs) are open-shell singlet multiradical states [20, 21, 22, 23, 24]. Also, the local aromaticity of such compounds is turned out to be well correlated to the benzenoid moieties in the resonance structures [25, 26].

Although the resonance structures with Clar’s sextet rule and aromaticity are useful for qualitatively estimating the open-shell character of the ground-state PAHs, we need a quantitative estimation scheme of the open-shell character and chemical design guidelines for tuning the open-shell character, which contribute to deepening the understanding of the electronic structures of these systems and also to realizing applications of open-shell based unique functionalities. In this chapter, we first provide a quantum-chemically well-defined open-shell character, i.e., diradical character [16, 18, 27, 28, 29, 30, 31], and clarify the physical and chemical meaning of this factor. Next, the relationships between the excitation energies/properties and diradical character are revealed based on the analysis of a simple two-site molecular model with two electrons in two active orbitals using the valence configuration interaction (VCI) method [7]. On the basis of this result, linear and nonlinear optical properties are investigated from the viewpoint of diradical character. Such analysis is also extended to asymmetric open-shell systems. Several realistic open-shell singlet molecular systems are also investigated from the viewpoint of the relationship between the diradical character and resonance structures.

## 2 Electronic Structures of Open-Shell Singlet Systems

### 2.1 Classification of Electronic States Based on Diradical Character

*α*and

*β*spins, e.g.,

*α*spin distributes mainly on the left-hand side, while the

*β*spin mainly on the right-hand side as increasing the bond distance. This picture (approximation) seems to be more intuitive than the symmetry-adapted approach, but this suffers from the intrinsic deficiency, i.e., spin contamination [16, 29], where high spin states such as triplet states are mixed in the singlet wavefunction. The bond dissociation process is qualitatively categorized into three regimes, i.e., stable bond regime (I), intermediate bond regime (II) and bond dissociation (weak bond) regime (III). As shown in later, these regimes are characterized by “diracial character”

*y*, which takes a value between 0 and 1: small

*y*(~0) for (I), intermediate

*y*for (II) and large

*y*(~1) for (III) (see Fig. 2). In other words, 1–

*y*indicates an “effective bond order” [29]. This description is employed in chemistry, while in physics, these three regimes are characterized by the degree of “electron correlation”: weak correlation regime (I), intermediate correlation regime (II) and strong correlation regime (III) (see Fig. 2). This physical picture is also described by the variation in the degree of delocalization of two electrons on two atomic sites: strong delocalization (weak localization) (I), intermediate delocalization (intermediate localization) (II) and weak delocalization (strong localization) (III). Namely, the effective repulsion interaction between two electrons means the electron correlation, so that the delocalization decreases (the localization increases) when the correlation increases. Namely, in physics, the bond dissociation limit is considered to be caused by the strong correlation limit (strong localization limit). Thus, the “diradical character” is a fundamental factor for describing the electronic states and could be a key factor bridging between chemical and physical concepts on the electronic structures [16, 18].

### 2.2 Schematic Diagram of Electronic Structure of a Two-Site Model

In this section, let us consider a one-dimensional (1D) homodinuclear molecule A–B with two electrons in two orbitals (HOMO and LUMO) in order to understand schematically its electronic structure, i.e., wavefunction [32]. In this case, the spatial distribution of the singlet wavefunction can be described on the (1*α*, 2*β*) plane, where 1*α* and 2*β* indicate the real coordinate of electron 1 with *α* spin and that of electron 2 with *β* spin, respectively. More exactly, the singlet wavefunction is also distributed on another plane (1*β*, 2*α*), but this is the same spatial distribution as that on (1*α*, 2*β*) plane. Thus, we can discuss the singlet wavefunction using only the distribution on the (1*α*, 2*β*) plane without loss of generality. Figure 3a shows the 1D two-electron system A–B and the 2D plane (1*α*, 2*β*), on which the spatial distribution of the singlet wavefunction is plotted. On the (1*α*, 2*β*) plane, the dotted lines represent the positions of nuclei A and B, and the diagonal dashed line indicates the Coulomb wall. The two electrons undergo large Coulomb repulsion near the Coulomb wall, while those receive attractive forces from nuclei A and B near the dotted lines. The covalent (or diradical) configuration (where mutually antiparallel spins are distributed on A and B, respectively) is described by the black dots symmetrically distributed with respect to the diagonal dashed line, while the zwitterionic configuration (where a pair of *α* and *β* spins is distributed on A or B) is done by the black dots on the diagonal dashed line.

*α*, 2

*β*) plane corresponds to \(\phi_{\text{H}} (1)\phi_{\text{H}} (2)\alpha (1)\beta (2)\), so that spatial part \(\phi_{\text{H}} (1)\phi_{\text{H}} (2)\) is a product of the HOMO(1) and HOMO(2) as shown in Fig. 3b. Apparently, the distribution of each black dot (intersection points of dotted lines) is found to be equal in the amplitude and phase. This implies that covalent (neutral) and ionic configurations are equally mixed in the HF singlet ground state wavefunction, which is a well-known feature of mean field approximation, i.e., no electron correlation. Next, we consider singly excited configuration from the HOMO to LUMO. The singly excited singlet Slater determinant is represented by

*λ*

^{2}, which is a weight of the doubly excited configuration in the ground state, is found to be able to change from 0 to 1/2, which indicates the change from the mean field wavefunction \(\psi_{\text{G}}^{{}}\) (MO limit) to the pure neutral (diradical) component (atomic orbital (AO) limit in the bond dissociation system). Accordingly, the second excited state \(\varPsi_{\text{f}}^{{}}\) changes from the mean field wavefunction \(\psi_{\text{D}}^{{}}\) to the pure ionic component. As a result, considering the bond dissociation model, the change of 2

*λ*

^{2}from 0 to 1 corresponds to the change from the stable bond region to the bond dissociation limit. Namely, the 2

*λ*

^{2}is regarded as the “diradical character”, which is indeed the original definition of the diradical character [27, 28, 29].

### 2.3 Broken-Symmetry Approach with Spin-Projection Scheme for Evaluation of Diradical Character

*χ*and

*η*, the ground state BS wavefunction is expressed as

*χ*and

*η*are represented by symmetry-adapted MOs \(\phi_{\text{H}}\) and \(\phi_{\text{L}}\) as [28, 29]

*θ*is a mixing parameter ranging from 0 to

*π*/2. For

*θ*= 0, \(\chi = \eta = \phi_{\text{H}}\), while

*θ*=

*π*/2, \(\chi = \frac{1}{\sqrt 2 }\left( {\phi_{\text{H}} + \phi_{\text{L}} } \right) \equiv a\) and \(\eta = \frac{1}{\sqrt 2 }\left( {\phi_{\text{H}} - \phi_{\text{L}} } \right) \equiv b\), where

*a*and

*b*are referred to as magnetic orbitals (localized natural orbitals (LNOs)) and are nearly equal to AO \(\varphi_{\text{A}}\) and \(\varphi_{\text{B}}\), respectively. Namely, the BS orbitals can represent the variation from the MO limit to the AO limit by changing

*θ*from 0 to

*π*/2. Using Eq. 8, the ground state BS wavefunction \(\varPsi^{\text{BS}} (\chi \bar{\eta })\) is expressed as [28, 29]

*χ*and

*η*. Although the BS wavefunction suffers from a spin contamination, which is known to sometimes give improper relative energies for different spin states and erroneous physicochemical properties [29, 33, 34], the BS approach has an advantage of being able to include partial electron-correlation, qualitatively correct singlet spatial distribution in the present case, by just using a simple single determinant calculation scheme instead of high-cost multi-reference calculation schemes. Indeed, Yamaguchi applied the perfect-pairing type spin-projection scheme to the BS solution and developed an easy evaluation method of diradical character

*y*[28, 29]. Using the overlap between

*χ*and

*η*, i.e., \(T \equiv \left\langle {\chi } \mathrel{\left | {\vphantom {\chi \eta }} \right. \kern-0pt} {\eta } \right\rangle = \cos^{2} \frac{\theta }{2} - \sin^{2} \frac{\theta }{2} = \cos \theta\), we rewrite Eq. 9 as

*y*, i.e., twice the weight of the doubly excitation configuration, we obtain the expression of diradical character in the PUHF formalism [28, 29]:

*n*-radical system, the perfect-pairing type (i.e., considering a doubly excitation from HONO −

*i*to LUNO +

*i*) spin-projected diradical characters and occupation numbers are defined as [28, 29]

*i*(\(n_{{{\text{LUNO + }}i}}\)) is given by \(1 - T_{i}\).

## 3 Electronic States of Two-Site Model by the Valence Configuration Interaction Method

### 3.1 Ground and Excited Electronic States and Diradical Character

*a*and

*b*, with the

*z*-component of spin angular momentum

*M*

_{s}= 0 (singlet and triplet), we can consider two neutral

*α*, 2

*β*) plane are described by \(a(1)b(2)\), \(b(1)a(2)\), \(a(1)a(2)\), and \(b(1)b(2)\), respectively (see

*n*1,

*n*2,

*i*1, and

*i*2, respectively, shown in Fig. 3a). The valence configuration interaction (VCI) matrix of the electronic Hamiltonian

*H*is represented by using the LNO basis [7, 35]:

*U*represents the difference between on- and neighbor-site Coulomb repulsions, referred to as effective Coulomb repulsion:

*f*is the Fock operator in the LNO representation] [16, 36].

We obtain the following four solutions by diagonalizing the CI matrix of Eq. 20 [4, 5, 7, 16, 18].

*S*

_{1g}has a larger weight of neutral determinant (the first term) than that of ionic one (the second term). The energy is

*S*

_{1g}, state

*S*

_{2g}has a larger weight of ionic determinant (the second term) than that of neutral one (the first term). The energy is

*κ*and

*η*are functions of |

*t*

_{ ab }/

*U*| [4, 5, 7, 16, 18], which indicates the ease of the electron transfer, i.e., the degree of delocalization, between atoms A and B. As seen from Fig. 5, as decreasing

*r*

_{t}, the

*κ*(the coefficient of the neutral determinant) increases toward \({1 \mathord{\left/ {\vphantom {1 {\sqrt 2 }}} \right. \kern-0pt} {\sqrt 2 }}\) at \(r_{t} = 0\), while the

*η*(the coefficient of the ionic determinant) decreases toward 0 at \(r_{t} = 0\). From this behavior, the mobility of electrons, i.e., the delocalization nature, between sites A and B is found to determine the relative neutral (covalent) and ionic natures of the state, i.e., the diradical nature.

*a*,

*b*} and symmetry-adapted MOs { \(\phi_{\text{H}}\), \(\phi_{\text{L}}\) }, i..e, \(a = \frac{1}{\sqrt 2 }\left( {\phi_{\text{H}} + \phi_{\text{L}} } \right)\) and \(b = \frac{1}{\sqrt 2 }\left( {\phi_{\text{H}} - \phi_{\text{L}} } \right)\), the lower singlet state Eq. 24a is also expressed by

*y*, which is defined as twice the weight of the doubly excitation configuration, \(2\zeta^{2} = 2(\kappa - \eta )^{2} = 1 - 4\kappa \eta\), is represented by

*y*as a function of \(\left| {{U \mathord{\left/ {\vphantom {U {t_{ab} }}} \right. \kern-0pt} {t_{ab} }}} \right|( \equiv {1 \mathord{\left/ {\vphantom {1 {r_{t} }}} \right. \kern-0pt} {r_{t} }})\) is shown in Fig. 6. As increasing \(\left| {{U \mathord{\left/ {\vphantom {U {t_{ab} }}} \right. \kern-0pt} {t_{ab} }}} \right|\),

*y*value is shown to increase from 0 to 1, which correspond to \(\left| {{U \mathord{\left/ {\vphantom {U {t_{ab} }}} \right. \kern-0pt} {t_{ab} }}} \right|\) ≤ ~1 (\(r_{t}\) ≥ ~1) and \(\left| {{U \mathord{\left/ {\vphantom {U {t_{ab} }}} \right. \kern-0pt} {t_{ab} }}} \right| \to \infty\) (\(r_{t} \to 0\)), respectively. From the physical meaning of the transfer integral

*t*

_{ ab }and the effective Coulomb repulsion

*U*, \(y \to 1\) at \(\left| {{U \mathord{\left/ {\vphantom {U {t_{ab} }}} \right. \kern-0pt} {t_{ab} }}} \right| \to \infty\) (\(r_{t} \to 0\)) implies the localization of electrons on each site, i.e., a pure diradical state, while \(y \to 0\) at \(\left| {{U \mathord{\left/ {\vphantom {U {t_{ab} }}} \right. \kern-0pt} {t_{ab} }}} \right|\) ≤ ~1 (\(r_{t}\) ≥ ~1) implies the delocalization of electrons over two sites, i.e., a closed-shell stable bond state. Namely, this represents that the diradical character

*y*indicates the degree of electron correlation \(\left| {{U \mathord{\left/ {\vphantom {U {t_{ab} }}} \right. \kern-0pt} {t_{ab} }}} \right|\) in the physical sense. On the other hand, this variation in delocalization over two sites according to the variation in

*y*substantiates the variation of diradical character during the bond dissociation of a homodinuclear system discussed in Sect. 2.1. Indeed, from Eq. 17, we obtain

*i*) and antibonding (LUNO +

*i*) orbitals [29]. This is demonstrated in Fig. 6 by the variation of 1–

*y*from 1 (stable bond region) to 0 (bond breaking region) with increasing the electron correlation \(\left| {{U \mathord{\left/ {\vphantom {U {t_{ab} }}} \right. \kern-0pt} {t_{ab} }}} \right|\). Namely,

*y*indicates the bond weakness in the chemical sense. In summary, the diradical character

*y*is a fundamental factor for describing electronic states and can bridge the two pictures for electronic states between physics, i.e., electron correlation, and chemistry, i.e., effective chemical bond.

### 3.2 Diradical Character Dependence of Excitation Energies and Properties

*R*

_{BA}\(\equiv\)

*R*

_{ bb }−

*R*

_{ aa }= \(\left( b \right.\left| r \right|\left. b \right) - \left( a \right.\left| r \right|\left. a \right)\) is an effective distance between the two radicals. In these formulae,

*U*and

*R*

_{BA}play roles for their units, energy and length, respectively. Except for Eq. 29, which includes the dimensionless direct exchange \(r_{K} ( \equiv 2K_{ab} /U)\), these quantities are as functions of

*y*. These dimensionless excitation energies (\(E_{{S_{{1{\text{u}}}} ,S_{{1{\text{g}}}} }}^{\text{DL}} \equiv E_{{S_{{1{\text{u}}}} ,S_{{1{\text{g}}}} }} /U\), \(E_{{S_{{2{\text{g}}}} ,S_{{1{\text{g}}}} }}^{\text{DL}} \equiv E_{{S_{{2{\text{g}}}} ,S_{{1{\text{g}}}} }} /U\)) and dimensionless transition moments squared (\((\mu_{{S_{{1{\text{g}}}} ,S_{{1{\text{u}}}} }}^{\text{DL}} )^{2} \equiv (\mu_{{S_{{1{\text{g}}}} ,S_{{1{\text{u}}}} }}^{{}} )^{2} /R_{\text{BA}}^{2}\), \((\mu_{{S_{{1{\text{u}}}} ,S_{{2{\text{g}}}} }}^{\text{DL}} )^{2} \equiv (\mu_{{S_{{1{\text{u}}}} ,S_{{2{\text{g}}}} }}^{{}} )^{2} /R_{\text{BA}}^{2}\)) are plotted as functions of

*y*(see Fig. 7b for

*r*

_{ K }= 0), where \(E_{{S_{{1{\text{u}}}} ,S_{{1{\text{g}}}} }}^{\text{DL}}\) approaches 1 −

*r*

_{ K }at \(y \to 1\).

Both the dimensionless transition moments squared, \((\mu_{{S_{{1{\text{g}}}} ,S_{{1{\text{u}}}} }}^{\text{DL}} )^{2}\) and \((\mu_{{S_{{1{\text{u}}}} ,S_{{2{\text{g}}}} }}^{\text{DL}} )^{2}\), monotonically increase toward 1 and decrease toward 0, respectively, from 0.5 at *y* = 0, as increasing *y* from 0 to 1. This is understood by the fact that the ground (\(S_{{1{\text{g}}}}\)) and the second (\(S_{{2{\text{g}}}}\)) excited states are correlated as described in Sect. 3.1 and become primary-diradical (neutral) and primary-ionic states as increasing the ground state diradical character *y*, while the first optically allowed excited state (\(S_{{1{\text{u}}}}\)) remains in a pure ionic state. Namely, as increasing *y*, the overlap between the ground (\(S_{{1{\text{g}}}}\)) and the first (\(S_{{1{\text{u}}}}\)) excited states, transition density corresponding to \((\mu_{{S_{{1{\text{g}}}} ,S_{{1{\text{u}}}} }}^{\text{DL}} )^{2}\) decreases, while that, transition density corresponding to \((\mu_{{S_{{1{\text{g}}}} ,S_{{1{\text{u}}}} }}^{\text{DL}} )^{2}\), between the first (\(S_{{1{\text{u}}}}\)) and second (\(S_{{2{\text{g}}}}\)) excited states increases. On the other hand, for *r* _{ K } = 0, with increasing *y*, both the dimensionless first and second excitation energies, \(E_{{S_{{1{\text{u}}}} ,S_{{1{\text{g}}}} }}^{\text{DL}}\) and \(E_{{S_{{2{\text{g}}}} ,S_{{1{\text{g}}}} }}^{\text{DL}}\), rapidly decrease in the small *y* region, and they gradually decrease toward 1 and then achieve a stationary value (1) from the intermediate to large *y* region. The reduction rate in the small *y* region is significant in \(E_{{S_{{1{\text{u}}}} ,S_{{1{\text{g}}}} }}^{\text{DL}}\) as compared with \(E_{{S_{{2{\text{g}}}} ,S_{{1{\text{g}}}} }}^{\text{DL}}\). It is also found that as increasing \(r_{K}\), the converged value of \(E_{{S_{{1{\text{u}}}} ,S_{{1{\text{g}}}} }}^{\text{DL}}\) is decreased, i.e., \(E_{{S_{{1{\text{u}}}} ,S_{{1{\text{g}}}} }}^{\text{DL}} \to 1 - r_{K}\) at \(y \to 1\) (see Eq. 29). Here, we consider the relationship between the first optically allowed excitation energy \(E_{{S_{{1{\text{u}}}} ,S_{{1{\text{g}}}} }}^{{}}\) and diradical character *y*. From Eq. 27, *y* tends to increase when *U* becomes large. Considering the *y* dependence of \(E_{{S_{{2{\text{g}}}} ,S_{{1{\text{g}}}} }}^{\text{DL}}\) (Fig. 7) and \(E_{{S_{{2{\text{g}}}} ,S_{{1{\text{g}}}} }} = UE_{{S_{{2{\text{g}}}} ,S_{{1{\text{g}}}} }}^{\text{DL}}\), it is predicted that the excitation energy \(E_{{S_{{ 1 {\text{u}}}} ,S_{{ 1 {\text{g}}}} }}\) decreases, reaches a stationary value, and for very large *U*, it increases again with increasing *y* values [16, 37]. Usually, the extension of *π*-conjugation length causes the decrease of the HOMO–LUMO gap (−2*t* _{ ab }) and the increase of *U*, so that the extension of the size of molecules with non-negligible diradical character *y* tends to decrease the first excitation energy in the relatively small *y* region, while tends to increase again in the intermediate/large *y* region. This behavior is contrast to the well-known feature that a closed-shell *π*-conjugated system exhibits a decrease of the excitation energy with increasing the *π*-conjugation length.

## 4 Asymmetric Open-Shell Singlet Systems

### 4.1 Ground/Excited Electronic States and Diradical Character Using the Valence Configuration Interaction Method

As explained in Sect. 3, the neutral (diradical) and ionic components in a wavefunction play a complementary role, so that the asymmetric charge distribution, referred to as, asymmetricity, tends to reduce the diradical character. This feature seems to be qualitatively correct, but “asymmetricity” and primary “ionic” contribution is not necessarily the same concept. In this section, we show the feature of the wavefunctions of the ground and excited states based on an asymmetric two-site model A^{·}–B^{·} with two electrons in two orbitals in order to clarify the effects of an asymmetric electronic distribution on the excitation energies and properties of open-shell molecular systems [38].

The asymmetric two-site model A^{·}–B^{·} is placed along the bond axis (*x*-axis). Using the AOs for A and B, i.e., \(\chi_{\text{A}}\) and \(\chi_{\text{B}}\), with overlap \(S_{\text{AB}}\), bonding and anti-bonding MOs, *g* and *u* can be defined as in the symmetric system:

*a*and

*b*,

*z*-component of spin angular momentum (

*M*

_{s}= 0, singlet and triplet) is expressed by [38],

*h*represents the one-electron core Hamiltonian difference, \(h \equiv h_{bb} - h_{aa}\), where \(h_{pp} \equiv \left\langle p \right|h(1)\left| p \right\rangle = \left\langle {\bar{p}} \right|h(1)\left| {\bar{p}} \right\rangle \le 0\) and \(h \ge 0\) (

*h*

_{ aa }≤

*h*

_{ bb }). Since the transfer integrals include the two-electron integral between the neutral and ionic determinants, there are two types of transfer integrals, e.g., \(t_{ab(aa)} \equiv \left\langle {a\bar{b}} \right|\hat{H}\left| {a\bar{a}} \right\rangle\) and \(t_{ab(bb)} \equiv \left\langle {a\bar{b}} \right|\hat{H}\left| {b\bar{b}} \right\rangle\), which are different since (

*ab*|

*aa*) \(\ne\) (

*ab*|

*bb*). Thus, the average transfer integral, \(t_{ab} \equiv (t_{ab(aa)} + t_{ab(bb)} )/2\), is introduced. In the present case,

*U*

_{ a }and

*U*

_{ b }represent effective Coulomb repulsions, \(U_{a} \equiv U_{aa} - U_{ab}\) and \(U_{b} \equiv U_{bb} - U_{ab}\), and we define the average effective Coulomb repulsion

*U*\([ \equiv (U_{a} + U_{b} )/2]\) [38].

*y*

_{S},

*r*

_{ h },

*r*

_{ U },

*r*

_{ tab }) = (0, 1, 1) [38]. Note here that this is not the diradical character for the asymmetric two-site model (referred to as

*y*

_{A}) and is referred to as “pseudo-diradical character”. The diradical character of the asymmetric two-site model is represented by

*y*

_{A}, which is a function of (

*r*

_{ t },

*r*

_{ K },

*r*

_{ h },

*r*

_{ U },

*r*

_{ tab }). For simplicity, we consider the case that the asymmetricity is caused by changing

*r*

_{ h }between 0 and 2 with keeping (

*r*

_{ U },

*r*

_{ tab }) = (1, 1), which means that the asymmetricity is governed by the difference of ionization potentials of the constitutive atoms A and B. The dimensionless Hamiltonian matrix, \({\mathbf{H}}_{\text{DL}}\)(\(\equiv {\mathbf{H}}/U\)), in the case of (

*r*

_{ U },

*r*

_{ tab }) = (1, 1) is expressed by [38]

**H**_{DL}are found to depend on the dimensionless quantities (

*r*

_{ t },

*r*

_{ K },

*r*

_{ h },

*r*

_{ U }

*r*

_{ tab }), i.e., (

*y*

_{S},

*r*

_{ K },

*r*

_{ h },

*r*

_{ U },

*r*

_{ tab }). The eigenvectors for state {

*j*} = {T, g, k, f} (T: triplet state, and g, k, f: singlet states) are represented by

*g*and

*u*) in Sect. 3.1, we can construct an alternative basis set {\(\psi_{\text{G}}\), \(\psi_{\text{S}}\), \(\psi_{\text{D}}\)} = {\(\psi (g\bar{g})\), \((\psi (g\bar{u}) + \psi (u\bar{g}))/\sqrt 2\), \(\psi (u\bar{u})\)} for the singlet states. By using this basis set, the singlet ground state is expressed by

*y*

_{A}of the two-site asymmetric model is defined, in the same way as in the symmetric model in Sect. 2.3, as the occupation number (

*n*

_{LUNO}) of the LUNO of the singlet ground state \(\varPsi_{\text{g}}\). The diradical character

*y*

_{A}is expressed by

*y*for symmetric systems:

*y*

_{S}and

*y*

_{A}relationship, which reveals that

*y*

_{A}is smaller than

*y*

_{S}, in particular for

*y*

_{S}~ 0.5 as increasing the asymmetricity

*r*

_{ h }. As seen from Fig. 8a, if

*y*

_{S}= 1 then

*y*

_{A}= 1 for

*r*

_{ h }< 1 but

*y*

_{A}= 0 for

*r*

_{ h }> 1, while

*y*

_{A}is close to ~0.134 for

*r*

_{ h }= 1 (

*r*

_{ K }= 0). This behavior corresponds to the exchange of the dominant configurations (neutral/ionic) in state g, i.e., \(P_{\text{N}} = \left| {C_{{a\bar{b},{\text{g}}}} } \right|^{2} + \left| {C_{{b\bar{a},{\text{g}}}} } \right|^{2}\) and \(P_{\text{I}} = \left| {C_{{a\bar{a},{\text{g}}}} } \right|^{2} + \left| {C_{{b\bar{b},{\text{g}}}} } \right|^{2}\), between

*r*

_{ h }< 1 and

*r*

_{ h }> 1 for

*y*

_{S}> 0 and

*r*

_{ K }= 0 (see Fig. 9).

### 4.2 Asymmetricity (*r* _{ h }) and Direct Exchange (*r* _{ K }) Dependence of Energies, Wavefunctions, and Diradical Character

As shown in Fig. 9, for *r* _{ K } = 0, the asymmetricity *r* _{ h } causes the exchange between the dominant configurations (neutral/ionic) in each state and the variation in the diradical character *y* _{A}. We here focus on the effects of *r* _{ K } on these variations. In order to capture the feature of *r* _{ K } effect, we consider the analytical expressions of energies and wavefunctions of each state {g, k, f} in the case of (*y* _{S}, *r* _{ U }, *r* _{ tab }) = (1, 1, 1) (*r* _{ h } > 0). The solutions are classified in the following three regions based on the amplitude relationship between \(r_{h}^{2}\) and \(1 - r_{K}\) [39].

For \(r_{h}^{2} < 1 - r_{K}\),

For \(r_{h}^{2} = 1 - r_{K}\),

*g*and

*k*are pure neutral (diradical) and ionic, while for \(r_{h}^{2} > 1 - r_{K}\), they are pure ionic and neutral (diradical), respectively. Namely, for

*y*

_{S}= 1, the diradical character

*y*

_{A}is abruptly reduced from 1 to \(1 - \frac{2A}{{1 + A^{2} }}\) when turning from \(r_{h}^{2} < 1 - r_{K}\) to \(r_{h}^{2} > 1 - r_{K}\). In the case of \(r_{h}^{2} = 1 - r_{K}\) at

*y*

_{S}\(\to\) 1, where

*E*

_{g}=

*E*

_{k},

*y*

_{A}asymptotically approaches \(1 - \frac{{\sqrt {1 + 2A^{2} } }}{{1 + A^{2} }}\) since the neutral (Eq. 44a) and ionic (Eq. 44e) components contribute to the wavefunction equivalently. At the same

*r*

_{ h }value, the

*y*

_{A}is shown to decrease in the intermediate

*y*

_{S}region with increasing

*r*

_{ K }up to \(1 - r_{h}^{2}\) (≥0), while further increase of

*r*

_{ K }is found to increase

*y*

_{A}again as seen from Eqs. 44e and 44h. Namely, the increase in

*r*

_{ K }operates similarly to asymmetricity

*r*

_{ h }for \(r_{h}^{2} < 1 - r_{K}\). This is also exemplified by the decrease of critical

*r*

_{ h }value (

*r*

_{ h c}), at which the exchange of the dominant configurations (neutral/ionic) in state g and k occurs, with increasing

*r*

_{ K }until \(1 - r_{{h{\text{ c}}}}^{2}\) (≥0) (see Eq. 44d), which is shown in

*r*

_{ K }dependence of

*P*

_{N}and

*P*

_{I}for

*g*and

*k*states (Fig. 9).

*r*

_{ h }dependence of the dimensionless excitation energies \(E_{\text{kg}}\) and \(E_{\text{fg}}\) (for a fixed

*y*

_{S}) with increasing

*r*

_{ K }are shown in Fig. 10. It is shown that

*E*

_{kg}and

*E*

_{fg}decrease and increase, respectively, with increasing \(r_{K}\) for \(r_{h}^{2} < 1 - r_{K}\), while that they increase with increasing \(r_{K}\) for \(r_{h}^{2} > 1 - r_{K}\) (see also Eqs. 44a–44h). The increase of

*r*

_{ K }is found to move the behaviors around

*r*

_{ h }= 1.0 of the excitation energies and transition moments to the lower

*r*

_{ h }region due to the displacement of the critical point

*r*

_{ h c}as shown in Fig. 9. Also, the increase of

*r*

_{ K }is turned out to decrease

*E*

_{kg}and |Δ

*µ*

_{ ii }| (

*i*= k, f), but increase |

*µ*

_{kf}| before ~

*r*

_{ h c}as predicted from the analytical expressions of excitation energies and wavefunctions for

*y*

_{S}= 1 (Eqs. 44a–44c). Indeed, the asymmetric distributions represented by the relative contributions of \(\left| {a\bar{a}} \right\rangle\) and \(\left| {b\bar{b}} \right\rangle\) are shown to decrease with increasing

*r*

_{ K }at the same

*r*

_{ h }, e.g., \(\left| {a\bar{a}} \right\rangle\):\(\left| {b\bar{b}} \right\rangle\) = 1:0 for

*r*

_{ K }= 0 vs. 1 + A:1–A for

*r*

_{ K }\(\ne\) 0 (see Eqs. 44a–44c), the feature of which decreases |Δ

*µ*

_{ ii }| (

*i*= k, f) and increases |

*µ*

_{kf}|.

*y*

_{S}causes the decrease of \(E_{\text{gT}}\) for \(r_{K} = 0\), (b) that the increase in

*r*

_{ K }stabilizes the triplet state, and (c) that for a given

*y*

_{ S }the increase of

*r*

_{ h }leads to the increase of the

*r*

_{ K }value giving a triplet ground state. As seen from Fig. 11, the singlet ground state

*r*

_{ K }(antiferromagnetic) region is broad in small

*y*

_{S}region and further broadens to larger

*y*

_{S}values with increasing

*r*

_{ h }. For \(r_{h} > 1\), the singlet ground state region is found to be widely extended over the whole

*y*

_{S}–

*r*

_{ K }region.

## 5 Relationship between Open-Shell Character and Optical Response Properties

*p*, which is defined by the difference between the induced dipole moment μ and permanent dipole moment μ

_{0}, is expanded by using the applied electric field

*F*[40, 41, 42]:

*F*

^{ i }(

*ω*

_{ l }) indicates the

*i*th component (

*i*=

*x*,

*y*,

*z*) of local electric field with frequency

*ω*

_{ l }. The coefficient of each term indicates the optical responsibility of the

*n*th order polarization: \(\alpha_{ij}^{{}}\), \(\beta_{ijk}^{{}}\) and \(\gamma_{ijkl}^{{}}\) are referred to as the polarizability, first hyperpolarizability, and second hyperpolarizability, respectively. These optical response properties are described by the electronic states of the atom/molecule and environmental effects, and their signs and amplitudes determines the characteristic of microscopic linear and nonlinear optical properties at the molecular scale. For example, the real and imaginary parts of \(\alpha_{ij}^{{}}\) describe the linear polarization and optical absorption, respectively, while those of \(\gamma_{ijkl}^{{}}\) are the off- and on-resonant third-order NLO properties, respectively, where the former and the latter typical phenomena are third-harmonic generation (THG), and two-photon absorption (TPA), respectively. As seen from Eq. 45, even-ordered coefficients such as \(\beta_{ijk}^{{}}\) vanish when the system has centrosymmetry, while odd-ordered coefficients such as \(\gamma_{ijkl}^{{}}\) generally have non-zero values regardless of the symmetry. The amplitude and sign of these coefficients are determined by the time-dependent perturbation formulae, which include excitation energies, transition moments and dipole moment differences, so that the molecular design for efficient NLO has been performed based on these perturbation expressions. For example, the polarizability, first hyperpolarizability, and second hyperpolarizability in the static limit (

*ω*

_{ i }= 0) are described as follows:

*n*th excited state; \(\mu_{mn}^{i}\) indicates the transition moment between the

*m*th and

*n*th states; \(\Delta \mu_{mm}^{i}\) indicates the dipole moment difference between the

*m*th excited state and the ground state (0). Applying these expressions to three singlet state model {g, k, f} for the symmetric two-site diradical model and using Eqs. 29–32, we obtain the analytical expressions of these response properties as functions of diradical character

*y*. For symmetric systems, the terms including dipole moment differences are vanished due to \(\Delta \mu_{mm}^{i}\) = 0 in Eq. 48, and takes the form (where the component index “

*i*” is omitted for simplicity):

*q*= 1–

*y*[4, 7]:

*γ*values, respectively. For

*r*

_{ K }= 0 (usual case for open-shell molecules with singlet ground states), the variations of dimensionless total

*γ*(

*γ*

^{DL}), as well as type II and III contributions (

*γ*

^{II DL}and

*γ*

^{III−2 DL}) as a function of

*y*are shown in Fig. 12. It is found that

*γ*

^{II DL}has a negative extremum in the small

*y*region, while

*γ*

^{III−2 DL}has a positive extremum in the intermediate

*y*region. Since the extremum amplitude of \(\gamma^{\text{III - 2 DL}}\) is shown to be much larger than that of \(\gamma^{\text{II DL}}\), the variation of total \(\gamma^{\text{DL}}\) with

*y*is found to be governed by that of \(\gamma^{\text{III - 2 DL}}\) and gives positive values in the whole

*y*region. This behavior of \(\gamma^{\text{III - 2 DL}}\) is understood by the variation in the numerator \((\mu_{\text{gk}}^{\text{DL}} )^{2} (\mu_{\text{kf}}^{\text{DL}} )^{2}\) and denominator \((E_{\text{kg}}^{\text{DL}} )^{2} E_{\text{fg}}^{\text{DL}}\) in the second term of Eq. 50 as a function of

*y*: the denominator and numerator approach infinity and a finite value, respectively, as \(y \to 0\), leading to \(\gamma^{\text{III - 2 DL}} \to 0\), while they do a finite value and 0, respectively, as \(y \to 1\), leading to \(\gamma^{\text{III - 2 DL}} \to 0\) again (see also Fig. 7b). Although both the denominator and numerator decrease with increasing

*y*from 0 to 1, the denominator decreases more rapidly in the small

*y*region than the numerator, which is the origin of the extremum of \(\gamma^{\text{III - 2 DL}}\) (~0.306) in the intermediate

*y*region (~0.243). The \(\left| {\gamma_{{}}^{\text{III - 2 DL}} } \right| > \left| {\gamma_{{}}^{\text{II DL}} } \right|\) except for

*y*~ 0 is understood by the fact that the numerator in the first term of Eq. 50, \((\mu_{\text{gk}}^{\text{DL}} )^{4}\), decreases more rapidly than that in the second term of Eq. 50, \((\mu_{\text{gk}}^{\text{DL}} )^{2} (\mu_{\text{kf}}^{\text{DL}} )^{2}\) (see also Fig. 7b). As shown in Fig. 7b \((\mu_{\text{gk}}^{\text{DL}} )^{2}\) and \((\mu_{\text{kf}}^{\text{DL}} )^{2}\) show decrease and increase, respectively, with increasing

*y*, so that \((\mu_{\text{gk}}^{\text{DL}} )^{2} (\mu_{\text{kf}}^{\text{DL}} )^{2}\) keeps a larger value than \((\mu_{\text{gk}}^{\text{DL}} )^{4}\) in the whole

*y*region. In summary, it turns out that the

*γ*values of open-shell singlet systems with intermediate diradical character tend to be significantly larger than those of closed-shell and pure diradcial systems. For asymmetric systems, we also have revealed remarkable enhancements of |

*γ*| and |

*β*| values in the intermediate diradical/ionic character region [39, 45]. These theoretical predictions pioneer a novel class of highly-efficient second- and third-order NLO substances, i.e., open-shell NLO systems, which outstrip traditional closed

^{shell}NLO systems.

## 6 Relationship between Open-Shell Character, Aromaticity, and Response Property

### 6.1 Indenofluorenes

*para*- and

*meta*-indenofluorenes (Fig. 13), which are

*π*-conjugated fused-ring systems with alternating structures composed of three six-membered and two five-membered rings synthesized by Haley’s and Tobe’s groups [49, 50, 51, 52]. Apparently, these are 20

*π*electron systems, so that they are regarded as anti-aromatic analogues of pentacene [49]. On the other hand, these systems exhibit pro-aromatic quinodimethane framework in the central region, which is predicted to exhibit open-shell singlet character [50, 52, 53]. Thus, as shown in Sect. 5, these systems will be appropriate model systems for clarifying the relationships of spatial contributions between the open-shell character, the aromaticity, and the second hyperpolarizability.

### 6.2 Structure, Odd Electron Density, Magnetic Shielding Tensor, and Hyperpolarizability Density

*para*- and

*meta*-type indenofluorenes are optimized with the U(R)B3LYP/6-311 + G** method under the symmetry constraints of

*C*

_{2h }for

*para*, and

*C*

_{2v }for

*meta*systems. The diradical character

*y*, unpaired(odd)-electron density, the magnetic shielding tensor component –

*σ*

_{ yy }and the second hyperpolarizabilities

*γ*are evaluated using the long-range corrected (LC) density functional theory (DFT) method, LC-UBLYP (range separating parameter μ = 0.33 bohr

^{−1}) method, with the 6–311 + G** basis set. Within the single determinantal UDFT scheme, the diradical character is defined as the occupation number of the LUNO of the unrestricted wavefunctions

*n*

_{LUNO}:

*μ*= 0.33) method is generally found to have smaller spin contamination than UHF and is found to reproduce well the diradical character and

*γ*values at the strong correlated level of theory like UCCSD(T) (see Sect. 7.1). The spatial contribution of diradical character is clarified using the odd-electron density

*ρ*

_{odd}at position

**r**, which is calculated using the frontier natural orbitals \(\phi_{\text{HONO}} ({\mathbf{r}})\) and \(\phi_{\text{LUNO}} ({\mathbf{r}})\) as follows [31]

*y*as expressed by

*z*–

*x*plane), the magnetic shielding tensor (–

*σ*

_{ yy }) is evaluated 1 Å above the center of each ring, which primarily reflects the contribution of the

*π*-electron ring current. Namely, more negative –

*σ*

_{ yy }values indicate more aromaticity, while more positive values more antiaromaticity due to the diatropic (paratropic) ring current in aromatic (antiaromatic) ring. The longitudinal component of the static second hyperpolarizabilities

*γ*

_{ zzzz }, which is along the spin polarization direction and thus reflects the polarization of odd electrons, is calculated by the forth-order differentiation of the total energy according to the static electric field, finite field (FF) method [55]. The spatial contribution of electrons to the

*γ*

_{ zzzz }can be analyzed by using

*γ*

_{ zzzz }density \(\rho_{zzz}^{(3)} (r)\), which is defined as [44]

*r*, and

*F*

_{z}is the

*z*component of the external electric field

*. This*

**F***γ*

_{ zzzz }value is obtained by

*γ*

_{ zzzz }density as

*γ*

_{ zzzz }densities with large amplitudes, separated by a large distance, contribute to the increase of |

*γ*

_{ zzzz }| values, and that the sign of the contribution is determined by the direction of the arrow drawn from positive to negative

*γ*

_{ zzzz }density: when the direction of the arrow coincides with (is opposite to) that of the coordinate axis, the contribution is positive (negative) in sign.

### 6.3 Diradical Character and Local Aromaticity of Indenofluorenes

*para*-quinodimethane framework, referred to as

*para*, exhibits negligible diradical character (<0.1), while the system involving

*meta*-quinodimethane framework, referred to as

*meta*, shows a larger value (

*y*= 0.645) [56]. This indicates that the

*para*system is classified into nearly closed-shell systems, while the

*meta*system is classified into intermediate singlet diradical system. These features are qualitatively understood based on their resonance structures with Clar’s sextets rule. Namely, the open-shell resonance structures for both systems exhibit a larger number of Clar’s sextets, (three benzene rings) than the closed-shell structures. This feature originates form the existence of the pro-aromatic quinodimethane structure in these systems. It is also noted that the

*meta*system exhibits a smaller number of Clar’s sextets in the closed-shell form than

*para*systems, i.e., one for

*meta*and two for

*para*, the difference of which indicates the relatively larger stability of the open-shell resonance structure in

*meta*system than in

*para*system, resulting in the larger diradical character of

*meta*system than that of

*para*system.

*σ*

_{ yy }) 1 Å above the center of each six- and five-membered ring plane (Fig. 15) [56]. For these systems, the middle three rings, the six-membered ring together with the adjoining two five-membered rings exhibit positive –

*σ*

_{ yy }(anti-aromatic), while the terminal benzene rings exhibit negative –

*σ*

_{ yy }(aromatic). On the other hand, it is found that the –

*σ*

_{ yy }values of the terminal benzene rings exhibit larger negative (aromaticity) (–20 ppm) for the

*para*system than for the

*meta*system (–11.1 ppm), while that of the anti-aromatic central six-membered ring is larger positive (anti-aromatic) (9.4 ppm) for the

*para*system than for the

*meta*system (0.2 ppm), which represents much reduced anti-aromatic or non-aromatic central six-membered ring. Considering the diradical characters

*y*= 0.072 for

*para*and 0.645 for

*meta*systems, it is found that the difference in the local aromaticity between the central and the terminal rings is much smaller in the intermediate diradical

*meta*system (|–

*σ*

_{ yy }(central) +

*σ*

_{ yy }(terminal)| = 11.3 ppm) than in the nearly closed-shell

*para*system (|–

*σ*

_{ yy }(central) +

*σ*

_{ yy }(terminal)| = 29.4 ppm). This feature is understood by comparing the number of the Clar’s sextets in the resonance structures (see Fig. 14). The diradical resonance structures are shown to exhibit the Clar’s sextets at all the six-membered rings, which contribute to the aromaticity at all the six-membered rings. In contrast, the closed-shell resonance structure of the

*para*system exhibits the Clar’s sextets at both the two terminal rings, while the

*meta*system does the Clar’s sextet at only one of the two terminal rings. This implies that the terminal six-membered rings of the nearly closed-shell

*para*system exhibit fully Clar’s sextet aromatic nature for all the resonance structures, while those of

*meta*system do less aromatic nature due to the both contribution of the fully aromatic nature in the diradical resonance structure and the half in the closed-shell resonance structure. Similarly, the fact that the

*meta*system has a larger contribution of the diradical resonance structure is found to lead to much reduced anti-aromatic or non-aromatic nature of the central six-membered ring, which exhibits a Clar’s sextet at the central benzene ring in the diradical resonance structure. Figure 15 also shows the spatial distributions of –

*σ*

_{ yy }with color contours, where the blue and yellow contours represent aromatic (with negative –

*σ*

_{ yy }) and anti-aromatic (with positive –

*σ*

_{ yy }) regions, respectively. The central benzene ring together with the adjoining two five-membered rings for

*para*system shows yellow contours (local anti-aromatic nature), while that of

*meta*system does almost white contours (local non-aromatic nature). Such spatial features of –

*σ*

_{ yy }maps give more detailed spatial contribution features of the local aromaticities in the indenofluorene series. The spatial correlation between the local aromaticity and the diradical character is clarified by examining the maps of odd (unpaired)-electron density distribution (Fig. 15). Large odd-electron densities are shown to be generally distributed around the zigzag-edge region of the five-membered rings. Since this feature is consistent with that in the diradical resonance structures, the odd-electron density maps also substantiate Clar’s sextet rule in these molecules. As seen from the odd-electron densities of the six-membered rings, the

*para*system exhibits odd-electron densities more significantly distributed at the central benzene rings than at the terminal ones, while the

*meta*system shows odd-electron densities more delocalized over both the central and terminal benzene rings. Although this distribution difference is not straightforwardly understood from the resonance structures, the primary odd-electron density distribution region well corresponds to the local anti- or weaker aromatic ones of the six-membered rings: the difference in the local aromaticity between the six-membered rings is more distinct in the

*para*system than in the

*meta*system. This indicates that for each indenofluorene system, the six-membered rings with larger odd-electron densities exhibit relatively anti-aromatic nature. This spatial correlation between the odd-electron density and local aromaticity is understood by the fact that the emergence of odd-electron density in the aromatic ring implies the partial destruction of the fully

*π*-delocalization over the ring, resulting in the reduction of aromaticity or in the emergence of anti-aromaticity.

*para*and

*meta*systems since these triplet states correspond to the pure diradical states. It is found that unlike the corresponding singlet systems (which exhibit anti-aromaticity in the central benzene rings; Fig. 15), all the benzene rings exhibit similar aromatic nature in both systems (Fig. 16), the feature of which is particularly different in the central benzene ring from that of the singlet

*para*system [56]. This is understood by the fact that the triplet states are described as pure diradical resonance structures for both the systems, which are stabilized by all the benzene rings with Clar’s sextet form, whereas the singlet systems have contribution of closed-shell resonance structures, which in particular reduce the aromaticity or increase the anti-aromaticity in the central six-membered ring (see Fig. 14). Furthermore, the feature of odd-electron density distribution of

*para*system in the triplet state differs from that in the singlet state (see Figs. 15, 16): (a) the odd-electron density amplitudes are much larger than those in the triplet state, and (b) the odd electron densities are negligible in the terminal benzene rings in the singlet state, while those are also observed in the terminal benzene rings in addition to the central one. Accordingly, by changing from singlet to triplet state, the central benzene ring drastically change from anti-aromatic to aromatic, and the terminal rings slightly reduce the aromaticity. These results indicate that there exists a correlation between the difference in the local aromatic nature and that in the amplitudes of odd-electron densities of the six-membered rings [56].

### 6.4 Second Hyperpolarizaibilities of Indenofluorene systems

*γ*,

*γ*

_{ zzzz }, which are the dominant components for these indenofluorene systems. Figure 17 shows the

*γ*

_{ zzzz }densities together with the

*γ*

_{ zzzz }values in singlet states. As expected from our

*y*–

*γ*correlation, the

*meta*system with the intermediate diradical character exhibits larger

*γ*

_{ zzzz }values than the

*para*system with much smaller diradical character. As seen from Figs. 5 and 16, the relative amplitudes of

*γ*

_{ zzzz }densities qualitatively accord with those of the odd electron densities. Indeed, for the

*para*system with slight diradical character, the

*γ*

_{ zzzz }density is primarily localized in the quinodimethane framework, while for

*meta*system with intermediate dirdadical character, that is not localized in the quinodimethane region, but spreads over the molecule. These results demonstrate the spatial-distribution similarity between the

*γ*

_{ zzzz }density, odd-electron density and –

*σ*

_{ yy }maps, especially around the six-membered ring regions.

Next, we consider spin state dependences of these quantities. For the *para* system with slight dirdadical character, the *γ* _{ zzzz } amplitude in the triplet state is somewhat smaller than that in the singlet state, \(\gamma_{zzzz}^{\text{triplet}} /\gamma_{zzzz}^{\text{singlet}}\) = 0.83, while for the *meta* system with the intermediate diradical character the *γ* _{ zzzz } amplitude in the triplet state shows significant reduction (only a half amplitude, \(\gamma_{zzzz}^{\text{triplet}} /\gamma_{zzzz}^{\text{singlet}}\) = 0.46) of the singlet state *γ* _{ zzzz } value. As seen from the *γ* _{ zzzz } density distribution maps for both systems (Fig. 17), the *γ* _{ zzzz } density distribution features are significantly different between the triplet and singlet states: the *γ* _{ zzzz } density distributions in the triplet states tend to spread over all the molecular frameworks, not localized at the central quinodimethane region. In particular, in the triplet states, the *γ* _{ zzzz } densities are shown to be significantly reduced as compared to the singlet states around the zigzag edge region of the five-membered rings, where the large odd-electrons are distributed. These features indicate that in the triplet states, the spatial contribution to the *γ* _{ zzzz } is not so correlated to the odd-electron density distributions unlike the intermediate open-shell singlet systems. This is understood by the Pauli effects [57]: the triplet diradical electrons are prohibited to be delocalized due to the Pauli principle and thus do not contribute to the enhancement of *γ* _{ zzzz }.

In summary, it is found that there exists strong correlation in the absolute values and the spatial distributions between the diradical character, local aromaticity and *γ* values. In this regard, the diradical character and the odd electron density distribution are qualitatively predicted with Clar’s sextets rule: the relative stability of the open-shell resonance structures varies based on the number of the Clar’s sextets in the closed-shell and the open-shell resonance structures. Also, the indenofluorene frameworks can vary the relative stability of the resonance structures, which is accompanied by the change of the geometry around the anti-aromatic five-membered rings.

## 7 Diradical Character and Optical Response Properties Calculated Using Broken-Symmetry Density Functional Theory Methods

### 7.1 Functional Dependence of Diradical Character for Polycyclic Hydrocarbons

*y*is usually defined by the occupation number of LUNO. This definition can be applied to highly correlated methods such as spin-unrestricted coupled-cluster with single, double, and perturbative triple excitations (UCCSD(T)), multi-reference Møller-Plesset perturbation (MRMP), and full CI. On the other hand, when applied to broken-symmetry methods, spin contamination effects must be considered for reproducing the highly correlated diradical character, e.g., UCCSD(T)

*y*value. The spin contamination effects on the diradical character is explicitly defined at UHF level of theory, so that approximate spin-projection schemes, for example, perfect-pairing type spin-projected HF (PUHF), Eq. 12, are applied to obtaining semi-quantitatively correct diradical character. Indeed, the PUHF

*y*values are known to reproduce those at full CI level of theory. On the other hand, for the BS DFT (UDFT) case, it is known that spin contamination effects are smaller than those at UHF level of theory, and depend on the exchange–correlation (xc)-functionals. In order to clarify the functional dependences of the diradical character and response property, polarizability, which reflects the description of excitation energies and transition properties, we examine these quantities for dicyclopenta-fused oligoacenes (DPA[

*N*],

*N*= 0–3) (Fig. 18) optimized by UB3LYP/6-3111G* method under constraint of

*D*

_{2h}symmetry. These molecules are known to be anti-aromatic 4

*nπ*systems with a wide range of open-shell characters depending on the oligomer size [25, 47]. Indeed, as seen from the resonance structures, DPA involves two types of spin polarizations, which are along the longitudinal (between the terminal five-membered rings) and lateral (which is observed in the middle region of oligoacenes) directions, respectively. However, in the considered oligomer size, the main open-shell character is described by the diradical character, which corresponds to the spin polarization between the terminal five-membered rings.

In general, conventional DFT methods, e.g., BLYP and B3LYP methods, are known to provide highly accurate results with less computational effort for geometry, electronic structure, reaction, etc., which demonstrates great successes in closed-shell based molecules in chemistry. On the other hand, the BLYP and B3LYP methods are known to have several drawbacks originating from their local character, e.g., underestimated band gap energy and too large (non)linear optical properties [58], and no description of weak-interaction, undershot charge-transfer excited state energy, etc. The local character of conventional xc-functionals in the DFT can be improved by the long-range corrected (LC-)DFT, e.g., LC-BLYP, method [59, 60, 61]. In such methods, the range-separation of the DFT exchange functional is realized by dividing the electron repulsion operator (1/*r* _{12}) into long- and short-range parts (the range separation is controlled by the range-separating parameter μ), is found to improve the description of NLO response properties for extended *π*-conjugated molecular systems [62, 63, 64, 65] and several open-shell singlet molecules [66]. In the case with a larger (smaller) range-separating parameter *μ*, the fraction of the HF exchange is larger (smaller) at a given *r* _{12}. The *μ* ^{−1} represents the delocalization length [68].

*y*at the UCCSD, LC-UBLYP, CAM-UBLYP, UBHandHLYP, and UB3LYP levels of approximation versus

*y*at PUHF level of approximation for DPA[

*N*] (

*N*= 0, 1, 2, 3). It is found that the UCCSD and LCUBLYP (

*μ*= 0.47) methods give slightly larger

*y*values at each size of DPA[

*N*] than PUHF

*y*value, while the UBHandHLYP gives a slightly smaller

*y*value than PUHF

*y*value at each

*N*. The LC-UBLYP (

*μ*= 0.33) and CAM-UB3LYP methods are shown to reproduce the PUHF

*y*values at large

*N*values (

*N*= 3, 2), while they are shown to result in smaller

*y*values than the PUHF

*y*values at small

*N*values (

*N*= 2, 0), e.g., about a half of the PUHF

*y*value at

*N*= 0. The B3LYP is found to give significantly undershot

*y*values than the PUHF

*y*values in the whole

*y*(PUHF) region. In summary, the inclusion of the HF exchange is important for well reproducing

*y*values at PUHF and strong-correlated UCCSD levels of theory. The UBHandHLYP, LC-UBLYP, and CAM-UB3LYP methods are found to work well for evaluating diradical character of open-shell singlet systems. For polarizabilities

*α*, it is found from Table 1 that the BHandHLYP and LC-UBLYP (

*μ*= 0.33, 0.47) methods well reproduce the UCCSD

*α*values at least up to

*N*= 2, while that the UBLYP, UB3LYP, and CAM-UB3LYP methods overestimate the UCCSD

*α*values. The overestimation is shown to be more emphasized for larger size systems.

### 7.2 Approximate Spin-Projection Scheme of Diradical Character and Optical Response Properties

#### 7.2.1 Calculation Methods and Model System

Although UDFT methods with several xc-functionals are found to reproduce semi-quantitatively the *y*–*γ* correlation obtained from strong-correlated quantum chemistry calculations [5, 6, 66, 69], the spin contamination effects involved in the broken-symmetry (BS) schemes often cause incorrect results on optimized molecular structures and magnetic properties of open-shell systems [29, 33, 34]. In this section, we show an approximate spin-projection scheme within the spin-unrestricted single-determinantal framework [31, 70]. This is based on a correction of the occupation numbers (diradical characters) and is expected to improve the description of the odd electron density [30, 71, 72] and the evaluation of the (hyper)polarizabilities of delocalized open-shell singlet molecules. The performance of this scheme is demonstrated by the static polarizability (*α*) and second hyperpolarizability (*γ*) of a typical open-shell singlet system, i.e., the *p*-quinodimethane (PQM) model using several xc-functionals are employed, ranging from the pure DFT BLYP, the hybrid B3LYP, and BHandHLYP functionals to the LC-UBLYP functional.

*N*-electron systems are calculated using the one-electron reduced density:

*k*th NO with occupation number {\(n_{k}\)} (

*n*

_{HONO-i }+

*n*

_{LUNO+i }= 2). Here, approximately removing spin contamination effects is performed by the perfect-pairing spin-projection scheme [28, 29] on the occupation numbers \(n_{k}\) (see also Eqs. 14, 15):

*T*

_{i}, the overlap between the corresponding orbital pairs, is expressed by

*i*,

*j*,

*k*,

*l*=

*x*,

*y*,

*z*) are expressed by [31]

*D*

_{2h}symmetry (

*R*

_{1}= 1.351 Å,

*R*

_{2}= 1.460 Å, and

*R*

_{3}= 1.346 Å) at UB3LYP/6-311G* level of approximation coincides with that at RB3LYP level, which implies that the equilibrium PQM exhibits the quinoid-like structure instead of diradical one. We consider a model with stretching

*R*

_{1}from 1.35 to 1.7 Å under the

*R*

_{2}=

*R*

_{3}= 1.4 Å constraint PQM, which can display a wide range of diradical character (

*y*) since the stretching

*R*

_{1}mainly causes the

*π*bond breaking. The diradical character

*y*at PUHF/6-31G* +

*p*(ζ = 0.0523) level ranges from 0.146 to 0.731 [6]. Here, we clarify the spin-projection effect on the diradical character dependence of

*α*and

*γ*using the several UDFT methods with 6-31G* +

*p*basis set. The UBLYP, UB3LYP, and UBHandHLYP xc-functionals, as well as the long-range-corrected, LC-UBLYP (

*μ*= 0.33 and 0.47 bohr

^{−1}), functional are employed for evaluating the longitudinal components of the static

*α*\(( \equiv \alpha_{xx} )\) and

*γ*\(( \equiv \gamma_{xxxx} )\). The non-spin-projected and ASP results are compared with the strong-correlated UCCSD(T) results.

#### 7.2.2 Polarizability

*α*values. Figure 21 displays the evolution of

*α*of PQM as a function of

*y*calculated by the non-spin projected (a) and ASP (b) UDFT methods, as well as the reference the UCCSD(T) method [70]. It is found that the UCCSD(T)

*α*value slightly increases with

*y*, attains a maximum (

*α*

_{max}= 185.4 a.u.) around

*y*= 0.34, and then decreases. Although the UBLYP functional without spin projection reproduces qualitatively the diradical character dependence of

*α*,

*α*

_{max}is significantly (~29%) overshot, while the

*y*value giving

*γ*

_{max}(

*y*

_{max}) is shifted toward a larger

*y*value (

*y*

_{max}= 0.58). It turns out that the increase of fraction of HF exchange reduces the overshot

*α*

_{max}and moves

*y*

_{max}to smaller values, e.g.,

*α*

_{max}= 214.9 a.u. at

*y*

_{max}= 0.34 (UB3LYP) vs.

*α*

_{max}= 239.6 a.u. at

*y*

_{max}= 0.58 (UBLYP). In the case of using UBHandHLYP, which involves 50% of HF exchange,

*α*

_{max}is shown to be located at the initial

*y*= 0.146 value, while

*α*is shown to decrease monotonically with

*y*. The differences in

*α*between the UCCSD(T) and UBHandHLYP are, however, found to be relatively small in the whole

*y*region. It is found that the

*y*dependence and the amplitude of

*α*at LC-UBLYP (

*μ*= 0.47) level are very similar to those at UBHandHLYP level though the LC-UBLYP (

*μ*= 0.47)

*α*is slightly smaller (within ~5%) than UBHandHLYP

*α*at each

*y*. The

*α*and

*y*

_{max}are found to increase as decreasing the range-separating parameter μ like the case of decreasing the fraction of HF exchange in the global hybrid functionals. It turns out that the

*α*values at LC-UBLYP (

*μ*= 0.33) level somewhat overshoot those at UCCSD(T) level in the small

*y*region (< 0.34), while that they well reproduce those for

*y*> 0.34. In summary, among the non-spin-projected UDFT methods, the LC-UBLYP (

*μ*= 0.33) method is found to best reproduce the amplitude and the variation of

*α*as a function of

*y*in the whole

*y*value region.

The approximate spin projection (Fig. 21b) is shown to cause an increase of *α* _{max} and a shift of *y* _{max} to larger *y*, e.g., (*y* _{max}, *α* _{max}) = (0.34, 214.9 a.u.) (UB3LYP) vs. (0.49, 239.8 a.u.) (ASP-UB3LYP). Thus, the ASP-UBLYP, -UB3LYP, -UBHandHLYP, and –LC-UBLYP (*μ* = 0.33) results are shown to go further apart from the reference UCCSD(T) result. It is found that the best agreement among the present methods is achieved by the ASP-LC-UBLYP (*μ* = 0.47) method, which decreases (enhances) the overshot (undershot) *α* in the small (intermediate and large) *y* region, and then semi-quantitatively reproduces the UCCSD(T) *y*–*α* curve, though the *y* _{max} is slightly smaller than that at UCCSD(T) level of approximation.

As seen from these results, the static correlation and the spin contamination effects in the UDFT treatments sensitively depend on the fraction of HF exchange as well as on its range of application, i.e., μ value in the LC-UBBLYP method. Namely, we should be careful about the application of the spin-projection corrections to the UDFT method in the case of calculating the polarizability. It is found that the LC-UBLYP (*μ* = 0.33) functional in the non-spin-projection scheme and the LC-UBLYP (*μ* = 0.47) functional in the ASP scheme well reproduce the UCCSD(T) *y*-*α* curve.

#### 7.2.3 Second Hyperpolarizability

*y*–

*γ*curves of PQM calculated with hybrid and LC functionals [70]. The UCCSD(T)

*y*–

*γ*curves show a bell-shape variation with a maximum

*γ*(

*γ*

_{max}) = 77500 a.u. around

*y*

_{max}= 0.49. As seen from Fig. 22a, for

*y*< 0.41, the UBLYP results coincide with the RBLYP results, which give much smaller

*γ*amplitudes, while for larger

*y*, the UBLYP results give significantly overshot behavior. The increase in fractions of HF exchange in xc-functionals moves the

*y*value at which the abrupt increase in

*γ*occurs toward a smaller

*y*value [70]. It is shown that the UBHandHLYP functional closely reproduces the UCCSD(T)

*γ*variation for

*y*> 0.4, while it overshoots the UCCSD(T)

*γ*values in the small

*y*value region (

*y*< ~0.34). The LC-UBLYP (

*μ*= 0.47) functional is found to provide a similar curve to UBHandHLYP, though the

*γ*values are found to be on average 13500 a.u. smaller. As a result, we find that LC-UBLYP (

*μ*= 0.33) functional best reproduces the UCCSD(T)

*y*–

*γ*curve, in particular for intermediate and large

*y*values though it reduces to the spin-restricted solution at

*y*= 0.146. In summary, it tuns out that tuning the HF exchange fraction in the hybrid UDFT methods or the range-separating parameter in the LC-UBLYP method can improve, to some degree, the agreement with the UCCSD(T) results, though the overshot behavior in the small

*y*value region cannot be fully corrected.

Let us consider the performance of ASP scheme. As seen from Fig. 22b, the significant improvement of *γ* is achieved in the small *y* value region. Namely, the ASP is shown to correct the overshot behavior of *γ* in the small *y* region, as well as the slightly undershot curve of *γ* in the intermediate and large *y* value region for DFT results (Fig. 22a, b). As a result, the ASP-UBLYP(μ = 0.47) is found to best reproduce both the whole *y*-*γ* curve and the (*y* _{max}, *γ* _{max}) values [=(0.49, 77500 a.u.) using UCCSD(T)]: (*y* _{max}, *γ* _{max}) = (0.49, 69400 a.u.) [LC-UBLYP (*μ* = 0.47)].

#### 7.2.4 (Hyper)polarizability Densities Using the ASP-LC-UBLYP (*μ* = 0.47) Method

In order to clarify the details of the ASP effects on the spatial electronic density, *α* and *γ*, we analyze the electronic density distributions as well as *α* and *γ* density distributions obtained by the ASP and non-spin-projected (NSP) UBLYP (*μ* = 0.47) xc-functional [70].

*y*= 0.257 (small), 0.491 (intermediate) and 0.731 (large). It is found that the increase (yellow) and decrease (blue) patterns of \(\Delta d({\mathbf{r}})\) are the same for any

*y*value, while that the amplitudes significantly decrease with

*y*. The increase of

*y*(bond weakness), in other words, a decrease of quinoid character, is shown to be caused by a stretching of the terminal CC bonds, so that variations of \(d({\mathbf{r}})\) in these bonding regions are expected by spin projection. Namely, the positive \(\Delta d({\mathbf{r}})\) in the

*R*

_{1}and

*R*

_{3}regions and the negative one in the

*R*

_{2}regions indicate that the spin contamination (primarily due to the mixing of the triplet component) emphasizes the bond dissociation nature, while the ASP favors the quinoid character, which corrects this excess bond dissociation features. Also, the reduction of the \(\Delta d({\mathbf{r}})\) amplitude in the large

*y*value region in spite of an increased spin contamination is predicted to be caused by the fact that in the bond dissociation limit region the electron density distributions in the singlet state become very similar to those in the triplet state.

The *α* densities (\(d^{{{\text{ASP}}(1)}} ({\mathbf{r}})\)) and their differences [\(\Delta d^{(1)} ({\mathbf{r}}) \equiv d^{{{\text{ASP}}(1)}} ({\mathbf{r}}) - d^{{{\text{NSP}}(1)}} ({\mathbf{r}})\)] are shown in Fig. 23b. We observe similar distribution pattern of *α* density for any *y* value, where the *π*-electrons distributed on both-end C atoms provide dominant positive contribution to *α*, while the *σ*-electrons do negative contributions with smaller amplitudes. The *π*-electron *α* density amplitudes decrease with *y* in agreement with *α*-*y* relationship (Fig. 21b). The *α* density differences \(\Delta d^{(1)} ({\mathbf{r}})\) are primarily localized on the C atoms with alternant sign, which exhibit the same pattern for any *y* value and increase *α* amplitudes. The \(\Delta d^{(1)} ({\mathbf{r}})\) is small for large *y*, as shown in the difference between the ASP- and NSP-LC-UBLYP (*μ* = 0.47) results (see Fig. 21).

Figure 23c shows the *γ* densities (\(d^{{{\text{ASP}}(3)}} ({\mathbf{r}})\)) and their differences [\(\Delta d^{(3)} ({\mathbf{r}}) \equiv d^{{{\text{ASP}}(3)}} ({\mathbf{r}}) - d^{{{\text{NSP}}(3)}} ({\mathbf{r}})\)]. Similar to the case of *α*, the distribution pattern is shown to be the same for any *y* value, and the primary positive contribution to *γ* is found to come from the *π*-electrons distributed on both-end C atoms. On the other hand, the positive and negative contributions in the middle benzene ring are shown to cancel with each other significantly. It is found that the amplitudes of the *π*-electron *γ* densities attain a maximum at *y* = 0.491 and then significantly decrease towards *y* = 0.731, the feature of which corresponds to the bell-shape variation of *γ* with *y* (see Fig. 22b). It turns out that the *γ* density differences \(\Delta d^{(3)} ({\mathbf{r}})\) are primarily distributed on all C atoms with alternant sign, while that the sign of the \(\Delta d^{(3)} ({\mathbf{r}})^{{}}\) pattern is inverted when going from at *y* = 0.257–0.491, the feature of which represents the sign change of the ASP correction to *γ* from low to large *y* region (see Fig. 22).

#### 7.2.5 Summary

As shown in this section, the degree of reproducibility of the UCCSD(T) (hyper)polarizabilities significantly depends on the fraction of HF exchange in the xc-functional or on the range-separating parameter μ in the LC-UBLYP functional. For the polarizability *α*, the LC-UBLYP (*μ* = 0.33) and ASP-LC-UBLYP (*μ* = 0.47) methods well reproduce *α*-*y* curve at the UCCSD(T) level in the whole *y* region. For the second hyperpolarizability *γ*, the ASP-LC-UBLYP (*μ* = 0.47) method semi-quantitatively reproduces the *y*–*γ* curve in the whole *y* region, and the LC-UBLYP (*μ* = 0.33) method also works well in the intermediate and large *y* regions.

## 8 Examples of Open-Shell Singlet Systems

In this section, we show several examples of open-shell singlet molecules based on polycyclic aromatic hydrocarbons (PAHs) together with their open-shell characters and hyperpolarizabilities. The architecture, edge shape, and size are found to significantly affect the diradical characters, the features of which are well understood based on the resonance structures with Clar’s sextet rule and correspond to the aromaticity of the rings involved in those molecules.

### 8.1 Diphenalenyl Diradicaloids

*n*= 1) is known as a thermally stable diradicaloid because of contributions from both resonance structures, quinoid and benzenoid resonance forms [10, 11, 12, 13, 14, 15, 17, 73]. Namely, the recovery of aromaticity in the central benzene ring in

*s*-indacene part of IDPL leads to the diradical structures. As a closed-shell reference, we consider a similar size condensed-ring compound, PY2, which is composed of two pyrene moieties. From comparison of the geometries (optimized at RB3LYP/6-31G** level) of these two compounds, the larger bond length alternations in phenalenyl ring are observed for PY2, the feature of which well reflects the Kekulé structures [74]. This large

*π*-electron delocalization in the phenarenyl rings contributes to stabilize the diradical form for IDPL. As expected from the resonance structures, the radical electrons are spin polarized between the both-end phenalenyl rings (see Fig. 24) and each up and down spin densities are delocalized in the phenalenyl ring though smaller spin polarization still exists in the phenalenyl ring. Although such spin density distribution is not observed in real singlet systems, this can be interpreted to indicate approximately the feature of spatial correlation between

*α*and

*β*spins. The diradical characters

*y*values calculated from Eq. 16 using HONO and LUNO of UNOs at UHF/6-31G* level for IDPL and PY2 are 0.7461 (intermediate diradical character) and 0.0 (closed-shell), respectively, the feature of which is consistent with the features of the spin density distributions and the resonance structures [74]. The

*γ*

_{ xxxx }(

*γ*) values for IDPL and PY2 are 2383 × 10

^{3}a.u. and 194 × 10

^{3}a.u., respectively, at (U)BHandHLYP/6-31G* level of approximation. This significant enhancement of

*γ*for IDPL as compared to that for PY2 (the ratio IDPL/PY2 = 12.3) is predicted to be caused by the intermediate diradical character (

*y*= 0.7461) for IDPL in contrast to the closed-shell PY2 (

*y*= 0.0) [74]. As seen from Fig. 25 (the

*γ*density distributions), for both systems, main contributions come from

*π*-electrons, whose contributions have opposite sign to those of

*σ*-electrons. For intermediate diradicaloid, IDPL, we observe extended positive and negative

*γ*densities distributed on the left and right phenalenyl ring regions, respectively, which cause a dominant positive contribution to

*γ*though small opposite (negative) contributions appear in the central region. Judging from the fact that the sites with dominant

*γ*density distributions on phenalenyl rings exhibit major spin density distributions, the spin-polarized

*π*-electrons concerning left- and right-hand phenalenyl rings are the origin of the enhancement of

*γ*values for these diradicaloids. Contrary to the dominant extended

*γ*densities with positive contributions for IDPL, the

*γ*density amplitudes for PY2 are smaller, and positive and negative

*γ*densities appear alternately in the bond-length alternated (Kekulé structured) region though both-end benzene ring regions provide positive contribution to

*γ*. This alternate change of sign of

*γ*densities for PY2 significantly cancels the positive contribution to

*γ*. In summary, the large enhancement of

*γ*density amplitudes and spatially well-separated positive and negative

*γ*densities on both end phenalenyl regions for IDPL turn out to be the origin of the significant enhancement of

*γ*value for diradical molecules with intermediate diradical character as compared to closed-shell

*π*-conjugated systems.

**1S**), and 1,2-bis(phenalen-2-yl)ethyne (

**2S**), which are different in their linked positions from each other (see Fig. 26) [75]. As a closed-shell reference, 1,2-bis(pyren-4-yl)ethyne (

**3S**) is also considered. Also, spin state dependences of

*γ*are also investigated for

**1S**and

**2S**. The diradical characters of these singlet compounds are listed in Table 2 together with their

*γ*values. It is found that

**1S**(singlet) has a n intermediate diradical character, while

**2S**(singlet) nearly pure diradicla character. Compound

**3S**has a smaller diradical character and are regarded as a closed-shell systems as expected. The pure diradical character for

**2S**(singlet) is caused by the linked position where no distribution (node line) of HSOMO of the phenalenyl ring. The ratios of

*γ*,

**1S**(singlet,

*y*= 0.6525)/

**3S**(

*y*= 0.1915) is 4.4 and

**1S**(singlet,

*y*= 0.6525)/

**2S**(singlet,

*y*= 0.9993) is 7.5. This feature is in agreement with the

*y*–

*γ*correlation, which states that singlet diradical systems with an intermediate diradical character tend to exhibit larger

*γ*value than the closed-shell systems with a similar

*π*-conjugation length. The spin state dependence is clarified by comparing singlet and triplet state results for

**1S**and

**2S**. In the intermediate diradical character region, the change from singlet (the lowest spin) to the triplet (the highest spin) state causes a significant reduction of

*γ*, while in the pure diradical character region, such significant reduction is not observed. This significant reduction of

*γ*in the triplet state in the intermediate diradical character region can be understood by the fact that in the triplet state the radical electron is localized in each atom due to the Pauli effect, resulting in suppressing the polarization over the two atoms. This feature is also similar to the case of the pure diradical state, where the radical electron is well localized on each atom, and is hard to contribute to polarization over the two atoms. Such remarkable spin state dependence of

*γ*is also a characteristic for diradicaloids with intermediate diradicla character.

System | | |
---|---|---|

| 0.6525 | 1568 |

| 0.9993 | 209 |

| 0.1915 | 360 |

| – | 461 |

| – | 687 |

*π*–

*π*distance (3.137 Å) less than a typical van der Waals distance (3.4 Å), large conductivity, as well as an absorption peak shifted extraordinarily to the low-energy region [10]. These features stem from the resonance structures of intra- and inter-molecular interactions of the unpaired electrons in 1D chain. In this section, we show the effects of intra- and inter-molecular interactions on the longitudinal

*γ*of IDPL and of its dimer in relation to their average diradical character. We examine a symmetric dimer model using the structure of IDPL monomer optimized by the UB3LYP/6-31G** method and the experimental interplanar distance of 3.137 Å [76]. As a reference, closed-shell dimer composed of PY2 (interplanar distance = 3.4 Å) is also considered. Table 3 lists the longitudinal BHandHLYP/6-31G*

*γ*values per monomer (

*γ*/

*N*,

*N*: the number of monomers) of IDPL monomer and dimer as well as PY2 monomer and dimer models. The interaction-induced increase ratio

*r*=

*γ*(dimer)

**/**[2 ×

*γ*(monomer)] is found to be larger for IDPL system (

*r*= 1.99) than for PY2 system (

*r*= 1.15). This significant increase is predicted to be caused by the covalent-like intermolecular interaction with open-shell singlet nature and intramolecular intermediate diradical interaction. Figure 28 shows that HONO and LUNO exhibit dominant distributions at the both-end phenalenyl rings in the dimer, while the HONO-1 and LUNO + 1 have dominant distributions at the cofacial phenalenyl rings in the middle region of the dimer. Since the occupation numbers of HONO (HONO-1) and LUNO (LUNO + 1) are related to the diradical character

*y*

_{0}(

*y*

_{1}), the primary diradical interaction for

*y*

_{0}(0.898) occurs between both end phenalenyl rings in the dimer, while that concerning

*y*

_{1}(0.770) is between cofacial phenalenyl rings in the middle region of the dimer. This relative amplitude of the diradical characters is understood by the fact that a pair of radicals with larger intersite distance gives larger

*y*values. In order to investigate further the spatial contribution of the diradical character, let us consider the spin polarization using Mulliken spin density distributions of the monomer and dimer of IDPL. From Fig. 29, both the monomer and the dimer, the primary

*α*and

*β*spin density distributions are separated into right- and left-hand side phenalenyl ring regions, respectively, though spin polarizations are observed in phenalenyl rings. For the dimer case, the spin density distributions on each cofacial phenalenyl ring (0.781) is smaller than that of the monomer (0.903), while that of the end-phenalenyl rings (0.910) is larger. This demonstrates a strong covalent-like intermolecular interaction, which leads to the smaller

*y*

_{1}(0.508) concerning the cofacial phenalenyl rings of the dimer than the

*y*

_{0}(0.770) of the monomer. Finally, we clarify the origin of the significant interaction-induced

*γ*increase in IDPL dimer by using the

*γ*density analysis (Fig. 30). As shown in Fig. 25, the large

*γ*(monomer) value stems from the extended positive and negative

*π*-electron

*γ*densities well-separated on the left- and right-hand side phenalenyl rings, respectively. This separation is also observed for each monomer building the dimer, while the \(\rho_{{}}^{(3)} ({\mathbf{r}})\) amplitude on the cofacial phenalenyl rings get smaller, leading therefore to positive and negative \(\rho_{{}}^{(3)} ({\mathbf{r}})\) difference on the left and right monomers, respectively. This feature corresponds to the suppression of the spin polarization between cofacial phenalenyl rings (Fig. 29) and the decrease in diradical character

*y*

_{1}as compared to

*y*

_{0}of the monomer. As a result, the strong covalent-like interaction (with intermediate diradical character) between the unpaired electrons of the cofacial phenalenyl rings provides a significant interaction-induced increase of

*γ*for the dimer, which is exemplified by the field-induced virtual charge transfer between both end phenalenyl rings of the dimer (Fig. 30). Recently, such covalent-like interaction between phenalneyl radicals in the real 1D column of

*π*-stacked phenalenyl aggregate is also found to exhibit strongly enhanced

*γ*amplitude in the stacking direction, and is predicted to very large macroscopic

*γ*value, which is comparable to that of polyacetylene [77].

### 8.2 Graphene Nanoflakes (GNFs)

#### 8.2.1 Rectangular GNFs

*X*,

*Y*], where

*X*and

*Y*denote the number of fused rings in the zigzag and armchair edges, respectively. It is well known that the open-shell character strongly depends on the edge shape and architectures of PAHs [1, 31, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103]. For example, in rectangular GNFs having zigzag and armchair edges, the spin polarization or odd electron density distributions appear only on the zigzag edges, while not on the armchair edges. This suggests that there is a strong correlation between the edge shape, diradical character

*y*and

*γ*values. Namely, in square PAH[3,3] (Fig. 31), the

*γ*

_{ yyyy }(diagonal component along its armchair edge) is expected to be larger than

*γ*

_{ xxxx }(diagonal component along its zigzag edge) due to the spin polarization between the mutually facing zigzag edges. In order to verify this prediction, we calculated the

*γ*components and their density distributions at UBHandHLYP/6-31G* level in its singlet and triplet states [93]. As seen from Fig. 31, PAH[3,3] has odd electron density distributions on the zigzag edges and an intermediate diradical character (

*y*

_{0}= 0.510), the feature of which leads to a relative enhancement of

*γ*

_{ xxxx }(34.1 × 10

^{3}a.u.) as compared to

*γ*

_{ yyyy }(145 × 10

^{3}a.u.). These results are in qualitative agreement with our prediction. Namely, the

*γ*is enhanced for the direction joining the radical sites. It is also found that a significant reduction of

*γ*

_{ yyyy }occurs by changing the spin state from the singlet (

*γ*

_{ yyyy }= 145 × 10

^{3}a.u.) to the triplet (

*γ*

_{ yyyy }= 35.0 × 10

^{3}a.u.) state due to the Pauli effect, while a negligible difference in

*γ*

_{ xxxx }. This is also an evidence that PAH[

*X*,

*Y*] belongs to the intermediate dirdadical character based NLO systems. Such edge shape effects are also observed in hexagonal GNFs (HGNFs) of similar size: the zigzag-edged form (Z-HGNF) presents intermediate tetraradical characters (

*y*

_{0}=

*y*

_{1}= 0.410) in contrast to its armchair-edged analogue (A-HGNF) (closed-shell) [94]. This difference is understood by the primary contributing resonance structures together with Clar’s sextet rule (see Fig. 32): for Z-HGNF, the number of Clar’s sextets in the closed-shell form (12), diradical form (12) and tetraradical form (12) are equal to each other, which implies that diradical and tetraradical contributions also exist with an equal weight to the closed-shell contribution, while for A-HGNF, only the closed-shell form contribution exist due to the largest number of Clar’s sextets. Reflecting these open-shell characters of these systems, the

*γ*

_{ xxxx }=

*γ*

_{ yyyy }values of Z-HGNF (139 × 10

^{4}a.u.) are shown to be more than three times as large as those of A-HGNF (41.7 × 10

^{4}a.u.).

*y*

_{ i }(

*i*= 0, 1) for rectangular GNFs PAH[

*X*,

*Y*], as well as the

*γ*

_{ xxxx }and

*γ*

_{ yyyy }values are shown in Table 4 [95, 96]. The

*y*

_{0}and

*y*

_{1}values are shown to increase with

*X*with keeping

*Y*constant and also with

*Y*with keeping

*X*constant. Also, the increase of

*y*

_{0}value precedes that in

*y*

_{1}value, i.e., the

*y*

_{1}begins to significantly increase after the

*y*

_{0}is close to 1. This feature indicates that for zigzag-edged rectangular GNFs, the extension of

*π*-conjugation enhances the open-shell character, leading to a multiradical state beyond the diradical state. For example, PAH[7,7] presents the significant multiple diradical characters,

*y*

_{0}= 1.000 and

*y*

_{1}= 0.899, which implies that this system exhibits a nearly pure tetraradical singlet nature. Next, we consider the multiradical character (

*y*

_{0},

*y*

_{1}) effects on the

*γ*

_{ xxxx }and

*γ*

_{ yyyy }values (see Table 4; Fig. 33). The variations of these quantities for PAH[

*X*,

*Y*] with

*Y*= 1, 3, 5, and 7 are plotted as functions of

*X*[95]. For all the systems, the

*γ*

_{ yyyy }values exhibit non-monotonic

*X*dependences, though the

*γ*

_{ xxxx }monotonically increase with

*X*. The maximum

*γ*

_{ yyyy }value for each family (PAH[

*X*,1], PAH[

*X*,3], PAH[

*X*,5], and PAH[

*X*,7] (1 ≤

*X*≤ 7)) is obtained at the intermediate

*y*

_{0}value, i.e.,

*y*

_{0}= 0.559 (PAH[1,6]),

*y*

_{0}= 0.510 (PAH[3,3]),

*y*

_{0}= 0.372 (PAH[2,5]), and

*y*

_{0}= 0.487 (PAH[2,7]). In addition, we observe the

*γ*enhancement in the intermediate diradical character

*y*

_{0}region, as well as in the intermediate second diradical character

*y*

_{1}region. The PAHs with

*Y*≥ 3 exhibit the second

*γ*

_{ yyyy }peak with increasing

*X*, which occurs at (

*y*

_{0},

*y*

_{1}) = (~1, 0.623) for PAH[5,6] and (~1, 0.763) for PAH[6,7] and (~1, 0.560) for PAH[3,7]. These features indicate that the

*y*–

*γ*correlation holds for

*y*

_{0}, as well as for

*y*

_{1}, although the

*γ*

_{ yyyy }peak at intermediate

*y*

_{1}value is smaller than that at intermediate

*y*

_{0}value. This reduced enhancement for intermediate

*y*

_{1}value is predicted to be associated with the corresponding larger excitation energy (originating form the smaller HOMO-1 − LUMO + 1 gap) than that concerning the intermediate

*y*

_{0}. Such multiradical character effect on

*γ*is furthermore clarified by using the odd electron and

*γ*density analyses. The odd electron densities for

*y*

_{0}and

*y*

_{1}, and

*γ*

_{ yyyy }density distributions of PAH[3,3] and PAH[6,7] are show in Fig. 34. For both systems,

*π*-electrons give primary positive and negative

*γ*

_{ yyyy }densities. The density distributions are well separated to around the bottom- and top-edges, respectively, and they rapidly decrease in amplitude toward the center region. For PAH[3,3], the

*γ*

_{ yyyy }densities are primarily distributed in the middle region of the both zigzag edges, which coincide with the region with large odd electron density distributions for

*y*

_{0}. In contrast, the primary amplitudes of

*γ*

_{ yyyy }densities for PAH[6,7] are located in the end zigzag-edge region, i.e., four-corner phenalenyl blocks, and this distribution feature is in good agreement with the odd electron density distribution for

*y*

_{1}. These results correspond to the fact that

*γ*

_{ yyyy }of PAH[3,3] is determined by the intermediate

*y*

_{0}value, while that of PAH[6,7] by the intermediate

*y*

_{1}value since the

*y*

_{0}~1 reduces the HONO–LUNO contribution. In summary, the first and the second

*γ*

_{ yyyy }peaks for PAH[

*X*,

*Y*], which appear at intermediate

*y*

_{0}and

*y*

_{1}values, respectively, are evidences of the multiradical effect (tetraradical in this case) on

*γ*.

| | | | | |
---|---|---|---|---|---|

1 | 1 | 0.000 | 0.000 | 0.032 | 0.032 |

2 | 1 | 0.050 | 0.016 | 0.305 | 0.090 |

3 | 1 | 0.149 | 0.022 | 1.52 | 0.160 |

4 | 1 | 0.282 | 0.034 | 7.48 | 0.561 |

5 | 1 | 0.419 | 0.068 | 19.8 | 0.730 |

6 | 1 | 0.559 | 0.118 | 47.3 | 0.748 |

7 | 1 | 0.696 | 0.183 | 99.3 | 0.667 |

1 | 3 | 0.037 | 0.012 | 0.051 | 1.73 |

2 | 3 | 0.217 | 0.022 | 0.409 | 4.30 |

3 | 3 | 0.510 | 0.053 | 3.41 | 14.5 |

4 | 3 | 0.806 | 0.137 | 15.0 | 7.33 |

5 | 3 | 0.922 | 0.259 | 41.2 | 5.51 |

6 | 3 | 0.972 | 0.407 | 79.3 | 7.33 |

7 | 3 | 0.995 | 0.560 | 123 | 9.29 |

1 | 5 | 0.069 | 0.014 | 0.068 | 11.5 |

2 | 5 | 0.372 | 0.057 | 0.928 | 80.3 |

3 | 5 | 0.808 | 0.100 | 5.36 | 50.0 |

4 | 5 | 0.953 | 0.233 | 21.9 | 25.2 |

5 | 5 | 0.989 | 0.420 | 49.9 | 40.5 |

6 | 5 | 0.999 | 0.623 | 82.5 | 57.2 |

7 | 5 | 0.999 | 0.790 | 152 | 56.0 |

1 | 7 | 0.094 | 0.026 | 0.086 | 37.3 |

2 | 7 | 0.487 | 0.114 | 1.13 | 316 |

3 | 7 | 0.921 | 0.152 | 7.08 | 102 |

4 | 7 | 0.999 | 0.304 | 27.5 | 88.8 |

5 | 7 | 0.999 | 0.534 | 53.1 | 197 |

6 | 7 | 1.000 | 0.763 | 100 | 215 |

7 | 7 | 1.000 | 0.899 | 220 | 147 |

#### 8.2.2 Rhombic and Bow-Tie GNFs

*y*

_{0}= 0.418) is found to be smaller than that for bow-tie GNF (

*y*

_{0}= 0.970). This difference indicates that there are unique structural dependences of open-shell characters of GNFs based on the linked form, and this is different from the conventional simple

*π*-conjugation size dependence of NLO properties. This structural dependence can be understood based on the resonance structures of these GNFs (Fig. 35): the rhombic GNF shows both closed-shell and diradical resonance forms, while there are no closed-shell forms for the bow-tie GNFs. It is also found that the number of Clar’s sextets in the diradical form of rhombic GNF is the same as that in the closed-shell form, while that it is less than that of diradical bow-tie GNF. This predicts that for the rhombic GNF, the thermal stabilities of the closed-shell and diradical forms are similar to each other, resulting in its intermediate diradical character.

#### 8.2.3 One-Dimensional (1D) GNFs Composed of Phenalenyl Radical Units

*γ*, we consider two types of GNFs composed of phenalenyl radiclal units, alternately linked (AL) and nonalternately linked (NAL) systems shown in Fig. 36, i.e., singlet and highest spin states of 1D conjugated systems composed of the linked

*N*phenalneyl units [2 ≤

*N*(even number) ≤ 10] [98, 99]. We first examine the two-unit systems. As seen from the resonance structures with Clar’s sextet rule, the AL system has both contributions from closed-shell and diradical forms, while the NAL system from only diradical form. Indeed, from Lieb’s theorem [104] and Ovchinnikov’s rule [105], the AL (NAL) system has the singlet (the highest) spin state as the ground state. Also, this difference is understood by the difference in the HOMO–LUMO gap, which is predicted by the interaction between the phenalenyl SOMOs [99]. As a result, the two-unit AL (2-AL) and 2-NAL exhibit

*y*

_{0}= 0.101 and 0.919, respectively, at LC-UBLYP/6-31G* level of approximation. Reflecting these diradical characters, in the singlet states, the

*γ*

_{ xxxx }of 2-AL (30.7 × 10

^{4}a.u.) is more than twice as large as that of 2-NAL (13.4 × 10

^{4}a.u.). Furthermore, when changing from singlet to triplet, the both

*γ*amplitudes are significantly reduced [

*γ*

_{ xxxx }(triplet)/

*γ*

_{ xxxx }(singlet) = 0.33 (2-AL) and 0.57 (2-NAL)], and the ratio of

*γ*

_{ xxxx }amplitude of 2-AL to that of 2-NAL system is also reduced [2.29 (singlet) \(\to\) 1.33 (triplet)].

*y*

_{ i }and

*γ*

_{ xxxx }for

*N*-AL and

*N*-NAL systems [99]. As seen from Fig. 37, the diradical characters of AL systems show slower increases with

*N*than NAL systems, while

*y*

_{ i }values for both systems show a systematic increase, which appears in the order of increasing

*i*. As a result, the relatively large size systems such as ten-unit AL system exhibit intermediate multiple diradical characters, i.e., intermediate multiradical nature, e.g., for 10-AL,

*y*

_{0}= 0.630,

*y*

_{1}= 0.383,

*y*

_{2}= 0.236,

*y*

_{3}= 0.153,

*y*

_{4}= 0.122. In contrast, in NAL systems, the

*y*

_{ i }values show abrupt transitions from almost 0 to almost 1 at every addition of a pair of units, due to the almost negligible orbital interaction. As a result, NAL systems are pure multiradical systems regardless of the number of units.

*γ*for AL and NAL systems in their singlet and highest spin states. Remarkable differences in the amplitudes and size dependences are observed between those systems. Namely, the AL systems show much larger

*γ*values together with much stronger size-dependent enhancement than the corresponding NAL systems. In particular, the

*γ*value of the10-AL system in the singlet state (13300 × 10

^{4}a.u.) is more than 30 times as large as that of the analogous NAL system (392 × 10

^{4}a.u.) and the

*γ*(

*N*= 10)/

*γ*(

*N*= 2) ratio for the singlet AL systems attains 433, which is more than ten times larger than that for the singlet NAL systems (38.4). In contrast, all AL systems exhibit a significant spin state dependence: the change from singlet to the highest spin states significantly reduce the

*γ*

_{ xxxx }values, e.g., 88% reduction at 10-AL system, whereas such change is negligible for NAL systems, e.g., 7% reduction at 10-NAL. As a result, in the highest spin state, size dependence of

*γ*is significantly reduced for NAL, e.g.,

*γ*

_{ xxxx }(

*N*= 10)/

*γ*

_{ xxxx }(

*N*= 2) attains 151 for AL versus 48 for NAL. Such spin state dependence indicates that the large

*γ*

_{ xxxx }enhancement rate with

*N*in singlet AL systems originates in their intermediate multiple diradical characters.

### 8.3 Asymmetric Open-Shell Singlet Systems

As shown in Sect. 4, asymmetric open-shell molecular systems have a potential for exhibiting further enhancement of amplitudes of hyperpolarizabilities by tuning the diradical character and asymmetricity [38, 39, 45, 106, 107]. On the other hand, there have been few realistic asymmetric systems with open-shell character, so that the molecular design and synthesis of such systems have been eagerly anticipated. Here, we show two types of asymmetric open-shell molecules, where the asymmetricity is induced by a static electric field application, and by donor/acceptor substitution.

*x*) direction ranges from 0.0 to 0.0077 a.u. (0.0–0.4 V/Å). The diradical character in the absence of the static field,

*y*

_{ F=0}= 0.717 (IDPL), is found to decrease as increasing the field amplitude, e.g., the

*y*

_{ F }(IDPL) amplitude goes down to 0.293 at

*F*= 0.0077 a.u. This is caused by the field-induced relative increase of the ionic component in the ground state of IDPL. Indeed, at

*F*= 0.0077 a.u., the charge transfer (CT) occurs from the right- to the left-hand side of IDPL and then leads to the

*x*component of the ground-state dipole moment (

*μ*

_{g x }= 9.88 a.u.), which is significantly larger than that in the pyrene rings of PY2 (

*μ*

_{g x }= 5.76 a.u.). On the other hand, the IDPL spin densities are reduced and asymmetrized. The evolution of

*γ*of IDPL and PY2 for the static fields ranging from 0.0 to 0.0077 a.u. is shown in Fig. 40. The

*γ*is much larger in IDPL than in PY2 in the whole

*F*region. The

*γ*values of both systems increase with

*F*, e.g.,

*γ*= 1.746 × 10

^{6}a.u. (IDPL) vs. 1.743 × 10

^{5}a.u. (PY2) at

*F*= 0.0 a.u. and

*γ*= 1.456 × 10

^{10}a.u. (IDPL) vs. 3.107 × 10

^{5}a.u. (PY2) at

*F*= 0.0077 a.u., and the enhancement ratio,

*γ*(IDPL)/

*γ*(PY2), increases with

*F*, e.g.,

*γ*(IDPL)/

*γ*(PY2) = 10 (

*F*= 0.0 a.u.) vs. 4.7 × 10

^{4}(

*F*= 0.0077 a.u.).

_{2})-acceptor (NO

_{2}) substitution into PAHs, i.e., DA-IDPL [(NO

_{2})

_{2}-IDPL-(NH

_{2})

_{2}] and DA-PY2 [(NO

_{2})

_{2}-PY2-(NH

_{2})

_{2}] (Fig. 39c, d) [106]. Such chemical modification is expected to achieve a similar situation to the static field application. Indeed, these substituted systems exhibit CTs from the right- to the left-hand side though the amounts of CT of DA-IDPL (0.162) and DA-PY2 (0.153) are smaller than those [0.427(IDPL) and 0.270(PY2)] at

*F*= 0.0077 a.u. From comparing the

*γ*values between substituted (Fig. 41) and field-application (Fig. 40) systems, the two donor–acceptor pairs substitution cause an effect on

*γ*comparable to a field of approximately 0.0060–0.0065 a.u.: the

*γ*of DA-IDPL becomes 8.373 × 10

^{7}a.u., in comparison to

*γ*= 5.501 × 10

^{7}–1.278 × 10

^{8}a.u. at

*F*= 0.0060–0.0065 a.u., respectively. However, electric field amplitudes of 0.0077 a.u. or smaller are not enough to reproduce the

*γ*value of DA-PY2 (9.546 × 10

^{5}a.u.), which indicates that the relationship between the effects of D/A pairs and external electric field is not universal and depends on the nature of the linker. Nevertheless, the

*γ*value of DA-IDPL (8.373 × 10

^{7}a.u.) is about 88 times larger than that of DA-PY2 (9.546 × 10

^{5}a.u.), the ratio of which is strongly enhanced as compared to the non-substituted case at

*F*= 0:

*γ*(IDPL)/

*γ*(PY2) = 10.

In summary, it is found that there is a gigantic enhancement of *γ* by applying an electric field (*F*) along the spin polarization direction to polycyclic aromatic diradicaloids with intermediate diradical character. Indeed, for IDPL, the enhancement with respect to the field-free case attains four orders of magnitude by applying an electric field of 0.0077 a.u., while a similar-size closed-shell analogue PY2 shows a weak field effect. Similar effects are achieved when substituting both end phenalenyl rings of IDPL by donor (NH_{2})/acceptor (NO_{2}) groups. In this case, DA-IDPL also exhibits a *γ* value more than two orders of magnitude larger than in the reference closed-shell PY2. Furthermore, the diradical character in this open-shell singlet system is reduced due to either the application of an electric field or the substitution by donor/acceptor groups. This behavior is an advantage towards improved thermal stability. The present results demonstrate that the introduction of asymmetricity into the open-shell molecular systems provide a new design guideline for further enhancement/tuning of the NLO responses.

## 9 Experimental Estimation of Diradical Character

*y*using the relationships between the excitation energies and

*y*value for two-site VCI diradical model, Eqs. 22, 23, 24b, 25b, and 27 [108]. The diradical character

*y*can be expressed by [108]

*y*(

*y*

^{exp}) values and theoretically calculated ones (

*y*

^{theor}). A strong correlation between the experimental and theoretical results for all compounds is observed in spite of the difference in the scales originating from several factors, e.g., including the effects of the environment (solvation or crystal packing), inconsistencies among the different experimental methods, and the approximate nature of the VCI model [7]. As a result, the theoretical relationship between the diradical character and measurable quantities (Eq. 64) is found to provide semiquantitative estimates (after scaling) for the diradical characters

## 10 Summary

## Notes

### Acknowledgements

This work has been supported by JSPS KAKENHI Grant Number JP25248007 in Scientific Research (A), Grant Number JP24109002 in Scientific Research on Innovative Areas “Stimuli-Responsive Chemical Species”, Grant Number JP15H00999 in Scientific Research on Innovative Areas “*π*-System Figuration”, and Grant Number JP26107004 in Scientific Research on Innovative Areas “Photosynergetics”. This is also partly supported by King Khalid University through a grant RCAMS/KKU/001-16 under the Research Center for Advanced Materials Science at King Khalid University, Kingdom of Saudi Arabia.

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