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

  • Masayoshi Nakano
Part of the following topical collections:
  1. Physical Organic Chemistry of Quinodimethanes


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.


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

1 Introduction

Recently, polycyclic aromatic hydrocarbons (PAHs) have attracted great attention from various science and engineering fields due to their unique electronic structures and fascinating physicochemical functionalities, e.g., low-energy gap between the singlet and triplet ground states [1, 2], geometrical dependences of open-shell character such as unpaired electron density distributions on the zigzag edges of acenes, which leads to the high reactivity on those region [2], significant near-infrared absorption [3], enhancement of nonlinear optical (NLO) properties including two-photon absorption [4, 5, 6, 7, 8, 9], and small stacking distance (less than van der Waals radius) and high electronic conductivity in ππ 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].
Fig. 1

Resonance structures of zethrene series (a), diphenalenyl compounds (b) and heptacene (c). Bold lines indicate the benzenoid and quinoid structures in (a) and (b), and Clar’s sextets in (c)

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

The simplest understanding of the open-shell character can be achieved by the single bond dissociation of a homodinuclear molecule (see Fig. 2), which is described by the highest occupied molecular orbital (HOMO) and the lowest unoccupied MO (LUMO) in the symmetry-adapted approach like restricted Hartree–Fock (RHF) method. Namely, the bond dissociation process is described by the decrease in the HOMO–LUMO gap, i.e., the correct wavefunction is described by the mixing between the HOMO (bonding) and LUMO (antibonding), and the wavefunction at the dissociation limit is composed of the equally weighted mixing of the HOMO and LUMO, which creates localized spatial distribution on each atom site and thus no distribution between the atoms. More precisely, as increasing the bond distance, the double excitation configuration from the HOMO to the LUMO becomes mixed into the doubly occupied configuration in the HOMO. On the other hand, in the spin-unrestricted (broken-symmetry) approach, the MO could have different spatial distribution for the 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].
Fig. 2

Bond dissociation process of a homodinuclear molecule, where the variations of the HOMO and LUMO levels in the symmetry-adapted approach as well as of the magnetic orbitals for the α and β spins in the broken-symmetry approach are also shown as a function of bond distance. The physical and chemical meanings of diradical character (y) are also shown in the three regimes (I)–(III) of the electronic states in the bond dissociation process

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 (1α, 2β) plane, where 1α and 2β 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 (1β, 2α), but this is the same spatial distribution as that on (1α, 2β) plane. Thus, we can discuss the singlet wavefunction using only the distribution on the (1α, 2β) plane without loss of generality. Figure 3a shows the 1D two-electron system A–B and the 2D plane (1α, 2β), on which the spatial distribution of the singlet wavefunction is plotted. On the (1α, 2β) 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.

We can here consider the spatial distribution of the singlet wavefunctions composed of the HOMO and LUMO. As shown in Fig. 3b, the HOMO and LUMO are represented by two while circles and a pair of white and black circles, respectively, where white and black indicate positive and negative phase of the MO. Using various electron configurations in the HOMO (\(\phi_{\text{H}}\)) and LUMO (\(\phi_{\text{L}}\)), we can describe the symmetry-adapted wavefunctions. For example, the double-occupied configuration in the HOMO gives the HF singlet ground state \(\psi_{\text{G}}^{{}}\), which is represented by the Slater determinant:
Fig. 3

Schematic diagram of 1D two-electron system A–B and the 2D (1α, 2β) plane (a) and the singlet spatial wavefunctions, \(\psi_{\text{G}}^{{}}\) (HF ground state determinant), \(\psi_{\text{S}}^{{}}\)(singly excited determinant), and \(\psi_{\text{D}}^{{}}\) (doubly excited determinant) on the (1α, 2β) plane with the HOMO and LUMO distributions (b)

$$\begin{aligned} \psi_{\text{G}} = \frac{1}{\sqrt 2 }\left| {\begin{array}{*{20}c} {\phi_{\text{H}} (1)\alpha (1)} & {\phi_{\text{H}} (1)\beta (1)} \\ {\phi_{\text{H}} (2)\alpha (2)} & {\phi_{\text{H}} (2)\beta (2)} \\ \end{array} } \right| \equiv \psi (\phi_{\text{H}} \bar{\phi }_{\text{H}} ) \\ = \frac{1}{\sqrt 2 }\phi_{\text{H}} (1)\phi_{\text{H}} (2)(\alpha (1)\beta (2) - \beta (1)\alpha (2)). \\ \end{aligned}$$
As mentioned before, the (1α, 2β) 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
$$\begin{aligned} \psi_{\text{S}} = \frac{1}{\sqrt 2 }\left\{ {\frac{1}{\sqrt 2 }\left| {\begin{array}{*{20}c} {\phi_{\text{H}} (1)\alpha (1)} & {\phi_{\text{L}} (1)\beta (1)} \\ {\phi_{\text{H}} (2)\alpha (2)} & {\phi_{\text{L}} (2)\beta (2)} \\ \end{array} } \right| + \frac{1}{\sqrt 2 }\left| {\begin{array}{*{20}c} {\phi_{\text{L}} (1)\alpha (1)} & {\phi_{\text{H}} (1)\beta (1)} \\ {\phi_{\text{L}} (2)\alpha (2)} & {\phi_{\text{H}} (2)\beta (2)} \\ \end{array} } \right|} \right\} \\ \equiv \frac{1}{\sqrt 2 }\left[ {\psi (\phi_{\text{H}} \bar{\phi }_{\text{L}} ) + \psi (\phi_{\text{L}} \bar{\phi }_{\text{H}} )} \right] \\ = \frac{1}{2}(\phi_{\text{H}} (1)\phi_{\text{L}} (2) + \phi_{\text{H}} (2)\phi_{\text{L}} (1))(\alpha (1)\beta (2) - \beta (1)\alpha (2)). \\ \end{aligned}$$
The spatial part is composed of two components, \(\phi_{\text{H}} (1)\phi_{\text{L}} (2)\) and \(\phi_{\text{H}} (2)\phi_{\text{L}} (1)\), which are needed to satisfy the symmetry for exchange between the real coordinates for electron 1 and 2. As shown in Fig. 3b, the spatial distribution of \(\psi_{\text{S}}\) is pure ionic, i.e., only diagonal distribution, and has a node line along the anti-diagonal line. The doubly excited Slater determinant is expressed as
$$\begin{aligned} \psi_{\text{D}}^{{}} = \frac{1}{\sqrt 2 }\left| {\begin{array}{*{20}c} {\phi_{\text{L}} (1)\alpha (1)} & {\phi_{\text{L}} (1)\beta (1)} \\ {\phi_{\text{L}} (2)\alpha (2)} & {\phi_{\text{L}} (2)\beta (2)} \\ \end{array} } \right| \equiv \psi (\phi_{\text{L}} \bar{\phi }_{\text{L}} ) \\ = \frac{1}{\sqrt 2 }\phi_{\text{L}} (1)\phi_{\text{L}} (2)(\alpha (1)\beta (2) - \beta (1)\alpha (2)) \\ \end{aligned}$$
In this case, the spatial distribution is described by \(\phi_{\text{L}} (1)\phi_{\text{L}} (2)\), which indicates the doubly occupied in the LUMO. The spatial distribution is the same as that of \(\psi_{\text{G}}^{{}}\) except for the phase, where \(\phi_{\text{L}} (1)\phi_{\text{L}} (2)\) has two node lines, i.e., neutral (covalent or diradical) and ionic distributions possess mutually opposite phase. Note here that the spatial parts of these wavefunctions are easily constructed from the HOMO and/or LUMO and that the symmetry of the spatial part for exchanging electron 1 and 2 is straightforward. For these Slater determinants, apparently, neutral and ionic distribution amplitudes in \(\psi_{\text{G}}^{{}}\) and \(\psi_{\text{D}}^{{}}\) are equal to each other, which indicates that the electron correlation is not considered. Next, we consider the effect of electron correlation on the spatial distribution of these wavefunctions.
In the ground state, the ionic distribution should be smaller than the neutral (covalent) distributions in order to more stabilize the ground state by avoiding the strong Coulomb repulsion on the ionic distribution. By mixing the spatial wavefunctions of \(\psi_{\text{G}}^{{}}\) and \(\psi_{\text{D}}^{{}}\), we can construct such wavefunction distribution as shown in Fig. 4. From symmetry, the HF ground state wavefunction \(\psi_{\text{G}}^{{}}\) and doubly excited wavefunction \(\psi_{\text{D}}^{{}}\) are correlated (mixed) with mutually opposite phase in the ground state and with the same phase in the second excited state, which leads to the increase (decrease) in the neutral component and decrease (increase) in the ionic component in the ground state g (the second excited state f). Note here that the first excited state k is not mixed with other wavefunctions and is a pure ionic state. Thus, the correct wavefunctions for states {g, k, f} are described by
$$\varPsi_{\text{g}}^{{}} = \sqrt {1 - \lambda^{2} } \psi_{\text{G}}^{{}} - \lambda \psi_{\text{D}}^{{}}$$
$$\varPsi_{\text{k}}^{{}} = \psi_{\text{S}}^{{}}$$
$$\varPsi_{\text{f}}^{{}} = \lambda \psi_{\text{G}}^{{}} + \sqrt {1 - \lambda^{2} } \psi_{\text{D}}^{{}}$$
Fig. 4

Schematic diagram of electron correlated wavefunctions for 1D two-electron system A–B on the (1α, 2β) plane: the ground state \(\varPsi_{\text{g}}\) (a), the first excited state \(\varPsi_{\text{k}}\) (b) and the second excited state \(\varPsi_{\text{f}}\) (c). The BS wavefunction described by the symmetry-adapted determinants, \(\psi_{\text{G}}\), \(\psi_{\text{T}}\), and \(\psi_{\text{D}}\), is also shown (d) (see Eq. 9)

The 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 2λ 2 from 0 to 1 corresponds to the change from the stable bond region to the bond dissociation limit. Namely, the 2λ 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

We consider the spin-unrestricted [broken-symmetry (BS)] ground state wavefunction using the symmetry-adapted wavefunctions. Using the BS HOMOs χ and η, the ground state BS wavefunction is expressed as
$$\begin{aligned} \varPsi^{\text{BS}} (\chi \bar{\eta }) = \frac{1}{\sqrt 2 }\left| {\begin{array}{*{20}c} {\chi (1)\alpha (1)} & {\chi (2)\alpha (2)} \\ {\eta (1)\beta (1)} & {\eta (2)\beta (2)} \\ \end{array} } \right| \\ = \frac{1}{\sqrt 2 }\left( {\chi (1)\eta (2)\alpha (1)\beta (2) - \eta (1)\chi (2)\beta (1)\alpha (2)} \right). \\ \end{aligned}$$
Here, the BS orbitals χ and η are represented by symmetry-adapted MOs \(\phi_{\text{H}}\) and \(\phi_{\text{L}}\) as [28, 29]
$$\chi = \cos\frac{\theta }{2}\phi_{\text{H}} + \sin\frac{\theta }{2}\phi_{\text{L}} ,\;{\text{and}}\; \, \eta = \cos\frac{\theta }{2}\phi_{\text{H}} - \sin\frac{\theta }{2}\phi_{\text{L}} \;$$
where 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]
$$\varPsi^{\text{BS}} (\chi \bar{\eta }) = \cos^{2} \frac{\theta }{2}\psi (\phi_{\text{H}} \bar{\phi }_{\text{H}} ) - \sqrt 2 \sin\frac{\theta }{2}\cos\frac{\theta }{2}\left[\frac{1}{\sqrt 2 }(\psi (\phi_{\text{H}} \bar{\phi }_{\text{L}} ) - \psi (\phi_{\text{L}} \bar{\phi }_{\text{H}} ))\right] - \sin^{2} \frac{\theta }{2}\psi (\phi_{\text{L}} \bar{\phi }_{\text{L}} )$$
where the first, second, and third terms involve the singlet ground state determinant \(\psi_{\text{G}} ( = \psi (\phi_{\text{H}} \bar{\phi }_{\text{H}} ))\) (Eq. 1), the triplet determinant \(\psi_{\text{T}} = \frac{1}{\sqrt 2 }(\psi (\phi_{\text{H}} \bar{\phi }_{\text{L}} ) - \psi (\phi_{\text{L}} \bar{\phi }_{\text{H}} ))\), and singlet double excited determinant \(\psi_{\text{D}} = \psi (\phi_{\text{L}} \bar{\phi }_{\text{L}} )\) (Eq. 3), respectively. As seen from Fig. 4d, the qualitatively correct spatial distribution of the singlet ground state wavefunction is built from superposition of \(\psi_{\text{G}}\) and \(\psi_{\text{D}}\), while the incorrect spin component (triplet) \(\psi_{\text{T}}\) is also mixed into the wavefunction. This triplet component, which is anti-symmetric with respect to the exchange of the real coordinate between electron 1 and 2, is schematically shown to asymmetrize the neutral components as shown in Fig. 4d. This is the reason why this wavefunction is called “broken symmetry” (neither symmetric nor anti-symmetric with respect to the exchange between electron 1 and 2), and is found to be made of broken-symmetry HOMOs 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
$$\varPsi^{\text{BS}} = \frac{1 + T}{2}\psi_{\text{G}} - \sqrt {\frac{{1 - T^{2} }}{2}} \psi_{\text{T}} - \frac{1 - T}{2}\psi_{\text{D}}$$
The perfect-pairing type spin-projection implies the removal of the second term from the BS wavefunction with keeping the weight ratio of the first and third terms, \([(1 + T)/(1 - T)]^{2}\). Thus, the spin-projected wavefunction is expressed by
$$\varPsi^{\text{PU}} = \frac{1 + T}{{\sqrt {2(1 + T^{2} )} }}\psi_{\text{G}} - \frac{1 - T}{{\sqrt {2(1 + T^{2} )} }}\psi_{\text{D}}$$
From the definition of the diradical character y, i.e., twice the weight of the doubly excitation configuration, we obtain the expression of diradical character in the PUHF formalism [28, 29]:
$$y^{\text{PU}} = 1 - \frac{2T}{{1 + T^{2} }}$$
Here, let us consider the one-electron reduced density using the BS wavefunction Eq. 7,
$$\begin{aligned} \rho ({\mathbf{r}}) = \left| {\chi ({\mathbf{r}})} \right|^{2} + \left| {\eta ({\mathbf{r}})} \right|^{2} = 2\cos^{2} \frac{\theta }{2}\left| {\phi_{\text{H}} ({\mathbf{r}})} \right|^{2} + 2\sin^{2} \frac{\theta }{2}\left| {\phi_{\text{L}} ({\mathbf{r}})} \right|^{2} \\ = (1 + T)\left| {\phi_{\text{H}} ({\mathbf{r}})} \right|^{2} + (1 - T)\left| {\phi_{\text{L}} ({\mathbf{r}})} \right|^{2} . \\ \end{aligned}$$
This equation indicates that \(\phi_{\text{H}} ({\mathbf{r}})\) and \(\phi_{\text{L}} ({\mathbf{r}})\) are the HONO (the highest occupied natural orbital) and LUNO (the lowest occupied natural orbital) of the BS solution with the occupation numbers of \(1 + T( \equiv n_{\text{HONO}} )\) and \(1 - T( \equiv n_{\text{LUNO}} )\), respectively. On the other hand, the occupation numbers of the HONO and LUNO of the spin-projected wavefunction Eq. 11 are expressed by
$$n_{\text{HONO}}^{\text{PU}} = \frac{{(1 + T)^{2} }}{{1 + T^{2} }} = \frac{{n_{\text{HONO}}^{ 2} }}{{1 + T^{2} }} = 2 - y^{\text{PU}}$$
$$n_{\text{LUNO}}^{\text{PU}} = \frac{{(1 - T)^{2} }}{{1 + T^{2} }} = \frac{{n_{\text{LUNO}}^{ 2} }}{{1 + T^{2} }} = y^{\text{PU}}$$
where \(n_{\text{HONO}} = 1 + T\) and \(n_{\text{LUNO}} = 1 - T\) are employed (see Eq. 13). This expression can be extended to a 2n-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]
$$y_{i}^{\text{PU}} = 1 - \frac{{2T_{i} }}{{1 + T_{i}^{2} }}$$
$$n_{{{\text{HONO - }}i}}^{\text{PU}} = 2 - y_{i}^{\text{PU}} ,\;{\text{and }}n_{{{\text{LUNO + }}i}}^{\text{PU}} = y_{i}^{\text{PU}}$$
where \(T_{i}\) is the overlap between the corresponding orbitals \(\chi_{i}\) and \(\eta_{i}\), and the occupation number of LUNO + 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

For a symmetric two-site diradical system with two electrons in two orbitals (LNOs), a and b, with the z-component of spin angular momentum M s = 0 (singlet and triplet), we can consider two neutral
$$\psi (a\bar{b}) = \frac{1}{\sqrt 2 }\left( {a(1)b(2)\alpha (1)\beta (2) - b(1)a(2)\beta (1)\alpha (2)} \right)$$
$$\psi (b\bar{a}) = \frac{1}{\sqrt 2 }\left( {b(1)a(2)\alpha (1)\beta (2) - a(1)b(2)\beta (1)\alpha (2)} \right)$$
and two ionic determinants:
$$\psi (a\bar{a}) = \frac{1}{\sqrt 2 }\left( {a(1)a(2)\alpha (1)\beta (2) - a(1)a(2)\beta (1)\alpha (2)} \right)$$
$$\psi (b\bar{b}) = \frac{1}{\sqrt 2 }\left( {b(1)b(2)\alpha (1)\beta (2) - b(1)b(2)\beta (1)\alpha (2)} \right)$$
The spatial distributions of these wavefunctions on the (1α, 2β) plane are described by \(a(1)b(2)\), \(b(1)a(2)\), \(a(1)a(2)\), and \(b(1)b(2)\), respectively (see n1, n2, i1, and i2, 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]:
$$\left( {\begin{array}{*{20}c} {\left\langle {a\bar{b}} \right|H\left| {a\bar{b}} \right\rangle } & {\left\langle {a\bar{b}} \right|H\left| {b\bar{a}} \right\rangle } & {\left\langle {a\bar{b}} \right|H\left| {a\bar{a}} \right\rangle } & {\left\langle {a\bar{b}} \right|H\left| {b\bar{b}} \right\rangle } \\ {\left\langle {b\bar{a}} \right|H\left| {a\bar{b}} \right\rangle } & {\left\langle {b\bar{a}} \right|H\left| {b\bar{a}} \right\rangle } & {\left\langle {b\bar{a}} \right|H\left| {a\bar{a}} \right\rangle } & {\left\langle {b\bar{a}} \right|H\left| {b\bar{b}} \right\rangle } \\ {\left\langle {a\bar{a}} \right|H\left| {a\bar{b}} \right\rangle } & {\left\langle {a\bar{a}} \right|H\left| {b\bar{a}} \right\rangle } & {\left\langle {a\bar{a}} \right|H\left| {a\bar{a}} \right\rangle } & {\left\langle {a\bar{a}} \right|H\left| {b\bar{b}} \right\rangle } \\ {\left\langle {b\bar{b}} \right|H\left| {a\bar{b}} \right\rangle } & {\left\langle {b\bar{b}} \right|H\left| {b\bar{a}} \right\rangle } & {\left\langle {b\bar{b}} \right|H\left| {a\bar{a}} \right\rangle } & {\left\langle {b\bar{b}} \right|H\left| {b\bar{b}} \right\rangle } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} 0 & {K_{ab} } & {t_{ab} } & {t_{ab} } \\ {K_{ab} } & 0 & {t_{ab} } & {t_{ab} } \\ {t_{ab} } & {t_{ab} } & U & {K_{ab} } \\ {t_{ab} } & {t_{ab} } & {K_{ab} } & U \\ \end{array} } \right)$$
Here, \(\left\langle {a\bar{b}} \right|H\left| {b\bar{b}} \right\rangle \equiv \int {\psi^{*} (a\bar{b})H} \psi (b\bar{b})d\tau_{1} d\tau_{2}\) and so on. The energy of the neutral determinant, \(\left\langle {a\bar{b}} \right|H\left| {a\bar{b}} \right\rangle = \left\langle {b\bar{a}} \right|H\left| {b\bar{a}} \right\rangle\), is taken as the energy origin (0). U represents the difference between on- and neighbor-site Coulomb repulsions, referred to as effective Coulomb repulsion:
$$\begin{aligned} U \equiv U_{aa} - U_{bb} \\ = \int {a^{*} (1)a(1)r_{12}^{ - 1} a^{*} (2)a(2)d{\mathbf{r}}_{1} d{\mathbf{r}}_{2} } - \int {b^{*} (1)b(1)r_{12}^{ - 1} b^{*} (2)b(2)d{\mathbf{r}}_{1} d{\mathbf{r}}_{2} } \\ = (aa|aa) - (bb|bb). \\ \end{aligned}$$
\(K_{ab}\) is a direct exchange integral [\(K_{ab} = (ab|ba)\) ≥ 0], and \(t_{ab}\) is a transfer integral [\(t_{ab} = \left\langle {a\bar{b}} \right|H\left| {b\bar{b}} \right\rangle\) \(= \left\langle a \right|f\left| b \right\rangle\) ≤ 0, where 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].

(A) Neutral triplet state (with u symmetry)
$$\varPsi_{{T_{{1{\text{u}}}} }} = \frac{1}{\sqrt 2 }\left( {\psi (a\bar{b}) - \psi (b\bar{a})} \right)\;{\text{with}}\;{\text{energy}}\;{}^{3}E_{{ 1 {\text{u}}}} = - K_{ab}$$
(B) Ionic singlet state (with u symmetry)
$$\varPsi_{{S_{{1{\text{u}}}} }} = \frac{1}{\sqrt 2 }\left( {\psi (a\bar{a}) - \psi (b\bar{b})} \right) \;{\text{with}}\;{}^{1}E_{{ 1 {\text{u}}}} = U - K_{ab}$$
(C) Lower singlet state (with g symmetry)
$$\varPsi_{{S_{{1{\text{g}}}} }} = \kappa \left( {\psi (a\bar{b}) + \psi (b\bar{a})} \right) + \eta \left( {\psi (a\bar{a}) + \psi (b\bar{b})} \right),$$
where \(2(\kappa^{2} + \eta^{2} ) = 1\) and \(\kappa > \eta > 0\). Thus, state S 1g has a larger weight of neutral determinant (the first term) than that of ionic one (the second term). The energy is
$${}^{1}E_{{1{\text{g}}}} = K_{ab} + \frac{{U - \sqrt {U^{2} + 16t_{ab}^{2} } }}{2}$$
(D) Higher singlet state (with g symmetry)
$$\varPsi_{{S_{{2{\text{g}}}} }} = - \eta \left( {\psi (a\bar{b}) + \psi (b\bar{a})} \right) + \kappa \left( {\psi (a\bar{a}) + \psi (b\bar{b})} \right),$$
where \(2(\kappa^{2} + \eta^{2} ) = 1\) and \(\kappa > \eta > 0\). In contrast to 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
$${}^{1}E_{{2{\text{g}}}} = K_{ab} + \frac{{U + \sqrt {U^{2} + 16t_{ab}^{2} } }}{2} .$$
Here, 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.
Fig. 5

Variations of \(\kappa\) and \(\eta\) as a function of r t

Using the relationship between BS orbitals {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
$$\varPsi_{{S_{{1{\text{g}}}} }} = (\kappa + \eta )\psi (g\bar{g}) + (\kappa - \eta )\psi (u\bar{u}) .$$
Thus, the diradical character 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
$$y = 1 - \frac{1}{{\sqrt {1 + \left( {\frac{U}{{4t_{ab} }}} \right)^{2} } }} = 1 - \frac{1}{{\sqrt {1 + \left( {\frac{1}{{4r_{t} }}} \right)^{2} } }}$$
The variation of 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
$$1 - y_{i} = \frac{{n_{{{\text{HONO}} - i}} - n_{{{\text{LUNO}} - i}} }}{2},$$
which represents that \(1 - y_{i}\) indicates the effective bond order concerned with bonding (HONO − 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.
Fig. 6

Variation of y as a function of \(\left| {{U \mathord{\left/ {\vphantom {U {t_{ab} }}} \right. \kern-0pt} {t_{ab} }}} \right|( \equiv {1 \mathord{\left/ {\vphantom {1 {r_{t} }}} \right. \kern-0pt} {r_{t} }})\)

3.2 Diradical Character Dependence of Excitation Energies and Properties

From Eqs. 2225b and 27, we obtain excitation energies (\(E_{{S_{{1{\text{u}}}} ,S_{{1{\text{g}}}} }}\), \(E_{{S_{{2{\text{g}}}} ,S_{{1{\text{g}}}} }}\)) and transition moments squared \(((\mu_{{S_{{1{\text{g}}}} ,S_{{1{\text{u}}}} }} )^{2} ,\;(\mu_{{S_{{1{\text{u}}}} ,S_{{2{\text{g}}}} }} )^{2} )\) (see Fig. 7a, b):
$$E_{{S_{{1{\text{u}}}} ,S_{{1{\text{g}}}} }} \equiv {}^{1}E_{{1{\text{u}}}} - {}^{1}E_{{1{\text{g}}}} = \frac{U}{2}\left\{ {1 - 2r_{K} + \frac{1}{{\sqrt {1 - \left( {1 - y} \right)^{2} } }}} \right\}$$
$$E_{{S_{{2{\text{g}}}} ,S_{{1{\text{g}}}} }} \equiv {}^{1}E_{{2{\text{g}}}} - {}^{1}E_{{1{\text{g}}}} = \frac{U}{{\sqrt {1 - \left( {1 - y} \right)^{2} } }}$$
$$(\mu_{{S_{{1{\text{g}}}} ,S_{{1{\text{u}}}} }}^{{}} )^{2} = \frac{{R_{\text{BA}}^{2} }}{2}\left\{ {1 - \sqrt {1 - \left( {1 - y} \right)^{2} } } \right\}$$
$$(\mu_{{S_{{1{\text{u}}}} ,S_{{2{\text{g}}}} }}^{{}} )^{2} = \frac{{R_{\text{BA}}^{2} }}{2}\left\{ {1 + \sqrt {1 - \left( {1 - y} \right)^{2} } } \right\} .$$
Fig. 7

a Electronic states of a two-site diradical model: three singlet states (S 1g, S 1u, S 2g) and a triplet state (T 1u). The excitation energies (\(E_{{S_{{1{\text{u}}}} ,S_{{1{\text{g}}}} }}\), \(E_{{S_{{2{\text{g}}}} ,S_{{1{\text{g}}}} }}\)) and transition moments (\(\mu_{{S_{{1{\text{g}}}} ,S_{{1{\text{u}}}} }}^{{}}\), \(\mu_{{S_{1u} ,S_{{2{\text{g}}}} }}^{{}}\)) are also shown. Note here that the transition between S 1g and S 2g is optically forbidden. b Diradical character dependences of 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 (\((\mu_{{S_{{1{\text{g}}}} ,S_{{1{\text{u}}}} }}^{\text{DL}} )^{2} \equiv (\mu_{{S_{{1{\text{g}}}} ,S_{{1{\text{u}}}} }}^{{}} )^{2} /R_{\text{BA}}^{2}\), \((\mu_{{S_{{1{\text{u}}}} ,S_{{2{\text{g}}}} }}^{\text{DL}} )^{2} \equiv (\mu_{{S_{{1{\text{u}}}} ,S_{{2{\text{g}}}} }}^{{}} )^{2} /R_{\text{BA}}^{2}\)) for r K  = 0

Here, 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 (−2t 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:

$$g = \frac{1}{{\sqrt {2(1 + S_{\text{AB}} )} }}(\chi_{\text{A}} + \chi_{\text{B}} ),{\text{ and}}\;u = \frac{1}{{\sqrt {2(1 - S_{\text{AB}} )} }}(\chi_{\text{A}} - \chi_{\text{B}} )$$
Note here that these are not the canonical MOs of the asymmetric systems when A ≠ B. Using these MOs, we can define the localized natural orbitals (LNOs), a and b,
$$a = \frac{1}{\sqrt 2 }(g + u),{\text{ and}}\;b \equiv \frac{1}{\sqrt 2 }(g - u)$$
which become the corresponding AOs, \(\chi_{\text{A}}\) and \(\chi_{\text{B}}\), at the dissociation limit. Using the LNOs, the VCI matrix for zero z-component of spin angular momentum (M s = 0, singlet and triplet) is expressed by [38],
$$\begin{aligned} \left( {\begin{array}{*{20}c} {\left\langle {a\bar{b}} \right|\hat{H}\left| {a\bar{b}} \right\rangle } & {\left\langle {a\bar{b}} \right|\hat{H}\left| {b\bar{a}} \right\rangle } & {\left\langle {a\bar{b}} \right|\hat{H}\left| {a\bar{a}} \right\rangle } & {\left\langle {a\bar{b}} \right|\hat{H}\left| {b\bar{b}} \right\rangle } \\ {\left\langle {b\bar{a}} \right|\hat{H}\left| {a\bar{b}} \right\rangle } & {\left\langle {b\bar{a}} \right|\hat{H}\left| {b\bar{a}} \right\rangle } & {\left\langle {b\bar{a}} \right|\hat{H}\left| {a\bar{a}} \right\rangle } & {\left\langle {b\bar{a}} \right|\hat{H}\left| {b\bar{b}} \right\rangle } \\ {\left\langle {a\bar{a}} \right|\hat{H}\left| {a\bar{b}} \right\rangle } & {\left\langle {a\bar{a}} \right|\hat{H}\left| {b\bar{a}} \right\rangle } & {\left\langle {a\bar{a}} \right|\hat{H}\left| {a\bar{a}} \right\rangle } & {\left\langle {a\bar{a}} \right|\hat{H}\left| {b\bar{b}} \right\rangle } \\ {\left\langle {b\bar{b}} \right|\hat{H}\left| {a\bar{b}} \right\rangle } & {\left\langle {b\bar{b}} \right|\hat{H}\left| {b\bar{a}} \right\rangle } & {\left\langle {b\bar{b}} \right|\hat{H}\left| {a\bar{a}} \right\rangle } & {\left\langle {b\bar{b}} \right|\hat{H}\left| {b\bar{b}} \right\rangle } \\ \end{array} } \right) \hfill \\ \, = \left( {\begin{array}{*{20}c} 0 & {K_{ab} } & {t_{ab(aa)} } & {t_{ab(bb)} } \\ {K_{ab} } & 0 & {t_{ab(aa)} } & {t_{ab(bb)} } \\ {t_{ab(aa)} } & {t_{ab(aa)} } & { - h + U_{a} } & {K_{ab} } \\ {t_{ab(bb)} } & {t_{ab(bb)} } & {K_{ab} } & {h + U_{b} } \\ \end{array} } \right). \hfill \\ \end{aligned}$$
Here, the matrix elements are similar to those of symmetric case Eq. 20. On the other hand, some additional and modified physical parameters are introduced to describe the asymmetric two-site system. For example, 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].
Similar to the symmetric diradical system in Sect. 3, we introduce dimensionless quantities [38],
$$\frac{{\left| {t_{ab} } \right|}}{U} \equiv r_{t} ( \ge 0),\;\;\frac{{2K_{ab} }}{U} \equiv r_{K} ( \ge 0),\;\;\frac{h}{U} \equiv r_{h} ( \ge 0),\;\frac{{U_{a} }}{{U_{b} }} \equiv r_{U} ( \ge 0),\;{\text{and}}\;\left| {\frac{{t_{ab(aa)} }}{{t_{ab(bb)} }}} \right| \equiv r_{tab} ( \ge 0)$$
We here introduce a parameter y S,
$$y_{\text{S}} = 1 - \frac{{4r_{t} }}{{\sqrt {1 + 16r_{t}^{2} } }}$$
which indicates the diradical character before introducing the asymmetricity, i.e., (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]
$$\varvec{H}_{\text{DL}} = \, \left( {\begin{array}{*{20}c} 0 & {\frac{{r_{K} }}{2}} & { - r_{t} } & { - r_{t} } \\ {\frac{{r_{K} }}{2}} & 0 & { - r_{t} } & { - r_{t} } \\ { - r_{t} } & { - r_{t} } & {1 - r_{h} } & {\frac{{r_{K} }}{2}} \\ { - r_{t} } & { - r_{t} } & {\frac{{r_{K} }}{2}} & {1 + r_{h} } \\ \end{array} } \right)$$
From this expression, the eigenvalues and eigenvectors of 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
$$\varPsi_{j} = C_{{a\bar{b},j}} \psi (a\bar{b}) + C_{{b\bar{a},j}} \psi (b\bar{a}) + C_{{a\bar{a},j}} \psi (a\bar{a}) + C_{{b\bar{b},j}} \psi (b\bar{b}) .$$
It is found that \(C_{{a\bar{b},j}} = C_{{b\bar{a},j}}\) and \(\left| {C_{{a\bar{a},j}} } \right| \ne \left| {C_{{b\bar{b},j}} } \right|\) for the asymmetric singlet states, while \(C_{{a\bar{b},{\text{T}}}} = - C_{{b\bar{a},{\text{T}}}} = 1/\sqrt 2\) and \(C_{{a\bar{a},{\text{T}}}} = C_{{b\bar{b},{\text{T}}}} = 0\) for the triplet state. Using the MOs (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
$$\varPsi_{\text{g}} = \xi \psi_{\text{G}} + \eta \psi_{\text{S}} - \zeta \psi_{\text{D}}$$
where the normalization condition, \(\xi^{2} + \eta^{2} + \zeta^{2} = 1\), is satisfied. By comparing Eq. 39 with 40, we obtain the relationships:
$$\xi = C_{{a\bar{b},{\text{g}}}} + \frac{1}{2}\left( {C_{{a\bar{a},{\text{g}}}} + C_{{b\bar{b},{\text{g}}}} } \right),\;\eta = \frac{1}{\sqrt 2 }\left( {C_{{a\bar{a},{\text{g}}}} - C_{{b\bar{b},{\text{g}}}} } \right),\;{\text{and}}\;\zeta = C_{{a\bar{b},{\text{g}}}} - \frac{1}{2}\left( {C_{{a\bar{a},{\text{g}}}} + C_{{b\bar{b},{\text{g}}}} } \right)$$
The diradical character 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
$$y_{\text{A}} \equiv n_{\text{LUNO}} = 1 - \left| {\xi - \zeta } \right|\sqrt {2 - \left( {\xi - \zeta } \right)^{2} } = 1 - \left| {C_{{a\bar{a},{\text{g}}}} + C_{{b\bar{b},{\text{g}}}} } \right|\sqrt {2 - (C_{{a\bar{a},{\text{g}}}} + C_{{b\bar{b},{\text{g}}}} )^{2} }$$
which reduces to the usual definition of y for symmetric systems:
$$y_{\text{A}} = y_{\text{S}} = 2\zeta^{2}$$
Figure 8 shows the 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).
Fig. 8

y S versus y A plots with r h  = 0.0–2.0 for r K  = 0.0 (a) and 0.8 (b)

Fig. 9

r h dependences of P N and P I for states g, k and f at y S = 0.6 for r K  = 0.0 (a), and 0.8 (b). The variations in y A for state g are also shown

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

$$E_{\text{g}} = \frac{{r_{K} }}{2}, \;\left| {\varPsi_{\text{g}} } \right\rangle = \frac{1}{\sqrt 2 }\left| G \right\rangle - \frac{1}{\sqrt 2 }\left| D \right\rangle = \frac{1}{\sqrt 2 }\left| {a\bar{b}} \right\rangle + \frac{1}{\sqrt 2 }\left| {b\bar{a}} \right\rangle ,y_{\text{A}} = 1,$$
$$\begin{aligned} E_{\text{k}} = 1 - \sqrt {r_{h}^{2} + \left( {\frac{{r_{K} }}{2}} \right)^{2} } , \left| {\varPsi_{\text{k}} } \right\rangle = \frac{1}{{\sqrt {2(1 + A^{2} )} }}\left( {\left| G \right\rangle + \sqrt 2 A\left| S \right\rangle + \left| D \right\rangle } \right) \\ = \frac{1}{{\sqrt {2(1 + A^{2} )} }}\left\{ {\left( {1 + A} \right)\left| {a\bar{a}} \right\rangle + \left( {1 - A} \right)\left| {b\bar{b}} \right\rangle } \right\}. \\ \end{aligned}$$
$$\begin{aligned} E_{\text{f}} = 1 + \sqrt {r_{h}^{2} + \left( {\frac{{r_{K} }}{2}} \right)^{2} }, \left| {\varPsi_{\text{f}} } \right\rangle = \frac{1}{{\sqrt {2(1 + B^{2} )} }}\left( {\left| G \right\rangle + \sqrt 2 B\left| S \right\rangle + \left| D \right\rangle } \right) \\ = \frac{1}{{\sqrt {2(1 + B^{2} )} }}\left\{ {\left( {1 + B} \right)\left| {a\bar{a}} \right\rangle + \left( {1 - B} \right)\left| {b\bar{b}} \right\rangle } \right\}. \\ \end{aligned}$$

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

$$E_{\text{g}} = E_{\text{k}} = \frac{{r_{K} }}{2}, \;E_{\text{f}} = 2 - \frac{{r_{K} }}{2}$$
For \(r_{h}^{2} > 1 - r_{K}\),
$$\begin{aligned} E_{\text{g}} = 1 - \sqrt {r_{h}^{2} + \left( {\frac{{r_{K} }}{2}} \right)^{2} }, \\ \left| {\varPsi_{\text{g}} } \right\rangle = \frac{1}{{\sqrt {2(1 + A^{2} )} }}\left( {\left| G \right\rangle + \sqrt 2 A\left| S \right\rangle + \left| D \right\rangle } \right) \\ = \frac{1}{{\sqrt {2(1 + A^{2} )} }}\left\{ {\left( {1 + A} \right)\left| {a\bar{a}} \right\rangle + \left( {1 - A} \right)\left| {b\bar{b}} \right\rangle } \right\}, \\ y_{\text{A}} = 1 - \frac{2A}{{1 + A^{2} }}. \\ \end{aligned}$$
$$\begin{aligned} E_{\text{k}} = \frac{{r_{K} }}{2}, \hfill \\ \left| {\varPsi_{\text{k}} } \right\rangle = \frac{1}{\sqrt 2 }\left| G \right\rangle - \frac{1}{\sqrt 2 }\left| D \right\rangle = \frac{1}{\sqrt 2 }\left| {a\bar{b}} \right\rangle + \frac{1}{\sqrt 2 }\left| {b\bar{a}} \right\rangle \hfill \\ \end{aligned}$$
$$\begin{aligned} E_{\text{f}} = 1 + \sqrt {r_{h}^{2} + \left( {\frac{{r_{K} }}{2}} \right)^{2} } \\ \left| {\varPsi_{\text{f}} } \right\rangle = \frac{1}{{\sqrt {2(1 + B^{2} )} }}\left( {\left| G \right\rangle + \sqrt 2 B\left| S \right\rangle + \left| D \right\rangle } \right) \\ = \frac{1}{{\sqrt {2(1 + B^{2} )} }}\left\{ {\left( {1 + B} \right)\left| {a\bar{a}} \right\rangle + \left( {1 - B} \right)\left| {b\bar{b}} \right\rangle } \right\}. \\ \end{aligned}$$
$$A \equiv \frac{{r_{K} + \sqrt {4r_{h}^{2} + r_{K}^{2} } }}{{2r_{h} }}\left( { > 0} \right){\text{ and}}\;B \equiv \frac{{r_{K} - \sqrt {4r_{h}^{2} + r_{K}^{2} } }}{{2r_{h} }} \left( { < 0} \right).$$
For \(r_{h}^{2} < 1 - r_{K}\), states 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).
The variations in 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. 44a44h). 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 |Δµ 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. 44a44c). 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. 44a44c), the feature of which decreases |Δµ ii | (i = k, f) and increases |µ kf|.
Fig. 10

r h dependence of the dimensionless excitation energies (E ij ), dimensionless dipole moment differences (Δµ ii ) and dimensionless transition moment amplitudes (|µ ij |) at y S = 0.6 for r K  = 0.0 (a) and 0.8 (b)

For the ground-state singlet–triplet energy gap, \(E_{\text{gT}}\)(\(\equiv E_{\text{g}} - E_{\text{T}}\)) (see Fig. 11), it is found (a) that the increase of 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 Sr K region.
Fig. 11

Dimensionless E gT contours on the y S-r K plane for r h  = 0.0 (a), 0.8 (b) and 1.4 (c). The variation from cold to warm color indicates that from negative to positive E gT values. The black solid line E gT contours range from −3.0 to 3.0 with division 0.2 and 0.0 contour is shown by a black dashed-dotted line. The black dashed lines represent the iso-y A lines

5 Relationship between Open-Shell Character and Optical Response Properties

The unique properties of excitation energies, transition moments, and dipole moment differences for open-shell singlet systems, i.e., their dependence on diradical character, cause a strong correlation of the optical response properties to the diradical character. This is understood by the perturbation analysis of those optical response properties. In general, the microscopic polarization p, which is defined by the difference between the induced dipole moment μ and permanent dipole moment μ0, is expanded by using the applied electric field F [40, 41, 42]:
$$\begin{aligned} p^{i} = \mu^{i} - \mu_{0}^{i} \\ = \sum\limits_{j} {\alpha_{ij} F^{j} (\omega_{1} )} + \sum\limits_{jk} {\beta_{ijk} F^{j} (\omega_{1} )F^{k} (\omega_{2} )} + \sum\limits_{jkl} {\gamma_{ijkl} F^{j} (\omega_{1} )F^{k} (\omega_{2} )F^{l} (\omega_{3} )} + \cdots \\ \end{aligned}$$
Here, F i (ω l ) indicates the ith component (i = x, y, z) of local electric field with frequency ω l . The coefficient of each term indicates the optical responsibility of the nth 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:
$$\alpha_{ii} = 2\sum\limits_{n \ne 0} {\frac{{(\mu_{0n}^{i} )^{2} }}{{E_{n0} }}} ,$$
$$\beta_{iii} = 3\sum\limits_{n \ne 0} {\frac{{(\mu_{0n}^{i} )^{2} \Delta \mu_{nn}^{i} }}{{E_{n0}^{2} }}} ,$$
$$\gamma_{iiii} = 4\left\{ {\sum\limits_{n \ne 0} {\frac{{(\mu_{0m}^{i} )^{2} (\bar{\mu }_{mm}^{i} )^{2} }}{{E_{m0}^{3} }}} } \right. - \sum\limits_{n,m \ne 0} {\frac{{(\mu_{0m}^{i} )^{2} (\mu_{n0}^{i} )^{2} }}{{E_{m0}^{{}} E_{n0}^{2} }}} \left.\,+\,{2\sum\limits_{m \ne n} {\frac{{\mu_{0m}^{i} \Delta \mu_{mm}^{i} \mu_{mn}^{i} \mu_{n0}^{i} }}{{E_{m0}^{2} E_{n0}^{{}} }} + \sum\limits_{m \ne n} {\frac{{\mu_{0m}^{i} \mu_{mn}^{i} \mu_{nq}^{i} \mu_{q0}^{i} }}{{E_{m0}^{2} E_{n0}^{{}} }}} } } \right\}$$
Here, \(E_{n0}^{{}}\) indicates the excitation energy of the nth excited state; \(\mu_{mn}^{i}\) indicates the transition moment between the mth and nth states; \(\Delta \mu_{mm}^{i}\) indicates the dipole moment difference between the mth 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. 2932, 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):
$$\gamma = -4\frac{{(\mu_{\text{gk}}^{{}} )^{4} }}{{(E_{\text{kg}}^{{}} )^{3} }} + 4\frac{{(\mu_{\text{gk}} )^{2} (\mu_{\text{kf}} )^{2} }}{{(E_{\text{kg}}^{{}} )^{2} E_{\text{fg}}^{{}} }}$$
and can be expressed as a function of effective bond order q = 1–y [4, 7]:
$$\begin{aligned} \frac{\gamma }{{\left( {{{R_{\text{BA}}^{4} } \mathord{\left/ {\vphantom {{R_{\text{BA}}^{4} } {U^{3} }}} \right. \kern-0pt} {U^{3} }}} \right)}} = \frac{{\gamma^{\text{II}} }}{{\left( {{{R_{\text{BA}}^{4} } \mathord{\left/ {\vphantom {{R_{\text{BA}}^{4} }{U^{3} }}} \right. \kern-0pt}{U^{3} }}} \right)}} + \frac{{\gamma^{{{\text{III}} - 2}} }}{{\left( {{{R_{\text{BA}}^{4} } \mathord{\left/ {\vphantom {{R_{\text{BA}}^{4} }{U^{3} }}} \right. \kern-0pt}{U^{3} }}} \right)}} \\ = - \frac{{8q^{4} }}{{\left( {1 + \sqrt {1 - q^{2} } } \right)^{2} \left( {1 - 2r_{K} + \frac{1}{{\sqrt {1 - q^{2} } }}} \right)^{3} }} + \frac{{4q^{2} }}{{\left( {1 - 2r_{K} + \frac{1}{{\sqrt {1 - q^{2} } }}} \right)^{2} \frac{1}{{\sqrt {1 - q^{2} } }}}}. \\ \end{aligned}$$
The first and the second terms, which are referred to as type II and III-2 virtual excitation processes [43, 44], respectively, are shown to be negative and positive contributions to total 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 closedshell NLO systems.
Fig. 12

Diradical character dependence of \(\gamma^{\text{DL}} ( = \gamma /(R_{\text{BA}}^{4} /U^{3} ))\), \(\gamma^{\text{II DL}} ( = \gamma^{\text{II}} /(R_{\text{BA}}^{4} /U^{3} ))\) and \(\gamma^{\text{III - 2 DL}} ( = \gamma^{\text{III - 2}} /(R_{\text{BA}}^{4} /U^{3} ))\) in the case of \(r_{K}\) = 0

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

6.1 Indenofluorenes

The relationship between open-shell character and other traditional chemical concepts like aromaticity is useful for deeper understanding of the open-shell singlet electronic structures, as well as for its application to constructing design guidelines for highly efficient functional molecular systems. Indeed, the chemical and physical tuning schemes of the diradical character have been obtained by revealing the relationships between the open-shell character and the traditional chemical concepts/indices that most chemists are familiar with [16, 18]. Among the chemical concepts/indices, “aromaticity” is one of the most essential and intuitive concepts relating to open-shell character for the chemists [46, 47, 48] since it is well-known that anti-aromatic systems have small energy gaps between the HOMO and the LUMO [47], which tend to increase the diradical character as shown in Eq. 27. Also, to clarify the structure–property relationship based on the diradical character and aromaticity, we have to reveal the spatial correlation between the open-shell character and aromaticity. In this section, we show the spatial correlation between the open-shell character, aromaticity, and the second hyperpolarizability (the third-order NLO response property at the molecular scale) by focusing on 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 20π 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.
Fig. 13

Molecular frameworks of para- and meta-type indenofluorenes (a and b, respectively)

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

The geometries of 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:
$$y = n_{\text{LUNO}} = 2 - n_{\text{HONO}} .$$
Note here that the spin-projection scheme (see Eqs. 14, 15) is not applied in this case since the LC-UBLYP (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]
$$\rho_{\text{odd}} ({\mathbf{r}}) = n_{\text{LUNO}} \left( {\left| {\phi_{\text{HONO}} ({\mathbf{r}})} \right|^{2} + \left| {\phi_{\text{LUNO}} ({\mathbf{r}})} \right|^{2} } \right)$$
This contributes to the diradical character y as expressed by
$$y = \frac{1}{2}\int {d{\mathbf{r}}\rho_{\text{odd}} ({\mathbf{r}})} = \frac{1}{2}\int {d{\mathbf{r}}\left[ {n_{\text{LUNO}} \left( {\left| {\phi_{\text{HONO}} ({\mathbf{r}})} \right|^{2} + \left| {\phi_{\text{LUNO}} ({\mathbf{r}})} \right|^{2} } \right)} \right]}$$
In order to estimate the aromaticity, we here employ the magnetic shielding tensor calculated using the gauge invariant atomic orbital (GIAO) method [54]. Since the examined systems have planar structures (zx 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]
$$\left. {\rho_{zzz}^{(3)} (r) = \frac{{\partial^{3} \rho (r)}}{{\partial F_{z}^{3} }}} \right|_{{{\mathbf{F}} = 0}}$$
Here, \(\rho (r)\) is the total electron density at the position r, and F z is the z component of the external electric field F. This γ zzzz value is obtained by γ zzzz density as
$$\gamma_{zzzz} = - \frac{1}{3!}\int {r_{z} \rho_{zzz}^{(3)} (r)} {\text{d}}r$$
This relationship indicates that a pair of positive and negative 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

The open-shell singlet character and resonance structure are shown in Fig. 14 for each indenofluorene system. It is found that the system involving 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.
Fig. 14

Resonance structures with Clar’s sextets (indicated by the delocalized benzene-ring forms) for para- (a) and meta- (b) type indenofluorenes. Diradical character y of each indenofluorene system is calculated at LC-UBLYP/6-311 + G**//U(R)B3LYP/6-311 + G** level of theory

The local aromaticity is clarified using the magnetic shielding tensor component (–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.
Fig. 15

σ yy Maps (left) and odd-electron densities (right) of para (a) and meta (b) systems in the singlet states calculated at the LC-UBLYP/6-311 + G** level of theory (contour values of 0.0004 a.u. (para) and 0.004 a.u. (meta) for the odd-electron density distributions)

In order to confirm further the correlation between odd-electron density and local aromaticity, let us consider the triplet states of the 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].
Fig. 16

σ yy Maps (left) and odd-electron densities (right) of para (a) and meta (b) systems in the triplet states calculated at the LC-UBLYP/6-311 + G** level of theory (contour value of 0.004 a.u. for the odd-electron density distributions)

6.4 Second Hyperpolarizaibilities of Indenofluorene systems

We here show the impact of the diradical character and the aromaticity on the molecular functionality, i.e., the longitudinal components of second hyperpolarizabilities 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 yγ 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.
Fig. 17

γ zzzz Density maps (contour value of 1000 a.u.) for para (a) and meta (b) systems in the singlet (left) and triplet (right) states calculated at the LC-UBLYP/6-311 + G** level of theory. The γ zzzz values are also shown

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

As shown in Eq. 51, diradical character 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 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.
Fig. 18

Structure of dicyclopenta-fused oligoacenes (DPA[N], N = 0–3) together with the resonance structures for N = 3

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].

Figure 19 shows the diradical character 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.
Fig. 19

Diradical character y at the UCCSD and UDFT levels of approximation versus y at PUHF level of approximation for DPA[N] (N = 0, 1, 2, 3). The 6-31G basis set is used

Table 1

Longitudinal polarizability α [a.u.] for DPA[N] up to N = 2



DPA [1]

DPA [2]





























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 yγ 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.

First, we briefly describe the approximate spin-projected method based on the NOs and occupation numbers obtained by spin-unrestricted single determinantal schemes. The chemicophysical properties like the optical responses of N-electron systems are calculated using the one-electron reduced density:
$$d({\mathbf{r}}) = \sum\limits_{i = 0}^{N/2 - 1} {[n_{{{\text{HONO}} - i}}^{{}} \phi_{{{\text{HONO}} - i}}^{*} ({\mathbf{r}})\phi_{{{\text{HONO}} - i}}^{{}} ({\mathbf{r}}) + n_{{{\text{LUNO}} + i}}^{{}} \phi_{{{\text{LUNO}} + i}}^{*} ({\mathbf{r}})\phi_{{{\text{LUNO}} + i}}^{{}} ({\mathbf{r}})]} ,$$
where {\(\phi_{k} ({\mathbf{r}})\)} is the kth 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):
$$n_{{{\text{HONO}} - i}}^{\text{ASP}} = \frac{{\left( {n_{{{\text{HONO}} - i}} } \right)^{2} }}{{1 + \left( {T_{i} } \right)^{2} }} \equiv 2 - y_{i}^{\text{ASP}} ,\;{\text{and}}\;n_{{{\text{LUNO}} + i}}^{\text{ASP}} = \frac{{\left( {n_{{{\text{LUNO}} + i}} } \right)^{2} }}{{1 + \left( {T_{i} } \right)^{2} }} \equiv y_{i}^{\text{ASP}}$$
T i, the overlap between the corresponding orbital pairs, is expressed by
$$T_{i}^{{}} = \frac{{n_{{{\text{HONO}} - i}}^{{}} - n_{{{\text{LUNO}} + i}}^{{}} }}{2} .$$
Here, \(y_{i}^{\text{ASP}}\) is the spin-projected diradical character [0 (closed-shell) ≤ \(y_{i}^{\text{ASP}}\) ≤ 1 (pure diradical)] defined by Yamaguchi [28, 29]. Using Eqs. 57 and 58, the approximate spin-projected (ASP) polarizability (\(\alpha_{ij}^{\text{ASP}}\)) and the ASP second hyperpolarizability (\(\gamma_{ijkl}^{\text{ASP}}\)) (i, j, k, l = x, y, z) are expressed by [31]
$$\alpha_{ij}^{\text{ASP}} = - \int {r_{i} d_{j}^{{{\text{ASP}}(1)}} ({\mathbf{r}})d{\mathbf{r}}}$$
$$\gamma_{ijkl}^{\text{ASP}} = - \frac{1}{3!}\int {r_{i} d_{jkl}^{{{\text{ASP}}(3)}} ({\mathbf{r}})d{\mathbf{r}}}$$
Here, \(d_{j}^{{{\text{ASP}}(1)}} ({\mathbf{r}})\) and \(d_{jkl}^{{{\text{ASP}}(3)}} ({\mathbf{r}})\) indicate the ASP polarizability and the ASP second hyperpolarizability densities [44], respectively:
$$d_{j}^{{{\text{ASP}}(1)}} ({\mathbf{r}}) = \left. {\frac{{\partial d^{\text{ASP}} ({\mathbf{r}})}}{{\partial F_{j} }}} \right|_{{{\mathbf{F}} = 0}}$$
$$d_{jkl}^{{{\text{ASP}}(3)}} ({\mathbf{r}}) = \left. {\frac{{\partial^{3} d^{\text{ASP}} ({\mathbf{r}})}}{{\partial F_{j} \partial F_{k} \partial F_{l} }}} \right|_{{{\mathbf{F}} = 0}}$$
Here, ASP one-electron reduced density is expressed, using Eqs. 56 and 57, as [31]
$$d^{\text{ASP}} ({\mathbf{r}}) = \sum\limits_{i = 0}^{N/2 - 1} {[n_{{{\text{HONO}} - i}}^{\text{ASP}} \phi_{{{\text{HONO}} - i}}^{*} ({\mathbf{r}})\phi_{{{\text{HONO}} - i}}^{{}} ({\mathbf{r}}) + n_{{{\text{LUNO}} + i}}^{\text{ASP}} \phi_{{{\text{LUNO}} + i}}^{*} ({\mathbf{r}})\phi_{{{\text{LUNO}} + i}}^{{}} ({\mathbf{r}})]} .$$
The PQM model, which is a prototypical open-shell singlet model molecule, has a contribution from mixing two resonance forms—the diradical (open-shell) and quinoid (closed-shell) forms (see Fig. 20). The optimized structure with 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.
Fig. 20

p-Quinodimethane (PQM) model together with the resonance structures

7.2.2 Polarizability

First, we consider 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.
Fig. 21

Diradical character versus α [a.u.] of the PQM model calculated by the non-spin-projected (a) and the approximate spin-projected (ASP) (b) UDFT methods in comparison with the UCCSD(T) reference method

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) yα 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

First, we show the non-spin-projected yγ curves of PQM calculated with hybrid and LC functionals [70]. The UCCSD(T) yγ 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) yγ 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.
Fig. 22

Diradical character versus γ [a.u.] of the PQM model calculated by the non-spin-projected (a) and the approximate spin-projected (ASP) (b) UDFT methods in comparison with the UCCSD(T) reference method

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].

Figure 23a shows the electron density differences [\(\Delta d({\mathbf{r}}) \equiv d^{\text{ASP}} ({\mathbf{r}}) - d^{\text{NSP}} ({\mathbf{r}})\)] for 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.
Fig. 23

Electron density differences [\(\Delta d({\mathbf{r}}) \equiv d^{\text{ASP}} ({\mathbf{r}}) - d^{\text{NSP}} ({\mathbf{r}})\)] (a), α densities [\(d^{{{\text{ASP}}(1)}} ({\mathbf{r}})\)] and the differences between ASP and NSP α densities [\(\Delta d^{(1)} ({\mathbf{r}}) \equiv d^{{{\text{ASP}}(1)}} ({\mathbf{r}}) - d^{{{\text{NSP}}(1)}} ({\mathbf{r}})\)] (b), and γ densities [\(d^{{{\text{ASP}}(3)}} ({\mathbf{r}})\)] and the ASP-NSP differences [\(\Delta d^{(3)} ({\mathbf{r}}) \equiv\) \(d^{{{\text{ASP}}(3)}} ({\mathbf{r}})\) \(- d^{{{\text{NSP}}(3)}} ({\mathbf{r}})\)] (c) at y = 0.257 (left), 0.491 (center) and 0.731 (right) at LC-UBLYP (µ = 0.47)/6-31G* + p(ζ = 0.0523) level of approximation. The yellow and blue surfaces indicate positive and negative densities, respectively, with iso-surfaces: ±0.0003 a.u. for \(\Delta \rho ({\mathbf{r}})\), ±0.1 a.u. for \(d^{{{\text{ASP}}(1)}} ({\mathbf{r}})\), ±0.01 a.u. for \(\Delta d^{(1)} ({\mathbf{r}})\), and ±100 a.u. for \(d^{{{\text{ASP}}(3)}} ({\mathbf{r}})\) and \(\Delta d^{(3)} ({\mathbf{r}})\)

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 yγ 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

As shown in Fig. 1b, diphenalenyl compound such as IDPL (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 × 103 a.u. and 194 × 103 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.
Fig. 24

Spin density distribution of IDPL at UBHandHLYP/6-31G* level of approximation. The yellow and blue meshes represent α and β spin densities with iso-surface 0.01 a.u., respectively. These spin densities represent spatial spin correlations between α and β spin

Fig. 25

γ density distributions for IDPL (a) and PY2 (b) at the BHandHLYP level. The yellow and blue meshes represent positive and negative γ densities with iso-surfaces with ±100 a.u

Another type of diphenalenyl diradicaloids, there are linked-type phenalenyl radical compounds, where two phenalenyl radical rings are bridged with acetylene or vinylene linkers. As examples, we consider acetylene-linked compounds, 1,2-bis(phenalen-1-ylidene)ethene (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 yγ 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.
Fig. 26

Structures of 1,2-bis(phenalen-1-ylidene)ethene (1S) and 1,2-bis(phenalen-2-yl)ethyne (2S), as well as the closed-shell reference 1,2-bis(pyren-4-yl)ethyne (3S). The molecular geometries are optimized at (U)B3LYP/6-31G** level of approximation under the constraint of C 2h symmetry. The coordinate axis is also shown

Table 2

Diradical character y calculated at PUHF/6-31G** level and longitudinal components of second hyperpolarizabilities γ(γ xxxx ) for 1S3S (Fig. 26) calculated by the BHandHLYP/6-31G* method [75]



γ [×103 a.u.]

1S (singlet)



2S (singlet)






1S (triplet)


2S (triplet)


The IDPL is known to make a needle-like one-dimensional (1D) crystal, where the monomer is arranged in a slipped stack form (see Fig. 27) [10]. This exhibits an unusually short ππ 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].
Fig. 27

Structures of IDPL dimer (R = H) (a) extracted from a single crystal (R = Ph) [10] and PY2 dimer (b). The average distance between cofacial phenalenyl rings of IDPL dimer is 3.137 Å, while that of PY2 dimer is fixed to be a typical van der Waals distance 3.4 Å. The structure of PY2 monomer is optimized by the RB3LYP/6-31G** method. The coordinate axis is also shown

Table 3

Longitudinal γ (γ zzzz ) values [×103 a.u.] per monomer (γ/N, N: the number of monomers) for IDPL monomer and dimer as well as PY2 monomer and dimer models (see Fig. 27) calculated by the BHandHLYP/6-31G* method [76]


IDPL dimer








$The γ values of IDPL systems are calculated by the UBHandHLYP/6-31G* method, while those of PY2 systems are done by the RBHandHLYP/6-31G* method

Fig. 28

UNO frontier orbitals (HONO-1, HONO, LUNO, and LUNO + 1) of the IDPL dimer model. The yellow and blue surfaces represent positive and negative MOs with iso-surfaces with ±0.025 a.u., respectively

Fig. 29

Mulliken spin densities of IDPL monomer and dimer at UBHandHLYP/6-31G* level of approximation. The white and black circles represent α and β spin densities, respectively

Fig. 30

γ (γ zzzz ) density distributions of the dimer (a) as well as γ density difference (\(\rho_{\text{diff}}^{ (3 )} ({\mathbf{r}}) = \rho_{\text{int}}^{ (3 )} ({\mathbf{r}}) - \rho_{\text{non - int}}^{ (3 )} ({\mathbf{r}})\)) (b) for the IDPL dimer model. The yellow and blue meshes represent positive and negative densities with iso-surface ±500 a.u., respectively

8.2 Graphene Nanoflakes (GNFs)

8.2.1 Rectangular GNFs

First, we consider rectangular graphene nanoflake PAH[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 × 103 a.u.) as compared to γ yyyy (145 × 103 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 × 103 a.u.) to the triplet (γ yyyy  = 35.0 × 103 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 × 104 a.u.) are shown to be more than three times as large as those of A-HGNF (41.7 × 104 a.u.).
Fig. 31

Odd electron (a) and γ (γ xxxx and γ yyyy ) density (b) distributions of PAH[3,3] (singlet) at UBHandHLYP/6-31G* level of approximation. y 0 (PUHF/6-31G*) and γ values are shown with Cartesian axes. Yellow and blue meshes indicate positive and negative densities (with contours of 0.003 a.u. for odd electron density and ±200 a.u. for γ density), respectively

Fig. 32

Primary resonance structures Clar’s sextets (solid benzene rings) in resonance structures of Z-HGNF (a) and of A-HGNF (b)

The size dependences of the diradical characters 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 yγ 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.
Table 4

Diradical characters y 0 and y 1 [-] (at PUHF/6-31G* level) and γ xxxx and γ yyyy values [×104 a.u.] (at UBHandHLYP/6-31G* level) for PAH[X,Y] (1 ≤ X ≤ 7, 1 ≤ Y ≤ 7) in the singlet states [95, 96]



y 0

y 1

γ yyyy

γ xxxx









































































































































































Fig. 33

Variation in diradical character y i (i = 0, 1) (at PUHF/6-31G* level) and γ (γ xxxx and γ yyyy ) (at ASP-UBHandHLYP/6-31G* level) for PAH[X,Y] with Y = 1 (a), 3 (b), 5 (c), and 7 (d) as a function of X

Fig. 34

Odd electron density [\(D_{{y_{0} }}^{\text{odd}} ({\mathbf{r}})\) and \(D_{{y_{1} }}^{\text{odd}} ({\mathbf{r}})\)] distributions (at ASP-LC-UBLYP (μ = 0.47)/6-31G* level) with the iso-surfaces of 0.0015 a.u. for \(D_{{y_{0} }}^{\text{odd}} ({\mathbf{r}})\) of PAH[3,3] (a), 0.0003 a.u. for \(D_{{y_{1} }}^{\text{odd}} ({\mathbf{r}})\) of PAH[3,3] (b), and 0.0025 a.u. for \(D_{{y_{0} }}^{\text{odd}} ({\mathbf{r}})\) (c) and \(D_{{y_{1} }}^{\text{odd}} ({\mathbf{r}})\) (d) of PAH[6,7]. γ yyyy density [\(\rho_{yyy}^{(3)} ({\mathbf{r}})\)] distributions (at ASP-LC-UBLYP (μ = 0.47)/6-31G* level) (e, f), where the yellow and blue meshes represent positive and negative densities with iso-surface ± 300 a.u., respectively

8.2.2 Rhombic and Bow-Tie GNFs

Another structural dependences of open-shell character is introduced for two types of GNFs composed of two phenalenyl rings, rhombic (a) and bow-tie (b) [97], where the two phenalenyl units are linked in a different manner, i.e., linked via their sides (a) and vertices (b) (see Fig. 35). Although both systems have open-shell singlet states, the diradical character (at PUHF/6-31G* level) for rhombic GNF (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.
Fig. 35

Resonance forms of rhombic (a) and bow-tie (b) GNFs

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

As another example of linked form dependence of open-shell character and 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 × 104 a.u.) is more than twice as large as that of 2-NAL (13.4 × 104 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)].
Fig. 36

Structures of alternately linked (AL) (a) and nonalternately linked (NAL) (b) systems together with their primary resonance structures

Next, we consider the size dependence of 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.
Fig. 37

Size dependences of the y i values (0 ≤ i ≤ 4) in N-AL (a) and N-NAL systems (b) (2 ≤ N ≤ 10), where N = 2n + 2. The y i values are calculated from the LC-UBLYP/6-31G* NO occupation numbers

Figure 38 shows the size dependences of 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 × 104 a.u.) is more than 30 times as large as that of the analogous NAL system (392 × 104 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.
Fig. 38

Size dependences of the longitudinal γ values in N-AL and N-NAL systems (2 ≤ N ≤ 10) in their singlet and highest spin [(N + 1)-multiplet] states. The γ values are calculated using the LC-UBLYP/6-31G* method

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.

First, we consider static electric field application to IDPL and PY2 (Fig. 39a, b) [106], the former and the latter of which are diradicaloid and closed-shell systems, respectively. The amplitude of the static electric field along the longitudinal (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 x  = 9.88 a.u.), which is significantly larger than that in the pyrene rings of PY2 (μ 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 × 106 a.u. (IDPL) vs. 1.743 × 105 a.u. (PY2) at F = 0.0 a.u. and γ = 1.456 × 1010 a.u. (IDPL) vs. 3.107 × 105 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 × 104 (F = 0.0077 a.u.).
Fig. 39

Molecular structures (grey carbon, blue nitrogen, red oxygen, white hydrogen) of IDPL (a), PY2 (b), DA-IDPL [(NO2)2-IDPL-(NH2)2] (c) and DA-PY2 [(NO2)2-PY2-(NH2)2] (d) (in absence of static electric field) optimized by the (U)B3LYP/6-31G* method. The diradical characters (y F=0) in absence of a field calculated by the LC-UBLYP/6-31G* method are also shown

Fig. 40

Static electric field F effect on γ xxxx [a.u.] of IDPL and PY2 at LC-UBLYP/6-31G* level of theory

Second, we consider donor (NH2)-acceptor (NO2) substitution into PAHs, i.e., DA-IDPL [(NO2)2-IDPL-(NH2)2] and DA-PY2 [(NO2)2-PY2-(NH2)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 × 107 a.u., in comparison to γ = 5.501 × 107–1.278 × 108 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 × 105 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 × 107 a.u.) is about 88 times larger than that of DA-PY2 (9.546 × 105 a.u.), the ratio of which is strongly enhanced as compared to the non-substituted case at F = 0: γ(IDPL)/γ(PY2) = 10.
Fig. 41

Comparison of γ values and diradical characters between nonsubstituted PY2 (closes-shell) and IDPL (diradicaloid), and donor (NH2)-acceptor (NO2) disubstituted PY2 and IDPL

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 (NH2)/acceptor (NO2) 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

As shown in previous sections, the diradical character is not an observable index, but a chemical/physical index for bond nature and electron correlation. From the experimental side, the diradical nature is qualitatively estimated by using the molecular structure, e.g., quinoid vs. benzenoid forms, optical absorption spectrum, singlet–triplet energy gap, and so on. However, the quantitative evaluation scheme of the diradcial character based on the experimental measurements has not been proposed yet. We have firstly proposed an approximate evaluation scheme of the 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]
$$y = 1 - \sqrt {1 - \left( {\frac{{{}^{1}E_{{ 1 {\text{u}}}} - {}^{3}E_{{ 1 {\text{u}}}} }}{{{}^{1}E_{{ 2 {\text{g}}}} - {}^{1}E_{{ 1 {\text{g}}}} }}} \right)^{2} } = 1 - \sqrt {1 - \left( {\frac{{\Delta E_{\text{S(u)}} - \Delta E_{\text{T}} }}{{\Delta E_{\text{S(g)}} }}} \right)^{2} }$$
where the energies in the first right-hand side (rhs) are concerned with the four electronic states shown in Sect. 3.1. \(\Delta E_{{{\text{S}}({\text{g}})}}\) \(( \equiv {}^{1}E_{{ 2 {\text{g}}}} - {}^{1}E_{{ 1 {\text{g}}}} )\), \(\Delta E_{{{\text{S}}({\text{u}})}}\) \(( \equiv {}^{1}E_{{ 1 {\text{u}}}} - {}^{1}E_{{ 1 {\text{g}}}} )\) and \(\Delta E_{\text{T}}\) \(( \equiv {}^{3}E_{{ 1 {\text{u}}}} - {}^{1}E_{{ 1 {\text{g}}}} )\) in the second rhs represent the excitation energies of the higher singlet state of g symmetry (two-photon allowed excited state), of the lower singlet state with u symmetry (one-photon allowed excited state), and of the triplet state with u symmetry, respectively. Namely, \(\Delta E_{{{\text{S}}({\text{u}})}}\) and \(\Delta E_{{{\text{S}}({\text{g}})}}\) are obtained from the lowest-energy peaks of the one- and two-photon absorption spectra, respectively, while \(\Delta E_{\text{T}}\) is obtained from phosphorescence and ESR measurement. Figure 42 shows the experimental 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
Fig. 42

The diradical character (y expr) deduced from experimetal results versus that theoretical one (y theor) obtained from PUHF/6-31G** calculations for several compounds [108]

10 Summary

This chapter focus on the theoretical aspects of electronic structures of open-shell molecular systems including symmetric and asymmetric di/multi-radicaloids and open-shell molecular aggregates from the viewpoint of diradical character. Also, on the basis of the two-electron two-site valence configuration interaction model, the excitation energies and properties (dipole moment differences and transition moments), as well as nonlinear optical (NLO) responses are found to be strongly correlated to the diradical character. In particular, we found (a) that the first optically allowed excitation energy tends to decrease and then increase as increasing the dirdaical character and (b) that the second hyperpolarizability (third-order NLO property at the molecular scale) is maximized in the intermediate diradical character region. These properties are useful for building the design guidelines for tuning the optical absorption spectra and NLO properties including two-photon absorption. Several realistic molecules and aggregates are designed based on these principles, and their electronic structures, as well as functionalities are analyzed by first-principles quantum chemical calculations. Recently, some of our designed molecules and a new class of stable open-shell singlet systems have been synthesized and their unique properties and high functionalities are confirmed by experiments [3, 9, 12, 13, 14, 15, 109, 110, 111, 112, 113, 114, 115, 116]. Our design principle for functional systems based on the open-shell character is illustrated in Fig. 43, which indicates that each category classified based on the diradical character, i.e., closes-shell, intermediate open-shell, and pure open-shell regimes, characterizes the functionalities (structure, reactivity, and property). If we could clarify the relationship between the diradical character and these functionalities, we could construct design guidelines for controlling such functionalities. In this point, this strategy has advantage over conventional one because the diradical character is an chemical index for effective bond nature in the ground state and can be easily connected to conventional chemical concepts, e.g., resonance structure and aromaticity, which most chemists are familiar with. Indeed, the second hyperpolarizability mentioned in this chapter is described, in principle, by excitation energies, transition properties and dipole moment differences for all the electronic states including ground and excited states, which are very complicated and are hard to be used for constructing useful design guidelines. Our strategy can clarify the relationship between hyperpolarizability and diradical character, and thus succeeds in presenting familiar design guidelines for enhancing/tuning the hyperpolarizbaility by tuning the molecular architecture, symmetry, aromaticity, substitution of atom species and so on based on the resonance structures and Clar’s sextet rule. Namely, such diradical character based method is useful for revealing the mechanism of the functionality and providing its simple physicochemical pictures, as well as for constructing practical molecular design guidelines for various functionalities including NLO properties, singlet fission and exciton migration [16, 18, 117, 118].
Fig. 43

Diradical character based molecular design for functional molecules



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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Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Department of Materials Engineering ScienceGraduate School of Engineering Science, Osaka UniversityToyonakaJapan
  2. 2.Center for Spintronics Research Network (CSRN)Graduate School of Engineering Science, Osaka UniversityToyonakaJapan

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