Theoretical Study on One- and Two-Photon Absorption Properties of π-Stacked Multimer Models of Phenalenyl Radicals

Abstract

Effects of the number of monomers (N) on the two-photon absorption (TPA) properties of π-stacked multimer models consisting of phenalenyl radicals were investigated theoretically. We conducted spectral simulations for the π-stacked N-mer models (N = 2, 4, and 6) with different stacking distances (d₁) and their alternation patterns (d₂/d₁). Excitation energies and transition dipole moments were calculated at the extended multi-configurational quasi-degenerate second-order perturbation theory (XMC-QDPT2) level based on the complete active space self-consistent field (CASSCF) wavefunctions with the active space orbitals constructed from the singly occupied molecular orbitals (SOMOs) of monomers. The TPA cross-section value per dimer unit at the first peak, originating from the electronic transition along the stacking direction, was predicted to increase significantly as the d₂/d₁ approaches one, as the d₁ decreases, and as the N increases from 2 to 6. These tendencies are similar to the calculation results for the static hyperpolarizabilities.

1. Introduction

Recently, stable open-shell singlet systems involving singlet diradicals and multiradicals have attracted intense attention as a novel class of functional molecular materials [1,2,3,4], although they usually exhibit high reactivity originating from partially or completely unpaired electrons. Molecular design for maintaining the high activities of unpaired electrons while increasing the kinetic and thermodynamic stabilities is indispensable for utilizing the singlet diradicals and multiradicals [1]. Understanding the structure–property relationships for open-shell singlet systems has been a hot topic in materials science. Diradical character (y) is a valuable chemical index characterizing the degree of open-shell in the singlet ground (S₀) state [5,6,7], taking a value from 0 (closed-shell state) to 1 (pure diradical state). Since the y of a molecular system can be easily evaluated by quantum chemical calculations, the structure–property relationships for open-shell singlet systems have been investigated in terms of y.

Nakano and coworkers theoretically predicted that the third-order nonlinear optical (NLO) properties of open-shell singlet systems can be characterized well by y [8,9,10,11,12]. They derived y-dependent expressions of the static second hyperpolarizabilities (γ) for symmetric and asymmetric diradical systems. They found that these quantities are enhanced for systems with intermediate y, compared with closed-shell or complete open-shell counterparts. They also predicted a similar y-dependence for the strength of two-photon absorption (TPA) for symmetric diradicaloids [13,14]. TPA is a third-order nonlinear optical (NLO) process and offers various attractive applications, such as three-dimensional (3D) optical storage, 3D printing, upconverted lasers, bioimaging, and photodynamic therapy [15,16,17,18,19,20,21,22,23]. Since these applications require molecular materials exhibiting the efficient TPA property at a given photon energy [24,25,26]. Various open-shell singlet systems with a wide range of y values were designed and synthesized, and their TPA cross-section values, ${\sigma }^{\left(2\right)}$, were examined.

Phenalenyl 1a (PLY (R = H); Figure 1a) is a neutral π-radical system showing high thermodynamical stability due to the delocalized nature of the singly occupied molecular orbital (SOMO) [27]. PLY has been employed as a building block for designing open-shell conjugated systems [28]. ${\sigma }^{\left(2\right)}$ values of the diphenalenyl diradical molecules called IDPL and NDPL, synthesized by Kubo and coworkers [29,30], were measured experimentally by Kamada and coworkers. The first TPA band was observed around 1300 nm with ${\sigma }^{\left(2\right)}$~330 GM for IDPL, and around 1610 nm with ${\sigma }^{\left(2\right)}$~2000 GM for NDPL (1 GM = 1 × 10⁻⁵⁰ cm⁴ s photon⁻¹) [31,32]. Further enhanced ${\sigma }^{\left(2\right)}$ values could be observed when both the one-photon and two-photon resonance conditions were nearly satisfied (i.e., $\left|E_{kg}-E_{fg}/2\right|~0$) [33], although the relationship between y and the increase in ${\sigma }^{\left(2\right)}$ in this region has not been clarified sufficiently.

Notably, several derivatives of 1a were reported to form stacked π-dimers and multimers in solution or crystalline phases [34,35,36,37]. For example, 1b (R = tBu) and 1c (R = C₆H₅) were reported to form anti-type π-dimers in the solid state with d~3.31 Å and ~3.02 Å, respectively. These experimental results indicate the existence of the multi-center two-electron covalent-like bonding interaction between the PLYs, known as the pancake bonding [38]. Furthermore, the formation of one-dimensional (1D) molecular aggregates composed of 1d (R = C₆F₅) with d~3.50 Å was reported [37].

Yoneda and coworkers performed the density functional theory (DFT) calculations for π-dimers and tetramers of 1a (R = H) with different stacking distances d in the singlet state [39]. They found that y of the dimer decreases as decreasing d, and static γ per monomer takes the maximum around d = 2.9 Å, at which y is in the intermediate region. They also predicted that the static γ per monomer was further enhanced in the tetramer. Salustro and coworkers evaluated static γ per unit of infinite 1D chains of 1a [40] by extending the coupled-perturbed Kohn–Sham (CPKS) analytic derivative method under the periodic boundary condition implemented in the CRYSTAL package [41], which was the first direct computation of the static γ of this type of 1D chain. In a series of theoretical studies, Matsui and coworkers investigated the effects of increasing the number of monomers (N) on the γ per unit based on the 1D chains of cyclic thiazyl radicals [42,43]. Recently, we examined the static γ per unit of finite and infinite 1D chains of 1a with stacking distance alternations (SDAs). We proposed a y-based index to describe the electronic structures of infinite 1D chains with different SDAs [44]. These studies predicted that the static third-order NLO properties of closely stacked 1D chains of π-radicals are comparable to those of π-conjugated polymers.

On the other hand, there have been a few theoretical studies focusing on the TPA properties of π-stacks of open-shell molecules. In contrast to usual closed-shell systems, calculating excitation energies and transition dipole moments between the electronic states for open-shell singlet systems is still challenging because of the multi-configurational character in the ground state. In this study, we investigated the effects of the number of monomers (N) on the TPA properties of π-stacked N-mer models consisting of 1a with different primary stacking distances (1a_N; see Figure 1). We attempted to apply the extended multi-configurational quasi-degenerate second-order perturbation theory (XMC-QDPT2) method [45,46] to evaluate the excitation energies and transition dipole moments. Although we performed calculations up to hexamer (N = 6) because of the high demands of computational resources in such calculations, the present study will provide valuable information for experimental investigations on the TPA properties of open-shell molecular assemblies.

2. Computational Details

2.1. Calculated Models and Calculation Methods

First, we optimized the geometry of monomer 1a at the UB3LYP/6-31G* level under the constraint of D_3h symmetry, the same level as the previous studies [39,44]. Then, we constructed the anti-type π-stacked N-mer models (N = 2, 4, and 6) consisting of 1a with different stacking distances d₁ (2.8 Å ≤ d₁ ≤ 4.0 Å, see Figure 1b), maintaining each monomer geometry frozen. For tetramer (N = 4), we also investigated the effects of stacking distance alternation between d₁ and d₂, shown in Figure 1c, by changing the ratio for SDA, d₂/d₁, from 1.0 to 2.0. We name the N-mer model with the ratio d₂/d₁ 1a_N(d₂/d₁) for simplicity.

For a given N-mer system consisting of monoradicals (N = 2m; m = 1, 2, …), we can define N/2 diradical characters, y_i (i = 0, 1, …, N/2–1; 1 ≥ y₀ ≥ y₁ ≥ … y_N/2–1 ≥ 0). We evaluated the y_i values of each system at the spin-projected (P)UHF/6-31G* level based on the following Yamaguchi’s formula that can efficiently remove the spin-contamination errors in the occupation numbers [5]:

$$y_i=1-{\displaystyle \frac{2T_i}{1+T_i^2}},$$

(1)

$$T_i={\displaystyle \frac{n_{\mathrm{H}\mathrm{O}\mathrm{N}\mathrm{O}-i}-n_{\mathrm{L}\mathrm{U}\mathrm{N}\mathrm{O}+\mathrm{i}}}{2}}$$

(2)

Here, T_i is the orbital overlap between corresponding orbital pairs, and n_HONO–i and n_LUNO+i are the occupation numbers of UHF natural orbitals (UNOs). The averaged diradical character y_av was calculated using the following formula [47,48]:

$$y_{\mathrm{a}\mathrm{v}}\left(N\right)=\sum _{i=0}^{{\displaystyle \frac{N}{2}}-1}{y}_i\left(N\right)$$

(3)

Note that there are N/2 radical orbital pairs in the N-mer. We also computed the standard deviation, y_SD, defined by the following equation [44]:

$$y_{\mathrm{S}\mathrm{D}}\left(N\right)=\sqrt{{\displaystyle \frac{1}{N/2}}\sum _{i=0}^{{\displaystyle \frac{N}{2}}-1}{\left\{y_i\left(N\right)-y_{\mathrm{a}\mathrm{v}}\left(N\right)\right\}}^2}$$

(4)

These y-based indices help characterize open-shell electronic structures of N-mers [44]. y_av is sensitive to the change in the primary stacking distance d₁ but insensitive to the change in the SDA ratio. y_SD can efficiently characterize the open-shell electronic structures of 1D chains of radicals with different SDA ratios. Note that $y_{\mathrm{a}\mathrm{v}}\left(4\right)=\left(y_0+y_1\right)/2$ and $y_{\mathrm{S}\mathrm{D}}\left(4\right)=\left(y_0-y_1\right)/2$ for N = 4. The occupation numbers of UNOs were calculated using the Gaussian 09 program package [49].

2.2. Excited State Calculations and Spectrum Simulations

Multi-reference methods should be used to examine the ground and excited states of diradical molecules and assemblies in principle for a balanced description of both the static and dynamical electron correlations. In this study, we employed the extended (X)MC-QDPT2 method implemented in the GAMESS-US program package (ver. 2014) [50] to calculate the excitation energies and transition dipole moments, although it is reported that the (multi-reference) second-order perturbation theory methods tend to overestimate the binding energy of PLY π-dimers in the S₀ state [51]. To express the open-shell nature in the S₀ state and two-photon transition of diradical systems, the two-electron and two-orbital complete active space [i.e., CAS(2,2)] is the minimal choice of the active space. For the dimer, the HOMO and LUMO constructed from the SOMOs of monomers are the minimal active orbitals. Accordingly, we performed state-averaged (SA-)CASSCF calculations based on the CAS(N, N) space, where the active orbitals are the frontier MOs constructed from the SOMOs of monomers. We solved for the low-lying three singlet states for N = 2 (all the electronic states derived from CAS(2,2) space) and for the low-lying ten singlet states with the A_g and B_u symmetries for N = 4 and 6. This symmetry constraint is expected to efficiently describe the one-photon absorption (OPA) and TPA peaks of N-mers concerning the intermonomer electronic transitions.

Although the basis sets including diffuse functions can improve the description of excitation energies and binding energies of π-dimers, we used the 6-31G* basis set since it is reported to describe the vertical excitation energies of several π-dimers with sufficient accuracy in combination with the MC-QDPT [52]. Then, we evaluated the excitation energies at the XMC-QDPT2 level with the intruder state avoidance parameter of 0.02. We should note that the transition dipole moments between the states were evaluated using the eigenvectors of the XMC-QDPT2 effective Hamiltonian.

The OPA cross-section, ${\sigma }^{\left(1\right)}\left(\omega \right)$, was evaluated by the following equation [53]:

$${\sigma }^{\left(1\right)}\left(\omega \right)={\displaystyle \frac{4{\pi }^2\omega }{cn\hslash }}〈{\left|{\mathsf{\mu }}_{fg}\right|}^2〉g\left(\omega \right)$$

(5)

Here, $c$ is the speed of light in vacuum, $\omega$ is the angular frequency of the incident light, $n$ is the refractive index of the medium (set to 1.0 in this study). ${\mathsf{\mu }}_{fg}$ is the transition dipole moment between the states $g$ and $f$. $〈\cdots 〉$ means the orientational average, i.e., $〈{\left|{\mathsf{\mu }}_{fg}\right|}^2〉=\left\{{\left({\mathsf{\mu }}_{fg}^X\right)}^2+{\left({\mathsf{\mu }}_{fg}^Y\right)}^2+{\left({\mathsf{\mu }}_{fg}^Z\right)}^2\right\}/3$ where ${\mu }_{fg}^{\alpha }$ represents the α-axis component of the transition dipole moment ${\mathsf{\mu }}_{fg}$. Since we solved only for the A_g and B_u states, the Z-axis component of ${\mathsf{\mu }}_{fg}$ is zero. We used the following Lorentzian function for the normalized shape function $g\left(\omega \right)$:

$$g\left(\omega \right)={\displaystyle \frac{1}{\pi }}{\displaystyle \frac{{\mathsf{\Gamma }}_{fg}}{{\left({\omega }_{fg}-\omega \right)}^2+{{\mathsf{\Gamma }}_{fg}}^2}}$$

(6)

where $E_{fg}=\hslash {\omega }_{fg}$ and ${\mathsf{\Gamma }}_{fg}$ is the damping parameter. This study employed $\hslash \mathsf{\Gamma }$ = 0.1 eV for all the damping parameters.

The TPA cross-section, ${\sigma }^{\left(2\right)}\left(\omega \right)$, was evaluated by the following equation [53,54]:

$${\sigma }^{\left(2\right)}\left(\omega \right)={\displaystyle \frac{4{\pi }^3{\omega }^2}{c^2{n}^2}}〈{\left|{\mathit{M}}_{fg}^{\left(2\right)}\right|}^2〉g\left(2\omega \right)$$

(7)

Here, ${\mathit{M}}_{fg}^{\left(2\right)}$ is the two-photon transition matrix element tensor with each component expressed as follows:

$$M_{fg,\alpha \beta }^{\left(2\right)}={\displaystyle \frac{1}{\hslash }}\sum _k\left[{\displaystyle \frac{{\mathsf{\mu }}_{fk}^{\alpha }{\mathsf{\mu }}_{kg}^{\beta }}{{\omega }_{kg}-\omega }}+{\displaystyle \frac{{\mathsf{\mu }}_{fk}^{\beta }{\mathsf{\mu }}_{kg}^{\alpha }}{{\omega }_{kg}-\omega }}\right]\left(\alpha ,\beta =X, Y, Z\right)$$

(8)

For the orientational average, $〈{\left|{{\mathit{M}}^{\left(2\right)}}_{fg}\right|}^2〉$, we assumed the linearly polarized incident light. For the actual calculation, we replaced ${\omega }_{kg}$ in the denominator with ${\omega }_{kg}-i{\mathsf{\Gamma }}_{kg}$. Again, we used the Lorentzian function for the normalized shape function $g\left(2\omega \right)$:

$$g\left(2\omega \right)={\displaystyle \frac{1}{\pi }} {\displaystyle \frac{{\mathsf{\Gamma }}_{fg}}{{\left({\omega }_{fg}-2\omega \right)}^2+{{\mathsf{\Gamma }}_{fg}}^2}}$$

(9)

3. Results and Discussion

3.1. Correlation Between Stacking Distances and Diradical Characters

Figure 2 shows the calculation results of y_av and y_SD as a function of d₁ (2.8 Å ≤ d₁ ≤ 4.0 Å) for each model. y_av decreased monotonically as the d₁ decreased, and took the intermediate values (~0.5) at d₁~3.0 Å. The ratio d₂/d₁ did not affect the results of y_av for 1a₄(d₂/d₁). In addition, y_av values of 1a₂, 1a₄(1.0), and 1a₆(1.0) for the considered d₁ were almost the same, although their open-shell characters are different. y_SD can characterize such differences in the open-shell characters between the models at each d₁. The calculated y_SD values of 1a₄(2.0) were almost zero regardless of d₁, whereas those of the other models increased monotonically with decreasing d₁. y_SD characterizes the effective orbital interaction between the dimer units. The value of y_i increases as the energy gap between the HOMO-i and LUMO+i decreases. The orbital interaction between the dimer units is weak for the tetramer when d₂ >> d₁. The HOMO and HOMO − 1 (LUMO and LUMO + 1) are nearly degenerated, resulting in y₀~y₁ (i.e., y_SD~0). As the d₂ approached d₁ (and as the d₁ became small), the orbital interaction between the dimer units resulted in the splitting of HOMO and HOMO − 1 (LUMO and LUMO + 1) and the increase in y_SD. y_SD of 1a₆(1.0) was slightly larger than 1a₄(1.0) at each d₁, but it is expected to converge to a certain value as increasing N [44].

3.2. One-Photon and Two-Photon Absorption Properties of Tetramers

Figure 3a shows the simulated OPA spectra for the tetramer models 1a₄(d₂/d₁) with d₁ = 3.0 Å. To compare the results with 1a₂, ${\sigma }^{\left(1\right)}\left(\omega \right)$ and ${\sigma }^{\left(2\right)}\left(\omega \right)$ values per dimer unit were presented. The OPA spectra were normalized to that at the first peak of the model 1a₄(1.0).

The first OPA and TPA peaks were red-shifted as the d₂/d₁ became small (i.e., d₂ approached d₁) and the d₁ became small. For example, the first OPA peak appeared at ~2.5 eV for 1a₂, ~2.4 eV for 1a₄(2.0), and ~1.9 eV for 1a₄(1.0). The intensity of the first OPA peak was almost unchanged by changing the d₂/d₁. However, the peak intensities per dimer of 1a₄(d₂/d₁) were stronger than that of 1a₂, suggesting the delocalization of wavefunction over the dimers in the excited state.

For the TPA spectra, only a single peak appeared at ~1.7 eV for 1a₂ because of the limited active space size [CAS(2,2)]. For the tetramer models 1a₄(d₂/d₁), enhanced TPA cross-section values (per dimer) were obtained as the incident photon energy approached the excitation energy of the OPA band. This is because both the one-photon and two-photon resonance conditions were nearly satisfied in this region [31,33]. However, the present level of approximation for the excited state calculations would not sufficiently describe the higher-lying excited states concerning the TPA transitions in this region. In the lower-energy region of incident photon energy (<1.7 eV), we obtained a single TPA peak at ~1.6 eV for 1a₄(2.0) and 1a₄(1.5). As the d₂/d₁ decreased, this peak position was red-shifted, and another peak appeared at ~1.4 eV. The intensity of the lower-energy peak at ~1.4 eV increased drastically as the d₂/d₁ approached one.

In

Table 1. Calculation results for excitation energies of 1a₄(d₂/d₁), $E_{fg}$ (in eV), at d₁ = 3.0 Å with different ratios d₂/d₁ = 1.0, 1.1, 1.2, 1.3, 1.5, and 2.0. Results for the 1a₂ are also provided for reference.

System	yav	ySD	${\mathit{E}}_{\mathit{f}\mathit{g}}$
			1Bu	2Ag	3Ag	4Ag	5Ag
1a4(1.0)	0.474	0.189	1.921	1.120	2.375	2.708	3.164
1a4(1.1)	0.482	0.126	2.076	1.273	2.369	2.727	2.999
1a4(1.2)	0.486	0.081	2.213	1.400	2.379	2.713	3.030
1a4(1.3)	0.488	0.051	2.298	1.465	2.388	2.760	3.063
1a4(1.5)	0.490	0.018	2.357	1.500	2.400	2.927	3.087
1a4(2.0)	0.490	0.001	2.381	1.501	2.410	3.091	3.271
1a2	0.490	–	2.573	3.452	–	–	–

and

Table 2. Calculation results for the Y-axis (stacking direction) component of transition dipole moment between the states i and j $\left|{\mathsf{\mu }}_{ij}^Y\right|$ (in Debye) for 1a₄(d₂/d₁) at d₁ = 3.0 Å with different ratios d₂/d₁ = 1.0, 1.1, 1.2, 1.3, 1.5, and 2.0. Results for the 1a₂ are also provided for reference.

System	yav	ySD	$\left|{\mathsf{\mu }}_{\mathit{i}\mathit{j}}^{\mathit{Y}}\right|$
			1Ag→1Bu	1Bu→2Ag	1Bu→3Ag	1Bu→4Ag	1Bu→5Ag
1a4(1.0)	0.474	0.189	11.6	1.0	1.1	21.2	2.3
1a4(1.1)	0.482	0.126	10.6	2.7	0.1	19.2	7.4
1a4(1.2)	0.486	0.081	10.2	2.1	0.4	14.8	12.1
1a4(1.3)	0.488	0.051	10.2	0.9	1.0	10.4	13.2
1a4(1.5)	0.490	0.018	10.1	0.1	1.4	4.1	13.7
1a4(2.0)	0.490	0.001	10.0	0.0	1.0	0.1	13.5
1a2	0.490	–	7.3	12.4	–	–	–

, we summarized the excitation energies, $E_{fg}$, and the Y-axis component of transition dipole moments, $\left|{\mathsf{\mu }}_{ij}^Y\right|$, respectively, crucial for describing these OPA and TPA peaks. The 1A_g→1B_u transition for all cases characterized the OPA peak. The $\left|{\mathsf{\mu }}_{ij}^Y\right|$ value for the 1A_g→1B_u transition of the tetramer models 1a₄(d₂/d₁) slightly increased as the d₂/d₁ decreased, the feature of which was reflected in the OPA peak intensities. The TPA peak at ~1.6 eV was primarily described by the virtual transition via 1A_g→1B_u→5A_g. The $\left|{\mathsf{\mu }}_{ij}^Y\right|$ value for the 1B_u→5A_g transition decreased whereas the $\left|{\mathsf{\mu }}_{ij}^Y\right|$ value for the 1B_u→4A_g transition increased as the d₂/d₁ decreased. Therefore, the virtual transition via 1A_g→1B_u→4A_g primarily described the lower-energy TPA peak at ~1.4 eV. The excitation energy of the 4A_g state was almost unchanged in the range of 1.0 ≤ d₂/d₁ ≤ 1.3, while the excitation energy of the 1B_u state decreased, which led to a decrease in $\left|E_{kg}-E_{fg}/2\right|$ (k: 1B_u state, f: 4A_g state) as the d₂/d₁ approached to one. These tendencies were attributed to the enhancement of ${\sigma }^{\left(2\right)}\left(\omega \right)$ values per dimer at ~1.4 eV.

In Tables S1–S4, we summarized the weights of electron configurations for these electronic states of 1a₄(1.0), 1a₄(1.2), 1a₄(1.5), and 1a₄(2.0) with d = 3.0 Å. We also plotted the HOMO-1, HOMO, LUMO, and LUMO+1 of 1a₄(1.0) with d = 3.0 Å (Figure S1). Note that the bonding/anti-bonding characters of these frontier MOs were unchanged for all the considered d₁ and d₂ values. For all these models, the 1A_g and 1B_u states were primarily described by the ground electron configuration and the HOMO–LUMO single excitation configuration, respectively. However, the 1A_g state of 1a₄(1.0) exhibited a relatively large weight for the HOMO–LUMO double excitation configuration. For 1a₄(1.5) and 1a₄(2.0), the 5A_g state was described by the linear combination of HOMO-1–LUMO and HOMO–LUMO+1 single excitation configurations. In contrast, the 4A_g state of these models was described by the linear combinations of the ground and several single and double excitation configurations. For 1a₄(1.2), the characters of 4A_g and 5A_g were reversed from those for 1a₄(1.5) and 1a₄(2.0). For 1a₄(1.0), the 4A_g state was described primarily by the ground and HOMO–LUMO double excitation configurations.

Next, we examined how the primary stacking distance d₁ and the ratio d₂/d₁ affect the intensity of the first TPA peak. Figure 4 shows the ${\sigma }^{\left(2\right)}\left(\omega \right)$ values per dimer for 1a₄(d₂/d₁) with different d₁ (2.8 Å ≤ d₁ ≤ 4.0 Å) as a function of the ratio d₂/d₁. The first TPA peak intensity increased as the d₁ decreased and d₂/d₁ approached one. Similar tendencies were observed for the calculation results of static γ per unit of finite and infinite 1D stacks of 1a [44]. The results suggest that not only the static γ but also ${\sigma }^{\left(2\right)}\left(\omega \right)$ the first TPA peak value is expected to be enhanced in the uniform 1D stacks of PLYs. Such enhancements of the third-order NLO responses are expected when d₁ is less than the van der Waals contact distance for the carbon atoms (3.4 Å).

In Figure S2, we plotted variations of the excitation energies and transition dipole moments of crucial transitions for 1a₄(d₂/d₁) with different d₁ (2.8 Å ≤ d₁ ≤ 4.0 Å) as a function of the ratio d₂/d₁. As the d₁ decreased, the excitation energy for the TPA target (4A_g or 5A_g) state increased more significantly than that for the OPA target (1B_u) state. As a result, the detuning energy, $\left|E_{kg}-E_{fg}/2\right|$, which inversely contributes to the ${\sigma }^{\left(2\right)}$ value at the peak, became small. The magnitude of the transition dipole moment for the 1A_g→1B_u transition increased as the d₁ decreased, whereas that for the 1B_u→4A_g(5A_g) transition showed somewhat complicated behaviors because the characters of 4A_g and 5A_g states exchanged at d₂/d₁ = 1.2–1.3.

Finally, we calculated the TPA property of the hexamer model 1a₆(1.0) to examine the effects of increasing N. Figure 5 shows the calculation results of ${\sigma }^{\left(2\right)}$ value per dimer at the first TPA peak as a function of y_av for the 1a₂, 1a₄(1.0), and 1a₆(1.0) (for the relationship between d₁ and y_av, see Figure 2a). The ${\sigma }^{\left(2\right)}$ values per dimer increased as the y_av decreased (i.e., as the d₁ decreased) for all the models. The enhancement of ${\sigma }^{\left(2\right)}$ values per dimer was more significant in N = 6 than in N = 4: At d₀ = 3.0 Å, ${\sigma }^{\left(2\right)}$ values per dimer of 1a₄(1.0) and 1a₆(1.0) were 5.1 and 8.4 times higher than that of 1a₂. Further enhancement of the ${\sigma }^{\left(2\right)}$ values per dimer unit can be expected when N approaches to ∞ (i.e., for the infinite 1D π-stacks), although the value will converge to a specific value as was observed for the calculation results of static γ per unit [44]. In Figure S3, we compared the excitation energies and transition dipole moments of crucial transitions for 1a_2, 1a₄(1.0), and 1a₆(1.0) as a function of y_av. As the N increased, the excitation energies for the OPA and TPA target states decreased, and the detuning energy, $\left|E_{kg}-E_{fg}/2\right|$, became small, which is considered the main cause of enhancing ${\sigma }^{\left(2\right)}$ values at the first peak.

4. Conclusions

In this study, we theoretically examined the low energy OPA and TPA properties of the π-stacked N-mer models, 1a_N(d₂/d₁) (N = 2, 4, and 6), consisting of unsubstituted phenalenyl radicals with different stacking distances d₁ and d₂. We conducted the CASSCF(N,N) and XMC-QDPT2/6-31G* calculations for the excitation energies and transition dipole moments that were used to simulate the OPA and TPA spectra of these models. The ${\sigma }^{\left(2\right)}$ value per dimer unit at the first peak, originating from the electronic transition along the stacking direction, was predicted to increase significantly as the d₂/d₁ approaches one, as the d₁ decreases, and as the N increases from 2 to 6. For the 1D chains of unsubstituted phenalenyl 1a, sufficiently enhanced TPA properties are expected for d₁ ≤ 3.2 Å. The appropriate d₁ value and d₂/d₁ ratio for achieving sufficiently enhanced TPA properties would depend on the monomer species.

Calculating the y-based indices, y_av, and y_SD, will help specify appropriate regions for these values. Of course, it is generally challenging to characterize the electronic structures and excitation properties of 1D π-stacks with only a few indices. Calculations and analyses with sufficient accuracies are necessary for predicting the OPA and TPA properties of more realistic calculation models involving substituent groups. However, we believe the present results provide essential information for the relationship between the structural features and linear and nonlinear optical responses of actual π-stacked 1D chains of open-shell molecular species.