Tuning the Inter-Chromophore Electronic Coupling in Perylene Diimide Dimers with Rigid Covalent Linkers

Abstract

The organic multi-chromophore system has been increasingly attractive due to the potential optoelectronic applications. The inter-chromophore electronic coupling (EC), i.e., J_Coul and J_CT, plays a critical role in determining the relaxation path of the excited state. However, the molecular designing strategy for effective tuning of inter-chromophore EC is still challenging. In this computational work, we designed a series of perylene diimides (PDI) covalent dimers with rigid linking cores containing thiophene (Th) or phenyl (Ph) fragments and performed corresponding theoretical investigation to analyze the inter-PDI electronic coupling. Vibrational analysis indicated that the minimized excited state structural relaxation (ES-SR) can ensure the rigid inter-PDI geometry pre-defined by the topological characteristic of linking cores, leading to comparable |J_Coul| on S₀ and S₁ states. The saddle-shaped linking cores allow collaborative tuning of inter-PDI dihedral (α) and slipping (θ) angles, leading to effective tuning of inter-PDI |J_Coul| = 0–1000 cm⁻¹. Our work provides a new molecular designing strategy for effective tuning of inter-chromophore EC for organic chromophores. By using a rigid inter-chromophore structure, the ignorable ES-SR allows simplified molecular designing without considering the plausible geometric difference between S₁ and S₀ states, which might be useful for future applications in organic optoelectronics.

1. Introduction

Inter-chromophore electronic coupling (EC) plays a key role in determining the excited state behavior of multi-chromophore systems and optoelectronic performance of organic semiconductors with ordered molecular packing [1,2,3,4,5,6,7]. In the weak coupling regime, photoinduced dynamics of chromophores can be dominated by the locally excited (LE) state. With enhanced inter-chromophore EC combining with further condition such as energy matching, the initially populated LE state may be obtained through charge transfer (CT) [8,9,10,11,12], symmetry-breaking charge separation (SB-CS) [13,14,15,16,17,18], singlet fission (SF) [19,20,21,22,23,24], and even formation of excimer/exciplex state [25,26,27,28,29,30,31]. Recently, photo-induced SB-CS [32,33,34] and SF [35,36,37] in covalent oligomers received great attention due to potential application in improving power conversion efficiency (PCE) of organic photovoltaics (OPV) [38,39,40,41,42,43]. For enabling efficient SB-CS and SF in organic multi-chromophore systems, a key task would be smart molecular designing to achieve delicate inter-chromophore EC [20,23,44], which remains challenging.

The inter-chromophore EC has been intensively discussed in the framework of the exciton theory pioneered by Davydov [45], Kasha [46,47,48], and Spano [49,50,51,52]. Briefly, inter-chromophore EC (J) can be described as the sum of Columbic coupling (J_Coul) and CT-mediated super-exchange (J_CT), i.e., J = J_Coul + J_CT, [4,51,52] in which J_Coul and J_CT correspond to the long-range dipole-dipole interaction and frontier orbital overlapping in short-range, respectively. In particular, J_Coul can be tunable by molecular designing as it is associated with inter-chromophore geometry, i.e., inter-chromophore distance (R), dihedral angle (α), and slipping angle (θ), which has been described by [45,48,53,54].

$$J_{\mathrm{C}\mathrm{o}\mathrm{u}\mathrm{l}}={\displaystyle \frac{1}{4\pi {\epsilon }_0}}{\displaystyle \frac{{\mu }^2\left(\cos \alpha -3{\cos }^2\theta \right)}{R^3}}$$

(1)

The definition of α and θ angles can be seen in the illustration of Figure S1. In comparison, J_CT is a short-range interaction originating from frontier orbital overlapping and associated with electron (t_e) and hole (t_h) transfer integral, as well as adiabatic energy of CT (E_CT) and the lowest-lying state (E_S1) [51,55,56]. Please note that J_CT can also be greatly affected by geometric parameters, such as inter-chromophore slipping in a co-facial dimer [51], but it relies on a short inter-chromophore distance (typically a few angstroms).

$$J_{\mathrm{C}\mathrm{T}}=-2{\displaystyle \frac{t_{\mathrm{e}}{t}_{\mathrm{h}}}{E_{\mathrm{C}\mathrm{T}}-E_{{\mathrm{S}}_1}}}$$

(2)

Unlike excimer/exciplex formation, which relies on strong EC [30,57,58], efficient SB-CS and SF require moderate EC, corresponding to the “degraded” structure rather than fully stacked inter-chromophore geometry, i.e., inter-chromophore geometry should be deviated from the co-facial stacking geometry to reduce EC [59,60], which is often accompanied by changes of inter-chromophore dihedral (α) and slip angles (θ). Therefore, enormous efforts in molecular designing have been made to achieve the “degraded” inter-chromophore geometry. For instance, taking advantage of excited state structural relaxation (S₁/S₀ ES-SR), multi-chromophore systems with flexible linkers have been widely employed for facilitating efficient SB-CS and SF [59,60,61,62,63,64]. However, considerable S₁/S₀ ES-SR also leads to unpredictable EC in the sense of molecular designing. Recently, we demonstrated a new molecular designing strategy for this issue [65]. By using a saddle-shaped cyclooctatetrathiophene (COTh) core and fused linking with perylene diimides (PDI), optimized inter-PDI EC (|J_Coul| ≈ 700 cm⁻¹) was achieved, by which efficient inter-PDI SF (τ_SF = ~150 ps and ~150% triplet yield) was facilitated. Considering the rigid COTh structure and fused linking with PDIs, the inter-PDI geometry that can greatly affect J_Coul can be defined by the topological characteristics of COTh. In this sense, we should be able to achieve fine-tuning of inter-PDI EC (J_Coul) only by modifying the structure of the rigid linking core, by which molecular designing of multi-chromophore systems can be simplified, i.e., only the geometry of the rigid core should be considered to facilitate the expected EC.

In this work, we further extended our molecular designing strategy for multi-chromophore systems. By using different rigid linkers with thiophene (Th) or phenyl (Ph) fragments, a series of PDI dimers was theoretically designed, leading to a broad tuning range of inter-PDI EC (0 < |J_Coul| < 1000 cm⁻¹). The vibrational analysis indicated that the S₁/S₀ ES-SR of rigid linking cores can be further suppressed by linking with PDI units, leading to comparable |J_Coul| on S₀ and S₁ states, for which inter-PDI geometry was pre-defined by rigid linking cores. Considering the saddle-shaped feature of involved linking cores, our strategy enabled collaborative tuning of inter-PDI dihedral (α) and slipping (θ) angles, which can greatly neutralize the extension of inter-PDI distance (R) and lead to effective tuning of inter-PDI |J_Coul|. Our work provides a new molecular designing strategy for fine-tuning of inter-chromophore EC for organic chromophores like PDI without disturbing S₁/S₀ ES-SR, which might be useful for future applications in organic optoelectronics.

2. Results and Discussion

2.1. Molecular Designing

In the present work, two structural features were considered for molecular designing of PDI dimers: (1) structure of rigid linking core; (2) linking manner between rigid core and PDI units. As illustrated in Figure 1a, we first modified the COTh (noted as Th₄ in this work) core to the Th₃ core, in which the structural symmetry can be reduced. The Me₄Th₃ and Me₄Si₂Th₃ cores were subsequently designed to further enhance the structural rigidity. Two PDI units can be further introduced on each Th-based core with fused (rigid, Figure 1a) and non-fused (flexible, Figure 1b) linking, respectively. In comparison between PDI dimers illustrated in Figure 1a,b, we can further verify the key role of pre-defining inter-PDI geometry by rigid linking cores. Similarly, we replaced all Th units on linking the core with phenyl units, leading to corresponding cores of Ph₄, Ph₃, Me₄Ph₃, and Me₄Si₂Ph₃. For each Ph-based core, PDI units can be fused to the linking core on meta- (Figure 1c) and para-position (Figure 1d) of the center cyclooctatetraene, leading to PDI dimers with variable EC due to changed inter-PDI distance (R), dihedral (α), and slipping angle (θ).

The optimized geometry on S₀ and S₁ states of the involved linking cores was calculated by using DFT and TDDFT approaches on B3LYP/6-311g** level, respectively. We employed the twisting dihedral angle (δ, as illustrated in Figure S2) for describing the saddle-shaped geometric feature of linking cores. As summarized in

Table 1. The DFT/TDDFT calculated twisting angle (δ) of the investigated linking cores on S₀/S₁ states and corresponding RMSD_S1/S0 with and without linking with PDI units.

Without PDI Units				With PDI Units
	δS0 (°)	δS1 (°)	RMSDS1/S0 (Å)		δS0 (°)	δS1 (°)	RMSDS1/S0 (Å)
Th4	47.7	22.9	0.51	-FPDI	63.9	50.2	0.23
				-PDI	47.1	33.9	0.24
Th3	29.8	1.4	0.42	-FPDI	76.5	70.6	0.06
				-PDI	58.9	57.4	0.02
Me4Th3	64.6	39.1	0.19	-FPDI	78.0	72.6	0.06
				-PDI	64.6	62.7	0.03
Me4Si2Th3	60.7	30.5	0.26	-FPDI	76.6	71.4	0.07
				-PDI	57.7	57.2	0.04
Ph4	61.2	42.4	0.45	-mFPDI	59.8	57.4	0.05
				-pFPDI	59.8	57.7	0.04
Ph3	60.1	34.7	0.21	-mFPDI	65.5	63.1	0.04
				-pFPDI	46.6	43.7	0.05
Me4Ph3	62.0	43.2	0.17	-mFPDI	66.3	64.4	0.04
				-pFPDI	60.1	57.6	0.03
Me4Si2Ph3	62.4	35.2	0.23	-mFPDI	64.2	57.5	0.08
				-pFPDI	61.9	60.6	0.06

, all investigated Th- and Ph-based cores exhibited greatly reduced δ_S1 than δ_S0, corresponding to pronounced S₁/S₀ ES-SR. However, we found that S₁/S₀ ES-SR of cores can be greatly suppressed by linking with PDI units, leading to comparably twisted (60 to 70°) δ_S1 and δ_S0 angles. The calculated root of the mean of squared displacement (RMSD_S1/S0) between S₁ and S₀ state also indicated less pronounced S₁/S₀ ES-SR of linking cores in PDI dimers, which might be explained by two aspects of factor [66,67]. Firstly, the steric effect induced by PDI units may suppress S₁/S₀ ES-SR of linking cores, which can be verified by lower core RMSD_S1/S0 in non-fused dimers (i.e., Th₃-PDI, etc.) than in corresponding fused dimers (i.e., Th₃-FPDI, etc.). Meanwhile, considering the higher S₀→S₁ energy gap of the investigated linking cores (>3 eV) than the PDI unit (<2.5 eV), the cores might not be greatly involved in the S₀→S₁ transition of the corresponding PDI dimers. The linking cores with minimized S₁/S₀ ES-SR in PDI dimers ensured that the inter-PDI geometry can be pre-defined by rigid cores for tuning the inter-PDI J_Coul.

2.2. Electronic Coupling

As described above, the inter-PDI EC in PDI dimers includes contributions of J_Coul and J_CT, in which J_Coul is highly dependent on the inter-PDI geometry, i.e., inter-PDI distance (R), dihedral angle (α), and slipping angle (θ). Meanwhile, J_CT is a short-range interaction that is dependent on overlapping and alignment of frontier orbitals. The term of (cosα − 3cos²θ) involved in Equation (1) of J_Coul can usually be defined as an angular factor (κ) which determines the positive and negative values of J_Coul. In exciton theory, J_Coul < 0 and J_Coul > 0 imply the J- (head-to-tail) and H-type (side-by-side) interaction among chromophores.

In contrast, J_CT is a sort of short-range interaction associated with frontier orbital overlapping, which is considerably dependent on inter-PDI geometry, only with short inter-PDI distance (typically a few angstroms). The J_Coul (Equation (1)) of all designed PDI dimers with S₀ and S₁ geometries was calculated together with corresponding J_CT (Equation (2)), leading to values of κ_S0, κ_S1, J_Coul^S0, J_Coul^S1, and J_CT summarized in

Table 2. Calculated angular factor (κ) and corresponding J_Coul of investigated PDI dimers on both S₀ and S₁ geometries, while J_CT of each PDI dimers was calculated as well.

	Noted as	κS0	κS1	JCoulS0 (cm−1)	JCoulS1 (cm−1)	JCT (cm−1)
Th4-FPDI	Th-FPDIs	−0.73	–0.54	−714	−698	852
Th3-FPDI		–0.93	–0.94	−699	−779	526
Me4Th3-FPDI		–0.92	–0.90	−721	−790	614
Me4Si2Th3-FPDI		–0.87	–0.85	−691	−686	829
Th4-PDI	Th-PDIs	–0.25	–0.36	−118	−187	102
Th3-PDI		–0.92	–1.15	−511	−677	45
Me4Th3-PDI		–0.41	–0.74	−170	−354	58
Me4Si2Th3-PDI		–0.18	–0.32	−67	−118	112
Ph4-mFPDI	Ph-mFPDIs	0.27	0.38	536	889	747
Ph3-mFPDI		–0.22	–0.12	−354	−229	452
Me4Ph3-mFPDI		0.05	0.08	−263	−164	411
Me4Si2Ph3-mFPDI		–0.03	0.12	−47	247	705
Ph4-pFPDI	Ph-pFPDIs	0.18	–0.10	−31	−25	112
Ph3-pFPDI		–0.37	–0.40	−106	−114	230
Me4Ph3-pFPDI		–0.14	–0.08	13	19	137
Me4Si2Ph3-pFPDI		–0.04	–0.05	−11	−13	121

. The geometric parameters of PDI dimers (α, θ, and R) for calculating J_Coul on S₀ and S₁ states can be seen in Table S1. The corresponding parameters for calculating inter-PDI J_CT were obtained by using the Multiwfn program [68,69], and are listed in Table S2.

As shown in Figure S3a, calculated J_CT of PDI dimers exhibited a jump-like dependence on inter-PDI distance, i.e., J_CT > 400 cm⁻¹ in the area of R = 6–9 Å and J_CT < 200 cm⁻¹ in the area of R = 9–12 Å, which is consistent with the short-range nature of J_CT. Meanwhile, dependence of J_CT on α and θ angles can be barely observed (Figure S3b,c), which is dramatically different from the case of slipping perylene dimers with 3.5 Å stacking distance [51]. Please note that the positive/negative sign of J_CT is highly dependent on the sign of t_h and t_e. However, a sign of t_h and t_e is determined by the phase assignment of the frontier orbital under a specific symmetry operation [52]. To avoid redundant discussion, we treated t_h and t_e with the definition by Wasielewski et al. [3], in which t_h and t_e can be defined with a different sign as moving a hole to one direction corresponds to moving an electron to the reverse direction.

We further discuss the calculated inter-PDI J_Coul that is highly dependent on inter-PDI geometry in designed covalent dimers. As shown in Figure 2a, PDI dimers with fused linking to Th-based cores (Th-FPDIs) exhibited large α angle (60–70°) and slightly slipping (θ ≈ 50°) inter-PDI geometry on the S₁ state, corresponding to a degraded stacking geometry with comparably strong inter-PDI EC (|J_Coul| = 700–800 cm⁻¹). In contrast, PDI dimers with flexible linking (Th-PDIs) exhibited greatly weakened EC (|J_Coul| = 100–700 cm⁻¹) than corresponding PDI dimers with fused linking. Compared to Th-FPDIs, inter-PDI geometry with non-fused linking can derive from the optimal geometry pre-defined by Th-based cores, leading to weakened EC. We further defined (J_Coul^S1 − J_Coul^S0)/J_Coul^S0 as an indicator for evaluating J_Coul changing between S₀ and S₁ geometry.

As shown in Figure 2b, PDI dimers with fused-linking exhibited minimized J_Coul changing (<10%), which is highly different from the Th-PDIs with up to 110% changing due to substantial S₁/S₀ ES-SR quantified by total reorganization energy of S₁→S₀ transition (Γ_S1→S0). In the sense of rational molecular designing, Th-FPDIs with nearly identical J_Coul on S₀ and S₁ states are highly appreciated as S₁/S₀ ES-SR might be less considered. Regarding PDI dimers with Ph-based cores, it can be seen that meta-fusing resulted in stronger inter-PDI EC (|J_Coul| up to 900 cm⁻¹) than corresponding Ph-pFPDIs (|J_Coul| = 10–100 cm⁻¹, Figure 2c). Meanwhile, Ph-mFPDIs exhibited larger S₁/S₀ ES-SR (Γ_S1→S0) than corresponding para-fused dimers, leading to S₁/S₀ changing of J_Coul up to 100% (Figure 2d). Thus, para-fused linking for Ph-based PDI dimers leads to weak inter-PDI EC, although it might be favored in the sense of rational molecular designing.

As described in Equation (1), angular factor (κ) and inter-PDI distance (R) play a central role in determining inter-PDI J_Coul, for which we plotted a contour map of κ/R³ as a function of κ and R. As can be seen in Figure 2e, Th-FPDI, Th-PDI, Ph-mFPDI, and Ph-pFPDI dimers are located in different areas of the κ–R map, leading to their own feature on absolute values and S₁/S₀ changing of inter-PDI J_Coul. The narrow κ (−0.6 to −0.9) and R (7–9 Å) distribution of Th-FPDI dimers resulted to considerable inter-PDI J_Coul. Meanwhile, non-fused linking and substantial S₁/S₀ ES-SR of Th-PDI dimers lead to κ ranging from −0.2 to −1.2 with slightly enlarged inter-PDI R (9.5–11.5 Å), corresponding to greatly reduced J_Coul than Th-FPDI dimers.

On the other hand, the weakened |J_Coul| of Ph-pFPDIs (R = 12–13 Å) in comparison with Ph-mFPDI dimers (R = 6–7 Å) can be attributed to the greatly enlarged inter-PDI R rather than changed κ. Meanwhile, it can be seen that each Ph-pFPDI dimer exhibited nearly identical κ and R on S₀ and S₁ states, indicating greatly suppressed S₁/S₀ ES-SR in comparison to Ph-mFPDI dimers. Our calculation on J_Coul of PDI dimers indicated that S₁/S₀ ES-SR might be a critical factor to be considered for rational molecular designing of multi-PDI systems, for which we further analyzed the origin of S₁/S₀ ES-SR in the sense of vibronic coupling effect in S₁→S₀ transition.

2.3. Vibrational Analysis

As discussed above, S₁/S₀ ES-SR is a critical factor for rational molecular designing of multi-PDI systems. In our recent works [31,70,71], we demonstrated that specific S₁/S₀ ES-SR motions can be associated with promoting vibrational modes, which can engage in S₁→S₀ electronic transition with high Huang-Rhys factor (S_k) and reorganization energy contribution (λ_k). For a multi-chromophore system, inter-chromophore J_Coul might also be sensitive to specific vibrational modes corresponding to collective motions that can vary inter-chromophore geometry [31,65]. To learn details on S₁/S₀ ES-SR of PDI dimers, we performed vibrational analysis on involved Th- and Ph-based cores, as well as corresponding PDI dimers. The Huang-Rhys factor (S_k) and reorganization energy contribution (λ_k) of each vibrational mode (k) engaged in S₁→S₀ electronic transition were calculated as S_k = (2ℏ)⁻¹ω_kΔQ_k² and λ_k = 2⁻¹ω_k²ΔQ_k² by using the MOMAP program [72,73,74,75] based on electronic structure calculation. The mode displacement (ΔQ_k) can be calculated with displacement components (ΔD_i) along internal coordinate i as ΔQ_k = ∑_iζ_kiΔD_i.

By summing the contribution of each mode, i.e., Γ_S1→S0 = Σ_kλ_k, the total reorganization energy of the S₁→S₀ transition (Γ_S1→S0) can be used for generalizing S₁/S₀ ES-SR of PDI dimers. As listed in

Table 3. Calculated total reorganization energy of S₁→S₀ transition (Γ_S1→S0) for linking cores and corresponding PDI dimers.

	ΓS1→S0 (cm−1)
	Th-based core	Th-FPDIs	Th-PDIs
Th4	6711	1864	2050
Th3	5296	1358	1691
Me4Th3	4904	1392	1694
Me4Si2Th3	4430	1388	1772
	Ph-based core	Ph-mFPDIs	Ph-pFPDIs
Ph4	5975	1282	1225
Ph3	5910	1309	1198
Me4Ph3	3896	1297	1125
Me4Si2Ph3	5742	1382	1219

, all Th- and Ph-based cores exhibited considerable Γ_S1→S0 (up to 7000 cm⁻¹). Meanwhile, the calculated Γ_S1→S0 of corresponding PDI dimers was observed to be greatly reduced to 1000–2000 cm⁻¹, indicating the suppressed λ_k of promoting modes associated with collective motion of PDI units, which is consistent with less pronounced S₁/S₀ ES-SR observed by RMSD_S1/S0. Since the collective motion of PDI units is largely correlated with low-frequency modes of Th- and Ph-cores, we could resolve S₁/S₀ ES-SR affecting inter-PDI geometry by extracting corresponding modes of Th- and Ph-cores. The calculated results of vibrational analysis for Th- and Ph-cores can be seen in Figure 3 and Figure 4, together with corresponding PDI dimers.

As shown in Figure 3, Γ_S1→S0 of Th₄ core was dominated by a promoting mode (S_k = 73.7 cm⁻¹, λ_k = 3020 cm⁻¹) at ω_k = 14.1 cm⁻¹ (noted as mode 1, blue colored), corresponding to planarized motion on S₁ state (see Figure S4). In the S₁→S₀ transition of Th₃, Me₄Th₃, and Me₄Si₂Th₃ cores, mode 1 might be suppressed due to the structural tension. The Me₄Th₃ and Me₄Si₂Th₃ cores exhibited an extra promoting mode with considerable λ_k contribution at ω_k = 100–140 cm⁻¹ (noted as mode 2, red colored), corresponding to twisting motion of Th units (see Figure S4). The vibrational analysis on PDI dimers indicated that λ_k contribution of modes 1 and 2 was greatly reduced in the S₁→S₀ transition of all Th-based PDI dimers, implying suppressed S₁/S₀ ES-SR associated with modes 1 and 2, which ensures the inter-PDI geometry pre-defined by linking cores.

Intriguingly, we observed a new promoting mode located in ω_k < 10 cm⁻¹ region (noted as mode 3, green colored) with considerable S_k and contribution to Γ_S1→S0 of Th-PDI dimers. As can be seen in Figure S4, mode 3 of Th-PDI dimers exhibited the largest displacement toward the opposite direction in the ending sides of PDI units, as well as the minimized displacement in spatial center of PDI units, corresponding to a rotational motion of PDI units along the linking bond between PDI and Th cores. Such a rotational motion can be greatly limited by fused linking in Th-FPDIs, for which inter-PDI geometry can be pre-defined by Th cores. However, with single bond linking, the presence of mode 3 can lead to considerable S₁/S₀ ES-SR of Th-PDI dimers, resulting in greatly relaxed inter-PDI geometry with undermined |J_Coul|.

We subsequently performed vibrational analysis on the S₁→S₀ transition of the Ph-based cores and corresponding PDI dimers. As shown in Figure 4, Ph-based cores exhibited very similar promoting modes of Th-based cores discussed above, i.e., mode 1 (ω_k = 48.1 cm⁻¹) for Ph₄ core and mode 2 (ω_k = 100–120 cm⁻¹) for Th₃, Me₄Th₃, and Me₄Si₂Th₃ cores, corresponding to planarized and Ph twisting motions (see Figure S5), respectively. With the fused linking, Ph-mFPDI and Ph-pFPDI dimers exhibited an ignorable contribution of low-frequency modes to Γ_S1→S0, indicating minimized S₁/S₀ ES-SR, which can ensure inter-PDI geometry pre-defined by linking cores. Please note that vibrational modes in the high-frequency regime (>500 cm⁻¹) can also contribute to Γ_S1→S0 as well. However, high-frequency modes usually correspond to localized motions without affecting inter-PDI geometry for J_Coul, which can be ignored in the discussion of the present work.

2.4. Further Discussion

As mentioned above, inter-PDI EC plays a key role in determining the excited-state relaxation path of multi-chromophore PDI model systems, in which J_Coul is highly sensitive to inter-PDI geometry, i.e., angular factor κ and inter-PDI distance R. As shown in Figure 5a, we plotted a contour map of κ as a function of inter-PDI α and θ angles. The κ of PDI dimers investigated in this work ranges from −1.0 to 0.5, leading to |J_Coul| tuning between 0 and 1000 cm⁻¹, which is a considerable tuning range and may benefit from our strategy of molecular designing. By using rigid cores and fused covalent linking, the inter-PDI geometry of PDI dimers can be pre-defined by the topological characteristics of linking cores. As a result, collaborative modification of both inter-PDI α and θ angles can be achieved, leading to effective tuning of J_Coul.

As shown in Figure 5d, collaborative changing of inter-PDI α (0°–90°) and θ (90°–0°) corresponds to a broad range of κ (1 to −3), which might lead to highly effective tuning of inter-PDI J_Coul. For instance, inter-PDI α and θ in our designed PDI dimers can be changed in a relatively small range, i.e., 45°–75° and 75°–45°, respectively. However, collaborative changing of inter-PDI α and θ leads to a broad κ range (0.5 to −1.0), which is broader than the theoretically maximized tuning range by α (κ = 1 to 0) and θ (κ = 1 to −0.2) individually. As a result, a broad κ range by collaborative changing of inter-PDI α and θ can enable highly effective tuning of inter-PDI J_Coul.

Last but not least, the plausible excited-state relaxation path of investigated PDI dimers can be discussed with the calculated J_Coul and J_CT. As shown in Figure 6, Th-FPDI dimers mostly exhibited large |J_Coul|, large |J_CT|, and small |J_Coul + J_CT|, which is consistent with the condition for plausible electronic state mixing described by Hariharan, Wasielewski, and co-workers [2,3,4,76]. In this type of PDI dimer, Th₄-FPDI has been experimentally demonstrated to undergo SF to generate ¹(TT) species by our ultrafast spectroscopic investigation [65], which can be regarded as the dephasing product of the mixed state. Furthermore, Ph-pFPDI dimers and most of Th-PDI dimers exhibited small |J_Coul|, small |J_CT|, and long inter-PDI distance, which are highly plausible to relax through incoherent SB-CS in a highly polar medium. For Ph₄-mFPDI with huge |J_Coul| (~900 cm⁻¹) and close inter-PDI stacking, the excimer-like state can be expected through an incoherent relaxation.

3. Calculational Methods

All electronic structure calculations of designed cores and PDI-based dimers were performed using the Gaussian 16 software package [77]. The geometric structure of the investigated emitters was optimized on both ground (S₀) and singlet (S₁) excited states at B3LYP/6-311G** level, while no imaginary frequency was found by frequency analysis. The DFT and TDDFT optimized geometries of PDI-based dimers with root of the mean of squared displacement were analyzed and rendered by using the VMD 1.9.3 program [78]. The molecular orbitals analysis of the corresponding excited states was performed by using the Multiwfn program [68,69]. The Huang-Rhys (HR) factor (S_k) and reorganization energy contribution (λ_k) of each vibrational mode were calculated by using MOMAP 2021A on the basis of frequency analysis of corresponding states [72,73,74,75]. We further estimated the reorganization energy contribution (λ_k) of each vibrational mode between the S₁ and S₀ states.

4. Conclusions

To summarize, we performed molecular designing of a series of PDI covalent dimers with rigid linking cores containing thiophene (Th) or phenyl (Ph) fragments and investigated the inter-PDI electronic coupling (J_Coul and J_CT). The vibrational analysis on S₁→S₀ transition indicated that the S₁/S₀ ES-SR of rigid linking cores can be further suppressed by linking with PDI units, leading to comparable |J_Coul| on S₀ and S₁ states, for which inter-PDI geometry can be pre-defined by rigid linking cores. Benefiting from the saddle-shaped feature of involved linking cores, our strategy enabled collaborative tuning of inter-PDI dihedral (α) and slipping (θ) angles, which can lead to effective tuning of inter-PDI |J_Coul| = 0–1000 cm⁻¹. Our work provides a new molecular designing strategy for fine-tuning of inter-chromophore EC for organic chromophores like PDI without disturbing S₁/S₀ ES-SR, which might be useful for future applications in organic optoelectronics.