Tuning Low-Lying Excited States and Optical Properties in IndenoFluorene Diradicaloids and Longitudinally Extended Derivatives: A Computational Perspective

Abstract

In this work, we have considered the family of indenofluorene (IF) and its longitudinally elongated variants fluorenofluorene and diindenoanthracene and investigated their low-lying excited states and optical properties via quantum-chemical studies at the density functional theory (DFT) level. Singlet ground-state diradicals exhibit distinct optical properties due to the presence of a low-lying state dominated by a doubly excited configuration (DE state), often below the lowest allowed singly excited state (SE state). IFs and their elongated derivatives, with tunable diradical character and both symmetric and nonsymmetric structures, provide an ideal platform for exploring DE state energy modulation and spectroscopic behavior. The study shows that absorption spectra simulated using time-dependent (TD) calculations based on unrestricted broken-symmetry antiparallel-spin reference configuration (TDUDFT) closely match the available experimental data. Additionally, it reveals distinct spectral behavior for symmetric and nonsymmetric derivatives, highlighting the role of lowest-lying weakly allowed excited states potentially promoting non-radiative deactivation pathways.

1. Introduction

The indenofluorene (IF) family is a fascinating class of organic materials that has garnered significant attention in recent years due to their unique electronic properties, potential applications in organic electronics and optoelectronics, and their role in advancing the understanding of open-shell systems [1,2,3,4,5]. IF features a distinctive 6-5-6-5-6 π-fused-ring system, forming five possible regioisomeric backbones with varying π-conjugation, aromaticity (aromatic vs. antiaromatic), and electronic structures (quinoidal or diradical) [2,6,7,8,9,10,11,12]. The extent of diradical character can be tuned by modifying the molecular structure, such as by introducing substituents or extending the π-conjugation. This tunability makes them valuable for studying the interplay between aromaticity, antiaromaticity, and diradical behavior.

Diradical species display two closely spaced spin states: an open-shell (OS) singlet spin state and a triplet spin state. The electronic ground state of such species is influenced by dynamic spin polarization, which often stabilizes the OS singlet spin state over the triplet state.

IFs have been structurally modified to tune the open-shell (OS) character and aromaticity of IF-type scaffolds and to adjust their optical and magnetic properties. One such modification is longitudinal extension of the indacene π-conjugated core by replacing the central benzene ring with naphthalene or anthracene units, leading to fluorenofluorene (FF) and diindenoanthracene (DIAn) [12,13,14,15,16].

Conjugated diradicals can be described by multiple resonance structures, some of which involve bond cleavage leading to diradical formation. These particular resonance forms play a significant role in stabilizing the diradical species, as they restore aromaticity [17]. Notably, these diradical resonance structures are distinguished by an increased number of Clar’s π-sextets. According to the proposed Ground State Stability (GSS) rule [18], an IF-type system adopts a triplet electronic ground state if the open-shell resonance form exhibits an increase of at least twice the number of Clar’s π-sextets compared to the closed-shell form. Otherwise, the ground state remains an open-shell singlet spin.

Diradicals with a singlet spin ground state exhibit unique optical properties, as demonstrated in several previous studies. In particular, their optoelectronic and luminescence behavior is influenced by the presence of low-lying, symmetry-forbidden states characterized by the doubly excited configuration involving the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), namely the HOMO,HOMO→LUMO,LUMO configuration (referred to as the DE state), which is often located below the lowest allowed excited state, dominated by the HOMO→LUMO singly excited configuration (referred to as the SE state) [19,20,21,22,23].

IFs and their longitudinally elongated derivatives offer tunable diradical characters and encompass symmetric and nonsymmetric structures, making them ideal for investigating the modulation of the DE state energy and spectroscopical activity.

Although numerous studies have explored IF derivatives [8,9,12,24], the role of the DE state has not been systematically examined across different regioisomers. Additionally, while many conjugated diradicals are symmetric, nonsymmetric diradicals have largely been overlooked. This paper aims to address this gap by considering a wide range of regioisomers, including nonsymmetric molecules. We performed quantum chemical calculations on IF and longitudinally extended derivatives, employing density functional theory (DFT) and focusing on the regioisomers with a singlet spin ground state, whose open-shell nature (OS) was identified at unrestricted DFT (UDFT) level as a more stable broken-symmetry (BS) solution. At the optimized OS geometries, time-dependent (TD) calculations based on unrestricted broken-symmetry antiparallel-spin reference configuration (TDUDFT) calculations were carried out to investigate low-lying excited states, as TDUDFT has proven effective in capturing DE states when the diradical character is significant [20,21,22].

2. Methods

The geometries of the investigated diradicals were optimized using the M062X functional [25], the def2SVP basis set [26] and the Grimme dispersion contribution D3 [27,28]. This functional has been shown to effectively reproduce the singlet-to-triplet energy difference in IF derivatives [29].

The descriptor $y_0$, which is a measure of the diradical character, was calculated by the Yamaguchi scheme [30,31] using the Unrestricted Hartree Fock (UHF) approach and def2SVP basis set (see computational details in the Supplementary Material).

The singlet OS geometries were determined when available, along with the equilibrium structures of the lowest triplet state. Minima were confirmed through vibrational frequency analysis. TDUB3LYP and TDRB3LYP calculations were carried out on optimized geometries to determine the electronic absorption spectra.

The effective exchange integral $J_{AB}$ between the radical sites A and B can be evaluated according to Yamaguchi’s procedure [32]:

$$J_{AB}={\displaystyle \frac{∆E\left(LS-HS\right)}{\left(S_{HS}^2-S_{LS}^2\right)}}={\displaystyle \frac{∆E\left(S-T\right)}{\left(S_{HS}^2-S_{LS}^2\right)}}$$

(1)

where $∆E$ is the energy difference between low-spin (LS) and high-spin (HS) states. LS is, in this case, the singlet spin state and HS is the triplet, while at the denominator the $S_{HS}^2$ and $S_{LS}^2$ values from calculations are reported.

The DE state characterizing diradicals with a singlet spin open-shell ground state can be captured by TDUB3LYP calculations, when the diradical character is large, since it corresponds to single excitations from strongly localized frontier MOs [20,22,33] (see the Supplementary Material for further details). The spectra were simulated neglecting vibronic progressions and using a Lorentzian broadening function, hwhm = 0.2 eV, superimposed to each computed intensity, chosen to facilitate the comparison with experimental spectra.

3. Results and Discussion

In the following, we consider all the IF derivatives displaying a singlet ground electronic state for which we determined the lowest excited states and compare, when available, the predicted and experimental spectra. For IF, FF, and DIAn regioisomers, we adopt the same nomenclature as ref. [18].

3.1. Equilibrium Structure and Molecular Orbitals of IF, FF, and DIAn

The regioisomers examined exhibit a wide range of diradical character, as shown in column two of

Table 1. Computed HOMO/LUMO gaps, excitation energies E of the DE state, oscillator strength f and character of the computed ground state (CS or OS), along with diradical character $y_0$ for the regioisomers of IF, FF, and DIAn considered in this study. From RM062X-D3/def2SVP calculations for CS structures and UM062X-D3/def2SVP for OS structures.

Compound	${\mathit{y}}_0$ This Work 1	Ground State Structure	$\mathsf{\Delta }\mathit{E}(\mathit{H}\mathit{O}\mathit{M}\mathit{O}/\mathit{L}\mathit{U}\mathit{M}\mathit{O})$/eV α Electrons	$\mathsf{\Delta }\mathit{E}(\mathit{H}\mathit{O}\mathit{M}\mathit{O}/\mathit{L}\mathit{U}\mathit{M}\mathit{O})$/eV β Electrons	E(DE State)/eV [f]
IF-1a	0.68	OS	3.62	3.62	1.00 [0.0102]
IF-1b	0.25	CS	4.35	-	-
IF-2b	0.29	CS	4.13	-	-
IF-2c	0.25	CS	4.21	-	-
FF-1a	0.80	OS	3.57	3.57	1.18 [0.0009]
FF-1b	0.43	CS	3.54	-	-
FF-3a	0.71	OS	3.69	3.69	1.18 [0.0000]
FF-3c	0.33	CS	3.58	-	-
DIAn-1a	0.87	OS	3.45	3.45	1.26 [0.0012]
DIAn-1b	0.68	OS	3.55	3.55	1.15 [0.0000]
DIAn-3a	0.77	OS	3.55	3.55	1.24 [0.0000]
DIAn-3c	0.58	OS	3.59	3.59	1.05 [0.0000]
Non symmetric isomers
FF-2a	0.41	CS	3.51	-	-
FF-2d	0.58	OS	3.62	3.55	0.96 [0.0021]
FF-4b	0.44	CS	3.45	-	-
DIAn-2b	0.74	OS	3.72	3.34	1.12 [0.0372]
DIAn-2d	0.65	OS	3.74	3.30	1.02 [0.0269]
DIAn-4b	0.73	OS	3.48	3.67	1.12 [0.0133]

¹ PUHF/def2SVP calculations @optimized UM062X-D3/def2SVP geometry (or RM062X-D3/def2SVP calculations when the OS singlet state could not be found).

. Consequently, an OS singlet structure could only be determined at the UM062X-D3/def2SVP level for a subset of the investigated diradicals, as indicated in column three of Table 1. Notably, in these cases, the computed frontier molecular orbitals exhibit a broken-symmetry character (see Figures S1–S5 and S9–S22), which emerges in UDFT calculations as a more appropriate representation of the enhanced diradical character. This localized BS nature is particularly evident in Figures S1, S3, S9, S11 and S14, while it appears to varying degrees in Figures S16–S22.

The HOMO/LUMO gap for CS structures was therefore computed at the RM062X-D3/def2SVP level, while for open-shell OS structures it was obtained at the UM062X-D3/def2SVP level. Two identical values are reported in Table 1 for symmetric molecules. For asymmetric structures, two distinct HOMO/LUMO gaps are reported in columns four and five, corresponding to the α and β electrons, respectively.

The optimized geometries, presented in Figures S1–S5 and S9–S22, reflect the dominant resonance structures. When an OS geometry is determined, the quinoidal character is less pronounced, and the computed structures generally show a restoration of aromaticity in the six-membered rings. In contrast, in CS optimized geometries, the quinoidal character associated with specific resonance structures is more prominent.

The computed molecular orbital shapes and energies are provided in Figures S1–S5 and S9–S22. The corresponding energy gaps, summarized in Table 1, range from 3.30 eV to 4.35 eV. Notably, within each diradical family (IF, FF, or DIAn), the energy gap varies depending on the regioisomer. However, the average energy gap tends to decrease as the molecular system becomes more extended. For example, the average HOMO/LUMO gap for IF is 4.08 eV, whereas for FF, it is 3.56 eV. While an extended conjugated core generally leads to a reduced energy gap, the OS or CS nature of the optimized geometry also plays a role. These combined effects make it challenging to establish a straightforward correlation between the computed data. However, when focusing on a specific regioisomer, such as “-1a,” we observe that the diradical character increases from 0.68 in IF-1a to 0.80 in FF-1a and 0.87 in DIAn-1a. Simultaneously, the HOMO/LUMO gap decreases progressively from 3.62 eV (IF-1a) to 3.57 eV (FF-1a) and 3.45 eV (DIAn-1a).

Finally, we note that the DE state could only be identified in calculations for systems exhibiting a medium-to-large diradical character, specifically when an OS structure was found. This explains why the DE state is not listed in Table 1 for regioisomers that only exhibit a CS ground state structure (see column three). TDUB3LYP/def2SVP calculations could be carried out only for the subset of systems with an OS ground-state geometry, as indicated in column six. The nature and excitation energies of the lowest-lying excited states are discussed in more detail in the following sections.

3.2. IF

Interestingly, all regioisomers of IF (see Figure 1) have been synthetized, either as ethynylated derivatives and/or stabilized by bulky groups (e.g., mesityls), and their optical properties measured. Thus, in this case we can directly compare observed and computed optical spectra (see Figure 2, Figure 3, Figures S6–S8, along with Tables S1–S5), considering a minor role of the substituents not directly affecting the conjugated framework.

In

Table 2. Computed values of the diradical descriptor $y_0$, singlet-triplet energy differences for the IF regioisomers of Figure 1, comparison with data from the literature, and Heisenberg effective exchange interactions.

Compound	${\mathit{y}}_0$ This Work 1 [ref. [18]] 2	$\mathsf{\Delta }\mathit{E}(\mathit{S}-\mathit{T})$/kcal/mol This Work 3 [ref. [18]] 4	$\mathsf{\Delta }\mathit{E}(\mathit{S}-\mathit{T})$/kcal/mol Experimental	${\mathit{J}}_{\mathit{A}\mathit{B}}$ kcal/mol
IF-1a	0.68 [0.53]	−1.24 [2.04]	−4.215 5	−1.03
IF-1b	0.25 [0.03]	−20.89 [−14.12]	-	−10.25
IF-2b	0.29 [0.07]	−15.23 [−10.00]	-	−7.48
IF-2c	0.25 [0.02]	−17.71 [−12.25]	-	−8.70
Non symmetric isomers
IF-2a	0.97 [1.00]	2.44 [3.65]	-	2.44

¹ PUHF/def2SVP calculations @optimized UM062X-D3/def2SVP geometry (or RM062X-D3/def2SVP calculations when the OS singlet state could not be found). ² Calculations @optimized ULCωPBE-D3BJ/def2-TZVPP geometry. ³ From UM062X-D3/def2SVP or RM062X-D3/def2SVP calculations when the OS singlet state could not be found. ⁴ From ULCωPBE-D3BJ/def2-TZVPP calculations. ⁵ From ref. [7].

, we compare our computed $y_0$ values and singlet-triplet energy differences ($\mathsf{\Delta }E(S-T)$) with those recently reported in ref. [18]. Our results are generally consistent; however, it is important to note that the computed $y_0$ values are highly sensitive to the calculated geometries and the theoretical approach used. In particular, UHF and UDFT methods may yield significantly different values, with strong dependence on the choice of functional [34]. Table 2 reveals that the $y_0\left(PUHF\right)$ values computed at UM062X-D3 geometries tend to be higher except for the system with the largest diradical character, IF-2a. Notably, our calculations predict IF-1a to have a singlet ground state, in contrast to ref. [18]. This result aligns with previously reported calculations and with the experimentally reported spectrum of a dimesityl IF-1a derivative [8]. The comparison between observed and computed absorption spectra (Figure 2) supports the attribution of a singlet ground state for IF-1a. The triplet absorption spectrum lacks low-energy transitions, with the lowest computed transition exceeding 1.5 eV. In contrast, the singlet absorption spectrum exhibits weak features at 1238 nm and 1235 nm (see Table S1 in the Supplementary Material), which align with the experimentally observed weak features in the 1200–1600 nm region [8,9]. Regarding the DE state, which is typically the lowest excited state in most diradicals with an inversion center, we note that the SE/DE state inversion does not occur in the present case. Consequently, the lowest excited state is formally the SE state dominated by the HOMO→LUMO excitation. However, this state is very weakly active, nearly degenerate with the DE state (Table S1), and accounts for the weak vibronic features observed in the 1200–1600 nm region [8,9].

All other symmetric IF isomers exhibit modest diradical character, and a more stable UDFT solution could not be identified. Due to the small diradical character, the DE state is expected to be higher in energy, with the low-energy absorption spectrum primarily dominated by the transition to the SE state or to other states dominated by singly excited configurations. A comparison with the observed spectrum is shown in Figure 3 for IF-2b [7]. Comparisons between the computed and observed spectra for the remaining regioisomers, including the helicoidal IF-2c, are presented in Figures S6–S8. Notably, a recent derivative of IF-2c was synthesized [35], and its spectrum closely resembles that of our computed model diradical.

3.3. FF

Concerning FF, we have included all the regioisomers that display a singlet ground state according to the GSS rule [18] (see Figure 4). Their singlet ground state is confirmed at the UM062X-D3/def2SVP level. Additionally, the ground state of isomer FF-1a is computed to be a singlet, with a closely lying upper triplet state (see

Table 3. Computed values of the diradical descriptor $y_0$, singlet-triplet energy differences for the FF regioisomers of Figure 4, comparison with data from the literature, and Heisenberg effective exchange interactions.

Compound	${\mathit{y}}_0$ This Work 1 [ref. [18]] 2	$\mathsf{\Delta }\mathit{E}(\mathit{S}-\mathit{T})$/kcal/mol This Work 3 [ref. [18]] 4	$\mathsf{\Delta }\mathit{E}(\mathit{S}-\mathit{T})$/kcal/mol Experimental	${\mathit{J}}_{\mathit{A}\mathit{B}}$ kcal/mol
FF-1a	0.80 [0.69]	−0.82 [2.04]	-	−0.74
FF-1b	0.43 [0.31]	−9.17 [−7.23]	−9.3 5	−4.49
FF-3a	0.71 [0.55]	−3.30 [−4.91]	-	−2.93
FF-3c	0.33 [0.28]	−9.32 [−8.04]	−12.6 ± 2.0 6	−4.59
Non symmetric isomers
FF-2a	0.41 [0.31]	−7.67 [−6.64]	-	−3.76
FF-2d	0.58 [0.43]	−4.76 [−5.32]	-	−3.44
FF-4b	0.44 [0.38]	−5.97 [−6.25]	-	−2.93

¹ PUHF/def2SVP calculations @optimized UM062X-D3/def2SVP geometry (or RM062X-D3/def2SVP calculations when the OS singlet state could not be found). ² Calculations @optimized ULCωPBE-D3BJ/def2-TZVPP geometry. ³ From UM062X-D3/def2SVP or RM062X-D3/def2SVP calculations when the OS singlet state could not be found. ⁴ From ULCωPBE-D3BJ/def2-TZVPP calculations. ⁵ From ref. [12]. ⁶ From ref. [16].

), and was thus included in the study. The simulated absorption spectra are collected in Figure 5 and Figures S24–S28, while computed excited states are enclosed in Tables S6–S12. Upon examining Table 3, we observe that the stability of the triplet state (relative to the lowest singlet state) computed at the UM062X-D3 level is generally lower compared to ref. [18], except for FF-3a. Interestingly, experimental data for the singlet-triplet energy difference are available for a dimesityl FF-1b derivative [12] and for a 4,11-di-t-butyl-1,8-dimesityl FF-3c derivative [16], which show good agreement with our calculations.

The calculations reveal that an OS singlet ground state exists for a greater number of isomers in FF compared to IF, specifically for FF-1a, FF-3a, and FF-2d, consistent with their more extended π-conjugation. In contrast, for FF-1b, FF-3c, FF-2b, and FF-4b, which display modest diradical character, only a CS ground state was found.

In addition to the singlet-triplet energy difference, absorption spectra have also been measured for the FF-1b and FF-3c isomers [12,36]. A comparison with the TDRB3LYP computed spectra in Figures S23 and S25 shows good agreement, revealing that the lowest energy observed absorption features in both cases are attributed to the transition to the SE state, dominated by the HOMO→LUMO excitation. Due to their modest diradical character, the DE state is expected to lie at a higher energy. However, the SE state is not the lowest, as a low-lying forbidden state, dominated by the HOMO-1→LUMO excitation (Tables S7 and S9), is computed for both isomers. This suggests that also these FF isomers are most likely non-emissive, as previously indicated for IF derivatives [10].

Figure 5. Comparison between (black dashed) the experimental spectrum of FF-1a adapted from ref. [37] and (green) the computed singlet absorption spectrum from TDUB3LYP/def2SVP calculations.

Figure 5. Comparison between (black dashed) the experimental spectrum of FF-1a adapted from ref. [37] and (green) the computed singlet absorption spectrum from TDUB3LYP/def2SVP calculations.

An experimental absorption spectrum is also available for the FF-1a isomer, and its comparison with the computed spectrum is presented in Figure 5. As previously observed [9], the low-energy absorption features in FF-1a appear blue-shifted relative to those in IF-1a, despite FF-1a having a more extended π-system. This seemingly counterintuitive trend was attributed to the interplay between diradical character, exchange integral, and HOMO–LUMO gap, a conclusion further supported by a complete-active-space configuration-interaction (CAS-CI) analysis with two electrons in two orbitals [9].

Beyond these factors, our calculations provide additional insight, revealing that both IF-1a and FF-1a exhibit a low-lying DE state. In IF-1a, this state is nearly degenerate with the SE state, while in FF-1a, it becomes the lowest energy excited state. This finding is particularly significant, as it enables the assignment of the two weakest absorption bands observed in FF-1a to transitions involving the DE and SE states (see Figure 5). The presence of a low-lying DE state in FF-1a suggests a potential impact on its photophysical behavior, possibly influencing its luminescence properties and nonradiative decay pathways.

The symmetric isomer FF-3a, has not been experimentally observed. However, the computed spectrum indicates the presence of a lowest-lying DE state (see Table S8 and Figure S24), reinforcing the tendency of the DE state to become the lowest excited state in diradicals with significant $y_0$ values.

Among the three nonsymmetric isomers, FF-2a and FF-4b exhibit small diradical character and a lowest excited state dominated by SE character, though weakly allowed (Figures S26 and S28; Tables S10 and S12). In contrast, FF-2d, with a notable diradical character ($y_0$ = 0.58), features a lowest-lying state with pronounced DE character. As shown in Table 1, for the three FF derivatives displaying a low-lying DE state, the associated intensity is influenced by the symmetry of the regioisomer and the asymmetric FF-2d isomer shows slightly larger oscillator strength compared to FF-3a and FF-1a.

3.4. DIAn

For DIAn, we included in our study all regioisomers predicted to have a singlet ground state according to the GSS rule [18] (see Figure 6), which was further confirmed at the UM062X-D3/def2SVP level. Additionally, DIAn-1a was considered in the study, as it is computed to have a singlet ground state with a nearly degenerate triplet state ($E(S-T)=-0.04$ kcal/mol; see

Table 4. Computed values of the diradical descriptor $y_0$, singlet-triplet energy differences for the DIAn regioisomers of Figure 6, comparison with data from the literature, and Heisenberg effective exchange interactions.

Compound	${\mathit{y}}_0$ This Work 1 [ref. [18]] 2	$\mathsf{\Delta }\mathit{E}(\mathit{S}-\mathit{T})$/kcal/mol This Work 3 [ref. [18]] 4	$\mathsf{\Delta }\mathit{E}(\mathit{S}-\mathit{T})$/kcal/mol Experimental	${\mathit{J}}_{\mathit{A}\mathit{B}}$ kcal/mol
DIAn-1a	0.87 [0.81]	−0.04 [3.05]	-	−0.04
DIAn-1b	0.68 [0.51]	−3.16 [-5.49]	−4.2 5	−2.66
DIAn-3a	0.77 [0.69]	−2.17 [-4.62]	-	−2.13
DIAn-3c	0.58 [0.47]	−4.92 [-6.91]	-	−4.07
Non symmetric isomers
DIAn-2b	0.74 [0.61]	−2.81 [−4.69]	-	−2.60
DIAn-2d	0.65 [0.49]	−4.15 [−5.75]	-	−3.48
DIAn-4b	0.73 [0.59]	−3.12 [−5.46]	-	−2.89

¹ PUHF/def2SVP calculations @optimized UM062X-D3/def2SVP geometry (or RM062X-D3/def2SVP calculations when the OS singlet state could not be found). ² Calculations @optimized ULCωPBE-D3BJ/def2-TZVPP geometry. ³ From UM062X-D3/def2SVP or RM062X-D3/def2SVP calculations when the OS singlet state could not be found. ⁴ From ULCωPBE-D3BJ/def2-TZVPP calculations. ⁵ From ref. [15].

). The calculations confirm an OS singlet ground state for all investigated DIAn isomers, with diradical character varying from 0.58 in DIAn-3c to 0.87 in DIAn-1a. The higher $y_0$ values and pronounced OS character align with the extended longitudinal conjugation of their core structures. The simulated absorption spectra are shown in Figure 7 and Figures S29–S34 while computed excited states are collected in Tables S13–S19.

Comparing the data in Table 3 and Table 4, we observe that as the longitudinal core extends, the singlet-triplet energy gap decreases, and the diradical character increases relative to FF derivatives. Unlike the trend seen in FF derivatives, the computed triplet state stability (relative to the singlet) at the UM062X-D3 level is generally higher than in ref. [18], leading to a reduced singlet-triplet energy gap.

The singlet-triplet energy difference has been experimentally obtained for a dimesityl and di(triisopropylsilyl)ethynyl (di(TIPS)ethynyl)-substituted DIAn-1b isomer [15]. Our computed value of −3.16 kcal/mol aligns well with the experimentally measured −4.2 kcal/mol.

The only available experimental spectrum is also for the same DIAn-1b derivative [13,14,15,38]. A comparison with the computed spectrum is presented in Figure 7. The calculations not only reproduce the spectral features but also identify the presence of a low-lying DE state, reinforcing the expected electronic structure of this diradical.

Although no experimental spectrum is available, the calculated excitation energies and simulated spectrum of DIAn-1a (Table S13, Figure S29) provide additional insights. Notably, a progressive blue shift is observed when moving from FF to DIAn derivatives, consistent with the blue shift observed when moving from IF-1a to FF-1a. Additionally, the DE state in DIAn-1a is computed to be slightly lower in energy than the SE state, further confirming the trend observed across these systems.

Furthermore, calculations reveal that all remaining symmetric and nonsymmetric DIAn isomers also exhibit a low-lying excited state with significant DE character, underscoring the trend across the series. The intensity associated with the DE state is highly influenced by the symmetry of the regioisomers. In some cases, this state is symmetry-forbidden. Consequently, asymmetric structures generally exhibit higher intensity for this state compared to symmetric ones. However, even among asymmetric diradicals, the intensity does not directly correlate with the diradical character, as evident from the data in column six of Table 1. Nevertheless, the larger intensity computed for the DE state of asymmetric DIAn regioisomers, compared to the corresponding FF regioisomers, suggests a correlation with the extension of the conjugated core and increased diradical character.

3.5. Heisenberg Effective Exchange Couplings

The family of IF derivatives discussed in this paper exhibits a wide range of diradical character, spanning from 0.25 to 0.97 (PUHF level). This significant variability suggests that for singlet spin ground states, the lowest-lying triplet state may be energetically accessible. In this context, it is valuable to examine the relationship between the Heisenberg effective exchange coupling J_AB (computed using Equation (1)) and the diradical character. The computed J_AB values are presented in Table 2, Table 3 and Table 4 while Figure 8 visualizes these data as a function of the computed diradical character. The plot reveals a strong correlation between J_AB couplings and the $y_0$ descriptor, offering a predictive framework for designing new magnetic molecules by tuning structural factors to control magnetic interactions.

4. Conclusions

IFs and their longitudinally extended derivatives offer tunable diradical character with both symmetric and nonsymmetric structures. This versatility makes them well-suited for studying the modulation of low-lying excited states and their spectroscopic behavior. To explore these effects, we conducted quantum-chemical calculations on the full set of singlet ground state IF, FF, and DIAn regioisomers.

Considering the frontier orbitals that contribute to the lowest-lying excitations, we found that an extended conjugated core generally leads to a reduced HOMO/LUMO energy gap, although the OS or CS nature of the optimized geometry also plays a role. These combined effects make it challenging to establish a straightforward correlation between orbital energies and diradical character. However, when focusing on a specific regioisomer, we observe that the diradical character increases in the series IF > FF > DIAn and simultaneously the HOMO/LUMO gap decreases progressively.

When examining the computed excitation energies, the results reveal that the lowest energy region of the absorption spectra is generally characterized by weak transitions. In systems with significant diradical character, these weak features are typically associated with the DE state. However, even when the diradical character is modest, the presence of low-lying forbidden states dominated by HOMO-1 → LUMO transitions suggests that these systems are unlikely to function as efficient emissive chromophores.

In the “-1b” series, the computed absorption spectra exhibit a red shift, as expected for an extended conjugated core, a trend that aligns well with experimental data. Additionally, calculations identify the DE state as the lowest excited state in both FF and DIAn, providing a plausible explanation for the weak low-energy features observed in the spectrum of DIAn-1b.

Conversely, in the “-1a” series, the lowest energy absorption band shifts to higher energies as the conjugated framework extends—a counterintuitive phenomenon previously rationalized by Nakano [9] and confirmed here also for DIAn-1a. At the same time, a lowest-lying DE state emerges, positioned on the red side of the SE state.

The intensity of the DE state is influenced by the symmetry of regioisomers, with asymmetric structures generally exhibiting higher intensity. However, this intensity does not strictly correlate with diradical character. For nonsymmetric systems with moderate to large diradical character, calculations predict that this lowest excited state remains predominantly DE-like but, due to the lack of symmetry, exhibits a mixed nature which can explain its slightly increased intensity. The higher intensity observed in asymmetric DIAn regioisomers compared to asymmetric FF regioisomers suggests a link to the conjugated core’s extension and increased diradical character.

Overall, the study highlights that the role of the DE state becomes more important for large diradical character, potentially enhancing non-radiative deactivation paths.