Computational design of boron-free triangular molecules with inverted singlet–triplet energy gap

Abstract

A novel, computationally designed, class of triangular-shape organic molecules with an inverted singlet–triplet (IST) energy gap is investigated with ab initio electronic structure methods. The considered molecular systems are cyclic oligomers and their common feature is electronic conjugation along the molecular rim. Vertical excitation energies from the electronic ground state to the lowest singlet and triplet excited states were computed, as well as vertical emission energies from these states to the ground state. The results underscore the significance of optimizing excited-state geometries to accurately describe the optoelectronic properties of IST molecules, in particular with respect to their application in OLEDs.

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1. Introduction

Within the domain of molecular physics, Hund's multiplicity rule, stating that triplet excited states should exhibit lower energy levels than their singlet counterparts with identical orbital configuration, has long served as a guiding principle. This fundamental principle has been a reliable rule governing the ordering of excited states in organic molecules and it was assumed that violations of Hund's multiplicity rule in the excited states of organic compounds are exceptionally rare. Recent computational and experimental developments, however, have confirmed the existence of energy inversion of the first singlet (S₁) and triplet (T₁) excited states, known as inverted singlet–triplet (IST) states, in a number of stable closed-shell organic molecules.^1–3

The significance of organic molecules featuring nearly degenerate or inverted S₁ and T₁ states extends well beyond theoretical chemistry. IST systems are of relevance when transitions from excited singlet states to triplet states are undesirable. This applies mainly to the development of chromophores for organic light-emitting diodes (OLEDs). Many organic chromophores currently under examination for OLED applications exhibit small positive S₁–T₁ energy gaps,^4–7 relying on reverse intersystem crossing (RISC) from T₁ to S₁, resulting in thermally activated delayed fluorescence (TADF) at room temperature.^8,9 The emergence of organic IST chromophores may open the door to a new generation of OLED devices, capitalizing on the potentially intense fluorescence from these inverted S₁ states and unlocking new possibilities in OLED technology.^10–12 Inverted singlet–triplet molecules also are of interest in photocatalysis,¹³ for example for the photocatalytic water-splitting process.^14,15

This resurgence of interest in the inversion of S₁ and T₁ excited-state energies was primarily initiated by the computational chemistry community. In recent years, researchers have explored design strategies for the engineering of IST molecules with theoretical and computational methods,^16–20 and the theoretical predictions were confirmed by spectroscopic measurements.^1,3,21,22 Molecular structures discovered by these explorations were predominantly nitrogen-doped phenalenes, such as cyclazine or heptazine derivatives,^1,2 along with related triangular^17,23 or hexagonal polycyclic aromatic hydrocarbons²⁴ with boron–nitride cores. In recent studies, also non-phenalene-based organic molecules were investigated, specifically focusing on non-alternant cyclic hydrocarbons.^25–30

Organic IST systems typically exhibit specific structural features and characteristics which contribute to their unique electronic properties. While the structural aspects may vary, there are four common themes:

1. Conjugated π-systems: IST molecules represent a sub-class of polycyclic aromatic systems with electronic conjugation along the molecular edges.

2. Rigid, planar geometry: IST molecules typically possess rigid internal structures built from nitrogen, boron, or boron–nitride insulating lattices.

3. Electron-donating and electron-withdrawing groups: IST molecules often incorporate electron-donating and electron-withdrawing groups or atoms which allows the fine-tuning of their electronic properties.

4. Heavy-atom substituents: The incorporation of heavy atoms, such as sulfur, may increase the spin–orbit coupling in the molecule and may enhance the quantum yield of RISC.

While the concept of organic IST chromophores appears promising, it is crucial to recognize an inherent challenge. Most IST compounds have thus far been found to exhibit minuscule oscillator strengths of the S₁–S₀ transition, resulting in low radiative decay rates.^16,18,19 The long radiative lifetime renders the fluorescence highly susceptible to competing non-radiative decay processes, compromising their emission efficiency.

In this study, we computationally explored the optoelectronic properties of a novel category of boron-free organic IST molecules, which are based on the triangle composed of pyran units (PX) as illustrated in Chart 1. Our findings suggest the potential to modify the inherently negative S₁–T₁ energy gap, the wavelength of S₁–S₀ fluorescence, and its intensity through chemical alterations of the parent compounds. The results discussed herein represent a preliminary step towards constructing IST systems based on the PX motif.

Chart 1 Triangular BCN (a) and boron-free (b) molecules (PX) considered in this work. X stands for O, NH, or S.

2. Computational methods

The ground-state equilibrium geometries of all compounds were in a first step optimized using density functional theory (DFT) employing the B3LYP functional,^31,32 augmented with Grimme's D3 dispersion correction.³³ The computation of the Hessian verified that the optimized stationary points represent energy minima. Vertical excitation energies of the lowest singlet and triplet states were computed with the second-order algebraic-diagrammatic construction (ADC(2)) method.^34–36 For consistency with the ADC(2) method employed for the excited-state calculations, the ground-state equilibrium geometries were re-optimized using the Møller–Plesset (MP2) method,³⁴ with the DFT equilibrium geometries serving as input. The ADC(2) method was also employed to determine the excited state (S₁ and T₁) equilibrium geometries. For all calculations, the correlation-consistent valence double-zeta cc-pVDZ basis set³⁵ was used. The calculations were performed with the Turbomole 7.3 program package.³⁶

Calculations benchmarking the ADC(2) method and the approximate coupled cluster singles and doubles (CC2) method³⁷ indicate that the accuracies of both methods are very similar.^38–43 Previous studies have established that only methods explicitly including double excitations can accurately reproduce negative singlet–triplet energy gaps.^1,2,38–42 The ADC(2) method chosen for this study represents a pragmatic compromise between accuracy and computational cost.^43,44 For cyclazine and heptazine, both ADC(2) and CC2 slightly overestimate the excitation energies, but provide a reasonably good estimate of the ST energy gap.

3. Results and discussion

3.1. Vertical excitation energies

The ground-state equilibrium geometries of BCN and PX with X = O (abbreviated PO in what follows), possess D_3h symmetry, a feature that profoundly influences their electronic properties. The highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) of both compounds are doubly degenerate. This orbital degeneracy gives rise to six valence excited states, three singlet states and three triplet states, with spatial symmetries A₁′, A₂′, and E′.

The vertical excitation energies of these lowest electronic states of the compounds shown in Chart 1 are listed in Table 1. Notably, in the symmetric systems BCN and PO (D_3h symmetry), the lowest excited singlet state is the nondegenerate ¹A₂′ (ππ*) state. It is interesting to note that the corresponding triplet state (³A₂′) is energetically higher by 0.312 eV in BCN and by 0.387 eV in PO (negative singlet–triplet splitting). The ³A₂′ state is, however, not the lowest triplet state in both molecules, because the singlet–triplet splitting of the E′ and A₁′ states is positive and much larger than that of the A₂′ states (see Table 1). As a result, the lowest triplet state has E′ symmetry in BCN, but A₁′ symmetry in PO. The D_3h symmetry is reduced in asymmetrically substituted molecules, such as PNH (X = NH) and PS (X = S), which are of C_2v and C_s symmetry, respectively. This reduction of symmetry removes the degeneracy of E′ states.

Table 1 Vertical excitation energies E (in electron volts), oscillator strengths (in parentheses) and the S₁–T₁ vertical energy gaps (Δ_ST) of the structures in Chart 1, computed at the ADC(2) level. Note that both components of degenerate E′ states are included in the table

BCN(D3h)		PO(D3h)		PNH(C2v)		PS(Cs)
State	E	State	E	State	E	State	E
Triplet states
^3E′	1.914	^3A1′	1.911	^3B1	1.734	^3A′′	1.550
^3E′	1.914	^3E′	1.937	^3A1	1.941	^3A′	1.552
^3A1′	1.934	^3E′	1.937	^3A1	2.111	^3A′	1.832
^3A2′	1.943	^3A2′	1.984	^3B1	2.371	^3A′′	2.137
Singlet states
^1A2′	1.631 (0.0)	^1A2′	1.597 (0.0)	^1B1	1.620 (0.037)	^1A′′	1.446 (0.023)
^1A1′	2.253 (0.0)	^1A1′	2.254 (0.0)	^1A1	2.398 (0.005)	^1A′	2.006 (0.017)
^1E′	2.370 (0.293)	^1E′	2.572 (0.468)	^1A1	2.659 (0.509)	^1A′	2.424 (0.400)
^1E′	2.370 (0.293)	^1E′	2.572 (0.468)	^1B1	2.699 (0.259)	^1A′′	2.549 (0.458)
ΔST	−0.283	ΔST	−0.314	ΔST	−0.114	ΔST	−0.104

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Remarkably, all the species included in Table 1 are IST systems, that is, the S₁–T₁ energy gap, defined as Δ_ST = E_S − E_T, is negative, where E_S and E_T are the lowest singlet and lowest triplet state energies. In the symmetric PO and BCN molecules, the S₀–S₁ transition is symmetry forbidden (oscillator strength f = 0). Breaking the trigonal symmetry axis by substituting one of the oxygen atoms of PO with a NH group (in PNH), or with sulfur (in PS), induces a nonzero transition dipole moment. The reduction of the symmetry is also accompanied by a decrease in the magnitude of the negative singlet–triplet energy gap from −0.283 for BCN to −0.104 for PS.

To gain deeper insights into this phenomenon, Fig. 1 depicts the electron density of molecular orbitals involved in the lowest-energy electronic transitions. The electron densities presented in Fig. 1 were computed assuming equal occupation of both components of the degenerate E′ HOMO and LUMO orbitals in BCN and PO. Asymmetric substitutions to the PX system remove orbital degeneracy, but still, the S₁ state in both (PNH and PS) molecules, contains comparable contributions from HOMO/LUMO and HOMO−1/LUMO+1 transitions. Thus for the sake of transparency, the assumption of equal occupation of HOMO and HOMO−1 as well as LUMO and LUMO+1 orbitals were used in computation of electron density involved in this electronic transition.

Fig. 1 Electron density of (near)degenerate HOMO and LUMO orbitals of the compounds presented in Chart 1.

Fig. 1 reveals that the distribution of the electron densities of HOMO and LUMO displays a characteristic pattern which has been found to be typical for IST systems.¹⁸ For the HOMO and, to a lesser extent, for the LUMO, the electronic charge distribution is expelled from the interior of the molecular framework. Instead, it is largely localized on alternating atoms along the rim. This distribution minimizes the exchange integral, which results in a small S₁–T₁ splitting. However, it also reduces the transition dipole moment and thus the oscillator strength of the S₀–S₁ transition. Chemical substitutions that reduce the symmetry, such as those in PNH and PS, increase the overlap between electron densities. As mentioned earlier, this affects both the S₁–T₁ energy gap and the oscillator strength of the S₀–S₁ transition.

3.2. Effect of additional symmetry reduction by chemical modification

While a negative Δ_ST represents an enticing prospect for applications in OLED devices, the low fluorescence intensity of the S₁ state poses a significant drawback. To delve deeper into the impact of symmetry reduction on the photophysical properties of IST molecules, an overview of the computed spectroscopic parameters for four specifically chosen chemically modified PX systems is provided in Table 2. Additionally, an extensive selection of chemically modified PX systems is presented in the ESI,† offering a broader exploration of the effects of symmetry alteration on their energy level structure.

Table 2 Vertical excitation energies (in eV) of the lowest excited singlet and triplet states, oscillator strengths for the S₀ → S₁ transition (in parentheses), and the respective S₁–T₁ energy gap (Δ_ST) of selected PX molecules, computed at the ADC(2) level of theory

State	PO-4N	PNH-4N	PS-2N	PNH-2NO2
T1	1.981	1.796	1.623	1.796
S1	1.953 (0.072)	1.761 (0.086)	1.519 (0.039)	1.792 (0.132)
ΔST	−0.028	−0.035	−0.104	−0.004

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Table 2 demonstrates that appropriate chemical modifications to the molecular structure can notably enhance the oscillator strength of the S₀–S₁ transition. However, this effect correlates strongly with a reduction in the magnitude of the negative singlet–triplet splitting: the greater the oscillator strength of the transition, the smaller the negative singlet–triplet energy gap. Bearing in mind that the ADC(2) method tends to overestimate this effect,^43,45 it can be concluded that the molecules listed in the table represent TADF systems with exceptionally small positive ST splitting.

The correlation between the oscillator strength and the singlet–triplet energy gap is revealed by the correlation diagram presented in Fig. 2, which showcases three families of PX (X = O, NH, S) systems featuring nitrogen substitutions at the molecular rim. Apart from the systems with symmetry-forbidden S₀–S₁ transitions (f = 0), a clear overall correlation between f and Δ_ST is discernible. On the other hand, the scattering of the symbols in the diagram reveals nuanced possibilities of manipulating Δ_ST and f by molecular symmetry reduction via CH/N substitutions at the rim. This variability highlights the flexibility inherent in designing PX systems tailored to exhibit specific desired properties, thereby opening up exciting possibilities for manipulating their photophysical functionality. Similar trends have been found for heptazine-based IST compounds.^3,16

Fig. 2 Correlation between the S₁–T₁ energy gap and the oscillator strength of the S₀ → S₁ transition for different carbon/nitrogen replacements at the molecular rim (PO - blue circles, PNH - red squares, PS - green triangles), computed at the ADC(2) level (see also Tables S1, S3 and S5 in the ESI†).

Another intriguing aspect of the computed photophysical properties of the PX systems is the correlation between the energy of the S₀–S₁ transition and the number of nitrogen atoms present in the molecular rim. Fig. 3 shows that an increased number of nitrogen atoms raises the energy of the S₀–S₁ transition and enhances the magnitude of the singlet–triplet inversion (for clarity, only nitrogen replacements maintaining the triple symmetry axis of the PX system are included in Fig. 3). This observation underscores the intricate relationship between the molecular composition and photophysical properties which can be exploited for the tailored design of OLED chromophores. The impact of the asymmetric CH/N substitutions at the molecular rim on the energy of the S₀–S₁ transition and its oscillator strength is additionally documented in the ESI.†

Fig. 3 Correlation between the energy of the S₀–S₁ transition (left axis, blue circles) and the S₁–T₁ energy gap (right axis, black diamonds), plotted against the number of nitrogen atoms present in the molecular rim (molecules PO, PO-3N, PO-6N, PO-9N from Table S1 of the ESI†), obtained with the ADC(2) method.

Previous work¹⁹ has established the effect of substitution sites of donors and acceptors on the oscillator strength and the ST energy gap of heptazine derivatives. According to these findings, electron donors attached at atoms which carry electron density in the LUMO orbital preserve the negative energy gap of the molecule, while electron acceptors on the same atoms decrease the gap or even make it positive. Conversely, electron acceptors attached to atoms which carry electron density in the HOMO preserve the negative ST gap, while the gap is reduced by substitution of electron donating groups at these atoms. This rule applies to the molecules considered in the present study (see electron densities in Fig. 1 and in the ESI,† Tables S2, S4 and S6).

3.3. Radiative emission properties

The spectral characteristics discussed so far were computed for the ground-state equilibrium geometries of the PX systems. However, for the characterization of radiative emissions (fluorescence, delayed fluorescence, and phosphorescence), it is crucial to consider the photophysical properties at the equilibrium geometries of the excited states (S₁ and T₁). The vertical excitation energies in Table 1 show that in the molecules with a three-fold symmetry axis (BCN and PO), the lowest excited singlet state S₁(A₂′) is well separated from higher excited singlet states. The two lowest excited triplet states (³A₁′ and ³E′), on the other hand, are nearly degenerate. While geometry optimization of the non-degenerate singlet state maintains D_3h symmetry, the potential-energy surface (PES) of the degenerate (E′) state may exhibit symmetry breaking due to the Jahn–Teller (JT) effect.

In the ADC(2) method as currently implemented in the TURBOMOLE package, the exploitation of symmetry is limited to Abelian symmetry groups. The highest Abelian subgroup of D_3h is C_2v. In C_2v symmetry, the A₂′ symmetry representation becomes B₁, and the two degenerate components of the E′ representation transform as A₁ and B₁, respectively. While the BCN molecule maintains D_3h symmetry when optimizing the geometry of the S₁(B₁) state, this symmetry is lost when the geometries of the ³B₁ and ³A₁ states are optimized. The Hessian computed at these stationary points indicates that the optimized ³A₁ state represents a local minimum, while the optimized ³B₁ state is a first-order saddle point of the JT-deformed two-dimensional PES of the ³E′ state. Its relative energy (0.24 eV) with respect to the minimum of the ³A₁ state represents the energy barrier for the so-called pseudo-rotation on the PES of the T₁ state.

A qualitatively similar energy-level scheme is obtained for PO. Both schemes are shown in Fig. 4. It is evident that the non-degenerate S₁(A₂′) state (¹B₁ symmetry in the C_2v point group) of both molecules exhibits rigidity, that is, the energy relaxation upon geometry optimization is small (0.040 eV and 0.045 eV, respectively) and does not break molecular symmetry. For the degenerate triplet state ³E′, on the contrary, the energy relaxation upon geometry optimization is much more pronounced (0.301 eV and 0.378 eV, respectively). A similarly pronounced relaxation of the energy is also observed in the ¹E′ state. In both E′ states, this effect is attributed to JT-induced geometric instability.

Fig. 4 Energy-level schemes of BCN (a) and PO (b), determined at the ADC(2) level. Solid lines denote the optimized energy level of the respective electronically excited states, while dashed (dotted) lines denote the vertical energy of the singlet (triplet) states computed at the equilibrium geometry of the electronic ground state. Up and down arrows denote vertical absorption and emission, respectively. Numbers denote energies in electron volts.

Several interesting conclusions emerge from Fig. 4. Particularly noteworthy is the singlet–triplet inversion measured by the vertical energy difference computed at the ground-state equilibrium geometry, which is remarkably large (−0.283 eV for BCN and −0.314 eV for PO).

However, the corresponding difference between the adiabatic (geometry-optimized) energies of singlet and triplet states, which represents the 0–0 spectroscopic transition energy, is much smaller (−0.022 eV and −0.007 eV, respectively). Furthermore, the difference between the vertical fluorescence (S₁–S₀) and phosphorescence (T₁–S₀) energies, representing the difference of the peak maxima of the fluorescence (E_fl) and phosphorescence (E_ph) spectra, respectively, becomes positive (0.312 eV and 0.370 eV, respectively).

A comparison of the level schemes of both molecules clearly shows that the removal of the central boron atom from BCN and the replacement of the remaining nitrogen atoms by oxygen atoms (PO) have a minor effect on the photophysics. Despite the removal of the central skeleton atom, the PO molecule exhibits rigidity which is similar to BCN. The inversion of the vertical singlet and triplet states is strongly tied to the electronic conjugation along the molecular rim.

Replacing one of the oxygen atoms in PO with the NH group conserves the number of electrons in the system, but reduces the molecular symmetry to C_2v or lower. The relevant energy-level schemes for PNH-4N (Table 2) and PNH-6N (Table S3 in the ESI†), where six nitrogen atoms are symmetrically distributed at the corners of pyran units (see ESI†), are shown in Fig. 5. It is evident that geometry optimization of the lowest singlet and triplet states of PNH-4N (both having the B₁ symmetry in the C_2v group) does not further lower the molecular symmetry and stabilizes both states by nearly the same amount of energy (0.076 eV for the ¹B₁ state and 0.069 for the ³B₁ state). The adiabatic energies of these states are inverted by merely −0.042 eV. The peak energy of phosphorescence is predicted to be only 0.03 eV lower than the peak energy of fluorescence.

Fig. 5 Energy-level schemes of PNH-4N (a) and PNH-6N (b) molecules determined at the ADC(2) level. Solid lines denote the optimized energy levels of the respective electronically excited states, and dashed (dotted) lines denote the vertical energy of the singlet (triplet) states computed at the equilibrium geometry of the ground state. Up and down arrows denote vertical absorption and emission, respectively. Numbers denote energy in electronvolts.

The symmetric distribution of six nitrogen atoms along the molecular rim of the PNH molecule (PNH-6N) maintains C_2v symmetry, but notably decreases the energy gap between the ³B₁ and ³A₁ states (Fig. 5b). While unconstrained geometry optimization of the lowest excited singlet state conserves C_2v symmetry, the optimization of the lowest triplet state results in symmetry lowering to C_s (only the molecular plane is conserved). The magnitude of the negative S₁–T₁ vertical energy gap (−0.230 eV) decreases to −0.103 eV for the adiabatic energies. The difference of the emission maxima from these states becomes positive (0.540 eV). Inspection of the equilibrium geometry of the ³A′ state (Table S7 of the ESI†) reveals that, apart from the JT distortion, a significant amount of single-double CC bond alternation occurs.

4. Summary

Pyran-substituted triangular aromatic hydrocarbons represent a new category of molecular systems that can be tailored to exhibit robustly negative singlet–triplet energy gaps. The results of this computational study confirm that previously discussed triangular carbon nitrides and triangular boron carbon nitrides are not the only stable organic molecules capable of displaying S₁–T₁ inversion.

A common feature of these IST systems is the electronic conjugation along the outer rim. This effect can be achieved by either filling the interior of an aromatic arene or azaarene molecule with an insulating motif (such as boron nitride) or by removing the central atom(s) altogether, as in the systems examined in this study. The triangular molecules investigated in this work can be considered as cyclic oligomers composed of the pyran units and its derivatives. This finding paves a new path for constructing IST systems utilizing organic molecular units.

In the computational literature the vertical energy gap between the S₁ and T₁ states, computed at the equilibrium geometry of the ground state, is taken as the definition of the ST gap. However, with respect to photophysical processes occurring in and between these states, such as intersystem crossing (ISC) and reverse-ISC (RISC), the adiabatic energies of these states are significant. The adiabatic energies determine the 0–0 line in the fluorescence and phosphorescence spectra.

Data availability

The data supporting this article have been included as part of the ESI.†

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The Authors would like to thank Wolfgang Domcke for fruitful discussions during the preparation of the manuscript. This research was funded by National Science Centre of Poland, grant number: 2020/39/B/ST4/01723. We gratefully acknowledge Polish high-performance computing infrastructure PLGrid (HPC Center: ACK Cyfronet AGH) for providing computer facilities and support within computational grant no. PLG/2024/017058.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4cp01658k

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