Optical versus fundamental gap and symmetry considerations

As recently emphasized by Bredas⁴⁵, and particularly appreciated in solid state physics^46,47, the notion of a “band gap” has often been used broadly to correlate experimental data obtained by different techniques, such as photoemission, photoconduction, or optical absorption, with the different underlying physical processes. One should, however, distinguish between the “optical gap”, E_opt, and the “fundamental gap”, E_g (Fig. 1b, c). The optical gap is the lowest-energy transition observed in the experimental absorption spectra corresponding to the vertical excitation energy from the ground state to the first dipole-allowed excited state. By contrast, the fundamental gap is defined as the difference between the energies of the first ionization potential and the first electron affinity (EA)^45,48.

A comparative discussion of the experimentally accessible values of E_opt and E_g on the basis of excitonic states must consider molecular symmetry arguments. For the hypothetical carbon allotrope carbyne (C ≡ C)_∞ with a sufficiently large number of carbon–carbon triple bonds, the derivatives with (C ≡ C)_n and (C ≡ C)_n+1 are indistinguishable in their properties. The corresponding point group, in the ideal (linear) conformation, is expected to be D_∞h with degenerate frontier molecular orbitals that possess Π_u and Π_g symmetries (Fig. 1b). A transition between these frontier orbitals will lead to a splitting of the excitation energy levels into three distinct electronic states: one degenerate state, Δ_u, as well as two non-degenerate states, Σ⁺_u and Σ⁻_u. The dipole-allowed representations (µ_xyz) as given by the D_∞h character table are Π_u and Σ⁺_u. Accordingly, the symmetry-allowed transition of lowest energy should populate the lowest-energy, dipole-allowed Σ⁺_u state, which should therefore be defined as the optical gap E_opt of carbyne (Fig. 1b). The lowest-lying, but dipole-forbidden transition to the Σ⁻_u state (which is hence undetectable in UV-Vis spectra) must be lower in energy than the fundamental gap E_g. The difference is the exciton binding energy, which is substantial in organic molecules and materials (Fig. 1b).

The presence of terminal substituents R in oligoynes R–(C ≡ C)_n–R (R[n]) perturbs the energy profile when compared with the parent compounds H–(C ≡ C)_n–H (H[n]) and therefore alters the symmetry rules. Computational analyses of oligoynes endcapped with two identical substituents CR₃ show the point group D₃ (owing to free rotation of the end groups), with frontier orbitals of E symmetry (Fig. 1c). Again, splitting of the excitation energy levels upon an electronic transition yields three electronic states, one degenerate state with E symmetry and two non-degenerate states A₁ and A₂, of which only A₂ and E are dipole-allowed representations (µ_xyz) according to the D₃ character table. Qualitatively similar to the situation in carbyne, the lowest-energy transition to the A₁ state remains symmetry-forbidden while transitions to the non-degenerate A₂ state are symmetry-allowed. Different from carbyne, however, the transition to the degenerate state E now becomes (weakly) allowed owing to the lower symmetry of the molecules (D₃ versus D_∞h). This should result in a series of weak absorption bands in the experimental UV-Vis spectra of oligoynes at lower energy and accordingly lead to a drastically altered optical gap E_opt than that estimated from the A₂ transitions (Fig. 1c). When the end groups impose an even lower symmetry (e.g., C₂ point group), the degeneracy of the frontier orbitals is marginally lifted, resulting in one main (bright) optical transition in experimental spectra along with several further weak transitions at lower energies (see Wakabayashi et al.³¹ for a detailed analysis of the relevant point groups). In any case, these bands are expected to disappear at extended chain lengths as the influence of the terminal substituents is decreased, but even minor deviations from an idealized linear conformation will render them weakly allowed²⁷.

These considerations motivated us to perform a detailed investigation of the solution-phase spectroscopic properties of oligoyne derivatives by experimental and computational means (a detailed description of the methodologies is given in the Methods section). We chose two series of oligoynes with terminal groups of different chemical nature and symmetry, namely the C₂-symmetric glycosylated oligoynes Glu[n] and the D₃-symmetric oligoynes Tr*[n] featuring triarylmethyl end groups (Fig. 1a)^17,22. The terminal groups are not electronically conjugated with the oligoyne segments in either series, and the observed photophysical properties arise from the oligoynes and not from end-group effects.

Steady-state UV-Vis spectroscopy

The UV-Vis absorptions of the oligoynes Glu[n] (in acetonitrile or dichloromethane solutions) and Tr*[n] (in a hexane solution) show a red-shift with increasing number of triple bonds n and display a characteristic vibronic fine structure starting approximately with Glu[4] and Tr*[4] (Fig. 2a, b; Supplementary Fig. 1). The spectra of the longer oligoynes display several well-defined maxima, among which the highest wavelength absorption, λ_main, exhibits the highest intensity (Table 1; Supplementary Table 1).

For example, λ_main for Glu[n] shifts from 240 nm for Glu[4] to 390 nm for Glu[12]. At the same time, the vibronic progressions (ΔE (S₀ → S_n) in Table 1) decrease from ~2000 cm⁻¹ for Glu[4] to ~1800 cm⁻¹ for Glu[12], reflecting the known trend of greater conjugation as oligoyne length is increased⁴⁹. Moreover, the molar absorption coefficient, ε_main, increases significantly with the number of C ≡ C bonds (Supplementary Table 1). A comparison of the two oligoynes series indicates that λ_main of the Tr*[n] series is slightly red-shifted by 10–30 nm compared with that of Glu[n]. The difference in absorption energies between the two series diminishes as a function of length¹³, and can likely be attributed to a combination of solvatochromic effects and to hyperconjugation, i.e., the orbital overlap between the conjugated π-system and the end groups. The latter is well documented in the literature⁴⁹, and particularly observed in the computational analysis of the orbitals participating in the main optical transitions of the Tr*[n] series of oligoynes (vide infra). All of these observations are in excellent agreement with previous reports on other series of oligoynes^{13,14,15,16,17,18}. A closer inspection of the absorption spectra of Glu[n] and Tr*[n] reveals additional absorption bands with weak intensities (λ_weak), at wavelengths significantly higher than λ_main (Fig. 2a, b, insets). The highest wavelength absorption maximum of this series of bands is red-shifted by >100 nm with respect to the main absorption maxima (Table 1). No significant changes are observed in temperature-dependent UV-Vis spectra, so that aggregation of the oligoynes in solution can be excluded as the origin of λ_weak (Supplementary Fig. 2).

As we will elucidate in the following sections, we infer that the S₁ state remains dark owing to the dipole-forbidden character of the S₀ → S₁ transition, in agreement with previous computational analyses and experimental studies of oligoynes^30,31,35. We hence attribute the series of “weak signals” to the spectroscopically weakly allowed S₀ → S_2/3 transition to the degenerate state E. In accordance with this interpretation, the amplitudes of λ_weak absorptions are two to three orders of magnitude lower relative to the main optical absorptions, λ_main, that can hence tentatively be assigned to a higher S₀ → S_n transition.

Fluorescence spectroscopy

Although the fluorescence of oligoynes is generally weak³⁸, we successfully recorded fluorescence and excitation spectra of the Tr*[n] series in hexane solutions (Fig. 2c). The fluorescence spectra are almost mirror images of the λ_main bands in the absorption spectra, and the corresponding excitation spectra are exact matches of the latter. Considering that the weakly allowed and the dipole-forbidden transitions are non-emissive and, in any case, too low in energy to contribute to the emission bands, the fluorescence originates only from the tentatively assigned S_n → S₀ transition. Notably, these oligoynes are thus apparently among the few examples where fluorescence does not obey Kasha’s rule, that is, emission occurring from the vibrationally relaxed lowest excited state⁵⁰. The large relevant energy gaps (e.g., ~8000 cm^–1 or 1 eV for Tr*[8]) and low oscillator strengths of the fundamental transitions render either thermal repopulation or reverse internal conversion processes unlikely, so that prompt fluorescence from the excited state remains as the presumed dominant deactivation pathway of photoexcited oligoynes⁵¹.

From the S₀ → S_n absorptions and shortest wavelength fluorescence, Stokes shifts as small as 280 cm⁻¹ are derived (Fig. 2c, Supplementary Fig. 3), which can be explained with the structural rigidity of oligoynes and little bond length changes upon excitation. Although small Stokes shifts are typically correlated with high fluorescence quantum yields, Φ_F, we observe quantum yields on the order of only Φ_F = 10⁻⁴ and negligible differences of the fluorescence quantum yields with increasing conjugation length (Supplementary Table 2). Because of the low quantum yields, multiple attempts to determine the fluorescence lifetime by means of either single photon counting or up-conversion were unsuccessful. The low quantum yields are not surprising given that fluorescence does not occur from the lowest electronic state, and rapid deactivation pathways such as internal conversion to the non-emissive S₁ state as well as intersystem crossing presumably contribute to a lowering of the S_n → S₀ fluorescence quantum yields (vide infra).

Time-resolved spectroscopy

Time-resolved absorption spectroscopy with Glu[n] (n = 6, 8, 10, 12; in DCM) and Tr*[n] (n = 4, 6, 8, 10; in hexane) at excitation wavelengths of 390 and 258 nm, respectively, show very similar spectroscopic features (Fig. 3, Supplementary Fig. 4–11) and dynamics (Table 1) for all molecules. The excitation wavelengths have been chosen to pump into either λ_main or λ_weak, and Supplementary Fig. 12 illustrates the lack of any meaningful dependence of the observed species and lifetimes on the excitation wavelength. For Glu[12] as a representative example, a broad photoinduced absorption (PIA) centered at 586 nm appears promptly after excitation (Fig. 3a). Within the first few picoseconds, this broad PIA narrows substantially, and a vibrational progression appears. Simultaneously, the center of the PIA undergoes a blue shift to 577 nm. Both effects can be related to the relaxation to S₁, consistent with the discussion of the steady-state UV-Vis and fluorescence spectra. On a longer timescale, we observe the rise of a second PIA at 464 nm, assigned to triplet-triplet absorptions T₁ → T_n upon intersystem crossing.

The underlying dynamics are well reproduced by means of deconvoluting the instrument response function and three exponential functions. We interpret the three time constants to describe the ultrafast transformation of S_n/S_2/3 into S₁, the intersystem crossing of the latter to yield T₁, and the subsequent decay to the ground state S₀. The assignment of the intermediate, evolution-associated spectroscopic features to the S₁ state are corroborated by the fact that the population of the state with time constants between 1 and 10 ps is, by far, too slow for a direct vertical excitation, which typically occurs within the instrument response function of the laser system (<1 ps). Moreover, previous studies of related systems by means of time-resolved infrared spectroscopy excluded processes such as intramolecular/vibrational relaxation and/or solvation processes³⁵.

Comparing the transient absorption spectra of Tr*[10] and Glu[10] indicates that the end groups have no major impact on these photo-initiated processes (Fig. 3b, c). In both series of oligoynes, the deactivation of S_n as well as S_2/3 and simultaneous population of S₁ is followed by intersystem crossing to T₁, which slows with increasing length of the oligoyne segment (and is slightly faster for Glu[n] compared with Tr*[n], possibly because of the higher excitation energy used in the latter series). The rapid transformation of the excited state (S_n) into a non-emissive, lower energy state (S₁) is likely to be nearly quantitative, a fact that is derived from the very low quantum yields of the competing fluorescence pathway (vide supra).

For Glu[8] and Glu[6] an additional broad peak is discernable as part of the S₁ → S_n transition that is subject to a slight blue-shift during intersystem crossing  (Supplementary Figs. 10–11). Fazzi et al.³⁴ have reported similar observations in their analysis of α,ω-dinaphthyloligoynes and attributed these peaks to the superposition of several transitions from the first singlet excited state. The observation of these S₁ → S_n transitions corroborates that the excitation of the shorter oligoynes at a lower energy than their most intense transition λ_main furnishes exactly the same excited state behavior as for the longer derivatives that are excited within λ_main. This, in turn, lends further support to our hypothesis that the main optical transition λ_main is not related the S₀ → S₁ transition. Instead, a higher, short-lived S_n excited state is populated by the pump pulse, which is rapidly deactivated to the S₁ state, which is then probed by transient absorption spectroscopy.

The same final, evolution-associated spectroscopic features assigned to the T₁ → T_n absorptions in the femtosecond transient absorption time window are also discernable by nanosecond spectroscopy, which we have performed in the absence and presence of oxygen (Supplementary Fig. 13–16). As a representative example, the differential absorption spectrum of Tr*[10] shows maxima at 440, 500, 560, and 600 nm, which have lifetimes of 7.7 µs in argon and 54 ns in oxygen-saturated solutions (Table 1). This difference in lifetimes indeed confirms that these features are related to a triplet excited state for which deactivation occurs via triplet-triplet energy transfer to molecular oxygen. The formation of singlet oxygen was independently confirmed in emission experiments, which show the characteristic signal at 1275 nm (Supplementary Fig. 17).

Both steady-state and time-resolved absorption spectroscopy hence provide unambiguous experimental evidence that the observed weak intensity absorptions at higher wavelengths, λ_weak, are intrinsic photophysical features of oligoynes.

Computational analysis of the optical transitions

The tentative assignment of the main (λ_main) and weakly allowed (λ_weak) optical absorptions of the two series of oligoynes Glu[n] and Tr*[n] to S₀ → S_n and S₀ → S_2/3 transitions, respectively, is corroborated by plotting the experimentally and computationally determined energies against the inverse number of carbon–carbon triple bonds n (Fig. 4). Both series each converge to virtually identical saturation values. The linear extrapolation of the S₀ → S_n transition energies (λ_main) for 1/n → 0 furnishes saturation values for the experimental optical transitions of the Glu[n] and Tr*[n] series of E_opt = 2.21 eV (λ_opt = 561 nm) and 2.28 eV (543 nm), respectively. These values compare very well to previously determined values of, e.g., E_opt = 2.18–2.19 eV (λ_opt = 565–570 nm) for ⁱPr₃Si-terminated oligoynes¹⁵, oligoynes with third-generation Frechet-type dendrons as end groups¹⁶, or oligoynes endcapped with rhenium complexes¹⁸. An extrapolation of the significantly red-shifted S₀ → S_2/3 transition energies (λ_weak) for 1/n → 0, however, results in saturation values of E_opt = 1.53 eV (λ_opt = 810 nm) for the Glu[n] series and E_opt = 1.64 eV (λ_opt = 756 nm) for the Tr*[n] series, which is ~0.6–0.7 eV lower than the E_opt value suggest by analysis of λ_main across the entire series of Tr*[n] (n = 2–22)¹⁷.

In good agreement with the experimental results, saturation values for the Glu[n] and Tr*[n] series of 2.59 eV (478 nm) and 2.50 eV (496 nm), respectively, are obtained from the computations of the S₀ → S_n transitions (λ_main), as well as 1.89 eV (656 nm) and 1.96 eV (634 nm) for the and S₀ → S_2/3 transitions (λ_weak), respectively. All trends and differences are hence fully reproduced, although the energies are slightly overestimated. Although the employed CAM-B3LYP functional provides good estimates for the excited state of π-conjugated chains⁵², it is known to have a tendency of over-localizing the electron density distribution^52,53 and, in turn, overestimating transition energies, especially when the effects of vibronic coupling are neglected⁵⁴. This is also the reason why the computations do not fully reproduce the minor differences between the Glu[n] and Tr*[n] series and the deviations from the linear extrapolation for short oligomers that can be attributed to hyperconjugation to the end groups in the Tr*[n] series (Supplementary Figs. 18 and 19)⁴⁹.

The good agreement between experimental and computational results for both series and both the S₀ → S_n (main) transitions as well as the S₀ → S_2/3 is relevant because it gives confidence in the computational determination of the S₀ → S₁ transitions that cannot be probed experimentally (Fig. 4b). Notably, the observed deviations are smaller for the S₀ → S_2/3 (weakly allowed) transitions because the quasi-forbidden state is significantly more compact than the bright state and does not benefit as much from a delocalization by hyperconjugation toward the end groups in the Tr*[n] series.

Using the values of λ_weak therefore suggests that the optical gap of oligoynes of finite length is significantly smaller than previously reported, converging to E_opt = 1.53 eV for the Glu[n] series and E_opt = 1.64 eV for the Tr*[n] series. As the S₀ → S_2/3 transitions should become symmetry-forbidden for a sufficiently large number of triple bonds in (C ≡ C)_n, it may be debatable whether this estimate also holds true for carbyne. In practice, however, even minor deviations from an idealized linear geometry will render S₀ → S_2/3 transitions partially allowed. The photophysical properties of carbyne, and its photochemistry, are therefore better described by the optical gap estimated based on extrapolations of the data presented herein using λ_weak, i.e., E_opt = 1.5–1.6 eV, about 0.6–0.7 eV smaller than previously inferred.

Moreover, the presented results provide an estimate for the fundamental gap of carbyne (E_g), which should be below the S₀ → S_2/3 but above the S₀ → S₁ transition energy (owing to the exciton binding energy). We conclude that its upper boundary should hence be E_g = 1.6 eV, as well. A lower boundary can be estimated from the extrapolation of the computational determination of the S₀ → S₁ transition energies for 1/n → 0, which furnishes saturation values of 1.75 and 1.84 eV for the Glu[n] and Tr*[n] series, that is, ~0.2 eV below the computationally determined saturation values for the S₀ → S_2/3 transition of 1.9–2.0 eV (at the given DFT level). It should be noted that the more accurate prediction method for saturation, reported by Meier et al.⁵⁵, has also been attempted but is unreliable owing to the limited number of experimental data points. Given the systematic and explicable overestimation of all transition energies of ~0.3 eV across both S₀ → S_2/3 and S₀ → S_n transitions and both series of molecules in the computations as compared to the respective experimental values, the lower boundary of the fundamental gap should hence be estimated to be on the order of E_g = 1.4–1.5 eV.

A systematic investigation of the ground-state and excited-state properties of two series of oligoynes has allowed us to determine the intrinsic photophysical properties of oligoynes, which are of particular interest as analogs for the elusive carbon allotrope carbyne. Our findings prove that the series of “weak absorptions”, λ_weak, that one can observe at significantly lower transition energies than the main optical absorptions, λ_main, are intrinsic photophysical features of oligoynes. Combining the experimental and computational results enabled us to assign λ_weak to the weakly allowed S₀ → S_2/3 transition to the degenerate state E. The main optical absorptions hence correspond to a higher-energy S₀ → S_n transition, whereas the lowest S₀ → S₁ transition remains dipole-forbidden. Moreover, our results require that both the optical gap (E_opt) and the fundamental gap (E_g) of oligoynes and carbyne are reassessed. We find that the optical gap of oligoynes converges to E_opt = 1.5–1.6 eV. These values are 0.6–0.7 eV smaller than previous estimates based on λ_main. Furthermore, the combined experimental and computational results provide an estimate of 1.4–1.6 eV for the fundamental gap (E_g) of a hypothetical sp-hybridized carbon allotrope carbyne. This value compares to ca. 5.5 eV in diamond (sp³)⁵⁶, 8.8 eV for polyethylene (CH₂–CH₂)_n (sp³)⁵⁷, and 1.8 eV in polyacetylene (CH=CH)_n (sp²)⁴⁴. The results of the present study provide more accurate and experimentally verified predictions for the photophysical properties of molecules with extended π-conjugated systems based on sp-hybridized carbons that will help guide the preparation of carbon-rich materials with tailored properties in the future.