Where, C is the effective electronic coupling matrix element and λi corresponds to the reorganization energy associated with the vibrational modes promoting non-radiative relaxation. Equation (1) formalises the energy-gap law, which predicts an increase in non-radiative decay rate with decreasing the energy-gap.

High-frequency molecular vibrations (1000 – 1600 cm-1) are ubiquitous in π-conjugated organic molecules and are strongly coupled to electronic excited states when they directly modulate the π-bond order4. This is particularly problematic when the energy gap is directly coupled to π-bonding alternations, such as the phenylene ring stretch mode6. On the other hand, low-frequency modes <500 cm-1 are associated with high-mass displacement and are structurally delocalised in nature. These modes contribute less to fast non-radiative decay, which is captured by the term in the Equation (1), representing the frequency of the promoting vibrational mode.

Fig. 1 illustrates these relationships for the widely studied organic semiconductor, pentacene. Using density-functional theory (see Methods), we calculated the vibrational modes of an isolated pentacene molecule. The most intense high-frequency mode at 1370 cm-1 corresponds to a C-C bond stretching vibration (Fig. 1a, top), while a representative low-frequency mode at 264 cm-1 is associated with backbone torsional motion (Fig. 1b, top). Assuming the same molecular structure, the internal conversion from the excited electronic state to the ground electronic state requires significantly fewer vibrational quanta in the case of a high-frequency vibration (Fig. 1a bottom), resulting in improved vibrational overlap and therefore an enhanced non-radiative decay relative to a low-frequency mediated internal conversion process (Fig. 1b bottom).

In Fig. 1c, we present the excited state Raman spectra of pentacene (blue spectrum) measured using impulsive vibrational spectroscopy7 (see below). The spectrum shows clearly that high-frequency modes (1160 and 1370 cm-1) couple efficiently to the excited state. We also plot in black circles the non-radiative decay constant for each normal mode, as calculated using Equation (1), considering that mode as the dominant deactivation pathway. It can be seen that the high-frequency modes lead to significantly faster rates of non-radiative recombination. For instance, taking DE = 14518 cm-1 (1.8 eV) and reorganisation energy = 1109 cm-1 (0.14 eV) (see SI section-12), for pentacene, the dominant 1370 cm-1 mode leads to a non-radiative rate ~1012 times faster that of the 256 cm-1 mode.

Exciton-vibrational coupling to high frequency modes as seen here for pentacene is generally observed for organic semiconductors8–12, and causes low-bandgap organic semiconductors to exhibit rapid non-radiative decay dynamics. This limits the efficiency of deep-red/NIR organic light emitting diode2 (OLEDs) and organic photovoltaics13 (OPVs). The key question is therefore, can we decouple the exciton from high frequency vibrations?

**Exceptions that appear to violate the Energy Gap Law **

There are a few selected examples of NIR organic emitters with high photoluminescence quantum efficiency (PLQE) and electroluminescence quantum efficiency (EQEEL). As examples, we focus here on two such systems, APDC-DTPA14–16 and TTM-Donor (Donor: 3NCz/3PCz17) (see Fig. 2a) . Both materials show luminescence from an intra-molecular charge transfer exciton14,17. We select APDC-DTPA as an example of a highly efficient NIR emitting thermally activated delayed fluorescence (TADF) system (PLQE = 63% at 693 nm at 10 weight percentage in solid state guest-host blend with TPBi)14. The key design feature of TADF systems as first developed by Adachi and co-workers18, is the introduction of donor-acceptor character so that the charge transfer exciton has spatially reduced electron-hole overlap that reduces the singlet-triplet exchange energy. This allows triplet excitons formed on the emitter molecules following electrical injection to be thermally upconverted into the bright singlet manifold, thus avoiding non-luminescent losses related to dark triplet excitons. While this strategy has been extremely successful, it is surprising that it should work so well, as the introduction of the CT character reduces the oscillator strength of the radiative transition. This is reflected in a reduced oscillator strength of the charge-transfer (CT) transition and a correspondingly, slow radiative emission rate19,20. A priori one may expect that the slow radiative rate would lead to reduced PLQE and EQEEL, especially in the NIR due to enhanced non-radiative decay mediated by exciton-vibrational coupling, as discussed above.

The second class of efficient NIR emitters that we study are a recently developed family of spin ½ radical molecular semiconductors. These are built with a spin radical unit (TTM) that acts as electron acceptor covalently linked to a donor unit, including an N-arylated carbazole moiety (TTM-3PCz and TTM-3NCz), and a triphenylamine moiety (TTM-TPA). The lowest optical excitation for absorption and emission is the CT transition within the spin doublet manifold, and with correct tuning of the CT energetics and overlap, efficient emission can be achieved21. For these materials, excitation within the doublet manifold avoids access to higher multiplicity spin states and eliminates the problems due to triplet exciton formation in conventional closed-shell organic emitters. These systems can show very high NIR PLQE, for example TTM-3NCz above 85% for solid state blends in OLED host CBP for emission at 710 nm17. We also report another novel derivative, TTM-TPA which has high NIR PLQE (24%, 800 nm, toluene). A key question is how these systems can achieve such high luminescence yield in the NIR and thus seemingly violate the energy gap law.

To answer these questions, we focus our study on TTM-3PCz, TTM-3NCz, TTM-TPA and APDC-DTPA and compared them with a range of conventional organic semiconductors including regioregular poly(3-hexylthiophene), rr-P3HT a well-studied semiconductor homopolymer which shows a low PLQE of <5 % at 680 nm; the laser dye rhodamine 6G (r6G), which emits brightly (PLQE 94%, 550 nm). We include also a higher energy gap green emitting TADF system, 4CzIPN18,22–25 (PLQE 94% at 550 nm), and some representative examples of low-band gap (NIR) n-type organic semiconductor molecules (ITIC26, IO-4Cl27, o-IDTBR28, Y5/BTP29, Y6/BTP-4F30, Y7/BTP-4Cl31, see Fig. 2) which are being extensively explored as non-fullerene acceptors (NFA) in organic photovoltaics. All of these NFA molecules have much higher non-radiative decay rates compared to similar energy-gap radical or TADF emitters (e.g.- TTM-3PCz, TTM-3NCz, APDC-DTPA and TTM-TPA) (see Extended Data Fig. 9)32.

**Broadband Impulsive Vibrational Spectroscopy**

We probe the vibrational coupling in the excited electronic state of these organic molecules by employing resonant impulsive vibrational spectroscopy (IVS)7. In IVS, an ultrafast pump pulse (sub-15 fs) resonant with the optical gap, impulsively generates vibrational coherence in the photo-excited state of a material, which evolves in time according to the underlying excited-state potential energy surface. The impulsive response of the system is recorded by a time-delayed probe pulse which is spectrally tuned to probe excited-state resonances. The so-obtained vibrational coherence manifests as oscillatory modulations superimposed on top of the sample’s transient population dynamics and provides direct access to the excited-state Raman spectrum of the material via Fourier transformation.

Fig. 2b shows the absorption spectra of the investigated organic molecules as well as the spectral range of the ultrafast pump pulse used in our IVS studies (grey rectangle, 8.8 fs, centred at 575 nm). In rr-P3HT, R6G and IO-4Cl (representative NFA molecule), the pump pulse is resonant with a π→π* transition. In contrast, in APDC-DTPA and TTM-3PCz the pump pulse is resonant with a charge-transfer transition. Specifically, in the TADF system APDC-DTPA, the electronic excitation promotes an electron from the (HOMO) centred at the triphenylamine (TPA) to the (LUMO) located at acenaphthene-based acceptor core (APDC)13. Similarly, for the spin radical system TTM-3PCz, the lowest energy transition corresponds to a doublet excitation (D0 → D1) from the 3PCz-centered HOMO to a TTM-centred SOMO17.

Following resonant impulsive excitation by the pump pulse, the early-time electronic population dynamics exhibit distinct oscillatory modulations across the entire visible probe region for all investigated molecules (Extended Data Fig. 1 for wavelength resolved analysis of the novel systems APDC-DTPA and TTM-3PCz). Fig. 2c displays the isolated excited-state vibrational coherences, and Fig. 2d shows the excited-state Raman spectrum of the corresponding time-domain data from Fig. 2c for each molecule (see Methods for details). In rr-P3HT, we observe a pronounced vibrational mode at 1441 cm-1 which is due to the C=C ring-stretching mode in this conjugated polymer system8. Rhodamine 6G displayed a series of high frequency modes (1356, 1504, 1647 cm-1) corresponding to localised C-C, C=C stretching motions. Similarly, all the NFA molecules studied here, show strong coupling to the high-frequency vibrational modes. As an example, we show data for IO-4Cl in Fig. 2, with data for all the other NFA molecules shown in Extended Data Fig. 2.

In contrast, the excited-state vibrational spectrum of APDC-DTPA exhibits vibrational activity only in the low frequency regime (183, 290, 324, 557, 587, 678, 735 cm-1), which is associated with more delocalised torsional modes in the system. Similarly, TTM-3PCz only showed one prominent mode (232 cm-1) which is in the range of frequencies associated with torsional motions of the TTM to 3PCz moiety (see Extended Data Fig. 3 for the complete analysis of low-frequency modes and theoretically calculated exciton-vibration coupling constants). In parallel, other radical and TADF systems studied also show high-frequency decoupled excited-state Raman spectra upon photo-exciting the charge-transfer band. (See Extended Data Fig. 4, for 4CzIPN, TTM-3NCz and Fig. 3 for TTM-TPA with further details)

These results suggest two different regimes for exciton-phonon coupling operate in the materials studied here. For the conjugated homopolymer (rr-P3HT), laser dye (rhodamine 6G) and all the NFAs studied (ITIC, IO-4Cl, o-IDTBR, Y-series NFAs) the photo-excited transition leads to formation of excitons coupled to high-frequency C-C and C=C stretching modes. In systems with a low energy-gap this strong coupling to high-frequency modes leads to high rates of non-radiative losses and low PLQEs for red and near-IR emitters (as discussed in Fig. 1). We note that while the NFAs have a donor-acceptor structural motif, due to coplanar geometry and strong electronic conjugation through the fused rings, the HOMO and LUMO strongly overlap in space so that the dipole oscillator strength of the lowest energy transition in these materials is very high, as required for their use in photovoltaics (see the frontier MOs in Supplementary Information section 10). We also note that in many of these molecules the exciton is delocalised across a large spatial extent (greater than for the APDC-DTPA and TTM-3PCz molecules), but that does not suppress coupling to high-frequency modes significantly nor the resultant non-radiative recombination.

In contrast, the electronic transitions involved in APDC-DTPA, 4CzIPN, TTM-3PCz, TTM-3NCz and TTM-TPA feature strong charge-transfer character and non-planar molecular geometry. This results in spatially separated and disjoint HOMO/SOMO or LUMO. Our results suggest this leads to the exciton being coupled primarily to low-frequency modes, which in turn leads to low rates of non-radiative recombination and hence high PLQE (as discussed in Fig. 1).

**Band Selective Impulsive Vibrational spectroscopy**

We note that in the TTM-donor type radical systems the higher-lying D0→D2 transition does not involve charge-transfer states17,35. Studying this transition therefore allows us to compare the vibrational coupling between charge-transfer and non-charge-transfer transitions on the same molecule. Here, we focus on the novel radical molecule TTM-TPA (Fig. 2a), which has similar donor-acceptor structural motif like TTM-3PCz and TTM-3NCz, but has a triphenylamine (TPA) group as electron donor in place of the N-aryl carbazole (PCz/NCz) group. As shown in Fig. 3a and similar to TTM-3PCz17, the lowest energy electronic transition (D0→D1) in TTM-TPA corresponds to a charge-transfer excitation from the TPA-centred HOMO (donor) to the TTM-centred SOMO (acceptor) as revealed by time-dependent density-functional theory (TDDFT) calculations (Extended Data Table 2). The second lowest energy transition (D0→D2) involves frontier molecular orbitals sitting predominantly on the TTM part (HOMO-2 to SOMO), corresponding to spatially-overlapped orbitals which is consistent with similar derivatives17,35. As illustrated in Fig. 3b, the charge-transfer character of the lowest energy absorption band (~700 nm, D0→D1) shows the expected solvatochromic red-shift, while the higher energy absorption band (~500 nm, D0→D2) is barely affected by solvent polarity. Since TPA has a higher-lying HOMO compared to 3PCz36, the charge-transfer transition is red-shifted while maintaining a nearly similar energy for the local exciton transition in TTM-TPA (Fig. 3b). This greater energy separation between charge-transfer and non-charge-transfer transitions allows us to compare their vibrational coupling more cleanly than would be possible in TTM-3PCz. We excited the charge-transfer state with a pump pulse centred at 725 nm (pulse P1, 12 fs, Fig. 3b), while the local exciton state could be excited with a pump pulse centred at 575 nm (Pulse P2, 15 fs, Fig. 3b).

Photo-excitation of TTM-TPA into D1 with pulse P1 yielded vibrational coherences similar to the previously observed charge-transfer exciton of TTM-3PCz and TTM-3NCz. Here, the excited-state impulsive vibrational spectrum is again dominated by low frequency modes (228 cm-1) with a minor contribution from high frequency modes in the range of 1100 – 1650 cm-1 (magenta, Fig. 3c). Photo-excitation into the non-charge-transfer exciton state via pulse P2, populated the D2 state which rapidly cools to the D1 state with a time constant of 670 fs (see Extended Data Fig. 5 for electronic and vibrational dynamics of D2→D1 cooling). Fig. 3c (purple) shows the corresponding vibrational spectrum obtained directly after photo-excitation into D2, which exhibits significantly enhanced coupling to high-frequency modes at 1272,1520 and 1565 cm-1, in stark contrast to the spectrum obtained for D1 (Fig. 3c, magenta; see Extended Data Fig. 6 for wavelength resolved analysis). Taken together, this selective photo-excitation reveals that the CT (D1) and non-CT exciton (D2) states exhibit large differences in the coupling to the vibrational modes even within the same molecule.

**First-Principles Modelling**

We performed first-principles spin-unrestricted DFT and TDDFT calculations to quantify the exciton-vibration coupling37–39 through the Huang-Rhys factor parameter (Sev, equation 2) for different electronic transitions.

We computed Sev(ω) for the normal modes of TTM-TPA associated with both the D0 → D1 (CT) and D0 →D2 (non-CT) excitations. This is shown in Fig. 3d, for the key high-frequency vibrational modes which are observed in the experimental data, 1273, 1522, 1561 and 1572 cm-1. The calculations show that for all high-frequency modes, the vibrational coupling is significantly reduced for the charge-transfer excitation (Fig. 3d, purple) compared to the local exciton (Fig. 3d, magenta), in line with the experimental observations

To better understand how these vibrational modes affect the electronic structure of TTM-TPA, we computed the exciton wavefunction (ρ) as well as the change in the wavefunction due to displacement along a normal mode, {Δρ}ω. In Fig. 3f-g we show these differential wavefunction plots {Δρ}ω, for the normal mode having frequency of 1561 cm-1, which is associated with C-C and C=C stretching vibrations localized primarily on the TTM moiety. This mode is strongly present in the experimental data (1565 cm-1, Fig. 3c) and also dominates the theoretically calculated exciton-vibrational coupling plot (Fig. 3d).

TDDFT calculations reveals that excitation to D2 localizes the wavefunction onto TTM (ρ, Fig. 3f, top), as expected for a non-CT local excitonic state35. The differential exciton density ({Δρ}1561, Fig. 3f, bottom) shows how the exciton density on the molecule varies due to perturbing the molecular geometry along the 1561 cm-1 mode. Displacement along this normal mode leads to large changes in the D2 exciton wavefunction, indicating strong coupling of the high-frequency vibrational modes to the non-CT exciton wavefunction. We then compare this to the exciton density arising from D0 →D1 (Fig. 3g, top), which leads to a delocalized wavefunction over the whole molecule and, with disjoint electron and hole densities. Critically, the differential exciton density upon perturbation along the 1561 cm-1 normal mode ({Δρ}1561, Fig. 3g, bottom) shows very little change for the D1 exciton, in marked contrast to the results for the D2 exciton. This shows the strong suppression of exciton-vibrational coupling for the CT-type D1 exciton. In Extended Data Fig. 7, similar results are presented for all other experimentally-obtained high-frequency vibrational modes.

Intuitively, a general proposition to explain these results can be understood as follows: The charge-transfer excitation in non-planar molecules provides spatially-separated electron (HOMOs/SOMOs) and hole (LUMOs) across the molecular backbone. Changes to the both electron and hole wavefunction simultaneously due to highly localized high-frequency carbon-carbon stretching motions therefore result in a smaller effect compared to planar excitonic systems, which exhibit strongly overlapping HOMOs and LUMOs with high electronic densities in the vicinity of these high-frequency nuclear oscillations.

An important point about TADF and radical emitters is that their assembly from donor and acceptor moieties allows independent optimisation of the hole and electron levels42. Empirical optimisation has steered design towards moieties with non-bonding character for HOMO and LUMO. In such systems the non-bonding character decouples these levels from the high frequency vibrational modes that modulate π-bond order, such as phenylic ring stretch modes. In both TADF and radical systems, triphenylamine (TPA) and N-aryl-carbazole (Cz) are widely adopted as donors18,19,42. These donor moieties have nitrogen pz orbital centred non-bonding type HOMO orbital. The degree of further localization of the non-bonding type HOMO on the nitrogen atom depends on the non-orthogonality of the nitrogen pz orbital to the adjacent π-systems imposed by steric hindrance43 (see Supplementary Information section 14). Additionally for the TTM-Donor type radical systems, the electron accepting SOMO level which is localised on TTM moiety’s central sp2 carbon atom has non-bonding character17. This is shown in Supplementary Information (section 14) where the SOMO of the TTM-Donor radicals couples less to the high frequency phenylic carbon-carbon stretching modes, due to its non-bonding character.