# Adaptable Range-Separated Hybrids for the Description of Excited States: Tuning the Range Separation Parameter on Effective Charge Transfer Distance

Tianhong Yan, Alessandro Bonardi, Carlo Adamo, Ilaria Ciofini

► To cite this version:

Tianhong Yan, Alessandro Bonardi, Carlo Adamo, Ilaria Ciofini. Adaptable Range-Separated Hybrids for the Description of Excited States: Tuning the Range Separation Parameter on Effective Charge Transfer Distance. Journal of Chemical Theory and Computation, 2025, 21 (4), pp.1892-1904. 10.1021/acs.jctc.4c01617 . hal-05346997

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Submitted on 4 Nov 2025

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# *Adaptable Range Separated Hybrids for the description of excited states :*
# tuning the range separation parameter on effective charge transfer distance

Tianhong Yan¹, Alessandro Bonardi¹, Carlo Adamo¹,* and Ilaria Ciofini¹,*

PSL University, Chimie ParisTech, CNRS, Institute of Chemistry for Life and Health Sciences, F-75005 Paris, France

## Abstract

In this contribution we describe a novel approach, rooted on Time Dependent Density Functional Theory (TD-DFT), enabling to adapt range separated hybrids (RSH) to correctly describe excited states of inter- and intra-molecular Charge Transfer (CT) character.

Contrary to previous works enforcing the fulfillment of Koopmans' theorem, here the range-split parameter of RSHs is tuned so to constrain it in the range of distances corresponding to the hole-electron separation occurring in target CT states for the molecule of interest. The procedure proposed, while not requiring a fit but only an estimate of the CT distances for all excited states (ES) of interest, is not based on empirical adjustment and enables to find a system dependent range separation parameter optimal for the treatment of CT states while not deteriorating its performances with respect to low Hartree-Fock exchange global hybrids for the description of excited states of more local character.

The results obtained for a series of CT compounds show the very good accuracy of this adaptative tuning procedure of RSHs and its potential for the exploration of the ES behavior of molecular systems.
# 1. Introduction

If Density Functional Theory (DFT)¹ and Time Dependent DFT (TD-DFT)²,³ have emerged among the methods of choice for the description of the reactivity and properties of the ground (GS) and excited states (ES) of chemically relevant systems,⁴ it is essentially thanks to i) their favorable accuracy to computational cost ratio; ii) their easiness in use and iii) their solid implementations in different computer codes allowing a wide spreading of the methods well beyond the community of theoretical chemists.⁵

One of the choices to be performed by the user, that is the choice of the density functional approximation (DFA), is nonetheless very relevant for the final result not only in terms of its accuracy but also, and more severely, of chemical relevance of the outcomes. This is particularly true when dealing with excited states. Examples of the influence of the DFA on the accuracy are provided by many valuable benchmark studies both for ground⁶,⁷,⁸ and excited state⁹,¹⁰,¹¹,¹² properties. Restricting to excited states, many studies have demonstrated that the choice of the DFA is particularly important when dealing with the description of the photophysical properties of systems presenting electronic excited states with significant charge transfer (CT) character.¹³,¹⁴,¹⁵,¹⁶,¹⁷ In this case, largely used DFA have been demonstrated to present serious weaknesses. Unfortunately, this type of excited states is actually target for many applications since Charge Transfer, and/or Charge Separation (CS) at the ES are relevant in many technological fields ranging from energy conversion (ex. photovoltaics¹⁸,¹⁹,²⁰), electronics (OLED materials²¹,²²,²³,²⁴,²⁵) but also in photocatalysis²⁶ or even health sciences (ex. in photodynamic therapy²⁷,²⁸,²⁹) to give few examples. For this reason, much effort is currently experimentally devoted to achieving a molecular design allowing to manipulate (that is control) CT states evolution. There is thus a quest for efficient yet robust theoretical methods enabling to accurately predict the energy of CT states to help in the in-silico design of novel molecule based devices.

Restricting to TD-DFT, several literature reports have contributed to point out how DFA relying on the use of local exchange and correlation functionals are particularly affected by errors such as self-interaction error (SIE) or the lack of the correct (-1/r) asymptotic behavior for the Kohn Sham potential and derivative discontinuities that determine a misleading description of CT states.³⁰,³¹,³²,³³,³⁴ This is particularly evident in the case of intermolecular CT excitations. Inclusion of exact exchange in the DFA, as in global hybrid functionals, do not enable to fully cure these problems while range-separated functionals (RSH) and, particularly, long range corrected functionals have emerged as a possible solution.³⁵,³⁶,³⁷

In this family of functionals, the exchange contribution is indeed computed relying on a Coulomb operator which is split into a short and a long range component as :

$$ \frac{1}{r_{12}} = \frac{1 - \operatorname{erf}(\gamma r_{12})}{r_{12}} + \frac{\operatorname{erf}(\gamma r_{12})}{r_{12}} \qquad (1) $$

In long range corrected (LC) RSH, the first term, dominating at short range, is computed by a DFT (GGA or hybrid) functional while the second, leading at long distances, is computed by an exact (Hartree Fock, HF) exchange term, thus restoring the correct long range asymptotic behavior. The total exchange and correlation energy is thus expressed as :

$$ E_{XC} = \alpha E_{X,HF}^{SR}(\gamma) + (1 - \alpha) E_{X,DFT}^{SR}(\gamma) + E_{X,HF}^{LR}(\gamma) + E_{C,DFT} \qquad (2) $$

Where α represents the exact HF fraction included at SR and γ is considered as a crucial adjustable parameter allowing setting the distance of switching between the SR and LR regimes. Although, previous studies on model systems have demonstrated how γ is not a system independent parameter
but actually a functional of the electron density,³⁸,³⁹ for current applications this range-split parameter is normally adjusted to minimize errors on specific dataset. Previous reports have also shown however that i) the range of $γs$ allowing to obtain the best results for ground and excited states properties are not necessarily the same and ii) that CT excited states are yet not very accurately described by LC functionals tuned on ground state properties.⁴⁰,⁴¹

For these reasons, different procedures have been set up to obtain a better description of ES using range separated hybrids modifying the SR percentage of HF exchange and/or the range-split parameter $γ$.⁴² These studies also well underlined the difficulty in finding a unique set of parameters allowing to recover, with the same accuracy, both ground and excited state molecular properties.

A substantial step forward for the performance of RSH in the evaluation of intermolecular CT energy was indeed provided by the method originally proposed by Stein, Kronik and Baer leading to the so called Optimally Tuned RSH (OT-RSH).⁴³ Using a RSH, where the correct asymptotic behavior is imposed by the presence of 100% HF exchange at large distances such as the one in eq. 2, these authors tuned the range-separation parameter, $γ$, so as to obey the Koopman theorem imposing that the energy of the highest occupied molecular orbital correspond to the ionization-potential. Imposing this condition for both the neutral and the anionic systems actually allows to recover both the ionization potential and the electron affinity, two key parameters in the evaluation of intermolecular CT excitations in Donor-Acceptor complexes. Further developments have also allowed to include the effect of the environment (solvent or solid matrix).⁴⁴

OT-RSH have since then emerged as an elegant and efficient way of reaching an accurate evaluation of CT excitation energies and are increasingly applied for the description of these systems both in solution and the solid state.⁴⁵ This tuning procedure, while being clearly dependent both on the nature of the system and its geometry, has the advantage of being non empirical and based on theoretical constraints. Nonetheless, one of the features often observed for RSH, that is an overestimation of ES energies and generally a degradation in the accuracy is not fully solved.¹²

In this work, we would like to propose a way to define a novel DFA that keeps the optimal accuracy of global hybrid for the description of local excited states and cure their misleading behavior leading to underestimated CT states energies applicable both to inter- and intra-molecular CT states. To this end, while keeping a strict respect of the exact asymptotic behavior, obtained by imposing full HF exchange at long range, we propose to tune the range-split parameter so to impose it to be in the range of distances of the hole-electron separation occurring in target CT state for the molecule of interest. This procedure, which does not necessitate a fit but only an evaluation of the CT distances for all ES of interest does not introduce empirically adjusted parameters and enables to find a system dependent range separation parameter optimal for the treatment of both CT states and LE excited states.

To test the accuracy of this adaptative tuning procedure of RSHs, we selected a series of compounds largely studied in literature (schematically depicted in Figure 1) presenting both intra- and intermolecular CT excitations.

In principle, the approach proposed can be used in conjunction with any exchange and correlation functional. In the present work, we have decided to focus on the use at SR of a DFA that has proven to provide good performances for local excitations, that is the PBE0 global hybrid.⁴⁶ The performance of the so obtained adaptative RSH have been compared to the one of the corresponding standard global hybrid (PBE0) and standard LC approaches (here LC-PBE⁴⁷).

The results obtained show an encouraging performance of the adaptative approach in the evaluation of the vertical excitation energies of CT excited states and, remarkably, the absence of a deterioration of more local ES thus providing an improvement with respect to standard LC approaches at negligible computational cost.
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Schematic representation of the systems investigated.</figcaption></figure>

The paper is structured as follows: after a detailed description of the adaptive tuning approach and of the computational details (Section 2 and 3, respectively), the results obtained are discussed in Section 4 before drawing some final conclusions and perspectives in Section 5.

## 2. The method

As mentioned in the introduction we aim at defining a novel long range corrected functional corresponding to the general expression (2) adjusting the range-split parameter.

In particular, to obtain the tuned range split parameter ($\gamma^*$) we first performed a calculation with the global hybrid used at SR and we evaluated the hole-electron separation for all excited states of interest by the mean of a density-based descriptor (the $D_{CT}$). To estimate the $D_{CT}$ value, in this paper we make use of the original formulation of the $D_{CT}$ index$^{48}$. To this end, the electronic density of the ground ($\rho^{GS}(\mathbf{r})$) and excited state ($\rho^{ES}(\mathbf{r})$) of interest are required to compute first the density change associated to the electronic transition $\Delta\rho(\mathbf{r})$:
$$ \Delta\rho(\mathbf{r}) = \rho^{ES}(\mathbf{r}) - \rho^{GS}(\mathbf{r}) \quad (3) $$

This latter is further split into two contributions ($\rho^+(\mathbf{r})$ and $\rho^-(\mathbf{r})$) corresponding to the region of increment and depletion of electron density upon excitation:

$$ \rho^{+}(\mathbf{r}) = \begin{cases} \Delta\rho(\mathbf{r}) & \text{if } \Delta\rho(\mathbf{r}) > 0 \\ 0 & \text{if } \Delta\rho(\mathbf{r}) \le 0 \end{cases} \quad (4) $$

$$ \rho^{-}(\mathbf{r}) = \begin{cases} \Delta\rho(\mathbf{r}) & \text{if } \Delta\rho(\mathbf{r}) < 0 \\ 0 & \text{if } \Delta\rho(\mathbf{r}) \ge 0 \end{cases} \quad (5) $$

$D_{CT}$ is then defined as the spatial distance between the two barycenters ($R^+$ and $R^-$) of these two (positive and negative) charge density distributions.

$$ D_{CT} = R^{+} - R^{-} \quad (6) $$

Although $D_{CT}$ itself has a vectorial character, and as proven useful to characterize CT in molecular systems,⁴⁹,⁵⁰ here we will make use of it only to quantify the CT extent (the hole-electron separation) and thus only the associated norm (hereafter labelled as $D_{CT}$) will be of interest. Unrelaxed densities for the ES have been used to evaluate the $D_{CT}$.

The $D_{CT}$ index has then been used to evaluate the $M_{AC}$ (Mulliken averaged configuration) index⁵¹,⁵² for each of the computed transitions. This index was developed with the aim of diagnostic the reliability of different DFA in describing CT processes and it allows to evaluate the associated energy on the basis of the original formulation of Mulliken.⁵³ Following this model, a lower bound energy for a net one-electron intermolecular charge transfer ($\omega_{CT}$) between a donor (D) and an acceptor (A) moiety is given by:

$$ \omega_{CT} = IP_{D} - EA_{A} - \frac{1}{R} \quad (8) $$

where $IP_D$, $EA_A$ are the ionization potential of the donor, the electron affinity of the acceptor and R is the distance between the donor and the acceptor, respectively.

Using a Koopmans-type approach, in the case of a TD-DFT calculation, the $M_{AC}$ index is approximating the $\omega_{CT}$ replacing R by the computed CT distance (given here by the $D_{CT}$) while $IP_D$ and $EA_A$ are expressed from the weighted average of the energies of the initial occupied ($\epsilon_i$) and final virtual ($\epsilon_a$) Kohn-Sham orbitals, respectively:

$$ M_{AC} = \frac{\sum_{ia} [c_{ia}^2 (\epsilon_a^{DFT-HF} - \epsilon_i^{DFT-HF})]}{\sum_{ia} c_{ia}^2} - \frac{1}{D_{CT}} \quad (8) $$

where the weights $c_{ia}$ are the CI coefficients obtained as solution of TD-DFT equations. Of note, deexcitation amplitudes are here negligible allowing the direct use of the CI coefficients. In eq. 8, the energies of the orbitals obtained using DFT theory ($\epsilon^{DFT-HF}$) are corrected with a single SCF cycle using Hartree-Fock theory. For more details on the $M_{AC}$ index we refer the reader to the original papers.⁵¹,⁵² By comparing the so computed $M_{AC}$ values with the transition energies computed at TD-DFT level of theory, one can determine whether the predicted energy with a given DFA is actually underestimated or not. In the first case we identify *unphysical states* while the second case hold for *real states*:

$$ \begin{aligned} E^{TD-DFT} &> M_{AC} \rightarrow \text{Real State,} \\ E^{TD-DFT} &\ll M_{AC} \rightarrow \text{Unphysical State.} \end{aligned} \quad (9) $$

In such a way we can pinpoint the CT states affected by error and decide to modify the range separation split parameter so that such transfer states are all described in the LR regime. It has nonetheless to be pointed out that in the case of intramolecular CT i) the $M_{AC}$ is only an approximation of the real CT energy and ii) the way this index is computed is affected by several approximations the
first a) being the use of orbital energies to estimate IP and EA and the second b) being the use of unrelaxed densities to estimate the D<sub>CT</sub>, a procedure known to provide overestimated CT distances.<sup>54</sup> From the above definition, it is also clear that the D<sub>CT</sub>, being the distance between the barycenters of charge distributions, actually represents an upper bound for the interelectronic distance of interest and associated with the CT states. At first, several attempts were performed using directly the D<sub>CT</sub> value to evaluate γ. Nonetheless, this choice seems not judicious since depletion of the overlap between hole and electron charge distributions starts to occur at much lower distances, depending on the spreading of these charges distributions. Here we decided to impose the range separation switch to occur at half of largest D<sub>CT</sub> value corresponding to unphysically predicted CT states (hereafter D<sub>CT</sub><sup>ghost-max</sup>) to avoid further system dependent adjustment. Finally our adjusted RSH thus reads :

$$E_{XC} = \alpha E_{X,HF}^{SR}(\gamma^*) + (1-\alpha) E_{X,DFT}^{SR}(\gamma^*) + E_{X,HF}^{LR}(\gamma^*) + E_{C,DFT} \quad (10)$$

$$\text{With } \gamma^* = \frac{1}{\frac{D_{CT}^{\text{ghost-max}}}{2}} = \frac{2}{D_{CT}^{\text{ghost-max}}}$$

A discussion on different possible choices of γ* is reported in Section 4.3

For clarity, the overall workflow corresponding to the tuning methodology is schematically reported in Figure 2.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Schematic workflow of the tuning methodology proposed to derive adaptable RSH. Optimization of the structure at the GS can be included (as in the present case) or not. Whenever the M<sub>AC</sub> index needs to be computed an estimate of orbital energies at HF level is also necessary.</figcaption></figure>

### 3. Computational details

To analyze the performances of our approach, the following DFAs have been compared: i) the PBE0 global hybrid casting a fixed percentage of HF exchange (25%) in the PBE<sup>55</sup> exchange correlation functional; ii) the standard LC-PBE functional corresponding at SR to the PBE GGA exchange functional (α=0 in eq. 2) and at LR to 100% HF exchange always in conjunction with the PBE correlation functional using its γ default value (γ=0.470); and iii) an adjusted RSH -hereafter labelled
as LC-PBE0\* functional constructed casting the PBE0 functional at SR ($α$=0.25) and 100% HF exchange at LR always in conjunction with the PBE correlation functional and using a system dependent $γ^*$ computed from the $D_{CT}$ evaluated at PBE0 level as described is Section 2.

Selected calculations were also performed using the “LC-PBE0” functional constructed casting the PBE functional at SR with an $α$=0.25 and 100% LR of HF exchange at LR always in conjunction with the PBE correlation functional and using a $γ$=0.470. For the sake of clarity, we would like to stress that this DFA does not correspond to the parametrizations previously reported in literature under the same acronym which is using different $α$ and $γ$ values.$^{40,41}$

Furthermore, since different basis sets were used for the different sets of molecules in order to allow direct comparison with previous literature reports, they will be explicitly mentioned in the Results and Discussion section. For all compounds but complex **1** (Figure 1), TD-DFT calculations have been performed on structures optimized, at the GS, at the same level of theory. Results obtained on structures optimized at PBE0 level using LC-PBE and LC-PBE0\* are reported and discussed in SI, since there are no noticeable changes in the conclusions both from a qualitative and quantitative point of view.

All DFT, TD-DFT and $D_{CT}$ calculations were performed with the Gaussian code$^{56}$ while the $M_{AC}$ index was evaluated with an in-house code.

**4. Results and Discussion**

In order to structure the discussion, we will present, in Sections 4.1, the results obtained for the simplest molecular system considered as prototypical example of intermolecular CT excitation (systems **1** in Figure 1) before moving, in Section 4.2 and 4.3, to the discussion of more chemically relevant molecular compounds for which accurate experimental photophysical data are available.

**4.1 Intermolecular CT excitations: a prototypical donor-acceptor system**

To test the performance of our approach in regard to the modelling of CT excitations, we first considered a model system (**1**, Figure 1) as prototypical example of inter-molecular charge transfer.

The molecular complex **1** (Figure 1) is a dimer constituted by an electron donor (D, here benzene) and an acceptor (A, here tetracyanoethylene -TCNE-). This system has been previously used to test the performance of OT-RSH since experimentally its optical absorption energy in the gas phase is available. For the sake of comparison with previous works, the molecular structure used by Baer and collaborators$^{43}$ imposing a D to A distance of 3.68 Å has been used in conjunction with the 6-311G(d) basis set. The results obtained for the first 3 excited states are collected in Table 1. To allow direct comparison with OT-RSH results, additional calculations were also performed with the cc-pVDZ basis set. Full data are reported in Table S.1, showing a negligible effect of the change in basis set on computed ES energy with a variation in the range of 0.06 - 0.08 eV for the CT ES state (ES1).

Let us first focus on the behavior of PBE0, LC-PBE and LC-PBE0. The three functionals provide a qualitative consistent picture : the first two excited states (of HOMO-LUMO and HOMO-1-LUMO character, respectively, refer to Figure S.1 for the corresponding Natural Transition Orbitals -NTO-)$^{57}$ are CT states with negligible oscillator strength while the third one is a $π-π^*$ excitation localized on the TCNE moiety predicted to display a significant absorbance (f>0.29 a.u.).

Nonetheless, the similarity between the three approaches ends here. As expected, the PBE0 functional predicts the two CT excitations to occur at much lower energy with respect to the RSHs significantly underestimating the CT energy while the RSHs are predicting a significantly higher energy for both the CT states and the Local Excited state localized on the TCNE moiety.

Of note, for each DFA, the LE state (Excited state 3 in Table 1), being fully localized on the TCNE moiety, occurs practically at the same energy both in the D-A dimer and in the isolated TCNE molecule (Table S.2). Inspection of the $M_{AC}$ index computed for the three functionals shows how, in
the case of PBE0, an underestimation of the CT states energy is predicted, justifying the use of the tuning leading to the LC-PBE0* approach. The value of 3.673 Å was thus used to determine the range separation parameter for the LC-PBE0* fixed to half of this value.

<table><caption>**Table 1.** Excitation energies (E in eV), oscillator strengths (f in a.u.) and charge transfer distances D<sub>CT</sub> in Å) and Mac index (in Å) computed for the benzene-TCNE complex (1 Figure 1) using different levels of theory.</caption><thead><tr><th rowspan="2">Excited State</th><th colspan="2">PBE0</th><th colspan="2">LC-PBE<br>(γ=0.470)</th><th colspan="2">LC-PBE0<br>(γ=0.470)</th><th colspan="2">LC-PBE0*<br>(γ=0.288)</th><th rowspan="2">Exp<sup>a</sup></th></tr><tr><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th><th>E/f</th></tr></thead><tbody><tr><td>1</td><td>2.08/0.00</td><td>3.654/4.22</td><td>4.08/0.00</td><td>3.622/4.23</td><td>4.19/0.00</td><td>3.623/4.20</td><td>3.60/0.00</td><td>3.624/4.21</td><td>3.59/0.02</td></tr><tr><td>Character</td><td></td><td>CT</td><td></td><td>CT</td><td></td><td>CT</td><td></td><td>CT</td><td></td></tr><tr><td>2</td><td>2.15/0.03</td><td><b>3.673</b>/4.25</td><td>4.09/0.03</td><td>3.642/4.26</td><td>4.21/0.03</td><td>3.643/4.23</td><td>3.63/0.03</td><td>3.644/4.24</td><td></td></tr><tr><td>Character</td><td></td><td>CT</td><td></td><td>CT</td><td></td><td>CT</td><td></td><td>CT</td><td></td></tr><tr><td>3</td><td>4.35/0.29</td><td>0.011/LE<sup>b</sup></td><td>4.82/0.34</td><td>0.029/LE<sup>b</sup></td><td>4.83/0.35</td><td>0.029/LE<sup>b</sup></td><td>4.67/0.33</td><td>0.035/LE<sup>b</sup></td><td></td></tr><tr><td>Character</td><td></td><td>LE (TCNE)</td><td></td><td>LE (TCNE)</td><td></td><td>LE (TCNE)</td><td></td><td>LE (TCNE)</td><td></td></tr></tbody></table>

In the case of the LC-PBE and LC-PBE0 γ is actually fixed to 0.470 Bohr<sup>-1</sup> corresponding to a distance of range separation split of 1.13 Å. This distance is significantly smaller than half of the CT distance (1.825 Å). As a consequence, the γ parameter for the tuned LC-PBE0* functional (0.288 Bohr<sup>-1</sup>) is sizably smaller than the one of LC-PBE and LC-PBE0. This allows to correct the CT state energy with respect to PBE0 but still providing a CT state lower in energy than the LC-PBE and LC-PBE0 ones and, overall, in better agreement with the experimental value (3.59 eV in gas phase<sup>43</sup>).

It is worth noting that the so optimized γ value is also smaller that the OT tuned value published by Baer tuning the BNL functional<sup>58,59</sup> in the original paper (0.333 Bohr<sup>-1</sup>).<sup>43</sup> This value tuned by Baer and collaborators actually corresponds to a distance of 1.60 Å, that is roughly 44% of the computed CT distance, further confirming the reliability of choosing a ½ scaling factor for γ with respect to the CT distance.

Finally, it should be noted that although in the case of LC-PBE0* we still obtain a M<sub>AC</sub> that is higher than the computed transition energy, the difference between the predicted TD-DFT CT energies and the M<sub>AC</sub> value is strongly reduced with respect to the corresponding PBE0 values and below the threshold allowing to identify it as a real state.

In Table SI.3 are reported the IP and EA computed as difference in total energies together with the HOMO and LUMO orbital energies of the neutral system 1. From these data one can easily see that all the LC approaches are actually providing a significant improvement of the orbital energies, the LC-PBE0* functional being the one yielding the orbitals energies closer to the IP and EA computed at the same level of theory. Indeed an absolute energy difference between IP (EA) computed as total energy or as orbital energies difference of 1.87 (1.71) eV, 0.63 (0.54) eV, 0.32 (0.29) eV at PBE0, LC-PBE and LC-PBE0*, respectively is computed demonstrating how the adjustment of the range separation parameter performed on the distance of CT is, as side effect, also accidentally recovering the Koopmans constrain on orbital energies.

The calculations have subsequently been repeated increasing the benzene-TCNE intermolecular distance of 0.5 Å and 1 Å, respectively, while keeping all other structural parameters unchanged. The evolution of the CT and LE excited state energies computed at PBE0, LC-PBE, and LC-PBE0* is graphically reported in Figure 3.

<footnote><sup>a</sup> experimental gas phase value from reference 43. <sup>b</sup>This transition is fully localized.</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Evolution of the CT (squares) and LE (full dot) energies (in eV) computed at the PBE0 (black), LC-PBE (red) and LC-PBE0\* (blue) level of theory as a function of the intermolecular distances (in Å) for the benzene-TCNE dimer. The value (1/γ in Å) used in the RSH functionals LC-PBE (red open circles) and LC-PBE0\* (blue open circles) are also reported.</figcaption></figure>

First of all, one can easily notice that all functionals recover the fact that the LE state (full dot), strictly localized on the TCNE moiety, is unaffected by the intermolecular D-A distance.

Concerning the CT states energies (squares), the PBE0, LC-PBE and LC-PBE0\* functionals actually show three different qualitative and quantitative behaviors. At PBE0, as expected from previous literature report,⁴⁸ the CT energy is decreasing with the increase of the intermolecular distance, going from 2.08 eV to 1.86 eV. The LC-PBE on the contrary shows CT energies that are slightly increasing with the increasing intermolecular distance, going from 4.08 eV to 4.39 eV. The LC-PBE0\* functional on the other hand shows a very small decrease in the CT energy (from 3.6 eV to 3.45 eV).

Of note, if one can question the necessity of adjusting the range-separation split value in the case of the system at 3.68 Å due to the relatively low difference in energy between the $M_{AC}$ index (4.2 eV) and the TD-DFT estimation (2.1 eV) at PBE0 level, when increasing the distance between the D and the A, the unphysical low energy of the first excited state more clearly appears with difference between the computed TD-DFT PBE0 energies and the corresponding $M_{AC}$ indexes of 2.8 eV and 3.2 eV, for an increase of 0.5 Å and 1 Å, respectively.

Overall, from the results obtained on this model intermolecular CT system we can conclude that while the nature of the excited states remains unchanged (first two states CT, the third LE) independently of the functional used, the LC-PBE0\* functional overall shows an encouraging behavior in terms of accuracy, both considering excitation and orbital energies.

## 4.2 Intramolecular CT in push-pull compounds

In order to test the performance of the adjustable RSH in the case of intramolecular CT, we considered a series of three compounds (NODSF (2-(p-nitrostyryl)-5-(p-methoxystyryl)furan), NODST (2-(p-nitrostyryl)-5-(p-methoxystyryl)thiophene) and NODSP(2-(p-nitrostyryl)-5-(p-methoxystyryl)pyridine), Figure 1). These molecules are push-pull stilbenoid compounds bearing a donor and an acceptor group linked by a conjugated bridge (D-π-A). In the three cases considered the bridge is an heteroaromatic ring namely: furane (NODSF), thiophene (NODST) or pyridine (NODSP). These three systems were chosen since accurate spectroscopic data are available and because it has been experimentally proposed that an excited state with sizable CT character can be populated in absorption, thanks to electronic delocalization.⁶⁰,⁶¹,⁶² This partial CT state can then evolve upon relaxation in a truly CT state responsible for the marked fluorosolvatochromic experimentally
observed. These systems are thus expected to be difficult test cases for global hybrids due to their expected overestimation of electron delocalization and underestimation of CT states energy. For each system, the first ten excited states were computed and the corresponding data are fully reported in Table S.4.

Here we will first focus on the NODSF molecule. In Table 2, the lowest three excited states responsible for the observed behavior in the visible range are reported. Independently of the functional used, all DFAs predict a very intense transition with sizable CT character as first excited state.

As evident from the computed D<sub>CT</sub> values and the NTOs associated for the first excited state (reported in Figure 4), compared with LC-PBE, the PBE0 functional predicts a first excited state with larger CT character stemming from a larger delocalization of the hole (extending up to the OMe substituent) and the electron (delocalized up to the nitro substituent). On the other hand, the picture provided by the LC-PBE functional is much more localized with a delocalization of the hole and the electron limited essentially to the heteroaromatic ring and the adjacent C=C double bonds.

<table><caption>**Table 2** Vertical transition energies (E, eV), oscillator strengths (f, a.u.), charge transfer distances (D<sub>CT</sub>, Å) and M<sub>AC</sub> index (in eV) computed for NODSF, NODST and NODSP (Figure 1) using different levels of theory and the 6-311G(d) basis set. When relevant the range separation parameter (γ, Bohr<sup>-1</sup>) is reported. The (PBE0) D<sub>CT</sub> value used to determine range separation split (γ*) used at LC-PBE0* level is reported in bold face.</caption><thead><tr><th rowspan="2">NODSF</th><th colspan="2">PBE0</th><th colspan="2">LC-PBE</th><th colspan="2">LC-PBE0*</th></tr><tr><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th></tr></thead><tbody><tr><td>ES1</td><td>2.60/1.03</td><td>7.533/5.14</td><td>3.77/1.27</td><td>2.383/2.29</td><td>2.93/1.26</td><td>5.011/4.50</td></tr><tr><td>ES2</td><td>3.45/0.43</td><td>3.987/4.84</td><td>4.34/0.00</td><td>0.744/LE</td><td>3.98/0.33</td><td>3.659/4.56</td></tr><tr><td>ES3</td><td>3.79/0.19</td><td><b>9.297</b>/7.27</td><td>4.84/0.01</td><td>0.592/LE</td><td>4.04/0.00</td><td>1.360/1.63</td></tr><tr><td>γ</td><td>--</td><td></td><td>0.470</td><td></td><td>0.114</td><td></td></tr><tr><td>Exp<sup>a</sup></td><td>2.87</td><td></td><td>2.87</td><td></td><td>2.87</td><td></td></tr><tr><th rowspan="2">NODST</th><th colspan="2">PBE0</th><th colspan="2">LC-PBE</th><th colspan="2">LC-PBE0*</th></tr><tr><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th></tr><tr><td>ES1</td><td>2.60/1.33</td><td>8.101/5.25</td><td>3.73/1.55</td><td>2.117/1.67</td><td>2.93/1.58</td><td>4.784/4.37</td></tr><tr><td>ES2</td><td>3.42/0.39</td><td>6.066/6.03</td><td>4.34/0.00</td><td>0.742/LE</td><td>3.99/0.14</td><td>5.344/5.85</td></tr><tr><td>ES3</td><td>3.70/0.10</td><td><b>9.173</b>/7.10</td><td>4.81/0.16</td><td>2.249/3.14</td><td>4.04/0.00</td><td>1.362/1.85</td></tr><tr><td>γ</td><td>--</td><td></td><td>0.470</td><td></td><td>0.115</td><td></td></tr><tr><td>Exp<sup>a</sup></td><td>2.89</td><td></td><td>2.89</td><td></td><td>2.89</td><td></td></tr><tr><th rowspan="2">NODSP</th><th colspan="2">PBE0</th><th colspan="2">LC-PBE</th><th colspan="2">LC-PBE0*</th></tr><tr><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th></tr><tr><td>ES1</td><td>2.86/0.15</td><td><b>8.154</b>/6.17</td><td>4.26/1.00</td><td>1.200/LE</td><td>3.44/0.65</td><td>4.548/5.29</td></tr><tr><td>ES2</td><td>3.53/0.95</td><td>5.086/6.04</td><td>4.34/0.00</td><td>0.728/LE</td><td>3.98/0.60</td><td>5.872/6.37</td></tr><tr><td>ES3</td><td>3.80/0.44</td><td>5.468/6.32</td><td>4.73/0.83</td><td>1.607/0.45</td><td>4.04/0.00</td><td>1.264/1.17</td></tr><tr><td>ES4</td><td>3.92/0.00</td><td>1.544/2.95</td><td>4.83/0.00</td><td>0.600/LE</td><td>4.09/0.68</td><td>3.410/4.79</td></tr><tr><td>γ</td><td>--</td><td></td><td>0.470</td><td></td><td>0.118</td><td></td></tr><tr><td>Exp<sup>a</sup></td><td>3.86</td><td></td><td>3.86</td><td></td><td>3.86</td><td></td></tr></tbody></table>

a Experimental absorption spectra recorded in isopentane, from reference 62.
Following the procedure described in section 2, we obtained the range-separation parameter $\gamma^*$ from the largest $D_{CT}$ value of 9.29 Å that we found for the third excited state at PBE0 level.

This transition is indeed clearly corresponding to an unphysical state, its $M_{AC}$ index being almost 3.5 eV higher than the predicted TD-DFT transition energy. In such a way a range separation parameter ($\gamma^*$) of 0.114 Bohr$^{-1}$ was obtained and applied in the LC-PBE0* functional. This latter yields an energy for the first excited state of 2.93 eV in better agreement with the experimental absorption maximum, recorded at 2.87 eV. At LC-PBE0* level the transition is predicted to possess a sizable CT character, with a CT distance of 5.0 Å, a value that is intermediate between the PBE0 (7.5 Å) and the LC-PBE (2.4 Å) predictions.

Indeed, inspection of the NTOs computed for the first excited state for the three functionals (reported in Figure 4) allows to clearly see that a larger delocalization of the electron is predicted at LC-PBE0*. Nonetheless, the hole is still quite localized on the bridging core as in the case of the LC-PBE. Analogous calculations were performed for NODST and NODSP (see Table S.4 for detailed information). The NODST molecule shows exactly the same behavior of NODSF in terms of character and energy of the transitions. Indeed, their predicted spectra (reported in panel a and b, respectively, of Figure 5) are very similar.

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Isocontour representation (isovalue 0.05 a.u.) of the NTOs (hole: left; electron: right) associated to the first excited state of NODSF computed at the PBE0, LC-PBE, and LC-PBE0* level.</figcaption></figure>

Again the first excited state is the one responsible for the observed absorption and shows sizable delocalization and CT character at PBE0 level, while being predicted to be much more localized and overall at higher energy when using the LC-PBE. For clarity NTOs corresponding to the first excited
state computed at PBE0, LC-PBE and LC-PBE0 are reported in Figure S.2. The hole electron distance associated to the first electronic transition of NODST (8.1 Å) is actually even larger than that computed (at the same level of theory) for NODSF (7.5 Å). The adjusted LC-PBE0* functional provides a description free of unphysical states and, again, to an intermediate hole-electron estimated distance (4.8 Å) when compared with PBE0 or LC-PBE.

The case of NODSP (Figure 5 panel c) is, on the other hand, slightly more complex and more relevant differences between the functionals are observed both in quantitative and qualitative terms. At PBE0, the first three ESs are all predicted to have a significant CT character (with D<sub>CTS</sub> ranging from 8.1 Å - ES1- to roughly 5 Å -ES2 and ES3-). The three transitions are energetically well-spaced (computed at 2.86, 3.53 and 3.80 eV, respectively) and they are all predicted to possess a non-negligible intensity, the most intense being the second one (ES2). This CT transition is mainly localized in the acceptor branch (i.e. that bearing the nitro group) as clearly visible from the computed NTOs, reported in Figure S.3. As a result, the spectra predicted show a maximum in the visible at 341 nm (basically corresponding to the energy of ES2) and a small shoulder redshifted, corresponding to the first excited state. Of note ES1 has a marked CT character and corresponds to a transition from the donor to the acceptor branch and it is predicted to occur at unphysically low energy since the difference between the computed M<sub>AC</sub> (6.2 eV) and its TD-DFT energy (2.9 eV) is extremely large.

<figure><img src="image_6.png" /><figcaption>**Figure 5** Absorption spectra computed at different levels of theory for a) NODSF, b) NODST and c) NODSP together with d) the computed charge transfer distance computed for their most intense transition. A gaussian broadening with FWHM= 0.6666 eV was applied to simulate the spectra from computed TD-DFT energies and oscillator strengths.</figcaption></figure>
<table><tr><td>c)</td><td>d)</td></tr></table>

The picture provided by LC-PBE is different. First of all, the ES energies are all strongly blue shifted and the low lying ESs have all a very limited CT character, with D<sub>CTS</sub> all below 1.6 Å basically corresponding in the case of ES1 to transition strongly localized on the pyridine bridge core and extending at maximum to the stilbenoid C=C bonds (refer to Figure S.3). Focusing on the first three ESs, only two (ES1 at 4.26 eV and ES3 at 4.73 eV) have sizable and similar intensity while ES2 is actually not visible in the spectra, possessing a null oscillator strength. As a consequence, the spectra predicted at LC-PBE level is characterized by a quite symmetric band with maximum at 280 nm, largely overestimating the experimental maximum observed at 321 nm.

In order to correct the erroneous PBE0 behavior, an adjusted LC-PBE0* functional was thus obtained considering, to define the range separation split value, the D<sub>CT</sub> value (8.1 Å) computed at the PBE0 level for the first excited state and corresponding to the largest D<sub>CT</sub> value. In such a way a range separation parameter (γ*) of 0.118 Bohr<sup>-1</sup> was obtained. By inspection of the low lying ESs computed at LC-PBE0* level one can notice that a significant change in ES character and ordering is obtained with respect to both the PBE0 and LC-PBE descriptions. First of all, there is a significant red shift of the absorption energies with respect to LC-PBE and a blue shift with respect to the PBE0 ones. The intense transitions correspond to ES1, ES2 and ES4 all possessing a similar intensity (f=0.6 a.u.) and occurring at 3.44, 3.98 and 4.09 eV respectively. All these transitions are characterized by a more marked delocalization with respect to the LC-PBE description but the delocalization is also significantly reduced with respect to the PBE0 description, none of the transitions corresponding to a full D to A CT. Furthermore, ES4 is very similar in character to the most intense transition computed at the PBE0 level. Overall, the computed spectrum is thus characterized by an absorption maximum at 310 nm (resulting from the convolution of the ES2 and ES4) and by an evident shoulder corresponding to the low lying ES1. Of note this spectrum is not only providing an estimate of the absorption maximum closer to the experimental one but also a better agreement in band shape with the experimental data.

Finally, since an estimate of the ground to excited state changes in dipole moment, derived by the solvatochromism in absorption, is experimentally available for NODSF and NODST<sup>62</sup>, these values were compared with the corresponding theoretical predictions (as reported in Table 3). Clearly, as a consequence of the overestimation of the hole-electron distance at the excited state, PBE0 is predicting a too large change in dipole moment with an error in predicted dipole change ranging from 5.4 to 7.2 D. On the other hand, the too localized character of the excited states predicted at LC-PBE level translates in an underestimation of the dipole moment changes of 3 and 2.5 D, respectively. Using the adjusted LC-PBE0* functionals a better agreement with the experimental data is obtained, with errors reducing to 0.6 and 1.18 D for NODSF and NODST, respectively.

<table><caption>**Table 3.** Changes in dipole moment (Δμ in Debye) between ground state and excited state computed and experimentally extrapolated (from ref. 60).</caption><thead><tr><th>Compound</th><th>NODSF</th><th>NODST</th><th>NODSP</th></tr></thead><tbody><tr><td>Δμ (exp.)</td><td>8.00</td><td>6.96</td><td>\</td></tr><tr><td>Δμ (PBE0)</td><td>13.40</td><td>14.18</td><td>24.24</td></tr><tr><td>Δμ (LC-PBE)</td><td>4.97</td><td>4.49</td><td>2.44</td></tr><tr><td>Δμ (LC-PBE0*)</td><td>8.60</td><td>8.14</td><td>10.36</td></tr></tbody></table>

Overall, these results, together with the better agreement with the experimental data in predicted absorption maxima for the three compounds, are thus confirming the good performance of the
proposed D<sub>CT</sub> based tuning procedure for the description of both the energy and the nature of the ES in systems giving rise to ES with relevant intramolecular CT character.

### 4.3 Intramolecular CT: prototypical Thermally Activated Delayed Fluorescence (TADF) emitters

Among organic compounds targeting intramolecular CT, those specifically designed as TADF emitters are particularly challenging both from an experimental and a modelling viewpoint.<sup>22,23,24,25,63,64,65,66</sup> In this case, the aim is indeed to design push-pull systems characterized by a first singlet excited state of CT nature and by a minimal singlet-triplet gap. This implies that the optimal target system is expected to display a CT singlet ES where the hole and the electron are localized on orbitals with minimal exchange interaction, thus resulting in a minimization of the singlet-triplet energy gap.

In practice this is chemically translating in π conjugated organic compounds with pseudo-orthogonal donor and acceptor moieties, reducing the overlap of the hole and the electron charge distributions at the ES. The erroneous behavior of DFAs in the description, at TD-DFT level, of CT ES is thus exalted in this type of molecules. Not surprisingly, several papers have focused on the development of efficient and -possibly- accurate approaches rooted on DFT to describe not only their CT singlet state energy but, more generally, to access to the singlet-triplet energy differences which are indeed the crucial quantity to predict their ability as TADF emitters.<sup>63,64,65,66</sup>

Here we have considered six systems (Figure 1) amongst the molecules developed by Adachi and collaborators<sup>22,23,24,25</sup> and lately subject of different theoretical studies<sup>63,64</sup> all pointing out the limitation of global and standard range separated hybrid in predicting their singlet excited state energies.

For these systems an estimate of the vertical absorption energy derived from experimental data is available allowing to directly evaluate the errors of the methods, in absence of other reference theoretical data, due to the large systems size. This direct comparison of vertical transitions and experimental absorption maxima thus clearly does not include the solvent effect nor the difference between vertical and adiabatic energy. The relevant computed data characterizing the first excited state are reported in Table 4.

<table><caption>**Table 4.** Vertical transition energies (E, eV), oscillator strengths (f, a.u.), charge transfer distances (D<sub>CT</sub>, Å) and M<sub>AC</sub> index (in eV) computed for PhCz, ACRFLCN, PXZ-OXD, PXZ-TAZ, PPZ-3TPT and PPZ-DPO (Figure 1) using different levels of theory and the 6-311G(d) basis set. When relevant the range separation parameter (γ, Bohr<sup>-1</sup>) is reported. Full data for the first 10 ES computed is given in Table S.5.</caption><thead><tr><th rowspan="2"></th><th colspan="2">PBE0</th><th colspan="2">LC-PBE</th><th colspan="2">LC-PBE0*</th></tr><tr><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th></tr></thead><tbody><tr><td><strong>PhCz</strong><br>Exp. 3.66</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ES1</td><td>4.10/0.03</td><td>1.412/LE</td><td>4.78/0.05</td><td>0.939/LE</td><td>4.53/0.05</td><td>1.174/LE</td></tr><tr><td>γ</td><td>--</td><td>--</td><td>0.470</td><td>0.470</td><td>0.232</td><td>0.232</td></tr><tr><td><strong>ACRFLCN</strong><br>Exp. 3.05</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ES1</td><td>2.75/0.00</td><td>3.840/4.64</td><td>4.75/0.52</td><td>0.566/LE</td><td>4.44/0.00</td><td>3.617/4.85</td></tr><tr><td>γ</td><td>--</td><td>--</td><td>0.470</td><td>0.470</td><td>0.276</td><td>0.276</td></tr><tr><td><strong>PXZ-OXD</strong><br>Exp. 3.18</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ES1</td><td>2.52/0.00</td><td>5.840/6.000</td><td>4.47/0.02</td><td>0.209/LE</td><td>2.94/0.10</td><td>4.987/5.70</td></tr><tr><td>γ</td><td>--</td><td>--</td><td>0.470</td><td>0.470</td><td>0.094</td><td>0.094</td></tr><tr><td><strong>PXZ-TAZ</strong><br>Exp. 3.33</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ES1</td><td>2.91/0.00</td><td>5.425/6.30</td><td>4.45/0.02</td><td>0.267/LE</td><td>3.35/0.04</td><td>4.578/5.96</td></tr><tr><td>γ</td><td>--</td><td>--</td><td>0.470</td><td>0.470</td><td>0.107</td><td>0.107</td></tr><tr><td><strong>PPZ-3TPT</strong><br>Exp. 3.34</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ES1</td><td>2.51/0.00</td><td>5.795/5.83</td><td>3.94/0.00</td><td>0.160/LE</td><td>3.00/0.00</td><td>4.849/5.57</td></tr><tr><td>γ</td><td>--</td><td>--</td><td>0.470</td><td>0.470</td><td>0.105</td><td>0.105</td></tr></tbody></table>
<table><thead><tr><th rowspan="2">PPZ-DPO<br>Exp. 2.78</th><th colspan="2">PBE0</th><th colspan="2">LC-PBE</th><th colspan="2">LC-PBE0*</th></tr><tr><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th><th>E/f</th><th>D<sub>CT</sub>/M<sub>AC</sub></th></tr></thead><tbody><tr><td>ES1</td><td>2.11/0.00</td><td>6.204/5.50</td><td>3.95/0.00</td><td>0.192/LE</td><td>2.59/0.00</td><td>5.270/5.32</td></tr><tr><td>γ</td><td>--</td><td>--</td><td>0.470</td><td>--</td><td>0.095</td><td>0.095</td></tr></tbody></table>

All ESs, independently of the level of theory, essentially correspond, in agreement with previous works, to a one electron excitation from the HOMO to the LUMO. The corresponding computed NTOs are reported in Figure S.4-S.9. Rapid inspection of the computed NTOs allows to show that only in the case of PhCz, the three approaches lead to the same qualitative picture, namely a transition largely localized on the carbazole unit, as also clearly reflected by the low values of the computed D<sub>CT</sub> all below 1.5 Å corresponding to ESs with negligible CT character. As a consequence the corresponding predicted energy already at the PBE0 level is well above the M<sub>AC</sub> descriptor thus not claiming for a correction to introduce the correct asymptotic behavior by the use of a range separated hybrid. Indeed, when this is done, the results obtained are, as expected, not particularly good, with errors comparable to those of LC-PBE. In other words, we are here introducing in the LC-PBE0* functional a larger fraction of HF exchange to describe an already local transition, thus deteriorating its energy with respect to the global hybrid behavior. Basically, since none of the transitions is actually spotted as ghost, we cannot use our approach to ameliorate the PBE0 results, and here only a tuning of the HF percentage (α) could have been useful (in agreement with previous work<sup>24</sup>).

The deterioration of the results obtained for ACRFLCN (10-phenyl-10H-spiro[acridine-9,9'-fluorene]-2',7'-dicarbonitrile), when moving from PBE0 to LC-PBE0*, on the other hand, is of very different nature. First of all, inspection of the computed NTOs (Figure S.5) shows how the nature of ES1 predicted at PBE0 and LC-PBE level is actually different: PBE0 predicts a charge transfer from the donor acridine core up to the cyano substituents of the fluorene acceptor while LC-PBE predicts a transition localized on the acceptor moiety. Nonetheless, due to the symmetric nature of the molecule, the D<sub>CT</sub> computed at PBE0 level and associated to this net CT is actually, by construction, underestimating of the hole-electron separation. As a result, the D<sub>CT</sub> value used to calibrate the range separation split is actually too low and as a consequence the correction with respect to the LC-PBE value only negligible. A simple geometrical estimate of the actual D-A distance on each of the non-symmetric unit, yields a value of more than 6 Å, much larger than the value used for the actual calculation of only 3.8 Å.

These two molecules are thus good examples of the limit of the approach here proposed which must be used only for systems for which the D<sub>CT</sub> provides a reliable estimate of the hole electron separation (that is not for symmetric systems) and, of course, only to correct unphysical CT states.

The last four molecules, (PXZ-OXD, PPZ-DPO, PXZ-TAZ, and PPZ-DPO) are on the other hand prototypical cases of D-A molecules showing a sizable CT character largely underestimated at PBE0 level as clearly evident from the M<sub>AC</sub> value reported in Table 4. In particular, in PXZ-OXD and PPZ-DPO the same 2,5-diphenyl-1,3,4-oxadiazole acceptor is combined with a phenoxazine (PXZ) and 5-phenyl-5,10, dihydrophenazine (PPZ) donor, respectively, while the 3,4,5-tripheny-4H-1,2,4-triazole acceptor is combined with the previously PXZ and PPZ donors to yield PXZ-TAZ and PPZ-3TPT, respectively (see Figure 1). Although confusing we have preferred to keep this nomenclature to be consistent with previous literature reports.

Inspection of the NTOs (Figure S.5 to S.9) and D<sub>CT</sub> (Table 4) computed for these systems shows how while the PBE0 approach predicts a quite long range CT transition (all D<sub>CT</sub> computed between 5.4 Å to 6.3 Å) from the donor core to the phenyl connector of the acceptor moiety, the LC-PBE functional provides a strongly localized description of the transitions (all D<sub>CT</sub> below 0.26 Å) centered on the donor moiety. The ES1 energies predicted at PBE0 level are all significantly underestimated, the differences between the M<sub>AC</sub> estimates and the TD-DFT energies being all well above 3.5 eV.
Therefore, for these molecules, the ES1 at PBE0 corresponds to unphysical CT states. Inspection of the first ten excited states computed for these systems (reported in Table S.5) nonetheless shows that (higher in energy) for each molecule there is also a transition possessing a larger D<sub>CT</sub> (in the 10-11 Å range) corresponding to a full Donor to Acceptor transfer and displaying an even larger underestimation of the transition energy. Therefore, as indicated in the Computational Details section we considered first, as D<sub>CT</sub> to set the range separation split in LC-PBE0*, the maximal D<sub>CT</sub> computed and thus corresponding to a full donor to acceptor CT. The results obtained are those reported in Table 4.

Clearly not only a net amelioration with respect to both the PBE0 underestimation and the LC-PBE overestimation of ES energy is obtained but also a very good agreement with the experimental data is reached, bringing the error on the standard TD-DFT range (that is below 0.25 eV). Furthermore, these corrections allow to reduce the differences between the M<sub>AC</sub> estimate and the TD-DFT energies below 3 eV thus in principle not needing for further iterations of the procedure. Nonetheless, to fully analyze the impact of different choices of D<sub>CT</sub> value to define the range separation split value, we decided to explore three possibilities :

1) The use of the D<sub>CT</sub> associated to the ES of interest (here the first), independently on its M<sub>AC</sub> value. This approach will be named as LC-PBE0* (D<sub>CT</sub><sup>target</sup>);

2) The use of the smallest D<sub>CT</sub> value associated to unphysical transitions (that is a transition for which the M<sub>AC</sub> to TD-DFT energy difference is larger than 3 eV). This approach will be named as LC-PBE0* (D<sub>CT</sub><sup>ghost-min</sup>);

3) The use of the D<sub>CT</sub> value associated to “most unphysical” transitions (that is to the transition for which the M<sub>AC</sub> to TD-DFT energy difference is the largest). This approach will be named as LC-PBE0* (D<sub>CT</sub><sup>ghost-max</sup>)

Of note the latter choice (LC-PBE0* (D<sub>CT</sub><sup>ghost-max</sup>)) actually fully coincides with the default choice since the unphysical transition possessing the largest D<sub>CT</sub> (our default choice) also shows the largest discrepancy between the computed TD-DFT energy and the M<sub>AC</sub> estimate. Full computed data on the first 10 ES are reported in Supporting Information (Table S.6 to S.9) while in Figure 6 we reported, for these three approaches, the computed difference (error) between the computed bright singlet with respect to the experimental absorption maxima.

**Figure 6.** Error (in eV) in computed singlet excited state energy with respect to the experimental data computed for the PXZ-OXD, PXZ-TAZ, PPZ-3TPT and PPZ-DPO using the PBE0; LC-PBE, and LC-PBE0* level and different range separation split values (see text for explanation). A negative(positive) value indicates an under(over)estimation of the ES energy
<figure><img src="image_7.png" /></figure>

From these data, it is clear that the use of D<sub>CT</sub><sup>ghost-max</sup> is actually yielding the best agreement with the experimental data although, not surprisingly, slightly systematically undercorrecting the PBE0 estimate. Indeed, the use of the maximal D<sub>CT</sub> value is actually introducing the full HF exchange at longest distances and thus the smallest correction with respect to the PBE0 value. A possible problem would actually be that some of the transitions (with unphysical character but smaller D<sub>CT</sub>) are not sufficiently corrected. This is not the case for the molecules analyzed here since, as already mentioned, the difference between the M<sub>AC</sub> estimate and the LC-PBE0* energies (computed using γ derived from D<sub>CT</sub><sup>ghost-max</sup>) are all below the threshold considered to evaluate them as unphysical states. Nonetheless, if this would happen the procedure can be re-iterated using the D<sub>CT</sub> estimated at this level of theory until the full disappearance of the unphysical states. On the other hand, the use of the smallest D<sub>CT</sub> value would certainly remove all unphysical states but will not allow for subsequent adjustment and is ultimately expected to provide a less efficient correction, leading to results always closer to the LC-PBE0 value.

Finally, in table SI.12 are reported the vertical excitations computed three selected systems using the B3LYP family. Also in this case, the LC-B3LYP* functional allows to correct the overestimation of the CT transition energies computed at standard LC level and consideration fully analogous to those pertinent for the PBE0 family can be drawn, thus stressing the general validity of the approach proposed.

**5. Conclusions**

The non empirically adjusted procedure here proposed to tune RSHs to deal with systems presenting inter- and/or intermolecular CT excitations has been proven to be able to systematically yield an accurate estimate of excitation energies, with a net amelioration of the performance with respect to both the underpinning global hybrid and the corresponding non optimized LR corrected RSH.

This procedure, that can be performed on the fly and practically with a very limited extra cost actually allows to recover an accuracy of 0.2 eV for intramolecular CT states and to cure both the overestimation of CT energies of standard LR corrected RSH and the underestimation of GH.

Beside the numerical accuracy, it has to be stressed that the adjustable RSHs also provide a correction to the overlocalized nature of the transition predicted by the LRC approaches, thus allowing not only to predict accurate energies but also properties of the CT excited states, such as dipole moments.
Although the use of specific descriptors to evaluate the hole-electron distance and thus to tune the range separation split (such as the D<sub>CT</sub> based on unrelaxed densities and the M<sub>AC</sub>) may introduce a certain degree of approximation and intrinsic limitations (for instance for the application to symmetric systems), the possibility of repeating the procedure of tuning in an iterative manner grants for a global improvement in the accuracy especially in the case of intramolecular CT and, more generally, for CT ES for which the term leading the erroneous behavior is the missing fulfillment of the exact asymptotic behavior, which is de-facto recovered by our tuning procedure. On the other hand, cases of CT for which the leading erroneous term would be an incorrect estimate of the IP/EA of the systems are expected to be still more problematic and only partially recovered by the present procedure. Work is currently in progress to allow further corrections in this direction.

**Supporting Information.** Energies, D<sub>CT</sub> and M<sub>AC</sub> descriptors, and NTO computed for all systems; Discussion on the effect of geometry: use of GS PBE0 structures vs consistently optimized geometries; Data for the BLYP family; structures (coordinated) used.

**Acknowledgements**

I.C. gratefully acknowledges fundings from the French National Agency (ANR) under the project TheoBioDy (ANR-21-CE29-0028). T.Y. acknowledges financial support from the China Scholarship Council (grant no. 202308440240). This work was also supported by the European Union (ERC, project MaMa, n. 101097351). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

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