Benchmarking Density Functionals for Ground- and Excited-State Reduction and Oxidation Potentials of Organic Photoredox Catalysts

Abstract

Photoredox catalysis enables chemical reactions under mild reaction conditions; single-electron transfer is a common key step. Quantum mechanical calculations has accelerated the increasing numbers of photoredox catalysts because of its straightforward implementation in many softwares. Unfortunately, computing the redox potentials of molecular excited states is difficult. While density functional theory (DFT) offers a cost-effective way to compute ground- and excited-state redox potentials, a benchmarking study identifying the best density functional has not yet been conducted. In this report, we evaluate 147 combinations of density functionals and basis sets (i.e., model chemistries) to compute the S₀, S₁, and T₁ reduction and oxidation potentials for nine organic photoredox catalysts (cyanoarenes, benzophenones, xanthenes, and acridinium) with experimentally determined ground- and excited-state redox potentials. We provide recommendations for predicting ground- and excited-state reduction and oxidation potentials. We find that the best model chemistry for excited-state reduction and oxidation potentials, with PBE0-D3BJ/6-311+G­(d,p) and N12-SX/6-311+G­(d,p) excelling for S₁ states and ωB97X/6-311+G­(d,p) and BHandH/6-311+G­(d,p) perform best for T₁ states. Guidance is provided for balancing accuracy and CPU time, especially for T₁ and S₁ redox potentials.

Introduction

Photoredox catalysis has emerged as an important methodology to achieve new chemical transformations under light irradiation and mild reaction conditions. , A typical reaction catalyzed by photoredox catalysts features a photoredox catalyst (PC) and a substrate. The PC absorbs light, which promotes an electronic transition to a molecular excited-state. Next, a single-electron transfer (SET) can occur from the PC to the substrate or from the substrate to the PC. , Organic PCs (OPCs) are especially attractive because they circumvent the need for precious metals (i.e., ruthenium and iridium), and substituents can be installed to tune molecular orbital energy levels. , It is essential to discover new molecules with the desired properties to extend the applicability of the OPCs. One property that can be tuned is the PC and/or substrate’s ground- and excited-state reduction and oxidation potentials. However, the chemical space of OPCs is vast, making the experimental trial-and-error discovery of novel OPCs laborious and costly. Computational approaches, such as quantum mechanical calculations, are crucial for the rational design and discovery of new OPCs and organic photoredox reactions by allowing the prediction of the ground- and excited-state structures and properties of the OPCs. Unfortunately, wave function-based methods are often too costly to implement for large OPCs (e.g., acridinium); DFT methods are advantageous because of their implementation in many quantum chemical software and relatively low computational cost.

Ground-state (GS) redox potential prediction has been extensively explored for organic molecules. Various methods have been benchmarked, including DFT, − semiempirical, , and a combined approach of DFT with machine learning (ML) corrections. , The B3LYP density functional has been extensively used to predict ground-state reduction and oxidation potentials. − Predicting excited-state (ES) redox potentials requires the computation of the corresponding GS redox potentials and 0–0 energies (E ^0–0). The E ^0–0 is the energy difference between the GS and ES energies at their respective minima, corrected for the difference in zero-point vibrational energy (ZPVE). Most density functionals have been developed, parametrized, and optimized for GS properties. As such, benchmarking density functionals for ES properties is crucial to ensure accurate results.

Jacquemin and co-workers benchmarked time-dependent DFT (TD-DFT) methods by evaluating the E ^0–0 of 40 π-conjugated molecules for which condensed-phase experimental 0–0 energies are available. , The density functionals with the lowest error of 0.22–0.23 eV were PBE0 and M06 for E ^0–0. The M06-2X and CAM-B3LYP methods provided the highest correlation between prediction and experiment for E ^0–0 with a R ² of 0.95 and mean absolute error (MAE) of 0.25 eV_. ,

In 2010, Pastore et al. benchmarked DFT and TD-DFT methods with the 6-31G­(d) basis set to predict S₀ and S₁ oxidation potentials for four triphenylamine dyes. MPW1K and CAM-B3LYP provided balanced descriptions of S₀ and S₁ oxidation potentials, with errors ranging from 0.2 to 0.3 eV. In 2018, McCarthy et al. predicted the T₁ reduction potentials of 14 phenoxazines using M06/6-311+G­(d,p). In 2018, Singh and co-workers computed the redox potentials of OPCs in the T₁ excited state using the B3LYP/6-311G­(d) model chemistry. In 2021, Choi and Kim computed the redox potentials for quinoxalines by performing geometry optimization with B3LYP/6-311++G­(d,p) density functional and energy refinement by spin-flip DFT; the mean signed deviations were −0.15 eV for ground state and 0.01 eV for S₁ excited state. Feher and co-workers recently benchmarked computational approaches to predict S₀ and S₁ redox potentials for 37 OPCs. They evaluated the performance of 15 different DFT functionals using a def2-TZVPP basis set for electronic energy and the def2-SVP basis set for optimization. The best results were achieved by combining M06-2X for GS potentials and using neural network ML to predict S₁ E ^0–0. The MAE for prediction of the reduction and oxidation potentials for S₀ was 0.2 V by applying a 0.2 V shift to correct systematic underestimation, and that for S₁ was 0.3 eV.

Despite previous efforts, further evaluation is required to assess the accuracy of DFT and TD-DFT functionals and basis sets in predicting GS and ES redox potentials for practical characterization of OPCs. In this study, we present an extensive benchmarking study of 147 combinations of density functionals and basis sets (i.e., model chemistries) to evaluate S₀, S₁, and T₁ reduction and oxidation potentials, as well as S₁ and T₁ E ^0–0 for nine experimentally measured OPCs (Scheme ). Given that ES redox potentials depend on GS redox potentials and the ES E ^0–0, we assessed the performance of these model chemistries for these properties. Additionally, we examine the role of basis set size and functional type on the predictive accuracy against experimental results.

1. Tested Sets of 9 OPCs.

Computational Details

We systematically evaluated the performance of 147 model chemistries, each formed by combining one of 21 DFT functionals (as listed in Table ) with seven basis sets: 6-31G­(d), ,,, 6-31+G­(d,p), 6-311+G­(d,p), , cc-pVDZ, ,, aug-cc-pVDZ, , cc-pVTZ, , and aug-cc-pVTZ. ,

1. DFT Functionals Details.

DFT functional	EXX % or range	class
SVWN	0	LSDA
SVWN-5	0	LSDA
M11-L	0	mGGA
BB95	0	mGGA
TPSSh ,	10	hybrid-mGGA
B3LYP-D3BJ ,	20	hybrid-GGA
mPW1PW91	25	hybrid-GGA
mPW1PBE	25	hybrid-GGA
PBE0-D3BJ ,	25	hybrid-GGA
M06	27	hybrid-mGGA
MPW1B95	31	hybrid-mGGA
BMK-D3BJ ,	42	hybrid-mGGA
MPW1K	43	hybrid-GGA
BHandH	50	hybrid-GGA
M06-2X	54	hybrid-mGGA
M06-HF	100	hybrid-mGGA
N12-SX	25	RSH GGA
MN12-SX	25	RSH mGGA
CAM-B3LYP-D3BJ ,	19–65	RSH GGA
ωB97X	15.77–100	RSH GGA
ωB97X-D	22.2–100	RSH GGA

We adapted the automated workflow developed for our VERDE materials database to perform conformational search and optimize GS and ES geometries with each model chemistry. The workflow is composed of four phases, illustrated in Scheme .

2. Automated Workflow Used to Compute the GS and ES Reduction and Oxidation Potential.

In phase 1, RDKit generates 3D coordinates from a simplified molecular-input line-entry system (SMILE) string, followed by a conformational search that produces four low-lying conformers minimized with the Universal Force Field. In phase 2, the lowest energy conformer is optimized with semiempirical quantum mechanical calculations (PM7 and RM1-D) and a single-point energy calculation with each model chemistry to determine the lowest energy conformer. In phase 3, optimization and frequency calculation with each model chemistry are performed in vacuo for the ground state and radical cation, followed by optimization with the implicit solvation with the polarizable continuum model (PCM) using the integral equation formalism variant IEFPCM^MeCN  for the S₀, S₁, T₁, radical cation, and radical anion. The optimizations were performed with DFT for S₀, T₁, radical cation, radical anion, and TD-DFT for S₁.

All single-point energy, optimization, and frequency calculations were carried out in Gaussian 16, except for RM1-D, which used GAMESS. We used the “ultrafine” pruned (99,590) grid for the calculations performed in Gaussian. The optimized geometries have been determined with a combination of EDIIS and CDIIS algorithms, with no damping or Fermi broadening. This is the default SCF procedure in Gaussian (SCF = Tight keyword). The analytic second derivatives were computed at the first step and every ten steps thereafter during the optimization. After converged optimization, frequency calculation was performed to confirm that all frequencies were positive.

The calculation results are extracted in phase 4, which includes the GS and ES geometries and the free and zero-point energies. These results are then used to compute the GS and ES reduction and oxidation potentials and S₁ and T₁ E ^0–0, detailed in the following subsections.

GS (S₀) Redox Potentials

The general approach for predicting GS redox potentials involves computing the free energy difference between the oxidized and reduced forms. This is related to the redox potential by eq , in our case referenced to SCE in MeCN. Some studies use a free energy cycle to transform gas-phase energies into solution-phase energies, which requires calculating each species’ solvation energy individually. , However, a more straightforward method is to calculate free energies with an implicit solvation model, which circumvents the calculation of gas-phase ionization potentials and gas-phase entropy requirements. , We adopted a straightforward computational approach involving the free energy difference calculation between the optimized reduced and oxidized forms and then dividing it by the number of electrons transferred (herein one electron) multiplied by the Faraday constant. For the GS reduction potential, the reduced form is the initial molecule plus one electron (a radical anion if the initial molecule is neutral), while the oxidized form is the original molecule.

For the GS oxidation potential, the reduced form is the original molecule and the oxidized form is the original molecule minus one electron (a radical cation if the initial molecule is neutral). The optimizations are performed with the implicit solvation with the PCM using the integral equation formalism variant IEFPCM^MeCN:

$$E_{1/2}=\frac{-({\mathrm{G}}_{\mathrm{reduced}}-{\mathrm{G}}_{\mathrm{oxidized}})}{{\mathrm{n}}_{\mathrm{e}}\mathcal{F}}-4.429$$ 1

0–0 Energies (E ^0–0)

To compute the E ^0–0, it is necessary to locate optimized geometries of the ground state (S₀) and excited state (i.e., S₁ and T₁) with the PCM implicit solvation using the integral equation formalism variant IEFPCM^MeCN. For S₀ and T₁, the optimization was performed with DFT and for the S₁, the optimization was performed with TD-DFT (td = root = 1). The adiabatic energy is the difference between the total electronic energies calculated for GS and ES at their respective optimized geometries, shown in eq :

$$E^{\mathrm{adia}}=E^{\mathrm{ES}}(R^{\mathrm{ES}})-E^{\mathrm{GS}}(R^{\mathrm{GS}})$$ 2

In addition, the ZPVE variation between GS and ES was computed, as shown in eq :

$$\mathrm{\Delta }{E}^{\mathrm{ZPVE}}=E^{\mathrm{ZPVE}}(R^{\mathrm{GS}})-E^{\mathrm{ZPVE}}(R^{\mathrm{ES}})$$ 3

Then, using eq , the E ^0–0 energies can be computed for each excited state (S₁ and T₁):

$$E^{0-0}=E^{\mathrm{adia}}-\mathrm{\Delta }{E}^{\mathrm{ZPVE}}$$ 4

ES (S₁ and T₁) Redox Potentials

The photoreducing or photooxidizing capabilities of the OPCs are quantified by their ES redox potentials. Usually, the ES redox potentials are calculated by combining the E ^0–0 energy with the GS redox potentials. The ES reduction potential (E _red ) and oxidation potential (E _ox ) are calculated using eq and , respectively:

$$E_{\mathrm{red}}^{*}=E_{\mathrm{red}}+E^{0-0}$$ 5

$$E_{\mathrm{ox}}^{*}=E_{\mathrm{ox}}-E^{0-0}$$ 6

In these calculations, the ES reduction potential is obtained by adding the GS reduction potential to the E ^0–0. Conversely, the ES oxidation potential is determined by subtracting the E ^0–0 energy from the GS oxidation potential.

Benchmark Set

These molecules have been employed as OPCs in a variety of reactions. The selection was based on their diversity in family type, molecular size, redox potentials, and specific ES involved in photoinduced electron transfers, also due to the availability of experimental values in the same solvent, acetonitrile, and redox potentials for both S₁ and T₁ ES. We compared our computed GS and ES redox potentials to those measured experimentally for four families of OPCs: cyanoarenes, xanthenes, acridiniums, and benzophenones (Scheme ). Experimentally measured values for the S₀, S₁, and T₁ redox potentials and corresponding ES redox potentials (i.e., S₁ and T₁) and E ^0–0 energies are summarized in Table .

2. Experimental GS and ES Redox Potentials and S₁ and T₁ E^0‑0 Energies .

molecule	E S 1	E T 1	E red	E ox	E red	E ox	E red	E ox
DCB	4.01	3.04	–1.46		2.55		1.58
DCN	3.57	2.41	–1.27		2.30		1.14
DCA	2.90	1.81	–0.91		1.99		0.90
RhB6G	2.32	2.09	–1.14	1.23	1.18	–1.09	0.95	–0.86
BP	3.22	3.00	–1.72	2.39	1.50	–0.83	1.28	–0.61
MK	2.98	2.70	–2.20	0.86	0.76	–2.12	0.48	–1.84
XO	3.40	3.22	–1.65	1.80	1.76	–1.61	1.57	–1.42
TXO	3.14	2.80	–1.62	1.69	1.52	–1.45	1.18	–1.11
Mes-Acr-Me+	2.68	2.37	–0.49	1.88	2.19	–0.80	1.88	–0.49

^aAll values are in electron volts (eV), and the potentials are measured against the standard calomel electrode (SCE) in acetonitrile. Reproduced from ref . Copyright 2016 American Chemical Society.

Cyanoarenes (DCB, DCN, and DCA) have been used for over 30 years as OPCs. The ES electron transfer occurs primarily through the S₁-state. , Rh6G-H ⁺ is part of the xanthene family; the SET occurs from the S₁ ES, and it has low intersystem crossing (ISC) efficiency (F_ISC = 0.01). Benzophenones (BP, MK, XO, and TXO) typically undergo rapid ISC, enabling SET from the T₁ excited state. The triplet state enables moderate activity as oxidants and is generally poor reductants, except for MK. Acridiniums are powerful oxidants in the S₁, with E _red greater than 2.0 eV. We chose Me-Acr-Me ⁺ because it is the most widely used acridinium OPC. ,

Results and Discussion

Our analysis is divided into sections, focusing on the overall trends for each computed property. Statistical analysis is employed to identify significant patterns. The output files and Cartesian coordinates for each molecule and model chemistry are available in the Supporting Information.

We first evaluate the role of the basis set and density functional in predicting GS redox potentials and the ES E ^0–0. Next, we assess the S₁ and T₁ redox potentials using the model chemistries that demonstrated accuracy in predicting both the GS redox potentials and ES E ^0–0, minimizing reliance on error cancellation. Finally, we analyze the CPU time of the top-performing model chemistries to identify the optimal model for ES redox potentials, balancing accuracy and computational cost.

Basis Set Effects

We evaluate the performance of seven basis sets, including Pople-type 6-31G­(d), 6-31+G­(d,p), and 6-311+G­(d,p), and Dunning-type cc-pVDZ, aug-cc-pVDZ, cc-pVTZ, and aug-cc-pVTZ, with or without diffuse functions. Figure presents the MAE for each model chemistry compared to experimental values, organized by increasing the exchange-correlation percentage (EXX) as detailed in Table . We set a maximum CPU time of 160 CPU days for optimization and frequency calculations. Figure presents the MAE for basis sets with successful convergence in at least five molecules for GS reduction and S₁ and T₁ E ^0–0 predictions. The MAE is based on at least three molecules for GS oxidation, as experimental data for cyanoarenes’ GS oxidation are not available in the literature.

1.

MAEs for GS reduction potential (a), GS oxidation potential (b), S₁ E ^0–0 (c), and T₁ E ^0–0 (d)_. The plot includes model chemistries with at least five molecules with completed jobs for GS reduction potential, S₁ E ^0–0 and T₁ E ^0–0, and at least three molecules with completed jobs for GS oxidation potential. The different shapes represent the basis sets: circles indicate Pople double-ζ basis sets (6-31G­(d), 6-31+G­(d,p)), squares indicate Pople triple-ζ basis set (6-311+G­(d,p)), crosses indicate Dunning double-ζ basis sets (cc-pVDZ, aug-cc-pVDZ), and triangles indicate Dunning triple-ζ basis sets (cc-pVTZ, aug-cc-pVTZ). The blue color represents basis sets with diffuse functions, while the orange color represents basis sets without diffuse functions.

For GS reduction potential predictions (Figure a), including diffuse functions leads to a notable reduction in MAE, except for SWVN, SVWN-5, M11-L, and MPW1K. Without diffuse functions, MAEs range from 0.10 to 0.53 eV; upon inclusion, the range narrows significantly to 0.05 to 0.23 eV. This improvement originates from the need to optimize radical anions in GS reduction, where diffuse functions mathematically account for electron density further from the nuclei, thus better describing electrons in nonbonding orbitals and anions. ,,, In general, no substantial improvement was observed when the basis set size was increased from double-ζ to triple-ζ.

We next examined the influence of basis sets on GS oxidation potential predictions (Figure b). For most model chemistries, MAEs vary considerably across basis sets for the same density functional, with differences ranging from 0.07 to 0.29 eV. No consistent trend is observed regarding the inclusion of diffuse functions. In some cases, the lowest MAE is obtained with a basis set lacking diffuse functions (e.g., BMK-D3BJ/6-31G­(d), MAE = 0.06 eV), while in others, basis sets with diffuse functions results in lower MAE (e.g., M06/aug-cc-pVTZ, MAE = 0.09 eV). For Pople basis sets, increasing from double-ζ to triple-ζ often reduces accuracy. In most cases, 6-31+G­(d,p) outperformed 6-311+G­(d,p), with MAE differences of 0.01 to 0.24 eV. No clear trend emerges between double- and triple-ζ for Dunning basis sets without diffuse functions. However, when diffuse functions are included, triple-ζ sets generally lead to lower MAEs, with improvements ranging from 0.01 to 0.13 eV. Exceptions include TPSSH, CAM-B3LYP-D3BJ, and M06-2X, where aug-cc-pVDZ has a lower MAE by 0.01 to 0.05 eV than aug-cc-pVTZ.

For S₁ E ^0–0 using aug-cc-pVDZ and aug-cc-pVTZ, 54% of S₁ calculations failed to converge within the 160 CPU-day limit. These augmented basis sets include diffuse functions on every atom for each symmetry function, significantly increasing CPU time. This issue is more pronounced with aug-cc-pVTZ, where only BMK-D3BJ achieved successful convergence for at least five molecules within the time limit. We compare BMK-D3BJ frequency calculation times using aug-cc-pVTZ and 6-311+G­(d,p), finding that aug-cc-pVTZ requires 13 to 32 times more CPU time for the same molecule. We then analyze the MAE for S₁ E ^0–0 predictions across different basis sets (Figure c). For a given density functional, MAEs differences range from 0.04 to 0.15 eV, depending on the basis set except for MPW1K, which shows a 0.20 eV variation. We found no correlation between MAE and a specific basis set. For example, N12-SX achieves its lowest MAE (0.20 eV) with aug-cc-pVDZ, while B3LYP-D3BJ has a low MAE with cc-pVDZ (0.22 eV). This suggests that the choice of basis set and density functional impacts the accuracy of the predictions, underscoring the importance of this benchmarking study.

For T₁ E ^0–0, the difference in the MAE across basis sets for the same functionals varies from 0.02 to 0.14 eV (Figure d). Functionals with 20 to 25% exact exchange (EXX) show minimal variation (0.02 to 0.04 eV), indicating that the choice of functional has a more significant impact on MAE than the basis set. In contrast, hybrid functionals with EXX > 42% and RSH show more considerable variations (0.05 to 0.14 eV), suggesting an increased sensitivity to basis set choice, with the lowest MAEs (0.08 to 0.20 eV) obtained using basis sets with diffuse functions. Notable exceptions include MPW1K, N12-SX, and MN12-SX, which have low MAEs with basis sets lacking diffuse functions. Overall, the effect of the basis set on T₁ E ^0–0 predictions depends on the density functional, with a higher EXX leading to a greater dependence on diffuse functions for improved accuracy.

Density Functionals

Next, we assess the role of density functionals in predicting GS redox potentials and ES E ^0–0, using the 6-311+G­(d,p) and aug-cc-pVDZ basis sets. We examine density functionals, including LSDA and meta-GGA, which omit EXX. We also include hybrid GGAs and meta-GGAs that incorporate EXX and range-separated hybrids (RSHs), which feature an increasing fraction of exact exchange as the interelectronic distance increases.

Figure shows the mean signed error (MSE) and MAE for each model chemistry, benchmarked against experimental data, and arranged in order of increasing EXX, as outlined in Table . The MAEs are depicted as dots. We include results for density functionals with at least five molecules showing successful convergence in the 160 CPU-day limit for GS reduction potential, S₁, and T₁ E ^0–0, and at least three molecules for GS oxidation potential. Our analysis explores whether the density functionals systematically overestimate or underestimate the predicted properties by examining the MSE. Additionally, we attempt to understand the role of the EXX percentage with MAE (relative to experimental values).

2.

MSEs and MAEs for the GS reduction potential (a), GS oxidation potential (b), S₁ E ^0–0 (c), and T₁ E ^0–0 (d). The plot includes model chemistries with at least five molecules with completed jobs for GS reduction potential, S₁ E ^0–0, and T₁ E ^0–0 and at least three molecules with completed jobs for GS oxidation potential. The bars represent the MSE values, while the dots indicate the MAEs. Results for the 6-311+G­(d,p) basis set are colored light blue, and those for the aug-cc-pVDZ basis set are colored dark blue.

Starting with the GS reduction potential, we find a consistent overestimation across 67% of the tested density functionals with positive MSEs ranging from 0.01 to 0.75 eV, irrespective of the basis set. Only three density functionals, BB95, BMK-D3BJ, and BHandH, show consistent MSE underestimation. The MAE range ranges from 0.05 to 0.13 eV for most functionals, with standard deviations from 0.02 to 0.13 eV. The lowest MAE of 0.05 eV is with B3LYP-D3BJ and BB95 with the aug-cc-pVDZ basis set and TPSSH with the 6-311+G­(d,p) basis set, along with MPW1B95 across both basis sets, with standard deviations between 0.02 and 0.05 eV. The BHandH, M11-L, M06-HF, SVWN, and SVWN-5 density functionals have higher MAEs ranging from 0.17 to 0.75 eV; we do not recommend using these density functionals for GS reduction potential predictions.

Almost all computed GS oxidation potentials are overestimated for both basis sets, with MSE values ranging from 0.02 to 0.82 eV. The only exception is TPSSH with aug-cc-pVDZ with an MSE of −0.10 eV. The MAE is consistently lower with the Dunning-type basis set, aug-cc-pVDZ. BHand is an exception; the Pople-type basis set 6-311+G­(d,p) is 0.01 eV lower than the Dunning type. The model chemistries with the lowest MAEs range from 0.10 to 0.15 eV and contain hybrid density functionals TPSSH, B3LYP-D3BJ, and BHandH with %EXX values of 10, 20, and 50%, respectively. For these model chemistries, the standard deviation ranges from 0.06 to 0.15 eV. We recommend using model chemistries with TPSSH, B3LYP-D3BJ, BHandH, N12-SX, and aug-cc-pVDZ or 6-311+G­(d,p) to predict GS oxidation potentials.

Next, we analyze the role of the density functionals in the prediction of S₁ E ^0–0. With density functionals with %EXX from 0 to 43%, the S₁ E ^0–0 is underestimated, with MSE ranging from −0.71 to −0.01 eV, except for BMK-D3BJ. For hybrid functionals with %EXX ≥ 50% or RSH density functionals, the S₁ E ^0–0 is overestimated, with MSE ranging from 0.03 to 0.47 eV, except for N12-SX/6-311+G­(d,p) model chemistry, which has an MSE of −0.05 eV. The model chemistries yielding the lowest MAEs (0.20 to 0.25 eV) include hybrid density functionals with %EXX between 20 and 25%, as well as the RSH functional N12-SX, which has a standard deviation of 0.15 to 0.23 eV across both basis sets. PBE0 is an exception, with less than five molecules converged with the aug-cc-pVDZ basis set. Thus, we recommend using the following hybrid density functionals for predicting S₁ E ^0–0: B3LYP-D3BJ (20% EXX); mPW1PBE, mPW1PW91, and PBE0-D3BJ (25% EXX); or the RSH functional N12-SX. We recommend using 6-311+G­(d,p) or aug-cc-pVDZ for the basis sets. However, we recommend the 6-311+G­(d,p) basis set for PBE0-D3BJ.

The choice of density functional also has an important effect on the S₁ geometry of BP. These functionals resulted in similar out-of-plane dihedral angles of 180° and 91°, indicating a twisted conformation between the two phenyl rings. This contrasts with the GS BP optimized geometries, where the dihedral angles typically range from 166° to 150°. For BP, these model chemistries yielded low errors between the predicted and experimental S₁ E ^0–0, ranging from 0.01 to 0.16 eV. In contrast, BHandH/6-311+G­(d,p) had the highest error of 0.67 eV, with dihedral angles that deviated substantially from the recommended functionals (150° and 166°). Figure illustrates the differences in the S₁ BP optimized geometry between the model chemistry with the lowest error for S₁ E ^0–0, B3LYP-D3BJ/aug-cc-pVDZ, which achieved an error of 0.04 eV.

3.

S₁ optimized structure with model chemistries with lowest and highest error between the predicted and experimental S₁ E ^0–0 of BP. The atoms are highlighted in blue for θ₁ and orange for θ₂.

The T₁ E ^0–0 is generally underestimated by density functionals, with MSEs ranging from −0.44 to −0.01 eV. The exceptions are M06-2X/6-311+G­(d,p), which has an MSE of 0.00 eV, and M06-HF, which shows positive MSEs of 0.26 and 0.28 eV with both basis sets. The density functionals with the lowest MAE for predicting T₁ E ^0–0 exhibit MAEs ranging from 0.15 to 0.21 eV, with standard deviations between 0.13 and 0.18 eV. These model chemistries include the hybrid density functionals BMK-D3BJ, BHandH, and M06-2X, with %EXX values of 42, 50, and 54, respectively, as well as the RSH functionals MN12-SX, CAM-B3LYP-D3BJ, ωB97X, and ωB97X-D. The MAE of both basis sets is similar to these density functionals, with an absolute difference of MAE ranging from 0.01 to 0.03 eV.

ES Redox Potentials

We established the optimal basis set and density functional choice for predicting the GS redox potentials and ES E ^0–0 energies. We now identify the model chemistry for predicting ES redox potentials with the lowest MAE. The ES reduction potential is calculated by adding the GS reduction potential to the ES E ^0–0 (eq ). In contrast, the ES oxidation potential is determined by subtracting the ES E ^0–0 from the GS oxidation potential (eq ). The model chemistries used to predict ES redox potentials were selected based on their ability to simultaneously minimize the MAE for both GS redox potentials and ES E ^0–0. The accuracy of the selected model chemistries will then be evaluated using MAE, and the CPU time will be assessed by measuring the time required for frequency calculations for the ES optimized geometry of TXO and BP for the selected model chemistries to predict reduction and oxidation potential, respectively. The top-performing methods will be identified based on a balance between accuracy and CPU time for frequency calculations. Finally, we will analyze the correlation between predicted ES potentials and experimental values by fitting a linear regression and determining the R ² value.

We begin by examining the S₁ reduction potential. The model chemistries that have the lowest MAE when predicting both the GS reduction potential and the S₁ E ^0–0 values are B3LYP-D3BJ, mPW1PBE, and N12-SX with both 6-311+G­(d,p) and aug-cc-pVDZ basis sets, PBE0-D3BJ/6-311+G­(d,p), and mPW1PW91/aug-cc-pVDZ. The S₁ reduction potentials of MAE range from 0.15 to 0.23 eV, with standard deviations between 0.09 and 0.17 eV. Among these, the N12-SX/aug-cc-pVDZ had the lowest MAE (0.15 eV), with standard deviations of 0.17 eV.

We note that cyanoarenes DCA and DCB exhibit absolute errors for S₁ reduction potential ranging from 0.26 to 0.44 eV and 0.35 to 0.52 eV, which were higher than the absolute errors observed for other molecules (ranging from 0.01 to 0.41 eV). The largest source of error arose from predicted S₁ E ^0–0. For DCA, S₁ E ^0–0 is underestimated, with errors ranging from −0.54 to −0.47 eV, while the GS reduction potential is overestimated, with errors ranging from 0.12 to 0.20 eV. These opposing errors partially cancel out because the two properties are summed to compute the ES reduction potential (eq ). For DCB, the GS reduction potential and S₁ E ^0–0 were overestimated, with ranges of −0.01 to 0.07 eV and 0.34 to 0.45 eV, respectively, which compounds the error.

Figure a compares the job CPU time of S₁ TXO. For N12-SX, B3LYP-D3BJ, and mPW1PBE, the Pople basis set is significantly more efficient, showing a 31 to 68% reduction in CPU time, respectively. The top 3 performers, based on both MAE and CPU time, were PBE0-D3BJ/6-311+G­(d,p), B3LYP/6-311+G­(d,p), and mPW1PW91/aug-cc-pVDZ, with MAEs of 0.16, 0.19, and 0.19 eV, respectively, and CPU times of 2, 3, and 5 h. Figure b compares the correlation between the predicted and experimental S₁ reduction potentials; the selected model chemistries have R ² ranging from 0.84 to 0.90, PBE0-D3BJ/6-311+G­(d,p) having the highest R ² of 0.90. Thus, we recommend the PBE0-D3BJ density functional with the 6-311+G­(d,p) basis set for predicting S₁ reduction potentials due to a MAE of 0.16 eV, low CPU time of 2 h, and high correlation between the predicted and experimental S₁ reduction potentials.

4.

CPU time and MAEs for computed S₁ reduction potential (a) and T₁ reduction potential (d). The bars represent the CPU time required to complete a vibrational frequency calculation of the ES optimized geometry of XO in hours. The black dots represent the MAE for each model chemistry in eV. Results for the 6-311+G­(d,p) basis set are shown in light blue, and those for the aug-cc-pVDZ basis set are displayed in dark blue. Comparison of experimental and predicted values of S₁ reduction potential (b), S₁ oxidation potential (c), T₁ reduction potential (e), and T₁ oxidation potential (f). The lines represent the line of best fit using linear equation.

We next evaluated the S₁ oxidation potential using two model chemistries: the hybrid GGA density functional B3LYP-D3BJ and the RSH density functional N12-SX, combined with the 6-311+G­(d,p) and aug-cc-pVDZ basis sets. These combinations are selected based on their low MAE for GS oxidation potential and S₁ E ^0–0. The MAEs for these model chemistries range from 0.08 to 0.17 eV, with standard deviations ranging from 0.04 to 0.26 eV. N12-SX/aug-cc-pVDZ achieved the lowest MAE at 0.08 eV. For the S₁ BP CPU time, B3LYP-D3BJ shows significantly greater CPU-time efficiency than N12-SX/6-311+G­(d,p), with a reduction from 10 to 6 h. Both density functionals have the same job CPU times of 15 h with the aug-cc-pVDZ basis sets. B3LYP-D3BJ, with either basis set, achieves an R ² of 0.99, while N12-SX/6-311+G­(d,p) yields an R ² of 0.82 (Figure c). However, N12-SX/6-311+G­(d,p) produced a lower MAE of 0.08 eV compared to B3LYP-D3BJ/6-311+G­(d,p), which had an MAE of 0.13 eV. Despite the higher CPU time and slightly lower R ², N12-SX/6-311+G­(d,p) allows for predicting S₁ oxidation potentials across the molecules. Therefore, we recommend this model chemistry for predicting S₁ oxidation potentials.

Next, we assessed the T₁ reduction potential. The density functionals with the lowest MAEs for GS reduction potential and T₁ E ^0–0, using the 6-311+G­(d,p) and aug-cc-pVDZ basis sets, were BMK-D3BJ, M06-2X, MN12-SX, CAM-B3LYP-D3BJ, ωB97X, and ωB97X-D. The MAEs for these model chemistries ranged from 0.10 to 0.18 eV, with standard deviations between 0.05 and 0.15 eV. The lowest MAE of 0.10 eV was obtained with ωB97X/6-311+G­(d,p) with a standard deviation of 0.09 eV. We compared the CPU time on TXO T₁ ES (Figure d). In most cases, for the same density functional, aug-cc-pVDZ was more expensive than 6-311+G­(d,p), with costs being 18 to 81% higher. MN12-SX is an exception; the 6-311+G­(d,p) basis set is 15% more expensive than aug-cc-pVDZ. Balancing errors and computational time, we selected the top-performing model chemistries: M06-2X/aug-cc-pVDZ, ωB97X/6-311+G­(d,p), and ωB97X-D/6-311+G­(d,p), with MAEs of 0.11, 0.10, and 0.12 eV and CPU times of 6, 6, and 4 h, respectively. Figure e shows the relationship between predicted and experimental T₁ reduction potentials for these model chemistries. The ωB97X/6-311+G­(d,p) model shows a strong correlation, with an R ² of 0.94, a MAE of 0.10 eV, and a relatively short computational time of 6 h. Thus, we recommend ωB97X/6-311+G­(d,p) for predicting the T₁ reduction potentials.

Conclusions

More than 14,000 quantum mechanical calculations were performed to identify the most accurate density functionals and basis sets for predicting the S₀, S₁, and T₁ reduction and oxidation potentials and the S₁ and T₁ E ^0–0 for nine OPCs. This study provides practical recommendations for the accurate and efficient prediction of redox properties in OPCs. We compared the predicted values to experimental data and assessed the performance using metrics, such as MAE, MSE, and CPU time. Since predicting ES redox potentials requires both GS redox potentials and ES E ^0–0 energies, the ideal strategy involves selecting a model chemistry that minimizes MAE across both properties.

For S₁ reduction potentials, the best-performing model chemistry is PBE0-D3BJ/6-311+G­(d,p), with an MAE of 0.16 eV, strong correlation to experiment (R ² = 0.90), and low CPU time (2 h). For S₁ oxidation, N12-SX/6-311+G­(d,p) offers the best balance of accuracy (MAE = 0.08 eV) and CPU time (10 h), with an R ² of 0.82.

For T₁ reduction potentials, ωB97X/6-311+G­(d,p) stood out for its low MAE (0.10 eV), high R ² (0.94), and modest CPU time (6 h). For T₁ oxidation, only BHandH achieved consistently low MAEs (0.24 and 0.26 eV) with both 6-311+G­(d,p) and aug-cc-pVDZ, respectively. Given its accuracy, reduced CPU time of 2 h, and strong correlation (R ² = 0.92), we recommend BHandH/6-311+G­(d,p) for T₁ oxidation predictions.

We systematically evaluated model chemistries to predict GS and ES redox potentials and ES E ^0–0 values and identified computational techniques to predict these properties with accuracy while balancing CPU time.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.5c02406.

• Optimized structures, log files, computed properties, MAE, and standard deviation for each model chemistry; nature of transition from S₀ to S₁ with PBE0-D3BJ/6-311+G­(d,p) and N12-SX/6-311+G­(d,p); and single occupied molecular orbital (SOMO) of triplet optimized geometries with ωB97X/6-311+G­(d,p) and BHandH/6-311+G­(d,p) (PDF)

The authors declare no competing financial interest.