# Benchmark Study on Phosphorescence Energies of Anthraquinone Compounds: Comparison between TDDFT and UDFT

^{*}

## Abstract

**:**

## 1. Introduction

_{1}) to the singlet ground state (S

_{0}), and the calculation of the triplet excited state is of particular importance in the simulation of the phosphorescence spectrum. The triplet excited states usually show the opposite electronic behavior when compared to the ground state, and the dynamic correlation should be included in the calculation of the triplet excited states. Thus, the coupled-cluster method, as a high level ab initio method that includes dynamic correlation, is usually adopted for the calculation of phosphorescence energy. Thiel and colleagues [16,17] tested singlet–triplet energy gaps for small organic molecules (limited to 15 atoms) with coupled cluster methods (CC3 [18] and CCSDR [19]). Although further calculations with complete active space with the second-order perturbation theory (CASPT2) [20,21] and coupled-cluster singles and doubles CC2 [22] can be extended to 40 atoms [23,24,25,26,27,28,29], these methods cannot overcome the “Index Wall” of high-level wave function theory (WFT). Density functional theory (DFT) [30,31,32] can obtain the energy and density of molecules in a way that is easy to calculate and DFT provides a balance between accuracy and computational cost for ground state calculations. Time-dependent density functional theory (TDDFT) [33,34,35,36,37] is the most widely used tool for calculating not only excitation energies, but also excited state properties, such as dipole moments and the emission spectrum, which is conducted with a computational cost between semiempirical methods and wave function theory. Latouche and colleagues benchmarked the phosphorescence spectra of transition metal complexes containing platinum and iridium with the unrestricted density functional theory (UDFT), time-dependent density functional theory (TDDFT), and the Tamm–Dancoff approximation (TDA) [38,39,40,41,42,43,44,45] methods. The results showed that UDFT and TDA performed better than TDDFT [46]. Recently, Ehara and colleagues used the symmetry-adapted cluster-configuration interaction (SAC-CI) [47] method and TDDFT to benchmark the geometric structures and phosphorescence energy of heterocyclic compounds; they concluded that both methods could provide accurate results in calculating phosphorescence energies of purely organic molecules [48]. Each of these methods has its advantages and limitations.

## 2. Computational Details

_{s}symmetry and the other molecules were in a C

_{1}symmetry. For the molecules with C

_{1}symmetry, the natural transition orbital (NTO) and the frontier orbital analyses were used to determine the wave functions of the lowest triplet excited state. Seven typical functionals of PEB, PBE0 [64], B3LYP [65], CAM-B3LYP [66], ωB97XD [67], M06-2X [68], and M06-HF [69,70]—ranging from GGA to meta-GGA functionals and cc-pVTZ—were used for the DFT calculations. In this work, TDDFT and UDFT were used to optimize the geometry and to compute the energy for the lowest triplet excited state, and thus the combination of the above two methods can yield the following groups of calculation schemes. If the geometries were optimized by UDFT and the excitation energies were computed with TDDFT, based on UDFT optimized geometries, then we denote this scheme as TD//UDFT. On the other hand, if both the optimized geometries and excitation energies were computed with UDFT, we denote this scheme as U//UDFT. In addition, there are other two kinds of schemes that were characterized as TD//TDDFT and U//TDDFT, in which the geometries were optimized by TDDFT and the excitation energies were computed with TDDFT and UDFT, respectively. The schemes are compared in Tables S18 and S19, and the results for TD//TDDFT and U//TDDFT were similar. All the geometry optimizations and single-point calculations with DFT were computed using Gaussian16 software [71]. The charge transfer characteristics and the stabilization energies were obtained using NBO Version 3.1 in Gaussian16 software. Besides DFT, the coupled-cluster method CC2 was used as the benchmark for calculation of the phosphorescence energies. In addition, all the CC2 results were obtained from the MOLPRO2021 package [72,73].

## 3. Results

#### 3.1. Geometry Optimization

#### 3.2. Phosphorescence Energy

^{2}values for the triplet states of AQs are around 2.0 for all of the functionals (as shown in Table S22). Thus, the effect of spin contamination on the results is small.

**Table 3.**The energy differences ΔE (in eV) relative to the CC2 for the phosphorescence energy of AQs, as calculated by U//UDFT with the different functionals (PBE, PBE0, B3LYP, CAM−B3LYP, ωB97XD, M06-2X, and M06-HF) at the cc-pVTZ level.

PBE | PBE0 | B3LYP | CAM-B3LYP | ωB97XD | M06-2X | M06-HF | |
---|---|---|---|---|---|---|---|

HA | 1.47 | −0.49 | −0.47 | −0.36 | −0.31 | −0.23 | −0.09 |

AAT | −0.40 | −0.41 | −0.38 | −0.34 | −0.3 | − 0.24 | −0.17 |

BP | −0.24 | −0.28 | −0.18 | −0.15 | −0.11 | −0.05 | −0.02 |

TX−BT | −0.35 | −0.34 | −0.32 | −0.30 | −0.23 | −0.07 | 0.14 |

TX-DBT | −0.62 | −0.42 | −0.44 | −0.21 | −0.14 | −0.12 | 0.06 |

BDBT | −0.37 | −0.28 | −0.22 | −0.14 | −0.10 | −0.03 | 0.75 |

FBDBT | −0.39 | −0.29 | −0.23 | −0.15 | −0.11 | −0.04 | 0.75 |

ClBDBT | −0.39 | −0.29 | −0.23 | −0.15 | −0.10 | −0.03 | 0.00 |

BrBDBT | −0.39 | −0.29 | −0.23 | −0.15 | −0.10 | −0.03 | 0.78 |

XA | −0.59 | −0.65 | −0.53 | −0.28 | −0.23 | −0.47 | 0.01 |

AR | 1.19 | −0.32 | −0.44 | −0.26 | −0.20 | −0.16 | 0.04 |

AD | 1.50 | −0.45 | −0.45 | −0.27 | −0.20 | −0.17 | −0.02 |

MUE ^{(a)} | 0.66 | 0.37 | 0.34 | 0.23 | 0.18 | 0.14 | 0.24 |

MUE ^{(b)} | 0.56 | 0.25 | 0.23 | 0.22 | 0.21 | 0.25 | 0.46 |

#### 3.3. Basis Sets Effect

## 4. Conclusions

## Supplementary Materials

^{2}values for the triplet states of AQs calculated by UDFT with different functionals (PBE, PBE0, B3LYP, CAM-B3LYP, ωB97XD, M062X, M06HF) at the cc-pVTZ level.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The MUE values obtained by different basis sets under the same phosphorescence energy calculation method and functionals. (

**a**–

**d**) represent the results obtained with the basis set of cc-pVDZ, cc-pVTZ, aug-cc-pVDZ, and aug-cc-pVTZ, respectively.

**Figure 2.**A comparison of the results of the BDBT, FBDBT, ClBDBT, and BrBDBT using four basis cc-pVDZ, cc-pVTZ, aug-cc-pVDZ, aug-cc-pVTZ and CC2.

**Figure 3.**Molecular orbital diagrams of the SOMO1 and the SOMO2 that were obtained by choosing aug-cc-pVDZ and aug-cc-pVTZ for the BDBT, FBDBT, ClBDBT and BrBDBT calculations.

**Table 1.**The root mean square deviation (RMSD) in Å of the two AQs geometries that were optimized by TDDFT and UDFT, respectively.

Molecular | RMSD | Molecular | RMSD |
---|---|---|---|

HA | 0.00 | FBDBT | 0.02 |

AAT | 0.05 | ClBDBT | 0.03 |

BP | 0.01 | BrBDBT | 0.03 |

TX-BT | 0.14 | XA | 0.03 |

TX-DBT | 0.01 | AR | 0.02 |

BDBT | 0.02 | AD | 0.01 |

**Table 2.**The energy differences ΔE (in eV) relative to CC2 for the phosphorescence energy of AQs were calculated by TD//UDFT with different functionals (PBE, PBE0, B3LYP, CAM-B3LYP, ωB97XD, M06-2X, and M06-HF) at the cc-pVTZ level.

CC2 | PBE | PBE0 | B3LYP | CAM-B3LYP | ωB97XD | M06-2X | M06-HF | |
---|---|---|---|---|---|---|---|---|

HA | 2.55 | −0.87 * | –0.80 | −0.75 | −0.68 | −0.57 | −0.41 | −0.25 * |

AAT | 2.03 | −0.75 | −0.67 | −0.64 | −0.53 | −0.46 | −0.33 | −0.11 |

BP | 2.50 | −0.62 | −0.39 | −0.34 | −0.27 | −0.18 | −0.11 | −0.26 |

TX−BT | 2.67 | −0.49 | −0.71 | −0.59 | −0.83 | −0.68 | −0.28 | −0.14 |

TX−DBT | 2.91 | −0.77 | −0.58 | −0.57 | −0.47 | −0.39 | −0.36 | −0.32 |

BDBT | 2.52 | −0.65 | −0.42 | −0.37 | −0.30 | −0.21 | −0.13 | −0.25 |

FBDBT | 2.56 | −0.65 | −0.44 | −0.39 | −0.33 | −0.23 | −0.15 | −0.27 |

ClBDBT | 2.52 | −0.66 | −0.43 | −0.39 | −0.31 | −0.22 | −0.14 | −0.26 |

BrBDBT | 2.51 | −0.66 | −0.43 | −0.38 | −0.31 | −0.21 | −0.14 | −0.25 |

XA | 3.16 | −0.68 * | −0.76 | −0.68 * | −0.71 | −0.60 | −0.55 | −0.59 |

AR | 2.83 | − 0.65 * | −0.63 | −0.59 | −0.54 | − 0.47 | −0.48 | −0.53 |

AD | 2.93 | −0.71 * | −0.64 | −0.60 | −0.56 | −0.48 | −0.44 | − 0.44 |

MUE^{(a)} | − | 0.68 | 0.57 | 0.53 | 0.49 | 0.39 | 0.29 | 0.31 |

^{(a)}MUE is the mean unsigned error between the calculated phosphorescence energy and CC2. * The symmetry of the lowest triplet excited state (T

_{1}) is different from that of the ground states, so the second triplet excited state (T

_{2}) is selected to obtain ΔE.

**Table 4.**The NBO analysis of the charge transfer and stabilization energy, E (2), of the AQs at the M06-HF/cc-pVTZ and M06-HF/aug-cc-pVTZ level.

cc-pVTZ | aug-cc-pVTZ | |||
---|---|---|---|---|

Charge Transfer | E (2) kcal/mol | Charge Transfer | E (2) kcal/mol | |

HA | σ* (C7-C10) → σ* (C3-C4) | 176.24 | σ* (C7-C10) → σ* (C3-C4) | 176.24 |

AAT | σ* (C8-C9) → σ* (C7-C10) | 105.28 | σ* (C8-C9) → σ* (C7-C10) | 105.28 |

BP | σ* (C19-C21) → σ* (C15-C17) | 271.00 | σ* (C19-C21) → σ* (C15-C17) | 271.00 |

TX-BT | σ* (C3-C4) → σ* (C5-C6) | 180.49 | σ* (C3-C4) → σ* (C5-C6) | 180.49 |

TX-DBT | σ* (C3-C4) → σ* (C7-C8) | 249.08 | σ* (C3-C4) → σ* (C7-C8) | 249.08 |

BDBT | σ* (C14-C15) → σ* (C11-C16) | 516.26 | σ* (C3-C4) → σ* (C1-C2) | 236.54 |

FBDBT | σ* (C14-C15) → σ* (C11-C16) | 307.23 | σ* (C11-C16) → σ* (C27-H31) | 1100.56 |

ClBDBT | σ* (C3-C4) → σ* (C1-C2) | 228.33 | σ* (C3-C4) → σ* (C1-C2) | 228.32 |

BrBDBT | σ* (C14-C15) → σ* (C11-C16) | 457.47 | σ* (C22-C23) → σ* (C25-C29) | 441.75 |

XA | σ* (C4-C5) → σ* (C1-C6) | 191.23 | σ* (C4-C5) → σ* (C1-C6) | 191.23 |

AR | σ* (C4-C5) →σ* (C1-C6) | 258.20 | σ* (C4-C5) →σ* (C1-C6) | 258.20 |

AD | σ* (C8-C9) →σ* (C4-C7) | 191.85 | σ* (C8-C9) →σ* (C4-C7) | 191.85 |

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## Share and Cite

**MDPI and ACS Style**

Guo, Y.; Zhang, L.; Qu, Z.
Benchmark Study on Phosphorescence Energies of Anthraquinone Compounds: Comparison between TDDFT and UDFT. *Molecules* **2023**, *28*, 3257.
https://doi.org/10.3390/molecules28073257

**AMA Style**

Guo Y, Zhang L, Qu Z.
Benchmark Study on Phosphorescence Energies of Anthraquinone Compounds: Comparison between TDDFT and UDFT. *Molecules*. 2023; 28(7):3257.
https://doi.org/10.3390/molecules28073257

**Chicago/Turabian Style**

Guo, Yujie, Lingyu Zhang, and Zexing Qu.
2023. "Benchmark Study on Phosphorescence Energies of Anthraquinone Compounds: Comparison between TDDFT and UDFT" *Molecules* 28, no. 7: 3257.
https://doi.org/10.3390/molecules28073257