## Introduction

The extension of Density functional theory (DFT) to the time-dependent domain, namely time-dependent density functional theory (TD-DFT) has been originally proposed by Runge and Gross 30 years ago.[1] Ten years latter, Casida developed an effective linear-response (LR) formalism for TD-DFT (so-called random-phase approximation or Casida's equations)[2] allowing to rapidly and efficiently determine the solution of the TD-DFT equations for molecules.[3] Thanks to these pioneering works, TD-DFT has become an extremely popular approach for modeling the energies, structures, and properties of electronically excited states (ES). Indeed, in 2011, more than 1000 papers devoted to TD-DFT have been published, surpassing by one order of magnitude the output measured one decade before. The applications of TD-DFT encompasses not only the simulation of vertical transition energies, but also the determination of ES structures and emission wavelengths, the computation of vibrationally resolved optical spectra, the estimation of atomic point charges and dipole moments, as well as the simulation of photochemical reactions. These remarkable successes, for a relatively young theory, are related to the valuable accuracy/effort ratio of TD-DFT and to the constant methodological developments in the field that aims to push the TD-DFT towards new frontiers. Like its parent DFT approach, TD-DFT is formally an exact theory, but the actual implementations require the selection of an exchange correlation functional (XCF) which concentrates the approximations of the model. In practice, one can state that LR-TD-DFT inherited from all DFT problems but also presents a few specific drawbacks. In addition, we underline that large efforts have been made to model environmental effects during TD-DFT simulations, notably within the well-known polarizable continuum model (PCM),[4] and the EFP[5-7] methods. One can distinguish several implicit approaches, namely, the classical LR,[8, 9] the corrected linear-response (cLR),[10] the VEM[11] and the state-specific (SS) approximations.[12] The three latter, more accurate and physically meaningful, approaches take into account the variations of the polarization of the solvent following the electronic density rearrangements of the solute. For each model, one generally distinguishes the equilibrium limit (that corresponds to slow phenomena in which the environment does fully adapt to the considered ES) and the nonequilibrium limit (corresponding to fast phenomena, e.g., absorption and fluorescence, during which only the electrons of the environment reorganize).

In this contribution, our goal is certainly not to describe the TD-DFT methodology—the interested reader could find excellent reviews, perspective articles, books, and special issues elsewhere[13-36]—nor to list all recent TD-DFT applications but rather to focus on extensive XCF benchmarks performed within the TD-DFT framework. Indeed, since the emergence of hybrid XCF,[37, 38] that contains a fraction of exact exchange (EXX) of Hartree-Fock (HF) form, a large number of new XCF have been proposed[37-52] in addition to the pure (that is EXX-free) LDA,[53, 54] GGA,[55-63] and *meta*-GGA[44, 64-68] functionals. In the ES modeling vein of research, one should certainly mention the developments of the so-called range-separated hybrids (RSHs)[69-79] that include an increasing fraction of EXX when the interelectronic distance increases. This allows to significantly improve the description of charge-transfer (CT) ES that otherwise suffers from the local nature of XCF. More recently, double-hybrid functionals,[80-83] that explicitly depend on the virtual orbitals, have been extended to TD-DFT by Grimme,[80, 84] who proposed to use a CIS(D) like perturbative correction for the TD-DFT energies, which significantly improved the description of states presenting a double-excited nature.[85, 86] Of course, the vast majority of these XCF has been designed (and often parametrized) to reproduce ground state (GS) rather than ES properties. This can be ascribed, on the one hand, to scientific common sense (all ES calculations imply a GS step, so optimizing a functional for ES at the cost of inaccurate GS properties is not very appealing) and, on the other hand, to the difficulty to obtain experimental or theoretical ES reference values that are accurate enough to allow XCF optimization. This is due to the rapidity of the ES phenomena that makes experimental determination a challenge, for example, it is difficult to measure bond lengths and valence angles in the ES. As most compounds exhibiting interesting ES features are large conjugated molecules, the experimental works are often performed in condensed phase, making the simulation of environment a further problem for computational chemists. Therefore, contrary to GS properties, for which many standard benchmark sets have been designed (e.g., G3 for thermochemistry,[87, 88] S22 and S66 for weak-interactions,[89-92] ISO34 and ISO22 for isomerization energies,[93] or sets for proton transfer,[94, 95] and GMTKN30 that gathers the majority of set developed for GS properties[96, 97]), a rather diverse palette of properties, training sets and calculation strategies have been proposed to evaluate the pros and cons of XCF within the TD-DFT framework. As we will see in the following, this sometimes leads to nonuniform conclusions. It is also important to underline that the performances of the reported XCF mentioned below correspond to their current implementation in the vast majority of actual TD-DFT codes. There are several methodological aspects related to the transfer of GS XCF to ES applications, a task far from being trivial and implying additional approximations (see the above-mentioned reviews), so that more complete or mathematically refined implementations of XCF to TD-DFT might lead to different conclusions.[98, 99]

In the following, we first present benchmarks performed for transition energies (both vertical and adiabatic phenomena) that are the most intensively studied properties. In that framework, the most significant studies are briefly summarized in Table 1. In that Table, like in the following of this review, we have mainly selected benchmarks that considered at least three XCF and a diverse set of compounds. Indeed, though one could find a number of studies that considered only one functional applied on a large set of molecules,[121-132] or tested a large number of XCF but on a rather specific family of compounds,[97, 133-148] these works are often rather specific and the obtained conclusions are sometimes difficult to generalize. We nevertheless briefly summarized a number of them in this review. We also discuss benchmarks that have been performed on other ES properties, that is, geometries, vibrational frequencies, dipole moments, oscillator strengths, vibronic couplings, etc. Eventually, we provide a few generic conclusions summarizing the works reported herein. We have chosen to classify benchmarks into several categories. Some works cannot enter univocally in one predefined box, and some choices have been to be performed, for example, we have decided to present cyanine (CT) benchmarks in the “theory” (“experiment”) sections, that are the most representative, though there exist several cyanine (CT) benchmarks using experimental (theoretical) references.

Ref. | Benchmark | States | Functionals | Basis set | Geometry | Solvent | |
---|---|---|---|---|---|---|---|

- Adapted and extended from Ref. [107]. For each contribution, we indicate the nature of the benchmark performed (vertical, or adiabatic, number of ES considered) as well as methodological details (basis set used during the TD step, geometry optimization scheme and use of solvation model.
^{a}No solvent model in the theory, but the experimental values have been shifted by a constant 0.15 eV (for all solvents) to include solvatochromism.^{b}142 singlet and 71 triplet ESs.^{c}Training set (84) is a subset of Thiel's set; test set (23) is a subset of Tozer's set.^{d}The authors transformed the experimental AFCP point into vertical corrections thanks to successive theoretical corrections, see text.^{e}Computed with the*E*^{vert-abso}PBE0/6-31G(d)//PBE/TZVP.^{f}Triplet ES.^{g}Comparisons with gas-phase experiments.^{h}Using Tozer's 58-state set that contains both experimental and theoretical reference, as well as a subset of Thiel's set.^{i}Majority of intermolecular and intramolecular CT states in this set.^{j}Multiplicity-changing valence and Rydberg excitation energies of atoms.^{k}Atomic spectra for Li to Ne atoms.^{l}63 singlet and 38 triplet states.^{m}45 singlet and 45 triplet states.
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[100] | E^{0-0}vs 0-0 | 34 | 6: | LDA BLYP BP86 PBE B3LYP PBE0 | aug-TZVPP | DFT/aug-TZVPP | None |

[101] | E^{0-0}vs 0-0 | 43 | 3: | BP86 B3LYP BHHLYP | TZVP | DFT/TZVP | Empiriciala |

[102] | E^{vert}vs λ_{max} | 34 | 12 | pure and hybrid XCF | 6-311+G(2df,p) | DFT | LR-PCM(neq) |

[103] | E^{vert}vsE^{vert} | 213b | 4: | BP86 B3LYP BHHLYP MR-DFT | TZVP | MP2/6-31G(d) | None |

[104] | E^{vert}vs λ_{max} | 118 | 5: | PBE, PBE0, LC-PBE, LC-ωPBE and CAM-B3LYP | 6-311+G(2d,p) | PBE0/6-311G(d,p) | LR-PCM(neq) |

[105] | E^{vert}vs Mixed | 59 | 3: | PBE B3LYP CAM-B3LYP | d-aug-cc-pVTZ/cc-pVTZ | Mixed | None |

[106] | E^{vert}vs Exp | 96 | 6: | LSDA PBE TPSS TPSSh B3LYP PBE0 | 6-311++G(3df,3pd) | DFT/6-311++G(3df,3pd) | None |

[107] | E^{vert}vsE^{vert} | 102 | 29 | pure and hybrid XCF | TZVP | MP2/6-31G(d) | None |

[107] | E^{vert}vs λ_{max} | 510 | 29 | pure and hybrid XCF | 6-311+G(2d,p) | PBE0/6-311G(d,p) | LR-PCM(neq) |

[108] | E^{vert}vsE^{vert} | 107c | 5: | LRC-ωPBEh (3 types), ωB97, ωB97X | aug-cc-pVTZ | DFT | None |

[86] | E^{0-0}vs AFCPd | 12 | 7: | BLYP B3LYP PBE38 BMK CAM-B3LYP B2GPPLYP | def2-TZVPP | PBE/TZVP | LR-PCM(neq)e |

[109] | E^{vert}vsE^{vert}f | 63 | 34 | pure and hybrid XCF | TZVP | MP2/6-31G(d) | None |

[110] | E^{vert}vs Exp | 69 | 26: | Pure, global and RSHs | 6-311(3+,3+)G(d,p) | MP2/6-311+G(d,p) | None |

[111] | E^{0-0}vs Exp | 17 | 4: | B3LYP CAM-B3LYP mCAM-B3LYP LC-BLYP | 6-311+G(d,p) | DFT/6-311+G(d,p) | None |

[112] | E^{0-0}vs 0-0 | 109 | 6: | LSDA PBE BP86 TPSS B3LYP PBE0 | def2-TZVP | B3LYP/def2-TZVP | Noneg |

[113] | E^{vert}vs Mixedh | 83 | 8: | Global and RSHs | Large | DFT and MP2 | None |

[114] | E^{vert}vs Exp | 41i | 11: | RSHs | cc-pV[D/T]Z and 6-311G(d,p) | Mixed | None |

[115] | E^{vert}vs ASj | 17 | 56: | Pure, global and RSHs | d-aug-cc-pVQZ-DK | N/A | None |

[116] | E^{vert}vs ASk | 18 | 31 | Hybrid XCF | exc-ETDZ | N/A | None |

[117] | E^{0-0}vs AFCP | 40 | 6: | B3LYP PBE0 M06 M06-2X CAM-B3LYP LC-PBE | 6-31+G(d) | DFT/6-31+G(d) | cLR-PCM(neq) |

[118] | E^{vert}vs Exp | 101l | 24: | Pure and hybrid XCF | 6-311++G(3df,3pd) | DFT/6-311++G(3df,3pd) | None |

[119] | E^{vert}vsE^{vert} | 90m | 3: | PBE B3LYP CAM-B3LYP | d-aug-cc-pVTZ/ aug-cc-pVTZ | MP2/6-31G(d) | None |

[120] | E^{vert}vs Exp | 69 | 30: | Pure, global and RSHs | 6-311(2+,2+)G(d,p) | MP2/6-311+G(d,p) | None |