# Is there an example where TD-DFT fails to provide sufficient accuracy for a small system, where RI-CC2 is sufficient?

Time-dependent density functional theory (TD-DFT) is considered to be more efficient (in terms of its accuracy/speed ratio) than alternatives such as the approximate coupled-cluster singles and doubles (CC2), even with resolution of identity which together forms RI-CC21.

Piacenza et al. (2008)2 highlight that one drawback of TD-DFT is that "for larger conjugated systems the $$\sigma\to\pi^\ast$$ and $$\pi\to\pi^\ast$$ transitions can become significantly wrong when calculated with TD-DFT" and so for such systems RI-CC2 should be used despite its taking slower.

However, is it always the case that for small(er) systems that RI-CC2 need not be used as TD-DFT can provide sufficient accuracy, and if not what is an example where the former prevails?

References

[1] Jacquemin, D. et al. (2015). 0–0 Energies Using Hybrid Schemes: Benchmarks of TD-DFT, CIS(D), ADC(2), CC2, and BSE/GW formalisms for 80 Real-Life Compounds. J Chem Theory Comput. 11(11):5340–5359.

[2] Piacenza, M., Zambianchi, M., Barbarella, G. et al. (2008). Theoretical study on oligothiophene N-succinimidyl esters: size and push–pull effects. Physical Chemistry Chemical Physics. 10(35):5363-5373.

• I have edited the title a bit. Feel free to rollback if you want. Basically, RI-CC2 is more accurate but more expensive, and certainly there's examples where TD-DFT is not accurate enough. So the question really is (in my mind), to provide an example of this. In addition to the title, I changed "what are some examples" with "what is an example", so that each person can answer with 1 example. You're also welcome to change that back too! I'm just trying to help get the un-answered questions answered now! – Nike Dattani May 30 '20 at 4:03
• @NikeDattani Thanks for improving the question, and yes that is what I meant :) – TheSimpliFire May 30 '20 at 8:58
• I'd like to leave this available for everyone else, but if there's no answer on the 6th day and you remind me (ping in chat), I can try my best to answer it. – Nike Dattani Jun 4 '20 at 14:35

The question cites a 2008 paper about fairly large molecules (e.g. bithiophene N-succinimidyl esters), in which TD-DFT gave significantly wrong results, so the authors recommended RI-CC2. The question then asks if there's any example where TD-DFT was insufficient for a smaller system. After searching the literature, I have found an example of a small molecule for which TD-DFT gave significantly worse results than RI-CC2: The molecule is C$$_2$$H$$_4$$.

Here I will make a condensed version of the relevant energies in their Table II:

$$\begin{array}{c c c c} \text{State} & \text{TD-DFT} & \text{RI-CC2} & \text{Benchmark}\\ \hline B_{1u} & 7.72 & 8.27 & 8.22\\ B_{1g} & 8.09 & 8.73 & 8.59\\ B_{3u} & 8.11 & 8.84 & 8.75\\ \end{array}$$

Here's some further explanations about the table:

• All energies are in eV.
• The benchmark was CC3, which the authors say they expect to be within about 0.1 eV of the exact truth (so whatever error is in these benchmarks, is negligible here).
• In every case RI-CC2 was closer to the benchmark by more than 0.5 eV.
• The functional used for TD-DFT was B3LYP. According to this figure one might be able to get 1.5 kcal/mol better by using a double-hybrid such as B2PLYP, but that's only 0.065 eV, which is negligible compared to the fact that the table shows TD-DFT being more than 0.5 eV worse than RI-CC2 in every case.
• I have only included the first 3 states from Table II of the original paper, you're welcome to see the whole table, and the observations above remain the same (in fact the TD-DFT error becomes much worse for some of the other states.

The bottom line is that if you have enough compute power available, coupled-cluster-based approaches will almost always beat DFT, except for multi-reference cases where coupled-cluster is known to fail, such as some molecules containing transition metals (where DFT can have an advantage, especially if the functionals are optimized for transition metal systems, such as some of the functionals that have come out of Don Truhlar's group).

• In bullet #4, you estimate the improvement of a double-hybrid over a hybrid by the GMTKN30 results. That test set does not contain excitation energies, which will show larger errors than ground state energies. TD-DHDF performance was cursorily evaluated in doi.org/10.1039/B902315A On its face, the estimation does not make sense because the same estimate gives a B3LYP error of less than 0.2 eV, which would be great, but is not found for the ethene example. – TAR86 Jun 11 '20 at 20:43
• @TAR86 Thanks for the useful comments! I knew that #4 was problematic because I was using results from DFT to make speculations about TD-DFT, however I chose it since I like that figure from a didactic perspective :) I think the conclusion is still transferable though: If B3LYP is wrong by 0.5 eV, then no presently known double-hybrid or double-meta-hybrid, or beyond-double-hybrid, will be able to compete with RI-CC2 (which is several times better). Maybe some functional can get lucky for this particular case, but you wouldn't know in advance: you'd be trying every functional until you match. – Nike Dattani Jun 11 '20 at 20:51

TD-DFT works best for valence-valence excitation energies, but does not very well for charge-transfer excitation (unless xc potentials are constructed to lower delocalization error).

Core-excitation energies are also not described will with TD-DFT.