What does “level of theory” mean?

I was reading this paper on the QM9 synthetic molecular database and their computed properties. It is stated that

All properties were calculated at the B3LYP/6-31G(2df,p) level of quantum chemistry.

I was wondering what this 'level of theory' exactly means.

• @NikeDattani This title edit doesn't really follow the spirit of the actual question. The question as stated isn't "What is a level of theory" but its what this level of theory means. – Tristan Maxson Oct 25 '20 at 18:14
• You're right: the question asks what "level of theory" means in the specific case of B3LYP/6-31G(2df,p), but it is equally easy to explain what "level of theory" means in general, which is what I did in this answer: mattermodeling.stackexchange.com/a/3621/5. That answer also answers the question of what it means for the specific case of B3LYP/6-31G(2df,p), as a special case of the more general answer. – Nike Dattani Oct 25 '20 at 18:45

Tristan's answer explains what B3LYP and 6-31G(2df,p) are. I agree with everything Tristan said, I will just write an answer that is a bit more generic: not specific to B3LYP and 6-31G(2df,p).

"Level of theory" in quantum chemistry, is a phrase indicating "how accurate" a calculation is. It is usually denoted in the form X/Y where X refers to how accurately the energy (or property) is calculated within the specific basis set being used, and Y refers to the basis set used (i.e. how the wavefunction is modeled). Here are some examples:

$$\begin{array}{lcc c} & \textrm{Accuracy within basis set used} & &\textrm{Basis set used}\\ \hline \textrm{B3LYP/6-31G(2df,p)} &\textrm{B3LYP} && \textrm{6-31G(2df,p)}\\ \textrm{CCSD(T)/cc-pVDZ} & \textrm{CCSD(T)} & &\textrm{cc-pVDZ}\\ \textrm{FCI/STO-3G} & \textrm{FCI} & &\textrm{STO-3G} \\ \textrm{MP2/def2-SVP } & \textrm{MP2} && \textrm{def2-SVP}\\ \end{array}$$

Warning: In this terminology, even if the "level of theory" is exact (i.e. FCI/CBS or "Full Configuration Interaction" in a "Complete Basis Set"), the energy or property being calculated is still not necessarily exact, because it does not make clear the level of treatment of relativistic, beyond-Born-Oppenheimer, hyperfine, electro-weak, and other effects. Within this notion of "level of theory", all that "exact" really means, is that the Schrödinger equation is being solved to full numerical convergence for the specific Hamiltonian being used (which could be non-relativistic, ignoring nuclear-electron correlation, or approximate in any of a number of different ways).

• Can we really call the method/function such as B3LYP its accuracy? Accuracy to experimental or accuracy to DFT? I am always skeptical of this sort of statement, except maybe in cases such as FCI where its normally fairly easy to argue its more accurate than lower levels. DFT functionals somewhat mess up this HF to MP2 to MP3 etc trend I feel. – Tristan Maxson Oct 25 '20 at 21:15
• Accuracy means difference between the result and the truth. In this case we're refering to how accurately the eigenvalue problem is being solved for a specific basis set. FCI will give the exact eigenvalue, CCSD(T) will be less accurate than FCI, and B3LYP will be even less accurate than CCSD(T), except in cases were CCSD(T) is not applicable. It's true that B3LYP is just the "method" that is being used to estimate the lowest eigenvalue (i.e. energy) and not precisely the "accuracy". However "level" of theory does imply some hierarchy in accuracy (top/bottom level = most/least accurate). – Nike Dattani Oct 25 '20 at 21:46

The DFT computational level is decided by the choosing of exchange-correlation functional, such as LDA and GGA. The following figure called Jacob's ladder lists five levels/generations of DFT exchange-correlation functionals, in which you can find B3LYP. In general, a higher exchange-correlation-functional level will give you more reliable results but with a more computational cost.

• Where did you get that image? Is there any source to know more about the state-of-the art of DFT functionals towards chemical acc? – TheVal Oct 25 '20 at 11:20
• This figure it from a fairly well known paper (I don't remember the name off the top of my head) but it should be cited. – Tristan Maxson Oct 25 '20 at 11:46
• Level of theory must also include basis set and any other relevant details. – Cody Aldaz Oct 25 '20 at 13:20
• @TheVal For your question "Is there any source to know more about the state-of-the art of DFT functionals towards chemical acc?", I suggest you search this site for answers about the state-of-the-art in DFT, then if you still don't have your answer, ask a separate question. The comments are not for asking unrelated questions like this. – Nike Dattani Oct 25 '20 at 17:30

B3LYP refers to the functional being used. It is a hybrid functional and it is very commonly used, but is often not sufficient and better functionals are available nowadays. It is constructed by a specific combination of HF, GGA, and LDA functionals.

6-31G(2df,p) is a Pople basis set, specifically a valence double-zeta polarized basis set. It has been modified to add 2 diffuse functions for d orbitals and 1 diffuse function for p orbitals on heavy atoms. It also has an extra diffuse functions for p orbitals on hydrogens. This basis set is okay for a lot of work, but there are alternatives nowadays as well that I believe function better.

Overall, this is a pretty standard level of theory for molecular computations, but I would not call it a very modern level of theory. That being said, this paper is from 2014 so I am unsure how these would fit in at that time.