# Cancellation of Errors in density functional approximations or any wavefunction based methods

I am curious to find references for when is it TRUE that the error cancellation in approximate methods like DFT/MP2/etc. are valid and when they fall apart.

For example: My understanding is (please argue against if you don't agree) that if an approximate method only captures interactions say X, Y, and Z (x/y/z could be lets say pairwise dispersion, hydrogen bonding, aromatic interaction), then the error cancellation would be valid across all the molecular systems that have X, Y, and Z interactions. However, let's say in reality (i.e. true physical interacting picture) there is X, Y, Z, and A (where A might be say many body dispersion or exact exchange), then is it true that the error cancellation of a method correctly capturing X, Y, and Z will not be valid for systems with X, Y, Z, A.

Mathematically, for say the energies of molecules 1 and 2, is the following true:

(X1 + Y1 + Z1) - (X2 + Y2 + Z2) is the same as (X1 + Y1 + Z1 + A1) - (X2 + Y2 + Z2 + A2)

where Xi,Yi/etc. may just be energy contributions of pairwise dispersion/hydrogen bonding/etc.

Papers supporting or arguing against error cancellation are all appreciated!

• Its slightly unclear to me how you are describing error cancellation. If $\Delta E_\text{true}=(X_2-X_1)+\delta$ where $\delta$ is an error term, this terms is small either when the neglected interaction is similar for the two molecules (i.e. $A_1 \approx A_2$) or when you neglect multiple interactions, but they have opposite effects on the energy (i.e. $A_1-A_2 \approx -(B_1-B_2)$).
– Tyberius
May 22, 2020 at 4:15
• Thanks for the comment and sorry for confusing. The overarching question is "when comparing relative energies between two molecules/solids, when does error cancellation in quantum chemical methods fail". Do you have a citation by any chance for your comment?
– gogo
May 22, 2020 at 12:39
• One example that comes to mind is related to electron-electron self-interaction error (SIE) with GGA functionals. If you compute formation or reaction energies in systems without a change in oxidation state, the SIE cancels out a fair bit. If you model an oxidation reaction, the different redox states of reactant and product make it so the SIE is not canceled out to such a degree and errors are large. This is a main motivating factor for using GGA+$U$ for oxidation energies of transition metal oxides, as discussed here. May 24, 2020 at 7:25
• That's a very good physical example.Thank you!
– gogo
May 24, 2020 at 22:44
• Does "Error Cancellation" need to be a tag? May 30, 2020 at 5:45