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

Question

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!

Answer 1

I don't know if I can answer the exact question here, but I can hopefully start a discussion in the right direction.

Error cancellation is pervasive throughout matter modeling, and in particular electronic structure calculations. Whenever we use an approximate method to predict, for example, the energy of some system, we are neglecting numerous possible interactions/factors that would be included in the exact formulation (e.g. the Schrodinger equation for nonrelativistic calculations). If the computed result is in good agreement with the exact result, what has happened is that the various error introduced by our approximation have effectively cancelled each other out. In this sense, the accuracy of approximate computational methods is entirely dependent on error cancellation. So asking when error cancellation holds for a given level of theory is essentially asking for when that method works at all.

So we can easily determine extent of error cancellation provided we have some sort of exact reference. Not as simple is determining what are the main sources of a particular instance of error cancellation. Often this just takes the form of heuristics developed over time by applying different methods to different types of systems. In a few cases, efforts have been made to actually justify theoretically/mathematically why a method performs well/poorly on certain systems or for certain properties.

A simple example is that Hartree-Fock can be used to predict surprisingly accurate ionization potentials (using Koopman's theorem) despite the simplicity of the method. The reason for this is that the calculation neglects both correlation energy and relaxation energy. Including correlation energy should stabilize the original molecule more than its ion, as the original molecule has more electrons to correlate. This would suggest that HF underestimates the ionization potential. However, allowing the orbitals of the ion to relax should stabilize it relative to the original molecule, which suggests that HF is overestimating the ionization potential. Since these two effects have opposite sign, neglecting them only introduces a small amount of error. (This argument is based on the one given in section 3.3 of Szabo and Ostlund's Modern Quantum Chemistry).