Exact double counting correction schemes from the flat plane condition

Since the inception of quantum embedding methods such as DFT+U and DMFT, numerous double counting correction schemes have been devised including the around mean field and fully localised limit schemes. However, these double counting correction schemes typically lack strong theoretical underpinnings and it is found that the predictive accuracy of DFT+U and DMFT methods depend strongly on the choice of double counting scheme.

We present our recent theoretical developments on the tilted plane condition [1], a generalisation of the well-known flat plane condition, which characterises the total energy curve, E[N,M], of a finite electronic system for all values of electron count N and magnetisation M. Using the tilted plane condition, we reverse engineer an exact double counting correction scheme for DFT+U [2], which may prove useful also in DMFT and DMET methods.

In the case of correlated d-orbital subspaces we furthermore reveal that the structure of the E[N,M] landscape depends strongly on the chemical coordination between the active subspace and its environment, so that for example a unique double counting correction scheme is required for octahedral, tetrahedral and square-planar coordinations. We thus show that our exact double counting correction scheme is necessarily dependent on the symmetry imposed degeneracies of the active subspace.