The second order contributions applied via the matrix K dictate the energetic cost of removing and adding electrons to each site as well as the correlational energy to move charge between sites. In principle, K includes both local and non-local components, as well as additive and non-additive contributions. Here we make a make a simpler approximation as used in QEq where only the diagonal components are used and treated as constants. Offdiagonal components (where ρ 0 i ̸ = ρ 0 j ) describe the kinetic-XC correlation between charge transfer sites, and could in principle be positive or negative in value as well as change in magnitude due to overlaps with neighboring fragments. However, ensuring that these values do not degrade the positive-definiteness of the overall energy (to ensure that we are always at an energy minimum) is not a simple task and it is unclear at this time how to restrict these off-diagonal terms within the parameterization process. We therefore assume that the non-additive nature of K can be mostly represented by the linear contributions in χ and choose to ignore them. Only diagonal components of K are therefore included in the model in which we denote by the diagonal matrix ∆η pol . The diagonal components of both J and K sum together to give what are termed atomic hardness values,