Observing the topological structure of the one-body unperturbed Hamiltonian in BW-s2(α), we can easily see that it is a one-dimensional vector space with a single, real coefficient. It is straightforward to assume that there exist further, linearly parameterised extensions to this choice of Ĥ(0) which retain size-consistency, regularity and size-extensivity at second order. To shorten the notation for this discussion, we will examine extensions R to the unperturbed Hamiltonian of BW-s2(α). Therefore, the constraints on R given by Eq. 36 as well as the overall previous discussion are simply that R be a Hermitian one-body operator which is traceless in the occupied space. Additionally, its presence should not break size-consistency or size-extensivity, which we will elaborate on later. One other desirable property of correlation methods (including MP2 and BW-s2(α)) is that the correlation energy itself should be invariant to independent unitary transformations of the occupied and virtual orbitals, if the amplitude equations are appropriate transformed. If the resulting regulariser (given by the sum of the operator defined in Eq. 37 and R) fulfils the aforementioned requirements, we will simply refer to it as valid.