Let us now consider how this structure of the space of all possible regularisers affects the underlying unperturbed Hamiltonians. We can create a (surjective) map from R to the spectral representation of the resulting unperturbed Hamiltonian, which is determined by the dressed orbitals φ ′ p as well as the dressed orbital energies ε ′ p . Since the space of all regularisers covers all possible Hermitian matrices by definition, the matrix elements of all Ĥ(0) can also form any Hermitian matrix. Thus, the space of all possible dressed orbital energies is simply R n and the space of all possible dressed orbitals can be generated as (U(n occ ) ⊕U(n vir )) φ p , where U(n) denotes the unitary group of size n. Note that two unperturbed Hamiltonians are linearly dependent only if both their dressed orbitals as well as their dressed orbital energies are identical up to permutation. The number of linearly independent regularisers is therefore not only infinite, it is uncountable! In order to create a computable second order perturbation theory, we have to introduce further constraints on each term in the regulariser.