We use â † p and âq to refer to the standard fermionic creation and annihilation operators, which create a formal excited state labelled by capital letters I, J, K, . . . when applied to a wavefunction. We write antisymmetrised electron repulsion integrals (ERIs) in 1212 notation as ⟨pq||rs⟩ and symmetric ERIs in 1122 notation as (pr|qs). Throughout this work, antisymmetrised ERIs refer to spin orbitals while symmetric ERIs use spatial orbitals, which are always assumed to be real. In all equations, we implicitly sum over all indices present on only one side of the equation.