where τα represents creation and annihilation operators for an α th -order electronic excitation. For example, τ ν i = ĉ † ν ĉi excites an electron from an occupied orbital i to an unoccupied orbital ν. Similarly, τ νυ ij = ĉ † ν ĉ † υ ĉi ĉj will excite two electrons i → ν and j → υ, respectively. The photonic excitation operator can be written in a simple idempotent form 163 as τn = |n⟩⟨0| for a finite number of Fock states 163 {|n⟩} = {|0⟩, |1⟩, ..., |N F ⟩}. The coupled excitation operator ταñ can be written, for example, as ĉ † ν ĉi |n⟩⟨0| for a single electron excitation coupled to an n th -level photonic excitation while ĉ † ν ĉ † υ ĉa ĉb |n⟩⟨0| will provide the double electron and n thlevel photonic excitations. Each of these excitation operators and one for every choice of n up to the photon level truncation N F with a unique cluster amplitude t.