where |0⟩ is the photon vacuum state of Ĥph (Eq. 42), a † is the photon creation operator (see Eq. 42 and Eq. 43) and c n is the expansion coefficients for the photon number states. The HF energy for the molecule-cavity hybrid system is then computed in the usual variational way by using the |Φ el+ph 0 ⟩ to sandwich Ĥpl (Eq. 106) as follows