If γ = ϕ , Eq. ( ) becomes identical to Eq. (11), which represents the free energy. Since the primary focus of this study is to optimize molecular geometries for a specific state, the energy for γ = ϕ is the most important energy to define rigorously. On the other hand, the energy of other states (γ ̸ = ϕ ) does not have a simple form and lacks straightforward physical interpretation. Therefore, transition energies should be evaluated as differences between two separate calculations. Finally, the state-averaged energy can be expressed as: