This trend can be accurately captured by the harmonic results based on the ground state CMES. In contrast, conventional classical treatment based on PES cannot describe tunneling, and at the zero-temperature limit, the particle always get trapped in either side of the well, leading to generally overestimated frequencies. Additionally, because of the wrong physical picture, the increasing trend is completely missed. For the 1 → 2 transition, the energy gap generally decreases as a becomes less negative but turns flat and starts to increase when it is about -4eV/Å 4 . This qualitative picture can be captured by our method based on the first excited state CMES, although we note that the quantitative agreement is not completely satisfactory, especially when a is a very negative number, CMES-MD could overestimate the gap by 100%. For the conventional classical treatment, although quantitatively it seems that it can slightly outperform CMES-MD in most of the double-well excited state region with the grey curve being closer to the red curve than the green curve, we note that it does not mean that conventional MD is better since its underlying trapped-particle picture is completely unphysical.