Here, N = K k=1 N k is total number of snapshots collected in all the simulations, and ∆U t (r) = U t (r) -U 0 (r). ω t (r n ) can be understood as the weighting factor for each configuration in a thermodynamic perturbation from the mixed reference potentials (U k ) to the target Hamiltonian (U t ). Therefore, this method is referred to as the multistate thermodynamic perturbation (MsTP). Thermodynamic properties that depend only on atomic coordinates under U t can thus be obtained as