under the constraint x i+1 n = x i n . This quantum Hamiltonian operator is made up of P group of beads (also known as replicas or time-slices), and each forms a classical representation of the original system. Beads of each atom are connected by harmonic potentials. In this way, a P -membered classical "necklace" model is constructed, and the simulation of quantum systems is transformed into an expanded classical calculation. m n is the mass of the particle, U is the potential energy, m i n and p i n are the fictitious mass of the ith bead of the nth particle and its corresponding momentum. This fictitious mass can be set to an arbitrary value, because this term disappears in the thermodynamical statistical average. The ω p = √ P /βh is the harmonic constraint connecting the two most adjacent beads. For the ideal quantum problem, the full quantum Hamiltonian Ĥ is established when P → ∞. The introduction of this isomorphic relationship also means the quantum mechanical sampling can be mapped onto classical simulations by either Monte Carlo (MC) or MD methods. The potential of mean force with a collection of PIMD samples can thus be written as