The Potential Energy Landscape Formalism for Quantum Liquids in the Continuous Ring-Polymer Limit

The potential energy landscape (PEL) formalism is a powerful tool within statistical mechanics to study the thermodynamic behavior of classical and quantum liquids as well as non-annealing glasses. In the quantum case, this theoretical framework is based on the path-integral formulation of statistical mechanics where the liquid/non-annealing glass is mapped onto a classical system of ring-polymers. In previous studies, and motivated by path-integral (PI) computer simulations, the PEL formalism was developed assuming that the ring-polymers were discrete, i.e., each ring-polymer was composed of a finite number of beads, nb. In this work, we revisit the PEL formalism for the case where the ring-polymers of the system are continuous, i.e., nb → ∞ (as required in the PI formulation of statistical mechanics). We find that in this limit, the PEL of the quantum liquid is unique and well-defined. The main PEL properties of the system, including the average inherent structure energy EIS, configurational entropy SIS, and vibrational Helmholtz free energy Fvib, are also well-defined (i.e., they converge). Notably, in the harmonic approximation of the PEL, Fvib can be expressed analytically; it is identical to the free energy of a multidimensional quantum harmonic oscillator with the corresponding frequencies given by the normal mode frequencies of the classical liquid counterpart (accessible from classical computer simulations). When applied to non-annealing glasses and crystals, this new version of the PEL formalism reduces to the so-called quasi-harmonic approximation, commonly used in the study of solids. Overall, the PEL formalism for nb → ∞ provides an alternative approach to its previous version (finite nb); it is also more elegant and intuitive, and simpler to apply. For a Gaussian and harmonic PEL, this formalism (nb → ∞) allows one to obtain the Helmholtz free energy of the quantum liquid using classical computer simulations, avoiding the more expensive PI computer simulations.