Probing the partition function for temperature-dependent potentials with nested sampling

Thermodynamic properties can, in principle, be derived from the partition function, which, in many-atom systems, is hard to evaluate as it involves a sum over the accessible microscopic states. Recently, the partition function has been computed via nested sampling, relying on Bayesian statistics, which is able to provide the density of states as a function of the energy in a single run, independently of the temperature. This appealing property is lost whenever the potential energy that appears in the partition function is temperature-dependent-for instance, in mean-field effective potential energies or the quantum partition function in the path-integral formalism. For these cases, nested sampling must be carried out at each temperature, which results in a massive increase in computational time. Here, we introduce and implement a new method based on an extended partition function where the temperature is considered an additional parameter to be sampled. The extended partition function can be evaluated by nested sampling in a single run, thereby restoring this highly desirable property even for temperature-dependent effective potential energies. We apply this original method to compute the quantum partition function for harmonic potentials and Lennard-Jones clusters at low temperatures and show that it outperforms the straightforward application of nested sampling for each temperature within several temperature ranges.