First-principles estimation of partition functions representing disordered lattices such as the cubic phases of Li2OHCl and Li2OHBr

Abstract

In order to develop computational methods that can simulate thermodynamic properties of disordered materials at a first-principles level, we investigate the use of a random set of configurations to evaluate the canonical partition function of lattice-based disordered systems. Testing the sampling method on the one- and two-dimensional Ising models indicates that for the ordered system at low temperature, convergence is achieved when the number of samples $\mathcal{S}$ is comparable to or larger than the number of configurations $\mathrm{\ensuremath{\Omega}}$, while for the partially disordered system at high temperature, convergence is achieved for smaller sample sizes as low as $\mathcal{S}\ensuremath{\approx}\mathrm{\ensuremath{\Omega}}/100$ or $\mathcal{S}\ensuremath{\approx}\mathrm{\ensuremath{\Omega}}/1000$. The sampling method is combined with first-principles calculations to examine the ordered $\ensuremath{\leftrightarrow}$ disordered phase transition for the Li ion electrolyte materials ${\mathrm{Li}}_{2}\mathrm{OHCl}$ and ${\mathrm{Li}}_{2}\mathrm{OHBr}$. Static-lattice internal energies and harmonic-phonon free energies were incorporated into the evaluation of the partition function. The evaluation of the partition function depends on the value of $\mathrm{\ensuremath{\Omega}}$ corresponding to the number of metastable states of the system. Accordingly, we developed a method of approximating $\mathrm{\ensuremath{\Omega}}$ using the properties of the sampled configurations. The results of the calculations are consistent with the experimental observation that the transition temperature for the orthorhombic $\ensuremath{\leftrightarrow}$ cubic phase transition is higher for ${\mathrm{Li}}_{2}\mathrm{OHCl}$ than for ${\mathrm{Li}}_{2}\mathrm{OHBr}$. We expect the sampling method to be generally useful for investigating the thermodynamic properties of other disordered-lattice systems. We also investigate a ``disordered-subspace function'' which is shown to satisfy inequality relationships with respect to the Helmholtz free energy.