The atomic radius as a fundamental chemical descriptor for the size of a chemical element is often used in physical chemistry. Here, a theoretical measure is proposed based on expectation values r4$$....

A common approach for computing free energy differences among multiple states is to build a perturbation graph connecting the states and compute free energy differences on all edges of the graph. Such perturbation graphs are often designed to have cycles. Because free energy is a function of states, the free energy around any cycle is zero, which we refer to as the cycle consistency condition. Since the cycle consistency condition relates free energy differences on the edges of a cycle, it could be used to improve the accuracy of free energy estimates. Here, we propose a Bayesian method called the coupled Bayesian multistate Bennett acceptance ratio (CBayesMBAR) that can properly couple the calculations of free energy differences on the edges of cycles in a principled way. We apply the CBayesMBAR to compute free energy differences among harmonic oscillators and relative protein–ligand binding free energies. In both cases, the CBayesMBAR provides more accurate results compared to methods that do not consider the cycle consistency condition. Additionally, it outperforms the cycle closure correction method that also uses cycle consistency conditions.