First-principles calculation of the entropy of liquids with a case study on sodium

Despite increasing demands for the thermodynamic data of liquids in a wide range of science and engineering fields, there is a still a considerable lack of reliable data over a wide range of temperature (T) and pressure conditions. The most significant obstacle is that there is no practical method to calculate the entropy (S) of liquids. This problem can be solved using the thermodynamic definition of entropy, i.e.S=∫Cdln⁡T, whereCis specific heat. The specific heat is calculated by the change in the internal energyUagainst the unit change ofT. Both quantities, i.e.UandT, are well defined in the molecular dynamics (MDs) simulations based on density functional theory. The reliability of the present method is entirely dependent on the accuracy of the specific heat of liquid, for which there is no standard model. The problem with liquids is that there are no eigenstates, based on which the standard procedures are constructed. The relationship betweenUandTis affected by the energy relaxation processes, the effect of which appears in theTdependence on the specific heat of liquids. This motivates us to conduct MD simulations by isolating the system from an external heat bath. In this paper, by applying this method to the liquid sodium, it is demonstrated that the experimentalTdependence of the isochoric specific heat is reproduced without any empirical parameter. On this basis, the entropy of the liquid Na is obtained with a good agreement with experimental values.