Computation of free energy of liquids and its application to melting of CO$_2$ and N

A computationally efficient method is proposed to compute the free energy of liquids with accuracy comparable to {\it ab initio} thermodynamic integration. The method has been applied to predict melting curves of CO$_2$ and N over a wide range of pressure using the solid-liquid phase coexistence approach. The calculated melting lines are compared with available experimental data and the crossing of the geotherm and melting line of CO$_2$ is determined.