Effect of the integration method on the accuracy and computational efficiency of free energy calculations using thermodynamic integration

Although calculations of free energy using molecular dynamics simulations have gained significant importance in the chemical and biochemical fields, they still remain quite computationally intensive. Furthermore, when using thermodynamic integration, numerical evaluation of the integral of the Hamiltonian with respect to the coupling parameter may introduce unwanted errors in the free energy. In this paper, we compare the performance of two numerical integration techniques-the trapezoidal and Simpsons rules-and propose a new method, based on the analytic integration of physically based fitting functions that are able to accurately describe the behavior of the data. We develop and test our methodology by performing detailed studies on two prototype systems, hydrated methane and hydrated methanol, and treat Lennard-Jones and electrostatic contributions separately. We conclude that the widely used trapezoidal rule may introduce systematic errors in the calculation, but these errors are reduced if Simpsons rule is employed, at least for the electrostatic component. Furthermore, by fitting thermodynamic integration data, we are able to obtain precise free energy estimates using significantly fewer data points (5 intermediate states for the electrostatic component and 11 for the Lennard-Jones term), thus significantly decreasing the associated computational cost. Our method and improved protocol were successfully validated by computing the free energy of more complex systems-hydration of 2-methylbutanol and of 4-nitrophenol-thus paving the way for widespread use in solvation free energy calculations of drug molecules.