Thermodynamic integration is a method used to compare the difference in free energy between two given states (e.g., A and B) whose potential energies and have different dependences on the spatial coordinates. Because the free energy of a system is not simply a function of the phase space coordinates of the system, but is instead a function of the Boltzmann-weighted integral over phase space (i.e. partition function), the free energy difference between two states cannot be calculated directly. In thermodynamic integration, the free energy difference is calculated by defining a thermodynamic path between the states and integrating over ensemble-averaged enthalpy changes along the path. Such paths can either be real chemical processes or alchemical processes. An example alchemical process is the Kirkwood's coupling parameter method.
Consider two systems, A and B, with potential energies and . The potential energy in either system can be calculated as an ensemble average over configurations sampled from a molecular dynamics or Monte Carlo simulation with proper Boltzmann weighting. Now consider a new potential energy function defined as:
Here, is defined as a coupling parameter with a value between 0 and 1, and thus the potential energy as a function of varies from the energy of system A for and system B for . In the canonical ensemble, the partition function of the system can be written as:
In this notation, is the potential energy of state in the ensemble with potential energy function as defined above. The free energy of this system is defined as:
If we take the derivative of F with respect to λ, we will get that it equals the ensemble average of the derivative of potential energy with respect to λ.
The change in free energy between states A and B can thus be computed from the integral of the ensemble averaged derivatives of potential energy over the coupling parameter . In practice, this is performed by defining a potential energy function , sampling the ensemble of equilibrium configurations at a series of values, calculating the ensemble-averaged derivative of with respect to at each value, and finally computing the integral over the ensemble-averaged derivatives.
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