Thermodynamic integration

A detailed description of thermodynamic integration is given in reference .

The free energy of a fully interacting system can be written as the sum of the free energy a non-interacting reference system and the difference in the free energy of the fully interacting system and the non-interacting system

$F_{1} = F_{0} + \Delta F_{0\rightarrow 1}$.

Using thermodynamic integration the free energy difference between the two systems is written as

$\Delta F_{0\rightarrow 1} = \int\limits_{0}^{1} d\lambda \langle U_{1}(\lambda) - U_{0}(\lambda) \rangle_{\lambda}$.

Here $U_{1}(\lambda)$ and $U_{0}(\lambda)$ describe the potential energies of a fully-interacting and a non-interacting reference system, respectively. The coupling strength of the systems is controlled via the coupling parameter $\lambda$. It is neccessary that the connection of the two systems via the coupling constant is reversible. The notation $\langle \ldots \rangle_{\lambda}$ denotes an ensemble average of a system driven by the following classical Hamiltonian

$H_{\lambda}= \lambda H_{1} + (1-\lambda) H_{0}$.

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