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 $U\_A$ and $U\_B$ 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 from the potential energy of just two coordinate sets (for state A and B respectively). 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.

Computational chemistry
Statistical mechanics

Derivation

Consider two systems, A and B, with potential energies $U\_A$ and $U\_B$. 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: :$U(\backslash lambda)\; =\; U\_A\; +\; \backslash lambda(U\_B\; -\; U\_A)$ Here, $\backslash lambda$ is defined as a coupling parameter with a value between 0 and 1, and thus the potential energy as a function of $\backslash lambda$ varies from the energy of system A for $\backslash lambda\; =\; 0$ and system B for $\backslash lambda\; =\; 1$. In the canonical ensemble, the partition function of the system can be written as: :$Q(N,\; V,\; T,\; \backslash lambda)\; =\; \backslash sum\_\; \backslash exp;\; href="/html/ALL/s/U\_s(\backslash lambda)/k\_T.html"\; ;"title="U\_s(\backslash lambda)/k\_T">U\_s(\backslash lambda)/k\_T$See also

* Free energy perturbation * Bennett acceptance ratio * Parallel temperingReferences