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. 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 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 ( λ ) = U A + λ ( U B − U A ) Here, λ is defined as a coupling parameter with a value between 0 and 1, and thus the potential energy as a function of λ varies from the energy of system A for λ = 0 and system B for λ = 1 . In the canonical ensemble, the partition function of the system can be written as:
Q ( N , V , T , λ ) = ∑ s exp [ − U s ( λ ) / k B T ] In this notation, U s ( λ ) is the potential energy of state s in the ensemble with potential energy function U ( λ ) as defined above. The free energy of this system is defined as:
F ( N , V , T , λ ) = − k B T ln Q ( N , V , T , λ ) ,
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.
Δ F ( A → B ) = ∫ 0 1 ∂ F ( λ ) ∂ λ d λ = − ∫ 0 1 k B T Q ∂ Q ∂ λ d λ = ∫ 0 1 k B T Q ∑ s 1 k B T exp [ − U s ( λ ) / k B T ] ∂ U s ( λ ) ∂ λ d λ = ∫ 0 1 ⟨ ∂ U ( λ ) ∂ λ ⟩ λ d λ = ∫ 0 1 ⟨ U B ( λ ) − U A ( λ ) ⟩ λ d 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 λ . In practice, this is performed by defining a potential energy function U ( λ ) , sampling the ensemble of equilibrium configurations at a series of λ values, calculating the ensemble-averaged derivative of U ( λ ) with respect to λ at each λ value, and finally computing the integral over the ensemble-averaged derivatives.
Umbrella sampling is a related free energy method. It adds a bias to the potential energy. In the limit of an infinite strong bias it is equivalent to thermodynamic integration.
