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.


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(\lambda) = U_A + \lambda(U_B - U_A) Here, \lambda is defined as a coupling parameter with a value between 0 and 1, and thus the potential energy as a function of \lambda varies from the energy of system A for \lambda = 0 and system B for \lambda = 1. In the canonical ensemble, the partition function of the system can be written as: :Q(N, V, T, \lambda) = \sum_ \exp U_s(\lambda)/k_T/math> In this notation, U_s(\lambda) is the potential energy of state s in the ensemble with potential energy function U(\lambda) as defined above. The free energy of this system is defined as: :F(N,V,T,\lambda)=-k_T \ln Q(N,V,T,\lambda), 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. :\begin \Delta F(A \rightarrow B) &= \int_0^1 \frac d\lambda \\ &= -\int_0^1 \frac \frac d\lambda \\ &= \int_0^1 \frac \sum_ \frac \exp U_s(\lambda)/k_T \frac d\lambda \\ &= \int_0^1 \left\langle\frac\right\rangle_ d\lambda \\ &= \int_0^1 \left\langle U_B(\lambda) - U_A(\lambda) \right\rangle_ d\lambda \end 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 \lambda. In practice, this is performed by defining a potential energy function U(\lambda), sampling the ensemble of equilibrium configurations at a series of \lambda values, calculating the ensemble-averaged derivative of U(\lambda) with respect to \lambda at each \lambda 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.{{cite journal , doi = 10.1021/ct050252w , pmid = 26626532, author = J Kästner, year = 2006 , title = QM/MM Free-Energy Perturbation Compared to Thermodynamic Integration and Umbrella Sampling: Application to an Enzymatic Reaction , journal = Journal of Chemical Theory and Computation , volume = 2 , issue = 2 , pages = 452–461 , display-authors=etal

See also

* Free energy perturbation * Bennett acceptance ratio * Parallel tempering


Computational chemistry Statistical mechanics