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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.


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(\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 In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat b ...
, 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 Umbrella sampling is a technique in computational physics and chemistry, used to improve sampling of a system (or different systems) where ergodicity is hindered by the form of the system's energy landscape. It was first suggested by Torrie an ...
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

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Free energy perturbation Free energy perturbation (FEP) is a method based on statistical mechanics that is used in computational chemistry for computing free energy differences from molecular dynamics or Metropolis Monte Carlo simulations. The FEP method was introduced ...
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Bennett acceptance ratio The Bennett acceptance ratio method (BAR) is an algorithm for estimating the difference in free energy between two systems (usually the systems will be simulated on the computer). It was suggested by Charles H. Bennett in 1976. Preliminaries Tak ...
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Parallel tempering Parallel tempering in physics and statistics, is a computer simulation method typically used to find the lowest free energy state of a system of many interacting particles at low temperature. That is, the one expected to be observed in reality. ...


References

Computational chemistry Statistical mechanics