Thermodynamic integration explained

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.^[1]

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( 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 ) = \sum _{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.

\begin{align} \Delta F(A → B) & = \int ^{1} _{0} (\partial F( λ ))/(\partial λ) d λ \\ & = - \int ^{1} _{0} (k _{B} T)/(Q) (\partial Q)/(\partial λ) d λ \\ & = \int ^{1} _{0} (k _{B} T)/(Q) \sum _{s} (1)/(k _{B} T) \exp[ - U _{s(λ)/k} _{B} T ] (\partial U _{s(λ)})/(\partial λ) d λ \\ & = \int ^{1} _{0} \left\langle (\partial U( λ ))/(\partial λ) \right\rangle _{λ} d λ \\ & = \int ^{1} _{0} \left\langle U _{B(λ)} - U _{A(λ)} \right\rangle _{λ} d λ \end{align}

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 .^[2] 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.^[3]

See also

• Free energy perturbation
• Bennett acceptance ratio
• Parallel tempering
• Alchemy

References

1. 10.1063/1.1749657. 1935JChPh...3..300K. Statistical Mechanics of Fluid Mixtures. 1935. Kirkwood. John G.. The Journal of Chemical Physics. 3. 5. 300–313.
2. Frenkel, Daan and Smit, Berend. Understanding Molecular Simulation: From Algorithms to Applications. Academic Press, 2007
3. 10.1021/ct050252w . 26626532. J Kästner. 2006 . QM/MM Free-Energy Perturbation Compared to Thermodynamic Integration and Umbrella Sampling: Application to an Enzymatic Reaction . Journal of Chemical Theory and Computation . 2 . 2 . 452–461 . etal.