Thermodynamic integration

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:

:$Q(N,\; V,\; T,\; \backslash lambda)\; =\; \backslash sum\_\{s\}\; \backslash exp\; [-U\_s(\backslash lambda)/k\_\{B\}T]$

In this notation, $U\_s(\backslash lambda)$ is the potential energy of state $s$ in the ensemble with potential energy function $U(\backslash lambda)$ as defined above. The free energy of this system is defined as:

:$F(N,V,T,\backslash lambda)=-k\_\{B\}T\; \backslash ln\; Q(N,V,T,\backslash lambda)$,

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.

:$\backslash begin\{align\}$

\Delta F(A \rightarrow B)

&= \int_0^1 \frac{\partial F(\lambda)}{\partial\lambda} d\lambda

\\

&= -\int_0^1 \frac{k_{B}T}{Q} \frac{\partial Q}{\partial\lambda} d\lambda

\\

&= \int_0^1 \frac{k_{B}T}{Q} \sum_{s} \frac{1}{k_{B}T} \exp[- U_s(\lambda)/k_{B}T ] \frac{\partial U_s(\lambda)}{\partial \lambda} d\lambda

\\

&= \int_0^1 \left\langle\frac{\partial U(\lambda)}{\partial\lambda}\right\rangle_{\lambda} d\lambda

\\

&= \int_0^1 \left\langle U_B(\lambda) - U_A(\lambda) \right\rangle_{\lambda} d\lambda

\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 $\backslash lambda$.Frenkel, Daan and Smit, Berend. Understanding Molecular Simulation: From Algorithms to Applications. Academic Press, 2007 In practice, this is performed by defining a potential energy function $U(\backslash lambda)$, sampling the ensemble of equilibrium configurations at a series of $\backslash lambda$ values, calculating the ensemble-averaged derivative of $U(\backslash lambda)$ with respect to $\backslash lambda$ at each $\backslash lambda$ 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}}

• Free energy perturbation
• Bennett acceptance ratio
• Parallel tempering

Category:Computational chemistry

Category:Statistical mechanics