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Thermodynamic integration
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<h2 id="derivation">Derivation</h2> <p>Consider two systems, A and B, with potential energies U A {\displaystyle U_{A}} and U B {\displaystyle 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: </p> U ( λ ) = U A + λ ( U B − U A ) {\displaystyle U(\lambda )=U_{A}+\lambda (U_{B}-U_{A})} <p>Here, λ {\displaystyle \lambda } is defined as a coupling parameter with a value between 0 and 1, and thus the potential energy as a function of λ {\displaystyle \lambda } varies from the energy of system A for λ = 0 {\displaystyle \lambda =0} and system B for λ = 1 {\displaystyle \lambda =1} . In the <a href="/facts/Canonical_ensemble/GIhyzkb8">canonical ensemble</a>, the partition function of the system can be written as: </p> Q ( N , V , T , λ ) = ∑ s exp [ − U s ( λ ) / k B T ] {\displaystyle Q(N,V,T,\lambda )=\sum _{s}\exp[-U_{s}(\lambda )/k_{B}T]} <p>In this notation, U s ( λ ) {\displaystyle U_{s}(\lambda )} is the potential energy of state s {\displaystyle s} in the ensemble with potential energy function U ( λ ) {\displaystyle U(\lambda )} as defined above. The free energy of this system is defined as: </p> F ( N , V , T , λ ) = − k B T ln Q ( N , V , T , λ ) {\displaystyle F(N,V,T,\lambda )=-k_{B}T\ln Q(N,V,T,\lambda )} , <p>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. </p> 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 {\displaystyle {\begin{aligned}\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{aligned}}} <p>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 {\displaystyle \lambda } .<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> In practice, this is performed by defining a potential energy function U ( λ ) {\displaystyle U(\lambda )} , sampling the ensemble of equilibrium configurations at a series of λ {\displaystyle \lambda } values, calculating the ensemble-averaged derivative of U ( λ ) {\displaystyle U(\lambda )} with respect to λ {\displaystyle \lambda } at each λ {\displaystyle \lambda } value, and finally computing the integral over the ensemble-averaged derivatives. </p><p><a href="/facts/Umbrella_sampling/sKy8pXYb">Umbrella sampling</a> 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.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Free_energy_perturbation/FYYd2CFT">Free energy perturbation</a></li> <li><a href="/facts/Bennett_acceptance_ratio/uE1LVxme">Bennett acceptance ratio</a></li> <li><a href="/facts/Parallel_tempering/REv1b1yF">Parallel tempering</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Kirkwood, John G. (1935). "Statistical Mechanics of Fluid Mixtures". The Journal of Chemical Physics. 3 (5): 300–313. Bibcode:1935JChPh...3..300K. doi:10.1063/1.1749657. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Frenkel, Daan and Smit, Berend. Understanding Molecular Simulation: From Algorithms to Applications. Academic Press, 2007 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>J Kästner; et al. (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. doi:10.1021/ct050252w. PMID 26626532. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>