Chemistry:Thermodynamic integration
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 [math]\displaystyle{ U_A }[/math] and [math]\displaystyle{ U_B }[/math] 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 [math]\displaystyle{ U_A }[/math] and [math]\displaystyle{ U_B }[/math]. 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:
- [math]\displaystyle{ U(\lambda) = U_A + \lambda(U_B - U_A) }[/math]
Here, [math]\displaystyle{ \lambda }[/math] is defined as a coupling parameter with a value between 0 and 1, and thus the potential energy as a function of [math]\displaystyle{ \lambda }[/math] varies from the energy of system A for [math]\displaystyle{ \lambda = 0 }[/math] and system B for [math]\displaystyle{ \lambda = 1 }[/math]. In the canonical ensemble, the partition function of the system can be written as:
- [math]\displaystyle{ Q(N, V, T, \lambda) = \sum_{s} \exp [-U_s(\lambda)/k_{B}T] }[/math]
In this notation, [math]\displaystyle{ U_s(\lambda) }[/math] is the potential energy of state [math]\displaystyle{ s }[/math] in the ensemble with potential energy function [math]\displaystyle{ U(\lambda) }[/math] as defined above. The free energy of this system is defined as:
- [math]\displaystyle{ F(N,V,T,\lambda)=-k_{B}T \ln Q(N,V,T,\lambda) }[/math],
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.
- [math]\displaystyle{ \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} }[/math]
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 [math]\displaystyle{ \lambda }[/math].[2] In practice, this is performed by defining a potential energy function [math]\displaystyle{ U(\lambda) }[/math], sampling the ensemble of equilibrium configurations at a series of [math]\displaystyle{ \lambda }[/math] values, calculating the ensemble-averaged derivative of [math]\displaystyle{ U(\lambda) }[/math] with respect to [math]\displaystyle{ \lambda }[/math] at each [math]\displaystyle{ \lambda }[/math] 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
References
- ↑ Kirkwood, John G. (1935). "Statistical Mechanics of Fluid Mixtures". The Journal of Chemical Physics 3 (5): 300–313. doi:10.1063/1.1749657. Bibcode: 1935JChPh...3..300K.
- ↑ Frenkel, Daan and Smit, Berend. Understanding Molecular Simulation: From Algorithms to Applications. Academic Press, 2007
- ↑ 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. doi:10.1021/ct050252w. PMID 26626532.
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