Dmrg of Heisenberg model

Thermodynamic integration is a method used to compare the difference in free energy between two given states (e.g., A and B) via sampling state configurations in a molecular dynamics or Metropolis Monte Carlo simulations. Because the free energy of a system is not simply a function of the phase space coordinates of the system, but is instead related to the canonical partition function Q(N,V,T), the free energy difference between two states cannot be calculated directly. In the thermodynamic integration approach, the free energy difference is calculated by defining a thermodynamic path between the states and integrating over enthalpy changes along the path. Such paths can either be real chemical processes or alchemical processes. A good example of the 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, the partition function of the system can be written as:

$Q(N,V,T,\lambda )=\sum _{s}\exp[-U_{s}(\lambda )/kT]$

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:

$F(N,V,T,\lambda )=-k_{B}T\ln Q(N,V,T,\lambda )$,

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.

$\Delta F(A\rightarrow B)=\int _{0}^{1}d\lambda {\frac {\partial F(\lambda )}{\partial \lambda }}=-\int _{0}^{1}d\lambda {\frac {k_{B}T}{Q}}{\frac {\partial Q}{\partial \lambda }}=\int _{0}^{1}d\lambda {\frac {k_{B}T}{Q}}\sum _{s}{\frac {1}{k_{B}T}}\exp[-U_{s}(\lambda )/k_{B}T]{\frac {\partial U(\lambda )}{\partial \lambda }}=\int _{0}^{1}d\lambda \left\langle {\frac {\partial U(\lambda )}{\partial \lambda }}\right\rangle _{\lambda }$

The change in free energy between states A and B can thus be computed from the integral of the ensemble average of the change in potential energy with the coupling parameter $\lambda$. In practice, this calculation is performed by first defining a potential energy function $U(\lambda )$, sampling a series of equilibrium configurations for different values of $\lambda$, finding the ensemble average of the derivative of the system energy with respect to $\lambda$ from the configurations in these separate equilibrium runs, then finally numerically computing the integral over all of these 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.

See also

• Free energy perturbation
• Bennett acceptance ratio