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Thermodynamic integration of free-energy gradients

In general, constrained molecular dynamics generates biased statistical averages. It can be shown that the correct average for a quantity $ a({\mathbf{\xi}})$ can be obtained using the formula:

$\displaystyle a({\mathbf{\xi}}) = \frac{\langle \vert{\mathbf{Z}}\vert^{-1/2} a(\xi^*) \rangle_{\xi^*}}{\langle \vert{\mathbf{Z}}\vert^{-1/2}\rangle_{\xi^*}},$ (6.27)

where $ \langle ... \rangle_{\xi^*}$ stands for the statistical average of the quantity enclosed in angular parentheses computed for a constrained ensemble and $ {\mathbf{Z}}$ is a mass metric tensor defined as:

$\displaystyle Z_{\alpha,\beta} ={\sum}_{i=1}^{3N} m_i^{-1} \nabla_i \xi_\alpha \cdot \nabla_i \xi_\beta, \, \alpha=1,...,r, \, \beta=1,...,r,$ (6.28)

It can be shown [66,68,67,70] that the free energy gradient can be computed using the equation:

$\displaystyle \left(\frac{\partial A}{\partial \xi_k}\right)_{\xi^*} = \frac{1}...
...m_i^{-1} \nabla_i \xi_j \cdot \nabla_i \vert{\mathbf{Z}}\vert ]\rangle_{\xi^*},$ (6.29)

where $ \lambda_{\xi_k}$ is the Lagrange multiplier associated with the parameter $ {\xi_k}$ used in the SHAKE algorithm [81]. The free-energy difference between states (1) and (2) can be computed by integrating the free-energy gradients over a connecting path:

$\displaystyle {\Delta}A_{1 \rightarrow 2} = \int_{{\xi(1)}}^{{\xi(2)}} \left ( \frac{\partial {A}} {\partial \xi} \right )_{\xi^*} \cdot d{\xi}.$ (6.30)

Note that as free-energy is a state quantity, the choice of path connecting (1) with (2) is irrelevant.

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