Adaptive Biasing Force Method

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Contents

• ABF:Description
• ABF:Controls
• ABF:Collective variables
• ABF:Post-processing
• ABF:Multiple walkers approach
• ABF:Utilities
• ABF:Examples

Description

Theory

The Adaptive Biasing Force (ABF) method calculates the derivative of the free energy (PMF) using the following formula:

$\frac{\partial A}{\partial \boldsymbol \xi} = - {\left \langle { \frac{d}{d t} \left( \bold {Z_{\xi}}^{-1} \frac{d \boldsymbol \xi}{d t} \right) } \right \rangle}_{\xi}$ .... (1)

where ξ is the set of collective variables, t is time, and Z_ξ is the matrix defined as

$\left [ \bold Z_{\xi} \right ]_{i,j} = \sum_{k=1}^{N}{ \frac{1}{m_k}\frac{\partial \xi_i}{\partial \bold x_k}\frac{\partial \xi_j}{\partial \bold x_k}}$ .... (2)

where m_k is the mass of atom with cartesian coordinate x_k.

The averages over ξ are collected from unconstrained molecular dynamics. The free energy derivatives are practically calculated over discretized ranges of collective variables (CVs). Each CV_i is discretized into M_i bins leading into one-dimensional bins for one CV, two-dimensional bins for two CVs, etc. In each such bin, the vector of PMF is accumulated using the formula (1). For one CV this can be written as

$\frac{\partial A}{\partial \xi} \bigg|_{(\xi_{i}+\xi_{i+1})/2} = - {\left \langle { \frac{d}{d t} \left( {Z_{\xi}}^{-1} \frac{d \xi}{d t} \right) } \right \rangle}_{ ( \xi_{i},\xi_{i+1} \rangle }$ .... (3)

In the ABF simulations, the estimated PMF is used to bias molecular dynamics simulations to improve sampling in the regions exhibiting large free energy barriers.