Adaptive Biasing Force Method: Difference between revisions
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<center><math>\frac{\partial G(\boldsymbol \xi^{*})}{\partial \boldsymbol \xi} = - {\langle \bold F_{ICF} \rangle}_{\boldsymbol \xi=\boldsymbol \xi^{*}} </math> ... (2)</center> | <center><math>\frac{\partial G(\boldsymbol \xi^{*})}{\partial \boldsymbol \xi} = - {\langle \bold F_{ICF} \rangle}_{\boldsymbol \xi=\boldsymbol \xi^{*}} </math> ... (2)</center> | ||
with the instantaneous collective force calculated from the time evolution of the collective variable: | with the instantaneous collective force calculated from the time evolution of the collective variable:<ref name="darve2008">Darve, E.; Rodríguez-Gómez, D.; Pohorille, A. Adaptive Biasing Force Method for Scalar and Vector Free Energy Calculations. ''J. Chem. Phys.'' '''2008''', ''128 (14)'', 144120. [https://doi.org/10.1063/1.2829861 doi:10.1063/1.2829861].</ref> | ||
<center><math>\bold F_{ICF} = \frac {d} {dt} \left( \bold Z^{-1} \frac{d \boldsymbol \xi} {dt} \right)</math> ... (3)</center> | <center><math>\bold F_{ICF} = \frac {d} {dt} \left( \bold Z^{-1} \frac{d \boldsymbol \xi} {dt} \right)</math> ... (3)</center> | ||
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<center><math>[Z]_{ij}=\sum_{k=1}^{N_{atoms}} \frac{1}{m_k} \frac{\partial \xi_i}{\partial \bold x_k} \frac{\partial \xi_j}{\partial \bold x_k}</math> ... (4)</center> | <center><math>[Z]_{ij}=\sum_{k=1}^{N_{atoms}} \frac{1}{m_k} \frac{\partial \xi_i}{\partial \bold x_k} \frac{\partial \xi_j}{\partial \bold x_k}</math> ... (4)</center> | ||
The analytical calculation of instantaneous collective force by Equation 3 requires the second derivatives of collective variables with respect to Cartesian coordinates. Since this can be prohibitive for complex collective variables such as the simple base-pair parameters, Equation 3 is evaluated numerically by a finite-difference approach | The analytical calculation of instantaneous collective force by Equation 3 requires the second derivatives of collective variables with respect to Cartesian coordinates. Since this can be prohibitive for complex collective variables such as the simple base-pair parameters, Equation 3 is evaluated numerically by a [[Adaptive Biasing Force Method#Instanteous Collective Forces|finite-difference approach]]. | ||
===Sampling Space Discretization=== | ===Sampling Space Discretization=== | ||
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where <math>\bold h</math> is the interval size also called a bin, <math>N_b</math> is the number of samples collected in a bin centered at <math>\boldsymbol \xi^{*}</math>, and <math>N_{corr}</math> is a statistical inefficiency due to correlation in time series. | where <math>\bold h</math> is the interval size also called a bin, <math>N_b</math> is the number of samples collected in a bin centered at <math>\boldsymbol \xi^{*}</math>, and <math>N_{corr}</math> is a statistical inefficiency due to correlation in time series. | ||
Therefore, each CV involved in ABF simulations must be discretized by specifying an interval in which the sampling is performed and the number of intervals (bins) for discretization, for further details, see [[ABF:Collective variables]]. The increasing number of bins improves the accuracy of Equation 5 but it also increases the noise because of a smaller number of samples collected in a bin. A reasonable compromise is the number of bins, which leads to 0.1 Å or 1-2° bin widths. | |||
===Adaptive Bias=== | ===Adaptive Bias=== | ||