Adaptive Biasing Force Method: Difference between revisions
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<center><math>\frac {d} {dt} \left( \bold Z^{-1} \frac{d \boldsymbol \xi} {dt} \right) \big|_{(t)} = \left( \bold Z^{-1}(t) \nabla \boldsymbol \xi (t) \right) \cdot \bold a(t) + \frac{d}{dt} \left( \bold Z^{-1}(t) \nabla \boldsymbol \xi (t)) \right) \cdot \bold v(t) = \bold F_{ICF,pot}(t) + \bold F_{ICF,kin}(t)</math> ... (9)</center> | <center><math>\frac {d} {dt} \left( \bold Z^{-1} \frac{d \boldsymbol \xi} {dt} \right) \big|_{(t)} = \left( \bold Z^{-1}(t) \nabla \boldsymbol \xi (t) \right) \cdot \bold a(t) + \frac{d}{dt} \left( \bold Z^{-1}(t) \nabla \boldsymbol \xi (t)) \right) \cdot \bold v(t) = \bold F_{ICF,pot}(t) + \bold F_{ICF,kin}(t)</math> ... (9)</center> | ||
The potential contribution <math>\bold F_{ICF,pot}(t)</math> | The potential contribution <math>\bold F_{ICF,pot}(t)</math> is given: | ||
<center><math>\bold a(t) = \frac{\bold v(t+dt/2) - \bold v(t-dt/2)}{dt}</math> ... ( | <center><math>\bold F_{ICF,pot}(t) = \left( \bold Z^{-1}(t) \nabla \boldsymbol \xi (t) \right) \cdot \frac{\bold v(t+dt/2) - \bold v(t-dt/2)}{dt}</math> ... (10)</center> | ||
It employs the accelerations <math>\bold a(t)</math> calculated back from the velocities: | |||
<center><math>\bold a(t) = \frac{\bold v(t+dt/2) - \bold v(t-dt/2)}{dt}</math> ... (11)</center>, | |||
which ensures that effect of SHAKE and other constraints on acceleration is properly counted (see [[Molecular_Dynamics_Algorithms|the leap-frog integration algorithm]] for further details). | |||
The kinetic contribution <math>\bold F_{ICF,kin}(t-dt/2)</math> is calculated from numerical differentiation: | The kinetic contribution <math>\bold F_{ICF,kin}(t-dt/2)</math> is calculated from numerical differentiation: | ||
<center><math>\bold F_{ICF,kin}(t-dt/2) = \frac {\bold Z^{-1}(t) \nabla \boldsymbol \xi (t) - \bold Z^{-1}(t-dt) \nabla \boldsymbol \xi (t-dt)}{dt} \bold v(t-dt/2) </math> ... ( | <center><math>\bold F_{ICF,kin}(t-dt/2) = \frac {\bold Z^{-1}(t) \nabla \boldsymbol \xi (t) - \bold Z^{-1}(t-dt) \nabla \boldsymbol \xi (t-dt)}{dt} \bold v(t-dt/2) </math> ... (12)</center> | ||
Finally, to get <math>\bold F_{ICF,kin}</math> at the same time as <math>\bold F_{ICF,kin}</math>, two values are averaged: | Finally, to get <math>\bold F_{ICF,kin}</math> at the same time as <math>\bold F_{ICF,kin}</math>, two values are averaged: | ||
<center><math>\bold F_{ICF,kin}(t) = \frac {\bold F_{ICF,kin}(t+dt/2) + \bold F_{ICF,kin}(t-dt/2) }{2}</math> ... ( | <center><math>\bold F_{ICF,kin}(t) = \frac {\bold F_{ICF,kin}(t+dt/2) + \bold F_{ICF,kin}(t-dt/2) }{2}</math> ... (13)</center> | ||
The algorithm uses a history of values collected in two consecutive time steps. But, the first result is available from the fourth time step to be compatible with the original algorithm. | The algorithm uses a history of values collected in two consecutive time steps. But, the first result is available from the fourth time step to be compatible with the original algorithm. | ||
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The original ABF algorithm (fmode=2) implements the algorithm presented by Darve ''at al''.<ref name="darve2008"/><ref name="darve2007">Darve, E. Thermodynamic Integration Using Constrained and Unconstrained Dynamics. In Free Energy Calculations Theory and Applications in Chemistry and Biology; Springer Series in CHEMICAL PHYSICS; Springer: Berlin, 2007; Vol. 86, pp 119–170.</ref>. This algorithm should not be used when SHAKE is enabled because the accelerations entering into the calculations are not corrected for constraints. The algorithm uses a history of values collected in four consecutive time steps. Thus, the first result is available from the fourth time step. | The original ABF algorithm (fmode=2) implements the algorithm presented by Darve ''at al''.<ref name="darve2008"/><ref name="darve2007">Darve, E. Thermodynamic Integration Using Constrained and Unconstrained Dynamics. In Free Energy Calculations Theory and Applications in Chemistry and Biology; Springer Series in CHEMICAL PHYSICS; Springer: Berlin, 2007; Vol. 86, pp 119–170.</ref>. This algorithm should not be used when SHAKE is enabled because the accelerations entering into the calculations are not corrected for constraints. The algorithm uses a history of values collected in four consecutive time steps. Thus, the first result is available from the fourth time step. | ||
<center><math>\frac {d} {dt} \left( \bold Z^{-1} \frac{d \boldsymbol \xi} {dt} \right) \big|_{(t)} = \frac{1}{2} \left( \frac{\bold p^+_{\xi}(t+dt) - \bold p^+_{\xi}(t) } {dt} + \frac{\bold p^-_{\xi}(t+dt) - \bold p^-_{\xi}(t) } {dt} \right)</math> ... ( | <center><math>\frac {d} {dt} \left( \bold Z^{-1} \frac{d \boldsymbol \xi} {dt} \right) \big|_{(t)} = \frac{1}{2} \left( \frac{\bold p^+_{\xi}(t+dt) - \bold p^+_{\xi}(t) } {dt} + \frac{\bold p^-_{\xi}(t+dt) - \bold p^-_{\xi}(t) } {dt} \right)</math> ... (14)</center> | ||
where | where | ||
<center><math>\bold p^+_{\xi}(t) = \bold Z^{-1}(t) \nabla \boldsymbol \xi(t) \cdot \left[ \bold v(t+dt/2) - \frac{dt}{6} \bold a(t+dt) \right]</math> ... ( | <center><math>\bold p^+_{\xi}(t) = \bold Z^{-1}(t) \nabla \boldsymbol \xi(t) \cdot \left[ \bold v(t+dt/2) - \frac{dt}{6} \bold a(t+dt) \right]</math> ... (15)</center> | ||
<center><math>\bold p^-_{\xi}(t) = \bold Z^{-1}(t) \nabla \boldsymbol \xi(t) \cdot \left[ \bold v(t-dt/2) + \frac{dt}{6} \bold a(t-dt) \right]</math> ... ( | <center><math>\bold p^-_{\xi}(t) = \bold Z^{-1}(t) \nabla \boldsymbol \xi(t) \cdot \left[ \bold v(t-dt/2) + \frac{dt}{6} \bold a(t-dt) \right]</math> ... (16)</center> | ||
==References== | ==References== | ||
<references /> | <references /> | ||