Adaptive Biasing Force Method: Difference between revisions
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====Simplified ABF Algorithm==== | ====Simplified ABF Algorithm==== | ||
The simplified ABF algorithm (fmode=1) uses the result of the product rule for derivatives applied on Equation 3: | The simplified ABF algorithm (fmode=1) uses the result of the product rule for derivatives applied on Equation 3 leading into the two distinct contributions into the instanteous collective forces: | ||
<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} + \bold F_{ICF,kin}</math> ... ( | <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 accelerations <math>\bold a(t)</math> | The potential contribution <math>\bold F_{ICF,pot}(t)</math> employs the accelerations <math>\bold a(t)</math> calculated back from the velocities. This 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): | ||
<center>\bold a(t) = \frac{\bold v(t+dt/2) - \bold v(t-dt/2)}{dt}<math> | <center><math>\bold a(t) = \frac{\bold v(t+dt/2) - \bold v(t-dt/2)}{dt}</math> ... (10)</center> | ||
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> ... (11)</center> | |||
Finaly, 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> ... (12)</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 accel | The accel | ||