Adaptive Biasing Force Method - Revision history

Durnik at 12:51, 2 August 2021

2021-08-02T12:51:26Z

← Older revision		Revision as of 12:51, 2 August 2021
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	<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>		<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,pot}</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> ... (13)</center>		<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>

Kulhanek at 13:19, 28 July 2021

2021-07-28T13:19:06Z

← Older revision		Revision as of 13:19, 28 July 2021
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	* [[ABF:Multiple walkers approach]]		* [[ABF:Multiple walkers approach]]
	* [[ABF:Examples]]		* [[ABF:Examples]]


	==Description==		==Description==

Kulhanek at 13:18, 28 July 2021

2021-07-28T13:18:58Z

← Older revision		Revision as of 13:18, 28 July 2021
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	* [[ABF:Collective variables]]		* [[ABF:Collective variables]]
	* [[ABF:Post-processing]]		* [[ABF:Post-processing]]
			* [[ABF:Multiple walkers approach]]
	* [[ABF:Examples]]		* [[ABF:Examples]]
	* [[ABF:Multiple walkers approach]]

	==Description==		==Description==

Kulhanek at 10:49, 25 July 2021

2021-07-25T10:49:31Z

← Older revision		Revision as of 10:49, 25 July 2021
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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> ~~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)~~:		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> ... (10)</center>		<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> ... (11)</center>		<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> ... (12)</center>		<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> ... (11)</center>		<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> ... (12)</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> ... (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> ... (13)</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> ... (16)</center>

	==References==		==References==
	<references />		<references />

Kulhanek at 09:19, 25 July 2021

2021-07-25T09:19:12Z

← Older revision		Revision as of 09:19, 25 July 2021
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	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 instantaneous collective forces:		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 instantaneous 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}(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> 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):		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):

Kulhanek at 08:51, 25 July 2021

2021-07-25T08:51:02Z

Kulhanek at 08:47, 25 July 2021

2021-07-25T08:47:51Z

← Older revision		Revision as of 08:47, 25 July 2021
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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> ... (11)</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 accelerations <math>\bold a(t)</math> ~~are~~ calculated from the velocities ~~to incorporate the~~ effect of SHAKE constraints in [[the leap-frog integration algorithm]]:		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

Kulhanek at 08:22, 25 July 2021

2021-07-25T08:22:55Z

← Older revision		Revision as of 08:22, 25 July 2021
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	ABF introduces an adaptive bias, which improves the sampling of rare events. The bias removes barriers or higher free energy regions in the space described by predefined collective variables <math>\boldsymbol \xi</math>. As a result, the system evolves by free diffusions alongside these collective variables. The bias is derived from the free energy, which is projected in the form of biasing force <math>\bold F_{bias}</math> to the Cartesian space and removed from force <math>\bold F_{pot}</math> originated from interatomic interaction potential <math>V(\bold R)</math>. The application of the bias thus leads to the modified equations of motions:		ABF introduces an adaptive bias, which improves the sampling of rare events. The bias removes barriers or higher free energy regions in the space described by predefined collective variables <math>\boldsymbol \xi</math>. As a result, the system evolves by free diffusions alongside these collective variables. The bias is derived from the free energy, which is projected in the form of biasing force <math>\bold F_{bias}</math> to the Cartesian space and removed from force <math>\bold F_{pot}</math> originated from interatomic interaction potential <math>V(\bold R)</math>. The application of the bias thus leads to the modified equations of motions:

	<center><math>m_{i} \frac { d^2 \bold r_i }{dt^2} = \bold F_{pot,i}(\bold r_i) - \bold F_{bias,i}(\bold r_i) = - \frac{\partial V(\bold R)} {\partial \bold r_i} - \frac{\partial G(\boldsymbol \xi)}{\partial \boldsymbol \xi} \frac{\partial \boldsymbol \xi}{\partial \bold r_i}</math> ... (8)</center>		<center><math>m_{i} \frac { d^2 \bold r_i }{dt^2} = \bold F_{pot,i}(\bold r_i) - \bold F_{bias,i}(\bold r_i) = - \frac{\partial V(\bold R)} {\partial \bold r_i} + \frac{\partial G(\boldsymbol \xi)}{\partial \boldsymbol \xi} \frac{\partial \boldsymbol \xi}{\partial \bold r_i}</math> ... (8)</center>

	where <math>m_i</math> is mass of atom i, <math>\bold r_i</math> is atom position, and <math>t</math> is time.		where <math>m_i</math> is mass of atom i, <math>\bold r_i</math> is atom position, and <math>t</math> is time.
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	===Instanteous Collective Forces===		===Instanteous Collective Forces===
	PMFLib implements two ~~approaches~~ to calculate Equation 3.		PMFLib implements two algorithms to calculate Equation 3: simplified and original ABF algorithms. The original ABF algorithm is theoretically more accurate than the simplified one but it works only for MD without constraints. Therefore, the simplified algorithm is the default one as it is numerically more robust at all MD setups.
	====Simplified Algorithm====
			====Simplified ABF Algorithm====
			The simplified ABF algorithm (fmode=1) uses the result of the product rule for derivatives applied on Equation 3:

			<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> ... (11)</center>

			The accelerations <math>\bold a(t)</math> are calculated from the velocities to incorporate the effect of SHAKE constraints in [[the leap-frog integration algorithm]]:

			<center>\bold a(t) = \frac{\bold v(t+dt/2) - \bold v(t-dt/2)}{dt}<math>



			The accel

	====Original ABF Algorithm====		====Original ABF Algorithm====
			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> ... (11)</center>

	<center><math>\frac {d} {dt} \left( \bold Z^{-1} \frac{d \boldsymbol \xi} {dt} \right) = \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> ... (11)</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> ... (12)</center>

	<center><math>\~~frac~~ {~~d} {dt~~} ~~\left~~( \bold Z^{-1} \~~frac{d~~ \boldsymbol \xi~~} {dt} \right~~) = \~~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> ... (11)</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> ... (13)</center>

	==References==		==References==
	<references />		<references />

Kulhanek at 20:01, 24 July 2021

2021-07-24T20:01:48Z

← Older revision		Revision as of 20:01, 24 July 2021
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	==Description==		==Description==
	===Free Energy from Unconstrained MD Simulations===		===Free Energy from Unconstrained MD Simulations===
	The Adaptive Biasing Force (ABF) ~~Method~~ calculates the free energy as a function of selected collective variables from unconstrained molecular dynamics (MD) simulations. The method does not provide the free energy directly, but instead, it provides the free energy gradient <math>\frac{\partial G(\boldsymbol \xi)}{\partial \boldsymbol \xi}</math>, which must be integrated to get the free energy:		The Adaptive Biasing Force (ABF) nethod calculates the free energy as a function of selected collective variables from unconstrained molecular dynamics (MD) simulations. The method does not provide the free energy directly, but instead, it provides the free energy gradient <math>\frac{\partial G(\boldsymbol \xi)}{\partial \boldsymbol \xi}</math>, which must be integrated to get the free energy:

	<center><math>\Delta G = G(\boldsymbol \xi_2) - G(\boldsymbol \xi_1) = \int_{\boldsymbol \xi_1}^{\boldsymbol \xi_2} \frac{\partial G(\boldsymbol \xi)} {\partial \boldsymbol \xi} \boldsymbol d \xi </math> ... (1)</center>		<center><math>\Delta G = G(\boldsymbol \xi_2) - G(\boldsymbol \xi_1) = \int_{\boldsymbol \xi_1}^{\boldsymbol \xi_2} \frac{\partial G(\boldsymbol \xi)} {\partial \boldsymbol \xi} \boldsymbol d \xi </math> ... (1)</center>
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	===Instanteous Collective Forces===		===Instanteous Collective Forces===
	~~TBA~~		PMFLib implements two approaches to calculate Equation 3.
			====Simplified Algorithm====

			====Original ABF Algorithm====

			<center><math>\frac {d} {dt} \left( \bold Z^{-1} \frac{d \boldsymbol \xi} {dt} \right) = \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> ... (11)</center>

			where

			<center><math>\frac {d} {dt} \left( \bold Z^{-1} \frac{d \boldsymbol \xi} {dt} \right) = \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> ... (11)</center>

	==References==		==References==
	<references />		<references />

Kulhanek at 15:16, 22 July 2021

2021-07-22T15:16:24Z

← Older revision		Revision as of 15:16, 22 July 2021
Line 8:		Line 8:
	* [[ABF:Collective variables]]		* [[ABF:Collective variables]]
	* [[ABF:Post-processing]]		* [[ABF:Post-processing]]
			* [[ABF:Examples]]
	* [[ABF:Multiple walkers approach]]		* [[ABF:Multiple walkers approach]]
	* [[ABF:Examples]]
	* [[ABF:Utilities]]

	==Description==		==Description==

← Older revision		Revision as of 08:51, 25 July 2021
Line 13:		Line 13:
	==Description==		==Description==
	===Free Energy from Unconstrained MD Simulations===		===Free Energy from Unconstrained MD Simulations===
	The Adaptive Biasing Force (ABF) ~~nethod~~ calculates the free energy as a function of selected collective variables from unconstrained molecular dynamics (MD) simulations. The method does not provide the free energy directly, but instead, it provides the free energy gradient <math>\frac{\partial G(\boldsymbol \xi)}{\partial \boldsymbol \xi}</math>, which must be integrated to get the free energy:		The Adaptive Biasing Force (ABF) method calculates the free energy as a function of selected collective variables from unconstrained molecular dynamics (MD) simulations. The method does not provide the free energy directly, but instead, it provides the free energy gradient <math>\frac{\partial G(\boldsymbol \xi)}{\partial \boldsymbol \xi}</math>, which must be integrated to get the free energy:

	<center><math>\Delta G = G(\boldsymbol \xi_2) - G(\boldsymbol \xi_1) = \int_{\boldsymbol \xi_1}^{\boldsymbol \xi_2} \frac{\partial G(\boldsymbol \xi)} {\partial \boldsymbol \xi} \boldsymbol d \xi </math> ... (1)</center>		<center><math>\Delta G = G(\boldsymbol \xi_2) - G(\boldsymbol \xi_1) = \int_{\boldsymbol \xi_1}^{\boldsymbol \xi_2} \frac{\partial G(\boldsymbol \xi)} {\partial \boldsymbol \xi} \boldsymbol d \xi </math> ... (1)</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 [[Adaptive Biasing Force Method#~~Instanteous~~ Collective Forces\|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#Instantaneous Collective Forces\|finite-difference approach]].

	===Sampling Space Discretization===		===Sampling Space Discretization===
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	===Adaptive Bias===		===Adaptive Bias===
	ABF introduces an adaptive bias, which improves the sampling of rare events. The bias removes barriers or higher free energy regions in the space described by predefined collective variables <math>\boldsymbol \xi</math>. As a result, the system evolves by free ~~diffusions~~ alongside these collective variables. The bias is derived from the free energy, which is projected in the form of biasing force <math>\bold F_{bias}</math> to the Cartesian space and removed from force <math>\bold F_{pot}</math> originated from ~~interatomic~~ interaction potential <math>V(\bold R)</math>. The application of the bias thus leads to the modified equations of motions:		ABF introduces an adaptive bias, which improves the sampling of rare events. The bias removes barriers or higher free energy regions in the space described by predefined collective variables <math>\boldsymbol \xi</math>. As a result, the system evolves by free diffusion alongside these collective variables. The bias is derived from the free energy, which is projected in the form of biasing force <math>\bold F_{bias}</math> to the Cartesian space and removed from force <math>\bold F_{pot}</math> originated from inter-atomic interaction potential <math>V(\bold R)</math>. The application of the bias thus leads to the modified equations of motions:

	<center><math>m_{i} \frac { d^2 \bold r_i }{dt^2} = \bold F_{pot,i}(\bold r_i) - \bold F_{bias,i}(\bold r_i) = - \frac{\partial V(\bold R)} {\partial \bold r_i} + \frac{\partial G(\boldsymbol \xi)}{\partial \boldsymbol \xi} \frac{\partial \boldsymbol \xi}{\partial \bold r_i}</math> ... (8)</center>		<center><math>m_{i} \frac { d^2 \bold r_i }{dt^2} = \bold F_{pot,i}(\bold r_i) - \bold F_{bias,i}(\bold r_i) = - \frac{\partial V(\bold R)} {\partial \bold r_i} + \frac{\partial G(\boldsymbol \xi)}{\partial \boldsymbol \xi} \frac{\partial \boldsymbol \xi}{\partial \bold r_i}</math> ... (8)</center>
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	Since the biasing force is not known prior to the simulation, it is calculated during the simulation and adaptively applied. To accelerate sampling, the biasing force is applied even if an inadequate number of samples is collected in a bin. In this case, the biasing force is scaled in the early stages to avoid artifacts from applications of overestimated biasing forces. The biasing force can also be smoothed to decrease noise in collected data. For further details, see feimode in [[ABF:Controls]].		Since the biasing force is not known prior to the simulation, it is calculated during the simulation and adaptively applied. To accelerate sampling, the biasing force is applied even if an inadequate number of samples is collected in a bin. In this case, the biasing force is scaled in the early stages to avoid artifacts from applications of overestimated biasing forces. The biasing force can also be smoothed to decrease noise in collected data. For further details, see feimode in [[ABF:Controls]].

	===~~Instanteous~~ Collective Forces===		===Instantaneous Collective Forces===
	PMFLib implements two algorithms to calculate Equation 3: simplified and original ABF algorithms. The original ABF algorithm is theoretically more accurate than the simplified one but it works only for MD without constraints. Therefore, the simplified algorithm is the default one as it is numerically more robust at all MD setups.		PMFLib implements two algorithms to calculate Equation 3: simplified and original ABF algorithms. The original ABF algorithm is theoretically more accurate than the simplified one but it works only for MD without constraints. Therefore, the simplified algorithm is the default one as it is numerically more robust at all MD setups.

	====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 leading into the two distinct contributions into the ~~instanteous~~ collective forces:		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 instantaneous 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}(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>
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	<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>		<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:		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> ... (12)</center>		<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>
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	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.

	~~The accel~~

	====Original ABF Algorithm====		====Original ABF Algorithm====