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'''Biased molecular dynamics''' (MD) refers to advanced [[:Category:Molecular dynamics|MD-simulation methods]] that introduce a ''bias potential''.
For instance, umbrella sampling{{cite|torrie:jcp:1977}} uses a bias potential to enhance the sampling of a geometric parameter in regions with low probability density, e.g., transition regions of chemical reactions.
The correct distributions are recovered afterward using an exact relation between the biased and unbiased systems. D. Frenkel and B. Smit provide a more detailed description of the method in Ref. {{cite|frenkel:ap-book:2002}}.
'''Biased MD''' can also be used to introduce soft geometric constraints in which the controlled geometric parameter is not strictly constant. Instead, it oscillates in a narrow interval of values.
In VASP, the bias potentials are supported in both the [[NVT ensemble|NVT]] and [[NpT ensemble|NpT]] MD simulations, regardless of the particular [[:Category:Thermostats|thermostat]] and/or [[PSTRESS|barostat]] setting.
Different types of potentials can be freely combined. The values of all collective variables defined in the {{FILE|ICONST}} file for each MD step are listed in the {{FILE|REPORT}} file. Check the lines after the string <tt>Metadynamics</tt>.
[[File:Bias potentials.png|thumb|Graphical representation of (a) harmonic, (b) Fermi function-shaped, and (c) and Gauss function-shaped bias potentials.  ]]
== Theoretical background ==


The probability density for a geometric parameter &xi; of the system driven by a Hamiltonian:
The probability density for a geometric parameter &xi; of the system driven by a Hamiltonian:
Line 17: Line 28:
and the probability density of &xi; in the biased ensemble is:
and the probability density of &xi; in the biased ensemble is:
:<math>
:<math>
\tilde{P}(\xi_i)= \frac{\int  \delta\Big(\xi(q)-\xi_i\Big) \exp\left\{-\tilde{H}(q,p)/k_B\,T\right\} dq\,dp}{\int  \exp\left\{-\tilde{H}(q,p)/k_B\,T\right\}dq\,dp} =  \langle\delta\Big(\xi(q)-\xi_i\Big)\rangle_{\tilde{H}}
\tilde{P}(\xi_i)= \frac{\int  \delta\Big(\xi(q)-\xi_i\Big) \exp\left\{-\tilde{H}(q,p)/k_B\,T\right\} dq\,dp}{\int  \exp\left\{-\tilde{H}(q,p)/k_B\,T\right\}dq\,dp} =  \langle\delta\Big(\xi(q)-\xi_i\Big)\rangle_{\tilde{H}}.
</math>
</math>
It can be shown that the biased and unbiased averages are related via a simple formula:
It can be shown that the biased and unbiased averages are related via a simple formula:
Line 31: Line 42:
\langle A \rangle_{H} =\frac{\langle A(q) \,\exp\left\{\tilde{V}(\xi)/k_B\,T\right\} \rangle_{\tilde{H}}}{\langle \exp\left\{\tilde{V}(\xi)/k_B\,T\right\} \rangle_{\tilde{H}}}.
\langle A \rangle_{H} =\frac{\langle A(q) \,\exp\left\{\tilde{V}(\xi)/k_B\,T\right\} \rangle_{\tilde{H}}}{\langle \exp\left\{\tilde{V}(\xi)/k_B\,T\right\} \rangle_{\tilde{H}}}.
</math>
</math>
Simulation methods such as umbrella sampling<ref name="Torrie77"/> use a bias potential to enhance sampling of &xi; in regions with low ''P''(&xi;<sub>''i''</sub>) such as transition regions of chemical
Simulation methods, e.g., umbrella sampling{{cite|torrie:jcp:1977}}, use a bias potential to enhance sampling of &xi; in regions with low ''P''(&xi;<sub>''i''</sub>), e.g., transition regions of chemical
reactions.
reactions.
The correct distributions are recovered afterwards using the equation for <math>\langle A \rangle_{H}</math> above.
The correct distributions are recovered afterward using the equation for <math>\langle A \rangle_{H}</math> above. Ref.{{cite|frenkel:ap-book:2002}} provides a more detailed description of the method.
 
A more detailed description of the method can be found in Ref.<ref name="FrenkelSmit"/>.
Biased molecular dynamics can be used also to introduce soft geometric constraints in which the controlled geometric parameter is not strictly constant, instead it oscillates in a narrow interval
of values.
The bias potentials are supported in both the NVT and NpT MD simulations regardless of the particular thermostat and/or barostat setting.
Different types of potentials can be freely combined.
[[File:Bias potentials.png|thumb|Graphical representation of (a) harmonic, (b) Fermi function-shaped, and (c) and Gauss function-shaped bias potentials. ]]


== Supported types of bias potentials ==
== Supported types of bias potentials ==
Presently, the following types of bias potential are supported:
Presently, the following types of bias potential are supported:


=== Harmonic potentials ===


*sum of Harmonic potentials (see curve (a) on the plot above)
:A sum of Harmonic potentials (see curve (a) on the plot above)
:<math>
::<math>
\tilde{V}(\xi_1,\dots,\xi_{M_8}) = \sum_{\mu=1}^{M}\frac{1}{2}\kappa_{\mu} (\xi_{\mu}(q)-\xi_{0\mu})^2 \;
\tilde{V}(\xi_1,\dots,\xi_{M_8}) = \sum_{\mu=1}^{M}\frac{1}{2}\kappa_{\mu} (\xi_{\mu}(q)-\xi_{0\mu})^2 \;
</math>
</math>
where the sum runs over all (<math>M_8</math>) coordinates the potential acts upon, which are defined in the {{FILE|ICONST}}-file by setting the <code>status</code> to 8.  
:where the sum runs over all (<math>M_8</math>) coordinates the potential acts upon. The coordinates are defined in the {{FILE|ICONST}} file by setting the <code>status</code> to 8. The parameters of the potential are the force constant <math>\kappa_{\mu}</math> ({{TAG|SPRING_K}}) and the minimum or the potential <math>\xi_{0\mu}</math> {{TAG|SPRING_R0}}. These must be set in the {{FILE|INCAR}} file. Optionally, it is possible to change the value of <math>\xi_{0\mu}</math> every MD step at a constant rate defined via the {{FILE|INCAR}} tag {{TAG|SPRING_V0}}.  
The parameters of the potential, <math>\kappa_{\mu}</math> (force constant) and <math>\xi_{0\mu}</math> (minimum of potential), are defined, respectively, via the keywords {{TAG|SPRING_K}} and {{TAG|SPRING_R0}} in {{FILE|INCAR}}.
{{NB|mind|The number of items defined via {{TAG|SPRING_K}}, {{TAG|SPRING_R0}}, and {{TAG|SPRING_V0}} must be equal to <math>M_8</math>. Otherwise, the calculation terminates with an error message.|:}}
Optionally, it is also possible to change the value of <math>\xi_{0\mu}</math> every MD step at a constant rate defined via the parameter {{TAG|SPRING_V0}}.  
:This form of bias potential is employed in several simulation protocols, such as the umbrella sampling{{cite|torrie:jcp:1977}}, umbrella integration, or steered MD, and is useful also in cases where the <math>\xi_{\mu}</math> values need to be restrained.  
The number of items defined via {{TAG|SPRING_K}}, {{TAG|SPRING_R0}}, and {{TAG|SPRING_V0}} must be equal to <math>M_8</math>, otherwise the calculation terminates with an error message.  
This form of bias potential is employed in several simulation protocols, such as the umbrella sampling<ref name="Torrie77"/>, umbrella integration, or steered MD, and is useful also in cases where the <math>\xi_{\mu}</math> values need restrained.  


*sum of Fermi-like step functions (see curve (b) on the plot above)
=== Step function ===
:<math>
 
:A sum of Fermi-like step functions (see curve (b) on the plot above)
::<math>
\tilde{V}(\xi_1,\dots,\xi_{M_4}) = \sum_{\mu=1}^{M_4}\frac{A_{\mu}}{1+\text{exp}\left [-D_{\mu}(\frac{\xi(q)}{\xi_{0\mu}} -1) \right ]} \;
\tilde{V}(\xi_1,\dots,\xi_{M_4}) = \sum_{\mu=1}^{M_4}\frac{A_{\mu}}{1+\text{exp}\left [-D_{\mu}(\frac{\xi(q)}{\xi_{0\mu}} -1) \right ]} \;
</math>
</math>
where the sum runs over all (<math>M_4</math>) coordinates the potential acts upon, which are defined in the {{FILE|ICONST}}-file by setting the <code>status</code> to 4.
:where the sum runs over all (<math>M_4</math>) coordinates the potential acts upon. The coordinates are defined in the {{FILE|ICONST}} file by setting the <code>status</code> to 4. The parameters of the potential are the height of the step (<math>A_{\mu}</math> set by {{TAG|FBIAS_A}}), the slope around the point <math>\xi_{0\mu}</math> (<math>D_{\mu}</math> set by {{TAG|FBIAS_D}}), and the position of the step (<math>\xi_{0\mu}</math> set by {{TAG|FBIAS_R0}}). These must be set in the {{FILE|INCAR}} file.  
The parameters of the potential, <math>A_{\mu}</math> (the height of step), <math>D_{\mu}</math> (controlling the slope around the point <math>\xi_{0\mu}</math>), and <math>\xi_{0\mu}</math> (position of the step), are defined, respectively,
{{NB|mind|The number of items defined via {{TAG|FBIAS_A}}, {{TAG|FBIAS_D}}, and {{TAG|FBIAS_R0}} must be equal to <math>M_4</math>. Otherwise, the calculation terminates with an error message.|:}}
via the keywords {{TAG|FBIAS_A}}, {{TAG|FBIAS_D}}, and {{TAG|FBIAS_R0}} in {{FILE|INCAR}}.
:This form of potential is suitable especially for imposing restrictions on the upper (or lower) limit of the value of <math>\xi</math>.  
The number of items defined via {{TAG|FBIAS_A}}, {{TAG|FBIAS_D}}, and {{TAG|FBIAS_R0}} must be equal to <math>M_4</math>, otherwise the calculation terminates with an error message.  
 
This form of potential is suitable especially for imposing restriction on the upper (or lower) limit of value of <math>\xi</math>.  
=== Gaussian potential ===


*sum of Gauss functions (see curve (b) on the plot above)
:A sum of Gauss functions (see curve (b) on the plot above)
:<math>
::<math>
\tilde{V}(\xi_1,\dots,\xi_{M})  = \sum_{\nu=1}^{N_5}h_{\nu}\text{exp}\left [-\frac{\sum_{\mu=1}^{M_5}(\xi_{\mu}(q)-\xi_{0\nu,\mu})^2}{2w_{\nu}^2}  \right ], \;
\tilde{V}(\xi_1,\dots,\xi_{M})  = \sum_{\nu=1}^{N_5}h_{\nu}\text{exp}\left [-\frac{\sum_{\mu=1}^{M_5}(\xi_{\mu}(q)-\xi_{0\nu,\mu})^2}{2w_{\nu}^2}  \right ], \;
</math>
</math>
where <math>N_5</math> is the number of Gaussian functions and  
:where <math>N_5</math> is the number of Gaussian functions and <math>M_5</math> is the number of coordinates the potential acts upon. The coordinates are defined in the {{FILE|ICONST}} file by setting the <code>status</code> to 5. The parameters of the potentials, <math>h_{\nu}</math>, <math>w_{\nu}</math>, and <math>\xi_{0\nu,\mu}</math> are defined in the {{FILE|PENALTYPOT}} file.  
<math>M_5</math> is the number of coordinates the potential acts upon.
 
The latter defined in the {{FILE|ICONST}}-file by setting the <code>status</code> to 5.
:This type of bias potential is primarily intended for use in metadynamics, but since Gaussians can be used as basis functions for more general shapes, they can also be used to prepare various atypically-shaped bias potentials.   
The parameters of the potentials, <math>h_{\nu}</math>, <math>w_{\nu}</math>, and <math>\xi_{0\nu,\mu}</math> are defined in the {{FILE|PENALTYPOT}}-file.  
The this type of bias potential is primarily intended for the use in metadynamics but since Gaussians can be used as basis functions for more general shapes, they can be used also to prepare various atypically-shaped bias potentials.   


== Examples of usage ==
== Examples of usage ==
As a concrete example, let us consider the nucleophile substitution reaction of CH<math>_3</math>Cl with Cl<math>^-</math>. The reactant is a weak vdW complex shown on the figure below, the corresponding {{FILE|POSCAR}}-file is  
Let us consider the nucleophile substitution reaction of CH<math>_3</math>Cl with Cl<math>^-</math>. The reactant is a weak van-der-Waals complex. The corresponding {{FILE|POSCAR}} file reads  


  vdW complex CH3Cl...Cl  
  vdW complex CH3Cl...Cl  
Line 94: Line 97:
  6.84157897  6.18713289  4.46842049
  6.84157897  6.18713289  4.46842049


* due to weak interactions between CH<math>_3</math>Cl and Cl<math>^-</math>, the complex can collapse at high temperature. This can be avoided by setting an upper bound for the length of the non-bonding Cl...C interactions, which can can conveniently be achieved by using a Fermi-shaped bias potential. To this end, we need to define the Cl...C distance (distance between the atoms 1 and 5) as a coordinate with status 4 in the {{FILE|ICONST}}-file:
Due to the weak interactions between CH<math>_3</math>Cl and Cl<math>^-</math>, the complex can collapse at high temperatures. This can be avoided by setting an upper bound for the length of the non-bonding Cl...C interactions.
This can be conveniently achieved by using a Fermi-like step-shaped bias potential. To this end, we need to define the Cl...C distance, i.e., the distance between the atoms 1 and 5, as a coordinate with status 4 in the {{FILE|ICONST}} file:


  R 1 5 4   
  R 1 5 4   


Next, we need to specify the bias potential parameters {{TAG|FBIAS_A}}, {{TAG|FBIAS_D}}, and {{TAG|FBIAS_R0}} in the {{FILE|ICONST}}-file. Since the bias potential acts only on one internal coordinate, we need to provide only one single number for each of these flags. As shown in Fig.3, the setting  
Next, we need to specify the bias potential parameters {{TAG|FBIAS_A}}, {{TAG|FBIAS_D}}, and {{TAG|FBIAS_R0}} in the {{FILE|INCAR}} file. Since the bias potential acts only on one internal coordinate (<math>M_4=1</math>), we need to provide only one number for each of the tags. The setting  


  FBIAS_A  = 1
  FBIAS_A  = 1
Line 104: Line 108:
  FBIAS_R0 = 3.5
  FBIAS_R0 = 3.5
    
    
ensures that repulsive bias forces steeply increase when the C...Cl distance is increased beyond about 3.2 A, which would in effect cause shortening of the distance in next MD step. Notice that the bias force is essentially negligible for distances below 3 A.  
ensures that repulsive bias forces steeply increase when the C...Cl distance is increased beyond about <math>3.2 \AA</math>. This causes a shortening of the distance in the next MD step. Notice that the bias force is essentially negligible for distances below <math>3 \AA</math>.  
A careful adjustment of <math>A</math> and <math>D</math> is needed to ensure that (i) the bias force is large enough to effectively limit the value of <math>\xi</math>, and (ii) the interval of <math>\xi</math> values for which the bias forces are significant is broad enough to avoid its overcoming via random fluctuations.       
A careful adjustment of {{TAG|FBIAS_A}} and {{TAG|FBIAS_D}} is needed to ensure that (i) the bias force is large enough to effectively limit the value of <math>\xi</math>, and (ii) the interval of <math>\xi</math> values for which the bias forces are significant is broad enough to avoid overcoming via random fluctuations.       
A suitable setting can be found by inspecting the  analytical expression for the potential from which it is obvious that the maximal bias force of <math>\frac{D\,A}{4\xi_0}</math> is exerted on the system at the point <math>\xi = \xi_{0}</math>.  
A suitable setting can be found by noting that the maximal bias force of <math>\frac{D\,A}{4\xi_0}</math> is exerted on the system at the point <math>\xi = \xi_{0}</math>. This can be seen by inspecting the analytical expression for the potential.
 
== How to ==


== Andersen thermostat ==
=== Biased MD run using the Andersen thermostat and a Gaussian bias potential ===


* For a biased molecular dynamics run with Andersen thermostat, one has to:
:For a biased MD run with Andersen thermostat, one has to:
#Set the standard MD-related tags: {{TAG|IBRION}}=0, {{TAG|TEBEG}}, {{TAG|POTIM}}, and {{TAG|NSW}}
:#Set the standard MD-related tags: {{TAG|IBRION}}=0, {{TAG|TEBEG}}, {{TAG|POTIM}}, and {{TAG|NSW}}
#Set {{TAG|MDALGO}}=1 ({{TAG|MDALGO}}=11 in VASP 5.x), and choose an appropriate setting for {{TAG|ANDERSEN_PROB}}
:#Set {{TAG|MDALGO}}=1 ({{TAG|MDALGO}}=11 in VASP 5.x), and choose an appropriate setting for {{TAG|ANDERSEN_PROB}}
#In order to avoid updating of the bias potential, set {{TAG|HILLS_BIN}}={{TAG|NSW}}
:#To avoid updating of the bias potential, set {{TAG|HILLS_BIN}}={{TAG|NSW}}
#Define collective variables in the {{FILE|ICONST}}-file, and set the <tt>STATUS</tt> parameter for the collective variables to 5
:#Define collective variables in the {{FILE|ICONST}} file, and set the <tt>STATUS</tt> parameter for the collective variables to 5
#Define the bias potential in the {{FILE|PENALTYPOT}}-file  
:#Define the bias potential in the {{FILE|PENALTYPOT}} file  


== Nose-Hoover thermostat ==
=== Biased MD run using the Nose-Hoover thermostat and a Gaussian bias potential ===


* For a biased molecular dynamics run with Nose-Hoover thermostat, one has to:
:For a biased MD run with Nose-Hoover thermostat, one has to:
#Set the standard MD-related tags: {{TAG|IBRION}}=0, {{TAG|TEBEG}}, {{TAG|POTIM}}, and {{TAG|NSW}}
:#Set the standard MD-related tags: {{TAG|IBRION}}=0, {{TAG|TEBEG}}, {{TAG|POTIM}}, and {{TAG|NSW}}
#Set {{TAG|MDALGO}}=2 ({{TAG|MDALGO}}=21 in VASP 5.x), and choose an appropriate setting for {{TAG|SMASS}}
:#Set {{TAG|MDALGO}}=2 ({{TAG|MDALGO}}=21 in VASP 5.x), and choose an appropriate setting for {{TAG|SMASS}}
#In order to avoid updating of the bias potential, set {{TAG|HILLS_BIN}}={{TAG|NSW}}
:#To avoid updating the bias potential, set {{TAG|HILLS_BIN}}={{TAG|NSW}}
#Define collective variables in the {{FILE|ICONST}}-file, and set the <tt>STATUS</tt> parameter for the collective variables to 5
:#Define collective variables in the {{FILE|ICONST}} file, and set the <tt>STATUS</tt> parameter for the collective variables to 5
#Define the bias potential in the {{FILE|PENALTYPOT}}-file
:#Define the bias potential in the {{FILE|PENALTYPOT}} file


The values of all collective variables for each MD step are listed in the {{FILE|REPORT}}-file, check the lines after the string <tt>Metadynamics</tt>.
The values of all collective variables for each MD step are listed in the {{FILE|REPORT}} file. Check the lines after the string <tt>Metadynamics</tt>.


== References ==
== References ==
<references>
<references/>
<ref name="Torrie77">[http://dx.doi.org/10.1016/0021-9991(77)90121-8 G. M. Torrie and J. P. Valleau, J. Comp. Phys. 23, 187 (1977).]</ref>
<ref name="FrenkelSmit">D. Frenkel and B. Smit, ''Understanding molecular simulations: from algorithms to applications'', Academic Press: San Diego, 2002.</ref>
</references>
----
----


[[Category:Molecular dynamics]][[Category:Biased molecular dynamics]][[Category:Theory]][[Category:Howto]]
[[Category:Molecular dynamics]][[Category:Biased molecular dynamics]][[Category:Theory]][[Category:Howto]]

Revision as of 09:32, 24 April 2023

Biased molecular dynamics (MD) refers to advanced MD-simulation methods that introduce a bias potential.

For instance, umbrella sampling[1] uses a bias potential to enhance the sampling of a geometric parameter in regions with low probability density, e.g., transition regions of chemical reactions. The correct distributions are recovered afterward using an exact relation between the biased and unbiased systems. D. Frenkel and B. Smit provide a more detailed description of the method in Ref. [2]. Biased MD can also be used to introduce soft geometric constraints in which the controlled geometric parameter is not strictly constant. Instead, it oscillates in a narrow interval of values.

In VASP, the bias potentials are supported in both the NVT and NpT MD simulations, regardless of the particular thermostat and/or barostat setting. Different types of potentials can be freely combined. The values of all collective variables defined in the ICONST file for each MD step are listed in the REPORT file. Check the lines after the string Metadynamics.

Graphical representation of (a) harmonic, (b) Fermi function-shaped, and (c) and Gauss function-shaped bias potentials.

Theoretical background

The probability density for a geometric parameter ξ of the system driven by a Hamiltonian:

with T(p), and V(q) being kinetic, and potential energies, respectively, can be written as:

The term stands for a thermal average of quantity X evaluated for the system driven by the Hamiltonian H.

If the system is modified by adding a bias potential acting on one or multiple selected internal coordinates of the system ξ=ξ(q), the Hamiltonian takes a form:

and the probability density of ξ in the biased ensemble is:

It can be shown that the biased and unbiased averages are related via a simple formula:

More generally, an observable :

can be expressed in terms of thermal averages within the biased ensemble:

Simulation methods, e.g., umbrella sampling[1], use a bias potential to enhance sampling of ξ in regions with low Pi), e.g., transition regions of chemical reactions. The correct distributions are recovered afterward using the equation for above. Ref.[2] provides a more detailed description of the method.

Supported types of bias potentials

Presently, the following types of bias potential are supported:

Harmonic potentials

A sum of Harmonic potentials (see curve (a) on the plot above)
where the sum runs over all () coordinates the potential acts upon. The coordinates are defined in the ICONST file by setting the status to 8. The parameters of the potential are the force constant (SPRING_K) and the minimum or the potential SPRING_R0. These must be set in the INCAR file. Optionally, it is possible to change the value of every MD step at a constant rate defined via the INCAR tag SPRING_V0.
Mind: The number of items defined via SPRING_K, SPRING_R0, and SPRING_V0 must be equal to . Otherwise, the calculation terminates with an error message.
This form of bias potential is employed in several simulation protocols, such as the umbrella sampling[1], umbrella integration, or steered MD, and is useful also in cases where the values need to be restrained.

Step function

A sum of Fermi-like step functions (see curve (b) on the plot above)
where the sum runs over all () coordinates the potential acts upon. The coordinates are defined in the ICONST file by setting the status to 4. The parameters of the potential are the height of the step ( set by FBIAS_A), the slope around the point ( set by FBIAS_D), and the position of the step ( set by FBIAS_R0). These must be set in the INCAR file.
Mind: The number of items defined via FBIAS_A, FBIAS_D, and FBIAS_R0 must be equal to . Otherwise, the calculation terminates with an error message.
This form of potential is suitable especially for imposing restrictions on the upper (or lower) limit of the value of .

Gaussian potential

A sum of Gauss functions (see curve (b) on the plot above)
where is the number of Gaussian functions and is the number of coordinates the potential acts upon. The coordinates are defined in the ICONST file by setting the status to 5. The parameters of the potentials, , , and are defined in the PENALTYPOT file.
This type of bias potential is primarily intended for use in metadynamics, but since Gaussians can be used as basis functions for more general shapes, they can also be used to prepare various atypically-shaped bias potentials.

Examples of usage

Let us consider the nucleophile substitution reaction of CHCl with Cl. The reactant is a weak van-der-Waals complex. The corresponding POSCAR file reads

vdW complex CH3Cl...Cl 
1.00000000000000
12.0000000000000000    0.0000000000000000    0.0000000000000000
0.0000000000000000    12.0000000000000000    0.0000000000000000
0.0000000000000000    0.0000000000000000    12.0000000000000000
C H Cl
1 3 2
cart
5.91331371  7.11364924  5.78037960
5.81982231  8.15982106  5.46969017
4.92222130  6.65954232  5.88978969
6.47810398  7.03808479  6.71586385
4.32824726  8.75151396  7.80743202
6.84157897  6.18713289  4.46842049

Due to the weak interactions between CHCl and Cl, the complex can collapse at high temperatures. This can be avoided by setting an upper bound for the length of the non-bonding Cl...C interactions. This can be conveniently achieved by using a Fermi-like step-shaped bias potential. To this end, we need to define the Cl...C distance, i.e., the distance between the atoms 1 and 5, as a coordinate with status 4 in the ICONST file:

R 1 5 4   

Next, we need to specify the bias potential parameters FBIAS_A, FBIAS_D, and FBIAS_R0 in the INCAR file. Since the bias potential acts only on one internal coordinate (), we need to provide only one number for each of the tags. The setting

FBIAS_A  = 1
FBIAS_D  = 50
FBIAS_R0 = 3.5
  

ensures that repulsive bias forces steeply increase when the C...Cl distance is increased beyond about . This causes a shortening of the distance in the next MD step. Notice that the bias force is essentially negligible for distances below . A careful adjustment of FBIAS_A and FBIAS_D is needed to ensure that (i) the bias force is large enough to effectively limit the value of , and (ii) the interval of values for which the bias forces are significant is broad enough to avoid overcoming via random fluctuations. A suitable setting can be found by noting that the maximal bias force of is exerted on the system at the point . This can be seen by inspecting the analytical expression for the potential.

How to

Biased MD run using the Andersen thermostat and a Gaussian bias potential

For a biased MD run with Andersen thermostat, one has to:
  1. Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW
  2. Set MDALGO=1 (MDALGO=11 in VASP 5.x), and choose an appropriate setting for ANDERSEN_PROB
  3. To avoid updating of the bias potential, set HILLS_BIN=NSW
  4. Define collective variables in the ICONST file, and set the STATUS parameter for the collective variables to 5
  5. Define the bias potential in the PENALTYPOT file

Biased MD run using the Nose-Hoover thermostat and a Gaussian bias potential

For a biased MD run with Nose-Hoover thermostat, one has to:
  1. Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW
  2. Set MDALGO=2 (MDALGO=21 in VASP 5.x), and choose an appropriate setting for SMASS
  3. To avoid updating the bias potential, set HILLS_BIN=NSW
  4. Define collective variables in the ICONST file, and set the STATUS parameter for the collective variables to 5
  5. Define the bias potential in the PENALTYPOT file

The values of all collective variables for each MD step are listed in the REPORT file. Check the lines after the string Metadynamics.

References