$\phantom{\rule{0.3em}{0ex}}$

COLLECTIVE VARIABLES MODULE

Reference manual for LAMMPS

Code version: 2018-12-21

Alejandro Bernardin, Jeﬀrey R. Comer, Giacomo Fiorin, Haohao Fu, Jérôme Hénin, Axel Kohlmeyer, Fabrizio Marinelli, Joshua V. Vermaas, Andrew D. White

### 1 Overview

In molecular dynamics simulations, it is often useful to reduce the large number of degrees of freedom of a physical system into few parameters whose statistical distributions can be analyzed individually, or used to deﬁne biasing potentials to alter the dynamics of the system in a controlled manner. These have been called ‘order parameters’, ‘collective variables’, ‘(surrogate) reaction coordinates’, and many other terms.

Here we use primarily the term ‘collective variable’, often shortened to colvar, to indicate any diﬀerentiable function of atomic Cartesian coordinates, ${\text{}x\text{}}_{i}$, with $i$ between $1$ and $N$, the total number of atoms:

 $\xi \left(t\right)\phantom{\rule{3.04074pt}{0ex}}=\xi \left(\text{}X\text{}\left(t\right)\right)\phantom{\rule{3.04074pt}{0ex}}=\xi \left({\text{}x\text{}}_{i}\left(t\right),{\text{}x\text{}}_{j}\left(t\right),{\text{}x\text{}}_{k}\left(t\right),\dots \right)\phantom{\rule{3.04074pt}{0ex}},\phantom{\rule{3.04074pt}{0ex}}\phantom{\rule{3.04074pt}{0ex}}1\le i,j,k\dots \le N$ (1)

This manual documents the collective variables module (Colvars), a software that provides an implementation for the functions $\xi \left(\text{}X\text{}\right)$ with a focus on ﬂexibility, robustness and high performance. The module is designed to perform multiple tasks concurrently during or after a simulation, the most common of which are:

• apply restraints or biasing potentials to multiple variables, tailored on the system by choosing from a wide set of basis functions, without limitations on their number or on the number of atoms involved;
• calculate potentials of mean force (PMFs) along any set of variables, using diﬀerent enhanced sampling methods, such as Adaptive Biasing Force (ABF), metadynamics, steered MD and umbrella sampling; variants of these methods that make use of an ensemble of replicas are supported as well;
• calculate statistical properties of the variables, such as running averages and standard deviations, correlation functions of pairs of variables, and multidimensional histograms: this can be done either at run-time without the need to save very large trajectory ﬁles, or after a simulation has been completed using VMD and the cv command.

Detailed explanations of the design of the Colvars module are provided in reference [1]. Please cite this reference whenever publishing work that makes use of this module.

### 2 A crash course

Suppose that we want to run a steered MD experiment where a small molecule is pulled away from a protein binding site. In Colvars terms, this is done by applying a moving restraint to the distance between the two objects. The conﬁguration will contain two blocks, one deﬁning the distance variable (see section 4 and 4.2.1), and the other the moving harmonic restraint (6.4).

indexFile index.ndx
colvar {
name dist
distance {
group1 { atomNumbersRange 42-55 }
group2 { indexGroup C-alpha_15-30 }
}
}

harmonic {
colvars dist
forceConstant 20.0
centers 4.0         # initial distance
targetCenters 15.0  # final distance
targetNumSteps 500000
}

Reading this input in plain English: the variable here named dist consists in a distance function between the centers of two groups: the ligand (atoms 42 to 55) and the $\alpha$-carbon atoms of residues 15 to 30 in the protein (selected from the index group “C-alpha_15-30”). To the “dist” variable, we apply a harmonic potential of force constant 20 (energy unit)/(length unit)${}^{2}$, initially centered around a value of 4 (length unit), which will increase to 15 (length unit) over 500,000 simulation steps.

The atom selection keywords is detailed in section 5. If the selection is too complex to implement only via internal keywords, an external index ﬁle may be created following the NDX format used in GROMACS:
http://manual.gromacs.org/documentation/current/user-guide/file-formats.html
or by using the group2ndx LAMMPS command.

### 3 General parameters

Here, we document the syntax of the commands and parameters used to set up and use the Colvars module in LAMMPS. One of these parameters is the conﬁguration ﬁle or the conﬁguration text for the module itself, whose syntax is described in 3.2 and in the following sections.

#### 3.1 LAMMPS keywords

To enable a Colvars-based calculation, the following line must be added to the LAMMPS conﬁguration ﬁle:

fix ID all colvars conﬁgﬁle keyword value pairs ...

where ID is a string that uniquely identiﬁes this ﬁx command inside a LAMMPS script, conﬁgﬁle is the name of the conﬁguration ﬁle for the Colvars module, followed by one or more of the following optional keywords with their corresponding arguments:

• input $⟨\phantom{\rule{0.3em}{0ex}}$Name or preﬁx of the input state ﬁle$\phantom{\rule{0.3em}{0ex}}⟩$
Context: Keyword of the fix colvars command
Acceptable values: string
Description: If a value is provided, it is interpreted as either the name of the input state ﬁle, or as the preﬁx of the ﬁle named input.colvars.state. This ﬁle contains information needed to continue a previous collective variables-based calculation, including the number of the last computed step (useful for time-dependent biases). The same information is also stored in the binary restart ﬁles written by LAMMPS, so this option is not needed when continuing a calculation from a LAMMPS restart.
• output $⟨\phantom{\rule{0.3em}{0ex}}$Preﬁx of the output state ﬁle$\phantom{\rule{0.3em}{0ex}}⟩$
Context: Keyword of the fix colvars command
Acceptable values: string
Default value: “out”
Description: If a value is provided, it is interpreted as the preﬁx to all output ﬁles that will be written by the Colvars module (see 3.4).
• unwrap $⟨\phantom{\rule{0.3em}{0ex}}$Whether to unwrap coordinates passed to the Colvars module$\phantom{\rule{0.3em}{0ex}}⟩$
Context: keyword of the fix colvars command
Acceptable values: “yes” or “no”
Default value: “yes”
Description: This keyword controls whether wrapped or unwrapped coordinates are passed to the Colvars module for calculation of the collective variables and of the resulting forces. The default is to use the image ﬂags to reconstruct the absolute atom positions: under this convention, centers of mass and centers of geometry are calculated as a weighted vector sum (see 5.3). Setting this to no will use the current local coordinates that are wrapped back into the simulation cell at each re-neighboring instead.
• seed $⟨\phantom{\rule{0.3em}{0ex}}$Seed for the random number generator$\phantom{\rule{0.3em}{0ex}}⟩$
Context: Keyword of the fix colvars command
Acceptable values: positive integer
Default value: 1966
Description: If deﬁned, the value of this keyword is provided as seed to the random number generator. This is only meaningful when the extendedLangevinDamping keyword is used (see 4.15).
• tstat $⟨\phantom{\rule{0.3em}{0ex}}$Thermostating ﬁx$\phantom{\rule{0.3em}{0ex}}⟩$
Context: Keyword of the fix colvars command
Acceptable values: string
Default value: NULL
Description: This keyword provides the ID of an applicable thermostating ﬁx command. This will be used to provide the Colvars module with the current thermostat target temperature when using a method that needs this information.

#### 3.2 Conﬁguration syntax

The syntax of the Colvars conﬁguration is “keyword value”, where the keyword and its value are separated by any white space. The following rules apply:

• keywords are case-insensitive (upperBoundary is the same as upperboundary and UPPERBOUNDARY): their string values are however case-sensitive (e.g. ﬁle names);
• a long value or a list of multiple values can be distributed across multiple lines by using curly braces, “{” and “}”: the opening brace “{” must occur on the same line as the keyword, following a space character or other white space; the closing brace “}” can be at any position after that;
• many keywords are nested, and are only meaningful within a speciﬁc context: for every keyword documented in the following, the “parent” keyword that deﬁnes such context is also indicated;
• if a keyword requiring a boolean value (yes|on|true or no|off|false) is provided without an explicit value, it defaults to ‘yes|on|true’; for example, ‘outputAppliedForce’ may be used as shorthand for ‘outputAppliedForce on’;
• the hash character # indicates a comment: all text in the same line following this character will be ignored.

#### 3.3 Global keywords

The following keywords are available in the global context of the Colvars conﬁguration, i.e. they are not nested inside other keywords:

• colvarsTrajFrequency $⟨\phantom{\rule{0.3em}{0ex}}$Colvar value trajectory frequency$\phantom{\rule{0.3em}{0ex}}⟩$
Context: global
Acceptable values: positive integer
Default value: 100
Description: The values of each colvar (and of other related quantities, if requested) are written to the ﬁle output.colvars.traj every these many steps throughout the simulation. If the value is 0, such trajectory ﬁle is not written. For optimization the output is buﬀered, and synchronized with the disk only when the restart ﬁle is being written.
• colvarsRestartFrequency $⟨\phantom{\rule{0.3em}{0ex}}$Colvar module restart frequency$\phantom{\rule{0.3em}{0ex}}⟩$
Context: global
Acceptable values: positive integer
Default value: restartFreq
Description: Allows to choose a diﬀerent restart frequency for the Colvars module. Redeﬁning it may be useful to trace the time evolution of those few properties which are not written to the trajectory ﬁle for reasons of disk space.
• indexFile $⟨\phantom{\rule{0.3em}{0ex}}$Index ﬁle for atom selection (GROMACS “ndx” format)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: global
Acceptable values: UNIX ﬁlename
Description: This option reads an index ﬁle (usually with a .ndx extension) as produced by the make_ndx tool of GROMACS. This keyword may be repeated to load multiple index ﬁles: the same group name cannot appear in multiple index ﬁles. In LAMMPS, the group2ndx command can be used to generate such ﬁle from existing groups. Note that the Colvars module reads the indices of atoms from the index ﬁle: therefore, the LAMMPS groups do not need to remain active during the simulation, and could be deleted right after issuing group2ndx. The names of index groups contained in this ﬁle can then be used to deﬁne atom groups with the indexGroup keyword. Other supported methods to select atoms are described in 5.
• smp $⟨\phantom{\rule{0.3em}{0ex}}$Whether SMP parallelism should be used$\phantom{\rule{0.3em}{0ex}}⟩$
Context: global
Acceptable values: boolean
Default value: on
Description: If this ﬂag is enabled (default), SMP parallelism over threads will be used to compute variables and biases, provided that this is supported by the LAMMPS build in use.

To illustrate the ﬂexibility of the Colvars module, a non-trivial setup is represented in Figure 1. The corresponding conﬁguration is given below. The options within the colvar blocks are described in 4, those within the harmonic and histogram blocks in 6. Note: except colvar, none of the keywords shown is mandatory.

colvar {
# difference of two distances
name d
width 0.2 # 0.2 Å of estimated fluctuation width
distance {
componentCoeff 1.0
group1 { atomNumbers 1 2 }
group2 { atomNumbers 3 4 5 }
}
distance {
componentCoeff -1.0
group1 { atomNumbers 7 }
group2 { atomNumbers 8 9 10 }
}
}

colvar {
name c
coordNum {
cutoff 6.0
group1 { atomNumbersRange 1-10 }
group2 { atomNumbersRange 11-20 }
}
}

harmonic {
colvars d c
centers 3.0 4.0
forceConstant 5.0
}

histogram {
colvars c
}

Section 4 explains how to deﬁne a colvar and its behavior, regardless of its speciﬁc functional form. To deﬁne colvars that are appropriate to a speciﬁc physical system, Section 5 documents how to select atoms, and section 4 lists all of the available functional forms, which we call “colvar components”. Finally, section 6 lists the available methods and algorithms to perform biased simulations and multidimensional analysis of colvars.

#### 3.4 Output ﬁles

During a simulation with collective variables deﬁned, the following three output ﬁles are written:

• A state ﬁle, named output.colvars.state; this ﬁle is in ASCII (plain text) format. This ﬁle is written at the end of the speciﬁed run.
This is the only Colvars output ﬁle needed to continue a simulation.
• If the parameter colvarsRestartFrequency is larger than zero, a restart ﬁle is written every that many steps: this ﬁle is fully equivalent to the ﬁnal state ﬁle. The name of this ﬁle is restart.colvars.state.
• If the parameter colvarsTrajFrequency is greater than 0 (default: 100), a trajectory ﬁle is written during the simulation: its name is output.colvars.traj; unlike the state ﬁle, it is not needed to restart a simulation, but can be used later for post-processing and analysis.

Other output ﬁles may also be written by speciﬁc methods, e.g. the ABF or metadynamics methods (6.1, 6.3). Like the trajectory ﬁle, they are needed only for analyzing, not continuing a simulation. All such ﬁles’ names also begin with the preﬁx output.

### 4 Deﬁning collective variables

A collective variable is deﬁned by the keyword colvar followed by its conﬁguration options contained within curly braces:

colvar {
name xi
$<$other options$>$
function_name {
$<$parameters$>$
$<$atom selection$>$
}
}

There are multiple ways of deﬁning a variable:

• The simplest and most common way way is using one of the precompiled functions (here called “components”), which are listed in section 4.1. For example, using the keyword rmsd (section 4.5.1) deﬁnes the variable as the root mean squared deviation (RMSD) of the selected atoms.
• A new variable may also be constructed as a linear or polynomial combination of the components listed in section 4.1 (see 4.11 for details).
• A user-deﬁned mathematical function of the existing components (see list in section 4.1), or of the atomic coordinates directly (see the cartesian keyword in 4.7.1). The function is deﬁned through the keyword customFunction (see 4.12 for details).

Choosing a component (function) is the only parameter strictly required to deﬁne a collective variable. It is also highly recommended to specify a name for the variable:

• name $⟨\phantom{\rule{0.3em}{0ex}}$Name of this colvar$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: string
Default value: colvar” + numeric id
Description: The name is an unique case-sensitive string which allows the Colvars module to identify this colvar unambiguously; it is also used in the trajectory ﬁle to label to the columns corresponding to this colvar.

#### 4.1 Choosing a function

In this context, the function that computes a colvar is called a component. A component’s choice and deﬁnition consists of including in the variable’s conﬁguration a keyword indicating the type of function (e.g. rmsd), followed by a deﬁnition block specifying the atoms involved (see 5) and any additional parameters (cutoﬀs, “reference” values, …). At least one component must be chosen to deﬁne a variable: if none of the keywords listed below is found, an error is raised.

The following components implement functions with a scalar value (i.e. a real number):

• distance: distance between two groups;
• distanceZ: projection of a distance vector on an axis;
• distanceXY: projection of a distance vector on a plane;
• distanceInv: mean distance between two groups of atoms (e.g. NOE-based distance);
• angle: angle between three groups;
• dihedral: torsional (dihedral) angle between four groups;
• dipoleAngle: angle between two groups and dipole of a third group;
• dipoleMagnitude magnitude of the dipole of a group of atoms;
• polarTheta: polar angle of a group in spherical coordinates;
• polarPhi: azimuthal angle of a group in spherical coordinates;
• coordNum: coordination number between two groups;
• selfCoordNum: coordination number of atoms within a group;
• hBond: hydrogen bond between two atoms;
• rmsd: root mean square deviation (RMSD) from a set of reference coordinates;
• eigenvector: projection of the atomic coordinates on a vector;
• orientationAngle: angle of the best-ﬁt rotation from a set of reference coordinates;
• orientationProj: cosine of orientationProj;
• spinAngle: projection orthogonal to an axis of the best-ﬁt rotation from a set of reference coordinates;
• tilt: projection on an axis of the best-ﬁt rotation from a set of reference coordinates;
• gyration: radius of gyration of a group of atoms;
• inertia: moment of inertia of a group of atoms;
• inertiaZ: moment of inertia of a group of atoms around a chosen axis;

Some components do not return scalar, but vector values:

• distanceVec: distance vector between two groups (length: 3);
• distanceDir: unit vector parallel to distanceVec (length: 3);
• cartesian: vector of atomic Cartesian coordinates (length: $N$ times the number of Cartesian components requested, X, Y or Z);
• distancePairs: vector of mutual distances (length: ${N}_{\mathrm{1}}×{N}_{\mathrm{2}}$);
• orientation: best-ﬁt rotation, expressed as a unit quaternion (length: 4).

The types of components used in a colvar (scalar or not) determine the properties of that colvar, and particularly which biasing or analysis methods can be applied.

What if “X” is not listed? If a function type is not available on this list, it may be possible to deﬁne it as a polynomial superposition of existing ones (see 4.11), a custom function (see 4.12).

In the rest of this section, all available component types are listed, along with their physical units and the ranges of values, if limited. Such limiting values can be used to deﬁne lowerBoundary and upperBoundary in the parent colvar.

For each type of component, the available conﬁgurations keywords are listed: when two components share certain keywords, the second component references to the documentation of the ﬁrst one that uses that keyword. The very few keywords that are available for all types of components are listed in a separate section 4.8.

#### 4.2 Distances

##### 4.2.1 distance: center-of-mass distance between two groups.

The distance {...} block deﬁnes a distance component between the two atom groups, group1 and group2.

• group1 $⟨\phantom{\rule{0.3em}{0ex}}$First group of atoms$\phantom{\rule{0.3em}{0ex}}⟩$
Context: distance
Acceptable values: Block group1 {...}
Description: First group of atoms.
• group2: analogous to group1
• forceNoPBC $⟨\phantom{\rule{0.3em}{0ex}}$Calculate absolute rather than minimum-image distance?$\phantom{\rule{0.3em}{0ex}}⟩$
Context: distance
Acceptable values: boolean
Default value: no
Description: By default, in calculations with periodic boundary conditions, the distance component returns the distance according to the minimum-image convention. If this parameter is set to yes, PBC will be ignored and the distance between the coordinates as maintained internally will be used. This is only useful in a limited number of special cases, e.g. to describe the distance between remote points of a single macromolecule, which cannot be split across periodic cell boundaries, and for which the minimum-image distance might give the wrong result because of a relatively small periodic cell.
• oneSiteTotalForce $⟨\phantom{\rule{0.3em}{0ex}}$Measure total force on group 1 only?$\phantom{\rule{0.3em}{0ex}}⟩$
Context: angle, dipoleAngle, dihedral
Acceptable values: boolean
Default value: no
Description: If this is set to yes, the total force is measured along a vector ﬁeld (see equation (15) in section 6.1) that only involves atoms of group1. This option is only useful for ABF, or custom biases that compute total forces. See section 6.1 for details.

The value returned is a positive number (in Å), ranging from $0$ to the largest possible interatomic distance within the chosen boundary conditions (with PBCs, the minimum image convention is used unless the forceNoPBC option is set).

##### 4.2.2 distanceZ: projection of a distance vector on an axis.

The distanceZ {...} block deﬁnes a distance projection component, which can be seen as measuring the distance between two groups projected onto an axis, or the position of a group along such an axis. The axis can be deﬁned using either one reference group and a constant vector, or dynamically based on two reference groups. One of the groups can be set to a dummy atom to allow the use of an absolute Cartesian coordinate.

• main $⟨\phantom{\rule{0.3em}{0ex}}$Main group of atoms$\phantom{\rule{0.3em}{0ex}}⟩$
Context: distanceZ
Acceptable values: Block main {...}
Description: Group of atoms whose position $\text{}r\text{}$ is measured.
• ref $⟨\phantom{\rule{0.3em}{0ex}}$Reference group of atoms$\phantom{\rule{0.3em}{0ex}}⟩$
Context: distanceZ
Acceptable values: Block ref {...}
Description: Reference group of atoms. The position of its center of mass is noted ${\text{}r\text{}}_{1}$ below.
• ref2 $⟨\phantom{\rule{0.3em}{0ex}}$Secondary reference group$\phantom{\rule{0.3em}{0ex}}⟩$
Context: distanceZ
Acceptable values: Block ref2 {...}
Default value: none
Description: Optional group of reference atoms, whose position ${\text{}r\text{}}_{2}$ can be used to deﬁne a dynamic projection axis: $\text{}e\text{}={\left(\parallel {\text{}r\text{}}_{2}-{\text{}r\text{}}_{1}\parallel \right)}^{-1}×\left({\text{}r\text{}}_{2}-{\text{}r\text{}}_{1}\right)$. In this case, the origin is ${\text{}r\text{}}_{m}=1∕2\left({\text{}r\text{}}_{1}+{\text{}r\text{}}_{2}\right)$, and the value of the component is $\text{}e\text{}\cdot \left(\text{}r\text{}-{\text{}r\text{}}_{m}\right)$.
• axis $⟨\phantom{\rule{0.3em}{0ex}}$Projection axis (Å)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: distanceZ
Acceptable values: (x, y, z) triplet
Default value: (0.0, 0.0, 1.0)
Description: The three components of this vector deﬁne a projection axis $\text{}e\text{}$ for the distance vector $\text{}r\text{}-{\text{}r\text{}}_{1}$ joining the centers of groups ref and main. The value of the component is then $\text{}e\text{}\cdot \left(\text{}r\text{}-{\text{}r\text{}}_{1}\right)$. The vector should be written as three components separated by commas and enclosed in parentheses.
• forceNoPBC: see deﬁnition of forceNoPBC in sec. 4.2.1 (distance component)
• oneSiteTotalForce: see deﬁnition of oneSiteTotalForce in sec. 4.2.1 (distance component)

This component returns a number (in Å) whose range is determined by the chosen boundary conditions. For instance, if the $z$ axis is used in a simulation with periodic boundaries, the returned value ranges between $-{b}_{z}∕2$ and ${b}_{z}∕2$, where ${b}_{z}$ is the box length along $z$ (this behavior is disabled if forceNoPBC is set).

##### 4.2.3 distanceXY: modulus of the projection of a distance vector on a plane.

The distanceXY {...} block deﬁnes a distance projected on a plane, and accepts the same keywords as the component distanceZ, i.e. main, ref, either ref2 or axis, and oneSiteTotalForce. It returns the norm of the projection of the distance vector between main and ref onto the plane orthogonal to the axis. The axis is deﬁned using the axis parameter or as the vector joining ref and ref2 (see distanceZ above).

• main: see deﬁnition of main in sec. 4.2.2 (distanceZ component)
• ref: see deﬁnition of ref in sec. 4.2.2 (distanceZ component)
• ref2: see deﬁnition of ref2 in sec. 4.2.2 (distanceZ component)
• axis: see deﬁnition of axis in sec. 4.2.2 (distanceZ component)
• forceNoPBC: see deﬁnition of forceNoPBC in sec. 4.2.1 (distance component)
• oneSiteTotalForce: see deﬁnition of oneSiteTotalForce in sec. 4.2.1 (distance component)

##### 4.2.4 distanceVec: distance vector between two groups.

The distanceVec {...} block deﬁnes a distance vector component, which accepts the same keywords as the component distance: group1, group2, and forceNoPBC. Its value is the 3-vector joining the centers of mass of group1 and group2.

• group1: see deﬁnition of group1 in sec. 4.2.1 (distance component)
• group2: analogous to group1
• forceNoPBC: see deﬁnition of forceNoPBC in sec. 4.2.1 (distance component)
• oneSiteTotalForce: see deﬁnition of oneSiteTotalForce in sec. 4.2.1 (distance component)

##### 4.2.5 distanceDir: distance unit vector between two groups.

The distanceDir {...} block deﬁnes a distance unit vector component, which accepts the same keywords as the component distance: group1, group2, and forceNoPBC. It returns a 3-dimensional unit vector $\mathbf{d}=\left({d}_{x},{d}_{y},{d}_{z}\right)$, with $|\mathbf{d}|=1$.

• group1: see deﬁnition of group1 in sec. 4.2.1 (distance component)
• group2: analogous to group1
• forceNoPBC: see deﬁnition of forceNoPBC in sec. 4.2.1 (distance component)
• oneSiteTotalForce: see deﬁnition of oneSiteTotalForce in sec. 4.2.1 (distance component)

##### 4.2.6 distanceInv: mean distance between two groups of atoms.

The distanceInv {...} block deﬁnes a generalized mean distance between two groups of atoms 1 and 2, weighted with exponent $1∕n$:

 ${d}_{\mathrm{1,2}}^{\left[n\right]}\phantom{\rule{3.04074pt}{0ex}}=\phantom{\rule{3.04074pt}{0ex}}{\left(\frac{1}{{N}_{\mathrm{1}}{N}_{\mathrm{2}}}\sum _{i,j}{\left(\frac{1}{\parallel {\mathbf{d}}^{ij}\parallel }\right)}^{n}\right)}^{-1∕n}$ (2)

where $\parallel {\mathbf{d}}^{ij}\parallel$ is the distance between atoms $i$ and $j$ in groups 1 and 2 respectively, and $n$ is an even integer.

• group1: see deﬁnition of group1 in sec. 4.2.1 (distance component)
• group2: analogous to group1
• oneSiteTotalForce: see deﬁnition of oneSiteTotalForce in sec. 4.2.1 (distance component)
• exponent $⟨\phantom{\rule{0.3em}{0ex}}$Exponent $n$ in equation 2$\phantom{\rule{0.3em}{0ex}}⟩$
Context: distanceInv
Acceptable values: positive even integer
Default value: 6
Description: Deﬁnes the exponent to which the individual distances are elevated before averaging. The default value of 6 is useful for example to applying restraints based on NOE-measured distances.

This component returns a number in Å, ranging from $0$ to the largest possible distance within the chosen boundary conditions.

#### 4.3 Angles

##### 4.3.1 angle: angle between three groups.

The angle {...} block deﬁnes an angle, and contains the three blocks group1, group2 and group3, deﬁning the three groups. It returns an angle (in degrees) within the interval $\left[0:180\right]$.

• group1: see deﬁnition of group1 in sec. 4.2.1 (distance component)
• group2: analogous to group1
• group3: analogous to group1
• forceNoPBC: see deﬁnition of forceNoPBC in sec. 4.2.1 (distance component)
• oneSiteTotalForce: see deﬁnition of oneSiteTotalForce in sec. 4.2.1 (distance component)

##### 4.3.2 dipoleAngle: angle between two groups and dipole of a third group.

The dipoleAngle {...} block deﬁnes an angle, and contains the three blocks group1, group2 and group3, deﬁning the three groups, being group1 the group where dipole is calculated. It returns an angle (in degrees) within the interval $\left[0:180\right]$.

• group1: see deﬁnition of group1 in sec. 4.2.1 (distance component)
• group2: analogous to group1
• group3: analogous to group1
• forceNoPBC: see deﬁnition of forceNoPBC in sec. 4.2.1 (distance component)
• oneSiteTotalForce: see deﬁnition of oneSiteTotalForce in sec. 4.2.1 (distance component)

##### 4.3.3 dihedral: torsional angle between four groups.

The dihedral {...} block deﬁnes a torsional angle, and contains the blocks group1, group2, group3 and group4, deﬁning the four groups. It returns an angle (in degrees) within the interval $\left[-180:180\right]$. The Colvars module calculates all the distances between two angles taking into account periodicity. For instance, reference values for restraints or range boundaries can be deﬁned by using any real number of choice.

• group1: see deﬁnition of group1 in sec. 4.2.1 (distance component)
• group2: analogous to group1
• group3: analogous to group1
• group4: analogous to group1
• forceNoPBC: see deﬁnition of forceNoPBC in sec. 4.2.1 (distance component)
• oneSiteTotalForce: see deﬁnition of oneSiteTotalForce in sec. 4.2.1 (distance component)

##### 4.3.4 polarTheta: polar angle in spherical coordinates.

The polarTheta {...} block deﬁnes the polar angle in spherical coordinates, for the center of mass of a group of atoms described by the block atoms. It returns an angle (in degrees) within the interval $\left[0:180\right]$. To obtain spherical coordinates in a frame of reference tied to another group of atoms, use the fittingGroup (5.2) option within the atoms block. An example is provided in ﬁle examples/11_polar_angles.in of the Colvars public repository.

• atoms $⟨\phantom{\rule{0.3em}{0ex}}$Atom group$\phantom{\rule{0.3em}{0ex}}⟩$
Context: polarPhi
Acceptable values: atoms {...} block
Description: Deﬁnes the group of atoms for the COM of which the angle should be calculated.

##### 4.3.5 polarPhi: azimuthal angle in spherical coordinates.

The polarPhi {...} block deﬁnes the azimuthal angle in spherical coordinates, for the center of mass of a group of atoms described by the block atoms. It returns an angle (in degrees) within the interval $\left[-180:180\right]$. The Colvars module calculates all the distances between two angles taking into account periodicity. For instance, reference values for restraints or range boundaries can be deﬁned by using any real number of choice. To obtain spherical coordinates in a frame of reference tied to another group of atoms, use the fittingGroup (5.2) option within the atoms block. An example is provided in ﬁle examples/11_polar_angles.in of the Colvars public repository.

• atoms $⟨\phantom{\rule{0.3em}{0ex}}$Atom group$\phantom{\rule{0.3em}{0ex}}⟩$
Context: polarPhi
Acceptable values: atoms {...} block
Description: Deﬁnes the group of atoms for the COM of which the angle should be calculated.

#### 4.4 Contacts

##### 4.4.1 coordNum: coordination number between two groups.

The coordNum {...} block deﬁnes a coordination number (or number of contacts), which calculates the function $\left(1-{\left(d∕{d}_{0}\right)}^{n}\right)∕\left(1-{\left(d∕{d}_{0}\right)}^{m}\right)$, where ${d}_{0}$ is the “cutoﬀ” distance, and $n$ and $m$ are exponents that can control its long range behavior and stiﬀness [2]. This function is summed over all pairs of atoms in group1 and group2:

 $C\left(\mathtt{group1},\mathtt{group2}\right)\phantom{\rule{3.04074pt}{0ex}}=\phantom{\rule{3.04074pt}{0ex}}\sum _{i\in \mathtt{group1}}\sum _{j\in \mathtt{group2}}\frac{1-{\left(|{\mathbf{x}}_{i}-{\mathbf{x}}_{j}|∕{d}_{0}\right)}^{n}}{1-{\left(|{\mathbf{x}}_{i}-{\mathbf{x}}_{j}|∕{d}_{0}\right)}^{m}}$ (3)

• group1: see deﬁnition of group1 in sec. 4.2.1 (distance component)
• group2: analogous to group1
• cutoff $⟨\phantom{\rule{0.3em}{0ex}}$“Interaction” distance (Å)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: coordNum
Acceptable values: positive decimal
Default value: 4.0
Description: This number deﬁnes the switching distance to deﬁne an interatomic contact: for $d\ll {d}_{0}$, the switching function $\left(1-{\left(d∕{d}_{0}\right)}^{n}\right)∕\left(1-{\left(d∕{d}_{0}\right)}^{m}\right)$ is close to 1, at $d={d}_{0}$ it has a value of $n∕m$ ($1∕2$ with the default $n$ and $m$), and at $d\gg {d}_{0}$ it goes to zero approximately like ${d}^{m-n}$. Hence, for a proper behavior, $m$ must be larger than $n$.
• cutoff3 $⟨\phantom{\rule{0.3em}{0ex}}$Reference distance vector (Å)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: coordNum
Acceptable values: (x, y, z)” triplet of positive decimals
Default value: (4.0, 4.0, 4.0)
Description: The three components of this vector deﬁne three diﬀerent cutoﬀs ${d}_{0}$ for each direction. This option is mutually exclusive with cutoff.
• expNumer $⟨\phantom{\rule{0.3em}{0ex}}$Numerator exponent$\phantom{\rule{0.3em}{0ex}}⟩$
Context: coordNum
Acceptable values: positive even integer
Default value: 6
Description: This number deﬁnes the $n$ exponent for the switching function.
• expDenom $⟨\phantom{\rule{0.3em}{0ex}}$Denominator exponent$\phantom{\rule{0.3em}{0ex}}⟩$
Context: coordNum
Acceptable values: positive even integer
Default value: 12
Description: This number deﬁnes the $m$ exponent for the switching function.
• group2CenterOnly $⟨\phantom{\rule{0.3em}{0ex}}$Use only group2’s center of mass$\phantom{\rule{0.3em}{0ex}}⟩$
Context: coordNum
Acceptable values: boolean
Default value: off
Description: If this option is on, only contacts between each atoms in group1 and the center of mass of group2 are calculated (by default, the sum extends over all pairs of atoms in group1 and group2). If group2 is a dummyAtom, this option is set to yes by default.
• tolerance $⟨\phantom{\rule{0.3em}{0ex}}$Pairlist control$\phantom{\rule{0.3em}{0ex}}⟩$
Context: coordNum
Acceptable values: decimal
Default value: 0.0
Description: This controls the pairlist feature, dictating the minimum value for each summation element in Eq. 3 such that the pair that contributed the summation element is included in subsequent simulation timesteps until the next pairlist recalculation. For most applications, this value should be small (eg. 0.001) to avoid missing important contributions to the overall sum. Higher values will improve performance by reducing the number of pairs that contribute to the sum. Values above 1 will exclude all possible pair interactions. Similarly, values below 0 will never exclude a pair from consideration. To ensure continuous forces, Eq. 3 is further modiﬁed by subtracting the tolerance and then rescaling so that each pair covers the range $\left[0,1\right]$.
• pairListFrequency $⟨\phantom{\rule{0.3em}{0ex}}$Pairlist regeneration frequency$\phantom{\rule{0.3em}{0ex}}⟩$
Context: coordNum
Acceptable values: positive integer
Default value: 100
Description: This controls the pairlist feature, dictating how many steps are taken between regenerating pairlists if the tolerance is greater than 0.

This component returns a dimensionless number, which ranges from approximately 0 (all interatomic distances are much larger than the cutoﬀ) to ${N}_{\mathtt{group1}}×{N}_{\mathtt{group2}}$ (all distances are less than the cutoﬀ), or ${N}_{\mathtt{group1}}$ if group2CenterOnly is used. For performance reasons, at least one of group1 and group2 should be of limited size or group2CenterOnly should be used: the cost of the loop over all pairs grows as ${N}_{\mathtt{group1}}×{N}_{\mathtt{group2}}$. Setting $\mathtt{tolerance}>0$ ameliorates this to some degree, although every pair is still checked to regenerate the pairlist.

##### 4.4.2 selfCoordNum: coordination number between atoms within a group.

The selfCoordNum {...} block deﬁnes a coordination number similarly to the component coordNum, but the function is summed over atom pairs within group1:

 $C\left(\mathtt{group1}\right)\phantom{\rule{3.04074pt}{0ex}}=\phantom{\rule{3.04074pt}{0ex}}\sum _{i\in \mathtt{group1}}\sum _{j>i}\frac{1-{\left(|{\mathbf{x}}_{i}-{\mathbf{x}}_{j}|∕{d}_{0}\right)}^{n}}{1-{\left(|{\mathbf{x}}_{i}-{\mathbf{x}}_{j}|∕{d}_{0}\right)}^{m}}$ (4)

The keywords accepted by selfCoordNum are a subset of those accepted by coordNum, namely group1 (here deﬁning all of the atoms to be considered), cutoff, expNumer, and expDenom.

• group1: see deﬁnition of group1 in sec. 4.4.1 (coordNum component)
• cutoff: see deﬁnition of cutoff in sec. 4.4.1 (coordNum component)
• cutoff3: see deﬁnition of cutoff3 in sec. 4.4.1 (coordNum component)
• expNumer: see deﬁnition of expNumer in sec. 4.4.1 (coordNum component)
• expDenom: see deﬁnition of expDenom in sec. 4.4.1 (coordNum component)
• tolerance: see deﬁnition of tolerance in sec. 4.4.1 (coordNum component)
• pairListFrequency: see deﬁnition of pairListFrequency in sec. 4.4.1 (coordNum component)

This component returns a dimensionless number, which ranges from approximately 0 (all interatomic distances much larger than the cutoﬀ) to ${N}_{\mathtt{group1}}×\left({N}_{\mathtt{group1}}-1\right)∕2$ (all distances within the cutoﬀ). For performance reasons, group1 should be of limited size, because the cost of the loop over all pairs grows as ${N}_{\mathtt{group1}}^{2}$.

##### 4.4.3 hBond: hydrogen bond between two atoms.

The hBond {...} block deﬁnes a hydrogen bond, implemented as a coordination number (eq. 3) between the donor and the acceptor atoms. Therefore, it accepts the same options cutoff (with a diﬀerent default value of 3.3 Å), expNumer (with a default value of 6) and expDenom (with a default value of 8). Unlike coordNum, it requires two atom numbers, acceptor and donor, to be deﬁned. It returns an adimensional number, with values between 0 (acceptor and donor far outside the cutoﬀ distance) and 1 (acceptor and donor much closer than the cutoﬀ).

• acceptor $⟨\phantom{\rule{0.3em}{0ex}}$Number of the acceptor atom$\phantom{\rule{0.3em}{0ex}}⟩$
Context: hBond
Acceptable values: positive integer
Description: Number that uses the same convention as atomNumbers.
• donor: analogous to acceptor
• cutoff: see deﬁnition of cutoff in sec. 4.4.1 (coordNum component)
Note: default value is 3.3 Å.
• expNumer: see deﬁnition of expNumer in sec. 4.4.1 (coordNum component)
Note: default value is 6.
• expDenom: see deﬁnition of expDenom in sec. 4.4.1 (coordNum component)
Note: default value is 8.

#### 4.5 Collective metrics

##### 4.5.1 rmsd: root mean square displacement (RMSD) from reference positions.

The block rmsd {...} deﬁnes the root mean square replacement (RMSD) of a group of atoms with respect to a reference structure. For each set of coordinates $\left\{{\mathbf{x}}_{1}\left(t\right),{\mathbf{x}}_{2}\left(t\right),\dots {\mathbf{x}}_{N}\left(t\right)\right\}$, the colvar component rmsd calculates the optimal rotation ${U}^{\left\{{\mathbf{x}}_{i}\left(t\right)\right\}\to \left\{{\mathbf{x}}_{i}^{\mathrm{\left(ref\right)}}\right\}}$ that best superimposes the coordinates $\left\{{\mathbf{x}}_{i}\left(t\right)\right\}$ onto a set of reference coordinates $\left\{{\mathbf{x}}_{i}^{\mathrm{\left(ref\right)}}\right\}$. Both the current and the reference coordinates are centered on their centers of geometry, ${\mathbf{x}}_{\mathrm{cog}}\left(t\right)$ and ${\mathbf{x}}_{\mathrm{cog}}^{\mathrm{\left(ref\right)}}$. The root mean square displacement is then deﬁned as:

 $\mathrm{RMSD}\left(\left\{{\mathbf{x}}_{i}\left(t\right)\right\},\left\{{\mathbf{x}}_{i}^{\mathrm{\left(ref\right)}}\right\}\right)\phantom{\rule{3.04074pt}{0ex}}=\phantom{\rule{3.04074pt}{0ex}}\sqrt{\frac{1}{N}\sum _{i=1}^{N}{\left|U\left({\mathbf{x}}_{i}\left(t\right)-{\mathbf{x}}_{\mathrm{cog}}\left(t\right)\right)-\left({\mathbf{x}}_{i}^{\mathrm{\left(ref\right)}}-{\mathbf{x}}_{\mathrm{cog}}^{\mathrm{\left(ref\right)}}\right)\right|}^{2}}$ (5)

The optimal rotation ${U}^{\left\{{\mathbf{x}}_{i}\left(t\right)\right\}\to \left\{{\mathbf{x}}_{i}^{\mathrm{\left(ref\right)}}\right\}}$ is calculated within the formalism developed in reference [3], which guarantees a continuous dependence of ${U}^{\left\{{\mathbf{x}}_{i}\left(t\right)\right\}\to \left\{{\mathbf{x}}_{i}^{\mathrm{\left(ref\right)}}\right\}}$ with respect to $\left\{{\mathbf{x}}_{i}\left(t\right)\right\}$.

• atoms $⟨\phantom{\rule{0.3em}{0ex}}$Atom group$\phantom{\rule{0.3em}{0ex}}⟩$
Context: rmsd
Acceptable values: atoms {...} block
Description: Deﬁnes the group of atoms of which the RMSD should be calculated. Optimal ﬁt options (such as refPositions and rotateReference) should typically NOT be set within this block. Exceptions to this rule are the special cases discussed in the Advanced usage paragraph below.
• refPositions $⟨\phantom{\rule{0.3em}{0ex}}$Reference coordinates$\phantom{\rule{0.3em}{0ex}}⟩$
Context: rmsd
Acceptable values: space-separated list of (x, y, z) triplets
Description: This option (mutually exclusive with refPositionsFile) sets the reference coordinates for RMSD calculation, and uses these to compute the roto-translational ﬁt. It is functionally equivalent to the option refPositions in the atom group deﬁnition, which also supports more advanced ﬁtting options.
• refPositionsFile $⟨\phantom{\rule{0.3em}{0ex}}$Reference coordinates ﬁle$\phantom{\rule{0.3em}{0ex}}⟩$
Context: rmsd
Acceptable values: UNIX ﬁlename
Description: This option (mutually exclusive with refPositions) sets the reference coordinates for RMSD calculation, and uses these to compute the roto-translational ﬁt. It is functionally equivalent to the option refPositionsFile in the atom group deﬁnition, which also supports more advanced ﬁtting options.

This component returns a positive real number (in Å).

##### 4.5.2 Advanced usage of the rmsd component.

In the standard usage as described above, the rmsd component calculates a minimum RMSD, that is, current coordinates are optimally ﬁtted onto the same reference coordinates that are used to compute the RMSD value. The ﬁt itself is handled by the atom group object, whose parameters are automatically set by the rmsd component. For very speciﬁc applications, however, it may be useful to control the ﬁtting process separately from the deﬁnition of the reference coordinates, to evaluate various types of non-minimal RMSD values. This can be achieved by setting the related options (refPositions, etc.) explicitly in the atom group block. This allows for the following non-standard cases:

1. applying the optimal translation, but no rotation (rotateReference off), to bias or restrain the shape and orientation, but not the position of the atom group;
2. applying the optimal rotation, but no translation (centerReference off), to bias or restrain the shape and position, but not the orientation of the atom group;
3. disabling the application of optimal roto-translations, which lets the RMSD component decribe the deviation of atoms from ﬁxed positions in the laboratory frame: this allows for custom positional restraints within the Colvars module;
4. ﬁtting the atomic positions to diﬀerent reference coordinates than those used in the RMSD calculation itself;
5. applying the optimal rotation and/or translation from a separate atom group, deﬁned through fittingGroup: the RMSD then reﬂects the deviation from reference coordinates in a separate, moving reference frame.

##### 4.5.3 eigenvector: projection of the atomic coordinates on a vector.

The block eigenvector {...} deﬁnes the projection of the coordinates of a group of atoms (or more precisely, their deviations from the reference coordinates) onto a vector in ${ℝ}^{3n}$, where $n$ is the number of atoms in the group. The computed quantity is the total projection:

 $p\left(\left\{{\mathbf{x}}_{i}\left(t\right)\right\},\left\{{\mathbf{x}}_{i}^{\mathrm{\left(ref\right)}}\right\}\right)\phantom{\rule{3.04074pt}{0ex}}=\phantom{\rule{3.04074pt}{0ex}}\sum _{i=1}^{n}{\mathbf{v}}_{i}\cdot \left(U\left({\mathbf{x}}_{i}\left(t\right)-{\mathbf{x}}_{\mathrm{cog}}\left(t\right)\right)-\left({\mathbf{x}}_{i}^{\mathrm{\left(ref\right)}}-{\mathbf{x}}_{\mathrm{cog}}^{\mathrm{\left(ref\right)}}\right)\right)\mathrm{,}$ (6)

where, as in the rmsd component, $U$ is the optimal rotation matrix, ${\mathbf{x}}_{\mathrm{cog}}\left(t\right)$ and ${\mathbf{x}}_{\mathrm{cog}}^{\mathrm{\left(ref\right)}}$ are the centers of geometry of the current and reference positions respectively, and ${\mathbf{v}}_{i}$ are the components of the vector for each atom. Example choices for $\left({\mathbf{v}}_{i}\right)$ are an eigenvector of the covariance matrix (essential mode), or a normal mode of the system. It is assumed that ${\sum }_{i}{\mathbf{v}}_{i}=0$: otherwise, the Colvars module centers the ${\mathbf{v}}_{i}$ automatically when reading them from the conﬁguration.

• atoms: see deﬁnition of atoms in sec. 4.5.1 (rmsd component)
• refPositions: see deﬁnition of refPositions in sec. 4.5.1 (rmsd component)
• refPositionsFile: see deﬁnition of refPositionsFile in sec. 4.5.1 (rmsd component)
• vector $⟨\phantom{\rule{0.3em}{0ex}}$Vector components$\phantom{\rule{0.3em}{0ex}}⟩$
Context: eigenvector
Acceptable values: space-separated list of (x, y, z) triplets
Description: This option (mutually exclusive with vectorFile) sets the values of the vector components.
• vectorFile $⟨\phantom{\rule{0.3em}{0ex}}$ﬁle containing vector components$\phantom{\rule{0.3em}{0ex}}⟩$
Context: eigenvector
Acceptable values: UNIX ﬁlename
Description: This option (mutually exclusive with vector) sets the name of a coordinate ﬁle containing the vector components; the ﬁle is read according to the same format used for refPositionsFile.
• differenceVector $⟨\phantom{\rule{0.3em}{0ex}}$The $3n$-dimensional vector is the diﬀerence between vector and refPositions$\phantom{\rule{0.3em}{0ex}}⟩$
Context: eigenvector
Acceptable values: boolean
Default value: off
Description: If this option is on, the numbers provided by vector are interpreted as another set of positions, ${\mathbf{x}}_{i}^{\prime }$: the vector ${\mathbf{v}}_{i}$ is then deﬁned as ${\mathbf{v}}_{i}=\left({\mathbf{x}}_{i}^{\prime }-{\mathbf{x}}_{i}^{\mathrm{\left(ref\right)}}\right)$. This allows to conveniently deﬁne a colvar $\xi$ as a projection on the linear transformation between two sets of positions, “A” and “B”. For convenience, the vector is also normalized so that $\xi =0$ when the atoms are at the set of positions “A” and $\xi =1$ at the set of positions “B”.

This component returns a number (in Å), whose value ranges between the smallest and largest absolute positions in the unit cell during the simulations (see also distanceZ). Due to the normalization in eq. 6, this range does not depend on the number of atoms involved.

##### 4.5.4 gyration: radius of gyration of a group of atoms.

The block gyration {...} deﬁnes the parameters for calculating the radius of gyration of a group of atomic positions $\left\{{\mathbf{x}}_{1}\left(t\right),{\mathbf{x}}_{2}\left(t\right),\dots {\mathbf{x}}_{N}\left(t\right)\right\}$ with respect to their center of geometry, ${\mathbf{x}}_{\mathrm{cog}}\left(t\right)$:

 ${R}_{\mathrm{gyr}}\phantom{\rule{3.04074pt}{0ex}}=\phantom{\rule{3.04074pt}{0ex}}\sqrt{\frac{1}{N}\sum _{i=1}^{N}{\left|{\mathbf{x}}_{i}\left(t\right)-{\mathbf{x}}_{\mathrm{cog}}\left(t\right)\right|}^{2}}$ (7)

This component must contain one atoms {...} block to deﬁne the atom group, and returns a positive number, expressed in Å.

• atoms: see deﬁnition of atoms in sec. 4.5.1 (rmsd component)

##### 4.5.5 inertia: total moment of inertia of a group of atoms.

The block inertia {...} deﬁnes the parameters for calculating the total moment of inertia of a group of atomic positions $\left\{{\mathbf{x}}_{1}\left(t\right),{\mathbf{x}}_{2}\left(t\right),\dots {\mathbf{x}}_{N}\left(t\right)\right\}$ with respect to their center of geometry, ${\mathbf{x}}_{\mathrm{cog}}\left(t\right)$:

 $I\phantom{\rule{3.04074pt}{0ex}}=\phantom{\rule{3.04074pt}{0ex}}\sum _{i=1}^{N}{\left|{\mathbf{x}}_{i}\left(t\right)-{\mathbf{x}}_{\mathrm{cog}}\left(t\right)\right|}^{2}$ (8)

Note that all atomic masses are set to 1 for simplicity. This component must contain one atoms {...} block to deﬁne the atom group, and returns a positive number, expressed in Å${}^{2}$.

• atoms: see deﬁnition of atoms in sec. 4.5.1 (rmsd component)

##### 4.5.6 dipoleMagnitude: dipole magnitude of a group of atoms.

The dipoleMagnitude {...} block deﬁnes the dipole magnitude of a group of atoms (norm of the dipole moment’s vector), being atoms the group where dipole magnitude is calculated. It returns the magnitude in elementary charge $e$ times (length unit).

• atoms: see deﬁnition of atoms in sec. 4.5.1 (rmsd component)

##### 4.5.7 inertiaZ: total moment of inertia of a group of atoms around a chosen axis.

The block inertiaZ {...} deﬁnes the parameters for calculating the component along the axis $\mathbf{e}$ of the moment of inertia of a group of atomic positions $\left\{{\mathbf{x}}_{1}\left(t\right),{\mathbf{x}}_{2}\left(t\right),\dots {\mathbf{x}}_{N}\left(t\right)\right\}$ with respect to their center of geometry, ${\mathbf{x}}_{\mathrm{cog}}\left(t\right)$:

 ${I}_{\mathbf{e}}\phantom{\rule{3.04074pt}{0ex}}=\phantom{\rule{3.04074pt}{0ex}}\sum _{i=1}^{N}{\left(\left({\mathbf{x}}_{i}\left(t\right)-{\mathbf{x}}_{\mathrm{cog}}\left(t\right)\right)\cdot \mathbf{e}\right)}^{2}$ (9)

Note that all atomic masses are set to 1 for simplicity. This component must contain one atoms {...} block to deﬁne the atom group, and returns a positive number, expressed in Å${}^{2}$.

• atoms: see deﬁnition of atoms in sec. 4.5.1 (rmsd component)
• axis $⟨\phantom{\rule{0.3em}{0ex}}$Projection axis (Å)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: inertiaZ
Acceptable values: (x, y, z) triplet
Default value: (0.0, 0.0, 1.0)
Description: The three components of this vector deﬁne (when normalized) the projection axis $\mathbf{e}$.

#### 4.6 Rotations

##### 4.6.1 orientation: orientation from reference coordinates.

The block orientation {...} returns the same optimal rotation used in the rmsd component to superimpose the coordinates $\left\{{\mathbf{x}}_{i}\left(t\right)\right\}$ onto a set of reference coordinates $\left\{{\mathbf{x}}_{i}^{\mathrm{\left(ref\right)}}\right\}$. Such component returns a four dimensional vector $\mathsf{q}=\left({q}_{0},{q}_{1},{q}_{2},{q}_{3}\right)$, with ${\sum }_{i}{q}_{i}^{2}=1$; this quaternion expresses the optimal rotation $\left\{{\mathbf{x}}_{i}\left(t\right)\right\}\to \left\{{\mathbf{x}}_{i}^{\mathrm{\left(ref\right)}}\right\}$ according to the formalism in reference [3]. The quaternion $\left({q}_{0},{q}_{1},{q}_{2},{q}_{3}\right)$ can also be written as $\left(cos\left(𝜃∕2\right),\phantom{\rule{0.3em}{0ex}}sin\left(𝜃∕2\right)\mathbf{u}\right)$, where $𝜃$ is the angle and $\mathbf{u}$ the normalized axis of rotation; for example, a rotation of 90${}^{\circ }$ around the $z$ axis is expressed as “(0.707, 0.0, 0.0, 0.707)”. The script quaternion2rmatrix.tcl provides Tcl functions for converting to and from a $4×4$ rotation matrix in a format suitable for usage in VMD.

As for the component rmsd, the available options are atoms, refPositionsFile and refPositions.

Note: refPositionsand refPositionsFile deﬁne the set of positions from which the optimal rotation is calculated, but this rotation is not applied to the coordinates of the atoms involved: it is used instead to deﬁne the variable itself.

• atoms: see deﬁnition of atoms in sec. 4.5.1 (rmsd component)
• refPositions: see deﬁnition of refPositions in sec. 4.5.1 (rmsd component)
• refPositionsFile: see deﬁnition of refPositionsFile in sec. 4.5.1 (rmsd component)
• closestToQuaternion $⟨\phantom{\rule{0.3em}{0ex}}$Reference rotation$\phantom{\rule{0.3em}{0ex}}⟩$
Context: orientation
Acceptable values: (q0, q1, q2, q3)” quadruplet
Default value: (1.0, 0.0, 0.0, 0.0) (“null” rotation)
Description: Between the two equivalent quaternions $\left({q}_{0},{q}_{1},{q}_{2},{q}_{3}\right)$ and $\left(-{q}_{0},-{q}_{1},-{q}_{2},-{q}_{3}\right)$, the closer to (1.0, 0.0, 0.0, 0.0) is chosen. This simpliﬁes the visualization of the colvar trajectory when sampled values are a smaller subset of all possible rotations. Note: this only aﬀects the output, never the dynamics.

Tip: stopping the rotation of a protein. To stop the rotation of an elongated macromolecule in solution (and use an anisotropic box to save water molecules), it is possible to deﬁne a colvar with an orientation component, and restrain it throuh the harmonic bias around the identity rotation, (1.0, 0.0, 0.0, 0.0). Only the overall orientation of the macromolecule is aﬀected, and not its internal degrees of freedom. The user should also take care that the macromolecule is composed by a single chain, or disable wrapAll otherwise.

##### 4.6.2 orientationAngle: angle of rotation from reference coordinates.

The block orientationAngle {...} accepts the same base options as the component orientation: atoms, refPositions, refPositionsFile. The returned value is the angle of rotation $𝜃$ between the current and the reference positions. This angle is expressed in degrees within the range [0${}^{\circ }$:180${}^{\circ }$].

##### 4.6.3 orientationProj: cosine of the angle of rotation from reference coordinates.

The block orientationProj {...} accepts the same base options as the component orientation: atoms, refPositions, refPositionsFile. The returned value is the cosine of the angle of rotation $𝜃$ between the current and the reference positions. The range of values is [-1:1].

##### 4.6.4 spinAngle: angle of rotation around a given axis.

The complete rotation described by orientation can optionally be decomposed into two sub-rotations: one is a “spin” rotation around e, and the other a “tilt” rotation around an axis orthogonal to e. The component spinAngle measures the angle of the “spin” sub-rotation around e.

• atoms: see deﬁnition of atoms in sec. 4.5.1 (rmsd component)
• refPositions: see deﬁnition of refPositions in sec. 4.5.1 (rmsd component)
• refPositionsFile: see deﬁnition of refPositionsFile in sec. 4.5.1 (rmsd component)
• axis $⟨\phantom{\rule{0.3em}{0ex}}$Special rotation axis (Å)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: tilt
Acceptable values: (x, y, z) triplet
Default value: (0.0, 0.0, 1.0)
Description: The three components of this vector deﬁne (when normalized) the special rotation axis used to calculate the tilt and spinAngle components.

The component spinAngle returns an angle (in degrees) within the periodic interval $\left[-180:180\right]$.

Note: the value of spinAngle is a continuous function almost everywhere, with the exception of conﬁgurations with the corresponding “tilt” angle equal to 180${}^{\circ }$ (i.e. the tilt component is equal to $-1$): in those cases, spinAngle is undeﬁned. If such conﬁgurations are expected, consider deﬁning a tilt colvar using the same axis e, and restraining it with a lower wall away from $-1$.

##### 4.6.5 tilt: cosine of the rotation orthogonal to a given axis.

The component tilt measures the cosine of the angle of the “tilt” sub-rotation, which combined with the “spin” sub-rotation provides the complete rotation of a group of atoms. The cosine of the tilt angle rather than the tilt angle itself is implemented, because the latter is unevenly distributed even for an isotropic system: consider as an analogy the angle $𝜃$ in the spherical coordinate system. The component tilt relies on the same options as spinAngle, including the deﬁnition of the axis e. The values of tilt are real numbers in the interval $\left[-1:1\right]$: the value $1$ represents an orientation fully parallel to e (tilt angle = 0${}^{\circ }$), and the value $-1$ represents an anti-parallel orientation.

• atoms: see deﬁnition of atoms in sec. 4.5.1 (rmsd component)
• refPositions: see deﬁnition of refPositions in sec. 4.5.1 (rmsd component)
• refPositionsFile: see deﬁnition of refPositionsFile in sec. 4.5.1 (rmsd component)
• axis: see deﬁnition of axis in sec. 4.6.4 (spinAngle component)

#### 4.7 Raw data: building blocks for custom functions

##### 4.7.1 cartesian: vector of atomic Cartesian coordinates.

The cartesian {...} block deﬁnes a component returning a ﬂat vector containing the Cartesian coordinates of all participating atoms, in the order $\left({x}_{1},{y}_{1},{z}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{x}_{n},{y}_{n},{z}_{n}\right)$.

• atoms $⟨\phantom{\rule{0.3em}{0ex}}$Group of atoms$\phantom{\rule{0.3em}{0ex}}⟩$
Context: cartesian
Acceptable values: Block atoms {...}
Description: Deﬁnes the atoms whose coordinates make up the value of the component. If rotateReference or centerReference are deﬁned, coordinates are evaluated within the moving frame of reference.

##### 4.7.2 distancePairs: set of pairwise distances between two groups.

The distancePairs {...} block deﬁnes a ${N}_{\mathrm{1}}×{N}_{\mathrm{2}}$-dimensional variable that includes all mutual distances between the atoms of two groups. This can be useful, for example, to develop a new variable deﬁned over two groups, by using the scriptedFunction feature.

• group1: see deﬁnition of group1 in sec. 4.2.1 (distance component)
• group2: analogous to group1
• forceNoPBC: see deﬁnition of forceNoPBC in sec. 4.2.1 (distance component)

This component returns a ${N}_{\mathrm{1}}×{N}_{\mathrm{2}}$-dimensional vector of numbers, each ranging from $0$ to the largest possible distance within the chosen boundary conditions.

#### 4.8 Shared keywords for all components

The following options can be used for any of the above colvar components in order to obtain a polynomial combination.

• name $⟨\phantom{\rule{0.3em}{0ex}}$Name of this component$\phantom{\rule{0.3em}{0ex}}⟩$
Context: any component
Acceptable values: string
Default value: type of component + numeric id
Description: The name is an unique case-sensitive string which allows the Colvars module to identify this component. This is useful, for example, when combining multiple components via a scriptedFunction.
• scalable $⟨\phantom{\rule{0.3em}{0ex}}$Attempt to calculate this component in parallel?$\phantom{\rule{0.3em}{0ex}}⟩$
Context: any component
Acceptable values: boolean
Default value: on, if available
Description: If set to on (default), the Colvars module will attempt to calculate this component in parallel to reduce overhead. Whether this option is available depends on the type of component: currently supported are distance, distanceZ, distanceXY, distanceVec, distanceDir, angle and dihedral. This ﬂag inﬂuences computational cost, but does not aﬀect numerical results: therefore, it should only be turned oﬀ for debugging or testing purposes.

#### 4.9 Periodic components

The following components returns real numbers that lie in a periodic interval:

• dihedral: torsional angle between four groups;
• spinAngle: angle of rotation around a predeﬁned axis in the best-ﬁt from a set of reference coordinates.

In certain conditions, distanceZ can also be periodic, namely when periodic boundary conditions (PBCs) are deﬁned in the simulation and distanceZ’s axis is parallel to a unit cell vector.

In addition, a custom scalar colvar may be periodic depending on its user-deﬁned expression. It will only be treated as such by the Colvars module if the period is speciﬁed using the period keyword, while wrapAround is optional.

The following keywords can be used within periodic components, or within custom variables (4.12)).

• period $⟨\phantom{\rule{0.3em}{0ex}}$Period of the component$\phantom{\rule{0.3em}{0ex}}⟩$
Context: distanceZ, custom colvars
Acceptable values: positive decimal
Default value: 0.0
Description: Setting this number enables the treatment of distanceZ as a periodic component: by default, distanceZ is not considered periodic. The keyword is supported, but irrelevant within dihedral or spinAngle, because their period is always 360 degrees.
• wrapAround $⟨\phantom{\rule{0.3em}{0ex}}$Center of the wrapping interval for periodic variables$\phantom{\rule{0.3em}{0ex}}⟩$
Context: distanceZ, dihedral, spinAngle, custom colvars
Acceptable values: decimal
Default value: 0.0
Description: By default, values of the periodic components are centered around zero, ranging from $-P∕2$ to $P∕2$, where $P$ is the period. Setting this number centers the interval around this value. This can be useful for convenience of output, or to set the walls for a harmonicWalls in an order that would not otherwise be allowed.

Internally, all diﬀerences between two values of a periodic colvar follow the minimum image convention: they are calculated based on the two periodic images that are closest to each other.

Note: linear or polynomial combinations of periodic components (see 4.11) may become meaningless when components cross the periodic boundary. Use such combinations carefully: estimate the range of possible values of each component in a given simulation, and make use of wrapAround to limit this problem whenever possible.

#### 4.10 Non-scalar components

When one of the following components are used, the deﬁned colvar returns a value that is not a scalar number:

• distanceVec: 3-dimensional vector of the distance between two groups;
• distanceDir: 3-dimensional unit vector of the distance between two groups;
• orientation: 4-dimensional unit quaternion representing the best-ﬁt rotation from a set of reference coordinates.

The distance between two 3-dimensional unit vectors is computed as the angle between them. The distance between two quaternions is computed as the angle between the two 4-dimensional unit vectors: because the orientation represented by $\mathsf{q}$ is the same as the one represented by $-\mathsf{q}$, distances between two quaternions are computed considering the closest of the two symmetric images.

Non-scalar components carry the following restrictions:

• Calculation of total forces (outputTotalForce option) is currently not implemented.
• Each colvar can only contain one non-scalar component.
• Binning on a grid (abf, histogram and metadynamics with useGrids enabled) is currently not implemented for colvars based on such components.

Note: while these restrictions apply to individual colvars based on non-scalar components, no limit is set to the number of scalar colvars. To compute multi-dimensional histograms and PMFs, use sets of scalar colvars of arbitrary size.

##### 4.10.1 Calculating total forces

In addition to the restrictions due to the type of value computed (scalar or non-scalar), a ﬁnal restriction can arise when calculating total force (outputTotalForce option or application of a abf bias). total forces are available currently only for the following components: distance, distanceZ, distanceXY, angle, dihedral, rmsd, eigenvector and gyration.

#### 4.11 Linear and polynomial combinations of components

To extend the set of possible deﬁnitions of colvars $\xi \left(\mathbf{r}\right)$, multiple components ${q}_{i}\left(\mathbf{r}\right)$ can be summed with the formula:

 $\xi \left(\mathbf{r}\right)=\sum _{i}{c}_{i}{\left[{q}_{i}\left(\mathbf{r}\right)\right]}^{{n}_{i}}$ (10)

where each component appears with a unique coeﬃcient ${c}_{i}$ (1.0 by default) the positive integer exponent ${n}_{i}$ (1 by default).

Any set of components can be combined within a colvar, provided that they return the same type of values (scalar, unit vector, vector, or quaternion). By default, the colvar is the sum of its components. Linear or polynomial combinations (following equation (10)) can be obtained by setting the following parameters, which are common to all components:

• componentCoeff $⟨\phantom{\rule{0.3em}{0ex}}$Coeﬃcient of this component in the colvar$\phantom{\rule{0.3em}{0ex}}⟩$
Context: any component
Acceptable values: decimal
Default value: 1.0
Description: Deﬁnes the coeﬃcient by which this component is multiplied (after being raised to componentExp) before being added to the sum.
• componentExp $⟨\phantom{\rule{0.3em}{0ex}}$Exponent of this component in the colvar$\phantom{\rule{0.3em}{0ex}}⟩$
Context: any component
Acceptable values: integer
Default value: 1
Description: Deﬁnes the power at which the value of this component is raised before being added to the sum. When this exponent is diﬀerent than 1 (non-linear sum), total forces and the Jacobian force are not available, making the colvar unsuitable for ABF calculations.

Example: To deﬁne the average of a colvar across diﬀerent parts of the system, simply deﬁne within the same colvar block a series of components of the same type (applied to diﬀerent atom groups), and assign to each component a componentCoeff of $1∕N$.

#### 4.12 Custom functions

Collective variables may be deﬁned by specifying a custom function as an analytical expression such as cos(x) + y^2. The expression is parsed by the Lepton expression parser (written by Peter Eastman), which produces eﬃcient evaluation routines for the function itself as well as its derivatives. The expression may use the collective variable components as variables, refered to as their name string. Scalar elements of vector components may be accessed by appending a 1-based index to their name. When implementing generic functions of Cartesian coordinates rather than functions of existing components, the cartesian component may be particularly useful. A scalar-valued custom variable may be manually deﬁned as periodic by providing the keyword period, and the optional keyword wrapAround, with the same meaning as in periodic components (see 4.9 for details). A vector variable may be deﬁned by specifying the customFunction parameter several times: each expression deﬁnes one scalar element of the vector colvar. This is illustrated in the example below.

colvar {
name custom

# A 2-dimensional vector function of a scalar x and a 3-vector r
customFunction cos(x) * (r1 + r2 + r3)
customFunction sqrt(r1 * r2)

distance {
name x
group1 { atomNumbers 1 }
group2 { atomNumbers 50 }
}
distanceVec {
name r
group1 { atomNumbers 10 11 12 }
group2 { atomNumbers 20 21 22 }
}
}

• customFunction $⟨\phantom{\rule{0.3em}{0ex}}$Compute colvar as a custom function of its components$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: string
Description: Deﬁnes the colvar as a scalar expression of its colvar components. Multiple mentions can be used to deﬁne a vector variable (as in the example above).
• customFunctionType $⟨\phantom{\rule{0.3em}{0ex}}$Type of value returned by the scripted colvar$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: string
Default value: scalar
Description: With this ﬂag, the user may specify whether the colvar is a scalar or one of the following vector types: vector3 (a 3D vector), unit_vector3 (a normalized 3D vector), or unit_quaternion (a normalized quaternion), or vector. Note that the scalar and vector cases are not necessary, as they are detected automatically.

#### 4.13 Deﬁning grid parameters

Many algorithms require the deﬁnition of boundaries and/or characteristic spacings that can be used to deﬁne discrete “states” in the collective variable, or to combine variables with very diﬀerent units. The parameters described below oﬀer a way to specify these parameters only once for each variable, while using them multiple times in restraints, time-dependent biases or analysis methods.

• width $⟨\phantom{\rule{0.3em}{0ex}}$Unit of the variable, or grid spacing$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: positive decimal
Default value: 1.0
Description: This number deﬁnes the eﬀective unit of measurement for the collective variable, and is used by the biasing methods for the following purposes. Harmonic (6.4), harmonic walls (6.6) and linear restraints (6.7) use it to set the physical unit of the force constant, which is useful for multidimensional restraints involving multiple variables with very diﬀerent units (for examples, $Å$ or degrees ${}^{\circ }$) with a single, scaled force constant. The values of the scaled force constant in the units of each variable are printed at initialization time. Histograms (6.9), ABF (6.1) and metadynamics (6.3) all use this number as the initial choice for the grid spacing along this variable: for this reason, width should generally be no larger than the standard deviation of the colvar in an unbiased simulation. Unless it is required to control the spacing, it is usually simplest to keep the default value of 1, so that restraint force constants are provided with their full physical unit.
• lowerBoundary $⟨\phantom{\rule{0.3em}{0ex}}$Lower boundary of the colvar$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: decimal
Description: Deﬁnes the lowest end of the interval of “relevant” values for the colvar. This number can be either a true physical boundary, or a user-deﬁned number. Together with upperBoundary and width, it is used to deﬁne a grid of values along the variable (not available for variables with vector values, 4.10). This option does not aﬀect dynamics: to conﬁne a colvar within a certain interval, use a harmonicWalls bias.
• upperBoundary $⟨\phantom{\rule{0.3em}{0ex}}$Upper boundary of the colvar$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: decimal
Description: Similarly to lowerBoundary, deﬁnes the highest possible or allowed value.
• hardLowerBoundary $⟨\phantom{\rule{0.3em}{0ex}}$Whether the lower boundary is the physical lower limit$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: boolean
Default value: off
Description: This option does not aﬀect simulation results, but enables some internal optimizations. Depending on its mathematical deﬁnition, a colvar may have “natural” boundaries: for example, a distance colvar has a “natural” lower boundary at 0. Setting this option instructs the Colvars module that the user-deﬁned lower boundary is “natural”. See Section 4.1 for the physical ranges of values of each component.
• hardUpperBoundary $⟨\phantom{\rule{0.3em}{0ex}}$Whether the upper boundary is the physical upper limit of the colvar’s values$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: boolean
Default value: off
Description: Analogous to hardLowerBoundary.
• expandBoundaries $⟨\phantom{\rule{0.3em}{0ex}}$Allow to expand the two boundaries if needed$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: boolean
Default value: off
Description: If deﬁned, biasing and analysis methods may keep their own copies of lowerBoundary and upperBoundary, and expand them to accommodate values that do not ﬁt in the initial range. Currently, this option is used by the metadynamics bias (6.3) to keep all of its hills fully within the grid. This option cannot be used when the initial boundaries already span the full period of a periodic colvar.

#### 4.14 Trajectory output

• outputValue $⟨\phantom{\rule{0.3em}{0ex}}$Output a trajectory for this colvar$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: boolean
Default value: on
Description: If colvarsTrajFrequency is non-zero, the value of this colvar is written to the trajectory ﬁle every colvarsTrajFrequency steps in the column labeled “$<$name$>$”.
• outputVelocity $⟨\phantom{\rule{0.3em}{0ex}}$Output a velocity trajectory for this colvar$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: boolean
Default value: off
Description: If colvarsTrajFrequency is deﬁned, the ﬁnite-diﬀerence calculated velocity of this colvar are written to the trajectory ﬁle under the label “v_$<$name$>$”.
• outputEnergy $⟨\phantom{\rule{0.3em}{0ex}}$Output an energy trajectory for this colvar$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: boolean
Default value: off
Description: This option applies only to extended Lagrangian colvars. If colvarsTrajFrequency is deﬁned, the kinetic energy of the extended degree and freedom and the potential energy of the restraining spring are are written to the trajectory ﬁle under the labels “Ek_$<$name$>$” and “Ep_$<$name$>$”.
• outputTotalForce $⟨\phantom{\rule{0.3em}{0ex}}$Output a total force trajectory for this colvar$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: boolean
Default value: off
Description: If colvarsTrajFrequency is deﬁned, the total force on this colvar (i.e. the projection of all atomic total forces onto this colvar — see equation (15) in section 6.1) are written to the trajectory ﬁle under the label “fs_$<$name$>$”. For extended Lagrangian colvars, the “total force” felt by the extended degree of freedom is simply the force from the harmonic spring. Note: not all components support this option. The physical unit for this force is the unit of energy speciﬁed by units, divided by the colvar unit U.
• outputAppliedForce $⟨\phantom{\rule{0.3em}{0ex}}$Output an applied force trajectory for this colvar$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: boolean
Default value: off
Description: If colvarsTrajFrequency is deﬁned, the total force applied on this colvar by Colvars biases are written to the trajectory under the label “fa_$<$name$>$”. For extended Lagrangian colvars, this force is actually applied to the extended degree of freedom rather than the geometric colvar itself. The physical unit for this force is the unit of energy speciﬁed by units divided by the colvar unit.

#### 4.15 Extended Lagrangian

The following options enable extended-system dynamics, where a colvar is coupled to an additional degree of freedom (ﬁctitious particle) by a harmonic spring. All biasing and conﬁning forces are then applied to the extended degree of freedom. The “actual” geometric colvar (function of Cartesian coordinates) only feels the force from the harmonic spring. This is particularly useful when combined with an ABF bias (6.1) to perform eABF simulations (6.2).

• extendedLagrangian $⟨\phantom{\rule{0.3em}{0ex}}$Add extended degree of freedom$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: boolean
Default value: off
Description: Adds a ﬁctitious particle to be coupled to the colvar by a harmonic spring. The ﬁctitious mass and the force constant of the coupling potential are derived from the parameters extendedTimeConstant and extendedFluctuation, described below. Biasing forces on the colvar are applied to this ﬁctitious particle, rather than to the atoms directly. This implements the extended Lagrangian formalism used in some metadynamics simulations [2].
• extendedFluctuation $⟨\phantom{\rule{0.3em}{0ex}}$Standard deviation between the colvar and the ﬁctitious particle (colvar unit)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: positive decimal
Description: Deﬁnes the spring stiﬀness for the extendedLagrangian mode, by setting the typical deviation between the colvar and the extended degree of freedom due to thermal ﬂuctuation. The spring force constant is calculated internally as ${k}_{B}T∕{\sigma }^{2}$, where $\sigma$ is the value of extendedFluctuation.
• extendedTimeConstant $⟨\phantom{\rule{0.3em}{0ex}}$Oscillation period of the ﬁctitious particle (fs)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: positive decimal
Default value: 200
Description: Deﬁnes the inertial mass of the ﬁctitious particle, by setting the oscillation period of the harmonic oscillator formed by the ﬁctitious particle and the spring. The period should be much larger than the MD time step to ensure accurate integration of the extended particle’s equation of motion. The ﬁctitious mass is calculated internally as ${k}_{B}T{\left(\tau ∕2\pi \sigma \right)}^{2}$, where $\tau$ is the period and $\sigma$ is the typical ﬂuctuation (see above).
• extendedTemp $⟨\phantom{\rule{0.3em}{0ex}}$Temperature for the extended degree of freedom (K)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: positive decimal
Default value: thermostat temperature
Description: Temperature used for calculating the coupling force constant of the extended variable (see extendedFluctuation) and, if needed, as a target temperature for extended Langevin dynamics (see extendedLangevinDamping). This should normally be left at its default value.
• extendedLangevinDamping $⟨\phantom{\rule{0.3em}{0ex}}$Damping factor for extended Langevin dynamics (ps${}^{-1}$)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: positive decimal
Default value: 1.0
Description: If this is non-zero, the extended degree of freedom undergoes Langevin dynamics at temperature extendedTemp. The friction force is minus extendedLangevinDamping times the velocity. This is useful because the extended dynamics coordinate may heat up in the transient non-equilibrium regime of ABF. Use moderate damping values, to limit viscous friction (potentially slowing down diﬀusive sampling) and stochastic noise (increasing the variance of statistical measurements). In doubt, use the default value.

#### 4.16 Backward-compatibility

• subtractAppliedForce $⟨\phantom{\rule{0.3em}{0ex}}$Do not include biasing forces in the total force for this colvar$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: boolean
Default value: off
Description: If the colvar supports total force calculation (see 4.10.1), all forces applied to this colvar by biases will be removed from the total force. This keyword allows to recover some of the “system force” calculation available in the Colvars module before version 2016-08-10. Please note that removal of all other external forces (including biasing forces applied to a diﬀerent colvar) is no longer supported, due to changes in the underlying simulation engines (primarily NAMD). This option may be useful when continuing a previous simulation where the removal of external/applied forces is essential. For all new simulations, the use of this option is not recommended.

#### 4.17 Statistical analysis

Run-time calculations of statistical properties that depend explicitly on time can be performed for individual collective variables. Currently, several types of time correlation functions, running averages and running standard deviations are implemented. For run-time computation of histograms, please see the histogram bias (6.9).

• corrFunc $⟨\phantom{\rule{0.3em}{0ex}}$Calculate a time correlation function?$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: boolean
Default value: off
Description: Whether or not a time correlaction function should be calculated for this colvar.
• corrFuncWithColvar $⟨\phantom{\rule{0.3em}{0ex}}$Colvar name for the correlation function$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: string
Description: By default, the auto-correlation function (ACF) of this colvar, ${\xi }_{i}$, is calculated. When this option is speciﬁed, the correlation function is calculated instead with another colvar, ${\xi }_{j}$, which must be of the same type (scalar, vector, or quaternion) as ${\xi }_{i}$.
• corrFuncType $⟨\phantom{\rule{0.3em}{0ex}}$Type of the correlation function$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: velocity, coordinate or coordinate_p2
Default value: velocity
Description: With coordinate or velocity, the correlation function ${C}_{i,j}\left(t\right)$ = $⟨\Pi \left({\xi }_{i}\left({t}_{0}\right),{\xi }_{j}\left({t}_{0}+t\right)\right)⟩$ is calculated between the variables ${\xi }_{i}$ and ${\xi }_{j}$, or their velocities. $\Pi \left({\xi }_{i},{\xi }_{j}\right)$ is the scalar product when calculated between scalar or vector values, whereas for quaternions it is the cosine between the two corresponding rotation axes. With coordinate_p2, the second order Legendre polynomial, $\left(3cos{\left(𝜃\right)}^{2}-1\right)∕2$, is used instead of the cosine.
• corrFuncNormalize $⟨\phantom{\rule{0.3em}{0ex}}$Normalize the time correlation function?$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: boolean
Default value: on
Description: If enabled, the value of the correlation function at $t$ = 0 is normalized to 1; otherwise, it equals to $⟨O\left({\xi }_{i},{\xi }_{j}\right)⟩$.
• corrFuncLength $⟨\phantom{\rule{0.3em}{0ex}}$Length of the time correlation function$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: positive integer
Default value: 1000
Description: Length (in number of points) of the time correlation function.
• corrFuncStride $⟨\phantom{\rule{0.3em}{0ex}}$Stride of the time correlation function$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: positive integer
Default value: 1
Description: Number of steps between two values of the time correlation function.
• corrFuncOffset $⟨\phantom{\rule{0.3em}{0ex}}$Oﬀset of the time correlation function$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: positive integer
Default value: 0
Description: The starting time (in number of steps) of the time correlation function (default: $t$ = 0). Note: the value at $t$ = 0 is always used for the normalization.
• corrFuncOutputFile $⟨\phantom{\rule{0.3em}{0ex}}$Output ﬁle for the time correlation function$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: UNIX ﬁlename
Default value: output.$<$name$>$.corrfunc.dat
Description: The time correlation function is saved in this ﬁle.
• runAve $⟨\phantom{\rule{0.3em}{0ex}}$Calculate the running average and standard deviation$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: boolean
Default value: off
Description: Whether or not the running average and standard deviation should be calculated for this colvar.
• runAveLength $⟨\phantom{\rule{0.3em}{0ex}}$Length of the running average window$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: positive integer
Default value: 1000
Description: Length (in number of points) of the running average window.
• runAveStride $⟨\phantom{\rule{0.3em}{0ex}}$Stride of the running average window values$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: positive integer
Default value: 1
Description: Number of steps between two values within the running average window.
• runAveOutputFile $⟨\phantom{\rule{0.3em}{0ex}}$Output ﬁle for the running average and standard deviation$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: UNIX ﬁlename
Default value: output.$<$name$>$.runave.traj
Description: The running average and standard deviation are saved in this ﬁle.

### 5 Selecting atoms

To deﬁne collective variables, atoms are usually selected as groups. Each group is deﬁned using an identiﬁer that is unique in the context of the speciﬁc colvar component (e.g. for a distance component, the two groups are group1 and group2). The identiﬁer is followed by a brace-delimited block containing selection keywords and other parameters, including an optional name:

• name $⟨\phantom{\rule{0.3em}{0ex}}$Unique name for the atom group$\phantom{\rule{0.3em}{0ex}}⟩$
Context: atom group
Acceptable values: string
Description: This parameter deﬁnes a unique name for this atom group, which can be referred to in the deﬁnition of other atom groups (including in other colvars) by invoking atomsOfGroup as a selection keyword.

#### 5.1 Atom selection keywords

Selection keywords may be used individually or in combination with each other, and each can be repeated any number of times. Selection is incremental: each keyword adds the corresponding atoms to the selection, so that diﬀerent sets of atoms can be combined. However, atoms included by multiple keywords are only counted once. Below is an example conﬁguration for an atom group called “atoms”. Note: this is an unusually varied combination of selection keywords, demonstrating how they can be combined together: most simulations only use one of them.

atoms {

# add atoms 1 and 3 to this group (note: the first atom in the system is 1)
atomNumbers {
1 3
}

# add atoms starting from 20 up to and including 50
atomNumbersRange 20-50

# add index group (requires a .ndx file to be provided globally)
indexGroup Water
}

The resulting selection includes atoms 1 and 3, those between 20 and 50, and those in the index group called “Water”. The indices of this group are read from the ﬁle provided by the global keyword indexFile.

The complete list of selection keywords available in LAMMPS is:

• atomNumbers $⟨\phantom{\rule{0.3em}{0ex}}$List of atom numbers$\phantom{\rule{0.3em}{0ex}}⟩$
Context: atom group
Acceptable values: space-separated list of positive integers
Description: This option adds to the group all the atoms whose numbers are in the list. The number of the ﬁrst atom in the system is 1: to convert from a VMD selection, use “atomselect get serial”.
• indexGroup $⟨\phantom{\rule{0.3em}{0ex}}$Name of index group to be used (GROMACS format)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: atom group
Acceptable values: string
Description: If the name of an index ﬁle has been provided by indexFile, this option allows to select one index group from that ﬁle: the atoms from that index group will be used to deﬁne the current group.
• atomsOfGroup $⟨\phantom{\rule{0.3em}{0ex}}$Name of group deﬁned previously$\phantom{\rule{0.3em}{0ex}}⟩$
Context: atom group
Acceptable values: string
Description: Refers to a group deﬁned previously using its user-deﬁned name. This adds all atoms of that named group to the current group.
• atomNumbersRange $⟨\phantom{\rule{0.3em}{0ex}}$Atoms within a number range$\phantom{\rule{0.3em}{0ex}}⟩$
Context: atom group
Acceptable values: $<$Starting number$>$-$<$Ending number$>$
Description: This option includes in the group all atoms whose numbers are within the range speciﬁed. The number of the ﬁrst atom in the system is 1.
• dummyAtom $⟨\phantom{\rule{0.3em}{0ex}}$Dummy atom position (Å)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: atom group
Acceptable values: (x, y, z) triplet
Description: Instead of selecting any atom, this option makes the group a virtual particle at a ﬁxed position in space. This is useful e.g. to replace a group’s center of geometry with a user-deﬁned position.

#### 5.2 Moving frame of reference.

The following options deﬁne an automatic calculation of an optimal translation (centerReference) or optimal rotation (rotateReference), that superimposes the positions of this group to a provided set of reference coordinates. This can allow, for example, to eﬀectively remove from certain colvars the eﬀects of molecular tumbling and of diﬀusion. Given the set of atomic positions ${\mathbf{x}}_{i}$, the colvar $\xi$ can be deﬁned on a set of roto-translated positions ${\mathbf{x}}_{i}^{\prime }=R\left({\mathbf{x}}_{i}-{\mathbf{x}}^{\mathrm{C}}\right)+{\mathbf{x}}^{\mathrm{ref}}$. ${\mathbf{x}}^{\mathrm{C}}$ is the geometric center of the ${\mathbf{x}}_{i}$, $R$ is the optimal rotation matrix to the reference positions and ${\mathbf{x}}^{\mathrm{ref}}$ is the geometric center of the reference positions.

Components that are deﬁned based on pairwise distances are naturally invariant under global roto-translations. Other components are instead aﬀected by global rotations or translations: however, they can be made invariant if they are expressed in the frame of reference of a chosen group of atoms, using the centerReference and rotateReference options. Finally, a few components are deﬁned by convention using a roto-translated frame (e.g. the minimal RMSD): for these components, centerReference and rotateReference are enabled by default. In typical applications, the default settings result in the expected behavior.

Warning on rotating frames of reference and periodic boundary conditions. rotateReference aﬀects coordinates that depend on minimum-image distances in periodic boundary conditions (PBC). After rotation of the coordinates, the periodic cell vectors become irrelevant: the rotated system is eﬀectively non-periodic. A safe way to handle this is to ensure that the relevant inter-group distance vectors remain smaller than the half-size of the periodic cell. If this is not desirable, one should avoid the rotating frame of reference, and apply orientational restraints to the reference group instead, in order to keep the orientation of the reference group consistent with the orientation of the periodic cell.

Warning on rotating frames of reference and ABF. Note that centerReference and rotateReference may aﬀect the Jacobian derivative of colvar components in a way that is not taken into account by default. Be careful when using these options in ABF simulations or when using total force values.

• centerReference $⟨\phantom{\rule{0.3em}{0ex}}$Implicitly remove translations for this group$\phantom{\rule{0.3em}{0ex}}⟩$
Context: atom group
Acceptable values: boolean
Default value: off
Description: If this option is on, the center of geometry of the group will be aligned with that of the reference positions provided by refPositions or refPositionsFile. Colvar components will only have access to the aligned positions. Note: unless otherwise speciﬁed, rmsd and eigenvector set this option to on by default.
• rotateReference $⟨\phantom{\rule{0.3em}{0ex}}$Implicitly remove rotations for this group$\phantom{\rule{0.3em}{0ex}}⟩$
Context: atom group
Acceptable values: boolean
Default value: off
Description: If this option is on, the coordinates of this group will be optimally superimposed to the reference positions provided by refPositions or refPositionsFile. The rotation will be performed around the center of geometry if centerReference is on, or around the origin otherwise. The algorithm used is the same employed by the orientation colvar component [3]. Forces applied to the atoms of this group will also be implicitly rotated back to the original frame. Note: unless otherwise speciﬁed, rmsd and eigenvector set this option to on by default.
• refPositions $⟨\phantom{\rule{0.3em}{0ex}}$Reference positions for ﬁtting (Å)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: atom group
Acceptable values: space-separated list of (x, y, z) triplets
Description: This option provides a list of reference coordinates for centerReference and/or rotateReference, and is mutually exclusive with refPositionsFile. If only centerReference is on, the list may contain a single (x, y, z) triplet; if also rotateReference is on, the list should be as long as the atom group, and its order must match the order in which atoms were deﬁned.
• refPositionsFile $⟨\phantom{\rule{0.3em}{0ex}}$File containing the reference positions for ﬁtting$\phantom{\rule{0.3em}{0ex}}⟩$
Context: atom group
Acceptable values: UNIX ﬁlename
Description: This option provides a list of reference coordinates for centerReference and/or rotateReference, and is mutually exclusive with refPositions. The acceptable ﬁle format is XYZ, which is read in double precision. Atomic positions are read diﬀerently depending on the following scenarios: (i) the ﬁle contains exactly as many records as the atoms in the group: all positions are read in sequence; (ii) (most common case) the ﬁle contains coordinates for the entire system: only the positions corresponding to the numeric indices of the atom group are read. In each case, atoms are read from the ﬁle in order of increasing number.
• fittingGroup $⟨\phantom{\rule{0.3em}{0ex}}$Use an alternate set of atoms to deﬁne the roto-translation$\phantom{\rule{0.3em}{0ex}}⟩$
Context: atom group
Acceptable values: Block fittingGroup { ... }
Default value: This group itself
Description: If either centerReference or rotateReference is deﬁned, this keyword deﬁnes an alternate atom group to calculate the optimal roto-translation. Use this option to deﬁne a continuous rotation if the structure of the group involved changes signiﬁcantly (a typical symptom would be the message “Warning: discontinuous rotation!”).

# Example: defining a group "atoms", with its coordinates expressed
# on a roto-translated frame of reference defined by a second group
atoms {

psfSegID PROT
atomNameResidueRange CA 41-48

centerReference yes
rotateReference yes
fittingGroup {
# define the frame by fitting the rest of the protein
psfSegID PROT PROT
atomNameResidueRange CA 1-40
atomNameResidueRange CA 49-100
}
refPositionsFile all.pdb # can be the entire system
}

The following two options have default values appropriate for the vast majority of applications, and are only provided to support rare, special cases.

• enableFitGradients $⟨\phantom{\rule{0.3em}{0ex}}$Include the roto-translational contribution to colvar gradients$\phantom{\rule{0.3em}{0ex}}⟩$
Context: atom group
Acceptable values: boolean
Default value: on
Description: When either centerReference or rotateReference is on, the gradients of some colvars include terms proportional to $\partial R∕\partial {\mathbf{x}}_{i}$ (rotational gradients) and $\partial {\mathbf{x}}^{\mathrm{C}}∕\partial {\mathbf{x}}_{i}$ (translational gradients). By default, these terms are calculated and included in the total gradients; if this option is set to off, they are neglected. In the case of a minimum RMSD component, this ﬂag is automatically disabled because the contributions of those derivatives to the gradients cancel out.
• enableForces $⟨\phantom{\rule{0.3em}{0ex}}$Apply forces from this colvar to this group$\phantom{\rule{0.3em}{0ex}}⟩$
Context: atom group
Acceptable values: boolean
Default value: on
Description: If this option is off, no forces are applied the atoms in the group. Other forces are not aﬀected (i.e. those from the MD engine, from other colvars, and other external forces). For dummy atoms, this option is off by default.

#### 5.3 Treatment of periodic boundary conditions.

In simulations with periodic boundary conditions, many of the implemented colvar components rely on the fact that each position within a group of atoms is at the nearest periodic image from the center of geometry of the group itself. However, due to the internal wrapping of individual atomic positions done by LAMMPS, this assumption is broken if the group straddles one of the unit cell’s boundaries. For this reason, within the Colvars module all coordinates are unwrapped by default to avoid discontinuities (see unwrap keyword in 3.1).

The user should determine whether maintaining the default value of unwrap, depending on the speciﬁcs of each system. In general, internal coordinate wrapping by LAMMPS does not aﬀect the calculation of colvars if each atom group satisﬁes one or more of the following:

1. it is composed by only one atom;
2. it is used by a colvar component which does not make use of its center of geometry, but only of pairwise distances (distanceInv, coordNum, hBond, alpha, dihedralPC);
3. it is used by a colvar component that ignores the ill-deﬁned Cartesian components of its center of mass (such as the $x$ and $y$ components of a membrane’s center of mass modeled with distanceZ).

#### 5.4 Performance of a Colvars calculation based on group size.

In simulations performed with message-passing programs (such as NAMD or LAMMPS), the calculation of energy and forces is distributed (i.e., parallelized) across multiple nodes, as well as over the processor cores of each node. When Colvars is enabled, certain atomic coordinates are collected on a single node, where the calculation of collective variables and of their biases is executed. This means that for simulations over large numbers of nodes, a Colvars calculation may produce a signiﬁcant overhead, coming from the costs of transmitting atomic coordinates to one node and of processing them.

Performance can be improved in multiple ways:

• The calculation of variables, components and biases can be distributed over the processor cores of the node where the Colvars module is executed. Currently, an equal weight is assigned to each colvar, or to each component of those colvars that include more than one component. The performance of simulations that use many colvars or components is improved automatically. For simulations that use a single large colvar, it may be advisable to partition it in multiple components, which will be then distributed across the available cores. In LAMMPS, this feature is supported automatically when LAMMPS is compiled with OpenMP support. If printed, the message “SMP parallelism is available.” indicates the availability of the option. If available, the option is turned on by default, but may be disabled using the keyword smp if required for debugging.
• As a general rule, the size of atom groups should be kept relatively small (up to a few thousands of atoms, depending on the size of the entire system in comparison). To gain an estimate of the computational cost of a large colvar, one can use a test calculation of the same colvar in VMD (hint: use the time Tcl command to measure the cost of running cv update).

### 6 Biasing and analysis methods

All of the biasing and analysis methods implemented recognize the following options:

• name $⟨\phantom{\rule{0.3em}{0ex}}$Identiﬁer for the bias$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar bias
Acceptable values: string
Default value: $<$type of bias$><$bias index$>$
Description: This string is used to identify the bias or analysis method in output messages and to name some output ﬁles.
• colvars $⟨\phantom{\rule{0.3em}{0ex}}$Collective variables involved$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar bias
Acceptable values: space-separated list of colvar names
Description: This option selects by name all the colvars to which this bias or analysis will be applied.
• outputEnergy $⟨\phantom{\rule{0.3em}{0ex}}$Write the current bias energy to the trajectory ﬁle$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar bias
Acceptable values: boolean
Default value: off
Description: If this option is chosen and colvarsTrajFrequency is not zero, the current value of the biasing energy will be written to the trajectory ﬁle during the simulation.

In addition, restraint biases (6.4, 6.6, 6.7, ...) and metadynamics biases (6.3) oﬀer the following optional keywords, which allow the use of thermodynamic integration (TI) to compute potentials of mean force (PMFs). In adaptive biasing force (ABF) biases (6.1) the same keywords are not recognized because their functionality is always included.

• writeTIPMF $⟨\phantom{\rule{0.3em}{0ex}}$Write the PMF computed by thermodynamic integration$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar bias
Acceptable values: boolean
Default value: off
Description: If the bias is applied to a variable that supports the calculation of total forces (see outputTotalForce and 4.10.1), this option allows calculating the corresponding PMF by thermodynanic integration, and writing it to the ﬁle output.$<$name$>$.ti.pmf, where $<$name$>$ is the name of the bias. The total force includes the forces applied to the variable by all bias, except those from this bias itself. If any bias applies time-dependent forces besides the one using this option, an error is raised.
• writeTISamples $⟨\phantom{\rule{0.3em}{0ex}}$Write the free-energy gradient samples$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar bias
Acceptable values: boolean
Default value: off
Description: This option allows to compute total forces for use with thermodynamic integration as done by the keyword writeTIPMF. The names of the ﬁles containing the variables’ histogram and mean thermodyunamic forces are output.$<$name$>$.ti.count and output.$<$name$>$.ti.grad, respectively: these can be used by abf_integrate or similar utility. This option on by default when writeTIPMF is on, but can be enabled separately if the bias is applied to more than one variable, making not possible the direct integration of the PMF at runtime. If any bias applies time-dependent forces besides the one using this option, an error is raised.

For a full description of the Adaptive Biasing Force method, see reference [4]. For details about this implementation, see references [5] and [6]. When publishing research that makes use of this functionality, please cite references [4] and [6].

An alternate usage of this feature is the application of custom tabulated biasing potentials to one or more colvars. See inputPrefix and updateBias below.

Combining ABF with the extended Lagrangian feature (4.15) of the variables produces the extended-system ABF variant of the method (6.2).

ABF is based on the thermodynamic integration (TI) scheme for computing free energy proﬁles. The free energy as a function of a set of collective variables $\text{}\xi \text{}={\left({\xi }_{i}\right)}_{i\in \left[1,n\right]}$ is deﬁned from the canonical distribution of $\text{}\xi \text{}$, $\mathsc{𝒫}\left(\text{}\xi \text{}\right)$:

 $A\left(\text{}\xi \text{}\right)=-\frac{1}{\beta }ln\mathsc{𝒫}\left(\text{}\xi \text{}\right)+{A}_{0}$ (11)

In the TI formalism, the free energy is obtained from its gradient, which is generally calculated in the form of the average of a force ${\text{}F\text{}}_{\xi }$ exerted on $\text{}\xi \text{}$, taken over an iso-$\text{}\xi \text{}$ surface:

 ${\text{}\nabla \text{}}_{\xi }A\left(\text{}\xi \text{}\right)={⟨-{\text{}F\text{}}_{\xi }⟩}_{\text{}\xi \text{}}$ (12)

Several formulae that take the form of (12) have been proposed. This implementation relies partly on the classic formulation [7], and partly on a more versatile scheme originating in a work by Ruiz-Montero et al. [8], generalized by den Otter [9] and extended to multiple variables by Ciccotti et al. [10]. Consider a system subject to constraints of the form ${\sigma }_{k}\left(\text{}x\text{}\right)=0$. Let ${\left({\text{}v\text{}}_{i}\right)}_{i\in \left[1,n\right]}$ be arbitrarily chosen vector ﬁelds (${ℝ}^{3N}\to {ℝ}^{3N}$) verifying, for all $i$, $j$, and $k$:

$\begin{array}{rcll}{\text{}v\text{}}_{i}\cdot \text{}{\nabla }_{\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}x}\phantom{\rule{0.3em}{0ex}}\text{}{\xi }_{j}& =& {\delta }_{ij}& \text{(13)}\text{}\text{}\\ {\text{}v\text{}}_{i}\cdot \text{}{\nabla }_{\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}x}\phantom{\rule{0.3em}{0ex}}\text{}{\sigma }_{k}& =& 0& \text{(14)}\text{}\text{}\end{array}$

then the following holds [10]:

 $\frac{\partial A}{\partial {\xi }_{i}}={⟨{\text{}v\text{}}_{i}\cdot \text{}{\nabla }_{\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}x}\phantom{\rule{0.3em}{0ex}}\text{}V-{k}_{B}T\text{}{\nabla }_{\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}x}\phantom{\rule{0.3em}{0ex}}\text{}\cdot {\text{}v\text{}}_{i}⟩}_{\text{}\xi \text{}}$ (15)

where $V$ is the potential energy function. ${\text{}v\text{}}_{i}$ can be interpreted as the direction along which the force acting on variable ${\xi }_{i}$ is measured, whereas the second term in the average corresponds to the geometric entropy contribution that appears as a Jacobian correction in the classic formalism [7]. Condition (13) states that the direction along which the total force on ${\xi }_{i}$ is measured is orthogonal to the gradient of ${\xi }_{j}$, which means that the force measured on ${\xi }_{i}$ does not act on ${\xi }_{j}$.

Equation (14) implies that constraint forces are orthogonal to the directions along which the free energy gradient is measured, so that the measurement is eﬀectively performed on unconstrained degrees of freedom.

In the framework of ABF, ${F}_{\xi }$ is accumulated in bins of ﬁnite size $\delta \xi$, thereby providing an estimate of the free energy gradient according to equation (12). The biasing force applied along the collective variables to overcome free energy barriers is calculated as:

 ${F}^{ABF}=\alpha \left({N}_{\xi }\right)×\text{}{\nabla }_{\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}x}\phantom{\rule{0.3em}{0ex}}\text{}\stackrel{̃}{A}\left(\text{}\xi \text{}\right)$ (16)

where $\text{}{\nabla }_{\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}x}\phantom{\rule{0.3em}{0ex}}\text{}\stackrel{̃}{A}$ denotes the current estimate of the free energy gradient at the current point $\text{}\xi \text{}$ in the collective variable subspace, and $\alpha \left({N}_{\xi }\right)$ is a scaling factor that is ramped from 0 to 1 as the local number of samples ${N}_{\xi }$ increases to prevent nonequilibrium eﬀects in the early phase of the simulation, when the gradient estimate has a large variance. See the fullSamples parameter below for details.

As sampling of the phase space proceeds, the estimate $\text{}{\nabla }_{\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}x}\phantom{\rule{0.3em}{0ex}}\text{}\stackrel{̃}{A}$ is progressively reﬁned. The biasing force introduced in the equations of motion guarantees that in the bin centered around $\text{}\xi \text{}$, the forces acting along the selected collective variables average to zero over time. Eventually, as the undelying free energy surface is canceled by the adaptive bias, evolution of the system along $\text{}\xi \text{}$ is governed mainly by diﬀusion. Although this implementation of ABF can in principle be used in arbitrary dimension, a higher-dimension collective variable space is likely to result in sampling diﬃculties. Most commonly, the number of variables is one or two.

##### 6.1.1 ABF requirements on collective variables

The following conditions must be met for an ABF simulation to be possible and to produce an accurate estimate of the free energy proﬁle. Note that these requirements do not apply when using the extended-system ABF method (6.2).

1. Only linear combinations of colvar components can be used in ABF calculations.
2. Availability of total forces is necessary. The following colvar components can be used in ABF calculations: distance, distance_xy, distance_z, angle, dihedral, gyration, rmsd and eigenvector. Atom groups may not be replaced by dummy atoms, unless they are excluded from the force measurement by specifying oneSiteTotalForce, if available.
3. Mutual orthogonality of colvars. In a multidimensional ABF calculation, equation (13) must be satisﬁed for any two colvars ${\xi }_{i}$ and ${\xi }_{j}$. Various cases fulﬁll this orthogonality condition:
• ${\xi }_{i}$ and ${\xi }_{j}$ are based on non-overlapping sets of atoms.
• atoms involved in the force measurement on ${\xi }_{i}$ do not participate in the deﬁnition of ${\xi }_{j}$. This can be obtained using the option oneSiteTotalForce of the distance, angle, and dihedral components (example: Ramachandran angles $\varphi$, $\psi$).
• ${\xi }_{i}$ and ${\xi }_{j}$ are orthogonal by construction. Useful cases are the sum and diﬀerence of two components, or distance_z and distance_xy using the same axis.
4. Mutual orthogonality of components: when several components are combined into a colvar, it is assumed that their vectors ${\text{}v\text{}}_{i}$ (equation (15)) are mutually orthogonal. The cases described for colvars in the previous paragraph apply.
5. Orthogonality of colvars and constraints: equation 14 can be satisﬁed in two simple ways, if either no constrained atoms are involved in the force measurement (see point 3 above) or pairs of atoms joined by a constrained bond are part of an atom group which only intervenes through its center (center of mass or geometric center) in the force measurement. In the latter case, the contributions of the two atoms to the left-hand side of equation 14 cancel out. For example, all atoms of a rigid TIP3P water molecule can safely be included in an atom group used in a distance component.

##### 6.1.2 Parameters for ABF

ABF depends on parameters from collective variables to deﬁne the grid on which free energy gradients are computed. In the direction of each colvar, the grid ranges from lowerBoundary to upperBoundary, and the bin width (grid spacing) is set by the width parameter. The following speciﬁc parameters can be set in the ABF conﬁguration block:

• name: see deﬁnition of name in sec. 6 (biasing and analysis methods)
• colvars: see deﬁnition of colvars in sec. 6 (biasing and analysis methods)
• fullSamples $⟨\phantom{\rule{0.3em}{0ex}}$Number of samples in a bin prior to application of the ABF$\phantom{\rule{0.3em}{0ex}}⟩$
Context: abf
Acceptable values: positive integer
Default value: 200
Description: To avoid nonequilibrium eﬀects due to large ﬂuctuations of the force exerted along the colvars, it is recommended to apply a biasing force only after a the estimate has started converging. If fullSamples is non-zero, the applied biasing force is scaled by a factor $\alpha \left({N}_{\xi }\right)$ between 0 and 1. If the number of samples ${N}_{\xi }$ in the current bin is higher than fullSamples, the factor is one. If it is less than half of fullSamples, the factor is zero and no bias is applied. Between those two thresholds, the factor follows a linear ramp from 0 to 1: $\alpha \left({N}_{\xi }\right)=\left(2{N}_{\xi }∕\mathrm{fullSamples}\right)-1$ .
• maxForce $⟨\phantom{\rule{0.3em}{0ex}}$Maximum magnitude of the ABF force$\phantom{\rule{0.3em}{0ex}}⟩$
Context: abf
Acceptable values: positive decimals (one per colvar)
Default value: disabled
Description: This option enforces a cap on the magnitude of the biasing force eﬀectively applied by this ABF bias on each colvar. This can be useful in the presence of singularities in the PMF such as hard walls, where the discretization of the average force becomes very inaccurate, causing the colvar’s diﬀusion to get “stuck” at the singularity. To enable this cap, provide one non-negative value for each colvar. The unit of force is the unit of energy speciﬁed by units divided by the colvar unit.
• hideJacobian $⟨\phantom{\rule{0.3em}{0ex}}$Remove geometric entropy term from calculated free energy gradient?$\phantom{\rule{0.3em}{0ex}}⟩$
Context: abf
Acceptable values: boolean
Default value: no
Description: In a few special cases, most notably distance-based variables, an alternate deﬁnition of the potential of mean force is traditionally used, which excludes the Jacobian term describing the eﬀect of geometric entropy on the distribution of the variable. This results, for example, in particle-particle potentials of mean force being ﬂat at large separations. Setting this parameter to yes causes the output data to follow that convention, by removing this contribution from the output gradients while applying internally the corresponding correction to ensure uniform sampling. It is not allowed for colvars with multiple components.
• outputFreq $⟨\phantom{\rule{0.3em}{0ex}}$Frequency (in timesteps) at which ABF data ﬁles are refreshed$\phantom{\rule{0.3em}{0ex}}⟩$
Context: abf
Acceptable values: positive integer
Default value: Colvars module restart frequency
Description: The ﬁles containing the free energy gradient estimate and sampling histogram (and the PMF in one-dimensional calculations) are written on disk at the given time interval.
• historyFreq $⟨\phantom{\rule{0.3em}{0ex}}$Frequency (in timesteps) at which ABF history ﬁles are accumulated$\phantom{\rule{0.3em}{0ex}}⟩$
Context: abf
Acceptable values: positive integer
Default value: 0
Description: If this number is non-zero, the free energy gradient estimate and sampling histogram (and the PMF in one-dimensional calculations) are appended to ﬁles on disk at the given time interval. History ﬁle names use the same preﬁx as output ﬁles, with “.hist” appended.
• inputPrefix $⟨\phantom{\rule{0.3em}{0ex}}$Filename preﬁx for reading ABF data$\phantom{\rule{0.3em}{0ex}}⟩$
Context: abf
Acceptable values: list of strings
Description: If this parameter is set, for each item in the list, ABF tries to read a gradient and a sampling ﬁles named $<$inputPrefix$>$.grad and $<$inputPrefix$>$.count. This is done at startup and sets the initial state of the ABF algorithm. The data from all provided ﬁles is combined appropriately. Also, the grid deﬁnition (min and max values, width) need not be the same that for the current run. This command is useful to piece together data from simulations in diﬀerent regions of collective variable space, or change the colvar boundary values and widths. Note that it is not recommended to use it to switch to a smaller width, as that will leave some bins empty in the ﬁner data grid. This option is NOT compatible with reading the data from a restart ﬁle (input keyword of the fix ID group-ID colvars command).
• applyBias $⟨\phantom{\rule{0.3em}{0ex}}$Apply the ABF bias?$\phantom{\rule{0.3em}{0ex}}⟩$
Context: abf
Acceptable values: boolean
Default value: yes
Description: If this is set to no, the calculation proceeds normally but the adaptive biasing force is not applied. Data is still collected to compute the free energy gradient. This is mostly intended for testing purposes, and should not be used in routine simulations.
• updateBias $⟨\phantom{\rule{0.3em}{0ex}}$Update the ABF bias?$\phantom{\rule{0.3em}{0ex}}⟩$
Context: abf
Acceptable values: boolean
Default value: yes
Description: If this is set to no, the initial biasing force (e.g. read from a restart ﬁle or through inputPrefix) is not updated during the simulation. As a result, a constant bias is applied. This can be used to apply a custom, tabulated biasing potential to any combination of colvars. To that eﬀect, one should prepare a gradient ﬁle containing the gradient of the potential to be applied (negative of the bias force), and a count ﬁle containing only values greater than fullSamples. These ﬁles must match the grid parameters of the colvars.

##### 6.1.3 Output ﬁles

The ABF bias produces the following ﬁles, all in multicolumn text format:

• output.grad: current estimate of the free energy gradient (grid), in multicolumn;
• output.count: histogram of samples collected, on the same grid;
• output.pmf: only for one-dimensional calculations, integrated free energy proﬁle or PMF.

If several ABF biases are deﬁned concurrently, their name is inserted to produce unique ﬁlenames for output, as in output.abf1.grad. This should not be done routinely and could lead to meaningless results: only do it if you know what you are doing!

If the colvar space has been partitioned into sections (windows) in which independent ABF simulations have been run, the resulting data can be merged using the inputPrefix option described above (a run of 0 steps is enough).

##### 6.1.4 Post-processing: reconstructing a multidimensional free energy surface

If a one-dimensional calculation is performed, the estimated free energy gradient is automatically integrated and a potential of mean force is written under the ﬁle name .pmf, in a plain text format that can be read by most data plotting and analysis programs (e.g. gnuplot).

In dimension 2 or greater, integrating the discretized gradient becomes non-trivial. The standalone utility abf_integrate is provided to perform that task. abf_integrate reads the gradient data and uses it to perform a Monte-Carlo (M-C) simulation in discretized collective variable space (speciﬁcally, on the same grid used by ABF to discretize the free energy gradient). By default, a history-dependent bias (similar in spirit to metadynamics) is used: at each M-C step, the bias at the current position is incremented by a preset amount (the hill height). Upon convergence, this bias counteracts optimally the underlying gradient; it is negated to obtain the estimate of the free energy surface.

abf_integrate is invoked using the command-line:
abf_integrate _file> [-n ] [-t ] [-m (0|1)] [-h _height>] [-f ]

The gradient ﬁle name is provided ﬁrst, followed by other parameters in any order. They are described below, with their default value in square brackets:

• -n: number of M-C steps to be performed; by default, a minimal number of steps is chosen based on the size of the grid, and the integration runs until a convergence criterion is satisﬁed (based on the RMSD between the target gradient and the real PMF gradient)
• -t: temperature for M-C sampling (unrelated to the simulation temperature) [500 K]
• -m: use metadynamics-like biased sampling? (0 = false) [1]
• -h: increment for the history-dependent bias (“hill height”) [0.01 kcal/mol]
• -f: if non-zero, this factor is used to scale the increment stepwise in the second half of the M-C sampling to reﬁne the free energy estimate [0.5]

Using the default values of all parameters should give reasonable results in most cases.

abf_integrate produces the following output ﬁles:

• _file>.pmf: computed free energy surface
• _file>.histo: histogram of M-C sampling (not usable in a straightforward way if the history-dependent bias has been applied)
• _file>.est: estimated gradient of the calculated free energy surface (from ﬁnite diﬀerences)
• _file>.dev: deviation between the user-provided numerical gradient and the actual gradient of the calculated free energy surface. The RMS norm of this vector ﬁeld is used as a convergence criteria and displayed periodically during the integration.

Note: Typically, the “deviation” vector ﬁeld does not vanish as the integration converges. This happens because the numerical estimate of the gradient does not exactly derive from a potential, due to numerical approximations used to obtain it (ﬁnite sampling and discretization on a grid).

#### 6.2 Extended-system Adaptive Biasing Force (eABF)

Extended-system ABF (eABF) is a variant of ABF (6.1) where the bias is not applied directly to the collective variable, but to an extended coordinate (“ﬁctitious variable”) $\lambda$ that evolves dynamically according to Newtonian or Langevin dynamics. Such an extended coordinate is enabled for a given colvar using the extendedLagrangian and associated keywords (4.15). The theory of eABF and the present implementation are documented in detail in reference [11].

Deﬁning an ABF bias on a colvar wherein the extendedLagrangian option is active will perform eABF; there is no dedicated option.

The extended variable $\lambda$ is coupled to the colvar $z=\xi \left(q\right)$ by the harmonic potential $\left(k∕2\right){\left(z-\lambda \right)}^{2}$. Under eABF dynamics, the adaptive bias on $\lambda$ is the running estimate of the average spring force:

 ${F}^{\mathrm{bias}}\left({\lambda }^{\ast }\right)={⟨k\left(\lambda -z\right)⟩}_{{\lambda }^{\ast }}$ (17)

where the angle brackets indicate a canonical average conditioned by $\lambda ={\lambda }^{\ast }$. At long simulation times, eABF produces a ﬂat histogram of the extended variable $\lambda$, and a ﬂattened histogram of $\xi$, whose exact shape depends on the strength of the coupling as deﬁned by extendedFluctuation in the colvar. Coupling should be somewhat loose for faster exploration and convergence, but strong enough that the bias does help overcome barriers along the colvar $\xi$.[11] Distribution of the colvar may be assessed by plotting its histogram, which is written to the output.zcount ﬁle in every eABF simulation. Note that a histogram bias (6.9) applied to an extended-Lagrangian colvar will access the extended degree of freedom $\lambda$, not the original colvar $\xi$; however, the joint histogram may be explicitly requested by listing the name of the colvar twice in a row within the colvars parameter of the histogram block.

The eABF PMF is that of the coordinate $\lambda$, it is not exactly the free energy proﬁle of $\xi$. That quantity can be calculated based on the CZAR estimator.

##### 6.2.1 CZAR estimator of the free energy

The corrected z-averaged restraint (CZAR) estimator is described in detail in reference [11]. It is computed automatically in eABF simulations, regardless of the number of colvars involved. Note that ABF may also be applied on a combination of extended and non-extended colvars; in that case, CZAR still provides an unbiased estimate of the free energy gradient.

CZAR estimates the free energy gradient as:

 ${A}^{\prime }\left(z\right)=-\frac{1}{\beta }\frac{dln\stackrel{̃}{\rho }\left(z\right)}{dz}+k\left({⟨\lambda ⟩}_{z}-z\right).$ (18)

where $z=\xi \left(q\right)$ is the colvar, $\lambda$ is the extended variable harmonically coupled to $z$ with a force constant $k$, and $\stackrel{̃}{\rho }\left(z\right)$ is the observed distribution (histogram) of $z$, aﬀected by the eABF bias.

Parameters for the CZAR estimator are:

• CZARestimator $⟨\phantom{\rule{0.3em}{0ex}}$Calculate CZAR estimator of the free energy?$\phantom{\rule{0.3em}{0ex}}⟩$
Context: abf
Acceptable values: boolean
Default value: yes
Description: This option is only available when ABF is performed on extended-Lagrangian colvars. When enabled, it triggers calculation of the free energy following the CZAR estimator.
• writeCZARwindowFile $⟨\phantom{\rule{0.3em}{0ex}}$Write internal data from CZAR to a separate ﬁle?$\phantom{\rule{0.3em}{0ex}}⟩$
Context: abf
Acceptable values: boolean
Default value: no
Description: When this option is enabled, eABF simulations will write a ﬁle containing the $z$-averaged restraint force under the name output.zgrad. The same information is always included in the colvars state ﬁle, which is suﬃcient for restarting an eABF simulation. These separate ﬁle is only useful when joining adjacent windows from a stratiﬁed eABF simulation, either to continue the simulation in a broader window or to compute a CZAR estimate of the PMF over the full range of the coordinate(s).

Similar to ABF, the CZAR estimator produces two output ﬁles in multicolumn text format:

• output.czar.grad: current estimate of the free energy gradient (grid), in multicolumn;
• output.czar.pmf: only for one-dimensional calculations, integrated free energy proﬁle or PMF.

The sampling histogram associated with the CZAR estimator is the $z$-histogram, which is written in the ﬁle output.zcount.

The metadynamics method uses a history-dependent potential [12] that generalizes to any type of colvars the conformational ﬂooding [13] and local elevation [14] methods, originally formulated to use as colvars the principal components of a covariance matrix or a set of dihedral angles, respectively. The metadynamics potential on the colvars $\text{}\xi \text{}=\left({\xi }_{1},{\xi }_{2},\dots ,{\xi }_{{N}_{\mathrm{cv}}}\right)$ is deﬁned as:

 ${V}_{\mathrm{meta}}\left(\text{}\xi \text{}\left(t\right)\right)\phantom{\rule{3.04074pt}{0ex}}=\phantom{\rule{3.04074pt}{0ex}}\sum _{{t}^{\prime }=\delta t,2\delta t,\dots }^{{t}^{\prime } (19)

where ${V}_{\mathrm{meta}}$ is the history-dependent potential acting on the current values of the colvars $\text{}\xi \text{}$, and depends only parametrically on the previous values of the colvars. ${V}_{\mathrm{meta}}$ is constructed as a sum of ${N}_{\mathrm{cv}}$-dimensional repulsive Gaussian “hills”, whose height is a chosen energy constant $W$, and whose centers are the previously explored conﬁgurations $\left(\text{}\xi \text{}\left(\delta t\right),\text{}\xi \text{}\left(2\delta t\right),\dots \right)$.

During the simulation, the system evolves towards the nearest minimum of the “eﬀective” potential of mean force $Ã\left(\text{}\xi \text{}\right)$, which is the sum of the “real” underlying potential of mean force $A\left(\text{}\xi \text{}\right)$ and the the metadynamics potential, ${V}_{\mathrm{meta}}\left(\text{}\xi \text{}\right)$. Therefore, at any given time the probability of observing the conﬁguration $\text{}{\xi }^{\ast }\text{}$ is proportional to $exp\left(-Ã\left(\text{}{\xi }^{\ast }\text{}\right)∕{\kappa }_{\mathrm{B}}T\right)$: this is also the probability that a new Gaussian “hill” is added at that conﬁguration. If the simulation is run for a suﬃciently long time, each local minimum is canceled out by the sum of the Gaussian “hills”. At that stage the “eﬀective” potential of mean force $Ã\left(\text{}\xi \text{}\right)$ is constant, and $-{V}_{\mathrm{meta}}\left(\text{}\xi \text{}\right)$ is an accurate estimator of the “real” potential of mean force $A\left(\text{}\xi \text{}\right)$, save for an additive constant:

 $A\left(\text{}\xi \text{}\right)\phantom{\rule{3.04074pt}{0ex}}\simeq \phantom{\rule{3.04074pt}{0ex}}-{V}_{\mathrm{meta}}\left(\text{}\xi \text{}\right)+K$ (20)

Assuming that the set of collective variables includes all relevant degrees of freedom, the predicted error of the estimate is a simple function of the correlation times of the colvars ${\tau }_{{\xi }_{i}}$, and of the user-deﬁned parameters $W$, ${\sigma }_{{\xi }_{i}}$ and $\delta t$ [15]. In typical applications, a good rule of thumb can be to choose the ratio $W∕\delta t$ much smaller than ${\kappa }_{\mathrm{B}}T∕{\tau }_{\text{}\xi \text{}}$, where ${\tau }_{\text{}\xi \text{}}$ is the longest among $\text{}\xi \text{}$’s correlation times: ${\sigma }_{{\xi }_{i}}$ then dictates the resolution of the calculated PMF.

##### 6.3.1 Basic syntax

To enable a metadynamics calculation, a metadynamics {...} block must be deﬁned in the Colvars conﬁguration ﬁle. Its mandatory keywords are colvars, which lists all the variables involved, hillWeight, which speciﬁes the weight parameter $W$, and hillWidth, which deﬁnes the Gaussian width $2{\sigma }_{\xi }$ as a number of grid points.

• name: see deﬁnition of name in sec. 6 (biasing and analysis methods)
• colvars: see deﬁnition of colvars in sec. 6 (biasing and analysis methods)
• outputEnergy: see deﬁnition of outputEnergy in sec. 6 (biasing and analysis methods)
• writeTIPMF: see deﬁnition of writeTIPMF in sec. 6 (biasing and analysis methods)
• writeTISamples: see deﬁnition of writeTISamples in sec. 6 (biasing and analysis methods)
• hillWeight $⟨\phantom{\rule{0.3em}{0ex}}$Height of each hill (unit of energy speciﬁed by units)$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: positive decimal
Description: This option sets the height $W$ of the Gaussian hills that are added during this run. Lower values provide more accurate sampling of the system’s degrees of freedom at the price of longer simulation times to complete a PMF calculation based on metadynamics.
• hillWidth $⟨\phantom{\rule{0.3em}{0ex}}$Width of a Gaussian hill, measured in number of grid points$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: positive decimal
Description: The value of this keyword is the Gaussian width $2{\sigma }_{{\xi }_{i}}$, expressed in number of grid points: the grid spacing is determined by width. The values of the parameter ${\sigma }_{{\xi }_{i}}$ for each variable ${\xi }_{i}$ inits physical units are printed by LAMMPS at initialization time. Values between 1 and 3 are recommended: smaller values fail to adequately interpolate each Gaussian [1], while larger values may be unable to account for steep free-energy gradients.
• newHillFrequency $⟨\phantom{\rule{0.3em}{0ex}}$Frequency of hill creation$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: positive integer
Default value: 1000
Description: This option sets the number of steps after which a new Gaussian hill is added to the metadynamics potential. The product of this number and the integration time-step deﬁnes the parameter $\delta t$ in eq. 19. Higher values provide more accurate statistical sampling, at the price of longer simulation times to complete a PMF calculation.

##### 6.3.2 Output ﬁles

When interpolating grids are enabled (default behavior), the PMF is written by default every colvarsRestartFrequency steps to the ﬁle output.pmf. The following two options allow to disable or control this behavior and to track statistical convergence:

• writeFreeEnergyFile $⟨\phantom{\rule{0.3em}{0ex}}$Periodically write the PMF for visualization$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: boolean
Default value: on
Description: When useGrids and this option are on, the PMF is written every colvarsRestartFrequency steps.
• keepFreeEnergyFiles $⟨\phantom{\rule{0.3em}{0ex}}$Keep all the PMF ﬁles$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: boolean
Default value: off
Description: When writeFreeEnergyFile and this option are on, the step number is included in the ﬁle name, thus generating a series of PMF ﬁles. Activating this option can be useful to follow more closely the convergence of the simulation, by comparing PMFs separated by short times.

Note: when Gaussian hills are deposited near the lowerBoundary or upperBoundary and interpolating grids are used (default behavior), their truncation can give rise to accumulating errors. In these cases, as a measure of fault-tolerance all Gaussian hills near the boundaries are included in the output state ﬁle, and are recalculated analytically whenever the variable falls outside the grid’s boundaries. (Such measure protects the accuracy of the calculation, and can only be disabled by hardLowerBoundary or hardUpperBoundary.) To avoid gradual loss of performance and growth of the state ﬁle, either one of the following solutions is recommended:

• enabling the option expandBoundaries, so that the grid’s boundaries are automatically recalculated whenever necessary; the resulting .pmf will have its abscissas expanded accordingly;
• applying a harmonicWalls bias with the wall locations well within the interval delimited by lowerBoundary and upperBoundary.

##### 6.3.3 Performance tuning

The following options control the computational cost of metadynamics calculations, but do not aﬀect results. Default values are chosen to minimize such cost with no loss of accuracy.

• useGrids $⟨\phantom{\rule{0.3em}{0ex}}$Interpolate the hills with grids$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: boolean
Default value: on
Description: This option discretizes all hills for improved performance, accumulating their energy and their gradients on two separate grids of equal spacing. Grids are deﬁned by the values of lowerBoundary, upperBoundary and width for each colvar. Currently, this option is implemented for all types of variables except the non-scalar types (distanceDir or orientation). If expandBoundaries is deﬁned in one of the colvars, grids are automatically expanded along the direction of that colvar.
• rebinGrids $⟨\phantom{\rule{0.3em}{0ex}}$Recompute the grids when reading a state ﬁle$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: boolean
Default value: off
Description: When restarting from a state ﬁle, the grid’s parameters (boundaries and widths) saved in the state ﬁle override those in the conﬁguration ﬁle. Enabling this option forces the grids to match those in the current conﬁguration ﬁle.

The following options deﬁne the conﬁguration for the “well-tempered” metadynamics approach [16]:

• wellTempered $⟨\phantom{\rule{0.3em}{0ex}}$Perform well-tempered metadynamics$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: boolean
Default value: off
Description: If enabled, this ﬂag causes well-tempered metadynamics as described by Barducci et al.[16] to be performed, rather than standard metadynamics. The parameter biasTemperature is then required.This feature was contributed by Li Li (Luthey-Schulten group, Departement of Chemistry, UIUC).
• biasTemperature $⟨\phantom{\rule{0.3em}{0ex}}$Temperature bias for well-tempered metadynamics$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: positive decimal
Description: When running metadynamics in the long time limit, collective variable space is sampled to a modiﬁed temperature $T+\Delta T$. In conventional metadynamics, the temperature “boost” $\Delta T$ would constantly increases with time. Instead, in well-tempered metadynamics $\Delta T$ must be deﬁned by the user via biasTemperature. The written PMF includes the scaling factor $\left(T+\Delta T\right)∕\Delta T$ [16]. A careful choice of $\Delta T$ determines the sampling and convergence rate, and is hence crucial to the success of a well-tempered metadynamics simulation.

The following options deﬁne metadynamics calculations with more than one replica:

• multipleReplicas $⟨\phantom{\rule{0.3em}{0ex}}$Multiple replicas metadynamics$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: boolean
Default value: off
Description: If this option is on, multiple (independent) replica of the same system can be run at the same time, and their hills will be combined to obtain a single PMF [17]. Replicas are identiﬁed by the value of replicaID. Communication is done by ﬁles: each replica must be able to read the ﬁles created by the others, whose paths are communicated through the ﬁle replicasRegistry. This ﬁle, and the ﬁles listed in it, are read every replicaUpdateFrequency steps. Every time the colvars state ﬁle is written (colvarsRestartFrequency), the ﬁle:
output.colvars.name.replicaID.state” is also written, containing the state of the metadynamics bias for replicaID. In the time steps between colvarsRestartFrequency, new hills are temporarily written to the ﬁle:
output.colvars.name.replicaID.hills”, which serves as communication buﬀer. These ﬁles are only required for communication, and may be deleted after a new MD run is started with a diﬀerent outputName.
• replicaID $⟨\phantom{\rule{0.3em}{0ex}}$Set the identiﬁer for this replica$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: string
Description: If multipleReplicas is on, this option sets a unique identiﬁer for this replica. All replicas should use identical collective variable conﬁgurations, except for the value of this option.
• replicasRegistry $⟨\phantom{\rule{0.3em}{0ex}}$Multiple replicas database ﬁle$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: UNIX ﬁlename
Default value: name.replica_files.txt
Description: If multipleReplicas is on, this option sets the path to the replicas’ database ﬁle.
• replicaUpdateFrequency $⟨\phantom{\rule{0.3em}{0ex}}$How often hills are communicated between replicas$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: positive integer
Default value: newHillFrequency
Description: If multipleReplicas is on, this option sets the number of steps after which each replica (re)reads the other replicas’ ﬁles. The lowest meaningful value of this number is newHillFrequency. If access to the ﬁle system is signiﬁcantly aﬀecting the simulation performance, this number can be increased, at the price of reduced synchronization between replicas. Values higher than colvarsRestartFrequency may not improve performance signiﬁcantly.
• dumpPartialFreeEnergyFile $⟨\phantom{\rule{0.3em}{0ex}}$Periodically write the contribution to the PMF from this replica$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: boolean
Default value: on
Description: When multipleReplicas is on, the ﬁle output.pmf contains the combined PMF from all replicas, provided that useGrids is on (default). Enabling this option produces an additional ﬁle output.partial.pmf, which can be useful to quickly monitor the contribution of each replica to the PMF.

##### 6.3.6 Compatibility and post-processing

The following options may be useful only for applications that go beyond the calculation of a PMF by metadynamics:

• name $⟨\phantom{\rule{0.3em}{0ex}}$Name of this metadynamics instance$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: string
Default value: meta” + rank number
Description: This option sets the name for this metadynamics instance. While it is not advisable to use more than one metadynamics instance within the same simulation, this allows to distinguish each instance from the others. If there is more than one metadynamics instance, the name of this bias is included in the metadynamics output ﬁle names, such as e.g. the .pmf ﬁle.
• keepHills $⟨\phantom{\rule{0.3em}{0ex}}$Write each individual hill to the state ﬁle$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: boolean
Default value: off
Description: When useGrids and this option are on, all hills are saved to the state ﬁle in their analytic form, alongside their grids. This makes it possible to later use exact analytic Gaussians for rebinGrids. To only keep track of the history of the added hills, writeHillsTrajectory is preferable.
• writeHillsTrajectory $⟨\phantom{\rule{0.3em}{0ex}}$Write a log of new hills$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: boolean
Default value: off
Description: If this option is on, a logﬁle is written by the metadynamics bias, with the name “output.colvars.$<$name$>$.hills.traj”, which can be useful to follow the time series of the hills. When multipleReplicas is on, its name changes to
output.colvars.$<$name$>$.$<$replicaID$>$.hills.traj”. This ﬁle can be used to quickly visualize the positions of all added hills, in case newHillFrequency does not coincide with colvarsRestartFrequency.

#### 6.4 Harmonic restraints

The harmonic biasing method may be used to enforce ﬁxed or moving restraints, including variants of Steered and Targeted MD. Within energy minimization runs, it allows for restrained minimization, e.g. to calculate relaxed potential energy surfaces. In the context of the Colvars module, harmonic potentials are meant according to their textbook deﬁnition:

 $V\left(\xi \right)=\frac{1}{2}k{\left(\frac{\xi -{\xi }_{0}}{{w}_{\xi }}\right)}^{2}$ (21)

Note that this diﬀers from harmonic bond and angle potentials in common force ﬁelds, where the factor of one half is typically omitted, resulting in a non-standard deﬁnition of the force constant.

The formula above includes the characteristic length scale ${w}_{\xi }$ of the colvar $\xi$ (keyword width) to allow the deﬁnition of a multi-dimensional restraint with a uniﬁed force constant:

 $V\left({\xi }_{1},\dots ,{\xi }_{M}\right)=\frac{1}{2}k\sum _{i=1}^{M}{\left(\frac{{\xi }_{i}-{\xi }_{0}}{{w}_{\xi }}\right)}^{2}$ (22)

If one-dimensional or homogeneous multi-dimensional restraints are deﬁned, and there are no other uses for the parameter ${w}_{\xi }$, width can be left at its default value of $1$.

A harmonic restraint is set up by a harmonic {...} block, which may contain the following keywords:

• name: see deﬁnition of name in sec. 6 (biasing and analysis methods)
• colvars: see deﬁnition of colvars in sec. 6 (biasing and analysis methods)
• outputEnergy: see deﬁnition of outputEnergy in sec. 6 (biasing and analysis methods)
• writeTIPMF: see deﬁnition of writeTIPMF in sec. 6 (biasing and analysis methods)
• writeTISamples: see deﬁnition of writeTISamples in sec. 6 (biasing and analysis methods)
• forceConstant $⟨\phantom{\rule{0.3em}{0ex}}$Scaled force constant (unit of energy speciﬁed by units)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: harmonic
Acceptable values: positive decimal
Default value: 1.0
Description: This option deﬁnes a scaled force constant $k$ for the harmonic potential (eq. 22). To ensure consistency for multidimensional restraints, it is divided internally by the square of the speciﬁc width of each variable (which is 1 by default). This makes all values eﬀectively dimensionless and of commensurate size. For instance, setting a scaled force constant of 10 kcal/mol acting on two variables, an angle with a width of 5 degrees and a distance with a width of 0.5 Å, will apply actual force constants of 0.4 kcal/mol$×$degree${}^{-2}$ for the angle and 40 kcal/mol/Å${}^{2}$ for the distance. The values of the actual force constants are always printed when the restraint is deﬁned.
• centers $⟨\phantom{\rule{0.3em}{0ex}}$Initial harmonic restraint centers$\phantom{\rule{0.3em}{0ex}}⟩$
Context: harmonic
Acceptable values: space-separated list of colvar values
Description: The centers (equilibrium values) of the restraint, ${\xi }_{0}$, are entered here. The number of values must be the number of requested colvars. Each value is a decimal number if the corresponding colvar returns a scalar, a “(x, y, z)” triplet if it returns a unit vector or a vector, and a “(q0, q1, q2, q3)” quadruplet if it returns a rotational quaternion. If a colvar has periodicities or symmetries, its closest image to the restraint center is considered when calculating the harmonic potential.

Tip: A complex set of restraints can be applied to a system, by deﬁning several colvars, and applying one or more harmonic restraints to diﬀerent groups of colvars. In some cases, dozens of colvars can be deﬁned, but their value may not be relevant: to limit the size of the colvars trajectory ﬁle, it may be wise to disable outputValue for such “ancillary” variables, and leave it enabled only for “relevant” ones.

##### 6.4.1 Moving restraints: steered molecular dynamics

The following options allow to change gradually the centers of the harmonic restraints during a simulations. When the centers are changed continuously, a steered MD in a collective variable space is carried out.

• targetCenters $⟨\phantom{\rule{0.3em}{0ex}}$Steer the restraint centers towards these targets$\phantom{\rule{0.3em}{0ex}}⟩$
Context: harmonic
Acceptable values: space-separated list of colvar values
Description: When deﬁned, the current centers will be moved towards these values during the simulation. By default, the centers are moved over a total of targetNumSteps steps by a linear interpolation, in the spirit of Steered MD. If targetNumStages is set to a nonzero value, the change is performed in discrete stages, lasting targetNumSteps steps each. This second mode may be used to sample successive windows in the context of an Umbrella Sampling simulation. When continuing a simulation run, the centers speciﬁed in the conﬁguration ﬁle $<$colvarsConfig$>$ are overridden by those saved in the restart ﬁle $<$colvarsInput$>$. To perform Steered MD in an arbitrary space of colvars, it is suﬃcient to use this option and enable outputAccumulatedWork and/or outputAppliedForce within each of the colvars involved.
• targetNumSteps $⟨\phantom{\rule{0.3em}{0ex}}$Number of steps for steering$\phantom{\rule{0.3em}{0ex}}⟩$
Context: harmonic
Acceptable values: positive integer
Description: In single-stage (continuous) transformations, deﬁnes the number of MD steps required to move the restraint centers (or force constant) towards the values speciﬁed with targetCenters or targetForceConstant. After the target values have been reached, the centers (resp. force constant) are kept ﬁxed. In multi-stage transformations, this sets the number of MD steps per stage.
• outputCenters $⟨\phantom{\rule{0.3em}{0ex}}$Write the current centers to the trajectory ﬁle$\phantom{\rule{0.3em}{0ex}}⟩$
Context: harmonic
Acceptable values: boolean
Default value: off
Description: If this option is chosen and colvarsTrajFrequency is not zero, the positions of the restraint centers will be written to the trajectory ﬁle during the simulation. This option allows to conveniently extract the PMF from the colvars trajectory ﬁles in a steered MD calculation.

Note on restarting moving restraint simulations: Information about the current step and stage of a simulation with moving restraints is stored in the restart ﬁle (state ﬁle). Thus, such simulations can be run in several chunks, and restarted directly using the same colvars conﬁguration ﬁle. In case of a restart, the values of parameters such as targetCenters, targetNumSteps, etc. should not be changed manually.

##### 6.4.2 Moving restraints: umbrella sampling

The centers of the harmonic restraints can also be changed in discrete stages: in this cases a one-dimensional umbrella sampling simulation is performed. The sampling windows in simulation are calculated in sequence. The colvars trajectory ﬁle may then be used both to evaluate the correlation times between consecutive windows, and to calculate the frequency distribution of the colvar of interest in each window. Furthermore, frequency distributions on a predeﬁned grid can be automatically obtained by using the histogram bias (see 6.9).

To activate an umbrella sampling simulation, the same keywords as in the previous section can be used, with the addition of the following:

• targetNumStages $⟨\phantom{\rule{0.3em}{0ex}}$Number of stages for steering$\phantom{\rule{0.3em}{0ex}}⟩$
Context: harmonic
Acceptable values: non-negative integer
Default value: 0
Description: If non-zero, sets the number of stages in which the restraint centers or force constant are changed to their target values. If zero, the change is continuous. Each stage lasts targetNumSteps MD steps. To sample both ends of the transformation, the simulation should be run for targetNumSteps $×$ (targetNumStages + 1).

##### 6.4.3 Changing force constant

The force constant of the harmonic restraint may also be changed to equilibrate [18].

• targetForceConstant $⟨\phantom{\rule{0.3em}{0ex}}$Change the force constant towards this value$\phantom{\rule{0.3em}{0ex}}⟩$
Context: harmonic
Acceptable values: positive decimal
Description: When deﬁned, the current forceConstant will be moved towards this value during the simulation. Time evolution of the force constant is dictated by the targetForceExponent parameter (see below). By default, the force constant is changed smoothly over a total of targetNumSteps steps. This is useful to introduce or remove restraints in a progressive manner. If targetNumStages is set to a nonzero value, the change is performed in discrete stages, lasting targetNumSteps steps each. This second mode may be used to compute the conformational free energy change associated with the restraint, within the FEP or TI formalisms. For convenience, the code provides an estimate of the free energy derivative for use in TI. A more complete free energy calculation (particularly with regard to convergence analysis), while not handled by the Colvars module, can be performed by post-processing the colvars trajectory, if colvarsTrajFrequency is set to a suitably small value. It should be noted, however, that restraint free energy calculations may be handled more eﬃciently by an indirect route, through the determination of a PMF for the restrained coordinate.[18]
• targetForceExponent $⟨\phantom{\rule{0.3em}{0ex}}$Exponent in the time-dependence of the force constant$\phantom{\rule{0.3em}{0ex}}⟩$
Context: harmonic
Acceptable values: decimal equal to or greater than 1.0
Default value: 1.0
Description: Sets the exponent, $\alpha$, in the function used to vary the force constant as a function of time. The force is varied according to a coupling parameter $\lambda$, raised to the power $\alpha$: ${k}_{\lambda }={k}_{0}+{\lambda }^{\alpha }\left({k}_{1}-{k}_{0}\right)$, where ${k}_{0}$, ${k}_{\lambda }$, and ${k}_{1}$ are the initial, current, and ﬁnal values of the force constant. The parameter $\lambda$ evolves linearly from 0 to 1, either smoothly, or in targetNumStages equally spaced discrete stages, or according to an arbitrary schedule set with lambdaSchedule. When the initial value of the force constant is zero, an exponent greater than 1.0 distributes the eﬀects of introducing the restraint more smoothly over time than a linear dependence, and ensures that there is no singularity in the derivative of the restraint free energy with respect to lambda. A value of 4 has been found to give good results in some tests.
• targetEquilSteps $⟨\phantom{\rule{0.3em}{0ex}}$Number of steps discarded from TI estimate$\phantom{\rule{0.3em}{0ex}}⟩$
Context: harmonic
Acceptable values: positive integer
Description: Deﬁnes the number of steps within each stage that are considered equilibration and discarded from the restraint free energy derivative estimate reported reported in the output.
• lambdaSchedule $⟨\phantom{\rule{0.3em}{0ex}}$Schedule of lambda-points for changing force constant$\phantom{\rule{0.3em}{0ex}}⟩$
Context: harmonic
Acceptable values: list of real numbers between 0 and 1
Description: If speciﬁed together with targetForceConstant, sets the sequence of discrete $\lambda$ values that will be used for diﬀerent stages.

#### 6.5 Computing the work of a changing restraint

If the restraint centers or force constant are changed continuosly (targetNumStages undeﬁned) it is possible to record the net work performed by the changing restraint:

• outputAccumulatedWork $⟨\phantom{\rule{0.3em}{0ex}}$Write the accumulated work of the changing restraint to the Colvars trajectory ﬁle$\phantom{\rule{0.3em}{0ex}}⟩$
Context: harmonic
Acceptable values: boolean
Default value: off
Description: If targetCenters or targetForceConstant are deﬁned and this option is enabled, the accumulated work from the beginning of the simulation will be written to the trajectory ﬁle (colvarsTrajFrequency must be non-zero). When the simulation is continued from a state ﬁle, the previously accumulated work is included in the integral. This option allows to conveniently extract the estimated PMF of a steered MD calculation (when targetCenters is used), or of other simulation protocols.

#### 6.6 Harmonic wall restraints

The harmonicWalls {...} bias is closely related to the harmonic bias (see 6.4), with the following two diﬀerences: (i) instead of a center a lower wall and/or an upper wall are deﬁned, outside of which the bias implements a half-harmonic potential;

 $V\left(\xi \right)=\left\{\begin{array}{cc}\frac{1}{2}k{\left(\frac{\xi -{\xi }_{\mathrm{upper}}}{{w}_{\xi }}\right)}^{2}\hfill & \mathrm{if}\phantom{\rule{1em}{0ex}}\xi >{\xi }_{\mathrm{upper}}\hfill \\ 0\hfill & \mathrm{if}\phantom{\rule{1em}{0ex}}{\xi }_{\mathrm{lower}}\le \xi \ge {\xi }_{\mathrm{upper}}\hfill \\ \frac{1}{2}k{\left(\frac{\xi -{\xi }_{\mathrm{lower}}}{{w}_{\xi }}\right)}^{2}\hfill & \mathrm{if}\phantom{\rule{1em}{0ex}}\xi <{\xi }_{\mathrm{lower}}\hfill \end{array}\right\$ (23)

where ${\xi }_{\mathrm{lower}}$ and ${\xi }_{\mathrm{upper}}$ are the lower and upper wall thresholds, respectively; (ii) because an interval between two walls is deﬁned, only scalar variables can be used (but any number of variables can be deﬁned, and the wall bias is intrinsically multi-dimensional).

Note: this bias replaces the keywords lowerWall, lowerWallConstant, upperWall and upperWallConstant deﬁned in the colvar context (see 4). These keywords are still supported, but are deprecated for future uses.

The harmonicWalls bias implements the following options:

• name: see deﬁnition of name in sec. 6 (biasing and analysis methods)
• colvars: see deﬁnition of colvars in sec. 6 (biasing and analysis methods)
• outputEnergy: see deﬁnition of outputEnergy in sec. 6 (biasing and analysis methods)
• writeTIPMF: see deﬁnition of writeTIPMF in sec. 6 (biasing and analysis methods)
• writeTISamples: see deﬁnition of writeTISamples in sec. 6 (biasing and analysis methods)
• lowerWalls $⟨\phantom{\rule{0.3em}{0ex}}$Position of the lower wall$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: Space-separated list of decimals
Description: Deﬁnes the values ${\xi }_{\mathrm{lower}}$ below which a conﬁning restraint on the colvar is applied to each colvar $\xi$.
• upperWalls $⟨\phantom{\rule{0.3em}{0ex}}$Position of the lower wall$\phantom{\rule{0.3em}{0ex}}⟩$
Context: colvar
Acceptable values: Space-separated list of decimals
Description: Deﬁnes the values ${\xi }_{\mathrm{upper}}$ above which a conﬁning restraint on the colvar is applied to each colvar $\xi$.
• forceConstant: see deﬁnition of forceConstant in sec. 6.4 (Harmonic restraints)
• lowerWallConstant $⟨\phantom{\rule{0.3em}{0ex}}$Force constant for the lower wall$\phantom{\rule{0.3em}{0ex}}⟩$
Context: harmonicWalls
Acceptable values: positive decimal
Default value: forceConstant
Description: When both sets of walls are deﬁned (lower and upper), this keyword allows setting diﬀerent force constants for them. The force constant reported in the output as “$k$”, and used in the change of force constant scheme, is the geometric mean of the two.
• upperWallConstant: analogous to lowerWallConstant
• targetForceConstant: see deﬁnition of targetForceConstant in sec. 6.4 (Harmonic restraints)
• targetForceConstant $⟨\phantom{\rule{0.3em}{0ex}}$Change the force constant(s) towards this value$\phantom{\rule{0.3em}{0ex}}⟩$
Context: harmonicWalls
Acceptable values: positive decimal
Description: This keyword allows changing either one or both of the wall force constants over time. In the case that lowerWallConstant and upperWallConstant have the same value, the behavior of this keyword is identical to the corresponding keyword in the harmonic restraint; otherwise, the change schedule is applied to the geometric mean of the two constant. When only one set of walls is deﬁned (lowerWall or upperWalls), only the respective force constant is changed. Note: if only one of the two force constants is meant to change over time, it is possible to use two instances of harmonicWalls, and apply the changing schedule only to one of them.
• targetNumSteps: see deﬁnition of targetNumSteps in sec. 6.4 (Harmonic restraints)
• targetForceExponent: see deﬁnition of targetForceExponent in sec. 6.4 (Harmonic restraints)
• targetEquilSteps: see deﬁnition of targetEquilSteps in sec. 6.4 (Harmonic restraints)
• targetNumStages: see deﬁnition of targetNumStages in sec. 6.4 (Harmonic restraints)
• lambdaSchedule: see deﬁnition of lambdaSchedule in sec. 6.4 (Harmonic restraints)
• outputAccumulatedWork: see deﬁnition of outputAccumulatedWork in sec. 6.4 (Harmonic restraints)

#### 6.7 Linear restraints

The linear restraint biasing method is used to minimally bias a simulation. There is generally a unique strength of bias for each CV center, which means you must know the bias force constant speciﬁcally for the center of the CV. This force constant may be found by using experiment directed simulation described in section 6.8. Please cite Pitera and Chodera when using [19].

• name: see deﬁnition of name in sec. 6 (biasing and analysis methods)
• colvars: see deﬁnition of colvars in sec. 6 (biasing and analysis methods)
• outputEnergy: see deﬁnition of outputEnergy in sec. 6 (biasing and analysis methods)
• forceConstant $⟨\phantom{\rule{0.3em}{0ex}}$Scaled force constant (unit of energy speciﬁed by units)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: linear
Acceptable values: positive decimal
Default value: 1.0
Description: This option deﬁnes a scaled force constant for the linear bias. To ensure consistency for multidimensional restraints, it is divided internally by the speciﬁc width of each variable (which is 1 by default), so that all variables are eﬀectively dimensionless and of commensurate size. The values of the actual force constants are always printed when the restraint is deﬁned.
• centers $⟨\phantom{\rule{0.3em}{0ex}}$Initial linear restraint centers$\phantom{\rule{0.3em}{0ex}}⟩$
Context: linear
Acceptable values: space-separated list of colvar values
Description: These are analogous to the centers keyword of the harmonic restraint. Although they do not aﬀect dynamics, they are here necessary to ensure a well-deﬁned energy for the linear bias.
• writeTIPMF: see deﬁnition of writeTIPMF in sec. 6 (biasing and analysis methods)
• writeTISamples: see deﬁnition of writeTISamples in sec. 6 (biasing and analysis methods)
• targetForceConstant: see deﬁnition of targetForceConstant in sec. 6.4 (Harmonic restraints)
• targetNumSteps: see deﬁnition of targetNumSteps in sec. 6.4 (Harmonic restraints)
• targetForceExponent: see deﬁnition of targetForceExponent in sec. 6.4 (Harmonic restraints)
• targetEquilSteps: see deﬁnition of targetEquilSteps in sec. 6.4 (Harmonic restraints)
• targetNumStages: see deﬁnition of targetNumStages in sec. 6.4 (Harmonic restraints)
• lambdaSchedule: see deﬁnition of lambdaSchedule in sec. 6.4 (Harmonic restraints)
• outputAccumulatedWork: see deﬁnition of outputAccumulatedWork in sec. 6.4 (Harmonic restraints)

#### 6.8 Adaptive Linear Bias/Experiment Directed Simulation

Experiment directed simulation applies a linear bias with a changing force constant. Please cite White and Voth [20] when using this feature. As opposed to that reference, the force constant here is scaled by the width corresponding to the biased colvar. In White and Voth, each force constant is scaled by the colvars set center. The bias converges to a linear bias, after which it will be the minimal possible bias. You may also stop the simulation, take the median of the force constants (ForceConst) found in the colvars trajectory ﬁle, and then apply a linear bias with that constant. All the notes about units described in sections 6.7 and 6.4 apply here as well. This is not a valid simulation of any particular statistical ensemble and is only an optimization algorithm until the bias has converged.

• name: see deﬁnition of name in sec. 6 (biasing and analysis methods)
• colvars: see deﬁnition of colvars in sec. 6 (biasing and analysis methods)
• centers $⟨\phantom{\rule{0.3em}{0ex}}$Collective variable centers$\phantom{\rule{0.3em}{0ex}}⟩$
Context: alb
Acceptable values: space-separated list of colvar values
Description: The desired center (equilibrium values) which will be sought during the adaptive linear biasing. The number of values must be the number of requested colvars. Each value is a decimal number if the corresponding colvar returns a scalar, a “(x, y, z)” triplet if it returns a unit vector or a vector, and a “q0, q1, q2, q3)” quadruplet if it returns a rotational quaternion. If a colvar has periodicities or symmetries, its closest image to the restraint center is considered when calculating the linear potential.
• updateFrequency $⟨\phantom{\rule{0.3em}{0ex}}$The duration of updates$\phantom{\rule{0.3em}{0ex}}⟩$
Context: alb
Acceptable values: An integer
Description: This is, $N$, the number of simulation steps to use for each update to the bias. This determines how long the system requires to equilibrate after a change in force constant ($N∕2$), how long statistics are collected for an iteration ($N∕2$), and how quickly energy is added to the system (at most, $A∕2N$, where $A$ is the forceRange). Until the force constant has converged, the method as described is an optimization procedure and not an integration of a particular statistical ensemble. It is important that each step should be uncorrelated from the last so that iterations are independent. Therefore, $N$ should be at least twice the autocorrelation time of the collective variable. The system should also be able to dissipate energy as fast as $N∕2$, which can be done by adjusting thermostat parameters. Practically, $N$ has been tested successfully at signiﬁcantly shorter than the autocorrelation time of the collective variables being biased and still converge correctly.
• forceRange $⟨\phantom{\rule{0.3em}{0ex}}$The expected range of the force constant in units of energy$\phantom{\rule{0.3em}{0ex}}⟩$
Context: alb
Acceptable values: A space-separated list of decimal numbers
Default value: 3 ${k}_{b}T$
Description: This is largest magnitude of the force constant which one expects. If this parameter is too low, the simulation will not converge. If it is too high the simulation will waste time exploring values that are too large. A value of 3 ${k}_{b}T$ has worked well in the systems presented as a ﬁrst choice. This parameter is dynamically adjusted over the course of a simulation. The beneﬁt is that a bad guess for the forceRange can be corrected. However, this can lead to large amounts of energy being added over time to the system. To prevent this dynamic update, add hardForceRange yes as a parameter
• rateMax $⟨\phantom{\rule{0.3em}{0ex}}$The maximum rate of change of force constant$\phantom{\rule{0.3em}{0ex}}⟩$
Context: alb
Acceptable values: A list of space-separated real numbers
Description: This optional parameter controls how much energy is added to the system from this bias. Tuning this separately from the updateFrequency and forceRange can allow for large bias changes but with a low rateMax prevents large energy changes that can lead to instability in the simulation.

#### 6.9 Multidimensional histograms

The histogram feature is used to record the distribution of a set of collective variables in the form of a N-dimensional histogram.

As with any other biasing and analysis method, when a histogram is applied to an extended-system colvar (4.15), it accesses the value of the ﬁctitious coordinate rather than that of the “true” colvar. A joint histogram of the “true” colvar and the ﬁctitious coordinate may be obtained by specifying the colvar name twice in a row in the colvars parameter: the ﬁrst instance will be understood as the “true” colvar, and the second, as the ﬁctitious coordinate.

A histogram block may deﬁne the following parameters:

• name: see deﬁnition of name in sec. 6 (biasing and analysis methods)
• colvars: see deﬁnition of colvars in sec. 6 (biasing and analysis methods)
• outputFreq $⟨\phantom{\rule{0.3em}{0ex}}$Frequency (in timesteps) at which the histogram ﬁles are refreshed$\phantom{\rule{0.3em}{0ex}}⟩$
Context: histogram
Acceptable values: positive integer
Default value: colvarsRestartFrequency
Description: The histogram data are written to ﬁles at the given time interval. A value of 0 disables the creation of these ﬁles (note: all data to continue a simulation are still included in the state ﬁle).
• outputFile $⟨\phantom{\rule{0.3em}{0ex}}$Write the histogram to a ﬁle$\phantom{\rule{0.3em}{0ex}}⟩$
Context: histogram
Acceptable values: UNIX ﬁlename
Default value: output.$<$name$>$.dat
Description: Name of the ﬁle containing histogram data (multicolumn format), which is written every outputFreq steps. For the special case of 2 variables, Gnuplot may be used to visualize this ﬁle.
• outputFileDX $⟨\phantom{\rule{0.3em}{0ex}}$Write the histogram to a ﬁle$\phantom{\rule{0.3em}{0ex}}⟩$
Context: histogram
Acceptable values: UNIX ﬁlename
Default value: output.$<$name$>$.dat
Description: Name of the ﬁle containing histogram data (OpenDX format), which is written every outputFreq steps. For the special case of 3 variables, VMD may be used to visualize this ﬁle.
• gatherVectorColvars $⟨\phantom{\rule{0.3em}{0ex}}$Treat vector variables as multiple observations of a scalar variable?$\phantom{\rule{0.3em}{0ex}}⟩$
Context: histogram
Acceptable values: UNIX ﬁlename
Default value: off
Description: When this is set to on, the components of a multi-dimensional colvar (e.g. one based on cartesian, distancePairs, or a vector of scalar numbers given by scriptedFunction) are treated as multiple observations of a scalar variable. This results in the histogram being accumulated multiple times for each simulation step). When multiple vector variables are included in histogram, these must have the same length because their components are accumulated together. For example, if $\xi$, $\lambda$ and $\tau$ are three variables of dimensions 5, 5 and 1, respectively, for each iteration 5 triplets $\left({\xi }_{i},{\lambda }_{i},\tau \right)$ ($i=1,\dots 5$) are accumulated into a 3-dimensional histogram.
• weights $⟨\phantom{\rule{0.3em}{0ex}}$Treat vector variables as multiple observations of a scalar variable?$\phantom{\rule{0.3em}{0ex}}⟩$
Context: histogram
Acceptable values: list of space-separated decimals
Default value: all weights equal to 1
Description: When gatherVectorColvars is on, the components of each multi-dimensional colvar are accumulated with a diﬀerent weight. For example, if $x$ and $y$ are two distinct cartesian variables deﬁned on the same group of atoms, the corresponding 2D histogram can be weighted on a per-atom basis in the deﬁnition of histogram.

##### 6.9.1 Grid deﬁnition for multidimensional histograms

Like the ABF and metadynamics biases, histogram uses the parameters lowerBoundary, upperBoundary, and width to deﬁne its grid. These values can be overridden if a conﬁguration block histogramGrid { } is provided inside the conﬁguration of histogram. The options supported inside this conﬁguration block are:

• lowerBoundaries $⟨\phantom{\rule{0.3em}{0ex}}$Lower boundaries of the grid$\phantom{\rule{0.3em}{0ex}}⟩$
Context: histogramGrid
Acceptable values: list of space-separated decimals
Description: This option deﬁnes the lower boundaries of the grid, overriding any values deﬁned by the lowerBoundary keyword of each colvar. Note that when gatherVectorColvars is on, each vector variable is automatically treated as a scalar, and a single value should be provided for it.
• upperBoundaries: analogous to lowerBoundaries
• widths: analogous to lowerBoundaries

#### 6.10 Probability distribution-restraints

The histogramRestraint bias implements a continuous potential of many variables (or of a single high-dimensional variable) aimed at reproducing a one-dimensional statistical distribution that is provided by the user. The $M$ variables $\left({\xi }_{1},\dots ,{\xi }_{M}\right)$ are interpreted as multiple observations of a random variable $\xi$ with unknown probability distribution. The potential is minimized when the histogram $h\left(\xi \right)$, estimated as a sum of Gaussian functions centered at $\left({\xi }_{1},\dots ,{\xi }_{M}\right)$, is equal to the reference histogram ${h}_{0}\left(\xi \right)$:

 $V\left({\xi }_{1},\dots ,{\xi }_{M}\right)=\frac{1}{2}k\int {\left(h\left(\xi \right)-{h}_{0}\left(\xi \right)\right)}^{2}\mathrm{d}\xi$ (24)
 $h\left(\xi \right)=\frac{1}{M\sqrt{2\pi {\sigma }^{2}}}\sum _{i=1}^{M}exp\left(-\frac{{\left(\xi -{\xi }_{i}\right)}^{2}}{2{\sigma }^{2}}\right)$ (25)

When used in combination with a distancePairs multi-dimensional variable, this bias implements the reﬁnement algorithm against ESR/DEER experiments published by Shen et al [21].

This bias behaves similarly to the histogram bias with the gatherVectorColvars option, with the important diﬀerence that all variables are gathered, resulting in a one-dimensional histogram. Future versions will include support for multi-dimensional histograms.

The list of options is as follows:

• name: see deﬁnition of name in sec. 6 (biasing and analysis methods)
• colvars: see deﬁnition of colvars in sec. 6 (biasing and analysis methods)
• outputEnergy: see deﬁnition of outputEnergy in sec. 6 (biasing and analysis methods)
• lowerBoundary $⟨\phantom{\rule{0.3em}{0ex}}$Lower boundary of the colvar grid$\phantom{\rule{0.3em}{0ex}}⟩$
Context: histogramRestraint
Acceptable values: decimal
Description: Deﬁnes the lowest end of the interval where the reference distribution ${h}_{0}\left(\xi \right)$ is deﬁned. Exactly one value must be provided, because only one-dimensional histograms are supported by the current version.
• upperBoundary: analogous to lowerBoundary
• width $⟨\phantom{\rule{0.3em}{0ex}}$Width of the colvar grid$\phantom{\rule{0.3em}{0ex}}⟩$
Context: histogramRestraint
Acceptable values: positive decimal
Description: Deﬁnes the spacing of the grid where the reference distribution ${h}_{0}\left(\xi \right)$ is deﬁned.
• gaussianSigma $⟨\phantom{\rule{0.3em}{0ex}}$Standard deviation of the approximating Gaussian$\phantom{\rule{0.3em}{0ex}}⟩$
Context: histogramRestraint
Acceptable values: positive decimal
Default value: 2 $×$ width
Description: Deﬁnes the parameter $\sigma$ in eq. 25.
• forceConstant $⟨\phantom{\rule{0.3em}{0ex}}$Force constant (unit of energy speciﬁed by units)$\phantom{\rule{0.3em}{0ex}}⟩$
Context: histogramRestraint
Acceptable values: positive decimal
Default value: 1.0
Description: Deﬁnes the parameter $k$ in eq. 24.
• refHistogram $⟨\phantom{\rule{0.3em}{0ex}}$Reference histogram ${h}_{0}\left(\xi \right)$$\phantom{\rule{0.3em}{0ex}}⟩$
Context: histogramRestraint
Acceptable values: space-separated list of $M$ positive decimals
Description: Provides the values of ${h}_{0}\left(\xi \right)$ consecutively. The mid-point convention is used, i.e. the ﬁrst point that should be included is for $\xi$ = lowerBoundary+width/2. If the integral of ${h}_{0}\left(\xi \right)$ is not normalized to 1, ${h}_{0}\left(\xi \right)$ is rescaled automatically before use.
• refHistogramFile $⟨\phantom{\rule{0.3em}{0ex}}$Reference histogram ${h}_{0}\left(\xi \right)$$\phantom{\rule{0.3em}{0ex}}⟩$
Context: histogramRestraint
Acceptable values: UNIX ﬁle name
Description: Provides the values of ${h}_{0}\left(\xi \right)$ as contents of the corresponding ﬁle (mutually exclusive with refHistogram). The format is that of a text ﬁle, with each line containing the space-separated values of $\xi$ and ${h}_{0}\left(\xi \right)$. The same numerical conventions as refHistogram are used.
• writeHistogram $⟨\phantom{\rule{0.3em}{0ex}}$Periodically write the instantaneous histogram $h\left(\xi \right)$$\phantom{\rule{0.3em}{0ex}}⟩$
Acceptable values: boolean
Default value: off
Description: If on, the histogram $h\left(\xi \right)$ is written every colvarsRestartFrequency steps to a ﬁle with the name output.$<$name$>$.hist.dat This is useful to diagnose the convergence of $h\left(\xi \right)$ against ${h}_{0}\left(\xi \right)$.

### 7 Syntax changes from older versions

The following is a list of syntax changes in Colvars since its ﬁrst release. Many of the older keywords are still recognized by the current code, thanks to speciﬁc compatibility code. This is not a list of new features: its primary purpose is to make you aware of those improvements that aﬀect the use of old conﬁguration ﬁles with new versions of the code.

Note: if you are using any of the NAMD and VMD tutorials:
https://www.ks.uiuc.edu/Training/Tutorials/
please be aware that several of these tutorials are not actively maintained: for those cases, this list will help you reconcile any inconsistencies.

• Colvars version 2016-06-09 or later (LAMMPS version 15Sep2016 or later).
The legacy keyword refPositionsGroup has been renamed fittingGroup for clarity (the legacy version is still supported).
• Colvars version 2016-08-10 or later (LAMMPS version 15Sep2016 or later).
“System forces” have been replaced by “total forces” (see for example outputTotalForce). See the following page for more information:
• Colvars version 2017-01-09 or later (LAMMPS version 10Mar2017 or later).
A new type of restraint, harmonicWalls (see 6.6), replaces and improves upon the legacy keywords lowerWall and upperWall: these are still supported as short-hands.
• Colvars version 2018-11-15 or later (LAMMPS version 23Nov2018 or later).
The global analysis keyword has been discontinued: speciﬁc analysis tasks are controlled directly by the keywords corrFunc and runAve, which continue to remain off by default.
• Deprecation warning for calculations including wall potentials.
The legacy keywords lowerWall and upperWall will stop having default values and will need to be set explicitly (preferably as part of the harmonicWalls restraint). When using an ABF bias, it is recommended to set the two walls equal to lowerBoundary and upperBoundary, respectively. When using a metadynamics bias, it is recommended to set the two walls within lowerBoundary and upperBoundary. This guarantees that the tails of each Gaussian hill are accounted in the region between the grid boundaries and the wall potentials. See also expandBoundaries for an automatic deﬁnition of the PMF grid boundaries.

Up-to-date documentation can always be accessed at:
https://colvars.github.io/colvars-refman-lammps/colvars-refman-lammps.html

### References

[1]   G. Fiorin, M. L. Klein, and J. Hénin. Using collective variables to drive molecular dynamics simulations. Mol. Phys., 111(22-23):3345–3362, 2013.

[2]   M. Iannuzzi, A. Laio, and M. Parrinello. Eﬃcient exploration of reactive potential energy surfaces using car-parrinello molecular dynamics. Phys. Rev. Lett., 90(23):238302, 2003.

[3]   E A Coutsias, C Seok, and K A Dill. Using quaternions to calculate RMSD. J. Comput. Chem., 25(15):1849–1857, 2004.

[4]   Eric Darve, David Rodríguez-Gómez, and Andrew Pohorille. Adaptive biasing force method for scalar and vector free energy calculations. J. Chem. Phys., 128(14):144120, 2008.

[5]   J. Hénin and C. Chipot. Overcoming free energy barriers using unconstrained molecular dynamics simulations. J. Chem. Phys., 121:2904–2914, 2004.

[6]   J. Hénin, G. Fiorin, C. Chipot, and M. L. Klein. Exploring multidimensional free energy landscapes using time-dependent biases on collective variables. J. Chem. Theory Comput., 6(1):35–47, 2010.

[7]   A. Carter, E, G. Ciccotti, J. T. Hynes, and R. Kapral. Constrained reaction coordinate dynamics for the simulation of rare events. Chem. Phys. Lett., 156:472–477, 1989.

[8]   M. J. Ruiz-Montero, D. Frenkel, and J. J. Brey. Eﬃcient schemes to compute diﬀusive barrier crossing rates. Mol. Phys., 90:925–941, 1997.

[9]   W. K. den Otter. Thermodynamic integration of the free energy along a reaction coordinate in cartesian coordinates. J. Chem. Phys., 112:7283–7292, 2000.

[10]   Giovanni Ciccotti, Raymond Kapral, and Eric Vanden-Eijnden. Blue moon sampling, vectorial reaction coordinates, and unbiased constrained dynamics. ChemPhysChem, 6(9):1809–1814, 2005.

[11]   Adrien Lesage, Tony Lelièvre, Gabriel Stoltz, and Jérôme Hénin. Smoothed biasing forces yield unbiased free energies with the extended-system adaptive biasing force method. J. Phys. Chem. B, 121(15):3676–3685, 2017.

[12]   A. Laio and M. Parrinello. Escaping free-energy minima. Proc. Natl. Acad. Sci. USA, 99(20):12562–12566, 2002.

[13]   Helmut Grubmüller. Predicting slow structural transitions in macromolecular systems: Conformational ﬂooding. Phys. Rev. E, 52(3):2893–2906, Sep 1995.

[14]   T. Huber, A. E. Torda, and W.F. van Gunsteren. Local elevation - A method for improving the searching properties of molecular-dynamics simulation. Journal of Computer-Aided Molecular Design, 8(6):695–708, DEC 1994.

[15]   G. Bussi, A. Laio, and M. Parrinello. Equilibrium free energies from nonequilibrium metadynamics. Phys. Rev. Lett., 96(9):090601, 2006.

[16]   Alessandro Barducci, Giovanni Bussi, and Michele Parrinello. Well-tempered metadynamics: A smoothly converging and tunable free-energy method. Phys. Rev. Lett., 100:020603, 2008.

[17]   P. Raiteri, A. Laio, F. L. Gervasio, C. Micheletti, and M. Parrinello. Eﬃcient reconstruction of complex free energy landscapes by multiple walkers metadynamics. J. Phys. Chem. B, 110(8):3533–9, 2005.

[18]   Yuqing Deng and Benoît Roux. Computations of standard binding free energies with molecular dynamics simulations. J. Phys. Chem. B, 113(8):2234–2246, 2009.

[19]   Jed W. Pitera and John D. Chodera. On the use of experimental observations to bias simulated ensembles. J. Chem. Theory Comput., 8:3445–3451, 2012.

[20]   A. D. White and G. A. Voth. Eﬃcient and minimal method to bias molecular simulations with experimental data. J. Chem. Theory Comput., ASAP, 2014.

[21]   Rong Shen, Wei Han, Giacomo Fiorin, Shahidul M Islam, Klaus Schulten, and Benoît Roux. Structural reﬁnement of proteins by restrained molecular dynamics simulations with non-interacting molecular fragments. PLoS Comput. Biol., 11(10):e1004368, 2015.