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 9.3.1. For example, using the keyword rmsd (section 9.3.1) 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 9.3.1 (see 9.3.5 for details).
• A user-defined mathematical function of the existing components (see list in section 9.3.1), or of the atomic coordinates directly (see the cartesian keyword in 9.3.1). The function is defined through the keyword customFunction (see 9.3.6) (see 9.3.6 for details).
• A user-defined Tcl function of the existing components (see list in section 9.3.1), or of the atomic coordinates directly (see the cartesian keyword in 9.3.1). The function is provided by a separate Tcl script, and referenced through the keyword scriptedFunction (see 9.3.7) (see 9.3.7 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 $<$ Name of this colvar $>$
  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.

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 9.4) 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 (see 9.3.1): distance between two groups;
• distanceZ (see 9.3.1): projection of a distance vector on an axis;
• distanceXY (see 9.3.1): projection of a distance vector on a plane;
• distanceInv (see 9.3.1): mean distance between two groups of atoms (e.g. NOE-based distance);
• angle (see 9.3.1): angle between three groups;
• dihedral (see 9.3.1): torsional (dihedral) angle between four groups;
• dipoleAngle (see 9.3.1): angle between two groups and dipole of a third group;
• polarTheta (see 9.3.1): polar angle of a group in spherical coordinates;
• polarPhi (see 9.3.1): azimuthal angle of a group in spherical coordinates;
• coordNum (see 9.3.1): coordination number between two groups;
• selfCoordNum (see 9.3.1): coordination number of atoms within a group;
• hBond (see 9.3.1): hydrogen bond between two atoms;
• rmsd (see 9.3.1): root mean square deviation (RMSD) from a set of reference coordinates;
• eigenvector (see 9.3.1): projection of the atomic coordinates on a vector;
• orientationAngle (see 9.3.1): angle of the best-fit rotation from a set of reference coordinates;
• orientationProj (see 9.3.1): cosine of orientationProj (see 9.3.1);
• spinAngle (see 9.3.1): projection orthogonal to an axis of the best-fit rotation from a set of reference coordinates;
• tilt (see 9.3.1): projection on an axis of the best-fit rotation from a set of reference coordinates;
• gyration (see 9.3.1): radius of gyration of a group of atoms;
• inertia (see 9.3.1): moment of inertia of a group of atoms;
• inertiaZ (see 9.3.1): moment of inertia of a group of atoms around a chosen axis;
• alpha (see 9.3.1): $\alpha$ -helix content of a protein segment.
• dihedralPC (see 9.3.1): projection of protein backbone dihedrals onto a dihedral principal component.

Some components do not return scalar, but vector values:

• distanceVec (see 9.3.1): distance vector between two groups (length: 3);
• distanceDir (see 9.3.1): unit vector parallel to distanceVec (length: 3);
• cartesian (see 9.3.1): vector of atomic Cartesian coordinates (length: $N$ times the number of Cartesian components requested, X, Y or Z);
• distancePairs (see 9.3.1): vector of mutual distances (length: $N_{\mathrm{1}}\times{}N_{\mathrm{2}}$ );
• orientation (see 9.3.1): 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 9.3.5), a custom function (see 9.3.6), or a scripted function (see 9.3.7).

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 (see 9.3.8) and upperBoundary (see 9.3.8) 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 9.3.2.

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

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

List of keywords (see also 9.3.5 for additional options):

• group1 $<$ First group of atoms $>$
  Context: distance
  Acceptable Values: Block group1 {...}
  Description: First group of atoms.

• group2: analogous to group1

• forceNoPBC $<$ Calculate absolute rather than minimum-image distance? $>$
  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 $<$ Measure total force on group 1 only? $>$
  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 (53) in section 9.5.1) that only involves atoms of group1. This option is only useful for ABF, or custom biases that compute total forces. See section 9.5.1 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).

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.

List of keywords (see also 9.3.5 for additional options):

• main $<$ Main group of atoms $>$
  Context: distanceZ
  Acceptable Values: Block main {...}
  Description: Group of atoms whose position ${\mbox{\boldmath {$r$}}}$ is measured.

• ref $<$ Reference group of atoms $>$
  Context: distanceZ
  Acceptable Values: Block ref {...}
  Description: Reference group of atoms. The position of its center of mass is noted ${\mbox{\boldmath {$r$}}}_1$ below.

• ref2 $<$ Secondary reference group $>$
  Context: distanceZ
  Acceptable Values: Block ref2 {...}
  Default Value: none
  Description: Optional group of reference atoms, whose position ${\mbox{\boldmath {$r$}}}_2$ can be used to define a dynamic projection axis: ${\mbox{\boldmath {$e$}}}=(\Vert {\mbox{\boldmath {$r$}}}_2 - {\mbox{\boldmath ... ...}_1\Vert)^{-1} \times ({\mbox{\boldmath {$r$}}}_2 - {\mbox{\boldmath {$r$}}}_1)$ . In this case, the origin is ${\mbox{\boldmath {$r$}}}_m = 1/2 ({\mbox{\boldmath {$r$}}}_1+{\mbox{\boldmath {$r$}}}_2)$ , and the value of the component is ${\mbox{\boldmath {$e$}}} \cdot ({\mbox{\boldmath {$r$}}}-{\mbox{\boldmath {$r$}}}_m)$ .

• axis $<$ Projection axis (Å) $>$
  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 ${\mbox{\boldmath {$e$}}}$ for the distance vector ${\mbox{\boldmath {$r$}}} - {\mbox{\boldmath {$r$}}}_1$ joining the centers of groups ref and main. The value of the component is then ${\mbox{\boldmath {$e$}}} \cdot ({\mbox{\boldmath {$r$}}}-{\mbox{\boldmath {$r$}}}_1)$ . The vector should be written as three components separated by commas and enclosed in parentheses.

• forceNoPBC: see definition of forceNoPBC in sec. 9.3.1 (distance component)

• oneSiteTotalForce: see definition of oneSiteTotalForce in sec. 9.3.1 (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).

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).

List of keywords (see also 9.3.5 for additional options):

• main: see definition of main in sec. 9.3.1 (distanceZ component)
• ref: see definition of ref in sec. 9.3.1 (distanceZ component)
• ref2: see definition of ref2 in sec. 9.3.1 (distanceZ component)
• axis: see definition of axis in sec. 9.3.1 (distanceZ component)
• forceNoPBC: see definition of forceNoPBC in sec. 9.3.1 (distance component)
• oneSiteTotalForce: see definition of oneSiteTotalForce in sec. 9.3.1 (distance component)

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.

List of keywords (see also 9.3.5 for additional options):

• group1: see definition of group1 in sec. 9.3.1 (distance component)
• group2: analogous to group1
• forceNoPBC: see definition of forceNoPBC in sec. 9.3.1 (distance component)
• oneSiteTotalForce: see definition of oneSiteTotalForce in sec. 9.3.1 (distance component)

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 $\mathbf{d} = (d_{x}, d_{y}, d_{z})$ , with $\vert\mathbf{d}\vert = 1$ .

List of keywords (see also 9.3.5 for additional options):

• group1: see definition of group1 in sec. 9.3.1 (distance component)
• group2: analogous to group1
• forceNoPBC: see definition of forceNoPBC in sec. 9.3.1 (distance component)
• oneSiteTotalForce: see definition of oneSiteTotalForce in sec. 9.3.1 (distance component)

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 $1/n$ :

$$d_{\mathrm{1,2}}^{[n]} \; = \; \left(\frac{1}{N_{\mathrm{1}}N_{\m... ...}\sum_{i,j} \left(\frac{1}{\Vert\mathbf{d}^{ij}\Vert}\right)^{n} \right)^{-1/n}$$ (36)

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

List of keywords (see also 9.3.5 for additional options):

• group1: see definition of group1 in sec. 9.3.1 (distance component)
• group2: analogous to group1
• oneSiteTotalForce: see definition of oneSiteTotalForce in sec. 9.3.1 (distance component)
• exponent $<$ Exponent $n$ in equation 36 $>$
  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.

distancePairs: set of pairwise distances between two groups.

The distancePairs {...} block defines a $N_{\mathrm{1}}\times{}N_{\mathrm{2}}$ -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.

List of keywords (see also 9.3.5 for additional options):

• group1: see definition of group1 in sec. 9.3.1 (distance component)
• group2: analogous to group1
• forceNoPBC: see definition of forceNoPBC in sec. 9.3.1 (distance component)

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

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 $(x_1, y_1, z_1, \cdots, x_n, y_n, z_n)$ .

List of keywords (see also 9.3.5 for additional options):

• atoms $<$ Group of atoms $>$
  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.

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 $[0:180]$ .

List of keywords (see also 9.3.5 for additional options):

• group1: see definition of group1 in sec. 9.3.1 (distance component)
• group2: analogous to group1
• group3: analogous to group1
• forceNoPBC: see definition of forceNoPBC in sec. 9.3.1 (distance component)
• oneSiteTotalForce: see definition of oneSiteTotalForce in sec. 9.3.1 (distance component)

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 $[0:180]$ .

List of keywords (see also 9.3.5 for additional options):

• group1: see definition of group1 in sec. 9.3.1 (distance component)
• group2: analogous to group1
• group3: analogous to group1
• forceNoPBC: see definition of forceNoPBC in sec. 9.3.1 (distance component)
• oneSiteTotalForce: see definition of oneSiteTotalForce in sec. 9.3.1 (distance component)

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 $[-180:180]$ . 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.

List of keywords (see also 9.3.5 for additional options):

• group1: see definition of group1 in sec. 9.3.1 (distance component)
• group2: analogous to group1
• group3: analogous to group1
• group4: analogous to group1
• forceNoPBC: see definition of forceNoPBC in sec. 9.3.1 (distance component)
• oneSiteTotalForce: see definition of oneSiteTotalForce in sec. 9.3.1 (distance component)

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 $[0:180]$ . To obtain spherical coordinates in a frame of reference tied to another group of atoms, use the fittingGroup (9.4.2) option within the atoms block. An example is provided in file examples/11_polar_angles.in of the Colvars public repository.

List of keywords (see also 9.3.5 for additional options):

• atoms $<$ Atom group $>$
  Context: polarPhi
  Acceptable Values: atoms {...} block
  Description: Defines the group of atoms for the COM of which the angle should be calculated.

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 $[-180:180]$ . 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 (9.4.2) option within the atoms block. An example is provided in file examples/11_polar_angles.in of the Colvars public repository.

List of keywords (see also 9.3.5 for additional options):

• atoms $<$ Atom group $>$
  Context: polarPhi
  Acceptable Values: atoms {...} block
  Description: Defines the group of atoms for the COM of which the angle should be calculated.

coordNum: coordination number between two groups.

The coordNum {...} block defines a coordination number (or number of contacts), which calculates the function $(1-(d/d_0)^{n})/(1-(d/d_0)^{m})$ , where $d_0$ is the ``cutoff'' distance, and $n$ and $m$ are exponents that can control its long range behavior and stiffness [45]. This function is summed over all pairs of atoms in group1 and group2:

$$C (\mathtt{group1}, \mathtt{group2}) \; = \; \sum_{i\in\mathtt{gr... ...}\vert/d_{0})^{n}}{ 1 - (\vert\mathbf{x}_{i}-\mathbf{x}_{j}\vert/d_{0})^{m} } }$$ (37)

List of keywords (see also 9.3.5 for additional options):

• group1: see definition of group1 in sec. 9.3.1 (distance component)

• group2: analogous to group1

• cutoff $<$ ``Interaction'' distance (Å) $>$
  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 $(1-(d/d_0)^{n})/(1-(d/d_0)^{m})$ 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 $<$ Reference distance vector (Å) $>$
  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 $<$ Numerator exponent $>$
  Context: coordNum
  Acceptable Values: positive even integer
  Default Value: 6
  Description: This number defines the $n$ exponent for the switching function.

• expDenom $<$ Denominator exponent $>$
  Context: coordNum
  Acceptable Values: positive even integer
  Default Value: 12
  Description: This number defines the $m$ exponent for the switching function.

• group2CenterOnly $<$ Use only group2's center of mass $>$
  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 $<$ Pairlist control $>$
  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. 37 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. 37 is further modified by subtracting the tolerance and then rescaling so that each pair covers the range $\left[0, 1\right]$ .

• pairListFrequency $<$ Pairlist regeneration frequency $>$
  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 $N_{\mathtt{group1}} \times N_{\mathtt{group2}}$ (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 $N_{\mathtt{group1}} \times N_{\mathtt{group2}}$ . Setting $\mathtt{tolerance} > 0$ ameliorates this to some degree, although every pair is still checked to regenerate the pairlist.

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:

$$C (\mathtt{group1}) \; = \; \sum_{i\in\mathtt{group1}}\sum_{j > i... ...}\vert/d_{0})^{n}}{ 1 - (\vert\mathbf{x}_{i}-\mathbf{x}_{j}\vert/d_{0})^{m} } }$$ (38)

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.

List of keywords (see also 9.3.5 for additional options):

• group1: see definition of group1 in sec. 9.3.1 (coordNum component)
• cutoff: see definition of cutoff in sec. 9.3.1 (coordNum component)
• cutoff3: see definition of cutoff3 in sec. 9.3.1 (coordNum component)
• expNumer: see definition of expNumer in sec. 9.3.1 (coordNum component)
• expDenom: see definition of expDenom in sec. 9.3.1 (coordNum component)
• tolerance: see definition of tolerance in sec. 9.3.1 (coordNum component)
• pairListFrequency: see definition of pairListFrequency in sec. 9.3.1 (coordNum component)

This component returns a dimensionless number, which ranges from approximately 0 (all interatomic distances much larger than the cutoff) to $N_{\mathtt{group1}} \times (N_{\mathtt{group1}} - 1) / 2$ (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 $N_{\mathtt{group1}}^2$ .

hBond: hydrogen bond between two atoms.

The hBond {...} block defines a hydrogen bond, implemented as a coordination number (eq. 37) 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).

List of keywords (see also 9.3.5 for additional options):

• acceptor $<$ Number of the acceptor atom $>$
  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. 9.3.1 (coordNum component)
  Note: default value is 3.3 Å.
• expNumer: see definition of expNumer in sec. 9.3.1 (coordNum component)
  Note: default value is 6.
• expDenom: see definition of expDenom in sec. 9.3.1 (coordNum component)
  Note: default value is 8.

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 $\{ \mathbf{x}_1(t), \mathbf{x}_2(t), \ldots \mathbf{x}_N(t) \}$ , the colvar component rmsd calculates the optimal rotation $U^{\{\mathbf{x}_{i}(t)\}\rightarrow\{\mathbf{x}_{i}^{\mathrm{(ref)}}\}}$ that best superimposes the coordinates $\{\mathbf{x}_{i}(t)\}$ onto a set of reference coordinates $\{\mathbf{x}_{i}^{\mathrm{(ref)}}\}$ . Both the current and the reference coordinates are centered on their centers of geometry, $\mathbf{x}_{\mathrm{cog}}(t)$ and $\mathbf{x}_{\mathrm{cog}}^{\mathrm{(ref)}}$ . The root mean square displacement is then defined as:

$${ \mathrm{RMSD}(\{\mathbf{x}_{i}(t)\}, \{\mathbf{x}_{i}^{\mathrm{... ...{(ref)}} - \mathbf{x}_{\mathrm{cog}}^{\mathrm{(ref)}} \right) \right\vert^{2} }$$ (39)

The optimal rotation $U^{\{\mathbf{x}_{i}(t)\}\rightarrow\{\mathbf{x}_{i}^{\mathrm{(ref)}}\}}$ is calculated within the formalism developed in reference [26], which guarantees a continuous dependence of $U^{\{\mathbf{x}_{i}(t)\}\rightarrow\{\mathbf{x}_{i}^{\mathrm{(ref)}}\}}$ with respect to $\{\mathbf{x}_{i}(t)\}$ .

List of keywords (see also 9.3.5 for additional options):

• atoms $<$ Atom group $>$
  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 $<$ Reference coordinates $>$
  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 (see 9.4.2) in the atom group definition, which also supports more advanced fitting options.

• refPositionsFile $<$ Reference coordinates file $>$
  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 (see 9.4.2) in the atom group definition, which also supports more advanced fitting options.

• refPositionsCol $<$ PDB column containing atom flags $>$
  Context: rmsd
  Acceptable Values: O, B, X, Y, or Z
  Description: If refPositionsFile is a PDB file that contains all the atoms in the topology, this option may be provided to set which PDB field is used to flag the reference coordinates for atoms.

• refPositionsColValue $<$ Atom selection flag in the PDB column $>$
  Context: rmsd
  Acceptable Values: positive decimal
  Description: If defined, this value identifies in the PDB column refPositionsCol of the file refPositionsFile which atom positions are to be read. Otherwise, all positions with a non-zero value are read.

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

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 (translateReference 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.

Path collective variables

An application of the rmsd component is "path collective variables",[12] which are implemented as Tcl-scripted combinations or RMSDs. The implementation is available as file colvartools/pathCV.tcl, and an example is provided in file examples/10_pathCV.namd of the Colvars public repository.

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 $\mathbb{R}^{3n}$ , where $n$ is the number of atoms in the group. The computed quantity is the total projection:

$${ p(\{\mathbf{x}_{i}(t)\}, \{\mathbf{x}_{i}^{\mathrm{(ref)}}\}) }... ...athrm{(ref)}} - \mathbf{x}_{\mathrm{cog}}^{\mathrm{(ref)}}) \right)\mathrm{,} }$$ (40)

where, as in the rmsd component, $U$ is the optimal rotation matrix, $\mathbf{x}_{\mathrm{cog}}(t)$ and $\mathbf{x}_{\mathrm{cog}}^{\mathrm{(ref)}}$ 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 $(\mathbf{v}_{i})$ 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.

List of keywords (see also 9.3.5 for additional options):

• atoms: see definition of atoms in sec. 9.3.1 (rmsd component)
• refPositions: see definition of refPositions in sec. 9.3.1 (rmsd component)
• refPositionsFile: see definition of refPositionsFile in sec. 9.3.1 (rmsd component)
• refPositionsCol: see definition of refPositionsCol in sec. 9.3.1 (rmsd component)
• refPositionsColValue: see definition of refPositionsColValue in sec. 9.3.1 (rmsd component)

• vector $<$ Vector components $>$
  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 $<$ file containing vector components $>$
  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. For a PDB file specifically, the components are read from the X, Y and Z fields. Note: The PDB file has limited precision and fixed-point numbers: in some cases, the vector components may not be accurately represented; a XYZ file should be used instead, containing floating-point numbers.

• vectorCol $<$ PDB column used to flag participating atoms $>$
  Context: eigenvector
  Acceptable Values: O or B
  Description: Analogous to atomsCol.

• vectorColValue $<$ Value used to flag participating atoms in the PDB file $>$
  Context: eigenvector
  Acceptable Values: positive decimal
  Description: Analogous to atomsColValue.

• differenceVector $<$ The $3n$ -dimensional vector is the difference between vector and refPositions $>$
  Context: eigenvector
  Acceptable Values: boolean
  Default Value: off
  Description: If this option is on, the numbers provided by vector or vectorFile are interpreted as another set of positions, $\mathbf{x}_{i}'$ : the vector $\mathbf{v}_{i}$ is then defined as $\mathbf{v}_{i} = \left(\mathbf{x}_{i}' - \mathbf{x}_{i}^{\mathrm{(ref)}}\right)$ . 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. 40, this range does not depend on the number of atoms involved.

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 $\{ \mathbf{x}_1(t), \mathbf{x}_2(t), \ldots \mathbf{x}_N(t) \}$ with respect to their center of geometry, $\mathbf{x}_{\mathrm{cog}}(t)$ :

$$R_{\mathrm{gyr}} \; = \; \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left\vert\mathbf{x}_{i}(t) - \mathbf{x}_{\mathrm{cog}}(t)\right\vert^{2} }$$ (41)

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

List of keywords (see also 9.3.5 for additional options):

• atoms: see definition of atoms in sec. 9.3.1 (rmsd component)

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 $\{ \mathbf{x}_1(t), \mathbf{x}_2(t), \ldots \mathbf{x}_N(t) \}$ with respect to their center of geometry, $\mathbf{x}_{\mathrm{cog}}(t)$ :

$$I \; = \; \sum_{i=1}^{N} \left\vert\mathbf{x}_{i}(t) - \mathbf{x}_{\mathrm{cog}}(t)\right\vert^{2}$$ (42)

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$^{2}$ .

List of keywords (see also 9.3.5 for additional options):

• atoms: see definition of atoms in sec. 9.3.1 (rmsd component)

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 $\{ \mathbf{x}_1(t), \mathbf{x}_2(t), \ldots \mathbf{x}_N(t) \}$ with respect to their center of geometry, $\mathbf{x}_{\mathrm{cog}}(t)$ :

$$I_{\mathbf{e}} \; = \; \sum_{i=1}^{N} \left(\left(\mathbf{x}_{i}(t) - \mathbf{x}_{\mathrm{cog}}(t)\right)\cdot\mathbf{e}\right)^{2}$$ (43)

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$^{2}$ .

List of keywords (see also 9.3.5 for additional options):

• atoms: see definition of atoms in sec. 9.3.1 (rmsd component)
• axis $<$ Projection axis (Å) $>$
  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 $\mathbf{e}$ .

orientation: orientation from reference coordinates.

The block orientation {...} returns the same optimal rotation used in the rmsd component to superimpose the coordinates $\{\mathbf{x}_{i}(t)\}$ onto a set of reference coordinates $\{\mathbf{x}_{i}^{\mathrm{(ref)}}\}$ . Such component returns a four dimensional vector $\mathsf{q} = (q_0, q_1, q_2, q_3)$ , with $\sum_{i} q_{i}^{2} = 1$ ; this quaternion expresses the optimal rotation $\{\mathbf{x}_{i}(t)\} \rightarrow \{\mathbf{x}_{i}^{\mathrm{(ref)}}\}$ according to the formalism in reference [26]. The quaternion $(q_0, q_1, q_2, q_3)$ can also be written as $\left(\cos(\theta/2), \, \sin(\theta/2)\mathbf{u}\right)$ , where $\theta$ 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 $4\times{}4$ rotation matrix in a format suitable for usage in VMD.

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

List of keywords (see also 9.3.5 for additional options):

• atoms: see definition of atoms in sec. 9.3.1 (rmsd component)
• refPositions: see definition of refPositions in sec. 9.3.1 (rmsd component)
• refPositionsFile: see definition of refPositionsFile in sec. 9.3.1 (rmsd component)

• refPositionsCol: see definition of refPositionsCol in sec. 9.3.1 (rmsd component)
• refPositionsColValue: see definition of refPositionsColValue in sec. 9.3.1 (rmsd component)

• closestToQuaternion $<$ Reference rotation $>$
  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 $(q_0, q_1, q_2, q_3)$ and $(-q_0, -q_1, -q_2, -q_3)$ , 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.

orientationAngle: angle of rotation from reference coordinates.

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

List of keywords (see also 9.3.5 for additional options):

• atoms: see definition of atoms in sec. 9.3.1 (rmsd component)
• refPositions: see definition of refPositions in sec. 9.3.1 (rmsd component)
• refPositionsFile: see definition of refPositionsFile in sec. 9.3.1 (rmsd component)

• refPositionsCol: see definition of refPositionsCol in sec. 9.3.1 (rmsd component)
• refPositionsColValue: see definition of refPositionsColValue in sec. 9.3.1 (rmsd component)

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, refPositionsCol and refPositionsColValue. The returned value is the cosine of the angle of rotation $\theta$ between the current and the reference positions. The range of values is [-1:1].

List of keywords (see also 9.3.5 for additional options):

• atoms: see definition of atoms in sec. 9.3.1 (rmsd component)
• refPositions: see definition of refPositions in sec. 9.3.1 (rmsd component)
• refPositionsFile: see definition of refPositionsFile in sec. 9.3.1 (rmsd component)
• refPositionsCol: see definition of refPositionsCol in sec. 9.3.1 (rmsd component)
• refPositionsColValue: see definition of refPositionsColValue in sec. 9.3.1 (rmsd component)

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.

List of keywords (see also 9.3.5 for additional options):

• atoms: see definition of atoms in sec. 9.3.1 (rmsd component)
• refPositions: see definition of refPositions in sec. 9.3.1 (rmsd component)
• refPositionsFile: see definition of refPositionsFile in sec. 9.3.1 (rmsd component)
• refPositionsCol: see definition of refPositionsCol in sec. 9.3.1 (rmsd component)
• refPositionsColValue: see definition of refPositionsColValue in sec. 9.3.1 (rmsd component)
• axis $<$ Special rotation axis (Å) $>$
  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 $[-180:180]$ .

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 $-1$ .

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 $\theta$ 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 $[-1:1]$ : the value $1$ represents an orientation fully parallel to e (tilt angle = 0$^\circ$ ), and the value $-1$ represents an anti-parallel orientation.

List of keywords (see also 9.3.5 for additional options):

• atoms: see definition of atoms in sec. 9.3.1 (rmsd component)
• refPositions: see definition of refPositions in sec. 9.3.1 (rmsd component)
• refPositionsFile: see definition of refPositionsFile in sec. 9.3.1 (rmsd component)
• refPositionsCol: see definition of refPositionsCol in sec. 9.3.1 (rmsd component)
• refPositionsColValue: see definition of refPositionsColValue in sec. 9.3.1 (rmsd component)
• axis: see definition of axis in sec. 9.3.1 (spinAngle component)

alpha: $\alpha$ -helix content of a protein segment.

The block alpha {...} defines the parameters to calculate the helical content of a segment of protein residues. The $\alpha$ -helical content across the $N+1$ residues $N_{0}$ to $N_{0}+N$ is calculated by the formula:

$${ \alpha\left( \mathrm{C}_{\alpha}^{(N_{0})}, \mathrm{O}^{(N_{0})... ...thrm{N}^{(N_{0}+N)}, \mathrm{C}_{\alpha}^{(N_{0}+N)} \right) } \; = \; \; \; \;$$ (44)

$$\; \; \; \; { \frac{1}{2(N-2)} \sum_{n=N_{0}}^{N_{0}+N-2} \mathrm... ...-4} \mathrm{hbf}\left( \mathrm{O}^{(n)}, \mathrm{N}^{(n+4)}\right) \mathrm{,} }$$

where the score function for the $\mathrm{C}_{\alpha} - \mathrm{C}_{\alpha} - \mathrm{C}_{\alpha}$ angle is defined as:

$${ \mathrm{angf}\left( \mathrm{C}_{\alpha}^{(n)}, \mathrm{C}_{\alp... ...eta_{0}\right)^{4} / \left(\Delta\theta_{\mathrm{tol}}\right)^{4}} \mathrm{,} }$$ (45)

and the score function for the $\mathrm{O}^{(n)} \leftrightarrow \mathrm{N}^{(n+4)}$ hydrogen bond is defined through a hBond colvar component on the same atoms.

List of keywords (see also 9.3.5 for additional options):

• residueRange $<$ Potential $\alpha$ -helical residues $>$
  Context: alpha
  Acceptable Values: ``$<$ Initial residue number$>$ -$<$ Final residue number$>$ ''
  Description: This option specifies the range of residues on which this component should be defined. The Colvars module looks for the atoms within these residues named ``CA'', ``N'' and ``O'', and raises an error if any of those atoms is not found.

• psfSegID $<$ PSF segment identifier $>$
  Context: alpha
  Acceptable Values: string (max 4 characters)
  Description: This option sets the PSF segment identifier for the residues specified in residueRange. This option is only required when PSF topologies are used.

• hBondCoeff $<$ Coefficient for the hydrogen bond term $>$
  Context: alpha
  Acceptable Values: positive between 0 and 1
  Default Value: 0.5
  Description: This number specifies the contribution to the total value from the hydrogen bond terms. 0 disables the hydrogen bond terms, 1 disables the angle terms.

• angleRef $<$ Reference $\mathrm{C}_{\alpha} - \mathrm{C}_{\alpha} - \mathrm{C}_{\alpha}$ angle $>$
  Context: alpha
  Acceptable Values: positive decimal
  Default Value: 88$^{\circ}$
  Description: This option sets the reference angle used in the score function (46).

• angleTol $<$ Tolerance in the $\mathrm{C}_{\alpha} - \mathrm{C}_{\alpha} - \mathrm{C}_{\alpha}$ angle $>$
  Context: alpha
  Acceptable Values: positive decimal
  Default Value: 15$^{\circ}$
  Description: This option sets the angle tolerance used in the score function (46).

• hBondCutoff $<$ Hydrogen bond cutoff $>$
  Context: alpha
  Acceptable Values: positive decimal
  Default Value: 3.3 Å
  Description: Equivalent to the cutoff option in the hBond component.

• hBondExpNumer $<$ Hydrogen bond numerator exponent $>$
  Context: alpha
  Acceptable Values: positive integer
  Default Value: 6
  Description: Equivalent to the expNumer option in the hBond component.

• hBondExpDenom $<$ Hydrogen bond denominator exponent $>$
  Context: alpha
  Acceptable Values: positive integer
  Default Value: 8
  Description: Equivalent to the expDenom option in the hBond component.

This component returns positive values, always comprised between 0 (lowest $\alpha$ -helical score) and 1 (highest $\alpha$ -helical score).

dihedralPC: protein dihedral pricipal component

The block dihedralPC {...} defines the parameters to calculate the projection of backbone dihedral angles within a protein segment onto a dihedral principal component, following the formalism of dihedral principal component analysis (dPCA) proposed by Mu et al.[70] and documented in detail by Altis et al.[2]. Given a peptide or protein segment of $N$ residues, each with Ramachandran angles $\phi_i$ and $\psi_i$ , dPCA rests on a variance/covariance analysis of the $4(N-1)$ variables $\cos(\psi_1), \sin(\psi_1), \cos(\phi_2), \sin(\phi_2) \cdots \cos(\phi_N), \sin(\phi_N)$ . Note that angles $\phi_1$ and $\psi_N$ have little impact on chain conformation, and are therefore discarded, following the implementation of dPCA in the analysis software Carma.[35]

For a given principal component (eigenvector) of coefficients $(k_i)_{1 \leq i \leq 4(N-1)}$ , the projection of the current backbone conformation is:

$$\xi = \sum_{n=1}^{N-1} k_{4n-3} \cos(\psi_n) + k_{4n-2} \sin (\psi_n) + k_{4n-1} \cos (\phi_{n+1}) + k_{4n} \sin(\phi_{n+1})$$ (46)

dihedralPC expects the same parameters as the alpha component for defining the relevant residues (residueRange and psfSegID) in addition to the following:

List of keywords (see also 9.3.5 for additional options):

• residueRange: see definition of residueRange in sec. 9.3.1 (alpha component)

• psfSegID: see definition of psfSegID in sec. 9.3.1 (alpha component)

• vectorFile $<$ File containing dihedral PCA eigenvector(s) $>$
  Context: dihedralPC
  Acceptable Values: file name
  Description: A text file containing the coefficients of dihedral PCA eigenvectors on the cosine and sine coordinates. The vectors should be arranged in columns, as in the files output by Carma.[35]

• vectorNumber $<$ File containing dihedralPCA eigenvector(s) $>$
  Context: dihedralPC
  Acceptable Values: positive integer
  Description: Number of the eigenvector to be used for this component.

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 or any user-supplied function provided by scriptedFunction (see 9.3.5).

• name $<$ Name of this component $>$
  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. It also defines the variable name representing the component's value in a customFunction (see 9.3.6) expression.

• scalable $<$ Attempt to calculate this component in parallel? $>$
  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.

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 or scripted 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 (9.3.6), or wthin scripted variables 9.3.7).

• period $<$ Period of the component $>$
  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 $<$ Center of the wrapping interval for periodic variables $>$
  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 9.3.5) 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.

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.

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.

Linear and polynomial combinations of components

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

$$\xi(\mathbf{r}) = \sum_i c_i [q_i(\mathbf{r})]^{n_i}$$ (47)

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 (48)) can be obtained by setting the following parameters, which are common to all components:

• componentCoeff $<$ Coefficient of this component in the colvar $>$
  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 $<$ Exponent of this component in the colvar $>$
  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 $1/N$ .

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 9.3.3 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 $<$ Compute colvar as a custom function of its components $>$
  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 $<$ Type of value returned by the scripted colvar $>$
  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.

Scripted functions

When scripting is supported (default in NAMD), a colvar may be defined as a scripted function of its components, rather than a linear or polynomial combination. When implementing generic functions of Cartesian coordinates rather than functions of existing components, the cartesian component may be particularly useful. A scalar-valued scripted 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 9.3.3 for details).

An example of elaborate scripted colvar is given in example 10, in the form of path-based collective variables as defined by Branduardi et al[12] (9.3.1).

• scriptedFunction $<$ Compute colvar as a scripted function of its components $>$
  Context: colvar
  Acceptable Values: string
  Description: If this option is specified, the colvar will be computed as a scripted function of the values of its components. To that effect, the user should define two Tcl procedures: calc_$<$ scriptedFunction$>$ and calc_$<$ scriptedFunction$>$ _gradient, both accepting as many parameters as the colvar has components. Values of the components will be passed to those procedures in the order defined by their sorted name strings. Note that if all components are of the same type, their default names are sorted in the order in which they are defined, so that names need only be specified for combinations of components of different types. calc_$<$ scriptedFunction$>$ should return one value of type $<$ scriptedFunctionType$>$ , corresponding to the colvar value. calc_$<$ scriptedFunction$>$ _gradient should return a Tcl list containing the derivatives of the function with respect to each component. If both the function and some of the components are vectors, the gradient is really a Jacobian matrix that should be passed as a linear vector in row-major order, i.e. for a function $f_i(x_j)$ : $\nabla_x f_1 \nabla_x f_2 \cdots$ .

• scriptedFunctionType $<$ Type of value returned by the scripted colvar $>$
  Context: colvar
  Acceptable Values: string
  Default Value: scalar
  Description: If a colvar is defined as a scripted function, its type is not constrained by the types of its components. 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 (a vector whose size is specified by scriptedFunctionVectorSize). Non-scalar values should be passed as space-separated lists.

• scriptedFunctionVectorSize $<$ Dimension of the vector value of a scripted colvar $>$
  Context: colvar
  Acceptable Values: positive integer
  Description: This parameter is only valid when scriptedFunctionType is set to vector. It defines the vector length of the colvar value returned by the function.

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 $<$ Unit of the variable, or grid spacing $>$
  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 (9.5.4), harmonic walls (9.5.6) and linear restraints (9.5.7) 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, $\AA$ 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 (9.5.9), ABF (9.5.1) and metadynamics (9.5.3) 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 $<$ Lower boundary of the colvar $>$
  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, 9.3.4). This option does not affect dynamics: to confine a colvar within a certain interval, use a harmonicWalls bias.

• upperBoundary $<$ Upper boundary of the colvar $>$
  Context: colvar
  Acceptable Values: decimal
  Description: Similarly to lowerBoundary, defines the highest possible or allowed value.

• hardLowerBoundary $<$ Whether the lower boundary is the physical lower limit $>$
  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 9.3.1 for the physical ranges of values of each component.

• hardUpperBoundary $<$ Whether the upper boundary is the physical upper limit of the colvar's values $>$
  Context: colvar
  Acceptable Values: boolean
  Default Value: off
  Description: Analogous to hardLowerBoundary.

• expandBoundaries $<$ Allow to expand the two boundaries if needed $>$
  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 (9.5.3) 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.

Trajectory output

• outputValue $<$ Output a trajectory for this colvar $>$
  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 $<$ Output a velocity trajectory for this colvar $>$
  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 $<$ Output an energy trajectory for this colvar $>$
  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 $<$ Output a total force trajectory for this colvar $>$
  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 (53) in section 9.5.1) 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 kcal/mol, divided by the colvar unit U.

• outputAppliedForce $<$ Output an applied force trajectory for this colvar $>$
  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 kcal/mol divided by the colvar unit.

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 (9.5.1) to perform eABF simulations (9.5.2).

• extendedLagrangian $<$ Add extended degree of freedom $>$
  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 [45]. The energy associated with the extended degree of freedom is reported under the MISC title in NAMD's energy output.

• extendedFluctuation $<$ Standard deviation between the colvar and the fictitious particle (colvar unit) $>$
  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 $<$ Oscillation period of the fictitious particle (fs) $>$
  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 $k_B T (\tau/2 \pi \sigma)^2$ , where $\tau$ is the period and $\sigma$ is the typical fluctuation (see above).

• extendedTemp $<$ Temperature for the extended degree of freedom (K) $>$
  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 $<$ Damping factor for extended Langevin dynamics (ps$^{-1}$ ) $>$
  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.

Backward-compatibility

• subtractAppliedForce $<$ Do not include biasing forces in the total force for this colvar $>$
  Context: colvar
  Acceptable Values: boolean
  Default Value: off
  Description: If the colvar supports total force calculation (see 9.3.4), 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.

Statistical analysis

When the global keyword analysis is defined in the configuration file, run-time calculations of statistical properties for individual colvars can be performed. At the moment, several types of time correlation functions, running averages and running standard deviations are available.

• corrFunc $<$ Calculate a time correlation function? $>$
  Context: colvar
  Acceptable Values: boolean
  Default Value: off
  Description: Whether or not a time correlaction function should be calculated for this colvar.

• corrFuncWithColvar $<$ Colvar name for the correlation function $>$
  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 $<$ Type of the correlation function $>$
  Context: colvar
  Acceptable Values: velocity, coordinate or coordinate_p2
  Default Value: velocity
  Description: With coordinate or velocity, the correlation function $C_{i,j}(t)$  = $\left\langle \Pi\left(\xi_{i}(t_{0}), \xi_{j}(t_{0}+t)\right) \right\rangle$ is calculated between the variables $\xi_{i}$ and $\xi_{j}$ , or their velocities. $\Pi(\xi_{i}, \xi_{j})$ 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, $(3\cos(\theta)^{2}-1)/2$ , is used instead of the cosine.

• corrFuncNormalize $<$ Normalize the time correlation function? $>$
  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 $\left\langle O\left(\xi_{i}, \xi_{j}\right) \right\rangle$ .

• corrFuncLength $<$ Length of the time correlation function $>$
  Context: colvar
  Acceptable Values: positive integer
  Default Value: 1000
  Description: Length (in number of points) of the time correlation function.

• corrFuncStride $<$ Stride of the time correlation function $>$
  Context: colvar
  Acceptable Values: positive integer
  Default Value: 1
  Description: Number of steps between two values of the time correlation function.

• corrFuncOffset $<$ Offset of the time correlation function $>$
  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 $<$ Output file for the time correlation function $>$
  Context: colvar
  Acceptable Values: UNIX filename
  Default Value: $<$ name$>$ .corrfunc.dat
  Description: The time correlation function is saved in this file.

• runAve $<$ Calculate the running average and standard deviation $>$
  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 $<$ Length of the running average window $>$
  Context: colvar
  Acceptable Values: positive integer
  Default Value: 1000
  Description: Length (in number of points) of the running average window.

• runAveStride $<$ Stride of the running average window values $>$
  Context: colvar
  Acceptable Values: positive integer
  Default Value: 1
  Description: Number of steps between two values within the running average window.

• runAveOutputFile $<$ Output file for the running average and standard deviation $>$
  Context: colvar
  Acceptable Values: UNIX filename
  Default Value: $<$ name$>$ .runave.dat
  Description: The running average and standard deviation are saved in this file.