$\phantom{\rule{0.17em}{0ex}}$
COLLECTIVE VARIABLES MODULE
Reference manual for VMD
Code version: 20220913
Updated versions of this manual: [GROMACS] [LAMMPS] [NAMD] [VMD]
Alejandro Bernardin, Haochuan Chen, Jeffrey R. Comer, Giacomo Fiorin, Haohao Fu,
Jérôme Hénin, Axel Kohlmeyer, Fabrizio Marinelli, Joshua V. Vermaas, Andrew D.
White
(PDF version)
Contents
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:
$$\xi \left(t\right)\phantom{\rule{0.28em}{0ex}}=\xi \left(\text{}X\text{}\right(t\left)\right)\phantom{\rule{0.28em}{0ex}}=\xi \left({\text{}x\text{}}_{i}\left(t\right),{\text{}x\text{}}_{j}\left(t\right),{\text{}x\text{}}_{k}\left(t\right),\dots \⁡\right)\phantom{\rule{0.28em}{0ex}},\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.28em}{0ex}}1\le i,j,k\dots \⁡\le N$$  (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
runtime without the need to save very large trajectory files, or after a simulation has been completed
(postprocessing).
Note: although restraints and PMF algorithms are primarily used during simulations, they are also available in
VMD to test a new input for a simulation, or to evaluate the relative free energy of a new structure based on data
from a previous calculation. Options that only have an effect during a simulation are also included for compatibility
purposes.
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, alongside any other publications for specific
features being, according to the usage summary printed when running a Colvarsenabled MD simulation or
analysis.
Using the Colvars module in VMD
Within VMD [2], the Colvars module can be accessed in two ways:
 Using the Colvars Dashboard, an intuitive interface to the Colvars module, to easily define and analyze
collective variables, but not biases (section 3).
 Using the fullfeatured Tcl scripting interface as documented in section 4.2; see in particular the
example in section 4.2.5.
2 Writing a Colvars configuration: a crash course
The Colvars configuration is a plain text file or string that defines collective variables, biases, and general
parameters of the Colvars module. It is passed to the module using backendspecific commands documented in
section 4. Writing the configuration fora collective variable in VMD is made much easier using the Dashboard and
its configuration editor (section 3). However, note that the Dashboard does not handle biases: if necessary, they
should be managed separately using the scripting interface.
Example: steering two groups of atoms away from each other.
Now let us look at a complete, nontrivial configuration. 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 5 and 5.2.1), and the other the moving harmonic restraint (7.5). Note that in VMD, no
biasing forces are applied, but biases may be useful in the context of an analysis script, e.g. to collect histograms or
to compute bias energies.
colvar {
name dist
distance {
group1 { atomNumbersRange 4255 }
group2 {
psfSegID PR
atomNameResidueRange CA 1530
}
}
}
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 $carbon atoms of residues
15 to 30 in the protein (segment name PR). To the “dist" variable, we apply a harmonic potential of force constant 20 energy
unit/length unit${}^{2}$,
initially centered around a value of 4 length unit, which will increase to 15 length unit over 500,000 simulation
steps.
The atom selection keywords are detailed in section 6.
Example: using multiple variables and multiple biasing/analysis methods together.
A more complex example configuration is included below, showing how a variable may be constructed by combining
multiple existing functions, and how multiple variables or multiple biases may be used concurrently. The colvar indicated
below as “$d$"
is defined as the difference between two distances (see 5.2): the first distance
(${d}_{1}$)
is taken between the center of mass of atoms 1 and 2 and that of atoms 3 to 5, the second
(${d}_{2}$)
between atom 7 and the center of mass of atoms 8 to 10 (see 6). The difference
$d={d}_{1}{d}_{2}$ is obtained by multiplying
the two by a coefficient $C=+1$ or
$C=1$, respectively (see 5.16).
The colvar called “$c$"
is the coordination number calculated between atoms 1 to 10 and atoms 11 to 20. A harmonic restraint (see 7.5) is applied
to both $d$ and
$c$: to allow using the
same force constant $K$,
both $d$ and
$c$ are scaled by their respective
fluctuation widths ${w}_{d}$ and
${w}_{c}$. A third colvar “alpha" is defined
as the $\alpha $helical content of residues 1
to 10 (see 5.7.1). The values of “$c$"
and “alpha" are also recorded throughout the simulation as a joint 2dimensional histogram (see 7.10).
colvar {
# difference of two distances
name d
width 0.2 # 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 110 }
group2 { atomNumbersRange 1120 }
tolerance 1.0e6
pairListFrequency 1000
}
}
colvar {
name alpha
alpha {
psfSegID PROT
residueRange 110
}
}
harmonic {
colvars d c
centers 3.0 4.0
forceConstant 5.0
}
histogram {
colvars c alpha
}
The Colvars Dashboard is a graphical interface for interactive visualization and refinement of collective
variables aided by molecular structures and trajectories. It is accessible in VMD's Main Menu under
“Extensions/Analysis/Colvars Dashboard". Throughout the interface, keyboard shortcuts for common operations are
indicated in square brackets. Context menus in the colvar and bias tables can be accessed using either right click or
Controlclick.
Here are the steps for a quick first tour of the Dashboard:
 load an MD trajectory into VMD;
 open the Dashboard;
 click “New" to create a new collective variable;
 in the Editor window, click “Apply" to accept the default template;
 in the Dashboard window, click “Show atoms" to display the two atom groups involved in this distance
coordinate;
 click “Timeline plot";
 click anywhere in the timeline plot to navigate in the trajectory.
Alternately, use the Automatic colvars features in the Actions tab.
Now, clicking “Edit" in the Dashboard window (or rightclikcing on a colvar in the list view), you can modify
the collective variable to reflect interesting geometric properties of the system. The power of the collective variables
approach lies in the variety of geometric functions (“components") and their combinations. The editor window
provides a number of helpers to make it easy and quick to define the most relevant variables. See section 3.5 for
details.
The Dashboard window displays a table listing currently defined variables, and their values for the current frame
indicated at the bottom of the window. By default the frame is updated to track VMD's currently displayed frame,
but that can be changed by toggling the “Track frame" checkbox, e.g. to animate the trajectory without recomputing
expensive variables. Vectorvalues variables can be expanded to list their scalar elements. This is necessary when
individual scalar quantities have to be selected for plotting. Other operations act on variables as a whole and ignore
specific selected scalar elements. Rightclicking in the table brings up a context menu with specific actions for the
selected variables.
Buttons above the table allow for general operations on the state of the Colvars module. Below the table, the
Actions tab allows operations on selected variables, while the Settings tab offers advanced settings for the various
visualizations.
If variables are modified, added or deleted by an external script, hit “Refresh" or press F5 to update the
displayed variables and values. Starting the Dashboard also enables trajectory animation using the left/right arrow
keys within VMD's graphical window. Atomic coordinates can be modified using VMD's “Mouse/Move"
functions, and the Colvars module can then be updated by pressing F5 directly from the graphical
window.
A dropdown list allows for changing the current unit system if no variables are defined (see section 4.1). When
changing units, it is strongly recommended to check the configuration for all colvars to ensure that all quantities are
expressed in the desired set of units.
The Colvars Module interacts with one VMD Molecule at a time. At the bottom of the Actions tab, a dropdown
lets the user change which VMD molecule is associated with the Colvars module. Internally, changing it means
recording the configuration of currently defined colvars, deleting the current instance of the Colvars module,
creating a new one linked to the target molecule, and applying the saved configuration. Beware of incompatible
colvar definitions, such as atom groups listing atom IDs that exist in one molecule, but not the other. Autoupdating
selections (see below) can be used to adapt the colvar definitions to a different system using VMD selection
texts.
3.3 Loading / Saving configuration files
This saves the current configuration of the colvars Module to a file. This includes comments found in the input,
general parameters of the Colvars Module, the collective variables themselves, and collective variable biases. The
resulting configuration file can be read by Colvars, either back in VMD, or within a supported MD
engine.
3.4 Automatic colvar generators
The Actions tab offers two buttons to generate sets of colvars from the current VMD session:
 "protein autocolvars" creates variables describing classic quantities used to describe protein
structures;
 "colvars from VMD labels" creates variables based on visible labels (Bonds, Angles, Dihedrals) for
the current molecule (see Molecule selector in the same tab). Such labels can be created by setting the
label mode with the key 2, 3, or 4, and picking atoms with the mouse.
These automatically generated colvars can be used asis, or customized using the configuration editor (Edit in
the context menu, or in the Actions tab).
3.5 The configuration editor
The configuration editor can be started with the “Edit" or “New" buttons. Using the “Edit" button, the
configuration of selected variables is loaded, and those variables will be replaced when applying the new
configuration.
The editor window offers links to online documentation, as well as helpers to write correct configuration
files.
As a first step, the most useful helper is the collection of template files. Some parameters that must
be supplied are indicated by the symbol @. Colvar templates can be inserted at the beginning of the
configuration, whereas “component" templates define basis functions that belong inside a colvar block.
Templates are indented using 4 spaces per level to indicate their position in the nested structure of the
configuration: general options, colvars and biases at level 0, bias and colvar parameters like components at
level 1, component parameters such as atom groups at level 2, and atom group parameters at level
3.
The next helper buttons allow importing atom selections from VMD, either typing a VMD atom selection text,
by copying the selection of an existing graphical representation, or by inserting the list of atoms currently labeled in
VMD using the “Pick atom" feature. Atom selections should be inserted within an atom group block, within a
component block (such as distance). By default atom selections are preceded with a comment line marking them
as autoupdating. This instructs the Dashboard to update the list of atoms whenever the configuration is
applied, that is, when it is edited, when the file is loaded by the Dashboard, or when changing the VMD
molecule linked to Colvars. This is useful when working across systems with different atom numberings,
but topologies that make the relevant atom groups identifiable using VMD selection texts. Special
fields (O, B, and user) may be used, as well as atom positions (e.g. z > 0). If the selection text is
modified manually, the atom list will be updated when applying the new configuration. This autoupdating
behavior can be disabled by removing the special comment line or altering the keywords “autoupdating
selection".
Note that atom lists are not autoupdated:
 when changing frame within the same molecule (animating a trajectory);
 when the configuration is read by the Colvars module outside of VMD (eg. within an MD engine).
The “Insert labeled..." button combined with the selection box allows for inserting components matching VMD's
geometry measurements: Bonds (distances), angles, and dihedrals. Hidden labels are not used for inserting
components.
3.6 Plotting and visualizing collective variables
Timeline plots
show the selected variables as a function of time. A vertical bar indicates the current frame, which
can be changed either using VMD's trajectory animation controls, or directly in the plot window by
clicking the mouse inside the graph, or using the keyboard left/right arrows. Shift+arrow steps by
10 frames for faster animation, and Ctrl+arrow steps over 50 frames. The Home/End keys jump to
the first and last frames, respectively. The up/down arrows operate a zoom/unzoom along the time
axis. Visible data can be fitted vertically using the h key. All data can be fitted horizontally using the h
key.
Pairwise scatterplots
are useful to identify correlation between variables. To create a pairwise plot, select exactly two scalar
variables (or scalar components of vector variables), and click “Pairwise plot". Frames are represented by
circles, and lines connect consecutive frames. The blue dot tracks the current frame: clicking a circle
jumps to the corresponding frame. Arrow keys and Home/End animate the trajectory as in the timeline
plot.
Histogram
displays a histogram of the selected scalar variable over the trajectory, using approximately the number of bins
specified in the Settings tab (adjusted after selecting a round number for the bin width). The current value of the
colvar is indicated by a vertical bar. Clicking anywhere in the plot jumps to the frame with the nearest
value of the colvar. Keyboard navigation of the trajectory is similar to the other plots, except that the
trajectory is traversed according to colvar values rather than frame numbers. This enables exploration
of frames corresponding to a neighborhood of colvar values, even when they are not contiguous in
time.
“Show atoms" creates representations of the atoms involved in the definition of the selected colvars. Each atom
group is shown in a different color. “Show gradient" is available for scalar variables only. It creates a
graphical representation of the atomic gradients of the selected variables, visualizing how the value
of the collective variable would vary in response to a change in atomic coordinates. The graphical
representation of the gradients is controlled by parameters in the Settings tab. Vectors representing the
gradient are rescaled as indicated by the radio buttons Max. vector norm and Scaling factor. Max. vector
norm rescales gradients so that the largest vector component of each colvar's gradient is represented
by an arrow of the specified length, in Å. Scaling factor rescales gradients by the specified factor,
divided by the colvar's width parameter (1 by default, see 5.19). This use of width makes it easier to
compare the gradients of collective variables that are not commensurate. The scaling factor has the unit
$\ast L\u2215(cv\u2215width)$, where
$L$ is the current length unit, and
$cv$ represents the natural unit of the
collective variable. By default $width$
is unity, but $(cv\u2215width)$ may be
seen as dimensionless if $width$
is expressed in cv units.
The Dashboard offers visualizations for specific types of variables:
 Variables that represent rotation operations (as unit quaternions), such as those based on an
orientation component, can be displayed by rightclicking on them in the colvar table. This creates
a representation of the rotation as an arrow representing its axis, and two arms separated by the angle
of the rotation, between 0 and 180 degrees.
 For variables that are computed based on volumetric maps, representations of the related maps can
be created using the "Show volmap" button, or again through a specific entry in the context menu
available via rightclick.
Parameters for finetuning these visualizations can be set in the Settings tab.
3.7 Monitoring colvar biases in the Dashboard
The main Dashboard window features a "Biases" tab that contains a list of currently defined biases, their current
energy, and the colvars that they depend upon. A new harmonic bias acting on a given set of colvars can be created
by selecting the colvars and opening the context menu in the colvar table (right click or controlclick). A button
enables creating a Timeline plot with the energies of currently selected biases, similar to the colvar Timeline plot.
Only one plot can be active at a time. Another button creates a visualization of the biasing forces computed by the
selected biases, similar to the gradient display for colvars. The graphical representation of forces is controlled by the
same parameters as the gradients, shown in the Settings tab. The difference is the scaling factor unit, which is
Ångström per Colvars force unit (that is, Ångström times Colvars length unit per Colvars energy
unit).
Note that the energy of historydependent biases such as ABF or metadynamics may not be updated in VMD.
However, values of the bias energy from a previous simulation can be displayed if a state file has been loaded with
cv load (section 4.2), or in the case of ABF, if inputPrefix was specified.
Hitting the Save button will save the configuration of any bias listed in this tab, except the harmonic
wall biases that are created automatically from a colvar for which the lower or upper wall options are
specified.
3.8 Plotting custom timedependent quantities as colvars
Sometimes, important timedependent quantities are available for a trajectory, but must be computed by an
external tool (e.g. energies from a simulation). These can be visualized alongside colvars by taking advantage of the
scripted colvar feature.
Such a custom variable can either be computed on the fly (by calling an external tool or a VMD feature) or
precomputed and cached. For example, we can use VMD to compute the protein solventaccessible surface area
(SASA) on the fly:
set sasa_sel [atomselect top "protein and noh"]
proc calc_user_colvar { x } {
# Follow current Colvars frame
$::sasa_sel frame [cv frame]
return [measure sasa 1 $::sasa_sel]
}
The Tcl script above should be entered into VMD's text console, or sourced from a file. Then we can define a
colvar that will be calculated by the scripted procedure above, by entering the following configuration as a new
colvar in the Dashboad Editor Window:
colvar {
name user
scriptedFunction user_colvar
distance {
# The distance component is just a placeholder
group1 {
dummyatom (0, 0, 0)
}
group2 {
dummyatom (0, 0, 0)
}
}
}
If the SASA computation is slow, it can be cached to make subsequent interaction responsive. To that effect, we
source the following script to precompute the SASA for every frame and store it in VMD's User field for atom
0.
set atom0_sel [atomselect top "index 0"]
for {set f 0} {$f < [molinfo top get numframes]} {incr f} {
$::sasa_sel frame $f
$::atom0_sel frame $f
$::atom0_sel set user [measure sasa 1 $::sasa_sel]
}
Then we use the following Tcl procedure to compute a custom variable that is simply equal to the User field for
atom 0 for the current frame:
proc calc_user_colvar { x } {
# Follow current Colvars frame
$::atom0_sel frame [cv frame]
return [$::atom0_sel get user]
}
This version results implementing extremely fast updates.
4 Enabling and controlling the Colvars module in VMD
Here, we document the syntax of the commands and parameters used to set up and use the Colvars module in
VMD. One of these parameters is the configuration file or the configuration text for the module itself, whose syntax
is described in 4.3 and in the following sections.
4.1 Units in the Colvars module
The “internal units" of the Colvars module are the units in which values are expected to be in the configuration
file, and in which collective variable values, energies, etc. are expressed in the output and colvars trajectory files.
Generally the Colvars module uses internally the same units as its backend MD engine, with the exception of
VMD, where different unit sets are supported to allow for easy setup, visualization and analysis of Colvars
simulations performed with any simulation engine.
Note that angles are expressed in degrees, and derived quantites such as force constants are based on degrees as
well. Some colvar components have default values, expressed in Ångström in this documentation. They are
converted to the current length unit as needed. Atomic coordinates read from XYZ files (and PDB files
where applicable) are expected to be expressed in Ångström, no matter what unit system is in use
by the backend (VMD) or the Colvars Module. They are converted internally to the current length
unit as needed. Note that force constants in harmonic and harmonicWalls biases (7.5) are rescaled
according to the width parameter of colvars, so that they are formally in energy units, although if
width is given its default value of 1.0, force constants are effectively expressed in energy unit/(colvar
unit)${}^{2}$.
To avoid errors due to reading configuration files written in a different unit system, it can be specified within the
input:
 Keyword units $\u27e8\phantom{\rule{0.17em}{0ex}}$Unit
system to be used$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: global
Acceptable values: string
Description: A string defining the units to be used internally by Colvars. Supported values are: real
(Å, kcal/mol), gromacs (nm, kJ/mol), metal (Å, eV), and electron (Bohr, Hartree). In VMD, the default
system of units for Colvars is VMD's native units: real (Å, kcal/mol). However, the units keyword
will switch to a different unit system than the current one if no colvars were defined before reading
the current configuration. If colvars are already defined, units will refuse changing the unit system
to avoid making the definition of those variables erroneous in the new system of units. If needed this
precaution can be overridden using the cv units scripting command (4.2).
4.2 Using the scripting interface to control the Colvars module
After the first initialization of the Colvars module, the internal state of Colvars objects may be queried or
modified in a VMD script:
cv $<$method$>$ arg1 arg2 ...
where $<$method$>$
is the name of a specific procedure and arg1, arg2, …are its required and/or optional arguments.
In VMD, the cv command is used by the Dashboard graphical interface,(3), but can be also used in scripts or
interactively from the commandline terminal (for example, in remote terminal sessions) or in the Tk
Console.
The most frequent uses of the scripting interface are discussed and exemplified in this section. For the full
reference documentation of all procedures supported by the Colvars scripting interface, see section 8 for the Tcl
command reference.
4.2.1 Setting up the Colvars module
The first step to using Colvars in VMD is choosing which “molecule" (i.e. which system): because VMD can
handle multiple “molecules", the Colvars module needs to remain attached to a specific VMD molecule. For
example:
will attach the Colvars module onto the molecule currently holding the “top" status (alternatively, you can refer to a
molecule by its numeric ID in lieu of top). All following invocations of the cv command will continue
operating on the same molecule, regardless of whether other molecules are loaded, or which one has the
“top" status. The cv molid command without argument will return the molid currently associated with
Colvars.
To define new collective variables and/or biases for immediate use in the current session, configuration can be
loaded from an external configuration file:
cv configfile "colvarsfile.in"
This can in principle be called at any time, if only flags internal to Colvars are being modified. In practice, when
new atoms or any new atomic properties (e.g. total forces are being requested), initialization steps are required that
are not carried out during a simulation. Therefore, it is good practice to change the Colvars configuration outside the
scope between segments of the same simulation.
To load the configuration directly from a string the “config" method may be used:
cv config "keyword { ... }"
This method is particularly useful to dynamically define the Colvars configuration within a VMD
script.
4.2.2 Using the Colvars version in scripts
The vast majority of the syntax in Colvars is backwardcompatible, adding keywords when new features are
introduced. However, when using multiple versions simultaneously it may be useful to test within the script whether
the version is recent enough to support the desired feature. The “version" can be used to get the Colvars version for
this use:
if { [cv version] >= "20200225" } {
cv config "(use a recent feature)"
}
4.2.3 Loading and saving the Colvars state and other information
After a configuration is fully defined, the “load" method may be used to load a state file from a previous
simulation that contains e.g. data from historydependent biases), to either continue that simulation or analyze its
results:
cv load "$<$oldjob$>$.colvars.state"
or more simply using the prefix of the state file itself:
The “save" method, analogous to “load", allows to save all restart information to a state file. This is normally
not required during a simulation if colvarsRestartFrequency is defined (either directly or indirectly by the VMD
restart frequency). Because not only a state file (used to continue simulations) but also other data files (used to
analyze the trajectory) are written, it is generally recommended to call the save method using a prefix, rather than a
complete file name:
4.2.4 Accessing atomic data
For computational efficiency the Colvars module keeps internal copies of the numeric IDs, masses, charges,
positions, and optionally total forces of the atoms requested for a Colvars computation. At each simulation step,
uptodate versions of these properties are copied from central memory of VMD into the internal memory of the
Colvars module. In a postprocessing workflow or outside a simulation (e.g. when using VMD), this copy can be
carried out as part of the update method:
which also performs the (re)computation of all variables and biases defined in Colvars.
For example, the current sequence of numeric IDs of the atoms processed by Colvars can be obtained
as:
and their current positions as:
This may prove useful to test the correctness of the coordinates passed to Colvars, particularly in regard to periodic
boundary conditions (6.3). There is currently no mechanism to modify the above fields via the scripting interface,
but such capability will be added in the future.
While running a simulation, or when setting one up in VMD, it is possible to examine all the forces that were
last applied by Colvars to the atoms, or are about to be applied:
where the length and order of this sequence matches that provided by the getatomids method. A simpler way of
testing the stability of a Colvars configuration before or during a simulation makes use of aggregated data, such as
the energy:
the rootmeansquare of the Colvars applied forces:
cv getatomappliedforcesrms
or the maximum norm of the applied forces:
cv getatomappliedforcesmax
which can be matched to a specific atom via its numeric ID obtained as:
cv getatomappliedforcesmaxid
See 8.1 for a complete list of scripting commands used to manage atomic data and runtime parameters of he
Colvars module.
4.2.5 Analyzing a trajectory in VMD
One of the typical uses of Colvars in VMD is computing the values of one or more variables along an existing
trajectory. This is most easily done using the Dashboard (section 3), but can also be done by a script, as in the
example below:
# Activate the module on the current VMD molecule
cv molid top
# Load a Colvars config file
cv configfile test.in
set out [open "test.colvars.traj" "w"]
# Write the labels to the file
puts nonewline ${out} [cv printframelabels]
for { set fr 0 } { ${fr} < [molinfo top get numframes] } { incr fr } {
# Point Colvars to this trajectory frame
cv frame ${fr}
# Recompute variables and biases (required in VMD)
cv update
# Print variables and biases to the file
puts nonewline ${out} [cv printframe]
}
close ${out}
4.2.6 Managing collective variables
After one or more collective variables are defined, they can be accessed with the following syntax.
cv colvar "$<$name$>$" $<$method$>$ arg1 arg2 ...
where “$<$name$>$"
is the name of the variable.
For example, to recompute the collective variable “xi" after a change in its parameters, the following command
can be used:
This ordinarily is not needed during a simulation run, where all variables are recomputed at every step (along with
biasing forces acting on them). However, when analyzing an existing trajectory, e.g. in VMD, a call to update is
generally required.
While in all typical cases all configuration of the variables is done with the “config" or “configfile" methods,
a limited set of changes can be enacted at runtime using:
cv colvar "$<$name$>$" modifycvcs arg1 arg2 ...
where each argument is a string passed to the function or functions that are used to compute the variable,
and are called colvar components, or CVCs (5.1). For example, a variable “DeltaZ" made of a single
“distanceZ" component can be made periodic with a period equal to the unit cell dimension along the
$Z$axis:
cv colvar "DeltaZ" modifycvcs "period Lz"
Please note that this option is currently limited to changing the values of the polynomial superposition parameters
componentCoeff, or of the componentExp to update on the fly, of period, wrapAround or forceNoPBC options for
components that support them.
If the variable is computed using many components, it is possible to selectively turn some of them on or
off:
cv colvar "$<$name$>$" cvcflags $<$flags$>$
where “$<$flags$>$"
is a list of 0/1 values, one per component. This is useful for example when scriptbased path collective variables in
Cartesian coordinates (5.10.3) are used, to minimize computational cost by disabling the computation of terms that
are very close to zero.
Important: None of the changes enacted by the “modifycvcs" or “cvcflags" methods will be saved to state
files, and will be lost when restarting a simulation, deleting the corresponding collective variable, or resetting the
module with the “delete" or “reset" methods.
4.2.7 Applying and analyzing forces on collective variables
As soon as a collective variable is up to date (during a MD run or after its “update" method has been called),
forces can be applied to it, e.g. as part of a custom restraint implemented by scriptedColvarForces:
cv colvar "xi" addforce $<$force$>$
where “$<$force$>$"
is a scalar or a vector (depending on the type of variable “xi") and is defined by the user's function. The force will
be physically applied to the corresponding atoms during the simulation after Colvars communicates all forces to the
rest of VMD. (In VMD, these forces will never have an effect.) Until then, the total force applied to “xi" from all
biases can be retrieved by:
cv colvar "xi" getappliedforce
(see also the use of the outputAppliedForce option).
To obtain the total force projected on the variable “xi":
cv colvar "xi" gettotalforce
Note that not all types of variable support this option, and the value of the total force may not be available
immediately: see outputTotalForce for more details.
See 8.2 for a complete list of scripting commands used to manage collective variables.
4.2.8 Managing collective variable biases
Because biases depend only upon data internal to the Colvars module (i.e. they do not need atomic coordinates
from VMD), it is generally easy to create them or update their configuration at any time. For example, given the
most current value of the variable “xi", an alreadydefined harmonic restraint on it named “h_xi" can be updated
as:
During a running simulation this step is not needed, because an automatic update of each bias is already carried
out.
4.2.9 Loading and saving the state of individual biases
Some types of bias are historydependent, and the magnitude of their forces depends not only on the values of
their corresponding variables, but also on previous simulation history. It is thus useful to load information from a
state file that contains information specifically for one bias only, for example:
cv bias "metadynamics1" load "old.colvars.state"
or alternatively, using the prefix of the file instead of its full name:
cv bias "metadynamics1" load "old"
A corresponding “save" function is also available:
cv bias "metadynamics1" save "new"
This pair of functions is also used internally by Colvars to implement e.g. multiplewalker metadynamics (7.4.7), but
they can be called from a scripted function to implement alternative coupling schemes.
See 8.3 for a complete list of scripting commands used to manage biases.
4.3 Configuration syntax used by the Colvars module
Configuration for the Colvars module is passed using an external file, or inlined as a string in a VMD script
using the Tcl command cv config "...". Configuration lines follow the format “keyword value" or “keyword {
... }", where the keyword and its value must be separated by one or more space characters. The following
formatting rules apply:
 Keywords are caseinsensitive; for example, upperBoundary is the same as upperboundary and
UPPERBOUNDARY); note that their string values are however still casesensitive (e.g. names of variables,
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 at least one
space character; the closing brace “}" may be at any position after that; any keywords following the
closing brace on the same line are not valid (they should appear instead on a different line).
 Many keywords are nested, and are only meaningful within the specific context of another keyword;
for example, the keyword name is available inside the block of the keyword colvar {...}, but not
outside of it; for every keyword documented in the following, the “parent" keyword that defines such
context is also indicated.
 If a keyword requiring a boolean value (yesontrue or noofffalse) is provided without an
explicit value, it defaults to ‘yesontrue'; 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.
 Outside of comments, only ASCII characters are allowed for defining keywords, and the only
whitespace characters supported are spaces, tabs and newlines: a warning will be printed upon
detection of nonASCII characters in a configuration line, which include both characters that are
visibly “special", as well as those with a very similar appearance to ASCII ones (for instance,
nonbreaking spaces); common ways to identify/remove nonASCII characters are using the Emacs
text editor, or using LC_ALL=C vi.
 Tcl syntax is generally not available inside a Colvars configuration file/string, but it is possible to use
Tcl variables or bracket expansion of commands when configuration is passed via the command cv
config "..."; for example, it is possible to convert the atom selection $sel into an atom group (see
6.1) using cv config "atomNumbers { [$sel get serial] }".
The following keywords are available in the global context of the Colvars configuration, i.e. they are not nested
inside other keywords:
 Keyword colvarsTrajFrequency $\u27e8\phantom{\rule{0.17em}{0ex}}$Colvar
value trajectory frequency$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 outputName.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.
 Keyword colvarsRestartFrequency $\u27e8\phantom{\rule{0.17em}{0ex}}$Colvar
module restart frequency$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: global
Acceptable values: positive integer
Default value: 0
Description: The state file and any other output files produced by Colvars are written every these
many steps (the trajectory file is still written every colvarsTrajFrequency steps). In VMD, the
simulation step is not progressed and this parameter is effectively ignored. It is generally a good idea
to leave this parameter at its default value, unless needed for special cases or to disable automatic
writing of output files altogether. Writing can still be invoked at any time via the command cv save.
 Keyword indexFile $\u27e8\phantom{\rule{0.17em}{0ex}}$Index
file for atom selection (GROMACS “ndx" format)$\phantom{\rule{0.17em}{0ex}}\u27e9$
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. A group
with the same name may appear multiple times, as long as it contains the same indices in identical
order each time: an error is raised otherwise. 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 6.
 Keyword smp $\u27e8\phantom{\rule{0.17em}{0ex}}$Whether
SMP parallelism should be used$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 VMD build in use.
A state file contains the information produced during a Colvarsbased simulation besides atomic data, which is
provided by VMD.
Because many of the methods implemented in Colvars are historydependent, this file is often needed to
continue a long simulation over consecutive runs. Such state file is written automatically at the end of any simulation
with Colvars, and contains data accumulated during that simulation along with the step number at the
end of it. The step number read from the state file is then used to control such timedependent biases:
because of this essential role, the step number internal to Colvars may not always match the step number
reported by the MD program that carried during the simulation (which may instead restart from zero each
time).
Depending on the configuration, a state file may need to be loaded issued at the beginning of a new simulation
when timedependent biasing methods are applied (moving restraints, metadynamics, ABF, ...). . After
initialization of the Colvars module, a state file may be loaded at any time with the Tcl command cv
load.
It is possible to load a state file even if the configuration has changed: for example, new variables may be defined
or restraints be added in between consecutive runs. For each newly defined variable or bias, no information will
be read from the state file if this is unavailable: such new objects will remain uninitialized until the
first compute step. Conversely, any information that the state file has about variables or biases that
are no longer defined is silently ignored. Because these checks are performed based on the names of
variables and biases, it is the user's responsibility to ensure that these definitions are consistent between
runs.
During a simulation with collective variables defined, the following three output files are written:
 A state file, named outputName.colvars.state; this file is in ASCII (plain text) format. This file is
written at the end of the specified run, but can also be written at any time with the command cv save
(4.2.3).
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
restartName.colvars.state.
 If the parameter colvarsTrajFrequency is greater than 0 (default value: 100 steps), a trajectory file,
named outputName.colvars.traj, is written during the simulation. Unlike the state file, this file is
not needed to restart a simulation, but can be used for postprocessing and analysis. The format of this
file is described in sec. 4.7.6.
 Additionally, certain features, when enabled, can emit output files with a specific purpose: for
example, potentials of mean force can be written to file to be analyzed or plotted. These files are
described in the respective sections, but as a general rule they all use names beginning with the
outputNameprefix.
Other output files may also be written by specific methods, e.g. the ABF or metadynamics methods (7.2, 7.4).
Like the trajectory file, they are needed only for analyzing, not continuing a simulation. All such files' names also
begin with the prefix outputName.
This section summarizes the file formats of various files that Colvars may be reading or writing.
4.7.1 Configuration and state files.
Configuration files are text files that are generally read as input by VMD, and may be optionally inlined in a
VMD script (see 4.2.1). Starting from version 20170201, changes in newline encodings are handled transparently,
i.e. it is possible to typeset a configuration file in Windows (CRLF newlines) and then use it with Linux or macOS
(LFonly newlines).
State files, although not written manually, follow otherwise the same text format as configuration
files.
For atom selections that cannot be specified only by using internal Colvars keywords, external index files may
also be used following the NDX format used in GROMACS:
[ group_1_name ]
i1 i2 i3 i4 ...
... ... iN
[ group_2_name ]
...
where i1 through iN are 1based indices. Each group name may not contain spaces or tabs: otherwise, a parsing error
will be raised.
Multiple index files may be provided to Colvars, each using the keyword indexFile. Two index files may
contain groups with the same names, however these must also represent identical atom selections, i.e. the same
sequence of indices including order.
Other than with GROMACS, an index group may also be generated from the VMD commandline interface,
using the helper function write_index_group provided in the colvartools folder:
source colvartools/write_index_group.tcl
set sel [atomselect top "resname XXX and not hydrogen"]
write_index_group indexfile.ndx $sel "Ligand"
4.7.3 XYZ coordinate files
XYZ coordinate files are text files with the extension “
.xyz". They are read by the Colvars module using an
internal reader, and expect the following format:
$N$ 
 Comment  line  
${E}_{1}$  ${x}_{1}$  ${y}_{1}$  ${z}_{1}$ 
${E}_{2}$  ${x}_{2}$  ${y}_{2}$  ${z}_{2}$ 
… 
${E}_{N}$  ${x}_{N}$  ${y}_{N}$  ${z}_{N}$ 

where $N$ is the number of atomic
coordinates in the file and ${E}_{i}$ is
the chemical element of the $i$th
atom. Because ${E}_{i}$
is not used in Colvars, any string that does not contain tabs or spaces is acceptable.
An XYZ file may contain either one of the following scenarios:
 The file contains as many coordinates as the atoms that they are being read for: all coordinates will
be read from the file following the same order as the atoms appear in the selection generated using
the keywords listed in section 6. (Note that the order is guaranteed only if a single type of selection
keyword is used one or more times, and not guaranteed when different types of selection keywords
are used.)
 The file contains more coordinates than needed, and it is assumed to contain coordinates for the entire
system: only coordinates that match the numeric indices of the selected atoms are read, in order of
increasing number.
XYZfile coordinates are read directly by Colvars and stored internally as doubleprecision floating point
numbers (unlike VMD's reader, which uses single precision).
4.7.4 PDB coordinate files
PDB coordinate files are read by the Colvars module using existing functionality in VMD, and therefore
follow the same format. The values of the atomic coordinates and other fields, such as occupancy or temperature
factors, are then communicated to Colvars by VMD.
PDB files may be used either as one of the available mechanisms to select atoms (see the atomsFile keyword),
or more frequently to read reference coordinates for leastsquares fit alignment (see the refPositionsFile
keyword).
To select atoms via the atomsFile keyword, the option atomsFile is required, and atoms are selected based on
either one of the following cases.
 All atoms with a nonzero value of the corresponding column are selected.
 If and only if atomsFile is provided, only atoms with a value of atomsCol equaling atomsColValue
are selected. This can be useful to reuse the same PDB file for generating multiple selections.
To read coordinates via the refPositionsFile keyword, there are four possible scenarios.
 The file contains as many coordinates as the atoms that they are being read for: all coordinates will
be read from the file following the same order as the atoms appear in the selection generated using
the keywords listed in section 6. (Note that the order is guaranteed only if a single type of selection
keyword is used one or more times, and not guaranteed when different types of selection keywords
are used.)
 The file contains more coordinates than needed, and it is assumed to contain coordinates for the entire
system: only coordinates that match the numeric indices of the selected atoms are read, in order of
increasing number.
 If the corresponding refPositionsCol keyword is specified, only positions with a nonzero value of
the column refPositionsCol are read. This is particularly useful when all atoms were selected via
atomsFile and atomsCol, which can then be the same as refPositionsCol for loading coordinates.
 This is a special case of the
previous point: if both refPositionsCol and refPositionsColValue are specified, only atoms for
which the refPositionsCol column as the value refPositionsColValue are read.
Due to the fixedprecision PDB format, it is not recommended to use PDB files to read coordinates when
precision is a concern, and the XYZ format (see 4.7.2) is recommended instead.
4.7.5 Grid files: multicolumn text format
Many simulation methods and analysis tools write files that contain functions of the collective variables
tabulated on a grid (e.g. potentials of mean force or multidimentional histograms) for the purpose of analyzing
results. Such files are produced by ABF (7.2), metadynamics (7.4), multidimensional histograms (7.10), as well as
any restraint with optional thermodynamic integration support (7.1).
In some cases, these files may also be read as input of a new simulation. Suitable input files for this purpose are
typically generated as output files of previous simulations, or directly by the user in the specific case of ensemblebiased
metadynamics (7.4.5). This section explains the “multicolumn" format used by these files. For a multidimensional
function $f({\xi}_{1}$,
${\xi}_{2}$,
…$)$ the
multicolumn grid format is defined as follows:
#  ${N}_{cv}$     
#  $min\left({\xi}_{1}\right)$  $width\left({\xi}_{1}\right)$  $npoints\left({\xi}_{1}\right)$  $periodic\left({\xi}_{1}\right)$ 
#  $min\left({\xi}_{2}\right)$  $width\left({\xi}_{2}\right)$  $npoints\left({\xi}_{2}\right)$  $periodic\left({\xi}_{2}\right)$ 
#  …  …  …  … 
#  $min\left({\xi}_{{N}_{cv}}\right)$  $width\left({\xi}_{{N}_{cv}}\right)$  $npoints\left({\xi}_{{N}_{cv}}\right)$  $periodic\left({\xi}_{{N}_{cv}}\right)$  

 ${\xi}_{1}^{1}$  ${\xi}_{2}^{1}$  …  ${\xi}_{{N}_{cv}}^{1}$  f(${\xi}_{1}^{1}$, ${\xi}_{2}^{1}$, …, ${\xi}_{{N}_{cv}}^{1}$)  
 ${\xi}_{1}^{1}$  ${\xi}_{2}^{1}$  …  ${\xi}_{{N}_{cv}}^{2}$  f(${\xi}_{1}^{1}$, ${\xi}_{2}^{1}$, …, ${\xi}_{{N}_{cv}}^{2}$) 
 …  …  …  …  … 


Lines beginning with the character “#" are the header of the file.
${N}_{cv}$
is the number of collective variables sampled by the grid. For each variable
${\xi}_{i}$,
$min\left({\xi}_{i}\right)$
is the lowest value sampled by the grid (i.e. the leftmost boundary of the grid along
${\xi}_{i}$),
$width\left({\xi}_{i}\right)$ is the width of each
grid step along ${\xi}_{i}$,
$npoints\left({\xi}_{i}\right)$ is the number
of points and $periodic\left({\xi}_{i}\right)$
is a flag whose value is 1 or 0 depending on whether the grid is periodic along
${\xi}_{i}$. In most
situations:
 $min\left({\xi}_{i}\right)$
is given by the lowerBoundary keyword of the variable ${\xi}_{i}$;
 $width\left({\xi}_{i}\right)$
is given by the width keyword;
 $npoints\left({\xi}_{i}\right)$
is calculated from the two above numbers and the upperBoundary keyword;
 $periodic\left({\xi}_{i}\right)$
is set to 1 if and only if ${\xi}_{i}$
is periodic and the grids' boundaries cover its period.
How the grid's boundaries affect the sequence of points depends on how the contents of the file were
computed. In many cases, such as histograms and PMFs computed by metadynamics (7.4.5), the values of
${\xi}_{i}$
in the first few columns correspond to the midpoints of the corresponing bins,
i.e. ${\xi}_{1}^{1}=min\left({\xi}_{i}\right)+width\left({\xi}_{i}\right)\u22152$.
However, there is a slightly different format in PMF files computed by ABF (7.2) or other biases that use thermodynamic
integration (7.1). In these cases, it is freeenergy gradients that are accumulated on an (npoints)long grid along each
variable $\xi $:
after these gradients are integrated, the resulting PMF is discretized on a slightly larger grid with (npoints+1) points
along $\xi $
(unless the interval is periodic). Therefore, the grid's outer edges extend by
$width\left({\xi}_{i}\right)\u22152$ above and below the specified
boundaries, so that for instance $min\left({\xi}_{i}\right)$ in the
header appears to be shifted back by $width\left({\xi}_{i}\right)\u22152$
compared to what would be expected. Please keep this difference in mind when comparing PMFs computed by
different methods.
After the header, the rest of the file contains values of the tabulated function
$f({\xi}_{1}$,
${\xi}_{2}$,
…${\xi}_{{N}_{cv}})$, one for each line.
The first ${N}_{cv}$ columns
contain values of ${\xi}_{1}$,
${\xi}_{2}$,
…${\xi}_{{N}_{cv}}$ and the last column contains
the value of the function $f$.
Points are sorted in ascending order with the fastestchanging values at the right (“Cstyle" order). Each sweep of the rightmost
variable ${\xi}_{{N}_{cv}}$
is terminated by an empty line. For two dimensional grid files, this allows quick visualization by programs such as
GNUplot.
Example 1: multicolumn text file for a onedimensional histogram with lowerBoundary = 15, upperBoundary = 48
and width = 0.1.

#  1     
#  15  0.1  330  0 

 15.05  6.14012e07    
 15.15  7.47644e07    
 …  …    
 47.85  1.65944e06    
 47.95  1.46712e06    


Example 2: multicolumn text file for a twodimensional histogram of two dihedral angles (periodic interval with
6${}^{\circ}$
bins):
    
#  2     
#  180.0  6.0  30  1 
#  180.0  6.0  30  1 

 177.0  177.0  8.97117e06   
 177.0  171.0  1.53525e06   
 …  …  …   
 177.0  177.0  2.44295606   

 171.0  177.0  2.04702e05   
 …  …  …   

4.7.6 Output trajectory files
The Colvars trajectory file (with a suffix .colvars.traj) is a plain text file (scientific notation with 14digit
precision) whose columns represent quantities such as colvar values, applied forces, or individual restraints' energies.
Under most scenarios, plotting or analyzing this file is straightforward: for example, the following contains a variable
“$A$" and the energy
of a restraint “$rA$":
# step A E_rA
0 1.42467449615693e+01 6.30982865292123e+02
100 1.42282559728026e+01 6.20640585041317e+02
…
Occasionally, if the Colvars configuration is changed midrun certain quantities may be
added or removed, changing the column layout. Labels in comment lines can assist in such
cases: for example, consider the trajectory above with the addition of a second variable,
“$B$", after
10,000 steps:
# step A E_rA
0 1.42467449615693e+01 6.30982865292123e+02
100 1.42282559728026e+01 6.20640585041317e+02
…
# step A B E_rA
10000 1.38136915830229e+01 9.99574098859265e01 4.11184644791030e+02
10100 1.36437184346326e+01 9.99574091957314e01 3.37726286543895e+02
Analyzing the above file with standard tools is possible, but laborious: a convenience script is
provided for this and related purposes. It may be used either as a commandline tool or as a Python
module:
>>> from plot_colvars_traj import Colvars_traj
>>> traj = Colvars_traj('test.colvars.traj')
>>> print(traj['A'].steps, traj['A'].values)
[ 0 100 ... 10000 10100] [14.246745 14.228256 ... 13.813692 13.643718]
>>> print(traj['B'].steps, traj['B'].values)
[10000 10100] [0.999574 0.9995741]
5 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 5.1. For example, using the keyword rmsd (section 5.5.1)
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 5.1 (see 5.16 for details).
 A userdefined mathematical function of the existing components (see list in section 5.1), or of the
atomic coordinates directly (see the cartesian keyword in 5.8.1). The function is defined through
the keyword customFunction (see 5.17 for details).
 A userdefined Tcl function of the existing components (see list in section 5.1), or of the atomic
coordinates directly (see the cartesian keyword in 5.8.1). The function is provided by a separate Tcl
script, and referenced through the keyword scriptedFunction (see 5.18 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:
 Keyword name $\u27e8\phantom{\rule{0.17em}{0ex}}$Name
of this colvar$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: string
Default value: “colvar" + numeric id
Description: The name is an unique casesensitive 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.
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 6) 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. NOEbased 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;
 eulerPhi: Roll angle of rotation from references coordinates;
 eulerTheta: Pitch angle of rotation from references coordinates;
 eulerPsi: Yaw angle of rotation from references 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;
 mapTotal: total value of a volumetric map;
 orientationAngle: angle of the bestfit rotation from a set of reference coordinates;
 orientationProj: cosine of orientationProj;
 spinAngle: projection orthogonal to an axis of the bestfit rotation from a set of reference coordinates;
 tilt: projection on an axis of the bestfit 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;
 alpha: $\alpha $helix
content of a protein segment.
 dihedralPC: projection of protein backbone dihedrals onto a dihedral principal component.
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: ${N}_{1}\times {N}_{2}$);
 orientation: bestfit 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 5.16), a custom function (see 5.17), or a scripted function (see
5.18).
In the rest of this section, all available component types are listed, along with their physical units and
their ranges of values, if limited. Such ranges are often used to define automatically default sampling
intervals, for example by setting the parameters 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
5.13.
5.2.1 distance: centerofmass distance between two groups.
The distance {...} block defines a distance component between the two atom groups, group1 and
group2.
List of keywords (see also 5.16 for additional options):
 Keyword group1 $\u27e8\phantom{\rule{0.17em}{0ex}}$First
group of atoms$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: distance
Acceptable values: Block group1 {...}
Description: First group of atoms.
 Keyword group2: analogous to group1
 Keyword forceNoPBC $\u27e8\phantom{\rule{0.17em}{0ex}}$Calculate
absolute rather than minimumimage distance?$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 minimumimage 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
minimumimage distance might give the wrong result because of a relatively small periodic cell.
 Keyword oneSiteTotalForce $\u27e8\phantom{\rule{0.17em}{0ex}}$Measure
total force on group 1 only?$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 (28) in
section 7.2) that only involves atoms of group1. This option is only useful for ABF, or custom biases
that compute total forces. See section 7.2 for details.
The value returned is a positive number (in length unit), 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).
5.2.2 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.
List of keywords (see also 5.16 for additional options):
 Keyword main $\u27e8\phantom{\rule{0.17em}{0ex}}$Main
group of atoms$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: distanceZ
Acceptable values: Block main {...}
Description: Group of atoms whose position $\text{}r\text{}$
is measured.
 Keyword ref $\u27e8\phantom{\rule{0.17em}{0ex}}$Reference
group of atoms$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword ref2 $\u27e8\phantom{\rule{0.17em}{0ex}}$Secondary
reference group$\phantom{\rule{0.17em}{0ex}}\u27e9$
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: $\text{}e\text{}={(\parallel {\text{}r\text{}}_{2}{\text{}r\text{}}_{1}\parallel )}^{1}\times ({\text{}r\text{}}_{2}{\text{}r\text{}}_{1})$.
In this case, the origin is ${\text{}r\text{}}_{m}=1\u22152({\text{}r\text{}}_{1}+{\text{}r\text{}}_{2})$,
and the value of the component is $\text{}e\text{}\cdot (\text{}r\text{}{\text{}r\text{}}_{m})$.
 Keyword axis $\u27e8\phantom{\rule{0.17em}{0ex}}$Projection
axis$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: distanceZ
Acceptable values: (x, y, z) triplet
Default value: (0.0, 0.0, 1.0)
Description: This vector will be normalized to 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 $\text{}e\text{}\cdot (\text{}r\text{}{\text{}r\text{}}_{1})$.
The vector should be written as three components separated by commas and enclosed in parentheses.
 Keyword forceNoPBC: see definition of forceNoPBC (distance component)
 Keyword oneSiteTotalForce: see definition of oneSiteTotalForce (distance component)
This component returns a number (in length unit) 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
${b}_{z}\u22152$ and
${b}_{z}\u22152$, where
${b}_{z}$ is the box
length along $z$
(this behavior is disabled if forceNoPBC is set).
5.2.3 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).
List of keywords (see also 5.16 for additional options):
 Keyword main: see definition of main (distanceZ component)
 Keyword ref: see definition of ref (distanceZ component)
 Keyword ref2: see definition of ref2 (distanceZ component)
 Keyword axis: see definition of axis (distanceZ component)
 Keyword forceNoPBC: see definition of forceNoPBC (distance component)
 Keyword oneSiteTotalForce: see definition of oneSiteTotalForce (distance component)
5.2.4 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 3vector joining the centers of mass of
group1 and group2.
List of keywords (see also 5.16 for additional options):
 Keyword group1: see definition of group1 (distance component)
 Keyword group2: analogous to group1
 Keyword forceNoPBC: see definition of forceNoPBC (distance component)
 Keyword oneSiteTotalForce: see definition of oneSiteTotalForce (distance component)
5.2.5 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 3dimensional unit vector
$d=({d}_{x},{d}_{y},{d}_{z})$, with
$\leftd\right=1$.
List of keywords (see also 5.16 for additional options):
 Keyword group1: see definition of group1 (distance component)
 Keyword group2: analogous to group1
 Keyword forceNoPBC: see definition of forceNoPBC (distance component)
 Keyword oneSiteTotalForce: see definition of oneSiteTotalForce (distance component)
5.2.6 distanceInv: mean distance between two groups of atoms.
The distanceInv {...} block defines a generalized mean distance between two groups of atoms 1 and 2, where each distance
is taken to the power $n$:
$${d}_{1,2}^{\left[n\right]}\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}{\left(\frac{1}{{N}_{1}{N}_{2}}\sum _{i,j}{d}_{ij}^{n}\right)}^{1\u2215n}$$  (2) 
where ${d}_{ij}$ 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 5.16 for additional options):
 Keyword group1: see definition of group1 (distance component)
 Keyword group2: analogous to group1
 Keyword oneSiteTotalForce: see definition of oneSiteTotalForce (distance component)
 Keyword exponent $\u27e8\phantom{\rule{0.17em}{0ex}}$Exponent
$n$
in equation 2$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 NOEmeasured distances.
This component returns a number ranging from $0$
to the largest possible distance within the chosen boundary conditions.
5.3.1 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
$[0:180]$.
List of keywords (see also 5.16 for additional options):
 Keyword group1: see definition of group1 (distance component)
 Keyword group2: analogous to group1
 Keyword group3: analogous to group1
 Keyword forceNoPBC: see definition of forceNoPBC (distance component)
 Keyword oneSiteTotalForce: see definition of oneSiteTotalForce (distance component)
5.3.2 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 $[0:180]$.
List of keywords (see also 5.16 for additional options):
 Keyword group1: see definition of group1 (distance component)
 Keyword group2: analogous to group1
 Keyword group3: analogous to group1
 Keyword forceNoPBC: see definition of forceNoPBC (distance component)
 Keyword oneSiteTotalForce: see definition of oneSiteTotalForce (distance component)
5.3.3 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
$[180:180]$. 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 5.16 for additional options):
 Keyword group1: see definition of group1 (distance component)
 Keyword group2: analogous to group1
 Keyword group3: analogous to group1
 Keyword group4: analogous to group1
 Keyword forceNoPBC: see definition of forceNoPBC (distance component)
 Keyword oneSiteTotalForce: see definition of oneSiteTotalForce (distance component)
5.3.4 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
$[0:180]$. To
obtain spherical coordinates in a frame of reference tied to another group of atoms, use the fittingGroup (6.2)
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 5.16 for additional options):
 Keyword atoms $\u27e8\phantom{\rule{0.17em}{0ex}}$Atom
group$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: polarPhi
Acceptable values: atoms {...} block
Description: Defines the group of atoms for the COM of which the angle should be calculated.
5.3.5 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
$[180:180]$. 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 (6.2) 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 5.16 for additional options):
 Keyword atoms $\u27e8\phantom{\rule{0.17em}{0ex}}$Atom
group$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: polarPhi
Acceptable values: atoms {...} block
Description: Defines the group of atoms for the COM of which the angle should be calculated.
5.4.1 coordNum: coordination number between two groups.
The coordNum {...} block defines a coordination number (or number of contacts), which calculates the function
$(1{(d\u2215{d}_{0})}^{n})\u2215(1{(d\u2215{d}_{0})}^{m})$, where
${d}_{0}$ is the “cutoff"
distance, and $n$
and $m$ are
exponents that can control its long range behavior and stiffness [3]. This function is summed over all pairs of atoms
in group1 and group2:
$$C(group1,group2)\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}\sum _{i\in group1}\sum _{j\in group2}\frac{1{\left(\right{x}_{i}{x}_{j}\u2215{d}_{0})}^{n}}{1{\left(\right{x}_{i}{x}_{j}\u2215{d}_{0})}^{m}}$$  (3) 
List of keywords (see also 5.16 for additional options):
 Keyword group1: see definition of group1 (distance component)
 Keyword group2: analogous to group1
 Keyword cutoff $\u27e8\phantom{\rule{0.17em}{0ex}}$“Interaction"
distance (length unit)$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: coordNum
Acceptable values: positive decimal
Default value: 4.0 Å
Description: This number defines the switching distance to define an interatomic contact: for $d\ll {d}_{0}$,
the switching function $(1{(d\u2215{d}_{0})}^{n})\u2215(1{(d\u2215{d}_{0})}^{m})$
is close to 1, at $d={d}_{0}$
it has a value of $n\u2215m$
($1\u22152$
with the default $n$
and $m$),
and at $d\gg {d}_{0}$
it goes to zero approximately like ${d}^{mn}$.
Hence, for a proper behavior, $m$
must be larger than $n$.
 Keyword cutoff3 $\u27e8\phantom{\rule{0.17em}{0ex}}$Reference
distance vector (length unit)$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword expNumer $\u27e8\phantom{\rule{0.17em}{0ex}}$Numerator
exponent$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: coordNum
Acceptable values: positive even integer
Default value: 6
Description: This number defines the $n$
exponent for the switching function.
 Keyword expDenom $\u27e8\phantom{\rule{0.17em}{0ex}}$Denominator
exponent$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: coordNum
Acceptable values: positive even integer
Default value: 12
Description: This number defines the $m$
exponent for the switching function.
 Keyword group2CenterOnly $\u27e8\phantom{\rule{0.17em}{0ex}}$Use
only group2's center of mass$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword tolerance $\u27e8\phantom{\rule{0.17em}{0ex}}$Pairlist
control$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 $\left[0,1\right]$.
 Keyword pairListFrequency $\u27e8\phantom{\rule{0.17em}{0ex}}$Pairlist
regeneration frequency$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 ${N}_{group1}\times {N}_{group2}$ (all distances
are less than the cutoff), or ${N}_{group1}$
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
${N}_{group1}\times {N}_{group2}$. Setting
$tolerance>0$
ameliorates this to some degree, although every pair is still checked to regenerate the pairlist.
5.4.2 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:
$$C\left(group1\right)\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}\sum _{i\in group1}\sum _{j>i}\frac{1{\left(\right{x}_{i}{x}_{j}\u2215{d}_{0})}^{n}}{1{\left(\right{x}_{i}{x}_{j}\u2215{d}_{0})}^{m}}$$  (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.
List of keywords (see also 5.16 for additional options):
 Keyword group1: see definition of group1 (coordNum component)
 Keyword cutoff: see definition of cutoff (coordNum component)
 Keyword cutoff3: see definition of cutoff3 (coordNum component)
 Keyword expNumer: see definition of expNumer (coordNum component)
 Keyword expDenom: see definition of expDenom (coordNum component)
 Keyword tolerance: see definition of tolerance (coordNum component)
 Keyword pairListFrequency: see definition of pairListFrequency (coordNum component)
This component returns a dimensionless number, which ranges from approximately 0 (all interatomic distances much larger
than the cutoff) to ${N}_{group1}\times ({N}_{group1}1)\u22152$
(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 ${N}_{group1}^{2}$.
5.4.3 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).
List of keywords (see also 5.16 for additional options):
 Keyword acceptor $\u27e8\phantom{\rule{0.17em}{0ex}}$Number
of the acceptor atom$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: hBond
Acceptable values: positive integer
Description: Number that uses the same convention as atomNumbers.
 Keyword donor: analogous to acceptor
 Keyword cutoff: see definition of cutoff (coordNum component)
Note: default value is 3.3 Å.
 Keyword expNumer: see definition of expNumer (coordNum component)
Note: default value is 6.
 Keyword expDenom: see definition of expDenom (coordNum component)
Note: default value is 8.
5.5.1 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
$\left\{{x}_{1}\right(t),{x}_{2}(t),\dots \⁡{x}_{N}(t\left)\right\}$, the colvar component rmsd
calculates the optimal rotation ${U}^{\left\{{x}_{i}\right(t\left)\right\}\to \left\{{x}_{i}^{\left(ref\right)}\right\}}$ that
best superimposes the coordinates $\left\{{x}_{i}\right(t\left)\right\}$
onto a set of reference coordinates $\left\{{x}_{i}^{\left(ref\right)}\right\}$.
Both the current and the reference coordinates are centered on their centers of geometry,
${x}_{cog}\left(t\right)$ and
${x}_{cog}^{\left(ref\right)}$. The
root mean square displacement is then defined as:
$$RMSD\left(\right\{{x}_{i}\left(t\right)\},\{{x}_{i}^{\left(ref\right)}\left\}\right)\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}\sqrt{\frac{1}{N}\sum _{i=1}^{N}{\leftU\left({x}_{i}\left(t\right){x}_{cog}\left(t\right)\right)\left({x}_{i}^{\left(ref\right)}{x}_{cog}^{\left(ref\right)}\right)\right}^{2}}$$  (5) 
The optimal rotation ${U}^{\left\{{x}_{i}\right(t\left)\right\}\to \left\{{x}_{i}^{\left(ref\right)}\right\}}$
is calculated within the formalism developed in reference [4], which guarantees a continuous dependence of
${U}^{\left\{{x}_{i}\right(t\left)\right\}\to \left\{{x}_{i}^{\left(ref\right)}\right\}}$ with respect
to $\left\{{x}_{i}\right(t\left)\right\}$.
List of keywords (see also 5.16 for additional options):
 Keyword atoms $\u27e8\phantom{\rule{0.17em}{0ex}}$Atom
group$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 rotateToReference) should typically NOT be set within this
block. Exceptions to this rule are the special cases discussed in the Advanced usage paragraph below.
 Keyword refPositions $\u27e8\phantom{\rule{0.17em}{0ex}}$Reference
coordinates$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: rmsd
Acceptable values: spaceseparated 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 rototranslational fit. See the equivalent option
refPositions within the atom group definition for details on acceptable formats and other features.
 Keyword refPositionsFile $\u27e8\phantom{\rule{0.17em}{0ex}}$Reference
coordinates file$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 rototranslational fit. See the equivalent option
refPositionsFile within the atom group definition for details on acceptable file formats and other
features.
 Keyword refPositionsCol $\u27e8\phantom{\rule{0.17em}{0ex}}$PDB
column containing atom flags$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword refPositionsColValue $\u27e8\phantom{\rule{0.17em}{0ex}}$Atom
selection flag in the PDB column$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 nonzero value are read.
 Keyword atomPermutation $\u27e8\phantom{\rule{0.17em}{0ex}}$Alternate
ordering of atoms for RMSD computation$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: rmsd
Acceptable values: List of atom numbers
Description: If defined, this parameter defines a reordering (permutation) of the 1based atom
numbers that can be used to compute the RMSD, typically due to molecular symmetry. This parameter
can be specified multiple times, each one defining a new permutation: the returned RMSD value is
the minimum over the set of permutations. For example, if the atoms making up the group are 6, 7, 8,
9, and atoms 7, 8, and 9 are invariant by circular permutation (as the hydrogens in a CH3 group), a
symmetryadapted RMSD would be obtained by adding:
atomPermutation 6 8 9 7
atomPermutation 6 9 7 8
This will result in these 2 atom orders being considered in addition to the order used when defining the
atom group. Note that this does not affect the leastsquares rototranslational fit. Therefore, this feature
is mostly useful when using custom fitting parameters within the atom group, such as fittingGroup,
or when fitting is disabled altogether. For details, see reference [5].
This component returns a positive real number (in length unit).
5.5.2 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 nonminimal RMSD values. This can be achieved by setting
the related options (
refPositions, etc.) explicitly in the atom group block. This allows for the following
nonstandard cases:
 applying the optimal translation, but no rotation (rotateToReference off), to bias or restrain the
shape and orientation, but not the position of the atom group;
 applying the optimal rotation, but no translation (centerToReference off), to bias or restrain the
shape and position, but not the orientation of the atom group;
 disabling the application of optimal rototranslations, which lets the RMSD component describe the
deviation of atoms from fixed positions in the laboratory frame: this allows for custom positional
restraints within the Colvars module;
 fitting the atomic positions to different reference coordinates than those used in the RMSD calculation
itself (by specifying refPositions or refPositionsFile within the atom group as well as within
the rmsd block);
 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 (see example in the section on fittingGroup).
5.5.3 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
${\mathbb{R}}^{3n}$, where
$n$ is the
number of atoms in the group. The computed quantity is the total projection:
$$p\left(\right\{{x}_{i}\left(t\right)\},\{{x}_{i}^{\left(ref\right)}\left\}\right)\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}\sum _{i=1}^{n}{v}_{i}\cdot \left(U\left({x}_{i}\right(t){x}_{cog}(t\left)\right)({x}_{i}^{\left(ref\right)}{x}_{cog}^{\left(ref\right)})\right),$$  (6) 
where, as in the rmsd component, $U$
is the optimal rotation matrix, ${x}_{cog}\left(t\right)$
and ${x}_{cog}^{\left(ref\right)}$
are the centers of geometry of the current and reference positions respectively, and
${v}_{i}$
are the components of the vector for each atom. Example choices for
$\left({v}_{i}\right)$ are an
eigenvector of the covariance matrix (essential mode), or a normal mode of the system. It is assumed that
${\sum \⁡}_{i}{v}_{i}=0$: otherwise, the Colvars
module centers the ${v}_{i}$
automatically when reading them from the configuration.
List of keywords (see also 5.16 for additional options):
 Keyword atoms: see definition of atoms (rmsd component)
 Keyword refPositions: see definition of refPositions (rmsd component)
 Keyword refPositionsFile: see definition of refPositionsFile (rmsd component)
 Keyword refPositionsCol: see definition of refPositionsCol (rmsd component)
 Keyword refPositionsColValue: see definition of refPositionsColValue (rmsd component)
 Keyword vector $\u27e8\phantom{\rule{0.17em}{0ex}}$Vector
components$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: eigenvector
Acceptable values: spaceseparated list of (x, y, z) triplets
Description: This option (mutually exclusive with vectorFile) sets the values of the vector components.
 Keyword vectorFile $\u27e8\phantom{\rule{0.17em}{0ex}}$file
containing vector components$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: eigenvector
Acceptable values: UNIX filename
Description: This option (mutually exclusive with vector) sets the name of an XYZ (4.7.3) or PDB
coordinate file containing the vector components. For a PDB file specifically, the components are read
from the X, Y and Z fields. Note: The PDB file has limited precision and fixedpoint numbers: in some
cases, the vector components may not be accurately represented; a XYZ file should be used instead,
containing floatingpoint numbers. Note: Reading data from a coordinate file may entail an automatic
unit conversion if the length unit currently used by the MD engine is not Å. If this is not the desired
behavior, this can be remedied using the normalizeVectoroption described below.
 Keyword vectorCol $\u27e8\phantom{\rule{0.17em}{0ex}}$PDB
column used to flag participating atoms$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: eigenvector
Acceptable values: O or B
Description: Analogous to atomsCol.
 Keyword vectorColValue $\u27e8\phantom{\rule{0.17em}{0ex}}$Value
used to flag participating atoms in the PDB file$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: eigenvector
Acceptable values: positive decimal
Description: Analogous to atomsColValue.
 Keyword normalizeVector $\u27e8\phantom{\rule{0.17em}{0ex}}$Normalize
the vector components when reading them?$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: eigenvector
Acceptable values: boolean
Default value: off
Description: If this flag is activated, the norm of the vector $\leftv\right=\sqrt{{\sum \⁡}_{i}{\left{v}_{i}\right}^{2}}$
is set equal to 1 by automatically rescaling all the components ${v}_{i}$;
alternatively, the value of $\leftv\right$
is printed.
 Keyword differenceVector $\u27e8\phantom{\rule{0.17em}{0ex}}$The
$3n$dimensional
vector is the difference between vector and refPositions$\phantom{\rule{0.17em}{0ex}}\u27e9$
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, ${x}_{i}^{\prime}$:
the vector ${v}_{i}$
is then defined as ${v}_{i}=\left({x}_{i}^{\prime}{x}_{i}^{\left(ref\right)}\right)$.
This allows to conveniently define a colvar $\xi $
as a projection on the linear transformation between two sets of positions, “A" and “B". If this flag is
on, the vector is normalized so that $\xi =0$
when the atoms are at the set of positions “A" and $\xi =1$
at the set of positions “B". Setting normalizeVectoron overrides this behavior.
5.5.4 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
$\left\{{x}_{1}\right(t),{x}_{2}(t),\dots \⁡{x}_{N}(t\left)\right\}$ with respect to their
center of geometry, ${x}_{cog}\left(t\right)$:
$${R}_{gyr}\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}\sqrt{\frac{1}{N}\sum _{i=1}^{N}{\left{x}_{i}\left(t\right){x}_{cog}\left(t\right)\right}^{2}}$$  (7) 
This component must contain one atoms {...} block to define the atom group, and returns a positive number,
expressed in length unit.
List of keywords (see also 5.16 for additional options):
 Keyword atoms: see definition of atoms (rmsd component)
5.5.5 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
$\left\{{x}_{1}\right(t),{x}_{2}(t),\dots \⁡{x}_{N}(t\left)\right\}$ with respect to their
center of geometry, ${x}_{cog}\left(t\right)$:
$$I\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}\sum _{i=1}^{N}{\left{x}_{i}\left(t\right){x}_{cog}\left(t\right)\right}^{2}$$  (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 length
unit${}^{2}$.
List of keywords (see also 5.16 for additional options):
 Keyword atoms: see definition of atoms (rmsd component)
5.5.6 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.
List of keywords (see also 5.16 for additional options):
 Keyword atoms: see definition of atoms (rmsd component)
5.5.7 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
$e$ of the moment of inertia of a group
of atomic positions $\left\{{x}_{1}\right(t),{x}_{2}(t),\dots \⁡{x}_{N}(t\left)\right\}$ with respect
to their center of geometry, ${x}_{cog}\left(t\right)$:
$${I}_{e}\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}\sum _{i=1}^{N}{\left(\left({x}_{i}\left(t\right){x}_{cog}\left(t\right)\right)\cdot e\right)}^{2}$$  (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 length
unit${}^{2}$.
List of keywords (see also 5.16 for additional options):
 Keyword atoms: see definition of atoms (rmsd component)
 Keyword axis $\u27e8\phantom{\rule{0.17em}{0ex}}$Projection
axis$\phantom{\rule{0.17em}{0ex}}\u27e9$
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
$e$.
The variables discussed in this section quantify the rotations of macromolecules (or other quasirigid objects)
from a given set of reference coordinates to the current coordinates. Such rotations are computed following the same
method used for bestfit RMSDs (see rmsd and fittingGroup). The underlying mathematical formalism is
described in reference [4], and the implementation in reference [1].
The first of the functions described is the orientation, which describes the full rotation as a unit quaternion
$q=({q}_{0},{q}_{1},{q}_{2},{q}_{3})$,
i.e. 4 numbers with one constraint (3 degrees of freedom). The quaternion
$q$
is one of only two representations that are both complete and accurate, the other being a
$3\times 3$ unit matrix with 3 independent
parameters. Although $q$
is used internally in the Colvars module for features such as the rmsd function and the fittingGroup
option, its direct use as a collective variable is more difficult, and mostly limited to fixed or moving
restraints.
The two functions orientationAngle and orientationProj, with the latter being
the cosine of the former, represent the amplitude of the full rotation described by
$q$,
regardless of the direction of its axis. As onedimensional scalar variables, both orientationAngle and
orientationProj are a much reduced simplification of the full rotation. However, they can be used in a variety of
methods including both restraints and PMF computations.
A slightly more complete parametrization is achieved by decomposing the full rotation into the two parameters,
tilt and spinAngle. These quantify the amplitudes of two independent subrotations away from a certain axis
$e$, and around the same
axis $e$, respectively.
The axis $e$ is
chosen by the user, and is by default the Z axis: under that choice, tilt is equivalent to the sine of the Euler “pitch" angle
$\mathit{\theta}$, and spinAngle to the sum
of the other two angles, $(\varphi +\psi )$.
This parameterization is mathematically well defined for almost all full rotations, including small ones when the
current coordinates are almost completely aligned with the reference ones. However, a mathematical singularity
prevents using the spinAngle function near configurations where the value of tilt tilt is 1 (i.e. a
180${}^{\circ}$ rotation around an
axis orthogonal to $e$).
For these reasons, tilt and spinAngle are useful when the allowed rotations are known to
have approximately the same axis, and differ only in the magnitude of the corresponding angle.
In this use case, spinAngle measures the angle of the subrotation around the chosen axis
$e$, whereas tilt measures
the dot product between $e$
and the actual axis of the full rotation.
Lastly, the traditional Euler angles are also available as the functions eulerPhi, eulerTheta and
eulerPsi. Altogether, these are sufficient to represent all three degrees of freedom of a full rotation.
However, they also suffer from the potential “gimbal lock" problem, which emerges whenever
$\mathit{\theta}\simeq \pm 9{0}^{\circ}$,
which includes also the case where the full rotation is small. Under such conditions, the angles
$\varphi $ and
$\psi $ are both
illdefined and cannot be used as collective variables. For these reasons, it is highly recommmended that Euler angles are
used only in simulations where their range of applicability is known ahead of time, and excludes configurations
where $\mathit{\theta}\simeq \pm 9{0}^{\circ}$
altogether.
5.6.1 orientation: orientation from reference coordinates.
The block orientation {...} returns the same optimal rotation used in the rmsd component to superimpose the coordinates
$\left\{{x}_{i}\right(t\left)\right\}$ onto a set of reference coordinates
$\left\{{x}_{i}^{\left(ref\right)}\right\}$. Such component returns a
four dimensional vector $q=({q}_{0},{q}_{1},{q}_{2},{q}_{3})$, with
${\sum \⁡}_{i}{q}_{i}^{2}=1$; this quaternion expresses the optimal
rotation $\left\{{x}_{i}\right(t\left)\right\}\to \left\{{x}_{i}^{\left(ref\right)}\right\}$ according to the formalism
in reference [4]. The quaternion $({q}_{0},{q}_{1},{q}_{2},{q}_{3})$
can also be written as $\left(\mathrm{cos}\⁡(\mathit{\theta}\u22152),\phantom{\rule{0.17em}{0ex}}\mathrm{sin}\⁡(\mathit{\theta}\u22152)u\right)$,
where $\mathit{\theta}$ is the angle and
$u$ 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 5.16 for additional options):
 Keyword atoms: see definition of atoms (rmsd component)
 Keyword refPositions: see definition of refPositions (rmsd component)
 Keyword refPositionsFile: see definition of refPositionsFile (rmsd component)
 Keyword refPositionsCol: see definition of refPositionsCol (rmsd component)
 Keyword refPositionsColValue: see definition of refPositionsColValue (rmsd component)
 Keyword closestToQuaternion $\u27e8\phantom{\rule{0.17em}{0ex}}$Reference
rotation$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 through 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.
5.6.2 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 $\mathit{\theta}$
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 5.16 for additional options):
 Keyword atoms: see definition of atoms (rmsd component)
 Keyword refPositions: see definition of refPositions (rmsd component)
 Keyword refPositionsFile: see definition of refPositionsFile (rmsd component)
 Keyword refPositionsCol: see definition of refPositionsCol (rmsd component)
 Keyword refPositionsColValue: see definition of refPositionsColValue (rmsd component)
5.6.3 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, refPositionsCol and refPositionsColValue. The returned value is the cosine of the
angle of rotation $\mathit{\theta}$
between the current and the reference positions. The range of values is [1:1].
List of keywords (see also 5.16 for additional options):
 Keyword atoms: see definition of atoms (rmsd component)
 Keyword refPositions: see definition of refPositions (rmsd component)
 Keyword refPositionsFile: see definition of refPositionsFile (rmsd component)
 Keyword refPositionsCol: see definition of refPositionsCol (rmsd component)
 Keyword refPositionsColValue: see definition of refPositionsColValue (rmsd component)
5.6.4 spinAngle: angle of rotation around a given axis.
The complete rotation described by orientation can optionally be decomposed into two subrotations: 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" subrotation around e.
List of keywords (see also 5.16 for additional options):
 Keyword atoms: see definition of atoms (rmsd component)
 Keyword refPositions: see definition of refPositions (rmsd component)
 Keyword refPositionsFile: see definition of refPositionsFile (rmsd component)
 Keyword refPositionsCol: see definition of refPositionsCol (rmsd component)
 Keyword refPositionsColValue: see definition of refPositionsColValue (rmsd component)
 Keyword axis $\u27e8\phantom{\rule{0.17em}{0ex}}$Special
rotation axis$\phantom{\rule{0.17em}{0ex}}\u27e9$
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
$1$.
5.6.5 tilt: cosine of the rotation orthogonal to a given axis.
The component tilt measures the cosine of the angle of the “tilt" subrotation, which combined with the “spin"
subrotation 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 $\mathit{\theta}$
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 antiparallel orientation.
List of keywords (see also 5.16 for additional options):
 Keyword atoms: see definition of atoms (rmsd component)
 Keyword refPositions: see definition of refPositions (rmsd component)
 Keyword refPositionsFile: see definition of refPositionsFile (rmsd component)
 Keyword refPositionsCol: see definition of refPositionsCol (rmsd component)
 Keyword refPositionsColValue: see definition of refPositionsColValue (rmsd component)
 Keyword axis: see definition of axis (spinAngle component)
5.6.6 eulerPhi: Roll angle from references coordinates.
Assuming the axes of the original frame are denoted as x, y, z and the axes of the rotated frame as X, Y, Z, the
line of nodes, N, can be defined as the intersection of the plane xy and XY. The axis perpendicular to N and z is
defined as P. In this case, as illustrated in the figure below, the complete rotation described by orientation can
optionally be decomposed into three Euler angles:
 the “roll" angle $\varphi $,
i.e. the rotation angle from the x axis to the N axis;
 the “pitch" angle $\mathit{\theta}$,
i.e. the rotation angle from the P axis to the Z axis, and
 the “yaw" angle $\psi $,
i.e. the rotation angle from the N axis to the X axis.
Although Euler angles are more straightforward to use than quaternions, they are also potentially subject to the
“gimbal lock" problem:
https://en.wikipedia.org/wiki/Gimbal_lock
which arises whenever $\mathit{\theta}\simeq \pm 9{0}^{\circ}$,
including the common case when the simulated coordinates are near the reference coordinates. Therefore, a safe
use of Euler angles as collective variables requires the use of restraints to avoid such singularities, such as done in
reference [6] and in the proteinligand binding NAMD tutorial.
The eulerPhi component accepts exactly the same options as orientation, and measures the
rotation angle from the x axis to the N axis. This angle is expressed in degrees within the periodic range
$[18{0}^{\circ}:18{0}^{\circ}]$.
List of keywords (see also 5.16 for additional options):
 Keyword atoms: see definition of atoms (rmsd component)
 Keyword refPositions: see definition of refPositions (rmsd component)
 Keyword refPositionsFile: see definition of refPositionsFile (rmsd component)

Keyword refPositionsCol:
see definition of refPositionsCol (rmsd component)

Keyword refPositionsColValue:
see definition of refPositionsColValue (rmsd component)
5.6.7 eulerTheta: Pitch angle from references coordinates.
This component accepts exactly the same options as orientation, and measures the
rotation angle from the P axis to the Z axis. This angle is expressed in degrees within the range
$[9{0}^{\circ}:9{0}^{\circ}]$.
Warning: When this angle reaches $9{0}^{\circ}$
or $9{0}^{\circ}$, the
definition of orientation by euler angles suffers from the gimbal lock issue. You may need to apply a restraint to keep
eulerTheta away from the singularities.
List of keywords (see also 5.16 for additional options):
 Keyword atoms: see definition of atoms (rmsd component)
 Keyword refPositions: see definition of refPositions (rmsd component)
 Keyword refPositionsFile: see definition of refPositionsFile (rmsd component)

Keyword refPositionsCol:
see definition of refPositionsCol (rmsd component)

Keyword refPositionsColValue:
see definition of refPositionsColValue (rmsd component)
5.6.8 eulerPsi: Yaw angle from references coordinates.
This component accepts exactly the same options as orientation, and measures the rotation
angle from the N axis to the X axis. This angle is expressed in degrees within the periodic range
$[18{0}^{\circ}:18{0}^{\circ}]$.
List of keywords (see also 5.16 for additional options):
 Keyword atoms: see definition of atoms (rmsd component)
 Keyword refPositions: see definition of refPositions (rmsd component)
 Keyword refPositionsFile: see definition of refPositionsFile (rmsd component)

Keyword refPositionsCol:
see definition of refPositionsCol (rmsd component)

Keyword refPositionsColValue:
see definition of refPositionsColValue (rmsd component)
5.7 Protein structure descriptors
5.7.1 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:
$$\begin{array}{rcll}\alpha \left({C}_{\alpha}^{\left({N}_{0}\right)},{O}^{\left({N}_{0}\right)},{C}_{\alpha}^{({N}_{0}+1)},{O}^{({N}_{0}+1)},\dots \⁡{N}^{({N}_{0}+5)},{C}_{\alpha}^{({N}_{0}+5)},{O}^{({N}_{0}+5)},\dots \⁡{N}^{({N}_{0}+N)},{C}_{\alpha}^{({N}_{0}+N)}\right)\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.28em}{0ex}}& & & \text{(10)}\text{}\text{}\\ \phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.28em}{0ex}}\frac{1}{2(N2)}\sum _{n={N}_{0}}^{{N}_{0}+N2}angf\left({C}_{\alpha}^{\left(n\right)},{C}_{\alpha}^{(n+1)},{C}_{\alpha}^{(n+2)}\right)\phantom{\rule{0.28em}{0ex}}+\phantom{\rule{0.28em}{0ex}}\frac{1}{2(N4)}\sum _{n={N}_{0}}^{{N}_{0}+N4}hbf\left({O}^{\left(n\right)},{N}^{(n+4)}\right),& & & \text{}\\ & & & \text{(11)}\text{}\text{}\end{array}$$
where the score function for the ${C}_{\alpha}{C}_{\alpha}{C}_{\alpha}$
angle is defined as:
$$angf\left({C}_{\alpha}^{\left(n\right)},{C}_{\alpha}^{(n+1)},{C}_{\alpha}^{(n+2)}\right)\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}\frac{1{\left(\mathit{\theta}({C}_{\alpha}^{\left(n\right)},{C}_{\alpha}^{(n+1)},{C}_{\alpha}^{(n+2)}){\mathit{\theta}}_{0}\right)}^{2}\u2215{\left(\mathrm{\Delta}{\mathit{\theta}}_{tol}\right)}^{2}}{1{\left(\mathit{\theta}({C}_{\alpha}^{\left(n\right)},{C}_{\alpha}^{(n+1)},{C}_{\alpha}^{(n+2)}){\mathit{\theta}}_{0}\right)}^{4}\u2215{\left(\mathrm{\Delta}{\mathit{\theta}}_{tol}\right)}^{4}},$$  (12) 
and the score function for the ${O}^{\left(n\right)}\leftrightarrow {N}^{(n+4)}$
hydrogen bond is defined through a hBond colvar component on the same atoms.
List of keywords (see also 5.16 for additional options):
 Keyword residueRange $\u27e8\phantom{\rule{0.17em}{0ex}}$Potential
$\alpha $helical
residues$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword psfSegID $\u27e8\phantom{\rule{0.17em}{0ex}}$PSF
segment identifier$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword hBondCoeff $\u27e8\phantom{\rule{0.17em}{0ex}}$Coefficient
for the hydrogen bond term$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword angleRef $\u27e8\phantom{\rule{0.17em}{0ex}}$Reference
${C}_{\alpha}{C}_{\alpha}{C}_{\alpha}$
angle$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: alpha
Acceptable values: positive decimal
Default value: 88${}^{\circ}$
Description: This option sets the reference angle used in the score function (12).
 Keyword angleTol $\u27e8\phantom{\rule{0.17em}{0ex}}$Tolerance
in the ${C}_{\alpha}{C}_{\alpha}{C}_{\alpha}$
angle$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: alpha
Acceptable values: positive decimal
Default value: 15${}^{\circ}$
Description: This option sets the angle tolerance used in the score function (12).
 Keyword hBondCutoff $\u27e8\phantom{\rule{0.17em}{0ex}}$Hydrogen
bond cutoff$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: alpha
Acceptable values: positive decimal
Default value: 3.3 Å
Description: Equivalent to the cutoff option in the hBond component.
 Keyword hBondExpNumer $\u27e8\phantom{\rule{0.17em}{0ex}}$Hydrogen
bond numerator exponent$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: alpha
Acceptable values: positive integer
Default value: 6
Description: Equivalent to the expNumer option in the hBond component.
 Keyword hBondExpDenom $\u27e8\phantom{\rule{0.17em}{0ex}}$Hydrogen
bond denominator exponent$\phantom{\rule{0.17em}{0ex}}\u27e9$
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).
5.7.2 dihedralPC: protein dihedral principal 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.[7] and documented in detail by Altis et al.[8]. Given a peptide or protein segment of
$N$ residues, each with
Ramachandran angles ${\varphi}_{i}$
and ${\psi}_{i}$, dPCA rests on a
variance/covariance analysis of the $4(N1)$
variables $\mathrm{cos}\⁡\left({\psi}_{1}\right),\mathrm{sin}\⁡\left({\psi}_{1}\right),\mathrm{cos}\⁡\left({\varphi}_{2}\right),\mathrm{sin}\⁡\left({\varphi}_{2}\right)\cdots \mathrm{cos}\⁡\left({\varphi}_{N}\right),\mathrm{sin}\⁡\left({\varphi}_{N}\right)$. Note
that angles ${\varphi}_{1}$
and ${\psi}_{N}$
have little impact on chain conformation, and are therefore discarded, following the implementation of dPCA in the
analysis software Carma.[9]
For a given principal component (eigenvector) of coefficients
${\left({k}_{i}\right)}_{1\le i\le 4(N1)}$, the
projection of the current backbone conformation is:
$$\xi =\sum _{n=1}^{N1}{k}_{4n3}\mathrm{cos}\⁡\left({\psi}_{n}\right)+{k}_{4n2}\mathrm{sin}\⁡\left({\psi}_{n}\right)+{k}_{4n1}\mathrm{cos}\⁡\left({\varphi}_{n+1}\right)+{k}_{4n}\mathrm{sin}\⁡\left({\varphi}_{n+1}\right)$$  (13) 
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 5.16 for additional options):
 Keyword residueRange: see definition of residueRange (alpha component)
 Keyword psfSegID: see definition of psfSegID (alpha component)
 Keyword vectorFile $\u27e8\phantom{\rule{0.17em}{0ex}}$File
containing dihedral PCA eigenvector(s)$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.[9]
 Keyword vectorNumber $\u27e8\phantom{\rule{0.17em}{0ex}}$File
containing dihedralPCA eigenvector(s)$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: dihedralPC
Acceptable values: positive integer
Description: Number of the eigenvector to be used for this component.
5.8 Raw data: building blocks for custom functions
5.8.1 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 $({x}_{1},{y}_{1},{z}_{1},\cdots \phantom{\rule{0.17em}{0ex}},{x}_{n},{y}_{n},{z}_{n})$.
List of keywords (see also 5.16 for additional options):
 Keyword atoms $\u27e8\phantom{\rule{0.17em}{0ex}}$Group
of atoms$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: cartesian
Acceptable values: Block atoms {...}
Description: Defines the atoms whose coordinates make up the value of the component. If rotateToReference,
centerToReference, or centerToOrigin are defined, coordinates are evaluated within the moving
frame of reference.
5.8.2 distancePairs: set of pairwise distances between two groups.
The distancePairs {...} block defines a
${N}_{1}\times {N}_{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 5.16 for additional options):
 Keyword group1: see definition of group1 (distance component)
 Keyword group2: analogous to group1
 Keyword forceNoPBC: see definition of forceNoPBC (distance component)
This component returns a ${N}_{1}\times {N}_{2}$dimensional
vector of numbers, each ranging from $0$
to the largest possible distance within the chosen boundary conditions.
5.9 Geometric path collective variables
The geometric path collective variables define the progress along a path,
$s$, and the distance
from the path, $z$.
These CVs are proposed by Leines and Ensing[10] , which differ from that[11] proposed by Branduardi et al., and utilize
a set of geometric algorithms. The path is defined as a series of frames in the atomic Cartesian coordinate space or the
CV space. $s$
and $z$ are
computed as
$$s=\frac{m}{M}\pm \frac{1}{2M}\left(\frac{\sqrt{{({v}_{1}\cdot {v}_{3})}^{2}\left{v}_{3}{}^{2}\right({v}_{1}{}^{2}{v}_{2}{}^{2})}({v}_{1}\cdot {v}_{3})}{{v}_{3}{}^{2}}1\right)$$  (14) 
$$z=\sqrt{{\left({v}_{1}+\frac{1}{2}\left(\frac{\sqrt{{({v}_{1}\cdot {v}_{3})}^{2}\left{v}_{3}{}^{2}\right({v}_{1}{}^{2}{v}_{2}{}^{2})}({v}_{1}\cdot {v}_{3})}{{v}_{3}{}^{2}}1\right){v}_{4}\right)}^{2}}$$  (15) 
where ${v}_{1}={s}_{m}z$
is the vector connecting the current position to the closest frame,
${v}_{2}=z{s}_{m1}$
is the vector connecting the second closest frame to the current position,
${v}_{3}={s}_{m+1}{s}_{m}$
is the vector connecting the closest frame to the third closest frame, and
${v}_{4}={s}_{m}{s}_{m1}$
is the vector connecting the second closest frame to the closest frame.
$m$ and
$M$ are the current index
of the closest frame and the total number of frames, respectively. If the current position is on the left of the closest reference
frame, the $\pm $ in
$s$ turns to
the positive sign. Otherwise it turns to the negative sign.
The equations above assume: (i) the frames are equidistant and (ii) the second and the third closest frames are
neighbouring to the closest frame. When these assumptions are not satisfied, this set of path CV should be used
carefully.
5.9.1 gspath: progress along a path defined in atomic Cartesian coordinate space.
In the gspath {...} and the gzpath {...} block all vectors, namely
$z$ and
${s}_{k}$
are defined in atomic Cartesian coordinate space. More specifically,
$z=\left[{r}_{1},\cdots \phantom{\rule{0.17em}{0ex}},{r}_{n}\right]$, where
${r}_{i}$ is the
$i$th atom specified in
the atoms block. ${s}_{k}=\left[{r}_{k,1},\cdots \phantom{\rule{0.17em}{0ex}},{r}_{k,n}\right]$,
where ${r}_{k,i}$ means
the $i$th atom
of the $k$th
reference frame.
List of keywords (see also 5.16 for additional options):
 Keyword atoms $\u27e8\phantom{\rule{0.17em}{0ex}}$Group
of atoms$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: gspath and gzpath
Acceptable values: Block atoms {...}
Description: Defines the atoms whose coordinates make up the value of the component.
 Keyword refPositionsCol $\u27e8\phantom{\rule{0.17em}{0ex}}$PDB
column containing atom flags$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: gspath and gzpath
Acceptable values: O, B, X, Y, or Z
Description: If refPositionsFileN 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.
 Keyword refPositionsFileN $\u27e8\phantom{\rule{0.17em}{0ex}}$File
containing the reference positions for fitting$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: gspath and gzpath
Acceptable values: UNIX filename
Description: The path is defined by multiple refPositionsFiles which are similiar to refPositionsFile
in the rmsd CV. If your path consists of $10$
nodes, you can list the coordinate file (in PDB or XYZ format) from refPositionsFile1 to refPositionsFile10.
 Keyword useSecondClosestFrame $\u27e8\phantom{\rule{0.17em}{0ex}}$Define
${s}_{m1}$
as the second closest frame?$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: gspath and gzpath
Acceptable values: boolean
Default value: on
Description: The definition assumes the second closest frame is neighbouring to the closest frame.
This is not always true especially when the path is crooked. If this option is set to on (default),
${s}_{m1}$
is defined as the second closest frame. If this option is set to off, ${s}_{m1}$
is defined as the left or right neighbouring frame of the closest frame.
 Keyword useThirdClosestFrame $\u27e8\phantom{\rule{0.17em}{0ex}}$Define
${s}_{m+1}$
as the third closest frame?$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: gspath and gzpath
Acceptable values: boolean
Default value: off
Description: The definition assumes the third closest frame is neighbouring to the closest frame. This
is not always true especially when the path is crooked. If this option is set to on, ${s}_{m+1}$
is defined as the third closest frame. If this option is set to off (default), ${s}_{m+1}$
is defined as the left or right neighbouring frame of the closest frame.
 Keyword fittingAtoms $\u27e8\phantom{\rule{0.17em}{0ex}}$The
atoms that are used for alignment$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: gspath and gspath
Acceptable values: Group of atoms
Description: Before calculating ${v}_{1}$,
${v}_{2}$,
${v}_{3}$
and ${v}_{4}$,
the current frame need to be aligned to the corresponding reference frames. This option specifies
which atoms are used to do alignment.
5.9.2 gzpath: distance from a path defined in atomic Cartesian coordinate space.
List of keywords (see also 5.16 for additional options):
 Keyword useZsquare $\u27e8\phantom{\rule{0.17em}{0ex}}$Compute
${z}^{2}$
instead of $z$$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: gzpath
Acceptable values: boolean
Default value: off
Description: $z$
is not differentiable when it is zero. This implementation workarounds it by setting the derivative of
$z$
to zero when $z=0$.
Another workaround is set this option to on, which computes ${z}^{2}$
instead of $z$,
and then ${z}^{2}$
is differentiable when it is zero.
The usage of gzpath and gspath is illustrated below:
colvar {
# Progress along the path
name gs
# The path is defined by 5 reference frames (from string00.pdb to string04.pdb)
# Use atomic coordinate from atoms 1, 2 and 3 to compute the path
gspath {
atoms {atomnumbers { 1 2 3 }}
refPositionsFile1 string00.pdb
refPositionsFile2 string01.pdb
refPositionsFile3 string02.pdb
refPositionsFile4 string03.pdb
refPositionsFile5 string04.pdb
}
}
colvar {
# Distance from the path
name gz
# The path is defined by 5 reference frames (from string00.pdb to string04.pdb)
# Use atomic coordinate from atoms 1, 2 and 3 to compute the path
gzpath {
atoms {atomnumbers { 1 2 3 }}
refPositionsFile1 string00.pdb
refPositionsFile2 string01.pdb
refPositionsFile3 string02.pdb
refPositionsFile4 string03.pdb
refPositionsFile5 string04.pdb
}
}
5.9.3 linearCombination: Helper CV to define a linear combination of other CVs
This is a helper CV which can be defined as a linear combination of other CVs. It maybe useful
when you want to define the gspathCV {...} and the gzpathCV {...} as combinations of other
CVs.
5.9.4 customColvar: Helper CV to define a mathematical expression as CV from other CVs
This is a helper CV which can be defined as a mathematical expression (see 5.17) of other CVs. It maybe useful
when you want to define the gspathCV {...}, the gzpathCV {...} and NeuralNetwork {...} as combinations
of other CVs.
5.9.5 gspathCV: progress along a path defined in CV space.
In the gspathCV {...} and the gzpathCV {...} block all vectors, namely
$z$ and
${s}_{k}$ are defined in CV space.
More specifically, $z=\left[{\xi}_{1},\cdots \phantom{\rule{0.17em}{0ex}},{\xi}_{n}\right]$,
where ${\xi}_{i}$ is the
$i$th CV.
${s}_{k}=\left[{\xi}_{k,1},\cdots \phantom{\rule{0.17em}{0ex}},{\xi}_{k,n}\right]$, where
${\xi}_{k,i}$ means
the $i$th CV of
the $k$th
reference frame. It should be note that these two CVs requires the pathFile option, which specifies a path file. Each
line in the path file contains a set of spaceseperated CV value of the reference frame. The sequence of reference
frames matches the sequence of the lines.
List of keywords (see also 5.16 for additional options):
 Keyword useSecondClosestFrame $\u27e8\phantom{\rule{0.17em}{0ex}}$Define
${s}_{m1}$
as the second closest frame?$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: gspathCV and gzpathCV
Acceptable values: boolean
Default value: on
Description: The definition assumes the second closest frame is neighbouring to the closest frame.
This is not always true especially when the path is crooked. If this option is set to on (default),
${s}_{m1}$
is defined as the second closest frame. If this option is set to off, ${s}_{m1}$
is defined as the left or right neighbouring frame of the closest frame.
 Keyword useThirdClosestFrame $\u27e8\phantom{\rule{0.17em}{0ex}}$Define
${s}_{m+1}$
as the third closest frame?$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: gspathCV and gzpathCV
Acceptable values: boolean
Default value: off
Description: The definition assumes the third closest frame is neighbouring to the closest frame. This
is not always true especially when the path is crooked. If this option is set to on, ${s}_{m+1}$
is defined as the third closest frame. If this option is set to off (default), ${s}_{m+1}$
is defined as the left or right neighbouring frame of the closest frame.
 Keyword pathFile $\u27e8\phantom{\rule{0.17em}{0ex}}$The
file name of the path file.$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: gspathCV and gzpathCV
Acceptable values: UNIX filename
Description: Defines the nodes or images that constitutes the path in CV space. The CVs of an image
are listed in a line of pathFile using spaceseperated format. Lines from top to button in pathFile
corresponds images from initial to last.
5.9.6 gzpathCV: distance from a path defined in CV space.
List of keywords (see also 5.16 for additional options):
 Keyword useZsquare $\u27e8\phantom{\rule{0.17em}{0ex}}$Compute
${z}^{2}$
instead of $z$$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: gzpathCV
Acceptable values: boolean
Default value: off
Description: $z$
is not differentiable when it is zero. This implementation workarounds it by setting the derivative of
$z$
to zero when $z=0$.
Another workaround is set this option to on, which computes ${z}^{2}$
instead of $z$,
and then ${z}^{2}$
is differentiable when it is zero.
The usage of gzpathCV and gspathCV is illustrated below:
colvar {
# Progress along the path
name gs
# Path defined by the CV space of two dihedral angles
gspathCV {
pathFile ./path.txt
dihedral {
name 001
group1 {atomNumbers {5}}
group2 {atomNumbers {7}}
group3 {atomNumbers {9}}
group4 {atomNumbers {15}}
}
dihedral {
name 002
group1 {atomNumbers {7}}
group2 {atomNumbers {9}}
group3 {atomNumbers {15}}
group4 {atomNumbers {17}}
}
}
}
colvar {
# Distance from the path
name gz
gzpathCV {
pathFile ./path.txt
dihedral {
name 001
group1 {atomNumbers {5}}
group2 {atomNumbers {7}}
group3 {atomNumbers {9}}
group4 {atomNumbers {15}}
}
dihedral {
name 002
group1 {atomNumbers {7}}
group2 {atomNumbers {9}}
group3 {atomNumbers {15}}
group4 {atomNumbers {17}}
}
}
}
5.10 Arithmetic path collective variables
The arithmetic path collective variable in CV space uses a similar formula as the one proposed by Branduardi[11] et al., except
that it computes $s$
and $z$
in CV space instead of RMSDs in Cartesian space. Moreover, this implementation allows different
coefficients for each CV components as described in [12]. Assuming a path is composed of
$N$ reference frames and defined
in an $M$dimensional CV
space, then the equations of $s$
and $z$ of
the path are
$$s=\frac{1}{N1}\frac{\sum _{i=0}^{N1}i\mathrm{exp}\⁡\left(\lambda \sum _{j=1}^{M}{c}_{j}^{2}{\left({x}_{j}{x}_{i,j}\right)}^{2}\right)}{\sum _{i=0}^{N1}\mathrm{exp}\⁡\left(\lambda \sum _{j=1}^{M}{c}_{j}^{2}{\left({x}_{j}{x}_{i,j}\right)}^{2}\right)}$$  (16) 
$$z=\frac{1}{\lambda}\mathrm{ln}\⁡\left(\sum _{i=0}^{N1}\mathrm{exp}\⁡\left(\lambda \sum _{j=1}^{M}{c}_{j}^{2}{\left({x}_{j}{x}_{i,j}\right)}^{2}\right)\right)$$  (17) 
where ${c}_{j}$ is the
coefficient(weight) of the $j$th
CV, ${x}_{i,j}$ is the value
of $j$th CV of
$i$th reference frame
and ${x}_{j}$ is the value of
$j$th CV of current frame.
$\lambda $ is a parameter to
smooth the variation of $s$
and $z$. It should be
noted that the index $i$
ranges from $0$ to
$N1$, and the definition
of $s$ is normalized
by $1\u2215(N1)$. Consequently,
the scope of $s$
is $[0:1]$.
5.10.1 aspathCV: progress along a path defined in CV space.
This colvar component computes the $s$
variable.
List of keywords (see also 5.16 for additional options):
 Keyword weights $\u27e8\phantom{\rule{0.17em}{0ex}}$Coefficients
of the collective variables$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: aspathCV and azpathCV
Acceptable values: spaceseparated numbers in a {...} block
Default value: {1.0 ...}
Description: Define the coefficients. The $j$th
value in the {...} block corresponds the ${c}_{j}$
in the equations.
 Keyword lambda $\u27e8\phantom{\rule{0.17em}{0ex}}$Smoothness
of the variation of $s$
and $z$$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: aspathCV and azpathCV
Acceptable values: decimal
Default value: inverse of the mean square displacements of successive reference frames
Description: The value of $\lambda $
in the equations.
 Keyword pathFile $\u27e8\phantom{\rule{0.17em}{0ex}}$The
file name of the path file.$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: aspathCV and azpathCV
Acceptable values: UNIX filename
Description: Defines the nodes or images that constitutes the path in CV space. The CVs of an image
are listed in a line of pathFile using spaceseperated format. Lines from top to button in pathFile
corresponds images from initial to last.
5.10.2 azpathCV: distance from a path defined in CV space.
This colvar component computes the $z$
variable. Options are the same as in 5.10.1.
The usage of azpathCV and aspathCV is illustrated below:
colvar {
# Progress along the path
name as
# Path defined by the CV space of two dihedral angles
aspathCV {
pathFile ./path.txt
weights {1.0 1.0}
lambda 0.005
dihedral {
name 001
group1 {atomNumbers {5}}
group2 {atomNumbers {7}}
group3 {atomNumbers {9}}
group4 {atomNumbers {15}}
}
dihedral {
name 002
group1 {atomNumbers {7}}
group2 {atomNumbers {9}}
group3 {atomNumbers {15}}
group4 {atomNumbers {17}}
}
}
}
colvar {
# Distance from the path
name az
azpathCV {
pathFile ./path.txt
weights {1.0 1.0}
lambda 0.005
dihedral {
name 001
group1 {atomNumbers {5}}
group2 {atomNumbers {7}}
group3 {atomNumbers {9}}
group4 {atomNumbers {15}}
}
dihedral {
name 002
group1 {atomNumbers {7}}
group2 {atomNumbers {9}}
group3 {atomNumbers {15}}
group4 {atomNumbers {17}}
}
}
}
5.10.3 Path collective variables in Cartesian coordinates
The path collective variables defined by Branduardi et al. [11] are based on RMSDs in Cartesian coordinates.
Noting ${d}_{i}$
the RMSD between the current set of Cartesian coordinates and those of image number
$i$ of the
path:
$$s=\frac{1}{N1}\frac{\sum _{i=1}^{N}(i1)\mathrm{exp}\⁡\left(\lambda {d}_{i}^{2}\right)}{\sum _{i=1}^{N}\mathrm{exp}\⁡\left(\lambda {d}_{i}^{2}\right)}$$  (18) 
$$z=\frac{1}{\lambda}\mathrm{ln}\⁡\left(\sum _{i=1}^{N}\mathrm{exp}\⁡(\lambda {d}_{i}^{2})\right)$$  (19) 
where $\lambda $ is
the smoothing parameter.
These coordinates are implemented as Tclscripted combinations of rmsd components. 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. It implements an optimization procedure, whereby the distance to a given image is only
calculated if its contribution to the sum is larger than a userdefined tolerance parameter. All distances are
calculated every freq timesteps to update the list of nearby images.
5.11 Dense neural network in CV space (MLCV)
This colvar component computes a nonlinear combination of other scalar colvar components, where the
transformation is defined by a dense neural network.[13] The network can be optimized using any framework, and
its parameters are provided to Colvars in plain text files, as detailed below. An example Python script to export the
parameters of a TensorFlow model is provided in colvartools/extract_weights_biases.py in the Colvars
source tree.
The output of the $j$th
node of a $k$th
layer that has ${N}_{k}$
nodes is computed by
$${y}_{k,j}={f}_{k}\left(\sum _{i=1}^{{N}_{k1}}{w}_{(k,j),(k1,i)}{y}_{k1,i}+{b}_{k,j}\right),$$  (20) 
where ${f}_{k}$ is the activation
function of the $k$th
layer, ${w}_{(k,j),(k1,i)}$ is the weight of
$j$th node with respect
to the $i$th output of
previous layer, and ${b}_{k,j}$
is the bias of $j$th
node of $k$th
layer.
List of keywords (see also 5.16 for additional options):
 Keyword output_component $\u27e8\phantom{\rule{0.17em}{0ex}}$The
$j$th
node of the output or the last layer$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: NeuralNetwork
Acceptable values: integer starting from 0
Description: The value of this option specifies the output node to be used as the value of this CV.
 Keyword layer$i$_WeightsFile $\u27e8\phantom{\rule{0.17em}{0ex}}$The
weights from layer $i1$
to layer $i$$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: NeuralNetwork
Acceptable values: UNIX filename
Description: The letter $i$
in this option needs to be replaced with the indexing number starting from 1, for example, layer1_WeightsFile
and layer2_WeightsFile. The value of this option specifies a plain text file containing the weights
from layer $i1$
to layer $i$.
In the file, the number at $k$th
column and $l$th
row represents the weight from node $k$
at layer $i1$
to node $l$
at layer $i$.
 Keyword layer$i$_BiasesFile $\u27e8\phantom{\rule{0.17em}{0ex}}$The
biases from layer $i1$
to layer $i$$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: NeuralNetwork
Acceptable values: UNIX filename
Description: The letter $i$
in this option needs to be replaced with the indexing number starting from 1, for example, layer1_BiasesFile
and layer2_BiasesFile. The value of this option specifies a plain text file containing the weights of
layer $i$.
The file should have only one column, where the number at $l$th
row represents the bias of node $l$
from layer $i1$
to layer $i$.
 Keyword layer$i$_activation $\u27e8\phantom{\rule{0.17em}{0ex}}$The
activation function from layer $i1$
to layer $i$$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: NeuralNetwork
Acceptable values: tanh, sigmoid, linear, relu, lrelu100, elu
Description: The letter $i$
in this option needs to be replaced with the indexing number starting from 1, for example, layer1_activation
and layer2_activation. The activation function from layer $i1$
to layer $i$.
Available choices are tanh, sigmoid, linear (identity), relu, lrelu100 (a leaky rely with coefficients
$1{0}^{2}$
and 1), and elu (with coefficient 1).
 Keyword layer$i$_custom_activation $\u27e8\phantom{\rule{0.17em}{0ex}}$An
alternative custom expression as the activation function from layer $i1$
to layer $i$$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: NeuralNetwork
Acceptable values: string
Description: Mathematical expression to define the activation function from layer $i1$
to layer $i$.
The input value must be written as x. For example, the ELU activation function can be expressed as
select(step(x), alpha*(exp(x)1), x). For details of the expression syntax, see customFunction.
This option is mutually exclusive with colvarNeuralNetworklayeri_activation
An example of configuration using NeuralNetwork is shown below:
colvar {
# Define a neural network with 2 layers
# The inputs are two torsion angles
# and the first node at the output layer is used as the final CV
name nn_output_1
NeuralNetwork {
output_component 0
layer1_WeightsFile dense_1_weights.txt
layer1_BiasesFile dense_1_biases.txt
layer1_activation tanh
layer2_WeightsFile dense_2_weights.txt
layer2_BiasesFile dense_2_biases.txt
layer2_activation tanh
# The component coefficient is used for normalization
componentCoeff 180.0
dihedral {
name 001
# normalization factor 1.0/180.0
componentCoeff 0.00555555555555555556
group1 {atomNumbers {5}}
group2 {atomNumbers {7}}
group3 {atomNumbers {9}}
group4 {atomNumbers {15}}
}
dihedral {
name 002
# normalization factor 1.0/180.0
componentCoeff 0.00555555555555555556
group1 {atomNumbers {7}}
group2 {atomNumbers {9}}
group3 {atomNumbers {15}}
group4 {atomNumbers {17}}
}
}
}
5.12 Volumetric mapbased variables
Volumetric maps of the Cartesian coordinates, typically defined as mesh grid along the three Cartesian axes, may
be used to define collective variables. Please cite [14] when using this implementation of collective variables based
on volumetric maps.
5.12.1 mapTotal: total value of a volumetric map
Given a function of the Cartesian coordinates
$\varphi \left(x\right)=\varphi (x,y,z)$, a mapTotal collective variable
component $\Phi \left(X\right)$ is defined as the sum of
the values of the function $\varphi \left(x\right)$ evaluated
at the coordinates of each atom, ${x}_{i}$:
$$\Phi \left(X\right)=\sum _{i=1}^{N}{w}_{i}\varphi \left({x}_{i}\right)$$  (21) 
where ${w}_{i}$
are weights assigned to each variable (1 by default). This formulation allows, for example,
to “count" the number of atoms within a region of space by using a positivevalued function
$\varphi \left(x\right)$, such as
for example the number of water molecules in a hydrophobic cavity [14].
Because the volumetric map itself and the atoms affected by it are defined externally to Colvars, this component
has a very limited number of keywords.
List of keywords (see also 5.16 for additional options):
 Keyword mapID $\u27e8\phantom{\rule{0.17em}{0ex}}$Specify
the numeric index of the volumetric map to use as a colvar$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: mapTotal
Acceptable values: nonnegative integer
Description: The value of this option specifies the numeric index of the volumetric map to use for this
collective variable component. The map referred to must be already loaded before using this option.
 Keyword atoms $\u27e8\phantom{\rule{0.17em}{0ex}}$Atom
group$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: mapTotal
Acceptable values: atoms {...} block
Description: If defined, this option allows defining the set of atoms to be aligned onto the volumetric
map following the syntax described in section 6. This also allows, for instance, to express the volumetric
map in a rotated frame of reference (see 6.2). In VMD, this definition is required, otherwise the
mapTotal value is identically zero.
 Keyword atoms $\u27e8\phantom{\rule{0.17em}{0ex}}$Weights
to assign to each atom$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: mapTotal
Acceptable values: list of spaceseparated decimals
Default value: all weights equal to 1
Description: If defined, the value of this option assigns weights ${w}_{i}$
to the individual atoms in a mapTotal variable. This option requires defining atoms explicitly, and the
number of weights must match the number of atoms.
5.12.2 Multiple volumetric maps collective variables
To study processes that involve changes in shape of a macromolecular aggregate (for example, deformations of
lipid membranes) it is useful to define collective variables based on more than one volumetric map at a time,
measuring the relative similarity with each map while still achieving correct thermodynamic sampling of each
state.
This is achieved by combining multiple mapTotal components, each based on a differentlyshaped volumetric map, into a single
collective variable $\xi $.
To track transitions between states, the contribution of each map to
$\xi $ should
be discriminated from the others, for example by assigning to it a different weight. The “MultiMap"
progress variable [14] uses a weight sum of these components, using linearlyincreasing weights:
$$\xi \left(X\right)=\sum _{k=1}^{K}k{\Phi}_{k}\left(X\right)=\sum _{k=1}^{K}k\sum _{i=1}^{N}{\varphi}_{k}\left({x}_{i}\right)$$  (22) 
where $K$ is the number of
maps employed and each ${\Phi}_{k}$
is a mapTotal component.
Here is a link to the MultiMap tutorial page: https://colvars.github.io/multimap/multimap.html
An example configuration for illustration purposes is also included below.
Example: transitions between macromolecular shapes using volumetric maps.
A series of map files, each representing a different shape, is loaded into NAMD:
mGridForce yes
for { set k 1 } { $k <= $K } { incr k } {
mGridForcePotFile Shape_$k map_$k.dx # Density map of the kth state
mGridForceFile Shape_$k atoms.pdb # PDB file used for atom selection
mGridForceCol Shape_$k O # Use the occupancy column of the PDB file atoms.pdb
mGridForceChargeCol Shape_$k B # Use beta as “charge" (default: electric charge)
mGridForceScale Shape_$k 0.0 0.0 0.0 # Do not use GridForces for this map
}
The GridForces maps thus loaded are then used to define the MultiMap collective variable, with coefficients
${\xi}_{k}=k$
[14]:
# Collect the definition of all components into one string
set components "
for { set k 1 } { $k <= $K } { incr k } {
set components "${components}
mapTotal {
mapName Shape_$k
componentCoeff $k
}
}
"
# Use this string to define the variable
cv config "
colvar {
name shapes
${components}
}"
The above example illustrates a use case where a weighted sum (i.e. a linear combination) is used to define a
single variable from multiple components. Depending on the problem under study, nonlinear functions may be
more appropriate. These may be defined a custom functions if implemented (see 5.17), or scripted functions (see
5.18).
5.13 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 usersupplied function provided by scriptedFunction.
 Keyword name $\u27e8\phantom{\rule{0.17em}{0ex}}$Name
of this component$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: any component
Acceptable values: string
Default value: type of component + numeric id
Description: The name is an unique casesensitive string which allows the Colvars module to identify
this component. This is useful, for example, when combining multiple components via a scriptedFunction.
 Keyword scalable $\u27e8\phantom{\rule{0.17em}{0ex}}$Attempt
to calculate this component in parallel?$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
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 bestfit 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 userdefined 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 (5.17), or wthin
scripted variables 5.18).
 Keyword period $\u27e8\phantom{\rule{0.17em}{0ex}}$Period
of the component$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword wrapAround $\u27e8\phantom{\rule{0.17em}{0ex}}$Center
of the wrapping interval for periodic variables$\phantom{\rule{0.17em}{0ex}}\u27e9$
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
$P\u22152$
to $P\u22152$,
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 5.16) 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.
5.15 Nonscalar components
When one of the following components are used, the defined colvar returns a value that is not a scalar number:
 distanceVec: 3dimensional vector of the distance between two groups;
 distanceDir: 3dimensional unit vector of the distance between two groups;
 orientation: 4dimensional unit quaternion representing the bestfit rotation from a set of reference
coordinates.
The distance between two 3dimensional unit vectors is computed as the angle between them. The distance between two
quaternions is computed as the angle between the two 4dimensional unit vectors: because the orientation represented by
$q$ is the same as the
one represented by $q$,
distances between two quaternions are computed considering the closest of the two symmetric images.
Nonscalar components carry the following restrictions:
 Calculation of total forces (outputTotalForce option) is currently not implemented.
 Each colvar can only contain one nonscalar 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 nonscalar components, no limit is set to the
number of scalar colvars. To compute multidimensional histograms and PMFs, use sets of scalar colvars of
arbitrary size.
5.15.1 Calculating total forces
In addition to the restrictions due to the type of value computed (scalar or nonscalar), 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.
5.16 Linear and polynomial combinations of components
To extend the set of possible definitions of colvars
$\xi \left(r\right)$, multiple
components ${q}_{i}\left(r\right)$
can be summed with the formula:
$$\xi \left(r\right)=\sum _{i}{c}_{i}{\left[{q}_{i}\right(r\left)\right]}^{{n}_{i}}$$  (23) 
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 (23)) can be obtained by setting the following parameters, which are common to
all components:
 Keyword componentCoeff $\u27e8\phantom{\rule{0.17em}{0ex}}$Coefficient
of this component in the colvar$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword componentExp $\u27e8\phantom{\rule{0.17em}{0ex}}$Exponent
of this component in the colvar$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 (nonlinear sum), total forces and the Jacobian force
are not available, making the colvar unsuitable for ABF calculations (eABF remains possible).
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 $1\u2215N$.
Collective variables may be defined by specifying a custom function as an analytical expression.
The expression is parsed by Lepton, the lightweight expression parser written by Peter Eastman
(https://simtk.org/projects/lepton). Lepton produces efficient evaluation routines for the function and its
derivatives.
 Keyword customFunction $\u27e8\phantom{\rule{0.17em}{0ex}}$Compute
colvar as a custom function of its components$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: string
Description: Mathematical expression to define the colvar as a closedform function of its colvar
components. See below for the detailed syntax of Lepton expressions. Multiple mentions of this
keyword can be used to define a vector variable (as in the example below).
 Keyword customFunctionType $\u27e8\phantom{\rule{0.17em}{0ex}}$Type
of value returned by the scripted colvar$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
The expression may use the collective variable components as variables, referred to by their userdefined name.
Scalar elements of vector components may be accessed by appending a 1based index to their name, as in the
example below. When implementing generic functions of Cartesian coordinates rather than functions of existing
components, the cartesian component may be particularly useful. A scalarvalued 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 5.14 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, as in this
example:
colvar {
name custom
# A 2dimensional vector function of a scalar x and a 3vector 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 }
}
}
Numeric constants may be given in either decimal or exponential form (e.g. 3.12e2). An expression may be
followed by definitions for intermediate values that appear in the expression, separated by semicolons. For example,
the expression:
a^2 + a*b + b^2; a = a1 + a2; b = b1 + b2
is exactly equivalent to:
(a1 + a2)^2 + (a1 + a2) * (b1 + b2) + (b1 + b2)^2.
The definition of an intermediate value may itself involve other intermediate values. All uses of a value must appear
before that value's definition.
Lepton supports the usual arithmetic operators +, , *, /, and ^ (power), as well as the following
functions:


sqrt  Square root 
exp  Exponential 
log  Natural logarithm 
erf  Error function 
erfc  Complementary error function 


sin  Sine (angle in radians) 
cos  Cosine (angle in radians) 
sec  Secant (angle in radians) 
csc  Cosecant (angle in radians) 
tan  Tangent (angle in radians) 
cot  Cotangent (angle in radians) 
asin  Inverse sine (in radians) 
acos  Inverse cosine (in radians) 
atan  Inverse tangent (in radians) 
atan2  Twoargument inverse tangent (in radians) 


sinh  Hyperbolic sine 
cosh  Hyperbolic cosine 
tanh  Hyperbolic tangent 


abs  Absolute value 
floor  Floor 
ceil  Ceiling 
min  Minimum of two values 
max  Maximum of two values 
delta  $delta\left(x\right)=1$ if $x=0$, 0 otherwise 
step  $step\left(x\right)=0$ if $x<0$, 1 if $x>=0$ 
select  $select(x,y,z)=z$ if $x=0$, $y$ otherwise 



When scripting is supported (default in VMD), 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
scalarvalued 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
5.14 for
details).
An example of elaborate scripted colvar is given in example 10, in the form of pathbased collective variables as
defined by Branduardi et al[11] (Section 5.10.3).
 Keyword scriptedFunction $\u27e8\phantom{\rule{0.17em}{0ex}}$Compute
colvar as a scripted function of its components$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 rowmajor order, i.e. for a function ${f}_{i}\left({x}_{j}\right)$:
${\nabla \⁡}_{x}{f}_{1}{\nabla \⁡}_{x}{f}_{2}\cdots \phantom{\rule{0.17em}{0ex}}$.
 Keyword scriptedFunctionType $\u27e8\phantom{\rule{0.17em}{0ex}}$Type
of value returned by the scripted colvar$\phantom{\rule{0.17em}{0ex}}\u27e9$
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).
Nonscalar values should be passed as spaceseparated lists.
 Keyword scriptedFunctionVectorSize $\u27e8\phantom{\rule{0.17em}{0ex}}$Dimension
of the vector value of a scripted colvar$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
5.19 Defining grid parameters for a colvar
Many algorithms require the definition of two boundaries and a bin width for each colvar, which are necessary to
compute discrete “states" for a collective variable's otherwise continuous values. The following keywords define
these parameters for a specific variable, and will be used by all bias that refer to that variable unless otherwise
specified.
 Keyword lowerBoundary $\u27e8\phantom{\rule{0.17em}{0ex}}$Lower
boundary of the colvar$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: decimal
Default value: natural boundary of the function
Description: Defines the lowest end of the interval of “relevant" values for the variable. This number
can be, for example, a true physical boundary imposed by the choice of function (e.g. the distance
function is always larger than zero): if this is the case, and only one function is used to define the
variable, the default value of this number is set to the lowest end of the range of values of that function,
if available (see Section 5.1). Alternatively, this value may be provided by the user, to represent for
example the leftmost point of a PMF calculation along this variable. In the latter case, it is the user's
responsibility to either (a) ensure the variable does not go significantly beyond the boundary (for
example by adding a harmonicWalls restraint, 7.7), or (b) instruct the code that this is a true physical
boundary by setting hardLowerBoundary.
 Keyword upperBoundary $\u27e8\phantom{\rule{0.17em}{0ex}}$Upper
boundary of the colvar$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: decimal
Default value: natural boundary of the function
Description: Similarly to lowerBoundary, defines the highest of the “relevant" values of the variable.
 Keyword width $\u27e8\phantom{\rule{0.17em}{0ex}}$grid
spacing, or unit of the variable$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: positive decimal
Default value: 1.0
Description: This number defines the width of a discrete “state" for a collective variable, and is
used by the many biasing methods to achieve different purposes. Histograms (7.10), ABF (7.2) and
metadynamics (7.4) all use this number as the initial choice for the grid spacing along this variable.
As a typical rule of thumb, width should be no larger than the standard deviation of the colvar in an
unbiased simulation (to characterize a local freeenergy minimum with at least two points).
Further, many restraints such as harmonic potentials (7.5), harmonic walls (7.7) and linear restraints
(7.8) also use this parameter to define the expected fluctuations of the colvar, alowing to express the
force constant in terms of this unit. This is most useful with multidimensional restraints acting on
variables that have very different units (for examples, working with length unit and degrees ${}^{\circ}$
simultaneously): a single force constant can be used for all, which is converted to the respective unit
of each variable when forces are applied (the are printed at initialization time.
 Keyword hardLowerBoundary $\u27e8\phantom{\rule{0.17em}{0ex}}$Whether
the lower boundary is the physical lower limit$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: boolean
Default value: provided by the component
Description: When the colvar has a “natural" boundary (for example, a distance colvar cannot go
below 0) this flag is automatically enabled. For more complex variable definitions, or when lowerBoundary
is provided directly by the user, it may be useful to set this flag explicitly. This option does not affect
simulation results, but enables some internal optimizations by letting the code know that the variable
is unable to cross the lower boundary, regardless of whether restraints are applied to it.
 Keyword hardUpperBoundary $\u27e8\phantom{\rule{0.17em}{0ex}}$Whether
the upper boundary is the physical upper limit of the colvar's values$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: boolean
Default value: provided by the component
Description: Analogous to hardLowerBoundary.
 Keyword expandBoundaries $\u27e8\phantom{\rule{0.17em}{0ex}}$Allow
to expand the two boundaries if needed$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: boolean
Default value: off
Description: If defined, lowerBoundary and upperBoundary may be automatically expanded to
accommodate colvar values that do not fit in the initial range. Currently, this option is used by the
metadynamics bias (7.4) to keep all of its hills fully within the grid. Enabling this option does not
affect any boundaries that are defined as “hard" (see above), or any boundaries that span the full period
of a periodic colvar.
 Keyword outputValue $\u27e8\phantom{\rule{0.17em}{0ex}}$Output
a trajectory for this colvar$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: boolean
Default value: on
Description: If colvarsTrajFrequency is nonzero, the value of this colvar is written to the trajectory
file every colvarsTrajFrequency steps in the column labeled “$<$name$>$".
 Keyword outputVelocity $\u27e8\phantom{\rule{0.17em}{0ex}}$Output
a velocity trajectory for this colvar$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: boolean
Default value: off
Description: If colvarsTrajFrequency is defined, the finitedifference calculated velocity of this
colvar are written to the trajectory file under the label “v_$<$name$>$".
 Keyword outputEnergy $\u27e8\phantom{\rule{0.17em}{0ex}}$Output
an energy trajectory for this colvar$\phantom{\rule{0.17em}{0ex}}\u27e9$
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$>$".
 Keyword outputTotalForce $\u27e8\phantom{\rule{0.17em}{0ex}}$Output
a total force trajectory for this colvar$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 (28) in section 7.2) 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 energy unit/(colvar unit).
 Keyword outputAppliedForce $\u27e8\phantom{\rule{0.17em}{0ex}}$Output
an applied force trajectory for this colvar$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 energy unit/(colvar unit).
The following options enable extendedsystem dynamics, where a colvar is coupled to an additional degree of
freedom (fictitious particle) by a harmonic spring. This extended coordinate masks the colvar and replaces it
transparently from the perspective of biasing and analysis methods. Biasing forces are then applied to the extended
degree of freedom, and 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 to perform eABF simulations
(7.3).
Note that for some biases (harmonicWalls, histogram), this masking behavior is controlled by the keyword
bypassExtendedLagrangian. Specifically for harmonicWalls, the default behavior is to bypass extended
Lagrangian coordinates and act directly on the actual colvars.
 Keyword extendedLagrangian $\u27e8\phantom{\rule{0.17em}{0ex}}$Add
extended degree of freedom$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 [3]. .
 Keyword extendedFluctuation $\u27e8\phantom{\rule{0.17em}{0ex}}$Standard
deviation between the colvar and the fictitious particle (colvar unit)$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 ${k}_{B}T\u2215{\sigma}^{2}$,
where $\sigma $
is the value of extendedFluctuation.
 Keyword extendedTimeConstant $\u27e8\phantom{\rule{0.17em}{0ex}}$Oscillation
period of the fictitious particle (fs)$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 ${k}_{B}T{(\tau \u22152\pi \sigma )}^{2}$,
where $\tau $
is the period and $\sigma $
is the typical fluctuation (see above).
 Keyword extendedTemp $\u27e8\phantom{\rule{0.17em}{0ex}}$Temperature
for the extended degree of freedom (K)$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword extendedLangevinDamping $\u27e8\phantom{\rule{0.17em}{0ex}}$Damping
factor for extended Langevin dynamics (ps${}^{1}$)$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: positive decimal
Default value: 1.0
Description: If this is nonzero, 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 nonequilibrium
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.
 Keyword reflectingLowerBoundary $\u27e8\phantom{\rule{0.17em}{0ex}}$Whether
the lower boundary reflects the extended Lagrangian particle$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: boolean
Default value: off
Description: This turns the specified lowerBoundary into a reflecting wall for the extended particle:
upon collision, the particle is reflected with opposite momentum.
 Keyword reflectingUpperBoundary $\u27e8\phantom{\rule{0.17em}{0ex}}$Whether
the upper boundary reflects the extended Lagrangian particle$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: boolean
Default value: off
Description: This turns the specified upperBoundary into a reflecting wall for the extended particle:
upon collision, the particle is reflected with opposite momentum.
5.22 Multiple timestep variables
 Keyword timeStepFactor $\u27e8\phantom{\rule{0.17em}{0ex}}$Compute
this colvar once in a certain number of timesteps$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: positive integer
Default value: 1
Description: Instructs this colvar to activate at a time interval equal to the base (MD) timestep times
timeStepFactor.[15] At other time steps, the value of the variable is not updated, and no biasing
forces are applied. Any forces exerted by biases are accumulated over the given time interval, then
applied as an impulse at the next update.
5.23 Backwardcompatibility
 Keyword subtractAppliedForce $\u27e8\phantom{\rule{0.17em}{0ex}}$Do
not include biasing forces in the total force for this colvar$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: boolean
Default value: off
Description: If the colvar supports total force calculation (see 5.15.1), 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 20160810. 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.
5.24 Statistical analysis
Runtime 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 runtime computation of histograms, please see the histogram bias
(7.10).
 Keyword corrFunc $\u27e8\phantom{\rule{0.17em}{0ex}}$Calculate
a time correlation function?$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: boolean
Default value: off
Description: Whether or not a time correlaction function should be calculated for this colvar.
 Keyword corrFuncWithColvar $\u27e8\phantom{\rule{0.17em}{0ex}}$Colvar
name for the correlation function$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: string
Description: By default, the autocorrelation 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 ${\xi}_{i}$.
 Keyword corrFuncType $\u27e8\phantom{\rule{0.17em}{0ex}}$Type
of the correlation function$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: velocity, coordinate or coordinate_p2
Default value: velocity
Description: With coordinate or velocity, the correlation function ${C}_{i,j}\left(t\right)$ =
$\u27e8\mathrm{\Pi}\left({\xi}_{i}\left({t}_{0}\right),{\xi}_{j}({t}_{0}+t)\right)\u27e9$
is calculated between the variables ${\xi}_{i}$
and ${\xi}_{j}$,
or their velocities. $\mathrm{\Pi}({\xi}_{i},{\xi}_{j})$
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, $(3\mathrm{cos}\⁡{\left(\mathit{\theta}\right)}^{2}1)\u22152$,
is used instead of the cosine.
 Keyword corrFuncNormalize $\u27e8\phantom{\rule{0.17em}{0ex}}$Normalize
the time correlation function?$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 $\u27e8O\left({\xi}_{i},{\xi}_{j}\right)\u27e9$.
 Keyword corrFuncLength $\u27e8\phantom{\rule{0.17em}{0ex}}$Length
of the time correlation function$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: positive integer
Default value: 1000
Description: Length (in number of points) of the time correlation function.
 Keyword corrFuncStride $\u27e8\phantom{\rule{0.17em}{0ex}}$Stride
of the time correlation function$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: positive integer
Default value: 1
Description: Number of steps between two values of the time correlation function.
 Keyword corrFuncOffset $\u27e8\phantom{\rule{0.17em}{0ex}}$Offset
of the time correlation function$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword corrFuncOutputFile $\u27e8\phantom{\rule{0.17em}{0ex}}$Output
file for the time correlation function$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: UNIX filename
Default value: outputName.$<$name$>$.corrfunc.dat
Description: The time correlation function is saved in this file.
 Keyword runAve $\u27e8\phantom{\rule{0.17em}{0ex}}$Calculate
the running average and standard deviation$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: boolean
Default value: off
Description: Whether or not the running average and standard deviation should be calculated for this
colvar.
 Keyword runAveLength $\u27e8\phantom{\rule{0.17em}{0ex}}$Length
of the running average window$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: positive integer
Default value: 1000
Description: Length (in number of points) of the running average window.
 Keyword runAveStride $\u27e8\phantom{\rule{0.17em}{0ex}}$Stride
of the running average window values$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: positive integer
Default value: 1
Description: Number of steps between two values within the running average window.
 Keyword runAveOutputFile $\u27e8\phantom{\rule{0.17em}{0ex}}$Output
file for the running average and standard deviation$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: UNIX filename
Default value: outputName.$<$name$>$.runave.traj
Description: The running average and standard deviation are saved in this file.
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 bracedelimited block containing selection keywords and other
parameters, including an optional name:
 Keyword name $\u27e8\phantom{\rule{0.17em}{0ex}}$Unique
name for the atom group$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
6.1 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: first atom in the system is 1)
atomNumbers {
1 3
}
# add atoms starting from 20 up to and including 50
atomNumbersRange 2050
# add all the atoms with occupancy 2 in the file atoms.pdb
atomsFile atoms.pdb
atomsCol O
atomsColValue 2.0
# add all the Calphas within residues 11 to 20 of segments "PR1" and "PR2"
psfSegID PR1 PR2
atomNameResidueRange CA 1120
atomNameResidueRange CA 1120
# 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, the
${C}_{\alpha}$ atoms
between residues 11 and 20 of the two segments PR1 and PR2, and those in the index group called “Water". The
indices of this group are read from the file provided by the global keyword indexFile.
In the current version, the Colvars module does not manipulate VMD atom selections directly: however, these
can be converted to atom groups within the Colvars configuration string, using selection keywords such as
atomNumbers. The complete list of selection keywords available in VMD is:
 Keyword atomNumbers $\u27e8\phantom{\rule{0.17em}{0ex}}$List
of atom numbers$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: atom group
Acceptable values: spaceseparated 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".
 Keyword indexGroup $\u27e8\phantom{\rule{0.17em}{0ex}}$Name
of index group to be used (GROMACS format)$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword atomsOfGroup $\u27e8\phantom{\rule{0.17em}{0ex}}$Name
of group defined previously$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: atom group
Acceptable values: string
Description: Refers to a group defined previously using its userdefined name. This adds all atoms of
that named group to the current group.
 Keyword atomNumbersRange $\u27e8\phantom{\rule{0.17em}{0ex}}$Atoms
within a number range$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: atom group
Acceptable values: $<$Starting
number$>$$<$Ending
number$>$
Description: This option includes in the group all atoms whose numbers are within the range
specified. The number of the first atom in the system is 1.
 Keyword atomNameResidueRange $\u27e8\phantom{\rule{0.17em}{0ex}}$Named
atoms within a range of residue numbers$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: atom group
Acceptable values: $<$Atom
name$>$
$<$Starting
residue$>$$<$Ending
residue$>$
Description: This option adds to the group all the atoms with the provided name, within residues in
the given range.
 Keyword psfSegID $\u27e8\phantom{\rule{0.17em}{0ex}}$PSF
segment identifier$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: atom group
Acceptable values: spaceseparated list of strings (max 4 characters)
Description: This option sets the PSF segment identifier for atomNameResidueRange. Multiple
values may be provided, which correspond to multiple instances of atomNameResidueRange, in order
of their occurrence. This option is only necessary if a PSF topology file is used.
 Keyword atomsFile $\u27e8\phantom{\rule{0.17em}{0ex}}$PDB
file name for atom selection$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: atom group
Acceptable values: UNIX filename
Description: This option selects atoms from the PDB file provided and adds them to the group
according to numerical flags in the column atomsCol. Note: the sequence of atoms in the PDB file
provided must match that in the system's topology.
 Keyword atomsCol $\u27e8\phantom{\rule{0.17em}{0ex}}$PDB
column to use for atom selection flags$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: atom group
Acceptable values: O, B, X, Y, or Z
Description: This option specifies which PDB column in atomsFile is used to determine which
atoms are to be included in the group.
 Keyword atomsColValue $\u27e8\phantom{\rule{0.17em}{0ex}}$Atom
selection flag in the PDB column$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: atom group
Acceptable values: positive decimal
Description: If defined, this value in atomsCol identifies atoms in atomsFile that are included in
the group. If undefined, all atoms with a nonzero value in atomsCol are included.
 Keyword dummyAtom $\u27e8\phantom{\rule{0.17em}{0ex}}$Dummy
atom position (length unit)$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 userdefined
position.
6.2 Moving frame of reference.
The following options define an automatic calculation of an optimal translation (centerToReference) or
optimal rotation (rotateToReference), that superimposes the positions of this group to a provided
set of reference coordinates. Alternately, centerToOrigin applies a translation to place the
geometric center of the group at (0, 0, 0). 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
${x}_{i}$, the colvar
$\xi $ can be defined on a set of
rototranslated positions ${x}_{i}^{\prime}=R({x}_{i}{x}^{C})+{x}^{ref}$.
${x}^{C}$ is the geometric
center of the ${x}_{i}$,
$R$ is the optimal rotation matrix
to the reference positions and ${x}^{ref}$
is the geometric center of the reference positions.
Components that are defined based on pairwise distances are naturally invariant under global rototranslations.
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 centerToReference and
rotateToReference options. Finally, a few components are defined by convention using a rototranslated frame
(e.g. the minimal RMSD): for these components, centerToReference and rotateToReference 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.
rotateToReference affects coordinates that depend on minimumimage distances in periodic boundary
conditions (PBC). After rotation of the coordinates, the periodic cell vectors become irrelevant: the rotated system is
effectively nonperiodic. A safe way to handle this is to ensure that the relevant intergroup distance
vectors remain smaller than the halfsize 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 centerToReference and rotateToReference 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.
 Keyword centerToReference $\u27e8\phantom{\rule{0.17em}{0ex}}$Implicitly
remove translations for this group$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 either refPositions or refPositionsFile. Colvar components will
only have access to the aligned positions. Note: unless otherwise specified, rmsd and eigenvector
set this option to onby default.
 Keyword centerToOrigin $\u27e8\phantom{\rule{0.17em}{0ex}}$Implicitly
remove translations for this group by keeping its center at the origin$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: atom group
Acceptable values: boolean
Default value: off
Description: This option implies centerToReference. If this option is on, coordinates from the
group will be translated so that the center of geometry of the group remains at (0, 0, 0), except if
fittingGroup is enabled. In that case, the translation applied is the translation that brings the center
of geometry of the fitting group to (0, 0, 0).
 Keyword rotateToReference $\u27e8\phantom{\rule{0.17em}{0ex}}$Implicitly
remove rotations for this group$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 either refPositions or refPositionsFile. The rotation will be
performed around the center of geometry if centerToReference is on, or around the origin otherwise.
The algorithm used is the same employed by the orientation colvar component [4]. 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 onby default.
 Keyword refPositions $\u27e8\phantom{\rule{0.17em}{0ex}}$Reference
positions for fitting (length unit)$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: atom group
Acceptable values: spaceseparated list of (x, y, z) triplets
Description: This option provides a list of reference coordinates for centerToReference and/or
rotateToReference, and is mutually exclusive with refPositionsFile. If only centerToReference
is on, the list may contain a single (x, y, z) triplet; if also rotateToReference is on, the list should
be as long as the atom group, and its order must match the order in which atoms were defined.
 Keyword refPositionsFile $\u27e8\phantom{\rule{0.17em}{0ex}}$File
containing the reference positions for fitting$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: atom group
Acceptable values: UNIX filename
Description: This option provides a list of reference coordinates for centerToReference and/or
rotateToReference, and is mutually exclusive with refPositions. The acceptable file format is
XYZ (4.7.3), which is read in double precision. Alternatively, a PDB file may be read (see 4.7.4) using
VMD's reader; however, due to the constraints of the PDB format PDB files are discouraged if the
precision of the reference coordinates is a concern.
 Keyword refPositionsCol $\u27e8\phantom{\rule{0.17em}{0ex}}$PDB
column containing atom flags$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: atom group
Acceptable values: O, B, X, Y, or Z
Description: Like atomsCol for atomsFile, indicates which column to use to identify the atoms in
refPositionsFile (if this is a PDB file).
 Keyword refPositionsColValue $\u27e8\phantom{\rule{0.17em}{0ex}}$Atom
selection flag in the PDB column$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: atom group
Acceptable values: positive decimal
Description: Analogous to atomsColValue, but applied to refPositionsCol.
 Keyword fittingGroup $\u27e8\phantom{\rule{0.17em}{0ex}}$Use
an alternate set of atoms to define the rototranslation$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: atom group
Acceptable values: Block fittingGroup { ... }
Default value: This atom group itself
Description: If either centerToReference or rotateToReference is defined, this keyword defines
an alternate atom group to calculate the optimal rototranslation. 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!"). Performance considerations: note that enabling this
option will result in projecting each of the atomic gradients of the colvar (e.g. the RMSD) onto each
the gradients of the rototranslation, which may be a computationally expensive operation: see the
closely related enableFitGradients for details.
The following example illustrates the use of fittingGroup as part of a Distance to Bound Configuration
(DBC) coordinate for use in ligand restraints for binding affinity calculations.[16] The group called
“atoms" describes coordinates of a ligand's atoms, expressed in a moving frame of reference tied to a
binding site (here within a protein). An optimal rototranslation is calculated automatically by fitting the
C${}_{\alpha}$ trace
of the rest of the protein onto the coordinates provided by a PDB file. To define a DBC coordinate, this atom group
would be used within an rmsd function.
# Example: defining a group "atoms" (the ligand) whose coordinates are expressed
# in a rototranslated frame of reference defined by a second group (the receptor)
atoms {
atomNumbers 1 2 3 4 5 6 7 # atoms of the ligand (1based)
centerToReference yes
rotateToReference yes
fittingGroup {
# define the frame by fitting alpha carbon atoms
# in 2 protein segments close to the site
psfSegID PROT PROT
atomNameResidueRange CA 140
atomNameResidueRange CA 59100
}
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.
 Keyword enableFitGradients $\u27e8\phantom{\rule{0.17em}{0ex}}$Include
the rototranslational contribution to colvar gradients$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: atom group
Acceptable values: boolean
Default value: on
Description: When either centerToReference or rotateToReference is on, the gradients of some
colvars include terms proportional to $\partial R\u2215\partial {x}_{i}$
(rotational gradients) and $\partial {x}^{C}\u2215\partial {x}_{i}$
(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; other
types of variable will require projecting each of the gradients of the variable onto each of the gradients
of the rototranslation (i.e. a $O\left({N}^{2}\right)$
loop). When fittingGroup is enabled, the computation is a $O(N\times M)$
loop for all variables, including RMSDs.
6.3 Treatment of periodic boundary conditions.
When periodic boundary conditions are defined, the Colvars module requires that the coordinates of each
molecular fragment are contiguous, without “jumps" when a fragment is partially wrapped near a
periodic boundary. The Colvars module relies on this assumption when calculating a group's center of
geometry, but the condition may fail if the group spans different molecules. In general, coordinate
wrapping does not affect the calculation of colvars if each atom group satisfies one or more of the
following:
 it is composed by only one atom;
 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);
 it is used by a colvar component that ignores the illdefined Cartesian components of its center of mass
(such as the $x$
and $y$
components of a membrane's center of mass modeled with distanceZ).
If none of these conditions are met, wrapping may affect the calculation of collective variables: a possible solution is to
use pbc wrap or pbc unwrap prior to processing a trajectory with the Colvars module.
6.4 Performance of a Colvars calculation based on group size.
In simulations performed with messagepassing 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. If printed, the message “SMP parallelism is available."
indicates the availability of the option (will be supported in a future release of VMD). 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).
7 Biasing and analysis methods
A biasing or analysis method can be applied to existing collective variables by using the following
configuration:
$<$biastype$>$ {
name $<$name$>$
colvars $<$xi1$>$ $<$xi2$>$ ...
$<$parameters$>$
}
The keyword $<$biastype$>$
indicates the method of choice. There can be multiple instances of the same method, e.g. using multiple harmonic
blocks allows defining multiple restraints.
All biasing and analysis methods implemented recognize the following options:
 Keyword name $\u27e8\phantom{\rule{0.17em}{0ex}}$Identifier
for the bias$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 the output, and to name
some output files. Tip: although a unique name is assigned automatically, you are strongly encouraged
to give a name to a bias that you may want to analyze later. For example, processing a trajectory
file (sec. 4.7.6) containing a harmonic restraint is much simpler with a clearly identifiable name, like
“smd" for a moving harmonic restraint or “us" for a static one.
 Keyword colvars $\u27e8\phantom{\rule{0.17em}{0ex}}$Collective
variables involved$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar bias
Acceptable values: spaceseparated list of colvar names
Description: This option selects by name all the variables to which this bias or analysis will be
applied.
 Keyword outputEnergy $\u27e8\phantom{\rule{0.17em}{0ex}}$Write
the current bias energy to the trajectory file$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword outputFreq $\u27e8\phantom{\rule{0.17em}{0ex}}$Frequency
(number of steps) at which output files are written$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar bias
Acceptable values: positive integer
Default value: colvarsRestartFrequency
Description: If this bias produces aggregated data that needs to be written to disk (for example, a
PMF), this number specifies the number of steps after which these data are written to files. A value
of zero disables writing files for this bias during the simulation (except for outputEnergy, which is
controlled by colvarsTrajFrequency). All output files are also written at the end of a simulation
run, regardless of the value of this number. In VMD, this option has no effect.
 Keyword bypassExtendedLagrangian $\u27e8\phantom{\rule{0.17em}{0ex}}$Apply
bias to actual colvars, bypassing extended coordinates$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar bias
Acceptable values: boolean
Default value: off
Description: This option is implemented by the harmonicWalls and histogram biases. It is only
relevant if the bias is applied to one or several extendedLagrangian colvars (5.21), for example
within an eABF (7.3) simulation. Usually, biases use the value of the extended coordinate as a proxy
for the actual colvar, and their biasing forces are applied to the extended coordinates as well. If
bypassExtendedLagrangian is enabled, the bias behaves as if there were no extended coordinates,
and accesses the value of the underlying colvars, applying any biasing forces along the gradients of
those variables.
 Keyword stepZeroData $\u27e8\phantom{\rule{0.17em}{0ex}}$Accumulate
data starting at step 0 of a simulation run$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar bias
Acceptable values: boolean
Default value: off
Description: This option is meaningful for biases that record and accumulate data during a simulation,
such as ABF (7.2), metadynamics (7.4), histograms (7.10) and in general any bias that accumulates
freeenergy samples with thermodynamic integration, or TI (7.1). When this option is disabled (default),
data will only be recorded into the bias after the first coordinate update: this is generally the correct
choice in simulation runs. Biasing energy and forces will always be computed for all active biases,
regardless of this option.
 Keyword scaledBiasingForce $\u27e8\phantom{\rule{0.17em}{0ex}}$Scale
biasing force by a factor in an external histogram$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar bias
Acceptable values: boolean
Default value: off
Description: If this option is set to on, the biasing force at each step will be scaled by a factor provided
in the grid of an external histogram. The histogram file is provided by scaledBiasingForceFactorsGrid.
It ought to be noted that if the variables are not in any grids of the histogram, the scaling factor is 1.0.
 Keyword scaledBiasingForceFactorsGrid $\u27e8\phantom{\rule{0.17em}{0ex}}$A
histogram file with the scaling factor of biasing force in each bin$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar bias
Acceptable values: string
Description: If scaledBiasingForce is set to on, this option accepts the filename of the histogram
file that contains the scaling factors. The histogram file is expected to be in “multicolumn" format
(4.7.5), similar to the lowdimensional the PMF files written by metadynamics and ABF.
7.1 Thermodynamic integration
The methods implemented here provide a variety of estimators of conformational freeenergies.
These are carried out at runtime, or with the use of postprocessing tools over the generated output
files. The specifics of each estimator are discussed in the documentation of each biasing or analysis
method.
A special case is the traditional thermodynamic integration (TI) method, used for example to compute potentials
of mean force (PMFs). Most types of restraints (7.5, 7.7, 7.8, ...) as well as metadynamics (7.4) can optionally use TI
alongside their own estimator, based on the keywords documented below.
 Keyword writeTIPMF $\u27e8\phantom{\rule{0.17em}{0ex}}$Write
the PMF computed by thermodynamic integration$\phantom{\rule{0.17em}{0ex}}\u27e9$
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
outputotalForce and 5.15.1), this option allows calculating the corresponding PMF by thermodynamic
integration, and writing it to the file outputName.$<$name$>$.ti.pmf,
where $<$name$>$
is the name of the bias and the contents of the file are in multicolumn text format (4.7.5). The total
force includes the forces applied to the variable by all bias, except those from this bias itself. If any
bias applies timedependent forces besides the one using this option, an error is raised.
 Keyword writeTISamples $\u27e8\phantom{\rule{0.17em}{0ex}}$Write
the freeenergy gradient samples$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 thermodynamic forces are outputName.$<$name$>$.ti.count
and outputName.$<$name$>$.ti.force,
respectively: these can be used by abf_integrate (see 7.2.4) or similar utility. Note that because the
.force file contains mean forces instead of freeenergy gradients, abf_integrate $<$filename$>$
s 1.0 should be used. This option is 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 timedependent forces besides the one using this option, an
error is raised.
In adaptive biasing force (ABF) (7.2) the above keywords are not recognized, because their functionality is
either included already (conventional ABF) or not available (extendedsystem ABF).
7.2 Adaptive Biasing Force
For a full description of the Adaptive Biasing Force method, see reference [17]. For details about this
implementation, see references [18] and [19]. When publishing research that makes use of this functionality,
please cite references [17] and [19].
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 (5.21) of the variables produces the extendedsystem
ABF variant of the method (7.3).
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
$\text{}\xi \text{}={\left({\xi}_{i}\right)}_{i\in [1,n]}$ is defined from the
canonical distribution of $\text{}\xi \text{}$,
$\mathcal{\mathcal{P}}\left(\text{}\xi \text{}\right)$:
$$A\left(\text{}\xi \text{}\right)=\frac{1}{\beta}\mathrm{ln}\⁡\mathcal{\mathcal{P}}\left(\text{}\xi \text{}\right)+{A}_{0}$$  (24) 
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:
$${\text{}\nabla \⁡\text{}}_{\xi}A\left(\text{}\xi \text{}\right)={\u27e8{\text{}F\text{}}_{\xi}\u27e9}_{\text{}\xi \text{}}$$  (25) 
Several formulae that take the form of (25) have been proposed. This implementation relies partly on the classic
formulation [20], and partly on a more versatile scheme originating in a work by RuizMontero et al. [21], generalized by
den Otter [22] and extended to multiple variables by Ciccotti et al. [23]. Consider a system subject to constraints of the
form ${\sigma}_{k}\left(\text{}x\text{}\right)=0$. Let
${\left({\text{}v\text{}}_{i}\right)}_{i\in [1,n]}$ be arbitrarily chosen
vector fields (${\mathbb{R}}^{3N}\to {\mathbb{R}}^{3N}$)
verifying, for all $i$,
$j$, and
$k$:
$$\begin{array}{rcll}{\text{}v\text{}}_{i}\cdot \text{}{\nabla \⁡}_{\phantom{\rule{0.17em}{0ex}}\phantom{\rule{0.17em}{0ex}}x}\phantom{\rule{0.17em}{0ex}}\text{}{\xi}_{j}& =& {\delta}_{ij}& \text{(26)}\text{}\text{}\\ {\text{}v\text{}}_{i}\cdot \text{}{\nabla \⁡}_{\phantom{\rule{0.17em}{0ex}}\phantom{\rule{0.17em}{0ex}}x}\phantom{\rule{0.17em}{0ex}}\text{}{\sigma}_{k}& =& 0& \text{(27)}\text{}\text{}\end{array}$$
then the following holds [23]:
$$\frac{\partial A}{\partial {\xi}_{i}}={\u27e8{\text{}v\text{}}_{i}\cdot \text{}{\nabla \⁡}_{\phantom{\rule{0.17em}{0ex}}\phantom{\rule{0.17em}{0ex}}x}\phantom{\rule{0.17em}{0ex}}\text{}V{k}_{B}T\text{}{\nabla \⁡}_{\phantom{\rule{0.17em}{0ex}}\phantom{\rule{0.17em}{0ex}}x}\phantom{\rule{0.17em}{0ex}}\text{}\cdot {\text{}v\text{}}_{i}\u27e9}_{\text{}\xi \text{}}$$  (28) 
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 [20]. Condition (26) 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 (27) 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 (25). The biasing force applied
along the collective variables to overcome free energy barriers is calculated as:
$${F}^{ABF}=\alpha \left({N}_{\xi}\right)\times \text{}{\nabla \⁡}_{\phantom{\rule{0.17em}{0ex}}\phantom{\rule{0.17em}{0ex}}x}\phantom{\rule{0.17em}{0ex}}\text{}\stackrel{~}{A}\left(\text{}\xi \text{}\right)$$  (29) 
where $\text{}{\nabla \⁡}_{\phantom{\rule{0.17em}{0ex}}\phantom{\rule{0.17em}{0ex}}x}\phantom{\rule{0.17em}{0ex}}\text{}\stackrel{~}{A}$
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
$\text{}{\nabla \⁡}_{\phantom{\rule{0.17em}{0ex}}\phantom{\rule{0.17em}{0ex}}x}\phantom{\rule{0.17em}{0ex}}\text{}\stackrel{~}{A}$ 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 higherdimension collective variable space is likely to be difficult to sample and visualize. Most commonly, the
number of variables is one or two, sometimes three.
7.2.1 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 extendedsystem ABF method
(7.3).
 Only linear combinations of colvar components can be used in ABF calculations.
 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.
 Mutual orthogonality of colvars. In a multidimensional ABF calculation, equation (26) 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 nonoverlapping 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.
 Mutual orthogonality of components: when several components are combined into a colvar, it is assumed that their
vectors ${\text{}v\text{}}_{i}$
(equation (28)) are mutually orthogonal. The cases described for colvars in the previous paragraph
apply.
 Orthogonality of colvars and constraints: equation 27 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 lefthand side of equation 27 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.
ABF depends on parameters from each collective variable to define the grid on which free energy gradients are
computed: see 5.19 for detauls. Other parameters to control the ABF runtime can be set in the ABF configuration
block:
 Keyword name: see definition of name (biasing and analysis methods)
 Keyword colvars: see definition of colvars (biasing and analysis methods)
 Keyword outputEnergy: see definition of outputEnergy (biasing and analysis methods)
 Keyword outputFreq: see definition of outputFreq (biasing and analysis methods)
 Keyword stepZeroData: see definition of stepZeroData (biasing and analysis methods)
 Keyword fullSamples $\u27e8\phantom{\rule{0.17em}{0ex}}$Number
of samples in a bin prior to application of the ABF$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 nonzero, 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: $\alpha \left({N}_{\xi}\right)=(2{N}_{\xi}\u2215fullSamples)1$
.
 Keyword maxForce $\u27e8\phantom{\rule{0.17em}{0ex}}$Maximum
magnitude of the ABF force$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 nonnegative value for each
colvar. The unit of force is energy unit/(colvar unit).
 Keyword hideJacobian $\u27e8\phantom{\rule{0.17em}{0ex}}$Remove
geometric entropy term from calculated free energy gradient?$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: abf
Acceptable values: boolean
Default value: no
Description: In a few special cases, most notably distancebased 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 particleparticle
potentials of mean force being flat at large separations. The Jacobian term is exactly represented in
equation (28) by the second term of the average, ${k}_{B}T\text{}{\nabla \⁡}_{\phantom{\rule{0.17em}{0ex}}\phantom{\rule{0.17em}{0ex}}x}\phantom{\rule{0.17em}{0ex}}\text{}\cdot {\text{}v\text{}}_{i}$.
Enabling the hideJacobian option causes the output data to follow the traditional potential of mean
force convention, by omitting this contribution from the measured free energy gradients. To ensure
uniform sampling despite the incomplete description of the free energy, an additional biasing force
that counteracts the Jacobian force is applied internally by the colvar.
 Keyword historyFreq $\u27e8\phantom{\rule{0.17em}{0ex}}$Frequency
(in timesteps) at which ABF history files are accumulated$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: abf
Acceptable values: positive integer
Default value: 0
Description: If this number is nonzero, the free energy gradient estimate and sampling histogram
(and the PMF in onedimensional calculations) are written to files on disk at the given time interval.
History file names use the same prefix as output files, with “.hist" appended (outputName.hist.pmf).
historyFreq must be a multiple of outputFreq.
 Keyword inputPrefix $\u27e8\phantom{\rule{0.17em}{0ex}}$Filename
prefix for reading ABF data$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 (cv load command).
 Keyword applyBias $\u27e8\phantom{\rule{0.17em}{0ex}}$Apply
the ABF bias?$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword updateBias $\u27e8\phantom{\rule{0.17em}{0ex}}$Update
the ABF bias?$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
The ABF bias produces the following files, all in multicolumn text format (4.7.5):
 outputName.grad: current estimate of the free energy gradient (grid), in multicolumn;
 outputName.count: histogram of samples collected, on the same grid;
 outputName.pmf: integrated free energy profile or PMF (for dimension 1, and dimension 2 or 3).
Also in the case of onedimensional calculations, the ABF bias can report its current energy via
outputEnergy; in higher dimensions, such computation is not implemented and the energy reported is
zero.
If several ABF biases are defined concurrently, their name is inserted to produce unique filenames for output, as
in outputName.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).
7.2.4 Multidimensional free energy surfaces
The ABF method only produces an estimate of the free energy gradient. The free energy surface itself can be
computed depending on the value of integrate and related options.
 Keyword integrate $\u27e8\phantom{\rule{0.17em}{0ex}}$Integrate
free energy surface from ABF gradients (dim <
3)$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: abf
Acceptable values: boolean
Default value: yes
Description: This option, active by default when the dimension of the colvar space is 3 or
less, enables the calculation of an integrated free energy surface every time ABF output files
are written. In dimension 2 or 3, integration is performed by solving a Poisson equation: [24]
$${\nabla \⁡}^{2}{A}_{t}=\nabla \⁡\cdot {G}_{t}$$  (30) 
wehere ${G}_{t}$ is the estimated
gradient at time $t$,
and ${A}_{t}$ is
corresponding free energy surface. The free energy surface is written under the file name .pmf,
in a plain text format (see 4.7.5) that can be read by most data plotting and analysis programs (e.g. Gnuplot).
Periodic boundary conditions are applied to periodic coordinates, and Neumann boundary conditions
otherwise (imposed free energy gradient at the boundary of the domain). The grid used for free energy
discretization is extended by one point along nonperiodic coordinates, but not along periodic coordinates. See
ref. [24] for details.
 Keyword integrateTol $\u27e8\phantom{\rule{0.17em}{0ex}}$Tolerance
for free energy integration$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: abf
Acceptable values: positive decimal
Default value: 1e6
Description: The conjugate gradients algorithm used to integrate the free energy surface is stopped when the
RMS error reaches integrateTol.
 Keyword
integrateMaxIterations $\u27e8\phantom{\rule{0.17em}{0ex}}$Maximum
iterations for free energy integration$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: abf
Acceptable values: integer
Default value: 10000
Description: The conjugate gradients algorithm used to integrate the free energy surface is stopped when the
number of iterations reaches integrateMaxIterations, unless the RMS error has reached integrateTol
before.
In dimension 4 or greater, integrating the discretized gradient becomes nontrivial. The standalone utility
abf_integrate is provided to perform that task. Because 4D ABF calculations are uncommon, this tool is
practically deprecated by the Poisson integration described above.
abf_integrate reads the gradient data and uses it to perform a MonteCarlo (MC) simulation in discretized
collective variable space (specifically, on the same grid used by ABF to discretize the free energy gradient). By
default, a historydependent bias (similar in spirit to metadynamics) is used: at each MC 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 commandline:
abf_integrate [n ] [t ] [m (01)] [h ] [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 MC 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 MC sampling; expects kcal/mol as energy unit (unrelated to the simulation
temperature) [500 K]
 s: scaling factor for the gradients; when using a histogram of total forces obtained from
outputTotalForce or the .force file written by writeTISamples, a scaling factor of 1 should be
used [1.0]
 m: use metadynamicslike biased sampling? (0 = false) [1]
 h: increment for the historydependent bias (“hill height") [0.01 kcal/mol]
 f: if nonzero, this factor is used to scale the increment stepwise in the second half of the MC
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:
 .pmf: computed free energy surface
 .histo: histogram of MC sampling (not usable in a straightforward way if the
historydependent bias has been applied)
 .est: estimated gradient of the calculated free energy surface (from finite
differences)
 .dev: deviation between the userprovided numerical gradient and the actual
gradient of the calculated free energy surface. The RMS norm of this vector field is used as a
convergence criterion and output periodically during 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). See Ref.[24] for
details.
7.3 Extendedsystem Adaptive Biasing Force (eABF)
Extendedsystem ABF (eABF) is a variant of ABF (7.2) 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 (5.21). The theory of eABF and the present
implementation are documented in detail in reference [25].
Defining an ABF bias on a colvar wherein the extendedLagrangian option is active will perform eABF
automatically; there is no dedicated option.
The extended variable $\lambda $ is
coupled to the colvar $z=\xi \left(q\right)$ by the
harmonic potential $(k\u22152){(z\lambda )}^{2}$. Under eABF
dynamics, the adaptive bias on $\lambda $
is the running estimate of the average spring force:
$${F}^{bias}\left({\lambda}^{\ast}\right)={\u27e8k(\lambda z)\u27e9}_{{\lambda}^{\ast}}$$  (31) 
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 $.[25]
Distribution of the colvar may be assessed by plotting its histogram, which is written to
the outputName.zcount file in every eABF simulation. Note that a histogram bias (7.10)
applied to an extendedLagrangian 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.
7.3.1 CZAR estimator of the free energy
The corrected zaveraged restraint (CZAR) estimator is described in detail in reference [25]. 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 nonextended colvars; in that case, CZAR still provides an unbiased
estimate of the free energy gradient.
CZAR estimates the free energy gradient as:
$${A}^{\prime}\left(z\right)=\frac{1}{\beta}\frac{d\mathrm{ln}\⁡\stackrel{~}{\rho}\left(z\right)}{dz}+k({\u27e8\lambda \u27e9}_{z}z).$$  (32) 
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:
 Keyword CZARestimator $\u27e8\phantom{\rule{0.17em}{0ex}}$Calculate
CZAR estimator of the free energy?$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: abf
Acceptable values: boolean
Default value: yes
Description: This option is only available when ABF is performed on extendedLagrangian colvars.
When enabled, it triggers calculation of the free energy following the CZAR estimator.
 Keyword writeCZARwindowFile $\u27e8\phantom{\rule{0.17em}{0ex}}$Write
internal data from CZAR to a separate file?$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 outputName.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). Important warning. Unbiased freeenergy estimators from eABF dynamics rely
on some form of sampling histogram. When running stratified (windowed) calculations this histogram
becomes discontinuous, and as a result the free energy gradient estimated by CZAR is inaccurate
at the window boundary, resulting in visible "blips" in the PMF. As a workaround, we recommend
manually replacing the two free energy gradient values at the boundary, either with the ABF values
from .grad files (accurate in the limit of tight coupling), or with values interpolated for the neighboring
values of the CZAR gradient.
Similar to ABF, the CZAR estimator produces two output files in multicolumn text format (4.7.5):
 outputName.czar.grad: current estimate of the free energy gradient (grid), in multicolumn;
 outputName.czar.pmf: only for onedimensional 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 outputName.zcount.
The metadynamics method uses a historydependent potential [26] that generalizes to any type of colvars the
conformational flooding [27] and local elevation [28] 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 $\text{}\xi \text{}=({\xi}_{1},{\xi}_{2},\dots \⁡,{\xi}_{{N}_{cv}})$
is defined as:
$${V}_{meta}\left(\text{}\xi \text{}\right(t\left)\right)\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}\sum _{{t}^{\prime}=\delta t,2\delta t,\dots}^{{t}^{\prime}<t}W\phantom{\rule{0.22em}{0ex}}\prod _{i=1}^{{N}_{cv}}\mathrm{exp}\⁡\left(\frac{{\left({\xi}_{i}\right(t){\xi}_{i}({t}^{\prime}\left)\right)}^{2}}{2{\sigma}_{{\xi}_{i}}^{2}}\right),$$  (33) 
where ${V}_{meta}$
is the historydependent potential acting on the current values of the colvars
$\text{}\xi \text{}$,
and depends only parametrically on the previous values of the colvars.
${V}_{meta}$ is constructed as a
sum of ${N}_{cv}$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 $\xc3\left(\text{}\xi \text{}\right)$,
which is the sum of the “real" underlying potential of mean force
$A\left(\text{}\xi \text{}\right)$ and the the
metadynamics potential, ${V}_{meta}\left(\text{}\xi \text{}\right)$.
Therefore, at any given time the probability of observing the configuration
$\text{}{\xi}^{\ast}\text{}$ is proportional
to $\mathrm{exp}\⁡\left(\xc3\left(\text{}{\xi}^{\ast}\text{}\right)\u2215{\kappa}_{B}T\right)$: 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
$\xc3\left(\text{}\xi \text{}\right)$ is constant, and
${V}_{meta}\left(\text{}\xi \text{}\right)$ is an estimator of the “real"
potential of mean force $A\left(\text{}\xi \text{}\right)$,
save for an additive constant:
$$A\left(\text{}\xi \text{}\right)\phantom{\rule{0.28em}{0ex}}\simeq \phantom{\rule{0.28em}{0ex}}{V}_{meta}\left(\text{}\xi \text{}\right)+K$$  (34) 
Such estimate of the free energy can be provided by enabling writeFreeEnergyFile.
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 userdefined
parameters $W$,
${\sigma}_{{\xi}_{i}}$ and
$\delta t$
[29]. In typical applications, a good rule of thumb can be to choose the ratio
$W\u2215\delta t$ much smaller
than ${\kappa}_{B}T\u2215{\tau}_{\text{}\xi \text{}}$, 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.
If the metadynamics parameters are chosen correctly, after an equilibration time,
${t}_{e}$,
the estimator provided by eq. 34 oscillates on time around the “real" free energy, thereby a
better estimate of the latter can be obtained as the time average of the bias potential after
${t}_{e}$
[30, 31]:
$$A\left(\text{}\xi \text{}\right)\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}\frac{1}{{t}_{tot}{t}_{e}}{\int}_{{t}_{e}}^{{t}_{tot}}{V}_{meta}(\text{}\xi \text{},t)dt$$  (35) 
where ${t}_{e}$ is
the time after which the bias potential grows (approximately) evenly during the simulation and
${t}_{tot}$ is the
total simulation time. The free energy calculated according to eq. 35 can thus be obtained averaging on time mutiple
timedependent free energy estimates, that can be printed out through the keyword keepFreeEnergyFiles. An
alternative is to obtain the free energy profiles by summing the hills added during the simulation; the hills trajectory
can be printed out by enabling the option writeHillsTrajectory.
7.4.1 Treatment of the PMF boundaries
In typical scenarios the Gaussian hills of a metadynamics potential are interpolated and summed together onto a
grid, which is much more efficient than computing each hill independently at every step (the keyword useGrids is
on by default). This numerical approximation typically yields neglibile errors in the resulting PMF [1]. However,
due to the finite thickness of the Gaussian function, the metadynamics potential would suddenly vanish each time a
variable exceeds its grid boundaries.
To avoid such discontinuity the Colvars metadynamics code will keep an explicit copy of each hill that straddles
a grid's boundary, and will use it to compute metadynamics forces outside the grid. This measure is taken to protect
the accuracy and stability of a metadynamics simulation, except in cases of “natural" boundaries (for example, the
$[0:180]$ interval
of an angle colvar) or when the flags hardLowerBoundary and hardUpperBoundary are explicitly set by the user.
Unfortunately, processing explicit hills alongside the potential and force grids could easily become inefficient,
slowing down the simulation and increasing the state file's size.
In general, it is a good idea to define a repulsive potential to avoid hills from coming too close to the grid's
boundaries, for example as a harmonicWalls restraint (see 7.7).
Example: Using harmonic walls to protect the grid's boundaries.
colvar {
name r
distance { ... }
upperBoundary 15.0
width 0.2
}
metadynamics {
name meta_r
colvars r
hillWeight 0.001
hillWidth 2.0
}
harmonicWalls {
name wall_r
colvars r
upperWalls 13.0
upperWallConstant 2.0
}
In the colvar r, the distance function used has a lowerBoundary automatically set to 0 by default, thus the keyword
lowerBoundary itself is not mandatory and hardLowerBoundary is set to yes internally. However, upperBoundary does
not have such a “natural" choice of value. The metadynamics potential meta_r will individually process any hill
whose center is too close to the upperBoundary, more precisely within fewer grid points than 6 times the Gaussian
$\sigma $
parameter plus one. It goes without saying that if the colvar r represents a distance between two freelymoving
molecules, it will cross this “threshold" rather frequently.
In this example, where the value of hillWidth ($2\sigma $)
amounts to 2 grid points, the threshold is 6+1 = 7 grid points away from upperBoundary. In explicit units, the width
of $r$ is
${w}_{r}=$ 0.2 Å, and the
threshold is 15.0  7$\times $0.2
= 13.6 Å.
The wall_r restraint included in the example prevents this: the position of its upperWall is 13 Å, i.e. 3 grid points
below the buffer's threshold (13.6 Å). For the chosen value of upperWallConstant, the energy of the wall_r bias
at r = ${r}_{upper}$ =
13.6 Å is:
$${E}^{\ast}=\frac{1}{2}k{\left(\frac{r{r}_{upper}}{{w}_{r}}\right)}^{2}=\frac{1}{2}2.0{\left(3\right)}^{2}=9\phantom{\rule{0.33em}{0ex}}kcal\u2215mol$$ 
which results in a relative probability $\mathrm{exp}\⁡({E}^{\ast}\u2215{\kappa}_{B}T)\simeq $
$3\times 1{0}^{7}$ that r
crosses the threshold. The probability that r exceeds upperBoundary, which is further away, has also become
vanishingly small. At that point, you may want to set hardUpperBoundary to yes for r, and let meta_r know that no
special treatment near the grid's boundaries will be needed.
What is the impact of the wall restraint onto the PMF? Not a very complicated one: the PMF reconstructed
by metadynamics will simply show a sharp increase in freeenergy where the wall potential kicks in
(r $>$
13 Å). You may then choose between using the PMF only up until that point and discard the rest, or subtracting the
energy of the harmonicWalls restraint from the PMF itself. Keep in mind, however, that the statistical convergence
of metadynamics may be less accurate where the wall potential is strong.
In summary, although it would be simpler to set the wall's position upperWall and the grid's boundary
upperBoundary to the same number, the finite width of the Gaussian hills calls for setting the former strictly within
the latter.
7.4.2 Required metadynamics keywords
To enable a metadynamicsbased calculation, a metadynamics {...} block must be included in the Colvars
configuration file.
By default, metadynamics bias energy and forces will be recorded onto a grid, the parameters of which can be
defined within the definition of each colvar, as described in 5.19.
Other required keywords will be specified within the metadynamics block: these
are colvars (the names of the variables involved), hillWeight (the weight parameter
$W$), and the
widths $2\sigma $
of the Gaussian hills in each dimension that can be given either as the single dimensionless parameter hillWidth, or
explicitly for each colvar with gaussianSigmas.
 Keyword name: see definition of name (biasing and analysis methods)
 Keyword colvars: see definition of colvars (biasing and analysis methods)
 Keyword outputEnergy: see definition of outputEnergy (biasing and analysis methods)
 Keyword outputFreq: see definition of outputFreq (biasing and analysis methods)
 Keyword writeTIPMF: see definition of writeTIPMF (biasing and analysis methods)
 Keyword writeTISamples: see definition of writeTISamples (biasing and analysis methods)
 Keyword stepZeroData: see definition of stepZeroData (biasing and analysis methods)
 Keyword hillWeight $\u27e8\phantom{\rule{0.17em}{0ex}}$Height
of each hill (energy unit)$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
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.
 Keyword hillWidth $\u27e8\phantom{\rule{0.17em}{0ex}}$Width
$2\sigma $
of a Gaussian hill, measured in number of grid points$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
Acceptable values: positive decimal
Description: This keyword sets the Gaussian width $2{\sigma}_{{\xi}_{i}}$
for all colvars, expressed in number of grid points, with the grid spacing along each colvar $\xi $
determined by the respective value of width. Values between 1 and 3 are recommended for this
option: smaller numbers will fail to adequately interpolate each Gaussian function [1], while larger
values may be unable to account for steep freeenergy gradients. The values of each halfwidth
${\sigma}_{{\xi}_{i}}$
in the physical units of ${\xi}_{i}$
are also printed by VMD at initialization time; alternatively, they may be set explicitly via gaussianSigmas.
 Keyword gaussianSigmas $\u27e8\phantom{\rule{0.17em}{0ex}}$Halfwidths
$\sigma $
of the Gaussian hill (one for each colvar)$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
Acceptable values: spaceseparated list of decimals
Description: This option sets the parameters ${\sigma}_{{\xi}_{i}}$
of the Gaussian hills along each colvar ${\xi}_{i}$,
expressed in the same unit of ${\xi}_{i}$.
No restrictions are placed on each value, but a warning will be printed if useGrids is on and the
Gaussian width $2{\sigma}_{{\xi}_{i}}$
is smaller than the corresponding grid spacing, $width\left({\xi}_{i}\right)$.
If not given, default values will be computed from the dimensionless number hillWidth.
 Keyword newHillFrequency $\u27e8\phantom{\rule{0.17em}{0ex}}$Frequency
of hill creation$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
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 timestep defines the
parameter $\delta t$
in eq. 33. Higher values provide more accurate statistical sampling, at the price of longer simulation
times to complete a PMF calculation. When analyzing data from a previous simulation in VMD, the
metadynamics potential does not need to be updated, and it is useful to set this number to 0.
When interpolating grids are enabled (default behavior), the PMF is written by default every
colvarsRestartFrequency steps to the file outputName.pmf in multicolumn text format (4.7.5). The following
two options allow to disable or control this behavior and to track statistical convergence:
 Keyword writeFreeEnergyFile $\u27e8\phantom{\rule{0.17em}{0ex}}$Periodically
write the PMF for visualization$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
Acceptable values: boolean
Default value: on
Description: When useGrids and this option are on, the PMF is written every outputFreq steps.
 Keyword keepFreeEnergyFiles $\u27e8\phantom{\rule{0.17em}{0ex}}$Keep
all the PMF files$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
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.
 Keyword writeHillsTrajectory $\u27e8\phantom{\rule{0.17em}{0ex}}$Write
a log of new hills$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
Acceptable values: boolean
Default value: off
Description: If this option is on, a file containing the Gaussian hills written by the metadynamics
bias, with the name:
“outputName.colvars.$<$name$>$.hills.traj",
which can be useful to postprocess the time series of the Gassian hills. Each line is written every
newHillFrequency, regardless of the value of outputFreq. When multipleReplicas is on, its name
is changed to:
“outputName.colvars.$<$name$>$.$<$replicaID$>$.hills.traj".
The columns of this file are the centers of the hills, ${\xi}_{i}\left({t}^{\prime}\right)$,
followed by the halfwidths, ${\sigma}_{{\xi}_{i}}$,
and the weight, $W$.
Note: prior to version 20200224, the fullwidth $2\sigma $
of the Gaussian was reported in lieu of $\sigma $.
7.4.4 Performance optimization
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.
 Keyword useGrids $\u27e8\phantom{\rule{0.17em}{0ex}}$Interpolate
the hills with grids$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
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 nonscalar types (distanceDir or orientation). If expandBoundaries is defined
in one of the colvars, grids are automatically expanded along the direction of that colvar.
 Keyword rebinGrids $\u27e8\phantom{\rule{0.17em}{0ex}}$Recompute
the grids when reading a state file$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
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.
 Keyword keepHills $\u27e8\phantom{\rule{0.17em}{0ex}}$Write
each individual hill to the state file$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
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.
7.4.5 EnsembleBiased Metadynamics
The ensemblebiased metadynamics (EBMetaD) approach [32] is designed to reproduce a target
probability distribution along selected collective variables. Standard metadynamics can be seen as a
special case of EBMetaD with a flat distribution as target. This is achieved by weighing the Gaussian
functions used in the metadynamics approach by the inverse of the target probability distribution:
$${V}_{EBmetaD}\left(\text{}\xi \text{}\right(t\left)\right)\phantom{\rule{0.28em}{0ex}}=\phantom{\rule{0.28em}{0ex}}\sum _{{t}^{\prime}=\delta t,2\delta t,\dots}^{{t}^{\prime}<t}\frac{W}{\mathrm{exp}\⁡\left({S}_{\rho}\right){\rho}_{exp}\left(\text{}\xi \text{}\right({t}^{\prime}\left)\right)}\phantom{\rule{0.22em}{0ex}}\prod _{i=1}^{{N}_{cv}}\mathrm{exp}\⁡\left(\frac{{\left({\xi}_{i}\right(t){\xi}_{i}({t}^{\prime}\left)\right)}^{2}}{2{\sigma}_{{\xi}_{i}}^{2}}\right),$$  (36) 
where ${\rho}_{exp}\left(\text{}\xi \text{}\right)$ is the target
probability distribution and ${S}_{\rho}=\int \⁡{\rho}_{exp}\left(\text{}\xi \text{}\right)\mathrm{log}\⁡{\rho}_{exp}\left(\text{}\xi \text{}\right)\phantom{\rule{0.17em}{0ex}}d\text{}\xi \text{}$
its corresponding differential entropy. The method is designed so that during the simulation the resulting distribution of the
collective variable $\text{}\xi \text{}$
converges to ${\rho}_{exp}\left(\text{}\xi \text{}\right)$.
A practical application of EBMetaD is to reproduce an “experimental" probability distribution, for example the
distance distribution between spectroscopic labels inferred from Förster resonance energy transfer (FRET) or
double electronelectron resonance (DEER) experiments [32].
The PMF along $\xi $
can be estimated from the bias potential and the target ditribution [32]:
$$A\left(\text{}\xi \text{}\right)\phantom{\rule{0.28em}{0ex}}\simeq \phantom{\rule{0.28em}{0ex}}{V}_{EBmetaD}\left(\text{}\xi \text{}\right){\kappa}_{B}T\mathrm{log}\⁡{\rho}_{exp}\left(\text{}\xi \text{}\right)$$  (37) 
and obtained by enabling writeFreeEnergyFile. Similarly to eq. 35, a more accurate estimate of the free
energy can be obtained by averaging (after an equilibration time) multiple timedependent free energy estimates (see
keepFreeEnergyFiles).
The following additional options define the configuration for the ensemblebiased metadynamics approach:
 Keyword ebMeta $\u27e8\phantom{\rule{0.17em}{0ex}}$Perform
ensemblebiased metadynamics$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
Acceptable values: boolean
Default value: off
Description: If enabled, this flag activates the ensemblebiased metadynamics as described by Marinelli
et al.[32]. The target distribution file, targetdistfile, is then required. The keywords lowerBoundary,
upperBoundary and width for the respective variables are also needed to set the binning (grid) of the
target distribution file.
 Keyword targetDistFile $\u27e8\phantom{\rule{0.17em}{0ex}}$Target
probability distribution file for ensemblebiased metadynamics$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
Acceptable values: multicolumn text file
Description: This file provides the target probability distribution, ${\rho}_{exp}\left(\text{}\xi \text{}\right)$,
reported in eq. 36. The latter distribution must be a tabulated function provided in a multicolumn text
format (see 4.7.5). The provided distribution is then normalized.
 Keyword ebMetaEquilSteps $\u27e8\phantom{\rule{0.17em}{0ex}}$Number
of equilibration steps for ensemblebiased metadynamics$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
Acceptable values: positive integer
Description: The EBMetaD approach may introduce large hills in regions with small values of
the target probability distribution (eq. 36). This happens, for example, if the probability distribution
sampled by a conventional molecular dynamics simulation is significantly different from the target
distribution. This may lead to instabilities at the beginning of the simulation related to large biasing
forces. In this case, it is useful to introduce an equilibration stage in which the bias potential gradually
switches from standard metadynamics (eq. 33) to EBmetaD (eq. 36) as $\lambda {V}_{meta}\left(\text{}\xi \text{}\right)+(1\lambda ){V}_{EBmetaD}\left(\text{}\xi \text{}\right)$,
where $\lambda =(ebMetaEquilStepsstep)\u2215ebMetaEquilSteps$
and step is the current simulation step number.
 Keyword targetDistMinVal $\u27e8\phantom{\rule{0.17em}{0ex}}$Minimum
value of the target distribution in reference to its maximum value$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
Acceptable values: positive decimal
Description: It is useful to set a minimum value of the target probability distribution to avoid values
of the latter that are nearly zero, leading to very large hills. This parameter sets the minimum value of
the target probability distribution that is expressed as a fraction of its maximum value: minimum value
= maximum value X targetDistMinVal. This implies that 0 1 and its default
value is set to 1/1000000. To avoid divisions by zero (see eq. 36), if targetDistMinVal is set as zero,
values of ${\rho}_{exp}$
equal to zero are replaced by the smallest positive value read in the same file.
As with standard metadynamics, multidimensional probability distributions can be targeted using a single
metadynamics block using multiple colvars and a multidimensional target distribution file (see 4.7.5). Instead,
multiple probability distributions on different variables can be targeted separately in the same simulation by
introducing multiple metadynamics blocks with the ebMeta option.
Example: EBmetaD configuration for a single variable.
colvar {
name r
distance {
group1 { atomNumbers 991 992 }
group2 { atomNumbers 1762 1763 }
}
upperBoundary 100.0
width 0.1
}
metadynamics {
name ebmeta
colvars r
hillWeight 0.01
hillWidth 3.0
ebMeta on
targetDistFile targetdist1.dat
ebMetaEquilSteps 500000
}
where targetdist1.dat is a text file in “multicolumn" format (4.7.5) with the same width as the variable r (0.1 in
this case):

#  1     
#  0.0  0.1  1000  0 

 0.05  0.0012 
 0.15  0.0014 
 …  … 
 99.95  0.0010 


Tip: Besides setting a meaninful value for targetDistMinVal, the exploration of unphysically low values
of the target distribution (which would lead to very large hills and possibly numerical instabilities)
can be also prevented by restricting sampling to a given interval, using e.g. harmonicWalls restraint
(7.7).
7.4.6 Welltempered metadynamics
The following options define the configuration for the “welltempered" metadynamics approach
[33]:
 Keyword wellTempered $\u27e8\phantom{\rule{0.17em}{0ex}}$Perform
welltempered metadynamics$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
Acceptable values: boolean
Default value: off
Description: If enabled, this flag causes welltempered metadynamics as described by Barducci et
al.[33] to be performed, rather than standard metadynamics. The parameter biasTemperature is then
required. This feature was contributed by Li Li (LutheySchulten group, Department of Chemistry,
UIUC).
 Keyword biasTemperature $\u27e8\phantom{\rule{0.17em}{0ex}}$Temperature
bias for welltempered metadynamics$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
Acceptable values: positive decimal
Description: When running metadynamics in the long time limit, collective variable space is sampled
to a modified temperature $T+\mathrm{\Delta}T$.
In conventional metadynamics, the temperature “boost" $\mathrm{\Delta}T$
would constantly increases with time. Instead, in welltempered metadynamics $\mathrm{\Delta}T$
must be defined by the user via biasTemperature. The written PMF includes the scaling factor
$(T+\mathrm{\Delta}T)\u2215\mathrm{\Delta}T$
[33]. A careful choice of $\mathrm{\Delta}T$
determines the sampling and convergence rate, and is hence crucial to the success of a welltempered
metadynamics simulation.
7.4.7 Multiplewalker metadynamics
Metadynamics calculations can be performed concurrently by multiple replicas that share a common
history. This variant of the method is called multiplewalker metadynamics
[34]: the Gaussian hills of all
replicas are periodically combined into a single biasing potential, intended to converge to a single
PMF.
In the implementation here described [1], replicas communicate through files. This arrangement allows
launching the replicas either (1) as a bundle (i.e. a single job in a cluster's queueing system) or (2) as fully
independent runs (i.e. as separate jobs for the queueing system). One advantage of the use case (1) is that an
identical Colvars configuration can be used for all replicas (otherwise, replicaID needs to be manually set to a
different string for each replica). However, the use case (2) is less demanding in terms of highperformance
computing resources: a typical scenario would be a computer cluster (including virtual servers from a cloud
provider) where not all nodes are connected to each other at high speed, and thus each replica runs on a small group
of nodes or a single node.
Whichever way the replicas are started (coupled or not), a shared filesystem is needed so that each replica can
read the files created by the others: paths to these files are stored in the shared file replicasRegistry. This file, and
those listed in it, are read every replicaUpdateFrequency steps. Each time the Colvars state file is written (for
example, colvarsRestartFrequency steps), the file named:
outputName.colvars.name.replicaID.state
is written as well; this file contains only the state of the metadynamics bias, which the other replicas will read in
turn. In between the times when this file is modified/replaced, new hills are also temporarily written to the file
named:
outputName.colvars.name.replicaID.hills
Both files are only used for communication, and may be deleted after the replica begins writing files with a new
outputName.
Example: Multiplewalker metadynamics with filebased communication.
metadynamics {
name mymtd
colvars x
hillWeight 0.001
newHillFrequency 1000
hillWidth 3.0
multipleReplicas on
replicasRegistry /sharedfolder/mymtdreplicas.txt
replicaUpdateFrequency 50000 # Best if larger than newHillFrequency
}
The following are the multiplewalkers related options:
 Keyword multipleReplicas $\u27e8\phantom{\rule{0.17em}{0ex}}$Enable
multiplewalker metadynamics$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
Acceptable values: boolean
Default value: off
Description: This option turns on multiplewalker communication between replicas.
 Keyword replicasRegistry $\u27e8\phantom{\rule{0.17em}{0ex}}$Multiple
replicas database file$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
Acceptable values: UNIX filename
Description: If multipleReplicas is on, this option sets the path to the replicas' shared database
file. It is best to use an absolute path (especially when running individual replicas in separate folders).
 Keyword replicaUpdateFrequency $\u27e8\phantom{\rule{0.17em}{0ex}}$How
often hills are shared between replicas$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
Acceptable values: positive integer
Description: If multipleReplicas is on, this option sets the number of steps after which each
replica tries to read the other replicas' files. On a networked file system, it is best to use a number of
steps that corresponds to at least a minute of wall time.
 Keyword replicaID $\u27e8\phantom{\rule{0.17em}{0ex}}$Set
the identifier for this replica (required only for independent jobs)$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
Acceptable values: string
Default value: replica index (only if MPI is used)
Description: If multipleReplicas is on, this option sets a unique identifier for this replicas. Specifying
this option is thus only required when the replicas are launched as independent computations: when
the replicas share a common parallel communication framework (i.e. they are all launched together
as a single messagepassing computation via MPI) the default value of this keyword is the replica's
numeric index (zerobased).
 Keyword writePartialFreeEnergyFile $\u27e8\phantom{\rule{0.17em}{0ex}}$Periodically
write the contribution to the PMF from this replica$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
Acceptable values: boolean
Default value: off
Description: If multipleReplicas is on, enabling this option produces an additional file outputName.partial.pmf,
which can be useful to monitor the contribution of each replica to the total PMF (which is written to
the file outputName.pmf). Note: the name of this file is chosen for consistency and convenience, but
its content is not a PMF and it is not expected to converge, even if the total PMF does.
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:
$$V\left(\xi \right)=\frac{1}{2}k{\left(\frac{\xi {\xi}_{0}}{{w}_{\xi}}\right)}^{2}$$  (38) 
There are two noteworthy aspects of this expression:
 Because the standard coefficient of $1\u22152$
of the harmonic potential is included, this expression differs from harmonic bond and angle potentials
historically used in common force fields, where the factor was typically omitted resulting in a nonstandard
definition of the force constant.
 The variable $\xi $
is not only centered at ${\xi}_{0}$,
but is also scaled by its characteristic length scale ${w}_{\xi}$
(keyword width). The resulting dimensionless variable $z=(\xi {\xi}_{0})\u2215{w}_{\xi}$
is typically easier to treat numerically: for example, when the forces typically experienced by $\xi $
are much smaller than $k\u2215{w}_{\xi}$
and $k$
is chosen equal to ${\kappa}_{B}T$
(thermal energy), the resulting probability distribution of $z$
is approximately a Gaussian with mean equal to 0 and standard deviation equal to 1.
This property can be used for setting the force constant in umbrellasampling ensemble runs: if the
restraint centers are chosen in increments of ${w}_{\xi}$,
the resulting distributions of $\xi $
are most often optimally overlapped. In regions where the underlying freeenergy landscape induces
highly skewed distributions of $\xi $,
additional windows may be added as needed, with spacings finer than ${w}_{\xi}$.
Beyond one dimension, the use of a scaled harmonic potential also allows a standard definition of a
multidimensional restraint with a unified force constant:
$$V({\xi}_{1},\dots \⁡,{\xi}_{M})=\frac{1}{2}k\sum _{i=1}^{M}{\left(\frac{{\xi}_{i}{\xi}_{0}}{{w}_{\xi}}\right)}^{2}$$  (39) 
If onedimensional or homogeneous multidimensional restraints are defined, and there are no other uses for the parameter
${w}_{\xi}$, widthcan be left at
its default value of $1$.
A harmonic restraint is defined by a harmonic {...} block, which may contain the following keywords:
 Keyword name: see definition of name (biasing and analysis methods)
 Keyword colvars: see definition of colvars (biasing and analysis methods)
 Keyword outputEnergy: see definition of outputEnergy (biasing and analysis methods)
 Keyword writeTIPMF: see definition of writeTIPMF (biasing and analysis methods)
 Keyword writeTISamples: see definition of writeTISamples (biasing and analysis methods)
 Keyword stepZeroData: see definition of stepZeroData (biasing and analysis methods)
 Keyword forceConstant $\u27e8\phantom{\rule{0.17em}{0ex}}$Scaled
force constant (energy unit)$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: harmonic
Acceptable values: positive decimal
Default value: 1.0
Description: This option defines a scaled force constant $k$
for the harmonic potential (eq. 39). 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, if this force constant is set to
the thermal energy ${\kappa}_{B}T$
(equal to $RT$
if molar units are used), then the amplitude of the thermal fluctuations of each variable $\xi $
will be on the order of its width, ${w}_{\xi}$.
This can be used to estimate the optimal spacing of umbrellasampling windows (under the assumption
that the force constant is larger than the curvature of the underlying free energy). The values of the
actual force constants $k\u2215{w}_{\xi}^{2}$
are always printed when the restraint is defined.
 Keyword centers $\u27e8\phantom{\rule{0.17em}{0ex}}$Initial
harmonic restraint centers$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: harmonic
Acceptable values: spaceseparated 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.
7.5.1 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.
 Keyword targetCenters $\u27e8\phantom{\rule{0.17em}{0ex}}$Steer
the restraint centers towards these targets$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: harmonic
Acceptable values: spaceseparated 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.
 Keyword targetNumSteps $\u27e8\phantom{\rule{0.17em}{0ex}}$Number
of steps for steering$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: harmonic
Acceptable values: positive integer
Description: In singlestage (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 multistage transformations, this sets the number of MD steps per stage.
 Keyword outputCenters $\u27e8\phantom{\rule{0.17em}{0ex}}$Write
the current centers to the trajectory file$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
7.5.2 Moving restraints: umbrella sampling
The centers of the harmonic restraints can also be changed in discrete stages: in this cases a onedimensional
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
7.10).
To activate an umbrella sampling simulation, the same keywords as in the previous section can be used, with the
addition of the following:
 Keyword targetNumStages $\u27e8\phantom{\rule{0.17em}{0ex}}$Number
of stages for steering$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: harmonic
Acceptable values: nonnegative integer
Default value: 0
Description: If nonzero, 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
$\times $
(targetNumStages + 1).
7.5.3 Changing force constant
The force constant of the harmonic restraint may also be changed to equilibrate [35].
 Keyword targetForceConstant $\u27e8\phantom{\rule{0.17em}{0ex}}$Change
the force constant towards this value$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 lambdaExponent 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, with the format:
colvars: Lambda= ***.** dA/dLambda= ***.**
A more complete free energy calculation (particularly with regard to convergence analysis), while
not handled by the Colvars module, can be performed by postprocessing 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.[35]
 Keyword lambdaExponent $\u27e8\phantom{\rule{0.17em}{0ex}}$Exponent
in the timedependence of the force constant$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 $:
${k}_{\lambda}={k}_{0}+{\lambda}^{\alpha}({k}_{1}{k}_{0})$,
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. To remove the restraint over time rather than introduce it, use the decoupling option.
 Keyword decoupling $\u27e8\phantom{\rule{0.17em}{0ex}}$Perform
a restraint decoupling transformation$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: harmonic
Acceptable values: boolean
Default value: off
Description: When this option is enabled, the restraint is decoupled by changing the force constant
from forceConstant to 0.0 over the course of the simulation, as specified by targetNumSteps and,
optionally, targetNumStages. The force constant is then varied as: ${k}_{\lambda}={(1\lambda )}^{\alpha}k$,
where $\alpha $
is the value of lambdaExponent. Setting lambdaExponent to a value greater than 1 (e.g. 4) ensures a
smoother dependence of the energy on $\lambda $
around ${k}_{\lambda}=0$.
 Keyword targetEquilSteps $\u27e8\phantom{\rule{0.17em}{0ex}}$Number
of steps discarded from TI estimate$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword lambdaSchedule $\u27e8\phantom{\rule{0.17em}{0ex}}$Schedule
of lambdapoints for changing force constant$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
7.6 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:
 Keyword outputAccumulatedWork $\u27e8\phantom{\rule{0.17em}{0ex}}$Write
the accumulated work of the changing restraint to the Colvars trajectory file$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 nonzero). 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.
7.7 Harmonic wall restraints
The harmonicWalls {...} bias is closely related to the harmonic bias (see 7.5), 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 halfharmonic potential;
$$V\left(\xi \right)=\{\begin{array}{cc}\frac{1}{2}k{\left(\frac{\xi {\xi}_{upper}}{{w}_{\xi}}\right)}^{2}\hfill & if\phantom{\rule{0.33em}{0ex}}\xi >{\xi}_{upper}\hfill \\ 0\hfill & if\phantom{\rule{0.33em}{0ex}}{\xi}_{lower}\le \xi \ge {\xi}_{upper}\hfill \\ \frac{1}{2}k{\left(\frac{\xi {\xi}_{lower}}{{w}_{\xi}}\right)}^{2}\hfill & if\phantom{\rule{0.33em}{0ex}}\xi <{\xi}_{lower}\hfill \end{array}$$  (40) 
where ${\xi}_{lower}$
and ${\xi}_{upper}$ 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
multidimensional).
Note: this bias replaces the keywords lowerWall, lowerWallConstant, upperWall and upperWallConstant
defined in the colvar context. Those keywords are deprecated.
The harmonicWalls bias implements the following options:
 Keyword name: see definition of name (biasing and analysis methods)
 Keyword colvars: see definition of colvars (biasing and analysis methods)
 Keyword outputEnergy: see definition of outputEnergy (biasing and analysis methods)
 Keyword writeTIPMF: see definition of writeTIPMF (biasing and analysis methods)
 Keyword writeTISamples: see definition of writeTISamples (biasing and analysis methods)
 Keyword stepZeroData: see definition of stepZeroData (biasing and analysis methods)
 Keyword lowerWalls $\u27e8\phantom{\rule{0.17em}{0ex}}$Position
of the lower wall(s)$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: Spaceseparated list of decimals
Description: Defines the values ${\xi}_{lower}$
below which a confining restraint on the colvar is applied to each colvar $\xi $.
 Keyword upperWalls $\u27e8\phantom{\rule{0.17em}{0ex}}$Position
of the upper wall(s)$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: colvar
Acceptable values: Spaceseparated list of decimals
Description: Defines the values ${\xi}_{upper}$
above which a confining restraint on the colvar is applied to each colvar $\xi $.
 Keyword forceConstant: see definition of forceConstant (Harmonic restraints)
 Keyword lowerWallConstant $\u27e8\phantom{\rule{0.17em}{0ex}}$Force
constant for the lower wall$\phantom{\rule{0.17em}{0ex}}\u27e9$
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. As with forceConstant, the specified constant is divided internally
by the square of the specific width of each variable (see also the equivalent keyword for the harmonic
restraint, forceConstant). The force constant reported in the output as “$k$",
and used in the change of force constant scheme, is the geometric mean of upperWallConstant and
upperWallConstant.
 Keyword upperWallConstant: analogous to lowerWallConstant
 Keyword targetForceConstant: see definition of targetForceConstant (harmonic restraints)
 Keyword targetForceConstant $\u27e8\phantom{\rule{0.17em}{0ex}}$Change
the force constant(s) towards this value$\phantom{\rule{0.17em}{0ex}}\u27e9$
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.
 Keyword targetNumSteps: see definition of targetNumSteps (harmonic restraints)
 Keyword lambdaExponent: see definition of lambdaExponent (harmonic restraints)
 Keyword targetEquilSteps: see definition of targetEquilSteps (harmonic restraints)
 Keyword targetNumStages: see definition of targetNumStages (harmonic restraints)
 Keyword lambdaSchedule: see definition of lambdaSchedule (harmonic restraints)
 Keyword outputAccumulatedWork: see definition of outputAccumulatedWork (harmonic
restraints)
 Keyword bypassExtendedLagrangian $\u27e8\phantom{\rule{0.17em}{0ex}}$Apply
bias to actual colvars, bypassing extended coordinates$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: harmonicWalls
Acceptable values: boolean
Default value: on
Description: This option behaves as bypassExtendedLagrangian for other biases, but it defaults
to on, unlike in the general case. Thus, by default, the harmonicWalls bias applies to the actual
colvars, so that the distribution of the colvar between the walls is unaffected by the bias, which then
applies a flatbottom potential as a function of the colvar value. This bias will affect the extended
coordinate distribution near the walls. If bypassExtendedLagrangian is disabled, harmonicWalls
applies a flatbottom potential as a function of the extended coordinate. Conversely, this bias will then
modify the distribution of the actual colvar value near the walls.
Example 1: harmonic walls for one variable with two different force constants.
harmonicWalls {
name mywalls
colvars dist
lowerWalls 22.0
upperWalls 38.0
lowerWallConstant 2.0
upperWallConstant 10.0
}
Example 2: harmonic walls for two variables with a single force constant.
harmonicWalls {
name mywalls
colvars phi psi
lowerWalls 180.0 0.0
upperWalls 0.0 180.0
forceConstant 5.0
}
The linear keyword defines a linear potential:
$$V\left(\xi \right)=k\left(\frac{\xi {\xi}_{0}}{{w}_{\xi}}\right)$$  (41) 
whose force is simply given by the constant $k\u2215{w}_{\xi}$
itself:
$$f\left(\xi \right)=k\u2215{w}_{\xi}$$  (42) 
This type of bias is therefore most useful in situations where a constant force is desired. As all other restraints, it can
be defined on one or more CVs, with each contribution added to the total potential and the parameters
${w}_{\xi}$
determining the relative magnitude for each.
Example: A possible use case of the linear bias is mimicking a constant electric field acting on a specific particle,
or the center of mass of many particles. In the following example, a linear restraint is applied on a distanceZ
variable (5.2.2), generating a constant force parallel to the Z axis of magnitude 5 energy unit/length
unit:
colvar {
name z
distanceZ {
...
}
}
linear {
colvars z
forceConstant 5.0
centers 0.0
}
Another useful application of a linear restraint is to enforce experimental constraints in a simulation, with a
lower nonequilibrium work than e.g. harmonic restraints [36]. 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
7.9.
 Keyword name: see definition of name (biasing and analysis methods)
 Keyword colvars: see definition of colvars (biasing and analysis methods)
 Keyword outputEnergy: see definition of outputEnergy (biasing and analysis methods)
 Keyword forceConstant $\u27e8\phantom{\rule{0.17em}{0ex}}$Scaled
force constant (energy unit)$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: linear
Acceptable values: Decimal
Default value: 1.0
Description: This option defines a scaled force constant $k$
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 $k\u2215{w}_{\xi}$
are always printed when the restraint is defined.
 Keyword centers $\u27e8\phantom{\rule{0.17em}{0ex}}$Initial
linear restraint centers$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: linear
Acceptable values: spaceseparated 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 welldefined energy for the linear bias.
 Keyword writeTIPMF: see definition of writeTIPMF (biasing and analysis methods)
 Keyword writeTISamples: see definition of writeTISamples (biasing and analysis methods)
 Keyword targetForceConstant: see definition of targetForceConstant (Harmonic restraints)
 Keyword targetNumSteps: see definition of targetNumSteps (Harmonic restraints)
 Keyword lambdaExponent: see definition of lambdaExponent (Harmonic restraints)
 Keyword targetEquilSteps: see definition of targetEquilSteps (Harmonic restraints)
 Keyword targetNumStages: see definition of targetNumStages (Harmonic restraints)
 Keyword lambdaSchedule: see definition of lambdaSchedule (Harmonic restraints)
 Keyword outputAccumulatedWork: see definition of outputAccumulatedWork (Harmonic
restraints)
7.9 Adaptive Linear Bias/Experiment Directed Simulation
Experiment directed simulation applies a linear bias with a changing force constant. Please cite White and Voth
[37] 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 7.8 and 7.5 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.
 Keyword name: see definition of name (biasing and analysis methods)
 Keyword colvars: see definition of colvars (biasing and analysis methods)
 Keyword centers $\u27e8\phantom{\rule{0.17em}{0ex}}$Collective
variable centers$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: alb
Acceptable values: spaceseparated 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.
 Keyword updateFrequency $\u27e8\phantom{\rule{0.17em}{0ex}}$The
duration of updates$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 ($N\u22152$),
how long statistics are collected for an iteration ($N\u22152$),
and how quickly energy is added to the system (at most, $A\u22152N$,
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 $N\u22152$,
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.
 Keyword forceRange $\u27e8\phantom{\rule{0.17em}{0ex}}$The
expected range of the force constant in units of energy$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: alb
Acceptable values: A spaceseparated 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
 Keyword rateMax $\u27e8\phantom{\rule{0.17em}{0ex}}$The
maximum rate of change of force constant$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: alb
Acceptable values: A list of spaceseparated 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.
7.10 Multidimensional histograms
The histogram feature is used to record the distribution of a set of collective variables in the form of a
Ndimensional histogram. Defining such a histogram is generally useful for analysis purposes, but it has no effect on
the simulation.
Example 1: the twodimensional histogram of a distance and an angle can be generated using the configuration
below. The histogram code requires that each variable is a scalar number that is confined within a predefined
interval. The interval's boundaries may be specified by hand (e.g. through lowerBoundary and upperBoundary in
the variable definition), or autodetected based on the type of function. In this example, the lower boundary for the
distance variable “r" is automatically set to zero, and interval for the threebody angle “theta" is
${0}^{\circ}$ and
$18{0}^{\circ}$: however,
that an upper boundary for the distance “r" still needs to be specified manually. The grid spacings for the two variables
are $0.2$ length
unitand $3.{0}^{\circ}$,
respectively.
colvar {
name r
width 0.2
upperBoundary 20.0
distance { ... }
}
colvar {
name theta
width 3.0
dihedral { ... }
}
histogram {
name hist2d
colvars r theta
}
Example 2: This example is similar to the previous one, but with the important difference that the parameters for the
histogram's grid are defined explicitly for this histogram instance. Therefore, this histogram's grid may differ from
the one defined from parameters embedded in the colvar { ... } block (for example, narrower intervals and finer
grid spacings may be selected).
colvar {
name r
upperBoundary 20.0
distance { ... }
}
colvar {
name theta
dihedral { ... }
}
histogram {
name hist2d
colvars r theta
histogramGrid {
widths 0.1 1.0
lowerBoundaries 2.0 30.0
upperBoundaries 10.0 90.0
}
}
The standard keywords below are used to control the histogram's computation and to select the variables that are
sampled. See also 7.10.1 for keywords used to define the grid, 7.10.2 for output parameters and 7.10.3 for more
advanced keywords.
 Keyword name: see definition of name (biasing and analysis methods)
 Keyword colvars: see definition of colvars (biasing and analysis methods)
 Keyword stepZeroData: see definition of stepZeroData (biasing and analysis methods)
7.10.1 Defining grids for multidimensional histograms
Grid parameters for the histogram may be provided at the level of the individual variables, or via a dedicated
configuration block histogramGrid { …} inside the configuration of this histogram. The options supported inside
this block are:
 Keyword lowerBoundaries $\u27e8\phantom{\rule{0.17em}{0ex}}$Lower
boundaries of the grid$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: histogramGrid
Acceptable values: list of spaceseparated 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.
 Keyword upperBoundaries: analogous to lowerBoundaries
 Keyword widths: analogous to lowerBoundaries
7.10.2 Output options for multidimensional histograms
The accumulated histogram is written in the Colvars state file, allowing for its accumulation across continued
runs. Additionally, the following files are written depending on the histogram's dimensionality:
 Keyword outputFreq: see definition of outputFreq (biasing and analysis methods)
 Keyword outputFile $\u27e8\phantom{\rule{0.17em}{0ex}}$Write
the histogram to a file$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: histogram
Acceptable values: UNIX filename
Default value: outputName.$<$name$>$.dat
Description: Name of the file containing histogram data (multicolumn format, see 4.7.5), which is
written every outputFreq steps. If outputFile is set to none, the file is not written.
 Keyword outputFileDX $\u27e8\phantom{\rule{0.17em}{0ex}}$Write
the histogram to a file$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: histogram
Acceptable values: UNIX filename
Default value: outputName.$<$name$>$.dx
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. This
file is written by default if the dimension is 3 or more (you cannot visualize it easily for dimensions 4
and above, but the DX format is still more compact than the multicolumn format). If outputFileDX
is set to none, the file is not written.
7.10.3 Histogramming vector variables
 Keyword gatherVectorColvars $\u27e8\phantom{\rule{0.17em}{0ex}}$Treat
vector variables as multiple observations of a scalar
variable?$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: histogram
Acceptable values: UNIX filename
Default value: off
Description: When this is set to on, the components of a multidimensional 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 or iteration of cv update). 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
$({\xi}_{i},{\lambda}_{i},\tau )$
($i=1,\dots \⁡5$)
are accumulated into a 3dimensional histogram.
 Keyword weights $\u27e8\phantom{\rule{0.17em}{0ex}}$Relative
contributions of each vector component to the
histogram$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: histogram
Acceptable values: list of spaceseparated decimals
Default value: all weights equal to 1
Description: When gatherVectorColvars is on, the components of each
multidimensional 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 peratom basis: to compute an electron density map, it is possible to use weights [$sel
get atomicnumber] in the definition of histogram.
As with any other biasing and analysis method, when a histogram is applied to an extendedsystem colvar
(5.21), it accesses the value of the extended coordinate rather than that of the actual colvar. This can be overridden
by enabling the bypassExtendedLagrangian option. A joint histogram of the actual colvar and the extended
coordinate may be collected by specifying the colvar name twice in a row in the colvars parameter (e.g. colvars
myColvar myColvar): the first instance will be understood as the actual colvar, and the second, as the extended
coordinate.
7.11 Probability distributionrestraints
The histogramRestraint bias implements a continuous potential of many variables (or of a single highdimensional
variable) aimed at reproducing a onedimensional statistical distribution that is provided by the user. The
$M$ variables
$({\xi}_{1},\dots \⁡,{\xi}_{M})$ 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 $({\xi}_{1},\dots \⁡,{\xi}_{M})$, is equal
to the reference histogram ${h}_{0}\left(\xi \right)$:
$$V({\xi}_{1},\dots \⁡,{\xi}_{M})=\frac{1}{2}k\int {\left(h\left(\xi \right){h}_{0}\left(\xi \right)\right)}^{2}d\xi $$  (43) 
$$h\left(\xi \right)=\frac{1}{M\sqrt{2\pi {\sigma}^{2}}}\sum _{i=1}^{M}\mathrm{exp}\⁡\left(\frac{{(\xi {\xi}_{i})}^{2}}{2{\sigma}^{2}}\right)$$  (44) 
When used in combination with a distancePairs multidimensional variable, this bias implements the
refinement algorithm against ESR/DEER experiments published by Shen et al [38].
This bias behaves similarly to the histogram bias with the gatherVectorColvars option, with the important
difference that all variables are gathered, resulting in a onedimensional histogram. Future versions will include
support for multidimensional histograms.
The list of options is as follows:
 Keyword name: see definition of name (biasing and analysis methods)
 Keyword colvars: see definition of colvars (biasing and analysis methods)
 Keyword outputEnergy: see definition of outputEnergy (biasing and analysis methods)
 Keyword lowerBoundary $\u27e8\phantom{\rule{0.17em}{0ex}}$Lower
boundary of the colvar grid$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 onedimensional histograms are supported
by the current version.
 Keyword upperBoundary: analogous to lowerBoundary
 Keyword width $\u27e8\phantom{\rule{0.17em}{0ex}}$Width
of the colvar grid$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: histogramRestraint
Acceptable values: positive decimal
Description: Defines the spacing of the grid where the reference distribution ${h}_{0}\left(\xi \right)$
is defined.
 Keyword gaussianSigma $\u27e8\phantom{\rule{0.17em}{0ex}}$Standard
deviation of the approximating Gaussian$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: histogramRestraint
Acceptable values: positive decimal
Default value: 2 $\times $
width
Description: Defines the parameter $\sigma $
in eq. 44.
 Keyword forceConstant $\u27e8\phantom{\rule{0.17em}{0ex}}$Force
constant (energy unit)$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: histogramRestraint
Acceptable values: positive decimal
Default value: 1.0
Description: Defines the parameter $k$
in eq. 43.
 Keyword refHistogram $\u27e8\phantom{\rule{0.17em}{0ex}}$Reference
histogram ${h}_{0}\left(\xi \right)$$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: histogramRestraint
Acceptable values: spaceseparated list of $M$
positive decimals
Description: Provides the values of ${h}_{0}\left(\xi \right)$
consecutively. The midpoint convention is used, i.e. the first point that should be included is for
$\xi $
= lowerBoundary+width/2. If the integral of ${h}_{0}\left(\xi \right)$
is not normalized to 1, ${h}_{0}\left(\xi \right)$
is rescaled automatically before use.
 Keyword refHistogramFile $\u27e8\phantom{\rule{0.17em}{0ex}}$Reference
histogram ${h}_{0}\left(\xi \right)$$\phantom{\rule{0.17em}{0ex}}\u27e9$
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 spaceseparated values of $\xi $
and ${h}_{0}\left(\xi \right)$.
The same numerical conventions as refHistogram are used.
 Keyword writeHistogram $\u27e8\phantom{\rule{0.17em}{0ex}}$Periodically
write the instantaneous histogram $h\left(\xi \right)$$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: metadynamics
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 outputName.$<$name$>$.hist.dat
This is useful to diagnose the convergence of $h\left(\xi \right)$
against ${h}_{0}\left(\xi \right)$.
7.12 Defining scripted biases
Rather than using the biasing methods described above, it is possible to apply biases provided at run time as a
Tcl script. This option, also available in NAMD, can be useful to test a new algorithm to be used in a MD
simulation.
 Keyword scriptedColvarForces $\u27e8\phantom{\rule{0.17em}{0ex}}$Enable
custom, scripted forces on colvars $\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: global
Acceptable values: boolean
Default value: off
Description: If this flag is enabled, a Tcl procedure named calc_colvar_forces accepting one
parameter should be defined by the user. It is executed at each timestep, with the current step number
as parameter, between the calculation of colvars and the application of bias forces. This procedure
may use the cv command to access the values of colvars (e.g. cv colvar xi value), apply forces
on them (cv colvar xi addforce $F) or add energy to the simulation system (cv addenergy $E),
effectively defining custom collective variable biases.
7.13 Performance of scripted biases
If concurrent computation over multiple threads is available (this is indicated by the message “SMP parallelism
is available." printed at initialization time), it is useful to take advantage of the scripting interface to combine many
components, all computed in parallel, into a single variable.
The default SMP schedule is the following:
 distribute the computation of all components across available threads;
 on a single thread, collect the results of multicomponent variables using polynomial combinations
(see 5.16), or custom functions (see 5.17), or scripted functions (see 5.18);
 distribute the computation of all biases across available threads;
 compute on a single thread any scripted biases implemented via the keyword
scriptedColvarForces.
 communicate on a single thread forces to VMD.
The following options allow to finetune this schedule:
 Keyword scriptingAfterBiases $\u27e8\phantom{\rule{0.17em}{0ex}}$Scripted
colvar forces need updated biases?$\phantom{\rule{0.17em}{0ex}}\u27e9$
Context: global
Acceptable values: boolean
Default value: on
Description: This flag specifies that the calc_colvar_forces procedure (last step in the list above)
is executed only after all biases have been updated (nexttolast step) For example, this allows using
the energy of a restraint bias, or the force applied on a colvar, to calculate additional scripted forces,
such as boundary constraints. When this flag is set to off, it is assumed that only the values of the
variables (but not the energy of the biases or applied forces) will be used by calc_colvar_forces:
this can be used to schedule the calculation of scripted forces and biases concurrently to increase
performance.
8 Scripting interface (Tcl): list of commands
This section lists all the commands used in VMD to control the behavior of the Colvars module from within a run
script.
8.1 Commands to manage the Colvars module
 cv addenergy
Add an energy to the MD engine (no effect in VMD)
Parameters
E : float  Amount of energy to add
 cv config
Read configuration from the given string
Parameters
conf : string  Configuration string
 cv configfile
Read configuration from a file
Parameters
conf_file : string  Path to configuration file
 cv delete
Delete this Colvars module instance (VMD only)
 cv featurereport
Return a summary of Colvars features used so far and their citations
Returns

report : string  Feature report and citations
 cv frame [frame]
Get or set current frame number (VMD only)
Returns

frame : integer  Frame number
 cv getatomappliedforces
Get the list of forces applied by Colvars to atoms
Returns

forces : array of arrays of floats  Atomic forces
 cv getatomappliedforcesmax
Get the maximum norm of forces applied by Colvars to atoms
Returns

force : float  Maximum atomic force
 cv getatomappliedforcesmaxid
Get the atom ID with the largest applied force
Returns

id : int  ID of the atom with the maximum atomic force
 cv getatomappliedforcesrms
Get the rootmeansquare norm of forces applied by Colvars to atoms
Returns

force : float  RMS atomic force
 cv resetatomappliedforces
Reset forces applied by Colvars to atoms
 cv getatomids
Get the list of indices of atoms used in Colvars
Returns

indices : array of ints  Atom indices
 cv getatomcharges
Get the list of charges of atoms used in Colvars
Returns

charges : array of floats  Atomic charges
 cv getatommasses
Get the list of masses of atoms used in Colvars
Returns

masses : array of floats  Atomic masses
 cv getatompositions
Get the list of cached positions of atoms used in Colvars
Returns

positions : array of arrays of floats  Atomic positions
 cv getatomtotalforces
Get the list of cached total forces of atoms used in Colvars
Returns

forces : array of arrays of floats  Atomic total foces
 cv getconfig
Get the module's configuration string read so far
Returns

conf : string  Current configuration string
 cv getenergy
Get the current Colvars energy
Returns

E : float  Amount of energy (internal units)
 cv getstepabsolute
Get the current step number of the simulation (including restarts)
Returns

step : int  Absolute step number
 cv getsteprelative
Get the current step number from the start of this job
Returns

step : int  Relative step number
 cv help [command]
Get the help string of the Colvars scripting interface
Returns

help : string  Help string
 cv languageversion
Get the C++ language version number
Returns

version : string  C++ language version
 cv list [param]
Return a list of all variables or biases
Returns

list : sequence of strings  List of elements
 cv listcommands
Get the list of script functions, prefixed with "cv_", "colvar_" or "bias_"
Returns

list : sequence of strings  List of commands
 cv listindexfiles
Get a list of the index files loaded in this session
 cv load
Load data from a state file into all matching colvars and biases
Parameters
prefix : string  Path to existing state file or input prefix
 cv loadfromstring
Load state data from a string into all matching colvars and biases
Parameters
buffer : string  String buffer containing the state information
 cv molid [molid]
Get or set the molecule ID on which Colvars is defined (VMD only)
Returns

molid : integer  Current molecule ID
 cv printframe
Return the values that would be written to colvars.traj
Returns

values : string  The values
 cv printframelabels
Return the labels that would be written to colvars.traj
Returns

Labels : string  The labels
 cv reset
Delete all internal configuration
 cv resetindexgroups
Clear the index groups loaded so far, allowing to replace them
 cv save
Change the prefix of all output files and save them
Parameters
prefix : string  Output prefix with trailing ".colvars.state" gets removed)
 cv savetostring
Write the Colvars state to a string and return it
Returns

state : string  The saved state
 cv units [units]
Get or set the current Colvars unit system
Returns

units : string  The current unit system
 cv update
Recalculate colvars and biases
 cv version
Get the Colvars Module version string
Returns

version : string  Colvars version
8.2 Commands to manage individual collective variables
 cv colvar name addforce
Apply the given force onto this colvar and return the same
Parameters
force : float or array  Applied force; must match colvar dimensionality
Returns

force : float or array  Applied force; matches colvar dimensionality
 cv colvar name communicateforces
Communicate bias forces from this colvar to atoms
 cv colvar name cvcflags
Enable or disable individual components by setting their active flags
Parameters
flags : integer array  Zero/nonzero value disables/enables the CVC
 cv colvar name delete
Delete this colvar, along with all biases that depend on it
 cv colvar name get
Get the value of the given feature for this colvar
Parameters
feature : string  Name of the feature
Returns

state : 1/0  State of the given feature
 cv colvar name getappliedforce
Return the total of the forces applied to this colvar
Returns

force : float  Applied force; matches the colvar dimensionality
 cv colvar name resetbiasforce
Return the total of the forces applied to this colvar
 cv colvar name getatomgroups
Return the atom indices used by this colvar as a list of lists
Returns

groups : array of arrays of ints  Atom indices
 cv colvar name getatomids
Return the list of atom indices used by this colvar
Returns

indices : array of ints  Atom indices
 cv colvar name getconfig
Return the configuration string of this colvar
Returns

conf : string  Current configuration string
 cv colvar name getgradients
Return the atomic gradients of this colvar
Returns

gradients : array of arrays of floats  Atomic gradients
 cv colvar name gettotalforce
Return the sum of internal and external forces to this colvar
Returns

force : float  Total force; matches the colvar dimensionality
 cv colvar name getvolmapids
Return the list of volumetric map indices used by this colvar
 cv colvar name help [command]
Get a help summary or the help string of one colvar subcommand
Returns

help : string  Help string
 cv colvar name modifycvcs
Modify configuration of individual components by passing string arguments
Parameters
confs : sequence of strings  New configurations; empty strings are skipped
 cv colvar name run_ave
Get the current running average of the value of this colvar
Returns

value : float or array  Averaged value; matches the colvar dimensionality
 cv colvar name set
Set the given feature of this colvar to a new value
Parameters
feature : string  Name of the feature
value : string  String representation of the new feature value
 cv colvar name state
Print a string representation of the feature state of this colvar
Returns

state : string  The feature state
 cv colvar name type
Get the type description of this colvar
Returns

type : string  Type description
 cv colvar name update
Recompute this colvar and return its uptodate value
Returns

value : float or array  Current value; matches the colvar dimensionality
 cv colvar name value
Get the current value of this colvar
Returns

value : float or array  Current value; matches the colvar dimensionality
 cv colvar name width
Get the width of this colvar
Returns

width : float  Value of the width
8.3 Commands to manage individual biases
 cv bias name bin
Get the current grid bin index (1D ABF only for now)
Returns

bin : integer  Bin index
 cv bias name bincount [index]
Get the number of samples at the given grid bin (1D ABF only for now)
Returns

samples : integer  Number of samples
 cv bias name binnum
Get the total number of grid points of this bias (1D ABF only for now)
Returns

Bins : integer  Number of grid points
 cv bias name delete
Delete this bias
 cv bias name energy
Get the current energy of this bias
Returns

E : float  Energy value
 cv bias name get
Get the value of the given feature for this bias
Parameters
feature : string  Name of the feature
Returns

state : 1/0  State of the given feature
 cv bias name getconfig
Return the configuration string of this bias
Returns

conf : string  Current configuration string
 cv bias name help [command]
Get a help summary or the help string of one bias subcommand
Returns

help : string  Help string
 cv bias name load
Load data into this bias
Parameters
prefix : string  Read from a file with this name or prefix
 cv bias name loadfromstring
Load state data into this bias from a string
Parameters
buffer : string  String buffer containing the state information
 cv bias name save
Save data from this bias into a file with the given prefix
Parameters
prefix : string  Prefix for the state file of this bias
 cv bias name savetostring
Save data from this bias into a string and return it
Returns

state : string  The bias state
 cv bias name set
Set the given feature of this bias to a new value
Parameters
feature : string  Name of the feature
value : string  String representation of the new feature value
 cv bias name share
Share bias information with other replicas (multiplewalker scheme)
 cv bias name state
Print a string representation of the feature state of this bias
Returns

state : string  String representation of the bias features
 cv bias name type
Print the type of this bias object
Returns

type : string  Type of this bias object (e.g. metadynamics)
 cv bias name update
Recompute this bias and return its uptodate energy
Returns

E : float  Energy value
9 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 20160609 or later (VMD version 1.9.3 or later).
The legacy keyword refPositionsGroup has been renamed fittingGroup for clarity (the legacy
version is still supported).
 Colvars version 20160810 or later (VMD version 1.9.3 or later).
“System forces" have been replaced by “total forces" (see for example outputTotalForce). See the
following page for more information:
https://colvars.github.io/READMEtotalforce.html
 Colvars version 20170109 or later (VMD version 1.9.4 or later).
A new type of restraint, harmonicWalls (see 7.7), replaces and improves upon the legacy keywords
lowerWall and upperWall: these are still supported as shorthands.
 Colvars version 20181115 or later (VMD version 1.9.4 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.
 Colvars version 20200225 or later (VMD version 1.9.4 or later).
The parameter hillWidth, expressing the Gaussian width $2\sigma $
in relative units (number of grid points), does not have a default value any more. A new alternative
parameter gaussianSigmas allows setting the $\sigma $
parameters explicitly for each variable if needed.
Furthermore, to facilitate the use of other analysis tools such as for example sum_hills: