MultiColvar

Oftentimes, when you do not need one of the collective variables described elsewhere in the manual, what you want instead is a function of a distribution of collective variables of a particular type. In other words, you would like to calculate a function something like this:

\[ s = \sum_i g[f(\{X\}_i)] \]

In this expression \(g\) is a function that takes in one argument and \(f\) is a function that takes a set of atomic positions as argument. The symbol \(\{X\}_i\) is used to indicate the fact that the function \(f\) is evaluated for a number of different sets of atoms. If you would just like to output the values of all the various \(f\) functions you should use the command DUMPMULTICOLVAR

This functionality is useful if you need to calculate a minimum distance or the number of coordination numbers greater than a 3.0.

To avoid duplicating the code to calculate an angle or distance many times and to make it easier to implement very complex collective variables PLUMED provides these sort of collective variables using so-called MultiColvars. MultiColvars are named in this way because a single PLUMED action can be used to calculate a number of different collective variables. For instance the DISTANCES action can be used to calculate the minimum distance, the number of distances less than a certain value, the number of distances within a certain range... A more detailed introduction to multicolvars is provided in this 10-minute video. Descriptions of the various multicolvars that are implemented in PLUMED 2 are given below:

ANGLES | Calculate functions of the distribution of angles . |

BOND_DIRECTIONS | Calculate the vectors connecting atoms that are within cutoff defined using a switching function. |

BRIDGE | Calculate the number of atoms that bridge two parts of a structure |

COORDINATIONNUMBER | Calculate the coordination numbers of atoms so that you can then calculate functions of the distribution of coordination numbers such as the minimum, the number less than a certain quantity and so on. |

DENSITY | Calculate functions of the density of atoms as a function of the box. This allows one to calculate the number of atoms in half the box. |

DISTANCES | Calculate the distances between one or many pairs of atoms. You can then calculate functions of the distribution of distances such as the minimum, the number less than a certain quantity and so on. |

FCCUBIC | Measure how similar the environment around atoms is to that found in a FCC structure. |

HBPAMM_SH | Number of HBPAMM hydrogen bonds formed by each hydrogen atom in the system |

INPLANEDISTANCES | Calculate distances in the plane perpendicular to an axis |

MOLECULES | Calculate the vectors connecting a pair of atoms in order to represent the orientation of a molecule. |

PLANES | Calculate the plane perpendicular to two vectors in order to represent the orientation of a planar molecule. |

Q3 | Calculate 3rd order Steinhardt parameters. |

Q4 | Calculate fourth order Steinhardt parameters. |

Q6 | Calculate sixth order Steinhardt parameters. |

SIMPLECUBIC | Calculate whether or not the coordination spheres of atoms are arranged as they would be in a simple cubic structure. |

TETRAHEDRAL | Calculate the degree to which the environment about ions has a tetrahedral order. |

TORSIONS | Calculate whether or not a set of torsional angles are within a particular range. |

XANGLES | Calculate the angles between the vector connecting two atoms and the x axis. |

XDISTANCES | Calculate the x components of the vectors connecting one or many pairs of atoms.You can then calculate functions of the distribution of values such as the minimum, the number less than a certain quantity and so on. |

XYDISTANCES | Calculate distance between a pair of atoms neglecting the z-component. |

XYTORSIONS | Calculate the torsional angle around the x axis from the positive y direction. |

XZDISTANCES | Calculate distance between a pair of atoms neglecting the y-component. |

XZTORSIONS | Calculate the torsional angle around the x axis from the positive z direction. |

YANGLES | Calculate the angles between the vector connecting two atoms and the y axis. |

YDISTANCES | Calculate the y components of the vectors connecting one or many pairs of atoms.You can then calculate functions of the distribution of values such as the minimum, the number less than a certain quantity and so on. |

YXTORSIONS | Calculate the torsional angle around the y axis from the positive x direction. |

YZDISTANCES | Calculate distance between a pair of atoms neglecting the x-component. |

YZTORSIONS | Calculate the torsional angle around the y axis from the positive z direction. |

ZANGLES | Calculate the angles between the vector connecting two atoms and the z axis. |

ZDISTANCES | Calculate the z components of the vectors connecting one or many pairs of atoms.You can then calculate functions of the distribution of values such as the minimum, the number less than a certain quantity and so on. |

ZXTORSIONS | Calculate the torsional angle around the z axis from the positive x direction. |

ZYTORSIONS | Calculate the torsional angle around the z axis from the positive y direction. |

To instruct PLUMED to calculate a multicolvar you give an instruction that looks something like this:

NAME <atoms involved> <parameters> <what am I calculating> TOL=0.001 LABEL=label

Oftentimes the simplest way to specify the atoms involved is to use multiple instances of the ATOMS keyword i.e. ATOMS1, ATOMS2, ATOMS3,... Separate instances of the quantity specified by NAME are then calculated for each of the sets of atoms. For example if the command issued contains the following:

DISTANCES ATOMS1=1,2 ATOMS2=3,4 ATOMS3=5,6

The distances between atoms 1 and 2, atoms 3 and 4, and atoms 5 and 6 are calculated. Obviously, generating this sort of input is rather tedious so short cuts are also available many of the collective variables. These are described on the manual pages for the actions.

After specifying the atoms involved you sometimes need to specify some parameters that required in the calculation. For instance, for COORDINATIONNUMBER - the number of atoms in the first coordination sphere of each of the atoms in the system - you need to specify the parameters for a switchingfunction that will tell us whether or not an atom is in the first coordination sphere. Details as to how to do this are provided on the manual pages.

One of the most important keywords for multicolvars is the TOL keyword. This specifies that terms in sums that contribute less than a certain value can be ignored. In addition, it is assumed that the derivative with respect to these terms are essentially zero. By increasing the TOL parameter you can increase the speed of the calculation. Be aware, however, that this increase in speed is only possible because you are lowering the accuracy with which you are computing the quantity of interest.

Once you have specified the base quantities that are to be calculated from the atoms involved and any parameters you need to specify what function of these base quantities is to be calculated. For most multicolvars you can calculate the minimum, the number less than a target value, the number within a certain range, the number more than a target value and the average value directly.

It is possible to use multicolvars to calculate complicated collective variables by exploiting the fact that the output from one multicolvar can be used as input to a second multicolvar. One simple way of exploiting this functionality is to filter the atoms based on the value they have for a symmetry function. For example you might want to consider only those atoms that with a COORDINATIONNUMBER higher that a certain threshold when calculating some particularly expensive symmetry function such as Q6. The following methods can thus all be used to filter the values of multicolvars in this way:

MFILTER_BETWEEN | This action can be used to filter the colvar values calculated by a mcolvso that one can compute the mean and so on for only those multicolvars within a certain range. |

MFILTER_LESS | This action can be used to filter the distribution of colvar values in a mcolvso that one can compute the mean and so on for only those multicolvars less than a tolerance. |

MFILTER_MORE | This action can be used to filter the distribution of colvar values in a mcolvso that one can compute the mean and so on for only those multicolvars more than a tolerance. |

An alternative way of filtering atoms is to consider only those atoms in a particular part of the simulation box. This can be done by exploiting the following methods

AROUND | This quantity can be used to calculate functions of the distribution of collective variables for the atoms that lie in a particular, user-specified part of of the cell. |

CAVITY | This quantity can be used to calculate functions of the distribution of collective variables for the atoms that lie in a box defined by the positions of four atoms. |

INCYLINDER | This quantity can be used to calculate functions of the distribution of collective variables for the atoms that lie in a particular, user-specified part of of the cell. |

INENVELOPE | This quantity can be used to calculate functions of the distribution of collective variables for the atoms that lie in a region where the density of a certain type of atom is high. |

INSPHERE | This quantity can be used to calculate functions of the distribution of collective variables for the atoms that lie in a particular, user-specified part of of the cell. |

TETRAHEDRALPORE | This quantity can be used to calculate functions of the distribution of collective variables for the atoms lie that lie in a box defined by the positions of four atoms at the corners of a tetrahedron. |

The idea with these methods is that function of the form:

\[ s = \sum_i w(\{X\}_i) g[f(\{X\}_i)] \]

can be evaluated where once again \(g\) is a function with one argument and \(g\) is a function of a set of atomic positions.

The difference from the more general function described earlier is that we now have a weight \(w\) which is again a function of the atomic positions. This weight varies between zero and one and it is this weight that is calculated in the list of filtering methods and volume methods described in the lists above.

In addition to these volume and filtering methods it is also possible to calculate the local average of a quantities in the manner described in [61] using the LOCAL_AVERAGE method. Furthermore, in many cases Q6, MOLECULES and PLANES the symmetry function being evaluated is a vector. You can thus construct a variety of novel collective variables by taking dot products of vectors on adjacent atoms as described below:

GRADIENT | Calculate the gradient of the average value of a multicolvar value |

INTERMOLECULARTORSIONS | Calculate torsion angles between vectors on adjacent molecules |

LOCAL_AVERAGE | Calculate averages over spherical regions centered on atoms |

LOCAL_Q3 | Calculate the local degree of order around an atoms by taking the average dot product between the \(q_3\) vector on the central atom and the \(q_3\) vector on the atoms in the first coordination sphere. |

LOCAL_Q4 | Calculate the local degree of order around an atoms by taking the average dot product between the \(q_4\) vector on the central atom and the \(q_4\) vector on the atoms in the first coordination sphere. |

LOCAL_Q6 | Calculate the local degree of order around an atoms by taking the average dot product between the \(q_6\) vector on the central atom and the \(q_6\) vector on the atoms in the first coordination sphere. |

MCOLV_COMBINE | Calculate linear combinations of multiple multicolvars |

MCOLV_PRODUCT | Calculate a product of multiple multicolvars |

NLINKS | Calculate number of pairs of atoms/molecules that are "linked" |

PAMM | Probabilistic analysis of molecular motifs. |

POLYMER_ANGLES | Calculate a function to investigate the relative orientations of polymer angles |

SMAC | Calculate a variant on the SMAC collective variable discussed in [47] |

The final set of functions that you can apply on multicolvars are functions that transform all the colvars calculated using a multicolvar using a function. This can be useful if you are calculating some complicated derived quantity of some simpler quantity. It is also useful if you are calculating a Willard Chandler surface or a histogram. The actions that you can use to perform these transforms are:

MTRANSFORM_BETWEEN | This action can be used to transform the colvar values calculated by a MultiColvar using a histogrambead |

MTRANSFORM_LESS | This action can be used to transform the colvar values calculated by a MultiColvar using a switchingfunction |

MTRANSFORM_MORE | This action can be used to transform the colvar values calculated by a MultiColvar using one minus a switchingfunction |

There may be occasions when you want add restraints on many collective variables. For instance if you are studying a cluster you might want to add a wall on the distances between each of the atoms and the center of mass of the cluster in order to prevent the cluster subliming. Alternatively, you may wish to insist that a particular set of atoms in your system all have a coordination number greater than 2. You can add these sorts of restraints by employing the following biases, which all act on the set of collective variable values calculated by a multicolvar. So for example the following set of commands:

COM ATOMS=1-20 LABEL=c1 DISTANCES GROUPA=c1 GROUPB=1-20 LABEL=d1 UWALLS DATA=d1 AT=2.5 KAPPA=0.2 LABEL=sr

creates the aforementioned set of restraints on the distances between the 20 atoms in a cluster and the center of mass of the cluster.

The list of biases of this type are as follows:

LWALLS | Add LOWER_WALLS restraints on all the multicolvar values |

UWALLS | Add UPPER_WALLS restraints on all the multicolvar values |

Notice that (in theory) you could also use this functionality to add additional terms to your force field or to implement your force field.

There may be occasions where you want to get information on all the individual colvar values that you have calculated. For example you might want to output the values of all the coordination numbers calculated by a COORDINATIONNUMBER action. You can thus use the following command to extract this sort of information, DUMPMULTICOLVAR.