LOCAL_AVERAGE

This is part of the multicolvar module |

Calculate averages over spherical regions centered on atoms

As is explained in this video certain multicolvars calculate one scalar quantity or one vector for each of the atoms in the system. For example COORDINATIONNUMBER measures the coordination number of each of the atoms in the system and Q4 measures the 4th order Steinhardt parameter for each of the atoms in the system. These quantities provide tell us something about the disposition of the atoms in the first coordination sphere of each of the atoms of interest. Lechner and Dellago [28] have suggested that one can probe local order in a system by taking the average value of such symmetry functions over the atoms within a spherical cutoff of each of these atoms in the systems. When this is done with Steinhardt parameters they claim this gives a coordinate that is better able to distinguish solid and liquid configurations of Lennard-Jones atoms.

You can calculate such locally averaged quantities within plumed by using the LOCAL_AVERAGE command. This command calculates the following atom-centered quantities:

\[ s_i = \frac{ c_i + \sum_j \sigma(r_{ij})c_j }{ 1 + \sum_j \sigma(r_{ij}) } \]

where the \(c_i\) and \(c_j\) values can be for any one of the symmetry functions that can be calculated using plumed multicolvars. The function \(\sigma( r_{ij} )\) is a switchingfunction that acts on the distance between atoms \(i\) and \(j\). Lechner and Dellago suggest that the parameters of this function should be set so that it the function is equal to one when atom \(j\) is in the first coordination sphere of atom \(i\) and is zero otherwise.

The \(s_i\) quantities calculated using the above command can be again thought of as atom-centred symmetry functions. They thus operate much like multicolvars. You can thus calculate properties of the distribution of \(s_i\) values using MEAN, LESS_THAN, HISTOGRAM and so on. You can also probe the value of these averaged variables in regions of the box by using the command in tandem with the AROUND command.

- Description of components

When the label of this action is used as the input for a second you are not referring to a scalar quantity as you are in regular collective variables. The label is used to reference the full set of quantities calculated by the action. This is usual when using MultiColvar functions. Generally when doing this the previously calculated multicolvar will be referenced using the DATA keyword rather than ARG.

This Action can be used to calculate the following scalar quantities directly. These quantities are calculated by employing the keywords listed below. These quantities can then be referenced elsewhere in the input file by using this Action's label followed by a dot and the name of the quantity. Some amongst them can be calculated multiple times with different parameters. In this case the quantities calculated can be referenced elsewhere in the input by using the name of the quantity followed by a numerical identifier e.g. *label*.lessthan-1, *label*.lessthan-2 etc. When doing this and, for clarity we have made the label of the components customizable. As such by using the LABEL keyword in the description of the keyword input you can customize the component name

Quantity | Keyword | Description |

between | BETWEEN | the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters. |

lessthan | LESS_THAN | the number of values less than a target value. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters. |

mean | MEAN | the mean value. The output component can be refererred to elsewhere in the input file by using the label.mean |

moment | MOMENTS | the central moments of the distribution of values. The second moment would be referenced elsewhere in the input file using label.moment-2, the third as label.moment-3, etc. |

morethan | MORE_THAN | the number of values more than a target value. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters. |

- Compulsory keywords

DATA | the labels of the action that calculates the multicolvars we are interested in |

NN | ( default=6 ) The n parameter of the switching function |

MM | ( default=12 ) The m parameter of the switching function |

D_0 | ( default=0.0 ) The d_0 parameter of the switching function |

R_0 | The r_0 parameter of the switching function |

- Options

NOPBC | ( default=off ) ignore the periodic boundary conditions when calculating distances |

SERIAL | ( default=off ) do the calculation in serial. Do not parallelize |

MEAN | ( default=off ) take the mean of these variables. The final value can be referenced using label.mean |

LOWMEM | ( default=off ) lower the memory requirements |

TOL | this keyword can be used to speed up your calculation. When accumulating sums in which the individual terms are numbers inbetween zero and one it is assumed that terms less than a certain tolerance make only a small contribution to the sum. They can thus be safely ignored as can the the derivatives wrt these small quantities. |

SWITCH | This keyword is used if you want to employ an alternative to the continuous swiching function defined above. The following provides information on the switchingfunction that are available. When this keyword is present you no longer need the NN, MM, D_0 and R_0 keywords. |

MORE_THAN | calculate the number of variables more than a certain target value. This quantity is calculated using \(\sum_i 1.0 - \sigma(s_i)\), where \(\sigma(s)\) is a switchingfunction. The final value can be referenced using label.more_than. You can use multiple instances of this keyword i.e. MORE_THAN1, MORE_THAN2, MORE_THAN3... The corresponding values are then referenced using label.more_than-1, label.more_than-2, label.more_than-3... |

LESS_THAN | calculate the number of variables less than a certain target value. This quantity is calculated using \(\sum_i \sigma(s_i)\), where \(\sigma(s)\) is a switchingfunction. The final value can be referenced using label.less_than. You can use multiple instances of this keyword i.e. LESS_THAN1, LESS_THAN2, LESS_THAN3... The corresponding values are then referenced using label.less_than-1, label.less_than-2, label.less_than-3... |

BETWEEN | calculate the number of values that are within a certain range. These quantities are calculated using kernel density estimation as described on histogrambead. The final value can be referenced using label.between. You can use multiple instances of this keyword i.e. BETWEEN1, BETWEEN2, BETWEEN3... The corresponding values are then referenced using label.between-1, label.between-2, label.between-3... |

HISTOGRAM | calculate a discretized histogram of the distribution of values. This shortcut allows you to calculates NBIN quantites like BETWEEN. |

MOMENTS | calculate the moments of the distribution of collective variables. The \(m\)th moment of a distribution is calculated using \(\frac{1}{N} \sum_{i=1}^N ( s_i - \overline{s} )^m \), where \(\overline{s}\) is the average for the distribution. The moments keyword takes a lists of integers as input or a range. Each integer is a value of \(m\). The final calculated values can be referenced using moment- \(m\). |

- Examples

This example input calculates the coordination numbers for all the atoms in the system. These coordination numbers are then averaged over spherical regions. The number of averaged coordination numbers that are greater than 4 is then output to a file.

COORDINATIONNUMBER SPECIES=1-64 D_0=1.3 R_0=0.2 LABEL=d1 LOCAL_AVERAGE ARG=d1 SWITCH={RATIONAL D_0=1.3 R_0=0.2} MORE_THAN={RATIONAL R_0=4} LABEL=la PRINT ARG=la.* FILE=colvar

This example input calculates the \(q_4\) (see Q4) vectors for each of the atoms in the system. These vectors are then averaged component by component over a spherical region. The average value for this quantity is then outputeed to a file. This calculates the quantities that were used in the paper by Lechner and Dellago [28]

Q4 SPECIES=1-64 SWITCH={RATIONAL D_0=1.3 R_0=0.2} LABEL=q4 LOCAL_AVERAGE ARG=q4 SWITCH={RATIONAL D_0=1.3 R_0=0.2} MEAN LABEL=la PRINT ARG=la.* FILE=colvar