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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.

To make the calculation of coordination numbers differentiable the following function is used:

\[ s = \frac{ 1 - \left(\frac{r-d_0}{r_0}\right)^n } { 1 - \left(\frac{r-d_0}{r_0}\right)^m } \]

Description of components

This Action can be used to calculate the following quantities by employing the keywords listed below. You must select which from amongst these quantities you wish to calculate - this command cannot be run unless one of the quantities below is being calculated.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.less_than-1, label.less_than-2 etc.

Quantity Keyword Description
more_than 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.
min MIN the minimum value. This is calculated using the formula described in the description of the keyword so as to make it continuous.
mean MEAN the mean value. The output component can be refererred to elsewhere in the input file by using the label.mean
less_than 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.
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.
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.
The atoms involved can be specified using
SPECIES this keyword is used for colvars such as coordination number. In that context it specifies that plumed should calculate one coordination number for each of the atoms specified. Each of these coordination numbers specifies how many of the other specified atoms are within a certain cutoff of the central atom.
Or alternatively by using
SPECIESA this keyword is used for colvars such as the coordination number. In that context it species that plumed should calculate one coordination number for each of the atoms specified in SPECIESA. Each of these cooordination numbers specifies how many of the atoms specifies using SPECIESB is within the specified cutoff
SPECIESB this keyword is used for colvars such as the coordination number. It must appear with SPECIESA. For a full explanation see the documentation for that keyword
Compulsory keywords
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
NUMERICAL_DERIVATIVES ( default=off ) calculate the derivatives for these quantities numerically
NOPBC ( default=off ) ignore the periodic boundary conditions when calculating distances
SERIAL ( default=off ) do the calculation in serial. Do not parallelize
VERBOSE ( default=off ) write a more detailed output
MEAN

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

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...
MIN calculate the minimum value. To make this quantity continuous the minimum is calculated using \( \textrm{min} = \frac{\beta}{ \log \sum_i \exp\left( \frac{\beta}{s_i} \right) } \) The value of \(\beta\) in this function is specified using (BETA= \(\beta\)) The final value can be referenced using label.min.
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

The following input tells plumed to calculate the coordination numbers of atoms 1-100 with themselves. The minimum coordination number is then calculated.

COORDINATIONNUMBER SPECIES=1-100 R_0=1.0 MIN={BETA=0.1}

The following input tells plumed to calculate how many atoms from 1-100 are within 3.0 of each of the atoms from 101-110. In the first 101 is the central atom, in the second 102 is the central atom and so on. The number of coordination numbers more than 6 is then computed.

COORDINATIONNUMBER SPECIESA=101-110 SPECIESB=1-100 R_0=3.0 MORE_THAN={RATIONAL R_0=6.0 NN=6 MM=12 D_0=0}