TETRAHEDRALPORE
 This is part of the multicolvar module

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.

Each of the base quantities calculated by a multicolvar can can be assigned to a particular point in three dimensional space. For example, if we have the coordination numbers for all the atoms in the system each coordination number can be assumed to lie on the position of the central atom. Because each base quantity can be assigned to a particular point in space we can calculate functions of the distribution of base quantities in a particular part of the box by using:

$\overline{s}_{\tau} = \frac{ \sum_i f(s_i) w(u_i,v_i,w_i) }{ \sum_i w(u_i,v_i,w_i) }$

where the sum is over the collective variables, $$s_i$$, each of which can be thought to be at $$(u_i,v_i,z_i)$$. The function $$(s_i)$$ can be any of the usual LESS_THAN, MORE_THAN, WITHIN etc that are used in all other multicolvars. Notice that here (at variance with what is done in AROUND) we have transformed from the usual $$(x_i,y_i,z_i)$$ position to a position in $$(u_i,v_i,z_i)$$. This is done using a rotation matrix as follows:

$\left( \begin{matrix} u_i \\ v_i \\ w_i \end{matrix} \right) = \mathbf{R} \left( \begin{matrix} x_i - x_o \\ y_i - y_o \\ z_i - z_o \end{matrix} \right)$

where $$\mathbf{R}$$ is a rotation matrix that is calculated by constructing a set of three orthonormal vectors from the reference positions specified by the user. Initially unit vectors are found by calculating the bisector, $$\mathbf{b}$$, and cross product, $$\mathbf{c}$$, of the vectors connecting atoms 1 and 2. A third unit vector, $$\mathbf{p}$$ is then found by taking the cross product between the cross product calculated during the first step, $$\mathbf{c}$$ and the bisector, $$\mathbf{b}$$. From this second cross product $$\mathbf{p}$$ and the bisector $$\mathbf{b}$$ two new vectors are calculated using:

$v_1 = \cos\left(\frac{\pi}{4}\right)\mathbf{b} + \sin\left(\frac{\pi}{4}\right)\mathbf{p} \qquad \textrm{and} \qquad v_2 = \cos\left(\frac{\pi}{4}\right)\mathbf{b} - \sin\left(\frac{\pi}{4}\right)\mathbf{p}$

In the previous function $$w(u_i,v_i,w_i)$$ measures whether or not the system is in the subregion of interest. It is equal to:

$w(u_i,v_i,w_i) = \int_{0}^{u'} \int_{0}^{v'} \int_{0}^{w'} \textrm{d}u\textrm{d}v\textrm{d}w K\left( \frac{u - u_i}{\sigma} \right)K\left( \frac{v - v_i}{\sigma} \right)K\left( \frac{w - w_i}{\sigma} \right)$

where $$K$$ is one of the kernel functions described on histogrambead and $$\sigma$$ is a bandwidth parameter. The values of $$u'$$ and $$v'$$ are found by finding the projections of the vectors connecting atoms 1 and 2 and 1 and 3 $$v_1$$ and $$v_2$$. This gives four projections: the largest two projections are used in the remainder of the calculations. $$w'$$ is calculated by taking the projection of the vector connecting atoms 1 and 4 on the vector $$\mathbf{c}$$. Notice that the manner by which this box is constructed differs from the way this is done in CAVITY. This is in fact the only point of difference between these two actions.

Examples

The following commands tell plumed to calculate the number of atom inside a tetrahedral cavity. The extent of the tetrahedral cavity is calculated from the positions of atoms 1, 4, 5, and 11, The final value will be labeled cav.

Click on the labels of the actions for more information on what each action computes
d1: DENSITY SPECIESthis keyword is used for colvars such as coordination number. =20-500
cav: TETRAHEDRALPORE DATAcompulsory keyword
The multicolvar that calculates the set of base quantities that we are interested
in =d1 ATOMSthe positions of four atoms that define spatial extent of the cavity. =1,4,5,11 SIGMAcompulsory keyword
the width of the function to be used for kernel density estimation =0.1


The following command tells plumed to calculate the coordination numbers (with other water molecules) for the water molecules in the tetrahedral cavity described above. The average coordination number and the number of coordination numbers more than 4 is then calculated. The values of these two quantities are given the labels cav.mean and cav.morethan

Click on the labels of the actions for more information on what each action computes
d1: COORDINATIONNUMBER SPECIESthis keyword is used for colvars such as coordination number. =20-500 R_0 could not find this keyword =0.1
cav: CAVITY DATAcompulsory keyword
The multicolvar that calculates the set of base quantities that we are interested
in =d1 ATOMSthe positions of four atoms that define spatial extent of the cavity. =1,4,5,11 SIGMAcompulsory keyword
the width of the function to be used for kernel density estimation =0.1 MEANtake the mean of these variables.  MORE_THANcalculate the number of variables more than a certain target value. ={RATIONAL R_0=4}

Glossary of keywords and components
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 of 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 it so that the user can set a particular label for each of the components. As such by using the LABEL keyword in the description of the keyword input you can customize the component name

 Quantity Keyword Description vmean VMEAN the norm of the mean vector. The output component can be referred to elsewhere in the input file by using the label.vmean 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 referred to elsewhere in the input file by using the label.mean 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. sum SUM the sum of values
The atoms involved can be specified using
 ATOMS the positions of four atoms that define spatial extent of the cavity. For more information on how to specify lists of atoms see Groups and Virtual Atoms
Compulsory keywords
 DATA The multicolvar that calculates the set of base quantities that we are interested in SIGMA the width of the function to be used for kernel density estimation KERNEL ( default=gaussian ) the type of kernel function to be used
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 use MPI LOWMEM ( default=off ) lower the memory requirements TIMINGS ( default=off ) output information on the timings of the various parts of the calculation OUTSIDE ( default=off ) calculate quantities for colvars that are on atoms outside the region of interest PRINT_BOX ( default=off ) write out the positions of the corners of the box to an xyz file VMEAN calculate the norm of the mean vector. The final value can be referenced using label.vmean. You can use multiple instances of this keyword i.e. VMEAN1, VMEAN2, VMEAN3... The corresponding values are then referenced using label.vmean-1, label.vmean-2, label.vmean-3... MEAN take the mean of these variables. The final value can be referenced using label.mean. You can use multiple instances of this keyword i.e. MEAN1, MEAN2, MEAN3... The corresponding values are then referenced using label.mean-1, label.mean-2, label.mean-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.lessthan. You can use multiple instances of this keyword i.e. LESS_THAN1, LESS_THAN2, LESS_THAN3... The corresponding values are then referenced using label.lessthan-1, label.lessthan-2, label.lessthan-3... 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.morethan. You can use multiple instances of this keyword i.e. MORE_THAN1, MORE_THAN2, MORE_THAN3... The corresponding values are then referenced using label.morethan-1, label.morethan-2, label.morethan-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 how many of the values fall in each of the bins of a histogram. This shortcut allows you to calculates NBIN quantities like BETWEEN. The final value can be referenced using label.histogram. You can use multiple instances of this keyword i.e. HISTOGRAM1, HISTOGRAM2, HISTOGRAM3... The corresponding values are then referenced using label.histogram-1, label.histogram-2, label.histogram-3... SUM calculate the sum of all the quantities. The final value can be referenced using label.sum. You can use multiple instances of this keyword i.e. SUM1, SUM2, SUM3... The corresponding values are then referenced using label.sum-1, label.sum-2, label.sum-3... FILE the file on which to write out the box coordinates UNITS ( default=nm ) the units in which to write out the corners of the box