Shortcut: LESS_THAN

Module	function
Description	Usage
Use a switching function to determine how many of the input variables are less than a certain cutoff.
output value	type
a function that is one if the input is less than a threshold	scalar/vector/matrix

Input

The arguments that serve as the input for this action are specified using one or more of the keywords in the following table.

Keyword	Type	Description
ARG	scalar/vector/matrix	the values input to this function

Further details and examples

Use a switching function to determine how many of the input variables are less than a certain cutoff.

This action takes one argument, $r$ and evaluates the following function:

$w(r) = s(r)$

The function $s(r)$ here is a switching function so the output value $w$ is a number between 0 and 1. Switching functions are typically used to determine if an input value is less than some cutoff. The value of $w$ the switches smoothly from one to zero as the input value $r$ crosses the threshold of interest.

The following example, shows how we can apply a switching function on the instantaneous value of the distance between atom 1 and atom 2.

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={RATIONAL R_0=0.2}

The output here is close to one when the distance between atoms 1 and 2 is less than 0.2 nm close to zero for values that are greater than 0.2.

Non rank zero arguments

Instead of passing a single scalar in the input to the LESS_THAN action you can pass a single vector as shown here:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d vector the DISTANCE for each set of specified atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMS1the pair of atom that we are calculating the distance between=1,2 ATOMS2the pair of atom that we are calculating the distance between=3,4 ATOMS3the pair of atom that we are calculating the distance between=5,6 ATOMS4the pair of atom that we are calculating the distance between=7,8
bThe LESS_THAN action with label b calculates the following quantities: Quantity    Type    Description  
b vector the vector obtained by doing an element-wise application of a function that is one if the input is less than a threshold to the input vectors: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={RATIONAL R_0=0.2}

The input to the LESS_THAN action here is a vector with four elements. The output from the action b is similarly a vector with four elements. In calculating the elements of this vector PLUMED applies the function described in the previous section on each of the distances in turn. The first element of b thus tells you if the distance between atoms 1 and 2 is less than 0.2 nm, the second element tells you if the distance between atoms 3 and 4 is less than 0.2 nm and so on.

You can use the commands in the above example input to evaluate the number of distances that are less than a cutoff as follows:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d vector the DISTANCE for each set of specified atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMS1the pair of atom that we are calculating the distance between=1,2 ATOMS2the pair of atom that we are calculating the distance between=3,4 ATOMS3the pair of atom that we are calculating the distance between=5,6 ATOMS4the pair of atom that we are calculating the distance between=7,8
bThe LESS_THAN action with label b calculates the following quantities: Quantity    Type    Description  
b vector the vector obtained by doing an element-wise application of a function that is one if the input is less than a threshold to the input vectors: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={RATIONAL R_0=0.2}
sThe SUM action with label s calculates the following quantities: Quantity    Type    Description  
s scalar the sum of all the elements in the input vector: SUMCalculate the sum of the arguments More details ARGthe values input to this function=b PERIODICif the output of your function is periodic then you should specify the periodicity of the function=NO
PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=s FILEthe name of the file on which to output these quantities=colvar

The final scalar that is output here is evaluated using the following summation:

$s = \sum_i s(d_i)$

where the sum over $i$ here runs over the four distances in the above expression. This scalar tells you the number of distances that are less than 0.2 nm.

Notice that you can do something similar with a matrix as input as shown below:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE_MATRIX action with label d calculates the following quantities: Quantity    Type    Description  
d matrix a matrix containing the weights for the bonds between each pair of atoms: DISTANCE_MATRIXCalculate a matrix of distances between atoms. This action has hidden defaults. More details GROUPAwhen you are calculating the adjacency matrix between two sets of atoms this keyword is used to specify the atoms along with the keyword GROUPB=1-10 GROUPBwhen you are calculating the adjacency matrix between two sets of atoms this keyword is used to specify the atoms along with the keyword GROUPA=11-20
d: DISTANCE_MATRIXCalculate a matrix of distances between atoms. This action uses the defaults shown here. More details GROUPAwhen you are calculating the adjacency matrix between two sets of atoms this keyword is used to specify the atoms along with the keyword GROUPB=1-10 GROUPBwhen you are calculating the adjacency matrix between two sets of atoms this keyword is used to specify the atoms along with the keyword GROUPA=11-20  NL_CUTOFF The cutoff for the neighbor list=0.0 NL_STRIDE The frequency with which we are updating the atoms in the neighbor list=1 CUTOFF ignore distances that have a value larger than this cutoff=-1
bThe LESS_THAN action with label b calculates the following quantities: Quantity    Type    Description  
b matrix the matrix obtained by doing an element-wise application of a function that is one if the input is less than a threshold to the input matrix: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={RATIONAL R_0=0.2}
sThe SUM action with label s calculates the following quantities: Quantity    Type    Description  
s scalar the sum of all the elements in the input matrix: SUMCalculate the sum of the arguments More details ARGthe values input to this function=b PERIODICif the output of your function is periodic then you should specify the periodicity of the function=NO
PRINTPrint quantities to a file. More details ARGthe labels of the values that you would like to print to the file=s FILEthe name of the file on which to output these quantities=colvar

This input tells PLUMED to calculate the 100 distances between the atoms in the two input groups. The final value that is printed to the colvar file then tells you how many of these distances are less than 0.2 nm.

Switching functions types

PLUMED allows you to use a range of different switching function types. Most of these require you to provide at least one input parameter $r_0$ . The switching function is then defined so that for $r \le d_0 \quad s(r)=1.0$ while for $r > d_0$ the function decays smoothly to 0. By changing the switching function you are using you change the decay that occurs in the switching function for $r$ values that are greater that $d_0$ .

You can specify the value at which the the switching function goes to zero by using the D_MAX keyword. PLUMED will ensure that the swtiching function is definitely zero for for input values that are greater than or equal to D_MAX by by stretching and shifting the function using the following transformations.

$s(r) = \frac{s'(r)-s'(d_{max})}{s'(0)-s'(d_{max})}$

The various expressions that are used for $s'(r)$ in the above expression are described in the following sections. Scaling the switching function so that it is exactly zero at $d_{max}$ using the transformation described above has been the default behaviour within PLUMED since version 2.2. If you wish to use a D_MAX parameter and the $s'(r)$ functions described in the following sections directly you can use the keyword NOSTRETCH. However, the advangage of performing the stretching and scaling in this way is that the function has no discontinuities. If you use the NOSTRETCH option the switching function may have a discontinuity at $d_{max}$ .

Rational switching function

The rational switching function is the most commonly used of the switching functions. The implementation of this function within PLUMED has been carefully optimised so calculations using this switching function should be fast.

For this function $s'(r)$ has the following functional form:

$s'(r) = \begin{cases} 1 & \textrm{if} \quad r<d_0 \\ \frac{ 1 - \left(\frac{ r - d_0 }{ r_0 }\right)^{n} }{ 1 - \left(\frac{ r - d_0 }{ r_0 }\right)^{m} } & \textrm{if} \quad d_0 < r < d_{max} \\ 0 & \textrm{otherwise}$

The following example illustrates how you can use the rational switching function is used in practice:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={RATIONAL R_0=0.5 D_0=0.1 NN=8 MM=16 D_MAX=1.0}

As was discussed in earlier parts of this page, you can also specify a rational switching function using the following input:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={RATIONAL R_0=0.2}

In this case, the paramers $d_0$ , $n$ and $m$ are set to their default values of 0, 6 and 12 respectively. Meanwhile, if D_MAX is unset the stretching and scaling that was described in the previous section is not performed.

Notice that you can choose to set a subset of the switching function parameters and to use the default values for the unset parameters by using an input like this one:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={RATIONAL R_0=0.2 NN=8}

The input above sets $n=8$ and $m=2n=16$ . Meanwhile, $d_0$ is set to its default value of 0.

Lastly, note that you can also specify the parameters for the rational switching function without using the SWITCH keyword as indicated below:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d D_0 The d_0 parameter of the switching function=0.1 R_0The r_0 parameter of the switching function=0.2 NN The n parameter of the switching function =6 MM The m parameter of the switching function; 0 implies 2*NN=12

The newer syntax using the SWITCH keyword is better than this option as there is no way to set the D_MAX parameter this way and because if you use this syntax you are forced to use the RATIONAL switching function type.

Expoential switching function

The way that the exponential switching function can be used is illustrated in the following example input:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={EXP D_0=0.1 R_0=0.2 D_MAX=1.0}

The $s'(r)$ that is used in this input is given by the following expression:

$s'(r) = \begin{cases} 1 & \textrm{if} \quad r<d_0 \\ \exp\left(-\frac{ r - d_0 }{ r_0 }\right) & \textrm{if} \quad d_0 < r < d_{max} \\ 0 & \textrm{otherwise}$

You can also specify that an exponential switching function is to be used by using the following input:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={EXP R_0=0.2}

For this input $d_0$ is set to its default value of 0. Furthermore, as D_MAX is unset the stretching and scaling that was described in the previous section is not performed.

Gaussian switching function

The way that the gaussian switching function can be used is illustrated in the following example input:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={GAUSSIAN D_0=0.1 R_0=0.2 D_MAX=1.0}

The $s'(r)$ that is used in this input is given by the following expression:

$s'(r) = \begin{cases} 1 & \textrm{if} \quad r<d_0 \\ \exp\left(-\frac{ (r - d_0)^2 }{ 2r_0^2 }\right) & \textrm{if} \quad d_0 < r < d_{max} \\ 0 & \textrm{otherwise}$

You can also specify that an gaussian switching function is to be used by using the following input:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={GAUSSIAN R_0=0.2}

For this input $d_0$ is set to its default value of 0. Furthermore, as D_MAX is unset the stretching and scaling that was described in the previous section is not performed.

Sketch-map switching function

You can specify that the sketch-map switching function is to be used by using the following input:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={SMAP D_0=1 R_0=4 A=3 B=2 D_MAX=10}

The $s'(r)$ that is used in this input is given by the following expression:

$s'(r) = \begin{cases} 1 & \textrm{if} \quad r<d_0 \\ \left[ 1 + ( 2^{a/b} -1 )\left( \frac{r-d_0}{r_0} \right)^a \right]^{-b/a} & \textrm{if} \quad d_0 < r < d_{max} \\ 0 & \textrm{otherwise}$

This type of swtiching function would be used with PLUMED's implementation of SKETCHMAP. If you are performing a sketch-map projection an input like the one below would be more common:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={SMAP R_0=4 A=3 B=2}

With this input $d_0$ is set to its default value of 0. Furthermore, as D_MAX is unset the stretching and scaling that was described in the previous section is not performed.

Q switching function

You can specify that the Q switching function is to be used by using the following input:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={Q D_0=0 REF=0.498366 BETA=50.0 LAMBDA=1.8 R_0=0.01 D_MAX=20}

The $s'(r)$ that is used in this input is given by the following expression:

$s'(r) = \begin{cases} 1 & \textrm{if} \quad r<d_0 \\ s(r) = \frac{1}{1 + \exp(\beta(r_{ij} - \lambda r_{ij}^0))} & \textrm{if} \quad d_0 < r < d_{max} \\ 0 & \textrm{otherwise}$

The value of $r_{ij}^0$ is specified in the above input by the REF keyword. Notice that you can also specify this type of switching function using the following input:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={Q REF=0.498366 R_0=0.01 }

With this input $_0$ , $\lambda$ and $\beta$ are set equal to their default values of 0, 1.8 and 50 nm $^{-1}$ respectively. Furthermore, as D_MAX is unset the stretching and scaling that was described in the previous section is not performed. These default values for $\lambda$ and $\beta$ are suitable if you are simulating all atom models. However, if you are using a coarse grainded models values of $\lambda=1.5$ and $\beta=50 nm^-1$ are more appropriate.

Cubic switching function

You can specify that the cubic swtiching function is to be used by using the following input:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={CUBIC D_0=0.1 D_MAX=0.5}

If this type of expression is used then the $s(r)$ is calculated as:

$$ s(r) = (y-1)^2(1+2y) \qquad \textrm{where} \quad y = \frac{d_0 - d_{max}}{d_0-d_{max}}$

No stretching is required for this type of switching function as its functional form ensures that it is zero at $d_{max}$ .

Tanh switching function

You can specify that the cubic swtiching function is to be used by using the following input:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={TANH D_0=0.1 R_0=0.1 D_MAX=1.0}

The $s'(r)$ that is used in this input is given by the following expression:

$s'(r) = \begin{cases} 1 & \textrm{if} \quad r<d_0 \\ 1 - \tanh\left( \frac{ r - d_0 }{ r_0 } \right) & \textrm{if} \quad d_0 < r < d_{max} \\ 0 & \textrm{otherwise}$

You can also specify that a tanh switching function is to be used by using the following input:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={TANH R_0=0.2}

For this input $d_0$ is set to its default value of 0. Furthermore, as D_MAX is unset the stretching and scaling that was described in the previous section is not performed.

Cosinus switching function

You can specify that the cosinum function is to be used by using the following input:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={COSINUS D_0=0.1 R_0=0.1}

The $s'(r)$ that is used in this input is given by the following expression:

$s(r) =\left\{\begin{array}{ll} 1 & \mathrm{if } r \leq d_0 \\ 0.5 \left( \cos ( \frac{ r - d_0 }{ r_0 } \pi ) + 1 \right) & \mathrm{if } d_0 < r\leq d_0 + r_0 \\ 0$

No stretching is required for this type of switching function as its functional form ensures that it is zero at $d_0+r_0$

Custom switching function

If you want to use a switching function that is different to all of the functions listed above you can use the cusom option that is illustrated below:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={CUSTOM FUNC=1/(1+x^6) D_0=0.1 R_0=0.1 D_MAX=10}

This option allows you to use the lepton library that is used to implement the CUSTOM action to evaluate your switching function. With this option you specify the functional form for $s'(r)$ by providing a function of x to the FUNC keyword. The x that enters the switching function definition that you provide is given by:

$x = \frac{r - d_0}{r_0}$

Notice, that you can also express your switching function in terms of $x^2$ as illustrated in the input below:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={CUSTOM FUNC=1/(1+x2^3) D_0=0.1 R_0=0.1 D_MAX=10}

which is equivalent to the earlier input with the CUSTOM switching function. However, using x2 in place of x is often more computationally efficient as you can avoid an expensive square root operation. Notice, lastly, that just as there is a MATHEVAL action that is equivalent to CUSTOM, there is also a matheval equivalent that you can use here:

Click on the labels of the actions for more information on what each action computes

dThe DISTANCE action with label d calculates the following quantities: Quantity    Type    Description  
d scalar the DISTANCE between this pair of atoms: DISTANCECalculate the distance/s between pairs of atoms. More details ATOMSthe pair of atom that we are calculating the distance between=1,2
sThe LESS_THAN action with label s calculates the following quantities: Quantity    Type    Description  
s scalar a function that is one if the input is less than a threshold: LESS_THANUse a switching function to determine how many of the input variables are less than a certain cutoff. More details ARGthe values input to this function=d SWITCHThis keyword is used if you want to employ an alternative to the continuous swiching function defined above={MATHEVAL FUNC=1/(1+x2^3) R_0=0.1}

For this input $d_0$ is set to its default value of 0. Furthermore, as D_MAX is unset the stretching and scaling that was described in the previous section is not performed.

[!CAUTION] With the default implementation CUSTOM is slower than other functions (e.g., it is slower than an equivalent RATIONAL function by approximately a factor 2). You can find information on how to improve its performance in the documenation for CUSTOM

Syntax

The following table describes the keywords and options that can be used with this action

Keyword	Type	Default	Description
ARG	input	none	the values input to this function
NN	compulsory	6	The n parameter of the switching function
MM	compulsory	0	The m parameter of the switching function; 0 implies 2*NN
D_0	compulsory	0.0	The d_0 parameter of the switching function
R_0	compulsory	none	The r_0 parameter of the switching function
SWITCH	optional	not used	This keyword is used if you want to employ an alternative to the continuous swiching function defined above
SQUARED	optional	false	is the input quantity the square of the value that you would like to apply the switching function to

Quantity	Type	Description
d	scalar	the DISTANCE between this pair of atoms