# pair_style cosine/squared command

## Syntax

``` LAMMPS
pair_style cosine/squared cutoff
```

-   cutoff = global cutoff for cosine-squared interactions (distance
    units)

``` LAMMPS
pair_coeff i j eps sigma
pair_coeff i j eps sigma cutoff
pair_coeff i j eps sigma wca
pair_coeff i j eps sigma cutoff wca
```

-   i,j = a particle type
-   eps = interaction strength, i.e. the depth of the potential minimum
    (energy units)
-   sigma = distance of the potential minimum from 0
-   cutoff = the cutoff distance for this pair type, if different from
    global (distance units)
-   wca = if specified a Weeks-Chandler-Andersen potential (with eps
    strength and minimum at sigma) is added, otherwise not

## Examples

``` LAMMPS
pair_style cosine/squared 3.0
pair_coeff * * 1.0 1.3
pair_coeff 1 3 1.0 1.3 2.0
pair_coeff 1 3 1.0 1.3 wca
pair_coeff 1 3 1.0 1.3 2.0 wca
```

## Description

Style *cosine/squared* computes a potential of the form

$$\begin{aligned}
E =
\begin{cases}
-\epsilon& \quad r < \sigma \\
-\epsilon\cos\left(\frac{\pi\left(r - \sigma\right)}{2\left(r_c - \sigma\right)}\right)^2&\quad \sigma \leq r < r_c \\
0& \quad r \geq r_c
\end{cases}
\end{aligned}$$

between two point particles, where ($\sigma, -\epsilon$) is the location
of the (rightmost) minimum of the potential, as explained in the syntax
section above.

This potential was first used in [(Cooke)](CKD) for a coarse-grained
lipid membrane model. It is generally very useful as a non-specific
interaction potential because it is fully adjustable in depth and width
while joining the minimum at (sigma, -epsilon) and zero at (cutoff, 0)
smoothly, requiring no shifting and causing no related artifacts, tail
energy calculations etc. This evidently requires *cutoff* to be larger
than *sigma*.

If the *wca* option is used then a Weeks-Chandler-Andersen potential
[(Weeks)](WCA) is added to the above specified cosine-squared potential,
specifically the following:

$$E = \epsilon \left[ \left(\frac{\sigma}{r}\right)^{12} -
                      2\left(\frac{\sigma}{r}\right)^6 + 1\right]
                      , \quad r < \sigma$$

In this case, and this case only, the $\sigma$ parameter can be equal to
*cutoff* ($\sigma =$ cutoff) which will result in ONLY the WCA potential
being used (and print a warning), so the minimum will be attained at
(sigma, 0). This is a convenience feature that enables a purely
repulsive potential to be used without a need to define an additional
pair style and use the hybrid styles.

The energy and force of this pair style for parameters epsilon = 1.0,
sigma = 1.0, cutoff = 2.5, with and without the WCA potential, are shown
in the graphs below:

![image](JPG/pair_cosine_squared_graphs.jpg){.align-center}

------------------------------------------------------------------------

## Mixing, shift, table, tail correction, restart, rRESPA info

Mixing is not supported for this style.

The *shift*, *table* and *tail* options are not relevant for this style.

This pair style writes its information to [binary restart
files](restart), so pair_style and pair_coeff commands do not need to be
specified in an input script that reads a restart file.

These pair styles can only be used via the *pair* keyword of the
[run_style respa](run_style) command. They do not support the *inner*,
*middle*, *outer* keywords.

------------------------------------------------------------------------

## Restrictions

The *cosine/squared* style is part of the EXTRA-PAIR package. It is only
enabled if LAMMPS is build with that package. See the [Build
package](Build_package) page for more info.

## Related commands

[pair_coeff](pair_coeff), [pair_style lj/cut](pair_lj)

## Default

none

::: {#CKD}
**(Cooke)** \"Cooke, Kremer and Deserno, Phys. Rev. E, 72, 011506
(2005)\"
:::

::: {#WCA}
**(Weeks)** \"Weeks, Chandler and Andersen, J. Chem. Phys., 54, 5237
(1971)\"
:::