# Potential truncations¶

## sharp¶

Create sharp truncation by setting the potential to zero for particle distances larger than the cutoff distance , i.e., by multiplying with a step function:

Both potential and force are discontinuous at the cutoff.

Example:

```
potential = potential:truncate({"sharp", cutoff = 4})
```

## shifted¶

Amend the sharp truncation by an energy shift such that the potential is continuous at the cutoff distance :

The force is not affected by this and remains discontinuous at the cutoff.

Example:

```
potential = potential:truncate({"shifted", cutoff = 2.5})
```

## force_shifted¶

Amend the energy-shifted potential by a linear term, effectively shifting the force to zero at the cutoff distance :

Both energy and force are continuous at the cutoff. Note that this modification globally tilts the potential with possible physical implications for, e.g., phase diagrams.

Example:

```
potential = potential:truncate({"force_shifted", cutoff = 2.5})
```

## smooth_r4¶

Truncate the potential such that it remains a C²-continuous
function at the cutoff, i.e., the force being continuously differentiable.
As a consequence, momentum and energy drift are drastically diminished
even from long runs using symplectic integrators such as
`halmd.mdsim.integrators.verlet`

.

The truncation is implemented by multiplication of the energy-shifted potential with the local smoothing function

where is the cutoff distance, and the parameter , which has the dimension of a length, controls the the range of smoothing. The C²-continuous truncated potential then reads

and the C¹-continuous force is

with the derivative of the smoothing function .

Example:

```
potential = potential:truncate({"smooth_r4", cutoff = 2.5, h = 0.005})
```

The following figure shows unmodified and C²-smooth variants of the Weeks-Chandler-Andersen potential, the repulsive part of the Lennard-Jones potential sharply cutoff at .

(Source code, png, hires.png, pdf)

The following figure shows the absolute value of the force.

(Source code, png, hires.png, pdf)