Local r⁴ smoothing¶

When passed to halmd.mdsim.forces.pair_trunc, this function transforms the potential into a C²-continuous function, and the force into a C¹-continuous function. The degree of smoothing is controlled with a dimensionless parameter.

The smoothing function is

$f\left(\xi\right) = \frac{\xi^4}{1 + \xi^4}, \qquad \xi = \frac{r - r_c}{h r_c},$

with smoothing parameter $h \ll 1$ and cutoff distance $r_c$ , and its derivative

$f^\prime\left(\xi\right) = 4 \frac{\xi^3}{\left(1 + \xi^4\right)^2}.$

The C²-continuous potential is

$V_{C^2}\left(r\right) = V\left(r\right) f\left(\frac{r - r_c}{h r_c}\right),$

and the C¹-continuous force is

$\lvert\vec{F}_{C^1}\left(\vec{r}\right)\rvert = \lvert\vec{F}\left(\vec{r}\right)\rvert f\left(\frac{r - r_c}{h r_c}\right) - \frac{1}{h r_c} V\left(r\right) f^\prime\left(\frac{r - r_c}{h r_c}\right).$

The following figure shows unmodified and C²-smooth variants of the Weeks-Chandler-Andersen potential, a purely repulsive form of the Lennard-Jones potential with cutoff $r_c = \sqrt[6]{2}$ .