Formulae and units¶

Lennard-Jones¶

Propagation laws¶

velocity-Verlet algorithm: $\begin{align*} \vec{r}_i(t+\Delta t) &= \vec{r}_i(t) + \vec{v}_i(t+\frac{\Delta t}{2})\Delta t \\ \vec{v}_i(t+\frac{\Delta t}{2}) &= \vec{v}_i(t) + \frac{\vec{F}_i(t)}{m}\frac{\Delta t}{2} \\ \vec{v}_i(t+\Delta t) &= \vec{v}_i(t+\frac{\Delta t}{2}) + \frac{\vec{F}_i(t+\Delta t)}{m}\frac{\Delta t}{2} \end{align*}$

Interaction laws¶

Lennard-Jones potential: $U(r_{ij}) = 4\epsilon \Bigl(\bigl(\frac{\sigma}{r_{ij}}\bigr)^{12} - \bigl(\frac{\sigma}{r_{ij}}\bigr)^6\Bigr)$
shifted, truncated Lennard-Jones potential: $U_s(r_{ij}) = \begin{cases} 4\epsilon\left(\left(\dfrac{\sigma}{r}\right)^{12} - \left(\dfrac{\sigma}{r}\right)^{6}\right)-U(r_c)\,, & r < r_c \,,\\ 0\,, & r > r_c \,. \end{cases}$
Lennard-Jones force: $\vec{F}(\vec{r}_{ij}) = \frac{48\epsilon}{\sigma} \Bigl(\bigl(\frac{\sigma}{r_{ij}}\bigr)^{13} - \frac{1}{2}\bigl(\frac{\sigma}{r_{ij}}\bigr)^7\Bigr) \hat{r}_{ij}$

Base MD units¶

unit of length: $[r] = \sigma$
unit of energy: $[E] = \epsilon$
unit of time: $[t] = \sqrt{\frac{m\sigma^2}{\epsilon}}$

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Formulae and units
- Lennard-Jones

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