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2.1 Units

To avoid the handling of small numbers, choose units appropriate to the model.

Lennard-Jones: Energy unit $E_{0}=\epsilon$; length $l_{0}=\sigma$. The pair energy is then

u_{LJ}^{*}=4\left[ r^{*-12}-r^{*-6}\right]

where $u^{*} \equiv u/\epsilon$ and $r^{*} \equiv r/\sigma$.

As the third mechanical unit, choose the atomic mass $m_{0}=1 AMU= 1.6606 \cdot 10^{-27} kg$.

The time unit is now the combination $t_{0}=\sqrt{m_{0} \sigma^{2}/\epsilon}$.

Electrical charge: best measured in multiples of the electron charge, $q_{0}=1.602 \cdot 10^{-19}  As$.

Number density: $\rho=N/V$ is a large number; therefore we reduce it by a suitable standard density, $\rho_{0}=1/\sigma^{3}$: $\rho^{*} \equiv N \sigma^{3}/V$.

Temperature: $T_{0}=\epsilon/k$

Hard spheres: no ``natural'' unit of energy; therefore choose self-consistent time unit $t_{0}=\sqrt{m_{0}d^{2}/kT}$.

For hard spheres of diameter $d_{0}=2  r_{0}$ the customary standard density is $\rho_{0}=\sqrt{2}/d_{0}^{3}$; thus: $\rho^{*}=N d_{0}^{3}/V\sqrt{2}$.
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F. J. Vesely / University of Vienna