Lennard Jones Potential
The Lennard Jones potential has several features which might make
it a suitable model for reality. This is best realised by looking
at the plot of the function:
(a) As the separation of the two atoms increases, the attraction
between them increases and tends to zero at infinite distance. This
is sensible, since as two atoms approach each other from a large
separation, their potential energy slowly drops as they are
attracted together. Mathematically this is seen by the fact that
both terms in the potential energy expression tend to zero as r
tends to infinity.
(b) There is a potential energy minima, which is the stable atomic
separation. We know that the atoms ARE attracted to each other by
van der Waals attractions, and so it makes sense that there will be
some fixed distance apart that they will remain. Mathematically,
this is as the turning point of the function, where the gradient is
equal to zero.
(c) As the separation of the atoms decreases further, the potential
rises sharply, which indicates that it is highly unfavourable for
the atoms to be squashed together further. This is seen in reality,
where two neutral atoms do not increasingly approach each other
indefinitely! Mathematically, this is the
$\left(\frac{\sigma}{r}\right)^{12}$ dominating the other term,
which leads to a very positive potential as $ r$ decreases.
The $W(r)$ potential curve differs from the Lennard-Jones potential
as it has a term to the power of 9 as opposed to 12. Consequently,
the curve still tends to zero at infinity, still has a potential
energy minima, and increases sharply with small r. Therefore it
could well yield a good match with reality with appropriate values
of the constants.