The natural question is: how do those two forces (attractive and repulsive) complement each other into one uniform system?
It turns out that attractive force can best be described by London dispersion forces: forces are induced by instantaneous electron density shift in atoms/molecules, which causes neutral atoms/molecules to have partially electronegative and electropositive sides.
This attractive energy can be described as:
U
a
(r)=-
3
2
α
1
α
2
I
1
I
2
I
1
+
I
2
·
1
6
r
The function can be simplified to:
U
a
(r)=-4ϵ
6
σ
r
Here, ϵ is the potential energy, required to infinitely separate two atoms from equilibrium distance
r
0
:
σ=
r
0
1/6
2
The attractive force is negligible at great distance, but becomes increasingly more significant as the atoms get closer.
Here is a typical shape of attractive energy potential with increasing distance (helium atoms are used for the example calculation):
On the other hand, the repulsive force is described the by Pauli exclusion principle—no two electrons can occupy the same orbitals in an atom. If the two atoms get too close, their electrons start to share the orbitals. and therefore this repulsive force becomes highly dominant. It is described by:
U
r
(r)=-4ϵ
12
σ
r
... which increases rapidly, as the atoms approach each other.
Here is the shape of the potential energy due to repulsive forces for helium atoms:
The total potential energy between the two molecules is:
Therefore, the total function of potential energy between the two atoms/molecules can be viewed.
However, the scale of interaction between the different molecules is not clear unless drawn within the same graph.
The Visualization of Interaction between the Particles
You can explore how two helium particles interact with each other in the following simulation. One particle can be adjusted at a fixed point, while the other interacts with the first one accordingly.
In this way, it is made sure that the particles will attract each other; however, the repulsive force prevents them from getting too close.