# Pair correlation function vs Bond length distribution of a van der Waal dimer

For a weakly bonded molecular dimer (when two atoms are interacting through van der Waal (vdW) interaction). Do the distribution of pair correlation function and vibrationally average bond-length distribution be same?

When we see a plot for the pair correlation function with interatomic distance (R) as the x-axis what does it stand for? Does it have one to one correspondence with the vibrationally average bond-length distribution?

• I don't fully understand your question. Are we talking about simulations of a single object or of an ensemble? In the latter case, the answer to your question would be "yes" in the ideal gas limit. – lr1985 Apr 19 '18 at 11:58
• Actually, I was thinking about a van der Waal system. Where two independent atoms can come close together to form a diatomic system. – Bikash Apr 19 '18 at 12:41
• What do you mean with "a van der Waal system"? So you have just two atoms in a box interacting through a Lennard-Jones system? – lr1985 Apr 20 '18 at 10:06
• Yes, exactly two atoms are interacting through a Lennard-Jones potential. – Bikash Apr 20 '18 at 11:29
• I think using the word "bond" is a bit of stretch, since in this context the latter is usually associated to covalently linked particles rather than particles interacting through a LJ-type of potential. The pair correlation function is defined as $P(r)/4\pi r^2 \rho$, where $P(r)$ is the "bond-length distribution", so the two things are connected, but not exactly the same. – lr1985 Apr 20 '18 at 12:20

Let's say you simulate two particles in a box with periodic boundary conditions. During the evolution you keep track of the relative distance between the two particles, $r$. You run enough to obtain the probability distribution $P(r)$, that is, the probability that the two particles are separated by a distance $r$. This is, I believe, what you would call the ''vibrationally average bond-length distribution''.
Now, if we take $P(r)$ and normalise it with respect to the case of non-interacting particles (ideal gas), we obtain the radial distribution function. Note that for two particles the following relation is exact:
$$g(r) = \exp\left(-\beta V(r)\right)$$
where $\beta = 1/k_BT$ and $V(r)$ is the pair potential.