In a molecular dynamics code, suppose, the distances are expressed in units of a characteristic length of the simulated system, $R_0$. In some papers it is written as, " distances are normalized by $R_0$", while in some other papers it is found as " distances are expressed in units of $R_0$". I want to understand what is the difference between the two statements ?
Does it have anything in common with normalization of a wave function ?
The two statements mean the same thing. It means that instead of reporting a distance $r$ in meters (or some other unit), all distances are reported as a dimensionless ratio $r/R_0$.
This has nothing in particular to do with the normalization of the wavefunction, beyond "division by a constant to set a standard scale." For a generic, unnormalized wavefunction $\psi(x)$ (which for simplicity we will assume is a function of one coordinate), we define the norm $Z=\int_{-\infty}^\infty dx |\psi|^2$, and define the normalized wavefunction as $\psi_{\rm norm}=\psi/\sqrt{Z}$.
