while the scale of a halo is set by its mass, the concentration is the most fundamental property. In this paper, the NFW form is used to fit the halo density profiles, which can be approximated by a simple formula with two free parameters, $$\label{eqn:nfw}
\frac {\rho(r)}{\rho_{\rm mean}} = \frac {\delta_{\rm c}}
{(r/r_s)(1+r/r_s)^2},$$ where $r_s$ is a scale radius and $\delta_{\rm c}$ is a characteristic density contrast. The concentration of a halo is defined as $c=R_{200}/r_s$, thus $\delta_{\rm c}$ and concentration are linked by the relation $$\label{deltac}
\delta_{\rm c}=\frac {200}{3} \frac{c^3}{[{\rm ln}(1+c)
-c(1+c)]}.$$ In our calculation of concentration from the simulation, the radius of a halo is divided in uniform logarithmic bins ($\Delta \log_{10}=0.1$), starting from the radius where the bin contains at least $20$ particles to $R_{\rm
  max}=R_{200}$ (or to $R_{\rm vir}$ in case halo masses are defined using the over-density $\Delta_{\rm    vir}$). Due to the relation of equation \[deltac\], at given halo mass $M_{200}$, there is a single free parameter in equation \[eqn:nfw\], which can be expressed as the concentration parameter $c$. The best-fit concentration parameter can be computed from simulation by minimizing the rms deviation $\sigma$ between the binned $\rho(r)$ and the NFW profile, $$\sigma^2 = \frac{1}{N_{\rm 