\sqrt{N_{\max}})$ (dot-dashed). Both hyperfine splittings, 1S and 2S, show reasonable convergence in the complete basis limit ($N_{\max}^{-1}=L_{\max}^{-1}=0$).

Table \[tab:model\_parametersII\] compares the spectroscopy obtained from different $N_{\max} = L_{\max}$ fits. While results from different $N_{\max} = L_{\max}$ are well converged, the r.m.s. deviation decreases as $N_{\max}=L_{\max}$ increases. In the present work, we adopt $N_{\max}=L_{\max}=32$ for our presented results, unless otherwise specified.

![The $N_{\max}$ convergence. The *left panel* compares the $J/\psi$ and $\eta_c$ mass eigenvalues as a function of $N_{\max}^{-1}$ ($N_{\max}=L_{\max}$, $m_j=0$) for fixed and refitted parameters. For the former (“fix-parameter”), parameters are the same for all $N_{\max}$, and are chosen to be the fitted values at $N_{\max}=32$. For the latter (“refit-parameter”), parameters are refitted for each $N_{\max}$. The *right panel* shows the hyperfine splittings $M_{\psi(nS)}-M_{\eta_c(nS)}$ as a function of $N_{\max}^{-1}$ ($N_{\max}=L_{\max}$, $m_j=0$) with fixed parameters. The PDG values are marked as crosses. In both figures, different fitting functions, $a+b/N_{\max}+c/N_{\max}^2$ (solid), $a+b \exp(-c N_{\max})$ (dashed), $a+b \exp(-c \sqrt{N_{\max}})$ (dot-dashed), are shown for the fix-parameter results. The refit-parameter results are simply connected by a straight line segments. []{data-label="fig:conv"}](Figures/charm_Nmax_convergence_refit "fig:"){width="48.00000%"} ![The $N_{\max}$ convergence. The *left panel* compares the $J/\psi$ and $\eta_c$ mass eigenvalues as a function of $N_{\max}^{-1}$ ($N_{\max}=L_{\max}$, $m_j=0$) for fixed and refitted 