With the experimental observable O exp i and the calculated value O cal i , by minimizing the square deviation χ $^{2}$( a ) = N ∑ i [ O exp i − O cal i ( a ) ω$_{i}$ ] 2 , (25) the ensemble of parameters a can be obtained. Furthermore, in order to balance the influence of different observables, the weight ω$_{i}$ is introduced for binding energies, charge radii, and empirical pairing gaps respectively. The corresponding weight ω$_{i}$ is roughly determined by the desired accuracy. Here the weights ω$_{i}$ are respectively 1 . 00 MeV for binding energies, 0 . 02 fm for charge radii, and 0 . 05 MeV for empirical pairing gaps. A new parameter set PC-PK1, which contains the nine coupling constants in Eq. (2) and the pairing strength in Eq. (20), is obtained and listed in Table I. By scaling the coupling constants in accordance with the QCD-based Lagrangian, the naturalness in effective theories can be investigated [12, 13]. According to the QCD-based Lagrangian [13], L ∼ − c$_{ln}$ [ ¯ ψψ f 2 $_{π}$Λ ] $^{l}$[ ∂ µ Λ ] n f 2 $_{π}$Λ $^{2}$, (26) with ψ the nucleon field, f$_{π}$ = 92 . 5 MeV the pion decay constant, and Λ = 770 MeV a generic QCD large-mass scale respectively, by taking into account the role of chiral symmetry in weakening N -body forces by Δ = l + n − 2 ⩾ 0 [26, 27], it has been found that six of the nine coupling constants in PC-LA and all of them in PC-F1 are natural, i.e., the QCD-scaled coupling constants c$_{ln}$ are of order unity [11, 12]. Similarly, the nine coupling constants of PC-PK1 are also tested for the naturalness and all the dimensionless coefficients c$_{ln}$ are of order 1, as shown in the last column of Table I, which indicates that all the coupling constants in PC-PK1 are natural. Tables II and III list respectively the binding energies and charge radii for nuclei selected in the determination of PC-PK1, PC-F1, PC-LA, and NL3* [28] effective interactions. The corresponding root mean square (rms) deviation Δ together with the root of relative square (rrs) deviation δ for the binding energy and charge radius are given in the last two rows of Table II and III, respectively. Compared with the other effective interactions, the newly obtained PC-PK1 provides a much better description for the experimental binding energies and the same good description for the charge radii. 9