This intuition is confirmed in Fig. . In Fig. (a), we plot the error in the energy of the ground state as a function of ζ. As we would guess, the larger ζ is, the less mixing there is between S 0 and S 1 , S 2 , and therefore the more accurate S 0 is. Next, in Fig. (b), we plot the asymptotic energy difference between E(S 2 ) (by DW-SA-cCASSCF(2,2)) and E(ne = 0) (the diabatic state whose number of electrons on the impurity is 0) as a function of ζ. We see clearly that E(S 2 ) is closer to E(ne = 0) when ζ is small and there is more mixing between S 0 and S 1 , S 2 during the orbital optimization. Lastly, as far as the smoothness is concerned, we find that surfaces are all smooth for ζ = 20, 30, 40, 50 for the one-site U = 0 model. (c), we can clearly see that the "fc" ground state energy recovers the RHF result. A similar anomaly is found in Fig. , where we plot the total number of electron on the impurity (⟨n tot ⟩) against the exact numerical renormalization group (NRG). In Fig. (a), for "pc", two excited state populations have a transition around ϵ d = -0.05: from 3 to 1 for S 1 and from 4 to 0 for S 2 .