shown, respectively. ‘[*b*]{}.’ and ‘[*a*]{}. [*b*]{}.’ represent the bonding and antibonding orbitals for an L dimer, respectively. ](fig5.eps){width="7.2cm"}

\[fig5\]

We set the values of $t_{lm}$ to the fitted results in Table \[table1\], while $\Delta_{\textrm{M}\sigma}$ and $\Delta_{\textrm{M}\pi}$ are adjusted by the conditions in eqs. (\[eq:orbene1\]) and (\[eq:orbene2\]) when the interaction parameters are varied. Then the band dispersions are unchanged within the paramagnetic metallic (PM) solution. Namely, the orbital occupations in the PM state, $\langle n_{i\uparrow}^\textrm{o} \rangle = \langle n_{i\downarrow}^\textrm{o} \rangle =
 \langle n_i^\textrm{o} \rangle/2$, are fixed at the values listed in Table \[table1\] as $ \langle n_i^\textrm{o} \rangle= \bar{n}_{\textrm{o}}$. As for the magnetic solutions, we relax the condition of fixed occupation and searched for self-consistent solutions of the lowest energy in an unrestricted manner within the periodicity of the supercell.

In the following, we set $U_{\textrm{M}\sigma} = U_{\textrm{M}\pi} = U'_{\textrm{M}} \equiv U_{\textrm{M}}$ for simplicity [@noteUparam] and vary $U_{\textrm{M}}$ and $U_{\textrm{L}}$ independently. In the whole parameter range we sought, the M$\pi$ orbital has a negligible spin moment, i.e., it is magnetically inactive. Then the parameters $U_{\textrm{M}}$ and $U_{\textrm{L}}$ control the correlation effect on the M$\sigma$ and L orbitals, respectively. One speculation we can make is the relation $U_{\textrm{M}} \gsim U_{\textrm{L}}$, considering the 