$d$ contribution to the M$\sigma$ orbital as well as its smaller spatial extent than the L orbital.

Before presenting the results, we remark about two typical AF solutions. The first is stabilized in the case of \[Au(tmdt)$_2$\] when $U_{\textrm{L}}$ is increased, whose nature was discussed in refs.  and . As shown in Fig. \[fig5\](b), spin moments appear only on L sites; therefore, this state is represented as L-AF. The spins are parallel within L1-L2 dimers connected by B bonds \[see Fig. \[fig3\](c)\] and antiparallel between dimers along interdimer bonds denoted as A, P, and Q in Fig. 3(a). Their ordering vector is $q_\textrm{mag}[\textrm{L}]=(\pi,0,0)$. This pattern corresponds to the dimer-AF spin order frequently appearing in 1/4-filled CTS under dimerization [@Seo_2004CR]. When the on-site Coulomb repulsion is sufficiently large, their staggered pattern can open a gap at $\epsilon_{\rm F}$ when the system is 1/4-filled in terms of either electrons or holes, with each dimer carrying an effective $S=$ 1/2. Such a case is considered as the dimer-Mott insulating state. In the case of the room-$T$ parameters for \[Au(tmdt)$_2$\], L bands are 3/4-filled, without pronounced mixing with other orbitals. Dimers are formed by intermolecular fMO sites, and spins on two L sites in 