and $n_{i}^\textrm{o}= n_{i\uparrow}^\textrm{o}+n_{i\downarrow}^\textrm{o}$ with an orbital index $\textrm{o}=$ M$\sigma$, L1, L2, or M$\pi$. The intraorbital on-site Coulomb interactions for the three kinds of fMO are denoted as $U_{\textrm{L}}$, $U_{\textrm{M}\sigma}$, and $U_{\textrm{M}\pi}$. As for the interorbital on-site interaction, we only include $U'_\textrm{M}$ between M$\sigma$ and M$\pi$ for simplicity, considering that these two orbitals share the spatial extent while they are separated from the L orbitals [@noteHund].

![image](fig2a.eps){width="7.5cm"} ![image](fig2b.eps){width="7.5cm"}\
![image](fig2c.eps){width="7.5cm"} ![image](fig2d.eps){width="7.5cm"}

\[fig2\]

               bond                        site pair            \[Ni(tmdt)$_2$\]   \[Au(tmdt)$_2$\]   \[Au(tmdt)$_2$\] (9 K)   \[Cu(tmdt)$_2$\]
  ------------------------------- ---------------------------- ------------------ ------------------ ------------------------ ------------------
            intra-mol.                       L1-L2                  -61 meV            -24 meV               -27 meV               -35 meV
                 A                        L1-L1, L2-L2            [**-86**]{}        [**-99**]{}           [**-98**]{}           [**-90**]{}
                 B                           L1-L2                [**221**]{}        [**207**]{}           [**222**]{}           [**250**]{}
                 Q                           L1-L2                [**131**]{}        [**109**]{}           [**129**]{}           [**134**]{}
                 P                           L1-L2                     33                 41                    46                    37
                 A                 L1-M$\sigma$, M$\sigma$-L2          37                 33                    41                    14
                 B                 L1-M$\sigma$, M$\sigma$-L2          27                 27                    33                    29
                 C                 L1-M$\sigma$, M$\sigma$-L2         -50                -25                   -29                   -24
                 R                 L1-M$\sigma$, M$\sigma$-L2          21                 22                    27                    14
            intra-mol.                L1-M$\pi$, M$\pi$-L2        [**-200**]{}       [**-138**]{}          [**-127**]{}          [**-188**]{}
                 B                    L1-M$\pi$, M$\pi$-L2             0                  30                    35                    20
                 C                    L1-M$\pi$, M$\pi$-L2            -22                 -5                   -22                    -7
                 Q                    L1-M$\pi$, M$\pi$-L2             5                  15                    14                    11
                 A                    M$\sigma$-M$\sigma$          [**82**]{}         [**95**]{}            [**98**]{}            [**96**]{}
                 A                       M$\pi$-M$\pi$                -13                -23                   -30                   -19
                 B                       M$\pi$-M$\pi$                 24                 37                    44                    50
                 A                      M$\pi$-M$\sigma$               53                -34                   -14                    -1
                                                                                                                              
         