write the part H$_{M}$ that specifies the extended molecule H = H$_{L}$ + H$_{R}$ + H$_{LM}$ + H$_{MR}$ + H$_{M}$ , H$_{L}$ = µ$_{L}$ −∞ ∑ l = − M$_{L}$ − 1 a † $_{l}$a$_{l}$ − ξ$_{L}$ −∞ ∑ l = − M$_{L}$ − 1 ( a † $_{l}$a$_{l}$$_{−}$$_{1}$ + h.c. ) , H$_{R}$ = µ$_{R}$ + ∞ ∑ r = M$_{R}$ +1 a † $_{r}$a$_{r}$ − ξ$_{R}$ + ∞ ∑ r = M$_{R}$ +1 ( a † $_{r}$a$_{r}$$_{+1}$ + h.c. ) , (19) H$_{LM}$ = − ξ$_{L}$ ( a † − $_{M$_{L}$}$a$_{−}$$_{M}$$_{L}$$_{−}$$_{1}$ + h.c. ) , H$_{MR}$ = − ξ$_{R}$ a † $_{M$_{R}$}$a$_{M}$$_{R}$$_{+1}$ + h.c. ) , H$_{M}$ = µ$_{L}$$^{−}$$^{M$_{L}$}$ ∑ l = − 1 a † $_{l}$a$_{l}$ − ξ$_{L}$$^{−}$$^{M$_{L}$}$$^{+1}$ ∑ l = − 1 ( a † $_{l}$a$_{l}$$_{−}$$_{1}$ + h.c. ) + µ$_{R}$ M$_{R}$ ∑ r =1 a † $_{r}$a$_{r}$ − ξ$_{R}$$^{M$_{R}$}$$^{−}$$^{1}$ ∑ r =1 ( a † $_{r}$a$_{r}$$_{+1}$ + h.c. ) − β$_{L}$ ( a † − $_{1}$a$_{0}$ + h.c. ) − β$_{R}$ ( a † $_{+1}$a$_{0}$ + h.c. ) (20) + J$_{L}$ a † $_{0}$a$_{0}$ − 1 2 ) ( a † − $_{1}$a$_{−}$$_{1}$ − 1 2 ) + J$_{R}$ ( a † $_{0}$a$_{0}$ − 1 2 ) ( a † $_{+1}$a$_{+1}$ − 1 2 ) + αa † $_{0}$a$_{0}$ . and In the above equation, the extended molecule comprises, besides the “molecule” itself, repre- sented by a single site (labeled by 0), parts of the left and right electrodes (sites − M$_{L}$, . . . , − 1 and 1 , . . . , M$_{R}$ , respectively). Importantly, the energy of the molecular level α can be tuned by direct molecular orbital gating, as demonstrated by a remarkable very recent exper- iment [41], or electrochemical gating [42–45] in a way that is similar to the usage of a “plunger” gate electrode in quantum-dot nanoelectronic devices [46, 47]. Unlike in quantum- dot based single-electron transistors, where the relevant α -range is of the order of a few meV [46, 47], in the aforementioned experiment [41] the energy of the molecular level α could be varied, remarkably, within much broader ranges: 1 . 1 eV ≲ ε$_{F}$ − α ≲ 1 . 9 eV for single- electron transistors based on 1,8-octanedithiol and 0 . 4 eV ≲ ε$_{F}$ − α ≲ 1 . 8 eV for those based on 1,4-benzenedithiole by varying the gate potential V$_{G}$ within − 3 . 3 V < V$_{G}$ < 0 and − 3 V < V$_{G}$ < 3 V , respectively. In the presently considered case of half filling, the number of sites of the extended molecule N = M$_{L}$ + M$_{R}$ + 1 must be even. This implies that M$_{L}$ is odd and M$_{R}$ is even or vice versa, and therefore it is impossible to consider perfectly symmetrical extended molecules. The numerical results presented below correspond to the 10