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write the part H$_{M}$ that specifies the extended molecule 

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and 

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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 

