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Why does $\ce{[C(NH3)2]}$ not exist as a complex? Here, $\ce{C}$ is carbon and the configuration of $\ce{C}$ in ground state is $\ce{[He] 2s^2 2p^2}$.

Can't the two $\ce{NH3}$ molecules pair the two $\ce{p}$ electrons of carbon and form coordinate bond with carbon?

Why does $\ce{C2}$ (one carbon atom bonded with other through quadruple bonds) not exist? Two $\ce{2p}$ orbitals of $\ce{C}$ should form π bonds and one $\ce{2p}$ and one $\ce{2s}$ should form σ bonds with the other $\ce{C}$ (without hybridisation).

I'd appreciate an answer for a beginner.

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Asking multiple unrelated questions at once is considered a bad practice, but let's forget that for a moment.

Your first molecule has more or less satisfactory electronic configuration, but that's not enough for the long-term survival. N would be more than willing to relinquish a proton, and C to accept it, and thus in no time you will have $\ce{H2N-CH2-NH2}$, which then may or may not transform into something else.

Your second molecule has been discussed here before on multiple occasions. Long story short, it does exist, but doesn't have a quadruple bond. (Such bonds are known, but not in $\ce{C2}$; they have a delta component, and that requires d orbitals). The problem with your description is that the sigma-oriented 2p orbital of one carbon will interact not just with the similar 2p, but also with the 2s orbital of the other. If not for this fact, it would work out pretty much as you expected.

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We may regard oxyanions as complexes with oxide or oxo ligands; indeed such an approach may offer advantages in understanding the electronic structure of some of these ions. In that sense carbonate ion is indeed a complex with carbon as a central atom.

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