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I heard that $\ce{H2}$ is the only example of s–s orbital overlap. Can anyone give an example which contradicts this statement?

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    $\begingroup$ Metallic Li will do. $\endgroup$ – Ivan Neretin Jun 3 at 14:31
  • $\begingroup$ Are you suggesting that 1A and 2A mettalic bonds are all masses of covalent compound? $\endgroup$ – NightKruger Jun 3 at 14:38
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    $\begingroup$ What's 1A and 2A? Also, no, a metal is definitely not a covalent compound. But there still is certain orbital overlap in it. $\endgroup$ – Ivan Neretin Jun 3 at 14:40
  • $\begingroup$ @IvanNeretin I meant the groups in modern periodic table. $\endgroup$ – NightKruger Jun 3 at 15:39
  • $\begingroup$ Oh, of course. Well, they all are as good as Li in this regard. $\endgroup$ – Ivan Neretin Jun 3 at 15:40
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You don't need particularly exotic things to get a counterexample. Metal–metal bonding in mercury(I) compounds is quite prevalent, where the electronic configuration is $\ce{Hg+}$: $\ce{[Xe](4f^14)(5d^10)(6s^1)}$, and the bonding involves 6s–6s overlap. The prototypical example is the $\ce{Hg2^2+}$ cation, which features in mercury(I) chloride $\ce{(Hg2^2+)(Cl^-)2}$; this cation is valence isoelectronic to $\ce{H2}$.

Fundamentally, it is a relativistic contraction of the 6s orbital which makes such bonding common in Hg(I): see also Why does mercury form polycations?. The bonding in alkali metal dimers is qualitatively similar, but significantly weaker.

If we play the "every bond has orbital overlap" card (i.e. not just covalent bonds, which I think is perfectly fair), then the delocalised bonding in Group 1 and 2 metals arises from overlap of ns orbitals on each atom to form bands, as has already been alluded to in the comments. Even simpler would be lithium hydride, which likely has some degree of 1s–2s orbital overlap, even though the bonding may be primarily ionic.

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Hydrogen forms, in addition to $\ce{H2}$, ion clusters of varying size. Following are two prominent examples.

The trihydrogen cation, $\ce{H3^+}$, is found in the interstellar medium and in the atmospheres of Jovian planets. The three-center, two-electron bond that binds this ion together simultaneously covers all three atoms and all three atom-atom linkages, giving a strongly stabilized structure.

The hexahydrogen ion, $\ce{H6^+}$, mentioned in the first reference above, has a turnstile structure in which three dihydrogen units are linked together with three mutually orthogonal orientations; alternatively it may be viewed as two tri-hydrogen triangles joined vertex to vertex. This species can be formed in solid hydrogen, where it migrates through an exchange mechanism similar to the Grotthus mechanism of proton conduction in water and other protic solvents.

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