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I know hydrogen is a non metal, but when I just study about some introductory elementary band theory I find the band structure of hydrogen has a half filled valence band just like alkali metals, and this should make it conductive. Is there some thing more complex about it or is my understanding is wrong? How can the non metallic nature of hydrogen be explained using band theory?

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Solid hydrogen is not made up from hydrogen atoms. It contains hydrogen molecules held together by relatively weak Van der Waals forces. In the hydrogen molecule the two $1s$ levels of the atoms form a bonding $\sigma$ molecular orbital that is full and an antibonding $\sigma^*$ orbital that is empty:

Hydrogen molecular energy levels

In the solid the $\sigma$ level broadens to form a full valence band and the $\sigma^*$ level broadens to form an empty conduction band, with a gap of around 15eV between the two bands. That is why solid hydrogen is an insulator.

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    $\begingroup$ Metallic hydrogen ostensibly does exist at extremely high pressures. $\endgroup$ – Emilio Pisanty May 21 at 4:49
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    $\begingroup$ Does the bands overlap when hydrogen becomes metallic... $\endgroup$ – Grace mars Mars May 21 at 5:05
  • $\begingroup$ @GracemarsMars yes. If you compress solid hydrogen you reduce the band gap. I don't think we have ever compressed it enough to make it metallic, though as I recall we have measured a decrease of the band gap to less than 1eV so we've managed to make solid hydrogen semiconducting. Everyone expects the band gap to fall to zero at higher pressure but those pressures are higher than we can currently achieve in the lab. $\endgroup$ – John Rennie May 21 at 5:25
  • $\begingroup$ @GracemarsMars The Wikipedia page on metallic hydrogen has a good listing of all the ways that people have tried to get to the pressures at which hydrogen might become metallic ─ as well as the reasons why each of them have been disputed. $\endgroup$ – Emilio Pisanty May 21 at 7:22
  • $\begingroup$ They have compressed it (in the form of $H_2 S$) enough to turn it into a BCS superconductor arxiv.org/abs/1506.08190 $\endgroup$ – tbt May 21 at 18:58
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In solid hydrogen the electrons are localised in H2 bonds. There are various crystal structures. The band structure of course depends on this. Upon very strong compression the crystal structure should change to a high coordination characteristic of a metal. The H2 bonds then disappear. So the behaviour is more complex than just a shift of bands. It is a structural and electronic phase transition. Search for metal-insulator transition.

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