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I have another question relating to band theory, wonder if anyone can help?

If I am correct, in band theory we have this idea of creating N different energy states for N atoms that are in a system (with a sigma and sigma* orbital being created for each orbital that the one atom had) The electrons from the N atoms populate these orbital from the lowest energy state, up. And with a large number of atoms (as you would expect in a normal system) you have so many energy states that this ends up with a wider number of states that we call a band. The 'band' that is full/almost full is the valence band and the band that is empty/almost empty is the conduction band.

As far as I know, for a metallic-type conductor we have the situation where the energy states of the valence band and the conduction band overlap so electrons can jump relatively freely (and hence allow for current flow to occur/conduct).

My question is: are the metallic and valence bands, sigma and sigma* bands? Or instead would they, for instance, be an s band and a p band? or a $p_x$ and a $p_y$ orbital? Or any of the above?

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    $\begingroup$ The solutions to Schroedingers equations in a Crystal are Bloch functions that extend throughout the crystal. There is no one-to-one correspondence to atomic orbitals. $\endgroup$
    – Jon Custer
    Jul 10, 2020 at 13:49

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As @JonCuster has already noted in the comments, the principled way to calculate energy bands for a crystal is by solving the Schrödinger equation in the periodic potential, exploiting its translational and other symmetries.

One of the approximate methods for the band structure calculation - the tight binding approximation - indeed constructs the bands by taking the orbitals of the separate atoms as zeroth approximation and adding perturbatively the hopping between these orbitals. Since this is a popular approach, one frequently discuss the band structure in these terms, although it is just an approximation.

When the tight-binding approximation is applicable, the bands can be traced to the original s, p, d, etc., although there is necessarily admixture of other orbitals. Which of them are conduction or valence bands depends on the material parameters: hopping integrals, lattice symmetry, etc.

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