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I understand that metals have overlapping of valence and conduction bands. But is this because there exists a partial conduction band within the top part of a metal valence band, or because the conduction band exists, but periodically in the valence region? Is this enabled by the formation of 'covalent' molecules in the metal structure matrix?

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I'm not sure that "overlapping of valence and conduction bands" is a particularly helpful way to think about metals. Band structure doesn't come with a set of labels saying "this is the valence band, this is the conduction band". As a first approximation we can think of the band structure as determined by the potential landscape of the atomic nuclei, and then fill it up with the appropriate number of electrons. If the Fermi level ends up in the middle of a band -- so that you have a Fermi surface in reciprocal space -- then you can expect metallic behaviour.

(This is essentially the nearly free electron gas model, incidentally, which is certainly an oversimplification of reality -- I can go into details about how if you like -- but is the simplest model capable of reproducing the qualitative features of band structures.)

As for the formation of "covalent molecules": this is not usually the way we would describe metallic bonding. When people talk about covalent bonds they are usually referring to models where electrons are relatively localised. Of course there are some advanced cases (I am thinking of things like graphene) where the boundaries are blurred somewhat, and indeed you can derive metallic band structure from the tight binding model which essentially starts from forming covalent bonds between adjacent pairs of atoms. But my point stands that most physicists would not describe the bonding in, e.g., Na as in any way covalent.

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So what's the difference between graphene and Na while they can both be metallic while graphene is well described by covalent bonding, and, Na not? –  Laurent Feb 1 '13 at 18:29
@Laurent in graphene there are electronic states that are localised between two nuclei, which is what we normally think of as a covalent bond, as well as states that are delocalised across the plane and those that are localised close to atomic nuclei. In sodium and other "typical" metals, ideally there are only delocalised states and states localised close to the nuclei, which is why we wouldn't normally describe this bonding as covalent. But at least in theory you don't have to decide beforehand what sort of bonding you have, just see what electronic states the Schrodinger equation gives you. –  Ant Feb 4 '13 at 18:03

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