Why Can Tight-Binding and DFT Models Assume Electrons are Non-Interacting?

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Tight-binding and density functional theory (DFT) models often assume electrons are non-interacting to simplify calculations and focus on the electronic structure of materials. This assumption allows for effective modeling of systems without the complexity of Coulombic interactions, which are indeed significant. The justification lies in the ability of these models to capture essential physical properties while treating electron-electron interactions in an averaged manner. DFT, for instance, expresses energy in terms of electron density, which can effectively account for interactions without explicitly modeling them. These approximations are valid within certain limits, enabling practical applications in material science.
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What is the justification behind tight-binding, nearly free electron drude model, to assume electrons as non-interacting? Because it is clearly not the case when considering coloumbic interactions, why can these models do this? Coloumbic interactions are strong so I wouldn't consider them to be idealized.

Or why is density functional theory valid? How is it still valid by expressing things in terms of its electron density while neglecting couloumbic interactions?
 
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Nevermind I understand now.
 
Thread 'Motional EMF in Faraday disc, co-rotating magnet axial mean flux'

Thread 'Motional EMF in Faraday disc, co-rotating magnet axial mean flux'

So here is the motional EMF formula. Now I understand the standard Faraday paradox that an axis symmetric field source (like a speaker motor ring magnet) has a magnetic field that is frame invariant under rotation around axis of symmetry. The field is static whether you rotate the magnet or not. So far so good. What puzzles me is this , there is a term average magnetic flux or "azimuthal mean" , this term describes the average magnetic field through the area swept by the rotating Faraday...
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