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I've recently learnt about Wannier functions and changing basis from wave-like 'Bloch functions' to seemingly more localised 'Wannier functions' and how one can think of electrons equally as delocalised waves in solids or as in more localised bonds.
To complete these equality, I was wondering if there is any obvious in the band structure of a covalent substance that causes the Wannier functions to be so directional like covalent bonds are?
The band structure is the relation between energy and momentum. It will take the actual crystal orbitals to specify the actual wave function. From these you can control the Wannier orbitals. However, band structure calculations are based on translational symmetry and in this formalism it is not possible to include local correlations that are important in directional, covalent bonds.
There is no direct correlation between the band structure of a covalent material and the directional character of the corresponding Wannier functions. The reason is quite general. There is no direct relation between the spectrum of the eigenvalues and the spatial properties of the eigenfunctions. The only thing which is possible to establish easily is how much the band structure differs from that of a free electron gas. However, it is enough to have a look at the band structure of a few insulators, semiconductors, $s-p$-bonded, and transition metals, to realize how difficult may be to establish a robust connection between band structure and spatial properties of wavefunctions, Wannier functions included.
If one takes into account that the characteristic signature of an insulator is the exponential decay of the Wannier functions, it should be clear that there is no chance of establishing a connection between this property of a linear combination of Bloch's states and their eigenvalues.
