Density functional theory crystal orbital structures

At present, the electronic structure of crystals, for the most part, has been calculated using the density-functional theory in a plane-wave (PW) basis set. The one-electron Bloch functions (crystal orbitals) calculated in the PW basis set are delocalized over the crystal and do not allow one to calculate the local characteristics of the electronic structure. As a consequence, the functions of the minimal valence basis set for atoms in the crystal should be constructed from the aforementioned Bloch functions. There exist several approaches to this problem. The most consistent approach was considered above and is associated with the variational method for constructing the Wannier-type atomic orbitals (WTAO) localized at atoms with the use of the calculated Bloch functions. Another two approaches use the so-called projection technique to connect the calculated in PW basis Bloch states with the atomic-like orbitals of the minimal basis set. 

The fourth aspect of molecular structure, after connectivity, symmetry and geometry, is electron density. This is at the heart of the concept of chemical bonding, and is important in the interpretation of data from both experiments and quantum mechanical calculations. On the one hand, high-resolution X-ray diffraction of well diffracting crystals can provide us with three-dimensional electron density maps (Section 10.9), and on the other ab initio molecular orbital theory and density functional theory allow us to simulate them directly using first-principles calculations (Section 3.6). Either way, we get information with a real physical meaning. 

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