One-Electron Electrostatic Hamiltonian

If H is the one-electron electrostatic Hamiltonian, based on the Born-Oppenheimer approximation, the solutions to Eq. [23] are called crystalline orbitals (CO). They are linear combinations of one-electron Bloch functions (Eq. [8]) 

Atomic orbitals (AO) and plane waves are common choices to represent Bloch functions. Both choices would be equivalent, in principle, if an infinite basis set was considered, but they are not equivalent in the practical case of a finite 

Any method of solution of Eq. [25] is specific of the kind of basis set used. In the remaining part of this chapter, we will always refer to the use of one-electron local basis sets within the linear combination of atomic orbitals (LCAO) method. Accordingly, nf AOs in the 0-cell are chosen and replicated in the other cells of the crystal to form the periodic component u r k) of nf Bloch functions. In particular, by denoting the p-th AO, with the origin at in the 0-cell, as x (r and the corresponding AO in a different cell, the g-cell, as x (r — r — g) or, equivalently, x (r — r ), the expression used for M (r k) consists of a linear combination of the equivalent AOs in all N cells of the crystal  

The translation invariance of k) is obvious because the sum is extended to all cells in the crystal. In fact, if a translation by lattice vector I is applied 

M (r k) is verified to be periodic throughout the direct lattice (the equivalence of the sum over lattice vectors m = g -I-1 and the sum over g originates from translation invariance and the periodic boundary conditions). 

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