Bloch orbital basis sets

The one-electron wave function in an extended solid can be represented with different basis sets. Discussed here are only two types, representing opposite extremes the plane-wave basis set (free-electron and nearly-free-electron models) and the Bloch sum of atomic orbitals basis set (LCAO method). A periodic solid may be considered constmcted by the coalescence of these isolated atoms into extended Bloch-wave functions. On the other hand, within the free-electron framework, in the limit of an infinitesimal periodic potential (V = 0), a Bloch-wave function becomes a simple... 

Quantum Chemistry Simulation. The calculations discussed in the preceding section were DFT calculations, in which the electron density of the molecule was represented by a linear combination of atomic orbitals. This technique can be adapted directly to the description of periodic systems by the inclusion of periodic potentials and adaptation of the atomic orbital basis set to the periodicity of the system using Bloch s theorem ... 

Realization with basis sets for Bloch orbital expansions that are physically, analytically and/or practically motivated, and also systematically improvable and testable ... 

This equation is similar to the one we have seen when considering the H repeat unit, but this time it is expressed on the a(n) basis set, not on the [H1 ( )] one. Just by drawing the [CO17] orbitals for k = 0 and k = it/2d, it is easy to see that the ET(k) vs. k curve (lowest dashed band on the top part of Figure 6.8) has a pseudo-sinusoidal shape similar to that of Figure 6.6. Similarly, the interactions between the cr (n) orbitals are described by the Bloch function ... 

For each t)q)e of atomic orbital in the basis set, which is the chemical point group, or lattice point, one defines a Bloch sum (also known as Bloch orbital or Bloch function). A Bloch sum is simply a linear combination of aU the atomic orbitals of that type, under the action of the infinite translation group. These Bloch sums are of the exact... 

Valdemoro and Rubio proposed a geminal approach to treat covalent crystals [85]. They worked in a minimal basis set of Bloch orbitals and constructed /c-veclor dependent geminals. 

The Bloch orbital can be expanded into a basis set of plane wave functions (2 " ). 

Alternatively, the basis set can be chosen as a set of nuclear-centred (Gaussian) basis functions, from which a set of Bloch orbitals can be constructed. 

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 ... 

Bloch functions and Bloch diagrams for the p AO basis sets, where the p-orbitals lie (a) perpendicular to and (b) parallel with the linear chain connecting the atoms in the crystal. The greater overlap in (b) implies a larger bandwidth for (b) than for (a). 

Any CO will be a linear combination of such Bloch functions, each corresponding to a given x- This is equivalent to the LCAO expansion for molecular orbitals, the only difference is that we have cleverly preorganized the atomic orbitals (of one type) into symmetry orbitals (Bloch functions). Hence, it is indeed appropriate to call this approach as the LCAO CO method Linear Combination of Atomic Orbitals - Crystal Orbitals), analogous to the LCAO MO (cf. p. 362). There is, however, a problem. Each CO should be a linear combination of the for various types of X and for various k. Only then would we have the full analogy a molecular orbital is a linear combination of all the atomic orbitals belonging to the atomic basis set. ... 

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