Bloch orbitals

In a crystal lattice where each atom contributes one atomic orbital, and where these orbitals are related to each other by the translations characteristic of the lattice, the molecular orbitals must belong to irreducible representations of the group of these translations and hence form so-called Bloch orbitals. 46)... 

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

The HF CO method is especially efficient if the Bloch orbitals are calculated in the form of a linear combination of atomic orbitals (LCAO)1 2 since in this case the large amount of experience collected in the field of molecular quantum mechanics can be used in crystal HF studies. The atomic basis orbitals applied for the above mentioned expansion are usually optimized in atoms and molecules. They can be Slater-type exponential functions if the integrals are evaluated in momentum space3 or Gaussian orbitals if one prefers to work in configuration space. The specific computational problems arising from the infinite periodic crystal potential will be discussed later. 

Crystal orbitals are built by combining different Bloch orbitals (which we will henceforth refer to as Bloch sums), which themselves are linear combinations of the atomic orbitals. There is one Bloch sum for every type of valence atomic orbital contributed by each atom in the basis. Thus, the two-carbon atom basis in diamond will produce eight Bloch sums - one for each of the s- and p-atomic orbitals. From these eight Bloch sums, eight COs are obtained, four bonding and four antibonding. For example, a Bloch sum of s atomic orbitals at every site on one of the interlocking FCC sublattices in the diamond structure can combine in a symmetric or antisymmetric fashion with the Bloch sum of s atomic orbitals at every site of the other FCC sublattice. 

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... 

Besides the mentioned aperiodicity problem the treatment of correlation in the ground state of a polymer presents the most formidable problem. If one has a polymer with completely filled valence and conduction bands, one can Fourier transform the delocalized Bloch orbitals into localized Wannier functions and use these (instead of the MO-s of the polymer units) for a quantum chemical treatment of the short range correlation in a subunit taking only excitations in the subunit or between the reference unit and a few neighbouring units. With the aid of the Wannier functions then one can perform a Moeller-Plesset perturbation theory (PX), or for instance, a coupled electron pair approximation (CEPA) (1 ), or a coupled cluster expansion (19) calculation. The long range correlation then can be approximated with the help of the already mentioned electronic polaron model (11). 

DOS = Density of states BO = Bloch orbital IBZ = Irreducible Brillouin zone BZ = Brillouin zone PZ = Primitive zone COOP = Crystal orbital overlap population CDW = Charge density wave MO = Molecular orbital DFT = Density functional theory HF = Hartree-Fock LAPW = Linear augmented plane wave LMTO = Linear muffin tin orbital LCAO = Linear combination of atomic orbitals. 

Hartree-Fock calculations on molecules commonly exploit the symmetry of the molecular point group to simplify calculations such studies on perfectly ordered bulk crystalline solids are possible if one exploits the translational symmetry of the crystalline lattice (see Ashcroft and Mermin, 1976) as well as the local symmetry of the unit cell. From orbitals centered on various nuclei within the unit cell of the crystal Bloch orbitals are generated, as given by the formula (in one dimension) ... 

Band-theoretical method using a LCAO expansion of the delocalized Bloch orbital (often with integral approximations)... 

Many materials can be studied within the Hartree-Fock scheme because the atomic, molecular or Bloch orbitals in the independent particle HF approximation are a very good first approximation to the actual electron-like quasiparticles of the system. The scheme breaks down in at least three special cases ... 

The simplest of all molecules is the ion H J since it possesses two nuclei and but one electron. If orbitals can be found to describe the states of the one electron in a two-centre field of this kind, it should be possible to develop an aufbau theory of diatomic molecules, exactly as in the one-centre instance (p. 57). And there is no reason why this should not be extended to a many-centre example. The one-electron orbitals, which extend over all nuclei, are called molecular orbitals or in the case of a crystal, crystal or Bloch orbitals (after Bloch who first used them). In the aufbau approach, the available... 

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. 

Besides Bloch orbitals, Wannier orbitals [76] are also widely used in solid state physics. They are defined as Fourier transformations of the Bloch orbitals, e.g. 

The calculation of the ground-state energy of the Wigner electron crystal necessitates the self-consistent solution of the Slater-Kohn-Sham equations for the Bloch orbitals of a single fully occupied energy band, since there is one electron per unit cell and one is concerned with the spin-polarized state [45], This was accomplished by standard computational routines for energy band-... 

Figure 25.10 Calculated eigenvalues and eigenstates lead to a series of vibrational states with quantum numbers (n, k) for the k = 0 state the Bloch orbitals of each branch were examined. At low energies, the respective orbitals are mainly confined around the fee, top, or hep sites of the Pt(l 11) surface. The figure shows for selected bands the localization/delocalization ofthe orbitals n = 1 (Fig. 25.9(a, b)) n = 3 (Fig. 25.9 (c, d)) n = 4 (Fig. 25.9 (e, f)) n = 26 (Fig. 25.9 (g, h)) n = 15 (Fig. 25.9 (k, I)), and n = 16 (Fig. 25.9 (m, n)), whereby the right sequence of graphs displays the probability density p(r) in transversal sections. Ten equidistant contour lines are used in each graph. After Badescu et al. [55].

The generalisation of this kind of procedure is the generation of so-called Bloch orbitals in solid-state theory, where translational symmetry will generate one symmetry orbital from amy orbital in the unit cell auid the variation problem is solved in terms of orbitals of the same symmetry type generated from different orbitals in the same unit cell. 

In order to make use of the fact that the orbitals of the unperturbed system provide good approximations to those of the perturbed system, we construct Wannier functions from the Bloch orbitals of the unperturbed state. The Bloch functions are given through... 

Alternatively stated, the Bloch theorem indicates that a crystalline orbital () for the nth band in the unit cell can be written as a wave-like part and a cell-periodic part ([Pg.114]

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