Crystalline orbitals

Morita, A., and Takahashi, K., Progr. Theoret. Phys. [Kyoto) 19, 257, Theory of cohesive energy of LiH crystal—the method of semilocalized crystalline orbitals."... 

When an external electric field is applied along the periodicity axis of the polymer, the potential becomes non periodic (Fig. 2), Bloch s theorem is no longer applicable and the monoelectronic wavefunctions can not be represented under the form of crystalline orbitals. In the simple case of the free electron in a one-dimensional box with an external electric field, the solutions of the Schrddinger equation are given as combinations of the first- and second-species Airy functions and do not show any periodicity [12-16],... 

The subscript labels a, b,... (i, j,...) correspond to unoccupied (occupied) bands. The Mulliken notation has been chosen to define the two-electron integrals between crystalline orbitals. Two recent studies demonstrate the nice converging behaviour of the different direct lattice sums involved in the evaluation of these two-electron integrals between crystalline orbitals [30]. According to Blount s procedure [31], the z-dipole matrix elements are defined by the following integration which is only non zero for k=k ... 

Figure 4 Conduction band levels and excitation levels of infinite periodic hydrogen chains by using different approximations of the polarization propagator. The left part refers to the crystalline orbital energy differences, namely, the Hartree-Fock excitation energies the right part refers to the random phase approximation results obtained by using 41 k-points in half the first Brillouin zone.

The first computational consideration is that of obtaining the solutions of the unperturbed problem, Eq. (15), and the approach taken in the present study is to utilize the Crystal program [1] as it has been successfully used for studies in molecular crystals [10-12,15], A given crystalline orbital, (k,r), such as that required for the matrix elements necessary given by the integral in Eq. (16), is expressed as a linear combination of Bloch functions, a ,(k) and atomic orbitals, (k,r) [1]... 

For any molecule, including polymers, the LCAO approximation and Bloch s theorem can be used to describe the delocalized crystalline orbitals as a periodic combination of functions centered... 

An alternative approach is to describe the valence electrons by localised functions. In the linear combination of crystalline orbitals (LCCO) which is similar to the LCAO approach to molecules, the valence electrons are represented by sums of atomic orbitals. As well as being in some respects more familiar, this method is capable of all-electron calculations and in principle should be better for properties which need a good description of core electrons. For some of these properties however, extensions to the plane wave method are available which can cope with inner electrons around nuclei of interest. At present, the performance of both approaches is comparable. 

Since the concepts of atoms and bonds are central to chemical understanding, approaches based on atom-additivity and bond-additivity are very appealing. Due to their simplicity, they were used in the early days for actual calculations, but nowadays they continue to be employed for interpretative purposes. Needless to say, their accuracy can be surpassed by methods based on quantum mechanics. As with field-free isolated molecules, early models used to estimate second- and third-order macroscopic nonlinear responses considered such simple schemes. In the following, we describe methods that treat either chemical bonds or atoms as the central quantities for evaluating the bulk NLO responses. The philosophy consists in incorporating in the description of these central constructs the effects of the surroundings. In this way the connection with more elaborate methods, such as the oriented gas model that focuses on one molecule with local field factor corrections, or with the crystalline orbital approach that reduces the system to its unit cell, is more obvious. In what follows, a selection of such schemes is analyzed and listed in Table VII. 

F(k) and S(k) are the Fock and overlap matrices between Bloch functions and C(k) collects the coefficients of the linear combinations of Bloch functions that provide the crystalline orbitals. 

SIC, or self-interaction corrected LDA [21, 22], and more recently the exact exchange (EXX) functionals [23], both of which include a dependence of the exchange functional on the occupied molecular or crystalline orbitals. 

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]

In principle, this electron correlation strategy is transferable from single molecules to solids, after the crystalline orbitals have been transformed to an equivalent set of well-localized functions (Wannier functions). Procedures for orbital localization have been proposed and implemented only recently, and the first MP2 calculations are becoming possible in the case of simple crystalline compounds. 

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])... 

Figure 10 Representations of bonding 4 +(r) and antibonding (r) n-crystalline orbitals at r in graphite. Black and gray circles represent the positive and negative signs of each AO in the linear combination, respectively.

Figure 11 Representation of bonding and antibonding re-crystalline orbitals at M in graphite.

Figure 12 Representation of bonding and antibonding n-crystalline orbitals at K in graphite.

Figure 13 Representation of bonding and antibonding Jt-crystalline orbitals at (, 0) in graphite.

Symmetry Adapted Crystalline Orbitals in SCF-LCAO Periodic Calculations. I. The Construction of the Symmetrized Orbitals. 

The quantum molecular model under Hiickel approximation is quite general as to allow the enlargement of the discourse from molecules to the crystalline orbitals the present discussion follows (Putz, 2006)... 

The connection between the orbital crystalline form (3.35) and the eigen-energies associated with Eq. (3.19) consists in customizations of the k values into the so-called k-points, which specifies the various types of crystalline orbitals. For example, one can immediately evaluate the crystalline orbitals corresponding to the extreme -points for the zone of continuous variation in Figure 3.4, namely =0 and k=n/a, which generates the crystalline orbital for the zone center (noted by F) with the superposition of atomic orbitals, all in phase (Putz, 2006) ... 

The visualization of these t5q)es of crystalline orbitals from the linear series of atomic orbitals in Eqs. (3.36) and (3.37), for various types of atomic orbitals, is shown in Table 3.2 (Putz, 2006). 

TABLE 3.2 Types of Crystalline Orbitals (CO) in the Center (X) and at the Frontier (T) of the k-Zone of the Crystalline Eigen-Function for the Various Types of Basic Atomic Orbitals (AO) after (Further Readings on Quantum Crystal 1940-1978)... 

Figure 3.6 illustrates the layout of these crystalline orbitals of bonding/ anti-bonding in relation with the associated eigen-energies of Figure 3.4 (Putz, 2006). 

This way, the crystalline orbital should be constructed from the atomic ones, (j)(r located at the distance respecting the origin of the reference system of (direct) lattice, so that also the Bloch theorems (3.88) and (3.89) to be respected for including the periodicity of the lattice in crystalline eigen-function. 

Under these conditions, the crystalline orbital can be written as a complete LCAO expansion of basic Bloch factors, see (Further Readings on Quantum Solid, 1936-1967)... 

The electrons migration between two solids in contact raises the question do the electrons of crystalline orbital s have amotion inside the solid body that they belong, in the absence of any contact potential ... 

If the electrons would have their own motion on the valence band (VB), they should be characterized also by a velocity. However, since the electrons have a quantum nature depending on their occupancy on Bloch-Schrodinger crystalline orbitals (3.101) they should be represented by the associated wave package, further characterized by the group velocity correlated with the quantum eneigy as such (Putz, 2006) ... 

By consideration of the hexagonal and trigonal structures we conclude the discussion of the structure definitions by space groups and Wyckoff positions. In the next chapter we consider the symmetry of crystalline orbitals, both canonical and localized. 

Itanslation and Space Symmetry of Crystalline Orbitals. Bloch Functions... 

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