Bloch basis functions

As applied to CPs, the EH method takes a set of basis orbitals for the atomic constituents of a unit cell, x, and forms the set of Bloch basis functions ... 

Both HF and DFT calculations can be performed. Supported DFT functionals include LDA, gradient-corrected, and hybrid functionals. Spin-restricted, unrestricted, and restricted open-shell calculations can be performed. The basis functions used by Crystal are Bloch functions formed from GTO atomic basis functions. Both all-electron and core potential basis sets can be used. 

The band structure and Bloch functions of metals have been extensively published. In particular, the results are compiled as standard tables. The book Calculated Electronic Properties of Metals by Moruzzi, Janak, and Williams (1978) is still a standard source, and a revised edition is to be published soon. Papaconstantopoulos s Handbook of the Band Structure of Elemental Solids (1986) listed the band structure and related information for 53 elements. In Fig. 4.14, the electronic structure of Pt is reproduced from Papaconstantopoulos s book. Near the Fermi level, the DOS of s and p states are much less than 1%. The d states are listed according to their symmetry properties in the cubic lattice (see Kittel, 1963). Type 2 includes atomic orbitals with basis functions xy, yz, xz], and type e, includes 3z - r-), (x - y ). The DOS from d orbitals comprises 98% of the total DOS at the Fermi level. 

Using the same type of LCAO expansion as for molecules (see eq. (2.8)), the crystal orbitals are expanded as Bloch sums of the basis function centred at site fi in cell / ... 

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... 

The muffin-tin orbital xa( — R) with its origin at a given site R contains a(— —, 1" — R) and from all other spheres at R R) A i,r — R ). As a final step from individual spheres to the periodic crystal we make the basis functions as a Bloch sum of xa ... 

For periodic systems, the GTO solution [24, 27] begins by defining Bloch-symmetrized gaussian basis functions for each primitive function Gy of... 

In the "tight-binding" or LCAO method [1.12] one uses as basis functions the eigenfunctions xn m of The bound states of the free atom, and the wave function for an electron in the solid is then expressed in terms of the Bloch sums... 

Alternatively, one may base the discussion on the Bloch waves constructed from the atom-centered basis functions. 

The discussion makes it obvious that it is a far from trivial problem how to treat r in an actual calculation. Therefore, we decided to first study the Hiickel-like model numerically, using two different approximations for r. In one case we used the full potential of Fig. 1(c), whereas in the other case we considered only the periodic part of Fig. 1(b). As discussed above, the latter case corresponds to use the separation of equations (13) and (15) applied on the Bloch waves formed by the basis functions (i.e., not the eigenfunctions). 

In our implementation of the method we consider infinite, periodic, isolated polymer chains. The periodicity is utilized in constructing Bloch functions from the basis functions of different unit cells. 

In this contribution we have concentrated on presenting some fundamental considerations concerning die flieoretical treatment of an infinite, periodic, polymeric chain being exposed to an external electrostatic field. The analysis of a simple Huckel-like model revealed that only under certain circumstances one can base the discussion on Bloch functions and substituting f with a derivative wifli respect to k. Thus, this was not the case when including the field directly in the calculations, but could, e.g., be used when using Wannier functions as basis functions which are continuous functions not only of r but also of k. 

The localised single-exdton wavefunctions are not eigenfunctions of the crystal Hamiltonian. A wavefunction appropriate to the crystal symmetry and the periodic potential can be found using the Bloch-wave ansatz. From the localised basis functions <1>, one obtains the delocalised wavefunction... 

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. 

The reason for this can be viewed in two ways. First, since the Bloch operator b.c. s effectively constrain the problem to a "box" of size A, the "translational component" of the eigenvalues will increase eventually with the square of the number of translational basis functions, m, and the series converges only as l/m. The second view is that the true wavefunctions will not have zero derivative at R = A, thus requiring many basis functions to permit ip to be adequately represented, in a mean square sense, by the basis. 

Bloch orbitals for the two extremes of the first Brillouin zone for a chain of Li atoms (2s AO basis functions) (a) k = Q, (b) k = 7tla. The shaded spheres represent the positive sign of the wavefunction, while the hollow spheres represent the negative sign of the wavefunction. 

Bloch diagram for the first Brillouin zone resulting from the overlap of Is AO basis functions for a ring of H atoms. The bandwidth is given by 4. ... 

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