Basis localized atomic-like orbitals

The choice of the basis set is of particular importance when treating periodic systems where a large variety of chemical bonding can be found. The following three approaches to the basis-set choice define three types of methods of the electronic-structure calculations in crystals [10] atomic-sphere (AS) methods, plane-wave (PW) methods, localized atomic-like orbitals (LCAO) methods. Each method has its advantages and disadvantages. 

The implementation of efficient EXX methods in local atomic-like basis set (e.g. Gaussian type orbital) for molecular systems is not a straightforward task due to several numerical shortcomings, still under investigations after ten years since the first implementation. To avoid these numerical shortcomings, several different approximations and other derivations have been presented in the last decade, which will be discussed in subsections 3.6 and 3.7. In Section 4 some of the main orbital-dependent implementations available for finite systems will be discussed. Finally conclusions and future perspectives are drawn in Section 5. 

At present, the electronic structure of crystals, for the most part, has been calculated using the density-functional theory in a plane-wave (PW) basis set. The one-electron Bloch functions (crystal orbitals) calculated in the PW basis set are delocalized over the crystal and do not allow one to calculate the local characteristics of the electronic structure. As a consequence, the functions of the minimal valence basis set for atoms in the crystal should be constructed from the aforementioned Bloch functions. There exist several approaches to this problem. The most consistent approach was considered above and is associated with the variational method for constructing the Wannier-type atomic orbitals (WTAO) localized at atoms with the use of the calculated Bloch functions. Another two approaches use the so-called projection technique to connect the calculated in PW basis Bloch states with the atomic-like orbitals of the minimal basis set. 

As has already been discussed above, the local basis states Rsu) play a key role in any local orbital scheme. In the RFPLO method they are calculated as the solutions of an atom-like single particle Dirac equation (33) in a spherical, orbital dependent potential (34) and XC-field (35). The atom-like potential U/ contains the spherically averaged crystal potential V/ around the lattice site s, which ensures that a Bloch sum of the core orbitals and the related core eigenvalues are very good approximations to the solutions of the true crystal Hamiltonian. For the calculation of valence states an additional attractive r -potential, acting on the large components only, is applied. 

Plane wave basis sets of reasonable size (of the order of 100 plane waves per atom) can be used in surface calculations of semiconductors like Si, Ge, GaAs (Hebenstreit et al., 1991) and s-p bonded metals like Al (Needs and Godfrey, 1990). For transition and noble metals the d-elcctrons which participate in the bonding are relatively localized and correspondingly feci a stronger potential than the s- and p-electrons. A plane-wave basis for such systems is unwieldy, and localized Gaussian orbitals of the form ... 

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