Potentials and Basis Sets in Solids

If we know the translational symmetry of an extended solid (Bloch s theorem) and also have a trustworthy strategy on how to deal with the nuclear potential and the electron-electron interactions (by, say, a semiempirical method or by DFT), we are ready to explicitly calculate the band structure of any real material. Although we have done this before for idealized systems (see sketches in Section 2.6), let us now attack the problem once again, but in more general terms. For real materials, one needs to solve SchrBdinger s equation using the true potential v r), namely. 

K is a reciprocal lattice vector (see Section 2.5), and the mixing coefficients Cn must be sought, either analytically or numerically. 

We reiterate that, just as in LCAO-MO theory, the originally unknown function tp k,r) is expressed by a set of known functions, but the latter are entirely delocalized and not localized as atomic orbitals. While this ansatz ultimately leads us away from the notion of a solid-state material being composed of atoms with their associated atomic functions (this ansatz is a very 

29) For a given atomic orbital, the number of nodes is - / - 1 where n and I stand for the main and angular-momentum quantum numbers. Thus, the 3s atomic orbital must have 3-0-1 = 2 nodes. 

Three different approaches have been followed to solve this clue, and they form the backbones of all existing band-structure methods in terms of their nuclear potentials. Somewhat simplified, one may either ignore the core functions (empirical tight-binding approaches), one may modify the potential, thereby also ignoring the core functions (pseudopotential approach), or one may modify the basis sets and split the functions into core and beyond-core functions (cellular approaches and successors) [210]. 

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FIGURE 34.4 Fukui functions determined using Equation 34.18. Long-dashed lines are determined using conventional calculations (no potential wall) with the aug-cc-pVTZ basis set dotted lines use conventional calculations with the optimally compact basis sets identified in Table 34.4 solid lines use the potential wall approach of Section 34.5, with the aug-cc-pVTZ basis. (Reprinted from Tozer, D J. and De Proft, F., J. Chem. Phys., 127, 034108, 2007. With permission.)... 

FICs are useful as electrochemical sensors, electrolytes and electrodes in batteries and in solid state displays (Farrington Briant, 1979 Ingram Vincent, 1984). If a FIC material containing mobile M ions separates two compositions with different activities of M, a potential is set up across the FIC that can be related to the difference in the chemical activities of M. By fixing the activity on one side, the unknown activity on the other can be determined. This principle forms the basis of a number of ion-selective electrodes LaFj doped with 5% SrF2 is used for monitoring fluoride ion concentration in drinking water. Similarly, calcia-stabilized-zirconia is used in cells of the type... 

Figure 1. Solid lines are contours of the potential energy surface for the H + H2 - H2 + H reaction. Broken lines are contours of the absorbing potential (which is zero in the central part of the interaction region and turned on at the edge) for three possible choices of it. The points are the grid points that constitute the basis set for the evaluation of the quantum trace, Eq. (2.5).

Similarly, expanding the KS potential in an LCAO expansion makes molecular density-functional calculations practical [9]. For metals and similar crystalline solids, it is best to expand the Kohn-Sham potential in momentum space via Fourier coefficients. For molecular solids various real-space method are under investigation. For molecules studied with the big, well-chosen Gaussian basis sets of quantum chemistry, it is undoubtedly best to expand the KS potential in linear-combination-of-Gaussian-type-orbital (LCGTO) form [10]. 

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... 

One important information coming from calculations is the structure of the oxide surface. Oxide surfaces are often heavily reconstructed or simply relaxed compared to the truncated bulk, and the experimental determination of the surface structure is often not easy. To this end, reliable classical potentials have been developed, in particular for the study of ionic crystals and of covalent solids. Nowadays, first principle band structure calculations making use of large supercells can also be used. These methods, although quite expensive from the point of view of the size of the calculations, provide results which are in excellent agreement with the experimental determinations. Band structure calculations, usually based on plane waves basis sets and on the density functional (DFT) approach represent the most appropriate eomputational... 

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