Supercell approach

With such calculations one can approach Hartree-Fock accuracy for a particular cluster of atoms. These calculations yield total energies, and so atomic positions can be varied and equilibrium positions determined for both ground and excited states. There are, however, drawbacks. First, Hartree-Fock accuracy may be insufficient, as correlation effects beyond Hartree-Fock may be of physical importance. Second, the cluster of atoms used in the calculation may be too small to yield an accurate representation of the defect. And third, the exact evaluation of exchange integrals is so demanding on computer resources that it is not practical to carry out such calculations for very large clusters or to extensively vary the atomic positions from calculation to calculation. Typically the clusters are too small for a supercell approach to be used. 

While the supercell approach works well for localized systems, it is typically necessary to consider a very large supercell. This results in a plane-wave basis replicating not only the relevant electronic states but also vacuum regions imposed by the supercell. A much more efficient method to implement for investigating the electronic structures of localized systems is to use real space methods such as the recursion methods [27] and the moments methods [28], These methods do not require symmetry and their cost grows linearly with the number of inequivalent atoms being considered. For these reasons, real space methods are very useful for a description of the electronic properties of complex systems, for which the usual k-space methods are either inapplicable or extremely costly. 

The interpretation of spectral properties in oxides such as NiO, in particular the valence-band photoemission spectra and inverse photoemission data (McKay and Henrich, 1984) has proved controversial. However, recent calculations using a supercell approach have given results for NiO in good agreement with such spectroscopic data (Norman and Freeman, 1986). These ealculations reconcile the band picture of Terakura et al. (1984a,b) and the experimental studies that have indicated large values ( 8 eV) for the intra-atomic Coulomb integral (Hufner et al., 1984 MeKay and Henrich, 1984 Sawatzky and Allen, 1984). 

One way of introducing the interactions with more distant solvent molecules is to use a supercell approach (see, e.g., ref. 1). Then, a finite system including the solute and a smaller number of solvent molecules is repeated periodically in all three directions. Due to the periodicity, electronic-structure calculations can be carried through, but the drawback is that also the solute is repeated periodically. This means that if the repeated unit is too small, interactions between the solute molecules become non-negligible and may affect the results. 

In their simulations, they use a supercell approach with the cell of Fig. 22 repeated periodically in all three dimensions (with, however, some further layers of Pt atoms). All atoms were treated quantum-mechanically within the density-functional formalism. 

The electronic structure of solids and surfaces is usually described in terms of band structure. To this end, a unit cell containing a given number of atoms is periodically repeated in three dimensions to account for the infinite nature of the crystalline solid, and the Schrodinger equation is solved for the atoms in the unit cell subject to periodic boundary conditions [40]. This approach can also be extended to the study of adsorbates on surfaces or of bulk defects by means of the supercell approach in which an artificial periodic structure is created where the adsorbate is translationally reproduced in correspondence to a given superlattice of the host. This procedure allows the use of efficient computer programs designed for the treatment of periodic systems and has indeed been followed by several authors to study defects using either density functional theory (DFT) and plane waves approaches [41 3] or Hartree-Fock-based (HF) methods with localized atomic orbitals [44,45]. 

In the present review we will discuss the current status of the quantum-chemical treatment of the adsorption of small molecules on oxide surfaces. We will limit our attention to oxide surfaces, because the problems encountered here are quite different from those connected with the treatment of metal surfaces. There are essentially two approaches to deal with a system that consists of a small molecule and an extended solid surface, i.e., a local process on a semi-infinite substrate. One way is the cluster approach described in the following in which a small cluster of atoms is cut out of the surface and the system molecule and cluster is treated as a supermolecule with the methods of molecular quantum chemistry. The alternative way is the supercell approach , in which the adsorbed molecule is repeated periodically on the surface, and the system surface with an ordered overlayer of adsorbed molecules is treated by means of periodic calculations. 

While plane wave basis sets have primarily been used for periodic systems, they can also be used for molecular species by using a supercell approach, where the molecule is placed in a sufficiently large unit cell such that it does not interact with its own image in the neighbouring cells. Placing a relatively small molecule in a large supercell to avoid self-interaction consequently requires many plane wave functions, and such cases are handled more efficiently by localized Gaussian functions. A three-dimensional periodic system, on the other hand, may be better described by a plane wave basis than with nuclear-centred basis functions. 

The model. Many different models can be proposed for the simulation of a single physical or chemical phenomenon. For example, a point defect in a crystalline system can be simulated either with a finite cluster with a defect at the center of the cluster and by assuming that the cluster is big enough and border effects are small, or with a periodic supercell approach, with the defects repeated periodically in such a way that the defect-defect interaction is small, if the supercell is big enough. 

Modeling different coverages is of interest in adsorption processes and can be achieved easily by enlarging the underlying surface unit cell (i.e., within a supercell approach), so that the density of adsorbed molecules can be increased or reduced. In the limit of low coverage, lateral interactions tend to vanish and adsorbed molecules can be considered as isolated. 

In the supercell approach, the defect is instead enclosed in a sufficiently large unit cell and periodically repeated throughout space. A common problem with both approaches is the availability of high-level quantum-mechanical periodic solutions, because, as already mentioned, it is difficult to go beyond the one-electron Hamiltonian approximations (HF and DFT), at present. 

In the following pages, we illustrate in more detail the supercell approach and discuss a few examples. 

The supercell approach consists of a periodic replica of the defect, which is enclosed in a large nonprimitive unit cell. A pictorial view (in 2-D) of the supercell approach is given in Figure 47, where, by starting from the perfect... 

Within the supercell approach, AE" should tend to a well-defined limit with increasing the supercell size ... 

A third example, carbon substitution in bulk silicon will compare the cluster and supercell approaches. 

As a last example, we consider another simple defect the carbon substitution in bulk silicon. " In this case, however, we will not only consider the convergence properties of the supercell approach but also compare the results of the cluster and supercell schemes. Calculations were performed at the HF level with a 6-21G basis set plus polarization fimctions for C and Si and a 2-lG basis set for H (the latter was used in the cluster calculations). 

The defect formation energies for the unrelaxed clusters are similar to those calculated by the supercell approach, even in the case of small clusters. The same screening mechanism, already discussed for the supercell models, is active also in the cluster calculations. 

These examples show that the supercell approach is an accurate and, in many cases, relatively cheap tool for the study of neutral defects in crystalline systems, once properly gauged with respect to supercell size. 

The supercell approach, as implemented in CRYSTAL, has been applied to the study of many different bulk and surface defective systems. These include Ca and Be substitution in bulk MgO, F-center in CaF2, Fe doped NiO, Li doped NiO, V doped Ti02, and Ti substitution in an all-silica Chabazite. An example of reactivity of a surface defect has been... 

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