Electronic structure supercell

Electronic structure methods for studies of nanostructures can be divided broadly into supercell methods and real-space methods. Supercell methods use standard k-space electronic structure techniques separating periodically repeated nanostructures by distances large enough to neglect their interactions. Direct space methods do not need to use periodic boundary conditions. Various electronic structure methods are developed and applied using both approaches. In this section we will shortly discuss few popular but powerful electronic structure methods the pseudopotential method, linear muffin-tin orbital and related methods, and tight-binding methods. 

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. 

All calculations in this study were implemented with the CASTEP package5, which is capable of simulating electronic structures for metals, insulators, or semiconductors. It is based on a supercell method, whereby all studies must be performed on a periodic system. Study of molecules is also possible by assuming that a molecule is put in a box and treated as a periodic system. Forces acting on atoms and stress on the unit cell can be calculated. These can be used to find the equilibrium structure. 

Use of the plane wave based electronic structure methods introduces two basic parameters the kinetic energy cutoff value, controlling the basis set quality, and the periodic unit-cell (supercell) size, present due to periodic nature of these approaches. Both of these parameters should be large enough to guarantee the convergence in the total energy and in all the physical quantities that are supposed to be determined from the simulation. 

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. 

Cucinotta et studied the equilibrium between the keto and enol tautomers of acetone, cf. Fig. 5, both in vacuum and in an aqueous solution. They used parameter-free, electronic-structure calculations for a periodically repeated supercell (see section IIB) with the Car-Parrinello approach (see, e.g., ref. 1). In static calculations for solely the acetone molecule they found a total-energy difference between the enol and keto forms of 11.8 kcal/mol. The barrier between the two forms was found to be around 58 kcal/mol (Fig. 5). 

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]. 

Since many experimental studies of 7-Fe were performed for 7-Fe particles in a Cu matrix (or Cu alloy, including Cu-Al) [113], [114], it is important to probe the electronic structure of the particle-matrix systems. Embedded-cluster methods are ideally taylored to treat small particles of a metal in a host matrix, a system that would require a very large supercell in band-structure calculations. DV calculations were performed for the 14-atom Fe particle in copper shown in Fig. 21 [118]. Spin-density contour maps were obtained to assess the polarization of the Cu matrix by the coherent magnetic 7-Fe particle. Examples are given in Figs. 22 and 23 for a Fe particle in Cu and 7-Fe in Cu with two substitutional Al. If the matrix is a Cu-Al alloy, this element is known to penetrate the Fe particle [114]. 

This chapter is structured as follows In Sect. 6.2, a basic introduction to molecular refinement is presented, stressing particularly relevant aspects. Section 6.3 reviews the recent work by Falklof et al., describing how the 2 x 2 x 2 supercell for the lysozyme structure was obtained. Section 6.4 reviews some modern advances in DFT, focusing on dispersion-corrected DFT, while Sect. 6.5 describes the effects of DFT optimization of atomic coordinates on the agreement between observed and calculated X-ray structure factors. The aim is to determine an optimal electronic-structure computational procedure for quantum protein refinement, and we consider only the effects of minor local perturbations to the existing protein model rather than those that would be produced by allowing full protein refinement. 

The LSGF method on the other hand is an order-IV method for calculation of the electronic system. It is based on a supercell (which may just be one unit cell) with periodic boundary conditions, see Fig.(4.5), and the concept of a Local Interaction Zone (LIZ), which is embedded in an effective medium, usually chosen to be the Coherent Potential Approximation medium (see next chapter). For each atom in the supercell, one uses the Dyson equation to solve the electronic structure problem as an impurity problem in the effective medium. The ASA is employed as well as the ASA+M correction described above. The total energy is defined to... 

We use a periodic supercell model based on the large unit cell (LUC) method [22] which is free from the limitations of different cluster models applicable mainly to ionic solids, e.g., alkali halides. The main computational equations for calculating the total energy of the crystal within the framework of the LUC have been given in Refs. [22-24]. Here, we shall outline some key elements of the method. The basic idea of the LUC is in computing the electronic structure of the unit cell extended in a special manner at k = 0 in the reduced Brillouin zone (BZ), which is equivalent to a band structure calculation at those BZ k points, which transform to the reduced BZ center on extending the unit cell [22]. The total energy of the crystal is... 

Let us now find out whether these classical enthalpies may be reproduced by electronic-structure calculations (VASP) on Sn/Zn supercells using ultra-soft pseudopotentials, plane-wave basis sets and the GGA. We therefore have to theoretically determine the total energies of all crystal structure types under consideration (a-Sn, j6-Sn, Zn) as a function of the composition SnxZni x by a variation of the available atomic sites in terms of Sn and Zn occupation, just as for the preceding oxynitrides (CoOi- N ). In the present case, supercells with a total of 16 atoms were generated, and nine different compositions per structure were numerically evaluated. Because this amounts to a significant computational task, the use of pseudopotentials is mandatory, and this also allows the rapid calculation of interatomic forces and stresses for structural... 

The 3D-RISM-MCSCF approach has been applied to carbon monoxide (CO) solute in ambient water [33]. Since it is known that the Hartree-Fock method predicts the electronic structure of CO in wrong character [167], the CASSCF method (2 core, 8 active orbitals, 10 electrons) in the basis sets of double zeta plus polarization (9s5pld/4s2pld) augmented with diffuse functions (s- and p-orbitals) was used. Water was described by the SPC/F model [127] and the site-centered local pseudopotential elaborated by Price and Halley for CP simulation [40]. The 3D-RISM/KH integral equations for the water distributions specified on a grid of 64 points in a cubic supercell of size 20 A were solved at each step of the SCF loop by using the method of modified direct inversion in the iterative subspace (MDIIS) [27, 29] (see Appendix). 

---