Electronic structure methods periodic density functional theory

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]. 

The purpose of this chapter will be to review the fundamentals of ab initio MD. We will consider here Density Functional Theory based ab initio MD, in particular in its Car-Parrinello version. We will start by introducing the basics of Density Functional Theory and the Kohn-Sham method, as the method chosen to perform electronic structure calculation. This will be followed by a rapid discussion on plane wave basis sets to solve the Kohn-Sham equations, including pseudopotentials for the core electrons. Then we will discuss the critical point of ab initio MD, i.e. coupling the electronic structure calculation to the ionic dynamics, using either the Born-Oppenheimer or the Car-Parrinello schemes. Finally, we will extend this presentation to the calculation of some electronic properties, in particular polarization through the modern theory of polarization in periodic systems. 

The reliability of method for electronic structure calculations and models used in zeolite modeling is briefly reviewed. Only traditional ab initio methods and methods based on the density functional theory are discussed. Periodic, cluster, and combined models are described and their suitability for investigation of various properties is discussed. This contribution is written for non-experts in computational chemistry. The author hopes that it will help them to gain a basic orientation in the field. 

Hybrid approaches combining ab-initio or DFT and semiempirical approaches have become popular. As an example, we can refer to LEDO (hmited expansion of differential overlap) densities application to the density-functional theory of molecules [262]. This LEDO-DFT method should be well suited to the electronic-structure calculations of large molecules and in the anthors opinion its extension to Bloch states for periodic structures is straightforward. In the next sections we discuss the extension of CNDO and INDO methods to periodic stmctures - models of an infinite crystal and a cyclic cluster. 

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