**LAPW (The linearized augmented plane wave method)**

As the typical ab-initio electronic structural technology with an equivalent computing efficiency, the linearized accelerated wave method was employed in the simulation of electronic characteristics of materials established on DFT. Thanks to the great precession, density function theory is commonly accepted. FPLAP is a complete electron algorithm whose uniformity includes all components, particularly transitional metals, and rare earths. (DFT) is one of the most detailed method for mechanical, structural and magnetic crystal calculations.

**3.14 Exchange correlation energy**

The key objective of new DFT was for the functional approximation of the Kohan Sham equation to be introduced. The LDA is the simplest among all of these methods. When integrating across all surfaces, the density of exchanges correlation is equal to each position at the same strong electron coordination, the strength of the exchange correlation may be attained.

**3.15 WIEN2K Package**

Basically, this WIEN2K package is a FORTRAN computer program designed for different quantum mechanical calculations of periodic solids. The first complete possible version has been developed about 20 years ago for the measurement of crystalline solids. Initially known as WIEN, the description was written by P at the Material Chemistry Institute at the University of Technology in Vienna. Blaha and Blaz Blaha. In 1990, this first publication of the code took place. Over time the enhancement of WIEN version to resolve the various parameters and measurements found by WIEN 93, WIEN95 and WIEN97 codes dramatically improved over time. WIEN2K’s package was used to solve the Kohan Sham equations, using a fully theoretically (linearized), increase in plane waves and local orbital structures (FP-LAPW+ lo) in keeping with Density Functional Theory (DFT). The latest updated edition of WIEN2K, one of the robust basic sets, has been used in the recent study in the estimation of various properties of compounds.

*Chapter 04*

**RESULTS AND DISCUSSION**

Current calculations are based on the complete structural and elastic, magnetic and electronic characteristics of both the XNNi_{3} Cubic Anti-perovskite (X = Sn, Cu, Mg) by using the plane wave (FPLAPW+ lo), theoretical linearized in functional density hypothesis as applicable under the WIEN2K code. The atoms X and N (0, 0, 0, 0) and X and N (0.5, 0.5, 0, 0, 0) are located in the optimum cubic structure (Pm3m (221)) of XNNi_{3} (X= Sn, Cu, Mg). At the heart of the bone, the N or transition metals are bound by three atoms of N (Nitrides). In figure 2.1 you can see the unit cell of basic cubic antipervoskite. The complete calculations of current work have been carried out with 3000 K-points to optimize one cell and with 250 K-points to optimize doubles cells. For the analysis of trade association potentials, GGA was used together with GGA+U. The 4f orbital N atom and 3d transition metal orbitals are passed by GGA plus U approximation. The structural and elastic properties are treated with single cell optimization, but by using a double-cell configuration, SCF (self-consistent field) the magnetic and electrostatic characteristics of such compounds are regulated.

**4.1 Ground state Structural properties**

The physical characteristics of the XNNi_{3} compounds (X= Sn, Cu, Mg) was determined using the state Birch Murnaghan equation by maximizing the unit cell of those compounds.

Approximations of GGA were considered by Birch-Murghanan state equation to measure a quantity of energy verses of the defined compound for the ferromagnetic, anti-ferromagnetic and paramagnetic properties. Specific structural parameters, such as lattice (a0), bulk moduli (Bo) and ground state energy (Eo) are required. Table 4.1 presents the calculated values for these parameters. More ionic radii are used by the following empirical formula to calculate lattice constants.

α, β and γ are ionic radius N (0.30Å), r Ni ionic radius of Ni (0.70Å) and rX (X = Sn, Cu and Mg) respectively, in these equations and they are 0.06741, 0.4905, and 1.2901 r N r N. Table 4.1 states that with the general gradient approximation the lattice constant was determined and analysis approach has an agreed consensus to check and other statistical tests. Table 4.1 concludes the measured performance bulk module B is the resistance of a volume-changing substance which is also shown in Table 4.1 for XNNi_{3} (X= Sn, Cu, and Mg) with optimization. Binding strength provides information on the stability and contrast of ionic radii for a given substance. We could conclude this XNNi_{3} is extra established than ZNNi_{3} from the above ionic radius values. Another valuable parameter is the connection frequency, which in one molecule is known as bond length or bond distance between two bond atoms. The energy relation force and bond separation is inverted and the shorter the length of contact is the strong connection, which plays a central character in antiperovskite symmetry. The frequency of the relations of X-Ni 2.668 (Å) and 2.668 (Å). The bond length between Å), 3.276 (Å), X-Ni 1.890(Å) and 1.890 (Å) is 3.271 (Å). The bond length between Å) and Å is 1.890 (Å). Mean binding length can be calculated for the crystal structure of the substance on the basis of its resistance factor.