**Fabrication of (ANNi _{3})(A=Sn, Cu Mg) based transition metal of Antiperovskites**

Rahman *et al.* (2009) have compared the iso-structural superconducting counterpart ZnNNi_{3} with the elastic optical and electronic characteristics of predicted XNNi_{3} (X= Sn, Cu, Mg). They used first principle functional density theory (DFT) with general gradient approximation (GGA). Specific elastic constants (C11), B mass module, compression k, G shear module and the partial density of the states and the whole band structures were investigated, along with the optical properties of XNNi_{3}. The structure of the three compounds of electronic band revealed a metallic character with a high density of the fermi states in which the Ni 3d states dominate like the ZnNi_{3}. Examination of Tc expressions using the parameter values available indicate that the three compounds are less likely to be superconductive.

*Chapter 03*

**THEORY AND CALCULATIONS**

**3.1 DFT (Density functional theory)**

The analysis of multi-body problems is an important activity in solid state physics. Therefore, material characteristics are split into two methods (simple and complex system). The solution of a simple system consisting of just few numbers of atoms and molecules can be found using the Schrödinger wave equation. Nevertheless, for numerous body problems (including a No of molecules and atoms), it becomes extremely difficult and time-consuming to find solutions to Schrödinger wave equation because every particle has three variables. Therefore it is possible to find the solution to such a complicated system using the most popular density functional theory.

The density function theory (DFT) that was suggested by Walter Kohan and Kohan-sham in 1964 is one of the most creative and powerful quantum mechanical techniques. DFT is used in many fields of physical science. From its outset, this DFT was engaged in condensation in materials science, high-pressure mechanics, mineralogy, solid state chemistry and more strong whole system fields. Current DFT simulation codes can be used to quantify wide infinite number of biological vibratory, chemical, electrical, spectroscopic, elastic and thermodynamic where Schrodinger empirical methods are not available because many body systems have taken more than one model into account. DFT utilizes the electron density relatively as a vector wave function reducing the No of variables to 3 and allows us to classify the substance’s properties easily. DFT is being used in many body problems, especially in atoms and molecules, and in condensed atomic, numerical and computational physical chemistry, for the purposes of explosion, energy content, and alloys magnetic properties (principal ground state).

**3.2 Many-body system**

A stable atom is both a positive and electron charged nucleus, which is an example of multi-particle configuration for each solid object. The Coulomb force interacts with each other including ions, nuclei and electrons. The ground level properties of the multi-particle network are very difficult to identify, because the characteristics of all systems are electron and nuclear places. For the calculation of multi-particle-structure and wave motion, Schrödinger wave equation in quantum mechanics is used. The Schrödinger equation can therefore be written for multi-particle systems as:

In the deficiency of any electrical or magnetic field, H is defined as the Hamiltonian operator. N electron and M nucleus are included in the device (Sun *et al.,* 2004). The wave function is represented by Ѱ and E represents system energy.

**3.3 The variation principle for the ground state1**

The principle of variation states that “the ground state energy is calculated from the minimum wave function Ψ”. It is written as follows: N electrons were expressed in a system of Ѱ→N and this law established a method to conclude energy for the ground state and ground state wave function Ψ_{0.} It means that ground state energy for many body systems is useful, and this variation theory helps us to solve problems, but due to its independent 3N coordinates several body disorders were not clarified by this concept.

**3.4 The Hartree-Fock approximation**

The equation (3.7) shows that the Schrodinger wave equation solution is not promising for the N electron system. Consequently, some other method needed to obtain an appropriate wave function to solve this problem. And Hartree-Fock is an approximation process in N number of electron system to determine wave function. Assume that Ψ_{0} (the ground state wave function) is estimated as N orthonormal spin orbital ‘sψi(x), anti-symmetric product, it is a product of a spatial orbital φk(r) and a spin function.

σ(s) = α(s) or β(s)These wave functions are generally linked with the Slater determinant, *φ _{SD }*as:

Such orbitals consist of spatial orbital *ϕ*(*r*) coordinate and spin function (α). Approximation of Hartree-Fock is the tool used to identify orthogonal orbital sψi functions that reduce energy in this determinant.

**3.5 The electron density**

The important number of DFT is electron density. It is described as “The electron density is the calculation of the probability that an electron is present as a particular position (x = r, s)” The electron density in quantum mechanical calculations is written as: ρ(r) the probability that any of the N-electrons will be located inside volume element d(r). The electron density is experimental and can be observed in comparison with the wave function. XRD is therefore used for electron density calculation.

**3.6 Thomas-Fermi model**

The atomic and molecular characteristics in a complex system first tried in 1927 to be calculated by Thomas and Fermi. The electron densities being implemented that fix the equation instead of the wave function and the Thomas-fermi formula has been developed. Thomas and Fermi found an atom a gas circle of N-electron around the strong nucleus and used quantitative statistical ideas. This equation cannot explain an atom’s exact energy, as T_{TF}[ρ(r)] is a uneven calculation based on real kinetic energy, and the connection force has also been overlooked. Their prediction therefore does not benefit realistic systems, but plays a significant role in density energy expression.

**3.7 Hohenberg and kohan theorems**

The physical character model of Thomas-Fermi (TF) did not explain of the material effectively, and energy representation is actually not supported by electron density. Therefore, a new theory, known as DFT, has been replaced by TF model. Two big theorems are the product of DFT. Hohenberg and Kohn (HK) proposed the theorem in 1964. DFT is a valid theory. It is proved. The electro-density of HK theorems is used to explore the system’s various physical properties rather than the wave function (Bannikov *et al.,* 2010).

**3.7.1 Hohenberg and Kohan First Theorem**

The first theorem in Hohenberg-kohan states that:

“Electron Density is the unique properties of the ground states of many body systems”

**3.7.2 Hohenberg and Kohan Second Theorem**

The second theorem of Hohen states:

“The FHK [ρ] is the useful function that supplies the system’s ground state energy and also provides the lowest amount of energy only where the input density is the true ground state density that is the prince of difference”.

**3.8 Kohan-Sham-Equation**

Equation of Kohan-Sham is a hypothetical particle system (typically electrons) containing the same mass in a given system of interacting components. This is also one of physically and especially in density functional theory of the electron-Schrodinger equations. But in 1965 Kohan and Sham suggested an equation called a Kohan-sham equation, with respect to the change of many systems for body interaction to a non-interacting system (Kohn *et al.,* 1965).

**3.9 The Exchange-correlation Approximations**

The concept of measuring Kohn-sham’s functional interchange correlation is not reflected. The evaluation of the functional exchange correlation helps resolve Kohn-Sham’s equation. These approaches are defined in significant detail below.

**3.10 LDA (Local density approximation)**

Local density approximation (LDA) is the basis for every consistent functional exchange connection. The cycle of all stable electron gas is at the core of this reproduction. This is an entity in which the electrons are traveling in a positive way so that the entire system is impartial.

The core indication of LDA is that EXC may be written in the custom below:

Here is a correlation of energy exchange with a uniform gas density ρ . The probability ρ of the electron at this position is measured for that particle energy. In the late 1920s, Bloch and the Dirac, which originally originated the interchange part, e.g., which defines the exchange energy of an electron into a uniform electron gas of certain densities, can also be divided into exchange contributions and association contributions (Kohn *et al.,* 1965).

**3.11 GGA (Generalized gradient approximation)**

This method not only provides reliable information on density, but also combines the density with knowledge about the charging density gradient In order to bear in mind that the actual electron density is not homogenous. This is the first logical step outside LDA to reach a GGA solution (Pardew *et al.,* 1996).

**3.12 GGA+U (Generalized gradient approximation with Hubbard potential)**

Calculations of initial DFT-oriented approximately are well known. (LDA) and (GGA) result in major defects, such as many compounds control reactions. The subsequent errors or defects are separate LDA and GGA that, due to electron transfer between different state and metal transition, such as metal and metal transition, and metal and oxygen transition, are not canceled during the energy conversion cycle. Complete solutions incorporating LDA and GGA include the new terms ” GGA+U (generalized gradient approximation with Hubbard potential). ‘This GGA+U concept tends to generalize d-state by explicably multiple-body speculations about LDA and GGA in general while maintaining normal Hamiltonian (LDA or GGA) DFT (Density Functional Theory) (Pardew* et al.,* 1996).