## Introduction:

The basic idea of the density functional theory started in the nineteen-twenties with the work on the uniform electron gas of Thomas and Fermi [1, 2], who came up with the idea that the energy of a system is given completely in terms of its electron density. However, only in 1964, with the publication of the Hohenberg and Kohn (H-K) theorems [3], and in 1965, with the derivation of the set of mono electronic equations with which one can obtain the ground-state density (Kohn-Sham equations) [4], further developments and practical applications of DFT methods have arisen. Essentially, the first H-K theorem demonstrates that the electron density uniquely determines the Hamiltonian operator and thus all the properties of the system, while the second one states that the functional pertaining to the ground-state energy of the system delivers the lowest energy if and only if the input density is the true ground state density (i.e. nothing but the variational principle). At present, the accuracy and efficiency of DFT-based methods depend on the basis-set for the expansion of the Kohn-Sham orbital’s, but particularly upon the quality of the applied exchange-correlation (XC) functional [5].

The main idea in density functional theory is to change the descriptor of the system from the wave function to the ground electron density [6].This is the remarkable theory that allows one to replace the complicated N-Electron wave function and the associated Schrödinger equation by much simpler electron density and its associated calculation scheme. The history begins with the works of Thomas and Fermi in the 1920s [7].In quantum mechanics we know that all information about a system is contained in system wave function Ѱ. Hence if you know Ѱ of the system you know everything about that system. Similarly the DFT teaches us that all information about system is coded in the system wave functional. The principles of DFT are conventionally developed by the attempts of theoreticians to solve many body Schrodinger equation.

HѰ=EѰ

H is the N-electron Hamiltonian, Ѱ is the N-electron antisymmetric wave function and E is the corresponding energy.

The first approximation may be considered as the one proposed by Hartree in 1928. He postulated that the N-electron wave function can be written as the simple product of N one electron wave function each of which verifies a one particle Schrödinger equation.

The DFT started with the landmark work of Hohenberg and Kohn. After the introduction of Hohenberg and Kohn theorem based DFT, the use of DFT give a major boost to the field of computational physics. Hohenberg and Kohn showed in their first theorem that the ground state properties of a many electron system are uniquely determined by an electron density the depend only three spatial coordinates [8].

## Thomas Fermi model

Some early attempts were made to calculate the kinetic and potential energy as functionals of only the density. The most well known approach is presented by Thomas and Fermi [9] .The name “functional” shows the function of a function, which is the electron density in this case. To solve the Schrödinger equation for the electronic motion states of solid crystals, Thomas suggested this basic concept of DFT (Thomas 1927). It is assumed that the external potential depends only on the distances from the nuclei and is therefore determined by the electron density and nuclear charge.

Based on these assumptions electron density is formulated as

Where

This is called first local density approximation (LDA). This approximation contains local density to demonstrate the approximate exchange–correlation functional that is why it is called the local density approximation. In the next year (1928), Fermi independently using Fermi statistics to find kinetic energy at the absolute zero point, completing what is now known as the Thomas–Fermi method [10].The appropriate known class of functional after LDA is local gradiant in electron density, this approach is generalized gradiant approximation (GGA).Generalized gradiant approximation have more physical information then LDA, it is more accurate. But sometimes it is not correct.

There present great number of different GGA functional. Most widely used functional involving solids are Perdew-Wang functional (PW91) and Perdew- Burke–Ernzerhof functional (PBE).

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It is significant to clarify which functional is used in calculation rather than simply saying DFT calculations [11].

## Hohenberg–Kohn Theorem

In 1964, the idea of the Thomas–Fermi method was console by a theorem called the Hohenberg–Kohn theorem. This theorem consists of the following two subsidiary theorems which states:

- The ground state electron density determines the potential of a system, v(r) within an additive constant.
- For any electron density the energy variational principle is always established.

These theorems can be treated as basic theorems of quantum theory based on electron density. The first theorem set the idea that external potentials can be represented not by a wavefuction but by electron density. This theorem confirms that external potentials and the Hamiltonian operator of ground electronic states can be determined by electron density.

The second theorem is about variational principle that the Hamiltonian operator shown by the electron density definitely has a solution of (local) minimum energy. To proof the theorem, it is required to consider both the theorems for the wave function. That is, if there is energy functional with external potential, it can be demonstrate that the electron density is uniquely determined to give minimum energy.

The Hamiltonian operator comprises the electron–electron interaction operator and kinetic energy operator as well as the external potential. Therefore, finding the external potential particularly from the electron density, it is appropriate to find the wave function which gives the lowest energy expectation value of the other two operators for the electron density. Therefore, it requires the universal functional which is

Therefore the survival of the universal functional gives the correspondence of the electron density with the wave function, which gives the lowest energy expectation value for the sum of the electron–electron interaction and kinetic energy operators [10].

## Kohn–Sham Method

Hohenberg and Kohn set the fundamental concept of quantum chemistry based in electron density but it didn’t give the complete information about electronic motion states. After one year in 1965 Kohn and Sham demonstrate a complete method for electronic motion states from this theorem called Kohn-Sham method. This theorem is variational approach which uses electron electron interaction potential to give lowest energy and analogous molecular orbital’s and orbital energies.

In this method, they used independent electron approximation of kinetic energy, same as Hartree-fock method, instead of using kinetic energy functional in Thomas-Fermi model. This alternation solved the weakest point in Thomas-Fermi model.

This method made possible to carry high speed quantitative electronic state calculation in solid state physics and chemistry through development of exchange correlation functional.

Kohn-Sham is formulation of Hohenberg and Kohn theorem because electronic structures are studied using one to one correspondence between external potential and electron density on basis of variational principle [10].

## Aims and Objectives:

Our purpose is to express the occurrence of metals when some alkaline based chalcogenides are doped in the transition metals. After the addition we will discuss the characteristic parameters of transition metals and study their electric and magnetic properties of these alkaline metals y using AB-Initio calculation.

## Research Methodology:

### Literature Review:

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