**2. Fundamentals of DFT**

The Schrodinger Equation [1] for a many body system may be simplified to Kohn-Sham equation, which is a single particle independent Schrodinger equation, and can be numerically solved with density functional theory. This computational process produces physical characteristics of solids; however, this hypothesis is based on electron density rather than wave functions, for which scientist Walter Kohn was given the Nobel Prize in 1998 [2]. Despite the fact that no exchange-correlation effects had been documented at the time, Thomas and Fermi claimed in 1927 that total density is the essential parameter in many body problems [3, 4]. The theorems of Hohenberg, Kohn, and Sham laid the groundwork for DFT in 1964, stating that the functional of a many-body problem's (non-degenerated) ground state electron charge density may completely characterize all properties in absence of magnetic field [5].

#### **2.1 The Hohenberg - Kohn (HK) theorems**

Hohenberg and Kohn [6] stated seemingly two simple theorems in 1964 that enabled the implementation of DFT.

**Theorem I:** The external potential, Vextðr !Þ is a unique functional of electron density ρ( r!), having a unique association among potential and electron density for a many body system; Vext( r!) ¼) <sup>ρ</sup>( r!), whereas this electron density can be used to describe the entire information of the system.

In order to establish a mathematical relation, let us assume external potentials as v rð Þ and v r<sup>0</sup> ð Þ, whereas the change between these potentials is always identical since the ground state electron density is comparable at entire parts of the crystal, that is, v(r<sup>0</sup> ) - v(r) = constant. According to theory, electrons move in a field produced by external potential Vext and interact with one-another in addition to their external potential, and the corresponding Hamiltonian of energy can be written as;

$$\mathbf{H} = \mathbf{T} + \mathbf{V}\_{\text{ext}} + \mathbf{U} \tag{1}$$

Where T, U, and Vext represents the K.E of electrons, coulomb interaction, and external potential respectively. Quantum mechanically the factors T, U, and Vext can be expressed as;

$$\mathbf{T} = \frac{1}{2} \int [\nabla \boldsymbol{\Psi}^\*(\mathbf{r}) \nabla \boldsymbol{\Psi}(\mathbf{r})] \mathbf{dr} \tag{2}$$

$$\mathbf{V} = \int [\mathbf{v}(\mathbf{r})\boldsymbol{\Psi}^\*(\mathbf{r})\boldsymbol{\Psi}(\mathbf{r})]d\mathbf{r} \tag{3}$$

*Fundamentals of Density Functional Theory: Recent Developments, Challenges and Future… DOI: http://dx.doi.org/10.5772/intechopen.99019*

$$\mathbf{U} = \frac{1}{2} \int \left[ \Psi^\* \left( \mathbf{r'} \right) \Psi^\* \left( \mathbf{r} \right) \Psi (\mathbf{r'}) \Psi (\mathbf{r}) \frac{\mathbf{1}}{|\mathbf{r} \cdot \mathbf{r'}|} \right] d\mathbf{r} d\mathbf{r'} \tag{4}$$

The solution of Hamiltonian for Eq. (1) can be expressed as;

$$\mathbf{H}\Psi(\mathbf{r}\_1, \mathbf{r}\_2, \dots \dots \dots \mathbf{r}\_N) = \mathbf{E}\Psi(\mathbf{r}\_1, \mathbf{r}\_2, \dots \dots \dots \mathbf{r}\_N) \tag{5}$$

The ψð Þ r1, r2, *:* ……… rN is a ground state N interacting particle's wave-function. Suppose an additional potential v0 (r) with changed Hamiltonian H0 and wavefunction ψ<sup>0</sup> ð Þr where the ground state density ρ(r) must remain the same for both cases. The Hamiltonian for this many-body system can be written as; H0 ψ<sup>0</sup> ¼ E<sup>0</sup> ψ0 .

Following a thorough exploration of the situation, established on v rð Þ‐v r<sup>0</sup> ð Þ¼ constant, it can be concluded, that ψ(r) and ψ<sup>0</sup> (r) are different; as a result, they both fulfill distinct Schrodinger wave equations. According to variational principle, it is an irrefutable reality that no wave function may produce energy lower than that of the ground stateψ (r) and this fact can be characterized as h j ψ Hj i ψ < ψ<sup>0</sup> h jH ψ<sup>0</sup> j i where E ¼ h j ψ Hj i ψ .

Employing essential property of ground state:

$$
\langle \Psi' | \mathbf{H} | \Psi' \rangle = \langle \Psi' | \mathbf{H}' | \Psi' \rangle + \int \left[ \mathbf{v} \left( \stackrel{\cdot}{\mathbf{r}} \right) \cdot \mathbf{v}' \left( \stackrel{\cdot}{\mathbf{r}} \right) \right] \rho \left( \stackrel{\cdot}{\mathbf{r}} \right) \mathbf{dr} \tag{6}
$$

Alternatively, by swapping;

$$
\langle \Psi | \mathbf{H}' | \Psi \rangle = \langle \Psi | \mathbf{H} | \Psi \rangle + \int \left[ \mathbf{v}' \left( \stackrel{\cdot}{\mathbf{r}} \right) \cdot \mathbf{v} \left( \stackrel{\cdot}{\mathbf{r}} \right) \right] \rho \left( \stackrel{\cdot}{\mathbf{r}} \right) \mathbf{dr} \tag{7}
$$

By adding above equations we get;

$$\mathbf{E} + \mathbf{E}' < \mathbf{E}' + \mathbf{E} \tag{8}$$

The Eq. (8) confirms clear disagreement, and two unlike potentials, v(r) as well as v<sup>0</sup> (r) will certainly provide different density ρ(r) and ρ<sup>0</sup> (r) respectively. As a result, details relating density and external potential are needed to determine the Hamiltonian information. Also, T and U are known for N-partials systems so ρ r !� � may be employed to find ground state H and E. The functional association of minimum energy state and corresponding resulting density is;

$$\mathbf{E}[\rho(\mathbf{r})] = \mathbf{T}[\rho(\mathbf{r}) + \mathbf{V}[\rho(\mathbf{r})] + \mathbf{U}[\rho(\mathbf{r})] \tag{9}$$

**Theorem II:** The true ground state density of an electron corresponds to electron density that minimizes the overall energy of the functional.

Consider, ρð Þr is the density which corresponds to ground state while ρ<sup>0</sup> (r) to any other state of a many-body system. The functional for total energy in this context is given as; E ρ<sup>0</sup> ½ �>E½ � ρ . Also, assume that F[ρ(r)] is a general functional that is valid for fixed electrons at all external potentials. Mathematically this can be written as;

$$\mathbf{F}\rho(\mathbf{r})] = \mathbf{T}[\rho(\mathbf{r})] + \mathbf{U}[\rho(\mathbf{r})] \tag{10}$$

Also,

$$\mathbf{E}[\rho(\mathbf{r})] = \int [\mathbf{v}(\mathbf{r})\rho(\mathbf{r})]d\mathbf{r} + \mathbf{F}[\rho(\mathbf{r})] \tag{11}$$

In order to have minimum energy functional, the corresponding density ρð Þr must be essentially a ground state density.

$$\mathbf{E}[\boldsymbol{\Psi}'] = (\boldsymbol{\Psi}', \mathbf{V}\boldsymbol{\Psi}') + (\boldsymbol{\Psi}', \mathbf{T} + \mathbf{U})\boldsymbol{\Psi}' \tag{12}$$

Assumingψ is ground state function associated to ρ(r) for external potential v(r), the ρ<sup>0</sup> ð Þr will correspond to higher energy in accordance with the variational principle.

$$\mathbf{E}[\boldsymbol{\Psi}'] = \int [\mathbf{v}(\mathbf{r})\rho'(\mathbf{r})]d\mathbf{r} + \mathbf{F}[\rho'(\mathbf{r})] \rhd \mathbf{E}[\boldsymbol{\Psi}] = [\mathbf{v}(\mathbf{r})\rho(\mathbf{r})]d\mathbf{r} + \mathbf{F}[\rho(\mathbf{r})] \tag{13}$$

As a result, provided the density functional is accurately described, one may easily compute the ground state density as well as energy in an identified external potential. Furthermore, it also demonstrates that ρð Þr minimize the energy functional E½ � ρð Þr .

#### **2.2 The Kohn-Sham (KH) equations**

The theorems given by Hohenberg-Kohn are exact; however not very useful in real calculations [6]. The equation given by Kohn-Sham [7] turned DFT into an applied tool. They converted the difficult problem of electrons interacting together in external effective potential (Vext) into the electrons that are non-interacting in Vext, and the total energy for a ground state of interacting electrons in fixed potential, v rð Þ is;

$$\mathbf{E}[\rho(\mathbf{r})] = \mathbf{V}[\rho] + \mathbf{U}[\rho] + \mathbf{G}[\rho] \tag{14}$$

Where universal density functional G[ρ] holds exchange-correlation, and is expressed as;

$$\mathbf{G}[\rho(\mathbf{r})] = \mathbf{T}\_s[\rho] + \mathbf{E}\_{\mathbf{xc}}[\rho] \tag{15}$$

$$\mathbf{E}[\rho(\mathbf{r})] = \mathbf{T}\_s[\rho] + \mathbf{V}[\rho] + \mathbf{U}[\rho] + \mathbf{E}\_{\mathbf{xc}}[\rho] \tag{16}$$

The kinetic energy for a many body system having non-interacting electrons is denoted by Ts½ � ρ , while V½ � ρ is the external potential produced by core having positive charge, U½ � ρ is coulomb potential as a result of electron–electron interactions, and Exc½ � ρ is the energy due to exchange-correlation effects.

$$\mathbf{T\_s[\rho(r)]} = -\frac{\hbar^2}{2m} \sum\_{\mathbf{i}}^{N} \int \boldsymbol{\varrho\_i^\*}(\mathbf{r}) \nabla^2 \boldsymbol{\varrho\_i}(\mathbf{r}) \mathbf{d}^3 \mathbf{r} = \mathbf{T} \left[ \boldsymbol{\varrho\_i} \sum (\rho) \right] \tag{17}$$

and

$$\mathbf{U}[\rho] = \frac{\mathbf{q}^2}{2} \int \left[ \frac{\rho(\mathbf{r})\rho(\mathbf{r}')}{|\mathbf{r} \cdot \mathbf{r}'|} \right] \mathbf{d} \mathbf{r} d\mathbf{r}' \tag{18}$$

$$\mathbf{V}[\rho] = \int \mathbf{v}(\mathbf{r})\rho(\mathbf{r})d\mathbf{r} \tag{19}$$

The exchange correlation energy Exc½ � ρ for a many-body system produced by ρ(r) is given by;

*Fundamentals of Density Functional Theory: Recent Developments, Challenges and Future… DOI: http://dx.doi.org/10.5772/intechopen.99019*

$$\mathbf{E\_{xc}}[\rho] = \int [\rho(\mathbf{r})\mathbf{e\_{xc}}\rho(\mathbf{r})]d\mathbf{r} \tag{20}$$

and

$$\mathbf{E\_{xc}}[\rho] = \mathbf{E\_{x}}[\rho]\_{\text{exchange}} + \mathbf{E\_{c}}[\rho]\_{\text{correlation}} \tag{21}$$

The Ex term denotes the reduction in energy as an outcome of antisymmetrization, and it may be represented through a single particle orbital as;

$$\mathbf{E\_x} = \int [\rho(\mathbf{r})\mathbf{e\_x}\rho(\mathbf{r})]d\mathbf{r} \tag{22}$$

and

$$\mathbf{E\_c = \int [\rho(\mathbf{r})\mathbf{e\_c}\rho(\mathbf{r})]d\mathbf{r}}\tag{23}$$

and

$$\varepsilon\_{\mathbf{x}}[\boldsymbol{\varrho}\_{\mathrm{i}}\boldsymbol{\varrho}(\mathbf{r})] = \frac{\mathbf{-q}^{2}}{\mathbf{r}} \sum\_{\mathbf{j},\mathbf{k}} \int \mathbf{d}^{3}\mathbf{r}'(\mathbf{r})\boldsymbol{\varrho}\_{\mathbf{k}}^{\*}\left(\mathbf{r}\right) \frac{\boldsymbol{\varrho}\_{\mathrm{j}}^{\*}\left(\mathbf{r}'\right)\boldsymbol{\varrho}\_{\mathrm{k}}^{\*}\left(\mathbf{r}'\right)\boldsymbol{\varrho}\_{\mathrm{k}}(\mathbf{r})}{|\mathbf{r} - \mathbf{r}'|} \tag{24}$$

Where the single term in the summation refers to the energy of a molecule 'j' at site 'r' in relation to a molecule 'k' at 'r<sup>0</sup> '. The system's energy is further reduced owing to mutual avoidance of the interacting particles, such as electrons that are anti-parallel and lower their energy by evenly arranging their moments. Kohn-Sham mapping of interacting and non-interacting system is shown in **Figure 1**.

$$\mathbf{e}\_{\mathbf{c}} = \sum\_{\mathbf{j} < \mathbf{k}} \frac{\mathbf{q}^2}{|\mathbf{r} \cdot \mathbf{r}'|} = \frac{\mathbf{q}^2}{2} \int \mathbf{d}^3 \mathbf{r} \int \left[ \frac{\rho(\mathbf{r})\rho(\mathbf{r}') \cdot \rho(\mathbf{r})\mathbf{\hat{s}}(\mathbf{r} \cdot \mathbf{r}')}{|\mathbf{r} \cdot \mathbf{r}'|} \right] \mathbf{d}^3 \mathbf{r}' \tag{25}$$

The energy of ground state may be obtained by differentiating Eq. (14) with respect to ρð Þr

$$\mathbf{O} = \frac{\delta \mathbf{E}[\rho]}{\delta \rho(\mathbf{r})} = \frac{\delta \mathbf{T}\_s[\rho]}{\delta \rho(\mathbf{r})} + \frac{\delta \mathbf{U}[\rho]}{\delta \rho(\mathbf{r})} + \frac{\delta \mathbf{V}[\rho]}{\delta \rho(\mathbf{r})} + \frac{\delta \mathbf{E}\_{\mathbf{xc}}[\rho]}{\delta \rho(\mathbf{r})} = \frac{\delta \mathbf{T}\_s[\rho]}{\delta \rho(\mathbf{r})} + \mathbf{v}(\mathbf{r}) + \mathbf{V}\_c(\mathbf{r}) + \mathbf{E}\_{\mathbf{xc}}(\mathbf{r})\tag{26}$$

By employing density ρs(r), the minimum state for a non-interacting many-body system is;

**Figure 1.** *Kohn-Sham mapping of interacting and non-interacting system.* *Density Functional Theory - Recent Advances, New Perspectives and Applications*

$$\mathbf{O} = \frac{\delta \mathbf{E}\_{\mathbf{s}}[\rho]}{\delta \rho\_{\mathbf{s}}(\mathbf{r})} = \frac{\delta \mathbf{T}\_{\mathbf{s}}[\rho]}{\delta \rho\_{\mathbf{s}}(\mathbf{r})} + \frac{\delta \mathbf{V}\_{\mathbf{s}}[\rho]}{\delta \rho\_{\mathbf{s}}(\mathbf{r})} + \frac{\delta \mathbf{V}\_{\mathbf{s}}[\rho]}{\delta \rho\_{\mathbf{s}}(\mathbf{r})} = \frac{\delta \mathbf{T}\_{\mathbf{s}}[\rho]}{\delta \rho\_{\mathbf{s}}(\mathbf{r})} + \mathbf{v}\_{\mathbf{s}}(\mathbf{r}) \tag{27}$$

Equating Eqs. (26) and (27), the potential Vs can be obtained as;

$$\mathbf{v}\_{\mathbf{s}} = \mathbf{V}(\mathbf{r}) + \mathbf{V}\_{\mathbf{c}}(\mathbf{r}) + \mathbf{V}\_{\mathbf{xc}}(\mathbf{r}) \tag{28}$$

The equation for a one-particle system that is non-interacting in potential vs(r) can be derived from the equation of interacting electrons of the system in the presence of v(r).

$$\left[\cdot \frac{\hbar^2}{2m}\nabla^2 + \mathbf{v}\_s(\mathbf{r})\right] \mathbf{q}\_k(\mathbf{r}) = \mathbf{E}\_k \mathbf{q}\_k(\mathbf{r})\tag{29}$$

The ρ(r) of an original system is replicated by orbitals, where fk is the kth orbital occupation, and can be expressed as;

$$\rho(\mathbf{r}) = \rho\_{\mathbf{s}}(\mathbf{r}) = \sum\_{k}^{N} \mathbf{f} |\varrho\_{k}(\mathbf{r})|^{2} \tag{30}$$

#### **2.3 Exchange-correlation potential**

The consequences of KS scheme revealed that the minimum energy state can be established by limiting energy of the energy functional, and it can be done using an agreeable solution of a set of single-particle equations. In the KS scheme, just one critical difficulty is that Exc (exchange-correlation energy) cannot be found exactly. If Exc is determined accurately, it is a precise solution for a many-body problem. There is currently no such exact solution exists, hence approximations are employed to estimate Exc with LDA and GGA being the most commonly used approximations.

### **3. Commonly used exchange-correlation approximations**

In this part, we will go through some of the major advances that lead to contemporary DFT in order to lay a foundation that will help us to comprehend both the theory's foundations and limits. Bloch (1929) was the first to write about the exchange contribution, and it has become well-known as a result of quantum Monte-Carlo simulations of uniform gases [8], which are parameterized in simple formulations [9, 10]. The Local Density Approximation (LDA) [11], proposed by Kohn and Sham, asserts that the exchange-correlation functional at any point in space is simply dependent on that location's spin density. LDA is quite correct for geometries, but it often over-binds atoms/molecules roughly by 1 eV per bond, rendering it ineffective for thermo-chemistry [12]. The Generalized Gradient Approximation (GGA) [13, 14] is an extension to the LDA component that includes terms that are dependent on density derivatives. Perdew was the first to apply real-space cutoffs to make GGAs, which led to the development of the PW86 functional model [13]. The PW91 functional [15] was the pinnacle of this comprehensive development, and it produces useful precision for binding energies, as proven in 1993 of around 6–10 kcal/mol [16]. PBE [17] is the most widely used GGA to investigate materials today, whereas BLYP [18] and Lee-Yang-Parr correlation [19] is the most generally employed GGA in chemistry. A hybrid GGA [20] is one that combines a normal GGA plus a Hartree-Fock component, in which the kinetic energy density is also employed to define the

*Fundamentals of Density Functional Theory: Recent Developments, Challenges and Future… DOI: http://dx.doi.org/10.5772/intechopen.99019*

GGA component. The GGA, Hartree-Fock, and kinetic energy density components are all present in a meta-hybrid, while hybrid or meta-hybrid component of a doublehybrid includes an involvement from second-order Moller-Plesset perturbation theory [21]. The Density Functional (DF) consists of a part of GGA, LDA, Hartree-Fock exchange or hybrids, and/or a meta-GGA, commonly known as the exchange-andcorrelation (XC) functional (meta-GGA or meta-hybrid). Furthermore, the addition of an orbital-dependent correlation, it may also be reliant on virtual Kohn-Sham orbitals (double-hybrids) [22]. A comparison of simplicity versus accuracy of existing approximations in DFT is shown in **Figure 2**.

The functionals currently utilized in DFT simulations constitute a natural hierarchy, and no systematic approach to the precise functional can be claimed. The available functional form is clearly improving, resulting in a considerably more accurate representation of ground state properties. The most important recent advancements are those that include the non-local aspect of the exchange potential in some way. **Table 1**, summarizes the present hierarchy.

**Figure 2.**

*A comparison of simplicity versus accuracy of existing approximations in DFT [23].*

**Table 1.** *Commonly used Exc functionals.*
