**2. Schrödinger equation**

It is a fundamental equation to be solved to describe the electronic structure of a system with several nuclei and electrons and for a non-relativistic quantum description of a molecular or crystalline system and which is written:

*Density Functional Theory - Recent Advances, New Perspectives and Applications*

$$H\Psi = \left(-\sum\_{i}^{n} \frac{\hbar^2 \nabla\_i^2}{2m} - \sum\_{I}^{N} \frac{\hbar^2 \nabla\_I^2}{2M} - \sum\_{i,I} \frac{Z\_I e^2}{|\overline{r\_i} - \overline{R\_I}|} + \sum\_{i$$

where H is the molecular Hamiltonian and Ψ is the wave function. It is therefore a question of seeking the solutions of this equation. We can write the Hamiltonian in the form:

$$H = T\_{\epsilon} + T\_{n} + V\_{\epsilon - \epsilon} + V\_{n-n} + V\_{n-\epsilon}.\tag{2}$$

We give the definition for each term:

*Te* ¼ �P*<sup>n</sup> i* ℏ2∇<sup>2</sup> *i* <sup>2</sup>*<sup>m</sup>* : The kinetic energy of n electrons of mass m. *Tn* ¼ �P*<sup>N</sup> I* ℏ2∇<sup>2</sup> <sup>2</sup>*<sup>M</sup>* : The kinetic energy of N nuclei of mass M. *Ve*�*<sup>e</sup>* <sup>¼</sup> <sup>P</sup> *<sup>i</sup>*<*<sup>j</sup> <sup>e</sup>*<sup>2</sup> ∣*ri* �*rj* <sup>∣</sup> : The electron–electron repulsive potential energy. *Ve*�*<sup>n</sup>* ¼ �<sup>P</sup> *i*,*I ZIe*<sup>2</sup> ∣*ri* �*RI* ∣ : The attractive potential energy nucleus-electron. *Vn*�*<sup>n</sup>* <sup>¼</sup> <sup>P</sup> *I* <*J ZIZJe*<sup>2</sup> ∣*RI* �*RJ* ∣ : The nucleus-nucleus repulsive potential energy.

For a system of N nuclei and n electrons, Schrödinger equation is too complex to be able to be solved analytically. The exact solution of this equation is only possible for the hydrogen atom and hydrogenoid systems. In order to simplify the solution of this equation, Max Born and Robert Oppenheimer [1] have proposed an approximation aiming to simplify it.

### **3. The Born-Oppenheimer approximation**

We consider that we can decouple the movement of electrons from that of nuclei, by considering that their movement of nuclei is much slower than that of electrons: we consider them as fixed in the study of the movement of the electrons of the molecule. The inter-nuclear distances are then treated as parameters. It has an immediate computational consequence, called an adiabatic hypothesis. It is in fact the same approximation and since the Oppenheimer approximation is still used in quantum chemistry, during chemical reactions or molecular vibrations, we can consider according to the classical Born-Oppenheimer approximation that the distribution of electrons (adapts) almost instantaneously, when from the relative motions of nuclei to the resulting Hamiltonian variation. This is due to the lower inertia of the electrons *M* ¼ 1800*me* then the electron wave function can therefore be calculated when we consider that the nuclei are immobile, from where

$$T\_n = 0; V\_{n-n} = constant,\tag{3}$$

and so the Hamiltonian becomes

$$H = T\_{\varepsilon} + V\_{\varepsilon-\varepsilon} + V\_{\varepsilon-n} + V\_{n-n}.\tag{4}$$

$$H = H\_{el} + V\_{n-n},\tag{5}$$

with *Hele*: electronic Hamiltonian which is equal to:

$$H\_{ele} = T\_{\epsilon} + V\_{\epsilon - \epsilon} + V\_{\epsilon - n}.\tag{6}$$

*The Density Functional Theory and Beyond: Example and Applications DOI: http://dx.doi.org/10.5772/intechopen.100618*

Therefore the Born Oppenheimer approximation gives us:

$$H = T\_{\epsilon} + V\_{\epsilon - \epsilon} + V\_{\epsilon - n}.\tag{7}$$

We use another notation to simplify the calculations

$$H = T + V\_{\text{ext}} + U.\tag{8}$$

$$T = T\_{\epsilon}; U = V\_{\epsilon - \epsilon} = V\_H; V\_{\text{ext}} = V\_{\epsilon - n}. \tag{9}$$

The Born-Oppenheimer approximation results in the Eq. (7) which keeps a very complex form: it always involves a wave function with several electrons. This approximation significantly reduces the degree of complexity but also the new wave function of the system depends on N bodies while other additional approximations are required to be able to effectively solve this equation. The remainder of this chapter will deal with approximations allowing to arrive at a solution of this equation within the framework of the density functional theory (DFT) and the random phase approximation (RPA).

### **4. Density Functional theory (DFT)**

Density Functional Theory is one of the most widely used methods for calculating the electronic structure of matter in both condensed matter physics and quantum chemistry. The DFT has become, over the last decades, a theoretical tool which has taken a very important place among the methods used for the description and the analysis of the physical and chemical properties for the complex systems, particularly for the systems containing a large number electrons. DFT is a reformulation of the N-body quantum problem and as the name suggests, it is a theory that only uses electron density as the fundamental function instead of the wave function as is the case in the method by Hartree and Hartree-Fock. The principle within the framework of the DFT is to replace the function of the multielectronic wave with the electronic density as a base quantity for the calculations. The formalism of the DFT is based on the two theorems of P. Hohenberg and W. Kohn [2].

#### **4.1 Hohenberg and Kohn theorems**

Hohenberg-Kohn (HK) reformulated the Schrödinger equation no longer in terms of wave functions but employing electron density, which can be defined for an N-electron system by:

$$n = 2N \int dr\_1 \int dr\_2 \dots \int dr\_{n-1} \Psi^\* \left( r\_1, r\_2, \dots, r\_{n-1}, r \right) \Psi(r\_1, r\_2, \dots, r\_{n-1}, r), \tag{10}$$

this equation depends only on the three position parameters r = (x, y, z), position vector of a given point in space. This approach is based on two theorems demonstrated by Hohenberg and Kohn.

**Theorem 1**: For any system of interacting particles in an external potential *Vext r* � �, the potential *Vext r* � � is only determined, except for an additive constant, by the electron density *<sup>n</sup>*<sup>0</sup> *<sup>r</sup>* � � in its ground state.

The first HK Theorem can be demonstrated very simply by using reasoning by the absurd. Suppose there can be two different external potentials *V*ð Þ<sup>1</sup> *ext* and *V*ð Þ<sup>2</sup> *ext* associated with the ground state density *n r* � �. These two potentials will lead to two different Hamiltonians *H*ð Þ<sup>1</sup> and *H*ð Þ<sup>2</sup> whose wave functions *ψ*ð Þ<sup>1</sup> and *ψ*ð Þ<sup>2</sup> describing the ground state are different. As described by the ground state of *H*ð Þ<sup>1</sup> we can therefore write that:

$$E^{(1)} = \left\langle \psi^{(1)} | H^{(1)} | \psi^{(1)} \right\rangle < \left\langle \psi^{(2)} | H^{(1)} | \psi^{(2)} \right\rangle. \tag{11}$$

This strict inequality is valid if the ground state is not degenerate which is supposed in the case of the approach of HK. The last term of the preceding expression can be written:

$$
\left\langle \Psi^{(2)} \vert H^{(1)} \vert \Psi^{(2)} \right\rangle = \left\langle \Psi^{(2)} \vert H^{(2)} \vert \Psi^{(2)} \right\rangle + \left\langle \Psi^{(2)} \vert H^{(1)} - H^{(2)} \vert \Psi^{(2)} \right\rangle,\tag{12}
$$

$$
\left\langle \psi^{(2)} \middle| H^{(1)} \middle| \psi^{(2)} \right\rangle = E^{(2)} + \int \left[ V\_{\text{ext}}^{(1)} \left( \stackrel{\leftarrow}{r} \right) - V\_{\text{ext}}^{(2)} \left( \stackrel{\leftarrow}{r} \right) \right] n\_0 \left( \stackrel{\leftarrow}{r} \right) d^3 r,\tag{13}
$$

$$E^{(1)} < E^{(2)} + \int \left[ V\_{\text{ext}}^{(1)} \left( \overleftarrow{r} \right) - V\_{\text{ext}}^{(2)} \left( \overleftarrow{r} \right) \right] n\_0 d^3 r. \tag{14}$$

It will also the same reasoning can be achieved by considering *E*ð Þ<sup>2</sup> instead of *E*ð Þ<sup>1</sup> . We then obtain the same equation as before, the symbols 1ð Þ and 2ð Þ being inverted:

$$E^{(2)} = \left\langle \psi^{(2)} | H^{(2)} | \psi^{(2)} \right\rangle < \left\langle \psi^{(1)} | H^{(2)} | \psi^{(1)} \right\rangle,\tag{15}$$

$$
\left\langle \boldsymbol{\psi^{(1)}} \middle| \boldsymbol{H^{(2)}} \middle| \boldsymbol{\psi^{(1)}} \right\rangle = \left\langle \boldsymbol{\psi^{(1)}} \middle| \boldsymbol{H^{(1)}} \middle| \boldsymbol{\psi^{(1)}} \right\rangle + \left\langle \boldsymbol{\psi^{(1)}} \middle| \boldsymbol{H^{(2)}} - \boldsymbol{H^{(1)}} \middle| \boldsymbol{\psi^{(1)}} \right\rangle,\tag{16}
$$

$$
\left\langle \psi^{(1)} \middle| H^{(2)} \middle| \psi^{(1)} \right\rangle = E^{(1)} + \int \left[ V\_{\text{ext}}^{(2)} \left( \stackrel{\leftarrow}{r} \right) - V\_{\text{ext}}^{(1)} \left( \stackrel{\leftarrow}{r} \right) \right] n\_0 \left( \stackrel{\leftarrow}{r} \right) d^3 r. \tag{17}
$$

$$E^{(2)} < E^{(1)} + \int \left[ V\_{\rm ext}^{(2)} - V\_{\rm ext}^{(1)} \right] n\_0 d^3 r,\tag{18}$$

we obtain the following contradictory equality:

$$E^{(1)} + E^{(2)} < E^{(1)} + E^{(2)}.\tag{19}$$

The initial hypothesis is therefore false; there cannot exist two external potentials differing by more than one constant leading at the same density of a non-degenerate ground state. This completes the demonstration.

) the external potential of the ground state is a density functional.

Since the fundamental energy of the system is uniquely determined by its density, then energy can be written as a density functional. By following reasoning

**Theorem 2**: The previous theorem only exposes the possibility of studying the system via density. It only allows knowledge of the density associated with the studied system. The Hohenberg-Kohn variational principle partially answers this problem:

a universal functional for the energy *E n*½ � can be defined in terms of the density. The exact ground state is the overall minimum value of this functional.

similar to that of the first part we show that the minimum of the functional corresponds to the energy of the ground state, indeed, the total energy can be written:

$$E\_{HK}[n] = \int n\left(\stackrel{\cdot}{r}\right) V\_{\text{ext}}\left(\stackrel{\cdot}{r}\right) d^3r + F\_{HK}[n],\tag{20}$$

*F n*½ � is a universal functional of n(r):

$$F\_{HK}[n] = T[n] + U[n]. \tag{21}$$

And the number of particles:

$$N = \int n\left(\stackrel{\leftarrow}{r}\right) dr.\tag{22}$$

Thus, we see that by minimizing the energy of the system with respect to the density we will obtain the energy and the density of the ground state. Despite all the efforts made to evaluate this functional E[n], it is important to note that no exact functional is yet known.

#### **4.2 Ansatz of Kohn-Sham**

Since the kinetic energy of a gas of interacting electrons being unknown, in this sense, Walter Kohn and Lu Sham [3] (KS) proposed in 1965 an ansatz which consists in replacing the system of electrons in interaction, impossible to solve analytically, by a problem of independent electrons evolving in an external potential. In the case of a system without interaction, the functional E[n] is reduced to kinetic energy and the interest of the reformulation introduced by Kohn and Sham is that we can now define a monoelectronic Hamiltonian and write the equations monoelectronic Kohn-Sham. According to KS the energy is written in the following form:

$$E\_{HK}[n] = T\_s[n] + \int V\_{ext}(r)n(r)d^3r + E\_{harmre}[n] + E\_{\infty}[n],\tag{23}$$

with the functional:

$$F\_{HK}[n] = E\_{HK}[n] - \int V\_{ext}\left(\stackrel{\smile}{r}\right) n\left(\stackrel{\smile}{r}\right) d^3r. \tag{24}$$

$$E\_{HK}[n] = T\_s[n] + E\_c[n] + E\_{hartree}[n] + E\_x[n] = T\_s[n] + E\_{hartree}[n] + E\_{xc}[n].\tag{25}$$

*Ts*½ � *n* : representing the kinetic energy of a fictitious gas of non-interacting electrons but of the same density is given by:

$$T\_s[n] = \sum\_i \int dr \Psi\_i^\* \left(\stackrel{\leftarrow}{r}\right) \frac{-\nabla^2}{2} \Psi\_i \left(\stackrel{\leftarrow}{r}\right). \tag{26}$$

$$E\_{\rm Hartree}[n] = \frac{e^2}{8\pi\varepsilon\_0} \int \frac{n\left(\stackrel{\leftarrow}{r}\right)n\left(\stackrel{\leftarrow}{r'}\right)}{|\stackrel{\leftarrow}{r} - \stackrel{\leftarrow}{r'}|} d^3r d^3r'.\tag{27}$$

*Ts*½ � *n* : the kinetic energy without interaction. *Exc*½ � *n* : the exchange-correlation energy. *Ehartree*½ � *n* : the electron–electron potential energy.

$$E\_{\rm xc}[n] = E\_{HK}[n] - \int V\_{\rm ext} \left( \stackrel{\frown}{r} \right) n \left( \stackrel{\frown}{r} \right) d^3 r - T\_s[n] - E\_{Hartree}[n]. \tag{28}$$

$$E\_{\infty}[n] = F[n] - T\_s[n] - E\_{harmree}[n].\tag{29}$$

Based on the second Hohenberg-Kohn theorem, which shows that the electron density of the ground state corresponds to the minimum of the total energy and on the condition of conservation of the number of particles

$$
\delta \mathcal{N} \left[ n \left( \stackrel{\leftarrow}{r} \right) \right] = \int \delta n \left( \stackrel{\leftarrow}{r} \right) dr = 0,\tag{30}
$$

So we have:

$$\delta \left\{ E\_{HK}[n] - \mu \left( \int n \left( \stackrel{\leftarrow}{r} \right) d^3 r - N \right) \right\} = \mathbf{0}.\tag{31}$$

$$\frac{\delta E\_{HK}[n]}{\delta n\left(\stackrel{\leftarrow}{r}\right)} = \mu,\tag{32}$$

$$\frac{\delta T\_s[n]}{\delta n\left(\stackrel{\frown}{r}\right)} + \nu^{\mathcal{G}\dagger}\left(\stackrel{\leftarrow}{r}\right) = \mu,\tag{33}$$

$$v\mathcal{U}\left(\stackrel{\leftarrow}{r}\right) = V\_{\text{ext}}\left(\stackrel{\leftarrow}{r}\right) + \frac{e^2}{8\pi\epsilon\_0} \left[\frac{n\left(\stackrel{\leftarrow}{r'}\right)}{|\stackrel{\leftarrow}{r} - \stackrel{\leftarrow}{r'}|}d^3r' + \frac{\delta E\_{\text{xc}}[n]}{\delta n\left(\stackrel{\leftarrow}{r}\right)}\right],\tag{34}$$

therefore, the kinetic energy without interaction *Ts*½ � *n* is determined by:

$$\frac{\delta T\_s[n]}{\delta n \left(\stackrel{\cdot}{r}\right)} = \left(3\pi^2 n\right)^{\frac{\zeta}{\zeta}} \frac{\hbar^2}{2m} = \frac{\hbar^2}{2m} k\_F^2. \tag{35}$$

Finally, the mono-electronic Hamiltonian of Kohn-sham in atomic unit is put in the form:

$$H = \frac{-\nabla^2}{2} + \nu^{\sharp \mathcal{F}}.\tag{36}$$

The Hamiltonian is iteratively computed, the self-consistency of a loop is reached when the variation of the calculated quantity is lower than the fixed convergence criterion. The wave functions are calculated by a conjugate gradient method (or equivalent). The density is built from the wave functions, convergence is reached when the density is sufficiently close to the density of the previous step. When seeking to optimize the atomic structure of the system, an additional loop is added. With each iteration of this loop, the atomic positions are changed. It is said that the system is minimized when the forces are lower than the convergence criterion on the amplitude of the forces.

*The Density Functional Theory and Beyond: Example and Applications DOI: http://dx.doi.org/10.5772/intechopen.100618*

#### **4.3 Expression of the exchange and correlation term**

As described above, DFT is at the stage of Kohn-Sham equations, a perfectly correct theory insofar as the electron density which minimizes the total energy is exactly the density of the system of N interacting electrons. However, DFT remains inapplicable because the exchange-correlation potential remains unknown. It is therefore necessary to approximate this exchange-correlation potential. Two types of approximations exist the local density approximation or LDA and the generalized gradient approximation or GGA as well as the derived methods which are based on a non-local approach.

#### *4.3.1 Local density approximation (LDA)*

In only one model case, that of the uniform gas of electrons (corresponding quite well to the electrons of the conduction band of a metal), we know the exact expressions or with an excellent approximation of the terms of exchange and correlation respectively. In this LDA (Local Density Approximation), the electron density is assumed to be locally uniform and the exchange-correlation functional is of the form:

$$E\_{\rm xc}^{LDA} = \int n\left(\stackrel{\leftarrow}{r}\right) \epsilon\_{\rm xc}^{homo}\left(n\left(\stackrel{\leftarrow}{r}\right)\right) dr. \tag{37}$$

$$
\varepsilon\_{\rm xc}^{homo}\left(n\left(\stackrel{\leftarrow}{r}\right)\right) = \varepsilon\_{\rm x}[n] + \varepsilon\_{\rm c}[n].\tag{38}
$$

The function of *εhomo xc n r* � � � � is determined from a quantum computation parameterization for a constant electron density *n r* � � <sup>¼</sup> *<sup>n</sup>*;

$$
\epsilon\_{\infty}^{homo}\left(n\left(\stackrel{\leftarrow}{r}\right)\right) = \frac{3}{4} \left(\frac{3n(r)}{\pi}\right)^{\frac{1}{3}}.\tag{39}
$$

Ceperley-Alder [4] numerically determined the contribution of the correlations. The search for analytical functions that come as close as possible to these results leads to the development of various functionalities with varying degrees of success. In general, the LDA approximation gives good results in describing the structural properties, i.e. it allows to determine the energy variations with the crystalline structure although it overestimates the cohesion energy, also concerning the mesh parameter for the majority of solids and good values of elastic constants like the isotropic modulus of compressibility. But this model remains insufficient in inhomogeneous systems.

#### *4.3.2 Generalized gradient approximation (GGA)*

To overcome the shortcomings of the LDA method, the generalized gradient approximation considers exchange-correlation functions depending not only on the density at each point, but also on its gradient [5], of the general form.

$$E\_{\mathbf{x}\prime}^{GGA}\left[n\_{\alpha},n\_{\beta}\right] = \int n\left(\stackrel{\leftarrow}{r}\right) \varepsilon\_{\mathbf{x}\prime}\left[n\_{\alpha},n\_{\beta},\nabla\_{n\_{\alpha}(r)}\nabla\_{n\_{\beta}\left(\stackrel{\leftarrow}{r}\right)}\right] dr,\tag{40}$$

*α* and *β* are spins, in this case again, a large number of expressions have been proposed for this factor *εxc* leading to so many functionals. In general, the GGA improves compared to the LDA a certain number of properties such as the total energy or the energy of cohesion, but does not lead to a precise description of all the properties of a semiconductor material namely its electronic properties.

#### *4.3.3 Functional hybrid HSE*

The functions of DFT have been proved to be quite useful in explaining a wide range of molecular characteristics. The long-term nature of the exchange interaction, and the resulting huge processing needs, are a key disadvantage for periodic systems. This is especially true for metallic systems that necessitate BZ sampling. A new hybrid functionality, recently proposed by Heyd et *al.* [6], addresses this problem by separating the description of the exchange and the interaction into a short and long part. The expression of the exchange-correlation energy in HSE03 is given by:

$$E\_{\rm xc}^{\rm HSEO3} = \frac{\mathbf{1}}{\mathbf{4}} E\_{\rm x}^{sr,\mu} + \frac{\mathbf{3}}{\mathbf{4}} E\_{\rm x}^{PBE,sr,\mu} + E\_{\rm x}^{PBE,lr,\mu} + E\_{\rm c}^{PBE}.\tag{41}$$

As can be seen from the Eq. (41) only the exchange component of the electron– electron interaction is split into a short and long (lr) range (sr) part. The full electron correlation is represented by the standard correlation portion of the density of the GGA functional. Note that the term hybrid refers to the combined use of the exact exchange energy of the Hartree-Fock model and the exchange-correlation energy at the DFT level. The construction of hybrid functionals has been a good advancement in the field of exchange-correlation energy processing by allowing an explicit incorporation of the nonlocal character through the use of the exact term of exchange energy.
