[feat]: Adding intuitive demonstration on how to perform potential co…

…rrection in crystal

docs/source/introduction.rst

@@ -34,21 +34,58 @@ this existing relationship, self-energy corrections performed in atoms could pro
 .. figure:: images/cdo_bands.png
    :width: 500
 
-   Orbital character for CdO valence bands. The character :math:`p` is represented
+   Fig 1. Orbital character for CdO valence bands. The character :math:`p` is represented
    in yellow and the character :math:`d` in a magnet [10]_.
 
 
 How to perform potential correction in crystals
 ###################################################
 
-Following the Slater half occupation procedure, a change in charge density is necessary to obtain the potential for 
-half occupation, however a change in charge density in a unit cell would result in an infinitely charged system, which would lead to a 
+In this section, calculations were developed using some approximations in order to demonstrate
+intuitively how the potential correction in crystals is made. To access the rigorous demonstration, consult the references [1]_ [2]_ . 
+
+Following the Slater half occupation procedure for atoms, a change in charge density is required
+to obtain the potential for half occupation and perform the consistent calculations using the Khon-Shan equation, as well as
+shown in Figure 2.
+
+.. figure:: images/slater-atoms.svg
+   :width: 400
+
+   Fig 2. Flowchart representing the Slater procedure in atoms
+
+
+
+Altough in extended systems like crystals a change in charge density in a unit cell would result in an infinitely charged system, which would lead to a 
 divergence in the Khon-Shan calculations. Furthermore, it would also be irrelevant to be able to modify only a finite amount of electrons in the crystal since
-the charge would become irrelevant to the infinite amount of electrons present in the system. To bypass
+the charge would become irrelevant to the infinite amount of electrons present in the system. To bypass this problem, it is necessary to find a new way
+to derive the half-occupied potential.
+
+Firstly, one have to define the system that corresponds to the semi-occupied potential for a solid. For an atom containing
+:math:`N` electrons in its ground state, the semi-occupied potential is defined as the potential of the atom with :math:`N-\frac{1}{2}` electrons. Similarly, we should 
+think consider that the semi-occupied potential of a solid would be the potential generated by a solid with :math:`M-\frac{1}{2}` electrons per primitive cell, where :math:`M` is the number of electrons
+of the unit cell in the ground state, as shown in Figure 3.
+
+.. figure:: images/semi-solid.svg
+   :width: 400
+
+   Fig 3. Scheme representing the unit cells of a solid that would generate the potential semi-occupied.
+
+So, to outline the solution, suppose one have :math:`N` independent charge distributions, where the :math:`mth` is given by:
+
+.. math::
+   \rho_{m}(\vec{r}) = (1+f_{m})n_{m}(\vec{r}) \\
+   n_{m}(\vec{r}) = -q \cdot \eta_{m} \\
+   \int \eta_{m}(\vec{r})d\vec{r} = 1 \\
+
+Where :math:`\rho` represents the density of the distribution :math:`m`, :math:`q` represents the charge of the electron, :math:`\eta` a normalized function in space and :math:`f` represents an occupancy factor that varies continuously from
+0(occupied) to -1(unoccupied)
+
+
+To bypass
 this problem, approximations are used in order to make it possible to calculate the potential for the half occupation of the
 crystal through other potentials [1]_ [2]_, as shown in the equation below: 
 
-.. math:: 
+.. math:: `
    V_{crystal}^{-1/2} = V_{crystal} - V_{1/2e}
 
 Where :math:`V_ {crystal}^{- 1/2}` is the potential of the half-occupied crystal, :math:`V_ {crystal}`