[feat]: Conclude demonstration of the section how to perform potentia…

…l correction in crystals

docs/source/introduction.rst

@@ -1,17 +1,17 @@
-=============
+##############
 Introduction
-=============
+##############
 
 What is minushalf?
-###################
+********************
 
 Minushalf is a CLI developed by GMSN that aims to automate 
 the application of the DFT -1/2 method. The commands available in this 
 CLI automate both the entire process and each of its steps in order to be 
 used by the user for any purposes.
 
 An intuitive explanation of the DFT -1/2 method
-##################################################
+************************************************
 
 DFT-1/2, an alternative way of referring to the LDA -1/2 [1]_ [2]_ and GGA -1/2 [2]_ techniques, 
 is a method that performs semiconductor band-gap calculations with precision close 
@@ -21,7 +21,7 @@ technique [3]_ [4]_ [5]_, formalized by Janak's theorem, to crystals using moder
 The Slater half-occupation scheme has already proven to be quite efficient for calculating atomic ionization 
 energies values close to the experimental [5]_. However, this technique cannot be applied blindly to
 extended systems like crystals, since the crystal is described by means of Bloch waves and removing the population
-of just one Bloch state would have no consequences[1]_. Moreover, removing the population of one Bloch State and set periodic
+of just one Bloch state would have no consequences [1]_. Moreover, removing the population of one Bloch State and set periodic
 conditions would result in a infinitely charged system.
 
 Thus, the proposed solution is to apply the Slater procedure to cystaline energy bands. 
@@ -39,7 +39,7 @@ this existing relationship, self-energy corrections performed in atoms could pro
 
 
 How to perform potential correction in crystals
-###################################################
+****************************************************
 
 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]_ . 
@@ -58,7 +58,7 @@ shown in Figure 2.
 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 this problem, it is necessary to find a new way
-to derive the half-occupied potential.
+to derive the semi-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 
@@ -80,16 +80,46 @@ So, to outline the solution, suppose one have :math:`N` independent charge distr
 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)
 
+Considering the charge density represented above, one can find the Coulomb potentials for each distribution
+by the Poison equation:
+
+.. math::
+   \nabla^{2}V_{m}(\vec{r}) = \frac{q\rho_{m}(\vec{r})}{\epsilon_{0}}
+
+Now, suppose another situation where we only alternate the occupation of the :math:`\alpha` level and the same charge distribution remains. In this scenario, the m-th
+potential is given by:
+
+.. math::
+   \nabla^{2}V_{m}{'}(\vec{r}) = \frac{q\rho_{m}{'}(\vec{r})}{\epsilon_{0}} \\
+   f_{i}=f_{i}{'}, i \neq \alpha
+   f_{\alpha} \neq f_{\alpha}{'}
 
-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: 
+Thus, one want to calculate the potential for all distributions, which is obtained by adding
+of the potential of all distributions, as shown in the equations below.
+
+.. math::
+   \nabla^{2}V(\vec{r}) = \frac{q\sum_{m=1}^{N}\rho_{m}}{\epsilon_{0}} \\
+   \nabla^{2}V{'}(\vec{r}) = \frac{q\sum_{m=1}^{N}\rho_{m}{'}}{\epsilon_{0}}
+
+Subtracting these two equations:
+
+.. math::
+   \nabla^{2}(V(\vec{r})-V{'}(\vec{r})) = \frac{q(f_{\alpha}-f_{\alpha}{'})n_{\alpha}(\vec{r})}{\epsilon_{0}}
 
-.. math:: `
+Using the above equation for the specific case of :math:`f_{\alpha} = 0` and :math:`f_{\alpha}{'} = -1/2`, the following equation is obtained:
+
+.. math::
+   \nabla^{2}(V(\vec{r})-V{'}(\vec{r})) = \frac{qn_{\alpha}(\vec{r})}{2\epsilon_{0}}\Rightarrow V{'}(\vec{r}) = V(\vec{r}) - V_{\alpha}^{f_{\alpha}=-1/2}
+
+
+Hence, using the equation above, one can calculate the potential semi-occupied from other potentials, which discards the need for
+modify the charge density. For a crystal, the equation is written as follows:
+
+.. 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}`
-is the potential of the crystal with the standard occupation and :math:`V_ {1 / 2e}` is the potential of the respective level
+Where :math:`V_ {crystal}^{- 1/2}` is the potential of the semi-occupied crystal, :math:`V_ {crystal}`
+is the potential of the crystal in the ground state and :math:`V_ {1 / 2e}` is the potential of the respective level
 occupied with half an electron. 
 
 To generate :math:`V_ {1 / 2e}`, the following equation is used for the atoms that compose the crystal [1]_ [2]_:
@@ -104,29 +134,31 @@ To generate :math:`V_ {1 / 2e}`, the following equation is used for the atoms th
    \end{matrix}\right.
 
 
-Where :math:`V_{atom}` is the potential of the atom with the standard occupation, :math:`V_{atom}^{f_{\alpha}=-1/2}`
+Where :math:`V_{atom}` is the potential of the atom in the ground state, :math:`V_{atom}^{f_{\alpha}=-1/2}`
 is the potential of the atom with the level :math:`\alpha` occupied, :math:`\theta (r)` is a cutting function,
 CUT is the radius of cut and A is a scale factor named amplitude.
 
-The need to have a cutting function is due to the fact that an artificial charged system is generated, therefore the correction
-in a cell it generates an  potential that reaches neighboring cells, which would lead to a divergence in the Khon-Shan calculations.
-It is worth mentioning that the values ​​for :math:`CUT` and :math:`A` must not be chosen arbitrarily, by means of variational 
+There is a problem with adding :math:`-V_ {1 / 2e}` to all the atoms of an infinite crystal: the potential will
+diverge. A :math:`-V_ {1 / 2e}` is a potential of an excess charge of 1/2 proton and has a tail of 0.5/r
+that cannot be summed in an infinite lattice. Therefore the tail has to be trimmed by a step function [2]_.
+Besides, it is worth mentioning that the values ​​for :math:`CUT` and :math:`A` must not be chosen arbitrarily, by means of variational 
 arguments it can be proved that the optimal values ​​for these parameters are those that maximize the Gap of the crystalline system [1]_ [2]_.
 
-Finally, since the atoms are repeated in each unit cell, the potential :math:`V_{1/2e}` is periodic, joining this
-information with the fact that :math:`V_{crystal}` is periodic, it has the implication that :math:`V_{crystal}^{-1/2}`
-is periodic, which implies that the boundary conditions remain periodic. 
+
+Finally, since the atoms repeats in each unit cell, the potential :math:`V_{1/2e}` is periodic, joining this
+information with the fact that :math:`V_{crystal}` is periodic, one can conclude that :math:`V_{crystal}^{-1/2}`
+is periodic, which implies that the boundary conditions remain periodic and the Khon-Shan calculations can be applied to the system. 
 
 
 Where to perform the half occupation? 
-############################################
+**************************************************
 
 There are two types of correction, simple and fractional, and they must be performed in the last valence band (:math:`VBM`) and the first conduction band (:math:`CBM`).
 The choice of which correction cannot be made blindly, it requires an analysis of the band's composition. To explain these two corrections, suppose that we have a matrix where the atoms 
 of the unit cell are represented as lines and the types of atomic orbitals :math:`(s, p, d, f ...)` as columns , each value `a_{ij}` 
 represents, in percentage, how much that orbital :math:`j` of a given atom :math:`i` contributes to the total module of the wave function. 
 
-.. math ::
+.. math::
    A = \begin{bmatrix} 
    a_{11} & a_{12} & \dots \\
    \vdots & \ddots & \\
@@ -135,48 +167,48 @@ represents, in percentage, how much that orbital :math:`j` of a given atom :math
 
 Where:
 
-.. math ::
+.. math::
    \sum_{i=1}^{N} \sum_{j=1}^{K} a_{ij} = 100
 
 
 Simple correction
-********************
+========================
 The simple correction method is applied when an index :math:`a_{ij}` mainly represents the
 composition of the band, so that the influence of the other orbitals is negligible.
 Thus, the correction of half an electron is done only in the orbital :math:`j` of the atom :math:`i`. 
 
 .. _frac_correction:
 
 Fractional correction
-************************
+=========================
 The fractional correction method is applied when different atomic orbitals have a significant influence
 in the composition of the band. To distribute the electron medium, a threshold is chosen
 :math:`\epsilon`, which represents the minimum value of :math:`a_{ij}` considered in the correction. Given these
 values, half an electron will be divided among the atoms, proportionally to the coefficient :math:`a_{ij}`.
 
 Is conduction band correction always necessary?
-********************************************************
+======================================================
 In many cases, the correction in the valence band already returns satisfactory and close enough to the 
 experimental results, which rules out the need for an additional correction in the conduction band. 
 
 Final considerations
-***********************
+=============================
 After applying the correction, the optimum cut and amplitude must be found for each corrected atom to, finally,
 we find the final value for the gap. 
 
 
 DFT -1/2 results
-###################
+*********************
 
-The results obtained by the application the method has the same precision [2]_ as the GW [9]_ algorithm ,as shown in Figure 2, considered 
+The results obtained by the application the method has the same precision [2]_ as the GW [9]_ algorithm considered 
 the state of the art for calculating the band-gap of semiconductors. In addition, the computational complexity of the method 
 is equivalent to calculating the Khon-Shan gap, which allows the technique to be applied to complex systems.
 
 
 
 
 References
-###########
+********************
 
 .. [1] L. G. Ferreira, M. Marques, and L. K. Teles, `Phys. Rev. B 78, 125116 (2008) <http://dx.doi.org/10.1103/PhysRevB.78.125116>`_.