Density Functional Theory
============================

:Date: 28 July 2019

Dependency
-------------

- `Schrodinger Equation`_
- `Born Oppenheimer Approximation`_
- Functional_

.. _Schrodinger Equation: 
.. _Born Oppenheimer Approximation:
.. _Functional:

What is the problem ?
------------------------

Calculate the approximate wavefunction and energy for atoms

Derivation
-------------

**Step 1** Hohenberg Kohn theorem

:math:`n(r)`: groundstate electron density

:math:`v(r)`: external potential

Then Functional_ :math:`v=v[n]`, meaning if we observe some :math:`n(r)`, then we know exactly what potential :math:`v(r)` generates such density distribution

**Step 2** Proof of **Step 1**

*TO BE ADDED*

**Step 3** The energy

.. math::
	\begin{align*}
	E[n] & = T+U+V = T[n]+U[n]+\int n(r)v(r) \; d^3r\\
	& where\\
	& T = \text{Kinetic energy of electrons}\\
	& U = \text{Electric potential}\\
	& V = \text{Total external potential}
	\end{align*}

**Step 4** Kohn-Sham approach to evaluate The energy

Introduce single particle orbitals :math:`\chi_i`

.. math::
	\begin{align*}
	& T=T_{\text{single particle}} + T_{\text{correlation}} = T_s+T_c \\
	& T_s = T_s\{\chi_i[n]\}\\
	& \text{Introduce:}\\
	& E_{xc} = (T-T_s)+(U-U_H)\\
	& \Rightarrow\\
	E[n] & = T_s\{\chi_i[n]\}+U_H+E_{xc}+V
	\end{align*}

The idea above is to leave all the non-trivial parts to :math:`E_{xc}=E_{x}+E_{c}`

- :math:`x` stands for exchange energy due to the Pauli Principle
- :math:`c` stands for correlation energy, :math:`T_c` is part of :math:`E_c`

We have

.. math::
	\begin{align*}
	& T_s = ...\text{ trivial expression we know} \\
	& U_H = ...\text{ trivial expression we know} \\
	& E_x = ???\text{ can only be calculated after approximation(step 5)} \\
	& E_c = ???\text{ can only be calculated after approximation(step 5)} 
	\end{align*}

**Step 5** Approximations to find :math:`E_{xc}`

*TO BE ADDED*

**Step 6** Get the groundstate energy by minimizing :math:`E[n]` wrt :math:`n`

**Kohn-Sham Equations**

**TO BE CONTINUED**