Hartree Fock Method
========================

:Date: 27 July 2019



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

Calculate the approximate wavefunction and energy for atoms

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

**Step 1** Approximate the wavefunction using Slater Determinant

.. math::
	\Psi = {1\over \sqrt{N!} } 
		\begin{vmatrix}
			\chi_1(x_1) & \chi_2(x_1) & ... & \chi_N(x_1)\\
			\chi_1(x_2) & \chi_2(x_2) & ... & \chi_N(x_2)\\
			...\\
			\chi_1(x_N) & \chi_2(x_N) & ... & \chi_N(x_N)
		\end{vmatrix}

**Step 2** The system Hamiltonian after Born Oppenheimer Approximation

.. math::
	\begin{align*}
	& H = \sum\limits_i h(i) + \sum\limits_{i<j} v(i,j) + V_{NN} \\
	& where \\
	& h(i) = -{1\over 2}\nabla_i^2 - \sum\limits_A {Z_A\over r_{iA}} \\
	& v(i,j) = {1\over r_{ij}}
	\end{align*}

**Step 3** The energy

.. math::
	\begin{align*}
		E & = \langle \Psi |H| \Psi \rangle \\
		  & = \sum\limits_i \langle i|h|i \rangle 
		  	 +{1\over 2} \sum\limits_{ij}[ii|jj]-[ij|ji] \\
	& where\\
	& \langle i|h|j \rangle = \int dx_1 \; \chi_i^* h \chi_j\\
	& [ij|kl] = \int dx_1 dx_2 \; \chi_i^* \chi_j {1\over r_{ij}} \chi_k^* \chi_l
	\end{align*}

**Step 4** Use Lagrange Multiplier to minimize the energy wrt changes in the orbitals :math:`\chi_i \to \chi_i+\delta \chi_i`

under constraint :math:`\langle i|j \rangle = \delta_{ij}`

.. math::
	\begin{align*}
	& L = E - \sum\limits_{ij} \epsilon_{ij} [\langle i|j \rangle - \delta_{ij}]\\
	& \delta L =0 \Rightarrow \boxed{
		f \chi_i = \epsilon_i \chi_i
	}\\
	& where \\
	& \text{Fock operator  } 
		f \equiv h + \sum\limits_j J_j - K_j\\
	& - where \\
	& \text{Coulomb operator  } 
		J_j(x_1) = \int dx_2 \; |\chi_j(x_2)|^2 {1\over r_{12}} \\
	& \text{Exchange operator  } 
		K_j(x_1) \chi_i(x_1) = \left[ \int dx_2\; \chi_j^*(x_2) {1\over r_{12}} \chi_i(x_2)\right] \chi_j(x_1) \\
	\end{align*}

**Step 5** Introducing basis functions to transform the above **integral equation** into a **matrix equation**

**TO BE CONTINUED**