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

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

Step 2 The system Hamiltonian after Born Oppenheimer Approximation

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

Step 3 The energy

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

Step 4 Use Lagrange Multiplier to minimize the energy wrt changes in the orbitals \(\chi_i \to \chi_i+\delta \chi_i\)

under constraint \(\langle i|j \rangle = \delta_{ij}\)

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

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

TO BE CONTINUED