A simple example for the application of the Hartree-Fock method?

Question

I'm currently reading a book about electronic structure, and I'm really interested in Hartree-Fock method. But I can only have an empirical understanding from the book. So I would like to ask for one or perhaps two simple, basic mathematical examples that I can work with to gain a better understanding of the method.

Answer 1

I would suggest that you try to write a small code that can perform a Hartree-Fock calculation.

1. Specify molecule and basis set.

2. Get the integrals. The integrals can be hard to calculate on your own so I would suggest to obtain those from different sources, see Programming Project #3: The Hartree-Fock Self-Consistent Field Method

3. Make a diagonalization of the overlap matrix, $S$, to get the transformation matrix, $ S^{-1/2} $. $$ S^{-1/2}=L_s\lambda_s^{-1/2}L_s^T $$ Here $ L_s $ is the eigenvectors of the overlap matrix, and $ \lambda_s $ is the diagonalized overlap matrix.

4. Make an initial guess for the density matrix, $D$. This can be done using the core Hamiltonian $H_{core}$ as an inital guess for the Fock matrix, $F$. The core Hamiltonian is given as the sum of the kinetic energy and electron nuclei coulomb interaction integrals. Now orthonormal Fock matrix can be constructed: $$ F'_0 = S^{-1/2,T}H_{\mathrm{core}}S^{-1/2} $$ This can now be diagonalized to obtain orbtial coefficients, $C'$: $$ F'_0C'_0=C'_0\epsilon_0 $$ These coefficient can be back transformed $C=S^{-1/2}C'$ and used to construct the density matrix: $$ D_{ij,0} = \sum_{ij}^{occ}C_{i,0}C_{j,0} $$ The inital energy can also be found as $E_{elec,0} = \sum_{ij}^{occ}D_{ij,0}(H_{ij,\mathrm{core}}+H_{ij,\mathrm{core}})$.

5. Build new Fock matrix. $$ F_{ij} = H_{ij,\mathrm{core}} + \sum_{kl}^{occ}D_{kl,0}(2(ij|kl)-(ik|jl))$$

6. Build new density matrix, in the same way as the inital density matrix was build. $$ F' = S^{-1/2,T}FS^{-1/2} $$ $$ F'C'=C'\epsilon $$ $$ D_{ij} = \sum_{ij}^{occ}C_{i}C_{j} $$ The energy can now be found as $E_{elec} = \sum_{ij}^{occ}D_{ij}(F_{ij}+H_{ij,\mathrm{core}})$. Return to step 5 until convergence is reached.