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

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.

• This post (v4) seems like a list question. Aug 4 '17 at 5:17
• @Qmechanic its a good question and I'd enjoy reading an answer and trying out the problem myself! Hopefully it's been sufficiently "de-listed" (de-listicized?)
– uhoh
Aug 4 '17 at 6:53

## 1 Answer

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.

• This is what I'd call a link-only answer, and therefore not a very good answer. If the link goes dead, this answer is useless. Can you take a little more time and actually state your answer here in your answer? Stackexchange answers should really be able to stand on their own. So maybe choose one (or two) from that list and explain it here. There is a tutorial for Mathjax. Put it between single dollar signs for in-line, and between pairs of dollar signs for centered. Thanks!
– uhoh
Aug 4 '17 at 7:02
• Thanks for your suggestion. And thank you very much for the link to the Mathjax tutorial. Aug 4 '17 at 13:02
• Holy granola what a transcription, very nice. +n! :)
– uhoh
Aug 4 '17 at 13:43