## Hartree-Fock method Replacing the Hartree products with [[Hartree method, from Hartree product to HF#^e9965a|Slater determinant]] and apply the self-consistency field calculation. We get the Hartree Fock method. The exchange effect on the Coulombic proportion is included for the interaction between each electron and the field by all other electrons. The main point of the Hartree-Fock is calculation of one electron integrals (kinetic of electron & interaction with nuclei) and two electrons integrals (Coulombic & exchange), cause one have to solve $\bra{\Psi_{SD}} \hat{H} \ket{\Psi_{SD}} $ So a general approach to do Hartree-Fock is from the Fock operator, $F_{\mu \nu}$ $\hat{F}_{\mu\nu}=\bra{\mu}-\frac{1}{2}\nabla^{2}\ket{\nu} - \sum_{k}^{nuclei} Z_{k} \bra{\mu} \frac{1}{r_{k}}\ket{\nu} + \sum_{\lambda\sigma} P_{\lambda\sigma} \left[ (\mu\nu|\lambda\sigma) - \frac{1}{2} (\mu\lambda|\nu\sigma) \right] $ ^b0b90c >[!Note] >Interpretation of this $F_{\mu\nu}$: >- $\mu$, $\nu$ are basis functions; >- $\bra{\mu}-\frac{1}{2}\nabla^{2}\ket{\nu}$ is the kinetic term; >- $\sum_{k}^{nuclei} Z_{k} \bra{\mu} \frac{1}{r_{k}}\ket{\nu}$ is the nuclei field; >- $(\mu\nu|\lambda\sigma)$ is the coulombic term; >- $- \frac{1}{2} (\mu\lambda|\nu\sigma)$ is the exchange interaction, the $\frac{1}{2}$ is because only $\uparrow\uparrow$ gets exchange E. With the Fock operator, we may write $\hat{F}\Psi_{SD}=\epsilon \Psi_{SD}$ Doing computation, we can get the electron distribution (density matrix), the update $\hat{F}$ and $\Psi_{SD}$, iterate, and finally solve this self-consistency field problem. >[!Notice] >What described here is when we solve the WF analytically, it could be extremely silly and computational intensive, but as an illustration it's okay. But we should do it in a more clear way, which will be discussed later. Before that, let's just see the problems and limitations of this method. - HF approximates the field by electoral as the static field. It does not have (full) correlation. (This is a big issue, although we have exchange interaction) - It uses the single electron Hamiltonian to represent, or say, approximate, many-electron system. Only when basis set with functions number $\rightarrow \infty$ can HF limit be accessed with an error -- the correlation error. ^ed2c7d - 4 index integral in secular equation (since in $\Psi$ itself we have 2, with e-e interaction and exchange, $\bra{\Psi} \hat{H} \ket{\Psi}$ becomes 4). $N^{4}$ is a large number of integration to be done. (scale with $N^{4}$, number of basis functions) ![[Drawing 2023-10-03 19.54.35.excalidraw.svg]] >[!Info] >See more on >- next part: [[Hartree-Fock-Roothaan method]] >- wiki: [HF](https://en.wikipedia.org/wiki/Hartree%E2%80%93Fock_method), [Fock matrix](https://en.wikipedia.org/wiki/Fock_matrix)