diff --git a/RapportStage/Rapport.tex b/RapportStage/Rapport.tex index c56a373..b741062 100644 --- a/RapportStage/Rapport.tex +++ b/RapportStage/Rapport.tex @@ -639,6 +639,7 @@ and \end{pmatrix}^2 \end{equation} \titou{T2: please define Wigner 3j symbols.} +\titou{STOPPED HERE.} We obtained Eq.~\eqref{eq:EUHF} for the general form of the wave function \eqref{eq:UHF_WF}, but to be associated with a physical wave function the energy needs to be stationary with respect to the coefficients. The general method is to use the Hartree-Fock self-consistent field method \cite{SzaboBook} to get the coefficients of the wave functions corresponding to physical solutions. We will work in a minimal basis, composed of a $Y_{00}$ and a $Y_{10}$ spherical harmonic, or equivalently a s and a p\textsubscript{z} orbital, to illustrate the difference between the RHF and UHF solutions. In this basis there is a shortcut to find the stationary solutions. One can define the one-electron wave functions $\phi(\theta)$ using a mixing angle between the two basis functions. Hence we just need to minimize the energy with respect to the two mixing angles $\chi_\alpha$ and $\chi_\beta$.