HF part for spherium

RapportStage/Rapport.tex

@@ -616,23 +616,22 @@ These one-electron orbitals are expanded in the basis of zonal spherical harmoni
 \begin{equation}
 	\phi_\sigma(\theta)=\sum_{\ell=0}^{\infty}C_{\sigma,\ell}Y_{\ell}(\theta)
 \end{equation}
-It is possible to obtain the formula for the ground state UHF energy in this basis set \cite{Loos_2009}:
-
+It is possible to obtain the formula for the HF energy in this basis set \cite{Loos_2009}:
 \begin{equation}
-E_{\text{UHF}} = T_{\alpha} + T_{\beta} + V
-\label{eq:EUHF}
+	E_{\text{HF}} = T_{\text{HF}} + V_{\text{HF}}
+\label{eq:EHF}
 \end{equation}
-with
-\begin{gather}
-	T_{\sigma} = \frac{1}{R^2} \sum_{\ell=0}^{\infty} C_{\sigma,\ell}^2 \, \ell(\ell+1)
-	\\
-	V = \frac{1}{R} \sum_{\ell_1,\ell_2,\ell_3,\ell_4=0}^{\infty} \sum_{L=0}^{\infty} 
-	(-1)^{\ell_3+\ell_4} v^\alpha_{\ell_1,\ell_2,L} v^\beta_{\ell_3,\ell_4,L} 
-\end{gather}
+where the kinetic and potential energies are
+\begin{align}
+	T_{\text{HF}} & = \sum_{\sigma=\alpha,\beta} \frac{1}{R^2} \sum_{\ell=0}^{\infty} C_{\sigma,\ell}^2 \, \ell(\ell+1)
+	&
+	V_{\text{HF}} & = \frac{1}{R} \sum_{L=0}^{\infty} 
+	v^\alpha_{L} v^\beta_{L} 
+\end{align}
 and
 \begin{equation}
-	v^\sigma_{\ell_1,\ell_2,L} 
-	= \sqrt{(2\ell_1+1)(2\ell_2+1)} C_{\sigma,\ell_1}C_{\sigma,\ell_2}
+	v^\sigma_{L} 
+	=  \sum_{\ell_1,\ell_2} \sqrt{(2\ell_1+1)(2\ell_2+1)} C_{\sigma,\ell_1}C_{\sigma,\ell_2}
 	\begin{pmatrix}
 		\ell_1 & \ell_2 & L 
 		\\
@@ -640,13 +639,18 @@ and
 	\end{pmatrix}^2 
 \end{equation}
 is expressed in terms of the Wigner 3j-symbols \cite{AngularBook}.
-\titou{STOPPED HERE.}
-
-Equation \eqref{eq:EUHF} is obtained from the general form of the wave function \eqref{eq:UHF_WF}, but to be associated with a physical wave function the energy has to be stationary with respect to the coefficients. The general method is to use a self-consistent field procedure \cite{SzaboBook} to get the coefficients of the wave functions corresponding to physical solutions. We will work in a minimal basis, composed of $Y_{0}$ and $Y_{1}$, or equivalently, a s and p\textsubscript{z} orbitals, 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 $\chi_\beta$.
 
+The general method is to use a self-consistent field procedure as described in Ref.~\cite{SzaboBook} to get the coefficients of the wave functions corresponding to stationary solutions with respect to the coefficients $C_{\sigma,\ell}$, i.e.,
+\begin{equation}
+	\pdv{E_{\text{HF}}}{C_{\sigma,\ell}} = 0.
+\end{equation}. 
+Here, we work in a minimal basis, composed of $Y_{0}$ and $Y_{1}$, or equivalently, a s and 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 
 \begin{equation}
 	\phi_\sigma(\theta)= \cos(\chi_\sigma)Y_{0}(\theta) + \sin(\chi_\sigma)Y_{1}(\theta)
 \end{equation}
+using a mixing angle between the two basis functions for each spin manifold. 
+Hence we just minimize the energy with respect to the two mixing angles $\chi_\alpha$ and $\chi_\beta$.
+\titou{STOPPED HERE.}
  
 The minimization gives the three following solutions valid for all value of $R$:
 \begin{itemize}