HF part for spherium
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@ -616,23 +616,22 @@ These one-electron orbitals are expanded in the basis of zonal spherical harmoni
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\begin{equation}
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\begin{equation}
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\phi_\sigma(\theta)=\sum_{\ell=0}^{\infty}C_{\sigma,\ell}Y_{\ell}(\theta)
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\phi_\sigma(\theta)=\sum_{\ell=0}^{\infty}C_{\sigma,\ell}Y_{\ell}(\theta)
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\end{equation}
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\end{equation}
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It is possible to obtain the formula for the ground state UHF energy in this basis set \cite{Loos_2009}:
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It is possible to obtain the formula for the HF energy in this basis set \cite{Loos_2009}:
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\begin{equation}
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\begin{equation}
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E_{\text{UHF}} = T_{\alpha} + T_{\beta} + V
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E_{\text{HF}} = T_{\text{HF}} + V_{\text{HF}}
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\label{eq:EUHF}
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\label{eq:EHF}
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\end{equation}
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\end{equation}
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with
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where the kinetic and potential energies are
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\begin{gather}
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\begin{align}
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T_{\sigma} = \frac{1}{R^2} \sum_{\ell=0}^{\infty} C_{\sigma,\ell}^2 \, \ell(\ell+1)
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T_{\text{HF}} & = \sum_{\sigma=\alpha,\beta} \frac{1}{R^2} \sum_{\ell=0}^{\infty} C_{\sigma,\ell}^2 \, \ell(\ell+1)
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\\
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&
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V = \frac{1}{R} \sum_{\ell_1,\ell_2,\ell_3,\ell_4=0}^{\infty} \sum_{L=0}^{\infty}
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V_{\text{HF}} & = \frac{1}{R} \sum_{L=0}^{\infty}
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(-1)^{\ell_3+\ell_4} v^\alpha_{\ell_1,\ell_2,L} v^\beta_{\ell_3,\ell_4,L}
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v^\alpha_{L} v^\beta_{L}
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\end{gather}
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\end{align}
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and
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and
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\begin{equation}
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\begin{equation}
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v^\sigma_{\ell_1,\ell_2,L}
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v^\sigma_{L}
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= \sqrt{(2\ell_1+1)(2\ell_2+1)} C_{\sigma,\ell_1}C_{\sigma,\ell_2}
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= \sum_{\ell_1,\ell_2} \sqrt{(2\ell_1+1)(2\ell_2+1)} C_{\sigma,\ell_1}C_{\sigma,\ell_2}
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\begin{pmatrix}
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\begin{pmatrix}
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\ell_1 & \ell_2 & L
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\ell_1 & \ell_2 & L
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\\
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\\
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@ -640,13 +639,18 @@ and
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\end{pmatrix}^2
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\end{pmatrix}^2
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\end{equation}
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\end{equation}
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is expressed in terms of the Wigner 3j-symbols \cite{AngularBook}.
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is expressed in terms of the Wigner 3j-symbols \cite{AngularBook}.
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\titou{STOPPED HERE.}
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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 $\chi_\beta$.
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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.,
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\begin{equation}
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\pdv{E_{\text{HF}}}{C_{\sigma,\ell}} = 0.
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\end{equation}.
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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
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\begin{equation}
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\begin{equation}
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\phi_\sigma(\theta)= \cos(\chi_\sigma)Y_{0}(\theta) + \sin(\chi_\sigma)Y_{1}(\theta)
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\phi_\sigma(\theta)= \cos(\chi_\sigma)Y_{0}(\theta) + \sin(\chi_\sigma)Y_{1}(\theta)
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\end{equation}
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\end{equation}
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using a mixing angle between the two basis functions for each spin manifold.
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Hence we just minimize the energy with respect to the two mixing angles $\chi_\alpha$ and $\chi_\beta$.
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\titou{STOPPED HERE.}
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The minimization gives the three following solutions valid for all value of $R$:
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The minimization gives the three following solutions valid for all value of $R$:
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\begin{itemize}
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\begin{itemize}
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