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HF part for spherium

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Pierre-Francois Loos 2020-07-30 16:08:43 +02:00
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RapportStage/Rapport.tex
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@ -650,16 +650,14 @@ Here, we work in a minimal basis, composed of $Y_{0}$ and $Y_{1}$, or equivalent
\end{equation} \end{equation}
using a mixing angle between the two basis functions for each spin manifold. 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 $\chi_\beta$. 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 $R$: This process provides the three following solutions valid for all value of $R$, which are respectively a minimum, a maximum and a saddle point of the HF equations:
\begin{itemize} \begin{itemize}
\item The two electrons are in the s orbital which is a RHF solution. This solution is associated with the energy $1/R^{2}$. \item The two electrons are in the s orbital which is a RHF solution. This solution is associated with the energy $1/R^{2}$.
\item The two electrons are in the p\textsubscript{z} orbital which is a RHF solution. This solution is associated with the energy $1/R^{2} + 1/R$ \item The two electrons are in the p\textsubscript{z} orbital which is a RHF solution. This solution is associated with the energy $1/R^{2} + 1/R$.
\item One electron is in the s orbital and the other is in the p\textsubscript{z} orbital which is a UHF solution. This solution is associated with the energy $2/R^{2}+ 29/(25R)$ \item One electron is in the s orbital and the other is in the p\textsubscript{z} orbital which is a UHF solution. This solution is associated with the energy $2/R^{2}+ 29/(25R)$.
\end{itemize} \end{itemize}
\titou{STOPPED HERE.}
Those solutions are respectively a minimum, a maximum and a saddle point of the HF equations.\\
In addition, there is also the well-known symmetry-broken UHF (sb-UHF) solution. For $R>3/2$ an other stationary UHF solution appears, this solution is a minimum of the HF equations. This solution corresponds to the configuration with the electron $\alpha$ on one side of the sphere and the electron $\beta$ on the opposite side and the configuration the other way round. The electrons can be on opposite sides of the sphere because the choice of p\textsubscript{z} as a basis function induced a privileged axis on the sphere for the electrons. This solution has the energy \eqref{eq:EsbUHF} for $R>3/2$. In addition, there is also the well-known symmetry-broken UHF (sb-UHF) solution. For $R>3/2$ an other stationary UHF solution appears, this solution is a minimum of the HF equations. This solution corresponds to the configuration with the electron $\alpha$ on one side of the sphere and the electron $\beta$ on the opposite side and the configuration the other way round. The electrons can be on opposite sides of the sphere because the choice of p\textsubscript{z} as a basis function induced a privileged axis on the sphere for the electrons. This solution has the energy \eqref{eq:EsbUHF} for $R>3/2$.