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Pierre-Francois Loos 2021-05-23 21:31:54 +02:00
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TrUEGs.tex
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@ -191,11 +191,11 @@ Note that this feature was first discovered by Seidl in Appendix A of Ref.~\onli
He also provided an estimate $R_\text{UEG} \approx \SI{-5.3}{\bohr}$ for the $^3P$ ground state, a rather good estimate that we refine here to $R_\text{UEG} \approx \SI{-5.32527}{\bohr}$. He also provided an estimate $R_\text{UEG} \approx \SI{-5.3}{\bohr}$ for the $^3P$ ground state, a rather good estimate that we refine here to $R_\text{UEG} \approx \SI{-5.32527}{\bohr}$.
For the $^1P$ states, the condition $c_0 + 2 c_1/3 + c_2/5 = 0$ [see Eq.~\eqref{eq:condition}] cannot be fulfilled and, hence, these states never exhibit uniform densities. For the $^1P$ states, the condition $c_0 + 2 c_1/3 + c_2/5 = 0$ [see Eq.~\eqref{eq:condition}] cannot be fulfilled and, hence, these states never exhibit uniform densities.
This further highlights the subtle balance that must be accomplished between the spin and spatial parts of the wave function [see Eq.~\eqref{eq:rho}] and can help us rationalizing why the $^3P$ states are TUEGs. This further highlights the subtle balance that must be accomplished between the spin and spatial parts of the wave function [see Eq.~\eqref{eq:rho}] and this can help us rationalizing why the $^3P$ states are TUEGs.
The $^3P$ spin wave function, $\chi_{^3P}$, defined in Eq.~\eqref{eq:spin_3P} has the natural tendency to pull apart the same-spin electron pair in accordance with the Pauli exclusion principle creating in the process a so-called Fermi hole. \cite{Boyd_1974,Giner_2016a} The $^3P$ spin wave function, $\chi_{^3P}$, defined in Eq.~\eqref{eq:spin_3P} has the natural tendency to pull apart same-spin electrons in accordance with the Pauli exclusion principle, creating in the process a so-called Fermi hole. \cite{Boyd_1974,Giner_2016a}
The same physical effect can be obtained by increasing the value of $R$ (\ie, $R \gg 0$). The same physical effect can be obtained by increasing the value of $R$ (\ie, $R \gg 0$).
In such a case, the two electrons localize (or ``crystallize'') on opposite side of the sphere to form a Wigner crystal. \cite{Wigner_1934} In such a case, the two electrons localize (or ``crystallize'') on opposite side of the sphere to minimize their repsulsion and they form a Wigner crystal. \cite{Wigner_1934}
Oppositely, when $R \ll 0$, the two electrons are attracted to each other to form a pair of tightly bound electrons that freely move on the sphere. \cite{Seidl_2007,Seidl_2010} Oppositely, when $R \ll 0$, the two electrons are attracted to each other to form a pair of tightly bound electrons that freely move on the sphere. \cite{Seidl_2007,Seidl_2010}
For certain $R$ values, the attractive effect stemming from the spatial part of the wave function exactly compensates the Pauli exclusion principle originating from the spin part of the wave function to make the total electron density uniform, hence producing TUEGs. For certain $R$ values, the attractive effect stemming from the spatial part of the wave function exactly compensates the Pauli exclusion principle originating from the spin part of the wave function to make the total electron density uniform, hence producing TUEGs.
In higher-energy excited states, the same-spin electrons are further away as compared to the ground state due to the larger number of nodes in the excited-state wave functions. In higher-energy excited states, the same-spin electrons are further away as compared to the ground state due to the larger number of nodes in the excited-state wave functions.
@ -211,9 +211,10 @@ Therefore, the magnitude of the attractive effect has to be larger to compensate
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For the IUEG and FUEGs, the density is uniform independently of the level of theory, \ie, the system has homogeneous density within the exact theory or any approximate methods (such as the HF approximation) unless the spin and/or spatial symmetry is broken. \cite{Fukutome_1981,Stuber_2003,VignaleBook} For the IUEG and FUEGs, the density is uniform independently of the level of theory, \ie, the system has homogeneous density within the exact theory or any approximate methods (such as the HF approximation) unless the spin and/or spatial symmetry is broken. \cite{Fukutome_1981,Stuber_2003,VignaleBook}
However, the value of $R_\text{UEG}$ is, \textit{a priori}, highly dependent of the level of theory for TUEGs. However, the value of $R_\text{UEG}$ is, \textit{a priori}, highly dependent on the level of theory for TUEGs.
Indeed, it is very unlikely that the exact theory and the HF approximation provide the same value of $R_\text{UEG}$ as the uniformity stems from the competition between Fermi effects originating from the antisymmetric nature of the wave function (which are well described at the HF level) and correlation effects (which are absent, by definition, at the HF level). Indeed, it is very unlikely that the exact theory and the HF approximation provide the same value of $R_\text{UEG}$ as the uniformity stems from the competition between Fermi effects originating from the antisymmetric nature of the wave function (which are well described at the HF level) and correlation effects (which are absent, by definition, at the HF level).
Actually, it is even possible for a system to be a TUEG within the exact treatment and being non-uniform for any $R$ values at the HF level, and this seems to be the case for the present two-electron example. Actually, it is even possible for a system to be a TUEG within the exact treatment and being non-uniform for all $R$ values at the HF level.
It seems to be the case for the present two-electron system.
Expanding the two HF orbitals of the $^3P$ ground state in a basis of zonal harmonics $Y_{\ell}(\theta) \equiv Y_{\ell 0}(\theta,\phi)$, we have not found any $R$ values for which the HF electron density, $\rho^\text{HF}(\theta)$, is uniform. Expanding the two HF orbitals of the $^3P$ ground state in a basis of zonal harmonics $Y_{\ell}(\theta) \equiv Y_{\ell 0}(\theta,\phi)$, we have not found any $R$ values for which the HF electron density, $\rho^\text{HF}(\theta)$, is uniform.
However, at $R \approx \SI{-7}{\bohr}$, $\rho^\text{HF}$ is \textit{locally} uniform around $\theta = \pi/2$ (\ie, in the $xy$ plane), as shown in Fig.~\ref{fig:HF}. However, at $R \approx \SI{-7}{\bohr}$, $\rho^\text{HF}$ is \textit{locally} uniform around $\theta = \pi/2$ (\ie, in the $xy$ plane), as shown in Fig.~\ref{fig:HF}.