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TrUEGs.bib
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@ -1,13 +1,49 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2021-03-23 15:51:47 +0100
%% Created for Pierre-Francois Loos at 2021-04-08 08:22:37 +0200
%% Saved with string encoding Unicode (UTF-8)
@book{VignaleBook,
address = {Cambridge, England},
author = {G. F. Giuliani and G. Vignale},
date-added = {2021-04-08 08:22:31 +0200},
date-modified = {2021-04-08 08:22:31 +0200},
keywords = {jellium},
publisher = {Cambridge University Press},
title = {Quantum Theory of Electron Liquid},
year = {2005}}
@inbook{Stuber_2003,
address = {Dordrecht},
author = {Stuber, J and Paldus, J},
booktitle = {Fundamental World of Quantum Chemistry: A Tribute to the Memory of Per-Olov L\"{o}wdin},
date-added = {2021-04-08 08:21:59 +0200},
date-modified = {2021-04-08 08:22:04 +0200},
editor = {Br\"{a}ndas, E J and Kryachko, E S},
pages = {67},
publisher = {Kluwer Academic},
title = {{Symmetry Breaking in the Independent Particle Model}},
volume = {1},
year = {2003}}
@article{Fukutome_1981,
author = {Fukutome, Hideo},
date-added = {2021-04-08 08:21:36 +0200},
date-modified = {2021-04-08 08:21:49 +0200},
doi = {10.1002/qua.560200502},
journal = {Int. J. Quantum Chem.},
number = {5},
pages = {955--1065},
title = {Unrestricted {Hartree-Fock} theory and its applications to molecules and chemical reactions},
volume = {20},
year = {1981},
Bdsk-Url-1 = {https://doi.org/10.1002/qua.560200502}}
@article{Boyd_1974,
abstract = {A new definition of the Fermi hole in many-electron systems is provided in terms of the distribution function falpha alpha (r12) of the interelectronic distance for electrons with parallel spins. By analogy with the Coulomb hole, the Fermi hole is defined as the difference between the values of f22(r12) derived from the Hartree-Fock and the Hartree wavefunctions. this definition, unlike previous ones, provides a simple picture of the Fermi hole as a function of r12. By assuming that the Hartree and Hartree-Fock orbitals are identical, an analytical formula is derived for the Fermi hole. Explicit calculations are presented for the 23S state of He and the ground state of Be. It is observed that the Fermi hole is remarkably similar in these two cases; and that the effects of Coulomb correlation are more long-ranged than those of Fermi correlation.},
author = {R J Boyd and C A Coulson},

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TrUEGs.nb
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TrUEGs.tex
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@ -65,7 +65,7 @@ Indeed, apart from very few exceptions, most density-functional approximations a
Thanks to the construction of exchange-correlation LDA functionals \cite{Slater_1951,Vosko_1980,Perdew_1981,Perdew_1992,Chachiyo_2016} which can be loosely seen as a one-to-one mapping between a given value of the electron density and the exchange-correlation energy of the UEG, one can then straightforwardly compute, within KS-DFT, the electronic ground-state energy and properties of any molecules or materials with, nonetheless, a certain degree of approximation inherently associated with the approximate nature of the exchange-correlation LDA functional.
One can also access excited states via the time-dependent version of DFT. \cite{Runge_1984,Casida_1995,Petersilka_1996,UllrichBook}
As commonly done, the LDA can be refined by adding up new ingredients, such as the gradient of the density $\nabla \rho$ [which defines the generalized gradient approximation (GGA)], \cite{Perdew_1986,Becke_1988,Lee_1988,Perdew_1996} the kinetic energy density $\tau$ (meta-GGA), \cite{Becke_1988b,Sun_2015} exact Hartree-Fock exchange (yielding the so-called hybrid functionals), \cite{Becke_1993a,Becke_1993b,Adamo_1999} and others.
As commonly done, the LDA can be refined by adding up new ingredients, such as the gradient of the density $\nabla \rho$ [which defines the generalized gradient approximation (GGA)], \cite{Perdew_1986,Becke_1988,Lee_1988,Perdew_1996} the kinetic energy density $\tau$ (meta-GGA), \cite{Becke_1988b,Sun_2015} exact Hartree-Fock (HF) exchange (yielding the so-called hybrid functionals), \cite{Becke_1993a,Becke_1993b,Adamo_1999} and others.
Each of these quantities defines a new rung of the well-known Jacob ladder of DFT \cite{Perdew_2001} that is supposed to bring electronic structure theory calculations from the evil Hartree world to the chemical accuracy heaven.
The UEG, also known as jellium in some context, \cite{Loos_2016} is a hypothetical infinite substance where an infinite number of electrons ``bathe'' in a (uniform) positively charged jelly of infinite volume. \cite{ParrBook,Loos_2016}
@ -78,7 +78,7 @@ In the following, this paradigm is named the infinite UEG (IUEG) for obvious rea
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Recently, it has been shown that one can create finite UEGs (FUEGs) by placing a finite number of electrons onto the surface of a sphere of radius $R$. \cite{Tempere_2002,Tempere_2007,Seidl_2007,Loos_2009a,Loos_2009c,Loos_2010e,Loos_2011b,Gill_2012,Loos_2018b}
Of course, FUEGs only appear for well-defined electron numbers and electronic states. \cite{Rogers_2016,Rogers_2017}
In particular, the spin-unpolarized ground state of $n$ electrons on a sphere has a homogeneous density for $n = 2(L+1)^2$ (where $L \in \mathbb{N}$) for any $R$ values, and this holds also within the Hartree-Fock approximation. \cite{Loos_2011b}
In particular, the spin-unpolarized ground state of $n$ electrons on a sphere has a homogeneous density for $n = 2(L+1)^2$ (where $L \in \mathbb{N}$) for any $R$ values, and this holds also within the HF approximation. \cite{Loos_2011b}
This property comes from the addition theorem of the spherical harmonics \cite{NISTbook} $Y_{\ell m}(\bm{\Omega})$ (which are the spatial orbitals of the system in this particular case):
\begin{equation}
\sum_{\ell=0}^L \sum_{m=-\ell}^{+\ell} Y_{\ell m}^*(\bm{\Omega}) Y_{\ell m}(\bm{\Omega}) = \frac{(L+1)^2}{2\pi^2}
@ -201,6 +201,11 @@ For certain $R$ values, the attractive effect stemming from the spatial part of
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.
Therefore, the magnitude of the attractive effect has to be larger to compensate it, which corresponds to more negative values of $R$.
While 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 but also within the HF approximation (unless the spin and/or spatial symmetry is broken \cite{Fukutome_1981,Stuber_2003,VignaleBook}), the value of $R_\text{UEG}$ is, \textit{a priori}, highly dependent of 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).
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.
%\titou{What about the nodes? Dyson orbitals? Cf Paola's paper.}
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