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Merge branch 'master' of https://git.irsamc.ups-tlse.fr/scemama/RSDFT-CIPSI-QMC

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Pierre-Francois Loos 2020-08-18 09:02:21 +02:00
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Manuscript/rsdft-cipsi-qmc.tex
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@ -118,10 +118,10 @@ to a loss of size consistency. \cite{Evangelisti_1983,Cleland_2010,Tenno_2017}
Another route to solve the Schr\"odinger equation is density-functional theory (DFT). \cite{Hohenberg_1964,Kohn_1999}
Present-day DFT calculations are almost exclusively done within the so-called Kohn-Sham (KS) formalism, \cite{Kohn_1965} which
transfers the complexity of the many-body problem to the exchange-correlation (xc) functional thanks to a judicious mapping between a non-interacting reference system and its interacting analog which both have exactly the same one-electron density.
transfers the complexity of the many-body problem to the \manu{universal and yet unknown} exchange-correlation (xc) functional thanks to a judicious mapping between a non-interacting reference system and its interacting analog which both have exactly the same one-electron density.
KS-DFT \cite{Hohenberg_1964,Kohn_1965} is now the workhorse of electronic structure calculations for atoms, molecules and solids thanks to its very favorable accuracy/cost ratio. \cite{ParrBook}
As compared to WFT, DFT has the indisputable advantage of converging much faster with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15,Loos_2019d,Giner_2020}
\titou{However, there is no systematic way of refining the approximation of the unknown exact xc functional.}
\manu{However, unlike WFT where many-body perturbation theory provides a precious tool to go toward the exact wave function, there is no systematic way to improve the approximated xc functionals toward the exact functional.}
Therefore, one faces, in practice, the unsettling choice of the \emph{approximate} xc functional. \cite{Becke_2014}
Moreover, because of the approximate nature of the xc functional, although the resolution of the KS equations is variational, the resulting KS energy does not have such property.
@ -194,8 +194,9 @@ orbitals.\cite{Filippi_2000,Scemama_2006,HaghighiMood_2017,Ludovicy_2019}
The description of electron correlation within DFT is very different
from correlated methods such as FCI or CC.
\titou{As mentioned above, within KS-DFT, one solves a mean-field problem with a modified potential
incorporating the effects of electron correlation, whereas in
\manu{As mentioned above, within KS-DFT, one solves a mean-field problem
with a modified potential incorporating the effects of electron correlation
while maintaining the exact ground state density, whereas in
correlated methods the real Hamiltonian is used and the
electron-electron interactions are considered.}
Nevertheless, as the orbitals are one-electron functions,
@ -355,7 +356,7 @@ energy is obtained as
E_0= \mel{\Psi^{\mu}}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi^{\mu}}+\bar{E}^{\text{sr},\mu}_{\text{Hxc}}[n_{\Psi^\mu}].
\end{equation}
\titou{Note that for $\mu=0$ the long-range interaction vanishes}, \ie,
\manu{Pour moi y'a pas de problèmes avec cette phrase. Note that for $\mu=0$ the long-range interaction vanishes}, \ie,
$w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus RS-DFT reduces to standard
KS-DFT and $\Psi^\mu$ is the KS determinant. For $\mu = \infty$, the long-range
interaction becomes the standard Coulomb interaction, \ie,