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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} 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 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} 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} 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} 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. 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 The description of electron correlation within DFT is very different
from correlated methods such as FCI or CC. 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 \manu{As mentioned above, within KS-DFT, one solves a mean-field problem
incorporating the effects of electron correlation, whereas in 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 correlated methods the real Hamiltonian is used and the
electron-electron interactions are considered.} electron-electron interactions are considered.}
Nevertheless, as the orbitals are one-electron functions, 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}]. 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} \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 $w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus RS-DFT reduces to standard
KS-DFT and $\Psi^\mu$ is the KS determinant. For $\mu = \infty$, the long-range KS-DFT and $\Psi^\mu$ is the KS determinant. For $\mu = \infty$, the long-range
interaction becomes the standard Coulomb interaction, \ie, interaction becomes the standard Coulomb interaction, \ie,