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

Manuscript/rsdft-cipsi-qmc.tex

@@ -35,6 +35,7 @@
 %  range-separated density functional theory and selected configuration interaction}
 
 \author{Anthony Scemama}
+\email{scemama@irsamc.ups-tlse.fr}
 \affiliation{\LCPQ}
 \author{Emmanuel Giner}
 \email{emmanuel.giner@lct.jussieu.fr}
@@ -68,40 +69,43 @@ exact solution is recovered.
 Hence, the accuracy of a FCI calculation can be systematically improved by increasing the size of the one-electron basis set.
 Nevertheless, its exponential scaling with the number of electrons and with the size of the basis is prohibitive for most chemical systems.
 In recent years, the introduction of new algorithms \cite{Booth_2009} and the
-revival \cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016}
+revival \cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016,Garniron_2018}
 of selected configuration interaction (sCI)
 methods \cite{Bender_1969,Huron_1973,Buenker_1974} pushed the limits of
-the sizes of the systems that could be computed at the FCI level.
+the sizes of the systems that could be computed at the FCI level. \cite{Booth_2010,Cleland_2010,Daday_2012,Chien_2018,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}
 However, the scaling remains exponential unless some bias is introduced leading
-to a loss of size consistency. % CITE CIPSI 3-CLASS AND INITIATOR APPROXIMATION.
+to a loss of size consistency. \cite{Evangelisti_1983,Cleland_2010,Tenno_2017}
 
 Diffusion Monte Carlo (DMC) is a numerical scheme to obtain
 the exact solution of the Schr\"odinger equation with a different
-constraint, imposing the solution to have the same nodal hypersurface
-as a given trial wave function.
+constraint. In DMC, the solution is imposed to have the same nodes (or zeroes) 
+as a given trial (approximate) wave function.
 Within this so-called \emph{fixed-node} (FN) approximation,
 the FN-DMC energy associated with a given trial wave function is an upper
 bound to the exact energy, and the latter is recovered only when the
 nodes of the trial wave function coincide with the nodes of the exact
 wave function.
 The polynomial scaling with the number of electrons and with the size
-of the trial wave function makes the FN-DMC method attractive.
+of the trial wave function makes the FN-DMC method particularly attractive.
 In addition, the total energies obtained are usually far below
 those obtained with the FCI method in computationally tractable basis
-sets because the constraints imposed by the fixed-node approximation
+sets because the constraints imposed by the FN approximation
 are less severe than the constraints imposed by the finite-basis
 approximation.
 
+\alert{However, it is usually harder to control the FN error in DMC, and this might affect energy differences such as atomization energies.
+Moreover, improving systematically the nodal surface of the trial wave function can be a tricky job as there is no variational principle for the nodes.}
+
 The qualitative picture of the electronic structure of weakly
 correlated systems, such as organic molecules near their equilibrium
 geometry, is usually well represented with a single Slater
 determinant. This feature is in part responsible for the success of
-density functional theory (DFT) and coupled cluster.
+density-functional theory (DFT) and coupled cluster.
 DMC with a single-determinant trial wave function can be used as a
 single-reference post-Hatree-Fock method, with an accuracy comparable
 to coupled cluster.\cite{Dubecky_2014,Grossman_2002}
 The favorable scaling of QMC, its very low memory requirements and
-its adequation with massively parallel architectures make it a
+its adequacy with massively parallel architectures make it a
 serious alternative for high-accuracy simulations on large systems.
 
 As it is not possible to minimize directly the FN-DMC energy with respect