Working on intro

Manuscript/rsdft-cipsi-qmc.bib

@@ -493,3 +493,107 @@ note={Gaussian Inc. Wallingford CT}
     eprint = {2002.03107},
     url = {https://arxiv.org/abs/2002.03107v2}
 }
+
+@article{Per_2012,
+    author = {Per, Manolo C. and Walker, Kelly A. and 
+    Russo, Salvy P.},
+    title = {{How Important is Orbital Choice in Single-Determinant Diffusion Quantum Monte
+    Carlo Calculations?}},
+    journal = {J. Chem. Theory Comput.},
+    volume = {8},
+    number = {7},
+    pages = {2255--2259},
+    year = {2012},
+    month = {Jul},
+    issn = {1549-9618},
+    publisher = {American Chemical Society},
+    doi = {10.1021/ct200828s}
+}
+
+@article{Petruzielo_2012,
+    author = {Petruzielo, F. R. and Toulouse, Julien and 
+    Umrigar, C. J.},
+    title = {{Approaching chemical accuracy with quantum Monte Carlo}},
+    journal = {J. Chem. Phys.},
+    volume = {136},
+    number = {12},
+    pages = {124116},
+    year = {2012},
+    month = {Mar},
+    issn = {0021-9606},
+    publisher = {American Institute of Physics},
+    doi = {10.1063/1.3697846}
+}
+
+@misc{Caffarel_2016_2,
+    title = {{Recent Progress in Quantum Monte Carlo}},
+    journal = {ACS Symp. Ser.},
+    year = {2016},
+    month = {Jan},
+    note = {[Online; accessed 6. Jul. 2020]},
+    url = {https://pubs.acs.org/doi/abs/10.1021/bk-2016-1234.ch002}
+}
+
+@article{Giner_2016,
+    author = {Giner, Emmanuel and Assaraf, Roland and 
+    Toulouse, Julien},
+    title = {{Quantum Monte Carlo with reoptimised perturbatively selected configuration-interaction
+    wave functions}},
+    journal = {Mol. Phys.},
+    volume = {114},
+    number = {7-8},
+    pages = {910--920},
+    year = {2016},
+    month = {Apr},
+    issn = {0026-8976},
+    publisher = {Taylor {\&} Francis},
+    doi = {10.1080/00268976.2016.1149630}
+}
+
+@article{Dash_2019,
+    author = {Dash, Monika and Feldt, Jonas and Moroni, Saverio and 
+    Scemama, Anthony and Filippi, Claudia},
+    title = {{Excited States with Selected Configuration Interaction-Quantum Monte Carlo:
+    Chemically Accurate Excitation Energies and Geometries}},
+    journal = {J. Chem. Theory Comput.},
+    volume = {15},
+    number = {9},
+    pages = {4896--4906},
+    year = {2019},
+    month = {Sep},
+    issn = {1549-9618},
+    publisher = {American Chemical Society},
+    doi = {10.1021/acs.jctc.9b00476}
+}
+
+@article{Dash_2018,
+    author = {Dash, Monika and Moroni, Saverio and Scemama, Anthony and 
+    Filippi, Claudia},
+    title = {{Perturbatively Selected Configuration-Interaction Wave Functions for Efficient
+    Geometry Optimization in Quantum Monte Carlo}},
+    journal = {J. Chem. Theory Comput.},
+    volume = {14},
+    number = {8},
+    pages = {4176--4182},
+    year = {2018},
+    month = {Aug},
+    issn = {1549-9618},
+    publisher = {American Chemical Society},
+    doi = {10.1021/acs.jctc.8b00393}
+}
+
+@article{Wang_2019,
+    author = {Wang, Ting and Zhou, Xiaojun and Wang, Fan},
+    title = {{Performance of the Diffusion Quantum Monte Carlo Method with a Single-Slater-Jastrow
+    Trial Wavefunction Using Natural Orbitals and Density Functional Theory Orbitals
+    on Atomization Energies of the Gaussian-2 Set}},
+    journal = {J. Phys. Chem. A},
+    volume = {123},
+    number = {17},
+    pages = {3809--3817},
+    year = {2019},
+    month = {May},
+    issn = {1089-5639},
+    publisher = {American Chemical Society},
+    doi = {10.1021/acs.jpca.9b01933}
+}

Manuscript/rsdft-cipsi-qmc.tex

@@ -54,7 +54,7 @@ solution of the Schrödinger equation with an approximate Hamiltonian
 expressed in a finite basis of Slater determinants.
 The FCI wave function can be interpreted as the exact solution of the
 true Hamiltonian obtained with the additional constraint that it
-can only span the space provided by a given basis. At the complete
+can only span the space provided by the basis. At the complete
 basis set (CBS) limit, the constraint vanishes and the exact solution
 is obtained.
 
@@ -79,25 +79,72 @@ as a given trial wave function. This approximation is known as the
 function coincide with the nodes of the exact wave function, the exact
 energy and wave function are obtained.
 The DMC method is attractive because its scaling is polynomial with
-the number of electrons and the size of the trial wave
+the number of electrons and with the size of the trial wave
 function. Moreover, the total energies obtained are usually below
 those obtained with FCI because the fixed-node approximation imposes
 less constraints on the solution than the finite-basis approximation.
 
+In many cases, the systems under study are well described by a single
+Slater determinant. Single-determinant DMC can be used as a post-Hatree-Fock
+single-reference method with an accuracy comparable to coupled cluster.
+The favorable scaling of QMC and its adequation with massively
+parallel architectures makes it an attractive alternative for large
+systems.
+It has been shown that the nodal surfaces obtained with
+Kohn-Sham determinants are in general better than those obtained with
+the Hartree-Fock determinant,\cite{Per_2012} and of comparable quality
+to those obtained with natural orbitals of single-determinant
+correlated calculations.\cite{Wang_2019} 
 However, the fixed-node approximation is much more difficult to
-control than the finite-basis approximation.
-
-
-% Single detrminant nodes improved with DFT
-% Jastrow optimization -> Better nodes
-% Combination CIPSI/DMC -> Water paper
+control than the finite-basis approximation, as it is not possible
+to minimize directly the DMC energy with respect to the variational
+parameters of the trial wave function.
+The conventional approach consists in multiplying the trial wave
+function by a positive function, the \emph{Jastrow factor}, taking
+account of the electron-electron cusp and the short-range correlation
+effects. The wave function is then re-optimized in the presence of the
+Jastrow factor and the nodal surface is expected to be improved.
+Using this technique, it has been shown that the chemical accuracy
+could be reached within DMC.\cite{Petruzielo_2012}
 
+Another approach consists in considering the DMC method as a
+\emph{post-FCI method}. The trial wave function is obtained by
+approaching the FCI with a selected configuration interaction
+method such as CIPSI for instance.\cite{Giner_2013,Caffarel_2016_2}
+When the basis set is increased, the trial wave function tends to the
+exact wave function, so the nodal surface can be systematically
+improved.\cite{Caffarel_2016}
+This technique has the advantage that using FCI nodes in a given basis
+set is well defined and has a unique solution. The optimization of the
+wave function is deterministic, so the calculations are reproducible
+and don't require the expertise of a QMC expert. However,
+this technique can't be applied to large systems because of the
+exponential scaling of the size of the wave function.  Extrapolation
+techniques have been applied to estimate the DMC energy of a FCI
+wave function in a large basis set,\cite{Scemama_2018} and other
+authors have used a combination of the two approaches where CIPSI
+trial wave functions are re-optimized under the presence of a Jastrow
+factor.\cite{Giner_2016,Dash_2018,Dash_2019}
 
 
 
 \section{Combining range-separated DFT with CIPSI}
 \label{sec:rsdft-cipsi}
 
+Starting from a Hartree-Fock determinant in a small basis set, 
+we have seen that we can systematically improve the trial wave
+function in two directions. The first one is by increasing the
+size of the atomic basis set, and the second one is by
+increasing the determinant expansion towards the FCI limit.
+A third direction of improvement exists. It is the path which connects
+the Hartree-Fock determinant to the Kohn-Sham determinant, which
+usually has a better nodal surface. This third path is obtained by
+the so-called \emph{range-separated} DFT framework which splits the
+electron-electron interaction in a long-range part and a short-range
+part. HELP MANU!
+
+
+
 \subsection{CIPSI}
 
 \emph{Configuration interaction using a perturbative selection made