saving work in intro

Manuscript/rsdft-cipsi-qmc.bib

@@ -1,13 +1,24 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-08-16 15:38:12 +0200 
+%% Created for Pierre-Francois Loos at 2020-08-16 22:14:09 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Bressanini_2012,
+	Author = {D. Bressanini},
+	Date-Added = {2020-08-16 22:13:51 +0200},
+	Date-Modified = {2020-08-16 22:14:05 +0200},
+	Doi = {10.1103/PhysRevB.86.115120},
+	Journal = {Phys. Rev. B},
+	Pages = {115120},
+	Title = {Implications of the two nodal domains conjecture for ground state fermionic wave functions},
+	Volume = {86},
+	Year = {2012}}
+
 @article{Giner_2013,
 	Author = {E. Giner and A. Scemama and M. Caffarel},
 	Date-Added = {2020-08-16 15:38:03 +0200},

Manuscript/rsdft-cipsi-qmc.tex

@@ -89,13 +89,13 @@ Solving the Schr\"odinger equation for atoms and molecules is a complex task tha
 In order to achieve this formidable endeavour, various strategies have been carefully designed and implemented in quantum chemistry software packages.
 
 One of this strategies consists in relying on wave function theory (WFT) and, in particular, on the full configuration interaction (FCI) method.
-However, FCI delivers only an approximate solution of the Schr\"odinger equation within a finite basis (FB) of one-electron basis functions.
+However, FCI delivers only an approximate solution of the Schr\"odinger equation within a finite basis (FB) of one-electron basis functions, the FB-FCI energy being an upper bound to the exact energy in accordance with the variational principle.
 The FB-FCI wave function and its corresponding energy form the eigenpair of an approximate Hamiltonian defined as
 the projection of the exact Hamiltonian onto the finite many-electron basis of
 all possible Slater determinants generated within this finite one-electron basis.
 The FB-FCI wave function can then be interpreted as a constrained solution of the
 true Hamiltonian forced to span the restricted space provided by the finite one-electron basis.
-In the complete basis set (CBS) limit, the constraint is lifted and the exact solution is recovered.
+In the complete basis set (CBS) limit, the constraint is lifted and the exact energy and wave function are recovered.
 Hence, the accuracy of a FB-FCI calculation can be systematically improved by increasing the size of the one-electron basis set.
 Nevertheless, the exponential growth of its computational scaling with the number of electrons and with the basis set size is prohibitive for most chemical systems.
 In recent years, the introduction of new algorithms \cite{Booth_2009,Xu_2018,Eriksen_2018,Eriksen_2019} and the
@@ -112,25 +112,31 @@ transfers the complexity of the many-body problem to the exchange-correlation (x
 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}
 However, one faces the unsettling choice of the \emph{approximate} xc functional which makes inexorably KS-DFT hard to systematically improve. \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.
 
 Diffusion Monte Carlo (DMC) is yet another numerical scheme to obtain
 the exact solution of the Schr\"odinger equation with a different
 constraint. \cite{Foulkes_2001,Austin_2012,Needs_2020} 
 In DMC, the solution is imposed to have the same nodes (or zeroes)
-as a given trial (approximate) wave function. \cite{Reynolds_1982,Ceperley_1991}
+as a given (approximate) antisymmetric trial wave function. \cite{Reynolds_1982,Ceperley_1991}
 Within this so-called 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 trial wave function in FN-DMC is then a key ingredient which dictates via the quality of its nodal surface the accuracy of the resulting energy and properties.
 The polynomial scaling of its computational cost with respect to the number of electrons and with the size
-of the trial wave function makes the FN-DMC method particularly attractive.\cite{Nakano_2020,Scemama_2013,Needs_2020,Kim_2018,Kent_2020}
+of the trial wave function makes the FN-DMC method particularly attractive.
+This favorable scaling, its very low memory requirements and
+its adequacy with massively parallel architectures make it a
+serious alternative for high-accuracy simulations of large systems. \cite{Nakano_2020,Scemama_2013,Needs_2020,Kim_2018,Kent_2020}
 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 FN approximation
+sets because the constraints imposed by the fixed-node approximation
 are less severe than the constraints imposed by the finite-basis
 approximation.
 
+
 %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
@@ -143,16 +149,13 @@ 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 theory.
+DFT and coupled cluster theory.
 DMC with a single-determinant trial wave function can be used as a
-single-reference post-Hatree-Fock method, with an accuracy comparable
+single-reference post-Hartree-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 adequacy with massively parallel architectures make it a
-serious alternative for high-accuracy simulations of large systems.
 
 As it is not possible to minimize directly the FN-DMC energy with respect
-to the variational parameters of the trial wave function, the
+to the linear and non-linear parameters of the trial wave function, the
 fixed-node approximation is much more difficult to control than the
 finite-basis approximation.
 The conventional approach consists in multiplying the trial wave
@@ -160,14 +163,13 @@ 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 within variational
 Monte Carlo (VMC) 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
+surface is expected to be improved. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}
+Using this technique, it has been shown that the chemical accuracy could be reached within
 FN-DMC.\cite{Petruzielo_2012}
 
 Another approach consists in considering the FN-DMC method as a
-\emph{post-FCI method}. The trial wave function is obtained by
-approaching the FCI with a selected configuration interaction (SCI)
-method such as CIPSI for instance.\cite{Giner_2013,Giner_2015,Caffarel_2016_2}
+``post-FCI'' method. The trial wave function is obtained by
+approaching the FCI with a SCI method such as CIPSI for instance.\cite{Giner_2013,Giner_2015,Caffarel_2016_2}
 When the basis set is enlarged, the trial wave function gets closer to
 the exact wave function, so we expect the nodal surface to be
 improved.\cite{Caffarel_2016}
@@ -191,6 +193,9 @@ calculations which can be reproduced systematically, and which is
 applicable for large systems with a multi-configurational character is
 still an active field of research. The present paper falls
 within this context.
+The central idea of the present work, and the launch-pad for the remainder of this study, is that one can combine the various strengths of these three philosophies in order to create a hybrid method with more attractive properties.
+In particular, we show here that one can combine FB-FCI and KS-DFT via the range separation (RS) of the interelectronic Coulomb operator to obtain accurate FN-DMC energies with compact multideterminant trial wave functions.
+
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -229,7 +234,7 @@ Nevertheless, even when using the exact exchange correlation potential at the
 CBS limit, a fixed-node error necessarily remains because the
 single-determinant ansätz does not have enough flexibility to describe the
 nodal surface of the exact correlated wave function of a generic $N$-electron
-system.
+system. \cite{Bressanini_2012}
 If one wants to recover the exact CBS limit, a multi-determinant parameterization
 of the wave functions is required.
 
@@ -548,17 +553,13 @@ Let us call $\Psi^J$ the linear combination of Slater determinant minimizing the
 \end{equation}
 Such a wave function $\Psi^J$ satisfies the generalized Hermitian eigenvalue equation
 \begin{equation}
-\toto{
  e^{J} \hat{H} \qty( e^{J} \Psi^J ) = E e^{2J} \Psi^J,
-}
 \label{eq:ci-j}
 \end{equation}
 but also the non-Hermitian \toto{transcorrelated eigenvalue problem \cite{many_things} MANU:CITATIONS}
 \begin{equation}
  \label{eq:transcor}
-\toto{
  e^{-J}  \hat{H} \qty( e^{J} \Psi^J) = E \Psi^J,
-}
 \end{equation}
 which is much easier to handle despite its non-Hermiticity.
 Of course, the FN-DMC energy of $\Phi$ depends therefore only on the nodes of $\Psi^J$.
@@ -571,7 +572,7 @@ function out of a large CIPSI calculation.  Within this set of determinants,
 we solve the self-consistent equations of RS-DFT [see Eq.~\eqref{rs-dft-eigen-equation}]
 with different values of $\mu$. This gives the CI expansions $\Psi^\mu$.
 Then, within the same set of determinants we optimize the CI coefficients in the presence of
-a simple one- and two-body Jastrow factor \toto{$e^J$ of the form $\exp(J_{eN} + J_{ee})$ with
+a simple one- and two-body Jastrow factor $e^J$ of the form $\exp(J_{eN} + J_{ee})$ with
 \begin{eqnarray}
   J_\text{eN} & = & - \sum_{A=1}^{\Nat} \sum_{i=1}^{\Nelec} \left( \frac{\alpha_A\, r_{iA}}{1 + \alpha_A\, r_{iA}} \right)^2
 \label{eq:jast-eN} \\
@@ -586,10 +587,10 @@ The optimal CI expansion $\Psi^J$ is obtained by sampling the matrix elements
 of the Hamiltonian ($\mathbf{H}$) and the overlap ($\mathbf{S}$), in the
 basis of Jastrow-correlated determinants $e^J D_i$:
 \begin{eqnarray}
-H_{ij} & = & \left \langle \frac{e^J D_i}{\Psi^J}\, \frac{\hat{H}\, (e^J D_j)}{\Psi^J}  \right \rangle \\ 
-S_{ij} & = & \left \langle \frac{e^J D_i}{\Psi^J}\, \frac{e^J D_j}{\Psi^J}  \right \rangle
+H_{ij} & = & \expval{ \frac{e^J D_i}{\Psi^J}\, \frac{\hat{H}\, (e^J D_j)}{\Psi^J}  }  \\ 
+S_{ij} & = & \expval{ \frac{e^J D_i}{\Psi^J}\, \frac{e^J D_j}{\Psi^J}  }
 \end{eqnarray}
-and solving Eq.~\eqref{eq:ci-j}.\cite{Nightingale_2001}}
+and solving Eq.~\eqref{eq:ci-j}.\cite{Nightingale_2001}
 
 We can easily compare $\Psi^\mu$ and  $\Psi^J$ as they are developed
 on the same Slater determinant basis.