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Manuscript/rsdft-cipsi-qmc.tex
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@ -94,7 +94,7 @@ In order to achieve this formidable endeavour, various strategies have been care
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
One of these strategies consists in relying on wave function theory \cite{Pople_1999} (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, the FB-FCI energy being an upper bound to the exact energy in accordance with the variational principle.
However, FCI delivers only an approximate solution of the Schr\"odinger equation within a finite basis (FB) of one-electron 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.
@ -121,17 +121,17 @@ Present-day DFT calculations are almost exclusively done within the so-called Ko
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.
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, and therefore in practice
one faces the unsettling choice of the \emph{approximate} xc functional. \cite{Becke_2014}}
\titou{However, there is no systematic way of refining the approximation of the unknown exact xc 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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Stochastic methods}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Diffusion Monte Carlo (DMC), which belongs to the family of stochastic methods, is yet another numerical scheme to obtain
Diffusion Monte Carlo (DMC) belongs to the family of stochastic methods and 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}
twist. \cite{Foulkes_2001,Austin_2012,Needs_2020}
In DMC, the solution is imposed to have the same nodes (or zeroes)
as a given (approximate) antisymmetric trial wave function. \cite{Reynolds_1982,Ceperley_1991}
Within this so-called fixed-node (FN) approximation,
@ -139,7 +139,7 @@ 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, which can be single- or multi-determinantal in nature depending on the type of correlation at play and the target accuracy, is then the key ingredient dictating, via the quality of its nodal surface, the accuracy of the resulting energy and properties.
The trial wave function, which can be single- or multi-determinantal in nature depending on the type of correlation at play and the target accuracy, is the key ingredient dictating, 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.
@ -155,12 +155,12 @@ However, because it is not possible to minimize directly the FN-DMC energy with
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, especially to compute energy differences.
The conventional approach consists in multiplying the trial wave
function by a positive function, the \emph{Jastrow factor}, taking
account of the bulk of the dynamical correlation.
The conventional approach consists in multiplying the determinantal part of the trial wave
function by a positive function, the Jastrow factor, which main assignment is to take into
account the bulk of the dynamical electron correlation and reduce the statistical fluctuation without altering the location of the nodes.
%electron-electron cusp and the short-range correlation effects.
The trial wave function is then re-optimized within variational
Monte Carlo (VMC) in the presence of the Jastrow factor and the nodal
The determinantal part of the trial wave function is then re-optimized within variational
Monte Carlo (VMC) in the presence of the Jastrow factor (which can also be simultaneously optimized) and the nodal
surface is expected to be improved. \cite{Umrigar_2005,Scemama_2006,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}
@ -192,7 +192,7 @@ Different molecular orbitals can be chosen:
Hartree-Fock (HF), Kohn-Sham (KS), natural orbitals (NOs) of a
correlated wave function, or orbitals optimized in the
presence of a Jastrow factor.
The nodal surfaces obtained with a KS determinant are in general
Nodal surfaces obtained with a KS determinant are in general
better than those obtained with a HF determinant,\cite{Per_2012} and
of comparable quality to those obtained with a Slater determinant
built with NOs.\cite{Wang_2019} Orbitals obtained in the presence