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Manuscript/rsdft-cipsi-qmc.tex
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@ -175,23 +175,21 @@ Nevertheless, as the orbitals are one-electron functions,
the procedure of orbital optimization in the presence of the
Jastrow factor can be interpreted as a self-consistent field procedure
with an effective Hamiltonian,\cite{Filippi_2000} similarly to DFT.
So DFT can be viewed as a very cheap way of introducing the effect of
correlation in the parameters determining the nodal surface. But in
the general case, even at the complete basis set limit a fixed-node
error will remain because the single-determinant ans\"atz does not
have enough freedom to describe the exact nodal surface.
If one wants to have to exact CBS limit, a multi-determinant
parameterization of the wave functions is required.
So \eg{KS-}DFT can be viewed as a very cheap way of introducing the effect of
correlation in the \eg{orbital }parameters determining the nodal surface \eg{of a single Slater determinant}.
\sout{But in the general case, even} \eg{Nevertheless, even using the exact exchange correlation potential at } the CBS limit, a fixed-node error \sout{will} \eg{necessary} remains because the single-determinant ans\"atz does not
have enough \sout{freedom} \eg{flexibility} to describe the \sout{exact} nodal surface \eg{of the exact correlated wave function of a generic $N$-electron system}.
If one wants to have to exact CBS limit, a multi-determinant parameterization of the wave functions is required.
\subsection{CIPSI}
Beyond the single-determinant representation, the best
multi-determinant wave function one can obtain is the FCI. FCI is
a \emph{post-Hartree-Fock} method, and there is a continuous
connection between the Hartree-Fock and FCI wave functions.
Multiple paths exist: one can for example use
CI methods increasing the maximum degree of excitation (CISD, CISDT,
CISDTQ, \emph{etc}), or use increasingly large complete active space
a \emph{post-Hartree-Fock} method, and \sout{there is a continuous} \eg{there exists several systematic improvements}
\sout{connections} between the Hartree-Fock and FCI wave functions:
\sout{Multiple paths exist: one can for example use}
increasing the maximum degree of excitation of CI methods (CISD, CISDT,
CISDTQ, \emph{etc}), or increasing the complete active space
(CAS) wave functions until all the orbitals are in the active space.
Selected CI methods take a shorter path between the Hartree-Fock
determinant and the FCI wave function by increasing iteratively the
@ -239,7 +237,7 @@ The parameter $\mu$ controls the range of the separation, and allows
to go continuously from the Kohn-Sham Hamiltonian ($\mu=0$) to
the FCI Hamiltinoan ($\mu = \infty$).
The universal density functional is decomposed as
\eg{To rigorously connect wave function theory and DFT,} the universal \eg{Levy-Lieb} density functional\cite{Lev-PNAS-79,Lie-IJQC-83} is decomposed as
\begin{equation}
\mathcal{F}[n] = \mathcal{F}^{\mathrm{lr},\mu}[n] + \bar{E}_{\mathrm{Hxc}}^{\mathrm{sr,}\mu}[n],
\label{Fdecomp}
@ -248,7 +246,7 @@ where $n$ is a one-particle density,
$\mathcal{F}^{\mathrm{lr},\mu}$ is a long-range universal density
functional and $\bar{E}_{\mathrm{Hxc}}^{\mathrm{sr,}\mu}$ is the
complementary short-range Hartree-exchange-correlation (Hxc) density
functional.
functional\cite{Savin_1996,Toulouse_2004}.
One obtains the following expression for the ground-state
electronic energy
\begin{equation}
@ -316,12 +314,12 @@ with the RS-DFT Hamiltonians,\cite{GinPraFerAssSavTou-JCP-18} as shown
in figure~\ref{fig:algo}. In the outer loop (red), a CIPSI selection
is performed with a RS-Hamiltonian parameterized using the current
density. An inner loop (blue) is introduced to accelerate the
calculation, in which the set of determinants is kept fixed, and only
\eg{convergence of the self-consistent} calculation, in which the set of determinants is kept fixed, and only
the diagonalization of the RS-Hamiltonian is performed iteratively.
The convergence of the algorithm was further improved
by introducing a direct inversion in the iterative subspace (DIIS)
step to extrapolate the density both in the outer and inner loops.
As always, the convergence criterion for CIPSI was set to $\EPT <
\sout{As always, } \eg{As mentioned above,} the convergence criterion for CIPSI was set to $\EPT <
1$~m$E_h$.
@ -418,20 +416,31 @@ correlation functionals of Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06}
values of $\mu$.}
\label{fig:f2-dmc}
\end{figure}
The water molecule was taken at the equilibrium
geometry,\cite{Caffarel_2016} and RSDFT-CIPSI wave functions were
\eg{The first question we would like to address is the quality of the nodes of the wave functions $\Psi^{\mu}$ obtained
with an intermediate range separation parameter $\mu$ (\textit{i.e.} $\mu > 0$ and $\mu < + \infty$).
Therefore, we computed the fixed node energy obtained with $\Psi^{\mu}$
without re optimizing any parameters having an impact on the nodes
(such as Slater determinant coefficients or orbitals),
and this for several values of $\mu$. We considered two weakly correlated molecular systems:
the water molecule and fluorine dimer, both studied near their equilibrium geometry\cite{Caffarel_2016}.
All RSDFT-CIPSI wave functions were obtained calculations using BFD pseudopotentials
and the corresponding double- and triple-zeta basis sets for the water molecule,
and double-zeta quality for the fluorine dimer.}
\sout{The water molecule was taken at the equilibrium
geometry,\cite{Caffarel_2016} and
generated with BFD pseudopotentials and the corresponding double-zeta
basis set using multiple values of the range-separation parameter
$\mu$. The convergence criterion for stopping the CIPSI calculation
was set to 1~m$E_h$ on the PT2 correction. Then, these wave functions
were used as trial wave functions for FN-DMC calculations, and the
corresponding energies are shown in table~\ref{tab:h2o-dmc} and
figure~\ref{fig:h2o-dmc}.
was set to 1~m$E_h$ on the PT2 correction.
Then, these wave functions
were used as trial wave functions for FN-DMC calculations,}
\eg{We report the values of the FN-DMC energies of the water molecule in table~\ref{tab:h2o-dmc} and
figure~\ref{fig:h2o-dmc}.}
Using FCI trial wave functions gives FN-DMC energies which are lower
\eg{From table~\ref{tab:h2o-dmc} and figure~\ref{fig:h2o-dmc} one can clearly observe that }using FCI trial wave functions gives FN-DMC energies which are lower
than the energies obtained with a single Kohn-Sham determinant:
3~m$E_h$ at the double-zeta level and 7~m$E_h$ at the triple-zeta
level. Interestingly, with the double-zeta basis one can obtain a
\eg{a gain of} 3~m$E_h$ at the double-zeta level and 7~m$E_h$ at the triple-zeta
level \eg{are obtained}. Interestingly, with the double-zeta basis one can obtain a
FN-DMC energy 2.5~m$E_h$ lower than the energy obtained with the FCI
trial wave function, using the RSDFT-CIPSI with a range-separation
parameter $\mu=1.75$. This can be explained by the inability of the
@ -448,6 +457,14 @@ shifted towards the FCI, which is consistent with the fact that
in the CBS limit we expect the minimum of the FN-DMC energy to be
obtained for the FCI wave function, at $\mu=\infty$.
\eg{To further study the behaviour of the FN-DMC energy as a function of $\mu$,
we report in table ???? and figure~\ref{fig:f2-dmc} the FN-DMC energies obtained
with a similar procedure for the fluorine dimer.
The global behaviour and shape of the curves show a very similar behaviour
with respect to that obtained on the water molecule: there exist an "optimal" value of $\mu$ which provides a lower FN-DMC energy than both the KS determinant (\textit{i.e.} $\mu = 0$) and the FCI wave function (\textit{i.e.} $\mu=\infty$).
Nevertheless, one can notice that the value of such optimal $\mu$ is sensibly larger in F$_2$ than H$_2$O: this is probably the signature of the fact that the average inter electronic distance in the valence is smaller in F$_2$ than in H$_2$O due to the larger nuclear charge and corresponding shrinking of the electronic density.
}
\begin{figure}
\centering