saving work

Manuscript/rsdft-cipsi-qmc.tex

@@ -272,7 +272,7 @@ improvements on the path from HF to FCI:
 i) increasing the maximum degree of excitation of CI methods (CISD, CISDT,
 CISDTQ, \ldots), or ii) expanding the size of a complete active space
 (CAS) wave function until all the orbitals are in the active space.
-Selected CI methods take a shortcut between the HF
+SCI methods take a shortcut between the HF
 determinant and the FCI wave function by increasing iteratively the
 number of determinants on which the wave function is expanded,
 selecting the determinants which are expected to contribute the most
@@ -281,7 +281,7 @@ extracted from the CI matrix expressed in the determinant subspace,
 and the FCI energy can be estimated by adding up to the variational energy 
 a second-order perturbative correction (PT2), $\EPT$.
 The magnitude of $\EPT$ is a measure of the distance to the FCI energy 
-and a diagnostic of the the quality of the wave function.
+and a diagnostic of the quality of the wave function.
 Within the CIPSI algorithm originally developed by Huron \textit{et al.} in Ref.~\onlinecite{Huron_1973} and efficiently implemented as described in Ref.~\onlinecite{Garniron_2019}, the PT2
 correction is computed simultaneously to the determinant selection at no extra cost. 
 $\EPT$ is then the sole parameter of the CIPSI algorithm and is chosen to be its convergence criterion.
@@ -598,10 +598,12 @@ were obtained by energy minimization with a single determinant.
 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} & = & \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}
+\begin{subequations}
+\begin{gather}
+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{gather}
+\end{subequations}
 and solving Eq.~\eqref{eq:ci-j}.\cite{Nightingale_2001}
 
 We can easily compare $\Psi^\mu$ and  $\Psi^J$ as they are developed
@@ -1064,7 +1066,7 @@ $\mu=0.5$~bohr$^{-1}$ with the quadruple-zeta basis set.
 \begin{figure*}
   \centering
   \includegraphics[width=\textwidth]{g2-dmc.pdf}
-  \caption{Errors in the FN-DMC atomization energies with the different
+  \caption{Errors in the FN-DMC atomization energies with various
     trial wave functions. Each dot corresponds to an atomization
     energy.
     The boxes contain the data between first and third quartiles, and
@@ -1090,7 +1092,7 @@ cancellations of errors.
 \begin{figure*}
   \centering
   \includegraphics[width=\textwidth]{g2-ndet.pdf}
-  \caption{Number of determinants in the different trial wave
+  \caption{Number of determinants for various trial wave
     functions. Each dot corresponds to an atomization energy.
     The boxes contain the data between first and third quartiles, and
     the line in the box represents the median. The outliers are shown
@@ -1132,8 +1134,8 @@ solution would have been the PBE single determinant.
 %%%%%%%%%%%%%%%%%%%%
 
 In the present work, we have shown that introducing short-range correlation via
-a range-separated Hamiltonian in a full CI expansion yields improved
-nodal surfaces, especially with small basis sets. The effect of sr-DFT
+a range-separated Hamiltonian in a FCI expansion yields improved
+nodal surfaces, especially with small basis sets. The effect of short-range DFT
 on the determinant expansion is similar to the effect of re-optimizing
 the CI coefficients in the presence of a Jastrow factor, but without
 the burden of performing a stochastic optimization.