Spin invariance

Manuscript/rsdft-cipsi-qmc.tex

@@ -607,31 +607,77 @@ Ref.~\onlinecite{Scemama_2015}).
 \end{table}
 
 In this section, we make a numerical verification that the produced
-wave functions are size-consistent. We have computed the energy of the
-dissocited fluorine dimer, where the two atoms are at a distance of 50~\AA.
-We expect that the energy of this system is equal to twice the energy
-of the fluorine atom.
-The data in table~\ref{tab:size-cons} shows that the proposed scheme
-provides size-consistent FN-DMC energies for all values of $\mu$.
+wave functions are size-consistent for a given range-separation
+parameter.
+We have computed the energy of the dissocited fluorine dimer, where
+the two atoms are at a distance of 50~\AA.  We expect that the energy
+of this system is equal to twice the energy of the fluorine atom.
+The data in table~\ref{tab:size-cons} shows that it is indeed the
+case, so we can conclude that the proposed scheme provides
+size-consistent FN-DMC energies for all values of $\mu$.
 
 
 \subsection{Spin-invariance}
 
-Closed-shell molecules usually dissociate into open-shell
+Closed-shell molecules often dissociate into open-shell
 fragments. To get reliable atomization energies, it is important to
 have a theory which is of comparable quality for open-shell and
-closed-shell systems.
+closed-shell systems. A good test is to check that all the components
+of a spin multiplet are degenerate.
 FCI wave functions are invariant with respect to the spin quantum
 number $m_s$, but the introduction of a
-Jastrow factor breaks this spin-invariance if the parameters 
+Jastrow factor introduces spin contamination if the parameters 
 for the same-spin electron pairs are different from those
 for the opposite-spin pairs.\cite{Tenno_2004}
-Again, using pseudo-potentials this error is transferred in the DMC
-calculation unless the determinant localization approximation is used.
+Again, when pseudo-potentials are used this tiny error is transferred
+in the FN-DMC energy unless the determinant localization approximation
+is used.
+
+Within DFT, the common density functionals make a difference for
+same-spin and opposite-spin interactions. As DFT is a
+single-determinant theory, the functionals are designed to work in
+with the highest value of $m_s$, and therefore different
+values of $m_s$ lead to different energies.
+So in the context of RS-DFT, the determinantal expansions will be
+impacted by this spurious effect, as opposed to FCI.
+
+\begin{table}
+  \caption{FN-DMC Energies of the triplet carbon atom (BFD-VDZ) with
+    different values of $m_s$.}
+  \label{tab:spin}
+  \begin{ruledtabular}
+    \begin{tabular}{cccc}
+      $\mu$ &  $m_s=1$          & $m_s=0$             & Spin-invariance error \\
+      \hline
+      0.00  &  $-5.416\,8(1)$  & $-5.414\,9(1)$     & $+0.001\,9(2)$ \\
+      0.25  &  $-5.417\,2(1)$  & $-5.416\,5(1)$     & $+0.000\,7(1)$ \\
+      0.50  &  $-5.422\,3(1)$  & $-5.421\,4(1)$     & $+0.000\,9(2)$ \\
+      1.00  &  $-5.429\,7(1)$  & $-5.429\,2(1)$     & $+0.000\,5(2)$ \\
+      2.00  &  $-5.432\,1(1)$  & $-5.431\,4(1)$     & $+0.000\,7(2)$ \\
+      5.00  &  $-5.431\,7(1)$  & $-5.431\,4(1)$     & $+0.000\,3(2)$ \\
+   $\infty$ &  $-5.431\,6(1)$  & $-5.431\,3(1)$     & $+0.000\,3(2)$ \\
+    \end{tabular}
+  \end{ruledtabular}
+\end{table}
+
+In this section, we investigate the impact of the spin contamination
+due to the short-range density functional on the FN-DMC energy. We have
+computed the energies of the carbon atom in its triplet state
+with BFD pseudo-potentials and the corresponding double-zeta basis
+set. The calculation was done with $m_s=1$ (3 $\uparrow$ electrons
+and 1 $\downarrow$ electrons) and with $m_s=0$ (2 $\uparrow$ and 2
+$\downarrow$ electrons).
+
+The results are presented in table~\ref{tab:spin}.
+Although using $m_s=0$ the energy is higher than with $m_s=1$, the
+bias is relatively small, more than one order of magnitude smaller
+than the energy gained by reducing the fixed-node error going from the single
+determinant to the FCI trial wave function. The highest bias, close to
+2~m\hartree, is obtained for $\mu=0$, but the bias decreases quickly
+below 1~m\hartree when $\mu$ increases. As expected, with $\mu=\infty$
+there is no bias (within the error bars), and the bias is not
+noticeable with $\mu=5$~bohr$^{-1}$.
 
-To check that the RSDFT-CIPSI are spin-invariant, we compute the
-FN-DMC energies of the ?? dimer with different values of the spin
-quantum number $m_s$.