Modifs Mimi (2)

Manuscript/QUEST_WIREs.tex

@@ -307,8 +307,7 @@ E_{\text{rPT2}}^{(0)} \approx E_{\text{rPT2}}^{(m)}$, and
 by using a common set of state-averaged natural orbitals with equal weights for the ground and excited states.
 %This last feature tends to make the values of $\alpha^{(0)}$ and $\alpha^{(m)}$ very close to each other, such that the error on the energy difference is decreased.
 
-
-Using Eq.~\eqref{eqx} the estimated error on the CIPSI energy is calculated as
+Using Eq.(\ref{eqx}) the estimated error on the CIPSI energy is calculated as
 \begin{equation}
   E_{\text{CIPSI}}^{(m)} - E_{\text{FCI}}^{(m)}
   = \qty(E_\text{var}^{(m)}+E_{\text{rPT2}}^{(m)}) - E_{\text{FCI}}^{(m)}
@@ -317,12 +316,16 @@ Using Eq.~\eqref{eqx} the estimated error on the CIPSI energy is calculated as
 and thus the extrapolated excitation energy associated with the $m$th
 state is given by
 \begin{equation}
-  \Delta E_{\text{FCI}}^{(m)}
-  = \qty[ E_\text{var}^{(m)} + E_{\text{rPT2}} + \qty(\alpha^{(m)}-1) E_{\text{rPT2}} ]
-  - \qty[ E_\text{var}^{(0)} + E_{\text{rPT2}} + \qty(\alpha^{(0)}-1) E_{\text{rPT2}} ]
-  + \mathcal{O}\qty[{E_{\text{rPT2}}^2 }]
+ \Delta E_{\text{FCI}}^{(m)}
+ = \qty[ E_\text{var}^{(m)} + E_{\text{rPT2}}^{(m)} + \qty(\alpha^{(m)}-1) E_{\text{rPT2}}^{(m)} ]
+ - \qty[ E_\text{var}^{(0)} + E_{\text{rPT2}}^{(0)} + \qty(\alpha^{(0)}-1) E_{\text{rPT2}}^{(0)} ].
 \end{equation}
-which evidences that the error in $\Delta E_{\text{FCI}}^{(m)}$ can be expressed as $\qty(\alpha^{(m)}-\alpha^{(0)}) E_{\text{rPT2}} + \mathcal{O}\qty[{E_{\text{rPT2}}^2}]$.
+The slopes $\alpha^{(m)}$ and $\alpha^{(0)}$ deviating only slightly from the unity, the error in
+$\Delta E_{\text{FCI}}^{(m)}$ can be expressed at leading order as $\qty(\alpha^{(m)}-\alpha^{(0)}) {\bar E}_{\text{rPT2}} + O\qty[{{\bar E}_{\text{rPT2}}^2}]$, where
+${\bar E}_{\text{rPT2}}$ is the averaged second-order correction,
+${\bar E}_{\text{rPT2}}=\qty(E_{\text{rPT2}}^{(m)}
++E_{\text{rPT2}}^{(0)})/2$.
+
 In the ideal case where one is able to fully correlate the CIPSI calculations associated with the ground and excited states, the fluctuations of
 $\Delta E_\text{CIPSI}^{(m)}(n)$ as a function of the iteration number $n$ would completely vanish and the exact excitation energy would be obtained from the first CIPSI iterations.
 Quite remarkably, in practice, numerical experience shows that the fluctuations with respect to the extrapolated value $\Delta E_\text{FCI}^{(m)}$ are small,