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\begin{document}

\title{Erratum: Spin-unrestricted random-phase approximation with range separation: Benchmark on atomization energies and reaction barrier heights [J. Chem. Phys. 142, 154123 (2015)]}

\author{Bastien Mussard}\email{bastien.mussard@upmc.fr}
\affiliation{Sorbonne Universit\'es, UPMC Univ Paris 06, Institut du Calcul et de la Simulation, F-75005, Paris, France}
\affiliation{Sorbonne Universit\'es, UPMC Univ Paris 06, UMR 7616, Laboratoire de Chimie Th\'eorique, F-75005 Paris, France}
\affiliation{CNRS, UMR 7616, Laboratoire de Chimie Th\'eorique, F-75005 Paris, France}
\author{Peter Reinhardt}
\affiliation{Sorbonne Universit\'es, UPMC Univ Paris 06, UMR 7616, Laboratoire de Chimie Th\'eorique, F-75005 Paris, France}
\affiliation{CNRS, UMR 7616, Laboratoire de Chimie Th\'eorique, F-75005 Paris, France}
\author{J\'anos G. \'Angy\'an}
\affiliation{CRM2, Institut Jean Barriol, Universit\'e de Lorraine, F-54506 Vand{\oe}uvre-l\'es-Nancy, France}
\affiliation{CRM2, Institut Jean Barriol, CNRS, F-54506 Vand{\oe}vre-l\'es-Nancy, France}
\author{Julien Toulouse}\email{julien.toulouse@upmc.fr}
\affiliation{Sorbonne Universit\'es, UPMC Univ Paris 06, UMR 7616, Laboratoire de Chimie Th\'eorique, F-75005 Paris, France}
\affiliation{CNRS, UMR 7616, Laboratoire de Chimie Th\'eorique, F-75005 Paris, France}

\maketitle

After publication of our paper\cite{MusReiAngTou-JCP-15},
we realized that the expressions for the dRPA-II and RPAx-II correlation energies,
shown in equations~(2) and~(4) of the paper\cite{MusReiAngTou-JCP-15},
should take the following forms (previously published in Ref.~[\onlinecite{AngLiuTouJan-JCTC-11}]):

\begin{widetext}
\begin{equation}
E_\c^\text{dRPA-II} = \frac{1}{2} \int_0^{1} \d \alpha \;\tr \left[
  \frac{1}{2}       \b{Q}_\alpha^\dRPA             \left(\b{A}^{\prime\II}_1+\b{B}^\II_1\right)
 +\frac{1}{2} \left(\b{Q}_\alpha^\dRPA\right)^{-1} \left(\b{A}^{\prime\II}_1-\b{B}^\II_1\right) - \b{A}^{\prime\II}_1
 \right],
\label{eq:ACdRPAII}
\end{equation}
and:
\begin{equation}
E_\c^\text{RPAx-II} = \frac{1}{4} \int_0^{1} \d \alpha \;\tr \left[
  \frac{1}{2}       \b{Q}_\alpha^\RPAx             \left(\b{A}^{\prime\II}_1+\b{B}^\II_1\right)
 +\frac{1}{2} \left(\b{Q}_\alpha^\RPAx\right)^{-1} \left(\b{A}^{\prime\II}_1-\b{B}^\II_1\right) - \b{A}^{\prime\II}_1
 \right],
\label{eq:ACRPAxII}
\end{equation}
\end{widetext}
where the matrix $\b{Q}_\alpha$ is defined as follow:

\begin{equation}
\b{Q}_\alpha=\left(\b{A}_\alpha-\b{B}_\alpha\right)^{1/2} \left(\b{M}_\alpha\right)^{-1/2} \left(\b{A}_\alpha-\b{B}_\alpha\right)^{1/2}
\end{equation}
with matrices $\b{A}_\alpha^\I$  and $\b{B}_\alpha^\I$  used to construct $\b{Q}_\alpha^\dRPA$
 and matrices $\b{A}_\alpha^\II$ and $\b{B}_\alpha^\II$ used to construct $\b{Q}_\alpha^\RPAx$.
Note that the matrices $\b{A}_\alpha^\I$, $\b{A}_\alpha^\II$, $\b{B}_\alpha^\I$, $\b{B}_\alpha^\II$ as well as $\b{M}_\alpha$
are defined in our paper\cite{MusReiAngTou-JCP-15}
but that, on the other hand,
the matrix $\b{A}^{\prime\II}_1$ appearing in Eqs.~(\ref{eq:ACdRPAII}) and~(\ref{eq:ACRPAxII}) of this erratum needs to be defined here:

\begin{equation}
\left(A_\alpha^{\prime\II} \right)_{ia,jb} = \alpha \; \bra{ib}\ket{aj}.
\end{equation}

It differs from $\b{A}^\II_\alpha$ in that it does not contain the differences of spin-orbital energies.

All results shown in the original paper were obtained using equations~(\ref{eq:ACdRPAII}) and~(\ref{eq:ACRPAxII}) of this erratum and are thus correct.

Note that the matrices $\b{Q}_\alpha$ are related to the matrices $\b{P}_{\c,\alpha}$ that appear in our paper\cite{MusReiAngTou-JCP-15} by $\b{P}_{\c,\alpha}=\b{Q}_\alpha-\b{I}$ (where $\b{I}$ is the identity matrix)
and that one can make the following approximations to the matrices $(\b{Q}_\alpha)^{-1}$, as explained in Ref.~[\onlinecite{AngLiuTouJan-JCTC-11}]:
\begin{align}
&\left(\b{Q}_\alpha^\dRPA\right)^{-1}=\left(\b{I}+\b{P}^\dRPA_{\c,\alpha}\right)^{-1}\approx\b{I}-\b{P}^\dRPA_{\c,\alpha}=2\;\b{I}-\b{Q}^\dRPA_{\alpha}
,\end{align}
and
\begin{align}
\qquad&\left(\b{Q}_\alpha^\RPAx\right)^{-1}=\left(\b{I}+\b{P}^\RPAx_{\c,\alpha}\right)^{-1}\approx\b{I}-\b{P}^\RPAx_{\c,\alpha}=2\; \b{I}-\b{Q}^\RPAx_{\alpha}
,\end{align}
which lead to the so-called ``IIa'' approximations to the dRPA-II and RPAx-II correlation energies:
% of Eqs.~(\ref{eq:ACdRPAII}) and (\ref{eq:ACRPAxII}) of this erratum:

\begin{align}
&E_\c^\text{dRPA-IIa} = \frac{1}{2} \int_0^{1} \d \alpha \;\tr \left[ \b{B}^\II_1 \; \b{P}_{\c,\alpha}^\dRPA \right],
\\
&E_\c^\text{RPAx-IIa} = \frac{1}{4} \int_0^{1} \d \alpha \;\tr \left[ \b{B}^\II_1  \; \b{P}_{\c,\alpha}^\RPAx \right].
\end{align}

These are the expressions that were erroneously shown in the original paper.

%\vspace*{1em}
%The interested reader might want to understand that the equations in our original paper involve matrices of the dimension of the number of single excitations, $N_\text{exc}\times N_\text{exc}$,
%but that equations~(1) to~(4) of our original paper are actually derived from expressions written with super-matrices of dimension $2 N_\text{exc}\times 2 N_\text{exc}$ (see Ref.~[\onlinecite{AngLiuTouJan-JCTC-11}]).
%These source expressions read:
%
%\begin{align}
%&E_\c^\text{dRPA-I\phantom{I}}  = \frac{1}{2} \int_0^{1} \d \alpha \;\Tr \left[
% \bb{V}  \bb{P}_{c,\alpha}^\dRPA
% \right],
%\label{eq:ACdRPAIbb}
%\\
%&E_\c^\text{dRPA-II} = \frac{1}{2} \int_0^{1} \d \alpha \;\Tr \left[
% \bb{W}  \bb{P}_{c,\alpha}^\dRPA
% \right],
%\label{eq:ACdRPAIIbb}
%\\
%&E_\c^\text{RPAx-I\phantom{I}}  = \frac{1}{2} \int_0^{1} \d \alpha \;\Tr \left[
% \bb{V}  \bb{P}_{c,\alpha}^\RPAx
% \right],
%\label{eq:ACRPAxIbb}
%\\
%&E_\c^\text{RPAx-II} = \frac{1}{4} \int_0^{1} \d \alpha \;\Tr \left[
% \bb{W}  \bb{P}_{c,\alpha}^\RPAx
% \right].
%\label{eq:ACRPAxIIbb}
%\end{align}
%
%The typos that appear in our original paper stem from the fact that we wanted Eqs.~(1) to~(4) to mirror as much as possible Eqs.~(11) to~(14).
%Unfortunatly, the effort to reduce the dimension of the matrices involved in the equations to solve,
%for example from $\bb{P}_{\c,\alpha}$ to $\b{P}_{c,\alpha}$,
%substantially changes the expressions of the dRPA-II and RPAx-II correlation energies, that are no longer easily comparable.
%
%As a consequence, one either has to consider similar expressions involving matrices of different dimensions
%(compare equations~(\ref{eq:ACdRPAIbb}) to~(\ref{eq:ACRPAxIIbb}) of this erratum and equations~(11) to~(14) of the original paper),
%or quite different expressions involving matrices sharing the same dimensions
%(see for example Eqs.~(\ref{eq:ACdRPAII}) and~(\ref{eq:ACRPAxII}) of this erratum and equations~(13) and~(14) of the original paper).

\bibliographystyle{jchemphys}

\begin{thebibliography}{10}

\bibitem{MusReiAngTou-JCP-15}
B. Mussard, P. Reinhardt, J.G. \'Angy\'an, J. Toulouse, J. Chem. Phys. {\bf 142}, 154123 (2015).

\bibitem{AngLiuTouJan-JCTC-11}
J.~G. \'Angy\'an, R.-F. Liu, J.~Toulouse and G.~Jansen, J. Chem. Theory Comput.
  {\bf 7}, 3116 (2011).

\end{thebibliography}

\end{document}

