correcting typos

Manuscript/ufGW.tex

@@ -343,7 +343,7 @@ Similarly notations will be employed for the $(\Ne\pm1)$-electron configurations
 
 In Fig.~\ref{fig:H2}, we report the variation of the quasiparticle energies of the four orbitals as functions of the internuclear distance $\RHH$. 
 One can easily diagnose two problematic regions showing obvious discontinuities around $\RHH = \SI{1.2}{\angstrom}$ for the LUMO$+1$ ($p = 3$) and $\RHH = \SI{0.5}{\angstrom}$ for the LUMO$+2$ ($p = 4$).
-As thoroughly explained in Ref.~\onlinecite{Veril_2018}, if one relies on the linearization of quasiparticle equation \eqref{eq:qp_eq} to compute the quasiparticle energies, \ie, $\eps{p}{\GW} = \eps{p}{\HF} + Z_{p} \SigC{p}(\eps{p}{\HF})$, these discontinuities are transformed into irregularities as the renormalization factor cancels out the singularities of the self-energy. 
+As thoroughly explained in Ref.~\onlinecite{Veril_2018}, if one relies on the linearization of the quasiparticle equation \eqref{eq:qp_eq} to compute the quasiparticle energies, \ie, $\eps{p}{\GW} = \eps{p}{\HF} + Z_{p} \SigC{p}(\eps{p}{\HF})$, these discontinuities are transformed into irregularities as the renormalization factor cancels out the singularities of the self-energy. 
 
 Figure \ref{fig:H2_zoom} shows the evolution of the quasiparticle energy, the energetically close-by satellites and their corresponding weights as functions of $\RHH$.
 Let us first look more closely at the region around $\RHH = \SI{1.2}{\angstrom}$ involving the LUMO$+1$ (left panel of Fig.~\ref{fig:H2_zoom}).
@@ -351,7 +351,7 @@ As one can see, an avoided crossing is formed between two solutions of the quasi
 Inspection of their corresponding eigenvectors reveals that the $(\Ne+1)$-electron determinants principally involved are the reference 1p determinant $\ket*{1\Bar{1}3}$ and an excited $(\Ne+1)$-electron determinant of configuration $\ket*{12\Bar{2}}$ that becomes lower in energy than the reference determinant for $\RHH > \SI{1.2}{\angstrom}$.
 By construction, the quasiparticle solution diabatically follows the reference determinant $\ket*{1\Bar{1}3}$ through the avoided crossing (thick lines in Fig.~\ref{fig:H2_zoom}) which is precisely the origin of the energetic discontinuity.
 
-As similar scenario is at play in the region around $\RHH = \SI{0.5}{\angstrom}$ for the LUMO$+2$ (right panel of Fig.~\ref{fig:H2_zoom}) but it now involves three solutions ($s = 5$, $s = 6$, and $s = 7$).
+A similar scenario is at play in the region around $\RHH = \SI{0.5}{\angstrom}$ for the LUMO$+2$ (right panel of Fig.~\ref{fig:H2_zoom}) but it now involves three solutions ($s = 5$, $s = 6$, and $s = 7$).
 The electronic configurations of the Slater determinant involved are the $\ket*{1\Bar{1}4}$ reference determinant as well as two external determinants of configuration $\ket*{1\Bar{?}?}$ and $\ket*{1\Bar{?}?}$.
 These forms two avoided crossings in rapid successions, which create two discontinuities in the energy surface (see Fig.~\ref{fig:H2}).
 In this region, although the ground-state wave function is well described by the $\Ne$-electron HF determinant, a situation that can be safely labeled as single-reference, one can see that the $(\Ne+1)$-electron state involves three Slater determinants and can then be labeled as a multi-reference (or strongly-correlated) situation with near-degenerate electronic configurations. 
@@ -413,7 +413,7 @@ Our investigations have shown that the following regularizer
 \begin{equation}
 	f_\eta(\Delta) = \frac{1-e^{-2\Delta^2/\eta^2}}{\Delta}
 \end{equation}
-derived from the (second-order) perturbative analysis of the similarity renormalization group equations \cite{Wegner_1994,Glazek_1994,White_2002,Evangelista_2014} is particularly convenient and effective in the present context.
+derived from the (second-order) perturbative analysis of the similarity renormalization group equations \cite{Wegner_1994,Glazek_1994,White_2002} by Evangelista \cite{Evangelista_2014} is particularly convenient and effective in the present context.
 Increasing $\eta$ gradually integrates out states with denominators $\Delta$ larger than $\eta$ while the states with $\Delta \ll \eta$ are not decoupled from the reference space, hence avoiding intruder state problems. \cite{Li_2019a}
 Of course, by construction, we have
 \begin{equation}
@@ -423,7 +423,7 @@ Of course, by construction, we have
 Figure \ref{fig:H2reg_zoom} compares the non-regularized and regularized quasiparticle energies in the two regions of interest for $\eta = 0.1$, $1$, and $10$.
 It clearly shows how the regularization of the $GW$ self-energy diabatically linked the two solutions to get rid of the discontinuities.
 However, this diabatization is more or less accurate depending on the value of $\eta$.
-For $\eta = 10$, the value is clearly too large which induces a large difference between the two sets of quasiparticle energies (purple curves).
+For $\eta = 10$, the value is clearly too large inducing a large difference between the two sets of quasiparticle energies (purple curves).
 For $\eta = 0.1$, we have the opposite scenario where the value is too small and some irregularities remain (green curves).
 We have found that $\eta = 1$ is a good compromise that does not alter significantly the quasiparticle energies while providing a smooth transition between the two solutions.