Done for good

dynker.tex

@@ -49,7 +49,7 @@ Linear response theory is a powerful approach that allows to directly access the
 From a practical point of view, these quantities are obtained by solving non-linear, frequency-dependent Casida-like equations in the space of single excitations and de-excitations \cite{Casida_1995}
 \begin{equation} \label{eq:LR}
 	\begin{pmatrix}
-		+\bR^{\sigma}(+\omega_S)	&	+\bC^{\sigma}(+\omega_S)
+		\bR^{\sigma}(\omega_S)	&	\bC^{\sigma}(\omega_S)
 		\\
 		-\bC^{\sigma}(-\omega_S)^*	&	-\bR^{\sigma}(-\omega_S)^*
 	\end{pmatrix}
@@ -102,7 +102,7 @@ where $\sigma = 1$ or $0$ for singlet and triplet excited states (respectively),
 \end{equation}
 are the usual two-electron integrals. \cite{Gill_1994}
 The launchpad of the present study is that, thanks to its non-linear nature stemming from its frequency dependence, a dynamical kernel potentially generates more than just single excitations.
-Unless otherwise stated, atomic units are used and we assume real quantities throughout this manuscript.
+Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{The concept of dynamical quantities}
@@ -116,7 +116,7 @@ In most cases, this can be done by solving a set of linear equations of the form
 	\label{eq:lin_sys}
 	\bA \cdot \bc = \omega \, \bc
 \end{equation}
-where $\omega$ is one of the optical excitation energies of interest and $\bc$ its transition vector .
+where $\omega$ is one of the optical excitation energies of interest and $\bc$ its transition vector.
 If we assume that the matrix $\bA$ is diagonalizable and of size $N \times N$, the \textit{linear} set of equations \eqref{eq:lin_sys} yields $N$ excitation energies.
 However, in practice, $N$ might be (very) large (\eg, equal to the total number of single and double excitations generated from a reference Slater determinant), and it might therefore be practically useful to recast this system as two smaller coupled systems, such that 
 \begin{equation}
@@ -164,7 +164,7 @@ Note that this \textit{exact} decomposition does not alter, in any case, the val
 
 How have we been able to reduce the dimension of the problem while keeping the same number of solutions?
 To do so, we have transformed a linear operator $\bA$ into a non-linear operator $\Tilde{\bA}_1(\omega)$ by making it frequency dependent.
-In other words, we have sacrificed the linearity of the system in order to obtain a new, non-linear systems of equations of smaller dimension [see Eq.~\eqref{eq:non_lin_sys}]. 
+In other words, we have sacrificed the linearity of the system in order to obtain a new, non-linear system of equations of smaller dimension [see Eq.~\eqref{eq:non_lin_sys}]. 
 This procedure converting degrees of freedom into frequency or energy dependence is very general and can be applied in various contexts. \cite{Sottile_2003,Garniron_2018,QP2}
 Thanks to its non-linearity, Eq.~\eqref{eq:non_lin_sys} can produce more solutions than its actual dimension. 
 However, because there is no free lunch, this non-linear system is obviously harder to solve than its corresponding linear analog given by Eq.~\eqref{eq:lin_sys}.
@@ -296,7 +296,7 @@ For the two-level model, the non-linear equations defined in Eq.~\eqref{eq:LR} p
 \begin{equation} \label{eq:H-M}
 	\bH_\text{D-TDHF}^{\sigma}(\omega) = 
 	\begin{pmatrix}
-		+R_\text{M}^{\sigma}(+\omega)	&	+C_\text{M}^{\sigma}(+\omega)
+		R_\text{M}^{\sigma}(\omega)		&	C_\text{M}^{\sigma}(\omega)
 		\\
 		-C_\text{M}^{\sigma}(-\omega)	&	-R_\text{M}^{\sigma}(-\omega)
 	\end{pmatrix}	
@@ -382,7 +382,7 @@ For this particular system, they showed that a BSE kernel based on the random-ph
 This issue was resolved in the subsequent work of Sangalli \textit{et al.} \cite{Sangalli_2011} via the design of a diagrammatic number-conserving approach based on the folding of the second-RPA Hamiltonian. \cite{Wambach_1988}
 Thanks to a careful diagrammatic analysis of the dynamical kernel, they showed that their approach produces the correct number of optically active poles, and this was further illustrated by computing the polarizability of two unsaturated hydrocarbon chains (\ce{C8H2} and \ce{C4H6}).
 Very recently, Loos and Blase have applied the dynamical correction to the BSE beyond the plasmon-pole approximation within a renormalized first-order perturbative treatment, \cite{Loos_2020e} generalizing the work of Rolhfing and coworkers on biological chromophores \cite{Ma_2009a,Ma_2009b} and dicyanovinyl-substituted oligothiophenes. \cite{Baumeier_2012b}
-They compiled a comprehensive set of vertical transitions in prototypical molecules, providing benchmark data and showing that the dynamical corrections can be sizable and improve the static BSE excitations considerably. \cite{Loos_2020e}
+They compiled a comprehensive set of vertical transitions in prototypical molecules, providing benchmark data and showing that the dynamical correction can be sizable (especially for $n \to \pi^*$ and $\pi \to \pi^*$ excitations) and improves the static BSE excitations considerably. \cite{Loos_2020e}
 Let us stress that, in all these studies, the TDA is applied to the dynamical correction (\ie, only the diagonal part of the BSE Hamiltonian is made frequency-dependent) and we shall do the same here.
 
 Within the so-called $GW$ approximation of MBPT, \cite{Aryasetiawan_1998,Onida_2002,Reining_2017,ReiningBook,Golze_2019} one can easily compute the quasiparticle energies associated with the valence and conduction orbitals. \cite{Hybertsen_1985a,Hybertsen_1986,vanSetten_2013,Bruneval_2016}
@@ -406,7 +406,7 @@ One can now build the dynamical BSE (dBSE) Hamiltonian \cite{Strinati_1988,Roman
 \begin{equation} \label{eq:HBSE}
 	\bH_{\dBSE}^{\sigma}(\omega) = 
 	\begin{pmatrix}
-		 +R_{\dBSE}^{\sigma}(+\omega)	&	+C_{\dBSE}^{\sigma}
+		 R_{\dBSE}^{\sigma}(\omega)	&	C_{\dBSE}^{\sigma}
 		\\
 		-C_{\dBSE}^{\sigma}	&	-R_{\dBSE}^{\sigma}(-\omega)
 	\end{pmatrix}
@@ -436,7 +436,7 @@ Within the usual static approximation, the BSE Hamiltonian is simply
 \begin{equation}
 	\bH_{\BSE}^{\sigma} = 
 	\begin{pmatrix}
-		 +R_{\BSE}^{\sigma} &	+C_{\BSE}^{\sigma}
+		 R_{\BSE}^{\sigma} &	C_{\BSE}^{\sigma}
 		\\
 		-C_{\BSE}^{\sigma}	&	-R_{\BSE}^{\sigma}
 	\end{pmatrix}
@@ -462,7 +462,7 @@ Indeed, diagonalizing the exact Hamiltonian \eqref{eq:H-exact} produces only two
 Therefore, there is the right number of singlet solutions but there is one spurious solution for the triplet manifold ($\omega_{2}^{\dBSE,\upup}$).
 It is worth mentioning that, around $\omega = \omega_1^{\dBSE,\sigma}$, the slope of the curves depicted in Fig.~\ref{fig:dBSE} is small, while the other solution, $\omega_2^{\dBSE,\sigma}$, stems from a pole and consequently the slope is very large around this frequency value.
 This makes this latter solution quite hard to locate with a method like Newton-Raphson (for example).
-Let us highlight the fact that, unlike in Ref.~\onlinecite{Loos_2020e} where dynamical effects have been shown to produce a systematic red-shifting of the static excitations, here we observe both blue- and red-shifted transitions (see values in square brackets in Table \ref{tab:BSE}).
+Let us highlight the fact that, unlike in Ref.~\onlinecite{Loos_2020e} where dynamical effects have been shown to produce a systematic red-shift of the static excitations, here we observe both blue- and red-shifted transitions (see values in square brackets in Table \ref{tab:BSE}).
 
 %%% TABLE III %%%
 \begin{table*}
@@ -506,9 +506,9 @@ Let us highlight the fact that, unlike in Ref.~\onlinecite{Loos_2020e} where dyn
 
 In the static approximation, only one solution per spin manifold is obtained by diagonalizing $\bH_{\BSE}^{\sigma}$ (see Fig.~\ref{fig:dBSE} and Table \ref{tab:BSE}).
 Therefore, the static BSE Hamiltonian misses the (singlet) double excitation (as it should), and it shows that the physical single excitation stemming from the dBSE Hamiltonian is indeed the lowest in energy for each spin manifold, \ie, $\omega_1^{\dBSE,\updw}$ and $\omega_1^{\dBSE,\upup}$.
-This can be further verify by switching off gradually the electron-electron interaction as one would do in the adiabatic connection formalism. \cite{Langreth_1975,Gunnarsson_1976}
+This can be further verified by switching off gradually the electron-electron interaction as one would do in the adiabatic connection formalism. \cite{Langreth_1975,Gunnarsson_1976}
 
-Enforcing the TDA, which corresponds to neglecting the coupling term between the resonant and anti-resonant parts of the dBSE Hamiltonian \eqref{eq:HBSE} does not change the situation in terms of spurious solutions: there is still one spurious excitation in the triplet manifold ($\omega_{2}^{\BSE,\upup}$), and the two solutions for the singlet manifold which corresponds to the single and double excitations.
+Enforcing the TDA, which corresponds to neglecting the coupling term between the resonant and anti-resonant parts of the dBSE Hamiltonian \eqref{eq:HBSE}, does not change the situation in terms of spurious solutions: there is still one spurious excitation in the triplet manifold ($\omega_{2}^{\BSE,\upup}$), and the two solutions for the singlet manifold which corresponds to the single and double excitations.
 However, it does increase significantly the static excitations while the magnitude of the dynamical corrections is not altered by the TDA.
 
 Another way to access dynamical effects while staying in the static framework is to use perturbation theory, \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b,Loos_2020e} a scheme we label as perturbative BSE (pBSE).
@@ -595,13 +595,13 @@ and
 \begin{equation}
 	Z_{p}^{\GF} = \qty( 1 - \left. \pdv{\SigGF{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1}
 \end{equation}
-The expression of the GF2 self-energy \eqref{eq:SigGF} can be easily obtained from its $GW$ counterpart \eqref{eq:SigGW} via the substitution $\Omega \to \e{c} - \e{v}$ and by dividing the numerator by a factor two. This shows that there is no screening within GF2 and BSE2, but that second-order exchange is properly taken into account. \cite{Zhang_2013,Loos_2018b}
+The expression of the GF2 self-energy \eqref{eq:SigGF} can be easily obtained from its $GW$ counterpart \eqref{eq:SigGW} via the substitution $\Omega \to \e{c} - \e{v}$ and by dividing the numerator by a factor two. This shows that there is no screening within GF2, but that second-order exchange is properly taken into account. \cite{Zhang_2013,Loos_2018b}
 
 The static Hamiltonian of BSE2 is just the usual TDHF Hamiltonian where one substitutes the HF orbital energies by the GF2 quasiparticle energies, \ie, 
 \begin{equation}
 	\bH_{\BSE2}^{\sigma} = 
 	\begin{pmatrix}
-		+R_{\BSE2}^{\sigma}	&	+C_{\BSE2}^{\sigma}
+		R_{\BSE2}^{\sigma}	&	C_{\BSE2}^{\sigma}
 		\\
 		-C_{\BSE2}^{\sigma}	&	-R_{\BSE2}^{\sigma}
 	\end{pmatrix}
@@ -620,9 +620,9 @@ The correlation part of the dynamical kernel for BSE2 is a bit cumbersome \cite{
 \begin{equation}
 	\bH_{\dBSE2}^{\sigma} = \bH_{\BSE2}^{\sigma} +
 	\begin{pmatrix}
-		 +R_{\dBSE2}^{\co,\sigma}(+\omega)	&	+R_{\dBSE2}^{\co,\sigma}
+		 R_{\dBSE2}^{\co,\sigma}(\omega)	&	C_{\dBSE2}^{\co,\sigma}
 		\\
-		-C_{\dBSE2}^{\co,\sigma}	&	-C_{\BSE2}^{\co,\sigma}(-\omega)
+		-C_{\dBSE2}^{\co,\sigma}	&	-R_{\BSE2}^{\co,\sigma}(-\omega)
 	\end{pmatrix}
 \end{equation}
 with 
@@ -710,7 +710,7 @@ However, one cannot access the double excitation.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 The take-home message of the present paper is that dynamical kernels have much more to give that one would think.
 In more scientific terms, dynamical kernels can provide, thanks to their frequency-dependent nature, additional excitations that can be associated to higher-order excitations (such as the infamous double excitations), an unappreciated feature of dynamical quantities.
-However, they sometimes give too much, and generate spurious excitations, \ie, excitation which does not corresponds to any physical excited state.
+However, they sometimes give too much, and generate spurious excitations, \ie, excitation which does not correspond to any physical excited state.
 The appearance of these factitious excitations is due to the approximate nature of the dynamical kernel.
 Moreover, because of the non-linear character of the linear response problem when one employs a dynamical kernel, it is computationally more involved to access these extra excitations.