fix problem

Manuscript/BSE_JPCL.tex

@@ -404,7 +404,7 @@ Taking the static limit, \ie, $W(\omega=0)$, for the screened Coulomb potential,
     \begin{pmatrix}
 		R & C  
 		\\
-		-C^*  & R^{*}
+		-C^*  & -R^{*}
 	\end{pmatrix}
     \begin{pmatrix}
 		X^m
@@ -427,7 +427,7 @@ with electron-hole ($eh$) eigenstates written as
 \end{equation}
 where $m$ indexes the electronic excitations. 
 The $\lbrace \phi_{i/a} \rbrace$ are, in the case of $G_0W_0$ and ev$GW$, the input (KS) eigenstates used to build the $GW$ self-energy. 
-The resonant and anti-resonant parts of the BSE Hamiltonian read
+The resonant and coupling parts of the BSE Hamiltonian read
 \begin{gather}
     R_{ai,bj} = \qty( \varepsilon_a^{\GW} - \varepsilon_i^{\GW} ) \delta_{ij} \delta_{ab} + \kappa (ia|jb) - W_{ij,ab},
     \\
@@ -446,7 +446,7 @@ $(ia|jb)$ bare Coulomb term defined as
 	\phi_i(\br) \phi_a(\br) v(\br-\br')
 	\phi_j(\br') \phi_b(\br'),
 \end{equation}  
-Neglecting the anti-resonant term $C$ in Eq.~\eqref{eq:BSE-eigen}, which is usually much smaller than its resonant counterpart $R$, leads to the well-known Tamm-Dancoff approximation.
+Neglecting the coupling term $C$ between the resonant term $R$ and anti-resonant term $-R^*$ in Eq.~\eqref{eq:BSE-eigen}, leads to the well-known Tamm-Dancoff approximation.
 As compared to TD-DFT,
 \begin{itemize}
     \item the $GW$ quasiparticle energies $\lbrace \varepsilon_{i/a}^{\GW} \rbrace$ replace the KS eigenvalues