From 60c8ade61587757fdd0a8992cd871d400af7e69f Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Wed, 27 May 2020 21:29:46 +0200 Subject: [PATCH] 1st sweep corrected lots of mistakes --- BSEdyn.tex | 196 +++++++++++++++++++++++++---------------------------- 1 file changed, 94 insertions(+), 102 deletions(-) diff --git a/BSEdyn.tex b/BSEdyn.tex index 319b866..57d2862 100644 --- a/BSEdyn.tex +++ b/BSEdyn.tex @@ -326,7 +326,11 @@ is the BSE kernel that takes into account the self-consistent variation of the H In Eqs.~\eqref{eq:G1} and \eqref{eq:G2}, the field operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add (respectively) an electron to the $N$-electron ground state $\ket{N}$ in space-spin-time positions ($\bx t$) and ($\bx't'$), while $T$ is the time-ordering operator. The resolution of the dynamical BSE equation \cite{Strinati_1988} starts with the expansion of $L_0$ and $L$ [see Eqs.~\eqref{eq:L0} and \eqref{eq:L}] over the complete orthonormalized set of $N$-electron excited states $\ket{N,s}$ (with $\ket{N,0} \equiv \ket{N}$). -In the optical limit of instantaneous electron-hole creation and destruction, imposing $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, one gets +In the optical limit of instantaneous electron-hole creation and destruction, imposing $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, and using the relation between the field operators in their time-dependent (Heisenberg) and time-independent (Schr\"{o}dinger) representations, \eg, +\begin{equation} + \hpsi(1) = e^{ i \hH t_1 } \hpsi(\bx_1) e^{-i \hH t_1 }, +\end{equation} +($\hH$ being the exact many-body Hamiltonian), one gets \begin{equation} \begin{split} iL(1,2; 1',2') @@ -343,13 +347,9 @@ where $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and \tchi_s(\bx_1,\bx_{2}) & = \mel{N,s}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{2})] }{N}. \end{align} \end{subequations} -The $\Om{s}{}$'s are the neutral excitation energies of interest. We have used the relation between the field operators in their time-dependent (Heisenberg) and time-independent (Schr\"{o}dinger) representations, e.g. -$$ -\hpsi(1) = e^{ i {\hat H} t_1 } \hpsi(\bx_1) e^{-i {\hat H} t_1 } -$$ -with $\hat H$ the exact many-body Hamiltonian. +The $\Om{s}{}$'s are the neutral excitation energies of interest. -Picking up the $e^{+i \Om{s}{} t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$, simplifying further by $\tchi_s(\bx_2,\bx_{2'})$ on both side of the BSE [see Eq.~\eqref{eq:BSE}], we are left with the search of the $e^{-i \Om{s}{} t_1 }$ Fourier component associated with the right-hand side of a modified dynamical BSE, which reads +Picking up the $e^{+i \Om{s}{} t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$, simplifying further by $\tchi_s(\bx_2,\bx_{2'})$ on both side of the BSE [see Eq.~\eqref{eq:BSE}], we seek the $e^{-i \Om{s}{} t_1 }$ Fourier component associated with the right-hand side of a modified dynamical BSE, which reads \begin{multline} \label{eq:BSE_2} \mel{N}{T [ \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}') ] } {N,s} e^{ - i \Om{s}{} t_1 } \theta ( \tau_{12} ) @@ -373,15 +373,9 @@ We now adopt the Lehman representation of the one-body Green's function in the q \begin{equation} \label{eq:G-Lehman} G(\bx_1,\bx_2 ; \omega) = \sum_p \frac{ \MO{p}(\bx_1) \MO{p}^*(\bx_2) } { \omega - \e{p} + i \eta \times \text{sgn} (\e{p} - \mu) }, \end{equation} -where $\mu$ is the chemical potential. -The $\e{p}$'s in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies (\ie, proper addition/removal energies) and the $\MO{p}$'s are their associated one-body (spin)orbitals. -%where the $\eps_{p}$'s are proper addition/removal energies such that -%\begin{equation} -% e^{i \hH \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N}, -%\end{equation} -%$\hH$ being the exact many-body Hamiltonian. -In the following, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals. -%\titou{namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. (T2: shall we really mention this here?)} +where $\eta$ is a positive infinitesimal and $\mu$ is the chemical potential. +The $\e{p}$'s in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies (\ie, proper addition/removal energies) and the $\MO{p}(\bx)$'s are their associated one-body (spin)orbitals. +In the following, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$ and $q$ indicate arbitrary orbitals. Projecting the Fourier component $L_0(\bx_1,4;\bx_{1'},3; \omega_1 = \Om{s}{} )$ onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$ yields \begin{multline} \label{eq:iL0bis} \iint d\bx_1 d\bx_{1'} \, \MO{a}^*(\bx_1) \MO{i}(\bx_{1'}) L_0(\bx_1,4;\bx_{1'},3; \Om{s}{}) @@ -401,7 +395,7 @@ For example, we have = - \qty( e^{ -i \Omega_s t^{65} } ) \sum_{pq} \MO{p}(\bx_6) \MO{q}^*(\bx_5) \mel{N}{\ha_q^{\dagger} \ha_p}{N,s} \\ - \times \qty[ \theta( \tau_{65} ) e^{- i ( \e{p} - \frac{\Om{s}{}}{2} ) \tau_{65} } + \theta( - \tau_{65} ) e^{ - i ( \e{q} + \frac{\Om{s}{}}{2}) \tau_{65} } ] + \times \qty[ \theta( \tau_{65} ) e^{- i ( \e{p} - \frac{\Om{s}{}}{2} ) \tau_{65} } + \theta( - \tau_{65} ) e^{ - i ( \e{q} + \frac{\Om{s}{}}{2}) \tau_{65} } ], \end{multline} with $t^{65} = (t_5 + t_6)/2$ and $\tau_{65} = t_6 -t_5$. %with a similar expression for $\mel{N}{T [\hpsi(\bx_3) \hpsi^{\dagger}(\bx_4)] }{N,s}$. @@ -426,20 +420,20 @@ The $GW$ quasiparticle energies $\eGW{p}$ are good approximations to the removal %Neglecting the $Y_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA). %Working out similar expressions for $\mel{N}{T [\hpsi(5) \hpsi^{\dagger}(5)] }{N,s}$ and $\mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})] }{N,s}$, -Substituting Eqs.~\eqref{eq:iL0bis},\eqref{eq:spectral65},\eqref{eq:Xi_GW} into Eq.~\eqref{eq:BSE_2}, and projecting onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$, one gets after a few tedious manipulations (see {\SI}) the dynamical BSE (dBSE): +Substituting Eqs.~\eqref{eq:iL0bis}, \eqref{eq:spectral65}, and \eqref{eq:Xi_GW} into Eq.~\eqref{eq:BSE_2}, and projecting onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$, one gets after a few tedious manipulations (see {\SI}) the dynamical BSE (dBSE): \begin{equation} \label{eq:BSE-final} \begin{split} - ( \eGW{a} - \eGW{i} - \Om{s}{} ) X_{ia}^{s} - & + \sum_{jb} \qty[ \ERI{ia}{jb} - \widetilde{W}_{ij,ab}(\Om{s}{}) ] X_{jb}^{s} \\ - & + \sum_{jb} \qty[ \ERI{ia}{bj} - \widetilde{W}_{ib,aj}(\Om{s}{}) ] Y_{jb}^{s} + ( \eGW{a} - \eGW{i} - \Om{s}{} ) X_{ia,s} + & + \sum_{jb} \qty[ \kappa \ERI{ia}{jb} - \widetilde{W}_{ij,ab}(\Om{s}{}) ] X_{jb,s} \\ + & + \sum_{jb} \qty[ \kappa \ERI{ia}{bj} - \widetilde{W}_{ib,aj}(\Om{s}{}) ] Y_{jb,s} = 0, \end{split} \end{equation} -with $X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$ and $Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$. -Neglecting the term $Y_{jb}^{s}$ in the dBSE, which is much smaller than $X_{jb}^{s}$, leads to the well-known Tamm-Dancoff approximation (TDA). +with $X_{jb,s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$ and $Y_{jb,s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$, and where $\kappa = 2 $ or $0$ for singlet and triplet excited states (respectively). +Neglecting the anti-resonant terms, $Y_{jb,s}$, in the dBSE, which are much smaller than their resonant counterparts, $X_{jb,s}$, leads to the well-known Tamm-Dancoff approximation (TDA). In Eq.~\eqref{eq:BSE-final}, \begin{equation} - \ERI{pq}{rs} = \iint d\br d\br' \, \MO{p}^*(\br) \MO{q}(\br) v(\br -\br') \MO{r}^*(\br') \MO{s}(\br'), + \ERI{ia}{jb} = \iint d\br d\br' \, \MO{i}^*(\br) \MO{a}(\br) v(\br -\br') \MO{j}^*(\br') \MO{b}(\br'), \end{equation} are the bare two-electron integrals in the spatial orbital basis $\lbrace \MO{p}(\br{}) \rbrace$, and \begin{multline} \label{eq:wtilde} @@ -450,8 +444,8 @@ are the bare two-electron integrals in the spatial orbital basis $\lbrace \MO{p} \end{multline} is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b} where $\Om{pq}{s} = \Om{s}{} - ( \eGW{q} - \eGW{p} )$ and \begin{equation} - W_{pq,rs}({\omega}) - = \iint d\br d\br' \, \MO{p}(\br) \MO{q}^*(\br) W(\br ,\br'; \omega) \MO{r}^*(\br') \MO{s}(\br'). + W_{ij,ab}({\omega}) + = \iint d\br d\br' \, \MO{i}(\br) \MO{j}^*(\br) W(\br ,\br'; \omega) \MO{a}^*(\br') \MO{b}(\br'). \end{equation} \xavier{A second coupled equation for the $(X_{ia}^{s}, Y_{ia}^{s} )$ vector can be obtained by projecting now onto the $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$ left-hand side and right-hand-side of the BSE, leading to : } @@ -469,7 +463,7 @@ In the present study, we consider the exact spectral representation of $W(\omega \\ \times \qty[ \frac{1}{ \omega-\Om{m}{\RPA} + i\eta } - \frac{1}{ \omega + \Om{m}{\RPA} - i\eta } ], \end{multline} -where +where $m$ labels single excitations, and \begin{equation} \label{eq:sERI} \sERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} (\bX{m}{\RPA} + \bY{m}{\RPA})_{ia} @@ -523,7 +517,7 @@ As a consequence, we observe a reduction of the electron-hole screening, \ie, an \subsection{Perturbative dynamical correction} %================================= -For a closed-shell system in a finite basis, to compute the BSE excitation energies, one must solve the following (non-linear) dynamical (\ie, frequency-dependent) response problem \cite{Strinati_1988} +From a more practical point of view, to compute the BSE excitation energies of a closed-shell system, one must solve the following (non-linear) dynamical (\ie, frequency-dependent) response problem \cite{Strinati_1988} \begin{equation} \label{eq:LR-dyn} \begin{pmatrix} @@ -544,20 +538,20 @@ For a closed-shell system in a finite basis, to compute the BSE excitation energ \bY{s}{} \\ \end{pmatrix}, \end{equation} -where the dynamical matrices $\bA{}$ and $\bB{}$, as well as $\bX{s}{}$, and $\bY{s}{}$, have the same size as their RPA counterparts. +where the dynamical matrices $\bA{}$ and $\bB{}$ have the same $\Nocc \Nvir \times \Nocc \Nvir$ size than their RPA counterparts. +Same comment applies to the eigenvectors $\bX{s}{}$, and $\bY{s}{}$ of length $\Nocc \Nvir$. Note that, due to its non-linear nature, Eq.~\eqref{eq:LR-dyn} may provide more than one solution for each value of $s$. \cite{Romaniello_2009b,Sangalli_2011,Martin_2016} -The BSE matrix elements read +Accordingly to Eq.~\eqref{eq:BSE-final}, the BSE matrix elements in Eq.~\eqref{eq:LR-dyn} read \begin{subequations} \begin{align} \label{eq:BSE-Adyn} - \A{ia,jb}{}(\Om{s}{}) & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + 2 \sigma \ERI{ia}{jb} - \tW{ij,ab}{}(\Om{s}{}), + \A{ia,jb}{}(\Om{s}{}) & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \kappa \ERI{ia}{jb} - \tW{ij,ab}{}(\Om{s}{}), \\ \label{eq:BSE-Bdyn} - \B{ia,jb}{}(\Om{s}{}) & = 2 \sigma \ERI{ia}{bj} - \tW{ib,aj}{}(\Om{s}{}), + \B{ia,jb}{}(\Om{s}{}) & = \kappa \ERI{ia}{bj} - \tW{ib,aj}{}(\Om{s}{}). \end{align} \end{subequations} -where $\sigma = 1 $ or $0$ for singlet and triplet excited states (respectively). %\begin{equation} % \ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}' %\end{equation} @@ -588,16 +582,16 @@ Now, let us decompose, using basic perturbation theory, the non-linear eigenprob \begin{pmatrix} \bA{(1)}(\Om{s}{}) & \bB{(1)}(\Om{s}{}) \\ -\bB{(1)}(\titou{-}\Om{s}{}) & -\bA{(1)}(\titou{-}\Om{s}{}) \\ - \end{pmatrix} + \end{pmatrix}, \end{multline} with \begin{subequations} \begin{align} \label{eq:BSE-A0} - \A{ia,jb}{(0)} & = \delta_{ij} \delta_{ab} \eGW{ia} + 2 \ERI{ia}{jb} - \W{ij,ab}{\text{stat}}, + \A{ia,jb}{(0)} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \kappa \ERI{ia}{jb} - \W{ij,ab}{\text{stat}}, \\ \label{eq:BSE-B0} - \B{ia,jb}{(0)} & = 2 \ERI{ia}{bj} - \W{ib,aj}{\text{stat}}. + \B{ia,jb}{(0)} & = \kappa \ERI{ia}{bj} - \W{ib,aj}{\text{stat}}. \end{align} \end{subequations} and @@ -615,7 +609,7 @@ where we have defined the static version of the screened Coulomb potential \label{eq:Wstat} \W{ij,ab}{\text{stat}} = W_{ij,ab}(\omega = 0) = \ERI{ij}{ab} - 4 \sum_m \frac{\sERI{ij}{m} \sERI{ab}{m}}{\OmRPA{m}{} - i \eta}. \end{equation} -According to perturbation theory, the $s$th BSE excitation energy and its corresponding eigenvector can then decomposed as +According to perturbation theory, the $s$th BSE excitation energy and its corresponding eigenvector can then expanded as \begin{subequations} \begin{gather} \Om{s}{} = \Om{s}{(0)} + \Om{s}{(1)} + \ldots, @@ -637,7 +631,7 @@ According to perturbation theory, the $s$th BSE excitation energy and its corres + \ldots. \end{gather} \end{subequations} -Solving the zeroth-order static problem yields +Solving the zeroth-order static problem \begin{equation} \label{eq:LR-BSE-stat} \begin{pmatrix} @@ -656,7 +650,8 @@ Solving the zeroth-order static problem yields \bY{s}{(0)} \\ \end{pmatrix}, \end{equation} -and, thanks to first-order perturbation theory, the first-order correction to the $s$th excitation energy is +yields the zeroth-order (static) $\Om{s}{(0)}$ excitation energies and their corresponding eigenvectors $\bX{s}{(0)}$ and $\bY{s}{(0)}$. +Thanks to first-order perturbation theory, the first-order correction to the $s$th excitation energy is \begin{equation} \label{eq:Om1} \Om{s}{(1)} = @@ -675,12 +670,12 @@ and, thanks to first-order perturbation theory, the first-order correction to th \bY{s}{(0)} \\ \end{pmatrix}. \end{equation} -From a practical point of view, if one enforces the TDA, we obtain the very simple expression +From a practical point of view, if one enforces the TDA for the dynamical correction (which we label dTDA in the following), we obtain the very simple expression \begin{equation} \label{eq:Om1-TDA} \Om{s}{(1)} = \T{(\bX{s}{(0)})} \cdot \bA{(1)}(\Om{s}{(0)}) \cdot \bX{s}{(0)}. \end{equation} -This correction can be renormalized by computing, at basically no extra cost, the renormalization factor which reads, in the TDA, +This correction can be renormalized by computing, at basically no extra cost, the renormalization factor which reads, in the dTDA, \begin{equation} \label{eq:Z} Z_{s} = \qty[ 1 - \T{(\bX{s}{(0)})} \cdot \left. \pdv{\bA{(1)}(\Om{s}{})}{\Om{s}{}} \right|_{\Om{s}{} = \Om{s}{(0)}} \cdot \bX{s}{(0)} ]^{-1}. @@ -705,7 +700,7 @@ In terms of computational cost, if one decides to compute the dynamical correcti These are then used to compute the first-order correction from Eq.~\eqref{eq:Om1} or Eq.~\eqref{eq:Om1-TDA}, which also require to construct and evaluate the dynamical part of the BSE Hamiltonian for each excitation one wants to dynamically correct. The static BSE Hamiltonian is computed once during the static BSE calculation and does not dependent on the targeted excitation. -Searching iteratively for the lowest eigenstates, via the Davidson algorithm for instance, can be performed in $\order*{\Norb^4}$ computational cost. +Searching iteratively for the lowest eigenstates, via Davidson's algorithm for instance, can be performed in $\order*{\Norb^4}$ computational cost. Constructing the static and dynamic BSE Hamiltonians is much more expensive as it requires the complete diagonalization of the $(\Nocc \Nvir \times \Nocc \Nvir)$ RPA linear response matrix [see Eq.~\eqref{eq:LR-RPA}], which corresponds to a $\order*{\Nocc^3 \Nvir^3} = \order*{\Norb^6}$ computational cost. Although it might be reduced to $\order*{\Norb^4}$ operations with standard resolution-of-the-identity techniques, \cite{Duchemin_2019,Duchemin_2020} this step is the computational bottleneck in our current implementation. @@ -717,8 +712,7 @@ All systems under investigation have close-shell singlet ground states. We then adopt a restricted formalism throughout this work. The $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies are done using a (restricted) HF starting point. Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} quasiparticle energies are employed as starting points to compute the BSE neutral excitations. -For both {\GOWO} and {\evGW}, the entire set of orbitals are corrected. -In the case of {\GOWO}, the quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation. +These quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation, and the entire set of orbitals is corrected. Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018}. As one-electron basis sets, we employ the augmented Dunning family (aug-cc-pVXZ) defined with cartesian Gaussian functions. Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations. @@ -743,9 +737,9 @@ All the BSE calculations have been performed with our locally developed $GW$ sof & \mc{3}{c}{aug-cc-pVTZ ($\Eg^{\GW} = 19.20$ eV)} & \mc{3}{c}{aug-cc-pVQZ ($\Eg^{\GW} = 19.00$ eV)} \\ \cline{2-4} \cline{5-7} \cline{8-10} - State & \tabc{$\Om{m}{\stat}$} & \tabc{$\Delta\Om{m}{\dyn}$(TDA)} & \tabc{$\Delta\Om{m}{\dyn}$} - & \tabc{$\Om{m}{\stat}$} & \tabc{$\Delta\Om{m}{\dyn}$(TDA)} & \tabc{$\Delta\Om{m}{\dyn}$} - & \tabc{$\Om{m}{\stat}$} & \tabc{$\Delta\Om{m}{\dyn}$(TDA)} & \tabc{$\Delta\Om{m}{\dyn}$} \\ + State & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$} + & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$} + & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$} \\ \hline $^1\Pi_g(n \ra \pis)$ & 10.18 & -0.41 & -0.43 & 10.42 & -0.42 & -0.40 & 10.52 & -0.43 & -0.40 \\ $^1\Sigma_u^-(\pi \ra \pis)$ & 9.95 & -0.44 & -0.44 & 10.11 & -0.45 & -0.45 & 10.20 & -0.45 & -0.45 \\ @@ -765,7 +759,7 @@ All the BSE calculations have been performed with our locally developed $GW$ sof %%% TABLE I %%% -\begin{squeezetable} +%\begin{squeezetable} \begin{table*} \caption{ Singlet excitation energies (in eV) for various molecules obtained with the aug-cc-pVTZ basis set at various levels of theory. @@ -774,36 +768,34 @@ All the BSE calculations have been performed with our locally developed $GW$ sof \label{tab:BigTabSi} } \begin{ruledtabular} - \begin{tabular}{llddddddddddddddd} - & & \mc{5}{c}{BSE@{\GOWO}@HF} & \mc{5}{c}{Wave function-based methods} & \mc{5}{c}{Density-based methods} \\ - \cline{3-7} \cline{8-12} \cline{13-17} - Mol. & State & \tabc{$\Eg^{\GW}$} & \tabc{$\Om{m}{\stat}$} & \tabc{$\Om{m}{\dyn}$} & \tabc{$\Delta\Om{m}{\dyn}$} & \tabc{$Z_{m}$} - & \tabc{CIS(D)} & \tabc{ADC(2)} & \tabc{CCSD} & \tabc{CC2} & \tabc{CC3} - & \tabc{B3LYP} & \tabc{PBE0} & \tabc{M06-2X} & \tabc{CAM-B3LYP} & \tabc{LC-$\omega$HPBE} \\ + \begin{tabular}{lldddddddddd} + & & \mc{5}{c}{BSE@{\GOWO}@HF} & \mc{5}{c}{Wave function-based methods} \\ %& \mc{5}{c}{Density-based methods} \\ + \cline{3-7} \cline{8-12} %\cline{13-17} + Mol. & State & \tabc{$\Eg^{\GW}$} & \tabc{$\Om{s}{\stat}$} & \tabc{$\Om{m}{\dyn}$} & \tabc{$\Delta\Om{s}{\dyn}$} & \tabc{$Z_{s}$} + & \tabc{CIS(D)} & \tabc{ADC(2)} & \tabc{CCSD} & \tabc{CC2} & \tabc{CC3} \\ +% & \tabc{B3LYP} & \tabc{PBE0} & \tabc{M06-2X} & \tabc{CAM-B3LYP} & \tabc{LC-$\omega$HPBE} \\ \hline \ce{HCl} & $^1\Pi$(CT) & 13.43 & 8.30 & 8.19 & -0.11 & 1.009 - & 6.07 & 7.97 & 7.91 & 7.96 & 7.84 - & 7.33 & 7.59 & 7.56 & 7.52 & 7.96 \\ + & 6.07 & 7.97 & 7.91 & 7.96 & 7.84 \\ +% & 7.33 & 7.59 & 7.56 & 7.52 & 7.96 \\ \\ \ce{H2O} & $^1B_1(n \ra 3s)$ & 13.58 & 8.09 & 8.00 & -0.09 & 1.007 - & 7.62 & 7.18 & 7.60 & 7.23 & 7.65 - & 6.92 & 7.18 & 7.46 & 7.13 & 7.50 \\ + & 7.62 & 7.18 & 7.60 & 7.23 & 7.65 \\ +% & 6.92 & 7.18 & 7.46 & 7.13 & 7.50 \\ & $^1A_2(n \ra 3p)$ & & 9.79 & 9.72 & -0.07 & 1.005 - & 9.41 & 8.84 & 9.36 & 8.89 & 9.43 - & 8.33 & 8.61 & 8.93 & 8.69 & 9.11 \\ + & 9.41 & 8.84 & 9.36 & 8.89 & 9.43 \\ +% & 8.33 & 8.61 & 8.93 & 8.69 & 9.11 \\ & $^1A_1(n \ra 3s)$ & & 10.42 & 10.35 & -0.07 & 1.006 - & 9.99 & 9.52 & 9.96 & 9.58 & 10.00 - & 9.08 & 9.37 & 9.64 & 9.28 & 9.65 \\ + & 9.99 & 9.52 & 9.96 & 9.58 & 10.00 \\ +% & 9.08 & 9.37 & 9.64 & 9.28 & 9.65 \\ \\ \ce{N2} & $^1\Pi_g(n \ra \pis)$ & 19.20 & 10.42 & 9.99 & -0.42 & 1.031 - & 9.66 & 9.48 & 9.41 & 9.44 & 9.34 - & 9.23 & \\ + & 9.66 & 9.48 & 9.41 & 9.44 & 9.34 \\ +% & 9.23 & \\ & $^1\Sigma_u^-(\pi \ra \pis)$ & & 10.11 & 9.66 & -0.45 & 1.029 - & 10.31 & 10.26 & 10.00 & 10.32 & 9.88 - & & \\ + & 10.31 & 10.26 & 10.00 & 10.32 & 9.88 \\ & $^1\Delta_u(\pi \ra \pis)$ & & 10.75 & 10.33 & -0.42 & 1.030 - & 10.85 & 10.79 & 10.44 & 10.86 & 10.29 - & \\ + & 10.85 & 10.79 & 10.44 & 10.86 & 10.29 \\ & $^1\Sigma_g^+$(R) & & 13.60 & 13.57 & -0.03 & 1.003 & 13.67 & 12.99 & 13.15 & 12.83 & 13.01 \\ & $^1\Pi_u$(R) & & 13.98 & 13.94 & -0.04 & 1.004 @@ -821,22 +813,22 @@ All the BSE calculations have been performed with our locally developed $GW$ sof & $^1\Pi$(R) & & 12.37 & 12.32 & -0.05 & 1.004 & 12.06 & 12.03 & 11.96 & 11.83 & 11.69 \\ \\ \ce{HNO} & $^1A''(n \ra \pis)$ & 11.71 & 2.46 & 1.98 & -0.48 & 1.035 - & 1.80 & 1.68 & 1.76 & 1.74 & 1.75 - & 1.55 & 1.51 & 0.99 & 1.51 & 1.46 \\ + & 1.80 & 1.68 & 1.76 & 1.74 & 1.75 \\ +% & 1.55 & 1.51 & 0.99 & 1.51 & 1.46 \\ & $^1A'$(R) & & 7.05 & 7.01 & -0.04 & 1.003 - & 5.81 & 5.73 & 6.30 & 5.72 & 6.26 - & 5.63 & 5.85 & 6.22 & 5.94 & 6.33 \\ + & 5.81 & 5.73 & 6.30 & 5.72 & 6.26 \\ +% & 5.63 & 5.85 & 6.22 & 5.94 & 6.33 \\ \\ %T2: check state ordering in BSE calculation \ce{C2H4} & $^1B_{3u}(\pi \ra 3s)$ & 11.49 & 7.64 & 7.62 & -0.03 & 1.004 - & 7.35 & 7.34 & 7.42 & 7.29 & 7.35 - & 6.63 & 6.88 & 6.94 & 6.93 & 7.57 \\ + & 7.35 & 7.34 & 7.42 & 7.29 & 7.35 \\ +% & 6.63 & 6.88 & 6.94 & 6.93 & 7.57 \\ & $^1B_{1u}(\pi \ra \pis)$ & & 8.18 & 8.03 & -0.15 & 1.022 - & 7.95 & 7.91 & 8.02 & 7.92 & 7.91 - & 8.06 & 7.51 & 7.50 & 7.46 & 7.64 \\ + & 7.95 & 7.91 & 8.02 & 7.92 & 7.91 \\ +% & 8.06 & 7.51 & 7.50 & 7.46 & 7.64 \\ & $^1B_{1g}(\pi \ra 3p)$ & & 8.29 & 8.26 & -0.03 & 1.003 - & 8.01 & 7.99 & 8.08 & 7.95 & 8.03 - & 7.18 & 7.45 & 7.47 & 7.54 & 8.15 \\ + & 8.01 & 7.99 & 8.08 & 7.95 & 8.03 \\ +% & 7.18 & 7.45 & 7.47 & 7.54 & 8.15 \\ \\ \ce{CH2O} & $^1A_2(n \ra \pis)$ & 12.00 & 5.03 & 4.68 & -0.35 & 1.027 & 4.04 & 3.92 & 4.01 & 4.07 & 3.97 \\ & $^1B_2(n \ra 3s)$ & & 7.87 & 7.85 & -0.02 & 1.001 & 6.64 & 6.50 & 7.23 & 6.56 & 7.18 \\ @@ -848,10 +840,10 @@ All the BSE calculations have been performed with our locally developed $GW$ sof \end{tabular} \end{ruledtabular} \end{table*} -\end{squeezetable} +%\end{squeezetable} %%% TABLE II %%% -\begin{squeezetable} +%\begin{squeezetable} \begin{table*} \caption{ Triplet excitation energies (in eV) for various molecules obtained with the aug-cc-pVTZ basis set at various levels of theory. @@ -859,22 +851,22 @@ All the BSE calculations have been performed with our locally developed $GW$ sof \label{tab:BigTabTr} } \begin{ruledtabular} - \begin{tabular}{llddddddddddddddd} - & & \mc{5}{c}{BSE@{\GOWO}@HF} & \mc{5}{c}{Wave function-based methods} & \mc{5}{c}{Density-based methods} \\ - \cline{3-7} \cline{8-12} \cline{13-17} + \begin{tabular}{lldddddddddd} + & & \mc{5}{c}{BSE@{\GOWO}@HF} & \mc{5}{c}{Wave function-based methods} \\%& \mc{5}{c}{Density-based methods} \\ + \cline{3-7} \cline{8-12} %\cline{13-17} Mol. & State & \tabc{$\Eg^{\GW}$} & \tabc{$\Om{m}{\stat}$} & \tabc{$\Om{m}{\dyn}$} & \tabc{$\Delta\Om{m}{\dyn}$} & \tabc{$Z_{m}$} - & \tabc{CIS(D)} & \tabc{ADC(2)} & \tabc{CCSD} & \tabc{CC2} & \tabc{CC3} - & \tabc{B3LYP} & \tabc{PBE0} & \tabc{M06-2X} & \tabc{CAM-B3LYP} & \tabc{LC-$\omega$HPBE} \\ + & \tabc{CIS(D)} & \tabc{ADC(2)} & \tabc{CCSD} & \tabc{CC2} & \tabc{CC3} \\ +% & \tabc{B3LYP} & \tabc{PBE0} & \tabc{M06-2X} & \tabc{CAM-B3LYP} & \tabc{LC-$\omega$HPBE} \\ \hline \ce{H2O} & $^3B_1(n \ra 3s)$ & 13.58 & 8.14 & 7.98 & -0.15 & 1.014 - & 7.25 & 6.86 & 7.20 & 6.91 & 7.28 - & 6.55 & 6.75 & 7.12 & 6.72 & 7.04 \\ + & 7.25 & 6.86 & 7.20 & 6.91 & 7.28 \\ +% & 6.55 & 6.75 & 7.12 & 6.72 & 7.04 \\ & $^3A_2(n \ra 3p)$ & & 9.97 & 9.89 & -0.07 & 1.008 - & 9.24 & 8.72 & 9.20 & 8.77 & 9.26 - & 8.22 & 8.45 & 8.77 & 8.54 & 8.92 \\ + & 9.24 & 8.72 & 9.20 & 8.77 & 9.26 \\ +% & 8.22 & 8.45 & 8.77 & 8.54 & 8.92 \\ & $^3A_1(n \ra 3s)$ & & 10.28 & 10.13 & -0.15 & 1.012 - & 9.54 & 9.15 & 9.49 & 9.20 & 9.56 - & 8.60 & 8.82 & 9.24 & 8.79 & 9.11 \\ + & 9.54 & 9.15 & 9.49 & 9.20 & 9.56 \\ +% & 8.60 & 8.82 & 9.24 & 8.79 & 9.11 \\ \\ \ce{N2} & $^3\Sigma_u^+(\pi \ra \pis)$ & 19.20 & 9.50 & 8.46 & -1.04 & 1.060 & 8.20 & 8.15 & 7.66 & 8.19 & 7.68 \\ & $^3\Pi_g(n \ra \pis)$ & & 9.85 & 9.27 & -0.58 & 1.050 & 8.33 & 8.20 & 8.09 & 8.19 & 8.04 \\ @@ -888,21 +880,21 @@ All the BSE calculations have been performed with our locally developed $GW$ sof & $^3\Sigma_u^+$(R) & & 11.48 & 11.38 & -0.10 & 1.010 & 10.98 & 10.83 & 10.71 & 10.60 & 10.45 \\ \\ \ce{HNO} & $^3A''(n \ra \pis)$ & 11.71 & 3.05 & 2.35 & -0.71 & 1.069 - & 0.91 & 0.78 & 0.85 & 0.84 & 0.88 - & -0.47 & -0.61 & 0.36 & -0.49 & -0.58 \\ + & 0.91 & 0.78 & 0.85 & 0.84 & 0.88 \\ +% & -0.47 & -0.61 & 0.36 & -0.49 & -0.58 \\ & $^3A'(\pi \ra \pis)$ & & 6.69 & 6.70 & 0.01 & 1.000 - & 5.72 & 5.46 & 5.49 & 5.44 & 5.59 - & 4.73 & 4.46 & 5.27 & 4.55 & 4.57 \\ + & 5.72 & 5.46 & 5.49 & 5.44 & 5.59 \\ +% & 4.73 & 4.46 & 5.27 & 4.55 & 4.57 \\ \\ \ce{C2H4} & $^3B_{1u}(\pi \ra \pis)$ & 11.49 & 6.54 & 5.85 & -0.69 & 1.065 - & 4.62 & 4.59 & 4.46 & 4.59 & 4.53 - & 4.07 & 3.84 & 4.54 & 3.92 & 3.55 \\ + & 4.62 & 4.59 & 4.46 & 4.59 & 4.53 \\ +% & 4.07 & 3.84 & 4.54 & 3.92 & 3.55 \\ & $^3B_{3u}(\pi \ra 3s)$ & & 7.61 & 7.55 & -0.06 & 1.008 - & 7.26 & 7.23 & 7.29 & 7.19 & 7.24 - & 6.54 & 6.74 & 6.90 & 6.83 & 7.41 \\ + & 7.26 & 7.23 & 7.29 & 7.19 & 7.24 \\ +% & 6.54 & 6.74 & 6.90 & 6.83 & 7.41 \\ & $^3B_{1g}(\pi \ra 3p)$ & & 8.34 & 8.31 & -0.03 & 1.003 - & 7.97 & 7.95 & 8.03 & 7.91 & 7.98 - & 7.14 & 7.34 & 7.46 & 7.45 & 7.53 \\ + & 7.97 & 7.95 & 8.03 & 7.91 & 7.98 \\ +% & 7.14 & 7.34 & 7.46 & 7.45 & 7.53 \\ \\ \ce{CH2O} & $^3A_2(n \ra \pis)$ & 12.00 & 5.53 & 5.05 & -0.47 & 1.049 & 3.58 & 3.46 & 3.56 & 3.59 & 3.57 \\ & $^3A_1(\pi \ra \pis)$ & & 8.15 & 7.32 & -0.83 & 1.067 & 6.27 & 6.20 & 5.97 & 6.30 & 6.05 \\ @@ -913,7 +905,7 @@ All the BSE calculations have been performed with our locally developed $GW$ sof \end{tabular} \end{ruledtabular} \end{table*} -\end{squeezetable} +%\end{squeezetable} %%% TABLE III %%% %\begin{table}