1st sweep corrected lots of mistakes
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BSEdyn.tex
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BSEdyn.tex
@ -326,7 +326,11 @@ is the BSE kernel that takes into account the self-consistent variation of the H
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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.
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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 $\ket{N,0} \equiv \ket{N}$).
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In the optical limit of instantaneous electron-hole creation and destruction, imposing $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, one gets
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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,
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\begin{equation}
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\hpsi(1) = e^{ i \hH t_1 } \hpsi(\bx_1) e^{-i \hH t_1 },
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\end{equation}
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($\hH$ being the exact many-body Hamiltonian), one gets
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\begin{equation}
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\begin{split}
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iL(1,2; 1',2')
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@ -343,13 +347,9 @@ where $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
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\tchi_s(\bx_1,\bx_{2}) & = \mel{N,s}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{2})] }{N}.
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\end{align}
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\end{subequations}
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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.
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$$
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\hpsi(1) = e^{ i {\hat H} t_1 } \hpsi(\bx_1) e^{-i {\hat H} t_1 }
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$$
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with $\hat H$ the exact many-body Hamiltonian.
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The $\Om{s}{}$'s are the neutral excitation energies of interest.
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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
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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
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\begin{multline} \label{eq:BSE_2}
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\mel{N}{T [ \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}') ] } {N,s} e^{ - i \Om{s}{} t_1 }
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\theta ( \tau_{12} )
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@ -373,15 +373,9 @@ We now adopt the Lehman representation of the one-body Green's function in the q
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\begin{equation} \label{eq:G-Lehman}
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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) },
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\end{equation}
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where $\mu$ is the chemical potential.
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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.
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%where the $\eps_{p}$'s are proper addition/removal energies such that
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%\begin{equation}
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% e^{i \hH \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N},
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%\end{equation}
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%$\hH$ being the exact many-body Hamiltonian.
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In the following, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
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%\titou{namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. (T2: shall we really mention this here?)}
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where $\eta$ is a positive infinitesimal and $\mu$ is the chemical potential.
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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.
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In the following, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$ and $q$ indicate arbitrary orbitals.
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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
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\begin{multline} \label{eq:iL0bis}
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\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}{})
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@ -401,7 +395,7 @@ For example, we have
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= - \qty( e^{ -i \Omega_s t^{65} } ) \sum_{pq} \MO{p}(\bx_6) \MO{q}^*(\bx_5)
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\mel{N}{\ha_q^{\dagger} \ha_p}{N,s}
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\\
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\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} } ]
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\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} } ],
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\end{multline}
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with $t^{65} = (t_5 + t_6)/2$ and $\tau_{65} = t_6 -t_5$.
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%with a similar expression for $\mel{N}{T [\hpsi(\bx_3) \hpsi^{\dagger}(\bx_4)] }{N,s}$.
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@ -426,20 +420,20 @@ The $GW$ quasiparticle energies $\eGW{p}$ are good approximations to the removal
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%Neglecting the $Y_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA).
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%Working out similar expressions for $\mel{N}{T [\hpsi(5) \hpsi^{\dagger}(5)] }{N,s}$ and $\mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})] }{N,s}$,
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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):
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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):
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\begin{equation} \label{eq:BSE-final}
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\begin{split}
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( \eGW{a} - \eGW{i} - \Om{s}{} ) X_{ia}^{s}
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& + \sum_{jb} \qty[ \ERI{ia}{jb} - \widetilde{W}_{ij,ab}(\Om{s}{}) ] X_{jb}^{s} \\
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& + \sum_{jb} \qty[ \ERI{ia}{bj} - \widetilde{W}_{ib,aj}(\Om{s}{}) ] Y_{jb}^{s}
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( \eGW{a} - \eGW{i} - \Om{s}{} ) X_{ia,s}
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& + \sum_{jb} \qty[ \kappa \ERI{ia}{jb} - \widetilde{W}_{ij,ab}(\Om{s}{}) ] X_{jb,s} \\
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& + \sum_{jb} \qty[ \kappa \ERI{ia}{bj} - \widetilde{W}_{ib,aj}(\Om{s}{}) ] Y_{jb,s}
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= 0,
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\end{split}
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\end{equation}
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with $X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$ and $Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$.
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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).
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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).
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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).
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In Eq.~\eqref{eq:BSE-final},
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\begin{equation}
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\ERI{pq}{rs} = \iint d\br d\br' \, \MO{p}^*(\br) \MO{q}(\br) v(\br -\br') \MO{r}^*(\br') \MO{s}(\br'),
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\ERI{ia}{jb} = \iint d\br d\br' \, \MO{i}^*(\br) \MO{a}(\br) v(\br -\br') \MO{j}^*(\br') \MO{b}(\br'),
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\end{equation}
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are the bare two-electron integrals in the spatial orbital basis $\lbrace \MO{p}(\br{}) \rbrace$, and
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\begin{multline} \label{eq:wtilde}
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@ -450,8 +444,8 @@ are the bare two-electron integrals in the spatial orbital basis $\lbrace \MO{p}
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\end{multline}
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is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b} where $\Om{pq}{s} = \Om{s}{} - ( \eGW{q} - \eGW{p} )$ and
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\begin{equation}
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W_{pq,rs}({\omega})
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= \iint d\br d\br' \, \MO{p}(\br) \MO{q}^*(\br) W(\br ,\br'; \omega) \MO{r}^*(\br') \MO{s}(\br').
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W_{ij,ab}({\omega})
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= \iint d\br d\br' \, \MO{i}(\br) \MO{j}^*(\br) W(\br ,\br'; \omega) \MO{a}^*(\br') \MO{b}(\br').
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\end{equation}
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\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 : }
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@ -469,7 +463,7 @@ In the present study, we consider the exact spectral representation of $W(\omega
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\\
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\times \qty[ \frac{1}{ \omega-\Om{m}{\RPA} + i\eta } - \frac{1}{ \omega + \Om{m}{\RPA} - i\eta } ],
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\end{multline}
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where
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where $m$ labels single excitations, and
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\begin{equation}
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\label{eq:sERI}
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\sERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} (\bX{m}{\RPA} + \bY{m}{\RPA})_{ia}
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@ -523,7 +517,7 @@ As a consequence, we observe a reduction of the electron-hole screening, \ie, an
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\subsection{Perturbative dynamical correction}
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%=================================
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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}
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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}
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\begin{equation}
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\label{eq:LR-dyn}
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\begin{pmatrix}
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@ -544,20 +538,20 @@ For a closed-shell system in a finite basis, to compute the BSE excitation energ
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\bY{s}{} \\
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\end{pmatrix},
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\end{equation}
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where the dynamical matrices $\bA{}$ and $\bB{}$, as well as $\bX{s}{}$, and $\bY{s}{}$, have the same size as their RPA counterparts.
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where the dynamical matrices $\bA{}$ and $\bB{}$ have the same $\Nocc \Nvir \times \Nocc \Nvir$ size than their RPA counterparts.
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Same comment applies to the eigenvectors $\bX{s}{}$, and $\bY{s}{}$ of length $\Nocc \Nvir$.
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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}
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The BSE matrix elements read
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Accordingly to Eq.~\eqref{eq:BSE-final}, the BSE matrix elements in Eq.~\eqref{eq:LR-dyn} read
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\begin{subequations}
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\begin{align}
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\label{eq:BSE-Adyn}
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\A{ia,jb}{}(\Om{s}{}) & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + 2 \sigma \ERI{ia}{jb} - \tW{ij,ab}{}(\Om{s}{}),
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\A{ia,jb}{}(\Om{s}{}) & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \kappa \ERI{ia}{jb} - \tW{ij,ab}{}(\Om{s}{}),
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\\
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\label{eq:BSE-Bdyn}
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\B{ia,jb}{}(\Om{s}{}) & = 2 \sigma \ERI{ia}{bj} - \tW{ib,aj}{}(\Om{s}{}),
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\B{ia,jb}{}(\Om{s}{}) & = \kappa \ERI{ia}{bj} - \tW{ib,aj}{}(\Om{s}{}).
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\end{align}
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\end{subequations}
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where $\sigma = 1 $ or $0$ for singlet and triplet excited states (respectively).
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%\begin{equation}
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% \ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}'
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%\end{equation}
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@ -588,16 +582,16 @@ Now, let us decompose, using basic perturbation theory, the non-linear eigenprob
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\begin{pmatrix}
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\bA{(1)}(\Om{s}{}) & \bB{(1)}(\Om{s}{}) \\
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-\bB{(1)}(\titou{-}\Om{s}{}) & -\bA{(1)}(\titou{-}\Om{s}{}) \\
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\end{pmatrix}
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\end{pmatrix},
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\end{multline}
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with
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\begin{subequations}
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\begin{align}
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\label{eq:BSE-A0}
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\A{ia,jb}{(0)} & = \delta_{ij} \delta_{ab} \eGW{ia} + 2 \ERI{ia}{jb} - \W{ij,ab}{\text{stat}},
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\A{ia,jb}{(0)} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \kappa \ERI{ia}{jb} - \W{ij,ab}{\text{stat}},
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\\
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\label{eq:BSE-B0}
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\B{ia,jb}{(0)} & = 2 \ERI{ia}{bj} - \W{ib,aj}{\text{stat}}.
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\B{ia,jb}{(0)} & = \kappa \ERI{ia}{bj} - \W{ib,aj}{\text{stat}}.
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\end{align}
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\end{subequations}
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and
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@ -615,7 +609,7 @@ where we have defined the static version of the screened Coulomb potential
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\label{eq:Wstat}
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\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}.
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\end{equation}
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According to perturbation theory, the $s$th BSE excitation energy and its corresponding eigenvector can then decomposed as
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According to perturbation theory, the $s$th BSE excitation energy and its corresponding eigenvector can then expanded as
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\begin{subequations}
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\begin{gather}
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\Om{s}{} = \Om{s}{(0)} + \Om{s}{(1)} + \ldots,
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@ -637,7 +631,7 @@ According to perturbation theory, the $s$th BSE excitation energy and its corres
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+ \ldots.
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\end{gather}
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\end{subequations}
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Solving the zeroth-order static problem yields
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Solving the zeroth-order static problem
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\begin{equation}
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\label{eq:LR-BSE-stat}
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\begin{pmatrix}
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@ -656,7 +650,8 @@ Solving the zeroth-order static problem yields
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\bY{s}{(0)} \\
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\end{pmatrix},
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\end{equation}
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and, thanks to first-order perturbation theory, the first-order correction to the $s$th excitation energy is
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yields the zeroth-order (static) $\Om{s}{(0)}$ excitation energies and their corresponding eigenvectors $\bX{s}{(0)}$ and $\bY{s}{(0)}$.
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Thanks to first-order perturbation theory, the first-order correction to the $s$th excitation energy is
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\begin{equation}
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\label{eq:Om1}
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\Om{s}{(1)} =
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@ -675,12 +670,12 @@ and, thanks to first-order perturbation theory, the first-order correction to th
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\bY{s}{(0)} \\
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\end{pmatrix}.
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\end{equation}
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From a practical point of view, if one enforces the TDA, we obtain the very simple expression
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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
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\begin{equation}
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\label{eq:Om1-TDA}
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\Om{s}{(1)} = \T{(\bX{s}{(0)})} \cdot \bA{(1)}(\Om{s}{(0)}) \cdot \bX{s}{(0)}.
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\end{equation}
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This correction can be renormalized by computing, at basically no extra cost, the renormalization factor which reads, in the TDA,
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This correction can be renormalized by computing, at basically no extra cost, the renormalization factor which reads, in the dTDA,
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\begin{equation}
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\label{eq:Z}
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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}.
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@ -705,7 +700,7 @@ In terms of computational cost, if one decides to compute the dynamical correcti
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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.
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The static BSE Hamiltonian is computed once during the static BSE calculation and does not dependent on the targeted excitation.
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Searching iteratively for the lowest eigenstates, via the Davidson algorithm for instance, can be performed in $\order*{\Norb^4}$ computational cost.
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Searching iteratively for the lowest eigenstates, via Davidson's algorithm for instance, can be performed in $\order*{\Norb^4}$ computational cost.
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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.
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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.
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@ -717,8 +712,7 @@ All systems under investigation have close-shell singlet ground states.
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We then adopt a restricted formalism throughout this work.
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The $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies are done using a (restricted) HF starting point.
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Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} quasiparticle energies are employed as starting points to compute the BSE neutral excitations.
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For both {\GOWO} and {\evGW}, the entire set of orbitals are corrected.
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In the case of {\GOWO}, the quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation.
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These quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation, and the entire set of orbitals is corrected.
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Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018}.
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As one-electron basis sets, we employ the augmented Dunning family (aug-cc-pVXZ) defined with cartesian Gaussian functions.
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Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
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@ -743,9 +737,9 @@ All the BSE calculations have been performed with our locally developed $GW$ sof
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& \mc{3}{c}{aug-cc-pVTZ ($\Eg^{\GW} = 19.20$ eV)}
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& \mc{3}{c}{aug-cc-pVQZ ($\Eg^{\GW} = 19.00$ eV)} \\
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\cline{2-4} \cline{5-7} \cline{8-10}
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State & \tabc{$\Om{m}{\stat}$} & \tabc{$\Delta\Om{m}{\dyn}$(TDA)} & \tabc{$\Delta\Om{m}{\dyn}$}
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& \tabc{$\Om{m}{\stat}$} & \tabc{$\Delta\Om{m}{\dyn}$(TDA)} & \tabc{$\Delta\Om{m}{\dyn}$}
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& \tabc{$\Om{m}{\stat}$} & \tabc{$\Delta\Om{m}{\dyn}$(TDA)} & \tabc{$\Delta\Om{m}{\dyn}$} \\
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State & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
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& \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
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& \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}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user