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BSE-PES.tex
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@ -182,10 +182,10 @@
\affiliation{\NEEL}
\begin{abstract}
The combination of the many-body Green's function $GW$ and the Bethe-Salpeter equation (BSE) formalisms has shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) for computing vertical transition energies and oscillator strengths in molecular systems.
The combination of the many-body Green's function $GW$ approximation and the Bethe-Salpeter equation (BSE) formalism has shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) for computing vertical transition energies and oscillator strengths in molecular systems.
The BSE formalism can also be employed to compute ground-state correlation energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT).
Here, we study the topology of the ground-state potential energy surfaces (PES) of several diatomic molecules near their equilibrium bond length.
Thanks to comparisons with state-of-art computational approaches, we show that ACFDT@BSE is surprisingly accurate, and can even compete with coupled cluster methods in terms of total energies and equilibrium bond distances for the systems considered here.
Thanks to comparisons with state-of-art computational approaches, we show that ACFDT@BSE is surprisingly accurate, and can even compete with coupled cluster methods in terms of total energies and equilibrium bond distances for the considered systems.
However, we sometimes observe unphysical irregularities on the ground-state PES in relation with the appearance of satellite resonances with a weight similar to that of the $GW$ quasiparticle peak.\\
\bigskip
\begin{center}
@ -212,7 +212,7 @@ A significant limitation of the BSE formalism, as compared to TD-DFT, lies in th
While calculations of the $GW$ quasiparticle energy ionic gradients is becoming increasingly popular,
\cite{Lazzeri_2008,Faber_2011b,Yin_2013,Faber_2015,Montserrat_2016,Zhenglu_2019} only one pioneering study of the excited-state BSE gradients has been published so far. \cite{Beigi_2003} In this seminar work devoted to small molecules (\ce{CO} and \ce{NH3}), only the BSE excitation energy gradients were calculated, while computing the Kohn-Sham (KS) LDA forces as its ground-state contribution.
In contrast to TD-DFT which relies on KS-DFT \cite{Hohenberg_1964,Kohn_1965,ParrBook} as its ground-state counterpart, the ground-state BSE energy is not a well-defined quantity, and no clear consensus has been found regarding its formal definition.
In contrast to TD-DFT which relies on KS-DFT \cite{Hohenberg_1964,Kohn_1965,ParrBook} as its ground-state analog, the ground-state BSE energy is not a well-defined quantity, and no clear consensus has been found regarding its formal definition.
Consequently, the BSE ground-state formalism remains in its infancy with very few available studies for atomic and molecular systems. \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020}
In the largest available benchmark study \cite{Holzer_2018} encompassing the total energies of the atoms \ce{H}--\ce{Ne}, the atomization energies of the 26 small molecules forming the HEAT test set, \cite{Harding_2008} and the bond lengths and harmonic vibrational frequencies of $3d$ transition-metal monoxides, the BSE correlation energy, as evaluated within the adiabatic-connection fluctuation-dissipation theorem (ACFDT) framework, \cite{Furche_2005} was mostly discarded from the set of tested techniques due to instabilities (negative frequency modes in the BSE polarization propagator) and replaced by an approximate (RPAsX) approach where the screened-Coulomb potential matrix elements was removed from the resonant electron-hole contribution. \cite{Maggio_2016,Holzer_2018}
Such a modified BSE polarization propagator was inspired by a previous study on the homogeneous electron gas. \cite{Maggio_2016}
@ -443,7 +443,7 @@ Finally, the infinitesimal $\eta$ is set to zero for all calculations.
The numerical integration required to compute the correlation energy along the adiabatic path [see Eq.~\eqref{eq:EcBSE}] is performed with a 21-point Gauss-Legendre quadrature.
Comparison with the so-called plasmon (or trace) formula \cite{Furche_2008} at the RPA level has confirmed the excellent accuracy of this quadrature scheme over $\IS$.
For comparison purposes, we have also computed the PES at the second-order M{\o}ller-Plesset (MP2), as well as with various increasingly accurate CC methods, namely, CC2 \cite{Christiansen_1995}, CCSD, \cite{Purvis_1982} and CC3. \cite{Christiansen_1995b}
For comparison purposes, we have also computed the PES at the second-order M{\o}ller-Plesset perturbation theory (MP2), as well as with various increasingly accurate CC methods, namely, CC2 \cite{Christiansen_1995}, CCSD, \cite{Purvis_1982} and CC3. \cite{Christiansen_1995b}
These calculations have been performed with DALTON \cite{dalton} and PSI4. \cite{Psi4}
The computational cost of these methods, in their usual implementation, scale as $\order*{N^5}$, $\order*{N^5}$, $\order*{N^6}$, and $\order*{N^7}$, respectively.
As shown in Refs.~\onlinecite{Hattig_2005c,Budzak_2017}, CC3 provides extremely accurate ground-state (and excited-state) geometries, and will be taken as reference in the present study.
@ -527,28 +527,28 @@ The error (in \%) compared to the reference CC3 values are reported in square br
\begin{tabular}{lrrrrrrrr}
& \mc{8}{c}{Equilibrium bond length $\Req$ (bohr) } \\
\cline{2-9}
Method & \tabc{\ce{H2}} & \tabc{\ce{LiH}} & \tabc{\ce{LiF}} & \tabc{\ce{HCl}} & \tabc{\ce{N2}} & \tabc{\ce{CO}} & \tabc{\ce{BF}} & \tabc{\ce{F2}} \\
Method & \tabc{\ce{H2}} & \tabc{\ce{LiH}} & \tabc{\ce{LiF}} & \tabc{\ce{HCl}} & \tabc{\ce{N2}} & \tabc{\ce{CO}} & \tabc{\ce{BF}} & \tabc{\ce{F2}} \\
\hline
CC3 & 1.402 & 3.019 & 2.963 & 2.403 & 2.075 & 2.136 & 2.390 & 2.663 \\
CCSD & 1.402[$+0.00\%$] & 3.020[$+0.03\%$] & 2.953[$-0.34\%$] & 2.398[$-0.21\%$] & 2.059[$-0.77\%$] & 2.118[$-0.84\%$] & 2.380[$-0.42\%$] & 2.621[$-1.58\%$] \\
CC2 & 1.391[$-0.78\%$] & 2.989[$-0.99\%$] & 2.982[$+0.64\%$] & 2.396[$-0.29\%$] & 2.106[$+1.49\%$] & 2.156[$+0.94\%$] & 2.393[$+0.13\%$] & 2.665[$+0.08\%$] \\
MP2 & 1.391[$-0.78\%$] & 3.008[$-0.36\%$] & 2.970[$+0.24\%$] & 2.395[$-0.33\%$] & 2.091[$+0.77\%$] & 2.137[$+0.05\%$] & 2.382[$-0.33\%$] & 2.634[$-1.09\%$] \\
BSE@{\GOWO}@HF & 1.399[$-0.21\%$] & 3.017[$-0.07\%$] & (2.974)[$+0.37\%$] & \gb{(2.408)} & \gb{(2.070)} & \gb{(2.130)} & \gb{(2.383)} & (2.640)[$-0.86\%$] \\
RPA@{\GOWO}@HF & 1.382[$-1.43\%$] & 2.997[$-0.73\%$] & (2.965)[$+0.07\%$] & \gb{(2.389)} & \gb{(2.043)} & \gb{(2.110)} & \gb{(2.367)} & (2.571)[$-3.45\%$] \\
BSE@{\GOWO}@HF & 1.399[$-0.21\%$] & 3.017[$-0.07\%$] & (2.974)[$+0.37\%$] & \gb{(2.408)} & 2.065[$-0.48\%$] & \gb{(2.130)} & \gb{(2.383)} & (2.640)[$-0.86\%$] \\
RPA@{\GOWO}@HF & 1.382[$-1.43\%$] & 2.997[$-0.73\%$] & (2.965)[$+0.07\%$] & \gb{(2.389)} & 2.043[$-1.54\%$] & \gb{(2.110)} & \gb{(2.367)} & (2.571)[$-3.45\%$] \\
RPAx@HF & 1.394[$-0.57\%$] & 3.011[$-0.26\%$] & 2.944[$-0.64\%$] & 2.391[$-0.50\%$] & 2.041[$-1.64\%$] & 2.104[$-1.50\%$] & 2.366[$-1.00\%$] & 2.565[$-3.68\%$] \\
RPA@HF & 1.386[$-1.14\%$] & 2.994[$-0.83\%$] & 2.946[$-0.57\%$] & 2.382[$-0.87\%$] & 2.042[$-1.59\%$] & 2.103[$-1.54\%$] & 2.364[$-1.09\%$] & 2.573[$-3.38\%$] \\
\\
\hline
& \mc{8}{c}{Correlation energy $-\Ec$ (millihartree)} \\
\cline{2-9}
Method & \tabc{\ce{H2}} & \tabc{\ce{LiH}} & \tabc{\ce{LiF}} & \tabc{\ce{HCl}} & \tabc{\ce{N2}} & \tabc{\ce{CO}} & \tabc{\ce{BF}} & \tabc{\ce{F2}} \\
Method & \tabc{\ce{H2}} & \tabc{\ce{LiH}} & \tabc{\ce{LiF}} & \tabc{\ce{HCl}} & \tabc{\ce{N2}} & \tabc{\ce{CO}} & \tabc{\ce{BF}} & \tabc{\ce{F2}} \\
\hline
CC3 & 40.382 & 69.974 & 383.686 & 382.188 & 494.393 & 477.580 & 447.472 & 668.875 \\
CCSD & 40.382[$+0.00\%$] & 69.845[$-0.18\%$] & 372.580[$-2.89\%$] & 370.764[$-2.99\%$] & 470.627[$-4.81\%$] & 455.214[$-4.68\%$] & 432.856[$-3.27\%$] & 644.001[$-3.72\%$] \\
CC2 & 33.259[$-17.64\%$] & 57.289[$-18.13\%$] & 376.712[$-1.82\%$] & 356.909[$-6.61\%$] & 488.017[$-1.29\%$] & 465.492[$-2.53\%$] & 427.285[$-4.51\%$] & 654.878[$-2.09\%$] \\
MP2 & 33.164[$-17.87\%$] & 57.930[$-17.21\%$] & 372.997[$-2.79\%$] & 355.652[$-6.94\%$] & 477.951[$-3.33\%$] & 455.020[$-4.72\%$] & 421.600[$-5.78\%$] & 644.349[$-3.67\%$] \\
BSE@{\GOWO}@HF & 46.498[$+15.15\%$] & 78.075[$+11.58\%$] & 388.907[$+1.36\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & 675.701[$+1.02\%$] \\
RPA@{\GOWO}@HF & 57.567[$+42.56\%$] & 101.092[$+44.47\%$] & 473.053[$+23.29\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & 794.324[$+18.76\%$] \\
BSE@{\GOWO}@HF & 46.498[$+15.15\%$] & 78.075[$+11.58\%$] & 388.907[$+1.36\%$] & xxx.xxx[$+0.00\%$] & 499.145[$+0.96\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & 675.701[$+1.02\%$] \\
RPA@{\GOWO}@HF & 57.567[$+42.56\%$] & 101.092[$+44.47\%$] & 473.053[$+23.29\%$] & xxx.xxx[$+0.00\%$] & 580.318[$+17.38\%$] & xxx.xxx[$+0.00\%$] & xxx.xxx[$+0.00\%$] & 794.324[$+18.76\%$] \\
RPAx@HF & 37.886[$-6.18\%$] & 65.203[$-6.82\%$] & 343.604[$-10.45\%$] & 344.249[$-9.93\%$] & 427.170[$-13.60\%$] & 416.315[$-12.83\%$] & 399.060[$-10.82\%$] & 586.090[$-12.38\%$] \\
RPA@HF & 57.332[$+41.98\%$] & 100.164[$+43.15\%$] & 465.905[$+21.43\%$] & 442.675[$+15.83\%$] & 569.384[$+15.17\%$] & 555.857[$+16.39\%$] & 537.685[$+20.16\%$] & 781.323[$+16.81\%$] \\
\end{tabular}
@ -557,7 +557,7 @@ The error (in \%) compared to the reference CC3 values are reported in square br
\end{squeezetable}
Let us start with the two smallest molecules, \ce{H2} and \ce{LiH}.
Their corresponding PES computed with the cc-pVQZ basis are reported in Fig.~\ref{fig:PES-H2-LiH}.
Their PES computed with the cc-pVQZ basis are reported in Fig.~\ref{fig:PES-H2-LiH}.
For \ce{H2}, we take as reference the full configuration interaction (FCI) energies \cite{QP2} and we also report the MP2 curve and its third-order variant (MP3), which improves upon MP2 towards FCI.
RPA@HF and RPA@{\GOWO}@HF yield almost identical results, and both significantly underestimate the FCI energy, while RPAx@HF and BSE@{\GOWO}@HF slightly over- and undershoot the FCI energy, respectively, RPAx@HF being the best match to FCI in the case of \ce{H2}.
Interestingly, the BSE@{\GOWO}@HF scheme yields a more accurate equilibrium bond length than any other method irrespectively of the basis set (see Table in the {\SI}).
@ -586,14 +586,15 @@ These values are reported in parenthesis in Table \ref{tab:Req}.
For the smooth PES where one can obtain both the genuine minimum and the fitted minimum (\ie, based on the Morse curve), this procedure has been shown to be very accurate with an error of the order of $10^{-3}$ bohr in most cases.
Let us now look at the isoelectronic series \ce{N2}, \ce{CO}, and \ce{BF}, which have a decreasing bond order (from triple to single bond).
The conclusions drawn for the previous systems also apply to these diatomic molecules.
In particular, the performance of BSE@{\GOWO}@HF is outstanding, as shown in Fig.~\ref{fig:PES-N2-CO-BF}, and systematically outperforms both CC2 and CCSD.
The conclusions drawn for the previous systems also apply to these molecules.
In particular, as shown in Fig.~\ref{fig:PES-N2-CO-BF}, the performance of BSE@{\GOWO}@HF is outstanding with an error of the order of $1\%$ on the correlation energy.
Importantly, it systematically outperforms both CC2 and CCSD.
One can notice some irregularities in the PES of \ce{BF} with the cc-pVDZ et cc-pVTZ basis sets (see the {\SI}).
The PES of \ce{N2} and \ce{CO} are smooth though, and yield accurate equilibrium bond lengths once again: at the BSE@{\GOWO}@HF/cc-pVQZ level of theory, we obtain \gb{$2.070$}, \gb{$2.130$}, and \gb{$2.383$} bohr for \ce{N2}, \ce{CO}, and \ce{BF}, respectively, which has to be compared with the CC3/cc-pVQZ values of $2.075$, $2.136$ and $2.390$ bohr, respectively.
The PES of \ce{N2} and \ce{CO} are smooth though, and yield accurate equilibrium bond lengths once again: at the BSE@{\GOWO}@HF/cc-pVQZ level of theory, we obtain $2.065$, \gb{$2.130$}, and \gb{$2.383$} bohr for \ce{N2}, \ce{CO}, and \ce{BF}, respectively, which has to be compared with the CC3/cc-pVQZ values of $2.075$, $2.136$ and $2.390$ bohr, respectively.
As a final example, we consider the \ce{F2} molecule, a notoriously difficult case to treat due to the weakness of its covalent bond (see Fig.~\ref{fig:PES-F2}), hence its relatively long equilibrium bond length ($2.663$ bohr at the CC3/cc-pVQZ level).
Similarly to what is observed for \ce{LiF} and \ce{BF}, there are irregularities near the minimum of the {\GOWO}-based curves.
However, BSE@{\GOWO}@HF is the closest to the CC3 curve, with an estimated bond length of $2.640$ bohr (via a Morse fit) at the BSE@{\GOWO}@HF/cc-pVQZ level.
However, BSE@{\GOWO}@HF is the closest to the CC3 curve, with an error on the correlation energy of $1\%$ and an estimated bond length of $2.640$ bohr (via a Morse fit) at the BSE@{\GOWO}@HF/cc-pVQZ level.
Note that, for this system, triplet (and then singlet) instabilities appear for quite short bond lengths.
However, around the equilibrium structure, we have not encountered any instabilities.
@ -605,11 +606,13 @@ In this Letter, we hope to have illustrated that the ACFDT@BSE formalism is a pr
To do so, we have shown that calculating the BSE correlation energy computed within the ACFDT framework yields extremely accurate PES around equilibrium.
%(Their accuracy near the dissociation limit remains an open question.)
We have illustrated this for 8 diatomic molecules for which we have also computed reference ground-state energies using coupled cluster methods (CC2, CCSD, and CC3).
For the larger systems considered here, we have observed that BSE@{\GOWO} recovers $99\%$ of the CC3 correlation energy.
Moreover, because triplet states do not contribute to the ACFDT correlation energy and singlet instabilities do not appear for weakly-correlated systems around their equilibrium structure, the present scheme does not suffer from singlet nor triplet instabilities.
However, we have also observed that, in some cases, unphysical irregularities on the ground-state PES due to the appearance of a satellite resonance with a weight similar to that of the $GW$ quasiparticle peak.
This shortcoming, which is entirely due to the quasiparticle nature of the underlying $GW$ calculation, has been thoroughly described in several previous studies.\cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020}
We believe that this central issue must be resolved if one wants to expand the applicability of the present method.
In the perspective of developing analytical nuclear gradients within the BSE@$GW$ formalism, we are currently investigating the accuracy of the ACFDT@BSE scheme for excited-state PES.
We hope to be able to report on this in the near future.
%In the perspective of developing analytical nuclear gradients within the BSE@$GW$ formalism, we are currently investigating the accuracy of the ACFDT@BSE scheme for excited-state PES.
%We hope to be able to report on this in the near future.
%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}

4
SI/BSE-PES-SI.tex
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@ -208,10 +208,10 @@ When irregularities appear in the PES, the values are reported in parenthesis an
& cc-pVQZ & 1.391 & 3.008 & 2.970 & 2.395 & 2.091 & 2.137 & 2.382 & 2.634 \\
BSE@{\GOWO}@HF & cc-pVDZ & 1.437 & 3.042 & 3.000 & 2.454 & 2.107 & 2.153 & 2.407 & (2.698) \\
& cc-pVTZ & 1.404 & 3.023 & (2.982) & 2.410 & 2.068 & 2.116 & (2.389) & (2.647) \\
& cc-pVQZ &\rb{1.399} &\rb{3.017} &\rb{(2.974)} &\gb{(2.408)} &\gb{(2.070)} &\gb{(2.130)} &\gb{(2.383)} &\rb{(2.640)}\\
& cc-pVQZ &\rb{1.399} &\rb{3.017} &\rb{(2.974)} &\gb{2.408} &\rb{2.065} &\gb{2.130} &\gb{2.383} &\rb{(2.640)}\\
RPA@{\GOWO}@HF & cc-pVDZ & 1.426 & 3.019 & 2.994 & 2.436 & 2.083 & 2.144 & 2.403 & (2.629) \\
& cc-pVTZ & 1.388 & 2.988 & (2.965) & 2.408 & 2.055 & 2.114 & (2.370) & (2.584) \\
& cc-pVQZ & 1.382 & 2.997 & (2.965) &\gb{(2.389)} &\gb{(2.045)} &\gb{(2.110)} &\gb{(2.367)} & (2.571) \\
& cc-pVQZ & 1.382 & 2.997 & (2.965) &\gb{2.389} & 2.043 &\gb{2.110} &\gb{2.367} & (2.571) \\
RPAx@HF & cc-pVDZ & 1.428 & 3.040 & 2.998 & 2.424 & 2.077 & 2.130 & 2.417 & 2.611 \\
& cc-pVTZ & 1.395 & 3.003 & 2.943 & 2.400 & 2.046 & 2.110 & 2.368 & 2.568 \\
& cc-pVQZ & 1.394 & 3.011 & 2.944 & 2.391 & 2.041 & 2.104 & 2.366 & 2.565 \\