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BSE-PES.tex
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@ -186,7 +186,8 @@ The combination of the many-body Green's function $GW$ approximation and the Bet
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 considered systems.
However, we sometimes observe unphysical irregularities on the ground-state PES in relation with \sout{the appearance of satellite resonances with a weight similar to that of the $GW$ quasiparticle peak} \xavier{discontinuities of a few $GW$ quasiparticle energies, questioning their identification with the solution of the quasiparticle equation with the largest spectral weight. } \\
However, we sometimes observe unphysical irregularities on the ground-state PES in relation with discontinuities of some $GW$ quasiparticle energies, questioning their identification as quasiparticle solution (\textit{i.e.}, the solution of the quasiparticle equation with the largest spectral weight).
\\
\bigskip
\begin{center}
\boxed{\includegraphics[width=0.5\linewidth]{TOC}}
@ -215,7 +216,7 @@ While calculations of the $GW$ quasiparticle energy ionic gradients is becoming
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}
Such a modified BSE polarization propagator was inspired by a previous study on the homogeneous electron gas (HEG). \cite{Maggio_2016}
With such an approximation, amounting to neglect excitonic effects in the electron-hole propagator, the question of using either KS-DFT or $GW$ eigenvalues in the construction of the propagator becomes further relevant, increasing accordingly the number of possible definitions for the ground-state correlation energy.
Finally, renormalizing or not the Coulomb interaction by the interaction strength $\IS$ in the Dyson equation for the interacting polarizability (see below) leads to two different versions of the BSE correlation energy, \cite{Holzer_2018} emphasizing further the lack of general agreement around the definition of the ground-state BSE energy.
@ -533,8 +534,8 @@ The error (in \%) compared to the reference CC3 values are reported in square br
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)} & 2.065[$-0.48\%$] & 2.134[$-0.09\%$] & 2.383[$-0.29\%$] & (2.640)[$-0.86\%$] \\
RPA@{\GOWO}@HF & 1.382[$-1.43\%$] & 2.997[$-0.73\%$] & (2.965)[$+0.07\%$] & \gb{(2.389)} & 2.043[$-1.54\%$] & 2.132[$-0.19\%$] & 2.367[$-0.96\%$] & (2.571)[$-3.45\%$] \\
BSE@{\GOWO}@HF & 1.399[$-0.21\%$] & 3.017[$-0.07\%$] & (2.974)[$+0.37\%$] & \gb{(2.408)} & 2.065[$-0.48\%$] & 2.134[$-0.09\%$] & 2.385[$-0.21\%$] & (2.640)[$-0.86\%$] \\
RPA@{\GOWO}@HF & 1.382[$-1.43\%$] & 2.997[$-0.73\%$] & (2.965)[$+0.07\%$] & \gb{(2.389)} & 2.043[$-1.54\%$] & 2.132[$-0.19\%$] & 2.365[$-1.05\%$] & (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[$-3.38\%$] \\
\\
@ -547,8 +548,8 @@ The error (in \%) compared to the reference CC3 values are reported in square br
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[$+0.00\%$] & 499.145[$+0.96\%$] & 481.151[$+0.75\%$] & 453.091[$+1.26\%$] & 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[$+17.38\%$] & 566.536[$+18.63\%$] & 545.543[$+21.92\%$] & 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[$+0.96\%$] & 481.151[$+0.75\%$] & 453.137[$+1.27\%$] & 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[$+17.38\%$] & 566.536[$+18.63\%$] & 545.546[$+21.92\%$] & 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}
@ -559,7 +560,7 @@ The error (in \%) compared to the reference CC3 values are reported in square br
Let us start with the two smallest molecules, \ce{H2} and \ce{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 \sout{underestimate} \xavier{overestimate} the FCI \xavier{correlation} energy, while RPAx@HF and BSE@{\GOWO}@HF slightly \xavier{over- and undershoot ??} the FCI energy, respectively, RPAx@HF being the best match to FCI in the case of \ce{H2}.
RPA@HF and RPA@{\GOWO}@HF yield almost identical results, and both significantly overestimate the FCI correlation 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}).
For example, BSE@{\GOWO}@HF/cc-pVQZ is only off by $0.003$ bohr as compared to FCI/cc-pVQZ, while RPAx@HF, MP2, and CC2 underestimate the bond length by $0.008$, $0.011$, and $0.011$ bohr, respectively.
The RPA-based schemes are much less accurate, with even shorter equilibrium bond lengths.
@ -573,7 +574,10 @@ The cases of \ce{LiF} and \ce{HCl} (see Fig.~\ref{fig:PES-LiF-HCl}) are chemical
For these partially ionic bonds, the performance of BSE@{\GOWO}@HF is terrific with an almost perfect match to the CC3 curve.
Maybe surprisingly, BSE@{\GOWO}@HF is on par with both CC2 and CCSD, and outperforms RPAx@HF by a big margin for these two molecules exhibiting charge transfer, the latter fact being also observed for the other diatomics discussed below.
Interestingly, while CCSD and CC2 systematically underestimates the total energy, the BSE@{\GOWO}@HF energy is always lower than the reference CC3 energy.
This observation is not only true for \ce{LiF} and \ce{HCl}, but holds for every single systems that is considered herein. \xavier{This is consistent with the study by Maggio and Kresse of the homogeneous electron gas (HEG)\cite{Maggio_2016} showing that BSE slightly overestimates the correlation energy as compared to QMC reference data. Similarly, the much larger overestimation of the correlation energy we observe at the RPA@$GW$ level was also observed for the HEG. Care must be taken however in drawing comparisons since the HEG studies were performed starting with LDA input eigenstates. }
This observation is not only true for \ce{LiF} and \ce{HCl}, but holds for every single systems that is considered herein.
Moreover, this is consistent with the study by Maggio and Kresse on the HEG showing that BSE slightly overestimates the correlation energy as compared to QMC reference data. \cite{Maggio_2016}
Similarly, the much larger overestimation of the correlation energy that we observe at the RPA@$GW$ level was also observed for the HEG.
Care must be taken however in drawing comparisons since the HEG study of Ref.~\onlinecite{Maggio_2016} was performed starting with LDA eigenstates.
For \ce{HCl}, the data reported in Table \ref{tab:Req} show that the BSE@{\GOWO}@HF equilibrium bond length is again in very good agreement with its CC3 counterpart.
In contrast to CCSD which is known to provide slightly shorter bond lengths, ACFDT@BSE underestimates the bond lengths by a few hundredths of bohr.
@ -583,20 +587,23 @@ Including a broadening via an increase of the $\eta$ value entering in the expre
%Note that these irregularities would be genuine discontinuities in the case of {\evGW}. \cite{Veril_2018}
When irregularities are present in the PES, we have fitted a Morse potential of the form $M(R) = D_0\qty{1-\exp[-\alpha\qty(R-\Req)]}^2$ to the PES in order to provide an estimate of the equilibrium bond length.
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. \xavier{ We note that these irregularities are much smaller than the differences between the BSE and the other RPA-like techniques (RPA, RPAx, RPA@$GW$) leaving BSE unambiguously more accurate. }
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.
We note that these irregularities are much smaller than the differences between the BSE and the other RPA-like techniques (RPA, RPAx, RPA@$GW$) leaving BSE unambiguously more accurate than these approaches.
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 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 $2.065$, $2.134$, and $2.383$ 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 $2.065$, $2.134$, and $2.385$ 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 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. \xavier{ This is an important outcome of the present study that the difficulties encountered at large interatomic distance, namely close to the dissociation limit, does not prevent the BSE approach to be potentially extremely useful and accurate in the vicinity of the equilibrium distance. Preliminary results indicate that no singlet instabilities could be seen in the vicinity of the lowest excited states minima. }
However, around the equilibrium structure, we have not encountered any instabilities.
This is an important outcome of the present study as the difficulties encountered at large interatomic distance (\ie, close to the dissociation limit) do not prevent the BSE approach to be potentially useful and accurate in the vicinity of equilibrium distances.
Furthermore, preliminary calculations could not detect any singlet instabilities in the vicinity of the lowest singlet excited-state minima.
\xavier{ Although we considered here only a limited set of compounds,
our correlation energy MAE (MSE) with BSE of xxx (xxx) as compared to CC3 are significantly smaller than the one obtained with MP2 and CCSD; xxx (xxx) }
@ -611,8 +618,9 @@ To do so, we have shown that calculating the BSE correlation energy computed wit
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 \sout{a satellite resonance with a weight similar to that of the $GW$ quasiparticle peak} \xavier{discontinuities as a function of bond length of a few $GW$ quasiparticle energies. Such an unphysical behaviour stems from defining the quasiparticle energy as the solution of the quasiparticle equation with the largest spectral weight in cases where several solutions can be found.}
This shortcoming \sout{, 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}
However, we have also observed that, in some cases, unphysical irregularities on the ground-state PES due to the appearance of discontinuities as a function of the bond length for some of the $GW$ quasiparticle energies.
Such an unphysical behaviour stems from defining the quasiparticle energy as the solution of the quasiparticle equation with the largest spectral weight in cases where several solutions can be found.
This shortcoming 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.