corrections Xav and Ivan
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BSE-PES.tex
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BSE-PES.tex
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@ -186,7 +186,7 @@ The combination of the many-body Green's function $GW$ approximation and the Bet
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The BSE formalism can also be employed to compute ground-state correlation energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT).
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Here, we study the topology of the ground-state potential energy surfaces (PES) of several diatomic molecules near their equilibrium bond length.
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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.
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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.}, as solution of the quasiparticle equation with the largest spectral weight).
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However, we sometimes observe unphysical irregularities on the ground-state PES in relation with difficulties in the identification of a few $GW$ quasiparticle energies.
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\\
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\bigskip
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\begin{center}
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@ -302,7 +302,7 @@ In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$
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In the absence of instabilities (\ie, when $\bA{\IS} - \bB{\IS}$ is positive-definite), \cite{Dreuw_2005} Eq.~\eqref{eq:LR} is usually transformed into an Hermitian eigenvalue problem of smaller dimension
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\begin{equation}
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\label{eq:small-LR}
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(\bA{\IS} - \bB{\IS})^{1/2} (\bA{\IS} + \bB{\IS}) (\bA{\IS} - \bB{\IS})^{1/2} \bZ{\IS} = (\bOm{\IS})^2 \bZ{\IS},
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(\bA{\IS} - \bB{\IS})^{1/2} (\bA{\IS} + \bB{\IS}) (\bA{\IS} - \bB{\IS})^{1/2} \bZ{\IS}_m = (\Om{m}{\IS})^2 \bZ{\IS}_m,
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\end{equation}
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where the excitation amplitudes are
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\begin{subequations}
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@ -525,32 +525,32 @@ The error (in \%) compared to the reference CC3 values are reported in square br
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}
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\label{tab:Req}
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\begin{ruledtabular}
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\begin{tabular}{lrrrrrrrr}
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\begin{tabular}{lllllllll}
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& \mc{8}{c}{Equilibrium bond length $\Req$ (bohr) } \\
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\cline{2-9}
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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}} \\
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\hline
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CC3 & 1.402 & 3.019 & 2.963 & 2.403 & 2.075 & 2.136 & 2.390 & 2.663 \\
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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[$-1.58\%$] \\
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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[$+0.08\%$] \\
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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[$-1.09\%$] \\
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BSE@{\GOWO}@HF & 1.399[$-0.21\%$] & 3.017[$-0.07\%$] & (2.974)[$+0.37\%$] & 2.400[$-0.12\%$] & 2.065[$-0.48\%$] & 2.134[$-0.09\%$] & 2.385[$-0.21\%$] & (2.640)[$-0.86\%$] \\
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RPA@{\GOWO}@HF & 1.382[$-1.43\%$] & 2.997[$-0.73\%$] & (2.965)[$+0.07\%$] & 2.370[$-1.50\%$] & 2.043[$-1.54\%$] & 2.132[$-0.19\%$] & 2.365[$-1.05\%$] & (2.571)[$-3.45\%$] \\
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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[$-3.68\%$] \\
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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[$-3.38\%$] \\
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CCSD & 1.402[$+0.0\%$] & 3.020[$+0.0\%$] & 2.953[$-0.3\%$] & 2.398[$-0.2\%$] & 2.059[$-0.8\%$] & 2.118[$-0.8\%$] & 2.380[$-0.4\%$] & 2.621[$-1.6\%$] \\
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CC2 & 1.391[$-0.8\%$] & 2.989[$-1.0\%$] & 2.982[$+0.6\%$] & 2.396[$-0.3\%$] & 2.106[$+1.5\%$] & 2.156[$+0.9\%$] & 2.393[$+0.1\%$] & 2.665[$+0.1\%$] \\
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MP2 & 1.391[$-0.8\%$] & 3.008[$-0.4\%$] & 2.970[$+0.2\%$] & 2.395[$-0.3\%$] & 2.091[$+0.8\%$] & 2.137[$+0.1\%$] & 2.382[$-0.3\%$] & 2.634[$-1.1\%$] \\
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BSE@{\GOWO}@HF & 1.399[$-0.2\%$] & 3.017[$-0.1\%$] & (2.974)[$+0.4\%$] & 2.400[$-0.1\%$] & 2.065[$-0.5\%$] & 2.134[$-0.1\%$] & 2.385[$-0.2\%$] & (2.640)[$-0.9\%$] \\
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RPA@{\GOWO}@HF & 1.382[$-1.4\%$] & 2.997[$-0.7\%$] & (2.965)[$+0.1\%$] & 2.370[$-1.5\%$] & 2.043[$-1.5\%$] & 2.132[$-0.2\%$] & 2.365[$-1.1\%$] & (2.571)[$-3.5\%$] \\
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RPAx@HF & 1.394[$-0.6\%$] & 3.011[$-0.3\%$] & 2.944[$-0.6\%$] & 2.391[$-0.5\%$] & 2.041[$-1.6\%$] & 2.104[$-1.5\%$] & 2.366[$-1.0\%$] & 2.565[$-3.7\%$] \\
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RPA@HF & 1.386[$-1.1\%$] & 2.994[$-0.8\%$] & 2.946[$-0.6\%$] & 2.382[$-0.9\%$] & 2.042[$-1.6\%$] & 2.103[$-1.5\%$] & 2.364[$-1.1\%$] & 2.573[$-3.4\%$] \\
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\hline
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& \mc{8}{c}{Correlation energy $-\Ec$ (millihartree)} \\
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\cline{2-9}
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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}} \\
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\hline
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CC3 & 40.382 & 69.974 & 383.686 & 382.188 & 494.393 & 477.580 & 447.472 & 668.875 \\
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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[$-3.72\%$] \\
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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[$-2.09\%$] \\
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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[$-3.67\%$] \\
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BSE@{\GOWO}@HF & 46.498[$+15.15\%$] & 78.075[$+11.58\%$] & 388.907[$+1.36\%$] & 385.864[$+0.96\%$] & 499.145[$+0.96\%$] & 481.151[$+0.75\%$] & 453.137[$+1.27\%$] & 675.701[$+1.02\%$] \\
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RPA@{\GOWO}@HF & 57.567[$+42.56\%$] & 101.092[$+44.47\%$] & 473.053[$+23.29\%$] & 451.232[$+18.07\%$] & 580.318[$+17.38\%$] & 566.536[$+18.63\%$] & 545.546[$+21.92\%$] & 794.324[$+18.76\%$] \\
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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[$-12.38\%$] \\
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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[$+16.81\%$] \\
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CC3 & 40.4 & 70.0 & 383.7 & 382.2 & 494.4 & 477.6 & 447.5 & 668.9 \\
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CCSD & 40.4[$+0.0\%$] & 69.8[$-0.2\%$] & 372.6[$-2.9\%$] & 370.8[$-3.0\%$] & 470.6[$-4.8\%$] & 455.2[$-4.7\%$] & 432.9[$-3.3\%$] & 644.0[$-3.7\%$] \\
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CC2 & 33.3[$-17.6\%$] & 57.3[$-18.1\%$] & 376.7[$-1.8\%$] & 356.9[$-6.6\%$] & 488.0[$-1.3\%$] & 465.5[$-2.5\%$] & 427.3[$-4.5\%$] & 654.9[$-2.1\%$] \\
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MP2 & 33.2[$-17.9\%$] & 57.9[$-17.2\%$] & 373.0[$-2.8\%$] & 355.7[$-6.9\%$] & 478.0[$-3.3\%$] & 455.0[$-4.7\%$] & 421.6[$-5.8\%$] & 644.3[$-3.7\%$] \\
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BSE@{\GOWO}@HF & 46.5[$+15.2\%$] & 78.1[$+11.6\%$] & 388.9[$+1.4\%$] & 385.9[$+1.0\%$] & 499.1[$+1.0\%$] & 481.2[$+0.8\%$] & 453.1[$+1.3\%$] & 675.7[$+1.0\%$] \\
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RPA@{\GOWO}@HF & 57.6[$+42.6\%$] & 101.1[$+44.5\%$]& 473.1[$+23.3\%$] & 451.2[$+18.1\%$]& 580.3[$+17.4\%$]& 566.5[$+18.6\%$]& 545.5[$+21.9\%$] & 794.3[$+18.8\%$] \\
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RPAx@HF & 37.9[$-6.2\%$] & 65.2[$-6.8\%$] & 343.6[$-10.5\%$] & 344.2[$-9.9\%$] & 427.2[$-13.6\%$]& 416.3[$-12.8\%$]& 399.1[$-10.8\%$] & 586.1[$-12.4\%$] \\
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RPA@HF & 57.3[$+42.0\%$] & 100.2[$+43.2\%$]& 465.9[$+21.4\%$] & 442.7[$+15.8\%$]& 569.4[$+15.2\%$]& 555.9[$+16.4\%$]& 537.7[$+20.2\%$] & 781.3[$+16.8\%$] \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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