From bbb2ce3bcc9667dec87071b93c169ae626a19bd4 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Thu, 9 Apr 2020 22:55:00 +0200 Subject: [PATCH] starting writing He and H2st --- Manuscript/FarDFT.bib | 28 +++++++++++++++++++++++++--- Manuscript/FarDFT.tex | 33 ++++++++++++++++++++++++++------- 2 files changed, 51 insertions(+), 10 deletions(-) diff --git a/Manuscript/FarDFT.bib b/Manuscript/FarDFT.bib index 07d552d..1ffce86 100644 --- a/Manuscript/FarDFT.bib +++ b/Manuscript/FarDFT.bib @@ -1,13 +1,35 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-04-09 21:27:50 +0200 +%% Created for Pierre-Francois Loos at 2020-04-09 22:26:01 +0200 %% Saved with string encoding Unicode (UTF-8) +@article{Becke_1988a, + Author = {A. D. Becke}, + Date-Added = {2020-04-09 22:22:05 +0200}, + Date-Modified = {2020-04-09 22:24:01 +0200}, + Doi = {10.1103/PhysRevA.38.3098}, + Journal = {Phys. Rev. A}, + Pages = {3098}, + Title = {Density-functional exchange-energy approximation with correct asymptotic behavior}, + Volume = {38}, + Year = {1988}} + +@article{Lee_1988, + Author = {C. Lee and W. Yang and R. G. Parr}, + Date-Added = {2020-04-09 22:20:57 +0200}, + Date-Modified = {2020-04-09 22:21:44 +0200}, + Doi = {10.1103/PhysRevB.37.785}, + Journal = {Phys. Rev. B}, + Pages = {785}, + Title = {Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density}, + Volume = {37}, + Year = {1988}} + @article{Burges_1995, Author = {A. Burgers and D. Wintgen and J.-M. Rost}, Date-Added = {2020-04-09 14:56:36 +0200}, @@ -177,10 +199,10 @@ Year = {2001}, Bdsk-Url-1 = {https://doi.org/10.1007/s002140100263}} -@article{Becke_1988, +@article{Becke_1988b, Author = {A. D. Becke}, Date-Added = {2020-03-30 09:58:02 +0200}, - Date-Modified = {2020-03-30 09:59:13 +0200}, + Date-Modified = {2020-04-09 22:24:05 +0200}, Doi = {10.1063/1.454033}, Journal = {J. Chem. Phys.}, Pages = {2547}, diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 337ddde..f88448a 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -311,15 +311,15 @@ where $\e{\ex}{\ew{}}(\n{}{})$ and $\e{\co}{\ew{}}(\n{}{})$ are the weight-depen %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Computational details} \label{sec:compdet} + The self-consistent GOK-DFT calculations have been performed in a restricted formalism with the \texttt{QuAcK} software, \cite{QuAcK} which is freely available on \texttt{github}, and where the present weight-dependent functionals have been implemented. For more details about the self-consistent implementation of GOK-DFT, we refer the interested reader to Ref.~\onlinecite{Loos_2020} where additional technical details can be found. For all calculations, we use the aug-cc-pVXZ (X = D, T, Q, and 5) Dunning family of atomic basis sets. \cite{Dunning_1989,Kendall_1992,Woon_1994} -Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} using 194 angular points (Lebedev grid) and a radial precision of $10^{-6}$. \cite{Becke_1988,Lindh_2001} +Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} using 194 angular points (Lebedev grid) and a radial precision of $10^{-6}$. \cite{Becke_1988b,Lindh_2001} This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities). Moreover, we restrict our study to the case of a two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$ with weight $1 - \ew{}$) and the first doubly-excited state ($I=1$ with weight $\ew{}$) are considered. Although one should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle, we will sometimes ``violate'' this variational constraint. Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a genuine saddle point of the KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b} -\titou{Moreover, the limits $\ew{} = 0$ and $\ew{} = 1$ are the only two weights for which there is no ghost-interaction error.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Results} @@ -390,7 +390,7 @@ and \end{align} \end{subequations} makes the ensemble almost perfectly linear (see Fig.~\ref{fig:Ew_H2}), and the excitation energy much more stable (with respect to $\ew{}$) and closer to the FCI reference (see Fig.~\ref{fig:Om_H2}). -As readily seen from Eq.~\eqref{eq:Cxw} and graphically illustrated in Fig.~\ref{fig:Cx_H2}, the weight-dependent correction does not affect the two ghost-interaction-free limits at $\ew{} = 0$ and $\ew{} = 1$, as $\Cx{\ew{}}$ reduces to $\Cx{}$ in these two limits. +As readily seen from Eq.~\eqref{eq:Cxw} and graphically illustrated in Fig.~\ref{fig:Cxw}, the weight-dependent correction does not affect the two ghost-interaction-free limits at $\ew{} = 0$ and $\ew{} = 1$, as $\Cx{\ew{}}$ reduces to $\Cx{}$ in these two limits. Maybe surprisingly, one would have noticed that we are not only using data from $0 \le \ew{} \le 1/2$, but we also consider ensemble energies for $1/2 < \ew{} \le 1$, which is deterred by the GOK variational principle. \cite{Gross_1988a} However, it is important to ensure that the weight-dependent functional does not alter the $\ew{} = 1$ limit, which is a genuine saddle point of the KS equations, as mentioned above. Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear. @@ -400,7 +400,7 @@ We shall come back to this point later on. \includegraphics[width=\linewidth]{Cxw} \caption{ $\Cx{\ew{}}/\Cx{\ew{}=0}$ as a function of $\ew{}$ [see Eq.~\eqref{eq:Cxw}] computed with the aug-cc-pVTZ basis set for the \ce{He} atom (blue) and the \ce{H2} molecule at $\RHH = 1.4$ bohr (red) and $\RHH = 3.7$ bohr (green). - \label{fig:Cx_H2} + \label{fig:Cxw} } \end{figure} @@ -689,6 +689,18 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{ \label{sec:H2st} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +To investigate the weight dependence of the xc functional in the strongly correlated regime, we now consider the \ce{H2} molecule in a stretched geometry ($\RHH = 3.7$ bohr). +For this particular geometry, the doubly-excited state becomes the lowest excited state. +We then follow the same protocol as in Sec.~\ref{sec:H2}, and designed a GIC-S functional for this system and the aug-cc-pVTZ basis set. +The weight-dependence of $\Cx{\ew{}}$ is illustrated in Fig.~\ref{fig:Cxw}. One clearly sees that the correction brought by GIC-S is much more subtile than at $\RHH = 1.4$ bohr, which means that the ensemble energy obtained with the Slater-Dirac functional is much more linear at $\RHH = 3.7$ bohr and the curvature more gentle. +Table \ref{tab:BigTab_H2st} reports, for the aug-cc-pVTZ basis set (which delivers converged results with respect to the size of the basis set), the same set of calculations as in Table \ref{tab:BigTab_H2}. +As a reference value, we have computed a FCI/aug-cc-pV5Z excitation energy of $8.69$ eV, which compares well with previous studies. \cite{Senjean_2015} +For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the excitation energy, the best match being obtained with HF exchange. +The GIC-S functional coupled or not with a correlation functional yield extremely stable excitation energies as a function of the weight, with only a few tenths of eV difference between the zero- and equi-weights limits. +Nonetheless, the excitation energy is still off by 3 eV. +The fundamental theoretical reason of such a poor agreement is not clear. +The fact that HF exchange yields better excitation energy hints at the effect of self-interaction error. + %%% TABLE I %%% \begin{table} \caption{ @@ -702,8 +714,8 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{ \tabc{x} & \tabc{c} & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\ \hline HF & & 19.09 & 6.59 & 12.92 & 6.52 \\ - HF & VWN5 & 19.40 & 6.54 & 13.02 & 6.49\\ - HF & eVWN5 & 19.59 & 6.72 & 13.11 & \\ + HF & VWN5 & 19.40 & 6.54 & 13.02 & 6.49 \\ + HF & eVWN5 & 19.59 & 6.72 & 13.11 & \\ S & & 5.31 & 5.60 & 5.46 & 5.56 \\ S & VWN5 & 5.34 & 5.57 & 5.46 & 5.52 \\ S & eVWN5 & 5.53 & 5.76 & 5.56 & 5.72 \\ @@ -715,7 +727,7 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{ B3 & LYP & & & & 5.55 \\ HF & LYP & & & & 6.68 \\ \hline - \mc{5}{l}{Accurate\fnm[1]} & 8.69 \\ + \mc{5}{l}{Accurate\fnm[1]} & 8.69 \\ \end{tabular} \end{ruledtabular} \fnt[1]{FCI/aug-cc-pV5Z calculation performed with QUANTUM PACKAGE. \cite{QP2}} @@ -727,6 +739,13 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{ \label{sec:He} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +As a final example, we consider the \ce{He} atom which can be seen as the limiting form of the \ce{H2} molecule for very short bond lengths. +In \ce{He}, the lowest doubly-excited state is extremely high in energy and lies in the continuum. \cite{Burges_1995} +In Ref.~\onlinecite{Burges_1995}, highly-accurate calculations estimates an excitation energy of $2.126$ hartree. +Nonetheless, it can be nicely described with a Gaussian basis set containing enough diffuse functions. +This is why we have considered for this particular example the d-aug-cc-pVQZ basis set which contains two sets of diffuse functions. +The excitation energies associated with this double excitation computed with various methods and combinations of xc functions are gathered in Table \ref{tab:BigTab_He}. + %%% TABLE I %%% \begin{table} \caption{