From 1a8cbe0ab4ec641c3eeae4e6c538bd08f9b2ed38 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sun, 26 Jan 2020 13:34:58 +0100 Subject: [PATCH] clean up intro and theory --- BSE-PES.bib | 40 ++++++++++++++++++++++++++++++++++++---- BSE-PES.tex | 30 +++++++++++++++++------------- 2 files changed, 53 insertions(+), 17 deletions(-) diff --git a/BSE-PES.bib b/BSE-PES.bib index 32b3964..0175e20 100644 --- a/BSE-PES.bib +++ b/BSE-PES.bib @@ -1,13 +1,44 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-01-26 11:17:45 +0100 +%% Created for Pierre-Francois Loos at 2020-01-26 13:32:50 +0100 %% Saved with string encoding Unicode (UTF-8) +@article{dalton, + Author = {Aidas, Kestutis and Angeli, Celestino and Bak, Keld L. and Bakken, Vebj{\o}rn and Bast, Radovan and Boman, Linus and Christiansen, Ove and Cimiraglia, Renzo and Coriani, Sonia and Dahle, P{\aa}l and Dalskov, Erik K. and Ekstr{\"o}m, Ulf and Enevoldsen, Thomas and Eriksen, Janus J. and Ettenhuber, Patrick and Fern{\'a}ndez, Berta and Ferrighi, Lara and Fliegl, Heike and Frediani, Luca and Hald, Kasper and Halkier, Asger and H{\"a}ttig, Christof and Heiberg, Hanne and Helgaker, Trygve and Hennum, Alf Christian and Hettema, Hinne and Hjerten{\ae}s, Eirik and H{\o}st, Stinne and H{\o}yvik, Ida-Marie and Iozzi, Maria Francesca and Jans{\'\i}k, Branislav and Jensen, Hans J{\o}rgen Aa. and Jonsson, Dan and J{\o}rgensen, Poul and Kauczor, Joanna and Kirpekar, Sheela and Kj{\ae}rgaard, Thomas and Klopper, Wim and Knecht, Stefan and Kobayashi, Rika and Koch, Henrik and Kongsted, Jacob and Krapp, Andreas and Kristensen, Kasper and Ligabue, Andrea and Lutn{\ae}s, Ola B. and Melo, Juan I. and Mikkelsen, Kurt V. and Myhre, Rolf H. and Neiss, Christian and Nielsen, Christian B. and Norman, Patrick and Olsen, Jeppe and Olsen, J{\'o}gvan Magnus H. and Osted, Anders and Packer, Martin J. and Pawlowski, Filip and Pedersen, Thomas B. and Provasi, Patricio F. and Reine, Simen and Rinkevicius, Zilvinas and Ruden, Torgeir A. and Ruud, Kenneth and Rybkin, Vladimir V. and Sa{\l}ek, Pawel and Samson, Claire C. M. and de Mer{\'a}s, Alfredo S{\'a}nchez and Saue, Trond and Sauer, Stephan P. A. and Schimmelpfennig, Bernd and Sneskov, Kristian and Steindal, Arnfinn H. and Sylvester-Hvid, Kristian O. and Taylor, Peter R. and Teale, Andrew M. and Tellgren, Erik I. and Tew, David P. and Thorvaldsen, Andreas J. and Th{\o}gersen, Lea and Vahtras, Olav and Watson, Mark A. and Wilson, David J. D. and Ziolkowski, Marcin and {\AA}gren, Hans}, + Date-Added = {2020-01-26 13:32:47 +0100}, + Date-Modified = {2020-01-26 13:32:47 +0100}, + Doi = {10.1002/wcms.1172}, + Issn = {1759-0884}, + Journal = {WIREs Comput. Mol. Sci.}, + Number = {3}, + Pages = {269--284}, + Title = {The Dalton Quantum Chemistry Program System}, + Url = {http://dx.doi.org/10.1002/wcms.1172}, + Volume = {4}, + Year = {2014}, + Bdsk-Url-1 = {http://dx.doi.org/10.1002/wcms.1172}} + +@article{Chr95, + Abstract = {An approximate coupled cluster singles and doubles model is presented, denoted CC2. The \{CC2\} total energy is of second-order M{\o}ller-Plesset perturbation theory (MP2) quality. The \{CC2\} linear response function is derived. Unlike MP2, excitation energies and transition moments can be obtained in CC2. A hierarchy of coupled cluster models, CCS, CC2, CCSD, CC3, \{CCSDT\} etc., is presented where \{CC2\} and \{CC3\} are approximate coupled cluster models defined by similar approximations. Higher levels give increased accuracy at increased computational effort. The scaling of CCS, CC2, CCSD, \{CC3\} and \{CCSDT\} is N4, N5, N6, \{N7\} and N8, respectively where N is th the number of orbitals. Calculations on Be, \{N2\} and \{C2H4\} are performed and results compared with those obtained in the second-order polarization propagator approach SOPPA. }, + Author = {Ove Christiansen and Henrik Koch and Poul J{\o}rgensen}, + Date-Added = {2020-01-26 13:24:49 +0100}, + Date-Modified = {2020-01-26 13:24:49 +0100}, + Doi = {http://dx.doi.org/10.1016/0009-2614(95)00841-Q}, + Issn = {0009-2614}, + Journal = {Chem. Phys. Lett.}, + Pages = {409--418}, + Title = {The Second-Order Approximate Coupled Cluster Singles and Doubles Model CC2}, + Url = {http://www.sciencedirect.com/science/article/pii/000926149500841Q}, + Volume = {243}, + Year = {1995}, + Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/000926149500841Q}, + Bdsk-Url-2 = {http://dx.doi.org/10.1016/0009-2614(95)00841-Q}} + @article{Duchemin_2019, Author = {Ivan Duchemin and Xavier Blase}, Date-Added = {2020-01-26 11:16:20 +0100}, @@ -21,15 +52,16 @@ Bdsk-Url-1 = {https://doi.org/10.1063/1.5090605}} @article{Furche_2005, - Author = {Filipp Furche, and Troy Van Voorhis}, + Author = {Filipp Furche and Troy {Van Voorhis}}, Date-Added = {2020-01-26 11:09:36 +0100}, - Date-Modified = {2020-01-26 11:10:10 +0100}, + Date-Modified = {2020-01-26 13:15:06 +0100}, Doi = {10.1063/1.1884112}, Journal = {J. Chem. Phys.}, Pages = {164106}, Title = {Fluctuation-dissipation theorem density- functional theory}, Volume = {122}, - Year = {2005}} + Year = {2005}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.1884112}} @article{Toulouse_2009, Author = {Julien Toulouse and Iann C. Gerber and Georg Jansen and Andreas Savin and Janos G. Angyan}, diff --git a/BSE-PES.tex b/BSE-PES.tex index 10bb93e..2b9690a 100644 --- a/BSE-PES.tex +++ b/BSE-PES.tex @@ -91,6 +91,8 @@ \newcommand{\BBSE}[2]{B_{#1}^{#2,\text{BSE}}} \newcommand{\ARPA}[2]{A_{#1}^{#2,\text{RPA}}} \newcommand{\BRPA}[2]{B_{#1}^{#2,\text{RPA}}} +\newcommand{\ARPAx}[2]{A_{#1}^{#2,\text{RPAx}}} +\newcommand{\BRPAx}[2]{B_{#1}^{#2,\text{RPAx}}} \newcommand{\G}[1]{G_{#1}} \newcommand{\LBSE}[1]{L_{#1}} \newcommand{\XiBSE}[1]{\Xi_{#1}} @@ -277,7 +279,7 @@ Finally, the static approximation is enforced, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{ %\label{sec:BSE_basis} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -For a closed-shell system, in order to compute the singlet BSE excitation energies (within the static approximation) of the physical system (\ie, $\IS = 1$) in a finite basis, one must solve the following linear response problem \cite{Casida,Dreuw_2005,Martin_2016} +For a closed-shell system, to compute the singlet BSE excitation energies (within the static approximation) of the physical system (\ie, $\IS = 1$) in a finite basis, one must solve the following linear response problem \cite{Casida,Dreuw_2005,Martin_2016} \begin{equation} \label{eq:LR} \begin{pmatrix} @@ -296,7 +298,7 @@ For a closed-shell system, in order to compute the singlet BSE excitation energi \end{pmatrix}, \end{equation} where $\Om{m}{\IS}$ is the $m$th excitation energy with eigenvector $\T{(\bX{\IS}_m \, \bY{\IS}_m)}$ at interaction strength $\IS$, and we have assumed real-valued spatial orbitals $\{\MO{p}(\br{})\}_{1 \le p \le \Norb}$. -The matrices $\bA{\IS}$, $\bB{\IS}$, $\bX{\IS}$, and $\bY{\IS}$ are all of size $\Nocc \Nvir \times \Nocc \Nvir$ where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$), respectively. +The matrices $\bA{\IS}$, $\bB{\IS}$, $\bX{\IS}$, and $\bY{\IS}$ are all of size $\Nocc \Nvir \times \Nocc \Nvir$ where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$ is the total number of spatial orbitals), respectively. In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals. In the absence of instabilities (\ie, $\bA{\IS} - \bB{\IS}$ is positive-definite), \cite{Dreuw_2005} Eq.~\eqref{eq:LR} is usually transformed into an Hermitian eigenvalue problem of smaller dimension @@ -353,10 +355,10 @@ where $\eHF{p}$ are the HF orbital energies. %\label{sec:BSE_energy} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The key quantity to define in the present context is the total ground-state BSE energy $\EBSE$. -Although this choice is not unique, \cite{Holzer_2018} we propose to define it as +Although this choice is not unique, \cite{Holzer_2018} we propose here to define it as \begin{equation} \label{eq:EtotBSE} - \EBSE = \Enuc + \EHF + \EcBSE + \EBSE = \Enuc + \EHF + \EcBSE, \end{equation} where $\Enuc$ and $\EHF$ are the nuclear repulsion energy and ground-state HF energy (respectively), and \begin{equation} @@ -387,17 +389,17 @@ is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\tA{ia,jb}{\IS} \end{equation} is the correlation part of the two-electron density matrix at interaction strength $\IS$, $\Tr$ denotes the matrix trace and $\T{}$ the matrix transpose. Note that the present definition of the BSE correlation energy [see Eq.~\eqref{eq:EcBSE}] has be labeled ``XBS'' for ``extended Bethe Salpeter'' by Holzer et al. \cite{Holzer_2018} +Contrary to DFT where the electron density is fixed along the adiabatic path, in the present formalism, the density is not maintain with respect to $\IS$. +Therefore, an additional contribution to Eq.~\eqref{eq:EcBSE} coming from the variation of the Green's function along the adiabatic connection should be added. +However, as commonly done within RPA and RPAx (\ie, RPA with exchange), \cite{Toulouse_2009, Toulouse_2010, Holzer_2018} we shall neglect it in the present study. Several important comments are in order here. -For spin-restricted closed-shell molecular systems around their equilibrium geometry (such as the ones studied here), it is rare to encounter singlet instabilities. -However, they may appear in the presence of strong correlation (\eg, when the bond is stretch). +For spin-restricted closed-shell molecular systems around their equilibrium geometry (such as the ones studied here), it is rare to encounter singlet instabilities as these systems can be labeled as weakly correlated. +However, singlet instabilities may appear in the presence of strong correlation (\eg, when the bond is stretched). In such a case, this hampers the use of Eq.~\eqref{eq:EcBSE}. -Triplet instabilities are much more common. -However, contrary to the plasmon formulation (an alternative expression of the BSE correlation energy), \cite{Schuck_Book, Gell-Mann_1957, Rowe_1968, Sawada_1957b, Li_2020} the triplet excitations do not contribute in the adiabatic connection formulation, which is an indisputable advantage of this approach. +Even for weakly correlated systems, triplet instabilities are much more common. +However, contrary to the plasmon formulation (an alternative expression of the BSE correlation energy), \cite{Schuck_Book, Gell-Mann_1957, Rowe_1968, Sawada_1957b, Li_2020} the triplet excitations do not contribute in the ACFDT formulation, which is an indisputable advantage of this approach. Indeed, although at the RPA level, the plasmon and adiabatic connection formulations are equivalent, \cite{Sawada_1957b, Fukuta_1964, Furche_2008} this is not the case at the BSE level. -Contrary to DFT where the electron density is fixed along the adiabatic path, in the present formalism (as well as in RPA and RPAx), the density is not maintain with respect to $\IS$. -Therefore, an additional contribution to Eq.~\eqref{eq:EcBSE} coming from the variation of the Green's function along the adiabatic connection should be added. -However, as commonly done, \cite{Toulouse_2009, Toulouse_2010, Holzer_2018} we shall neglect it in the present study. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\section{Computational details} @@ -416,16 +418,18 @@ However, we have found that the conclusions drawn in the present study hold with Because Eq.~\eqref{eq:EcBSE} requires the entire BSE singlet excitation spectrum for each quadrature point, we perform several complete diagonalization of the $\Nocc \Nvir \times \Nocc \Nvir$ BSE linear response matrix [see Eq.~\eqref{eq:small-LR}], which corresponds to a $\order{\Nocc^3 \Nvir^3} = \order{\Norb^6}$ computational cost. This step is, by far, the computational bottleneck in our current implementation. -However, we are currently pursuing different avenues to lower this cost by computing the two-electron density matrix of Eq.~\eqref{eq:2DM} via a quadrature in frequency space. +However, we are currently pursuing different avenues to lower this cost by computing the two-electron density matrix of Eq.~\eqref{eq:2DM} via a quadrature in frequency space. \cite{Duchemin_2019,Duchemin_2020} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\section{Potential energy surfaces} %\label{sec:PES} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In order to illustrate the performance of the BSE-based adiabatic connection formulation, we have computed the ground-state PES of several diatomic closed-shell molecules around their equilibrium geometry: \ce{H2}, \ce{LiH}, \ce{LiF}, \ce{N2}, \ce{CO}, \ce{BF}, \ce{F2}, and \ce{HCl}. -The PES of these molecules for various methods and Dunning's triple-$\zeta$ basis cc-pVTZ are represented in Fig.~\ref{fig:PES}, and the comuted equilibrium distances are gathered in Table \ref{tab:Req}. +The PES of these molecules for various methods and Dunning's triple-$\zeta$ basis cc-pVTZ are represented in Fig.~\ref{fig:PES}, and the computed equilibrium distances are gathered in Table \ref{tab:Req}. Additional graphs for other basis sets can be found in the {\SI}. +For comparison purposes, we have also computed the PES at the MP2, CC2 \cite{Christiansen_1995}, CCSD, \cite{Purvis_1982} and CC3 \cite{Christiansen_1995b} levels of theory using DALTON. \cite{dalton} + %%% FIG 1 %%% \begin{figure*} \includegraphics[width=0.45\linewidth]{H2_GS_VTZ}