From eeb99ea40c56f727114c82e4aafad6818f11f17b Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sat, 25 Jan 2020 23:20:22 +0100 Subject: [PATCH] reworked intro --- BSE-PES.tex | 107 +++++++++++++++++++++++++++++----------------------- 1 file changed, 59 insertions(+), 48 deletions(-) diff --git a/BSE-PES.tex b/BSE-PES.tex index 9e55b4d..94e0923 100644 --- a/BSE-PES.tex +++ b/BSE-PES.tex @@ -183,46 +183,49 @@ However, we also observe, in some cases, unphysical irregularities on the ground %\label{sec:intro} %%%%%%%%%%%%%%%%%%%%%%%% -The features of ground- and excited-state potential energy surfaces (PES) are critical for the faithful description and a deeper understanding of photochemical and photophysical processes. \cite{Bernardi_1996,Olivucci_2010,Robb_2007} -For example, chemoluminescence, fluorescence and other related processes are associated with geometric relaxation of excited states, and structural changes upon electronic excitation. \cite{Navizet_2011} -Reliable predictions of these mechanisms which have attracted much experimental and theoretical interest lately require exploring the ground- and excited-state PES. -From a theoretical point of view, the accurate prediction of excited electronic states remains a challenge, \cite{Gonzales_2012, Loos_2020a} especially for large systems where state-of-the-art computational techniques (such as multiconfigurational methods \cite{Andersson_1990,Andersson_1992,Roos_1996,Angeli_2001}) cannot be afforded. -For such systems, one has to rely on more approximate, yet computationally cheaper approaches. \cite{Grimme_2004a,Ghosh_2018} +%The features of ground- and excited-state potential energy surfaces (PES) are critical for the faithful description and a deeper understanding of photochemical and photophysical processes. \cite{Bernardi_1996,Olivucci_2010,Robb_2007} +%For example, chemoluminescence, fluorescence and other related processes are associated with geometric relaxation of excited states, and structural changes upon electronic excitation. \cite{Navizet_2011} +%Reliable predictions of these mechanisms which have attracted much experimental and theoretical interest lately require exploring the ground- and excited-state PES. +%From a theoretical point of view, the accurate prediction of excited electronic states remains a challenge, \cite{Gonzales_2012, Loos_2020a} especially for large systems where state-of-the-art computational techniques (such as multiconfigurational methods \cite{Andersson_1990,Andersson_1992,Roos_1996,Angeli_2001}) cannot be afforded. +%For such systems, one has to rely on more approximate, yet computationally cheaper approaches. \cite{Grimme_2004a,Ghosh_2018} -For the last two decades, time-dependent density-functional theory (TD-DFT) \cite{Casida} has been the go-to method to compute absorption and emission spectra in large molecular systems. -At a relatively low computational cost, TD-DFT can provide accurate vertical and adiabatic transition energies for low-lying excited states of organic molecules with a typical error of $0.2$--$0.4$ eV. \cite{Loos_2019b} -Similar to density-functional theory (DFT), \cite{Hohenberg_1964,Kohn_1965,ParrBook} TD-DFT is an in-principle exact theory which formal foundation relies on the Runge-Gross theorem. \cite{Runge_1984} -The Kohn-Sham (KS) formalism of TD-DFT transfers the complexity of the many-body problem to the exchange-correlation (xc) functional thanks to a judicious mapping between a time-dependent non-interacting reference system and its interacting analog which have both the exact same electronic density. -In TD-DFT, the excitation energies are obtained as the poles of the ground-state frequency-dependent linear response function, which is obtained by solving a Dyson equation in which the exchange and correlation effects are cast in the xc kernel. -This xc kernel plays for the excited states the same role as the xc functional for the ground state. -Hence, the PES for the excited states can be easily and efficiently obtained as a function of the molecular geometry by simply adding the ground-state DFT energy to the excitation energy of the selected state. -One of the strongest assets of TD-DFT is the availability of first- and second-order analytic nuclear gradients (\ie, the first derivatives of the excited-state energy with respect to the nuclear displacements), which enables the exploration of excited-state PES.\cite{Furche_2002} +%For the last two decades, time-dependent density-functional theory (TD-DFT) \cite{Casida,Runge_1984} has been the go-to method to compute absorption and emission spectra in large molecular systems. +%At a relatively low computational cost, TD-DFT can provide accurate vertical and adiabatic transition energies for low-lying excited states of organic molecules with a typical error of $0.2$--$0.4$ eV. \cite{Loos_2019b} +%Similar to density-functional theory (DFT), \cite{Hohenberg_1964,Kohn_1965,ParrBook} TD-DFT is an in-principle exact theory which formal foundation relies on the Runge-Gross theorem. \cite{Runge_1984} +%The Kohn-Sham (KS) formalism of TD-DFT transfers the complexity of the many-body problem to the exchange-correlation (xc) functional thanks to a judicious mapping between a time-dependent non-interacting reference system and its interacting analog which have both the exact same electronic density. +%In TD-DFT, the excitation energies are obtained as the poles of the ground-state frequency-dependent linear response function, which is obtained by solving a Dyson equation in which the exchange and correlation effects are cast in the exchange-correlation (xc) kernel. +%This xc kernel plays for the excited states the same role as the xc functional for the ground state. +%Hence, the PES for the excited states can be easily and efficiently obtained as a function of the molecular geometry by simply adding the ground-state DFT energy to the excitation energy of the selected state. +%One of the strongest assets of TD-DFT is the availability of first- and second-order analytic nuclear gradients (\ie, the first derivatives of the excited-state energy with respect to the nuclear displacements), which enables the exploration of ground- and excited-state PES. \cite{Furche_2002} -Although the exact spectrum of the interacting system can be obtained in principle, both the ground-state xc potential and the xc kernel (which enters the definition of the response function) have to be approximated in practice. -This has major consequences, and it is well documented that, TD-DFT fails badly to model, for examples, charge transfer,\cite{Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004,Maitra_2017} Rydberg \cite{Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tozer_2003} and doubly excited states. \cite{Levine_2006,Tozer_2000,Elliott_2011} -An important point from a practical point of view is that the TD-DFT xc kernel is usually considered as static instead of being frequency dependent. \cite{Romaniello_2009a,Sangalli_2011} -One key consequence of this so-called adiabatic approximation (based on the assumption that the density varies slowly with time) is that double excitations are completely absent from the TD-DFT spectra. \cite{Levine_2006,Tozer_2000,Huix-Rotllant_2010,Elliott_2011} -Yet another problem is the choice of the xc functionals as the quality of excitation energies are substantially dependent on this choice. +%Although the exact spectrum of the interacting system can be obtained in principle, both the ground-state xc potential and the xc kernel (which enters the definition of the response function) have to be approximated in practice. +%This has major consequences, and it is well documented that, TD-DFT fails badly to model, for examples, charge transfer,\cite{Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004,Maitra_2017} Rydberg \cite{Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tozer_2003} and doubly excited states. \cite{Levine_2006,Tozer_2000,Elliott_2011} +%An important point from a practical point of view is that the TD-DFT xc kernel is usually considered as static instead of being frequency dependent. \cite{Romaniello_2009a,Sangalli_2011} +%One key consequence of this so-called adiabatic approximation (based on the assumption that the density varies slowly with time) is that double excitations are completely absent from the TD-DFT spectra. \cite{Levine_2006,Tozer_2000,Huix-Rotllant_2010,Elliott_2011} +%Yet another problem is the choice of the xc functionals as the quality of excitation energies are substantially dependent on this choice. -With a similar computational cost, the many-body Green's function Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} is a very valuable alternative to TD-DFT with early \textit{ab initio} calculations in condensed matter physics appearing at the end of the 90's. \cite{Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} +With a similar computational cost to time-dependent density-functional theory (TD-DFT), \cite{Casida,Runge_1984} the many-body Green's function Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} is a valuable alternative with early \textit{ab initio} calculations in condensed matter physics appearing at the end of the 90's. \cite{Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} In the past few years, BSE has gained momentum for the study of molecular systems \cite{Ma_2009,Pushchnig_2002,Tiago_2003,Tiago_2008,Sai_2008,Palumno_2009,Rocca_2010,Sharifzadeh_2012,Cudazzo_2012,Boulanger_2014,Ljungberg_2015,Hirose_2015,Cocchi_2015,Ziaei_2017,Abramson_2017} and is now a serious candidate as a computationally inexpensive method that can effectively model excited states with a typical error of $0.1$--$0.3$ eV according to large and systematic benchmark calculations. \cite{Jacquemin_2015,Bruneval_2015,Blase_2016,Jacquemin_2016,Hung_2016,Hung_2017,Krause_2017,Jacquemin_2017,Blase_2018} One of the main advantage of BSE compared to TD-DFT is that it allows a faithful description of charge-transfer states. \cite{Lastra_2011,Blase_2011b,Baumeier_2012,Duchemin_2012,Cudazzo_2013,Ziaei_2016} Moreover, when performed on top of a (partially) self-consistently {\evGW} calculation, \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011,Rangel_2016,Kaplan_2016,Gui_2018} BSE@{\evGW} has been shown to be weakly dependent on its starting point (\ie, on the xc functional selected for the underlying DFT calculation). \cite{Jacquemin_2016,Gui_2018} -However, similar to TD-DFT, the static version of BSE cannot describe multiple excitations. \cite{Romaniello_2009a,Sangalli_2011} +However, similar to adiabatic TD-DFT, \cite{Levine_2006,Tozer_2000,Huix-Rotllant_2010,Elliott_2011} the static version of BSE cannot describe multiple excitations. \cite{Romaniello_2009a,Sangalli_2011} -A significant limitation of the BSE formalism as compared to TD-DFT lies in the lack of analytic forces for the excited states, preventing efficient applications to the study of photoluminescence, photocatalysis, etc. While calculations of the {\GW} quasiparticle energies ionic gradients is becoming very popular, +A significant limitation of the BSE formalism, as compared to TD-DFT, \cite{Furche_2002} lies in the lack of analytic forces for both the ground and excited states, preventing efficient applications to the study of chemoluminescence, fluorescence and other related processes associated with geometric relaxation of ground and excited states, and structural changes upon electronic excitation. \cite{Navizet_2011} +While calculations of the {\GW} quasiparticle energies ionic gradients is becoming 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 study devoted to small molecules (\ce{CO} and \ce{NH3}), only the BSE excitation energy gradients were calculated, while computing the KS-DFT (LDA) forces as ground-state analogue. -Contrary to TD-DFT, the ground-state correlation energy calculated at the BSE level remains in its infancy with very few available studies for atomic and molecular systems. \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020} +Contrary to TD-DFT, the BSE ground-state correlation energy is not a well-defined quantity, and remains in its infancy with very few available studies for atomic and molecular systems. \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020} As a matter of fact, in the largest recent available benchmark study of the 26 small molecules forming the HEAT test set, \cite{Holzer_2018} the BSE correlation energy, as evaluated within the adiabatic-connection fluctuation-dissipation (ACFDT) framework, 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} 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 coupling parameter $\lambda$ in the Dyson equation for the interacting polarizability leads to two different versions of the BSE correlation energy. Here, in analogy to the random-phase approximation (RPA)-type formalismes, \cite{Furche_2008, Angyan_2011, Holzer_2018} the ground-state BSE energy is calculated in the adiabatic-connection fluctuation-dissipation theorem framework. -Embracing this definition, the purpose of the present study is to investigate the quality of ground--state PES near equilibrium obtained within the BSE approach. +Embracing this definition, the purpose of the present study is to investigate the quality of ground-state PES near equilibrium obtained within the BSE approach of several diatomic molecules. The location of the minima on the ground-state PES is of particular interest. This study is a first preliminary step towards the development of analytical nuclear gradients within the BSE@{\GW} formalism. +Thanks to comparison with both similar and state-of-art computational approaches, we show that the present ACFDT-based approach is surprisingly accurate, and can even compete with high-order coupled cluster (CC) methods. +However, we also observe, in some cases, unphysical irregularities on the ground-state PES, which are due to the appearance of a satellite resonance with a weight similar to that of the $GW$ quasiparticle peak. \cite{Loos_2018, Veril_2018} %The paper is organized as follows. %In Sec.~\ref{sec:theo}, we introduce the equations behind the BSE formalism. @@ -292,7 +295,7 @@ For a closed-shell system, in order to compute the singlet BSE excitation energi \bY{\IS}_m \\ \end{pmatrix}, \end{equation} -where $\Om{m}{\IS}$ is the $m$th excitation energy with eigenvector $\T{(\bX{\IS}_m \, \bY{\IS}_m)}$ at interpolation strength $\IS$, and we have assumed real-valued spatial orbitals $\{\MO{p}(\br{})\}_{1 \le p \le \Norb}$. +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. 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. @@ -370,6 +373,7 @@ is the ground-state BSE correlation energy computed in the adiabatic connection \end{equation} is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\tA{ia,jb}{\IS} = \IS \ERI{ia}{bj}$], \begin{equation} +\label{eq:2DM} \bP{\IS} = \begin{pmatrix} \bY{\IS} \T{(\bY{\IS})} & \bY{\IS} \T{(\bX{\IS})} \\ @@ -389,9 +393,11 @@ For spin-restricted closed-shell molecular systems around their equilibrium geom However, they may appear in the presence of strong correlation (\eg, when the bond is stretch). 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 the ACFDT appraoch. -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. - +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. +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} @@ -401,20 +407,42 @@ All the preliminary {\GW} calculations performed to obtain the screened Coulomb Perturbative {\GW} (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} calculations are employed as starting point to compute the BSE neutral excitations. These will be labeled as BSE@{\GOWO}. In the case of {\GOWO}, the quasiparticle energies have been obtained by linearizing the non-linear, frequency-dependent quasiparticle equation. -Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018,Veril_2018}. +Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018, Veril_2018}. Finally, the infinitesimal $\eta$ has been set to zero for all calculations. The numerical integration required to compute the correlation energy along the adiabatic path has been performed with a 21-point Gauss-Legendre quadrature. As one-electron basis sets, we employ the Dunning family (cc-pVXZ) defined with cartesian gaussian functions. Unless, otherwise stated, the frozen-core approximation has not been enforced in our calculations in order to provide a fairer comparison between methods. However, we have found that the conclusions drawn in the present study hold within the frozen-core approximation (see {\SI}). -Because Eq.~\eqref{eq:EcBSE} requires the entire BSE excitation spectrum (both singlet and triplet), we perform a 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. +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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\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}. +Additional graphs for other basis sets can be found in the {\SI}. + +%%% FIG 1 %%% +\begin{figure*} + \includegraphics[width=0.45\linewidth]{H2_GS_VTZ} + \includegraphics[width=0.45\linewidth]{LiH_GS_VTZ} + \includegraphics[width=0.45\linewidth]{LiF_GS_VTZ} + \includegraphics[width=0.45\linewidth]{N2_GS_VTZ} + \includegraphics[width=0.45\linewidth]{CO_GS_VTZ} + \includegraphics[width=0.45\linewidth]{BF_GS_VTZ} + \includegraphics[width=0.45\linewidth]{F2_GS_VTZ} + \includegraphics[width=0.45\linewidth]{HCl_GS_VTZ} +\caption{ +PES of the ground state of diatomic molecules around their respective equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set. +Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}. +\label{fig:PES} +} +\end{figure*} +%%% %%% %%% %%% TABLE I %%% \begin{table*} @@ -484,24 +512,6 @@ Equilibrium distances (in bohr) of the ground state of diatomic molecules obtain \end{ruledtabular} \end{table*} - -%%% FIG 1 %%% -\begin{figure*} - \includegraphics[width=0.45\linewidth]{H2_GS_VTZ} - \includegraphics[width=0.45\linewidth]{LiH_GS_VTZ} - \includegraphics[width=0.45\linewidth]{LiF_GS_VTZ} - \includegraphics[width=0.45\linewidth]{N2_GS_VTZ} - \includegraphics[width=0.45\linewidth]{CO_GS_VTZ} - \includegraphics[width=0.45\linewidth]{BF_GS_VTZ} - \includegraphics[width=0.45\linewidth]{F2_GS_VTZ} - \includegraphics[width=0.45\linewidth]{HCl_GS_VTZ} -\caption{ -PES of the ground state of diatomic molecules around their respective equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set. -Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}. -\label{fig:PES} -} -\end{figure*} -%%% %%% %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\subsection{Hydrogen molecule} %\label{sec:H2} @@ -526,10 +536,11 @@ Additional graphs for other basis sets and within the frozen-core approximation %%%%%%%%%%%%%%%%%%%%%%%% \section*{Supporting Information} %%%%%%%%%%%%%%%%%%%%%%%% -See {\SI} for \titou{bla bla bla.} +See {\SI} for additional potential energy curves with other basis sets and within the frozen-core approximation. %%%%%%%%%%%%%%%%%%%%%%%% \begin{acknowledgements} +PFL would like to thank Anthony Scemama for technical assistance. This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005. Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged. This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''}.