From 8b87e97f23a94d3bb697e19eb7138f35ef59d6cc Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 5 May 2020 13:57:08 +0200 Subject: [PATCH] Clean up up to results --- Manuscript/FarDFT.tex | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 64ac8fe..589705b 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -185,12 +185,12 @@ In other words, memory effects are absent from the xc functional which is assume (the xc energy is in fact an xc action, not an energy functional). \cite{Vignale_2008} Third and more importantly in the present context, a major issue of TD-DFT actually originates directly from the choice of the xc functional, and more specifically, the possible (not to say likely) substantial variations in the quality of the excitation energies for two different choices of xc functionals. -Because its popularity, approximate TD-DFT has been studied in excruciated details by the community, and some researchers have quickly unveiled various theoretical and practical deficiencies. +Because of its popularity, approximate TD-DFT has been studied in excruciated details by the community, and some researchers have quickly unveiled various theoretical and practical deficiencies. For example, TD-DFT has problems with charge-transfer \cite{Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004,Maitra_2017} and Rydberg \cite{Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tozer_2003} excited states (the excitation energies are usually drastically underestimated) due to the wrong asymptotic behaviour of the semi-local xc functional. The development of range-separated hybrids provides an effective solution to this problem. \cite{Tawada_2004,Yanai_2004} From a practical point of view, the TD-DFT xc kernel is usually considered as static instead of being frequency dependent. 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,Elliott_2011} -Although these double excitations are usually experimentally dark (which means they usually cannot be observed in photo-absorption spectroscopy), these states play, indirectly, a key role in many photochemistry mechanisms. \cite{Boggio-Pasqua_2007} They are, moreover, a real challenge for high-level computational methods. \cite{Loos_2018,Loos_2019,Loos_2020b} +Although these double excitations are usually experimentally dark (which means that they usually cannot be observed in photo-absorption spectroscopy), these states play, indirectly, a key role in many photochemistry mechanisms. \cite{Boggio-Pasqua_2007} They are, moreover, a real challenge for high-level computational methods. \cite{Loos_2018,Loos_2019,Loos_2020b} One possible solution to access double excitations within TD-DFT is provided by spin-flip TD-DFT which describes double excitations as single excitations from the lowest triplet state. \cite{Huix-Rotllant_2010,Krylov_2001,Shao_2003,Wang_2004,Wang_2006,Minezawa_2009} However, spin contamination might be an issue. \cite{Huix-Rotllant_2010} @@ -220,8 +220,8 @@ Electrons restricted to remain on the surface of a $\cD$-sphere (where $\cD$ is %did in our previous work we should motivate the use of FUEGs for %developing weight-dependent functionals.} Very recently, \cite{Loos_2020} two of the present authors have taken advantages of these FUEGs to construct a local, weight-dependent correlation functional specifically designed for one-dimensional many-electron systems. -Unlike any standard functional, this first-rung functional incorporates derivative discontinuities thanks to its natural weight dependence, and has shown to deliver accurate excitation energies for both single and double excitations. -Extending this methodology to more realistic (atomic and molecular) systems, we combine here these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA correlation functional for ensembles, which is specifically designed to compute double excitations within GOK-DFT, and automatically incorporates the infamous ensemble derivative contribution to the excitation energies through its explicit ensemble weight dependence. \cite{Levy_1995, Perdew_1983} +Unlike any standard functional, this first-rung functional automatically incorporates ensemble derivative contributions thanks to its natural weight dependence, \cite{Levy_1995, Perdew_1983} and has shown to deliver accurate excitation energies for both single and double excitations. +Extending this methodology to more realistic (atomic and molecular) systems, we combine here these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA correlation functional for ensembles, which is specifically designed to compute double excitations within GOK-DFT. The paper is organised as follows. In Sec.~\ref{sec:theo}, the theory behind GOK-DFT is presented. @@ -558,7 +558,7 @@ For all calculations, we use the aug-cc-pVXZ (X = D, T, Q, and 5) Dunning family Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} using 194 angular points (Lebedev grid) and a radial precision of $10^{-7}$. \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 three-state ensemble (\ie, $\nEns = 3$) where the ground state ($I=0$ with weight $1 - \ew{1} - \ew{2}$), the lowest singly-excited state ($I=1$ with weight $\ew{1}$), as well as the lowest doubly-excited state ($I=2$ with weight $\ew{2}$) are considered. +Moreover, we restrict our study to the case of a three-state ensemble (\ie, $\nEns = 3$) where the ground state ($I=0$ with weight $1 - \ew{1} - \ew{2}$), a singly-excited state ($I=1$ with weight $\ew{1}$), as well as the lowest doubly-excited state ($I=2$ with weight $\ew{2}$) are considered. Assuming that the singly-excited state is lower in energy than the doubly-excited state (which is not always the case as one would notice later), one should have $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1 - \ew{2})/2$ to ensure the GOK variational principle. %Taking a generic two-electron system as an example, the individual one-electron densities read %\begin{subequations} @@ -599,7 +599,7 @@ The pure-state limit, $\ew{1} = 0 \land \ew{2} = 1$, is nonetheless of particula \section{Results and Discussion} \label{sec:res} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -In this Section, we propose a two-step procedure to design, first, a weight- and system-dependent local exchange functional in order to remove the curvature of the ensemble energy. +In this Section, we propose a two-step procedure to design, first, a weight- and system-dependent local exchange functional in order to remove some of the curvature of the ensemble energy. Second, we describe the construction of a universal, weight-dependent local correlation functional based on FUEGs. This procedure is applied to various two-electron systems in order to extract excitation energies associated with doubly-excited states.