diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 4e37cb1..c1292eb 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -161,14 +161,14 @@ In the present article, we discuss the construction of first-rung (\textit{i.e.} %%%%%%%%%%%%%%%%%%%% \section{Introduction} Time-dependent density-functional theory (TD-DFT) has been the dominant force in the calculation of excitation energies of molecular systems in the last two decades.\cite{Casida_1995,Ulrich_2012,Loos_2020a} -\titou{At a moderate computational cost} (at least compared to the other excited-state \textit{ab initio} methods), TD-DFT can provide accurate transition energies for low-lying excited states of organic molecules (see, for example, Ref.~\onlinecite{Dreuw_2005} and references therein). +At a moderate computational cost (at least compared to the other excited-state \textit{ab initio} methods), TD-DFT can provide accurate transition energies for low-lying excited states of organic molecules (see, for example, Ref.~\onlinecite{Dreuw_2005} and references therein). \titou{Importantly, within the widely-used adiabatic approximation, setting up a TD-DFT calculation for a given system is an almost pain-free process from a user perspective as the only (yet essential) input variable is the choice of the ground-state exchange-correlation (xc) functional.} Similar to density-functional theory (DFT), \cite{Hohenberg_1964,Kohn_1965,ParrBook} TD-DFT is an in-principle exact theory which formal foundations relie on the Runge-Gross theorem. \cite{Runge_1984} -The Kohn-Sham (KS) \titou{formulation} of TD-DFT transfers the +The Kohn-Sham (KS) formulation of TD-DFT transfers the complexity of the many-body problem to the xc functional thanks to a judicious mapping between a time-dependent non-interacting reference system and its interacting analog \titou{which have both @@ -235,23 +235,19 @@ Unless otherwise stated, atomic units are used throughout. \section{Theory} \label{sec:theo} -Let us consider a GOK ensemble of $\nEns$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\nEns-1)}$, and (normalised) monotonically decreasing weights $\bw = (\ew{0},\ldots,\ew{M-1})$, \ie, $\sum_{I=0}^{\nEns-1} \ew{I} = 1$, and $\ew{0} \ge \ldots \ge \ew{\nEns-1}$. -\manu{For clarity, I usually exclude $\ew{0}$ from $\bw$ so that $\bw$ -only contains the weights that are allowed to vary independently. One -should write explicitly $\ew{0}=1-\sum_{I=1}^{\nEns-1} \ew{I}$ and -define $\bw$ as $\bw = (\ew{1},\ldots,\ew{M-1})$} +Let us consider a GOK ensemble of $\nEns$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\nEns-1)}$, and (normalised) monotonically decreasing weights $\bw = (\ew{1},\ldots,\ew{M-1})$, \ie, $\ew{0}=1-\sum_{I=1}^{\nEns-1} \ew{I}$, and $\ew{0} \ge \ldots \ge \ew{\nEns-1}$. The corresponding ensemble energy \begin{equation} \E{}{\bw} = \sum_{I=0}^{\nEns-1} \ew{I} \E{}{(I)} \end{equation} -fulfils \manu{can be obtained from?} the variational principle +can be obtained from the variational principle as follows\cite{Gross_1988a} \begin{eqnarray}\label{eq:ens_energy} \E{}{\bw} = \min_{\hGam{\bw}} \Tr[\hGam{\bw} \hH], \end{eqnarray} where $\hH = \hT + \hWee + \hVne$ contains the kinetic, electron-electron and nuclei-electron interaction potential operators, -respectively, $\Tr$ denotes the trace and $\hGam{\bw}$ is a trial +respectively, $\Tr$ denotes the trace, and $\hGam{\bw}$ is a trial density matrix operator of the form \begin{eqnarray} \hGam{\bw} = \sum_{I=0}^{\nEns - 1} \ew{I} \dyad*{\overline{\Psi}^{(I)}}, @@ -259,7 +255,7 @@ density matrix operator of the form where $\lbrace \overline{\Psi}^{(I)} \rbrace_{0 \le I \le \nEns-1}$ is a set of $\nEns$ orthonormal trial wave functions. The lower bound of Eq.~\eqref{eq:ens_energy} is reached when the set of wave functions correspond to the exact eigenstates of $\hH$, \ie, $\lbrace \overline{\Psi}^{(I)} \rbrace_{0 \le I \le \nEns-1} = \lbrace \Psi^{(I)} \rbrace_{0 \le I \le \nEns-1}$. Multiplet degeneracies can be easily handled by assigning the same -weight to the degenerate states \cite{Gross_1988b}. +weight to the degenerate states. \cite{Gross_1988b} One of the key feature of the GOK ensemble is that individual excitation energies can be extracted from the ensemble energy via differentiation with respect to individual weights: \begin{equation}\label{eq:diff_Ew} @@ -274,54 +270,49 @@ Turning to GOK-DFT, the extension of the Hohenberg--Kohn theorem to ensembles al where $\vne(\br{})$ is the external potential and $\F{}{\bw}[\n{}{}]$ is the universal ensemble functional (the weight-dependent analog of the Hohenberg--Kohn universal functional for ensembles). In the KS formulation, this functional can be decomposed as -\begin{equation} - \F{}{\bw}[\n{}{}] - = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}] - = \Tr[ \hgam{\bw} \hT ] + \Tr[ \hgam{\bw} \hWee ], -\end{equation} -\manu{The above equation is wrong (the correlation is missing) and the -notations are ambiguous. I should also say that Tim does not like the -original separation into H and xc. I propose the following reformulation -to get everyone satisfied. I also reorganized the theory for clarity. +%\begin{equation} +% \F{}{\bw}[\n{}{}] +% = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}] +% = \Tr[ \hgam{\bw} \hT ] + \Tr[ \hgam{\bw} \hWee ], +%\end{equation} +%\manu{The above equation is wrong (the correlation is missing) and the +%notations are ambiguous. I should also say that Tim does not like the +%original separation into H and xc. I propose the following reformulation +%to get everyone satisfied. I also reorganized the theory for clarity. \begin{equation}\label{eq:FGOK_decomp} \F{}{\bw}[\n{}{}] - = \Tr[ \hgamdens{\bw} \hT ]+ \E{\Ha}{}[\n{}{}]+\E{\xc}{\bw}[\n{}{}], + = \Tr{ \hgamdens{\bw} \hT }+ \E{\Ha}{}[\n{}{}]+\E{\xc}{\bw}[\n{}{}], \end{equation} -} where -\manuf{$\Tr[ \hgamdens{\bw} \hT ]=\Ts{\bw}[\n{}{}]$} is the noninteracting ensemble kinetic energy functional, +$\Tr{ \hgamdens{\bw} \hT } =\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kinetic energy functional, \begin{equation} \hgam{\bw}[n] = \sum_{I=0}^{\nEns-1} \ew{I} \dyad{\Det{I}{\bw}[n]} \end{equation} -is the \manuf{KS density-functional} density matrix operator, and $\lbrace +is the KS density-functional density matrix operator, and $\lbrace \Det{I}{\bw}[n] \rbrace_{0 \le I \le \nEns-1}$ are single-determinant -wave functions (or configuration state functions). \manuf{Their -dependence on the density -is determined from the ensemble density -constraint +wave functions (or configuration state functions). +Their dependence on the density is determined from the ensemble density constraint \begin{equation} -\sum_{I=0}^{\nEns-1} \ew{I} n_{\Det{I}{\bw}[n]}(\br)=n(\br). + \sum_{I=0}^{\nEns-1} \ew{I} \n{\Det{I}{\bw}[n]}{}(\br) = \n{}{}(\br). \end{equation} -Note that the original decomposition \cite{Gross_1988b} shown in Eq.~(\ref{eq:FGOK_decomp}), where the +Note that the original decomposition \cite{Gross_1988b} shown in Eq.~\eqref{eq:FGOK_decomp}, where the conventional (weight-independent) Hartree functional \beq -\E{\Ha}{}[\n{}{}]=\frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}' + \E{\Ha}{}[\n{}{}]=\frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}' \eeq is separated from the (weight-dependent) exchange-correlation (xc) functional, is formally exact. In practice, the use of such a decomposition might be problematic as inserting an ensemble density into $\E{\Ha}{}[\n{}{}]$ -causes the infamous ghost-interaction error \cite{Gidopoulos_2002, -Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}. The latter should in -principle be removed by the exchange component of the ensemble xc -functional $\E{\xc}{\bw}[\n{}{}]\equiv -\E{\ex}{\bw}[\n{}{}]+\E{\co}{\bw}[\n{}{}]$, as readily seen from the -exact expression +causes the infamous ghost-interaction error. \cite{Gidopoulos_2002,Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} +The latter should in principle be removed by the exchange component of the ensemble xc functional +$\E{\xc}{\bw}[\n{}{}] \equiv \E{\ex}{\bw}[\n{}{}] + \E{\co}{\bw}[\n{}{}]$, +as readily seen from the exact expression \beq -\E{\ex}{\bw}[\n{}{}]=\sum_{I=0}^{\nEns-1} \ew{I}\bra{\Det{I}{\bw}[n]}\hat{W}_{\rm ee}\ket{\Det{I}{\bw}[n]} --\E{\Ha}{}[\n{}{}]. + \E{\ex}{\bw}[\n{}{}] + = \sum_{I=0}^{\nEns-1} \ew{I}\mel{\Det{I}{\bw}[\n{}{}]}{\hWee}{\Det{I}{\bw}[\n{}{}]} - \E{\Ha}{}[\n{}{}]. \eeq -The minimum in Eq.~(\ref{eq:Ew-GOK}) is reached when the density $n$ +The minimum in Eq.~\eqref{eq:Ew-GOK} is reached when the density $n$ equals the exact ensemble one \beq\label{eq:nw} n^{\bw}(\br)=\sum_{I=0}^{\nEns-1} @@ -341,7 +332,7 @@ result, the orbitals $\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le \nOrb}$ from which the KS wavefunctions $\left\{\Det{I}{\bw}\left[n^{\bw}\right]\right\}_{0\leq -I\leq M-1}$ are constructed can be obtained by solving the following ensemble KS equation +I\leq \nEns-1}$ are constructed can be obtained by solving the following ensemble KS equation \begin{equation} \label{eq:eKS} \qty{ \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{\bw}]}{\n{}{}(\br{})}} \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}), @@ -355,7 +346,7 @@ where $\hHc(\br{}) = -\nabla^2/2 + \vne(\br{})$, and + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}. \end{equation} The ensemble density can be obtained directly (and exactly, if no -approximation is made) from those orbitals: +approximation is made) from these orbitals, \ie, \beq\label{eq:ens_KS_dens} \n{}{\bw}(\br{})=\sum_{I=0}^{\nEns-1} \ew{I}\left(\sum_{p}^{\nOrb} \ON{p}{(I)} [\MO{p}{\bw}(\br{})]^2\right), @@ -363,19 +354,19 @@ approximation is made) from those orbitals: where $\ON{p}{(I)}$ denotes the occupation of $\MO{p}{\bw}(\br{})$ in the $I$th KS wave function $\Det{I}{\bw}\left[n^{\bw}\right]$. Turning to the excitation energies, they can be extracted from the -density-functional ensemble as follows [see Eqs. ({\ref{eq:diff_Ew}}) -and ({\ref{eq:Ew-GOK}}) and Refs. -\cite{Gross_1988b,Deur_2019}]: +density-functional ensemble as follows [see Eqs.~\eqref{eq:diff_Ew} +and \eqref{eq:Ew-GOK} and Refs.~\onlinecite{Gross_1988b,Deur_2019}]: \beq -\label{eq:dEdw} -\Omega^{(I)}= \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}}, + \label{eq:dEdw} + \Omega^{(I)}= \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}}, \eeq where \begin{equation} \label{eq:KS-energy} \Eps{I}{\bw} = \sum_{p}^{\nOrb} \ON{p}{(I)} \eps{p}{\bw} \end{equation} -is the energy of the $I$th KS state.\\ +is the energy of the $I$th KS state. + Equation \eqref{eq:dEdw} is our working equation for computing excitation energies from a practical point of view. Note that the individual KS densities $\n{\Det{I}{\bw}\left[n^{\bw}\right]}{}(\br{})=\sum_{p}^{\nOrb} @@ -383,14 +374,14 @@ $\n{\Det{I}{\bw}\left[n^{\bw}\right]}{}(\br{})=\sum_{p}^{\nOrb} not necessarily match the \textit{exact} (interacting) individual-state densities $n_{\Psi_I}(\br)$ as the non-interacting KS ensemble is expected to reproduce the true interacting ensemble density $\n{}{\bw}(\br{})$ defined in Eq.~\eqref{eq:nw}, and not each individual density. Nevertheless, these densities can still be extracted in principle -exactly from the KS ensemble as shown by Fromager. -\cite{Fromager_2020}.\\ +exactly from the KS ensemble as shown by Fromager. \cite{Fromager_2020} + In the following, we will work at the (weight-dependent) LDA level of approximation, \ie \beq \E{\xc}{\bw}[\n{}{}] &\overset{\rm LDA}{\approx}& -\int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{} +\int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{}, \\ \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})} &\overset{\rm LDA}{\approx}& @@ -403,11 +394,8 @@ We will also adopt the usual decomposition, and write down the weight-dependent where $\e{\ex}{\bw{}}(\n{}{})$ and $\e{\co}{\bw{}}(\n{}{})$ are the weight-dependent density-functional exchange and correlation energies per particle, respectively. -}%%%%%% end manuf +The explicit construction of these functionals is discussed at length in Sec.~\ref{sec:res}. -\manu{Maybe we should say a little bit more about how we will design -such approximations, or just say the design of these functionals will be -presented in the following...} %%%%%%%%%%%%%%%% %%%%%%% Manu: stuff that I removed from the first version %%%%% \iffalse%%%% @@ -477,8 +465,7 @@ is the Hxc potential, with \section{Computational details} \label{sec:compdet} -The self-consistent GOK-DFT calculations \manuf{[see Eqs.~(\ref{eq:eKS}) -and (\ref{eq:ens_KS_dens})]} 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. +The self-consistent GOK-DFT calculations [see Eqs.~\eqref{eq:eKS} and \eqref{eq:ens_KS_dens}] 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_1988b,Lindh_2001} @@ -490,7 +477,7 @@ Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a g it stands a little bit beyond the theory discussed previously. What you are looking at in the range $1/2\leq \ew{}\leq 1$ are, indeed, other stationary points (than the minimizing ones) of the density matrix -operator functional in Eq.~(\ref{eq:min_KS_DM}). I would say that we +operator functional in Eq.~\eqref{eq:min_KS_DM}. I would say that we look at these solutions for analysis purposes. I personally never looked (formally) at these solutions and their physical meaning. One should clearly mention that applying GOK-DFT in this range of weights would simply @@ -541,6 +528,10 @@ As anticipated, $\E{}{\ew{}}$ is far from being linear, which means that the exc Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $8$ eV from $\ew{} = 0$ to $1/2$. Note that the exact xc ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of $\ew{}$. +\titou{The eDFT purist is going to surprised to see that we left out the singly-excited states $\sigma_g \sigma_u$ from the ensemble as this state is lower in energy than the doubly-excited state of configuration $1\sigma_u^2$. +As we wish to stick with a restricted formalism, the single excitation is naturally left out of the ensemble. +However, as a sanity check, we have tried to introduce the single excitations as well.} + \begin{figure} \includegraphics[width=\linewidth]{Ew_H2} \caption{ @@ -762,7 +753,7 @@ As shown in Fig.~\ref{fig:Ew_H2}, the GIC-SeVWN5 is slightly less concave than i For a more qualitative picture, Table \ref{tab:BigTab_H2} reports excitation energies for various methods and basis sets. In particular, we report the excitation energies obtained with GOK-DFT in the zero-weight limit (\ie, $\ew{} = 0$) and for the equi-ensemble -(\ie, $\ew{} = 1/2$). \manu{Maybe we should refer to Eq.~(\ref{eq:dEdw}) for clarity.} +(\ie, $\ew{} = 1/2$). \manu{Maybe we should refer to Eq.~\eqref{eq:dEdw} for clarity.} For comparison purposes, we also report the linear interpolation method (LIM) excitation energies, \cite{Senjean_2015,Senjean_2016} a pragmatic way of getting weight-independent excitation energies defined as