diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index b79e508..eef09af 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -220,7 +220,11 @@ In the KS formulation of eDFT, the universal ensemble functional (the weight-dep \begin{equation} \F{}{\bw}[\n{}{}] = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}], \end{equation} -where $\Ts{\bw}[\n{}{}]$ and $\E{\Hxc}{}[\n{}{}]$ are the noninteracting ensemble kinetic energy functional and ensemble Hartree-exchange-correlation (Hxc) functional, respectively with +where +\begin{equation} + \Ts{\bw}[\n{}{}] = +\end{equation} +and \begin{equation} \label{eq:exc_def} \begin{split} @@ -230,6 +234,7 @@ where $\Ts{\bw}[\n{}{}]$ and $\E{\Hxc}{}[\n{}{}]$ are the noninteracting ensembl & = \frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}'+ \int \e{\xc}{\bw}[\n{}{}(\br{})] \n{}{}(\br{}) d\br{}. \end{split} \end{equation} +are the noninteracting ensemble kinetic energy functional and ensemble Hartree-exchange-correlation (Hxc) functional, respectively. Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$ causes the infamous ghost-interaction error (GIC) \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} in eDFT, which is supposed to be cancelled by the weight-dependent xc functional $\E{\xc}{\bw}[\n{}{}]$. From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain \cite{Gross_1988b,Deur_2019} @@ -245,14 +250,15 @@ where $\Eps{I}{\bw} = \sum_{p}^{\Norb} \ON{p}{(I)} \eps{p}{\bw}$, $\eps{p}{\bw}$ \end{equation} where $\hHc(\br{}) = -\frac{\nabla^2}{2} + \vext(\br{})$, $\MO{p}{\bw}(\br{})$ is a KS orbital, $\ON{p}{(I)}$ its occupancy for the state $I$, and $\n{}{\bw} = \sum_{I=0}^{\Nens-1} \ew{I} \n{}{(I)}$ is the ensemble density. Equation \eqref{eq:dEdw} is our working equation for computing excitation energies. - + %%%%%%%%%%%%%%%%%% %%% FUNCTIONAL %%% %%%%%%%%%%%%%%%%%% \section{Functional} \label{sec:func} The present work deals with the explicit construction of the (reduced) LDA xc functional $\e{\xc}{\bw}[\n{}{}]$ defined in Eq.~\eqref{eq:exc_def}. -Here, we restrict our study to the case of a two-state ensemble (\ie, $\Nens = 2$) where both the ground state ($I=0$) and the first doubly-excited state ($I=1$) are considered. +Here, we restrict our study to the case of a two-state ensemble (\ie, $\Nens = 2$) where both the ground state ($I=0$) and the first doubly-excited state ($I=1$) are considered. +Thus, we have $0 \le \ew{} \le 1/2$. The generalisation to a larger number of states (in particular the inclusion of the first singly-excited state) is trivial and left for future work. We adopt the usual decomposition, and write down the weight-dependent xc functional as @@ -447,8 +453,12 @@ For the sake of clarity, the explicit expression of the VWN5 functional is not r Equation \eqref{eq:becw} can be recast \begin{equation} \label{eq:eLDA} +\begin{split} \be{\xc}{\ew{}}(\n{}{}) - = \e{\xc}{\LDA}(\n{}{}) + \ew{} \qty[\e{\xc}{(1)}(\n{}{}) - \e{\xc}{(0)}(\n{}{})], + & = \e{\xc}{\LDA}(\n{}{}) + \ew{} \qty[\e{\xc}{(1)}(\n{}{}) - \e{\xc}{(0)}(\n{}{})] + \\ + & = \e{\xc}{\LDA}(\n{}{}) + \ew{} \pdv{\e{\xc}{\ew{}}(\n{}{})}{\ew{}} +\end{split} \end{equation} which nicely highlights the centrality of the LDA in the present eDFA. In particular, $\be{\xc}{(0)}(\n{}{}) = \e{\xc}{\LDA}(\n{}{})$. @@ -456,7 +466,7 @@ Consequently, in the following, we name this correlation functional ``eLDA'' as Also, we note that, by construction, \begin{equation} \label{eq:dexcdw} - \left. \pdv{\be{\xc}{\ew{}}[\n{}{}]}{\ew{I}}\right|_{\n{}{} = \n{}{\ew{}}(\br)} = \be{\xc}{(I)}[\n{}{\ew{}}(\br)] - \be{\xc}{(0)}[\n{}{\ew{}}(\br)]. + \left. \pdv{\be{\xc}{\ew{}}[\n{}{}]}{\ew{}}\right|_{\n{}{} = \n{}{\ew{}}(\br)} = \be{\xc}{(1)}[\n{}{\ew{}}(\br)] - \be{\xc}{(0)}[\n{}{\ew{}}(\br)]. \end{equation} This embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles (GACE) @@ -487,17 +497,7 @@ The bonding and antibonding orbitals of the \ce{H2} molecule are given by \end{subequations} where $\AO{A}$ and $\AO{B}$ are the two contracted Gaussian basis functions centred on each of the nucleus, and $S_{AB} = \braket{\AO{A}}{\AO{B}}$. -As reference results, we consider CID (configuration interaction with doubles) computed in the same (minimal) basis set. -The CID energies of the ground state and doubly-excited states are provided by the eigenvalues of the following CID matrix: -\begin{equation} - \bH_\CID = - \begin{pmatrix} - \E{\HF}{(0)} & \eK{12} - \\ - \eK{12} & \E{\HF}{(1)} - \end{pmatrix}, -\end{equation} -with +The HF energies of the ground state and the doubly-excited states are \begin{subequations} \begin{align} \label{eq:HF0} @@ -507,7 +507,7 @@ with \E{\HF}{(1)} & = 2 \eHc{2} + 2 \eJ{22} - \eK{22}, \end{align} \end{subequations} -and +with \begin{subequations} \begin{align} \eHc{p} & = \int \MO{p}{}(\br{}) \qty[-\frac{\nabla^2}{2} + \vext(\br{})] \MO{p}{}(\br{})d\br{}, @@ -518,8 +518,27 @@ and \end{align} \end{subequations} Note that, in the HF case, there is no self-interaction error as $\eJ{pp} = \eK{pp}$. +We also define the HF excitation energy as $\Ex{\HF}{(1)} = \E{\HF}{(1)} - \E{\HF}{(0)}$. +The HF orbital energies are +\begin{subequations} +\begin{align} + \eps{1}{\HF} & = \eHc{1} + 2\eJ{11} - \eK{11}, + \\ + \eps{2}{\HF} & = \eHc{2} + 2\eJ{12} - \eK{12}. +\end{align} +\end{subequations} -The CID energies are explicitly given by +As reference results, we consider CID (configuration interaction with doubles) computed in the same (minimal) basis set. +The CID energies of the ground state and doubly-excited states are provided by the eigenvalues of the following CID matrix: +\begin{equation} + \bH_\CID = + \begin{pmatrix} + \E{\HF}{(0)} & \eK{12} + \\ + \eK{12} & \E{\HF}{(1)} + \end{pmatrix}, +\end{equation} +These CID energies are explicitly given by \begin{subequations} \begin{align} \E{\CID}{(0)} & = \frac{\E{\HF}{(0)} + \E{\HF}{(1)}}{2} - \frac{1}{2} \sqrt{\qty(\E{\HF}{(1)} - \E{\HF}{(0)})^2 + 4 \eK{12}^2}, @@ -549,6 +568,14 @@ with \n{}{(1)}(\br{}) & = 2 \MO{2}{2}(\br{}), \end{align} Note that, contrary to the HF case, self-interaction is present in LDA. +The KS orbital energies are given by +\begin{subequations} +\begin{align} + \eps{1}{\LDA} & = \eHc{1} + 2\eJ{11} + \ldots, + \\ + \eps{2}{\LDA} & = \eHc{2} + 2\eJ{12} + \ldots. +\end{align} +\end{subequations} At the eLDA, we have \begin{subequations} @@ -562,12 +589,24 @@ At the eLDA, we have \end{subequations} with $\be{\xc}{(0)}(\n{}{}) \equiv \e{\xc}{\LDA}(\n{}{})$ and $\be{\xc}{(1)}(\n{}{}) = \e{\xc}{\LDA}(\n{}{}) + \e{\xc}{(1)}(\n{}{}) - \e{\xc}{(0)}(\n{}{})$. -%\titou{Note that we do not consider symmetry-broken solutions.} - Interestingly here, there is a strong connection between the LDA and eLDA excitation energies: \begin{equation} - \Ex{\eLDA}{(1)} = \Ex{\LDA}{(1)} + \int \qty( \e{\xc}{(1)} - \e{\xc}{(0)} )[\n{}{(1)}(\br{})] \n{}{(1)}(\br{}) d\br{}. +\begin{split} + \Ex{\eLDA}{(1)} + & = \Ex{\LDA}{(1)} + \int \qty( \e{\xc}{(1)} - \e{\xc}{(0)} )[\n{}{(1)}(\br{})] \n{}{(1)}(\br{}) d\br{}. + \\ + & = \Ex{\LDA}{(1)} + \int \left. \pdv{\e{\xc}{\ew{}}[\n{}{}]}{\ew{}} \right|_{\n{}{} = \n{}{(1)}(\br{})} \n{}{(1)}(\br{}) d\br{}. +\end{split} \end{equation} +The KS orbital energies are given by +\begin{subequations} +\begin{align} + \eps{1}{\eLDA} & = \eHc{1} + 2\eJ{11} + \ldots, + \\ + \eps{2}{\eLDA} & = \eHc{2} + 2\eJ{12} + \ldots. +\end{align} +\end{subequations} + These equations can be combined to define three ensemble energies \begin{subequations} @@ -583,16 +622,21 @@ These equations can be combined to define three ensemble energies \end{align} \end{subequations} which are all, by construction, linear with respect to $\ew{}$. +Excitation energies can be easily extracted from these formulae via differenciation with respect to $\ew{}$. -These energies given in Eqs.~\eqref{eq:EwHF}, \eqref{eq:EwLDA} and \eqref{eq:EweLDA} can also be obtained directly from the ensemble density $\n{}{\ew{}} = (1-\ew{}) \n{}{(0)} + \ew{} \n{}{(1)}$. +Similar energies than the ones given in Eqs.~\eqref{eq:EwHF}, \eqref{eq:EwLDA} and \eqref{eq:EweLDA} can also be obtained directly from the ensemble density +\begin{equation} + \n{}{\ew{}} = (1-\ew{}) \n{}{(0)} + \ew{} \n{}{(1)}. +\end{equation} (This is what one would do in practice, \ie, by performing a KS ensemble calculation.) -We will label these energies as $\bE{}{\ew{}}$. +We will label these energies as $\bE{}{\ew{}}$ to avoid confusion. \begin{widetext} For HF, we have \begin{equation} +\label{eq:bEwHF} \begin{split} \bE{\HF}{\ew{}} - & = \int \hHc(\br{}) \n{}{\ew{}}(\br{}) d\br{} + & = \titou{\int \hHc(\br{}) \n{}{\ew{}}(\br{}) d\br{}} + \frac{1}{2} \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}' \\ & = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2} @@ -602,9 +646,10 @@ For HF, we have which is clearly quadratic with respect to $\ew{}$ due to the ghost interaction error in the Hartree term. In the case of the LDA, it reads \begin{equation} +\label{eq:bEwLDA} \begin{split} \bE{\LDA}{\ew{}} - & = \int \hHc(\br{}) \n{}{\ew{}}(\br{}) d\br{} + & = \titou{\int \hHc(\br{}) \n{}{\ew{}}(\br{}) d\br{}} + \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}' + \int \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] \n{}{\ew{}}(\br{}) d\br{} \\ @@ -618,9 +663,10 @@ In the case of the LDA, it reads which is also clearly quadratic with respect to $\ew{}$ because the (weight-independent) LDA functional cannot compensate the ``quadraticity'' of the Hartree term. For eLDA, the ensemble energy can be decomposed as \begin{equation} +\label{eq:bEweLDA} \begin{split} \bE{\eLDA}{\ew{}} - & = \int \hHc(\br{}) \n{}{\ew{}}(\br{}) d\br{} + & = \titou{\int \hHc(\br{}) \n{}{\ew{}}(\br{}) d\br{}} + \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}' + \int \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] \n{}{\ew{}}(\br{}) d\br{} \\ @@ -646,12 +692,22 @@ which \textit{could} be linear with respect to $\ew{}$ if the weight-dependent x This would be, for example, the case with the exact xc functional. \end{widetext} +Extracting excitation energies from Eqs.~\eqref{eq:bEwHF}, \eqref{eq:bEwLDA} and \eqref{eq:bEweLDA} is more tricky. +To do so, we will employ Eq.~\eqref{eq:dEdw}. +The derivative discontinuity, modelled by the last term of the RHS of Eq.~\eqref{eq:dEdw} and only non-zero in the case of an explicitly weight-dependent functional, is straightforward to compute in our case [see Eq.~\eqref{eq:dexcdw}]. + +\begin{align} + \Eps{0}{\ew{}} & = 2 \eHc{1} + \ldots, + \\ + \Eps{1}{\ew{}} & = 2 \eHc{2} + \ldots. +\end{align} + %%%%%%%%%%%%%%%%%% %%% CONCLUSION %%% %%%%%%%%%%%%%%%%%% \section{Conclusion} \label{sec:ccl} -As concluding remarks, we would like to say that what we have done is awesome. +As concluding remarks, we would like to say that what we have done, we think, is awesome. %%%%%%%%%%%%%%%%%%%%%%%% %%% ACKNOWLEDGEMENTS %%%