Ts

Manuscript/FarDFT.tex

@@ -39,10 +39,11 @@
 % operators
 \newcommand{\hHc}{\Hat{h}}
 \newcommand{\hT}{\Hat{T}}
+\newcommand{\hWee}{\Hat{W}_\text{ee}}
+\newcommand{\hGam}[1]{\Hat{\Gamma}^{#1}}
 \newcommand{\bH}{\boldsymbol{H}}
 \newcommand{\hVext}{\Hat{V}_\text{ext}}
 \newcommand{\vext}{v_\text{ext}}
-\newcommand{\hWee}{\Hat{W}_\text{ee}}
 
 % functionals, potentials, densities, etc
 \newcommand{\F}[2]{F_{#1}^{#2}}
@@ -85,9 +86,10 @@
 \newcommand{\eK}[1]{K_{#1}}
 \newcommand{\eF}[2]{F_{#1}^{#2}}
 \newcommand{\ON}[2]{f_{#1}^{#2}}
+\newcommand{\Det}[2]{\Phi_{#1}^{#2}}
 
 % Numbers
-\newcommand{\Nens}{M}
+\newcommand{\nEns}{M}
 \newcommand{\Nel}{N}
 \newcommand{\Norb}{K}
 
@@ -187,9 +189,9 @@ Unless otherwise stated, atomic units are used throughout.
 \label{sec:theo}
 As mentioned above, eDFT for excited states is based on the GOK variational principle \cite{Gross_1988a} which states that a variational principle holds for the ensemble energy
 \begin{equation}
-	\E{}{\bw} = \sum_{I=0}^{\Nens-1} \ew{I} \E{}{(I)}
+	\E{}{\bw} = \sum_{I=0}^{\nEns-1} \ew{I} \E{}{(I)}
 \end{equation}
-built from an 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 $\ew{0} \ge \ldots \ge \ew{\Nens-1}$.
+built from an 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 $\ew{0} \ge \ldots \ge \ew{\nEns-1}$.
 Multiplet degeneracies can be easily handled by assigning the same weight to the degenerate states.
 
 One of the key feature of GOK-DFT in the present context is that one can easily extract individual excitation energies from the ensemble energy via differentiation with respect to individual weights:
@@ -206,9 +208,15 @@ In GOK-DFT, one defines a universal (weight-dependent) ensemble functional $\F{}
 where $\vext(\br{})$ is the external potential.
 In the KS formulation of GOK-DFT, the universal ensemble functional (the weight-dependent analog of the Hohenberg-Kohn universal functional for ensembles) is decomposed as
 \begin{equation}
-	\F{}{\bw}[\n{}{}] = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}],
+	\F{}{\bw}[\n{}{}] 
+	= \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}]
+	= \Tr[ \hGam{\bw} \hT ] + \Tr[ \hGam{\bw} \hWee ],
 \end{equation}
-where $\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kinetic energy functional and 
+where $\hT$ and $\hWee$ are the kinetic and electron-electron interaction potential operators, respectively, $\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kinetic energy functional,
+\begin{equation}
+	\hGam{\bw} = \sum_{I=0}^{\nEns} \ew{I} \dyad{\Det{I}{\bw}}
+\end{equation}
+is the density matrix operator, $\Det{I}{\bw}$ are single-determinant wave functions (or configuration state functions) built with KS orbitals $\MO{p}{\bw}(\br{})$, and 
 \begin{equation}
 \label{eq:exc_def}
 \begin{split}
@@ -232,7 +240,7 @@ From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain
 where 
 \begin{align}
 	\label{eq:nw}
-	\n{}{\bw}(\br{}) & = \sum_{I=0}^{\Nens-1} \ew{I} \n{}{(I)}(\br{}),
+	\n{}{\bw}(\br{}) & = \sum_{I=0}^{\nEns-1} \ew{I} \n{}{(I)}(\br{}),
 	\\
 	\label{eq:nI}
 	\n{}{(I)}(\br{}) & = \sum_{p}^{\Norb} \ON{p}{(I)} [\MO{p}{\bw}(\br{})]^2
@@ -267,7 +275,7 @@ Note that, although we have dropped the weight-dependency in the individual dens
 \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.
 
@@ -481,7 +489,7 @@ This embedding procedure can be theoretically justified by the generalised adiab
 \label{eq:GACE}
 	\E{\xc}{\bw}[\n{}{}] 
 	= \E{\xc}{}[\n{}{}] 
-	+ \sum_{I=1}^{\Nens-1} \int_0^{\ew{I}} \pdv{\E{\xc}{(0,\ldots,0,\xi,\ew{I},\ldots,\ew{\Nens-1})}[\n{}{}]}{\xi} d\xi,
+	+ \sum_{I=1}^{\nEns-1} \int_0^{\ew{I}} \pdv{\E{\xc}{(0,\ldots,0,\xi,\ew{I},\ldots,\ew{\nEns-1})}[\n{}{}]}{\xi} d\xi,
 \end{equation}
 (where $\E{\xc}{}[\n{}{}]$ is the usual ground-state xc functional) originally derived by Franck and Fromager. \cite{Franck_2014} 
 Within this in-principle-exact formalism, the (weight-dependent) xc energy of the ensemble is constructed from the (weight-independent) ground-state functional.