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Pierre-Francois Loos 2019-11-28 12:16:46 +01:00
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Manuscript/FarDFT.tex
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@ -700,17 +700,10 @@ For eLDA, the ensemble energy can be decomposed as
\end{equation} \end{equation}
which \textit{could} be linear with respect to $\ew{}$ if the weight-dependent xc functional compensates exactly the quadratic terms in the Hartree term. which \textit{could} be linear with respect to $\ew{}$ if the weight-dependent xc functional compensates exactly the quadratic terms in the Hartree term.
This would be, for example, the case with the exact xc functional. 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. 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}. To do so, we will employ Eq.~\eqref{eq:dEdw}.
The two first terms are simply The two first terms are simply $\Eps{0}{\ew{}} = 2 \eps{1}{\ew{}}$, $\Eps{1}{\ew{}} = 2 \eps{2}{\ew{}}$, and the HF, LDA and eLDA weight-dependent orbital energies are
\begin{align}
\Eps{0}{\ew{}} & = 2 \eps{1}{\ew{}},
&
\Eps{1}{\ew{}} & = 2 \eps{2}{\ew{}},
\end{align}
and the HF, LDA and eLDA weight-dependent orbital energies are
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\eps{1}{\ew{},\HF} \eps{1}{\ew{},\HF}
@ -723,42 +716,36 @@ and the HF, LDA and eLDA weight-dependent orbital energies are
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\begin{split}
\eps{1}{\ew{},\LDA} \eps{1}{\ew{},\LDA}
& = \eHc{1} + 2(1-\ew{}) \eJ{11} + 2\ew{} \eJ{12} & = \eHc{1} + 2(1-\ew{}) \eJ{11} + 2\ew{} \eJ{12}
+ \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
+ \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] } \n{}{(0)}(\br{}) d\br{},
\\ \\
& + \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] } \n{}{(0)}(\br{}) d\br{},
\end{split}
\\
\begin{split}
\eps{2}{\ew{},\LDA} \eps{2}{\ew{},\LDA}
& = \eHc{2} + 2(1-\ew{}) \eJ{12} + 2 \ew{} \eJ{22} & = \eHc{2} + 2(1-\ew{}) \eJ{12} + 2 \ew{} \eJ{22}
\\ + \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
& + \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{}, + \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{},
\end{split}
\end{align} \end{align}
\end{subequations} \end{subequations}
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\begin{split}
\eps{1}{\ew{},\eLDA} \eps{1}{\ew{},\eLDA}
& = \eHc{1} + (1-\ew{})(2\eJ{11} - \eK{11}) + \ew{}(2\eJ{12} - \eK{12}) & = \eHc{1} + (1-\ew{})(2\eJ{11} - \eK{11}) + \ew{}(2\eJ{12} - \eK{12})
+ \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
+ \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] } \n{}{(0)}(\br{}) d\br{},
\\ \\
& + \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] } \n{}{(0)}(\br{}) d\br{}, \eps{2}{\ew{},\eLDA}
\end{split} & = \eHc{2} + (1-\ew{})(2\eJ{12} - \eK{12}) + \ew{}(2\eJ{22} - \eK{22})
\\ + \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{})
\begin{split} + \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{},
\eps{2}{\ew{},\eLDA} & = \eHc{2} + (1-\ew{})(2\eJ{12} - \eK{12}) + \ew{}(2\eJ{22} - \eK{22})
\\
& + \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} + \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{},
\end{split}
\end{align} \end{align}
\end{subequations} \end{subequations}
respectively. respectively.
The derivative discontinuity is modelled by the last term of the RHS of Eq.~\eqref{eq:dEdw}. The derivative discontinuity is modelled by the last term of the RHS of Eq.~\eqref{eq:dEdw}.
Note that this contribution is only non-zero in the case of an explicitly weight-dependent functional [see Eq.~\eqref{eq:dexcdw}]. Note that this contribution is only non-zero in the case of an explicitly weight-dependent functional [see Eq.~\eqref{eq:dexcdw}].
\end{widetext}