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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}
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.
\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 two first terms are simply
\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
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{subequations}
\begin{align}
\eps{1}{\ew{},\HF}
@ -723,42 +716,36 @@ and the HF, LDA and eLDA weight-dependent orbital energies are
\begin{subequations}
\begin{align}
\begin{split}
\eps{1}{\ew{},\LDA}
& = \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}
& = \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{})} + \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{},
\end{split}
+ \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{}{(1)}(\br{}) d\br{},
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
\begin{split}
\eps{1}{\ew{},\eLDA}
& = \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{},
\end{split}
\\
\begin{split}
\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}
\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{})} \n{}{\ew{}}(\br{})
+ \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] } \n{}{(1)}(\br{}) d\br{},
\end{align}
\end{subequations}
respectively.
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}].
\end{widetext}