loos/FarDFT
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

theory done except problem with KS eigval

Browse Source
This commit is contained in:
Pierre-Francois Loos 2019-11-21 23:01:10 +01:00
parent ffd0dfd4e1
commit 20b2b3f3cf
1 changed files with 11 additions and 10 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

21
Manuscript/FarDFT.tex
View File

@ -190,19 +190,16 @@ Unless otherwise stated, atomic units are used throughout.
\label{sec:theo}
As mentioned above, eDFT is based on the so-called Gross-Oliveria-Kohn (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 normalized, monotonically decreasing weights $\bw = (\ew{0},\ldots,\ew{M-1})$, \ie,
\begin{align}
& \sum_{I=0}^{\Nens-1} \ew{I} = 1,
&
& \ew{0} \ge \ldots \ge \ew{\Nens-1}.
\end{align}
built from an ensemble of $\Nens$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\Nens-1)}$, and (normalized) monotonically decreasing weights $\bw = (\ew{0},\ldots,\ew{M-1})$, \ie, $\sum_{I=0}^{\Nens-1} \ew{I} = 1$, and $\ew{0} \ge \ldots \ge \ew{\Nens-1}$.
Degeneracies can be easily handled.
One of the key feature of eDFT in the present context is that one can easily extract individual excitation energies from the ensemble energy via differentiation with respect to individual weights:
\begin{equation}
\pdv{\E{}{\bw}}{\ew{I}} = \E{}{(I)} - \E{}{(0)},
\end{equation}
where we used the fact that $\ew{0} = 1 - \sum_{I \ne 0} \ew{I}$.
where the weights are normalised by setting $\ew{0} = 1 - \sum_{I \ne 0} \ew{I}$.
In GOK-DFT, one defines a universal (weight-dependent) ensemble functional $\F{}{\bw}[\n{}{}]$ such that
\begin{equation}
@ -224,7 +221,7 @@ 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}
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 $\e{\xc}{\bw}[\n{}{}]$.
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}
\begin{equation}
@ -233,7 +230,11 @@ From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain
= \E{}{(I)} - \E{}{(0)}
= \eps{I}{\bw} - \eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}},
\end{equation}
where $\eps{I}{\bw}$ is the $I$th KS orbital energy.
\titou{where $\eps{I}{\bw}$ is the $I$th KS orbital energy (T2: wrong!)} given by the ensemble KS equation
\begin{equation}
\qty( -\frac{\nabla^2}{2} + \vext(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})}) \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}),
\end{equation}
(where $\MO{p}{\bw}(\br{})$ is a KS orbital) 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.
%%%%%%%%%%%%%%%%%%