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

Manu: II

Browse Source
This commit is contained in:
Emmanuel Fromager 2020-05-09 10:48:47 +02:00
parent da6b34ebe1
commit 81f32550d3
1 changed files with 9 additions and 8 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

17
Manuscript/FarDFT.tex
View File

@ -275,7 +275,7 @@ Turning to GOK-DFT, the extension of the Hohenberg--Kohn theorem to ensembles al
\end{equation} \end{equation}
where $\vne(\br{})$ is the external potential and $\F{}{\bw}[\n{}{}]$ is the universal ensemble functional where $\vne(\br{})$ is the external potential and $\F{}{\bw}[\n{}{}]$ is the universal ensemble functional
(the weight-dependent analog of the Hohenberg--Kohn universal functional for ensembles). (the weight-dependent analog of the Hohenberg--Kohn universal functional for ensembles).
In the KS formulation, this functional is decomposed as In the KS formulation\cite{Gross_1988b}, this functional can be decomposed as
\begin{equation}\label{eq:FGOK_decomp} \begin{equation}\label{eq:FGOK_decomp}
\F{}{\bw}[\n{}{}] \F{}{\bw}[\n{}{}]
= \Tr{ \hgamdens{\bw} \hT }+ \E{\Ha}{}[\n{}{}]+\E{\xc}{\bw}[\n{}{}], = \Tr{ \hgamdens{\bw} \hT }+ \E{\Ha}{}[\n{}{}]+\E{\xc}{\bw}[\n{}{}],
@ -285,9 +285,9 @@ $\Tr{ \hgamdens{\bw} \hT } =\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kin
\begin{equation} \begin{equation}
\hgam{\bw}[n] = \sum_{I=0}^{\nEns-1} \ew{I} \dyad{\Det{I}{\bw}[n]} \hgam{\bw}[n] = \sum_{I=0}^{\nEns-1} \ew{I} \dyad{\Det{I}{\bw}[n]}
\end{equation} \end{equation}
is the KS density-functional density matrix operator, and $\lbrace is the density-functional KS density matrix operator, and $\lbrace
\Det{I}{\bw}[n] \rbrace_{0 \le I \le \nEns-1}$ are single-determinant \Det{I}{\bw}[n] \rbrace_{0 \le I \le \nEns-1}$ are single-determinant
wave functions (or configuration state functions). wave functions (or configuration state functions\cite{Gould_2017}).
Their dependence on the density is determined from the ensemble density constraint Their dependence on the density is determined from the ensemble density constraint
\begin{equation} \begin{equation}
\sum_{I=0}^{\nEns-1} \ew{I} \n{\Det{I}{\bw}[n]}{}(\br) = \n{}{}(\br). \sum_{I=0}^{\nEns-1} \ew{I} \n{\Det{I}{\bw}[n]}{}(\br) = \n{}{}(\br).
@ -324,7 +324,7 @@ GOK variational principle, \cite{Gross_1988b,Senjean_2015}
\left(\hT+\hVne\right)]+\E{\Ha}{}[\n{\hGam{\bw}}{}]+\E{\xc}{\bw}[\n{\hGam{\bw}}{}]\right\}, \left(\hT+\hVne\right)]+\E{\Ha}{}[\n{\hGam{\bw}}{}]+\E{\xc}{\bw}[\n{\hGam{\bw}}{}]\right\},
\eeq \eeq
where $\n{\hGam{\bw}}{}(\br)=\sum_{I=0}^{\nEns - 1} where $\n{\hGam{\bw}}{}(\br)=\sum_{I=0}^{\nEns - 1}
\ew{I}\n{\overline{\Psi}^{(I)}}{}$ is the trial ensemble density. As a \ew{I}\n{\overline{\Psi}^{(I)}}{}$ is a trial ensemble density. As a
result, the orbitals result, the orbitals
$\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le $\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le
\nOrb}$ from which the KS \nOrb}$ from which the KS
@ -351,7 +351,7 @@ where $\ON{p}{(I)}$ denotes the occupation of $\MO{p}{\bw}(\br{})$ in
the $I$th KS wave function $\Det{I}{\bw}\left[n^{\bw}\right]$. Turning the $I$th KS wave function $\Det{I}{\bw}\left[n^{\bw}\right]$. Turning
to the excitation energies, they can be extracted from the to the excitation energies, they can be extracted from the
density-functional ensemble as follows [see Eqs.~\eqref{eq:diff_Ew} density-functional ensemble as follows [see Eqs.~\eqref{eq:diff_Ew}
and \eqref{eq:Ew-GOK} and Refs.~\onlinecite{Gross_1988b,Deur_2019}]: and \eqref{eq:min_KS_DM} and Refs.~\onlinecite{Gross_1988b,Deur_2019}]:
\beq \beq
\label{eq:dEdw} \label{eq:dEdw}
\Omega^{(I)}= \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}}, \Omega^{(I)}= \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}},
@ -372,15 +372,16 @@ densities $n_{\Psi_I}(\br)$ as the non-interacting KS ensemble is expected to re
Nevertheless, these densities can still be extracted in principle Nevertheless, these densities can still be extracted in principle
exactly from the KS ensemble as shown by Fromager. \cite{Fromager_2020} exactly from the KS ensemble as shown by Fromager. \cite{Fromager_2020}
In the following, we will work at the (weight-dependent) LDA In the following, we will work at the (weight-dependent) \manu{ensemble}
LDA \manu{(eLDA)}
level of approximation, \ie level of approximation, \ie
\beq \beq
\E{\xc}{\bw}[\n{}{}] \E{\xc}{\bw}[\n{}{}]
&\overset{\rm LDA}{\approx}& &\overset{\rm \manu{e}LDA}{\approx}&
\int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{}, \int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{},
\\ \\
\fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})} \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
&\overset{\rm LDA}{\approx}& &\overset{\rm \manu{e}LDA}{\approx}&
\left. \pdv{\e{\xc}{\bw{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{}(\br{})} \n{}{}(\br{}) + \e{\xc}{\bw{}}(\n{}{}(\br{})). \left. \pdv{\e{\xc}{\bw{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{}(\br{})} \n{}{}(\br{}) + \e{\xc}{\bw{}}(\n{}{}(\br{})).
\eeq \eeq
We will also adopt the usual decomposition, and write down the weight-dependent xc functional as We will also adopt the usual decomposition, and write down the weight-dependent xc functional as