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

clean up results

Browse Source
This commit is contained in:
Pierre-Francois Loos 2020-02-22 09:50:35 +01:00
parent b08bd66499
commit 4d9eb557f2
3 changed files with 6517 additions and 2865 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

9233
FarDFT.nb
View File

File diff suppressed because it is too large Load Diff

40
Manuscript/FarDFT.bib
View File

@ -1,13 +1,37 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-11-25 22:20:38 +0100 %% Created for Pierre-Francois Loos at 2020-02-22 09:50:18 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{QP2,
Author = {Y. Garniron and K. Gasperich and T. Applencourt and A. Benali and A. Fert{\'e} and J. Paquier and B. Pradines and R. Assaraf and P. Reinhardt and J. Toulouse and P. Barbaresco and N. Renon and G. David and J. P. Malrieu and M. V{\'e}ril and M. Caffarel and P. F. Loos and E. Giner and A. Scemama},
Date-Added = {2020-02-22 09:29:06 +0100},
Date-Modified = {2020-02-22 09:29:06 +0100},
Doi = {10.1021/acs.jctc.9b00176},
Journal = {J. Chem. Theory Comput.},
Pages = {3591},
Title = {Quantum Package 2.0: A Open-Source Determinant-Driven Suite Of Programs},
Volume = {15},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00176}}
@article{Perdew_1981,
Author = {J. P. Perdew and A. Zunger},
Date-Added = {2020-02-22 09:26:50 +0100},
Date-Modified = {2020-02-22 09:27:38 +0100},
Doi = {10.1103/PhysRevB.23.5048},
Journal = {Phys. Rev. B},
Pages = {5048},
Title = {Self-interaction correction to density-functional approximations for many-electron systems},
Volume = {23},
Year = {1981},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.23.5048}}
@article{Avery_1993, @article{Avery_1993,
Author = {J. Avery}, Author = {J. Avery},
Date-Added = {2019-11-23 21:56:19 +0100}, Date-Added = {2019-11-23 21:56:19 +0100},
@ -6025,20 +6049,6 @@
Title = {QMC=Chem}, Title = {QMC=Chem},
Year = 2017} Year = 2017}
@misc{QP,
Author = {A. Scemama and T. Applencourt and Y. Garniron and E. Giner and G. David and M. Caffarel},
Date-Added = {2018-10-24 22:38:52 +0200},
Date-Modified = {2018-10-24 22:38:52 +0200},
Doi = {10.5281/zenodo.200970},
Month = {Dec},
Note = {\url{https://github.com/LCPQ/quantum_package}},
Publisher = {Zenodo},
Title = {Quantum Package v1.0},
Url = {https://github.com/LCPQ/quantum_package},
Year = {2016},
Bdsk-Url-1 = {https://github.com/LCPQ/quantum_package},
Bdsk-Url-2 = {http://dx.doi.org/10.5281/zenodo.200970}}
@article{Ralphs_2013, @article{Ralphs_2013,
Author = {K Ralphs and G Serna and L R Hargreaves and M A Khakoo and C Winstead and V McKoy}, Author = {K Ralphs and G Serna and L R Hargreaves and M A Khakoo and C Winstead and V McKoy},
Date-Added = {2018-10-24 22:38:52 +0200}, Date-Added = {2018-10-24 22:38:52 +0200},

109
Manuscript/FarDFT.tex
View File

@ -224,11 +224,11 @@ is the density matrix operator, $\Det{I}{\bw}$ are single-determinant wave funct
& = \E{\Ha}{}[\n{}{}] + \E{\xc}{\bw}[\n{}{}] & = \E{\Ha}{}[\n{}{}] + \E{\xc}{\bw}[\n{}{}]
\\ \\
& = \frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}' & = \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{}. + \int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{}.
\end{split} \end{split}
\end{equation} \end{equation}
is the ensemble Hartree-exchange-correlation (Hxc) functional. is the ensemble Hartree-exchange-correlation (Hxc) functional.
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 (GIE) \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} From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain \cite{Gross_1988b,Deur_2019}
\begin{equation} \begin{equation}
@ -262,7 +262,7 @@ where $\hHc(\br{}) = -\nabla^2/2 + \vext(\br{})$, and
& = \fdv{\E{\Ha}{\bw}[\n{}{}]}{\n{}{}(\br{})} + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})} & = \fdv{\E{\Ha}{\bw}[\n{}{}]}{\n{}{}(\br{})} + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
\\ \\
& = \frac{1}{2} \int \frac{\n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{}' & = \frac{1}{2} \int \frac{\n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{}'
+ \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{}))
\end{split} \end{split}
\end{equation} \end{equation}
is the Hxc potential. is the Hxc potential.
@ -481,7 +481,8 @@ Consequently, in the following, we name this weight-dependent xc functional ``eL
Also, we note that, by construction, Also, we note that, by construction,
\begin{equation} \begin{equation}
\label{eq:dexcdw} \label{eq:dexcdw}
\left. \pdv{\be{\xc}{\ew{}}[\n{}{}]}{\ew{}}\right|_{\n{}{} = \n{}{\ew{}}(\br)} = \be{\xc}{(1)}[\n{}{\ew{}}(\br)] - \be{\xc}{(0)}[\n{}{\ew{}}(\br)]. \left. \pdv{\be{\xc}{\ew{}}[\n{}{}]}{\ew{}}\right|_{\n{}{} = \n{}{\ew{}}(\br)}
= \be{\xc}{(1)}(\n{}{\ew{}}(\br)) - \be{\xc}{(0)}(\n{}{\ew{}}(\br)).
\end{equation} \end{equation}
This embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles (GACE) This embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles (GACE)
@ -498,7 +499,7 @@ In the case of a homogeneous system (or equivalently within the LDA), substituti
%%% TABLE I %%% %%% TABLE I %%%
\begin{table*} \begin{table*}
\caption{ \caption{
Total energies (in hartree) and excitation energies (in \titou{hartree}) of \ce{H2} with $\RHH = 1.4$ bohr for various methods with the STO-3G minimal basis. Total energies (in hartree) and excitation energies (in hartree) of \ce{H2} with $\RHH = 1.4$ bohr for various methods with the STO-3G minimal basis.
\label{tab:Energies} \label{tab:Energies}
} }
\begin{ruledtabular} \begin{ruledtabular}
@ -514,7 +515,7 @@ Total energies (in hartree) and excitation energies (in \titou{hartree}) of \ce{
accurate\fnm[1] & & & & \\ accurate\fnm[1] & & & & \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{FCI/aug-cc-pV5Z excitation energies computer with QUANTUM PACKAGE. \cite{QP2}} \fnt[1]{FCI/cc-pV5Z excitation energies computed with QUANTUM PACKAGE. \cite{QP2}}
\end{table*} \end{table*}
%%% %%% %%% %%% %%% %%% %%% %%%
@ -556,8 +557,8 @@ Ensemble energies (in hartree) of \ce{H2} with $\RHH = 1.4$ bohr as a function o
Here, we consider as testing ground the minimal-basis \ce{H2} molecule. Here, we consider as testing ground the minimal-basis \ce{H2} molecule.
We select STO-3G as minimal basis, and study the behaviour of the total energy of \ce{H2} as a function of the internuclear distance $\RHH$ (in bohr). We select STO-3G as minimal basis, and study the behaviour of the total energy of \ce{H2} as a function of the internuclear distance $\RHH$ (in bohr).
This minimal-basis example is quite pedagogical as the molecular orbitals are fixed by symmetry. This minimal-basis example is quite pedagogical as the molecular orbitals are fixed by symmetry.
We have then access to the individual densities of the ground and doubly-excited states (which is not usually possible in practice). We have then access to the individual densities of the ground and doubly-excited states (which is not straightforward in practice).
Therefore, thanks to the spatial symmetry and the minimal basis, the individual densities extracted from the ensemble density are equal to the \textit{exact} individual densities. Moreover, thanks to the spatial symmetry and the minimal basis, the individual densities extracted from the ensemble density are equal to the \textit{exact} individual densities.
In other words, there is no density-driven error and the only error that we are going to observe is the functional-driven error (and this is what we want to study). In other words, there is no density-driven error and the only error that we are going to observe is the functional-driven error (and this is what we want to study).
The bonding and antibonding orbitals of the \ce{H2} molecule are given by The bonding and antibonding orbitals of the \ce{H2} molecule are given by
@ -568,7 +569,7 @@ The bonding and antibonding orbitals of the \ce{H2} molecule are given by
\MO{2}{}(\br{}) & = \qty[ \AO{A}(\br{}) - \AO{B}(\br{}) ]/\sqrt{2(1 - S_{AB})}, \MO{2}{}(\br{}) & = \qty[ \AO{A}(\br{}) - \AO{B}(\br{}) ]/\sqrt{2(1 - S_{AB})},
\end{align} \end{align}
\end{subequations} \end{subequations}
where $\AO{A}$ and $\AO{B}$ are the two contracted Gaussian basis functions centred on each of the nucleus, and $S_{AB} = \braket{\AO{A}}{\AO{B}}$. where $\AO{A}$ and $\AO{B}$ are the two contracted Gaussian basis functions centred on each of the nucleus, and $S_{AB} = \braket{\AO{A}}{\AO{B}}$ is the overlap between these two basis functions.
The HF energies of the ground state and the doubly-excited states are The HF energies of the ground state and the doubly-excited states are
\begin{subequations} \begin{subequations}
@ -591,7 +592,10 @@ with
\end{align} \end{align}
\end{subequations} \end{subequations}
Note that, in the HF case, there is no self-interaction error as $\eJ{pp} = \eK{pp}$. Note that, in the HF case, there is no self-interaction error as $\eJ{pp} = \eK{pp}$.
We also define the HF excitation energy as $\Ex{\HF}{(1)} = \E{\HF}{(1)} - \E{\HF}{(0)}$. We also define the HF excitation energy as
\begin{equation}
\Ex{\HF}{(1)} = \E{\HF}{(1)} - \E{\HF}{(0)}.
\end{equation}
%The HF orbital energies are %The HF orbital energies are
%\begin{subequations} %\begin{subequations}
%\begin{align} %\begin{align}
@ -601,7 +605,7 @@ We also define the HF excitation energy as $\Ex{\HF}{(1)} = \E{\HF}{(1)} - \E{\H
%\end{align} %\end{align}
%\end{subequations} %\end{subequations}
As reference results, we consider CID (configuration interaction with doubles) computed in the same (minimal) basis set. It is also instructive to consider the CID (configuration interaction with doubles) excitation energies computed in the same (minimal) basis set.
The CID energies of the ground state and doubly-excited states are provided by the eigenvalues of the following CID matrix: The CID energies of the ground state and doubly-excited states are provided by the eigenvalues of the following CID matrix:
\begin{equation} \begin{equation}
\bH_\CID = \bH_\CID =
@ -609,7 +613,7 @@ The CID energies of the ground state and doubly-excited states are provided by t
\E{\HF}{(0)} & \eK{12} \E{\HF}{(0)} & \eK{12}
\\ \\
\eK{12} & \E{\HF}{(1)} \eK{12} & \E{\HF}{(1)}
\end{pmatrix}, \end{pmatrix}.
\end{equation} \end{equation}
These CID energies are explicitly given by These CID energies are explicitly given by
\begin{subequations} \begin{subequations}
@ -624,23 +628,23 @@ and the CID excitation energy reads
\Ex{\CID}{(1)} = \sqrt{\qty(\E{\HF}{(1)} - \E{\HF}{(0)})^2 + 4 \eK{12}^2} \ge \Ex{\HF}{(1)}. \Ex{\CID}{(1)} = \sqrt{\qty(\E{\HF}{(1)} - \E{\HF}{(0)})^2 + 4 \eK{12}^2} \ge \Ex{\HF}{(1)}.
\end{equation} \end{equation}
At the (ground-state) LDA level (\ie, we only consider ground-state functionals), these energies reads At the (ground-state) LDA level (\ie, we only consider ground-state functionals), the individual energies reads
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:LDA0} \label{eq:LDA0}
\E{\LDA}{(0)} & = 2 \eHc{1} + 2 \eJ{11} + \int \e{\xc}{\LDA}[\n{}{(0)}(\br{})] \n{}{(0)}(\br{}) d\br{}, \E{\LDA}{(0)} & = 2 \eHc{1} + 2 \eJ{11} + \int \e{\xc}{\LDA}(\n{}{(0)}(\br{})) \n{}{(0)}(\br{}) d\br{},
\\ \\
\label{eq:LDA1} \label{eq:LDA1}
\E{\LDA}{(1)} & = 2 \eHc{2} + 2 \eJ{22} + \int \e{\xc}{\LDA}[\n{}{(1)}(\br{})] \n{}{(1)}(\br{}) d\br{}, \E{\LDA}{(1)} & = 2 \eHc{2} + 2 \eJ{22} + \int \e{\xc}{\LDA}(\n{}{(1)}(\br{})) \n{}{(1)}(\br{}) d\br{},
\end{align} \end{align}
\end{subequations} \end{subequations}
with with
\begin{align} \begin{align}
\n{}{(0)}(\br{}) & = 2 [\MO{1}{\ew{}}(\br{})]^2, \n{}{(0)}(\br{}) & = 2 \MO{1}{2}(\br{}),
& &
\n{}{(1)}(\br{}) & = 2 [\MO{2}{\ew{}}(\br{})]^2, \n{}{(1)}(\br{}) & = 2 \MO{2}{2}(\br{}),
\end{align} \end{align}
Note that, contrary to the HF case, self-interaction is present in LDA. Note that, contrary to the HF case, self-interaction is present in LDA. \cite{Perdew_1981}
%The KS orbital energies are given by %The KS orbital energies are given by
%\begin{subequations} %\begin{subequations}
%\begin{align} %\begin{align}
@ -657,10 +661,10 @@ At the eLDA, we have
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:eLDA0} \label{eq:eLDA0}
\E{\eLDA}{(0)} & = 2 \eHc{1} + 2 \eJ{11} + \int \be{\xc}{(0)}[\n{}{(0)}(\br{})] \n{}{(0)}(\br{}) d\br{}, \E{\eLDA}{(0)} & = 2 \eHc{1} + 2 \eJ{11} + \int \be{\xc}{(0)}(\n{}{(0)}(\br{})) \n{}{(0)}(\br{}) d\br{},
\\ \\
\label{eq:eLDA1} \label{eq:eLDA1}
\E{\eLDA}{(1)} & = 2 \eHc{2} + 2 \eJ{22} + \int \be{\xc}{(1)}[\n{}{(1)}(\br{})] \n{}{(1)}(\br{}) d\br{}, \E{\eLDA}{(1)} & = 2 \eHc{2} + 2 \eJ{22} + \int \be{\xc}{(1)}(\n{}{(1)}(\br{})) \n{}{(1)}(\br{}) d\br{},
\end{align} \end{align}
\end{subequations} \end{subequations}
with $\be{\xc}{(0)}(\n{}{}) \equiv \e{\xc}{\LDA}(\n{}{})$ and $\be{\xc}{(1)}(\n{}{}) = \e{\xc}{\LDA}(\n{}{}) + \e{\xc}{(1)}(\n{}{}) - \e{\xc}{(0)}(\n{}{})$. with $\be{\xc}{(0)}(\n{}{}) \equiv \e{\xc}{\LDA}(\n{}{})$ and $\be{\xc}{(1)}(\n{}{}) = \e{\xc}{\LDA}(\n{}{}) + \e{\xc}{(1)}(\n{}{}) - \e{\xc}{(0)}(\n{}{})$.
@ -669,20 +673,11 @@ Interestingly here, there is a strong connection between the LDA and eLDA excita
\begin{equation} \begin{equation}
\begin{split} \begin{split}
\Ex{\eLDA}{(1)} \Ex{\eLDA}{(1)}
& = \Ex{\LDA}{(1)} + \int \qty( \e{\xc}{(1)} - \e{\xc}{(0)} )[\n{}{(1)}(\br{})] \n{}{(1)}(\br{}) d\br{}. & = \Ex{\LDA}{(1)} + \int \qty[ \e{\xc}{(1)}(\n{}{(1)}(\br{})) - \e{\xc}{(0)}(\n{}{(1)}(\br{})) ] \n{}{(1)}(\br{}) d\br{}.
\\ \\
& = \Ex{\LDA}{(1)} + \int \left. \pdv{\e{\xc}{\ew{}}[\n{}{}]}{\ew{}} \right|_{\n{}{} = \n{}{(1)}(\br{})} \n{}{(1)}(\br{}) d\br{}. & = \Ex{\LDA}{(1)} + \int \left. \pdv{\e{\xc}{\ew{}}(\n{}{})}{\ew{}} \right|_{\n{}{} = \n{}{(1)}(\br{})} \n{}{(1)}(\br{}) d\br{}.
\end{split} \end{split}
\end{equation} \end{equation}
The KS orbital energies are given by
%\begin{subequations}
%\begin{align}
% \eps{1}{\eLDA} & = \eHc{1} + 2\eJ{11} + \ldots,
% \\
% \eps{2}{\eLDA} & = \eHc{2} + 2\eJ{12} + \ldots.
%\end{align}
%\end{subequations}
These equations can be combined to define three ensemble energies These equations can be combined to define three ensemble energies
\begin{subequations} \begin{subequations}
@ -698,14 +693,16 @@ These equations can be combined to define three ensemble energies
\end{align} \end{align}
\end{subequations} \end{subequations}
which are all, by construction, linear with respect to $\ew{}$. which are all, by construction, linear with respect to $\ew{}$.
Excitation energies can be easily extracted from these formulae via differenciation with respect to $\ew{}$. Excitation energies can be easily extracted from these formulae via differentiation with respect to $\ew{}$.
Note that this is not how one would do in a ``practical'' ensemble calculation as one does not have (usually) access to the individual densities.
However, for pedagogical purposes and to study the magnitude of the ghost-interaction error, it is interesting to defined them.
Similar energies than the ones given in Eqs.~\eqref{eq:EwHF}, \eqref{eq:EwLDA} and \eqref{eq:EweLDA} can also be obtained directly from the ensemble density Alternatively to Eqs.~\eqref{eq:EwHF}, \eqref{eq:EwLDA} and \eqref{eq:EweLDA}, one can obtain ensemble energies directly from the ensemble density
\begin{equation} \begin{equation}
\n{}{\ew{}} = (1-\ew{}) \n{}{(0)} + \ew{} \n{}{(1)}. \n{}{\ew{}} = (1-\ew{}) \n{}{(0)} + \ew{} \n{}{(1)}.
\end{equation} \end{equation}
(This is what one would do in practice, \ie, by performing a KS ensemble calculation.) (This is what one would do in practice, \ie, by performing a KS ensemble calculation.)
We will label these energies as $\tE{}{\ew{}}$ to avoid confusion. We will label these energies as $\tE{}{\ew{}}$ to avoid confusion with the expressions reported in Eqs.~\eqref{eq:EwHF}, \eqref{eq:EwLDA} and \eqref{eq:EweLDA}.
\begin{widetext} \begin{widetext}
For HF, we have For HF, we have
\begin{equation} \begin{equation}
@ -729,13 +726,13 @@ In the case of the LDA, it reads
& = \Ts{\ew{}}[\n{}{\ew{}}(\br{})] & = \Ts{\ew{}}[\n{}{\ew{}}(\br{})]
+ \int \vext(\br{}) \n{}{\ew{}}(\br{}) d\br{} + \int \vext(\br{}) \n{}{\ew{}}(\br{}) d\br{}
+ \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}' + \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
+ \int \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] \n{}{\ew{}}(\br{}) d\br{} + \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{\ew{}}(\br{}) d\br{}
\\ \\
& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2} & = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
+ 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12} + 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12}
\\ \\
& + (1-\ew{}) \int \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] \n{}{(0)}(\br{}) d\br{} & + (1-\ew{}) \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
+ \ew{} \int \e{\xc}{\LDA}[\n{}{\ew{}}(\br{})] \n{}{(1)}(\br{}) d\br{}, + \ew{} \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{},
\end{split} \end{split}
\end{equation} \end{equation}
which is also clearly quadratic with respect to $\ew{}$ because the (weight-independent) LDA functional cannot compensate the ``quadraticity'' of the Hartree term. which is also clearly quadratic with respect to $\ew{}$ because the (weight-independent) LDA functional cannot compensate the ``quadraticity'' of the Hartree term.
@ -747,24 +744,24 @@ For eLDA, the ensemble energy can be decomposed as
& = \Ts{\ew{}}[\n{}{\ew{}}(\br{})] & = \Ts{\ew{}}[\n{}{\ew{}}(\br{})]
+ \int \vext(\br{}) \n{}{\ew{}}(\br{}) d\br{} + \int \vext(\br{}) \n{}{\ew{}}(\br{}) d\br{}
+ \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}' + \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}'
+ \int \be{\xc}{\ew{}}[\n{}{\ew{}}(\br{})] \n{}{\ew{}}(\br{}) d\br{} + \int \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) \n{}{\ew{}}(\br{}) d\br{}
\\ \\
& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2} & = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
+ 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12} + 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12}
\\ \\
& + (1-\ew{})^2 \int \be{\xc}{(0)}[\n{}{\ew{}}(\br{})] \n{}{(0)}(\br{}) d\br{} & + (1-\ew{})^2 \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
+ \ew{}^2 \int \be{\xc}{(1)}[\n{}{\ew{}}(\br{})] \n{}{(1)}(\br{}) d\br{} + \ew{}^2 \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
\\ \\
& + (1-\ew{})\ew{} \int \be{\xc}{(0)}[\n{}{\ew{}}(\br{})] \n{}{(1)}(\br{}) d\br{} & + (1-\ew{})\ew{} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
+ \ew{}(1-\ew{}) \int \be{\xc}{(1)}[\n{}{\ew{}}(\br{})] \n{}{(0)}(\br{}) d\br{} + \ew{}(1-\ew{}) \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}
\\ \\
& = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2} & = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2}
+ (1-\ew{})^2 \qty[ 2\eJ{11} + \int \be{\xc}{(0)}[\n{}{\ew{}}(\br{})] \n{}{(0)}(\br{}) d\br{} ] + (1-\ew{})^2 \qty[ 2\eJ{11} + \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ]
+ \ew{}^2 \qty[ 2\eJ{22} + \int \be{\xc}{(1)}[\n{}{\ew{}}(\br{})] \n{}{(1)}(\br{}) d\br{} ] + \ew{}^2 \qty[ 2\eJ{22} + \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{} ]
\\ \\
& + 2 (1-\ew{})\ew{} \qty[ 2\eJ{12} & + 2 (1-\ew{})\ew{} \qty[ 2\eJ{12}
+ \frac{1}{2} \int \be{\xc}{(0)}[\n{}{\ew{}}(\br{})] \n{}{(1)}(\br{}) d\br{} + \frac{1}{2} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}
+ \frac{1}{2} \int \be{\xc}{(1)}[\n{}{\ew{}}(\br{})] \n{}{(0)}(\br{}) d\br{} ], + \frac{1}{2} \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ],
\end{split} \end{split}
\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.
@ -772,7 +769,7 @@ This would be, for example, the case with the exact xc functional.
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 $\Eps{0}{\ew{}} = 2 \eps{1}{\ew{}}$, $\Eps{1}{\ew{}} = 2 \eps{2}{\ew{}}$, and the HF, LDA and eLDA weight-dependent orbital energies are 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, respectively,
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\eps{1}{\ew{},\HF} \eps{1}{\ew{},\HF}
@ -788,12 +785,12 @@ The two first terms are simply $\Eps{0}{\ew{}} = 2 \eps{1}{\ew{}}$, $\Eps{1}{\ew
\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{}) + \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{}, + \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) } \n{}{(0)}(\br{}) d\br{},
\\ \\
\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{})} \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{align} \end{align}
\end{subequations} \end{subequations}
@ -802,20 +799,22 @@ The two first terms are simply $\Eps{0}{\ew{}} = 2 \eps{1}{\ew{}}$, $\Eps{1}{\ew
\eps{1}{\ew{},\eLDA} \eps{1}{\ew{},\eLDA}
& = \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{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) + \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{}, + \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) } \n{}{(0)}(\br{}) d\br{},
\\ \\
\eps{2}{\ew{},\eLDA} \eps{2}{\ew{},\eLDA}
& = \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{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) + \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{}, + \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) } \n{}{(1)}(\br{}) d\br{}.
\end{align} \end{align}
\end{subequations} \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} \end{widetext}
The derivative discontinuity is modelled by the last term of the right-hand-side 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}].
Numerical results are reported in Table \ref{tab:Energies}.
%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%% CONCLUSION %%% %%% CONCLUSION %%%
%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%