expand Ew
This commit is contained in:
parent
b194b39fce
commit
e59eea0736
@ -1,7 +1,7 @@
|
|||||||
%% 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 2019-11-23 21:56:54 +0100
|
%% Created for Pierre-Francois Loos at 2019-11-25 22:20:38 +0100
|
||||||
|
|
||||||
|
|
||||||
%% Saved with string encoding Unicode (UTF-8)
|
%% Saved with string encoding Unicode (UTF-8)
|
||||||
@ -17,7 +17,8 @@
|
|||||||
Pages = {2406--2412},
|
Pages = {2406--2412},
|
||||||
Title = {Selected applications of hyperspherical harmonics in quantum theory},
|
Title = {Selected applications of hyperspherical harmonics in quantum theory},
|
||||||
Volume = {97},
|
Volume = {97},
|
||||||
Year = {1993}}
|
Year = {1993},
|
||||||
|
Bdsk-Url-1 = {https://doi.org/10.1021/j100112a048}}
|
||||||
|
|
||||||
@book{AveryBook,
|
@book{AveryBook,
|
||||||
Address = {Dordrecht},
|
Address = {Dordrecht},
|
||||||
@ -31,14 +32,12 @@
|
|||||||
@article{Loos_2019,
|
@article{Loos_2019,
|
||||||
Author = {Loos, Pierre-Fran{\c c}ois and Boggio-Pasqua, Martial and Scemama, Anthony and Caffarel, Michel and Jacquemin, Denis},
|
Author = {Loos, Pierre-Fran{\c c}ois and Boggio-Pasqua, Martial and Scemama, Anthony and Caffarel, Michel and Jacquemin, Denis},
|
||||||
Date-Added = {2019-11-21 21:56:17 +0100},
|
Date-Added = {2019-11-21 21:56:17 +0100},
|
||||||
Date-Modified = {2019-11-21 21:56:23 +0100},
|
Date-Modified = {2019-11-25 22:14:42 +0100},
|
||||||
Doi = {10.1021/acs.jctc.8b01205},
|
Doi = {10.1021/acs.jctc.8b01205},
|
||||||
Eprint = {https://doi.org/10.1021/acs.jctc.8b01205},
|
|
||||||
Journal = {J. Chem. Theory Comput.},
|
Journal = {J. Chem. Theory Comput.},
|
||||||
Number = {3},
|
Number = {3},
|
||||||
Pages = {1939--1956},
|
Pages = {1939--1956},
|
||||||
Title = {Reference Energies for Double Excitations},
|
Title = {Reference Energies for Double Excitations},
|
||||||
Url = {https://doi.org/10.1021/acs.jctc.8b01205},
|
|
||||||
Volume = {15},
|
Volume = {15},
|
||||||
Year = {2019},
|
Year = {2019},
|
||||||
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b01205}}
|
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b01205}}
|
||||||
@ -1935,10 +1934,10 @@
|
|||||||
@article{Boggio-Pasqua_2007,
|
@article{Boggio-Pasqua_2007,
|
||||||
Author = {{Boggio-Pasqua}, Martial and Bearpark, Michael J. and Robb, Michael A.},
|
Author = {{Boggio-Pasqua}, Martial and Bearpark, Michael J. and Robb, Michael A.},
|
||||||
Date-Added = {2018-10-24 22:38:52 +0200},
|
Date-Added = {2018-10-24 22:38:52 +0200},
|
||||||
Date-Modified = {2018-10-24 22:38:52 +0200},
|
Date-Modified = {2019-11-25 22:15:16 +0100},
|
||||||
Doi = {10.1021/jo070452v},
|
Doi = {10.1021/jo070452v},
|
||||||
Issn = {0022-3263, 1520-6904},
|
Issn = {0022-3263, 1520-6904},
|
||||||
Journal = {The Journal of Organic Chemistry},
|
Journal = {J. Org. Chem.},
|
||||||
Language = {en},
|
Language = {en},
|
||||||
Month = jun,
|
Month = jun,
|
||||||
Number = {12},
|
Number = {12},
|
||||||
|
@ -600,8 +600,7 @@ For HF, we have
|
|||||||
\end{split}
|
\end{split}
|
||||||
\end{equation}
|
\end{equation}
|
||||||
which is clearly quadratic with respect to $\ew{}$ due to the ghost interaction error in the Hartree term.
|
which is clearly quadratic with respect to $\ew{}$ due to the ghost interaction error in the Hartree term.
|
||||||
|
In the case of the LDA, it reads
|
||||||
For LDA, we have
|
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\begin{split}
|
\begin{split}
|
||||||
\bE{\LDA}{\ew{}}
|
\bE{\LDA}{\ew{}}
|
||||||
@ -613,12 +612,11 @@ For LDA, we have
|
|||||||
+ 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.
|
||||||
|
For eLDA, the ensemble energy can be decomposed as
|
||||||
For eLDA, we have
|
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\begin{split}
|
\begin{split}
|
||||||
\bE{\eLDA}{\ew{}}
|
\bE{\eLDA}{\ew{}}
|
||||||
@ -634,9 +632,17 @@ For eLDA, we have
|
|||||||
\\
|
\\
|
||||||
& + (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}
|
||||||
|
+ (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{} ]
|
||||||
|
\\
|
||||||
|
& + 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}{(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 the weight if the weight-dependent xc functional is very well constructed.
|
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}
|
\end{widetext}
|
||||||
|
|
||||||
@ -645,7 +651,7 @@ This would be, for example, the case with the exact xc functional.
|
|||||||
%%%%%%%%%%%%%%%%%%
|
%%%%%%%%%%%%%%%%%%
|
||||||
\section{Conclusion}
|
\section{Conclusion}
|
||||||
\label{sec:ccl}
|
\label{sec:ccl}
|
||||||
As concluding remarks, we would like to say that, what we have done is awesome.
|
As concluding remarks, we would like to say that what we have done is awesome.
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
%%% ACKNOWLEDGEMENTS %%%
|
%%% ACKNOWLEDGEMENTS %%%
|
||||||
|
Loading…
Reference in New Issue
Block a user