From e59eea073678661c04bf9f4a7f1a6d899c519906 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 25 Nov 2019 22:21:10 +0100 Subject: [PATCH] expand Ew --- Manuscript/FarDFT.bib | 13 ++++++------- Manuscript/FarDFT.tex | 20 +++++++++++++------- 2 files changed, 19 insertions(+), 14 deletions(-) diff --git a/Manuscript/FarDFT.bib b/Manuscript/FarDFT.bib index bbb3834..9d58519 100644 --- a/Manuscript/FarDFT.bib +++ b/Manuscript/FarDFT.bib @@ -1,7 +1,7 @@ %% This BibTeX bibliography file was created using BibDesk. %% 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) @@ -17,7 +17,8 @@ Pages = {2406--2412}, Title = {Selected applications of hyperspherical harmonics in quantum theory}, Volume = {97}, - Year = {1993}} + Year = {1993}, + Bdsk-Url-1 = {https://doi.org/10.1021/j100112a048}} @book{AveryBook, Address = {Dordrecht}, @@ -31,14 +32,12 @@ @article{Loos_2019, 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-Modified = {2019-11-21 21:56:23 +0100}, + Date-Modified = {2019-11-25 22:14:42 +0100}, Doi = {10.1021/acs.jctc.8b01205}, - Eprint = {https://doi.org/10.1021/acs.jctc.8b01205}, Journal = {J. Chem. Theory Comput.}, Number = {3}, Pages = {1939--1956}, Title = {Reference Energies for Double Excitations}, - Url = {https://doi.org/10.1021/acs.jctc.8b01205}, Volume = {15}, Year = {2019}, Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b01205}} @@ -1935,10 +1934,10 @@ @article{Boggio-Pasqua_2007, Author = {{Boggio-Pasqua}, Martial and Bearpark, Michael J. and Robb, Michael A.}, 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}, Issn = {0022-3263, 1520-6904}, - Journal = {The Journal of Organic Chemistry}, + Journal = {J. Org. Chem.}, Language = {en}, Month = jun, Number = {12}, diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index d5f1d6f..b79e508 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -600,8 +600,7 @@ For HF, we have \end{split} \end{equation} which is clearly quadratic with respect to $\ew{}$ due to the ghost interaction error in the Hartree term. - -For LDA, we have +In the case of the LDA, it reads \begin{equation} \begin{split} \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} \\ & + (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{equation} 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, we have +For eLDA, the ensemble energy can be decomposed as \begin{equation} \begin{split} \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{} + \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{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. \end{widetext} @@ -645,7 +651,7 @@ This would be, for example, the case with the exact xc functional. %%%%%%%%%%%%%%%%%% \section{Conclusion} \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 %%%