loos/srDFT_G2
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

manuscript

Browse Source
This commit is contained in:
Emmanuel Giner 2019-03-23 20:36:51 +01:00
parent 31e2137439
commit a93b7b43bb
2 changed files with 6 additions and 6 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

BIN
Manuscript/G2-srDFT.pdf
View File

Binary file not shown.

12
Manuscript/G2-srDFT.tex
View File

@ -237,9 +237,9 @@ Also, as demonstrated in the appendix B of \cite{GinPraFerAssSavTou-JCP-18}, $\w
\subsubsection{Definition of a valence effective interaction}
As the average inter electronic distances are very different between the valence electrons and the core electrons, it can be advantageous to define an effective interaction taking into account only for the valence electrons which are the most important in most of the chemical processes.
According to \eqref{eq:def_weebasis} and \eqref{eq:expectweeb}, the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\psibasis$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define first the following function $\fbasisval$ satisfying
According to \eqref{eq:def_weebasis} and \eqref{eq:expectweeb}, the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\psibasis$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\fbasisval$ satisfying
\begin{equation}
\label{eq:expectweeb}
\label{eq:expectweebval}
\elemm{\psibasis}{\weeopbasisval}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasisval,
\end{equation}
where $\weeopbasisval$ is the valence coulomb operator defined as
@ -248,15 +248,15 @@ where $\weeopbasisval$ is the valence coulomb operator defined as
\weeopbasisval = \frac{1}{2}\,\, \sum_{ijkl\,\,\in\,\,\basisval} \,\, \vijkl \,\, \aic{k}\aic{l}\ai{j}\ai{i},
\end{aligned}
\end{equation}
$\basisval$ is a given set of molecular orbitals associated to the valence space which will be defined later on,
and the function $\fbasisval$
$\basisval$ is a given set of molecular orbitals associated to the valence space which will be defined later on.
Following the spirit of \eqref{eq:fbasis}, the function $\fbasisval$ can be defined as
\begin{equation}
\label{eq:fbasis}
\begin{aligned}
\fbasisval = \sum_{ijklmn\,\,\in\,\,\basisval} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1} \phix{i}{1} \phix{j}{2}.
\fbasisval = \sum_{(ij)\,\,\in\,\,\basis} \,\, \sum_{(klmn)\,\,\in\,\,\basisval} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1} \phix{i}{1} \phix{j}{2}.
\end{aligned}
\end{equation}
To define
It is important to notice the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\basis$, and the $(k,l,m,n)$, which span only the valence space $\basisval$. With such a definition, one can show (see annex) that $\fbasisval$ fulfills \eqref{eq:expectweebval} and tends to the exact interaction $1/r_{12}$ in the limit of a complete basis set $\basis$, whatever the choice of subset $\basisval$.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
%%%%%%%%%%%%%%%%%%%%%%%%