working on equations
This commit is contained in:
Emmanuel Giner
parent
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commit
8bdbfa5d11
@ -66,6 +66,7 @@
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\citation{TouColSav-PRA-04,GoriSav-PRA-06,PazMorGorBac-PRB-06}
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\citation{FerGinTou-JCP-18}
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\citation{GritMeePer-PRA-18}
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\citation{CarTruGag-JPCA-17}
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\bibdata{srDFT_SCNotes,srDFT_SC}
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\bibcite{Thom-PRL-10}{{1}{2010}{{Thom}}{{}}}
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\bibcite{ScoTho-JCP-17}{{2}{2017}{{Scott\ and\ Thom}}{{}}}
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@ -75,6 +76,23 @@
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\bibcite{DeuEmiYumShePie-JCP-19}{{6}{2019}{{Deustua\ \emph {et~al.}}}{{Deustua, Yuwono, Shen,\ and\ Piecuch}}}
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\bibcite{QiuHenZhaScu-JCP-17}{{7}{2017}{{Qiu\ \emph {et~al.}}}{{Qiu, Henderson, Zhao,\ and\ Scuseria}}}
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\bibcite{QiuHenZhaScu-JCP-18}{{8}{2018}{{Qiu\ \emph {et~al.}}}{{Qiu, Henderson, Zhao,\ and\ Scuseria}}}
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\newlabel{sec:mur}{{II\tmspace +\thinmuskip {.1667em}C}{4}{}{section*.7}{}}
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\newlabel{eq:weelr}{{11}{4}{}{equation.2.11}{}}
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\newlabel{eq:def_mur}{{12}{4}{}{equation.2.12}{}}
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\newlabel{eq:cbs_mu}{{14}{4}{}{equation.2.14}{}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {D}Approximation for $\mathaccentV {bar}916{E}^\mathcal {B}[{n}({\bf r})]$ : link with RSDFT}{4}{section*.8}}
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Generic form and properties of the approximations for functionals $\mathaccentV {bar}916{E}^\mathcal {B}[{n}({\bf r})]$ }{4}{section*.9}}
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\newlabel{eq:def_ecmdpbebasis}{{15}{4}{}{equation.2.15}{}}
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\newlabel{eq:def_ecmdpbe}{{16}{4}{}{equation.2.16}{}}
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\newlabel{eq:lim_muinf}{{19}{4}{}{equation.2.19}{}}
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\newlabel{eq:lim_ebasis}{{20}{4}{}{equation.2.20}{}}
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Introduction of the effective spin-density}{4}{section*.10}}
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {3}Requirement for $\Psi _{}^{\mathcal {B}}$ for size extensivity}{4}{section*.11}}
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\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{4}{section*.13}}
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\bibcite{GomHenScu-JCP-19}{{9}{2019}{{Gomez, Henderson,\ and\ Scuseria}}{{}}}
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\bibcite{WerKno-JCP-88}{{10}{1988}{{Werner\ and\ Knowles}}{{}}}
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\bibcite{KnoWer-CPL-88}{{11}{1988}{{Knowles\ and\ Werner}}{{}}}
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@ -92,19 +110,6 @@
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\bibcite{BytRue-CP-09}{{23}{2009}{{Bytautas\ and\ Ruedenberg}}{{}}}
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\bibcite{GinSceCaf-CJC-13}{{24}{2013}{{Giner, Scemama,\ and\ Caffarel}}{{}}}
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\bibcite{CafGinScemRam-JCTC-14}{{25}{2014}{{Caffarel\ \emph {et~al.}}}{{Caffarel, Giner, Scemama,\ and\ Ram{\'\i }rez-Sol{\'\i }s}}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {C}Definition of an range-separation parameter varying in real space}{4}{section*.7}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {D}Approximation for $\mathaccentV {bar}916{E}^\mathcal {B}[{n}({\bf r})]$ : link with RSDFT}{4}{section*.8}}
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\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{4}{section*.13}}
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@ -113,6 +118,10 @@
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@ -126,10 +135,6 @@
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@ -141,6 +146,10 @@
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@ -152,28 +161,25 @@
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Phys. Chem. A}\ }\textbf {\bibinfo {volume} {121}},\ \bibinfo {pages} {5540}
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@ -12657,4 +12657,3 @@ eprint = {https://doi.org/10.1021/acs.jpclett.9b01176}
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doi = {10.1103/PhysRevA.98.062510},
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@ -4,9 +4,9 @@
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\BOOKMARK [1][-]{section*.4}{Theory}{section*.2}% 4
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\BOOKMARK [2][-]{section*.5}{Basic formal equations}{section*.4}% 5
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\BOOKMARK [2][-]{section*.6}{Definition of an effective interaction within B}{section*.4}% 6
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\BOOKMARK [2][-]{section*.7}{Definition of an range-separation parameter varying in real space}{section*.4}% 7
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\BOOKMARK [2][-]{section*.7}{Definition of a range-separation parameter varying in real space}{section*.4}% 7
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\BOOKMARK [2][-]{section*.8}{Approximation for B[n\(r\)] : link with RSDFT}{section*.4}% 8
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\BOOKMARK [3][-]{section*.9}{Generic form of the approximations for functionals B[n\(r\)]}{section*.8}% 9
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\BOOKMARK [3][-]{section*.9}{Generic form and properties of the approximations for functionals B[n\(r\)] }{section*.8}% 9
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\BOOKMARK [3][-]{section*.10}{Introduction of the effective spin-density}{section*.8}% 10
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\BOOKMARK [3][-]{section*.11}{Requirement for B for size extensivity}{section*.8}% 11
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\BOOKMARK [1][-]{section*.12}{Results}{section*.2}% 12
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@ -368,7 +368,7 @@ As it was shown in \cite{GinPraFerAssSavTou-JCP-18}, the effective interaction $
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\lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|}.
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\end{equation}
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The condition of equation \eqref{eq:cbs_wbasis} is fundamental as it guarantees the good behaviour of all the theory in the limit of a complete basis set.
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\subsection{Definition of an range-separation parameter varying in real space}
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\subsection{Definition of a range-separation parameter varying in real space}
|
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\label{sec:mur}
|
||||
As the effective interaction within a basis set $\wbasis$ is non divergent, one can fit such a function with a long-range interaction defined in the framework of RSDFT which depends on the range-separation parameter $\mu$
|
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\begin{equation}
|
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@ -377,6 +377,7 @@ As the effective interaction within a basis set $\wbasis$ is non divergent, one
|
||||
\end{equation}
|
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As originally proposed in \cite{GinPraFerAssSavTou-JCP-18}, we introduce a range-separation parameter $\murpsi$ varying in real space
|
||||
\begin{equation}
|
||||
\label{eq:def_mur}
|
||||
\murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal
|
||||
\end{equation}
|
||||
such that
|
||||
@ -391,17 +392,19 @@ Because of the very definition of $\wbasis$, one has the following properties at
|
||||
which is fundamental to guarantee the good behaviour of the theory at the CBS limit.
|
||||
|
||||
\subsection{Approximation for $\efuncden{\denr}$ : link with RSDFT}
|
||||
\subsubsection{Generic form of the approximations for functionals $\efuncden{\denr}$}
|
||||
As originally proposed in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis set functional $\efuncden{\denr}$ by using the so-called multi-determinant correlation functional (ECMD)\cite{TouGorSav-TCA-05}.
|
||||
Here, we extend the recent work\cite{LooPraSceTouGin-JCPL-19} and propose to use a PBE-like functional which uses the total density $\denr$, the spin polarisation $\xi(\br{}) = n_{\alpha}(\br{}) - n_{\beta}(\br{})$, reduced gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$ and the on-top pair density $n^{2}(\br{})$ taken from a given wave function $\psibasis$.
|
||||
\subsubsection{Generic form and properties of the approximations for functionals $\efuncden{\denr}$ }
|
||||
As originally proposed and motivated in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis set functional $\efuncden{\denr}$ by using the so-called multi-determinant correlation functional (ECMD) introduced by Toulouse and co-workers\cite{TouGorSav-TCA-05}.
|
||||
Following the recent work of some of the present authors\cite{LooPraSceTouGin-JCPL-19}, we propose to use a PBE-like functional which uses the total density $\denr$, the spin polarisation $\xi(\br{}) = n_{\alpha}(\br{}) - n_{\beta}(\br{})$, reduced gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$ and the on-top pair density $n^{2}(\br{})$ taken from a given wave function $\psibasis$.
|
||||
Therefore, we take the following form for the approximation of $\efuncden{\denr}$:
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
\label{eq:def_ecmdpbebasis}
|
||||
\efuncdenpbe{\argebasis} = &\int d\br{} \,\denr \\ & \ecmd(\argrebasis)
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
where $\ecmd(\argebasis)$ is the correlation energy density defined as
|
||||
\begin{equation}
|
||||
\label{eq:def_ecmdpbe}
|
||||
\ecmd(\argebasis) = \frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{1+ \mu^3 \beta(\argepbe)}
|
||||
\end{equation}
|
||||
with
|
||||
@ -409,15 +412,27 @@ with
|
||||
\beta(\argepbe) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{n^{2}/\den},
|
||||
\end{equation}
|
||||
and where $\varepsilon_{\text{c,PBE}}(\argepbe)$ is the usual PBE correlation density\cite{PerBurErn-PRL-96}.
|
||||
As initially proposed by some of the authors~\cite{FerGinTou-JCP-18}, such a correlation energy density admits the two following limits
|
||||
The function $\ecmd(\argebasis)$ have been originally proposed by some of the authors~\cite{FerGinTou-JCP-18}, in order to fulfill the two following limits
|
||||
\begin{equation}
|
||||
\lim_{\mu \rightarrow 0} \ecmd(\argebasis) = \varepsilon_{\text{c,PBE}}(\argepbe)
|
||||
\end{equation}
|
||||
which can be qualified as the weak correlation regime, and
|
||||
\begin{equation}
|
||||
\label{eq:lim_muinf}
|
||||
\lim_{\mu \rightarrow \infty} \ecmd(\argebasis) = \frac{1}{\mu^3} n^{2}.
|
||||
\lim_{\mu \rightarrow \infty} \ecmd(\argebasis) = \frac{1}{\mu^3} n^{2} + o(\frac{1}{\mu^5})
|
||||
\end{equation}
|
||||
As it was previously shown\cite{TouColSav-PRA-04, GoriSav-PRA-06,PazMorGorBac-PRB-06}, the condition \eqref{eq:lim_muinf} is exact for the ECMD in the limit of large $\mu$ provided that $n^{2}$ is the \textit{exact} on-top pair density of the system. Therefore, in the present work we will approximate the \textit{exact} on-top pair density of the system by that computed by an approximated wave function $\psibasis$. In the context of RSDFT, some of the present authors have illustrated in Ref.~\cite{FerGinTou-JCP-18} that the on-top pair density plays a crucial role when reaching strong correlation limit, which have been also found in a related context by Pernal and co-workers\cite{GritMeePer-PRA-18}.
|
||||
which, as it was previously shown\cite{TouColSav-PRA-04, GoriSav-PRA-06,PazMorGorBac-PRB-06} by various authors, is the exact expression for the ECMD in the limit of large $\mu$ provided that $n^{2}$ is the \textit{exact} on-top pair density of the system.
|
||||
In the context of RSDFT, some of the present authors have illustrated in Ref.~\cite{FerGinTou-JCP-18} that the on-top pair density involved in eq. \eqref{eq:def_ecmdpbe} plays a crucial role when reaching strong correlation limit. The importance of the on-top pair density in the strong correlation regime have been also acknowledged by Pernal and co-workers\cite{GritMeePer-PRA-18} and Gagliardi and co-workers\cite{CarTruGag-JPCA-17}.
|
||||
Of course, the \textit{exact} on-top pair density of a system is rarely affordable and therefore, in the present work, we will approximate it by that computed by an approximated wave function $\psibasis$.
|
||||
|
||||
Within the definition of \eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, the approximated complementary basis set functionals $\efuncdenpbe{\argebasis}$ satisfies two important properties.
|
||||
Because of the properties \eqref{eq:cbs_mu} and \eqref{eq:lim_muinf}, $\efuncdenpbe{\argebasis}$ vanishes when reaching the complete basis set limit, whatever the wave function $\psibasis$ used to define the range separation parameter $\mu_{\Psi^{\basis}}$:
|
||||
\begin{equation}
|
||||
\label{eq:lim_ebasis}
|
||||
\lim_{\basis \rightarrow \text{CBS}} \efuncdenpbe{\argebasis} = 0\quad \forall\, \psibasis.
|
||||
\end{equation}
|
||||
which guarantees an unaltered limit when reaching the CBS limit.
|
||||
Also, because of eq. \eqref{eq:lim_muinf}, $\efuncdenpbe{\argebasis}$ vanishes for systems with vanishing on-top pair density, which guarantees the good limit in the case of stretched H$_2$ and for one-electron system.
|
||||
\subsubsection{Introduction of the effective spin-density}
|
||||
\subsubsection{Requirement for $\wf{}{\Bas}$ for size extensivity}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
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