working on equations ...
This commit is contained in:
Emmanuel Giner
parent
7dba9d8a2a
commit
e1fc184d3d
@ -67,20 +67,26 @@
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\citation{FerGinTou-JCP-18}
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\citation{CarTruGag-JPCA-17}
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\bibcite{Thom-PRL-10}{{1}{2010}{{Thom}}{{}}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {C}Definition of a range-separation parameter varying in real space}{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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\@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 $\mathaccentV {bar}916{E}^\mathcal {B}[{n}({\bf r})]$ }{4}{section*.9}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {D}Generic form and properties of the approximations for $\mathaccentV {bar}916{E}^\mathcal {B}[{n}({\bf r})]$ }{4}{section*.8}}
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Generic form of the approximated functionals}{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}Properties of approximated functionals}{4}{section*.10}}
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\newlabel{eq:lim_ebasis}{{22}{4}{}{equation.2.22}{}}
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@ -92,24 +98,34 @@
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\bibcite{WerKno-JCP-88}{{10}{1988}{{Werner\ and\ Knowles}}{{}}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {E}Approximations for the strong correlation regime}{5}{section*.11}}
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {1}Requirements: separability of the energies and $S_z$ invariance}{5}{section*.12}}
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Condition for the functional $\mathaccentV {bar}916{E}_{\text {PBE}}^\mathcal {B}[{n},\xi ,s,n^{(2)},\mu _{\Psi ^{\mathcal {B}}}]$ to obtain $S_z$ invariance}{5}{section*.13}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {F}Requirement on $\Psi ^{\mathcal {B}}$ for the extensivity of $\mu ({\bf r};\Psi _{}^{\mathcal {B}})$}{5}{section*.15}}
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Requirement for $\Psi _{}^{\mathcal {B}}$ for size extensivity}{5}{section*.17}}
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\@writefile{toc}{\contentsline {section}{\numberline {IV}Conclusion}{5}{section*.19}}
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\newlabel{sec:conclusion}{{IV}{5}{}{section*.19}{}}
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\bibcite{WhiHac-JCP-1969}{{13}{1969}{{Whitten\ and\ Hackmeyer}}{{}}}
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Introduction of the effective spin-density}{5}{section*.10}}
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\newlabel{fig:N2_avtz}{{2}{5}{N$_2$, aug-cc-pvtz: Comparison between the near FCI and corrected near FCI energies and the estimated exact one}{figure.2}{}}
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\newlabel{conv_He_table}{{I}{6}{Total energies (in Hartree) for HF and $E$ in aug-cc-pvdz for the He atom, F$_2$ (with F-F=1.411 angstroms) and the super non interacting system He--F$_2$}{table.1}{}}
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@ -132,11 +148,11 @@
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@ -157,13 +173,13 @@
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@ -5,9 +5,15 @@
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\BOOKMARK [3][-]{section*.11}{Requirement for B for size extensivity}{section*.8}% 11
|
||||
\BOOKMARK [1][-]{section*.12}{Results}{section*.2}% 12
|
||||
\BOOKMARK [1][-]{section*.13}{Conclusion}{section*.2}% 13
|
||||
\BOOKMARK [2][-]{section*.8}{Generic form and properties of the approximations for B[n\(r\)] }{section*.4}% 8
|
||||
\BOOKMARK [3][-]{section*.9}{Generic form of the approximated functionals}{section*.8}% 9
|
||||
\BOOKMARK [3][-]{section*.10}{Properties of approximated functionals}{section*.8}% 10
|
||||
\BOOKMARK [2][-]{section*.11}{Approximations for the strong correlation regime}{section*.4}% 11
|
||||
\BOOKMARK [3][-]{section*.12}{Requirements: separability of the energies and Sz invariance}{section*.11}% 12
|
||||
\BOOKMARK [3][-]{section*.13}{Condition for the functional PBEB[n,,s,n\(2\),B] to obtain Sz invariance}{section*.11}% 13
|
||||
\BOOKMARK [3][-]{section*.14}{Functionals for strong correlation}{section*.11}% 14
|
||||
\BOOKMARK [2][-]{section*.15}{Requirement on B for the extensivity of \(r;B\)}{section*.4}% 15
|
||||
\BOOKMARK [3][-]{section*.16}{Introduction of the effective spin-density}{section*.15}% 16
|
||||
\BOOKMARK [3][-]{section*.17}{Requirement for B for size extensivity}{section*.15}% 17
|
||||
\BOOKMARK [1][-]{section*.18}{Results}{section*.2}% 18
|
||||
\BOOKMARK [1][-]{section*.19}{Conclusion}{section*.2}% 19
|
||||
|
||||
@ -69,8 +69,11 @@
|
||||
\newcommand{\efuncdenpbe}[1]{\bar{E}_{\text{PBE}}^\Bas[#1]}
|
||||
\newcommand{\ecompmodel}[0]{\bar{E}^\Bas[\denmodel]}
|
||||
\newcommand{\argepbe}[0]{\den,\xi,s}
|
||||
\newcommand{\argebasis}[0]{\den,\xi,s,n^{2},\mu_{\Psi^{\basis}}}
|
||||
\newcommand{\argrebasis}[0]{\denr,\xi(\br{}),s,n^{2}(\br{}),\mu_{\Psi^{\basis}}(\br{})}
|
||||
\newcommand{\argebasis}[0]{\den,\xi,s,\ntwo,\mu_{\Psi^{\basis}}}
|
||||
\newcommand{\argecmd}[0]{\den,\xi,s,\ntwo,\mu}
|
||||
\newcommand{\argepbeueg}[0]{\den,\xi,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
|
||||
\newcommand{\argepbeuegspin}[0]{\den,\Xi,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
|
||||
\newcommand{\argrebasis}[0]{\denr,\xi(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})}
|
||||
\newcommand{\ecmubis}[0]{\bar{E}_{\text{c,md}}^{\text{sr}}[\denr;\,\mu]}
|
||||
\newcommand{\ecmubisldapbe}[0]{\bar{E}_{\text{c,md}\,\text{PBE}}^{\text{sr}}[\denr;\,\mu]}
|
||||
\newcommand{\ecmuapprox}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mu]}
|
||||
@ -102,6 +105,7 @@
|
||||
% effective interaction
|
||||
\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
|
||||
\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{\Bas})}
|
||||
\newcommand{\ntwo}[0]{n^{(2)}}
|
||||
\newcommand{\mur}[0]{\mu({\bf r})}
|
||||
\newcommand{\murr}[1]{\mu({\bf r}_{#1})}
|
||||
\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})}
|
||||
@ -215,6 +219,7 @@
|
||||
\newcommand{\br}[1]{{\mathbf{r}_{#1}}}
|
||||
\newcommand{\bx}[1]{\mathbf{x}_{#1}}
|
||||
\newcommand{\dbr}[1]{d\br{#1}}
|
||||
\newcommand{\PBEspin}{PBEspin}
|
||||
|
||||
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}
|
||||
@ -393,10 +398,10 @@ Because of the very definition of $\wbasis$, one has the following properties at
|
||||
\end{equation}
|
||||
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 and properties of the approximations for $\efuncden{\denr}$ }
|
||||
\subsection{Generic form and properties of the approximations for $\efuncden{\denr}$ }
|
||||
\subsubsection{Generic form of the approximated functionals}
|
||||
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 density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$ and the on-top pair density $n^{2}(\br{})$. In the present work, unless explicitly stated the quantities $\denr$, $\xi(\br{})$, $s(\br{})$ and $n^{2}(\br{})$ will be computed from the wave function $\psibasis$ used to define $\murpsi$.
|
||||
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 density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$ and the on-top pair density $n^{2}(\br{})$. In the present work, the quantities $\denr$, $\xi(\br{})$, $s(\br{})$ and $n^{2}(\br{})$ are be computed from the same wave function $\psibasis$ used to define $\murpsi$.
|
||||
The generic form for the approximations to $\efuncden{\denr}$ proposed here reads
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
@ -404,37 +409,111 @@ The generic form for the approximations to $\efuncden{\denr}$ proposed here read
|
||||
\efuncdenpbe{\argebasis} = &\int d\br{} \,\denr \\ & \ecmd(\argrebasis)
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
where $\ecmd(\argebasis)$ is the ECMD correlation energy density defined as
|
||||
where $\ecmd(\argecmd)$ is the ECMD correlation energy density defined as
|
||||
\begin{equation}
|
||||
\label{eq:def_ecmdpbe}
|
||||
\ecmd(\argebasis) = \frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{1+ \mu^3 \beta(\argepbe)}
|
||||
\ecmd(\argecmd) = \frac{\varepsilon_{\text{c,PBE}}(\argepbe)}{1+ \mu^3 \beta(\argepbe)}
|
||||
\end{equation}
|
||||
with
|
||||
\begin{equation}
|
||||
\label{eq:def_beta}
|
||||
\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 energy density\cite{PerBurErn-PRL-96}.
|
||||
The actual functional form of $\ecmd(\argebasis)$ have been originally proposed by some of the present authors in the context of RSDFT~\cite{FerGinTou-JCP-18} in order to fulfill the two following limits
|
||||
The actual functional form of $\ecmd(\argecmd)$ have been originally proposed by some of the present authors in the context of RSDFT~\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),
|
||||
\lim_{\mu \rightarrow 0} \ecmd(\argecmd) = \varepsilon_{\text{c,PBE}}(\argepbe),
|
||||
\end{equation}
|
||||
which can be qualified as the weak correlation regime, and
|
||||
which can be qualified as the weak correlation regime, and the large $\mu$ limit
|
||||
\begin{equation}
|
||||
\label{eq:lim_mularge}
|
||||
\ecmd(\argecmd) = \frac{1}{\mu^3} n^{2} + o(\frac{1}{\mu^5}),
|
||||
\end{equation}
|
||||
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.~\onlinecite{FerGinTou-JCP-18} that the on-top pair density involved in eq. \eqref{eq:def_ecmdpbe} plays a crucial role when reaching the strong correlation regime. 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}.
|
||||
Also, $\ecmd(\argecmd) $ vanishes when $\ntwo$ vanishes
|
||||
\begin{equation}
|
||||
\label{eq:lim_n2}
|
||||
\lim_{\ntwo \rightarrow 0} \ecmd(\argecmd) = 0
|
||||
\end{equation}
|
||||
which is exact for systems with vanishing spin density, such as the totally dissociated H$_2$ which is the archetype of strongly correlated systems.
|
||||
Of course, as all RSDFT functionals the function $\ecmd(\argecmd)$ vanishes when $\mu \rightarrow \infty$
|
||||
\begin{equation}
|
||||
\label{eq:lim_muinf}
|
||||
\lim_{\mu \rightarrow \infty} \ecmd(\argebasis) = \frac{1}{\mu^3} n^{2} + o(\frac{1}{\mu^5}),
|
||||
\lim_{\mu \rightarrow \infty} \ecmd(\argecmd) = 0.
|
||||
\end{equation}
|
||||
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 the strong correlation regime. 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}.
|
||||
For equation \eqref{eq:lim_muinf} to be exact, the \textit{exact} on-top pair density $n^{2}$ of the physical system is needed, which is of course rarely affordable and therefore, in the present work, it will be approximated by that computed by an approximated wave function $\psibasis$.
|
||||
%For equation \eqref{eq:lim_muinf} to be exact, the \textit{exact} on-top pair density $n^{2}$ of the physical system is needed, which is of course rarely affordable and therefore, in the present work, it will be approximated 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 $\mu_{\Psi^{\basis}}$:
|
||||
\subsubsection{Properties of approximated functionals}
|
||||
Within the definition of \eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, the approximated complementary basis set functionals $\efuncdenpbe{\argecmd}$ satisfies two important properties.
|
||||
Because of the properties \eqref{eq:cbs_mu} and \eqref{eq:lim_muinf}, $\efuncdenpbe{\argecmd}$ 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.
|
||||
\lim_{\basis \rightarrow \text{CBS}} \efuncdenpbe{\argecmd} = 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.
|
||||
Also, the $\efuncdenpbe{\argecmd}$ 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.
|
||||
Such a property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ (see equation \eqref{eq:wbasis}) together with the condition \eqref{eq:lim_muinf}, ii) the fact that the $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes (see equation \eqref{eq:lim_n2}).
|
||||
|
||||
\subsection{Approximations for the strong correlation regime}
|
||||
\subsubsection{Requirements: separability of the energies and $S_z$ invariance}
|
||||
An important requirement for any electronic structure method is the extensivity of the energy, \textit{i. e.} the additivity of the energies in the case of non interacting fragments, which is particularly important to avoid any ambiguity in computing interaction energies.
|
||||
When two subsystems $A$ and $B$ dissociate in closed shell systems, as in the case of weak interactions for instance, a simple HF wave function leads to extensive energies.
|
||||
When the two subsystems dissociate in open shell systems, such as in covalent bond breaking, it is well known that the HF approach fail and an alternative is to use a CASSCF wave function which, provided that the active space has been properly chosen, leads to additives energies.
|
||||
Another important requirement is the independence of the energy with respect to the $S_z$ component of a given spin state.
|
||||
Such a property is also important in the context of covalent bond breaking where the ground state of the super system $A+B$ is in general of low spin while the ground states of the fragments $A$ and $B$ are in high spin which can have multiple $S_z$ components.
|
||||
|
||||
\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain $S_z$ invariance}
|
||||
A sufficient condition to achieve $S_z$ invariance is to eliminate all dependency to any quantity related to $S_z$, which is the spin density $s(\b{})$ in the case of the definition $\ecmd(\argecmd)$.
|
||||
The spin density is involved in the usual PBE correlation functional $\varepsilon_{\text{c,PBE}}(\argepbe)$ which contributes to the definition of $\ecmd(\argecmd)$ (see equation \eqref{eq:def_ecmdpbe}). A possible way to eliminate the $S_z$ dependency would be then to simply set $\xi(\br{})=0$, but this would lower the accuracy of the usual PBE correlation functional $\varepsilon_{\text{c,PBE}}(\argepbe)$. Therefore, we use the proposal by Scuseria and co-workers\cite{GarBulHenScu-PCCP-15} which introduce an effective spin density depending on the on-top pair density and the total density
|
||||
\begin{equation}
|
||||
\label{eq:def_effspin}
|
||||
\Xi(n,\ntwo) =
|
||||
\begin{cases}
|
||||
\sqrt{ n^2 - 4 \ntwo }. & \text{if $n^2 - 4 \ntwo > 0$,} \\
|
||||
0 & \text{otherwise.}
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
The definition of equation \eqref{eq:def_effspin} is exact when $n$ and $\ntwo$ are obtained from a single Slater determinant.
|
||||
With this definition, the $\Xi(n,\ntwo)$ depends only on $S_z$ invariants quantities, which naturally makes it $S_z$ invariant.
|
||||
We also propose
|
||||
|
||||
\subsubsection{Functionals for strong correlation}
|
||||
The first one, referred as the \PBEspin functional, is a natural extension with effective spin density of the previously introduced PBE\cite{LooPraSceTouGin},
|
||||
\begin{equation}
|
||||
\label{eq:def_pbeueg}
|
||||
\begin{aligned}
|
||||
\efuncdenpbe{\argepbeuegspin} = &\int d\br{} \,\denr \\ & \ecmd(\argepbeuegspin)
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
where $\ntwo_{\text{UEG}}$ is the on-top pair density of the UEG defined as
|
||||
\begin{equation}
|
||||
\label{eq:def_n2ueg}
|
||||
\ntwo_{\text{UEG}}(n,\xi) = n^2(1-\xi)g_0(n).
|
||||
\end{equation}
|
||||
Therefore, such a functional used the on-top pair density of the UEG computed with the total density and effective spin density which depends on the on-top pair density.
|
||||
|
||||
\subsection{Requirement on $\psibasis$ for the extensivity of $\murpsi$}
|
||||
\begin{table*}
|
||||
\caption{Total energies (in Hartree) for HF and $E$ in aug-cc-pvdz for the He atom, F$_2$ (with F-F=1.411 angstroms) and the super non interacting system He--F$_2$. }
|
||||
\begin{tabular}{lcc}
|
||||
%\hline
|
||||
System & HF & $E$ \\
|
||||
\hline
|
||||
F$_2$ & -2.85570466771188 & -0.0112667838948910 \\
|
||||
He & -198.698792752661 & -0.1596345827582842 \\
|
||||
He $\ldots$ F$_2$ & -201.554497420371 & -0.1709013666531826 \\
|
||||
\hline
|
||||
Error to additivity & 1.2 $\times 10^{-12}$ & 7 $\times 10^{-15}$ \\
|
||||
\end{tabular}
|
||||
\label{conv_He_table}
|
||||
\end{table*}
|
||||
|
||||
|
||||
In the case of the present basis set correction, as $\murpsi$ is a local quantity, one of the necessary but not sufficient requirements for extensivity is that $\murpsi$ must be the same on the system $A$ that in the subsystem $A$ of the super system $A+B$ in the limit of non interacting fragments.
|
||||
|
||||
|
||||
|
||||
\subsubsection{Introduction of the effective spin-density}
|
||||
\subsubsection{Requirement for $\wf{}{\Bas}$ for size extensivity}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
Loading…
Reference in New Issue
Block a user