changes in theory
This commit is contained in:
Julien Toulouse
parent
f58e01e0e8
commit
e0afadbf98
@ -6,7 +6,7 @@
|
||||
%Control: page (0) single
|
||||
%Control: year (1) truncated
|
||||
%Control: production of eprint (0) enabled
|
||||
\begin{thebibliography}{65}%
|
||||
\begin{thebibliography}{68}%
|
||||
\makeatletter
|
||||
\providecommand \@ifxundefined [1]{%
|
||||
\@ifx{#1\undefined}
|
||||
@ -531,15 +531,34 @@
|
||||
{author} {\bibfnamefont {E.}~\bibnamefont {Giner}}, \ and\ \bibinfo {author}
|
||||
{\bibfnamefont {J.}~\bibnamefont {Toulouse}},\ }\href@noop {} {\bibinfo
|
||||
{journal} {arXiv:1910.12238}\ }\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Levy}(1979)}]{Lev-PNAS-79}%
|
||||
\BibitemOpen
|
||||
\bibfield {journal} { }\bibfield {author} {\bibinfo {author} {\bibfnamefont
|
||||
{M.}~\bibnamefont {Levy}},\ }\href@noop {} {\bibfield {journal} {\bibinfo
|
||||
{journal} {Proc. Natl. Acad. Sci. U.S.A.}\ }\textbf {\bibinfo {volume}
|
||||
{76}},\ \bibinfo {pages} {6062} (\bibinfo {year} {1979})}\BibitemShut
|
||||
{NoStop}%
|
||||
\bibitem [{\citenamefont {Lieb}(1983)}]{Lie-IJQC-83}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~H.}\ \bibnamefont
|
||||
{Lieb}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Int. J.
|
||||
Quantum Chem.}\ }\textbf {\bibinfo {volume} {{24}}},\ \bibinfo {pages} {24}
|
||||
(\bibinfo {year} {1983})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Kato}(1957)}]{Kat-CPAM-57}%
|
||||
\BibitemOpen
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
|
||||
{Kato}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Comm.
|
||||
Pure Appl. Math.}\ }\textbf {\bibinfo {volume} {10}},\ \bibinfo {pages} {151}
|
||||
(\bibinfo {year} {1957})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Toulouse}, \citenamefont {Gori-Giorgi},\ and\
|
||||
\citenamefont {Savin}(2005)}]{TouGorSav-TCA-05}%
|
||||
\BibitemOpen
|
||||
\bibfield {journal} { }\bibfield {author} {\bibinfo {author} {\bibfnamefont
|
||||
{J.}~\bibnamefont {Toulouse}}, \bibinfo {author} {\bibfnamefont
|
||||
{P.}~\bibnamefont {Gori-Giorgi}}, \ and\ \bibinfo {author} {\bibfnamefont
|
||||
{A.}~\bibnamefont {Savin}},\ }\href@noop {} {\bibfield {journal} {\bibinfo
|
||||
{journal} {Theor. Chem. Acc.}\ }\textbf {\bibinfo {volume} {114}},\ \bibinfo
|
||||
{pages} {305} (\bibinfo {year} {2005})}\BibitemShut {NoStop}%
|
||||
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
|
||||
{Toulouse}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
|
||||
{Gori-Giorgi}}, \ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
|
||||
{Savin}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Theor.
|
||||
Chem. Acc.}\ }\textbf {\bibinfo {volume} {114}},\ \bibinfo {pages} {305}
|
||||
(\bibinfo {year} {2005})}\BibitemShut {NoStop}%
|
||||
\bibitem [{\citenamefont {Perdew}, \citenamefont {Burke},\ and\ \citenamefont
|
||||
{Ernzerhof}(1996)}]{PerBurErn-PRL-96}%
|
||||
\BibitemOpen
|
||||
|
||||
@ -83,16 +83,16 @@
|
||||
|
||||
%pbeuegxiHF
|
||||
\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas}
|
||||
\newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{HF}}^{\basis}}
|
||||
\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{HF}}^{\basis}(\br{})}
|
||||
\newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu^{\text{HF},\basis}}
|
||||
\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu^{\text{HF},\basis}(\br{})}
|
||||
%pbeuegxiCAS
|
||||
\newcommand{\pbeuegxi}{\text{PBE-UEG-}\zeta\text{-CAS}^\Bas}
|
||||
\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
|
||||
\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
|
||||
\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
|
||||
\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
|
||||
%pbeuegXiCAS
|
||||
\newcommand{\pbeuegXi}{\text{PBE-UEG-}\tilde{\zeta}}
|
||||
\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
|
||||
\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
|
||||
\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
|
||||
\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
|
||||
%pbeontxiCAS
|
||||
\newcommand{\pbeontxi}{\text{PBE-ot-}\zeta}
|
||||
\newcommand{\argpbeontxi}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
|
||||
@ -111,9 +111,9 @@
|
||||
\newcommand{\argepbe}[0]{\den,\zeta,s}
|
||||
\newcommand{\argebasis}[0]{\den,\zeta,s,\ntwo,\mu_{\Psi^{\basis}}}
|
||||
\newcommand{\argecmd}[0]{\den,\zeta,s,\ntwo,\mu}
|
||||
\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
|
||||
\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}}
|
||||
\newcommand{\argepbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
|
||||
\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
|
||||
\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}}
|
||||
\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})}
|
||||
\newcommand{\argrebasisab}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})}
|
||||
|
||||
@ -129,17 +129,17 @@
|
||||
|
||||
% 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{\ntwohf}[0]{n^{(2),\text{HF}}}
|
||||
\newcommand{\ntwophi}[0]{n^{(2)}_{\phi}}
|
||||
\newcommand{\ntwoextrap}[0]{\mathring{n}^{(2)}_{\psibasis}}
|
||||
\newcommand{\ntwoextrapcas}[0]{\mathring{n}^{(2)\,\basis}_{\text{CAS}}}
|
||||
\newcommand{\murpsi}[0]{\mu_{\wf{}{\Bas}}({\bf r})}
|
||||
\newcommand{\ntwo}[0]{n_{2}}
|
||||
\newcommand{\ntwohf}[0]{n_2^{\text{HF}}}
|
||||
\newcommand{\ntwophi}[0]{n_2^{{\phi}}}
|
||||
\newcommand{\ntwoextrap}[0]{\mathring{n}_{2}^{\psibasis}}
|
||||
\newcommand{\ntwoextrapcas}[0]{\mathring{n}_2^{\text{CAS},\basis}}
|
||||
\newcommand{\mur}[0]{\mu({\bf r})}
|
||||
\newcommand{\murr}[1]{\mu({\bf r}_{#1})}
|
||||
\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})}
|
||||
\newcommand{\murpsival}[0]{\mu_{\text{val}}({\bf r};\wf{}{\Bas})}
|
||||
\newcommand{\murrval}[1]{\mu_{\text{val}}({\bf r}_{#1})}
|
||||
\newcommand{\murpsival}[0]{\mu_{\wf{}{\Bas}}^{\text{val}}({\bf r})}
|
||||
\newcommand{\murrval}[1]{\mu^{\text{val}}({\bf r}_{#1})}
|
||||
\newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}}
|
||||
|
||||
|
||||
@ -149,9 +149,9 @@
|
||||
\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
|
||||
\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
|
||||
\newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)}
|
||||
\newcommand{\twodmrpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
|
||||
\newcommand{\twodmrdiagpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rr{1}{2})}
|
||||
\newcommand{\twodmrdiagpsival}[0]{ \ntwo_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})}
|
||||
\newcommand{\twodmrpsi}[0]{ \ntwo^{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
|
||||
\newcommand{\twodmrdiagpsi}[0]{ n_{2,{\wf{}{\Bas}}}(\rr{1}{2})}
|
||||
\newcommand{\twodmrdiagpsival}[0]{n_{2,\wf{}{\Bas}}^{\text{val}}(\rr{1}{2})}
|
||||
\newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]}
|
||||
\newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}}
|
||||
\newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]}
|
||||
@ -239,6 +239,7 @@
|
||||
\newcommand{\Cor}{\mathcal{C}}
|
||||
|
||||
% operators
|
||||
\renewcommand{\d}{\text{d}}
|
||||
\newcommand{\hT}{\Hat{T}}
|
||||
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
|
||||
\newcommand{\f}[2]{f_{#1}^{#2}}
|
||||
@ -325,56 +326,56 @@ Then, in Section \ref{sec:results} we apply the method to the calculation of the
|
||||
\section{Theory}
|
||||
\label{sec:theory}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
As the theoretical framework of the basis set correction has been exposed in details in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we briefly recall the main equations and concepts needed for this study in sections \ref{sec:basic}, \ref{sec:wee} and \ref{sec:mur}.
|
||||
More specifically, in section \ref{sec:basic} we recall the basic mathematical framework of the present theory by introducing the density functional complementary to a basis set $\Bas$.
|
||||
Then in section \ref{sec:wee} we introduce an effective non divergent interaction in a basis set $\Bas$, which leads us to the definition of an effective range separation parameter varying in space in section \ref{sec:mur}.
|
||||
Then, in section \ref{sec:functional} we expose the new approximated functionals complementary to a basis set $\Bas$ based on RSDFT functionals. The generic form of such functionals is exposed in section \ref{sec:functional_form}, their properties in the context of the basis set correction is discussed in \ref{sec:functional_prop}, and the requirements for strong correlation is discussed in section \ref{sec:requirements}. Then, the actual form of the functionals used in this work are introduced in section \ref{sec:final_def_func}.
|
||||
As the theory of the basis-set correction has been exposed in details in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we only briefly recall the main equations and concepts needed for this study in Section \ref{sec:basic}, \ref{sec:wee}, and \ref{sec:mur}. More specifically, in Section \ref{sec:basic} we recall the basic mathematical framework of the present theory by introducing the density functional complementary to a basis set $\Bas$. In Section \ref{sec:wee} we introduce the effective non-divergent interaction in the basis set $\Bas$, which leads us to the definition of the effective local range-separation parameter in Section \ref{sec:mur}. Then, in Section \ref{sec:functional} we expose the new approximate complementary functionals based on RSDFT. The generic form of such functionals is exposed in Section \ref{sec:functional_form}, their properties in the context of the basis-set correction are discussed in Section \ref{sec:functional_prop}, and the specific requirements for strong correlation are discussed in Section \ref{sec:requirements}. Finally, the actual form of the functionals used in this work are introduced in Section \ref{sec:final_def_func}.
|
||||
|
||||
\subsection{Basic formal equations}
|
||||
\label{sec:basic}
|
||||
The exact ground state energy $E_0$ of a $N-$electron system can be obtained by an elegant mathematical framework connecting WFT and DFT, that is the Levy-Lieb constrained search formalism which reads
|
||||
|
||||
The exact ground-state energy $E_0$ of a $N$-electron system can in principle be obtained in DFT by a minimization over $N$-electron density $\denr$
|
||||
\begin{equation}
|
||||
\label{eq:levy}
|
||||
E_0 = \min_{\denr} \bigg\{ F[\denr] + (v_{\text{ne}} (\br{}) |\denr) \bigg\},
|
||||
E_0 = \min_{\den} \bigg\{ F[\den] + \int \d \br{} v_{\text{ne}} (\br{}) \denr \bigg\},
|
||||
\end{equation}
|
||||
where $(v_{ne}(\br{})|\denr)$ is the nuclei-electron interaction for a given density $\denr$ and $F[\denr]$ is the so-called Levy-Liev universal density functional
|
||||
where $v_{ne}(\br{})$ is the nuclei-electron potential, and $F[\den]$ is the universal Levy-Lieb density functional written with the constrained search formalism as~\cite{Lev-PNAS-79,Lie-IJQC-83}
|
||||
\begin{equation}
|
||||
\label{eq:levy_func}
|
||||
F[\denr] = \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi}.
|
||||
F[\den] = \min_{\Psi \rightarrow \den} \elemm{\Psi}{\kinop +\weeop }{\Psi},
|
||||
\end{equation}
|
||||
The minimizing density $n_0$ of equation \eqref{eq:levy} is the exact ground state density.
|
||||
Nevertheless, in practical calculations the minimization is performed over the set $\setdenbasis$ which are the densities representable in a basis set $\Bas$ and we assume from thereon that the densities used in the equations belong to $\setdenbasis$.
|
||||
where the notation $\Psi \rightarrow \den$ means that the wave function $\Psi$ yields the density $n$. The minimizing density $n_0$ in Eq.~\eqref{eq:levy} is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densities ``representable in a basis set $\Bas$'', i.e. densities coming from wave functions expandable in the Hilbert space generated by the basis set $\Bas$. In the following, we always implicitly consider only such densities representable in $\Bas$. With this restriction, Eq.~\eqref{eq:levy} gives then an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a fast convergence with the size of the basis set, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is a very good approximation to the exact ground-state energy: $E_0^\Bas \approx E_0$.
|
||||
|
||||
In the present context it is important to notice that in order to recover the \textit{exact} ground state energy, the wave functions $\Psi$ involved in the definition of eq. \eqref{eq:levy_func} must be developed in a complete basis set.
|
||||
An important step proposed originally by some of the present authors in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}
|
||||
was to propose to split the minimization in the definition of $F[\denr]$ using $\wf{}{\Bas}$ which are wave functions developed in $\basis$
|
||||
In the present context, it is important to notice that in the definition of Eq.~\eqref{eq:levy_func} the wave functions $\Psi$ involved have no restriction to a finite basis set, i.e. they should be expanded in a complete basis set. In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, it was then propose to decompose $F[\den]$ as
|
||||
\begin{equation}
|
||||
\label{eq:def_levy_bas}
|
||||
F[\denr] = \min_{\wf{}{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\denr},
|
||||
F[\den] = \min_{\wf{}{\Bas} \rightarrow \den} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den},
|
||||
\end{equation}
|
||||
which leads to the following definition of $\efuncden{\denr}$ which is the density functional complementary to the basis set $\Bas$
|
||||
where $\wf{}{\Bas}$ are wave functions expandable in the Hilbert space generated by $\basis$, and $\efuncden{\den}$ is the density functional complementary to the basis set $\Bas$ defined as
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
\efuncden{\denr} =& \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi} \\
|
||||
&- \min_{\Psi^{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}.
|
||||
\efuncden{\den} = \min_{\Psi \rightarrow \den} \elemm{\Psi}{\kinop +\weeop }{\Psi}
|
||||
- \min_{\Psi^{\Bas} \rightarrow \den} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}.
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
Therefore thanks to eq. \eqref{eq:def_levy_bas} one can properly connect the DFT formalism with the basis set error in WFT calculations. In other terms, the existence of $\efuncden{\denr}$ means that the correlation effects not taken into account in $\basis$ can be formulated as a density functional.
|
||||
Introducing the decomposition in Eq. \eqref{eq:def_levy_bas} back into Eq.~\eqref{eq:levy}, we arrive at the following expression for $E_0^\Bas$
|
||||
\begin{eqnarray}
|
||||
\label{eq:E0basminPsiB}
|
||||
E_0^\Bas &=& \min_{\Psi^{\Bas}} \bigg\{ \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den_{{\Psi^{\Bas}}}}
|
||||
\nonumber\\
|
||||
&&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \int \d \br{} v_{\text{ne}} (\br{}) \den_{\Psi^{\Bas}}(\br{}) \bigg\},
|
||||
\end{eqnarray}
|
||||
where the minimization is only over wave functions $\wf{}{\Bas}$ restricted to the basis set $\basis$ and $\den_{{\Psi^{\Bas}}}(\br{})$ refers to the density extracted from $\wf{}{\Bas}$. Therefore, with Eq.~\eqref{eq:E0basminPsiB}, one can properly combine a WFT calculation in a finite basis set with a density functional accounting for the correlation effects not included in the basis set.
|
||||
|
||||
Assuming that the density $\denFCI$ associated to the ground state FCI wave function $\psifci$ is a good approximation of the exact density, one obtains the following approximation for the exact ground state energy (see equations 12-15 of Ref. \onlinecite{GinPraFerAssSavTou-JCP-18})
|
||||
As a simple non-self-consistent version of this approach, we can approximate the minimizing wave function $\Psi^{\Bas}$ in Eq.~\eqref{eq:E0basminPsiB} by the ground-state FCI wave function $\psifci$ within $\Bas$, and we then obtain the following approximation for the exact ground-state energy (see Eqs. (12)-(15) of Ref. \onlinecite{GinPraFerAssSavTou-JCP-18})
|
||||
\begin{equation}
|
||||
\label{eq:e0approx}
|
||||
E_0 = \efci + \efuncbasisFCI
|
||||
E_0 \approx E_0^\Bas \approx \efci + \efuncbasisFCI,
|
||||
\end{equation}
|
||||
where $\efci$ is the ground state FCI energy within $\Bas$. As it was originally shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Ref. \onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis set incompleteness errors, a large part of which originates from the lack of cusp in any wave function developed in an incomplete basis set.
|
||||
The whole purpose of this paper is to determine approximations for $\efuncbasisFCI$ which are suited for treating strong correlation regimes. The two requirement for such conditions are that i) it must provide size extensive energies, ii) it is invariant of the $S_z$ component of a given spin multiplicity.
|
||||
where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Ref. \onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for treating strong correlation regimes. Two requirements on the approximations for this purpose are i) size consistency and ii) spin-multiplet degeneracy.
|
||||
|
||||
\subsection{Definition of an effective interaction within $\Bas$}
|
||||
\label{sec:wee}
|
||||
As it was originally shown by Kato\cite{kato}, the cusp in the exact wave function originates from the divergence of the coulomb interaction at the coalescence point. Therefore, a cusp less wave function $\wf{}{\Bas}$ could also be obtained from a Hamiltonian with a non divergent electron-electron interaction. In other words, the impact of the incompleteness of a finite basis set can be understood as the removal of the divergence of the usual coulomb interaction at the electron coalescence point.
|
||||
As originally shown by Kato\cite{Kat-CPAM-57}, the cusp in the exact wave function originates from the divergence of the Coulomb interaction at the coalescence point. Therefore, a cuspless wave function $\wf{}{\Bas}$ could also be obtained from a Hamiltonian with a non-divergent electron-electron interaction. In other words, the impact of the incompleteness of a finite basis set can be understood as the removal of the divergence of the usual Coulomb electron-electron interaction at the coalescence point.
|
||||
|
||||
As it was originally derived in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} (see section D and annexes), one can obtain an effective non divergent interaction, here referred as $\wbasis$, which reproduces the expectation value of the coulomb operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite spin part of the electron-electron interaction.
|
||||
|
||||
More specifically, the effective interaction associated to a given wave function $\wf{}{\Bas}$ is defined as
|
||||
As originally derived in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} (Section D and Appendices), one can obtain an effective non-divergent electron-electron interaction, here referred to as $\wbasis$, which reproduces the expectation value of the Coulomb electron-electron interaction operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite-spin part of the electron-electron interaction. More specifically, the effective electron-electron interaction associated to a given wave function $\wf{}{\Bas}$ is defined as
|
||||
\begin{equation}
|
||||
\label{eq:wbasis}
|
||||
\wbasis =
|
||||
@ -384,53 +385,55 @@ More specifically, the effective interaction associated to a given wave function
|
||||
\infty, & \text{otherwise,}
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
where $\twodmrdiagpsi$ is the opposite spin two-body density associated to $\wf{}{\Bas}$
|
||||
where $\twodmrdiagpsi$ is the opposite-spin pair density associated with $\wf{}{\Bas}$
|
||||
\begin{equation}
|
||||
\twodmrdiagpsi = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
|
||||
\end{equation}
|
||||
$\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated two-body tensor, $\SO{p}{}$ are the spatial orthonormal orbitals,
|
||||
and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated tensor in a basis of spatial orthonormal orbitals $\{\SO{p}{}\}$, and $\fbasis$ is
|
||||
\begin{equation}
|
||||
\label{eq:fbasis}
|
||||
\fbasis
|
||||
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
|
||||
\end{equation}
|
||||
and $\V{pq}{rs}=\langle pq | rs \rangle$ are the usual two-electron Coulomb integrals.
|
||||
with the usual two-electron Coulomb integrals $\V{pq}{rs}=\langle pq | rs \rangle$.
|
||||
With such a definition, one can show that $\wbasis$ satisfies
|
||||
\begin{equation}
|
||||
\int \int \dr{1} \dr{2} \wbasis \twodmrdiagpsi = \int \int \dr{1} \dr{2} \frac{\twodmrdiagpsi}{|\br{1}-\br{2}|}.
|
||||
\end{equation}
|
||||
As it was shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessary finite at coalescence for an incomplete basis set, and tends to the regular coulomb interaction in the limit of a complete basis set for any choice of wave function $\psibasis$, that is
|
||||
\begin{eqnarray}
|
||||
\frac{1}{2}\iint \dr{1} \dr{2} \wbasis \twodmrdiagpsi =
|
||||
\nonumber\\
|
||||
\frac{1}{2} \iint \dr{1} \dr{2} \frac{\twodmrdiagpsi}{|\br{1}-\br{2}|}.
|
||||
\end{eqnarray}
|
||||
As shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, the effective interaction $\wbasis$ is necessarily finite at coalescence for an incomplete basis set, and tends to the usual Coulomb interaction in the CBS limit for any choice of wave function $\psibasis$, i.e.
|
||||
\begin{equation}
|
||||
\label{eq:cbs_wbasis}
|
||||
\lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|}\quad \forall\,\psibasis.
|
||||
\lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|},\quad \forall\,\psibasis.
|
||||
\end{equation}
|
||||
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.
|
||||
The condition in Eq.~\eqref{eq:cbs_wbasis} is fundamental as it guarantees the correct behavior of the theory in the CBS limit.
|
||||
|
||||
\subsection{Definition of a range-separation parameter varying in real space}
|
||||
\subsection{Definition of a local range-separation parameter}
|
||||
\label{sec:mur}
|
||||
\subsubsection{General definition}
|
||||
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 $\mu$
|
||||
As the effective interaction within a basis set, $\wbasis$, is non divergent, it ressembles the long-range interaction used in RSDFT
|
||||
\begin{equation}
|
||||
\label{eq:weelr}
|
||||
w_{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}}.
|
||||
w_\text{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}},
|
||||
\end{equation}
|
||||
As originally proposed in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we use a range-separation parameter $\murpsi$ varying in real space
|
||||
where $\mu$ is the range-separation parameter. As originally proposed in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we make the correspondance between these two interactions by using the local range-separation parameter $\murpsi$
|
||||
\begin{equation}
|
||||
\label{eq:def_mur}
|
||||
\murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal
|
||||
\murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal,
|
||||
\end{equation}
|
||||
such that
|
||||
such that the interactions coincide at the electron-electron colescence point for each $\br{}$
|
||||
\begin{equation}
|
||||
w_{ee}^{\lr}(\murpsi;0) = \wbasiscoal \quad \forall \, \br{}.
|
||||
w_\text{ee}^{\lr}(\murpsi;0) = \wbasiscoal, \quad \forall \, \br{}.
|
||||
\end{equation}
|
||||
Because of the very definition of $\wbasis$, one has the following properties at the CBS limit (see \eqref{eq:cbs_wbasis})
|
||||
Because of the very definition of $\wbasis$, one has the following property in the CBS limit (see Eq.~\eqref{eq:cbs_wbasis})
|
||||
\begin{equation}
|
||||
\label{eq:cbs_mu}
|
||||
\lim_{\Bas \rightarrow \text{CBS}} \murpsi = \infty\quad \forall \,\psibasis,
|
||||
\lim_{\Bas \rightarrow \text{CBS}} \murpsi = \infty, \quad \forall \,\psibasis,
|
||||
\end{equation}
|
||||
which is fundamental to guarantee the good behaviour of the theory at the CBS limit.
|
||||
which is again fundamental to guarantee the correct behavior of the theory in the CBS limit.
|
||||
|
||||
\subsubsection{Frozen core density approximation}
|
||||
\subsubsection{Frozen-core approximation}
|
||||
As all WFT calculations for the purpose of that work are performed within the frozen core approximation, we use the valence-only versions of the various quantities needed for the complementary basis set functional introduced in Ref. \cite{LooPraSceTouGin-JCPL-19}.
|
||||
We split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\Cor$ and $\BasFC$ are the sets of core and active MOs, respectively)
|
||||
and define the valence only range separation parameter
|
||||
@ -538,17 +541,17 @@ Based on this reasoning, a similar approach has been used in the context of mult
|
||||
In practice, these approaches introduce the effective spin polarisation
|
||||
\begin{equation}
|
||||
\label{eq:def_effspin}
|
||||
\tilde{\zeta}(n,\ntwo_{\psibasis}) =
|
||||
\tilde{\zeta}(n,\ntwo^{\psibasis}) =
|
||||
% \begin{cases}
|
||||
\sqrt{ n^2 - 4 \ntwo_{\psibasis} }
|
||||
\sqrt{ n^2 - 4 \ntwo^{\psibasis} }
|
||||
% 0 & \text{otherwise.}
|
||||
% \end{cases}
|
||||
\end{equation}
|
||||
which uses the on-top pair density $\ntwo_{\psibasis}$ of a given wave function $\psibasis$.
|
||||
which uses the on-top pair density $\ntwo^{\psibasis}$ of a given wave function $\psibasis$.
|
||||
|
||||
The advantages of this approach are at least two folds: i) the effective spin polarisation $\tilde{\zeta}$ is $S_z$ invariant, ii) it introduces an indirect dependency on the on-top pair density of the wave function $\psibasis$ which usually improves the treatment of strong correlation.
|
||||
Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $n^2 - 4 \ntwo_{\psibasis}<0$ and also
|
||||
the formula of equation \eqref{eq:def_effspin} is exact only when the density $n$ and on-top pair density $\ntwo_{\psibasis}$ are obtained from a single determinant\cite{PerSavBur-PRA-95}, but it is applied to multi configurational wave functions.
|
||||
Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $n^2 - 4 \ntwo^{\psibasis}<0$ and also
|
||||
the formula of equation \eqref{eq:def_effspin} is exact only when the density $n$ and on-top pair density $\ntwo^{\psibasis}$ are obtained from a single determinant\cite{PerSavBur-PRA-95}, but it is applied to multi configurational wave functions.
|
||||
|
||||
An alternative to eliminate the $S_z$ dependency would be to simply set $\zeta(\br{})=0$, but this would lower the accuracy of the usual correlation functional, such as the PBE correlation functional used here $\varepsilon_{\text{c,PBE}}(\argepbe)$. Nevertheless, as the spin polarisation usually tries to mimic the on-top pair density and the function $\ecmd(\argecmd)$ explicitly depends on the on-top pair density (see equations \eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}), we propose here to use the $\ecmd$ functional with \textit{a zero spin polarisation}. This ensures a $S_z$ invariance and, as will be numerically shown, very weakly affect the accuracy of the functional.
|
||||
|
||||
@ -571,14 +574,14 @@ Regarding the spin polarisation that enters into $\varepsilon_{\text{c,PBE}}(\ar
|
||||
Regarding the approximation to the \textit{exact} on-top pair density entering in equation \eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads
|
||||
\begin{equation}
|
||||
\label{eq:def_n2ueg}
|
||||
\ntwo_{\text{UEG}}(n,\zeta,\br{}) = n(\br{})^2\big(1-\zeta(\br{})\big)g_0\big(n(\br{})\big)
|
||||
\ntwo^{\text{UEG}}(n,\zeta,\br{}) = n(\br{})^2\big(1-\zeta(\br{})\big)g_0\big(n(\br{})\big)
|
||||
\end{equation}
|
||||
where the pair-distribution function $g_0(n)$ is taken from equation (46) of Ref. \onlinecite{GorSav-PRA-06}. As some spin polarization appear in equation \eqref{eq:def_n2ueg}, we use the effective spin density $\tilde{\zeta}$ of equation \eqref{eq:def_effspin} in order to ensure $S_z$ invariance. Notice that, as we use a CASSCF wave function and $\tilde{\zeta}$ as spin polarization, the $\ntwo_{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function as $\tilde{\zeta}$ depends on the on-top pair density.
|
||||
where the pair-distribution function $g_0(n)$ is taken from equation (46) of Ref. \onlinecite{GorSav-PRA-06}. As some spin polarization appear in equation \eqref{eq:def_n2ueg}, we use the effective spin density $\tilde{\zeta}$ of equation \eqref{eq:def_effspin} in order to ensure $S_z$ invariance. Notice that, as we use a CASSCF wave function and $\tilde{\zeta}$ as spin polarization, the $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function as $\tilde{\zeta}$ depends on the on-top pair density.
|
||||
|
||||
Another approach to approximate of the exact on top pair density consists in taking advantage of the on-top pair density of the wave function $\psibasis$. Following the work of some of the previous authors\cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density $\ntwoextrap$ as
|
||||
\begin{equation}
|
||||
\label{eq:def_n2extrap}
|
||||
\ntwoextrap(\ntwo_{\psibasis},\mu,\br{}) = \ntwo_{\wf{}{\Bas}}(\br{}) \bigg( 1 + \frac{2}{\sqrt{\pi}\murpsi} \bigg)^{-1}
|
||||
\ntwoextrap(\ntwo^{\psibasis},\mu,\br{}) = \ntwo^{\wf{}{\Bas}}(\br{}) \bigg( 1 + \frac{2}{\sqrt{\pi}\murpsi} \bigg)^{-1}
|
||||
\end{equation}
|
||||
which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density proposed by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT.
|
||||
When using $\ntwoextrap(\ntwo,\mu,\br{})$ in a functional, we will refer simply refer it as "ot".
|
||||
|
||||
Loading…
Reference in New Issue
Block a user