846 lines
48 KiB
TeX
846 lines
48 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
|
|
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
|
|
|
|
\usepackage{mathpazo,libertine}
|
|
\usepackage[normalem]{ulem}
|
|
\newcommand{\alert}[1]{\textcolor{red}{#1}}
|
|
\definecolor{darkgreen}{RGB}{0, 180, 0}
|
|
\newcommand{\beurk}[1]{\textcolor{darkgreen}{#1}}
|
|
\newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}}
|
|
\usepackage{xspace}
|
|
|
|
\usepackage{hyperref}
|
|
\hypersetup{
|
|
colorlinks=true,
|
|
linkcolor=blue,
|
|
filecolor=blue,
|
|
urlcolor=blue,
|
|
citecolor=blue
|
|
}
|
|
\newcommand{\cdash}{\multicolumn{1}{c}{---}}
|
|
\newcommand{\mc}{\multicolumn}
|
|
\newcommand{\fnm}{\footnotemark}
|
|
\newcommand{\fnt}{\footnotetext}
|
|
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
|
|
\newcommand{\mr}{\multirow}
|
|
\newcommand{\SI}{\textcolor{blue}{supporting information}}
|
|
|
|
% second quantized operators
|
|
\newcommand{\psix}[1]{\hat{\Psi}\left({\bf X}_{#1}\right)}
|
|
\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)}
|
|
\newcommand{\ai}[1]{\hat{a}_{#1}}
|
|
\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}}
|
|
\newcommand{\vijkl}[0]{V_{ij}^{kl}}
|
|
\newcommand{\phix}[2]{\phi_{#1}(\bfr{#2})}
|
|
\newcommand{\phixprim}[2]{\phi_{#1}(\bfr{#2}')}
|
|
|
|
|
|
%operators
|
|
\newcommand{\elemm}[3]{{\ensuremath{\bra{#1}{#2}\ket{#3}}\xspace}}
|
|
\newcommand{\ovrlp}[2]{{\ensuremath{\langle #1|#2\rangle}\xspace}}
|
|
|
|
%\newcommand{\ket}[1]{{\ensuremath{|#1\rangle}\xspace}}
|
|
%\newcommand{\bra}[1]{{\ensuremath{\langle #1|}\xspace}}
|
|
|
|
%
|
|
|
|
|
|
% energies
|
|
\newcommand{\Ec}{E_\text{c}}
|
|
\newcommand{\EPT}{E_\text{PT2}}
|
|
\newcommand{\EsCI}{E_\text{sCI}}
|
|
\newcommand{\EDMC}{E_\text{DMC}}
|
|
\newcommand{\EexFCI}{E_\text{exFCI}}
|
|
\newcommand{\EexFCIbasis}{E_\text{exFCI}^{\Bas}}
|
|
\newcommand{\EexFCIinfty}{E_\text{exFCI}^{\infty}}
|
|
\newcommand{\EexDMC}{E_\text{exDMC}}
|
|
\newcommand{\Ead}{\Delta E_\text{ad}}
|
|
\newcommand{\efci}[0]{E_{\text{FCI}}^{\Bas}}
|
|
\newcommand{\emodel}[0]{E_{\model}^{\Bas}}
|
|
\newcommand{\emodelcomplete}[0]{E_{\model}^{\infty}}
|
|
\newcommand{\efcicomplete}[0]{E_{\text{FCI}}^{\infty}}
|
|
\newcommand{\ecccomplete}[0]{E_{\text{CCSD(T)}}^{\infty}}
|
|
\newcommand{\ecc}[0]{E_{\text{CCSD(T)}}^{\Bas}}
|
|
\newcommand{\efuncbasisfci}[0]{\bar{E}^\Bas[\denfci]}
|
|
\newcommand{\efuncbasis}[0]{\bar{E}^\Bas[\den]}
|
|
\newcommand{\efuncden}[1]{\bar{E}^\Bas[#1]}
|
|
\newcommand{\ecompmodel}[0]{\bar{E}^\Bas[\denmodel]}
|
|
\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]}
|
|
\newcommand{\ecmuapproxmur}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mur]}
|
|
\newcommand{\ecmuapproxmurfci}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denfci;\,\mur]}
|
|
\newcommand{\ecmuapproxmurmodel}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denmodel;\,\mur]}
|
|
\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\Bas,\wf{}{\Bas}}[\denmodel]}
|
|
\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\Bas,\wf{}{\Bas}}[\den]}
|
|
\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\Bas,\wf{}{\Bas}}[\den]}
|
|
\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\Bas,\wf{}{\Bas}}[\den]}
|
|
\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\wf{}{\Bas})\right)}
|
|
\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\wf{}{\Bas})\right)}
|
|
\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)}
|
|
|
|
|
|
|
|
|
|
% numbers
|
|
\newcommand{\rnum}[0]{{\rm I\!R}}
|
|
\newcommand{\bfr}[1]{{\bf x}_{#1}}
|
|
\newcommand{\bfrb}[1]{{\bf r}_{#1}}
|
|
\newcommand{\dr}[1]{\text{d}\bfr{#1}}
|
|
\newcommand{\rr}[2]{\bfr{#1}, \bfr{#2}}
|
|
\newcommand{\rrrr}[4]{\bfr{#1}, \bfr{#2},\bfr{#3},\bfr{#4} }
|
|
|
|
|
|
|
|
% 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{\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{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}}
|
|
|
|
|
|
\newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
|
|
\newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
|
|
\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]{ n^{(2)}_{\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]}
|
|
\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})}
|
|
\newcommand{\ontoppsi}[1]{ n^{(2)}_{\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
|
|
\newcommand{\wbasiscoalval}[1]{W_{\wf{}{\Bas}}^{\text{val}}({\bf r}_{#1})}
|
|
\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
|
|
|
|
|
|
|
|
\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$}
|
|
\newcommand{\ra}{\rightarrow}
|
|
\newcommand{\De}{D_\text{e}}
|
|
|
|
% MODEL
|
|
\newcommand{\model}[0]{\mathcal{Y}}
|
|
|
|
% densities
|
|
\newcommand{\denmodel}[0]{\den_{\model}^\Bas}
|
|
\newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})}
|
|
\newcommand{\denfci}[0]{\den_{\psifci}}
|
|
\newcommand{\denhf}[0]{\den_{\text{HF}}^\Bas}
|
|
\newcommand{\denrfci}[0]{\denr_{\psifci}}
|
|
\newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\Bas({\bf r})}
|
|
\newcommand{\dencipsi}[0]{{n}_{\text{CIPSI}}^\Bas}
|
|
\newcommand{\den}[0]{{n}}
|
|
\newcommand{\denval}[0]{{n}^{\text{val}}}
|
|
\newcommand{\denr}[0]{{n}({\bf r})}
|
|
\newcommand{\onedmval}[0]{\rho_{ij,\sigma}^{\text{val}}}
|
|
|
|
% wave functions
|
|
\newcommand{\psifci}[0]{\Psi^{\Bas}_{\text{FCI}}}
|
|
\newcommand{\psimu}[0]{\Psi^{\mu}}
|
|
% operators
|
|
\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\Bas}
|
|
\newcommand{\kinop}[0]{\hat{T}}
|
|
|
|
\newcommand{\weeopbasisval}[0]{\hat{W}_{\text{ee}}^{\Basval}}
|
|
\newcommand{\weeop}[0]{\hat{W}_{\text{ee}}}
|
|
|
|
|
|
% units
|
|
\newcommand{\IneV}[1]{#1 eV}
|
|
\newcommand{\InAU}[1]{#1 a.u.}
|
|
\newcommand{\InAA}[1]{#1 \AA}
|
|
|
|
|
|
% methods
|
|
\newcommand{\FCI}{\text{FCI}}
|
|
\newcommand{\CCSDT}{\text{CCSD(T)}}
|
|
|
|
\newcommand{\Nel}{N}
|
|
|
|
\newcommand{\n}[2]{n_{#1}^{#2}}
|
|
\newcommand{\E}[2]{E_{#1}^{#2}}
|
|
\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
|
|
\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
|
|
\newcommand{\W}[2]{W_{#1}^{#2}}
|
|
\newcommand{\SO}[2]{\phi_{#1}(\bx{#2})}
|
|
|
|
\newcommand{\modX}{\text{X}}
|
|
\newcommand{\modY}{Y}
|
|
|
|
% basis sets
|
|
\newcommand{\Bas}{\mathcal{B}}
|
|
\newcommand{\Basval}{\mathcal{B}_\text{val}}
|
|
\newcommand{\Val}{\mathcal{V}}
|
|
\newcommand{\Cor}{\mathcal{C}}
|
|
|
|
% operators
|
|
\newcommand{\hT}{\Hat{T}}
|
|
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
|
|
\newcommand{\f}[2]{f_{#1}^{#2}}
|
|
\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
|
|
|
|
% coordinates
|
|
\newcommand{\br}[1]{\mathbf{r}_{#1}}
|
|
\newcommand{\bx}[1]{\mathbf{x}_{#1}}
|
|
\newcommand{\dbr}[1]{d\br{#1}}
|
|
\newcommand{\dbx}[1]{d\bx{#1}}
|
|
|
|
\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}
|
|
|
|
\begin{document}
|
|
|
|
\title{A Density-Based Basis Set Correction For Wave Function Theory}
|
|
|
|
\author{Bath\'elemy Pradines}
|
|
\affiliation{\LCPQ}
|
|
\author{Anthony Scemama}
|
|
\affiliation{\LCPQ}
|
|
\author{Julien Toulouse}
|
|
\affiliation{\LCT}
|
|
\author{Pierre-Fran\c{c}ois Loos}
|
|
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
|
|
\affiliation{\LCPQ}
|
|
\author{Emmanuel Giner}
|
|
\affiliation{\LCT}
|
|
|
|
\begin{abstract}
|
|
We report a universal density-based basis set incompleteness correction that can be applied to any wave function theory method.
|
|
\end{abstract}
|
|
|
|
\maketitle
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Introduction}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
Contemporary quantum chemistry has developed in two directions --- wave function theory (WFT) \cite{Pop-RMP-99} and density-functional theory (DFT). \cite{Koh-RMP-99}
|
|
Although both spring from the same Schr\"odinger equation, each of these philosophies has its own advantages and shortcomings.
|
|
|
|
WFT is attractive as it exists a well-defined path for systematic improvement.
|
|
For example, the coupled cluster (CC) family of methods offers a powerful WFT approach for the description of weakly correlated systems and is well regarded as the gold standard of quantum chemistry.
|
|
By increasing the excitation degree of the CC expansion, one can systematically converge, for a given basis set, to the exact, full configuration interaction (FCI) limit, although the computational cost associated with such improvement is usually pricey.
|
|
One of the most fundamental drawback of conventional WFT methods is the slow convergence of energies and properties with respect to the size of the one-electron basis set.
|
|
This undesirable feature was put into light by Kutzelnigg more than thirty years ago, \cite{Kut-TCA-85}
|
|
who proposed, to palliate this, to introduce explicitly the interelectronic distance $r_{12} = \abs{\br{1} - \br{2}}$ as a basis function. \cite{Kut-TCA-85, KutKlo-JCP-91, TerKloKut-JCP-91, KloKut-JCP-91, KloRohKut-CPL-91, NogKut-JCP-94}
|
|
The resulting F12 methods yields a prominent improvement of the energy convergence, and achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12}
|
|
For example, at the CCSD(T) level, it is advertised that one can obtain quintuple-zeta quality correlation energies with a triple-zeta basis, \cite{TewKloNeiHat-PCCP-07} although computational overheads are introduced by the large auxiliary basis used to resolve three- and four-electron integrals.
|
|
|
|
Present-day DFT calculations are almost exclusively done within the so-called Kohn-Sham (KS) formalism, which corresponds to an exact dressed one-electron theory. \cite{KohSha-PR-65}
|
|
DFT's attractivity originates from its very favorable cost/efficient ratio as it can provide accurate energies and properties at a relatively low computational cost.
|
|
Thanks to this, KS-DFT \cite{HohKoh-PR-64, KohSha-PR-65} has become the workhorse of electronic structure calculations for atoms, molecules and solids. \cite{ParYan-BOOK-89}
|
|
To obtain accurate results within DFT, one only requires an exchange and correlation functionals, which can be classified in various families depending on their physical input quantities. \cite{Bec-JCP-14}
|
|
Although there is no clear way on how to systematically improve density-functional approximations (DFAs), climbing the Jacob's ladder of DFT is potentially the most satisfactory way forward (or upward in that case). \cite{PerSch-AIPCP-01, PerRuzTaoStaScuCso-JCP-05}
|
|
%The local-density approximation (LDA) sits on the first rung of the Jacob's ladder and only uses as input the electron density $n$. \cite{Dir-PCPRS-30, VosWilNus-CJP-80}
|
|
%The generalized-gradient approximation (GGA) corresponds to the second rung and adds the gradient of the electron density $\nabla n$ as an extra ingredient.\cite{Bec-PRA-88, PerWan-PRA-91, PerBurErn-PRL-96}
|
|
In the present context, one of the interesting feature of density-based methods is their much faster convergence with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15}
|
|
|
|
Progress toward unifying these two approaches are on-going.
|
|
Using accurate and rigorous WFT methods, some of us have developed radical generalisations of DFT that are free of the well-known limitations of conventional DFT.
|
|
In that respect range-separated DFT (RS-DFT) is particularly promising as it allows to perform multi-configurational DFT calculations within a rigorous mathematical framework.
|
|
Range-separated hybrids, i.e.~single-determinant approximations of RS-DFT, correct for the wrong long-range behavior of the usual hybrid approximations thanks to the inclusion of the long-range part of the Hartree-Fock (HF) exchange.
|
|
|
|
|
|
|
|
The present manuscript is organised as follows.
|
|
Unless otherwise stated, atomic used are used.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Theory}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
The basis set correction employed here relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the non-completeness of the one-electron basis set.
|
|
Here, we provide the main working equations.
|
|
We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for further details about the formal derivation of the theory.
|
|
|
|
|
|
%=================================================================
|
|
%\subsection{Correcting the basis set error of a general WFT model}
|
|
%=================================================================
|
|
Let us assume we have both the density $\n{\modX}{\Bas}$ and energy $\E{\modX}{\Bas}$ of a $\Nel$-electron system described by a method $\modX$ in an incomplete basis set $\Bas$.
|
|
According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\n{\modX}{\Bas}$ is a resonable approximation of the \textit{exact} ground state density $\n{}{}$, one may approximate the \textit{exact} ground state energy as
|
|
\begin{equation}
|
|
\label{eq:e0basis}
|
|
\E{}{}
|
|
\approx \E{\modX}{\Bas}
|
|
+ \bE{}{\Bas}[\n{\wf{\modX}{\Bas}}{}],
|
|
\end{equation}
|
|
where
|
|
\begin{equation}
|
|
\label{eq:E_funcbasis}
|
|
\bE{}{\Bas}[\n{}{}]
|
|
= \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}}
|
|
- \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
|
|
\end{equation}
|
|
is the complementary density functional defined in Eq.~(8) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, and $\hT$ and $\hWee{}$ are the kinetic and interelectronic repulsion operators, respectively.
|
|
In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Nel$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively.
|
|
Both wave functions yield the same target density $\n{}{}$.
|
|
|
|
%\alert{Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\n{\FCI}{\Bas}$ coincides with the exact ground state density, which in general is a reasonable approximation as the density converges rapidly with the basis set.}
|
|
|
|
An important aspect of such theory is that, in the limit of a complete basis set $\Bas$ (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that
|
|
\begin{equation}
|
|
\label{eq:limitfunc}
|
|
\lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\wf{\modX}{\Bas}}{}] ) = \E{\modX}{} \approx E,
|
|
\end{equation}
|
|
where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set.
|
|
In the case of $\modX = \FCI$, we have as strict equality as $\E{\FCI}{} = E$.
|
|
Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the method $\modX$.
|
|
|
|
%Here we propose to generalize such approach to a general WFT model, referred here as $\model$, projected in a basis set $\Bas$ which must provides a density $\denmodel$ and an energy $\emodel$.
|
|
%As any wave function model is necessary an approximation to the FCI model, one can write
|
|
%\begin{equation}
|
|
% \efci \approx \emodel
|
|
%\end{equation}
|
|
%and
|
|
%\begin{equation}
|
|
% \denfci \approx \denmodel
|
|
%\end{equation}
|
|
%and by defining the energy provided by the model $\model$ in the complete basis set
|
|
%\begin{equation}
|
|
% \emodelcomplete = \lim_{\Bas \rightarrow \infty} \emodel\,\, ,
|
|
%\end{equation}
|
|
%we can then write
|
|
%\begin{equation}
|
|
% \emodelcomplete \approx \emodel + \ecompmodel
|
|
%\end{equation}
|
|
%which verifies the correct limit since
|
|
%\begin{equation}
|
|
% \lim_{\Bas \rightarrow \infty} \ecompmodel = 0\,\, .
|
|
%\end{equation}
|
|
|
|
|
|
%=================================================================
|
|
%\subsection{Basis set correction for the CIPSI algorithm and the CCSD(T) ansatz}
|
|
%=================================================================
|
|
%In this work we propose to apply the basis set correction to a selected CI algorithm, namely the CIPSI algorithm, and to the CCSD(T) ansatz in %order to speed-up the basis set convergence of these models.
|
|
|
|
%=================================================================
|
|
%\subsubsection{Basis set correction for the CCSD(T) energy}
|
|
%=================================================================
|
|
%The CCSD(T) method is a very popular WFT approach which is known to provide very good estimation of the correlation energies for weakly correlated systems, whose wave function are dominated by the HF Slater determinant.
|
|
%Defining $\ecc$ as the CCSD(T) energy obtained in $\Bas$, in the present notations we have
|
|
%\begin{equation}
|
|
% \emodel = \ecc \,\, .
|
|
%\end{equation}
|
|
%In the context of the basis set correction, one needs to choose a density as the density of the model $\denmodel$, and we chose here the HF density
|
|
%\begin{equation}
|
|
% \denmodel = \denhf \,\, .
|
|
%\end{equation}
|
|
%Such a choice can be motivated by the fact that the correction to the HF density brought by the excited Slater determinants are at least of second-order in perturbation theory.
|
|
%Therefore, we approximate the complete basis set CCSD(T) energy $\ecccomplete$ by
|
|
%\begin{equation}
|
|
% \ecccomplete \approx \ecc + \efuncden{\denhf} \,\, .
|
|
%\end{equation}
|
|
|
|
%=================================================================
|
|
%\subsubsection{Correction of the CIPSI algorithm}
|
|
%=================================================================
|
|
%The CIPSI algorithm approximates the FCI wave function through an iterative selected CI procedure, and the FCI energy through a second-order multi-reference perturbation theory.
|
|
%The CIPSI algorithm belongs to the general class of methods build upon selected CI\cite{bender,malrieu,buenker1,buenker-book,three_class_CIPSI,harrison,hbci}
|
|
%which have been successfully used to converge to FCI correlation energies, one-body properties, and nodal surfaces.\cite{three_class_CIPSI,Rubio198698,cimiraglia_cipsi,cele_cipsi_zeroth_order,Angeli2000472,canadian,atoms_3d,f2_dmc,atoms_dmc_julien,GinTewGarAla-JCTC-18,SceBenJacCafLoo-JCP-18,LooSceBloGarCafJac-JCTC-18,LooBogSceCafJAc-JCTC-19}
|
|
%The CIPSI algorithm used in this work uses iteratively enlarged selected CI spaces and
|
|
%Epstein--Nesbet\cite{epstein,nesbet} multi-reference perturbation theory. Within a basis set $\Bas$, the CIPSI energy is
|
|
%\begin{align}
|
|
% E_\mathrm{CIPSI}^{\Bas} &= E_\text{v} + E^{(2)} \,\,,
|
|
%\end{align}
|
|
%where $E_\text{v}$ is the variational energy
|
|
%\begin{align}
|
|
% E_\text{v} &= \min_{\{ c_{\rm I}\}} \frac{\elemm{\Psi^{(0)}}{\hat{H}}{\Psi^{(0)}} }{\ovrlp{\Psi^{(0)}}{\Psi^{(0)}}}\,\,,
|
|
%\end{align}
|
|
%where the reference wave function $\ket{\Psi^{(0)}} = \sum_{{\rm I}\,\in\,\mathcal{R}} \,\,c_{\rm I} \,\,\ket{\rm I}$ is expanded in Slater determinants I within the CI reference space $\mathcal{R}$, and $E^{(2)}$ is the second-order energy correction
|
|
%\begin{align}
|
|
% E^{(2)} &= \sum_{\kappa} \frac{|\elemm{\Psi^{(0)}}{\hat{H}}{\kappa}|^2}{E_\text{v} - \elemm{\kappa}{H}{\kappa}} = \sum_{\kappa} \,\, e_{\kappa}^{(2)} \,\, ,
|
|
%\end{align}
|
|
%where $\kappa$ denotes a determinant outside $\mathcal{R}$.
|
|
%To reduce the cost of the evaluation of the second-order energy correction, the semi-stochastic multi-reference approach
|
|
%of Garniron \textit{et al.} \cite{stochastic_pt_yan} was used, adopting the technical specifications recommended in that work.
|
|
%The CIPSI energy is systematically refined by doubling the size of the CI reference space at each iteration, selecting
|
|
%the determinants $\kappa$ with the largest $\vert e_{\kappa}^{(2)} \vert$.
|
|
%In order to reach a faster convergence of the estimation of the FCI energy, we use the extrapolated FCI energy (exFCI) proposed by Holmes \textit{et al}\cite{HolUmrSha-JCP-17} which we refer here as $\EexFCIbasis$.
|
|
%
|
|
%In the context of the basis set correction, we use the following conventions
|
|
%\begin{equation}
|
|
% \emodel = \EexFCIbasis
|
|
%\end{equation}
|
|
%\begin{equation}
|
|
% \denmodelr = \dencipsir
|
|
%\end{equation}
|
|
%where the density $\dencipsir$ is defined as
|
|
%\begin{equation}
|
|
% \dencipsi = \sum_{ij \in \Bas} \elemm{\Psi^{(0)}}{\aic{i}\ai{j}}{\Psi^{(0)}} \phi_i(\bfrb{} ) \phi_j(\bfrb{} ) \,\, ,
|
|
%\end{equation}
|
|
%and $\phi_i(\bfrb{} )$ are the spin orbitals in the MO basis evaluated at $\bfrb{}$. As it was shown in \onlinecite{GinPraFerAssSavTou-JCP-18} that the CIPSI density converges rapidly with the size of $\Psi^{(0)}$ for weakly correlated systems, $\dencipsir$ can be thought as a reasonable approximation of the FCI density $\denfci$.
|
|
%
|
|
%Finally, we approximate complete basis set exFCI energy $\EexFCIinfty$ as
|
|
%\begin{equation}
|
|
% \EexFCIinfty \approx \EexFCIbasis + \efuncden{\dencipsi}
|
|
%\end{equation}
|
|
|
|
|
|
|
|
%=================================================================
|
|
%\subsection{General scheme for the approximation of the unknown complementary functional $\efuncbasis$}
|
|
%=================================================================
|
|
However, in addition of being unknown, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
|
|
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ following a two-step procedure which guarantees the correct behaviour in the limit $\Bas \to \infty$ [see Eq.~\eqref{eq:limitfunc}].
|
|
First, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation parameter $\mu(\br{})$ varying in space. %(see Sec.~\ref{sec:weff}).
|
|
Then, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al.} \cite{TouGorSav-TCA-05}, that we evaluate at $\n{\modX}{\Bas}$ with $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) .
|
|
|
|
|
|
%=================================================================
|
|
%\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\Bas$}
|
|
%\label{sec:weff}
|
|
%=================================================================
|
|
One of the consequences of the incompleteness of $\Bas$ is that $\wf{}{\Bas}$ does not have a cusp (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points.
|
|
As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could also originate from a Hamiltonian with a non-divergent Coulomb interaction.
|
|
Therefore, the impact of the incompleteness of $\Bas$ can be viewed as a removal of the divergence of the Coulomb interaction at $r_{12} = 0$.
|
|
The present paragraph briefly describes how to obtain an effective interaction $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
|
|
|
|
%----------------------------------------------------------------
|
|
%\subsubsection{General definition of an effective interaction for the basis set $\Bas$}
|
|
%----------------------------------------------------------------
|
|
Consider the Coulomb operator projected in $\Bas$
|
|
\begin{equation}
|
|
\begin{aligned}
|
|
\weeopbasis = \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i},
|
|
\end{aligned}
|
|
\end{equation}
|
|
where the indices run over all orthonormal spin-orbitals in $\Bas$ and $\vijkl$ are the usual two-electron Coulomb integrals.
|
|
Consider now the expectation value of $\weeopbasis$ over a general wave function $\wf{}{\Bas}$ belonging to the $\Nel$-electron Hilbert space spanned by $\Bas$.
|
|
One can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that such an expectation value can be rewritten as an integral over the two-electron spin and space coordinates:
|
|
\begin{equation}
|
|
\label{eq:expectweeb}
|
|
\mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} = \frac{1}{2} \iint \f{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \dbx{1} \dbx{2}
|
|
\end{equation}
|
|
where
|
|
\begin{multline}
|
|
\label{eq:fbasis}
|
|
\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})
|
|
\\
|
|
= \sum_{ijklmn \in \Bas} \vijkl \Gam{mn}{pq}[\wf{}{\Bas}] \SO{n}{2} \SO{m}{1} \SO{i}{1} \SO{j}{2},
|
|
\end{multline}
|
|
and
|
|
\begin{equation}
|
|
\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\wf{}{\Bas}},
|
|
\end{equation}
|
|
is the two-body density tensor of $\wf{}{\Bas}$, $\bfr{} = \qty(\br,\sigma)$ collects the space and spin variables, $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$.
|
|
Then, consider the expectation value of the exact Coulomb operator over $\wf{}{\Bas}$
|
|
\begin{equation}
|
|
\label{eq:expectwee}
|
|
\mel*{\wf{}{\Bas}}{\weeop}{\wf{}{\Bas}} = \frac{1}{2} \iint r_{12}^{-1} \twodmrdiagpsi \dr{1} \dr{2}
|
|
\end{equation}
|
|
where $\n{\wf{}{\Bas}}{(2)}$ is the two-body density associated with $\wf{}{\Bas}$.
|
|
Because $\wf{}{\Bas}$ belongs to $\Bas$, such an expectation value coincides with the expectation value of $\weeopbasis$
|
|
\begin{equation}
|
|
\mel*{\wf{}{\Bas}}{\weeopbasis}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\weeop}{\wf{}{\Bas}},
|
|
\end{equation}
|
|
which can be rewritten as:
|
|
\begin{multline}
|
|
\label{eq:int_eq_wee}
|
|
\iint \wbasis \twodmrdiagpsi \dr{1} \dr{2}
|
|
\\
|
|
= \iint r_{12}^{-1} \twodmrdiagpsi \dr{1} \dr{2}.
|
|
\end{multline}
|
|
where we introduced $\wbasis$
|
|
\begin{equation}
|
|
\label{eq:def_weebasis}
|
|
\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})}{\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})},
|
|
\end{equation}
|
|
which is the effective interaction in the basis set $\Bas$.
|
|
|
|
As already discussed in \onlinecite{GinPraFerAssSavTou-JCP-18}, such an effective interaction is symmetric, \textit{a priori} non translational nor rotational invariant if the basis set $\Bas$ does not have such symmetries and is necessary \textit{finite} at the electron coalescence point for an incomplete basis set $\Bas$.
|
|
Also, as demonstrated in the appendix B of \onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ tends to the regular coulomb interaction $r_{12}^{-1}$ for all points $(\bx{1},\bx{2})$ and any choice of $\wf{}{\Bas}$ in the limit of a complete basis set.
|
|
|
|
|
|
%----------------------------------------------------------------
|
|
\subsubsection{Definition of a valence effective interaction}
|
|
%----------------------------------------------------------------
|
|
As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a general subset of molecular orbitals.
|
|
We then split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively, and $\Cor \bigcap \Val = \O$.
|
|
|
|
According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\fbasisval$ satisfying
|
|
\begin{equation}
|
|
\label{eq:expectweebval}
|
|
\mel*{\wf{}{\Bas}}{\weeopbasisval}{\wf{}{\Bas}} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasisval,
|
|
\end{equation}
|
|
where $\weeopbasisval$ is the valence coulomb operator defined as
|
|
\begin{equation}
|
|
\begin{aligned}
|
|
\weeopbasisval = \frac{1}{2}\,\, \sum_{ijkl\,\,\in\,\,\Basval} \,\, \vijkl \,\, \aic{k}\aic{l}\ai{j}\ai{i}\,\,\,,
|
|
\end{aligned}
|
|
\end{equation}
|
|
and $\Basval$ is the subset of molecular orbitals for which we want to define the expectation value, which will be typically the all MOs except those frozen.
|
|
Following the spirit of \eqref{eq:fbasis}, the function $\fbasisval$ can be defined as
|
|
\begin{equation}
|
|
\label{eq:fbasisval}
|
|
\begin{aligned}
|
|
\fbasisval = \sum_{ij\,\,\in\,\,\Bas} \,\, \sum_{klmn\,\,\in\,\,\Basval} & \vijkl \,\, \gammaklmn{\wf{}{\Bas}} \\& \phix{n}{2} \phix{m}{1} \phix{i}{1} \phix{j}{2}.
|
|
\end{aligned}
|
|
\end{equation}
|
|
|
|
Then, the effective interaction associated to the valence $\wbasisval$ is simply defines as
|
|
\begin{equation}
|
|
\label{eq:def_weebasis}
|
|
\wbasisval = \frac{\fbasisval}{\twodmrdiagpsival},
|
|
\end{equation}
|
|
where $\twodmrdiagpsival$ is the two body density associated to the valence electrons:
|
|
\begin{equation}
|
|
\twodmrdiagpsival = \sum_{klmn\,\,\in\,\,\Basval} \gammamnkl[\wf{}{\Bas}] \,\, \phix{m}{1} \phix{n}{2} \phix{k}{1} \phix{l}{2} .
|
|
\end{equation}
|
|
It is important to notice in \eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\Bas$, and the $(k,l,m,n)$, which span only the valence space $\Basval$. Only 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 $\Bas$, whatever the choice of subset $\Basval$.
|
|
|
|
|
|
\subsubsection{Definition of a range-separation parameter varying in space}
|
|
To be able to approximate the complementary functional $\efuncbasis$ thanks to functionals developed in the field of RSDFT, we fit the effective interaction with a long-range interaction having a range-separation parameter \textit{varying in space}.
|
|
More precisely, if we define the value of the interaction at coalescence as
|
|
\begin{equation}
|
|
\label{eq:def_wcoal}
|
|
\wbasiscoal{} = W_{\wf{}{\Bas}}(\bfr{},\bar{{\bf x}}_{}).
|
|
\end{equation}
|
|
where $(\bfr{},\bar{{\bf x}}_{})$ means a couple of anti-parallel spins at the same point in $\bfrb{}$,
|
|
we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction:
|
|
\begin{equation}
|
|
\wbasiscoal{} = w^{\text{lr},\murpsi}(\bfrb{},\bfrb{})
|
|
\end{equation}
|
|
where the long-range-like interaction is defined as:
|
|
\begin{equation}
|
|
w^{\text{lr},\mur}(\bfrb{1},\bfrb{2}) = \frac{ 1 }{2} \bigg( \frac{\text{erf}\big( \murr{1} \, r_{12}\big)}{r_{12}} + \frac{\text{erf}\big( \murr{2} \, r_{12}\big)}{ r_{12}}\bigg).
|
|
\end{equation}
|
|
The equation \eqref{eq:def_wcoal} is equivalent to the following condition for $\murpsi$:
|
|
\begin{equation}
|
|
\label{eq:mu_of_r}
|
|
\murpsi = \frac{\sqrt{\pi}}{2} \, \wbasiscoal{} \, .
|
|
\end{equation}
|
|
As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as
|
|
\begin{equation}
|
|
\label{eq:mu_of_r_val}
|
|
\murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, .
|
|
\end{equation}
|
|
An important point to notice is that, in the limit of a complete basis set $\Bas$, as
|
|
\begin{equation}
|
|
\begin{aligned}
|
|
&\lim_{\Bas \rightarrow \infty}\wbasis = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\\
|
|
&\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, ,
|
|
\end{aligned}
|
|
\end{equation}
|
|
one has
|
|
\begin{equation}
|
|
\begin{aligned}
|
|
&\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = +\infty\,\,, \\
|
|
&\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,,
|
|
\end{aligned}
|
|
\end{equation}
|
|
and therefore
|
|
\begin{equation}
|
|
\label{eq:lim_mur}
|
|
\begin{aligned}
|
|
&\lim_{\Bas \rightarrow \infty} \murpsi = +\infty \,\, \\
|
|
&\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
|
|
\end{aligned}
|
|
\end{equation}
|
|
|
|
\subsection{Approximations for the complementary functional $\ecompmodel$}
|
|
\subsubsection{General scheme}
|
|
\label{sec:ecmd}
|
|
In \onlinecite{GinPraFerAssSavTou-JCP-18} the authors have proposed to approximate the complementary functional $\efuncbasis$ by using a specific class of SRDFT energy functionals, namely the ECMD whose general definition is\cite{TouGorSav-TCA-05}:
|
|
\begin{equation}
|
|
\begin{aligned}
|
|
\label{eq:ec_md_mu}
|
|
\ecmubis = & \min_{\Psi \rightarrow \denr}\elemm{\Psi}{\kinop +\weeop}{\Psi}\\-\;&\elemm{\psimu[\denr]}{\kinop+\weeop}{\psimu[\denr]},
|
|
\end{aligned}
|
|
\end{equation}
|
|
where the wave function $\psimu[\denr]$ is defined by the constrained minimization
|
|
\begin{equation}
|
|
\label{eq:argmin}
|
|
\psimu[\denr] = \arg \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop + \weeopmu}{\Psi},
|
|
\end{equation}
|
|
where $\weeopmu$ is the long-range electron-electron interaction operator
|
|
\begin{equation}
|
|
\label{eq:weemu}
|
|
\weeopmu = \frac{1}{2} \iint \text{d}{\bf r}_1 \text{d}{\bf r}_2 \; w^{\text{lr},\mu}(|{\bf r}_1 - {\bf r}_2|) \hat{n}^{(2)}({\bf r}_1,{\bf r}_2),
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\label{eq:erf}
|
|
w^{\text{lr},\mu}(|{\bf r}_1 - {\bf r}_2|) = \frac{\text{erf}(\mu |{\bf r}_1 - {\bf r}_2|)}{|{\bf r}_1 - {\bf r}_2|},
|
|
\end{equation}
|
|
and the pair-density operator $\hat{n}^{(2)}({\bf r}_1,{\bf r}_2) =\hat{n}({\bf r}_1) \hat{n}({\bf r}_2) - \delta ({\bf r}_1-{\bf r}_2) \hat{n}({\bf r}_1)$.
|
|
The ECMD functionals admit two limits as function of $\mu$
|
|
\begin{equation}
|
|
\label{eq:large_mu_ecmd}
|
|
\lim_{\mu \rightarrow \infty} \ecmubis = 0 \quad \forall\,\,\denr
|
|
\end{equation}
|
|
\begin{equation}
|
|
\label{eq:small_mu_ecmd}
|
|
\lim_{\mu \rightarrow 0} \ecmubis = E_{\text{c}}[\denr]\quad \forall\,\,\denr
|
|
\end{equation}
|
|
where $E_{\text{c}}[\denr]$ is the usual universal correlation functional defined in the Kohn-Sham DFT.
|
|
These functionals differ from the standard RSDFT correlation functional by the fact that the reference is not the Kohn-Sham Slater determinant but a multi determinant wave function, which makes them much more adapted in the present context where one aims at correcting the general multi-determinant WFT model.
|
|
|
|
The general scheme for estimating $\ecompmodel$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$ labelled by ECMD-$\mathcal{X}$.
|
|
Such a functional of the density $\denr$ (and potentially its derivatives $\nabla \denr$) is defined for any value of the range-separation parameter $\mu$.
|
|
A general scheme to approximate $\ecompmodel$ is to use $\ecmuapprox$ with the $\mur$ defined in \eqref{eq:def_weebasis} and to evaluate it at the density defined by the model $\denmodel$
|
|
\begin{equation}
|
|
\label{eq:approx_ecfuncbasis}
|
|
\ecompmodel \approx \ecmuapproxmurmodel
|
|
\end{equation}
|
|
Therefore, any approximated ECMD can be used to estimate $\ecompmodel$.
|
|
It is important to notice that in the limit of a complete basis set, according to equations \eqref{eq:lim_mur} and \eqref{eq:large_mu_ecmd} one has
|
|
\begin{equation}
|
|
\lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
|
|
\end{equation}
|
|
for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
|
|
|
|
\subsubsection{LDA approximation for the complementary functional}
|
|
As done in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, one can define an LDA-like approximation for $\ecompmodel$ as
|
|
\begin{equation}
|
|
\label{eq:def_lda_tot}
|
|
\ecompmodellda = \int \, \text{d}{\bf r} \,\, \denmodelr \,\, \emuldamodel\,,
|
|
\end{equation}
|
|
where $\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}(n,\mu)$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}. In practice, for open-shell systems, we use the spin-polarized version of this functional (i.e., depending on the spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.
|
|
|
|
\subsubsection{New PBE interpolated ECMD functional}
|
|
The LDA-like functional defined in \eqref{eq:def_lda_tot} relies only on the transferability of the physics of UEG which is certainly valid for large values of $\mu$ but which is known to over correlate for small values of $\mu$.
|
|
In order to correct such a defect, we propose here a new ECMD functional inspired by the recently proposed functional of some of the present authors\cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional when $\mu \rightarrow 0$ and the exact behaviour which is known when $\mu \rightarrow \infty$.
|
|
|
|
Thanks to the study of the behaviour in the large $\mu$ limit of the various quantities appearing in the ECMD\cite{TouColSav-PRA-04,GoriSav-PRA-06,PazMorGori-PRB-06}, one can have an analytical expression of $\ecmubis$ in that regime
|
|
\begin{equation}
|
|
\label{eq:ecmd_large_mu}
|
|
\ecmubis = \frac{2\sqrt{\pi}\left(1 - \sqrt{2}\right)}{3\,\mu^3} \int \text{d}{\bf r} \,\, n^{(2)}({\bf} r)
|
|
\end{equation}
|
|
where $ n^{(2)}({\bf r}) $ is the \textit{exact} on-top pair density for the ground state of the system.
|
|
As the exact ground state on-top pair density $n^{(2)}({\bf} r)$ is not known, we propose here to approximate it by that of the UEG at the density of the system:
|
|
\begin{equation}
|
|
\label{eq:ueg_ontop}
|
|
n^{(2)}({\bf} r) \approx n^{(2)}_{\text{UEG}}(n_{\uparrow}({\bf} r) , \, n_{\downarrow}({\bf} r))
|
|
\end{equation}
|
|
where $n_{\uparrow}({\bf} r)$ and $ n_{\downarrow}({\bf} r)$ are, respectively, the up and down spin densities of the physical system at ${\bf} r$, $n^{(2)}_{\text{UEG}}(n_{\uparrow} , n_{\downarrow})$ is the UEG on-top pair density
|
|
\begin{equation}
|
|
\label{eq:ueg_ontop}
|
|
n^{(2)}_{\text{UEG}}(n_{\uparrow} , n_{\downarrow}) = 4\, n_{\uparrow} \, n_{\downarrow} \, g_0(n_{\uparrow},\, n_{\downarrow})
|
|
\end{equation}
|
|
and $g_0(n_{\uparrow} ,\, n_{\downarrow})$ is the correlation factor of the UEG whose parametrization can be found in equation (46) of \onlinecite{GorSav-PRA-06}.
|
|
|
|
As the form in \eqref{eq:ecmd_large_mu} diverges for small values of $\mu$ as $1/\mu^3$, we follow the work proposed in \cite{FerGinTou-JCP-18} and interpolate between the large-$\mu$ limit and the $\mu = 0$ limit where the $\ecmubis$ reduces to the Kohn-Sham correlation functional (see equation \eqref{eq:small_mu_ecmd}), for which we take the PBE approximation as in \cite{FerGinTou-JCP-18}.
|
|
More precisely, we propose the following expression for the
|
|
\begin{equation}
|
|
\label{eq:ecmd_large_mu}
|
|
\ecmubis = \int \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(n({\bf} r),\nabla n({\bf} r);\,\mu)
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\label{eq:epsilon_cmdpbe}
|
|
\bar{e}_{\text{c,md}}^\text{PBE}(n,\nabla n;\,\mu) = \frac{e_c^{PBE}(n,\nabla n)}{1 + \beta_{\text{c,md}\,\text{PBE}}(n,\nabla n;\,\mu)\mu^3 }
|
|
\end{equation}
|
|
\begin{equation}
|
|
\label{eq:epsilon_cmdpbe}
|
|
\beta(n,\nabla n;\,\mu) = \frac{3 e_c^{PBE}(n,\nabla n)}{2\sqrt{\pi}\left(1 - \sqrt{2}\right)n^{(2)}_{\text{UEG}}(n_{\uparrow} , n_{\downarrow})}.
|
|
\end{equation}
|
|
|
|
Therefore, we propose this approximation for the complementary functional $\ecompmodel$:
|
|
\begin{equation}
|
|
\label{eq:def_lda_tot}
|
|
\ecompmodelpbe = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(n({\bf r}),\nabla n({\bf r});\,\mur)
|
|
\end{equation}
|
|
|
|
\subsection{Valence-only approximation for the complementary functional}
|
|
We now introduce a valence-only approximation for the complementary functional which is needed to correct for frozen core WFT models.
|
|
Defining the valence one-body spin density matrix as
|
|
\begin{equation}
|
|
\begin{aligned}
|
|
\onedmval[\wf{}{\Bas}] & = \elemm{\wf{}{\Bas}}{a^{\dagger}_{i,\sigma} a_{j,\sigma}}{\wf{}{\Bas}} \qquad \text{if }(i,j)\in \Basval \\
|
|
& = 0 \qquad \text{in other cases}
|
|
\end{aligned}
|
|
\end{equation}
|
|
then one can define the valence density as:
|
|
\begin{equation}
|
|
\denval_{\sigma}({\bf r}) = \sum_{i,j} \onedmval[\wf{}{\Bas}] \phi_i({\bf r}) \phi_j({\bf r})
|
|
\end{equation}
|
|
Therefore, we propose the following valence-only approximations for the complementary functional
|
|
\begin{equation}
|
|
\label{eq:def_lda_tot}
|
|
\ecompmodelldaval = \int \, \text{d}{\bf r} \,\, \denval({\bf r}) \,\, \emuldaval\,,
|
|
\end{equation}
|
|
\begin{equation}
|
|
\label{eq:def_lda_tot}
|
|
\ecompmodelpbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval)
|
|
\end{equation}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Results}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\subsection{Comparison between the CIPSI and CCSD(T) models in the case of C$_2$, N$_2$, O$_2$, F$_2$}
|
|
We begin the investigation of the behavior of the basis-set correction by the study of the atomization energies of the C$_2$, N$_2$, O$_2$, F$_2$ homo-nuclear diatomic molecules in the Dunning cc-pVXZ and cc-pCVXZ (X=D,T,Q,5) using both the CIPSI algorithm and the CCSD(T). All through this work, we follow the frozen core (FC) convention of Klopper \textit{et. al}\cite{HauKlo-JCP-12} which consists in all-electron calculations for Li-Be, a He core for B-Na atoms and a Ne core for the Al-Cl series. In the context of the DFT correction for the basis-set, this implies that, for a given system in a given basis set $\Bas$, the set of valence orbitals $\Basval$ involved in the definition of the valence interaction $\wbasisval$ and density $\onedmval$ refers to all MOs except the core.
|
|
%\subsubsection{CIPSI calculations and the basis-set correction}
|
|
%All CIPSI calculations were performed in two steps. First, a CIPSI calculation was performed until the zeroth-order wave function reaches $10^6$ Slater determinants, from which we extracted the natural orbitals. From this set of natural orbitals, we performed CIPSI calculations until the $\EexFCIbasis$ reaches about $0.1$ mH convergence for each systems. Such convergence criterion is more than sufficient for the CIPSI densities $\dencipsi$.
|
|
%Regarding the wave function $\wf{}{\Bas}$ chosen to define the local range-separation parameter $\mur$, we take a single Slater determinant built with the natural orbitals of the first CIPSI calculation.
|
|
\subsubsection{CCSD(T) calculations and the basis-set correction}
|
|
|
|
\begin{table*}
|
|
\caption{
|
|
\label{tab:diatomics}
|
|
Dissociation energy ($\De$) in kcal/mol of the \ce{C2}, \ce{O2}, \ce{N2} and \ce{F2} molecules computed with various methods and basis sets.
|
|
}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{llddddd}
|
|
& & \mc{4}{c}{Dunning's basis set}
|
|
\\
|
|
\\
|
|
\cline{3-6}
|
|
Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$E_{QZ5Z}^{\infty}$}
|
|
\\
|
|
\\
|
|
\hline
|
|
\ce{C2} & (FC)FCIQMC & 130.0(1) & 139.9(3) & 143.3(2) & & 144.9 \\
|
|
& (FC)FCIQMC+F12 & 142.3 & 145.3 & & & \\
|
|
\hline
|
|
& ex (FC)FCI & 132.0 & 140.3 & 143.6 & 144.3 & \\
|
|
\hline
|
|
& ex (FC)FCI+LDA-val & 143.0 & 145.4 & 146.4 & 146.0 & \\
|
|
& ex (FC)FCI+PBE-val & 147.4 & 146.1 & 146.4 & 145.9 & \\
|
|
& exFCI+PBE-on-top-val & 143.3 & 144.7 & 145.7 & 145.6 & \\
|
|
%%%%%%%% \hline
|
|
%%%%%%%% & ex (FC)FCI+LDA & 141.9 & 142.8 & 145.8 & 146.2 & \\
|
|
%%%%%%%% & ex (FC)FCI+PBE & 146.1 & 143.9 & 145.9 & 145.12 & \\
|
|
%%%%%%%% \hline
|
|
%%%%%%%% & exFCI+PBE-on-top& 142.7 & 142.7 & 145.3 & 144.9 & \\
|
|
\\
|
|
\cline{3-6}
|
|
& & \tabc{cc-pCVDZ} & \tabc{cc-pCVTZ} & \tabc{cc-pCVQZ} & \tabc{cc-pCV5Z} & \tabc{ $E_{CQZC5Z}^{\infty}$ }
|
|
\\
|
|
\\
|
|
\hline
|
|
%%%%%%%% & ex (FC)FC-FCI & 130.5 & 140.5 & 143.8 & 144.9 & 147.1 \\
|
|
%%%%%%%% \hline
|
|
%%%%%%%% & ex (FC)FCI+LDA & 140.9 & 145.7 & 146.6 & 146.4 & \\
|
|
%%%%%%%% & ex (FC)FCI+LDA-val & 141.3 & 145.6 & 146.5 & 146.4 & \\
|
|
%%%%%%%% \hline
|
|
%%%%%%%% & ex (FC)FCI+PBE & 144.5 & 145.9 & 146.4 & 146.3 & \\
|
|
%%%%%%%% & ex (FC)FCI+PBE -val & 145.2 & 145.9 & 146.4 & 146.3 & \\
|
|
|
|
\hline
|
|
& ex FCI & 131.0 & 141.5 & 145.1 & 146.1 & 147.1 \\
|
|
\hline
|
|
& ex FCI+LDA & 141.4 & 146.7 & 147.8 & 147.6 & \\
|
|
%%%%%%%% & ex FCI+LDA-val & 141.8 & 146.6 & 147.7 & 147.6 & \\
|
|
\hline
|
|
& ex FCI+PBE & 145.1 & 147.0 & 147.7 & 147.5 & \\
|
|
%%%%%%%% & ex FCI+PBE-val & 145.7 & 147.0 & 147.6 & 147.5 & \\
|
|
\\
|
|
\\
|
|
& & \mc{4}{c}{Dunning's basis set}
|
|
\\
|
|
\cline{3-6}
|
|
Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$E_{QZ5Z}^{\infty}$}
|
|
\\
|
|
\\
|
|
\ce{N2} & ex (FC)FCI & 201.1 & 217.1 & 223.5 & 225.7 & 227.8 \\
|
|
\hline
|
|
& ex (FC)FCI+LDA-val & 217.9 & 225.9 & 228.0 & 228.6 & \\
|
|
& ex (FC)FCI+PBE-val & 227.7 & 227.8 & 228.3 & 228.5 & \\
|
|
& exFCI+PBE-on-top-val & 224.8 & 226.7 & 228.3 & 228.3 & \\
|
|
\hline
|
|
\hline
|
|
& (FC)CCSD(T) & 199.9 & 216.3 & 222.8 & 225.0 & 227.2 \\
|
|
\hline
|
|
%%%%%%%%& ex (FC)CCSD(T)+LDA & 214.7 & 221.9 & ----- & ----- & \\
|
|
%%%%%%%%& ex (FC)CCSD(T)+PBE & 223.4 & 224.3 & ----- & ----- & \\
|
|
& ex (FC)CCSD(T)+LDA-val & 216.3 & 224.8 & 227.2 & 227.8 & \\
|
|
& ex (FC)CCSD(T)+PBE-val & 225.9 & 226.7 & 227.5 & 227.8 & \\
|
|
\hline
|
|
%%%%%%%%& ex (FC)FCI+LDA & 216.4 & 223.1 & 227.9 & 228.1 & \\
|
|
%%%%%%%%& ex (FC)FCI+PBE & 225.4 & 225.6 & 228.2 & 227.9 & \\
|
|
%%%%%%%% & exFCI+PBE-on-top& 222.3 & 224.6 & 227.7 & 227.7 & \\
|
|
\\
|
|
\\
|
|
\cline{3-6}
|
|
& & \tabc{cc-pCVDZ} & \tabc{cc-pCVTZ} & \tabc{cc-pCVQZ} & \tabc{cc-pCV5Z} & \tabc{$E_{QZ5Z}^{\infty}$}
|
|
\\
|
|
\\
|
|
%%%%%%%% & ex (FC)FCI & 201.7 & 217.9 & 223.7 & 225.7 & 228.8 \\
|
|
%%%%%%%%\hline
|
|
%%%%%%%% & ex (FC)FCI+LDA & 217.5 & 226.2 & 228.4 & 228.5 & \\
|
|
%%%%%%%% & ex (FC)FCI+LDA-val & 218.5 & 226.3 & 228.4 & 228.0 & \\
|
|
%%%%%%%%\hline
|
|
%%%%%%%% & ex (FC)FCI+PBE & 225.8 & 227.6 & 228.4 & 228.3 & \\
|
|
%%%%%%%% & ex (FC)FCI+PBE-val & 227.5 & 227.7 & 228.4 & 228.0 & \\
|
|
\hline
|
|
& ex FCI & 202.2 & 218.5 & 224.4 & 226.6 & 228.8 \\
|
|
\hline
|
|
& ex FCI+LDA & 218.0 & 226.8 & 229.1 & 229.4 & \\
|
|
%%%%%%%% & ex FCI+LDA-val & 219.1 & 226.9 & 229.0 & 228.9 & \\
|
|
\hline
|
|
& ex FCI+PBE & 226.4 & 228.2 & 229.1 & 229.2 & \\
|
|
%%%%%%%% & ex FCI+PBE-val & 228.0 & 228.2 & 229.1 & 228.9 & \\
|
|
\\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\fnt[1]{Results from Ref.~\onlinecite{BytLaiRuedenJCP05}.}
|
|
\fnt[2]{Results from Ref.~\onlinecite{PetTouUmr-JCP-12}.}
|
|
\end{table*}
|
|
|
|
|
|
\begin{table*}
|
|
\caption{
|
|
\label{tab:diatomics}
|
|
Dissociation energy ($\De$) in kcal/mol of the \ce{C2}, \ce{O2}, \ce{N2} and \ce{F2} molecules computed with various methods and basis sets.
|
|
}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{llddddd}
|
|
\\
|
|
\cline{3-6}
|
|
Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$E_{QZ5Z}^{\infty}$}
|
|
\\
|
|
\ce{O2} & exFCI & 105.2 & 114.5 & 118.0 &119.1 & 120.0 \\
|
|
\hline
|
|
%%%%%%%% & exFCI+LDA & 111.8 & 117.2 & 120.0 &119.9 & \\
|
|
& exFCI+LDA-val & 112.4 & 118.4 & 120.2 &120.4 & \\
|
|
%%%%%%%%\hline
|
|
%%%%%%%% & exFCI+PBE & 115.9 & 118.4 & 120.1 &119.9 & \\
|
|
& exFCI+PBE-val & 117.2 & 119.4 & 120.3 &120.4 & \\
|
|
\hline
|
|
& (FC)CCSD(T) & 103.9 & 113.6 & 117.1 & 118.6 & 120.0 \\
|
|
& ex (FC)CCSD(T)+LDA-val & 110.6 & 117.2 & 119.2 & 119.8 & \\
|
|
& ex (FC)CCSD(T)+PBE-val & 115.1 & 118.0 & 119.3 & 119.8 & \\
|
|
\hline
|
|
%%%%%%%% & exFCI+PBE-on-top & 115.0 & 118.4 & 120.2 & & \\
|
|
%%%%%%%% & exFCI+PBE-on-top-val & 116.1 & 119.4 & 120.5 & & \\
|
|
\\
|
|
Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$E_{QZ5Z}^{\infty}$}
|
|
\\
|
|
\ce{F2} & exFCI & 26.7 & 35.1 & 37.1 & 38.0 & 39.0 \\
|
|
\hline
|
|
%%%%%%%% & exFCI+LDA & 30.8 & 37.0 & 38.7 & 38.7 & \\
|
|
& exFCI+LDA-val & 30.4 & 37.2 & 38.4 & 38.9 & \\
|
|
%%%%%%%% \hline
|
|
%%%%%%%% & exFCI+PBE & 33.3 & 37.8 & 38.8 & 38.7 & \\
|
|
& exFCI+PBE -val & 33.1 & 37.9 & 38.5 & 38.9 & \\
|
|
\hline
|
|
%%%%%%%% & exFCI+PBE-on-top& 32.1 & 37.5 & 38.7 & 38.7 & \\
|
|
%%%%%%%% & exFCI+PBE-on-top-val & 32.4 & 37.8 & 38.8 & 38.8 & \\
|
|
\hline
|
|
& (FC)CCSD(T) & 25.7 & 34.4 & 36.5 & 37.4 & 38.2 \\
|
|
& ex (FC)CCSD(T)+LDA-val & 29.2 & 36.5 & 37.2 & 38.2 & \\
|
|
& ex (FC)CCSD(T)+PBE-val & 31.5 & 37.1 & 37.8 & 38.2 & \\
|
|
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\fnt[1]{Results from Ref.~\onlinecite{BytLaiRuedenJCP05}.}
|
|
\fnt[2]{Results from Ref.~\onlinecite{PetTouUmr-JCP-12}.}
|
|
\end{table*}
|
|
|
|
|
|
|
|
|
|
%
|
|
\bibliography{G2-srDFT}
|
|
|
|
\end{document}
|