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Manuscript/G2-srDFT.bib
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@ -10,6 +10,16 @@ year = {1973},
pages = {5745}
}
@ARTICLE{malrieu,
author = {B. Huron and J.P. Malrieu and P. Rancurel},
journal = {J. Chem. Phys.},
volume = {58},
year = {1973},
pages = {5745},
doi ={10.1063/1.1679199}
}
@ARTICLE{bender,
author = {C. F. Bender and E. R. Davidson},
keywords = {},
@ -53,21 +63,6 @@ pages = {5745}
isbn = {},
}
@ARTICLE{harrison,
author = {R. J. Harrison},
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number = {},
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publisher = {},
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}
@article{Rubio198698,
title = "Convergence of a multireference second-order mbpt method (CIPSI) using a zero-order wavefunction derived from an \{MS\} \{SCF\} calculation ",
@ -109,6 +104,74 @@ pages = {5745}
pages={117-128}
}
@article{canadian,
author = {Emmanuel Giner and Anthony Scemama and Michel Caffarel},
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}
@article{atoms_3d,
author = "Scemama, A. and Applencourt, T. and Giner, E. and Caffarel, M.",
title = "Accurate nonrelativistic ground-state energies of 3d transition metal atoms",
journal = "J. Chem. Phys.",
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}
@article{f2_dmc,
author = "Giner, Emmanuel and Scemama, Anthony and Caffarel, Michel",
title = "Fixed-node diffusion Monte Carlo potential energy curve of the fluorine molecule F2 using selected configuration interaction trial wavefunctions",
journal = "J. Chem. Phys.",
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}
@article{atoms_dmc_julien,
author = {Emmanuel Giner and Roland Assaraf and Julien Toulouse},
title = {Quantum Monte Carlo with reoptimised perturbatively selected configuration-interaction wave functions},
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}^M
@article{Angeli2000472,
title = "On a mixed Møller–Plesset Epstein–Nesbet partition of the Hamiltonian to be used in multireference perturbation configuration interaction ",
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@ -130,6 +193,34 @@ pages = {5745}
pages={259-264}
}
@ARTICLE{harrison,
author = {R. J. Harrison},
keywords = {},
journal = {J. Chem. Phys.},
volume = {94},
number = {},
year = {1991},
pages = {5021},
publisher = {},
url = {},
doi = {},
abstract = {},
numpages ={},
}
@article{hbci,
author = {Holmes, Adam A. and Tubman, Norm M. and Umrigar, C. J.},
title = {Heat-Bath Configuration Interaction: An Efficient Selected Configuration Interaction Algorithm Inspired by Heat-Bath Sampling},
journal = {J. Chem. Theory Comput.},
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URL = {https://doi.org/10.1021/acs.jctc.6b00407}
}
@article{three_class_CIPSI,
title = "Convergence of an improved CIPSI algorithm",
journal = "Chem. Phys.",
@ -7498,7 +7589,7 @@ doi = {10.1142/S1793984412300063}
year = {2014}
}
% QMC@home for noncovalent interactions
% QMC home for noncovalent interactions
@article{KorLucGri-JPCA-08,
author = {M. Korth and A. L\"uchow and S. Grimme},
journal = {J. Phys. Chem. A},
@ -14822,3 +14913,114 @@ pages = {084101},
year = {2008}
}
@article{FerGinTou-JCP-18,
author = {Ferté,Anthony and Giner,Emmanuel and Toulouse,Julien },
title = {Range-separated multideterminant density-functional theory with a short-range correlation functional of the on-top pair density},
journal = {The Journal of Chemical Physics},
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year = {2019},
doi = {10.1063/1.5082638},
URL = {https://doi.org/10.1063/1.5082638},
eprint = {https://doi.org/10.1063/1.5082638}
}
@article{GoriSav-PRA-06,
title = {Properties of short-range and long-range correlation energy density functionals from electron-electron coalescence},
author = {Gori-Giorgi, Paola and Savin, Andreas},
journal = {Phys. Rev. A},
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numpages = {9},
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month = {Mar},
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doi = {10.1103/PhysRevA.73.032506},
url = {https://link.aps.org/doi/10.1103/PhysRevA.73.032506}
}
@article{PazMorGori-PRB-06,
title = {Local-spin-density functional for multideterminant density functional theory},
author = {Paziani, Simone and Moroni, Saverio and Gori-Giorgi, Paola and Bachelet, Giovanni B.},
journal = {Phys. Rev. B},
volume = {73},
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numpages = {9},
year = {2006},
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doi = {10.1103/PhysRevB.73.155111},
url = {https://link.aps.org/doi/10.1103/PhysRevB.73.155111}
}
@article{GinTewGarAla-JCTC-18,
author = {Giner, Emmanuel and Tew, David P. and Garniron, Yann and Alavi, Ali},
title = {Interplay between Electronic Correlation and Metal–Ligand Delocalization in the Spectroscopy of Transition Metal Compounds: Case Study on a Series of Planar Cu2+ Complexes},
journal = {Journal of Chemical Theory and Computation},
volume = {14},
number = {12},
pages = {6240-6252},
year = {2018},
doi = {10.1021/acs.jctc.8b00591},
note ={PMID: 30347156},
URL = {https://doi.org/10.1021/acs.jctc.8b00591},
eprint = {https://doi.org/10.1021/acs.jctc.8b00591}
}
@article{LooBogSceCafJAc-JCTC-19,
author = {Loos, Pierre-François and Boggio-Pasqua, Martial and Scemama, Anthony and Caffarel, Michel and Jacquemin, Denis},
title = {Reference Energies for Double Excitations},
journal = {Journal of Chemical Theory and Computation},
volume = {15},
number = {3},
pages = {1939-1956},
year = {2019},
doi = {10.1021/acs.jctc.8b01205},
URL = {https://doi.org/10.1021/acs.jctc.8b01205},
eprint = {https://doi.org/10.1021/acs.jctc.8b01205}
}
@article{LooSceBloGarCafJac-JCTC-18,
author = {Loos, Pierre-François and Scemama, Anthony and Blondel, Aymeric and Garniron, Yann and Caffarel, Michel and Jacquemin, Denis},
title = {A Mountaineering Strategy to Excited States: Highly Accurate Reference Energies and Benchmarks},
journal = {Journal of Chemical Theory and Computation},
volume = {14},
number = {8},
pages = {4360-4379},
year = {2018},
doi = {10.1021/acs.jctc.8b00406},
note ={PMID: 29966098},
URL = {https://doi.org/10.1021/acs.jctc.8b00406},
eprint = {https://doi.org/10.1021/acs.jctc.8b00406}
}
@article{HolUmrSha-JCP-17,
author = {Holmes,Adam A. and Umrigar,C. J. and Sharma,Sandeep },
title = {Excited states using semistochastic heat-bath configuration interaction},
journal = {The Journal of Chemical Physics},
volume = {147},
number = {16},
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URL = {https://doi.org/10.1063/1.4998614},
eprint = {https://doi.org/10.1063/1.4998614}
}
@article{stochastic_pt_yan,
author = {Yann Garniron and Anthony Scemama and Pierre-François Loos and Michel Caffarel},
title = {Hybrid stochastic-deterministic calculation of the second-order perturbative contribution of multireference perturbation theory},
journal = {J. Chem. Phys.},
volume = {147},
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doi = {10.1063/1.4992127},
URL = {https://doi.org/10.1063/1.4992127}
}

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Manuscript/G2-srDFT.tex
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@ -39,9 +39,12 @@
%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
@ -53,20 +56,29 @@
\newcommand{\EDMC}{E_\text{DMC}}
\newcommand{\EexFCI}{E_\text{exFCI}}
\newcommand{\EexFCIbasis}{E_\text{exFCI}^{\basis}}
\newcommand{\EexFCIinfty}{E_\text{exFCI}^{\infty}}
\newcommand{\EexDMC}{E_\text{exDMC}}
\newcommand{\Ead}{\Delta E_\text{ad}}
\newcommand{\efci}[0]{E_{\text{FCI}}^{\basis}}
\newcommand{\emodel}[0]{E_{\model}^{\basis}}
\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)}}^{\basis}}
\newcommand{\efuncbasisfci}[0]{\bar{E}^\basis[\denfci]}
\newcommand{\efuncbasis}[0]{\bar{E}^\basis[\den]}
\newcommand{\efuncden}[1]{\bar{E}^\basis[#1]}
\newcommand{\ecompmodel}[0]{\bar{E}^\basis[\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{\efuncbasislda}[0]{\bar{E}_{\text{LDA}}^{\basis,\psibasis}[\den]}
\newcommand{\efuncbasisldaval}[0]{\bar{E}_{\text{LDA, val}}^{\basis,\psibasis}[\den]}
\newcommand{\efuncbasispbe}[0]{\bar{E}_{\text{PBE}}^{\basis,\psibasis}[\den]}
\newcommand{\efuncbasispbeval}[0]{\bar{E}_{\text{PBE, val}}^{\basis,\psibasis}[\den]}
\newcommand{\ecmuapproxmurmodel}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denmodel;\,\mur]}
\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\basis,\psibasis}[\den]}
\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\basis,\psibasis}[\den]}
\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\basis,\psibasis}[\den]}
\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\basis,\psibasis}[\den]}
\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\psibasis)\right)}
\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\psibasis)\right)}
@ -120,10 +132,17 @@
\newcommand{\basis}[0]{\mathcal{B}}
\newcommand{\basisval}[0]{\mathcal{B}_\text{val}}
% MODEL
\newcommand{\model}[0]{\mathcal{Y}}
% densities
\newcommand{\denmodel}[0]{\den_{\model}^\basis}
\newcommand{\denmodelr}[0]{\den_{\model}^\basis ({\bf r})}
\newcommand{\denfci}[0]{\den_{\psifci}}
\newcommand{\denhf}[0]{\den_{\text{HF}}^\basis}
\newcommand{\denrfci}[0]{\denr_{\psifci}}
\newcommand{\dencipsi}[0]{{n}_{\text{CIPSI}}^\basis({\bf r})}
\newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\basis({\bf r})}
\newcommand{\dencipsi}[0]{{n}_{\text{CIPSI}}^\basis}
\newcommand{\den}[0]{{n}}
\newcommand{\denval}[0]{{n}^{\text{val}}}
\newcommand{\denr}[0]{{n}({\bf r})}
@ -185,29 +204,131 @@
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The DFT basis-set correction in a nutshell}
The basis-set correction investigated here proposes to use the RSDFT formalism to capture a part of the short-range correlation effects missing from the description of the WFT in a finite basis set.
Here, we briefly explain the working equations and notations needed for this work, and the interested reader can find the detailed formal derivation of the theory in \cite{GinPraFerAssSavTou-JCP-18}.
Here, we briefly explain the working equations and notations needed for this work, and the interested reader can find the detailed formal derivation of the theory in \onlinecite{GinPraFerAssSavTou-JCP-18}.
\subsection{The basic concepts}
Consider an incomplete basis-set $\basis$ for which we assume to have accurate approximations of both the FCI density $\denfci$ and energy $\efci$. According to equation (15) of \cite{GinPraFerAssSavTou-JCP-18}, one can approximate the exact ground state energy $E_0$ as
\subsection{Correcting the basis set error of a general WFT model}
Consider a $N-$electron physical system described in an incomplete basis-set $\basis$ and for which we assume to have both the FCI density $\denfci$ and energy $\efci$. Assuming that $\denfci$ is a good approximation of the \textit{exact} ground state density, according to equation (15) of \onlinecite{GinPraFerAssSavTou-JCP-18}, one can approximate the exact ground state energy $E_0$ as
\begin{equation}
\label{eq:e0basis}
E_0 \approx \efci + \efuncbasisfci
\end{equation}
where $\efuncbasis$ is the complementary density functional defined in equation (8) of \cite{GinPraFerAssSavTou-JCP-18}
where $\efuncbasis$ is the complementary density functional defined in equation (8) of \onlinecite{GinPraFerAssSavTou-JCP-18}
\begin{equation}
\begin{aligned}
\label{eq:E_funcbasis}
\efuncbasisfci = & \min_{\Psi \rightarrow \denfci} \elemm{\Psi}{\kinop + \weeop}{\Psi} \\&- \min_{\psibasis \rightarrow \denfci} \elemm{\psibasis}{\kinop + \weeop}{\psibasis},
\efuncbasis = & \min_{\Psi \rightarrow \den} \elemm{\Psi}{\kinop + \weeop}{\Psi} \\&- \min_{\psibasis \rightarrow \den} \elemm{\psibasis}{\kinop + \weeop}{\psibasis},
\end{aligned}
\end{equation}
where $\Psi$ is a general wave function being obtained in a complete basis. Provided that functional $\efuncbasis$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\denfci$ coincides with the exact ground state density, which in general is a reasonable approximation as the density converges rapidly with the basis set.
$\psibasis$ is a wave function obtained from the $N-$electron Hilbert space spanned by $\basis$, $\Psi$ is a general $N-$electron wave function being obtained in a complete basis, and both wave functions $\psibasis$ and $\Psi$ yield the same target density $\den$.
Provided that the functional $\efuncbasis$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\denfci$ 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 a theory is that, in the limit of a complete basis set $\basis$ (which we refer as $\basis \rightarrow \infty$), the functional $\efuncbasis$ tends to zero
\begin{equation}
\label{eq:limitfunc}
\lim_{\basis \rightarrow \infty} \efuncbasis = 0\qquad \forall \,\, \den\,\, ,
\end{equation}
which implies that the exact ground state energy coincides with the FCI energy in complete basis set (which we refer as $\efcicomplete$)
\begin{equation}
\label{eq:limitfunc}
\lim_{\basis \rightarrow \infty} \efci + \efuncbasisfci = \efcicomplete\,\,.
\end{equation}
Here we propose to generalize such approach to a general WFT model, referred here as $\model$, projected in a basis set $\basis$ 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_{\basis \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_{\basis \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 $\basis$, 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 $\basis$, the CIPSI energy is
\begin{align}
E_\mathrm{CIPSI}^{\basis} &= 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 \basis} \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 $\efuncbasis$}
The functional $\efuncbasis$ is not universal as it depends on the basis set $\basis$ used and a simple analytical form for such a functional is of course not known.
Following the work of \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\efuncbasis$ in two-steps using the RS-DFT formalism. First, we define a real-space representation of the coulomb interaction projected in $\basis$, which is then fitted with a long-range interaction thanks to a range-separation parameter $\mu(r)$ varying in space (see \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 the density $\denmodel$ provided by the model (see \ref{sec:ecmd}) and with the range-separation parameter $\mu(r)$ varying in space.
The functional $\efuncbasisfci$ is not universal as it depends on the basis set $\basis$ used. A simple analytical form for such a functional is of course not known and we approximate it in two-steps. First, we define a real-space representation of the coulomb interaction projected in $\basis$, which is then fitted with a long-range interaction thanks to a range-separation parameter $\mu(r)$ varying in space (see \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{Toulouse2005_ecmd}, that we evaluate at the FCI density $\denfci$ (see \ref{sec:ecmd}) and with the range-separation parameter $\mu(r)$ varying in space.
\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\basis$}
\label{sec:weff}
One of the consequences of the use of an incomplete basis-set $\basis$ is that the wave function does not present a cusp near the electron coalescence point, which means that all derivatives of the wave function are continuous. As the exact electronic cusp originates from the divergence of the coulomb interaction at the electron coalescence point, a cusp-free wave function could also come from a non-divergent electron-electron interaction. Therefore, the impact of the incompleteness of a finite basis-set $\basis$ can be thought as a cutting of the divergence of the coulomb interaction at the electron coalescence point.
The present paragraph briefly describes how to obtain an effective interaction $\wbasis$ which:
@ -224,7 +345,7 @@ Consider the coulomb operator projected in the basis-set $\basis$
\end{equation}
where the indices run over all orthonormal spin-orbitals in $\basis$ and $\vijkl$ are the usual coulomb two-electron integrals.
Consider now the expectation value of $\weeopbasis$ over a general wave function $\psibasis$ belonging to the $N-$electron Hilbert space spanned by the basis set $\basis$.
After a few mathematical work (see appendix A of \cite{GinPraFerAssSavTou-JCP-18} for a detailed derivation), such an expectation value can be rewritten as an integral over $\rnum^6$:
After a few mathematical work (see appendix A of \onlinecite{GinPraFerAssSavTou-JCP-18} for a detailed derivation), such an expectation value can be rewritten as an integral over $\rnum^6$:
\begin{equation}
\label{eq:expectweeb}
\elemm{\psibasis}{\weeopbasis}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasis,
@ -273,8 +394,8 @@ where we introduced $\wbasis$
\end{equation}
which is the effective interaction in the basis set $\basis$.
As already discussed in \cite{GinPraFerAssSavTou-JCP-18}, such an effective interaction is symmetric, \textit{a priori} non translational nor rotational invariant if the basis set $\basis$ does not have such symmetries and is necessary \textit{finite} at the electron coalescence point for an incomplete basis set $\basis$.
Also, as demonstrated in the appendix B of \cite{GinPraFerAssSavTou-JCP-18}, $\wbasis$ tends to the regular coulomb interaction $1/r_{12}$ for all points in $(\bfr{1},\bfr{2})$ in the limit of a complete basis set $\basis$.
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 $\basis$ does not have such symmetries and is necessary \textit{finite} at the electron coalescence point for an incomplete basis set $\basis$.
Also, as demonstrated in the appendix B of \onlinecite{GinPraFerAssSavTou-JCP-18}, $\wbasis$ tends to the regular coulomb interaction $1/r_{12}$ for all points in $(\bfr{1},\bfr{2})$ in the limit of a complete basis set $\basis$.
\subsubsection{Definition of a valence effective interaction}
@ -313,7 +434,7 @@ It is important to notice in \eqref{eq:fbasisval} the difference between the set
\subsubsection{Definition of a range-separation parameter varying in space}
To be able to approximate the complementary functional $\efuncbasisfci$ 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}.
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}
@ -339,9 +460,10 @@ As we defined an effective interaction for the valence electrons, we also introd
\murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, .
\end{equation}
\subsection{Approximations for the complementary functional}
\subsection{Approximations for the complementary functional $\ecompmodel$}
\subsubsection{General scheme}
In \cite{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:
\label{sec:ecmd}
In \onlinecite{GinPraFerAssSavTou-JCP-18} the authors have proposed to approximate the complementary functional $\ecompmodel$ 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}
@ -367,27 +489,38 @@ and the pair-density operator $\hat{n}^{(2)}({\bf r}_1,{\bf r}_2) =\hat{n}({\bf
These functionals differ from the standard RSDFT correlation functional by the fact that the reference is not the Konh-Sham determinant but a multi determinant wave function, which makes them much more adapted in the present context where one aims at correcting the FCI energy.
The general scheme for estimating $\efuncbasis$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$ labelled by ECMD-$\mathcal{X}$.
The general scheme for estimating $\ecompmodel$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$ labelled by ECMD-$\mathcal{X}$.
Such a functional which might depend on 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 $\efuncbasisfci$ is to use $\ecmuapprox$ with the $\mur$ defined in \eqref{eq:def_weebasis} and the FCI density $\denfci$
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 FCI density $\denfci$
\begin{equation}
\label{eq:approx_ecfuncbasis}
\efuncbasisfci \approx \ecmuapproxmurfci
\ecompmodel \approx \ecmuapproxmurmodel
\end{equation}
Therefore, any approximated ECMD can be used to estimate $\efuncbasisfci$.
Therefore, any approximated ECMD can be used to estimate $\ecompmodel$.
It is important to notice that in the limit of a complete basis set, as
\begin{equation}
\lim_{\basis \rightarrow \infty} \wbasiscoalval{} = +\infty \quad \forall\,\, \psibasis\,\,\text{and}\,\,\,{\bf r}\,,
\end{equation}
the local range separation parameter $\murpsi$ (or $\murpsival$) tends to infinity and therefore
\begin{equation}
\lim_{\basis \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
\end{equation}
which is a condition required by the exact theory (see \eqref{eq:limitfunc}).
Also, it means that one recovers a WFT model in the limit of a complete basis set, whatever the choice of $\psibasis$, functional ECMD or density used.
\subsubsection{LDA approximation for the complementary functional}
Therefore, one can define an LDA-like functional for $\efuncbasis$ as
Therefore, one can define an LDA-like functional for $\ecompmodel$ as
\begin{equation}
\label{eq:def_lda_tot}
\efuncbasislda = \int \, \text{d}{\bf r} \,\, \denr \,\, \emulda\,,
\ecompmodellda = \int \, \text{d}{\bf r} \,\, \denr \,\, \emulda\,,
\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{pbeontop} which interpolates between the usual PBE correlation functional when $\mu \rightarrow 0$ and the exact behaviour which is known when $\mu \rightarrow \infty$.
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 $\mu \rightarrow \infty$.
The exact behaviour of the $\ecmubis$ is known in the large $\mu$ limit\cite{pbeontop}:
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)
@ -398,14 +531,14 @@ As the exact ground state on-top pair density $n^{(2)}({\bf} r)$ is not known, w
\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
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})
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 \cite{ueg_ontop}.
As such a form diverges for small values of $\mu$ as $1/\mu^3$, we follow the work proposed in \cite{pbeontop} and interpolate with the Kohn-Sham correlation functional at $\mu = 0$.
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, 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}
@ -418,13 +551,13 @@ with
\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})}.
\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 $\efuncbasisfci$:
Therefore, we propose this approximation for the complementary functional $\ecompmodel$:
\begin{equation}
\label{eq:def_lda_tot}
\efuncbasispbe = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(n({\bf r}),\nabla n({\bf r});\,\mur)
\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}
@ -443,30 +576,21 @@ then one can define the valence density as:
Therefore, we propose the following valence-only approximations for the complementary functional
\begin{equation}
\label{eq:def_lda_tot}
\efuncbasisldaval = \int \, \text{d}{\bf r} \,\, \denval({\bf r}) \,\, \emuldaval\,,
\ecompmodelldaval = \int \, \text{d}{\bf r} \,\, \denval({\bf r}) \,\, \emuldaval\,,
\end{equation}
\begin{equation}
\label{eq:def_lda_tot}
\efuncbasispbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval)
\ecompmodelpbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval)
\end{equation}
\subsection{Generalization for the correction of any model of WFT}
The theory proposed in \cite{GinPraFerAssSavTou-JCP-18} was derived for a FCI or CIPSI wave function model, and we propose here to generalize the basis-set correction to any wave function model.
\subsection{Correcting a general wave function molde $\mathcal{Y}$}
Consider a general wave function model $\mathcal{Y}$ which provides
\begin{itemize}
\item a variational energy $E_v$
\item a non-variational energy $E_p$
\item a variational wave function $\Psi_v$
\item a non-variational wave function $\Psi_v$
\end{itemize}
\subsection{Generalization of the basis-set correction to any model of WFT}
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The case of C$_2$, N$_2$, O$_2$, F$_2$ and the impact of the lack of basis functions adapted to core correlation }
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 the CIPSI algorithm to obtain reliable estimate of $\efci$ and $\denfci$.
\subsubsection{CIPSI calculations }
\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 $\basis$, the set of valence orbitals $\basisval$ 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 $\dencipsi$.
Therefore, from now on, we assume that
\begin{equation}
@ -477,7 +601,7 @@ and that
\denrfci \approx \dencipsi.
\end{equation}
Regarding the wave function 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{Treating the valence electrons}
\subsubsection{CCSD(T) calculations and the basis-set correction}
\begin{table*}
\caption{
@ -544,12 +668,12 @@ Regarding the wave function chosen to define the local range-separation paramete
& exFCI+PBE-on-top-val & 224.8 & 226.7 & 228.3 & 228.3 & \\
\hline
\hline
& (FC)CCSD(T) & 199.9 & 216.3 & ----- & ----- & 227.4 \\
& (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 & ----- & ----- & \\
& ex (FC)CCSD(T)+PBE-val & 225.9 & 226.7 & ----- & ----- & \\
& 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 & \\
@ -602,27 +726,29 @@ Regarding the wave function chosen to define the local range-separation paramete
%%%%%%%% & 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 & ----- & ----- & 120.5 \\
& ex (FC)CCSD(T)+LDA-val & 110.6 & 117.2 & ----- & ----- & \\
& ex (FC)CCSD(T)+PBE-val & 115.1 & 118.0 & ----- & ----- & \\
& (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 & & \\
\\
\ce{F2} & exFCI & 26.7 & 35.1 & 37.1 & ---- & ---- \\
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 & ---- & \\
& 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 & ---- & \\
& 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 & ----- & ----- & 38.8 \\
& ex (FC)CCSD(T)+LDA-val & 29.2 & 36.5 & ----- & ----- & \\
& ex (FC)CCSD(T)+PBE-val & 31.5 & 37.1 & ----- & ----- & \\
& (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}
@ -634,6 +760,6 @@ Regarding the wave function chosen to define the local range-separation paramete
%
%\bibliography{G2-srDFT}
\bibliography{G2-srDFT}
\end{document}