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Pierre-Francois Loos 2020-01-06 15:33:44 +01:00
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Manuscript/srDFT_SC.tex
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@ -434,6 +434,7 @@ Because of the very definition of $\wbasis$, one has the following property in t
which is again fundamental to guarantee the correct behavior of the theory in the CBS limit. which is again fundamental to guarantee the correct behavior of the theory in the CBS limit.
\subsubsection{Frozen-core approximation} \subsubsection{Frozen-core approximation}
\label{sec:FC}
As all WFT calculations in this work are performed within the frozen-core approximation, we use a valence-only version of the various quantities needed for the complementary density functional introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ and $\BasFC$ are the sets of core and active orbitals, respectively, and define the valence-only local range-separation parameter as As all WFT calculations in this work are performed within the frozen-core approximation, we use a valence-only version of the various quantities needed for the complementary density functional introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ and $\BasFC$ are the sets of core and active orbitals, respectively, and define the valence-only local range-separation parameter as
\begin{equation} \begin{equation}
\label{eq:def_mur_val} \label{eq:def_mur_val}
@ -637,16 +638,16 @@ The performance of each of these four functionals is tested below.
\subsection{Computational details} \subsection{Computational details}
The purpose of the present paper being the study of the basis-set correction in regimes of both weak and/or strong correlation, we study the potential energy curves up to the dissociation limit of a \ce{H10} chain with equally-spaced atoms and the \ce{N2}, \ce{O2}, and \ce{F2} diatomics. In a given basis set, in order to compute the approximation of the exact ground-state energy using Eq.~\eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the basis-set correction $\efuncbasisFCI$. The purpose of the present paper being to investigate the performance of the density-based basis-set correction in regimes of both weak and strong correlation, we study the potential energy curves up to the dissociation limit of a \ce{H10} chain with equally-spaced atoms and the \ce{N2}, \ce{O2}, and \ce{F2} diatomics. For a given basis set, in order to compute the approximate exact ground-state energy in Eq.~\eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the basis-set correction $\efuncbasisFCI$.
In the case of the \ce{N2}, \ce{O2}, and \ce{F2} molecules for the aug-cc-pVXZ (X=D,T), approximations to the FCI energies are obtained using converged frozen-core ($1s$ orbitals are kept frozen) selected CI calculations and the extrapolation scheme for the perturbative correction of Umrigar \textit{et. al.} (see Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more details) using the latest version of \textsc{QUANTUM PACKAGE} \cite{QP2} (exFCI), and the correlation energy extrapolation by intrinsic scaling\cite{BytNagGorRue-JCP-07} (CEEIS) in the case of \ce{F2} for the cc-pVXZ (X=D,T,Q) basis set. The estimated exact potential energy curves are obtained from experimental data in Ref.~\onlinecite{LieCle-JCP-74a} for the \ce{N2} and \ce{O2} molecules, and from extrapolated CEEIS calculations in the case of \ce{F2}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be on the order of $0.5$ mHa. For the \ce{N2}, \ce{O2}, and \ce{F2} molecules, we also performed single-point exFCI calculations in the aug-cc-pVQZ basis set at the equilibrium geometry to obtain estimations of the FCI dissociation energies in these basis sets. In the case of the \ce{N2}, \ce{O2}, and \ce{F2} molecules for the aug-cc-pVDZ and aug-cc-pVTZ basis sets, approximations to the FCI energies are obtained using frozen-core selected CI calculations followed by the extrapolation scheme proposed by Umrigar and coworkers (see Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more details). All these calculations are performed with the latest version of \textsc{QUANTUM PACKAGE}, \cite{QP2} and will be labeled as exFCI in the following. In the case of \ce{F2}, the correlation energy extrapolation by intrinsic scaling \cite{BytNagGorRue-JCP-07} (CEEIS) are used as reference for the cc-pVXZ (X $=$ D, T, and Q) basis set. The estimated exact potential energy curves are obtained from experimental data \cite{LieCle-JCP-74a} for the \ce{N2} and \ce{O2} molecules, and from extrapolated CEEIS calculations in the case of \ce{F2}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be of the order of $0.5$ mHa. For the \ce{N2}, \ce{O2}, and \ce{F2} molecules, we also performed single-point exFCI calculations in the aug-cc-pVQZ basis set at the equilibrium geometry to obtain reliable estimates of the FCI/CBS dissociation energy.
In the case of the \ce{H10} chain, the approximation to the FCI energies together with the estimated exact potential energy curves are obtained from the data of Ref.~\onlinecite{h10_prx} where the authors performed MRCI+Q calculations with a minimal valence active space as reference (see below for the description of the active space). In the case of the \ce{H10} chain, the approximation to the FCI energies together with the estimated exact potential energy curves are obtained from the data of Ref.~\onlinecite{h10_prx} where the authors performed MRCI+Q calculations with a minimal valence active space as reference (see below for the description of the active space).
Regarding the complementary density functional, we first perform full-valence complete-active-space self-consistent-field (CASSCF) calculations with the GAMESS-US software\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-like quantities involved in the functional [density $n(\br{})$, spin polarization $\zeta(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$] together with the local range-separation function $\mu(\br{})$ of Eq.~\eqref{eq:def_mur} are calculated with this full-valence CASSCF wave function. The CASSCF calculations have been performed with the following active spaces: (10e,10o) for \ce{H10}, (10e,8o) for \ce{N2}, (12e,8o) for \ce{O2}, and (14e,8o) for \ce{F2}. Regarding the complementary density functional, we first perform full-valence CASSCF calculations with the GAMESS-US software\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-related quantities involved in the functional [density $n(\br{})$, spin polarization $\zeta(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$] together with the local range-separation function $\mu(\br{})$ of Eq.~\eqref{eq:def_mur} are calculated with this full-valence CASSCF wave function. The CASSCF calculations have been performed with the following active spaces: (10e,10o) for \ce{H10}, (10e,8o) for \ce{N2}, (12e,8o) for \ce{O2}, and (14e,8o) for \ce{F2}.
Also, as the frozen-core approximation is used in all our selected CI calculations, we use the corresponding valence-only complementary density functionals. Therefore, all density-like quantities exclude any contribution from the $1s$ core orbitals, and the range-separation function is taken as the one defined in Eq.~\eqref{eq:def_mur_val}. Also, as the frozen-core approximation is used in all our selected CI calculations, we use the corresponding valence-only complementary density functionals (see Subsec.~\ref{sec:FC}). Therefore, all density-related quantities exclude any contribution from the $1s$ core orbitals, and the range-separation function follows the definition given in Eq.~\eqref{eq:def_mur_val}.
Regarding the computational cost of the present approach, it should be stressed (see supplementary information) that the basis set correction represents, for all systems and basis sets studied here, a much smaller computational cost than any of the selected CI calculations. We thus believe that this approach is a significant step towards the routine calculation of near-CBS energetic quantities in strongly correlated systems. Regarding the computational cost of the present approach, it should be stressed (see {\SI} for additional details) that the basis set correction represents, for all systems and basis sets studied here, a much smaller computational cost than any of the selected CI calculations. We thus believe that this approach is a significant step towards the routine calculation of near-CBS energetic quantities in strongly correlated systems.
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\begin{table*} \begin{table*}
@ -708,7 +709,7 @@ Regarding the computational cost of the present approach, it should be stressed
\subsection{H$_{10}$ chain} \subsection{H$_{10}$ chain}
The study of the \ce{H10} chain with equally distant atoms is a good prototype of strongly-correlated systems as it consists in the simultaneous breaking of 10 covalent $\sigma$ bonds which all interact with each other. Also, being a relatively small system, benchmark calculations at near-CBS values can be obtained (see Ref.~\onlinecite{h10_prx} for a detailed study of this problem). The study of the \ce{H10} chain with equally distant atoms is a good prototype of strongly-correlated systems as it consists in the simultaneous breaking of 10 covalent $\sigma$ bonds which all interact with each other. Also, being a relatively small system, benchmark calculations at near-CBS values can be obtained (see Ref.~\onlinecite{h10_prx} for a detailed study of this system).
We report in Fig.~\ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X $=$ D, T, and Q) basis sets for different levels of approximation. The computation of the atomization energies $D_0$ for each level of theory is reported in Table \ref{tab:d0}. A general trend that can be observed from these data is that, in a given basis set, the quality of the potential energy curves are globally improved by adding the basis-set correction, independently of the approximation level of $\efuncbasis$. Also, no erratic behavior is found when stretching the bonds, which shows that the present procedure (\textit{i.e.} the determination of the range-separation parameter and the definition of the functionals) is robust when reaching the strong-correlation regime. We report in Fig.~\ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X $=$ D, T, and Q) basis sets for different levels of approximation. The computation of the atomization energies $D_0$ for each level of theory is reported in Table \ref{tab:d0}. A general trend that can be observed from these data is that, in a given basis set, the quality of the potential energy curves are globally improved by adding the basis-set correction, independently of the approximation level of $\efuncbasis$. Also, no erratic behavior is found when stretching the bonds, which shows that the present procedure (\textit{i.e.} the determination of the range-separation parameter and the definition of the functionals) is robust when reaching the strong-correlation regime.
In other words, smooth potential energy surfaces are obtained with the present basis-set correction. In other words, smooth potential energy surfaces are obtained with the present basis-set correction.
@ -765,8 +766,8 @@ This could have significant implications for the construction of more robust fam
\includegraphics[width=0.45\linewidth]{data/F2/DFT_vqzE_relat.eps} \includegraphics[width=0.45\linewidth]{data/F2/DFT_vqzE_relat.eps}
\includegraphics[width=0.45\linewidth]{data/F2/DFT_vqzE_relat_zoom.eps} \includegraphics[width=0.45\linewidth]{data/F2/DFT_vqzE_relat_zoom.eps}
\caption{ \caption{
Potential energy curves of the \ce{F2} molecule calculated with CEEIS$^1$ and basis-set corrected CEEIS$^1$ using the cc-pVDZ (top), cc-pVTZ (middle) and cc-pVQZ (bottom) basis sets. The estimated exact energies are based on fit on valence-only extrapolated CEEIS data obtained from Ref.~\onlinecite{BytNagGorRue-JCP-07}. \\ Potential energy curves of the \ce{F2} molecule calculated with CEEIS$^1$ and basis-set corrected CEEIS$^1$ using the cc-pVDZ (top), cc-pVTZ (middle) and cc-pVQZ (bottom) basis sets. The estimated exact energies are based on fit on valence-only extrapolated CEEIS data obtained from Ref.~\onlinecite{BytNagGorRue-JCP-07}.
$^1$: CEEIS calculations obtained from non-relativistic calculations of Ref.~\onlinecite{BytNagGorRue-JCP-07}. The non-relativistic CEEIS energies have been extracted from Ref.~\onlinecite{BytNagGorRue-JCP-07}.
\label{fig:F2}} \label{fig:F2}}
\end{figure*} \end{figure*}
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