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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.
\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
\begin{equation}
\label{eq:def_mur_val}
@ -637,16 +638,16 @@ The performance of each of these four functionals is tested below.
\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).
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*}
@ -708,7 +709,7 @@ Regarding the computational cost of the present approach, it should be stressed
\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.
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_zoom.eps}
\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}. \\
$^1$: CEEIS calculations obtained from non-relativistic calculations of 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}.
The non-relativistic CEEIS energies have been extracted from Ref.~\onlinecite{BytNagGorRue-JCP-07}.
\label{fig:F2}}
\end{figure*}
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