RSDFT
This commit is contained in:
parent
b3406fe888
commit
ee6a62a5d4
@ -136,11 +136,10 @@
|
||||
|
||||
\begin{abstract}
|
||||
We report a universal density-based basis set incompleteness correction that can be applied to any wave function method.
|
||||
The present correction relies on short-range correlation density functionals (with multi-determinant reference) from range-separated density-functional theory (RS-DFT) to estimate the basis set incompleteness error and appropriately vanishes in the complete basis set (CBS) limit.
|
||||
Contrary to conventional RS-DFT schemes which require an \textit{ad hoc} range separation \textit{parameter} $\mu$, the key ingredient here is a range separation \textit{function} $\mu(\bf{r})$ which automatically adapts to the spatial non-homogeneity of the basis set incompleteness error.
|
||||
As illustrative examples, we show how this density-based correction allows us to obtain CCSD(T) atomization energies near the CBS limit for the G2-1 set of molecules with compact Gaussian basis sets.
|
||||
For example, while CCSD(T)/cc-pVTZ yields a mean absolute deviation (MAD) of 7.79 kcal/mol compared to CCSD(T)/CBS atomization energies, the CCSD(T)+LDA and CCSD(T)+PBE corrected methods return MAD of 2.89 and 2.46 kcal/mol (respectively) with the same basis.
|
||||
These values drop below 1 {\kcal} with the cc-pVQZ basis set.
|
||||
The present correction, which appropriately vanishes in the complete basis set (CBS) limit, relies on short-range correlation density functionals (with multi-determinant reference) from range-separated density-functional theory (RS-DFT) to estimate the basis set incompleteness error.
|
||||
Contrary to conventional RS-DFT schemes which require an \textit{ad hoc} range-separation \textit{parameter} $\mu$, the key ingredient here is a range-separation \textit{function} $\mu(\bf{r})$ which automatically adapts to the spatial non-homogeneity of the basis set incompleteness error.
|
||||
As illustrative examples, we show how this density-based correction allows us to obtain CCSD(T) correlation energies near the CBS limit for the G2-1 set of molecules with compact Gaussian basis sets.
|
||||
For example, while CCSD(T)/cc-pVTZ yields a mean absolute deviation (MAD) of 6.06 kcal/mol compared to CCSD(T)/CBS atomization energies, the CCSD(T)+LDA and CCSD(T)+PBE corrected methods return MAD of 1.19 and 0.85 kcal/mol (respectively) with the same basis.
|
||||
\end{abstract}
|
||||
|
||||
\maketitle
|
||||
@ -256,8 +255,8 @@ and $\Gam{pq}{rs} = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}
|
||||
\end{equation}
|
||||
and $\V{pq}{rs}$ are the usual two-electron Coulomb integrals.
|
||||
Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems do not have any basis set correction.
|
||||
\PFL{I don't agree with this. There must be a correction for one-electron system.
|
||||
However, it does not come from the e-e cusp but from the e-n cusp.}
|
||||
%\PFL{I don't agree with this. There must be a correction for one-electron system.
|
||||
%However, it does not come from the e-e cusp but from the e-n cusp.}
|
||||
With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
|
||||
\begin{equation}
|
||||
\label{eq:int_eq_wee}
|
||||
@ -275,9 +274,9 @@ An important quantity to define in the present context is the value of the effec
|
||||
\label{eq:wcoal}
|
||||
\W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}),
|
||||
\end{equation}
|
||||
which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{}(\br{},\br{})$ is non vanishing.
|
||||
|
||||
Of course, there exists \textit{a priori} an infinite set of functions in $\mathbb{R}^6$ satisfying \eqref{eq:int_eq_wee}, but thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
|
||||
which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{}(\br{})$ is non vanishing.
|
||||
%Of course, there exists \textit{a priori} an infinite set of functions in $\mathbb{R}^6$ satisfying \eqref{eq:int_eq_wee}, but
|
||||
Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
|
||||
\begin{equation}
|
||||
\label{eq:lim_W}
|
||||
\lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = r_{12}^{-1}\
|
||||
@ -290,7 +289,7 @@ for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any $
|
||||
%\subsection{Range-separation function}
|
||||
%=================================================================
|
||||
|
||||
\alert{Because the Coulomb operator within a basis set $\Bas$ is a non divergent two-electron interaction, we can straightforwardly link the present theory with the RS-DFT which uses the so-called long-range interaction which are smooth bounded two-electron operators.}
|
||||
Because $\W{\Bas}{}(\br{1},\br{2})$ is a non-divergent two-electron interaction, it can be straightforwardly linked to RS-DFT which employs smooth long-range operators.
|
||||
Although this choice is not unique, we choose here the range-separation function
|
||||
\begin{equation}
|
||||
\label{eq:mu_of_r}
|
||||
@ -300,9 +299,9 @@ such that the long-range interaction
|
||||
\begin{equation}
|
||||
\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{\Bas}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{\Bas}{}(\br{2}) r_{12}]}{ r_{12}} }
|
||||
\end{equation}
|
||||
\PFL{This expression looks like a cheap spherical average.
|
||||
What about $\rsmu{\Bas}{}(\br{1},\br{2}) = \sqrt{\rsmu{\Bas}{}(\br{1}) \rsmu{\Bas}{}(\br{2})}$ and a proper spherical average to get $\rsmu{\Bas}{}(r_{12})$?}
|
||||
coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{})$.
|
||||
%\PFL{This expression looks like a cheap spherical average.
|
||||
%What about $\rsmu{\Bas}{}(\br{1},\br{2}) = \sqrt{\rsmu{\Bas}{}(\br{1}) \rsmu{\Bas}{}(\br{2})}$ and a proper spherical average to get $\rsmu{\Bas}{}(r_{12})$?}
|
||||
|
||||
%=================================================================
|
||||
%\subsection{Short-range correlation functionals}
|
||||
@ -327,7 +326,7 @@ with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
|
||||
% \w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}.
|
||||
%\end{equation}
|
||||
%is the long-range part of the Coulomb operator.
|
||||
The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following limiting forms:
|
||||
The ECMD functionals admit, for any density $\n{}{}(\br{})$, the following two limiting forms
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
\label{eq:large_mu_ecmd}
|
||||
@ -338,21 +337,20 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following l
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
|
||||
The choice of ECMD in the present scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [Eq.~\eqref{eq:E_funcbasis}] and that of the ECMD functionals [Eq.~\eqref{eq:ec_md_mu}].
|
||||
The choice of ECMD in the present scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [Eq.~\eqref{eq:E_funcbasis}] and the ECMD functionals [Eq.~\eqref{eq:ec_md_mu}].
|
||||
Indeed, provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) \approx \W{\Bas}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\Bas}{}}$ coincides with $\wf{}{\Bas}$.
|
||||
%The ECMD functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function.
|
||||
%This makes them particularly well adapted to the present context where one aims at correcting a general WFT method.
|
||||
|
||||
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{\Bas}{}(\br{})$.
|
||||
Therefore, we define the LDA version of $\bE{}{\Bas}[\n{}{}]$ as
|
||||
Therefore, following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{\Bas}{}(\br{})$.
|
||||
The LDA version of $\bE{}{\Bas}[\n{}{}]$ is defined as
|
||||
\begin{equation}
|
||||
\label{eq:def_lda_tot}
|
||||
\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{},
|
||||
\end{equation}
|
||||
where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the short-range ECMD per particle of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
|
||||
where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the short-range reduced (i.e.~per electron) correlation energy of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
|
||||
|
||||
The short-range LDA correlation functional defined in Eq.~\eqref{eq:def_lda_tot} relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$.
|
||||
In order to correct such a defect, we propose here a new ECMD functional inspired by the recent functional proposed by some of the present authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{\PBE}{}(\n{}{},\nabla \n{}{})$ for $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
|
||||
In order to correct such a defect, we propose here a new ECMD functional inspired by the recent functional proposed by some of the authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{\PBE}{}(\n{}{},\nabla \n{}{})$ for $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
|
||||
\begin{subequations}
|
||||
\begin{gather}
|
||||
\label{eq:epsilon_cmdpbe}
|
||||
@ -362,7 +360,7 @@ In order to correct such a defect, we propose here a new ECMD functional inspire
|
||||
\beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{2}{\UEG}(\n{}{})}.
|
||||
\end{gather}
|
||||
\end{subequations}
|
||||
The difference between the ECMD PBE functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe} is that we approximate here the \textit{exact} ground-state on-top pair density of the system $\n{2}{}(\br{})$ by its UEG version, i.e.~$\n{2}{}(\br{}) \approx \n{2}{\UEG}(\br{}) = \n{}{}(\br{})^2 g_0(\n{}{}(\br{}) )$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
|
||||
The difference between the ECMD PBE functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe} is that we approximate here the \textit{exact} ground-state on-top pair density by its UEG version, i.e.~$\n{2}{}(\br{}) \approx \n{2}{\UEG}(\br{}) = \n{}{}(\br{})^2 g_0(\n{}{}(\br{}) )$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
|
||||
This represents a major computational saving without loss of performance as we eschew the computation of $\n{2}{}(\br{})$.
|
||||
Therefore, the PBE complementary functional reads
|
||||
\begin{equation}
|
||||
@ -523,25 +521,24 @@ iii) it vanishes in the limit of a complete basis set, hence guaranteeing an una
|
||||
%\subsection{Comparison between the CIPSI and CCSD(T) models in the case of C$_2$, N$_2$, O$_2$, F$_2$}
|
||||
We begin our investigation of the performance of the basis set correction by computing the atomization energies of \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} obtained with Dunning's cc-pVXZ basis sets (X $=$ D, T, Q and 5).
|
||||
In the case of \ce{C2} and \ce{N2}, we also perform calculations with the cc-pCVXZ family.
|
||||
\ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2 set, whereas \ce{C2} already contains a non-negligible amount of strong correlation. \cite{BooCleThoAla-JCP-11}
|
||||
In a second time, we compute the entire atomization energies of the G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family of basis sets.
|
||||
\ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2-1 set, whereas \ce{C2} already contains a non-negligible amount of strong correlation. \cite{BooCleThoAla-JCP-11}
|
||||
In a second time, we compute the entire correlation energies of the G2-1 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family of basis sets.
|
||||
This molecular set has been exhausively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08,Gro-JCP-09,FelPet-JCP-09,NemTowNee-JCP-10,FelPetHil-JCP-11,PetTouUmr-JCP-12,FelPet-JCP-13,KesSylKohTewMar-JCP-18}).
|
||||
%The reference values for the atomization energies are extracted from Ref.~\onlinecite{HauKlo-JCP-12} and corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
|
||||
As a method $\modY$ we employ either CCSD(T) or exFCI.
|
||||
Here, exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
|
||||
We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
|
||||
In the case of the CCSD(T) calculations, we have $\modZ = \HF$ as we use the restricted open-shell Hartree-Fock (ROHF) one-electron density to compute the complementary energy.
|
||||
\titou{For exFCI, we use the density of a converged variational wave function.}
|
||||
For the definition of the interaction, we use a single Slater determinant built in the ROHF basis for the CCSD(T) calculations, and built with the natural orbitals of the converged variational wave function for the exFCI calculations.
|
||||
In the case of the CCSD(T) calculations, we have $\modZ = \HF$ as we use the restricted open-shell HF (ROHF) one-electron density to compute the complementary energy.
|
||||
\titou{For exFCI, we use the density of a converged variational wave function.
|
||||
For the definition of the interaction, we use a single Slater determinant built in the ROHF basis for the CCSD(T) calculation, and built with the natural orbitals of the converged variational wave function for the exFCI calculations.}
|
||||
The CCSD(T) calculations have been performed with Gaussian09 with standard threshold values. \cite{g09}
|
||||
RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2}
|
||||
For the numerical quadrature, we employ the SG-2 grid. \cite{DasHer-JCC-17}
|
||||
Except for the carbon dimer where we have taken the experimental equilibrium bond length (\InAA{1.2425}), all geometries have been extracted from Ref.~\onlinecite{HauJanScu-JCP-09} and have been obtained at the B3LYP/6-31G(2df,p) level of theory.
|
||||
Frozen core calculations are defined as such: an \ce{He} core is frozen from \ce{Li} to \ce{Ne}, while a \ce{Ne} core is frozen from \ce{Na} to \ce{Ar}.
|
||||
In the context of the basis set correction, the set of valence spinorbitals $\val$ involved in the definition of the effective interaction refers to the non-frozen spinorbitals.
|
||||
The ``valence'' correction was used consistently when the FC approximation was applied.
|
||||
In order to estimate the complete basis set (CBS) limit for each model, we employed the two-point extrapolation proposed in Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98} for the correlation energies.
|
||||
We refer to these atomization energies as $\CBS$.
|
||||
In the context of the basis set correction, the set of spinorbitals $\BasFC$ involved in the definition of the effective interaction refers to the non-frozen spinorbitals.
|
||||
The FC density-based correction was used consistently when the FC approximation was applied in WFT methods.
|
||||
In order to estimate the complete basis set (CBS) limit for each model, we employed the two-point extrapolation proposed in Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98} for the correlation energies, and we refer to these as $\CBS$.
|
||||
|
||||
%\subsection{Convergence of the atomization energies with the WFT models }
|
||||
As the exFCI calculations were converged with a precision of about 0.1 {\kcal}, we can consider these atomization energies as near-FCI values.
|
||||
@ -564,12 +561,12 @@ Such weak sensitivity to the approximated functionals in the DFT part when reach
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section*{Supporting information}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
See {\SI} for raw data associated with the G2 atomization energies.
|
||||
See {\SI} for raw data associated with the G2-1 correlation energies.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{acknowledgements}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
The authors would like to thank... nobody.
|
||||
The authors would like to thank the \textit{Centre National de la Recherche Scientifique} (CNRS) for funding.
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\end{acknowledgements}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
Loading…
Reference in New Issue
Block a user