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Manuscript/srDFT_SC.tex
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@ -88,7 +88,7 @@
\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)}
\newcommand{\ecmd}[0]{\bar{\varepsilon}_{\text{c,md}}^{\text{sr},\text{PBE}}}
\newcommand{\psibasis}[0]{\Psi^{\basis}}
\newcommand{\BasFC}{\mathcal{A}}
\newcommand{\BasFC}{\mathcal{V}}
%pbeuegxiHF
%\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas}
@ -437,7 +437,7 @@ which is again fundamental to guarantee the correct behavior of the theory in th
\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 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'' (or no-core) version of the various quantities needed for the complementary 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 valence (\ie, no-core) orbitals, respectively, and define the valence-only local range-separation parameter as
\begin{equation}
\label{eq:def_mur_val}
\murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval{},
@ -462,7 +462,7 @@ is the valence-only effective interaction and
\twodmrdiagpsival
= \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}.
\end{gather}
One would note the restrictions of the sums to the set of active orbitals in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}.
One would note the restrictions of the sums to the set of valence orbitals in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}.
It is also noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$.
\subsection{General form of the complementary functional}
@ -570,10 +570,7 @@ An alternative way to eliminate the $S_z$ dependency is to simply set $\zeta=0$,
\subsubsection{Size consistency}
cite~\cite{Sav-CP-09} here
Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb{R}^3$ [see Eq.~\eqref{eq:def_ecmdpbebasis}] which involves only local quantities [$n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$, and $\mu(\br{})$], in the case of non-overlapping fragments \ce{A\bond{...}B}, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem \ce{A} and the other one from the region of subsystem \ce{B}. Therefore, a sufficient condition for size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem \ce{A\bond{...}B}. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\Psi_{\ce{A\bond{...}B}}^{\basis} = \Psi_{\ce{A}}^{\basis} \Psi_{\ce{B}}^{\basis}$.
We refer the interested reader to the {\SI} for a detailed proof of the latter statement.
Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb{R}^3$ [see Eq.~\eqref{eq:def_ecmdpbebasis}] which involves only local quantities [$n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$, and $\mu(\br{})$], in the case of non-overlapping fragments $\text{A}+\text{B}$, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem \ce{A} and the other one from the region of subsystem \ce{B}. Therefore, a sufficient condition for size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem $\text{A}+\text{B}$. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\ket{\Psi_{\text{A}+\text{B}}^{\basis}} = \ket{\Psi_{\ce{A}}^{\basis}} \otimes \ket{\Psi_{\ce{B}}^{\basis}}$. We refer the interested reader to Appendix~\ref{sizeconsistency} for a detailed proof and discussion of the latter statement.
In the case where the two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, a properly chosen CASSCF wave function is sufficient to recover this property. The underlying active space must however be chosen in such a way that it leads to size-consistent energies in the limit of dissociated fragments.
@ -716,7 +713,7 @@ Regarding the complementary functional, we first perform full-valence CASSCF cal
Also, as the frozen-core approximation is used in all our selected CI calculations, we use the corresponding valence-only complementary functionals (see Subsec.~\ref{sec:FC}). Therefore, all density-related quantities exclude any contribution from the $1s$ core orbitals, and the range-separation parameter follows the definition given in Eq.~\eqref{eq:def_mur_val}.
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.
Regarding the computational cost of the present approach, it should be stressed (see Appendix~\ref{computational} 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.
@ -825,22 +822,21 @@ Also, it is shown that the basis-set correction gives substantial differential c
\appendix
\section{Size consistency of the basis-set correction}
\label{sizeconsistency}
\subsection{Sufficient condition for size consistency}
The basis-set correction is expressed as an integral in real space
\begin{equation}
\label{eq:def_ecmdpbebasis}
\label{eq:def_ecmdpbebasisAnnex}
\begin{aligned}
& \efuncdenpbe{\argebasis} = \\ & \int \text{d}\br{} \,\denr \ecmd(\argrebasis),
\end{aligned}
\end{equation}
where all the local quantities $\argrebasis$ are obtained from the same wave function $\Psi$. In the limit of two non-overlapping and non-interacting dissociated fragments $\text{A}+\text{B}$, this integral can be rewritten as the sum of the integral over the region $\Omega_\text{A}$ and the integral over the region $\Omega_\text{B}$
\begin{equation}
\label{eq:def_ecmdpbebasis}
\label{eq:def_ecmdpbebasisAB}
\begin{aligned}
& \efuncdenpbeAB{\argebasis} = \\ & \int_{\Omega_\text{A}} \text{d}\br{} \,\denr \ecmd(\argrebasis) \\ & + \int_{\Omega_\text{B}} \text{d}\br{} \,\denr \ecmd(\argrebasis).
\end{aligned}
@ -928,38 +924,36 @@ with $\mu_{\text{X}}(\bfr{}) = (\sqrt{\pi}/2) f_{\text{X}}(\bfr{},\bfr{})/n_{2,\
In conclusion, if the wave function of the supersystem $\text{A}+\text{B}$ is multiplicative separable, all local quantities used in the basis-set correction functional are intensive and therefore the basis-set correction is size consistent.
\section{Computational considerations}
\section{Efficient computation of the basis-set correction for a CASSCF wave function}
\label{computational}
The computational cost of the present approach is driven by two quantities: the computation of the on-top pair density and the $\murpsibas$ on the real-space grid. Within a blind approach, for each grid point the computational cost is of order $n_{\Bas}^4$ and $n_{\Bas}^6$ for the on-top pair density $n_{2,\wf{\Bas}{}}(\br{})$ and the local range separation parameter $\murpsibas$, respectively.
Nevertheless, using CASSCF wave functions to compute these quantities leads to significant simplifications which can substantially reduce the CPU time.
The computational cost of the basis-set correction is determined by the calculation of the on-top pair density $n_{2}(\br{})$ and the local range-separation parameter $\mur$ on the real-space grid. For a general multideterminant wave function, the computational cost is of order $O(N_\text{grid}N_{\Bas}^4)$ where $N_\text{grid}$ is the number of grid points and $N_{\Bas}$ is the number of basis functions.\cite{LooPraSceTouGin-JCPL-19} For a CASSCF wave function, a significant reduction of the scaling of the computational cost can be achieved.
\subsection{Computation of the on-top pair density for a CASSCF wave function}
\subsection{Computation of the on-top pair density}
Given a generic wave function developed on a basis set with $n_{\Bas}$ basis functions, the evaluation of the on-top pair density is of order $\left(n_{\Bas}\right)^4$.
Nevertheless, assuming that the wave function $\Psi^{\Bas}$ is of CASSCF type, a lot of simplifications happen.
If the active space is referred as the set of spatial orbitals $\mathcal{A}$ which are labelled by the indices $t,u,v,w$, and the doubly occupied orbitals are the set of spatial orbitals $\mathcal{C}$ labeled by the indices $i,j$, one can write the on-top pair density of a CASSCF wave function as
For a CASSCF wave function $\Psi$, the occupied orbitals can be partitioned into a set of active orbitals $\mathcal{A}$ and a set of inactive (doubly occupied) orbitals $\mathcal{I}$. The CASSCF on-top pair density can then be written as
\begin{equation}
\label{def_n2_good}
n_{2,\wf{\Bas}{}}(\br{}) = n_{2,\mathcal{A}}(\br{}) + n_{\mathcal{C}}(\br{}) n_{\mathcal{A}}(\br{}) + \left( n_{\mathcal{C}}(\br{})\right)^2
n_{2}(\br{}) = n_{2,\mathcal{A}}(\br{}) + n_{\mathcal{A}}(\br{}) n_{\mathcal{I}}(\br{}) + \frac{n_{\mathcal{I}}(\br{})^2}{2},
\end{equation}
where
where $n_{2,\mathcal{A}}(\br{})$ is the purely active part of the on-top pair density
\begin{equation}
\label{def_n2_act}
n_{2,\mathcal{A}}(\br{}) = \sum_{t,u,v,w \, \in \mathcal{A}} 2 \mel*{\wf{}{\Bas}}{ \aic{t_\downarrow}\aic{u_\uparrow}\ai{v_\uparrow}\ai{w_\downarrow}}{\wf{}{\Bas}} \phi_t (\br{}) \phi_u (\br{}) \phi_v (\br{}) \phi_w (\br{})
n_{2,\mathcal{A}}(\br{}) = \sum_{pqrs \in \mathcal{A}} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
\end{equation}
is the purely active part of the on-top pair density,
$n_{\mathcal{A}}(\br{})$ is the active part of the density
\begin{equation}
n_{\mathcal{C}}(\br{}) = \sum_{i\, \in \mathcal{C}} \left(\phi_i (\br{}) \right)^2,
n_{\mathcal{A}}(\br{}) = \sum_{pq\, \in \mathcal{A}} \phi_p (\br{}) \phi_q (\br{})
\mel*{\wf{}{}}{ \aic{p_\uparrow}\ai{q_\uparrow} + \aic{p_\downarrow}\ai{q_\downarrow} }{\wf{}{}},
\end{equation}
and
and $n_{\mathcal{I}}(\br{})$ is the inactive part of the density
\begin{equation}
n_{\mathcal{A}}(\br{}) = \sum_{t,u\, \in \mathcal{A}} \phi_t (\br{}) \phi_u (\br{})
\mel*{\wf{}{\Bas}}{ \aic{t_\downarrow}\ai{u_\downarrow} + \aic{t_\uparrow}\ai{u_\uparrow}}{\wf{}{\Bas}}
n_{\mathcal{I}}(\br{}) = 2 \sum_{p\, \in \mathcal{I}} \phi_p (\br{})^2.
\end{equation}
is the purely active one-body density.
Written as in eq. \eqref{def_n2_good}, the leading computational cost is the evaluation of $n_{2,\mathcal{A}}(\br{})$ which, according to eq. \eqref{def_n2_act}, scales as $\left( n_{\mathcal{A}}\right) ^4$ where $n_{\mathcal{A}}$ is the number of active orbitals which is much smaller than the number of basis functions $n_{\Bas}$. Therefore, the final computational scaling of the on-top pair density for a CASSCF wave function over the whole real-space grid is of $\left( n_{\mathcal{A}}\right) ^4 n_G$, where $n_G$ is the number of grid points.
The leading computational cost is the evaluation of $n_{2,\mathcal{A}}(\br{})$ on the grid which, according to Eq.~\eqref{def_n2_act}, scales as $O(N_\text{grid} N_\mathcal{A}^4)$ where $N_{\mathcal{A}}$ is the number of active orbitals which is much smaller than the number of basis functions $N_{\Bas}$.
\subsection{Computation of the local range-separation parameter}
\subsection{Computation of $\murpsibas$}
At a given grid point, the computation of $\murpsibas$ needs the computation of $f_{\wf{}{}}(\bfr{},\bfr{}) $ defined in eq. \eqref{eq:def_f} and the on-top pair density defined in eq. \eqref{eq:def_n2}. In the previous paragraph we gave an explicit form of the on-top pair density in the case of a CASSCF wave function with a computational scaling of $\left( n_{\mathcal{A}}\right)^4$. In the present paragraph we focus on simplifications that one can obtain for the computation of $f_{\wf{}{}}(\bfr{},\bfr{}) $ in the case of a CASSCF wave function.
One can rewrite $f_{\wf{}{}}(\bfr{},\bfr{}) $ as