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srDFT_SC/Revised_Manuscript/SI/srDFT_SC-SI.tex

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start revised manuscript
2020-03-24 15:04:08 +01:00
\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
%\documentclass[aip,jcp,noshowkeys]{revtex4-1}
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\usepackage[normalem]{ulem}
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filecolor=blue,
urlcolor=blue,
citecolor=blue
}
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\mr}{\multirow}
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% second quantized operators
\newcommand{\psix}[1]{\hat{\Psi}\left({\bf X}_{#1}\right)}
\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)}
\newcommand{\ai}[1]{\hat{a}_{#1}}
\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}}
\newcommand{\vijkl}[0]{V_{ij}^{kl}}
\newcommand{\phix}[2]{\phi_{#1}(\bfr{#2})}
\newcommand{\phixprim}[2]{\phi_{#1}(\bfr{#2}')}
\newcommand{\CBS}{\text{CBS}}
%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
\newcommand{\Ec}{E_\text{c}}
\newcommand{\EPT}{E_\text{PT2}}
\newcommand{\EsCI}{E_\text{sCI}}
\newcommand{\EDMC}{E_\text{DMC}}
\newcommand{\EexFCI}{E_\text{exFCI}}
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\newcommand{\EexDMC}{E_\text{exDMC}}
\newcommand{\Ead}{\Delta E_\text{ad}}
\newcommand{\efci}[0]{E_{\text{FCI}}^{\Bas}}
\newcommand{\emodel}[0]{E_{\model}^{\Bas}}
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\newcommand{\ecccomplete}[0]{E_{\text{CCSD(T)}}^{\infty}}
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\newcommand{\psibasis}[0]{\Psi^{\basis}}
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%pbeuegxiHF
\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas}
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%pbeuegxiCAS
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\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),n_{2}^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
%pbeuegXiCAS
\newcommand{\pbeuegXi}{\text{PBE-UEG-}\tilde{\zeta}}
\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,n_{2}^{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
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%pbeontxiCAS
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%pbeontXiCAS
\newcommand{\pbeontXi}{\text{PBE-ot-}\tilde{\zeta}}
\newcommand{\argpbeontXi}[0]{\den,\tilde{\zeta},s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
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%pbeont0xiCAS
\newcommand{\pbeontns}{\text{PBE-ot-}0\zeta}
\newcommand{\argpbeontns}[0]{\den,0,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
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%%%%%% arguments
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\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argepbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}}
\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s(\br{}),\ntwo(\br{}),\mu(\br{})}
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% numbers
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\newcommand{\bfr}[1]{{\bf r}_{#1}}
\newcommand{\dr}[1]{\text{d}\bfr{#1}}
\newcommand{\rr}[2]{\bfr{#1}, \bfr{#2}}
\newcommand{\rrrr}[4]{\bfr{#1}, \bfr{#2},\bfr{#3},\bfr{#4} }
% effective interaction
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\newcommand{\murpsibas}[0]{\mu_{\wf{}{\Bas}}({\bf r})}
\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{A+B})}
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\newcommand{\ntwo}[0]{n_{2}}
\newcommand{\ntwohf}[0]{n^{(2),\text{HF}}}
\newcommand{\ntwophi}[0]{n^{(2)}_{\phi}}
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\newcommand{\ntwoextrapcas}[0]{\mathring{n}^{(2)\,\basis}_{\text{CAS}}}
\newcommand{\mur}[0]{\mu({\bf r})}
\newcommand{\murr}[1]{\mu({\bf r}_{#1})}
\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})}
\newcommand{\murpsival}[0]{\mu_{\text{val}}({\bf r};\wf{}{\Bas})}
\newcommand{\murrval}[1]{\mu_{\text{val}}({\bf r}_{#1})}
\newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}}
\newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
\newcommand{\wbasiscoal}[0]{W_{\wf{}{\Bas}}(\bfr{},\bfr{})}
\newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
\newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)}
\newcommand{\twodmrpsi}[0]{ n^{2,\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
\newcommand{\twodmrdiagpsi}[0]{ n_{2,{\wf{}{\Bas}}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsitot}[0]{ n_{2,\wf{}{A+B}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsiaa}[0]{ n_{2,\wf{}{AA}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsiaad}[0]{ n_{2,\wf{}{AA}}(\rr{}{})}
\newcommand{\twodmrdiagpsibb}[0]{ n_{2,\wf{}{BB}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsibbd}[0]{ n_{2,\wf{}{BB}}(\rr{}{})}
\newcommand{\twodmrdiagpsiab}[0]{ n_{2\wf{}{AB}}(\rr{1}{2})}
\newcommand{\twodmrdiagpsival}[0]{ n_{2\wf{}{\Bas},\,\text{val}}(\rr{1}{2})}
\newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]}
\newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}}
\newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]}
%\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})}
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\newcommand{\wbasiscoalval}[1]{W_{\wf{}{\Bas}}^{\text{val}}({\bf r}_{#1})}
\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$}
\newcommand{\ra}{\rightarrow}
\newcommand{\De}{D_\text{e}}
% MODEL
\newcommand{\model}[0]{\mathcal{Y}}
% densities
\newcommand{\denmodel}[0]{\den_{\model}^\Bas}
\newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})}
\newcommand{\denfci}[0]{\den_{\psifci}}
\newcommand{\denFCI}[0]{\den^{\Bas}_{\text{FCI}}}
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\newcommand{\denr}[0]{{n}({\bf r})}
\newcommand{\onedmval}[0]{\rho_{ij,\sigma}^{\text{val}}}
% wave functions
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\newcommand{\psimu}[0]{\Psi^{\mu}}
% operators
\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\Bas}
\newcommand{\kinop}[0]{\hat{T}}
\newcommand{\weeopbasisval}[0]{\hat{W}_{\text{ee}}^{\Basval}}
\newcommand{\weeop}[0]{\hat{W}_{\text{ee}}}
% units
\newcommand{\IneV}[1]{#1 eV}
\newcommand{\InAU}[1]{#1 a.u.}
\newcommand{\InAA}[1]{#1 \AA}
% methods
\newcommand{\UEG}{\text{UEG}}
\newcommand{\LDA}{\text{LDA}}
\newcommand{\PBE}{\text{PBE}}
\newcommand{\FCI}{\text{FCI}}
\newcommand{\CCSDT}{\text{CCSD(T)}}
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\newcommand{\V}[2]{V_{#1}^{#2}}
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\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}}
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\newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
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% basis sets
\newcommand{\setdenbasis}{\mathcal{N}_{\Bas}}
\newcommand{\Bas}{\mathcal{B}}
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% operators
\newcommand{\hT}{\Hat{T}}
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
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% coordinates
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\newcommand{\bx}[1]{\mathbf{x}_{#1}}
\newcommand{\dbr}[1]{d\br{#1}}
\newcommand{\PBEspin}{PBEspin}
\newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}
\begin{document}
\title{A density-based basis-set correction for weak and strong correlation}
\begin{abstract}
\end{abstract}
\maketitle
\section{Size consistency of the basis-set correction}
\subsection{Sufficient condition for size consistency}
The basis-set correction is expressed as an integral in real space
\begin{equation}
\label{eq:def_ecmdpbebasis}
\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}
\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}
\end{equation}
Therefore, a sufficient condition to obtain size consistency is that all the local quantities $\argrebasis$ are \textit{intensive}, i.e. that they \textit{locally} coincide in the supersystem $\text{A}+\text{B}$ and in each isolated fragment $\text{X}=\text{A}$ or $\text{B}$. Hence, for $\br{} \in \Omega_\text{X}$, we should have
\begin{subequations}
\begin{equation}
n_\text{A+B}(\br{}) = n_\text{X}(\br{}),
\label{nAB}
\end{equation}
\begin{equation}
\zeta_\text{A+B}(\br{}) = \zeta_\text{X}(\br{}),
\label{zAB}
\end{equation}
\begin{equation}
s_\text{A+B}(\br{}) = s_\text{X}(\br{}),
\label{sAB}
\end{equation}
\begin{equation}
n_{2,\text{A+B}}(\br{}) = n_{2,\text{X}}(\br{}),
\label{n2AB}
\end{equation}
\begin{equation}
\mu_{\text{A+B}}(\br{}) = \mu_{\text{X}}(\br{}),
\label{muAB}
\end{equation}
\end{subequations}
where the left-hand-side quantities are for the supersystem and the right-hand-side quantities for an isolated fragment. Such conditions can be difficult to fulfil in the presence of degeneracies since the system X can be in a different mixed state (i.e. ensemble) in the supersystem $\text{A}+\text{B}$ and in the isolated fragment~\cite{Sav-CP-09}. Here, we will consider the simple case where the wave function of the supersystem is multiplicatively separable, i.e.
\begin{equation}
\ket{\wf{\text{A}+\text{B}}{}} = \ket{\wf{\text{A}}{}} \otimes \ket{\wf{\text{B}}{}},
\end{equation}
where $\otimes$ is the antisymmetric tensor product. In this case, it is easy to shown that Eqs.~(\ref{nAB})-(\ref{sAB}) are valid, as well known, and it remains to show that Eqs.~(\ref{n2AB}) and~(\ref{muAB}) are also valid. Before showing this, we note that even though we do not explicity consider the case of degeneracies, the lack of size consistency which could arise from spin-multiplet degeneracies can be avoided by the same strategy used for imposing the energy independence on $S_z$, i.e. by using the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ or a zero spin polarization $\zeta(\br{}) = 0$. Moreover, the lack of size consistency which could arise from spatial degeneracies (e.g., coming from atomic p states) can also be avoided by selecting the same member of the ensemble in the supersystem and in the isolated fragement. This applies to the systems treated in this work.
\subsection{Intensivity of the on-top pair density and of the local range-separation parameter}
The on-top pair density can be written in an orthonormal spatial orbital basis $\{\SO{p}{}\}$ as
\begin{equation}
\label{eq:def_n2}
n_{2{}}(\br{}) = \sum_{pqrs \in \Bas} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
\end{equation}
with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$. As the summations run over all orbitals in the basis set $\Bas$, $n_{2{}}(\br{})$ is invariant to orbital rotations and can thus be expressed in terms of localized orbitals. For two non-overlapping fragments $\text{A}+\text{B}$, the basis set can then partitioned into orbitals localized on the fragment A and orbitals localized on B, i.e. $\Bas=\Bas_\text{A}\cup \Bas_\text{B}$. Therefore, we see that the on-top pair density of the supersystem $\text{A}+\text{B}$ is additively separable
\begin{equation}
\label{eq:def_n2}
n_{2,\text{A}+\text{B}}(\br{}) = n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{}),
\end{equation}
where $n_{2,\text{X}}(\br{})$ is the on-top pair density of the fragment X
\begin{equation}
\label{eq:def_n2}
n_{2,\text{X}}(\br{}) = \sum_{pqrs \in \Bas_\text{X}} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}.
\end{equation}
This shows that the on-top pair density is a local intensive quantity.
The local range-separation parameter is defined by
\begin{equation}
\label{eq:def_murAnnex}
\mur = \frac{\sqrt{\pi}}{2} \frac{f(\bfr{},\bfr{})}{n_{2}(\br{})},
\end{equation}
where
\begin{equation}
\label{eq:def_f}
f(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }.
\end{equation}
Again, $f(\bfr{},\bfr{})$ is invariant to orbital rotations and can be expressed in terms of orbitals localized on the fragments A and B. In the limit of infinitely separated fragments, the Coulomb interaction vanishes between A and B and therefore any two-electron integral $\V{pq}{rs}$ involving orbitals on both $A$ and $B$ vanishes. We thus see that the quantity $f(\bfr{},\bfr{})$ of the supersystem $\text{A}+\text{B}$ is additively separable
\begin{equation}
\label{eq:def_fa+b}
f_{\text{A}+\text{B}}(\bfr{},\bfr{}) = f_{\text{A}}(\bfr{},\bfr{}) + f_{\text{B}}(\bfr{},\bfr{}),
\end{equation}
with
\begin{equation}
\label{eq:def_fX}
f_\text{X}(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas_\text{X}} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }.
\end{equation}
So, $f(\bfr{},\bfr{})$ is a local intensive quantity.
As a consequence, the local range-separation parameter of the supersystem $\text{A}+\text{B}$ is
\begin{equation}
\label{eq:def_murAB}
\mu_{\text{A}+\text{B}}(\bfr{}) = \frac{\sqrt{\pi}}{2} \frac{f_{\text{A}}(\bfr{},\bfr{}) + f_{\text{B}}(\bfr{},\bfr{})}{n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{})},
\end{equation}
which gives
\begin{equation}
\label{eq:def_murABsum}
\mu_{\text{A}+\text{B}}(\bfr{}) = \mu_{\text{A}}(\bfr{}) + \mu_{\text{B}}(\bfr{}),
\end{equation}
with $\mu_{\text{X}}(\bfr{}) = (\sqrt{\pi}/2) f_{\text{X}}(\bfr{},\bfr{})/n_{2,\text{X}}(\br{})$. The local range-separation parameter is thus a local intensive quantity.
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}
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.
\subsection{Computation of the on-top pair density for a CASSCF wave function}
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
\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
\end{equation}
where
\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{})
\end{equation}
is the purely active part of the on-top pair density,
\begin{equation}
n_{\mathcal{C}}(\br{}) = \sum_{i\, \in \mathcal{C}} \left(\phi_i (\br{}) \right)^2,
\end{equation}
and
\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}}
\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.
\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
\begin{equation}
\label{eq:f_good}
f_{\wf{}{}}(\bfr{},\bfr{}) = \sum_{r,s \in \Bas} \mathcal{V}_r^s(\bfr{}) \, \mathcal{N}_{r}^s(\bfr{})
\end{equation}
where
\begin{equation}
\mathcal{V}_r^s(\bfr{}) = \sum_{p,q \in \Bas} V_{pq}^{rs} \phi_p(\br{}) \phi_q(\br{})
\end{equation}
and
\begin{equation}
\mathcal{N}_{r}^s(\bfr{}) = \sum_{p,q \in \Bas} \Gam{pq}{rs} \phi_p(\br{}) \phi_q(\br{}) .
\end{equation}
\textit{A priori}, for a given grid point, the computational scaling of eq. \eqref{eq:f_good} is of $\left(n_{\Bas}\right)^4$ and the total computational cost over the whole grid scales therefore as $\left(n_{\Bas}\right)^4 n_G$.
In the case of a CASSCF wave function, it is interesting to notice that $\Gam{pq}{rs}$ vanishes if one index $p,q,r,s$ does not belong
to the set of of doubly occupied or active orbitals $\mathcal{C}\cup \mathcal{A}$. Therefore, at a given grid point, the matrix $\mathcal{N}_{r}^s(\bfr{})$ has only at most $\left(n_{\mathcal{A}}+n_{\mathcal{C}}\right)^2$ non-zero elements, which is usually much smaller than $\left(n_{\Bas}\right)^2$.
As a consequence, in the case of a CASSCF wave function one can rewrite $f_{\wf{}{}}(\bfr{},\bfr{})$ as
\begin{equation}
f_{\wf{}{}}(\bfr{},\bfr{}) = \sum_{r,s \in \mathcal{C}\cup\mathcal{A}} \mathcal{V}_r^s(\bfr{}) \, \mathcal{N}_{r}^s(\bfr{}).
\end{equation}
Therefore the final computational cost of $f_{\wf{}{}}(\bfr{},\bfr{})$ is dominated by that of $\mathcal{V}_r^s(\bfr{})$, which scales as $\left(n_{\mathcal{A}}+n_{\mathcal{C}}\right)^2 \left( n_{\Bas} \right)^2 n_G$, which is much weaker than $\left(n_{\Bas}\right)^4 n_G$.
\bibliography{../srDFT_SC}
\end{document}
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