Similar to other electron correlation methods, many-body perturbation theory methods based on Green functions, such as the so-called $GW$ approximation, suffer from the usual slow convergence of energetic properties with respect to the size of the one-electron basis set.
This displeasing feature is due to lack of explicit electron-electron terms modeling the infamous Kato electron-electron cusp and the correlation Coulomb hole around it.
Here, we propose a computationally efficient density-based basis set correction based on short-range correlation density functionals which significantly speeds up the convergence of energetics towards the complete basis set limit.
The performance of this density-based correction is illustrated by computing the ionization potentials of the twenty smallest atoms and molecules of the GW100 test set at the perturbative $GW$ (or $G_0W_0$) level using increasingly large basis sets.
We also compute the ionization potentials of the five canonical nucleobases (adenine, cytosine, thymine, guanine, and uracil) and show that, here again, a significant improvement is obtained.
The purpose of many-body perturbation theory (MBPT) based on Green functions is to solve the formidable many-body problem by adding the electron-electron Coulomb interaction perturbatively starting from an independent-particle model. \cite{Martin_2016}
In this approach, the \textit{screening} of the Coulomb interaction is an essential quantity. \cite{Aryasetiawan_1998, Onida_2002, Reining_2017}
The so-called {\GW} approximation is the workhorse of MBPT and has a long and successful history in the calculation of the electronic structure of solids. \cite{Aryasetiawan_1998, Onida_2002, Reining_2017}
{\GW} is getting increasingly popular in molecular systems \cite{Blase_2011, Faber_2011, Bruneval_2012, Bruneval_2015, Bruneval_2016, Bruneval_2016a, Boulanger_2014, Blase_2016, Li_2017, Hung_2016, Hung_2017, vanSetten_2015, vanSetten_2018, Ou_2016, Ou_2018, Faber_2014} thanks to efficient implementation relying on plane waves \cite{Marini_2009, Deslippe_2012, Maggio_2017} or local basis functions. \cite{Blase_2011, Blase_2018, Bruneval_2016, vanSetten_2013, Kaplan_2015, Kaplan_2016, Krause_2017, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b}
which connects the Green function $\G{}{}$, its non-interacting version $\G{0}{}$, the irreducible vertex function $\Gam{}{}$, the irreducible polarizability $\Po{}{}$, the dynamically-screened Coulomb interaction $\W{}{}$
and the self-energy $\Sig{}{}$, where $\vc{}{}$ is the bare Coulomb interaction, $\delta(12)$ is the Dirac delta function \cite{NISTbook} and $1$ is a composite coordinate gathering space, spin, and time variables $(\br{1},\sigma_1,t_1)$.
Although obviously starting-point dependent, \cite{Bruneval_2013, Jacquemin_2016, Gui_2018} it has been widely used in the literature to study solids, atoms and molecules. \cite{Bruneval_2012, Bruneval_2013, vanSetten_2015, vanSetten_2018}
For finite systems such as atoms and molecules, partially \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011} or fully self-consistent \cite{Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b}{\GW} methods have shown great promise. \cite{Ke_2011, Blase_2011, Faber_2011, Koval_2014, Hung_2016, Blase_2018, Jacquemin_2017}
Similar to other electron correlation methods, MBPT methods suffer from the usual slow convergence of energetic properties with respect to the size of the one-electron basis set.
This can be tracked down to the lack of explicit electron-electron terms modeling the infamous electron-electron coalescence point (also known as Kato cusp \cite{Kato_1957}) and, more specifically, the Coulomb correlation hole around it.
Pioneered by Hylleraas \cite{Hylleraas_1929} in the 1930's and popularized in the 1990's by Kutzelnigg and coworkers \cite{Kutzelnigg_1985, Noga_1994, Kutzelnigg_1991} (and subsequently others \cite{Kong_2012, Hattig_2012, Tenno_2012a, Tenno_2012b, Gruneis_2017}), the so-called F12 methods overcome this slow convergence by employing geminal basis functions that closely resemble the correlation holes in electronic wave functions.
F12 methods are now routinely employed in computational chemistry and provide robust tools for electronic structure calculations where small basis sets may be used to obtain near complete basis set (CBS) limit accuracy. \cite{Tew_2007}
The basis-set correction presented here follow a different route, and relies on the range-separated density-functional theory (RS-DFT) formalism to capture, thanks to a short-range correlation functional, the missing part of the short-range correlation effects. \cite{Giner_2018}
As shown in recent studies on both ground- and excited-state properties, \cite{Loos_2019, Giner_2019} similar to F12 methods, it significantly speeds up the convergence of energetics towards the CBS limit while avoiding the usage of the large auxiliary basis sets that are used in F12 methods to avoid the numerous three- and four-electron integrals. \cite{Kong_2012, Hattig_2012, Tenno_2012a, Tenno_2012b, Gruneis_2017, Barca_2016, Barca_2017, Barca_2018}
Explicitly correlated F12 correction schemes have been derived for second-order Green function methods (GF2) \cite{SzaboBook, Casida_1989, Casida_1991, Stefanucci_2013, Ortiz_2013, Phillips_2014, Phillips_2015, Rusakov_2014, Rusakov_2016, Hirata_2015, Hirata_2017, Loos_2018} by Ten-no and coworkers \cite{Ohnishi_2016, Johnson_2018} and Valeev and coworkers. \cite{Pavosevic_2017, Teke_2019}
In the present manuscript, we illustrate the performance of the density-based basis set correction developed in Refs.~\onlinecite{Giner_2018, Loos_2019, Giner_2019} on ionization potentials obtained within {\GOWO}.
Note that the present basis set correction can be straightforwardly applied to other properties (\textit{e.g.}, electron affinities and fundamental gap), as well as other flavors of (self-consistent) {\GW} or Green function-based methods, such as GF2 (and its higher-order variants).
Following Ref.~\onlinecite{Giner_2018}, we start by defining, for a $\Ne$-electron system with nuclei-electron potential $\vne(\br{})$, the approximate ground-state energy for one-electron densities $\n{}{}$ which are ``representable'' in a finite basis set $\Bas$
where $\cD^\Bas$ is the set of $\Ne$-representable densities which can be extracted from a wave function $\wf{}{\Bas}$ expandable in the Hilbert space generated by $\Bas$.
is the exact Levy-Lieb universal density functional, \cite{Levy_1979, Levy_1982, Lieb_1983} where the notation $\wf{}{}\rightsquigarrow\n{}{}$ in Eq.~\eqref{eq:E0B} states that $\wf{}{}$ yields the one-electron density $\n{}{}$.
In the present work, instead of using wave-function methods for calculating $\F{}{\Bas}[\n{}{}]$, we use Green-function methods.
We assume that there exists a functional $\Omega^\Bas[\G{}{\Bas}]$ of $\Ne$-representable one-electron Green functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$ and yielding the density $\n{}{}$ which gives $\F{}{\Bas}[\n{}{}]$ at a stationary point
The reason why we use a stationary condition rather than a minimization condition is that only a stationary property is generally known for functionals of the Green function.
For example, we can choose for $\Omega^\Bas[\G{}{}]$ a Klein-like energy functional (see, \textit{e.g.}, Refs.~\onlinecite{Stefanucci_2013, Martin_2016, Dahlen_2005, Dahlen_2005a, Dahlen_2006})
In Eq.~\eqref{eq:OmegaB}, $\Phi_\Hxc^\Bas[\G{}{}]$ is a Hartree-exchange-correlation ($\Hxc$) functional of the Green function such that its functional derivatives yields the $\Hxc$ self-energy in the basis
where the stationary point is searched over $\Ne$-representable one-electron Green functions $\G{}{\Bas}(\br{},\br{}',\omega)$ representable in the basis set $\Bas$.
The stationary condition from Eq.~\eqref{eq:E0BGB} is
and $\bSig{}{\Bas}$ is a frequency-independent local self-energy coming from the functional derivative of the complementary basis-correction density functional
The solution of the Dyson equation \eqref{eq:Dyson} gives the Green function $\G{}{\Bas}(\br{},\br{}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bSig{}{\Bas}[\n{}{}]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bSig{}{\Bas}$.
Of course, in the CBS limit, the basis-set correction vanishes and the Green function becomes exact, \textit{i.e.},
For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\br{},\br{}')=\Sig{\Hx}{\Bas}(\br{},\br{}')$ is the $\HF$ nonlocal self-energy, and if the reference is Kohn-Sham ($\KS$), $\Sig{\text{ref}}{\Bas}(\br{},\br{}')=\pot{\Hxc}{\Bas}(\br{})\delta(\br{}-\br{}')$ is the local $\Hxc$ potential.
The one-electron energies $\e{p}$ and their corresponding (real-valued) orbitals $\MO{p}(\br{})$ (which defines the basis set $\Bas$) are then $\KS$ energies and orbitals.
and the transition densities $(\bX_m+\bY_m)_{ia}$ originating from a (direct) random-phase approximation (RPA) calculation \cite{Casida_1995, Dreuw_2005}
Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{m}$ which represent the poles of the screened Coulomb interaction $\W{}{}(\omega)$.
The {\GOWO} quasiparticle energies $\eGOWO{p}$ are provided by the solution of the (non-linear) quasiparticle equation \cite{Hybertsen_1985a, vanSetten_2013, Veril_2018}
The frequency-independent local self-energy $\bSig{}{\Bas}[\n{}{}](\br{},\br{}')=\bpot{}{\Bas}[\n{}{}](\br{})\delta(\br{}-\br{}')$ originates from the functional derivative of complementary basis-correction density functionals $\bpot{}{\Bas}[\n{}{}](\br{})=\delta\bE{}{\Bas}[\n{}{}]/\delta\n{}{}(\br{})$.
Here, we employ two types of complementary, short-range correlation functionals $\bE{}{\Bas}[\n{}{}]$: a short-range local-density approximation ($\srLDA$) functional with multideterminant reference \cite{Toulouse_2005, Paziani_2006} and a short-range Perdew-Burke-Ernzerhof ($\srPBE$) correlation functional \cite{Ferte_2019, Loos_2019} which interpolates between the usual PBE functional \cite{Perdew_1996} at $\mu=0$ and the exact large-$\mu$ behavior~\cite{Toulouse_2004, Gori-Giorgi_2006, Paziani_2006} using the on-top pair density from the uniform-electron gas. \cite{Loos_2019}
Additionally to the one-electron density calculated from the HF or KS orbitals, these RS-DFT functionals require a range-separation function $\rsmu{}{\Bas}(\br{})$ which automatically adapts to the spatial inhomogeneity of the basis-set incompleteness error and is computed using the HF or KS opposite-spin pair-density matrix in the basis set $\Bas$.
We refer the interested reader to Refs.~\onlinecite{Giner_2018, Loos_2019, Giner_2019} where our procedure is thoroughly detailed and the explicit expressions of these two short-range correlation functionals are provided.
As evidenced by Eq.~\eqref{eq:QP-corrected}, the present basis set correction is a non-self-consistent, \textit{post}-{\GW} correction.
Although outside the scope of this study, various other strategies can be potentially designed, for example, within linearized {\GOWO} or self-consistent {\GW} calculations.
IPs (in eV) of the 20 smallest molecules of the GW100 set computed at the {\GOWO}@HF level of theory with various basis sets and corrections.
The mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the {\GOWO}@HF/CBS values are also reported.
IPs (in eV) of the 20 smallest molecules of the GW100 set computed at the {\GOWO}@PBE0 level of theory with various basis sets and corrections.
The mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the {\GOWO}@PBE0/CBS values are also reported.
All the geometries have been extracted from the GW100 set. \cite{vanSetten_2015}
Unless otherwise stated, all the {\GOWO} calculations have been performed with the MOLGW software developed by Bruneval and coworkers. \cite{Bruneval_2016a}
The HF, PBE, and PBE0 calculations as well as the srLDA and srPBE basis set corrections have been computed with Quantum Package, \cite{QP2} which by default uses the SG-2 quadrature grid for the numerical integrations.
The {\GOWO} quasiparticle energies have been obtained ``graphically'', \textit{i.e.}, by solving the non-linear, frequency-dependent quasiparticle equation \eqref{eq:QP-G0W0} (without linearization).
Moreover, the infinitesimal $\eta$ in Eq.~\eqref{eq:SigC} has been set to zero.
Compared to the conventional $\order*{\Nbas^6}$ computational cost of {\GW} (where $\Nbas$ is the number of basis functions), the present basis set correction represents a marginal additional cost as further discussed in Refs.~\onlinecite{Loos_2019, Giner_2019}.
Note, however, that the formal $\order*{\Nbas^6}$ cost of {\GW} can be significantly reduced thanks to resolution-of-the-identity techniques \cite{vanSetten_2013, Bruneval_2016, Duchemin_2017} and other tricks. \cite{Rojas_1995, Duchemin_2019}
IP (in eV) of the water molecule computed at the {\GOWO} (black circles), {\GOWO}+srLDA (red squares), and {\GOWO}+srPBE (blue diamonds) levels of theory with increasingly large Dunning's basis sets \cite{Dunning_1989} (cc-pVDZ, cc-pVTZ, cc-pVQZ, and cc-pV5Z) with two different starting points: HF (left) and PBE0 (right).
In this section, we study a subset of atoms and molecules from the GW100 test set. \cite{vanSetten_2015}
In particular, we study the 20 smallest molecules of the GW100 set, a subset that we label as GW20.
This subset has been recently considered by Lewis and Berkelbach to study the effect of vertex corrections to $\W{}{}$ on IPs of molecules. \cite{Lewis_2019a}
Later in this section, we also study the five canonical nucleobases (adenine, cytosine, thymine, guanine, and uracil) which are also part of the GW100 test set.
The IPs of the GW20 set obtained at the {\GOWO}@{\HF} and {\GOWO}@{\PBEO} levels with increasingly larger Dunning's basis sets cc-pVXZ (X $=$ D, T, Q, and 5) are reported in Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0}, respectively.
The corresponding statistical deviations (with respect to the CBS values) are also reported: mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX).
These reference CBS values have been obtained with the usual X$^{-3}$ extrapolation procedure using the three largest basis sets. \cite{Bruneval_2012}
The convergence of the IP of the water molecule with respect to the basis set size is depicted in Fig.~\ref{fig:IP_G0W0_H2O}.
Additional graphs reporting the convergence of the IPs of each molecule of the GW20 subset at the {\GOWO}@{\HF} and {\GOWO}@{\PBEO} levels are reported in the {\SI}.
Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0} (as well as Fig.~\ref{fig:IP_G0W0_H2O}) clearly evidence that the present basis set correction significantly increases the rate of convergence of IPs.
At the {\GOWO}@{\HF} (see Table \ref{tab:GW20_HF}), the MAD of the conventional calculations (\textit{i.e}, without basis set correction) is roughly divided by two each time one increases the basis set size (MADs of $0.60$, $0.24$, $0.10$, and $0.05$ eV going from cc-pVDZ to cc-pV5Z) with maximum errors higher than $1$ eV for molecules such as \ce{HF}, \ce{H2O}, and \ce{LiF} with the smallest basis set.
For each basis set, the correction brought by the short-range correlation functionals reduces by roughly half or more the MAD, RMSD, and MAX compared to the correction-free calculations.
For example, we obtain MADs of $0.27$, $0.12$, $0.04$, and $0.01$ eV at the {\GOWO}@HF+srPBE level with increasingly larger basis sets.
In a nutshell, the present basis set correction provides cc-pVQZ quality results at the cc-pVTZ level.
Besides, it allows to reach chemical accuracy with the quadruple-$\zeta$ basis set, an accuracy that could not be reached even with the cc-pV5Z basis set for the conventional calculations.
It is worth pointing out that, for ground-state properties such as atomization and correlation energies, the density-based correction brought a more significant basis set reduction.
For example, we evidenced in Ref.~\onlinecite{Loos_2019} that quintuple-$\zeta$ quality atomization and correlation energies are recovered with triple-$\zeta$ basis sets.
The potential reasons for this could be: i) potential-based DFT corrections are usually less accurate than the ones based directly on energies, \cite{Kim_2013} and ii) because the present scheme only corrects the basis set incompleteness error originating from the electron-electron cusp, some incompleteness remains at the HF or KS level. \cite{Adler_2007}
IPs (in eV) of the five canonical nucleobases (adenine, cytosine, thymine, guanine, and uracil) computed at the {\GOWO}@PBE level of theory for various basis sets.
Error (in eV) with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values for the IPs of the five canonical nucleobases (adenine, cytosine, thymine, guanine, and uracil) computed at the {\GOWO}@PBE level of theory for various basis sets.
In order to check the transferability of the present observations to larger systems, we have computed the values of the IPs of the five canonical nucleobases (adenine, cytosine, thymine, guanine, and uracil) at the {\GOWO}@PBE level of theory with a different basis set family. \cite{Weigend_2003a, Weigend_2005a}
The numerical values are reported in Table \ref{tab:DNA_IP}, and their error with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values \cite{vanSetten_2015} (obtained via extrapolation of the def2-TZVP and def2-QZVP results) are shown in Fig.~\ref{fig:DNA_IP}.
The CCSD(T)/def2-TZVPP computed by Krause \textit{et al.}\cite{Krause_2015} as well as the experimental results extracted from Ref.~\onlinecite{vanSetten_2015} are reported for comparison purposes.
For these five systems, the IPs are all of the order of $8$ or $9$ eV with an amplitude of roughly $1$ eV between the smallest basis set (def2-SVP) and the CBS value.
The conclusions that we have drawn in the previous subsection do apply here as well.
For the smallest double-$\zeta$ basis def2-SVP, the basis set correction reduces by roughly half an eV the basis set incompleteness error.
In the present manuscript, we have shown that the density-based basis set correction developed by some of the authors in Ref.~\onlinecite{Giner_2018} and applied recently to ground- and excited-state properties \cite{Loos_2019, Giner_2019} can also be successfully applied to Green function methods such as {\GW}.
In particular, we have evidenced that the present basis set correction (which relies on LDA- or PBE-based short-range correlation functionals) significantly speeds up the convergence of IPs for small and larger molecules towards the CBS limit.
As mentioned earlier, the present basis set correction can be straightforwardly applied to other properties of interest such as electron affinities or fundamental gap.
It is also applicable to other flavors of {\GW} such as the partially self-consistent {\evGW} or {\qsGW} methods.
We are currently investigating the performance of the present approach within linear response theory in order to speed up the convergence of excitation energies obtained within the RPA and Bethe-Salpeter equation (BSE) \cite{Strinati_1988, Leng_2016, Blase_2018} formalisms.
