Similarly to other electron correlation methods, many-body perturbation theory methods, 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 due to the lack of explicit electron-electron terms modeling the infamous electron-electron cusp.
Here, we propose a density-based basis set correction based on short-range correlation density functionals which significantly speed up the convergence of energetics towards the complete basis set limit.
The purpose of many-body perturbation theory (MBPT) is to solve the formidable many-body problem by adding the electron-electron Coulomb interaction perturbatively starting from an independent-particle model. \cite{MarReiCep-BOOK-16}
In MBPT, the \textit{screening} of the Coulomb interaction is a central quantity, and is responsible for a rich variety of phenomena that would be otherwise absent (such as quasiparticle satellites and lifetimes). \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} and is getting increasingly popular in molecular systems \cite{Blase_2011, Faber_2011, Bruneval_2012, Bruneval_2013, 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 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}
The $GW$ approximation stems from the acclaimed Hedin's equations \cite{Hedin_1965}
which connects the Green's function $G$, its non-interacting version $G_0$, the irreducible vertex function $\Gamma$, the irreducible polarizability $P$, the dynamically-screened Coulomb interaction $W$
and the self-energy $\Sigma$, where $v$ is the bare Coulomb interaction, $\delta(12)$ is Dirac's delta function \cite{NISTbook} and $(1)$ is a composite coordinate gathering spin, space and time variables $(\sigma_1,\br{1},t_1)$.
Depending on the degree of self-consistency one is willing to perform, there exists several types of $GW$ calculations. \cite{Loos_2018}
The simplest and most popular variant of $GW$ is perturbative $GW$, or {\GOWO}. \cite{Hybertsen_1985a, Hybertsen_1986}
Although obviously starting-point dependent, 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 or fully self-consistent $GW$ methods have shown great promise. \cite{Ke_2011, Blase_2011, Faber_2011, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b, 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{Kat-CPAM-57}) and, more specifically, the Coulomb correlation hole around it.
Pioneered by Hyllerras \cite{Hyl-ZP-29} in the 1930's and popularized in the 1990's by Kutzelnigg and coworkers \cite{Kut-TCA-85, NogKut-JCP-94, KutKlo-JCP-91} (and subsequently others \cite{KonBisVal-CR-12, HatKloKohTew-CR-12, TenNog-WIREs-12, GruHirOhnTen-JCP-17}), 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{TewKloNeiHat-PCCP-07}
The basis-set correction presented here follow a different avenue, 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{GinPraFerAssSavTou-JCP-18}
As shown in recent studies on both ground- and excited-state properties, \cite{LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} similar to F12 methods, it significantly speeds up the convergence of energetics towards the CBS limit while avoiding the usage of large auxiliary basis sets.
Explicitly correlated F12 correction schemes have been derived for second-order Green's 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 on ionization potentials (IPs) obtained within {\GOWO}.
Note that the the present basis set correction can be straightforwardly applied to other properties (e.g., electron affinities and fundamental gap), as well as other flavours of $GW$ or Green's function-based methods, such as GF2 (and its higher-order variants).
Moreover, we are currently investigating the performances of the present approach for linear response theory, in order to speed up the convergence of excitation energies obtained within the random-phase approximation (RPA) \cite{Dreuw_2005} and Bethe-Salpeter equation (BSE) formalism. \cite{Strinati_1988, Leng_2016, Blase_2018}
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we start by defining, for a $N$-electron system with nuclei-electron potential $v_\text{ne}(\b{r})$, the approximate ground-state energy for one-electron densities $n$ which are ``representable'' in a finite basis set ${\cal B}$
where ${\cal D}^{\cal B}$ is the set of $N$-representable densities which can be extracted from a wave function $\Psi^{\cal B}$ expandable in the Hilbert space generated by ${\cal B}$. In this expression, $F[n]=\min_{\Psi\to n}\bra{\Psi}\hat{T}+\hat{W}_\text{ee}\ket{\Psi}$ is the exact Levy-Lieb universal density functional, where $\hat{T}$ and $\hat{W}_\text{ee}$ are the kinetic and electron-electron interaction operators, which is then decomposed as
and $\bar{E}^{\cal B}[n]$ is the complementary basis-correction density functional. In the present work, instead of using wave-function methods for calculating $F^{\cal B}[n]$, we reexpress it with a contrained search over $N$-representable one-electron Green's functions $G^{\cal B}(\b{r},\b{r}',\omega)$ representable in the basis set ${\cal B}$
where $\Omega^{\cal B}[G]$ is chosen to be a Klein-like energy functional of the Green's function (see, e.g., Refs.~\onlinecite{SteLee-BOOK-13,MarReiCep-BOOK-16,DahLee-JCP-05,DahLeeBar-IJQC-05,DahLeeBar-PRA-06})
where $(G_\text{s}^{\cal B})^{-1}$ is the projection into ${\cal B}$ of the inverse free-particle Green's function $(G_\text{s})^{-1}(\b{r},\b{r}',\omega)=(\omega+(1/2)\nabla_\b{r}^2)\delta(\b{r}-\b{r}')$ and we have used the notation $\Tr[A B]=1/(2\pi i)\int_{-\infty}^{+\infty}\!\d\omega\, e^{i \omega0^+}\!\iint\!\d\b{r}\d\b{r}' A(\b{r},\b{r}',\omega) B(\b{r}',\b{r},\omega)$. In Eq.~(\ref{OmegaB}), $\Phi_\text{Hxc}^{\cal B}[G]$ is a Hartree-exchange-correlation (Hxc) functional of the Green's functional such as its functional derivatives yields the Hxc self-energy in the basis: $\delta\Phi_\text{Hxc}^{\cal B}[G]/\delta G(\b{r},\b{r}',\omega)=\Sigma_\text{Hxc}^{\cal B}[G](\b{r},\b{r}',\omega)$. Inserting Eqs.~(\ref{Fn}) and~(\ref{FBn}) into Eq.~(\ref{E0B}), we finally arrive at
where the minimization is over $N$-representable one-electron Green's functions $G^{\cal B}(\b{r},\b{r}',\omega)$ representable in the basis set ${\cal B}$.
$(G_\text{0})^{-1}(\b{r},\b{r}',\omega)=(\omega+(1/2)\nabla_\b{r}^2+ v_\text{ne}(\b{r})+\lambda)\delta(\b{r}-\b{r}')$ with the chemical potential $\lambda$, and $\bar{\Sigma}^{\cal B}$ is a frequency-independent local self-energy coming from functional derivative of the complementary basis-correction density functional
with $\bar{v}^{\cal B}[n](\b{r})=\delta\bar{E}^{\cal B}[n]/\delta n(\b{r})$. The solution of the Dyson equation~(\ref{Dyson}) gives the Green's function $G^{\cal B}(\b{r},\b{r}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bar{\Sigma}^{\cal B}[n]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bar{\Sigma}^{\cal B}$. Of course, in the complete-basis-set (CBS) limit, the basis-set correction vanishes, $\bar{\Sigma}^{{\cal B}\to\text{CBS}}=0$, and the Green's function becomes exact, $G^{{\cal B}\to\text{CBS}}=G$.
where $(G_\text{ref}^{\cal B})^{-1}=(G_\text{0}^{\cal B})^{-1}-\Sigma_\text{ref}^{\cal B}$. For example, if the reference is Hartree-Fock (HF), $\Sigma_\text{ref}^{\cal B}(\b{r},\b{r}')=\Sigma_\text{Hx,HF}^{\cal B}(\b{r},\b{r}')$ is the HF nonlocal self-energy, and if the reference is Kohn-Sham, $\Sigma_\text{ref}^{\cal B}(\b{r},\b{r}')= v_\text{Hxc}^{\cal B}(\b{r})\delta(\b{r}-\b{r}')$ is the local Hxc potential.
Here, we provide self-contained summary of the main equations and quantities behind {\GOWO} and {\evGW}.
More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}.
For a given (occupied or virtual) orbital $p$, the correlation part of the self-energy is conveniently split in its hole (h) and particle (p) contributions
are obtained via the contraction of the bare two-electron integrals \cite{Gill_1994}$(pq|rs)$ and the transition densities $(\bX+\bY)_{ia}^{x}$ originating from a random phase approximation (RPA) calculation \cite{Casida_1995, Dreuw_2005}
Because of sum rules, \cite{Martin_1959, Baym_1961, Baym_1962, vonBarth_1996} the other solutions, known as satellites, share the remaining weight.
In a well-behaved case (belonging to the weakly correlated regime), the QP weight is much larger than the sum of the satellite weights, and of the order of $0.7$-$0.9$.
Within the linearized version of {\GOWO}, one assumes that
Unless otherwise stated, in the remaining of this paper, the {\GOWO} QP energies are determined via the linearized method.
In the case of {\evGW}, the QP energy, $\eGW{p}$, are obtained via Eq.~\eqref{eq:QP-G0W0}, which has to be solved self-consistently due to the QP energy dependence of the self-energy [see Eq.~\eqref{eq:SigC}]. \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011}
At least in the weakly correlated regime where a clear QP solution exists, we believe that, within {\evGW}, the self-consistent algorithm should select the solution of the QP equation \eqref{eq:QP-G0W0} with the largest renormalization weight $\Z{p}(\eGW{p})$.
All the geometries have been extracted from the GW100 set. \cite{vanSetten_2015}
Unless otherwise stated, all the {\GOWO} calculations have been performed with MOLGW 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.
Frozen-core (FC) calculations are systematically performed.
The FC density-based correction is used consistently with the FC approximation in the {\GOWO} calculations.
The {\GOWO} quasiparticle energies have been obtained ``graphically'', i.e., by solving the non-linear, frequency-dependent quasiparticle equation (without linearization).
Moreover, the infinitesimal $\eta$ has been set to zero.
IPs (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 (cc-pVDZ, cc-pVTZ, cc-pVQZ and cc-pV5Z) with two different starting points: HF (left) and PBE0 (right).
The thick black line represents the CBS value obtained by extrapolation (see text for more details).
Error (in eV) with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values fort 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.
PFL would like to thank Fabien Bruneval for technical assistance. He also would like to thank Arjan Berger and Pina Romaniello for stimulating discussions.
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''}.
