Periodic Implementation of the Random Phase Approximation with Numerical Atomic Orbitals and Dual Reciprocal Space Grids

Abstract

The random phase approximation (RPA) has emerged as a prominent first-principles method in material science, particularly to study the adsorption and chemisorption of small molecules on surfaces. However, its widespread application is hampered by its relatively high computational cost. Here, we present a well-parallelised implementation of the RPA with localized atomic orbitals and pair-atomic density fitting, which is especially suitable for studying two-dimensional systems. Through a dual k -grid scheme, we achieve fast and reliable convergence of RPA correlation energies to the thermodynamic limit. We demonstrate the efficacy of our implementation through an application to the adsorption of CO on MgO(001) using PBE input orbitals (RPA@PBE). Our calculated adsorption energy is in excellent agreement with previously published RPA@PBE studies, but, as expected, overestimates the experimentally available adsorption energies as well as recent CCSD­(T) results.

1. Introduction

The Random Phase Approximation (RPA) is a well-established method in quantum chemistry and solid-state physics. Originally developed by Bohm and Pines while working on plasmonic oscillation in the jellium model, − it is equivalent to approximating the correlation energy through a summation of ring diagrams within diagrammatic perturbation theory. − Equivalently, it can be derived from the Klein functional using a noninteracting single particle Green function − or within the Adiabatic-Connection (AC) Fluctuation Dissipation theorem (ACFDT). , Ring diagrams describe one of the most important signatures of electron correlation, the screening of electron–electron interactions at large interelectronic distances. Therefore, the RPA is especially accurate for reactions dominated by the difference in the long-range correlation energies of the reactants, as caused, for instance, by noncovalent interactions. −

The RPA exchange-correlation (xc) energy is fully nonlocal and depends on virtual orbitals, placing itself at the top of Perdew’s Jacob’s ladder of density functional theory (DFT). , Therefore, it is computationally more involved than generalized gradient approximations (GGA). While density functionals including empirical dispersion corrections , are cheaper options for treating noncovalent interactions and give accurate results in many scenarios, they may break down in highly anisotropic or highly polarizable systems. Nonlocal dispersion corrections are more advanced alternatives but are also computationally more involved.

Compared to wave function-based methods, the RPA includes interactions present both in second-order Møller–Plesset perturbation theory (MP2) and Coupled Cluster (CC) theory. − The development of CC implementations for periodic systems is a relatively recent development − and applications of these methods to the adsorption of CO on the MgO(001) surface , demonstrate their immense potential in applications to molecule–surface interactions. As shown by Scuseria, Henderson, and co-workers, , RPA can be seen as a simplification of coupled cluster with single and double excitations (CCSD), including fewer interaction channels but having the advantage of much lower computational cost.

RPA and MP2 both share the second-order ring diagram, but this term diverges in strongly polarizable small band gap systems or metals in the thermodynamic limit. While MP2 is increasingly applied to solids − and these shortcomings might be alleviated using regularization techniques, the RPA is widely applicable without such corrections. ,, To combine the benefits of MP2 and RPA, the addition of second-order screened exchange (SOX) corrections has also been explored extensively for molecules − and homogeneous electron gases. , SOX corrections come however with increased computational cost and tend to deteriorate the good description of stretched bonds within the RPA, − , which is important to describe transition states.

The many beneficial characteristics of RPA are key factors behind its increasing popularity for applications in heterogeneous catalysis. Here, the rate-determining step is the chemisorption of a reactant (typically a small molecule) on a metal surface, followed by its dissociation into reactive fragments. Modeling such molecule–surface interactions accurately requires a reliable description of long-range dispersion forces and reaction barrier heights. In many instances, GGAs are not suitable to study these processes, , and the RPA promises much higher accuracy. , For this reason the RPA has been applied to model a wide array of molecule–surface interactions − and is predicted to play an increasingly prominent role in computational catalysis in the future. , More advanced ACFDT methods like adiabatic xc kernel methods − or σ-functionals − can boost the accuracy of the RPA at similar computational cost and their implementations can be realized through minor modifications of the underlying RPA algorithms. We focus here on the efficient implementation of the RPA, which is an important stepping stone to these more advanced methods.

While being significantly cheaper than wave function-based methods, RPA calculations are computationally much more demanding than conventional DFT. In their canonical formulations , RPA calculations scale as N _A N _k with N _A being the number of atoms in the unit cell and N _k being the number of k -points. Applications of the RPA and other post-SCF methods to heterogeneous catalysis therefore often rely on embedding approaches − and further algorithmic developments are needed to make full RPA calculations routine for molecule–surface interactions. Algorithms based on the space-time formulation , of the GW approximations lower the scaling of RPA calculations to formally N _A N _k . − In practice, the time-determining step of these calculations is the calculation of the RPA polarizability, which can be done with subquadratic scaling, and therefore the scaling is subquadratic for practical system sizes. − These algorithms come with a higher prefactor than canonical implementations, and for this reason, they are useful for large k -grids and large unit cells. Important potential use-cases include Moiré superlattices, point-defects, or molecule–surface interactions with low coverages.

Many molecule–surface interactions can however be modeled with relatively small unit cells for which canonical RPA implementations are more suitable. Here, it is important to account for the reduced 2-dimensional (2D) periodicity of the system. In plane-wave-based implementations, the 2D geometry is replicated periodically along the nonperiodic spatial direction, and large replica separations are required to prevent unphysical interactions between them. − The number of plane waves required to reach a given energy cutoff increases significantly with the lattice constant, making such calculations computationally demanding. To address this, plane-wave simulations of molecule–surface interactions often employ truncated Coulomb potentials. , Despite these techniques, relatively large replica separations of 10–20 Å are still necessary. This approach also introduces challenges in Brillouin zone integration, necessitating even denser k -point grids. ,

For low-dimensional systems, atomic orbital-based implementations are more suitable since they can naturally describe the nonperiodic behavior of the wave function. The most commonly employed localized basis sets for post-self-consistent field (SCF) calculations in periodic systems are Gaussian-type orbitals (GTO) − as implemented for instance in pySCF ,,− or CP2K ,− and numerical atomic orbitals (NAO) as implemented in FHI-AIMS ,,, or ABACUS. − NAO basis sets are both compact and highly flexible and can be used to numerically represent GTOs, Slater-type orbitals (STO), or molecular DFT orbitals. Choosing STOs as their functional form, the exponential decay of the wave function far from a finite system can be better represented. This property is important when modeling 2D systems, and is difficult to achieve with plane waves. In practice, this enables energies closer to the infinite basis set limit for the same number of basis functions.

This work reports a novel NAO-based RPA implementation in the BAND module of the Amsterdam modeling suite (AMS). Our implementation relies on the pair-atomic density fitting (PADF) approximation , (also known as local RI ,,, or concentric atomic density fitting − ) which expands products of atomic basis functions in an auxiliary set of functions centered on the same atoms as the basis function product to be expanded. ,− PADF has been used extensively within the space-time method to realize low-scaling implementations of RPA and GW for finite , and periodic systems, , and for periodic Hartree–Fock. ,, Here, we use it only to reduce the size of involved tensors and consequently memory demands.

Inspired by the HF implementation of Irmler et al., we introduce a scheme to dampen the Coulomb potential at long distances depending on the size of the employed k -grid. Using mono- and bilayer hexagonal boron nitride (h-BN) as well as the adsorption of CO on monolayer MgO(001) as test systems, we demonstrate this method to be numerically stable and leading to a rapid convergence to the thermodynamic limit in combination with a dual k -grid scheme (inspired by the staggered-mesh method , ) which we introduce here as well. We also describe a parallelization scheme, which handles reciprocal space and frequency grid integration, achieving near-perfect parallel efficiency for thousands of cores on multiple nodes. Finally, as a practical application for our implementation, we calculate the adsorption energy of CO on the MgO(001) surface and demonstrate good agreement of our result to previous periodic RPA calculations.

2. Theory

2.1. RPA Correlation Energy from Adiabatic Connection

From the ACFDT, the RPA Correlation energy can be written as ,

$$E_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}^{\mathrm{R}\mathrm{P}\mathrm{A}}=-\frac{1}{2\pi }{\int }_0^{\infty }\mathrm{d}\omega \mathrm{Tr}[\sum _{n=1}\frac{1}{n}{[Z(\mathit{r},{\mathit{r}}^{\prime },i\omega )]}^n-Z(\mathit{r},{\mathit{r}}^{\prime },i\omega )]$$ 1

with

$$Z(\mathit{r},{\mathit{r}}^{\prime },i\omega )=\int \mathrm{d}{r}^{″}{P}^{(0)}(\mathit{r},{\mathit{r}}^{″},i\omega )v({\mathit{r}}^{″},{\mathit{r}}^{\prime })$$ 2

and the trace operator for a generic two variable function A( r , r ′) defined as

$$\mathrm{Tr}[A(\mathit{r},{\mathit{r}}^{\prime })]=\int \mathrm{d}r\mathrm{d}{r}^{\prime }A(\mathit{r},{\mathit{r}}^{\prime })\delta (\mathit{r}-{\mathit{r}}^{\prime })$$ 3

Here, P ⁽⁰⁾( r , r ′, iω) is the irreducible RPA polarizability, and $v(\mathit{r},{\mathit{r}}^{\prime })=\frac{1}{|\mathit{r}-{\mathit{r}}^{\prime }|}$ is the Coulomb potential. By following the derivation given in the appendix, we can expand this equation in a basis F = {f( r , q )} with q being differences between points in the first Brillouin zone (1BZ). We obtain

$$E_{\mathrm{corr}}^{\mathrm{RPA}}=-\frac{1}{2\pi }\sum _{\mathit{q}\in 1\mathrm{BZ}}{\int }_0^{\infty }\mathrm{d}\omega [\log \{\det [\mathbf{1}-\mathbf{Z}(\mathit{q},i\omega )]\}-\mathrm{Tr}[\mathbf{Z}(\mathit{q},i\omega )]]$$ 4

where Z = v ^1/2 P ⁽⁰⁾ v ^1/2, and the matrix elements of P ⁽⁰⁾ are

$$P_{\alpha \beta }^{(0)}(\mathit{q},i\omega )=⟨f_{\alpha }(\mathit{r},\mathit{q})|P^{(0)}(\mathit{r},{\mathit{r}}^{\prime },i\omega )|f_{\beta }({\mathit{r}}^{\prime },\mathit{q})⟩$$ 5

The matrix v ^1/2( q ) is the square root of the Coulomb potential with matrix elements

$$v_{\alpha \beta }(\mathit{q})=⟨f_{\alpha }(\mathit{r},\mathit{q})|\frac{1}{|\mathit{r}-{\mathit{r}}^{\prime }|}|f_{\beta }({\mathit{r}}^{\prime },\mathit{q})⟩$$ 6

In general, for both and , matrix elements between fit functions from different q and q ′ points can be defined. However, these off-diagonal elements are zero, making P _αβ ( q ,ω) and v _αβ( q ) diagonal in q . We therefore have simplified the notation and use only one q value to index these matrix elements.

2.2. Bloch Summation Functions RI

After having summarized the key relations to obtain E _corr , we will show in the following how the matrix elements of P ⁽⁰⁾ can be calculated. We will refer to the basis F as fit set, to distinguish it from the primary basis set that is used to expand the orbitals in and which is labeled as X. We define X = {χ_μ( r , k _n )} where the k _n vector defines the basis functions at its specific reciprocal point in the first Brillouin zone. The set of all k -points will be called K _G . Similarly, the fit set F includes functions {f _α ( r , q _n )} which depend on all the reciprocal vectors q _n ≡ k _m – k _l , obtained as differences between all pairs k _m , k _l ∈ K _G . The set of all q -vectors defines another grid Q _G . Both types of basis functions are related to their real-space analogues via Fourier transforms,

$${\chi }_{\mu }(\mathit{r},\mathit{k})=\sum _{\mathit{R}}{\chi }_{\mu }(\mathit{r}-\mathit{R})e^{i\mathit{k}·\mathit{R}}$$ 7

and

$$f_{\alpha }(\mathit{r},\mathit{q})=\sum _{\mathit{R}}{f}_{\alpha }(\mathit{r}-\mathit{R})e^{i\mathit{q}·\mathit{R}}$$ 8

where the vectors R enumerate the unit cells. The fit set is related to the primary basis through the PADF equations (also known as local RI ,,, ) as

$${\chi }_{\mu \in A}(\mathit{r}-\mathit{R}){\chi }_{\nu \in B}(\mathit{r}-{\mathit{R}}^{\prime })\approx \sum _{\alpha \in A}{C}_{\nu \mu \alpha }^{\mathit{R}\prime ,\mathit{R}}{f}_{\alpha }(\mathit{r}-\mathit{R})+\sum _{\beta \in B}{C}_{\mu \nu \beta }^{\mathit{R},\mathit{R}\prime }{f}_{\beta }(\mathit{r}-{\mathit{R}}^{\prime })$$ 9

The fit functions on the r.h.s. are thus centered on the same two atoms with indices A and B as the two primary basis functions. Utilizing this definition, we next consider the following product in reciprocal space:

$$\begin{gathered} {\chi }_{\mu }^{*}(\mathit{r},\mathit{k}){\chi }_{\nu }(\mathit{r},\mathit{k}+\mathit{q})=\sum _{\mathit{R}}{\chi }_{\mu }(\mathit{r}-\mathit{R})e^{-i\mathit{k}·\mathit{R}}\sum _{{\mathit{R}}^{\prime }}{\chi }_{\nu }(\mathit{r}-{\mathit{R}}^{\prime })e^{i(\mathit{k}+\mathit{q})·\mathit{R}\prime } \\ \begin{array}{l}=\sum _{{\mathit{RR}}^{\prime }}{e}^{i(\mathit{k}+\mathit{q})·\mathit{R}\prime }{e}^{-i\mathit{k}·\mathit{R}}{\chi }_{\mu }(\mathit{r}-\mathit{R}){\chi }_{\nu }(\mathit{r}-{\mathit{R}}^{\prime })\end{array} \\ \approx \sum _{{\mathit{RR}}^{\prime }}{e}^{i(\mathit{k}+\mathit{q})·\mathit{R}\prime }{e}^{-i\mathit{k}·\mathit{R}}[\sum _{\alpha \in A}{C}_{\nu \mu \alpha }^{\mathit{R}\prime ,\mathit{R}}{f}_{\alpha }(\mathit{r}-\mathit{R})+\sum _{\beta \in B}{C}_{\mu \nu \beta }^{\mathit{R},\mathit{R}\prime }{f}_{\beta }(\mathit{r}-{\mathit{R}}^{\prime })] \end{gathered}$$ 10

Combining phase factors and employing the translational symmetry of fit coefficients, we obtain

$$\begin{array}{l}{\chi }_{\mu }^{*}(\mathit{r},\mathit{k}){\chi }_{\nu }(r,\mathit{k}+\mathit{q})\approx \sum _{\alpha \in A}\sum _{{\mathit{RR}}^{\prime }}{e}^{i(\mathit{k}+\mathit{q})·\mathit{R}\prime }{e}^{-\mathit{ik}·\mathit{R}}{C}_{\nu \mu \alpha }^{\mathit{R}\prime ,\mathit{R}}{f}_{\alpha }(\mathit{r}-\mathit{R})+\sum _{\beta \in B}\sum _{{\mathit{RR}}^{\prime }}{e}^{i(\mathit{k}+\mathit{q})\mathit{R}\prime }{e}^{-\mathit{ik}·\mathit{R}}{C}_{\mu \nu \beta }^{\mathit{R},\mathit{R}\prime }{f}_{\beta }(r-{\mathit{R}}^{\prime })\\ =\sum _{\alpha \in A}\sum _{{\mathit{R}}^{\prime }}{e}^{i(\mathit{k}+\mathit{q})·(\mathit{R}-\mathit{R}\prime )}{C}_{\nu \mu \alpha }^{0,(\mathit{R}\prime -\mathit{R})}\sum _R{e}^{\mathit{iq}·\mathit{R}}{f}_{\alpha }(\mathit{r}-\mathit{R})+\sum _{\beta \in B}\sum _{\mathit{R}}{C}_{\mu \nu \beta }^{0,(\mathit{R}\prime -\mathit{R})}{e}^{-\mathit{ik}·(\mathit{R}-\mathit{R}\prime )}\sum _{{\mathit{R}}^{\prime }}{e}^{\mathit{iq}·\mathit{R}\prime }{f}_{\beta }(\mathit{r}-{\mathit{R}}^{\prime })\end{array}$$ 11

and by further defining ΔR = R – R ′, we get

$${\chi }_{\mu }^{*}(\mathit{r},\mathit{k}){\chi }_{\nu }(\mathit{r},\mathit{k}+\mathit{q})\approx \sum _{\alpha \in A}\sum _{\mathbf{\Delta }\mathit{R}}{e}^{i(\mathit{k}+\mathit{q})·\mathbf{\Delta }\mathit{R}}{C}_{\nu \mu \alpha }^{0,\mathbf{\Delta }\mathit{R}}\sum _{\mathit{R}}{e}^{\mathit{iq}·\mathit{R}}{f}_{\alpha }(\mathit{r}-\mathit{R})+\sum _{\beta \in B}\sum _{\mathbf{\Delta }\mathit{R}}{e}^{-\mathit{ik}·\mathbf{\Delta }\mathit{R}}{C}_{\mu \nu \beta }^{0,-\mathbf{\Delta }\mathit{R}}\sum _{{\mathit{R}}^{\prime }}{e}^{\mathit{iq}·\mathit{R}\prime }{f}_{\beta }(\mathit{r}-{\mathit{R}}^{\prime })$$ 12

Defining

$$C_{\nu \mu \alpha }(\mathit{k})=\sum _{\mathbf{\Delta }\mathit{R}}{C}_{\nu \mu \alpha }^{0,\mathbf{\Delta }\mathit{R}}{e}^{\mathit{ik}·\mathbf{\Delta }\mathit{R}}$$ 13

the k -dependent PADF equations become

$$\begin{gathered} {\chi }_{\mu }^{*}(\mathit{r},\mathit{k}){\chi }_{\nu }(\mathit{r},\mathit{k}+\mathit{q})\approx \sum _{\alpha \in A}{C}_{\nu \mu \alpha }(\mathit{k}+\mathit{q})f_{\alpha }(\mathit{q},\mathit{r})+\sum _{\beta \in B}{C}_{\mu \nu \beta }^{*}(\mathit{k})f_{\beta }(\mathit{q},\mathit{r}) \\ =\sum _{\gamma }{C}_{\gamma }^{\mu \nu }(\mathit{k}+\mathit{q},\mathit{k})f_{\gamma }(\mathit{q},\mathit{r}) \end{gathered}$$ 14

where we have combined the separate summations over α and β for notational convenience. In eq , the structure of pair fitting is maintained: fit functions are still located on the same atoms as the basis set products. The notion of ‘same atoms’ implies that two atoms are considered equivalent as long as they are invariant by a direct lattice vector translation. P ⁽⁰⁾ can be expressed in terms of Kohn–Sham (KS) states ψ and KS eigenvalues ϵ as

$$\begin{array}{l}{P}^{(0)}(\mathit{r},{\mathit{r}}^{\prime },i\omega )=\sum _{n,m}\sum _{\mathit{k}+\mathit{q},\mathit{k}}{\zeta }_{\mathit{k}}{\zeta }_{\mathit{q}}\\ \frac{(o_m^{\mathit{k}+\mathit{q}}-o_n^{\mathit{k}}){\psi }_m^{*}(\mathit{r},\mathit{k}+\mathit{q}){\psi }_n(\mathit{r},\mathit{k}){\psi }_n^{*}({\mathit{r}}^{\prime },\mathit{k}){\psi }_m({\mathit{r}}^{\prime },\mathit{k}+\mathit{q})}{{ϵ}_m(\mathit{k}+\mathit{q})-{ϵ}_n(\mathit{k})-i\omega },\end{array}$$ 15

where m and n denote band indices. We do not consider the case of partially filled bands and assume spin-compensation in this work. Therefore, the occupation factors o are either 2 for valence, and 0 for conduction bands. The weights ζ _k and ζ _q account for the sampling of K and Q spaces over all Born–von Karman states in the first Brillouin zone and include the normalization factors of the Bloch summation functions. Transforming eq into the basis of KS states and adopting the usual convention of denoting virtual orbitals with the index a and occupied ones with i

$${\psi }_a^{*}(\mathit{r},\mathit{k}){\psi }_i(\mathit{r},\mathit{k}+\mathit{q})\approx \sum _{\gamma }{\mathcal{C}}_{\gamma }^{\mathit{ai}}(\mathit{k}+\mathit{q},\mathit{k})f_{\gamma }(\mathit{q},\mathit{r})$$ 16

and substituting eq in eq we finally obtain the expression

$$P_{\gamma \gamma \prime }^{(0)}(\mathit{q},i\omega )=-2\sum _{\mathit{k}}{\zeta }_{\mathit{k}}{\zeta }_{\mathit{q}}\sum _{i,a}\frac{{\mathcal{C}}_{\gamma }^{\mathit{ai}}(\mathit{k}+\mathit{q},\mathit{k}){\mathcal{C}}_{\gamma \prime }^{*\mathit{ai}}(\mathit{k}+\mathit{q},\mathit{k})}{{ϵ}_{a,\sigma }(\mathit{k})-{ϵ}_{i,\sigma }(\mathit{k}+\mathit{q})-i\omega }+c.c..$$ 17

Additionally, the Coulomb potential can be obtained as

$$\begin{array}{l}{ṽ}_{\gamma \gamma \prime }(\mathit{q},{\mathit{q}}^{\prime })=\int \mathrm{d}r\mathrm{d}{r}^{\prime }\frac{f_{\gamma }^{*}({\mathit{q}}^{\prime },{\mathit{r}}^{\prime })f_{\gamma \prime }(\mathit{q},\mathit{r})}{|\mathit{r}-{\mathit{r}}^{\prime }|}\\ =\int \mathrm{d}r\mathrm{d}{r}^{\prime }\sum _{\mathit{R},{\mathit{R}}^{\prime }}\frac{f_{\gamma }({\mathit{r}}^{\prime }-{\mathit{R}}^{\prime })e^{-i\mathit{q}\prime ·\mathit{R}\prime }{f}_{\gamma \prime }(\mathit{r}-\mathit{R})e^{i\mathit{q}·\mathit{R}}}{|\mathit{r}-{\mathit{r}}^{\prime }|}\\ =\sum _{\mathbf{\Delta }\mathit{R}}\int \mathrm{d}r\mathrm{d}{r}^{\prime }\frac{f_{\gamma }({\mathit{r}}^{\prime }-\mathit{R}-\mathbf{\Delta }\mathit{R})f_{\gamma \prime }(\mathit{r}-\mathit{R})e^{-i\mathit{q}\prime ·\mathbf{\Delta }\mathit{R}}}{|\mathit{r}-{\mathit{r}}^{\prime }|}\sum _{\mathit{R}}{e}^{i(\mathit{q}-\mathit{q}\prime )·\mathit{R}}\\ =N_{\mathit{R}}·{\delta }_{\mathit{q},\mathit{q}\prime }·\sum _{\mathbf{\Delta }\mathit{R}}⟨f_{\gamma }(\mathit{r}-\mathbf{\Delta }\mathit{R})\left|\frac{1}{|\mathit{r}-{\mathit{r}}^{\prime }|}\right|f_{\gamma \prime }({\mathit{r}}^{\prime })⟩e^{-i\mathit{q}\prime ·\mathbf{\Delta }\mathit{R}}\end{array}$$ 18

where the N _R -factor can be set to one to yield the RPA correlation energy per unit cell.

2.3. Periodic Projector Method

PADF-based implementations of perturbation-theoretical methods relying on virtual orbitals often suffer from stronger numerical instabilities than methods utilizing only occupied orbitals. To overcome this issue which arises from linear dependencies in the primary basis, we generalize the projector method of ref to periodic systems. Starting from the k -point specific overlap matrix S _μν ( k ) = ⟨χ_μ (r, k ) |χ_ν(r, k )⟩ we build a projector T( k ) by diagonalizing S

$$S(\mathit{k})=U(\mathit{k})D(\mathit{k})U^{†}(\mathit{k})$$ 19

and subsequently removing the subspace spanned by eigenvectors corresponding to eigenvalues smaller than a user-specified threshold ϵ _d

$$R_{\mathit{ij}}(\mathit{k})={\delta }_{\mathit{ij}}\mathrm{\Theta }(D_{\mathit{ii}}(\mathit{k})-{ϵ}_d)$$ 20

$${\mathrm{R̃}}_{\mathit{ij}}=\prod _{\mathit{k}}{R}_{\mathit{ij}}(\mathit{k})$$ 21

$$T(\mathit{k})=U(\mathit{k})\mathrm{R̃}{U}^{†}(\mathit{k})$$ 22

where Θ is the Heaviside function. The R( k ) matrices are combined via matrix multiplication, hence, whenever a matrix element from any R( k ) is zero, it will be zero for the cumulative projector. This allows removing the same atomic specific linear combinations from every k point. Eventually T( k ) is then used to regularize the KS orbital coefficients as

$${\mathrm{c̃}}_{\mu }^i(\mathit{k})=\sum _{\nu }{T}_{\mu \nu }(\mathit{k})c_{\nu }^i(\mathit{k})$$ 23

These coefficients are then used to transform eq to eq . Numerical issues can become more pronounced with increasing system and basis set size, necessitating the use of this projector method. Of course, this issue is strongly related to the quality of the auxiliary fit set. When large fit sets are used, a small ϵ _d will suffice, while a smaller fit set necessitates a larger value of ϵ _d .

3. Implementation

3.1. Restricted Bloch Summation

The periodic RI-RPA eqs eqs , , and but also the q -dependent Coulomb overlap integrals eq , formally rely on Bloch summations to map between two infinite spaces. In practice, the reciprocal space grid will however be a finite sampling of the infinite number of k -points in the first Brillouin zone prescribed by the Born–von Karman boundary conditions. This unavoidable limitation demands additional considerations on the representation of such quantities. Since the set of Bloch states is obtained from its real-space analog via a discrete Fourier transform of the unit cell coordinate R , a finite k -grid implies that only a limited amount of unit cells can be considered. In other words, for a regular k -sampling along the reciprocal lattice vectors, there will be a maximum representable distance in real space. Starting from the minimum increment between k -points min­[k _l – k _n ] = q _min along each reciprocal lattice vector direction i, the ‘Nyquist’ distance R _max , on the corresponding real lattice vector direction, is defined by

$$q_{\min }^i·R_{\max }^i=2\pi$$ 24

R _max , defines an n-dimensional parallelepiped (n = 3 for bulk). Within this parallelepiped, a new real space grid is defined, containing as many unit cells as the number of k -points with coordinates R _x . To ensure that the Fourier transform is invertible, it is necessary to restrict every Bloch summation to cells lying only within the grid induced by the Nyquist vectors R _max . Beyond R _max , every function represented through Q _G is not properly defined, unless it reaches zero within the limits of R _max . If not, trying to represent them with a unsuitable Q _G would introduce spurious long-range effects. On the other hand, due to the noninvertibility of the Fourier transform, functions that do not decay to zero within R _max would artificially repeat if transformed back to real space beyond R _max . This induces undesirable boundary effects for nonconverged k -grids. To ensure a proper decay within R _max , Spencer and Alavi suggested to attenuate the Coulomb potential by setting v( r ₁, r ₂) to zero beyond R _max . Here, we instead introduce a spherical Fermi–Dirac function

$$\theta (|{\mathit{r}}_2-{\mathit{r}}_1|,r_0,\beta )=\frac{1}{1+e^{\beta (|{\mathit{r}}_2-{\mathit{r}}_1|+r_0)}}$$ 25

and damp the Coulomb potential according to

$$v_{\theta }({\mathit{r}}_1,{\mathit{r}}_2)=v({\mathit{r}}_1,{\mathit{r}}_2)\theta (|{\mathit{r}}_2-{\mathit{r}}_1|,r_0,\beta )$$ 26

Similar schemes to damp the Coulomb potential have previously been used in periodic HF calculations.

As illustrated in Figure a, the parameters of θ directly depend on the chosen k -grid. If not specified otherwise, we choose the damping radius r ₀ as 50% of R _c , the radius of the largest circle which fits the parallelepiped defined by R _max . The decay parameter β is chosen to reduce the damping function to 0.1% within a distance of 1.4 r ₀. The corresponding Fermi-Dirac function is shown in Figure b. In general, we will refer to the practice of damping the Coulomb potential depending on the k -grid as AUTO damping. In this way, increasing the k -grid will increase r ₀ and β. In the limit of an infinite k -grid, we would sample the infinite number of unit cells.

1.

Pictorial representation of the damping of the Coulomb potential. In part (a), a 7 × 7 grid in k -space defines a 7 × 7 grid of cells in real space, which defines a parallelepiped. The blue circle (full line) represents the largest circle with radius R_c centred around the central highlighted cell, which fits inside this parallelepiped. The Coulomb potential is dampened by the Fermi-Dirac function shown in part (b), whose decay parameter is chosen so that it decays to a value of 0.001 at 0.7R _c= 1.4r ₀. This behaviour is also illustrated by the shaded area in part (a).

3.2. Dual Grids for Enhanced Sampling Around the Γ-Point

Due to the slow convergence of RPA correlation energies, regular sampling of the first Brillouin zone often requires unpractically large k -grids to reach satisfactory accuracy. − This is caused by the divergence of the Coulomb potential at the Γ-point and can be mitigated by truncating the Coulomb interaction as described in the previous section. For a nonconverged k -grid, such a truncation would however be artificial and undesirable. To overcome this issue, we enhance the sampling of Q _G around q ≡ Γ by generating a regular grid in q , removing q ≡ Γ, and adding additional points in the vicinity of Γ as can be seen Figure A. This approach is inspired by the staggered mesh method of Lin and co-workers. , Differences lie in the fact that while we are using the same grids for occupied and virtual orbitals, we have an independent q -grid augmenting our regular k -grid. This allows us to introduce a flexible displacement parameter in q to treat the integrable divergence at the Γ-point.

2.

In (A) we show the Q _G grid. The additional sampling near Γ is represented in blue, the neglected Γ point with a red cross, and the other regular points in black. In (B) we show the K _G grid. The regular set K _reg, which is sampled at first is represented in green, in black all the other points which are added accordingly to Q _G . Using () from green points in (B) with black points in (A) will transform to a green point k _j . The additional blue points in Q _G create the surrounding black points in (B).

Considering the reciprocal lattice vectors b _i as basis for the reciprocal space, these additional points have coordinates x = ∑ _i ± d b _i , where x is position of the additional point in Q _G . According to the dimension of the system, we add 2,4, and 8 points for respectively 1D, 2D and 3D systems. With this procedure, we sample k -space, by generating a regular k -grid K _reg first and then combining it with the increments Q _G as

$${\mathit{k}}_j+\mathit{q}={\mathit{k}}_i$$ 27

In this way, all k _i which augment K _reg, will be used to generate the final K _G , which is schematically presented in Figure B.

With the right choice of Q _G , the size of K _G increases linearly with Q _G , because many target points k _i will already be part of the regular set K _reg. The additional sampling increases the size of the K _reg grid exactly by a multiplicative factor of N _add+1. For instance, for a 2D system, where 4 additional points are added in Q _G around Γ, a regular grid K _reg of 10 × 10 k -points, results in a K _G of (10 × 10) × 5. This scaling however only affects the memory demands of the calculations. As can be seen from eq , the computational effort to evaluate the polarizability in reciprocal space scales as the product of the k and q integration grids, which is K _reg × Q _G .

In the worst-case scenario, using other choices of Q _G , such as generic nonregular grids, can make the size of K _G scale quadratically with respect to Q _G . From eq , it is evident that this occurs when none of the k _j points belong to the previous K _reg sampling. Other samplings around Γ, improving on a regular grid, also make the size of K _G scale linearly with respect to Q _G , but the number of additional points influences the prefactor of the memory scaling. Since now the minimum increment between two k -points is determined by the distance of the additional sampling around q = Γ, also the radius of the Coulomb potential damping increases. In particular, for the results that will be presented further in this article, the distance $d^x=\frac{1}{10}{d}_{\mathrm{reg}}^x$ where d _reg ^x is the directional increment between the initial regular grid in q .

3.3. RPA Algorithm

Here, we discuss the design of our RPA algorithm which is summarized in algorithm 1. The decisive part is the calculation of the RPA integrand ε_ω,q for all q and ω in their respective grids which enters eq through

$$E_{\mathrm{corr}}^{\mathrm{RPA}}\approx \sum _{\omega }\sum _{\mathit{q}\in Q_G}{\epsilon }_{\omega ,\mathit{q}}{\zeta }_{\mathit{q}}{\zeta }_{\omega }$$ 28

where ζ _q and ζ_ω denote the integration weights for q and ω-integration, respectively. In our implementation, the q -integration for a specific ω is achieved through subsequent summations at every cycle, obtaining in the end the intermediate e _ω. The integration of e _ω is performed at the end of both loops.

To parallelise () efficiently, the loops over ω and q group all the concurring processes N _p in smaller ScaLapack contexts. This technique allows each ScaLapack context to be composed of n _{q ,ω} = N _p /N _g. ω N _{g.  q} processes where each handles a portion of the whole workload, consisting of couples of q and ω. Each group will have a separate ScaLapack context to enable distributed algebra. In practice, considering sequential routines proportionally more efficient than their distributed version, one would prefer to have only one single process per group, without actually adopting any parallelization in the algebra. However, this is not always achievable due to memory constraints, since storing all the intermediate matrices at the same time can become prohibitive. Apart from the communication that occurs within ScaLapack, the only communication that is needed consists in gathering e _ω, which later is integrated only in the main node with the respective weights. The procedure is illustrated in Figure .

3.

Schematic overview of the parallelization strategy adopted in our implementation.

The fit coefficients are transformed to the MO basis on the fly when needed for the corresponding polarizability terms. Therefore, some of these transformations have to be performed multiple times. However, the alternative of storing them all, even on disk, is prohibitive due to memory constraints. Nevertheless, for some architectures with faster read and write to disk, or for small calculations in which the memory is not a constraint, these become viable options. For larger systems, storing the PADF fit coefficients becomes the memory bottleneck.

4. Results

In the following, we report tests of several aspects of our RPA implementation. In all calculations, we employ STO-type basis sets represented in the NAO-form. These range from double-ζ (DZ) to quadruple-ζ quality (QZ), referred to as DZP, TZ2P, and QZ4P, respectively, where the suffix xP indicates the number of polarization functions. We note that some integrals, such as kinetic energy integrals or overlap integrals, could in principle be evaluated analytically for STOs. However, the computational overhead of evaluating them numerically when they are represented as NAOs is completely negligible. All calculations used a modified Gauss–Legendre frequency grid of 32 points, according to the prescription of ref . All other computational details are provided in connection with the rest of the results.

4.1. Coulomb Potential Truncation

As a first test of our implementation, we assess how the damping of the Coulomb potential affects the accuracy of our results, and how it influences convergence of the RPA correlation energies with respect to the size of the k -grid for two-dimensional hexagonal boron nitride (h-BN). Figure shows the absolute errors of the RPA correlation energy for different r ₀ in the Fermi–Dirac damping function for different regular k -grids, distinguished by different colors. We have chosen the infinite k -grid limit, obtained from fitting the AUTO damping scheme with an inverse linear fit with respect to the number of k -points as reference value. We have used a constant decay parameter β, obtained according to the rules depicted in Sec. 3.1 in all calculations. The error in the RPA correlation energy decreases with the number of k -points, as long as the damping radius r ₀ is smaller than the radius of the circle confined by the parallelogram defined by R ^max. Larger values of r ₀ lead to uncontrolled errors in the correlation energy.

4.

Effect of Fermi–Dirac damping on RPA@PBE correlation energy convergence for hexagonal boron nitride (STO–DZ basis) The y-axis shows the absolute deviation from a reference RPA energy, extrapolated via an inverse-linear fit in the infinite Q _G limit using the AUTO damping scheme. The x-axis varies the Fermi–Dirac damping parameter r ₀. Colors denote different Q _G grids, with vertical lines indicating the largest inscribed circle radius in the corresponding R _x grids. All curves use the same β parameter, as defined along with the AUTO damping settings in Section .

Moreover, as long as r ₀ is smaller than R ^max, we observe convergence of the RPA correlation energy to a short-range separated value. For instance, choosing a fixed damping radius of r ₀ = 40 Bohr, we obtain almost the same RPA correlation energy with the 30 × 30 and 50 × 50 k -grids. A fixed damping radius would correspond to a fixed short-range separated RPA calculation. This is not desirable, since RPA correlation is important in the long-range. Moreover, the artificial convergence comes from restricting the RPA correlation energy to the short-range, leading to potentially large errors in absolute correlation energies.

The automatic damping we use in this work avoids such an artificial truncation of the Coulomb potential outside the finite resolution prescribed by the finite k -grid. The stars in Figure show the convergence of the RPA correlation energy with respect to the k -grid and associated damping distance to the true undamped result. For this test, we used a damping distance r ₀ equal to 75% of R _c , with a decay speed that reached a damping factor of 0.1% at a distance of 1.3r ₀. Lastly, we bring to the attention the large deviations from the fully converged result, which in the best cases is of the order of 1 kcal mol^–1. Leaving out that we are looking at an absolute energy and that energy differences will be subjected to error cancellation, the root cause is the difficulty of regular grids to treat the integration at the Γ point. In section , we will show how the use of dual grids, described in section improves the k -point convergence.

4.2. Damping Distance Effects on Fluoro-Polyacetylene

To further study the effects of the damping of the Coulomb potential on RPA correlation energies we analyzed the RPA integrand ε_ω,q of fluoro-polyacetylene (whose structure is shown in Figure for different Fermi–Dirac damping distances r ₀ using a decay parameter β reaching a damping of 0.1% at distance of 1.3 r ₀ using regular 1D grids K _G and Q _G of 128 k -points. The results, displayed in Figure , demonstrate that the damping of the Coulomb potential, or, more generally, the effect of a finite grid of unit cells in real space, do not solely affect the integrand at the Γ-point ( q ≡ 0) but modifies ε_ω,q globally.

5.

in (A) Fluoro-polyacetylene geometry with its RPA correlation energy integrand ϵ_ω,q for a specific imaginary frequency, iω = 8 hartree, using the DZ basis set of Band. The different numbers in the legend denote the different values of r ₀ in Bohr. In (B) Its Fourier analysis for the same frequency and same basis. The black line represents a calculation performed with 1000 k -points and AUTO range separation.

We notice that ε_ω,q tends to converge to a definite value at the Γ-point for increasing damping distances, while the behavior away from it is more subtle. Guided by the oscillating trend of the curves in the left panel of Figure , we performed a Fourier analysis of the same function presented in the right panel of Figure , where, given the parity of the function, only the positive axis is shown. The different lines exhibit unique spikes in increasing order, according to the respective damping radii.

Even though the functions ε_ω,q are very different for different damping distances, the RPA results reported in Table remain stable. This suggests that for this system, to some extent, the effect of the damping cancels out in the oscillations shown in Figure . It is clear that the damping, which undoubtedly guarantees a regular convergence to the fully periodic RPA Correlation energy, affects the RPA integrand. As shown by the Fourier analysis, this is mitigated only when the damping parameter goes to infinity, which then comes with the known problem of the Γ-point divergence of the Coulomb potential. Damping of the Coulomb potential facilitates convergence to the limit of infinite k -grid sampling, but it does not necessarily guarantee fast convergence. As shown in the next section, this can be achieved with the dual grid approach.

1. RPA Correlation Energies for Fluoro-Polyacetylene (kcal mol^–1) using Different Bond Lengths between the Two Carbon Atoms (Indicated in the Parentheses), and Different Damping Distances r ₀ (First Column, in Bohr) .

r 0	Ec RPA (2.47 Å)	Ec RPA (2.67 Å)	Δ
10	–333.408	–339.461	–6.052
20	–331.902	–338.277	–6.374
30	–331.731	–338.145	–6.414
40	–331.694	–338.113	–6.418
50	–331.683	–338.100	–6.417

^aThe RPA contribution to the energy difference between the equilibrium and stretched geometries (kcal mol^–1) is reported in the last column, using the DZ basis set.

4.3. Validation of Damping Approach and the Reciprocal Space Convergence

To validate the combined use of damping and dual-grid techniques, we calculate the MgO–CO adsorption energy for a single MgO layer at 100% coverage with a Mg–C distance of 2.479 Å, as well as the interaction energy of an AA’-stacked h-BN bilayer at an interlayer distance of 3.45 Å. Enhanced grids were used, adding sampling at 1/10 of the regular k -grid step.

Figure a shows the CO adsorption energy on the MgO monolayer for different k -meshes relative to its value calculated with a 17 × 17 k -mesh. The data demonstrates that k -point convergence is both rapid and that the rate of convergence is independent of the basis set: lines of the same color (DZP vs TZ2P) for the same decay parameter r ₀ nearly overlap. This observation is consistent with previous plane-wave results. Figure b shows again the RPA contribution to the CO adsorption energy on the MgO monolayer for all different k -meshes. The data clearly shows that the same thermodynamic limit is reached, independent of the precise value of r ₀. The convergence rate is excellent, and independent of r ₀, already a 9 × 9 k -mesh is sufficient to converge the RPA contribution to the adsorption energy with 0.05 kcal mol^–1. Convergence becomes faster when r ₀ is chosen as a larger fraction of R _c , since more and more of the Coulomb potential is accounted for for a given k -grid (r ₀ = R _c would correspond to zero damping). Tables S4 and S6 in the Supporting Information demonstrate that absolute correlation energies converge equally fast.

6.

(a) Absolute deviation of the RPA contribution to the adsorption energy with respect to its value calculated with a 17 × 17 k -mesh, for different decay parameters and basis sets. (b) RPA contributions to the interaction energy for different decay parameters.

For the h-BN bilayer, Figure shows the differences in RPA adsorption energies (DZP vs TZ2P) at various k -grids relative to a 15 × 15 reference. All values are very close to zero, demonstrating that convergence is independent of the basis set. In all of the following calculations, we use a damping of r ₀ = 0.5R _c , since it seems to guarantee both a rapid and smooth convergence to the thermodynamic limit.

7.

Differences between the RPA contributions to the adsorption energy for the h-BN bilayer for different k -point meshes relative to the values obtained with a 15 × 15 grid calculated with the DZP and TZ2P basis sets for different Coulomb damping parameters. The t parameter prescribes a damping factor of 0.1% at r ₀ + 0.5t.

4.4. Parallel Performance

In this section, we discuss the parallel efficiency of our algorithm. All calculations discussed here have been performed on AMD Genoa 9654 nodes with 192 cores, 2GB of DRAM per core, and 2.4 GHz clock speed.

In Figure , we report the strong scaling of the algorithm through a test on a 1D fluoro-polyacetylene chain. It can be seen that the algorithm scales almost perfectly with the number of nodes. The timings for the computationally most intensive part of the algorithm, involving loops over q and ω, are in almost perfect agreement with the theoretical speedup, represented through the dashed line. Nevertheless, we find a small deviation of increasing relative magnitude. This is likely due to a nonperfect balance with respect to the q variable, which involves evaluating the Coulomb potential. In practice, if a specific q point is duplicated over different nodes, multiple evaluations of the potential for the same point are performed. For the overall speedup of the algorithm, the considerations made for the q , ω loop apply as well, but the deviations discussed before are larger. This is due to the necessary preparation steps before the start of the main RPA loop. These steps are carried out separately on each node to avoid the transfer of very large tensors.

8.

Percentage speed up of RPA correlation energy calculation for fluoro-poliacetylene.

As a second test, we show in Figure the scaling for different sizes of Q _G and K _G for h-BN using regular k -grids. Absolute timings are shown in Table . Here, the scaling is again close to optimal. The deviation of the 8 × 8 cases showing a superperfect scaling is due to an artifact in the reference 2 × 192 cores case, normalized to 100%. Some operations are independent of the k -grid size and therefore are more relevant for the calculation with the smallest grid, which consequently appears faster. These deviations are negligible and do not affect our conclusions.

9.

Percentage speed up of RPA correlation energy calculation by reducing the K _G size h-BN.

2. RPA Calculation Timings for Different Parts of the Code: the Innermost q , ω Loop, the Total RPA Routine Time, and the One of the Total Calculation Including the DFT Reference .

# nodes	$\sqrt{K_G}$	q , ω-loop [s]	RPA [s]	RPA + SCF [s]
2	8	35	45	115
2	16	478	491	577
2	32	7422	7456	7698
4	8	17	23	102
4	16	244	254	336
4	32	3753	3789	4019
8	8	8	15	95
8	16	121	136	216
8	32	1891	1932	2053

^aResults have been evaluated on hBN for different regular Q _G /K _G sizes.

4.5. Adsorption CO on MgO

As a practical application of our algorithm, we calculate the adsorption energy of CO on a MgO(001) surface at the RPA@PBE level of theory. Also known as the ‘hydrogen molecule of surface science, the adsorption of CO on MgO(001) is one of the most widely studied molecule–surface interactions. ,,,,,− While MgO is relevant as a catalyst for a wide variety of applications, − the adsorption of a small molecule on metal oxide surfaces is prototypical for many problems in heterogeneous catalysis and surface science. − Results of RPA@PBE calculations with a fully periodic implementation as well as finite cluster approach have been reported previously. With −1.66 kcal mol^–1, the RPA@PBE adsorption energy reported by Bajdich et al. is significantly higher (less negative) than experimental and recent embedded CCSD­(T) results ,, which all agree on an adsorption energy of around −4.5 kcal mol^–1.

Here, we focus on the convergence with respect to K _g and Q _G sampling, basis set convergence, the convergence with respect to the coverage θ, and convergence with the number of MgO layers. As commonly done in the calculation of adsorption energies, , all calculations are counterpoise-corrected to account for basis set superposition errors (BSSE), as this is crucial to obtain reliable results. We do so by using the full basis of the dimer also when computing the energy for the noninteracting systems, yielding:

$$E_c^{\mathrm{ads}}=E_c^{\mathrm{full}\mathrm{basis}}(\mathrm{MgO}+\mathrm{CO})-E_c^{\mathrm{full}\mathrm{basis}}(\mathrm{CO})-E_c^{\mathrm{full\; basis}}(\mathrm{MgO})$$ 29

Analyzing the BSSE explicitly via the evaluation of

$$\mathrm{BSSE}=E_c^{\mathrm{full\; basis}}(\mathrm{MgO})+E_c^{\mathrm{full\; basis}}(\mathrm{CO})-E_c(\mathrm{MgO})-E_c(\mathrm{CO})$$ 30

is, however, complicated. Using the projector method, the number of functions projected out from the primary basis set of each subsystem does depend on the basis functions present on other subsystems. This gives an additional effect beyond the traditional definition of BSSE, which, for larger basis sets, where more functions are projected out, can be rather pronounced.

The system we study is shown in Figure . It involves a slab composed of a variable number of MgO layers, and another slab consisting of a single layer of CO molecules of variable density. The MgO surface is modeled through a rock-salt lattice with a lattice constant of 2.105 Å, and the distance between the C and O atoms in the CO molecule is 1.134 Å. The distance between the surface and the CO molecule is the experimental distance of 2.479 Å, and the molecule is oriented perpendicular to the slab. All calculations employ dual grids, with an enhancement factor of 0.1. The K _g grid consists of a regular number of k -points, and the total number of k -points is 5 times larger than the size of K _G .

10.

Geometry of Carbon Oxide on MgO adsorption for 100% coverage case.

4.5.1. Convergence with the Size of Primary and Auxiliary Basis

Using PADF, the auxiliary basis used to expand the AO products in eq significantly affects the accuracy of correlation energies, especially for larger molecules and basis sets, and their convergence with respect to this parameter is decisive for reliable results. In Table , we show the RPA contribution to the counterpoise-corrected adsorption energy of CO on a 2-layer MgO slab at 100% coverage for the DZP, TZ2P, and QZ4P basis sets for auxiliary fit sets of variable size. For this test, we employ a 9 × 9 k -grid. The threshold parameter for the projector method () is ϵ _d = 10^–3.

3. Total RPA@PBE Correlation Energies Per Unit Cell and Contributions to Adsorption Energy in kcal mol^–1 for 2 MgO Layers and 100% Coverage for Different Basis Sets and Auxiliary Basis Sets Using a 9 × 9 k -Grid .

		N bas	N aux	E c (MgO-CO)	E c (CO)	E c (MgO)	ΔE c
DZP	100	T1	873	–1033.20	–316.23	–710.36	–6.61
		T2	1477	–1033.21	–316.23	–710.38	–6.60
TZ2P	172	T1	873	–1219.42	–374.14	–837.54	–7.75
		T2	1477	–1218.62	–374.07	–836.87	–7.67
		T3	2956	–1217.88	–374.06	–836.29	–7.53
		T4	4070	–1220.08	–374.19	–838.31	–7.53
QZ4P	273	T1	873	–1439.00	–439.05	–988.74	–11.22
		T2	1477	–1431.92	–438.39	–983.11	–10.42
		T3	2956	–1419.13	–437.99	–972.57	–8.56
		T4	4070	–1418.40	–437.98	–972.01	–8.36

^a N _bas denotes the number of primary basis functions remaining after applying the projector method.

Different auxiliary basis sets are denoted as T1 to T4. Their generation has been described in ref . Importantly, they are not automatically generated from the primary basis. T1 contains auxiliary basis functions up to l _max = 4, T2 and T3 contain auxiliary basis functions up to l _max = 6 (As also reflected by the numbers of auxiliary basis functions in each system shown in Table , T3 is composed of a much larger number of fit functions for each angular momentum), and T4 contains additional basis functions with l _max = 7. Even though the maximum angular momentum in our primary basis sets is l _max = 3, angular momenta larger than 2l _max in the auxiliary basis can be important. , For the DZP basis set, even absolute correlation energies are already converged with the T1 auxiliary basis. The TZ2P calculations show a larger dependence on the auxiliary basis, and the T3 set is necessary for the convergence of the correlation energy difference. For the QZ4P basis set, correlation energies are not converged with T1 and T2, and even the T4 auxiliary basis changes the relative correlation energy by 0.2 kcal mol^–1 compared to T3. The following DZP and TZ2P calculations are all performed with the T2 fit set, and the TZ2P results are adjusted for the resulting fit error of 0.14 kcal mol^–1 in the RPA contribution to the adsorption energy.

After assessing the influence of the auxiliary basis, we also quantify the impact of the threshold for the projector method. We do this here for the TZ2P basis set and the T3 auxiliary basis. The results in Table reveal a pronounced sensitivity of absolute RPA correlation energies to this value. This is expected, since a smaller number of N _mod translates into a larger primary basis. The contribution to the adsorption energy is relatively stable with respect to this parameter, but increases for ϵ _d = 1 × 10^–4. This can be fixed using a larger auxiliary basis set. Repeating the same calculation with the T4 auxiliary basis set again reduces ΔE _c to −7.50 kcal mol^–1. This demonstrates that the convergence of a calculation with respect to the size of the auxiliary basis is heavily influenced by the threshold used for the projector method. In all of the following calculations, we will use ϵ _d = 1 × 10^–3.

4. Total RPA@PBE Correlation Energies Per Unit Cell and Contributions to Adsorption Energy in kcal mol^–1 for 2 MgO Layers and 100% Coverage Calculated with the TZ2P Basis Set and T3 Auxiliary Basis Set .

ϵ d	N mod	E c (MgO–CO)	E c (CO)	E c (MgO)	ΔE c
1 × 10–3	12	–1217.88	–374.06	–836.29	–7.53
5 × 10–4	8	–1227.49	–375.01	–844.97	–7.51
1 × 10–4	4	–1234.80	–377.40	–850.05	–7.35

^a N _mod denotes the number of primary basis functions that have been projected out, and ϵ _d is the value of the threshold for the projector method.

Using the numbers that are converged with respect to the auxiliary basis set, we extrapolate the RPA contribution to the adsorption energy to the complete basis set (CBS) limit. We assume a linear dependence of the basis set error on the inverse number of basis functions, as often done in plane wave calculations. As shown in Figure , the linear fit including all three basis sets (DTQ) captures the trend in the calculated values quite well. From this fit, we obtain an extrapolated RPA contribution to the adsorption energy of −9.25 ± 0.29 kcal mol^–1, which is about 1 kcal mol^–1 lower than the QZ result. We assume the standard fit error (intercept) of 0.29 kcal mol^–1 to represent the uncertainty of our extrapolation.

11.

Inverse linear fit of the RPA contribution to the adsorption energy in kcal mol^–1 of CO on a 2-layer MgO slab against the size of the single particle basis. The red area is the confidence interval of the fit.

4.6. K _G /Q _G Convergence

Figure shows the convergence with respect to the number of k -points of the RPA contribution to the adsorption energy of CO on a 2-layer MgO slab at 100% coverage for the DZP and TZ2P basis sets. As for the previous calculations involving the monolayer MgO slab, the convergence with the size of the k -grid is rapid, and independent of the employed basis set. For both basis sets, convergence within 0.02 kcal mol^–1 is reached already for the 9 × 9 grid. In practical calculations, the independence of the convergence rate with respect to the thermodynamic limit is particularly useful, as it allows for converging the RPA correlation energies with respect to the k -grid in a small basis set, and subsequently to converge them with respect to the single-particle basis size using a coarser k -grid.

12.

RPA@PBE contribution to adsorption energy in kcal mol^–1 for 2 MgO layers and 100% coverage for the DZP and TZ2P basis sets with different k -grids. The shaded areas highlight the regions where the correlation energy is converged within 0.05 kcal mol^–1.

4.6.1. Number of Layers

Next, we evaluate the RPA contribution to the adsorption energy as a function of the number of MgO layers, using both the DZP basis set. As shown in Table S13 in the Supporting Information, this contribution consistently remains at −6.60 kcal mol^–1, regardless of the number of layers, with the same behavior observed for the TZ2P basis set. This is in agreement with previous correlated calculations for the same system, where convergence of the MP2 contribution to the adsorption energy has been obtained already with 2 MgO layers as well.

4.6.2. Convergence with Respect to Coverage

Finally, we investigate the convergence of the adsorption energy with respect to the coverage of the Mg atoms, using a slab geometry with two layers. To this end, we calculate the CO-MgO adsorption energies for the cases of 100, 50, and 25% coverage, and show the results in Table . We obtain an adsorption energy of −0.84 kcal mol^–1 at 100% coverage. In the Supporting Information (S.3.2.2), we also show that the differences between the 100 and 50% coverage cases are exactly the same for the DZP and TZ2P basis sets, from which we conclude that the change of the adsorption energy with respect to the coverage is basis set independent. We therefore apply a correction evaluated with the DZP basis set to the TZ2P result at 50% coverage to extrapolate it to the 25% coverage case. The adsorption energies for the 25 and 50% cases are relatively close, suggesting that the convergence with respect to the CO saturation is achieved already at the 50% case. For this reason, we decided to use the average between the interaction energies obtained for the 50 and 25% coverages as the error bar for the low coverage limit, amounting to

$${ϵ}_{\mathrm{cov}}=\frac{E_{50\%}-E_{25\%}}{2}=0.12{\mathrm{kcal}\mathrm{mol}}^{-1}$$ 31

while the low-coverage limit is assumed to be reached at 25% coverage, obtaining the coverage correction as

$${\mathrm{\Delta }}_{\mathrm{cov}}=E_{100\%}^{2l}-E_{25\%}^{2l}=0.92{\mathrm{kcal}\mathrm{mol}}^{-1}$$ 32

5. RPA@PBE Interaction Energies of CO with a 2-Layer MgO Slab for Different k Meshes and Coverages in kcal mol^–1 Using TZ2P and DZP Basis Sets .

	$\sqrt[\dim ]{n_k}$
coverage	5	7	9	11
100	–1.10	–0.90	–0.85	–0.84
50			0.33	0.32
25*	0.07	0.08

^aThe TZ2P result for 25% coverage has been extrapolated by adding the difference between the TZ2P and DZP result for 50% and 25% coverage to the TZ2P result at 25% coverage, as shown in the supporting information in section S.3.2.2.

4.6.3. Final Estimate and Discussion

To obtain the final adsorption energy, we combine the CBS-limit extrapolated RPA contribution from Figure with the PBE and the HF@PBE contributions. We calculated these numbers using 4 MgO layers and the QZ4P basis set, which ensures convergence with respect to both parameters, and we obtain (See Table S14 for the raw data)

$$\begin{gathered} E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{100\%}={E_c^{\mathrm{R}\mathrm{P}\mathrm{A}}}^{\mathrm{CBS}}+E_{\mathrm{H}\mathrm{F}}^{\mathrm{QZ}4\mathrm{P}}+E_{\mathrm{P}\mathrm{B}\mathrm{E}}^{\mathrm{QZ}4\mathrm{P}}-E_{\mathrm{XC}@\mathrm{P}\mathrm{B}\mathrm{E}}^{\mathrm{QZ}4\mathrm{P}} \\ =-9.25{\mathrm{kcal}\mathrm{mol}}^{-1}-3.01{\mathrm{kcal}\mathrm{mol}}^{-1}-3.49{\mathrm{kcal}\mathrm{mol}}^{-1}+13.16{\mathrm{kcal}\mathrm{mol}}^{-1} \\ =-2.59{\mathrm{kcal}\mathrm{mol}}^{-1} \end{gathered}$$ 33

The basis set extrapolation error, which we estimated above to be 0.29 kcal mol^–1, is the only source of uncertainty in this number. To give the final estimate of the MgO(001)-CO adsorption energy, we use as a reference the RPA@PBE/BSSE adsorption energy at 100% coverage. As mentioned, we apply the constant shift evaluated in eq to correct for the low coverage limit. Since we consider errors arising from the number of layers and the reciprocal space convergence negligible, we combine the uncertainties from the extrapolation to the CBS error and of the coverage correction to estimate the total uncertainty as $e=\sqrt{{ϵ}_{\mathrm{bas}}^2+{ϵ}_{\mathrm{cov}}^2}=0.31{\mathrm{kcal}\mathrm{mol}}^{-1}$ . The final estimate for the RPA@PBE adsorption energy of CO on the MgO surface in the low coverage limit becomes

$$E_{\mathrm{ads}}^{\mathrm{CO}-\mathrm{MgO}}\approx E_{\mathrm{ads}}^{100\%}+{\mathrm{\Delta }}_{\mathrm{cov}}\pm e=-1.67\pm 0.31{\mathrm{kcal}\mathrm{mol}}^{-1}$$ 34

This result is in excellent agreement with the periodic RPA@PBE calculations by Bajdich et al. who obtained an adsorption energy of −1.66 kcal mol^–1 for the same system. Our calculations confirm that RPA@PBE overestimates both the experimental adsorption energy of −4.57 kcal mol^–1, and recent (embedded) CCSD­(T) results obtained by different authors. ,, This is expected since RPA@PBE is known to overestimate (underestimate the magnitude of) noncovalent binding energies. ,, We assume that better agreement can be reached by evaluating the RPA adsorption energy with orbitals obtained from a hybrid DFT calculation. Using a hybrid scheme where the RPA correlation energy is obtained with exact Fock exchange from a self-consistent HF calculation, Bajdich et al. obtained an adsorption energy of −7.14 kcal mol^–1. It is further known that RPA@PBE0 gives more negative noncovalent molecular interaction energies than RPA@PBE. Both findings suggest that RPA@PBE0 will yield more negative energies for the adsorption of small molecules on transition metal surfaces than RPA@PBE. Such calculations require the fully self-consistent evaluation of periodic HF exchange. We are planning to report the results of such calculations shortly.

5. Conclusions

In this work, we reported the implementation of the RPA correlation with localized atomic orbitals and pair-atomic density fitting in the BAND module of AMS. We have introduced an algorithm, inspired by the staggered mesh method of Lin and co-workers , which improves the treatment of the Γ-point divergence. This allowed us to achieve fast and reliable convergence to the infinite k -grid limit. A complete account of this algorithm and a detailed benchmarks will be presented in future work. By distributing the evaluation of the most computationally involved steps of the RPA energy evaluation over multiple ScaLapack contexts for all possible pairs of q and ω, our code achieves almost perfect parallel efficiency.

We have demonstrated the efficacy of our implementation through an application to the adsorption of CO on MgO(001). After careful extrapolation to the complete basis set and infinite k -grid limit we obtain a final adsorption energy of −1.67 ± 0.31 kcal mol^–1 in the low-coverage limit, in excellent agreement with a previous calculation. As could be expected, ,, this value underestimates both the experimental adsorption energy of −4.57 kcal mol^–1, and recent (embedded) CCSD­(T) results. ,, The remaining observed discrepancies can possibly be addressed by evaluating the RPA correlation energies with orbitals obtained from a hybrid DFT calculation. Another promising option to improve over the RPA would be to extend our current algorithm to σ-functionals, − which can be achieved through minor modifications at the same computational cost. Also renormalized adiabatic xc kernel methods have been shown to provide more accurate results than the RPA at comparable computational cost. −

The developments presented here lay the groundwork for further extensions. In particular, we currently focus on extending our algorithm to metallic systems, which would enable the calculation of accurate energy profiles for the dissociative chemisorption of small molecules on transition metal surfaces. This process is decisive in heterogeneous catalysis, but difficult to model with (semi)­local density functionals or even advanced embedded wave function methods. , On the other hand, the RPA has recently been shown to give accurate reaction barrier heights within chemical accuracy for the challenging cases of H₂+Cu­(111) , and H₂+Al­(110). The widespread availability of efficient RPA implementations with localized AOs could pave the way for more applications of the RPA to such systems in the future. , Another important application would be the calculation of potential energy surfaces for the adsorption of graphene on metal surfaces, which allows for the tuning of its band gap with potential applications in catalysis. The competition between chemi- and physisorption induced by the interplay of covalent and dispersive interactions between both surfaces is well described by the RPA ,, but less so by most van der Waals density functionals.

Furthermore, we are currently working on evaluating the RPA polarizability in imaginary time. , Combined with PADF, this should lead to an RPA algorithm that scales cubically in the number of atoms ,,, and only linearly in the number of k -points. , This could be important for applications in heterogeneous catalysis requiring large unit cells. Together, these advancements should position the RPA as a versatile and accurate tool for computational materials science, offering a pathway for tackling increasingly challenging problems in heterogeneous catalysis.

Appendix A RPA Equations in a Basis

In this appendix, we discuss how the expression for the RPA correlation energy can be expressed in a nonorthogonal auxiliary basis set. Given an operator A(r,r′) in a L ²-Hilbert space, we may write

$$\begin{array}{ll}A(r,r^{\prime })=\sum _{\alpha \beta \gamma \delta }{S}_{\alpha \gamma }^{-1}⟨f_{\gamma }(r)|Â|f_{\delta }(r)⟩S_{\delta \beta }^{-1}{f}_{\alpha }(r)f_{\beta }^{*}(r^{\prime })& (35)\\ =\sum _{\alpha \beta }{A}_{\alpha \beta }{f}_{\alpha }(r)f_{\beta }^{*}(r^{\prime })& (36)\end{array}$$

Where we have defined

$$S_{\alpha \gamma }={\int }_{{\mathbb{R}}^3}{f}_{\alpha }^{*}(r)f_{\gamma }(r)\mathrm{d}r$$ 37

and

$${Ã}_{\alpha \gamma }=⟨f_{\alpha }|Â|f_{\gamma }⟩={\int }_{{\mathbb{R}}^3}{f}_{\alpha }^{*}(r)A(r,r^{\prime })f_{\gamma }(r^{\prime })\mathrm{d}r\mathrm{d}{r}^{\prime }$$ 38

The set of functions f _α ∈F is assumed to be a basis of the Hilbert space in which A(r,r′) is defined. We then obtain

$$\begin{array}{l}Z(r,r^{″},\omega )={\int }_{{\mathbb{R}}^3}{P}^0(r,r^{\prime },\omega )v(r^{\prime },r^{″})\mathrm{d}{r}^{\prime }\\ ={\int }_{{\mathbb{R}}^3}\mathrm{d}{r}^{\prime }\sum _{\alpha \gamma }{P}_{\alpha \gamma }^0(\omega )f_{\alpha }(r)f_{\gamma }^{*}(r^{\prime })\sum _{\delta \beta }{v}_{\delta \beta }{f}_{\delta }(r^{\prime })f_{\beta }^{*}(r^{″})\\ =\sum _{\alpha \gamma }\sum _{\delta \beta }{P}_{\alpha \gamma }^0(\omega )S_{\gamma \delta }{v}_{\delta \beta }{f}_{\alpha }(r)f_{\beta }^{*}(r^{″})\\ =\sum _{\alpha \beta }{[P^0(\omega )\mathit{Sv}]}_{\alpha \beta }{f}_{\alpha }(r)f_{\beta }^{*}(r^{″})\\ =\sum _{\alpha \beta }{[P^0(\omega )S{S}^{-1}ṽS^{-1}]}_{\alpha \beta }{f}_{\alpha }(r)f_{\beta }^{*}(r^{″})\\ =\sum _{\alpha \beta }{[P^0(\omega )ṽS^{-1}]}_{\alpha \beta }{f}_{\alpha }(r)f_{\beta }^{*}(r^{″})\end{array}$$ 39

for the product of polarizability and Coulomb potential. The trace becomes

$$\begin{array}{l}\mathrm{Tr}[Z(r,r^{\prime },\omega )]={\int }_{{\mathbb{R}}^3}{\int }_{{\mathbb{R}}^3}\mathrm{d}{r}^{\prime }\mathrm{d}r\sum _{\alpha \beta }{Z}_{\alpha \beta }{f}_{\beta }^{*}(r^{\prime })f_{\alpha }(r)\delta (r,r^{\prime })\\ ={\int }_{{\mathbb{R}}^3}{\int }_{{\mathbb{R}}^3}\mathrm{d}{r}^{\prime }\mathrm{d}r\sum _{\alpha \beta }{[P^0ṽS^{-1}]}_{\alpha \beta }{f}_{\beta }^{*}(r^{\prime })f_{\alpha }(r)\delta (r,r^{\prime })\\ =\sum _{\alpha \beta }{[P^0ṽS^{-1}]}_{\alpha \beta }{S}_{\beta \alpha }\\ =\mathrm{Tr}[{\mathbf{P}}^0ṽ]\\ =\mathrm{Tr}[{\mathbf{ṽ}}^{1/2}{P}^0{ṽ}^{1/2}]\end{array}$$ 40

We can notice that a similar cancellation of the overlap matrix S also occurs in the power series term,

$$\frac{1}{n}\mathrm{Tr}(Z{(r,r,\omega )}^{\prime n})=\frac{1}{n}\mathrm{Tr}({[{\mathbf{P}}^0ṽ]}^n)$$ 41

The RPA equation then becomes

$$\begin{array}{l}{E}_{\mathrm{corr}}^{\mathrm{RPA}}=-\frac{1}{2\pi }{\int }_0^{\infty }\mathrm{d}\omega \mathrm{Tr}\left[\sum _{n=1}\frac{1}{n}[Z(r,r^{\prime }){]}^n-Z(r,r^{\prime })\right]\\ =-\frac{1}{2\pi }{\int }_0^{\infty }\mathrm{d}\omega [\sum _{n=1}\frac{1}{n}\mathrm{Tr}({[{\mathbf{P}}^0ṽ]}^n)-\mathrm{Tr}({\mathbf{P}}^{0}v)]\\ =-\frac{1}{2\pi }{\int }_0^{\infty }\mathrm{d}\omega [\log (\det (\mathbf{1}-\mathbf{Z}))-\mathrm{Tr}(\mathbf{Z})]\end{array}$$ 42

where the matrix Z, by using the cyclic property of the trace operation is defined as $\mathbf{Z}={\mathbf{ṽ}}^{1/2}{P}_0{ṽ}^{1/2}$ . We remark that the discussion presented here assumes the basis F to be complete when representing both the Coulomb potential and the polarizability. In practice, however, this can not be achieved and an error deriving from the incompleteness will always be present.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.5c00751.

• Additional benchmarks for molecular chains and noble gas crystals, and all energies for MgO–CO (PDF)

The authors declare no competing financial interest.