Applying the Coupled-Cluster Ansatz to Solids and Surfaces in the Thermodynamic Limit

PHYSICAL REVIEW X 8, 021043 (2018)

Thomas Gruber, Ke Liao, and Theodoros Tsatsoulis Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany

Felix Hummel Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria

Andreas Grüneis* Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria and Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany

(Received 2 November 2017; revised manuscript received 19 February 2018; published 14 May 2018)

Modern electronic structure theories can predict and simulate a wealth of phenomena in surface science and solid-state physics. In order to allow for a direct comparison with experiment, such ab initio predictions have to be made in the thermodynamic limit, substantially increasing the computational cost of many- electron wave-function theories. Here, we present a method that achieves thermodynamic limit results for solids and surfaces using the “gold standard” coupled cluster ansatz of quantum chemistry with unprecedented efficiency. We study the energy difference between carbon diamond and graphite crystals, adsorption energies of water on h-BN, as well as the cohesive energy of the Ne solid, demonstrating the increased efficiency and accuracy of coupled cluster theory for solids and surfaces.

DOI: 10.1103/PhysRevX.8.021043 Subject Areas: Chemical Physics, Computational Physics, Condensed Matter Physics

I. INTRODUCTION periodic condensed-matter systems with high accuracy [3–13]. Modern ab initio methods to solve the electronic Electronic correlation is, for the most part, a short-ranged Schrödinger equation for real solids and molecules such phenomenon. The proper description of the wave-function as density functional theory or wave-function-based meth- shape at short interelectronic distances allows for capturing ods are becoming increasingly accurate and efficient [1–4]. the largest fraction of the correlation energy in solids However, in contrast to molecular systems, properties of [2,14]. Significant progress has been achieved for many- solids and surfaces need to be calculated in the thermo- electron wave-function-based theories by exploiting the dynamic limit. The convergence towards the thermody- locality of electronic correlation in large molecules and namic limit with respect to the number of particles is very solids. The development of so-called local correlation slow, often exceeding the computational resources of even methods and embedding theories has improved their modern supercomputers. This is particularly the case for computational efficiency considerably [15–24]. However, many-electron wave-function-based theories that allow for theories that approximate long-range correlation effects a systematic improvability upon the description of the such as van der Waals interactions must carefully be electronic correlation energy. Nonetheless, these methods checked for convergence with respect to the employed are becoming increasingly popular in theoretical physics as cutoff parameters to allow for accurate and predictive well as in chemistry to treat electronic correlation in ab initio studies of real materials. This is of particular importance in condensed-matter systems where the accu- mulation of weak van der Waals interactions can become a *[email protected] non-negligible contribution to the property of interest, as, for example, in the case of the energy difference between Published by the American Physical Society under the terms of carbon diamond and graphite or the adsorption of a water the Creative Commons Attribution 4.0 International license. h Further distribution of this work must maintain attribution to molecule on an -BN sheet. Pairwise additive interatomic the author(s) and the published article’s title, journal citation, van der Waals interactions cause a 1=N convergence of the and DOI. electronic correlation energy per unit cell in insulating

2160-3308=18=8(2)=021043(8) 021043-1 Published by the American Physical Society THOMAS GRUBER et al. PHYS. REV. X 8, 021043 (2018) three-dimensional systems, where N is the number of In the above equation, G corresponds to a plane-wave explicitly correlated atoms. However, in general, the exact vector that is defined as G ¼ g þ Δk, where g is a form of these scaling laws depends on the dimensionality reciprocal lattice vector and Δk is the difference between and electronic response properties of the system; e.g., the any two Bloch wave vectors that are conventionally chosen adsorption energy of molecules on two-dimensional insu- to sample the first Brillouin zone. Note that vðGÞ is the lating surfaces exhibits a 1=N2 convergence. Moreover, we Coulomb kernel in reciprocal space that diverges at G ¼ 0, note that collective phenomena such as plasmons in making it numerically necessary to disregard this contri- metallic systems can also modify the observed scaling bution to the sum, as indicated by the apostrophe. Thus, laws [25]. For these reasons, robust and reliable approx- SðGÞ is the partial functional derivative of the correlation imations to long-range correlation effects are nontrivial. energy with respect to vðGÞ, and we will return to its Many-body methods such as coupled cluster or con- explicit definition later, as well as in Ref. [28] (see also figuration interaction theory can describe both long- and Ref. [27]). short-ranged electronic correlation effects with high accu- The thermodynamic limit is approached as N → ∞, racy. However, the scaling of the computational complexity where N is the number of particles in the simulation cell of these theories with respect to system size is either of a while the density is kept constant. Finite-size errors are high-order polynomial or an even exponential form. defined as the difference between the thermodynamic limit Therefore, it is difficult to treat long-range correlation and the finite simulation cell results. For electronic corre- effects in a computationally efficient manner using these lation energies obtained using many-electron perturbation 1=N theories. This has led to the development of various theories, these errors typically decay as as a conse- techniques that partition the correlation problem according quence of long-range interatomicP van der Waals forces.R In to a predefined criterion such as the distance between the thermodynamic limit, G of Eq. (1) is replaced by G. electron pairs or fragment size. Local theories employ Therefore, finite-size errors in the correlation energy of correlation energy expressions that depend on localized periodic systems originate from two sources [29]: (i) quad- electron pairs, making it possible to treat long-distance rature errors in the summation over G, and (ii) the slow Sð Þ pairs using computationally more efficient yet less accurate convergence of G with respect to the employed super- k theories. Embedding theories typically aim at combining cell size or -point mesh. In the following, we discuss how the computational efficiency of mean field theories for the to reduce both errors substantially. long range, with the high accuracy of wave-function-based We first seek to discuss finite-size errors originating from the quadrature in the summation over G. These contribu- methods applied to small fragments only. In this work, we ¼ 0 introduce an efficient method that seamlessly integrates tions can be partitioned into the G volume element long-range correlation effects for solids without any pre- contribution and the remaining terms. We note that, as a result of the Coulomb divergence, the integrable contribu- defined criteria such as cutoff distance or fragment size. tion of Sð0Þvð0Þ to the correlation energy is usually Our approach is inspired by structure factor interpolation neglected in computer implementations of Eq. (1) [11,27]. techniques as performed in the field of quantum However, this is the dominant contribution to the finite-size Monte Carlo theory [26]. However, in coupled cluster error of the correlation energy of insulators. A Taylor theory, the structure factor, being the functional derivative expansion of SðGÞ around G ¼ 0 shows that SðGÞ exhibits of the total energy with respect to the Coulomb kernel, is a quadratic behavior close to zero, explaining the 1=N not directly available. Instead, we seek to interpolate the decay of the finite-size error for three-dimensional insula- partial functional derivative of the coupled cluster corre- tors [27]. An estimate of Sð0Þvð0Þ can be obtained by lation energy expression with respect to the Coulomb spherically averaging SðGÞ and interpolating around kernel. The interpolation scheme is chosen such that it is G ¼ 0. Subsequently, the interpolated function is multi- directly transferable to systems with arbitrary dimensions plied by the analytic Coulomb kernel and integrated over a including solids and surfaces. Because of the adverse sphere around G ¼ 0, yielding an estimate of Sð0Þvð0Þ scaling of the computational complexity in coupled cluster [27]. However, this approach is not well defined for theories, the proposed method allows for reducing the anisotropic systems because it requires a spherical cutoff computational cost by several orders of magnitude without parameter. In this work, we propose to interpolate SðGÞ compromising accuracy compared to previous studies [4]. using a tricubic interpolation without spherical averaging. Once obtained, the interpolation of SðGÞ and the analytic II. THEORY expression for the Coulomb kernel allows for integrating over G on a very fine grid, simulating the thermodynamic The electronic correlation energy can be calculated in a limit integration. This approach accounts for the Sð0Þvð0Þ plane-wave basis set using the following expression [27]: contribution to the correlation energy and reduces quadrature errors originating from too coarse a Brillouin zone X 0 sampling. We refer to coupled cluster correlation energies Ec ¼hΨ0jH − E0jΨi¼ vðGÞSðGÞ: ð1Þ G obtained using this interpolation strategy as CC-FS.

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introduces quadrature errors that cause the slow convergence of SðGÞ towards the thermodynamic limit. In the case of semiconductors or metals, these errors can become significant because 1=ðϵi þ ϵj − ϵa − ϵbÞ varies strongly depending on ki, kj, ka, and kb. In particular, materials with a Dirac cone at the Fermi surface such as graphene exhibit a large variation of the denominator between zero and several eV depending on k. As a result, Eq. (2) needs to be calculated using a finer k-point mesh to reduce quadrature errors. In this work, we show that the above quadrature errors can be substantially reduced by calculat- ing and averaging SðGÞ for a set of shifted k-point meshes. Note that the vectors G are not affected by shifting the employed k-mesh because G depends only on the difference between any two Bloch wave vectors Δk. We replace SðGÞ in Eq. (1) with an average obtained for Nt different k FIG. 1. Partial derivative of correlation energy with respect to meshes shifted from Γ by ti such that the Coulomb kernel for graphite. Slices and isosurface of SðGÞ for carbon graphite in the ABC stacking. Darker colors indicate 1 XNt S¯ðGÞ¼ S ðGÞ: ð3Þ more negative values. White corresponds to zero. The blue N ti isosurface is a hexagonally shaped torus, and it reflects the t i¼1 anisotropic shape for SðGÞ. Black dots represent the sampling points using a 4 × 4 × 4 k-point mesh. The shifts ti are chosen such that they sample the first Brillouin zone uniformly. Coupled cluster theory calculations for different shifts can be performed independently To illustrate the importance of the interpolation method, from each other, and the computational complexity scales we consider the following example. Figure 1 shows slices N Sð Þ only linearly with respect to t. Coupled cluster theory and an isosurface of the interpolated G for carbon energies that have been obtained using this twist-averaging graphite in the ABC stacking. Black dots indicate sampling Sð Þ technique are referred to as CC-TA or CC-TA-FS if the points of G obtained using coupled cluster singles and interpolation method has been employed as well. 4 4 4 k doubles theory and a × × -point mesh. This figure We note that quantum Monte Carlo (QMC) methods Sð Þ ¼ 0 illustrates that G is very anisotropic around G . employ finite-size corrections that share similarities with 4 4 4 k Furthermore, we show that even a × × -point mesh the methods outlined above [29–31]. However, QMC sampling, indicated by the black dots, corresponds to a methods such as diffusion Monte Carlo are real-space relatively coarse grid, causing non-negligible quadrature theories that provide estimates of total energies rather than errors. We return to the discussion of the results for the partitioning the energy into a Hartree-Fock and an elec- correlation energy later. tronic correlation contribution. An advantage of the parti- We now turn to the discussion of finite-size errors in tioning ansatz is that Hartree-Fock energy contributions can semiconductors and metals. We stress that small gap be converged to the thermodynamic limit independently systems suffer from a relatively slow convergence of from the correlation energy at little extra computational Sð Þ G with respect to the studied system size. This behavior cost. Consequently, finite-size corrections are only required Sð Þ can be understood by considering the definition of G in for the comparatively smaller correlation energy contribu- second-order Møller-Plesset perturbation (MP2) theory, tions. In passing, we note that auxiliary-field quantum Monte Carlo theory employs finite-size corrections that are X X Γabð Þ ij G based on parametrized density functionals obtained from SðGÞ¼ ; ð2Þ ϵi þ ϵj − ϵa − ϵb finite, uniform, electron gas simulation cells [32]. Such ki;kj;ka ni;nj;na;nb corrections work for solids but have not yet been applied to surfaces or molecular crystals, where they are expected to where ϵi correspond to one-electron energies usually be less accurate. obtained from Hartree-Fock theory. The indices i, j and a, b label occupied and virtual orbitals, respectively, and are understood to be a shorthand for the Bloch wave vector III. RESULTS ki and a band index ni. Because of momentum conserva- We now turn to the discussion of the results obtained tion, kb can be calculated from the other Bloch wave using the methods outlined above. The present computa- ab vectors in the above equation. Note that Γij ðGÞ is defined tions were performed using the VASP code [33,34] and the in Ref. [28]. The summation over Bloch vectors in Eq. (2) projector augmented-wave method [35]. The coupled

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FIG. 2. Convergence of the energy difference between the carbon diamond and graphite using CCSD and 2 atomic unit cells with respect to the number of natural orbitals used. The (T) correlation energy contribution has been added to the CCSD energy using 16 natural orbitals per atom. cluster theory calculations were partly performed using the newly developed cc4s code [36] interfaced with VASP and employing the automated tensor contraction engine CTF [37]. More technical details are outlined in Ref. [28]. As a first application, we investigate the carbon diamond FIG. 3. Energy difference between carbon diamond and graph- and graphite crystals. Before discussing the thermody- ite phases. The dotted line represents the linear fit of the namic limit convergence, we seek to address the conver- uncorrected values with shifts. (T) corrections (black) are on 4 4 4 k gence of the calculated correlation energy differences with top of CCSD-TA-FS energy obtained using a × × -point mesh (red). Zero-point energies are included, and they stabilize respect to the employed orbital basis. We employ MP2 graphite compared to diamond by 9 meV=atom. natural orbitals that are obtained using a procedure outlined in Ref. [38]. Figure 2 shows the convergence of coupled cluster singles and doubles (CCSD) and perturbative triples rapidly convergent correlation energies for both carbon (T) correlation energy differences with respect to the diamond and graphite. CCSD-TA-FS correlation energies number of bands using a 2 × 2 × 2 k-point mesh, respec- obtained using a 2 × 2 × 2 k mesh only deviate from tively. We find that calculations using 16 orbitals per carbon extrapolated CCSD thermodynamic limit energies by atom yield an energy difference that agrees to within approximately 60 meV=atom. We note that correlation 4 meV=atom compared to results obtained using 40 natural energies obtained using the same k mesh and CCSD-TA orbitals per atom. The (T) correction to CCSD converges theory exhibit finite-size errors on the scale of even faster with respect to the number of orbitals and is 200–300 meV=atom. The CCSD(T) correlation energies fortuitously close to zero in the case of the 2 × 2 × 2 k- are obtained using twist averaging for the (T) contribution point mesh. We stress that natural orbitals allow for a and adding the correction to the CCSD-TA-FS result significantly more systematic truncatability and improved obtained using a 4 × 4 × 4 k-point mesh. This allows for basis-set incompleteness error cancellation between differ- investigating the finite-size errors of the (T) correction ent systems compared to virtual Hartree-Fock or density independently from finite-size errors of CCSD theory. We functional theory orbitals. Achieving the same level of find that (T) converges rapidly with respect to the employed accuracy requires several hundred virtual Hartree-Fock k-mesh size, reflecting its short-ranged nature. orbitals per atom. Convergence with respect to other The bottom panel in Fig. 3 shows the differences of the computational parameters has been checked and is dis- total energies of both carbon allotropes, including zero- cussed in Ref. [28]. point corrections retrieved as a function of the employed k- We now turn to the discussion of finite-size errors in total point mesh. Results obtained using CCSD theory without correlation energies. The top and middle panels in Fig. 3 finite-size corrections are depicted by the blue line and show CCSD correlation energies retrieved as a function of oscillate strongly with increasing k-point mesh density. By the number of k points of graphite and diamond, respec- using too-coarse k-point meshes, we can predict graphite to tively. We note that twist averaging (TA) is necessary for be more stable than diamond, whereas denser k-point CCSD correlation energies to achieve a smooth 1=N meshes predict diamond to be the more stable allotrope. convergence to the thermodynamic limit, in particular, Employing the averaging over different shifts yields for graphite. Accounting for quadrature errors by means CCSD-TA results that converge significantly smoother of the tricubic interpolation method (CCSD-TA-FS) yields with increasing k-point mesh density as shown by the

021043-4 APPLYING THE COUPLED-CLUSTER ANSATZ TO SOLIDS … PHYS. REV. X 8, 021043 (2018) green line. Furthermore, performing the newly proposed results obtained using the incremental method to within tricubic interpolation and integration in addition to the twist 3 meV=atom [6,10]. Furthermore, CCSD(T) predicts a averaging referred to as CCSD-TA-FS yields rapidly cohesive energy of 30 meV=atom, which is in good convergent energy differences shown by the red line. We agreement with experimental estimates of 27 meV=atom note that CCSD-TA-FS using a 2 × 2 × 2 k-point mesh is as corrected for zero-point fluctuations [6]. close to the thermodynamic limit as CCSD-TA using a As a final demonstration of the applicability of the 4 × 4 × 4 k-point mesh. Since the computational complex- proposed method to reach the thermodynamic limit, we 4 ity scales at least as OðNkÞ with respect to the number of k study the adsorption energy of a single water molecule on points, this corresponds to a reduction in the computational an h-BN sheet. The same system has recently been studied cost by 3 orders of magnitude. From these calculations, we using diffusion Monte Carlo (DMC), the random-phase conclude that CCSD theory predicts diamond to be more approximation (RPA), and dispersion functionals [40,41], stable than graphite by 22 meV=atom, including zero-point as well as molecular MP2 [42] and periodic coupled cluster energies. We have also performed perturbative triples theory [43], demonstrating the need for reliable methods calculations and added the corresponding correlation that can account for long-range van der Waals interactions energy correction to our CCSD findings. CCSD(T) theory and also to provide benchmark data. Furthermore, the predicts graphite to be slightly less stable than diamond by recent work of Al-Hamdani et al. [40] illustrates the 7 meV=atom, including zero-point corrections. We note importance of long-range correlation effects that account that our CCSD(T) results agree with experimental findings for approximately 25% of the reference adsorption energy to within the observed precision of CCSD(T) for similar computed in a (4 × 4) unit cell of h-BN. Figure 4 shows applications, which is generally better than 1 kcal=mol calculated adsorption energies at the level of RPA (43 meV=atom). The difference in the experimental Gibbs plus second-order screened exchange, MP2, CCSD, and free energy of carbon diamond and graphite at room CCSD(T) theories retrieved as a function of the number of temperature has been reported to be 25 meV=atom [39], atoms in the h-BN sheet. Convergence with respect to other predicting graphite to be more stable than diamond. computational parameters has been checked and is dis- Having demonstrated the ability of the proposed method cussed in Ref. [28]. Using MP2 theory, it is possible to to correct for finite-size errors on the scale of about study very large systems [40], and we find that the MP2 100 meV=atom, we now study the thermodynamic limit adsorption energy converges slowly to a thermodynamic of the cohesive energy of the weakly bound neon solid. In limit value of 119 meV. We note that finite-size errors this case, we need to correct for finite-size errors on the for adsorption energies on two-dimensional insulators are 2 scale of a few meV/atom. The dominant contribution to the expected to decay as 1=N , which is the predicted scaling attractive long-range interatomic interaction of neon atoms from pairwise additive van der Waals interactions [40]. originates from van der Waals forces. Table I summarizes Applying the proposed finite-size correction to MP2 theory MP2, CCSD, and CCSD(T) cohesive energies obtained for the (4 × 4) unit cell h-BN sheet with 32 atoms yields an with and without the proposed finite-size correction. The correction yields MP2 and CCSD cohesive energies using 3 × 3 × 3 k meshes that deviate from the thermodynamic limit results by approximately 1–2 meV=atom, whereas the uncorrected estimates deviate by 2–4 meV=atom. Although the finite-size errors are small on an absolute scale, we stress that the corresponding relative finite-size errors of the cohesive energy are non-negligible. Our best estimates for the MP2, CCSD, and CCSD(T) cohesive energies using finite-size corrections and a 4 × 4 × 4 k mesh agree with

TABLE I. Cohesive energies of solid neon obtained using MP2, CCSD, and CCSD(T) theory. The summarized results have been extrapolated to the complete basis set limit using pseudized aug- cc-pV(D,T)Z basis sets and corrected for basis-set superposition errors using counterpoise corrections. All units are in meV=atom. FIG. 4. Adsorption energy of a single water molecule on an k mesh MP2 MP2-FS CCSD CCSD-FS CCSD(T)-FS h-BN sheet. The energies are retrieved as a function of atoms in the sheet using MP2, RPA þ SOSEX, CCSD, and CCSD(T) 2 × 2 × 2 −5 25 4 36 47 theory. FS indicates that finite-size corrections are included. The 3 × 3 × 3 13 17 16 21 32 results have been obtained using pseudized aug-cc-pVTZ basis 4 × 4 × 4 17 17 19 19 30 sets and corrected for basis-set superposition errors using Ref. [10] 19 22 27 counterpoise corrections.

021043-5 THOMAS GRUBER et al. PHYS. REV. X 8, 021043 (2018) adsorption energy of 113 meV, in close agreement with the coupled cluster theory for solids, yielding excited-state thermodynamic limit result. We observe a similar speed-up structure factors that are expected to exhibit similar finite- in convergence using RPA þ SOSEX-FS theory, illustrat- size errors [46]. In metallic systems, the structure factor is ing the transferability of the proposed method. The water still algebraic around G ¼ 0. We are therefore confident adsorption on the 32-atom h-BN sheet can also be studied that the proposed methods can also be transferred to the using the more sophisticated CCSD theory [43]. CCSD study of such systems, and we expect that the outlined with and without finite-size corrections yields an adsorp- twist-averaging methodology will be of significant impor- tion energy of 83 meVand 68 meV, respectively. The finite- tance when approaching the thermodynamic limit. We note, size corrections of MP2 and CCSD theory agree to within a however, that the perturbative triples (T) contribution few meV. However, we note that CCSD theory underbinds requires methodological improvements when applied to the water molecule. We estimate the (T) contribution using metals. the 18-atom cell only and find that CCSD(T) theory yields adsorption energy of 102 meV and 87 meV with and ACKNOWLEDGMENTS without finite-size corrections, respectively. The DMC adsorption energy was reported to be 84 meV without This project has received funding from the European finite-size corrections and agrees well with our CCSD(T) Research Council (ERC) under the European Unions results using the same 32-atom h-BN sheet, disregarding Horizon 2020 research and innovation program (Grant finite-size corrections. Agreement No 715594). The computational results pre- sented have been achieved in part using the Vienna Scientific Cluster (VSC). IV. 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