Quartic scaling MP2 for solids: A highly parallelized algorithm in the plane wave basis

Toward accurate modeling of structure and energetics of bulk hexagonal boron nitride

Materials that exhibit both strong covalent and weak van der Waals interactions pose a considerable challenge to many computational methods, such as DFT. This makes assessing the accuracy of calculated properties, such as exfoliation energies in layered materials like hexagonal boron nitride (h-BN) problematic, when experimental data are not available. In this paper, we investigate the accuracy of equilibrium lattice constants and exfoliation energy calculation for various DFT-based computational approaches in bulk h-BN. We contrast these results with available experiments and reference fixed-node diffusion quantum Monte Carlo (QMC) results. From our reference QMC calculation, we obtained an exfoliation energy of - 33 ± 2 meV/atom (-0.38 ± 0.02 J/m2 ).

Plane Waves Versus Correlation-Consistent Basis Sets: A Comparison of MP2 Non-Covalent Interaction Energies in the Complete Basis Set Limit

Second-order Møller-Plesset perturbation theory (MP2) is the most expedient wave function-based method for considering electron correlation in quantum chemical calculations and, as such, provides a cost-effective framework to assess the effects of basis sets on correlation energies, for which the complete basis set (CBS) limit can commonly only be obtained via extrapolation techniques. Software packages providing MP2 energies are commonly based on atom-centered bases with innate issues related to possible basis set superposition errors (BSSE), especially in the case of weakly bonded systems. Here, we present noncovalent interaction energies in the CBS limit, free of BSSE, for 20 dimer systems of the S22 data set obtained via a highly parallelized MP2 implementation in the plane-wave pseudopotential molecular dynamics package CPMD. The specificities related to plane waves for accurate and efficient calculations of gas-phase energies are discussed, and results are compared to the localized (aug-)cc-pV[D,T,Q,5]Z correlation-consistent bases as well as their extrapolated CBS estimates. We find that the BSSE-corrected aug-cc-pV5Z basis can provide MP2 energies highly consistent with the CBS plane wave values with a minimum mean absolute deviation of ∼0.05 kcal/mol without the application of any extrapolation scheme. In addition, we tested the performance of 13 different extrapolation schemes and found that the X-3 expression applied to the (aug-)cc-pVXZ bases provides the smallest deviations against CBS plane wave values if the extrapolation sequence is composed of points D and T, while ( X + 1 2 ) - 4 performs slightly better for TQ and Q5 extrapolations. Also, we propose A ( X - 1 2 ) - 3 + B ( X + 1 2 ) - 4 as a reliable alternative to extrapolate total energies from the DTQ, TQ5, or DTQ5 data points. In spite of the general good agreement between the values obtained from the two types of basis sets, it is noticed that differences between plane waves and (aug-)cc-pVXZ basis sets, extrapolated or not, tend to increase with the number of electrons, thus raising the question of whether these discrepancies could indeed limit the attainable accuracy for localized bases in the limit of large systems.

Direct Determination of Optimal Real-Space Orbitals for Correlated Electronic Structure of Molecules

We demonstrate how to determine numerically nearly exact orthonormal orbitals that are optimal for the evaluation of the energy of arbitrary (correlated) states of atoms and molecules by minimization of the energy Lagrangian. Orbitals are expressed in real space using a multiresolution spectral element basis that is refined adaptively to achieve the user-specified target precision while avoiding the ill-conditioning issues that plague AO basis set expansions traditionally used for correlated models of molecular electronic structure. For light atoms, the orbital solver, in conjunction with a variational electronic structure model [selected Configuration Interaction (CI)] provides energies of comparable precision to a state-of-the-art atomic CI solver. The computed electronic energies of atoms and molecules are significantly more accurate than the counterparts obtained with the orbital sets of the same rank expanded in Gaussian AO bases, and can be determined even when linear dependence issues preclude the use of the AO bases. It is feasible to optimize more than 100 fully correlated numerical orbitals on a single computer node, and significant room exists for additional improvement. These findings suggest that real-space orbital representations might be the preferred alternative to AO representations for high-end models of correlated electronic states of molecules and materials.

Further investigations into a Laplace MP2 method using range separated Coulomb potential and orbital selective virtuals: Multipole correction, OSV extrapolation, and critical assessment

We report further investigations to aid the development of a Laplace MP2 (second-order Møller Plesset) method with a range separated Coulomb potential partitioned into short- and long-range parts. The implementation of the method extensively uses sparse matrix algebra, density fitting techniques for the short-range part, and a Fourier transformation in spherical coordinates for the long-range part of the potential. Localized molecular orbitals are employed for the occupied space, whereas virtual space is described by orbital specific virtual orbitals (OSVs) associated with localized molecular orbitals. The Fourier transform is deficient for very large distances between localized occupied orbitals, and a multipole expansion for widely separated pairs is introduced for the direct MP2 contribution, which is applicable also to non-Coulombic potentials that do not satisfy the Laplace equation. For the exchange contribution, an efficient screening of contributing localized occupied pairs is employed, which is discussed more completely here. To mitigate errors due to the truncation of OSVs, a simple and efficient extrapolation procedure is used to obtain results close to MP2 for the full basis set of atomic orbitals Using a suitable set of default parameters, the accuracy of the approach is demonstrated. The current implementation of the approach is not very efficient, and the aim of this paper is to introduce and critically discuss ideas that can have more general applicability beyond MP2 calculations for large molecules.

Accurate thermochemistry of covalent and ionic solids from spin-component-scaled MP2

We study the performance of spin-component-scaled second-order Møller-Plesset perturbation theory (SCS-MP2) for the prediction of the lattice constant, bulk modulus, and cohesive energy of 12 simple, three-dimensional covalent and ionic semiconductors and insulators. We find that SCS-MP2 and the simpler scaled opposite-spin MP2 (SOS-MP2) yield predictions that are significantly improved over the already good performance of MP2. Specifically, when compared to experimental values with zero-point vibrational corrections, SCS-MP2 (SOS-MP2) yields mean absolute errors of 0.015 (0.017) Å for the lattice constant, 3.8 (3.7) GPa for the bulk modulus, and 0.06 (0.08) eV for the cohesive energy, which are smaller than those of leading density functionals by about a factor of two or more. We consider a reparameterization of the spin-scaling parameters and find that the optimal parameters for these solids are very similar to those already in common use in molecular quantum chemistry, suggesting good transferability and reliable future applications to surface chemistry on insulators.

Cluster-in-Molecule Method Combined with the Domain-Based Local Pair Natural Orbital Approach for Electron Correlation Calculations of Periodic Systems

The cluster-in-molecule (CIM) method was extended to systems with periodic boundary conditions (PBCs) in a previous work (PBC-CIM) [J. Chem. Theory Comput.2019, 15, 2933], which is able to compute the electronic structures of periodic systems at second-order Møller-Plesset perturbation theory (MP2) and coupled cluster singles and doubles (CCSD) levels. However, the high computational costs of CCSD with respect to the size of clusters limit the usage of PBC-CIM to crystals with small or medium unit cells. In this work, we further develop the PBC-CIM method by employing the domain-based local pair natural orbital (DLPNO) methods for the electron correlation calculations of clusters to reduce the computational costs. The combined approach allows CCSD with perturbative triples, denoted as CCSD(T), to be computationally available for accurate descriptions of periodic systems. The distant-pair correction is also implemented to improve the accuracy of PBC-CIM. As in the molecular cases, the distant pair correction significantly improves the accuracy of various PBC-CIM methods with few additional costs. The PBC-CIM-DLPNO-CCSD(T) approach has been applied to investigate the optimized lattice parameter of the cubic LiCl crystal and two adsorption problems (CO on the NaCl(100) surface and H₂O on the h-BN surface). The results show that the CIM-DLPNO-CCSD(T) method offers accurate and efficient descriptions for the studied systems. Another application to the cohesive energy of the acetic acid crystal reveals that large basis sets are necessary for reliable calculations on the cohesive energies of molecular crystals.

Integral-Direct Hartree-Fock and Møller-Plesset Perturbation Theory for Periodic Systems with Density Fitting: Application to the Benzene Crystal

We present an algorithm and implementation of integral-direct, density-fitted Hartree-Fock (HF) and second-order Møller-Plesset perturbation theory (MP2) for periodic systems. The new code eliminates the formerly prohibitive storage requirements and allows us to study systems 1 order of magnitude larger than before at the periodic MP2 level. We demonstrate the significance of the development by studying the benzene crystal in both the thermodynamic limit and the complete basis set limit, for which we predict an MP2 cohesive energy of -72.8 kJ/mol, which is about 10-15 kJ/mol larger in magnitude than all previously reported MP2 calculations. Compared to the best theoretical estimate from literature, several modified MP2 models approach chemical accuracy in the predicted cohesive energy of the benzene crystal and hence may be promising cost-effective choices for future applications on molecular crystals.

The binding of atomic hydrogen on graphene from density functional theory and diffusion Monte Carlo calculations

In this work, density functional theory (DFT) and diffusion Monte Carlo (DMC) methods are used to calculate the binding energy of a H atom chemisorbed on the graphene surface. The DMC value of the binding energy is about 16% smaller in magnitude than the Perdew–Burke–Ernzerhof (PBE) result. The inclusion of exact exchange through the use of the Heyd–Scuseria–Ernzerhof functional brings the DFT value of the binding energy closer in line with the DMC result. It is also found that there are significant differences in the charge distributions determined using PBE and DMC approaches.

Surface science using coupled cluster theory via local Wannier functions and in-RPA-embedding: The case of water on graphitic carbon nitride

A first-principles study of the adsorption of a single water molecule on a layer of graphitic carbon nitride is reported employing an embedding approach for many-electron correlation methods. To this end, a plane-wave based implementation to obtain intrinsic atomic orbitals and Wannier functions for arbitrary localization potentials is presented. In our embedding scheme, the localized occupied orbitals allow for a separate treatment of short-range and long-range correlation contributions to the adsorption energy by a fragmentation of the simulation cell. In combination with unoccupied natural orbitals, the coupled cluster ansatz with single, double, and perturbative triple particle-hole excitation operators is used to capture the correlation in local fragments centered around the adsorption process. For the long-range correlation, a seamless embedding into the random phase approximation yields rapidly convergent adsorption energies with respect to the local fragment size. Convergence of computed binding energies with respect to the virtual orbital basis set is achieved employing a number of recently developed techniques. Moreover, we discuss fragment size convergence for a range of approximate many-electron perturbation theories. The obtained benchmark results are compared to a number of density functional calculations.

Toward Laplace MP2 method using range separated Coulomb potential and orbital selective virtuals

We report the development of a new Laplace MP2 (second-order Møller-Plesset) implementation using a range separated Coulomb potential, partitioned into short- and long-range parts. The implementation heavily relies on the use of sparse matrix algebra, density fitting techniques for the short-range Coulomb interactions, while a Fourier transformation in spherical coordinates is used for the long-range part of the potential. Localized molecular orbitals are employed for the occupied space, whereas orbital specific virtual orbitals associated with localized molecular orbitals are obtained from the exchange matrix associated with specific localized occupied orbitals. The range separated potential is crucial to achieve efficient treatment of the direct term in the MP2, while extensive screening is employed to reduce the expense of the exchange contribution in MP2. The focus of this paper is on controllable accuracy and linear scaling of the data entering the algorithm.

Cerium Oxides without U: The Role of Many-Electron Correlation

Electron transfer with changing occupation in the 4f subshell poses a considerable challenge for quantitative predictions in quantum chemistry. Using the example of cerium oxide, we identify the main deficiencies of common parameter-dependent one-electron approaches, such as density functional theory (DFT) with a Hubbard correction, or hybrid functionals. As a response, we present the first benchmark of ab initio many-electron theory for electron transfer energies and lattice parameters under periodic boundary conditions. We show that the direct random phase approximation clearly outperforms all DFT variations. From this foundation, we, then, systematically improve even further. Periodic second-order Møller-Plesset perturbation theory meanwhile manages to recover standard hybrid functional values. Using these approaches to eliminate parameter bias allows for highly accurate benchmarks of strongly correlated materials, the reliable assessment of various density functionals, and functional fitting via machine-learning.

Staggered Mesh Method for Correlation Energy Calculations of Solids: Second-Order Møller-Plesset Perturbation Theory

The calculation of the MP2 correlation energy for extended systems can be viewed as a multidimensional integral in the thermodynamic limit, and the standard method for evaluating the MP2 energy can be viewed as a trapezoidal quadrature scheme. We demonstrate that the existing analysis neglects certain contributions due to the nonsmoothness of the integrand and may significantly underestimate finite-size errors. We propose a new staggered mesh method, which uses two staggered Monkhorst-Pack meshes for occupied and virtual orbitals, respectively, to compute the MP2 energy. The staggered mesh method circumvents a significant error source in the standard method in which certain quadrature nodes are always placed on points where the integrand is discontinuous. One significant advantage of the proposed method is that there are no tunable parameters, and the additional numerical effort needed can be negligible compared to the standard MP2 calculation. Numerical results indicate that the staggered mesh method can be particularly advantageous for quasi-1D systems as well as quasi-2D and 3D systems with certain symmetries.

Focal-point approach with pair-specific cusp correction for coupled-cluster theory

We present a basis set correction scheme for the coupled-cluster singles and doubles (CCSD) method. The scheme is based on employing frozen natural orbitals (FNOs) and diagrammatically decomposed contributions to the electronic correlation energy, which dominate the basis set incompleteness error (BSIE). As recently discussed in the work of Irmler et al. [Phys. Rev. Lett. 123, 156401 (2019)], the BSIE of the CCSD correlation energy is dominated by the second-order Møller-Plesset (MP2) perturbation energy and the particle-particle ladder term. Here, we derive a simple approximation to the BSIE of the particle-particle ladder term that effectively corresponds to a rescaled pair-specific MP2 BSIE, where the scaling factor depends on the spatially averaged correlation hole depth of the coupled-cluster and first-order pair wavefunctions. The evaluation of the derived expressions is simple to implement in any existing code. We demonstrate the effectiveness of the method for the uniform electron gas. Furthermore, we apply the method to coupled-cluster theory calculations of atoms and molecules using FNOs. Employing the proposed correction and an increasing number of FNOs per occupied orbital, we demonstrate for a test set that rapidly convergent closed and open-shell reaction energies, atomization energies, electron affinities, and ionization potentials can be obtained. Moreover, we show that a similarly excellent trade-off between required virtual orbital basis set size and remaining BSIEs can be achieved for the perturbative triples contribution to the CCSD(T) energy employing FNOs and the (T*) approximation.

Doubly Hybrid Functionals Close to Chemical Accuracy for Both Finite and Extended Systems: Implementation and Test of XYG3 and XYGJ-OS

While being widely used to understand the chemical reactions in heterogeneous catalysis or other multidisciplinary systems, a great challenge that semilocal and hybrid density functional approximations (DFAs) are facing is to deliver a uniformly accurate description for both finite and extended systems. Herein, we perform reliable and well-converged periodic calculations of two doubly hybrid approximations (DHAs), XYG3 and XYGJ-OS, and demonstrate that the good accuracy of DHAs achieved for molecules is transferable to the semiconductors and insulators. Such an accuracy is not only for energetic properties but also for the first- and second-order response properties, which is general for different kinds of chemical environments, including simple cubic bulks, perovskite-type transition metal oxides like TiO2, and heterogeneous systems like CO adsorption on the NaCl(100) surface. The present finding has strengthened the predictive power of DFT, which not only will inspire the future development of the top-rung DFAs but also will boost their applications in multidisciplinary studies with high accuracy and efficiency.

Linear-Scaling Open-Shell MP2 Approach: Algorithm, Benchmarks, and Large-Scale Applications

A linear-scaling local second-order Møller–Plesset (MP2) method is presented for high-spin open-shell molecules based on restricted open-shell (RO) reference functions. The open-shell local MP2 (LMP2) approach inherits the iteration- and redundancy-free formulation and the completely integral-direct, OpenMP-parallel, and memory and disk use economic algorithms of our closed-shell LMP2 implementation. By utilizing restricted local molecular orbitals for the demanding integral transformation step and by introducing a novel long-range spin-polarization approximation, the computational cost of RO-LMP2 approaches that of closed-shell LMP2. Extensive benchmarks were performed for reactions of radicals, ionization potentials, as well as spin-state splittings of carbenes and transition-metal complexes. Compared to the conventional MP2 reference for systems of up to 175 atoms, local errors of at most 0.1 kcal/mol were found, which are well below the intrinsic accuracy of MP2. RO-LMP2 computations are presented for challenging protein models of up to 601 atoms and 11 000 basis functions, which involve either spin states of a complexed iron ion or a highly delocalized singly occupied orbital. The corresponding runtimes of 9–15 h obtained with a single, many-core CPU demonstrate that MP2, as well as spin-scaled MP2 and double-hybrid density functional methods, become widely accessible for open-shell systems of unprecedented size and complexity.

Local embedding of coupled cluster theory into the random phase approximation using plane waves

We present an embedding approach to treat local electron correlation effects in periodic environments. In a single consistent framework, our plane wave based scheme embeds a local high-level correlation calculation [here, Coupled Cluster (CC) theory], employing localized orbitals, into a low-level correlation calculation [here, the direct Random Phase Approximation (RPA)]. This choice allows for an accurate and efficient treatment of long-range dispersion effects. Accelerated convergence with respect to the local fragment size can be observed if the low-level and high-level long-range dispersions are quantitatively similar, as is the case for CC in RPA. To demonstrate the capabilities of the introduced embedding approach, we calculate adsorption energies of molecules on a surface and in a chabazite crystal cage, as well as the formation energy of a lattice impurity in a solid at the level of highly accurate many-electron perturbation theories. The absorption energy of a methane molecule in a zeolite chabazite is converged with an error well below 20 meV at the CC level. As our largest periodic benchmark system, we apply our scheme to the adsorption of a water molecule on titania in a supercell containing more than 1000 electrons.

Efficient Treatment of Correlation Energies at the Basis-Set Limit by Monte Carlo Summation of Continuum States

The calculation of electron correlation is vital for the description of atomistic phenomena in physics, chemistry, and biology. However, accurate wavefunction-based methods exhibit steep scaling and often sluggish convergence with respect to the basis set at hand. Because of their delocalization and ease of extrapolation to the basis-set limit, plane waves would be ideally suited for the calculation of basis-set limit correlation energies. However, the routine use of correlated wavefunction approaches in a plane-wave basis set is hampered by prohibitive scaling due to a large number of virtual continuum states and has not been feasible for all but the smallest systems, even if substantial computational resources are available and methods with comparably beneficial scaling, such as the Møller-Plesset perturbation theory to second order (MP2), are used. Here, we introduce a stochastic sampling of the MP2 integrand based on Monte Carlo summation over continuum orbitals, which allows for speedups of up to a factor of 1000. Given a fixed number of sampling points, the resulting algorithm is dominated by a flat scaling of ∼ O ( N 2 ) . Absolute correlation energies are accurate to <0.1 kcal/mol with respect to conventional calculations for several hundreds of electrons. This allows for the calculation of unbiased basis-set limit correlation energies for systems containing hundreds of electrons with unprecedented efficiency gains based on a straightforward treatment of continuum contributions.

Bootstrap Embedding For Large Molecular Systems

Recent developments in quantum embedding theories have provided attractive approaches to correlated calculations for large systems. In this work, we extend our previous work [J. Chem. Theory Comput. 2019, 15, 4497-4506; J. Phys. Chem. Lett. 2019, 10, 6368-6374] on bootstrap embedding (BE) to enable correlated ab initio calculations at the coupled cluster with singles and doubles (CCSD) level for large molecules. We introduce several new algorithmic developments that significantly reduce the computational cost of BE, while maintaining its accuracy. The resulting implementation scales as O(N³) for the integral transform and O(N) for the CCSD calculation. Numerical results on a series of conjugated molecules suggest that BE with reasonably sized fragments can recover more than 99.5% of the total correlation energy of a full CCSD calculation, while the required computational resources (time and storage) compare favorably to one popular local correlation scheme: domain localized pair natural orbital (DLPNO). The largest BE calculation in this work involves ∼2900 basis functions and can be performed on a single node with 16 CPU cores and 64 GB of memory in a few days. We anticipate that these developments represent an important step toward the application of BE to solve practical problems.

Q-MP2-OS: Møller-Plesset Correlation Energy by Quadrature

We present a quadrature-based algorithm for computing the opposite-spin component of the MP2 correlation energy which scales quadratically with basis set size and is well-suited to large-scale parallelization. The key ideas, which are rooted in the earlier work of Hirata and co-workers, are to abandon all two-electron integrals, recast the energy as a seven-dimensional integral, approximate that integral by quadrature, and employ a cutoff strategy to minimize the number of intermediate quantities. We discuss our implementation in detail and show that it parallelizes almost perfectly on 840 cores for cyclosporine (a molecule with roughly 200 atoms), exhibits [Formula: see text] scaling for a sequence of polyglycines, and is principally limited by the accuracy of its quadrature.

Efficient and accurate description of adsorption in zeolites

Accurate theoretical methods are needed to correctly describe adsorption on solid surfaces or in porous materials. The random phase approximation (RPA) with singles corrections scheme and the second order Møller-Plesset perturbation theory (MP2) are two schemes, which offer high accuracy at affordable computational cost. However, there is little knowledge about their applicability and reliability for different adsorbates and surfaces. Here, we calculate adsorption energies of seven different molecules in zeolite chabazite to show that RPA with singles corrections is superior to MP2, not only in terms of accuracy but also in terms of computer time. Therefore, RPA with singles is a suitable scheme for obtaining highly accurate adsorption energies in porous materials and similar systems.

Duality of Ring and Ladder Diagrams and Its Importance for Many-Electron Perturbation Theories

We present a diagrammatic decomposition of the transition pair correlation function for the uniform electron gas. We demonstrate explicitly that ring and ladder diagrams are dual counterparts that capture significant long- and short-ranged interelectronic correlation effects, respectively. Our findings help to guide the further development of approximate many-electron theories and reveal that the contribution of the ladder diagrams to the electronic correlation energy can be approximated in an effective manner using second-order perturbation theory. We employ the latter approximation to reduce the computational cost of coupled cluster theory calculations for insulators and semiconductors by 2 orders of magnitude without compromising accuracy.

Optimized Pair Natural Orbitals for the Coupled Cluster Methods

We present the coupled-cluster singles and doubles method formulated in terms of truncated pair natural orbitals (PNO) that are optimized to minimize the effect of truncation. Compared to the standard ground-state PNO coupled-cluster approaches, in which truncated PNOs derived from first-order Møller-Plesset (MP1) amplitudes are used to compress the CC wave operator, the iteratively optimized PNOs ("iPNOs") offer moderate improvement for small PNO ranks but rapidly increase their effectiveness for large PNO ranks. The error introduced by PNO truncation in the CCSD energy is reduced by orders of magnitude in the asymptotic regime, with an insignificant increase in PNO ranks. The effect of PNO optimization is particularly effective when combined with Neese's perturbative correction for the PNO incompleteness of the CCSD energy. The use of the perturbative correction in combination with the PNO optimization procedure seems to produce the most precise approximation to the canonical CCSD energies for small and large PNO ranks. For the standard benchmark set of noncovalent binding energies, remarkable improvements with respect to the standard PNO approach range from a factor of 3 with PNO truncation threshold τ_PNO = 10^-6 (with the maximum PNO truncation error in the binding energy of only 0.1 kcal/mol) to more than 2 orders of magnitude with τ_PNO = 10^-9.

Lowering of the complexity of quantum chemistry methods by choice of representation

The complexity of the standard hierarchy of quantum chemistry methods is not invariant to the choice of representation. This work explores how the scaling of common quantum chemistry methods can be reduced using real-space, momentum-space, and time-dependent intermediate representations without introducing approximations. We find the scalings of exact Gaussian basis Hartree-Fock theory, second-order Møller-Plesset perturbation theory, and coupled cluster theory (specifically, linearized coupled cluster doubles and the distinguishable cluster approximation with doubles) to be O(N³), O(N³), and O(N⁵), respectively, where N denotes the system size. These scalings are not asymptotic and hold over all ranges of N.

Laplace transformed MP2 for three dimensional periodic materials using stochastic orbitals in the plane wave basis and correlated sampling

We present an implementation and analysis of a stochastic high performance algorithm to calculate the correlation energy of three-dimensional periodic systems in second-order Møller-Plesset perturbation theory (MP2). In particular we measure the scaling behavior of the sample variance and probe whether this stochastic approach is competitive if accuracies well below 1 meV per valence orbital are required, as it is necessary for calculations of adsorption, binding, or surface energies. The algorithm is based on the Laplace transformed MP2 (LTMP2) formulation in the plane wave basis. The time-dependent Hartree-Fock orbitals, appearing in the LTMP2 formulation, are stochastically rotated in the occupied and unoccupied Hilbert space. This avoids a full summation over all combinations of occupied and unoccupied orbitals, as inspired by the work of Neuhauser, Rabani, and Baer [J. Chem. Theory Comput. 9, 24 (2013)]. Additionally, correlated sampling is introduced, accelerating the statistical convergence significantly.

Optimization of the linear-scaling local natural orbital CCSD(T) method: Redundancy-free triples correction using Laplace transform

An improved algorithm is presented for the evaluation of the (T) correction as a part of our local natural orbital (LNO) coupled-cluster singles and doubles with perturbative triples [LNO-CCSD(T)] scheme [Z. Rolik et al., J. Chem. Phys. 139, 094105 (2013)]. The new algorithm is an order of magnitude faster than our previous one and removes the bottleneck related to the calculation of the (T) contribution. First, a numerical Laplace transformed expression for the (T) fragment energy is introduced, which requires on average 3 to 4 times fewer floating point operations with negligible compromise in accuracy eliminating the redundancy among the evaluated triples amplitudes. Second, an additional speedup factor of 3 is achieved by the optimization of our canonical (T) algorithm, which is also executed in the local case. These developments can also be integrated into canonical as well as alternative fragmentation-based local CCSD(T) approaches with minor modifications. As it is demonstrated by our benchmark calculations, the evaluation of the new Laplace transformed (T) correction can always be performed if the preceding CCSD iterations are feasible, and the new scheme enables the computation of LNO-CCSD(T) correlation energies with at least triple-zeta quality basis sets for realistic three-dimensional molecules with more than 600 atoms and 12 000 basis functions in a matter of days on a single processor.