Scalable Electron Correlation Methods. 3. Efficient and Accurate Parallel Local Coupled Cluster with Pair Natural Orbitals (PNO-LCCSD)

57Fe Mössbauer parameters from domain based local pair-natural orbital coupled-cluster theory

We report on applications of the domain based local pair-natural orbital (PNO) coupled-cluster method within the singles and doubles approximation (DLPNO-CCSD) to the calculation of ⁵⁷Fe isomer shifts and quadrupole splittings in a small training set of iron complexes consisting of large molecular ligands and iron atoms in varying charge, spin, and oxidation states. The electron densities and electric field gradients needed for these calculations were obtained within the recently implemented analytic derivative scheme. A method for the direct treatment of scalar relativistic effects in the calculation of effective electron densities is described by using the first-order Douglas-Kroll-Hess Hamiltonian and a Gaussian charge distribution model for the nucleus. The performance of DLPNO-CCSD is compared with four modern-day density functionals, namely, RPBE, TPSS, B3LYP, and B2PLYP, as well as with the second-order Møller-Plesset perturbation theory. An excellent correlation between the calculated electron densities and the experimental isomer shifts is attained with the DLPNO-CCSD method. The correlation constant a obtained from the slope of the linear correlation plot is found to be ≈-0.31 a.u.³ mm s^-1, which agrees very well with the experimental calibration constant α = -0.31 ± 0.04 a.u.³ mm s^-1. This value of a is obtained consistently using both nonrelativistic and scalar relativistic DLPNO-CCSD electron densities. While the B3LYP and B2PLYP functionals achieve equally good correlation between theory and experiment, the correlation constant a is found to deviate from the experimental value. Similar trends are observed also for quadrupole splittings. The value of the nuclear quadrupole moment for ⁵⁷Fe is estimated to be 0.15 b at the DLPNO-CCSD level. This is consistent with previous results and is here supported by a higher level of theory. The DLPNO-CCSD results are found to be insensitive to the intrinsic approximations in the method, in particular the PNO occupation number truncation error, while the results obtained with density functional theory (DFT) are found to depend on the choice of the functional. In a statistical sense, i.e., on the basis of the linear regression analysis, however, the accuracies of the DFT and DLPNO-CCSD results can be considered comparable.

Performance of Coupled-Cluster Singles and Doubles on Modern Stream Processing Architectures

We develop a new implementation of coupled-cluster singles and doubles (CCSD) optimized for the most recent graphical processing unit (GPU) hardware. We find that a single node with 8 NVIDIA V100 GPUs is capable of performing CCSD computations on roughly 100 atoms and 1300 basis functions in less than 1 day. Comparisons against massively parallel implementations of CCSD suggest that more than 64 CPU-based nodes (each with 16 cores) are required to match this performance.

Double hybrid DFT calculations with Slater type orbitals

On a comprehensive database with 1,644 datapoints, covering several aspects of main-group as well as of transition metal chemistry, we assess the performance of 60 density functional approximations (DFA), among them 36 double hybrids (DH). All calculations are performed using a Slater type orbital (STO) basis set of triple-ζ (TZ) quality and the highly efficient pair atomic resolution of the identity approach for the exchange- and Coulomb-term of the KS matrix (PARI-K and PARI-J, respectively) and for the evaluation of the MP2 energy correction (PARI-MP2). Employing the quadratic scaling SOS-AO-PARI-MP2 algorithm, DHs based on the spin-opposite-scaled (SOS) MP2 approximation are benchmarked against a database of large molecules. We evaluate the accuracy of STO/PARI calculations for B3LYP as well as for the DH B2GP-PLYP and show that the combined basis set and PARI-error is comparable to the one obtained using the well-known def2-TZVPP Gaussian-type basis set in conjunction with global density fitting. While quadruple-ζ (QZ) calculations are currently not feasible for PARI-MP2 due to numerical issues, we show that, on the TZ level, Jacob's ladder for classifying DFAs is reproduced. However, while the best DHs are more accurate than the best hybrids, the improvements are less pronounced than the ones commonly found on the QZ level. For conformers of organic molecules and noncovalent interactions where very high accuracy is required for qualitatively correct results, DHs provide only small improvements over hybrids, while they still excel in thermochemistry, kinetics, transition metal chemistry and the description of strained organic systems.

Near-Linear Scaling in DMRG-Based Tailored Coupled Clusters: An Implementation of DLPNO-TCCSD and DLPNO-TCCSD(T)

We present a new implementation of density matrix renormalization group based tailored coupled clusters method (TCCSD), which employs the domain-based local pair natural orbital approach (DLPNO). Compared to the previous local pair natural orbital (LPNO) version of the method, the new implementation is more accurate, offers more favorable scaling, and provides more consistent behavior across the variety of systems. On top of the singles and doubles, we include the perturbative triples correction (T), which is able to retrieve even more dynamic correlation. The methods were tested on three systems: tetramethyleneethane, oxo-Mn(Salen), and iron(II)-porphyrin model. The first two were revisited to assess the performance with respect to LPNO-TCCSD. For oxo-Mn(Salen), we retrieved between 99.8 and 99.9% of the total canonical correlation energy which is an improvement of 0.2% over the LPNO version in less than 63% of the total LPNO runtime. Similar results were obtained for iron(II)-porphyrin. When the perturbative triples correction was employed, irrespective of the active space size or system, the obtained energy differences between two spin states were within the chemical accuracy of 1 kcal/mol using the default DLPNO settings.

Scalable Electron Correlation Methods. 7. Local Open-Shell Coupled-Cluster Methods Using Pair Natural Orbitals: PNO-RCCSD and PNO-UCCSD

We present well-parallelized local implementations of high-spin open-shell coupled cluster methods with single and double excitations (CCSD) using pair natural orbitals (PNOs). The methods are based on the spin-orbital coupled cluster theory using restricted open-shell Hartree-Fock (ROHF) reference functions. Two variants, namely, PNO-UCCSD and PNO-RCCSD are implemented and compared. In PNO-UCCSD, the coupled cluster amplitudes are spin-unrestricted, while in PNO-RCCSD the linear terms are spin-adapted by a spin-projection approach as described in J. Chem. Phys. 1993, 99, 5219-5227. Near linear scaling of the computational cost with the number of correlated electrons is achieved by applying domain and pair approximations. The PNOs are spin-independent and obtained using a semicanonical spin-restricted MP2 approximation with large domains of projected atomic orbitals (PAOs). The pair approximations of our previously described closed-shell PNO-LCCSD method are carefully revised so that they are compatible to the UCCSD theory, and PNO-UCCSD or PNO-RCCSD calculations for closed-shell molecules yield exactly the same results as corresponding spin-free closed-shell PNO-LCCSD calculations. The convergence of the results with respect to the thresholds and options that control the domain and pair approximations is demonstrated. It is found that large domains are required for the single excitations in open-shell calculations in order to obtain converged results. In general, the errors of relative energies caused by the local approximations can be reduced to below 1 kcal mol^-1, even for difficult cases. Presently, PNO-RCCSD and PNO-UCCSD calculations for molecules with 100-200 atoms and augmented triple-ζ basis sets can be carried out in a few hours of elapsed time using ∼100 CPU cores. In addition, the program is also capable of performing distinguishable cluster (PNO-RDCSD and PNO-UDCSC) calculations. The present work is a critical step in developing fully local open-shell PNO-RCCSD(T)-F12 methods.

The Molpro quantum chemistry package

Molpro is a general purpose quantum chemistry software package with a long development history. It was originally focused on accurate wavefunction calculations for small molecules but now has many additional distinctive capabilities that include, inter alia, local correlation approximations combined with explicit correlation, highly efficient implementations of single-reference correlation methods, robust and efficient multireference methods for large molecules, projection embedding, and anharmonic vibrational spectra. In addition to conventional input-file specification of calculations, Molpro calculations can now be specified and analyzed via a new graphical user interface and through a Python framework.

A multireference coupled-electron pair approximation combined with complete-active space perturbation theory in local pair-natural orbital framework

The Complete-Active Space Second-order Perturbation Theory (CASPT2) has been one of the most widely-used methods for reliably calculating electronic structures of multireference systems. Because of its lowest level treatment of dynamic correlation, it has a high computational feasibility; however, its accuracy in some cases falls short of needs. Here, as a simple yet higher-order alternative, we introduce a hybrid theory of the CASPT2 and a multireference variant of the Coupled-Electron Pair Approximation (CEPA), which is a class of high level correlation theory. A central feature of our theory (CEPT2) is to use the two underlying theories for describing different divisions of correlation components based on the full internal contraction framework. The external components, which usually give a major contribution to the dynamic correlation, are intensively described using the CEPA Ansatz, while the rests are treated at the CASPT2 level. Furthermore, to drastically reduce the computational demands, we have incorporated the pair-natural orbital (PNO) method into our multireference implementations. This development, thus, requires highly complex derivations and coding, while it has been largely facilitated with an automatic expression and code generation technique. To highlight the accuracy of the CEPT2 approach and to assess the errors caused by the PNO truncation, benchmark calculations are shown on small- to medium-size molecules, illustrating the high accuracy of the present CEPT2 model. By tightening the truncation thresholds, the PNO-CEPT2 energy converges toward the canonical counterpart and is more accurate than that of PNO-CASPT2 as long as the same truncation thresholds are used.

Linear scaling perturbative triples correction approximations for open-shell domain-based local pair natural orbital coupled cluster singles and doubles theory [DLPNO-CCSD(T0/T)]

The coupled cluster method with single-, double-, and perturbative triple excitations [CCSD(T)] is considered to be one of the most reliable quantum chemistry theories. However, the steep scaling of CCSD(T) has limited its application to small or medium-sized systems for a long time. In our previous work, the linear scaling domain based local pair natural orbital CCSD variant (DLPNO-CCSD) has been developed for closed-shell and open-shell. However, it is known from extensive benchmark studies that triple-excitation contributions are important to reach chemical accuracy. In the present work, two linear scaling (T) approximations for open-shell DLPNO-CCSD are implemented and compared: (a) an algorithm based on the semicanonical approximation, in which off-diagonal Fock matrix elements in the occupied space are neglected [referred to as DLPNO-(T₀)]; and (b) an improved algorithm in which the triples amplitudes are computed iteratively [referred to as DLPNO-(T)]. This work is based on the previous open-shell DLPNO-CCSD algorithm [M. Saitow et al., J. Chem. Phys. 146, 164105 (2017)] as well as the iterative (T) correction for closed-shell systems [Y. Guo et al., J. Chem. Phys. 148, 011101 (2018)]. Our results show that the new open-shell perturbative corrections, DLPNO-(T₀/T), can predict accurate absolute and relative correlation energies relative to the canonical reference calculations with the same basis set. The absolute energies from DLPNO-(T) are significantly more accurate than those of DLPNO-(T₀). The additional computational effort of DLPNO-(T) relative to DLPNO-(T₀) is a factor of 4 on average. We report calculations on systems with more than 4000 basis functions.

Domain-based local pair natural orbital methods within the correlation consistent composite approach

Ab initio composite approaches have been utilized to model and predict main group thermochemistry within 1 kcal mol^-1 , on average, from well-established reliable experiments, primarily for molecules with less than 30 atoms. For molecules of increasing size and complexity, such as biomolecular complexes, composite methodologies have been limited in their application. Therefore, the domain-based local pair natural orbital (DLPNO) methods have been implemented within the correlation consistent composite approach (ccCA) framework, namely DLPNO-ccCA, to reduce the computational cost (disk space, CPU (central processing unit) time, memory) and predict energetic properties such as enthalpies of formation, noncovalent interactions, and conformation energies for organic biomolecular complexes including one of the largest molecules examined via composite strategies, within 1 kcal mol^-1 , after calibration with 119 molecules and a set of linear alkanes. © 2019 Wiley Periodicals, Inc.

Comprehensive Benchmark Results for the Domain Based Local Pair Natural Orbital Coupled Cluster Method (DLPNO-CCSD(T)) for Closed- and Open-Shell Systems

In this study we examine the accuracy of domain-based local pair natural orbital coupled cluster theory with single, double, and perturbative triple excitations (DLPNO-CCSD(T)) on a large benchmark data set. To this end, we use the recently published GMTKN55 superset of molecules that contains 1505 relative energies and 2462 single-point calculations. To our knowledge this is the most comprehensive benchmark evaluation of any highly correlated wave function based ab initio method to date. In the first part of the study, canonical CCSD(T) reference calculations were carried out on the entire test set in order to guarantee that the reference data are of uniform quality. Second, DLPNO-CCSD(T) calculations were carried out under identical conditions. The main finding is that with the exception of two data sets, all data sets have a MAD of 0.4 kcal/mol or less and the majority of sets have a MAD of less than 0.2 kcal/mol. For open shells, the accuracy of the DLPNO calculations was significantly improved through an iterative version of the triples correction.

Principal Domains in Local Correlation Theory

The computational efficiency of local correlation methods is strongly dependent on the size of the domain of functions used to expand local correlating orbitals such as orbital specific or pair natural orbitals. Here, we define a principal domain of order m as the subset of m one-particle functions that provides the best support for a given n-electron wave function by maximizing the partial trace of the one-body reduced density matrix. Principal domains maximize the overlap between the wave function and its approximant for two-electron systems and are the domain selection equivalent of Löwdin's natural orbitals. We present an efficient linear scaling greedy algorithm for obtaining principal domains of projected atomic orbitals and demonstrate its utility in the context of the pair natural orbital local correlation theory. We numerically determine thresholds such that the projected atomic orbital domain error is an order of magnitude smaller than the pair natural orbital truncation error.

Fantasy versus reality in fragment-based quantum chemistry

Since the introduction of the fragment molecular orbital method 20 years ago, fragment-based approaches have occupied a small but growing niche in quantum chemistry. These methods decompose a large molecular system into subsystems small enough to be amenable to electronic structure calculations, following which the subsystem information is reassembled in order to approximate an otherwise intractable supersystem calculation. Fragmentation sidesteps the steep rise (with respect to system size) in the cost of ab initio calculations, replacing it with a distributed cost across numerous computer processors. Such methods are attractive, in part, because they are easily parallelizable and therefore readily amenable to exascale computing. As such, there has been hope that distributed computing might offer the proverbial "free lunch" in quantum chemistry, with the entrée being high-level calculations on very large systems. While fragment-based quantum chemistry can count many success stories, there also exists a seedy underbelly of rarely acknowledged problems. As these methods begin to mature, it is time to have a serious conversation about what they can and cannot be expected to accomplish in the near future. Both successes and challenges are highlighted in this Perspective.

A Linear-Scaling Method for Noncovalent Interactions: An Efficient Combination of Absolutely Localized Molecular Orbitals and a Local Random Phase Approximation Approach

A novel method for the accurate and efficient calculation of interaction energies in weakly bound complexes composed of a large number of molecules is presented. The new ALMO+RPAd method circumvents the prohibitive scaling of coupled cluster singles and doubles while still providing similar accuracy across a diverse range of weakly bound chemical systems. Linear-scaling procedures for the Fock build are given utilizing absolutely localized molecular orbitals (ALMOs), resulting in the a priori exclusion of basis set superposition errors. A bespoke data structure and algorithm using density fitting are described, leading to linear scaling for the storage and computation of the two-electron integrals. Electron correlation is included through a new, linear-scaling pairwise local random phase approximation approach, including exchange interactions, and decomposed into purely dispersive excitations (RPAxd). Collectively, these allow meaningful decomposition of the interaction energy into physically distinct contributions: electrostatic, polarization, charge transfer, and dispersion. Comparison with symmetry-adapted perturbation theory shows good qualitative agreement. Tests on various dimers and the S66 benchmark set demonstrate results within 0.5 kcal mol^-1 of coupled cluster singles and doubles results. On a large cluster of water molecules, we achieve calculations involving over 3500 orbital and 12,000 auxiliary basis functions in under 10 min on a single CPU core.

Analytical Energy Gradients for the Cluster-in-Molecule MP2 Method and Its Application to Geometry Optimizations of Large Systems

An efficient analytical energy gradient algorithm for the cluster-in-molecule (CIM) second order Møller-Plesset perturbation theory (MP2) method is presented. In our algorithm, the gradient contributions from the nonseparable term of the two-body density matrix on a given atom is extracted from calculations on a cluster constructed for this atom. The other terms in the CIM-MP2 energy gradient expression are evaluated by constructing the density matrices of the whole system with the contributions from all clusters constructed. For basis sets with diffuse functions, tight CIM parameters are necessary to obtain accurate gradients. Benchmark calculations show that the CIM-MP2 method can accurately reproduce the conventional MP2 gradients and geometries for larger systems. The optimized structure of a 174-atom oligopeptide using the CIM-MP2 method with the cc-pVDZ basis set is in good agreement with the corresponding crystal structure. The present CIM-MP2 gradient program can be used for optimizing the geometries of large systems with hundreds of atoms on ordinary workstations.

Diagrammatic Coupled Cluster Monte Carlo

We propose a modified coupled cluster Monte Carlo algorithm that stochastically samples connected terms within the truncated Baker-Campbell-Hausdorff expansion of the similarity-transformed Hamiltonian by construction of coupled cluster diagrams on the fly. Our new approach-diagCCMC-allows propagation to be performed using only the connected components of the similarity-transformed Hamiltonian, greatly reducing the memory cost associated with the stochastic solution of the coupled cluster equations. We show that for perfectly local, noninteracting systems diagCCMC is able to represent the coupled cluster wavefunction with a memory cost that scales linearly with system size. The favorable memory cost is observed with the only assumption of fixed stochastic granularity and is valid for arbitrary levels of coupled cluster theory. Significant reduction in memory cost is also shown to smoothly appear with dissociation of a finite chain of helium atoms. This approach is also shown not to break down in the presence of strong correlation through the example of a stretched nitrogen molecule. Our novel methodology moves the theoretical basis of coupled cluster Monte Carlo closer to deterministic approaches.

Perturbative triples correction to domain-based local pair natural orbital variants of Mukherjee's state specific coupled cluster method

In this article we report an implementation of the perturbative triples correction to Mukherjee's state-specific multireference coupled cluster method based on the domain-based pair natural orbital approach (DLPNO-MkCC). We tested the performance of DLPNO-MkCCSD(T) in calculations involving tetramethyleneethane and isomers of naphthynes. These tests show that more than 97% of triples energy was recovered with respect to the canonical MkCCSD(T) method, which together with the DLPNO-MkCCSD part accounts for about 99.70-99.85% of the total correlation energy. The applicability of the method was demonstrated on calculations of singlet-triplet gaps for several large systems: triangulene, dynemicin A, and a beryllium complex.

Chemistry and Quantum Mechanics in 2019: Give Us Insight and Numbers

This Perspective revisits Charles Coulson’s famous statement from 1959 “give us insight not numbers” in which he pointed out that accurate computations and chemical understanding often do not go hand in hand. We argue that today, accurate wave function based first-principle calculations can be performed on large molecular systems, while tools are available to interpret the results of these calculations in chemical language. This leads us to modify Coulson’s statement to “give us insight and numbers”. Examples from organic, inorganic, organometallic and surface chemistry as well as molecular magnetism illustrate the points made.

Scalable Electron Correlation Methods. 6. Local Spin-Restricted Open-Shell Second-Order Møller-Plesset Perturbation Theory Using Pair Natural Orbitals: PNO-RMP2

We present a (near) linear scaling implementation of high-spin open-shell Møller-Plesset perturbation theory using pair natural orbitals (PNO-RMP2). The theory is based on a new variant of open-shell MP2 which is fully spin-adapted and uses a single set of spin-free amplitudes, as in closed-shell MP2. This method, denoted SROMP2, is invariant to unitary orbital transformations within the closed, open, and virtual orbital subspaces. Accordingly, only a single set of PNOs per spatial orbital pair is needed, and the efficiency is similar to closed-shell calculations. The PNOs are obtained using a semicanonical approximation with large domains of projected atomic orbitals (PAOs). Linear scaling is achieved provided that the open-shell orbitals are local, and distant pairs are treated by multipole approximations. The method is efficiently parallelized. The convergence of ionization and reaction energies as a function of the PAO and PNO domain sizes is demonstrated and found to be very similar as for closed-shell calculations. The suitability of the PNOs for explicitly correlated PNO-RCCSD-F12 calculations is also tested. So far, this method is only simulated using a conventional program with appropriate projections to the PAO and PNO subspaces. It is demonstrated for radical stabilization energies as well as ionization potentials that the errors caused by the local domain approximations with our default thresholds are negligible.

Assessment of DFT for endohedral complexes' dipole moment: PNO-LCCSD-F12 as a reference method

We present a systematic evaluation of the performance of a wide range of exchange-correlation functionals and related dispersion correction schemes for the computation of dipole moments of endohedral complexes, formed through the encapsulation of an AB molecule (AB = LiF, HCl) inside carbon nanotubes (CNTs) of different diameter. The consistency and accuracy of (i) generalized gradient approximation, (ii) meta GGA, (iii) global hybrid, and (iv) range-separated hybrid density functionals are assessed. In total, 37 density functionals are tested. The results obtained using the highly accurate pair natural orbitals based explicitly correlated local coupled cluster singles doubles (PNO-LCCSD-F12) method of Werner and co-workers [Schwilk et al., J. Chem. Theory Comput., 2017, 13, 3650; Ma et al., J. Chem. Theory Comput., 2017, 13, 4871] with the aug-cc-pVTZ basis set serve as a reference. The static electric dipole moment is computed via the finite field response or, when possible, as the expectation value of the dipole operator. Among others, it is shown that functionals belonging to the class of range-separated hybrids, provide results closest to the coupled cluster reference data. In particular, the ωB97X as well as the M11 functional may be considered as a promising choice for computing electric properties of noncovalent endohedral complexes. On the other hand, the worst performance was found for the functionals which do not include the Hartree-Fock exchange. The analysis of both the coupled cluster and the DFT results indicates a strong coupling of dispersion and polarization that may also explain why lower level DFT methods, as well as Hartree-Fock and MP2, cannot yield dipole moments beyond a qualitative agreement with the higher order reference data. Interestingly, the much smaller and less systematically constructed basis sets of Pople of moderate size provide results of accuracy at least comparable with the extended Dunning's aug-cc-pVTZ basis set.

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.

Harmonizing accuracy and efficiency: A pragmatic approach to fragmentation of large molecules

Fragmentation methods offer an attractive alternative for ab initio treatment of large molecules and molecular clusters. However, balancing the accuracy and efficiency of these methods is a tight-rope-act. With this in view, we present an algorithm for automatic molecular fragmentation within Molecular Tailoring Approach (MTA) achieving this delicate balance. The automated code is tested out on a variety of molecules and clusters at the Hartree-Fock (HF)- and Møller-Plesset second order perturbation theory as well as density functional theory employing augmented Dunning basis sets. The results show remarkable accuracy and efficiency vis-à-vis the respective full calculations. Thus the present work forms an important step toward the development of an MTA-based black box code for implementation of HF as well as correlated quantum chemical calculations on large molecular systems.