Analytical Gradient Using Cluster-in-Molecule RI-MP2 Method for the Geometry Optimizations of Large Systems

We present an efficient analytical energy gradient algorithm for the cluster-in-molecule resolution-of-identity second-order Møller-Plesset perturbation (CIM-RI-MP2) method based on the Lagrange multiplier method. Our algorithm independently constructs the Lagrangian formalism within each cluster, avoiding the solution of the coupled-perturbed Hartree-Fock (CPHF) equation for the whole system. Due to this feature, the computational cost of the CIM-RI-MP2 gradients is much lower than that of other local MP2 algorithms. Benchmark calculations of several molecules containing up to 312 atoms demonstrate the general applicability of our CIM-RI-MP2 gradient algorithm. The optimized structure of a 244-atom molecule using the CIM-RI-MP2 method with the cc-pVDZ basis set is in good agreement with the corresponding crystal structure. A single-point gradient calculation conducted for a molecular cage containing 972 atoms and 9612 basis functions takes 48 h on 25 nodes, utilizing a total of 600 CPU cores. The present CIM-RI-MP2 gradient program is applicable for obtaining the optimized geometries of large systems with hundreds of atoms.

The p-block challenge: assessing quantum chemistry methods for inorganic heterocycle dimerizations

The elements of the p-block of the periodic table are of high interest in various chemical and technical applications like frustrated Lewis-pairs (FLP) or opto-electronics. However, high-quality benchmark data to assess approximate density functional theory (DFT) for their theoretical description are sparse. In this work, we present a benchmark set of 604 dimerization energies of 302 "inorganic benzenes" composed of all non-carbon p-block elements of main groups III to VI up to polonium. This so-called IHD302 test set comprises two classes of structures formed by covalent bonding and by weaker donor-acceptor (WDA) interactions, respectively. Generating reliable reference data with ab initio methods is challenging due to large electron correlation contributions, core-valence correlation effects, and especially the slow basis set convergence. To compute reference values for these dimerization reactions, after thorough testing, we applied a computational protocol using state-of-the-art explicitly correlated local coupled cluster theory termed PNO-LCCSD(T)-F12/cc-VTZ-PP-F12(corr.). It includes a basis set correction at the PNO-LMP2-F12/aug-cc-pwCVTZ level. Based on these reference data, we assess 26 DFT methods in combination with three different dispersion corrections and the def2-QZVPP basis set, five composite DFT approaches, and five semi-empirical quantum mechanical methods. For the covalent dimerizations, the r2SCAN-D4 meta-GGA, the r2SCAN0-D4 and ωB97M-V hybrids, and the revDSD-PBEP86-D4 double-hybrid functional are found to be the best-performing methods among the evaluated functionals of the respective class. However, since def2 basis sets for the 4th period are not associated to relativistic pseudo-potentials, we obtained significant errors in the covalent dimerization energies (up to 6 kcal mol-1) for molecules containing p-block elements of the 4th period. Significant improvements were achieved for systems containing 4th row elements by using ECP10MDF pseudopotentials along with re-contracted aug-cc-pVQZ-PP-KS basis sets introduced in this work with the contraction coefficients taken from atomic DFT (PBE0) calculations. Overall, the IHD302 set represents a challenge to contemporary quantum chemical methods. This is due to a large number of spatially close p-element bonds which are underrepresented in other benchmark sets, and the partial covalent bonding character for the WDA interactions. The IHD302 set may be helpful to develop more robust and transferable approximate quantum chemical methods in the future.

Local Second-Order Møller-Plesset Theory with a Single Threshold Using Orthogonal Virtual Orbitals: Theory, Implementation, and Assessment

It has long been clear that electron correlation methods exhibit unphysical compute scalings with molecular size, which has motivated the development of local correlation methods to discard effectively zero contributions in a controlled way to yield an approximate correlation energy. The ideal local correlation method should have a single numerical threshold that controls the dropping of terms with the ability to have that threshold set small enough so that the correlation energy is reproduced to enough significant figures such that the result is chemically identical. This work reports such a method for the second-order Møller-Plesset (MP2) theory. The theory, implementation, and testing of this local MP2 theory are reported. Thresholds ranging from 10-5 to 10-8 and basis sets ranging from split valence plus polarization through to quadruple-ζ are assessed for local MP2 calculations on a range of molecules, including linear chains and molecules with two- and three-dimensional character. The implementation is shared memory parallel via OpenMP and yields roughly 50% parallel efficiency with 16 cores for a large job. Considerable efforts were made to minimize memory demands, which increased as thresholds were tightened. A variety of relative energy calculations are presented as a function of threshold to provide some guidance to users on how to obtain adequate precision at a low compute cost. It is particularly clear that derivative properties require tighter thresholds in order to achieve an adequate precision.

Accurate Calculation of Isomerization and Conformational Energies of Larger Molecules Using Explicitly Correlated Local Coupled Cluster Methods in Molpro and ORCA

An overview of the approximations in the explicitly correlated local coupled cluster methods PNO-LCCSD(T)-F12 in Molpro and DLPNO-CCSD(T)F12 in ORCA is given. Options to select the domains of projected atomic orbitals (PAOs), pair natural orbitals (PNOs), and triples natural orbitals (TNOs) in both programs are described and compared in detail. The two programs are applied to compute isomerization and conformational energies of the ISOL24 and ACONFL test sets, where the former is part of the GMTKN55 benchmark suite. Thorough studies of basis set effects are presented for selected systems. These revealed large intramolecular basis set superposition effects that make it practically impossible to reliably determine the complete basis set (CBS) limits without including explicitly correlated terms. The latter strongly reduce the basis set dependence and at the same time also errors caused by the local domain approximations. On the basis of these studies, the PNO-LCCSD(T)-F12 method is applied to determine new reference energies for the above-mentioned benchmark sets. We are confident that our results should agree within a few tenths of a kcal mol-1 with the (unknown) CCSD(T)/CBS values, which therefore allowed us to define computational settings for accurate explicitly correlated local coupled cluster methods with moderate computational effort. With these protocols, especially PNO-LCCSD(T)-F12b/AVTZ', reliable reference values for comprehensive benchmark sets can be generated efficiently. This can significantly advance the development and evaluation of the performance of approximate electronic structure methods, especially improved density functional approximations or machine learning approaches.

Reduced Scaling Real-Time Coupled Cluster Theory

Real-time coupled cluster (CC) methods have several advantages over their frequency-domain counterparts, namely, response and equation of motion CC theories. Broadband spectra, strong fields, and pulse manipulation allow for the simulation of complex spectroscopies that are unreachable using frequency-domain approaches. Due to the high-order polynomial scaling, the required numerical time propagation of the CC residual expressions is a computationally demanding process. This scaling may be reduced by local correlation schemes, which aim to reduce the size of the (virtual) orbital space by truncation according to user-defined parameters. We present the first application of local correlation to real-time CC. As in previous studies of locally correlated frequency-domain CC, traditional local correlation schemes are of limited utility for field-dependent properties; however, a perturbation-aware scheme proves promising. A detailed analysis of the amplitude dynamics suggests that the main challenge is a strong time dependence of the wave function sparsity.

Corrigendum: Coupled cluster theory on modern heterogeneous supercomputers

[This corrects the article DOI: 10.3389/fchem.2023.1154526.].

Coupled cluster theory on modern heterogeneous supercomputers

This study examines the computational challenges in elucidating intricate chemical systems, particularly through ab-initio methodologies. This work highlights the Divide-Expand-Consolidate (DEC) approach for coupled cluster (CC) theory-a linear-scaling, massively parallel framework-as a viable solution. Detailed scrutiny of the DEC framework reveals its extensive applicability for large chemical systems, yet it also acknowledges inherent limitations. To mitigate these constraints, the cluster perturbation theory is presented as an effective remedy. Attention is then directed towards the CPS (D-3) model, explicitly derived from a CC singles parent and a doubles auxiliary excitation space, for computing excitation energies. The reviewed new algorithms for the CPS (D-3) method efficiently capitalize on multiple nodes and graphical processing units, expediting heavy tensor contractions. As a result, CPS (D-3) emerges as a scalable, rapid, and precise solution for computing molecular properties in large molecular systems, marking it an efficient contender to conventional CC models.

A local pair-natural orbital-based complete-active space perturbation theory using orthogonal localized virtual molecular orbitals

The multireference second-order perturbation theory (CASPT2) is known to deliver a quantitative description of various complex electronic states. Despite its near-size-consistent nature, the applicability of the CASPT2 method to large, real-life systems is mostly hindered by large computational and storage costs for the two-external tensors, such as two-electron integrals, amplitudes, and residuum. To this end, Menezes and co-workers developed a reduced-scaling CASPT2 scheme by incorporating the local pair-natural orbital (PNO) representation of the many-body wave functions using non-orthonormal projected atomic orbitals (PAOs) into the CASPT theory [F. Menezes et al., J. Chem. Phys. 145, 124115 (2016)]. Alternatively, in this paper, we develop a new PNO-based CASPT2 scheme using the orthonormal localized virtual molecular orbitals (LVMOs) and assess its performance and accuracy in comparison with the conventional PAO-based counterpart. Albeit the compactness, the LVMOs were considered to perform somewhat poorly compared to PAOs in the local correlation framework because they caused enormously large orbital domains. In this work, we show that the size of LVMO domains can be rendered comparable to or even smaller than that of PAOs by the use of the differential overlap integrals for domain construction. Optimality of the MOs from the CASSCF treatment is a key to reducing the LVMO domain size for the multireference case. Due to the augmented Hessian-based localization algorithm, an additional computational cost for obtaining the LVMOs is relatively minor. We demonstrate that the LVMO-based PNO-CASPT2 method is routinely applicable to large, real-life molecules such as Menshutkin SN2 reaction in a single-walled carbon nanotube reaction field.

DLPNO-MP2 second derivatives for the computation of polarizabilities and NMR shieldings

We present a derivation and efficient implementation of the formally complete analytic second derivatives for the domain-based local pair natural orbital second order Møller-Plesset perturbation theory (MP2) method, applicable to electric or magnetic field-response properties but not yet to harmonic frequencies. We also discuss the occurrence and avoidance of numerical instability issues related to singular linear equation systems and near linear dependences in the projected atomic orbital domains. A series of benchmark calculations on medium-sized systems is performed to assess the effect of the local approximation on calculated nuclear magnetic resonance shieldings and the static dipole polarizabilities. Relative deviations from the resolution of the identity-based MP2 (RI-MP2) reference for both properties are below 0.5% with the default truncation thresholds. For large systems, our implementation achieves quadratic effective scaling, is more efficient than RI-MP2 starting at 280 correlated electrons, and is never more than 5-20 times slower than the equivalent Hartree-Fock property calculation. The largest calculation performed here was on the vancomycin molecule with 176 atoms, 542 correlated electrons, and 4700 basis functions and took 3.3 days on 12 central processing unit cores.

PNO++: Perturbed Pair Natural Orbitals for Coupled Cluster Linear Response Theory

Reduced-scaling methods are needed to make accurate and systematically improvable coupled cluster linear response methods for the calculation of molecular properties tractable for large molecules. In this paper, we examine the perturbed pair natural orbital-based PNO++ approach that creates an orbital space optimized for response properties derived from a lower-cost field-perturbed density matrix. We analyze truncation errors in correlation energies, dynamic polarizabilities, and specific rotations from a coupled cluster singles and doubles (CCSD) reference. We find that incorporating a fixed number of orbitals from the pair natural orbital (PNO) space into the PNO++ method-a new method presented here, the "combined PNO++" approach-recovers accuracy in the CCSD correlation energy while preserving the well-behaved convergence behavior of the PNO++ method for linear response properties.

Scalable Electron Correlation Methods. 8. Explicitly Correlated Open-Shell Coupled-Cluster with Pair Natural Orbitals PNO-RCCSD(T)-F12 and PNO-UCCSD(T)-F12

We present explicitly correlated open-shell pair natural orbital local coupled-cluster methods, PNO-RCCSD(T)-F12 and PNO-UCCSD(T)-F12. The methods are extensions of our previously reported PNO-R/UCCSD methods (J. Chem. Theory Comput., 2020, 16, 3135-3151, https://pubs.acs.org/doi/10.1021/acs.jctc.0c00192) with additions of explicit correlation and perturbative triples corrections. The explicit correlation treatment follows the spin-orbital CCSD-F12b theory using Ansatz 3*A, which is found to yield comparable or better basis set convergence than the more rigorous Ansatz 3C in computed ionization potentials and reaction energies using double- to quaduple-ζ basis sets. The perturbative triples correction is adapted from the spin-orbital (T) theory to use triples natural orbitals (TNOs). To address the coupling due to off-diagonal Fock matrix elements, the local triples amplitudes are iteratively solved using small domains of TNOs, and a semicanonical (T0) domain correction with larger domains is applied to reduce the domain errors. The performance of the methods is demonstrated through benchmark calculations on ionization potentials, radical stabilization energies, reaction energies of fragmentations and rearrangements in radical cations, and spin-state energy differences of iron complexes. For a few test sets where canonical calculations are feasible, PNO-RCCSD(T)-F12 results agree with the canonical ones to within 0.4 kcal mol^-1, and this maximum error is reduced to below 0.2 kcal mol^-1 when large local domains are used. For larger systems, results using different thresholds for the local approximations are compared to demonstrate that 1 kcal mol^-1 level of accuracy can be achieved using our default settings. For a couple of difficult cases, it is demonstrated that the errors from individual approximations are only a fraction of 1 kcal mol^-1, and the overall accuracy of the method does not rely on error compensations. In contrast to canonical calculations, the use of spin-orbitals does not lead to a significant increase of computational time and memory usage in the most expensive steps of PNO-R/UCCSD(T)-F12 calculations. The only exception is the iterative solution of the (T) amplitudes, which can be avoided without significant errors by using a perturbative treatment of the off-diagonal coupling, known as (T1) approximation. For most systems, even the semicanonical approximation (T0) leads only to small errors in relative energies. Our program is well parallelized and capable of computing accurate correlation energies for molecules with 100-200 atoms using augmented triple-ζ basis sets in less than a day of elapsed time on a small computer cluster.

Explicitly correlated coupled cluster method for accurate treatment of open-shell molecules with hundreds of atoms

We present a near-linear scaling formulation of the explicitly correlated coupled-cluster singles and doubles with the perturbative triples method [CCSD(T)_F12¯] for high-spin states of open-shell species. The approach is based on the conventional open-shell CCSD formalism [M. Saitow et al., J. Chem. Phys. 146, 164105 (2017)] utilizing the domain local pair-natural orbitals (DLPNO) framework. The use of spin-independent set of pair-natural orbitals ensures exact agreement with the closed-shell formalism reported previously, with only marginally impact on the cost (e.g., the open-shell formalism is only 1.5 times slower than the closed-shell counterpart for the C₁₆₀H₃₂₂ n-alkane, with the measured size complexity of ≈1.2). Evaluation of coupled-cluster energies near the complete-basis-set (CBS) limit for open-shell systems with more than 550 atoms and 5000 basis functions is feasible on a single multi-core computer in less than 3 days. The aug-cc-pVTZ DLPNO-CCSD(T)_F12¯ contribution to the heat of formation for the 50 largest molecules among the 348 core combustion species benchmark set [J. Klippenstein et al., J. Phys. Chem. A 121, 6580-6602 (2017)] had root-mean-square deviation (RMSD) from the extrapolated CBS CCSD(T) reference values of 0.3 kcal/mol. For a more challenging set of 50 reactions involving small closed- and open-shell molecules [G. Knizia et al., J. Chem. Phys. 130, 054104 (2009)], the aug-cc-pVQ(+d)Z DLPNO-CCSD(T)_F12¯ yielded a RMSD of ∼0.4 kcal/mol with respect to the CBS CCSD(T) estimate.

Implementation of the iterative triples model CC3 for excitation energies using pair natural orbitals and Laplace transformation techniques

We present a pair natural orbital (PNO)-based implementation of CC3 excitation energies, which extends our previously published state-specific PNO ansatz for the solution of the excited state eigenvalue problem to methods including connected triple excitations. A thorough analysis of the equations for the excited state triples amplitudes is presented from which we derive a suitable state-specific triple natural orbital basis for the excited state triples amplitudes, which performs equally well for local and non-local excitations. The accuracy of the implementation is evaluated using a large and diverse test set. We find that for states with small contributions from double excitations, a T0 approximation to PNO-CC3 yields accurate results with a mean absolute error (MAE) for T_PNO = 10^-7 in the range of 0.02 eV. However, for states with larger double excitation contributions, the T0 approximation is found to yield significantly less accurate results, while the Laplace-transformed variant of PNO-CC3 shows a uniform accuracy for singly and doubly excited states (MAE and maximum error of 0.01 eV and 0.07 eV for T_PNO = 10^-7, respectively). Finally, we apply PNO-CC3 to the calculation of the first excited state of berenil at a S₁ minimum geometry, which is shown to be close to a conical intersection. This calculation in the aug-cc-pVTZ basis set (more than 1300 basis functions) is the largest calculation ever performed with CC3 on excitation energies.

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.

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.

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.

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.

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.

Crystal Field Splittings in Lanthanide Complexes: Inclusion of Correlation Effects beyond Second Order Perturbation Theory

State-averaged complete active space self-consistent field (CASSCF) calculations and a subsequent spin-orbit calculation mixing the CASSCF wave functions (CASSCF/state-interaction with spin-orbit coupling) is the conventional approach used for ab initio calculations of crystal-field splittings and magnetic properties of lanthanide complexes. However, this approach neglects dynamical correlation. Complete active space second-order perturbation theory (CASPT2) can be used to account for dynamical correlation but suffers from the well-known problems of multireference perturbation theory, e.g., intruder state problems. Variational multireference configuration interaction (MRCI) calculations do not show these problems but are usually not feasible due to the large size of real lanthanide complexes. Here, we present a quasi-local projected internally contracted MRCI approach which makes MRCI calculations of lanthanide complexes feasible and allows assessing the influence of dynamical correlation beyond second-order perturbation theory. We apply the method to two well-studied molecules, namely, [Er{N(SiMe₃)₂}₃] and {C(NH₂)₃}₅[Er(CO₃)₄]·11H₂O.

A pair natural orbital based implementation of CCSD excitation energies within the framework of linear response theory

We present a pair natural orbital (PNO)-based implementation of coupled cluster singles and doubles (CCSD) excitation energies that builds upon the previously proposed state-specific PNO approach to the excited state eigenvalue problem. We construct the excited state PNOs for each state separately in a truncated orbital specific virtual basis and use a local density-fitting approximation to achieve an at most quadratic scaling of the computational costs for the PNO construction. The earlier reported excited state PNO construction is generalized such that a smooth convergence of the results for charge transfer states is ensured for general coupled cluster methods. We investigate the accuracy of our implementation by applying it to a large and diverse test set comprising 153 singlet excitations in organic molecules. Already moderate PNO thresholds yield mean absolute errors below 0.01 eV. The performance of the implementation is investigated through the calculations on alkene chains and reveals an at most cubic cost-scaling for the CCSD iterations with the system size.

Combining Accuracy and Efficiency: An Incremental Focal-Point Method Based on Pair Natural Orbitals

In this work, we present a new pair natural orbitals (PNO)-based incremental scheme to calculate CCSD(T) and CCSD(T0) reaction, interaction, and binding energies. We perform an extensive analysis, which shows small incremental errors similar to previous non-PNO calculations. Furthermore, slight PNO errors are obtained by using T_PNO = T_TNO with appropriate values of 10^-7 to 10^-8 for reactions and 10^-8 for interaction or binding energies. The combination with the efficient MP2 focal-point approach yields chemical accuracy relative to the complete basis-set (CBS) limit. In this method, small basis sets (cc-pVDZ, def2-TZVP) for the CCSD(T) part are sufficient in case of reactions or interactions, while some larger ones (e.g., (aug)-cc-pVTZ) are necessary for molecular clusters. For these larger basis sets, we show the very high efficiency of our scheme. We obtain not only tremendous decreases of the wall times (i.e., factors >10²) due to the parallelization of the increment calculations as well as of the total times due to the application of PNOs (i.e., compared to the normal incremental scheme) but also smaller total times with respect to the standard PNO method. That way, our new method features a perfect applicability by combining an excellent accuracy with a very high efficiency as well as the accessibility to larger systems due to the separation of the full computation into several small increments.

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.

Correcting density-driven errors in projection-based embedding

Projection-based embedding provides a simple and numerically robust framework for multiscale wavefunction-in-density-functional-theory (WF-in-DFT) calculations. The approach works well when the approximate DFT is sufficiently accurate to describe the energetics of the low-level subsystem and the coupling between subsystems. It is also necessary that the low-level DFT produces a qualitatively reasonable description of the total density, and in this work, we study model systems where delocalization error prevents this from being the case. We find substantial errors in embedding calculations on open-shell doublet systems in which self-interaction errors cause spurious delocalization of the singly occupied orbital. We propose a solution to this error by evaluating the DFT energy using a more accurate self-consistent density, such as that of Hartree-Fock (HF) theory. These so-called WF-in-(HF-DFT) calculations show excellent convergence towards full-system wavefunction calculations.

SparseMaps--A systematic infrastructure for reduced-scaling electronic structure methods. IV. Linear-scaling second-order explicitly correlated energy with pair natural orbitals

We present a formulation of the explicitly correlated second-order Møller-Plesset (MP2-F12) energy in which all nontrivial post-mean-field steps are formulated with linear computational complexity in system size. The two key ideas are the use of pair-natural orbitals for compact representation of wave function amplitudes and the use of domain approximation to impose the block sparsity. This development utilizes the concepts for sparse representation of tensors described in the context of the domain based local pair-natural orbital-MP2 (DLPNO-MP2) method by us recently [Pinski et al., J. Chem. Phys. 143, 034108 (2015)]. Novel developments reported here include the use of domains not only for the projected atomic orbitals, but also for the complementary auxiliary basis set (CABS) used to approximate the three- and four-electron integrals of the F12 theory, and a simplification of the standard B intermediate of the F12 theory that avoids computation of four-index two-electron integrals that involve two CABS indices. For quasi-1-dimensional systems (n-alkanes), the ON DLPNO-MP2-F12 method becomes less expensive than the conventional ON(5) MP2-F12 for n between 10 and 15, for double- and triple-zeta basis sets; for the largest alkane, C200H402, in def2-TZVP basis, the observed computational complexity is N(∼1.6), largely due to the cubic cost of computing the mean-field operators. The method reproduces the canonical MP2-F12 energy with high precision: 99.9% of the canonical correlation energy is recovered with the default truncation parameters. Although its cost is significantly higher than that of DLPNO-MP2 method, the cost increase is compensated by the great reduction of the basis set error due to explicit correlation.

Sparse maps--A systematic infrastructure for reduced-scaling electronic structure methods. II. Linear scaling domain based pair natural orbital coupled cluster theory

Domain based local pair natural orbital coupled cluster theory with single-, double-, and perturbative triple excitations (DLPNO-CCSD(T)) is a highly efficient local correlation method. It is known to be accurate and robust and can be used in a black box fashion in order to obtain coupled cluster quality total energies for large molecules with several hundred atoms. While previous implementations showed near linear scaling up to a few hundred atoms, several nonlinear scaling steps limited the applicability of the method for very large systems. In this work, these limitations are overcome and a linear scaling DLPNO-CCSD(T) method for closed shell systems is reported. The new implementation is based on the concept of sparse maps that was introduced in Part I of this series [P. Pinski, C. Riplinger, E. F. Valeev, and F. Neese, J. Chem. Phys. 143, 034108 (2015)]. Using the sparse map infrastructure, all essential computational steps (integral transformation and storage, initial guess, pair natural orbital construction, amplitude iterations, triples correction) are achieved in a linear scaling fashion. In addition, a number of additional algorithmic improvements are reported that lead to significant speedups of the method. The new, linear-scaling DLPNO-CCSD(T) implementation typically is 7 times faster than the previous implementation and consumes 4 times less disk space for large three-dimensional systems. For linear systems, the performance gains and memory savings are substantially larger. Calculations with more than 20 000 basis functions and 1000 atoms are reported in this work. In all cases, the time required for the coupled cluster step is comparable to or lower than for the preceding Hartree-Fock calculation, even if this is carried out with the efficient resolution-of-the-identity and chain-of-spheres approximations. The new implementation even reduces the error in absolute correlation energies by about a factor of two, compared to the already accurate previous implementation.