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

Local Wave Function Embedding: Correlation Regions in PNO-LCCSD(T)-F12 Calculations

Many chemical reactions affect only a rather small number of bonds, leaving the largest part of the chemical and geometrical structure of the molecules nearly unchanged. In this work we extended the previously proposed region method [J. Chem. Phys. 128, 144106 (2008)] to PNO-LCCSD(T)-F12. Using this method, we investigate whether accurate reaction energies for larger systems can be obtained by correlating only the electrons in a region of localized molecular orbitals close to the reaction center at high-level (PNO-LCCSD(T)-F12). The remainder is either treated at lower level (PNO-LMP2-F12) or left uncorrelated (Hartree-Fock frozen core). It is demonstrated that indeed the computed reaction energies converge rather quickly with the size of the correlation regions toward the results of the full calculations. Typically, 2-3 bonds from the reacting atoms need to be included to reproduce the results of the full calculations to within ±0.2 kcal/mol. We also computed spin-state energy differences in a large transition metal complex, where a factor of 15 in computation time could be saved, still yielding a result that is within ±0.1 kcal/mol of the one obtained in a full PNO-LCCSD(T)-F12 calculation.

Improved CPS and CBS Extrapolation of PNO-CCSD(T) Energies: The MOBH35 and ISOL24 Data Sets

Computation of heats of reaction of large molecules is now feasible using the domain-based pair natural orbital (PNO)-coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] theory. However, to obtain agreement within 1 kcal/mol of experiment, it is necessary to eliminate basis set incompleteness error, which comprises both the AO basis set error and the PNO truncation error. Our investigation into the convergence to the canonical limit of PNO-CCSD(T) energies with the PNO truncation threshold T shows that errors follow the model E ( T ) = E + A T 1 / 2 . Therefore, PNO truncation errors can be eliminated using a simple two-point CPS extrapolation to the canonical limit so that subsequent CBS extrapolation is not limited by the residual PNO truncation error. Using the ISOL24 and MOBH35 data sets, we find that PNO truncation errors are larger for molecules with significant static correlation and that it is necessary to use very tight thresholds of T = 10 - 8 to ensure that errors do not exceed 1 kcal/mol. We present a lower-cost extrapolation scheme that uses information from small basis sets to estimate the PNO truncation errors for larger basis sets. In this way, the canonical limit of CCSD(T) calculations on sizable molecules with large basis sets can be reliably estimated in a practical way. Using this approach, we report near complete basis set (CBS)-CCSD(T) reaction energies for the full ISOL24 and MOBH35 data sets.

Possible Roles of Transition Metal Cations in the Formation of Interstellar Benzene via Catalytic Acetylene Cyclotrimerization

Polycyclic aromatic hydrocarbons (PAHs) are ubiquitous interstellar molecules. However, the formation mechanisms of PAHs and even the simplest cyclic aromatic hydrocarbon, benzene, are not yet fully understood. Recently, we reported the statistical and dynamical properties in the reaction mechanism of Fe+-catalyzed acetylene cyclotrimerization, whereby three acetylene molecules are directly converted to benzene. In this study, we extended our previous work and explored the possible role of the complex of other 3d transition metal cations, TM+ (TM = Sc, Ti, Mn, Co, and Ni), as a catalyst in acetylene cyclotrimerization. Potential energy profiles for bare TM+-catalyst (TM = Sc and Ti), for TM+NC--catalyst (TM = Sc, Ti, Mn, Co, and Ni), and for TM+-(H2O)8-catalyst (TM = Sc and Ti) systems were obtained using quantum chemistry calculations, including the density functional theory levels. The calculation results show that the scandium and titanium cations act as efficient catalysts in acetylene cyclotrimerization and that reactants, which contain an isolated acetylene and (C2H2)2 bound to a bare (ligated) TM cation (TM = Sc and Ti), can be converted into a benzene-metal-cation product complex without an entrance barrier. We found that the number of electrons in the 3d orbitals of the transition metal cation significantly contributes to the catalytic efficiency in the acetylene cyclotrimerization process. On-the-fly Born-Oppenheimer molecular dynamics (BOMD) simulations of the Ti+-NC- and Ti+-(H2O)8 complexes were also performed to comprehensively understand the nuclear dynamics of the reactions. The computational results suggest that interstellar benzene can be produced via acetylene cyclotrimerization reactions catalyzed by transition metal cation complexes.

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.

Excited-state van der Waals potential energy surfaces for the NO A2Σ+ + CO2X1Σg+ collision complex

Excited state van der Waals (vdW) potential energy surfaces (PESs) of the NO A2Σ+ + CO2X1Σg+ system are thoroughly investigated using coupled cluster theory and complete active space perturbation theory to second order (CASPT2). First, it is shown that pair natural orbital coupled cluster singles and doubles with perturbative triples yields comparable accuracy compared to CCSD(T) for molecular properties and vdW-minima at a fraction of computational cost of the latter. Using this method in conjunction with highly diffuse basis sets and counterpoise correction for basis set superposition error, the PESs for different intermolecular orientations are investigated. These show numerous vdW-wells, interconnected for all geometries except one, with a maximum depth of up to 830 cm-1; considerably deeper than those on the ground state surface. Multi-reference effects are investigated with CASPT2 calculations. The long-range vdW-surfaces support recent experimental observations relating to rotational energy transfer due the anisotropy in the potentials.

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.

A combined first- and second-order optimization method for improving convergence of Hartree–Fock and Kohn–Sham calculations

In this work, we investigate the optimization of Hartree–Fock (HF) orbitals with our recently proposed combined first- and second-order (SO-SCI) method, which was originally developed for multi-configuration self-consistent field (MCSCF) and complete active space SCF (CASSCF) calculations. In MCSCF/CASSCF, it unites a second-order optimization of the active orbitals with a Fock-based first-order treatment of the remaining closed-virtual orbital rotations. In the case of the single-determinant wavefunctions, the active space is replaced by a preselected “second-order domain,” and all rotations involving orbitals in this subspace are treated at second-order. The method has been implemented for spin-restricted and spin-unrestricted Hartree–Fock (RHF, UHF), configuration-averaged Hartree–Fock (CAHF), as well as Kohn–Sham (KS) density functional theory (RKS, UKS). For each of these cases, various choices of the second-order domain have been tested, and appropriate defaults are proposed. The performance of the method is demonstrated for several transition metal complexes. It is shown that the SO-SCI optimization provides faster and more robust convergence than the standard SCF procedure but requires, in many cases, even less computation time. In difficult cases, the SO-SCI method not only speeds up convergence but also avoids convergence to saddle-points. Furthermore, it helps to find spin-symmetry broken solutions in the cases of UHF or UKS. In the case of CAHF, convergence can also be significantly improved as compared to a previous SCF implementation. This is particularly important for multi-center cases with two or more equal heavy atoms. The performance is demonstrated for various two-center complexes with different lanthanide atoms.

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.

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.

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.