Sparse tensor framework for implementation of general local correlation methods

Code generation in ORCA: progress, efficiency and tight integration

An improved version of ORCA's automated generator environment (ORCA-AGE II) is presented. The algorithmic improvements and the move to C++ as the programming language lead to a performance gain of up to two orders of magnitude compared to the previously developed PYTHON toolchain. Additionally, the restructured modular design allows for far more complex code engines to be implemented readily. Importantly, we have realised an extremely tight integration with the ORCA host program. This allows for a workflow in which only the wavefunction Ansatz is part of the source code repository while all actual high-level code is generated automatically, inserted at the appropriate place in the host program before it is compiled and linked together with the hand written code parts. This construction ensures longevity and uniform code quality. Furthermore the new developments allow ORCA-AGE II to generate parallelised production-level code for highly complex theories, such as fully internally contracted multireference coupled-cluster theory (fic-MRCC) with an enormous number of contributing tensor contractions. We also discuss the automated implementation of nuclear gradients for arbitrary theories. All these improvements enable the implementation of theories that are too complex for the human mind and also reduce development times by orders of magnitude. We hope that this work enables researchers to concentrate on the intellectual content of the theories they develop rather than be concerned with technical details of the implementation.

Automatic derivation of many-body theories based on general Fermi vacua

This paper describes Wick&d, an implementation of the algebra of second-quantized operators normal ordered with respect to general correlated references and the corresponding Wick theorem [D. Mukherjee, Chem. Phys. Lett. 274, 561 (1997) and W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 107, 432 (1997)]. Wick&d employs a compact representation of operators and a backtracking algorithm to efficiently evaluate Wick contractions. Since Wick&d can handle both fully and partially contracted terms, it can be applied to both projective and Fock-space many-body formalisms. To demonstrate the usefulness of  Wick&d, we use it to evaluate the single-reference coupled cluster equations up to octuple excitations and report an automated derivation and implementation of the second-order driven similarity renormalization group multireference perturbation theory.

Full Configuration Interaction Quantum Monte Carlo treatment of fragments embedded in a periodic mean field

We present an embedded fragment approach for high-level quantum chemical calculations on local features in periodic systems. The fragment is defined as a set of localized orbitals (occupied and virtual) corresponding to a converged periodic Hartree-Fock solution. These orbitals serve as the basis for the in-fragment post-Hartree Fock treatment. The embedding field for the fragment, consisting of the Coulomb and exchange potential from the rest of the crystal, is included in the fragment's one-electron Hamiltonian. As an application of the embedded fragment approach we investigate the performanceof full configuration interaction quantum Monte Carlo (FCIQMC) with the adaptive shift. As the orbital choice we use the natural orbitals from the distinguishable cluster method with singles and doubles. FCIQMC is a stochastic approximation to the full CI method and can be routinely applied to much larger active spaces than the latter. This makes this method especially attractive in the context of open shell defects in crystals, where fragments of adequate size can be ratherlarge. As a test case we consider dissociation of a fluorine atom from a fluorographane surface. This process poses a challenge for high-level electronic structure models as both the static and dynamic correlations are essential here. Furthermore the active space for an adequate fragment (32 electrons in 173 orbitals) is already quite large even for FCIQMC. Despite this, FCIQMC delivers accurate dissociation and total energies.

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.

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.

Multireference Electron Correlation Methods: Journeys along Potential Energy Surfaces

Multireference electron correlation methods describe static and dynamical electron correlation in a balanced way and, therefore, can yield accurate and predictive results even when single-reference methods or multiconfigurational self-consistent field theory fails. One of their most prominent applications in quantum chemistry is the exploration of potential energy surfaces. This includes the optimization of molecular geometries, such as equilibrium geometries and conical intersections and on-the-fly photodynamics simulations, both of which depend heavily on the ability of the method to properly explore the potential energy surface. Because such applications require nuclear gradients and derivative couplings, the availability of analytical nuclear gradients greatly enhances the scope of quantum chemical methods. This review focuses on the developments and advances made in the past two decades. A detailed account of the analytical nuclear gradient and derivative coupling theories is presented. Emphasis is given to the software infrastructure that allows one to make use of these methods. Notable applications of multireference electron correlation methods to chemistry, including geometry optimizations and on-the-fly dynamics, are summarized at the end followed by a discussion of future prospects.

An optimized twist angle to find the twist-averaged correlation energy applied to the uniform electron gas

We explore an alternative to twist averaging in order to obtain more cost-effective and accurate extrapolations to the thermodynamic limit (TDL) for coupled cluster doubles (CCD) calculations. We seek a single twist angle to perform calculations at, instead of integrating over many random points or a grid. We introduce the concept of connectivity, a quantity derived from the nonzero four-index integrals in an MP2 calculation. This allows us to find a special twist angle that provides appropriate connectivity in the energy equation, which yields results comparable to full twist averaging. This special twist angle effectively makes the finite electron number CCD calculation represent the TDL more accurately, reducing the cost of twist-averaged CCD over N_s twist angles from N_s CCD calculations to N_s MP2 calculations plus one CCD calculation.

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.

Understanding non-covalent interactions in larger molecular complexes from first principles

Non-covalent interactions pervade all matter and play a fundamental role in layered materials, biological systems, and large molecular complexes. Despite this, our accumulated understanding of non-covalent interactions to date has been mainly developed in the tens-of-atoms molecular regime. This falls considerably short of the scales at which we would like to understand energy trends, structural properties, and temperature dependencies in materials where non-covalent interactions have an appreciable role. However, as more reference information is obtained beyond moderately sized molecular systems, our understanding is improving and we stand to gain pertinent insights by tackling more complex systems, such as supramolecular complexes, molecular crystals, and other soft materials. In addition, accurate reference information is needed to provide the drive for extending the predictive power of more efficient workhorse methods, such as density functional approximations that also approximate van der Waals dispersion interactions. In this perspective, we discuss the first-principles approaches that have been used to obtain reference interaction energies for beyond modestly sized molecular complexes. The methods include quantum Monte Carlo, symmetry-adapted perturbation theory, non-canonical coupled cluster theory, and approaches based on the random-phase approximation. By considering the approximations that underpin each method, the most accurate theoretical references for supramolecular complexes and molecular crystals to date are ascertained. With these, we also assess a handful of widely used exchange-correlation functionals in density functional theory. The discussion culminates in a framework for putting into perspective the accuracy of high-level wavefunction-based methods and identifying future challenges.

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.

Novel strategy to implement active-space coupled-cluster methods

A new approach is presented for the efficient implementation of coupled-cluster (CC) methods including higher excitations based on a molecular orbital space partitioned into active and inactive orbitals. In the new framework, the string representation of amplitudes and intermediates is used as long as it is beneficial, but the contractions are evaluated as matrix products. Using a new diagrammatic technique, the CC equations are represented in a compact form due to the string notations we introduced. As an application of these ideas, a new automated implementation of the single-reference-based multi-reference CC equations is presented for arbitrary excitation levels. The new program can be considered as an improvement over the previous implementations in many respects; e.g., diagram contributions are evaluated by efficient vectorized subroutines. Timings for test calculations for various complete active-space problems are presented. As an application of the new code, the weak interactions in the Be dimer were studied.

Exfoliation Energy of Black Phosphorus Revisited: A Coupled Cluster Benchmark

Black phosphorus (black-P) consists of phosphorene sheets, stacked by van der Waals dispersion. In a recent study based on periodic local second-order Møller-Plesset perturbation theory (LMP2) with higher-order corrections evaluated on finite clusters, we obtained a value of -151 meV/atom for the exfoliation energy. This is almost twice as large as another recent theoretical result (around -80 meV/atom) obtained with quantum Monte Carlo (QMC). Here, we revisit this system on the basis of the recently implemented, periodically embedded ring coupled cluster (rCCD) model instead of LMP2. Higher-order coupled cluster corrections on top of rCCD are obtained from finite clusters by utilizing our new "unit-cell-in-cluster" scheme. Our new value of -92 meV/atom is noticeably lower than that based on LMP2 and in reasonably close agreement with the QMC result. However, in contrast to QMC, no strong effect from the second-neighbor and farther layers in black-P are observed in our calculations.

The distinguishable cluster approach from a screened Coulomb formalism

The distinguishable cluster doubles equations have been derived starting from an effective screened Coulomb formalism and a particle-hole symmetric formulation of the Fock matrix. A perturbative triples correction to the distinguishable cluster with singles and doubles (DCSD) has been introduced employing the screened integrals. It is shown that the resulting DCSD(T) method is more accurate than DCSD for reaction energies and is less sensitive to the static correlation than coupled cluster with singles and doubles with a perturbative triples correction.

Electron-correlated fragment-molecular-orbital calculations for biomolecular and nano systems

Recent developments in the fragment molecular orbital (FMO) method for theoretical formulation, implementation, and application to nano and biomolecular systems are reviewed. The FMO method has enabled ab initio quantum-mechanical calculations for large molecular systems such as protein-ligand complexes at a reasonable computational cost in a parallelized way. There have been a wealth of application outcomes from the FMO method in the fields of biochemistry, medicinal chemistry and nanotechnology, in which the electron correlation effects play vital roles. With the aid of the advances in high-performance computing, the FMO method promises larger, faster, and more accurate simulations of biomolecular and related systems, including the descriptions of dynamical behaviors in solvent environments. The current status and future prospects of the FMO scheme are addressed in these contexts.

An efficient linear-scaling CCSD(T) method based on local natural orbitals

An improved version of our general-order local coupled-cluster (CC) approach [Z. Rolik and M. Kállay, J. Chem. Phys. 135, 104111 (2011)] and its efficient implementation at the CC singles and doubles with perturbative triples [CCSD(T)] level is presented. The method combines the cluster-in-molecule approach of Li and co-workers [J. Chem. Phys. 131, 114109 (2009)] with frozen natural orbital (NO) techniques. To break down the unfavorable fifth-power scaling of our original approach a two-level domain construction algorithm has been developed. First, an extended domain of localized molecular orbitals (LMOs) is assembled based on the spatial distance of the orbitals. The necessary integrals are evaluated and transformed in these domains invoking the density fitting approximation. In the second step, for each occupied LMO of the extended domain a local subspace of occupied and virtual orbitals is constructed including approximate second-order Mo̸ller-Plesset NOs. The CC equations are solved and the perturbative corrections are calculated in the local subspace for each occupied LMO using a highly-efficient CCSD(T) code, which was optimized for the typical sizes of the local subspaces. The total correlation energy is evaluated as the sum of the individual contributions. The computation time of our approach scales linearly with the system size, while its memory and disk space requirements are independent thereof. Test calculations demonstrate that currently our method is one of the most efficient local CCSD(T) approaches and can be routinely applied to molecules of up to 100 atoms with reasonable basis sets.