Speeding up Hartree-Fock and Kohn-Sham calculations with first-order corrections

Linear-Scaling Local Natural Orbital CCSD(T) Approach for Open-Shell Systems: Algorithms, Benchmarks, and Large-Scale Applications

The extension of the highly optimized local natural orbital (LNO) coupled cluster (CC) with single-, double-, and perturbative triple excitations [LNO-CCSD(T)] method is presented for high-spin open-shell molecules based on restricted open-shell references. The techniques enabling the outstanding efficiency of the closed-shell LNO-CCSD(T) variant are adopted, including the iteration- and redundancy-free second-order Møller-Plesset and (T) formulations as well as the integral-direct, memory- and disk use-economic, and OpenMP-parallel algorithms. For large molecules, the efficiency of our open-shell LNO-CCSD(T) method approaches that of its closed-shell parent method due to the application of restricted orbital sets for demanding integral transformations and a novel approximation for higher-order long-range spin-polarization effects. The accuracy of open-shell LNO-CCSD(T) is extensively tested for radicals and reactions thereof, ionization processes, as well as spin-state splittings, and transition-metal compounds. At the size range where the canonical CCSD(T) reference is accessible (up to 20-30 atoms), the average open-shell LNO-CCSD(T) correlation energies are found to be 99.9 to 99.95% accurate, which translates into average absolute deviations of a few tenths of kcal/mol in the investigated energy differences already with the default settings. For more extensive molecules, the local errors may grow, but they can be estimated and decreased via affordable systematic convergence studies. This enables the accurate modeling of large systems with complex electronic structures, as illustrated on open-shell organic radicals and transition-metal complexes of up to 179 atoms as well as on challenging biochemical systems, including up to 601 atoms and 11,000 basis functions. While the protein models involve difficulties for local approximations, such as the spin states of a bounded iron ion or an extremely delocalized singly occupied orbital, the corresponding single-node LNO-CCSD(T) computations were feasible in a matter of days with 10s to 100 GB of memory use. Therefore, the new LNO-CCSD(T) implementation enables highly accurate computations for open-shell systems of unprecedented size and complexity with widely accessible hardware.

Analytic gradients for local density fitting Hartree-Fock and Kohn-Sham methods

We present analytic gradients for local density fitting Hartree-Fock (HF) and hybrid Kohn-Sham (KS) density functional methods. Due to the non-variational nature of the local fitting algorithm, the method of Lagrange multipliers is used to avoid the solution of the coupled perturbed HF and KS equations. We propose efficient algorithms for the solution of the arising Z-vector equations and the gradient calculation that preserve the third-order scaling and low memory requirement of the original local fitting algorithm. In order to demonstrate the speed and accuracy of our implementation, gradient calculations and geometry optimizations are presented for various molecular systems. Our results show that significant speedups can be achieved compared to conventional density fitting calculations without sacrificing accuracy.

Performance of Multilevel Methods for Excited States

The performance of multilevel quantum chemical approaches, which utilize an atom-based system partitioning scheme to model various electronic excited states, is studied. The considered techniques include the mechanical-embedding (ME) of “our own N-layered integrated molecular orbital and molecular mechanics” (ONIOM) method, the point charge embedding (PCE), the electronic-embedding (EE) of ONIOM, the frozen density-embedding (FDE), the projector-based embedding (PbE), and our local domain-based correlation method. For the investigated multilevel approaches, the second-order algebraic-diagrammatic construction [ADC(2)] approach was utilized as the high-level method, which was embedded in either Hartree–Fock or a density functional environment. The XH-27 test set of Zech et al. [J. Chem. Theory Comput., 2018, 14, 402829906111] was used for the assessment, where organic dyes interact with several solvent molecules. With the selection of the chromophores as active subsystems, we conclude that the most reliable approach is local domain-based ADC(2) [L-ADC(2)], and the least robust schemes are ONIOM-ME and ONIOM-EE. The PbE, FDE, and PCE techniques often approach the accuracy of the L-ADC(2) scheme, but their precision is far behind. The results suggest that a more conservative subsystem selection algorithm or the inclusion of subsystem charge-transfers is required for the atom-based cost-efficient methods to produce high-accuracy excitation energies.

Straightforward and Accurate Automatic Auxiliary Basis Set Generation for Molecular Calculations with Atomic Orbital Basis Sets

Density fitting (DF), also known as the resolution of the identity (RI), is a widely used technique in quantum chemical calculations with various types of atomic basis sets─Gaussian-type orbitals, Slater-type orbitals, as well as numerical atomic orbitals─to speed up density functional, Hartree-Fock (HF), and post-HF calculations. Traditionally, custom auxiliary basis sets are hand-optimized for each orbital basis set; however, some automatic schemes have also been suggested. In this work, we propose a simple yet numerically stable automated scheme for forming auxiliary basis sets with the help of a pivoted Cholesky decomposition, which is applicable to any type of atomic basis function. We exemplify the scheme with proof-of-concept calculations with Gaussian basis sets and show that the proposed approach leads to negligible DF/RI errors in HF and second-order Møller-Plesset (MP2) total energies of the non-multireference part of the W4-17 test set when used with orbital basis sets of at least polarized triple-ζ quality. The results are promising for future applications employing Slater-type orbitals or numerical atomic orbitals, as well as schemes based on the use of local fitting approaches. Global fitting approaches can also be used, in which case the high-angular-momentum functions produced by the present scheme can be truncated to minimize the computational cost.

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.