Atomic orbital-based SOS-MP2 with tensor hypercontraction. II. Local tensor hypercontraction

High-Performance Multi-GPU Analytic RI-MP2 Energy Gradients

This article presents a novel algorithm for the calculation of analytic energy gradients from second-order Møller-Plesset perturbation theory within the Resolution-of-the-Identity approximation (RI-MP2), which is designed to achieve high performance on clusters with multiple graphical processing units (GPUs). The algorithm uses GPUs for all major steps of the calculation, including integral generation, formation of all required intermediate tensors, solution of the Z-vector equation and gradient accumulation. The implementation in the EXtreme Scale Electronic Structure System (EXESS) software package includes a tailored, highly efficient, multistream scheduling system to hide CPU-GPU data transfer latencies and allows nodes with 8 A100 GPUs to operate at over 80% of theoretical peak floating-point performance. Comparative performance analysis shows a significant reduction in computational time relative to traditional multicore CPU-based methods, with our approach achieving up to a 95-fold speedup over the single-node performance of established software such as Q-Chem and ORCA. Additionally, we demonstrate that pairing our implementation with the molecular fragmentation framework in EXESS can drastically lower the computational scaling of RI-MP2 gradient calculations from quintic to subquadratic, enabling further substantial savings in runtime while retaining high numerical accuracy in the resulting gradients.

Tensor Hypercontraction of Cluster Perturbation Theory: Quartic Scaling Perturbation Series for the Coupled Cluster Singles and Doubles Ground-State Energies

Even though cluster perturbation theory has been shown to be a robust noniterative alternative to coupled cluster theory, it is still plagued by high order polynomial computational scaling and the storage of higher order tensors. We present a proof-of-concept strategy for implementing a cluster perturbation theory ground-state energy series for the coupled cluster singles and doubles energy with N4 computational scaling using tensor hypercontraction (THC). The reduction in computational scaling by two orders is achieved by decomposing two electron repulsion integrals, doubles amplitudes and multipliers, as well as selected double intermediates to the THC format. Using the outlined strategy, we showcase that the THC pilot implementations retain numerical accuracy to within 1 kcal/mol relative to corresponding conventional and density fitting implementations, and we empirically verify the N4 scaling.

Low-Scaling Algorithm for the Random Phase Approximation Using Tensor Hypercontraction with k-point Sampling

We present a low-scaling algorithm for the random phase approximation (RPA) with k-point sampling in the framework of tensor hypercontraction (THC) for electron repulsion integrals (ERIs). The THC factorization is obtained via a revised interpolative separable density fitting (ISDF) procedure with a momentum-dependent auxiliary basis for generic single-particle Bloch orbitals. Our formulation does not require preoptimized interpolating points or auxiliary bases, and the accuracy is systematically controlled by the number of interpolating points. The resulting RPA algorithm scales linearly with the number of k-points and cubically with the system size without any assumption on sparsity or locality of orbitals. The errors of ERIs and RPA energy show rapid convergence with respect to the size of the THC auxiliary basis, suggesting a promising and robust direction to construct efficient algorithms of higher order many-body perturbation theories for large-scale systems.

Tensor-Hypercontracted MP2 First Derivatives: Runtime and Memory Efficient Computation of Hyperfine Coupling Constants

We employ our recently introduced tensor-hypercontracted (THC) second-order Møller-Plesset perturbation theory (MP2) method [Bangerter, F. H., Glasbrenner, M., Ochsenfeld, C. J. Chem. Theory Comput. 2021, 17, 211-221] for the computation of hyperfine coupling constants (HFCCs). The implementation leverages the tensor structure of the THC factorized electron repulsion integrals for an efficient formation of the integral-based intermediates. The computational complexity of the most expensive and formally quintic scaling exchange-like contribution is reduced to effectively subquadratic, by making use of the intrinsic, exponentially decaying coupling between tensor indices through screening based on natural blocking. Overall, this yields an effective subquadratic scaling with a low prefactor for the presented THC-based AO-MP2 method for the computation of isotropic HFCCs on DNA fragments with up to 500 atoms and 5000 basis functions. Furthermore, the implementation achieves considerable speedups with up to a factor of roughly 600-1000 compared to previous implementations [Vogler, S., Ludwig, M., Maurer, M., Ochsenfeld, C. J. Chem. Phys. 2017, 147, 024101] for medium-sized organic radicals, while also significantly reducing storage requirements.

Do Double-Hybrid Functionals Benefit from Regularization in the PT2 Term? Observations from an Extensive Benchmark

We put to the test a recent suggestion [Shee, J., et al. J. Phys. Chem. Lett. 2021, 12 (50), 12084-12097] that MP2 regularization might improve the performance of double-hybrid density functionals. Using the very large and chemically diverse GMTKN55 benchmark, we find that κ-regularization is indeed beneficial at lower percentages of Hartree-Fock exchange, especially if spin-component scaling is not applied [such as in B2GP-PLYP or ωB97M(2)]. This benefit dwindles for DSD and DOD functionals and vanishes entirely in the ∼70% HF exchange region optimal for them.

Near-Exact CCSDT Energetics from Rank-Reduced Formalism Supplemented by Non-iterative Corrections

We introduce a non-iterative energy correction, added on top of the rank-reduced coupled-cluster method with single, double, and triple substitutions, that accounts for excitations excluded from the parent triple excitation subspace. The formula for the correction is derived by employing the coupled-cluster Lagrangian formalism, with an additional assumption that the parent excitation subspace is closed under the action of the Fock operator. Owing to the rank-reduced form of the triple excitation amplitudes tensor, the computational cost of evaluating the correction scales as N⁷, where N is the system size. The accuracy and computational efficiency of the proposed method is assessed for both total and relative correlation energies. We show that the non-iterative correction can fulfill two separate roles. If the accuracy level of a fraction of kJ/mol is sufficient for a given system, the correction significantly reduces the dimension of the parent triple excitation subspace needed in the iterative part of the calculations. Simultaneously, it enables reproducing the exact CCSDT results to an accuracy level below 0.1 kJ/mol, with a larger, yet still reasonable, dimension of the parent excitation subspace. This typically can be achieved at a computational cost only several times larger than required for the CCSD(T) method. The proposed method retains the black-box features of the single-reference coupled-cluster theory; the dimension of the parent excitation subspace remains the only additional parameter that has to be specified.

A diagrammatic approach for automatically deriving analytical gradients of tensor hyper-contracted electronic structure methods

We introduce a diagrammatic approach to facilitate the automatic derivation of analytical nuclear gradients for tensor hyper-contraction (THC) based electronic structure methods. The automatically derived gradients are guaranteed to have the same scaling in terms of both operation count and memory footprint as the underlying energy calculations, and the computation of a gradient is roughly three times as costly as the underlying energy. The new diagrammatic approach enables the first cubic scaling implementation of nuclear derivatives for THC tensors fitted in molecular orbital basis (MO-THC). Furthermore, application of this new approach to THC-MP2 analytical gradients leads to an implementation, which is at least four times faster than the previously reported, manually derived implementation. Finally, we apply the new approach to the 14 tensor contraction patterns appearing in the supporting subspace formulation of multireference perturbation theory, laying the foundation for developments of analytical nuclear gradients and nonadiabatic coupling vectors for multi-state CASPT2.

Low-Scaling Tensor Hypercontraction in the Cholesky Molecular Orbital Basis Applied to Second-Order Møller-Plesset Perturbation Theory

We employ various reduced scaling techniques to accelerate the recently developed least-squares tensor hypercontraction (LS-THC) approximation [Parrish, R. M., Hohenstein, E. G., Martínez, T. J., Sherrill, C. D. J. Chem. Phys. 137, 224106 (2012)] for electron repulsion integrals (ERIs) and apply it to second-order Møller-Plesset perturbation theory (MP2). The grid-projected ERI tensors are efficiently constructed using a localized Cholesky molecular orbital basis from density-fitted integrals with an attenuated Coulomb metric. Additionally, rigorous integral screening and the natural blocking matrix format are applied to reduce the complexity of this step. By recasting the equations to form the quantized representation of the 1/r operator Z into the form of a system of linear equations, the bottleneck of inverting the grid metric via pseudoinversion is removed. This leads to a reduced scaling THC algorithm and application to MP2 yields the (sub-)quadratically scaling THC-ω-RI-CDD-SOS-MP2 method. The efficiency of this method is assessed for various systems including DNA fragments with over 8000 basis functions and the subquadratic scaling is illustrated.

Double hybrid DFT calculations with Slater type orbitals

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

Reduced scaling extended multi-state CASPT2 (XMS-CASPT2) using supporting subspaces and tensor hyper-contraction

We present a reduced scaling formulation of the extended multi-state CASPT2 (XMS-CASPT2) method, which is based on our recently developed state-specific CASPT2 (SS-CASPT2) formulation using supporting subspaces and tensor hyper-contraction. By using these two techniques, the off-diagonal elements of the effective Hamiltonian can be computed with only O(N³) operations and O(N²) memory, where N is the number of basis functions. This limits the overall computational scaling to O(N⁴) operations and O(N²) memory. Thus, excited states can now be obtained at the same reduced (relative to previous algorithms) scaling we achieved for SS-CASPT2. In addition, we also investigate how the energy denominators can be factorized with the Laplace quadrature when some of the denominators are negative, which is critical for excited state calculations. An efficient implementation of the method has been developed using graphical processing units while also exploiting spatial sparsity in tensor operations. We benchmark the accuracy of the new method by comparison to non-THC formulated XMS-CASPT2 for the excited states of various molecules. In our tests, the THC approximation introduces negligible errors (≈0.01 eV) compared to the non-THC reference method. Scaling behavior and computational timings are presented to demonstrate performance. The new method is also interfaced with quantum mechanics/molecular mechanics (QM/MM). In an example study of green fluorescent protein, we show how the XMS-CASPT2 potential energy surfaces and excitation energies are affected by increasing the size of the QM region up to 278 QM atoms with more than 2300 basis functions.

TeraChem: Accelerating electronic structure and ab initio molecular dynamics with graphical processing units

Developed over the past decade, TeraChem is an electronic structure and ab initio molecular dynamics software package designed from the ground up to leverage graphics processing units (GPUs) to perform large-scale ground and excited state quantum chemistry calculations in the gas and the condensed phase. TeraChem's speed stems from the reformulation of conventional electronic structure theories in terms of a set of individually optimized high-performance electronic structure operations (e.g., Coulomb and exchange matrix builds, one- and two-particle density matrix builds) and rank-reduction techniques (e.g., tensor hypercontraction). Recent efforts have encapsulated these core operations and provided language-agnostic interfaces. This greatly increases the accessibility and flexibility of TeraChem as a platform to develop new electronic structure methods on GPUs and provides clear optimization targets for emerging parallel computing architectures.

A Quadratic Pair Atomic Resolution of the Identity Based SOS-AO-MP2 Algorithm Using Slater Type Orbitals

We report a production level implementation of pair atomic resolution of the identity (PARI) based second-order Møller-Plesset perturbation theory (MP2) in the Slater type orbital (STO) based Amsterdam Density Functional (ADF) code. As demonstrated by systematic benchmarks, dimerization and isomerization energies obtained with our code using STO basis sets of triple-ζ-quality show mean absolute deviations from Gaussian type orbital, canonical, basis set limit extrapolated, global density fitting (DF)-MP2 results of less than 1 kcal/mol. Furthermore, we introduce a quadratic scaling atomic orbital based spin-opposite-scaled (SOS)-MP2 approach with a very small prefactor. Due to a worst-case scaling of [Formula: see text], our implementation is very fast already for small systems and shows an exceptionally early crossover to canonical SOS-PARI-MP2. We report computational wall time results for linear as well as for realistic three-dimensional molecules and show that triple-ζ quality calculations on molecules of several hundreds of atoms are only a matter of a few hours on a single compute node, the bottleneck of the computations being the SCF rather than the post-SCF energy correction.

Q-MP2-OS: Møller-Plesset Correlation Energy by Quadrature

We present a quadrature-based algorithm for computing the opposite-spin component of the MP2 correlation energy which scales quadratically with basis set size and is well-suited to large-scale parallelization. The key ideas, which are rooted in the earlier work of Hirata and co-workers, are to abandon all two-electron integrals, recast the energy as a seven-dimensional integral, approximate that integral by quadrature, and employ a cutoff strategy to minimize the number of intermediate quantities. We discuss our implementation in detail and show that it parallelizes almost perfectly on 840 cores for cyclosporine (a molecule with roughly 200 atoms), exhibits [Formula: see text] scaling for a sequence of polyglycines, and is principally limited by the accuracy of its quadrature.