Economical quasi-Newton unitary optimization of electronic orbitals.

We present an efficient quasi-Newton orbital solver optimized to reduce the number of gradient evaluations and other computational steps of comparable cost. The solver optimizes orthogonal orbitals by sequences of unitary rotations generated by the (preconditioned) limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm equipped with trust-region step restriction. The low-rank structure of the L-BFGS inverse Hessian is exploited when solving the trust-region problem. The efficiency of the proposed "Quasi-Newton Unitary Optimization with Trust-Region" (QUOTR) solver is compared to that of the standard Roothaan-Hall approach accelerated by the Direct Inversion of Iterative Subspace (DIIS), and other exact and approximate Newton solvers for mean-field (Hartree-Fock and Kohn-Sham) problems.

Dirac-Coulomb-Breit Molecular Mean-Field Exact-Two-Component Relativistic Equation-of-Motion Coupled-Cluster Theory.

We present a relativistic equation-of-motion coupled-cluster with single and double excitation formalism within the exact two-component framework (X2C-EOM-CCSD), where both scalar relativistic effects and spin-orbit coupling are variationally included at the reference level. Three different molecular mean-field treatments of relativistic corrections, including the one-electron, Dirac-Coulomb, and Dirac-Coulomb-Breit Hamiltonian, are considered in this work. Benchmark calculations include atomic excitations and fine-structure splittings arising from spin-orbit coupling. Comparison with experimental values and relativistic time-dependent density functional theory is also carried out. The computation of the oscillator strength using the relativistic X2C-EOM-CCSD approach allows for studies of spin-orbit-driven processes, such as the spontaneous phosphorescence lifetime.

Relativistic Coupled Cluster with Completely Renormalized and Perturbative Triples Corrections.

We have implemented noniterative triples corrections to the energy from coupled-cluster with single and double excitations (CCSD) within the 1-electron exact two-component (1eX2C) relativistic framework. The effectiveness of both the CCSD(T) and the completely renormalized (CR) CC(2,3) approaches are demonstrated by performing all-electron computations of the potential energy curves and spectroscopic constants of copper, silver, and gold dimers in their ground electronic states. Spin-orbit coupling effects captured via the 1eX2C framework are shown to be crucial for recovering the correct shape of the potential energy curves, and the correlation effects due to triples in these systems change the dissociation energies by about 0.1-0.2 eV or about 4-7%. We also demonstrate that relativistic effects and basis set size and contraction scheme are significantly more important in Au2 than in Ag2 or Cu2.

3-center and 4-center 2-particle Gaussian AO integrals on modern accelerated processors.

We report an implementation of the McMurchie-Davidson (MD) algorithm for 3-center and 4-center 2-particle integrals over Gaussian atomic orbitals (AOs) with low and high angular momenta l and varying degrees of contraction for graphical processing units (GPUs). This work builds upon our recent implementation of a matrix form of the MD algorithm that is efficient for GPU evaluation of 4-center 2-particle integrals over Gaussian AOs of high angular momenta (l ≥ 4) [A. Asadchev and E. F. Valeev, J. Phys. Chem. A 127, 10889-10895 (2023)]. The use of unconventional data layouts and three variants of the MD algorithm allow for the evaluation of integrals with double precision and sustained performance between 25% and 70% of the theoretical hardware peak. Performance assessment includes integrals over AOs with l ≤ 6 (a higher l is supported). Preliminary implementation of the Hartree-Fock exchange operator is presented and assessed for computations with up to a quadruple-zeta basis and more than 20 000 AOs. The corresponding C++ code is part of the experimental open-source LibintX library available at https://github.com/ValeevGroup/libintx.

Many-Body Quantum Chemistry on Massively Parallel Computers.

The deployment of many-body quantum chemistry methods onto massively parallel high-performance computing (HPC) platforms is reviewed. The particular focus is on highly accurate methods that have become popular in predictive description of chemical phenomena, such as the coupled-cluster method. The account of relevant literature is preceded by a discussion of the modern and near-future HPC landscape and the relevant computational traits of the many-body methods, in their canonical and reduced-scaling formulations, that underlie the challenges in their HPC realization.

From NWChem to NWChemEx: Evolving with the Computational Chemistry Landscape.

Since the advent of the first computers, chemists have been at the forefront of using computers to understand and solve complex chemical problems. As the hardware and software have evolved, so have the theoretical and computational chemistry methods and algorithms. Parallel computers clearly changed the common computing paradigm in the late 1970s and 80s, and the field has again seen a paradigm shift with the advent of graphical processing units. This review explores the challenges and some of the solutions in transforming software from the terascale to the petascale and now to the upcoming exascale computers. While discussing the field in general, NWChem and its redesign, NWChemEx, will be highlighted as one of the early codesign projects to take advantage of massively parallel computers and emerging software standards to enable large scientific challenges to be tackled.

High-Performance Evaluation of High Angular Momentum 4-Center Gaussian Integrals on Modern Accelerated Processors.

We present a high-performance evaluation method for 4-center 2-particle integrals over Gaussian atomic orbitals with high angular momenta (l ≥ 4) and arbitrary contraction degrees on graphical processing units (GPUs) and other accelerators. The implementation uses the matrix form of McMurchie-Davidson recurrences. Evaluation of the four-center integrals over four l = 6 (i) Gaussian AOs in double precision (FP64) on an NVIDIA V100 GPU outperforms the reference implementation of the Obara-Saika recurrences (Libint) running on a single Intel Xeon core by more than a factor of 1000, easily exceeding the 73:1 ratio of the respective hardware peak FLOP rates while reaching almost 50% of the V100 peak. The approach can be extended to support AOs with even higher angular momenta; for lower angular momenta (l ≤ 3), additional improvements will be reported elsewhere. The implementation is part of an open-source LibintX library freely available at github.com:ValeevGroup/LibintX.

SparseMaps-A systematic infrastructure for reduced-scaling electronic structure methods. VI. Linear-scaling explicitly correlated N-electron valence state perturbation theory with pair natural orbital.

In this work, a linear scaling explicitly correlated N-electron valence state perturbation theory (NEVPT2-F12) is presented. By using the idea of a domain-based local pair natural orbital (DLPNO), computational scaling of the conventional NEVPT2-F12 is reduced to near-linear scaling. For low-lying excited states of organic molecules, the excitation energies predicted by DLPNO-NEVPT2-F12 are as accurate as the exact NEVPT2-F12 results. Some cluster models of rhodopsin are studied using the new algorithm. Our new method is able to study systems with more than 3300 basis functions and an active space containing 12 π-electrons and 12 π-orbitals. However, even larger calculations or active spaces would still be feasible.

Distributed memory, GPU accelerated Fock construction for hybrid, Gaussian basis density functional theory.

With the growing reliance of modern supercomputers on accelerator-based architecture such a graphics processing units (GPUs), the development and optimization of electronic structure methods to exploit these massively parallel resources has become a recent priority. While significant strides have been made in the development GPU accelerated, distributed memory algorithms for many modern electronic structure methods, the primary focus of GPU development for Gaussian basis atomic orbital methods has been for shared memory systems with only a handful of examples pursing massive parallelism. In the present work, we present a set of distributed memory algorithms for the evaluation of the Coulomb and exact exchange matrices for hybrid Kohn-Sham DFT with Gaussian basis sets via direct density-fitted (DF-J-Engine) and seminumerical (sn-K) methods, respectively. The absolute performance and strong scalability of the developed methods are demonstrated on systems ranging from a few hundred to over one thousand atoms using up to 128 NVIDIA A100 GPUs on the Perlmutter supercomputer.

Toward the Minimal Floating Operation Count Cholesky Decomposition of Electron Repulsion Integrals.

As quantum chemistry calculations deal with molecular systems of increasing size, the memory requirement to store electron-repulsion integrals (ERIs) greatly outpaces the physical memory available in computing hardware. The Cholesky decomposition of ERIs provides a convenient yet accurate technique to reduce the storage requirement of integrals. Recent developments of a two-step algorithm have drastically reduced the memory operation (MOP) count, leaving the floating operation (FLOP) count as the last frontier of cost reduction in the Cholesky ERI algorithm. In this report, we introduce a dynamic integral tracking, reusing, and compression/elimination protocol embedded in the two-step Cholesky ERI method. Benchmark studies suggest that this technique becomes particularly advantageous when the basis set consists of many computationally expensive high-angular-momentum basis functions. With this dynamic-ERI improvement, the Cholesky ERI approach proves to be a highly efficient algorithm with minimal FLOP and MOP count.

Quantum simulation of electronic structure with a transcorrelated Hamiltonian: improved accuracy with a smaller footprint on the quantum computer.

Quantum simulations of electronic structure with a transformed Hamiltonian that includes some electron correlation effects are demonstrated. The transcorrelated Hamiltonian used in this work is efficiently constructed classically, at polynomial cost, by an approximate similarity transformation with an explicitly correlated two-body unitary operator. This Hamiltonian is Hermitian, includes no more than two-particle interactions, and is free of electron-electron singularities. We investigate the effect of such a transformed Hamiltonian on the accuracy and computational cost of quantum simulations by focusing on a widely used solver for the Schrödinger equation, namely the variational quantum eigensolver method, based on the unitary coupled cluster with singles and doubles (q-UCCSD) Ansatz. Nevertheless, the formalism presented here translates straightforwardly to other quantum algorithms for chemistry. Our results demonstrate that a transcorrelated Hamiltonian, paired with extremely compact bases, produces explicitly correlated energies comparable to those from much larger bases. For the chemical species studied here, explicitly correlated energies based on an underlying 6-31G basis had cc-pVTZ quality. The use of the very compact transcorrelated Hamiltonian reduces the number of CNOT gates required to achieve cc-pVTZ quality by up to two orders of magnitude, and the number of qubits by a factor of three.

Combined Relativistic Ab Initio Multireference and Experimental Study of the Electronic Structure of Terbium Luminescent Compound.

A new terbium (III) luminescent compound {[Tb2(PDC)2(ox)(H2O)4](H2O)2}n was synthesized by the self-assembly of Tb3+ ions with 3,5-pyridinedicarboxylate (PDC) and oxalate (ox) ligands and characterized by fluorescence spectroscopy and single-crystal X-ray diffraction. The density functional theory (DFT) and high-level correlated ab initio wave function methods with Spin-Orbit Coupling correction (CASSCF/SO and CAS-NEVPT2/SOC) were successfully applied to predict the absorption and emission spectra of this strongly correlated lanthanide system in excellent agreement with the experimental results.

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.

Direct determination of optimal pair-natural orbitals in a real-space representation: The second-order Moller-Plesset energy.

An efficient representation of molecular correlated wave functions is proposed, which features regularization of the Coulomb electron-electron singularities via the F12-style explicit correlation and a pair-natural orbital factorization of the correlation components of the wave function expressed in the real space. The pair-natural orbitals are expressed in an adaptive multiresolution basis and computed directly by iterative variational optimization. The approach is demonstrated by computing the second-order Moller-Plesset energies of small- and medium-sized molecules. The resulting MRA-PNO-MP2-F12 method allows for the first time to compute correlated wave functions in a real-space representation for systems with dozens of atoms (as demonstrated here by computations on alkanes as large as C10H22), with precision exceeding what is achievable with the conventional explicitly correlated MP2 approaches based on the atomic orbital representations.

Efficient evaluation of exact exchange for periodic systems via concentric atomic density fitting.

The evaluation of the exact [Hartree-Fock (HF)] exchange operator is a crucial ingredient for the accurate description of the electronic structure in periodic systems through ab initio and hybrid density functional approaches. An efficient formulation of periodic HF exchange in a linear combination of atomic orbitals representation presented here is based on the concentric atomic density fitting approximation, a domain-free local density fitting approach in which the product of two atomic orbitals is approximated using a linear combination of fitting basis functions centered at the same nuclei as the AOs in that product. A significant reduction in the computational cost of exact exchange is demonstrated relative to the conventional approach due to avoiding the need to evaluate four-center two-electron integrals, with sub-millihartree/atom errors in absolute HF energies and good cancellation of fitting errors in relative energies. The novel aspects of the evaluation of the Coulomb contribution to the Fock operator, such as the use of real two-center multipole expansions and spheropole-compensated unit cell densities, are also described.

Massively Parallel Quantum Chemistry: A high-performance research platform for electronic structure.

The Massively Parallel Quantum Chemistry (MPQC) program is a 30-year-old project that enables facile development of electronic structure methods for molecules for efficient deployment to massively parallel computing architectures. Here, we describe the historical evolution of MPQC's design into its latest (fourth) version, the capabilities and modular architecture of today's MPQC, and how MPQC facilitates rapid composition of new methods as well as its state-of-the-art performance on a variety of commodity and high-end distributed-memory computer platforms.

Explicitly correlated renormalized second-order Green's function for accurate ionization potentials of closed-shell molecules.

We present an energy-dependent explicitly correlated (F12) formalism for the nondiagonal renormalized second-order (NR2) Green's function method of closed-shell molecules. For a test set of 21 small molecules, the mean basis set error in IP computed using NR2-F12 with aug-cc-pVTZ basis is 0.028 eV, compared to 0.044 eV for NR2 with aug-cc-pV5Z basis. Similarly, for a set of 24 medium-sized organic electron acceptor molecules (OAM24), the mean basis set errors are 0.015 eV for NR2-F12 with aug-cc-pVTZ basis compared to 0.067 eV for NR2 with aug-cc-pVQZ basis. Hence, NR2-F12 facilitates accurate calculation of IP at a lower cost compared to the NR2 method. NR2-F12 has O(N6)/O(N5)noniterative/iterative costs with system size. At a small basis, the performance of NR2-F12 for 21 small molecules and OAM24 dataset is comparable to equation-of-motion ionized coupled-cluster singles and doubles, whose cost is iterativeO(N6).

A new near-linear scaling, efficient and accurate, open-shell domain-based local pair natural orbital coupled cluster singles and doubles theory.

The Coupled-Cluster expansion, truncated after single and double excitations (CCSD), provides accurate and reliable molecular electronic wave functions and energies for many molecular systems around their equilibrium geometries. However, the high computational cost, which is well-known to scale as O(N6) with system size N, has limited its practical application to small systems consisting of not more than approximately 20-30 atoms. To overcome these limitations, low-order scaling approximations to CCSD have been intensively investigated over the past few years. In our previous work, we have shown that by combining the pair natural orbital (PNO) approach and the concept of orbital domains it is possible to achieve fully linear scaling CC implementations (DLPNO-CCSD and DLPNO-CCSD(T)) that recover around 99.9% of the total correlation energy [C. Riplinger et al., J. Chem. Phys. 144, 024109 (2016)]. The production level implementations of the DLPNO-CCSD and DLPNO-CCSD(T) methods were shown to be applicable to realistic systems composed of a few hundred atoms in a routine, black-box fashion on relatively modest hardware. In 2011, a reduced-scaling CCSD approach for high-spin open-shell unrestricted Hartree-Fock reference wave functions was proposed (UHF-LPNO-CCSD) [A. Hansen et al., J. Chem. Phys. 135, 214102 (2011)]. After a few years of experience with this method, a few shortcomings of UHF-LPNO-CCSD were noticed that required a redesign of the method, which is the subject of this paper. To this end, we employ the high-spin open-shell variant of the N-electron valence perturbation theory formalism to define the initial guess wave function, and consequently also the open-shell PNOs. The new PNO ansatz properly converges to the closed-shell limit since all truncations and approximations have been made in strict analogy to the closed-shell case. Furthermore, given the fact that the formalism uses a single set of orbitals, only a single PNO integral transformation is necessary, which offers large computational savings. We show that, with the default PNO truncation parameters, approximately 99.9% of the total CCSD correlation energy is recovered for open-shell species, which is comparable to the performance of the method for closed-shells. UHF-DLPNO-CCSD shows a linear scaling behavior for closed-shell systems, while linear to quadratic scaling is obtained for open-shell systems. The largest systems we have considered contain more than 500 atoms and feature more than 10 000 basis functions with a triple-ζ quality basis set.

Explicitly correlated N-electron valence state perturbation theory (NEVPT2-F12).

In this work, explicitly correlated second order N-electron valence state perturbation theory (NEVPT2-F12) has been derived and implemented for the first time. The NEVPT2-F12 algorithm presented here is based on a fully internally contracted wave function and includes the correction of semi-internal excitation subspaces. The algorithm exploits the resolution of identity (RI) approximation to improve the computational efficiency. The overall O(N5) scaling of the computational effort is documented. In Sec. III, the dissociation processes of diatomic molecules and the singlet-triplet gap of several systems are studied. For all relative energies studied in this work, the errors with respect to the complete basis set (CBS) limit for the NEVPT2-F12 method are within 1 kcal/mol. For moderately sized active spaces, the computational cost of a RI-NEVPT2-F12 correlation energy calculation for each root is comparable to a closed-shell RI-MP2-F12 calculation on the same system.