Accelerating seminumerical Fock-exchange calculations using mixed single- and double-precision arithmethic.

We investigate the applicability of single-precision (fp32) floating point operations within our linear-scaling, seminumerical exchange method sn-LinK [Laqua et al., J. Chem. Theory Comput. 16, 1456 (2020)] and find that the vast majority of the three-center-one-electron (3c1e) integrals can be computed with reduced numerical precision with virtually no loss in overall accuracy. This leads to a near doubling in performance on central processing units (CPUs) compared to pure fp64 evaluation. Since the cost of evaluating the 3c1e integrals is less significant on graphic processing units (GPUs) compared to CPU, the performance gains from accelerating 3c1e integrals alone is less impressive on GPUs. Therefore, we also investigate the possibility of employing only fp32 operations to evaluate the exchange matrix within the self-consistent-field (SCF) followed by an accurate one-shot evaluation of the exchange energy using mixed fp32/fp64 precision. This still provides very accurate (1.8 µEh maximal error) results while providing a sevenfold speedup on a typical "gaming" GPU (GTX 1080Ti). We also propose the use of incremental exchange-builds to further reduce these errors. The proposed SCF scheme (i-sn-LinK) requires only one mixed-precision exchange matrix calculation, while all other exchange-matrix builds are performed with only fp32 operations. Compared to pure fp64 evaluation, this leads to 4-7× speedups for the whole SCF procedure without any significant deterioration of the results or the convergence behavior.

An improved molecular partitioning scheme for numerical quadratures in density functional theory.

We present a modification to Becke's molecular partitioning scheme [A. D. Becke, J. Chem. Phys. 88, 2547 (1988)] that provides substantially better accuracy for weakly bound complexes and allows for a faster and linear scaling grid generation without introducing a cutoff error. We present the accuracy of our new partitioning scheme for atomization energies of small molecules and for interaction energies of van der Waals complexes. Furthermore, the efficiency and scaling behavior of the grid generation are demonstrated for large molecular systems with up to 1707 atoms.

Communication: Density functional theory model for multi-reference systems based on the exact-exchange hole normalization.

The correct description of multi-reference electronic ground states within Kohn-Sham density functional theory (DFT) requires an ensemble-state representation, employing fractionally occupied orbitals. However, the use of fractional orbital occupation leads to non-normalized exact-exchange holes, resulting in large fractional-spin errors for conventional approximative density functionals. In this communication, we present a simple approach to directly include the exact-exchange-hole normalization into DFT. Compared to conventional functionals, our model strongly improves the description for multi-reference systems, while preserving the accuracy in the single-reference case. We analyze the performance of our proposed method at the example of spin-averaged atoms and spin-restricted bond dissociation energy surfaces.

Efficient Exploitation of Numerical Quadrature with Distance-Dependent Integral Screening in Explicitly Correlated F12 Theory: Linear Scaling Evaluation of the Most Expensive RI-MP2-F12 Term.

We present a linear scaling atomic orbital based algorithm for the computation of the most expensive exchange-type RI-MP2-F12 term by employing numerical quadrature in combination with CABS-RI to avoid six-center-three-electron integrals. Furthermore, a robust distance-dependent integral screening scheme, based on integral partition bounds [Thompson, T. H.; Ochsenfeld, C. J. Chem. Phys. 2019, 150, 044101], is used to drastically reduce the number of the required three-center-one-electron integrals substantially. The accuracy of our numerical quadrature/CABS-RI approach and the corresponding integral screening is thoroughly assessed for interaction and isomerization energies across a variety of numerical integration grids. Our method outperforms the standard density fitting/CABS-RI approach with errors below 1 µEh even for small grid sizes and moderate screening thresholds. The choice of the grid size and screening threshold allows us to tailor our ansatz to a desired accuracy and computational efficiency. We showcase the approach's effectiveness for the chemically relevant system valinomycin, employing a triple-ζ F12 basis set combination (C54H90N6O18, 5757 AO basis functions, 10,266 CABS basis functions, 735,783 grid points). In this context, our ansatz achieves higher accuracy combined with a 135× speedup compared to the classical density fitting based variant, requiring notably less computation time than the corresponding RI-MP2 calculation. Additionally, we demonstrate near-linear scaling through calculations on linear alkanes. We achieved an 817-fold acceleration for C80H162 and an extrapolated 28,765-fold acceleration for C200H402, resulting in a substantially reduced computational time for the latter─from 229 days to just 11.5 min. Our ansatz may also be adapted to the remaining MP2-F12 terms, which will be the subject of future work.

High-pressure synthesis and characterization of Li2Ca3[N2]3--an uncommon metallic diazenide with [N2]2- ions.

Dinitrogen (N2) ligation is a common and well-characterized structural motif in bioinorganic synthesis. In solid-state chemistry, on the other hand, homonuclear dinitrogen entities as structural building units proved existence only very recently. High-pressure/high-temperature (HP/HT) syntheses have afforded a number of binary diazenides and pernitrides with [N2](2­) and [N2](4­) ions, respectively. Here, we report on the HP/HT synthesis of the first ternary diazenide. Li2Ca3[N2]3 (space group Pmma, no. 51, a = 4.7747(1), b = 13.9792(4), c = 8.0718(4) Å, Z = 4, wRp = 0.08109) was synthesized by controlled thermal decomposition of a stoichiometric mixture of lithium azide and calcium azide in a multianvil device under a pressure of 9 GPa at 1023 K. Powder X-ray diffraction analysis reveals strongly elongated N­N bond lengths of dNN = 1.34(2)­1.35(3) Å exceeding those of previously known, binary diazenides. In fact, the refined N­N distances in Li2Ca3[N2]3 would rather suggest the presence of [N2](3·­) radical ions. Also, characteristic features of the N­N stretching vibration occur at lower wavenumbers (1260­1020 cm(­1)) than in the binary phases, and these assignments are supported by first-principles phonon calculations. Ultimately, the true character of the N2 entity in Li2Ca3[N2]3 is probed by a variety of complementary techniques, including electron diffraction, electron spin resonance spectroscopy (ESR), magnetic and electric conductivity measurements, as well as density-functional theory calculations (DFT). Unequivocally, the title compound is shown to be metallic containing diazenide [N2](2­) units according to the formula (Li(+))2(Ca(2+))3([N2](2­))3·(e(­))2.

Highly Efficient and Accurate Computation of Multiple Orbital Spaces Spanning Fock Matrix Elements on Central and Graphics Processing Units for Application in F12 Theory.

We employ our recently published highly efficient seminumerical exchange (sn-LinK) [Laqua, H.; Thompson, T. H.; Kussmann, J.; Ochsenfeld, C. J. Chem. Theory Comput. 2020, 16, 1456-1468] and integral-direct resolution of the identity Coulomb (RI-J) [Kussmann, J.; Laqua, H.; Ochsenfeld, C. J. Chem. Theory Comput. 2021, 17, 1512-1521] methods to significantly accelerate the computation of the demanding multiple orbital spaces spanning Fock matrix elements present in R12/F12 theory on central and graphics processing units. The errors introduced by RI-J and sn-LinK into the RI-MP2-F12 energy are thoroughly assessed for a variety of basis sets and integration grids. We find that these numerical errors are always below "chemical accuracy" (∼1 mH) even for the coarsest settings and can easily be reduced below 1 µH by employing only moderately large integration grids and RI-J basis sets. Since the number of basis functions of the multiple orbital spaces is notably larger compared with conventional Hartree-Fock theory, the efficiency gains from the superior basis scaling of RI-J and sn-LinK (O(Nbas2) instead of O(Nbas4) for both) are even more significant, with maximum speedup factors of 37 000 for RI-J and 4500 for sn-LinK. In total, the multiple orbital spaces spanning Fock matrix evaluation of the largest tested structure using a triple-ζ F12 basis set (5058 AO basis functions, 9267 CABS basis functions) is accelerated over 1575× using CPUs and over 4155× employing GPUs.

Efficient Method for the Computation of Frozen-Core Nuclear Gradients within the Random Phase Approximation.

A method for the evaluation of analytical frozen-core gradients within the random phase approximation is presented. We outline an efficient way to evaluate the response of the density of active electrons arising only when introducing the frozen-core approximation and constituting the main difficulty, together with the response of the standard Kohn-Sham density. The general framework allows to extend the outlined procedure to related electron correlation methods in the atomic orbital basis that require the evaluation of density responses, such as second-order Møller-Plesset perturbation theory or coupled cluster variants. By using Cholesky decomposed densities─which reintroduce the occupied index in the time-determining steps─we are able to achieve speedups of 20-30% (depending on the size of the basis set) by using the frozen-core approximation, which is of similar magnitude as for molecular orbital formulations. We further show that the errors introduced by the frozen-core approximation are practically insignificant for molecular geometries.

Accelerating Hybrid Density Functional Theory Molecular Dynamics Simulations by Seminumerical Integration, Resolution-of-the-Identity Approximation, and Graphics Processing Units.

The computationally very demanding evaluation of the 4-center-2-electron (4c2e) integrals and their respective integral derivatives typically represents the major bottleneck within hybrid Kohn-Sham density functional theory molecular dynamics simulations. Building upon our previous works on seminumerical exact-exchange (sn-LinK) [Laqua, H., Thompsons, T. H., Kussmann, J., Ochsenfeld, C., J. Chem. Theory Comput. 2020, 16, 1465] and resolution-of-the-identity Coulomb (RI-J) [Kussmann, J., Laqua, H., Ochsenfeld, C., J. Chem. Theory Comput. 2021, 17, 1512], the expensive 4c2e integral evaluation can be avoided entirely, resulting in a highly efficient electronic structure theory method, allowing for fast ab initio molecular dynamics (AIMD) simulations even with large basis sets. Moreover, we propose to combine the final self-consistent field (SCF) step with the subsequent nuclear forces evaluation, providing the forces at virtually no additional cost after a converged SCF calculation, reducing the total runtime of an AIMD simulation by about another 25%. In addition, multiple independent MD trajectories can be computed concurrently on a single node, leading to a greatly increased utilization of the available hardware─especially when combined with graphics processing unit acceleration─improving the overall throughput by up to another 5 times in this way. With all of those optimizations combined, our proposed method provides nearly 3 orders of magnitude faster execution times than traditional 4c2e integral-based methods. To demonstrate the practical utility of the approach, quantum-mechanical/molecular-mechanical dynamics simulations on double-stranded DNA were performed, investigating the relative hydrogen bond strength between adenine-thymine and guanine-cytosine base pairs. In addition, this illustrative application also contains a general accuracy assessment of the introduced approximations (integration grids, resolution-of-the-identity) within AIMD simulations, serving as a protocol on how to apply these new methods to practical problems.

Lagrangian-Based Minimal-Overhead Batching Scheme for the Efficient Integral-Direct Evaluation of the RPA Correlation Energy.

A highly memory-efficient integral-direct random phase approximation (RPA) method based on our ω-CDGD-RI-RPA method [Graf, D. J. Chem. Theory Comput. 2018, 14, 2505] is presented that completely alleviates the memory bottleneck of storing the multidimensional three-center integral tensor, which severely limited the tractable system sizes. Based on a Lagrangian formulation, we introduce an optimized batching scheme over the auxiliary and basis-function indices, which allows to compute the optimal number of batches for a given amount of system memory, while minimizing the batching overhead. Thus, our optimized batching constitutes the best tradeoff between program runtime and memory demand. Within this batching scheme, the half-transformed three-center integral tensor BiµM is recomputed for each batch of auxiliary and basis functions. This allows the computation of systems that were out of reach before. The largest system within this work consists of a DNA fragment comprising 1052 atoms and 11 230 basis functions calculated on a single node, which emphasizes the new possibilities of our integral-direct RPA method.

Highly Efficient Resolution-of-Identity Density Functional Theory Calculations on Central and Graphics Processing Units.

We present an efficient method to evaluate Coulomb potential matrices using the resolution of identity approximation and semilocal exchange-correlation potentials on central (CPU) and graphics processing units (GPU). The new GPU-based RI-algorithm shows a high performance and ensures the favorable scaling with increasing basis set size as the conventional CPU-based method. Furthermore, our method is based on the J-engine algorithm [White; , Head-Gordon, J. Chem. Phys. 1996, 7, 2620], which allows for further optimizations that also provide a significant improvement of the corresponding CPU-based algorithm. Due to the increased performance for the Coulomb evaluation, the calculation of the exchange-correlation potential of density functional theory on CPUs quickly becomes a bottleneck to the overall computational time. Hence, we also present a GPU-based algorithm to evaluate the exchange-correlation terms, which results in an overall high-performance method for density functional calculations. The algorithms to evaluate the potential and nuclear derivative terms are discussed, and their performance on CPUs and GPUs is demonstrated for illustrative calculations.

Range-Separated Density-Functional Theory in Combination with the Random Phase Approximation: An Accuracy Benchmark.

A formulation of range-separated random phase approximation (RPA) based on our efficient ω-CDGD-RI-RPA [J. Chem. Theory Comput. 2018, 14, 2505] method and a large scale benchmark study are presented. By application to the GMTKN55 data set, we obtain a comprehensive picture of the performance of range-separated RPA in general main group thermochemistry, kinetics, and noncovalent interactions. The results show that range-separated RPA performs stably over the broad range of molecular chemistry included in the GMTKN55 set. It improves significantly over semilocal DFT but it is still less accurate than modern dispersion corrected double-hybrid functionals. Furthermore, range-separated RPA shows a faster basis set convergence compared to standard full-range RPA making it a promising applicable approach with only one empirical parameter.

Highly Efficient, Linear-Scaling Seminumerical Exact-Exchange Method for Graphic Processing Units.

We present a highly efficient and asymptotically linear-scaling graphic processing unit accelerated seminumerical exact-exchange method (sn-LinK). We go beyond our previous central processing unit-based method (Laqua, H.; Kussmann, J.; Ochsenfeld, C. J. Chem. Theory Comput. 2018, 14, 3451-3458) by employing our recently developed integral bounds (Thompson, T. H.; Ochsenfeld, C. J. Chem. Phys. 2019, 150, 044101) and high-accuracy numerical integration grid (Laqua, H.; Kussmann, J.; Ochsenfeld, C. J. Chem. Phys. 2018, 149, 204111). The accuracy is assessed for several established test sets, providing errors significantly below 1mEh for the smallest grid. Moreover, a comprehensive performance analysis for large molecules between 62 and 1347 atoms is provided, revealing the outstanding performance of our method, in particular, for large basis sets such as the polarized quadruple-zeta level with diffuse functions.

Efficient and Linear-Scaling Seminumerical Method for Local Hybrid Density Functionals.

Local hybrid functionals, that is, functionals with local dependence on the exact exchange energy density, generalize the popular class of global hybrid functionals and extend the applicability of density functional theory to electronic structures that require an accurate description of static correlation. However, the higher computational cost compared to conventional Kohn-Sham density functional theory restrained their widespread application. Here, we present a low-prefactor, linear-scaling method to evaluate the local hybrid exchange-correlation potential as well as the corresponding nuclear forces by employing a seminumerical integration scheme. In the seminumerical scheme, one integration in the electron repulsion integrals is carried out analytically and the other one is carried out numerically, employing an integration grid. A high computational efficiency is achieved by combining the preLinK method [ J. Kussmann and C. Ochsenfeld, J. Chem. Phys. 2013 138, 134114 ] with explicit screening of integrals for batches of grid points to minimize the screening overhead. This new method, denoted as preLinX, provides an 8-fold performance increase for a DNA fragment containing four base pairs as compared to existing S- and P-junction-based methods. In this way, our method allows for the evaluation of local hybrid functionals at a cost similar to that of global hybrid functionals. The linear-scaling behavior, efficiency, accuracy, and multinode parallelization of the presented method is demonstrated for large systems containing more than 1000 atoms.