Coulomb Matrices Research Articles

Neural Network Model for Predicting the Atomization Energy of Multi-Atomic Molecules Based on Sorted Coulomb Matrices

Neural Network Model for Predicting the Atomization Energy of Multi-Atomic Molecules Based on Sorted Coulomb Matrices

VASE: A High-Entropy Alloy Short-Range Order Structural Descriptor for Machine Learning.

The short-range order (SRO) structure in high-entropy alloys (HEAs) is closely associated with many properties, which can be studied through density functional theory (DFT) calculations. Atomic-scale modeling and calculations require substantial computational resources, and machine learning can provide rapid estimations of DFT results. To describe SRO information in HEAs, a new descriptor based on Voronoi Analysis and Shannon Entropy (VASE) is proposed. Based on Voronoi analysis, the Shannon entropy is introduced to directly characterize atomic spatial arrangement information except for composition and atomic interactions, which is necessary for describing the disorder atomic occupancy in HEAs. The new descriptor is used for predicting the formation energy of FeCoNiAlTiCu system based on machine learning model, which is more accurate than other descriptors (Coulomb matrices, partial radial distribution functions, and Voronoi analysis). Moreover, the model trained based on VASE descriptors exhibits the best predictive performance for unrelaxed structures (24.06 meV/atom). The introduction of Shannon entropy provides an effective representation of atomic arrangement information in HEAs, which is a powerful tool for investigating the SRO phenomena.

Accelerating Density Matrix Embedding with Stochastic Density Fitting Theory: An Application to Hydrogen Bonded Clusters.

In this work, we demonstrate how using semistochastic density fitting (ss-DF) can accelerate self-consistent density matrix embedding theory (DMET) calculations by reducing the number of auxiliary orbitals in the three-indexed DF integrals. This reduction results in significant time savings when building the Hartree-Fock (HF) Coulomb and Exchange Matrices and in transforming integrals from the atomic orbital (AO) basis to the embedding orbital (EO) basis. We apply ss-DF to a range of hydrogen-bonded clusters to showcase its effectiveness. First, we examine how the amount of deterministic space impacts the quality of the calculation in a (H2O)10 cluster. Next, we test the computational efficiency of ss-DF compared to deterministic DF (d-DF) in water clusters containing 6-30 water molecules using a triple-ζ basis set. Finally, we perform numerical structural optimizations on water and hydrogen fluoride clusters, revealing that DMET can recover weak interactions using a back-transformed energy formula. This work demonstrates the potential of using stochastic resolution of identity in quantum embedding theories and highlights its capability to recover weak interactions effectively.

Efficient interpolation of molecular properties across chemical compound space with low-dimensional descriptors

We demonstrate accurate data-starved models of molecular properties for interpolation in chemical compound spaces with low-dimensional descriptors. Our starting point is based on three-dimensional, universal, physical descriptors derived from the properties of the distributions of the eigenvalues of Coulomb matrices. To account for the shape and composition of molecules, we combine these descriptors with six-dimensional features informed by the Gershgorin circle theorem. We use the nine-dimensional descriptors thus obtained for Gaussian process regression based on kernels with variable functional form, leading to extremely efficient, low-dimensional interpolation models. The resulting models trained with 100 molecules are able to predict the product of entropy and temperature (S × T) and zero point vibrational energy (ZPVE) with the absolute error under 1 kcal mol−1 for >78 % and under 1.3 kcal mol−1 for >92 % of molecules in the test data. The test data comprises 20 000 molecules with complexity varying from three atoms to 29 atoms and the ranges of S × T and ZPVE covering 36 kcal mol−1 and 161 kcal mol−1, respectively. We also illustrate that the descriptors based on the Gershgorin circle theorem yield more accurate models of molecular entropy than those based on graph neural networks that explicitly account for the atomic connectivity of molecules.

Machine Learning Models for Predicting Zirconocene Properties and Barriers.

Zr metallocenes have significant potential to be highly tunable polyethylene catalysts through modification of the aromatic ligand framework. Here we report the development of multiple machine learning models using a large library (>700 systems) of DFT-calculated zirconocene properties and barriers for ethylene polymerization. We show that very accurate machine learning models are possible for HOMO-LUMO gaps of precatalysts but the performance significantly depends on the machine learning algorithm and type of featurization, such as fingerprints, Coulomb matrices, smooth overlap of atomic positions, or persistence images. Surprisingly, the description of the bonding hapticity, the number of direct connections between Zr and the ligand aromatic carbons, only has a moderate influence on the performance of most models. Despite robust models for HOMO-LUMO gaps, these types of machine learning models based on structure connectivity type features perform poorly in predicting ethylene migratory insertion barrier heights. Therefore, we developed several relatively robust and accurate machine learning models for barrier heights that are based on quantum-chemical descriptors (QCDs). The quantitative accuracy of these models depends on which potential energy surface structure QCDs were harvested from. This revealed a Hammett-type principle to naturally emerge showing that QCDs from the π-coordination complexes provide much better descriptions of the transition states than other potential-energy structures. Feature importance analysis of the QCDs provides several fundamental principles that influence zirconocene catalyst reactivity.

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.

Hybrid localized graph kernel for machine learning energy-related properties of molecules and solids.

Nowadays, the coupling of electronic structure and machine learning techniques serves as a powerful tool to predict chemical and physical properties of a broad range of systems. With the aim of improving the accuracy of predictions, a large number of representations for molecules and solids for machine learning applications has been developed. In this work we propose a novel descriptor based on the notion of molecular graph. While graphs are largely employed in classification problems in cheminformatics or bioinformatics, they are not often used in regression problem, especially of energy-related properties. Our method is based on a local decomposition of atomic environments and on the hybridization of two kernel functions: a graph kernel contribution that describes the chemical pattern and a Coulomb label contribution that encodes finer details of the local geometry. The accuracy of this new kernel method in energy predictions of molecular and condensed phase systems is demonstrated by considering the popular QM7 and BA10 datasets. These examples show that the hybrid localized graph kernel outperforms traditional approaches such as, for example, the smooth overlap of atomic positions and the Coulomb matrices.

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.

Variational fitting of the Fock exchange potential with modified Cholesky decomposition.

The variational fitting of the Fock potential employing localized molecular orbitals requires either the inversion of the local two-center Coulomb matrices or alternatively the solution of corresponding linear equation systems with these matrices. In both cases, the method of choice is the Cholesky decomposition of the formally positive definite local two-center Coulomb matrices. However, due to finite-precision round-off errors, the local Coulomb matrices may be indefinite, and thus, the Cholesky decomposition is not applicable. To overcome this problem, we propose to make use of a modified Cholesky decomposition based on the indefinite factorization of local two-center Coulomb matrices. To this end, the working equations for the use of the modified Cholesky decomposition within the variational fitting of the Fock potential are presented. Benchmark calculations with global and range-separated hybrid functionals show that the proposed method can improve considerably the workload balance in parallel calculations.

A linear scaling hierarchical block low-rank representation of the electron repulsion integral tensor.

Efficient representations of the electron repulsion integral (ERI) tensor and fast algorithms for contractions with the ERI tensor often employ a low-rank approximation of the tensor or its sub-blocks. Such representations include density fitting (DF), the continuous fast multipole method (CFMM), and, more recently, hierarchical matrices. We apply the H2 hierarchical matrix representation to the ERI tensor with Gaussian basis sets to rapidly calculate the Coulomb matrices in Hartree-Fock and density functional theory calculations. The execution time and storage requirements of the hierarchical matrix approach and the DF approach are compared. The hierarchical matrix approach has very modest storage requirements, allowing large calculations to be performed in memory without recomputing ERIs. We interpret the hierarchical matrix approach as a multilevel, localized DF method and also discuss the close relationship between the hierarchical matrix approaches with CFMM. Like CFMM, the hierarchical matrix approach is asymptotically linear scaling, but the latter requires severalfold less memory (or severalfold less computation, if quantities are computed dynamically) due to being able to efficiently employ low-rank approximations for far more blocks.

Wasserstein metric for improved quantum machine learning with adjacency matrix representations

We study the Wasserstein metric to measure distances between molecules represented by the atom index dependent adjacency ‘Coulomb’ matrix, used in kernel ridge regression based supervised learning. Resulting machine learning models of quantum properties, a.k.a. quantum machine learning models exhibit improved training efficiency and result in smoother predictions of energies related to molecular distortions. We first illustrate smoothness for the continuous extraction of an atom from some organic molecule. Learning curves, quantifying the decay of the atomization energy’s prediction error as a function of training set size, have been obtained for tens of thousands of organic molecules drawn from the QM9 data set. In comparison to conventionally used metrics (L 1 and L 2 norm), our numerical results indicate systematic improvement in terms of learning curve off-set for random as well as sorted (by norms of row) atom indexing in Coulomb matrices. Our findings suggest that this metric corresponds to a favorable similarity measure which introduces index-invariance in any kernel based model relying on adjacency matrix representations.

Techniques for high-performance construction of Fock matrices.

This paper presents techniques for Fock matrix construction that are designed for high performance on shared and distributed memory parallel computers when using Gaussian basis sets. Four main techniques are considered. (1) To calculate electron repulsion integrals, we demonstrate batching together the calculation of multiple shell quartets of the same angular momentum class so that the calculation of large sets of primitive integrals can be efficiently vectorized. (2) For multithreaded summation of entries into the Fock matrix, we investigate using a combination of atomic operations and thread-local copies of the Fock matrix. (3) For distributed memory parallel computers, we present a globally accessible matrix class for accessing distributed Fock and density matrices. The new matrix class introduces a batched mode for remote memory access that can reduce the synchronization cost. (4) For density fitting, we exploit both symmetry (of the Coulomb and exchange matrices) and sparsity (of 3-index tensors) and give a performance comparison of density fitting and the conventional direct calculation approach. The techniques are implemented in an open-source software library called GTFock.

A reduced Hartree–Fock model of slice-like defects in the Fermi sea

Studying the electronic structure of defects in materials is an important subject in condensed matter physics. From a mathematical point of view, nonlinear mean-field models of localized defects in insulators are well understood. We present here a mean-field model to study a particular instance of extended defects in metals. These extended defects typically correspond to taking out a slice of finite width in the three-dimensional homogeneous electron gas. We work in the framework of the reduced Hartree–Fock model with either Yukawa or Coulomb interactions. Using techniques developed in Frank et al (2011 Phys. Rev. Lett. 106 150402; Frank et al 2013 Duke Math. J. 162 435–95) to study local perturbations of the free-electron gas, we show that our model admits minimizers, and that Yukawa ground state energies and density matrices converge to ground state Coulomb energies and density matrices as the Yukawa parameter tends to zero. These minimizers are unique for Yukawa interactions, and are characterized by a self-consistent equation. We moreover present numerical simulations where we observe Friedel oscillations in the total electronic density.

Applying machine learning techniques to predict the properties of energetic materials

We present a proof of concept that machine learning techniques can be used to predict the properties of CNOHF energetic molecules from their molecular structures. We focus on a small but diverse dataset consisting of 109 molecular structures spread across ten compound classes. Up until now, candidate molecules for energetic materials have been screened using predictions from expensive quantum simulations and thermochemical codes. We present a comprehensive comparison of machine learning models and several molecular featurization methods - sum over bonds, custom descriptors, Coulomb matrices, Bag of Bonds, and fingerprints. The best featurization was sum over bonds (bond counting), and the best model was kernel ridge regression. Despite having a small data set, we obtain acceptable errors and Pearson correlations for the prediction of detonation pressure, detonation velocity, explosive energy, heat of formation, density, and other properties out of sample. By including another dataset with ≈300 additional molecules in our training we show how the error can be pushed lower, although the convergence with number of molecules is slow. Our work paves the way for future applications of machine learning in this domain, including automated lead generation and interpreting machine learning models to obtain novel chemical insights.

Calculation of screened Coulomb potential matrices and its application to He bound and resonant states

We present two analytical methods, Taylor expansion and Gegenbauer expansion, to efficiently and accurately calculate the two-electron screened Coulomb potential matrix elements with Slater-type configuration-interaction basis functions. The former permits great advantages in fast computation of the potential matrices at small screening parameters and the latter allows accurate calculation of the matrices at all screening parameters. The bound and resonant states of a He atom embedded in the screening environment are calculated by employing the variational and complex-scaling methods, respectively, and the results are compared with other theoretical predictions. The expectation values of some physical quantities for He ground state are compared with the recent calculation of Ancarani and Rodriguez [Phys. Rev. A 89, 012507 (2014)] and extended to stronger screening environment. The energies and widths for the doubly excited resonant states are in good agreement with previous calculations, while the interelectronic angle $arccos\ensuremath{\langle}cos({\ensuremath{\theta}}_{12})\ensuremath{\rangle}$ show significant discrepancies with the Feshbach projection calculation of Ord\'o\~nez-Lasso et al. [Phys. Rev. A 88, 012702 (2013)]. The expectation values of $\ensuremath{\langle}{\mathbf{p}}_{\mathbf{1}}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{p}}_{\mathbf{2}}\ensuremath{\rangle}$ are also calculated for the resonant states investigated here. We conclude that the present methods in the framework of complex scaling enable us to get reliable energy, width, and other physical quantities of the resonant states in a variety of screening conditions.

Large-scale Dirac–Fock–Breit method using density fitting and 2-spinor basis functions

We present an efficient theory and algorithm for computing four-component relativistic Dirac-Fock wave functions using the Coulomb, Gaunt, and full Breit interactions. Our implementation is based on density fitting, and is routinely applicable to systems with 100 atoms and a few heavy elements. The small components are expanded using 2-spinor basis functions. We show that the factorization of 3-index half-transformed integrals before building Coulomb and exchange matrices is essential for efficient evaluation of the Fock matrix. With the Coulomb interaction, the computational cost for evaluating the Fock operator has been found to be only 70-90 times that in the non-relativistic density-fitted Hartree-Fock method. The prefactors have been 170 and 350-450 for the Gaunt and Breit interactions, respectively. The largest molecule to which we have applied the Dirac-Fock-Coulomb method is an Ac(III) motexafin complex (130 atoms, 556 electrons, 1289 basis functions), for which one self-consistent iteration takes around 1100 s using 1024 CPU cores. In addition, we have found that, while the standard fitting basis sets are accurate for Dirac-Fock-Coulomb calculations, their accuracy is very poor for Dirac-Fock-Gaunt and Breit calculations. We report a prototype of accurate fitting basis sets for these cases.

Efficient Structure Optimization with Second-Order Many-Body Perturbation Theory: The RIJCOSX-MP2 Method.

Efficient energy calculations and structure optimizations employing second-order Møller-Plesset perturbation theory (MP2) are presented. The application of the RIJCOSX approximation, which involves different approximations for the formation of the Coulomb- and exchange-type matrices, to MP2 theory is demonstrated. The RIJCOSX approximation incorporates the 'resolution of the identity' approximation in terms of a Split-RI-J variant for the evaluation of the Coulomb matrices and a seminumeric exchange treatment via the 'chain-of-spheres' algorithm for the formation of the exchange-type matrices. Beside the derivation of the working equations, the RIJCOSX-MP2 method is benchmarked against the original MP2 and the already highly efficient RI-MP2 method. Energies as well as gradients are computed employing various basis sets and are compared to the conventional MP2 results concerning accuracy and total wall clock times. Speedups of typically a factor of 5-7 in comparison to MP2 can be obeserved for the largest basis set employed in our study. Total energies are reproduced with an average error of ≤0.8 kcal/mol and minimum energy geometries differ by ∼0.1 pm in bond lengths and typically ∼0.2 degrees in bond angles. The RIJCOSX-MP2 gradient parallelizes with a speedup of 8.2 on 10 processors. The algorithms are implemented into the ORCA electronic structure package.

Direct calculation of the Coulomb matrix: Slater-type orbitals

It is proved that the evaluation of the Coulomb potential and the calculation of its matrix elements can be carried out in separate steps whose costs formally increase as the square of the number of basis functions. The resulting method for computing the Coulomb matrix is reported, and its main features are tested with a trial program for Slater functions. A comparison of the Coulomb matrices obtained with this method and those computed from the repulsion integrals shows that the current procedure is potentially exact, highly accurate in practice, and much less expensive. The effects of basis product cutoff and long-range separation have been analyzed finding that the method tends to linear scaling with the size of the system. Moreover, the storage requirements are very low since two-electron integrals are completely absent, and it is well suited to be used in the density functional context.

Automatic Selection of Integral Thresholds by Extrapolation in Coulomb and Exchange Matrix Constructions.

We present a method to compute Coulomb and exchange matrices with predetermined accuracy as measured by a matrix norm. The computation of these matrices is fundamental in Hartree-Fock and Kohn-Sham electronic structure calculations. We show numerically that, when modern algorithms for Coulomb and exchange matrix evaluation are applied, the Euclidean norm of the error matrix ε is related to the threshold value τ as ε = cτ(α). The presented extrapolation method automatically selects the integral thresholds so that the Euclidean norm of the error matrix is at the requested accuracy. This approach is demonstrated for a variety of systems, including protein-like systems, water clusters, and graphene sheets. The proposed method represents an important step toward complete error control throughout the self-consistent field calculation as described in [J. Math. Phys. 2008, 49, 032103].

Poisson-transformed density fitting in relativistic four-component Dirac–Kohn–Sham theory

We present recent developments in the implementation of the density fitting approach for the Coulomb interaction within the four-component formulation of relativistic density functional theory [Belpassi et al., J. Chem. Phys. 124, 124104 (2006)]. In particular, we make use of the Poisson equation to generate suitable auxiliary basis sets and simplify the electron repulsion integrals [Manby and Knowles, Phys. Rev. Lett. 87, 163001 (2001)]. We propose a particularly simple and efficient method for the generation of accurate Poisson auxiliary basis sets, based on already available standard Coulomb fitting sets. Just as is found in the nonrelativistic case, we show that the number of standard auxiliary fitting functions that need to be added to the Poisson-generated functions in order to achieve a fitting accuracy equal or, in some cases, better than that of the standard procedure is remarkably small. The efficiency of the present implementation is demonstrated in a detailed study of the spectroscopic properties and energetics of several gold containing systems, including the Au dimer and the CsAu molecule. The extraction reaction of a H(2)O molecule from a Au(H(2)O)(9) (+) cluster is also calculated as an example of mixed heavy-light-atom molecular systems. The scaling behavior of the algorithm implemented is illustrated for some closed shell gold clusters up to Au(5) (+). The increased sparsity of the Coulomb matrices involved in the Poisson fitting is identified, as are potential computational applications and the use of the Poisson fitting for the relativistic exchange-correlation problem.