Electron Repulsion Integrals Research Articles

Implementation of analytic gradients for CCSD and EOM-CCSD using Cholesky decomposition of the electron-repulsion integrals and their derivatives: Theory and benchmarks.

We present a general formulation of analytic nuclear gradients for the coupled-cluster with single and double substitution (CCSD) and equation-of-motion (EOM) CCSD energies computed using Cholesky decomposition (CD) representations of the electron repulsion integrals. By rewriting the correlated energy and response equations such that the storage of the largest four-index intermediates is eliminated, CD leads to a significant reduction in disk storage requirements, reduced I/O penalties, and an improved parallel performance. CD thus extends the scope of the systems that can be treated by (EOM-)CCSD methods, although analytic gradients in the framework of CD are needed to extend the applicability of (EOM-)CCSD methods in the context of geometry optimizations. This paper presents a formulation of analytic (EOM-)CCSD gradient within the CD framework and reports on the salient details of the corresponding implementation. The accuracy and the capabilities of analytic CD-based (EOM-)CCSD gradients are illustrated by benchmark calculations and several illustrative examples.

Analytical dipole moments and dipole polarizabilities of doublet radicals using Cholesky representation of constrained variational linear response to Fock-space multi-reference coupled-cluster method with single and double substitutions

We present dipole moments and dipole polarizabilities of doublet radicals computed analytically using constrained variational approach (CVA) based Fock-space (FS) multi-reference (MR) coupled-cluster (CC) method with single and double substitutions (SD) after Cholesky decomposition (CD) of electron repulsion integrals (ERI’s) in molecular orbital (MO) basis. We present our results computed with CD tolerance of 10-1,10-2 and 10-3 and test their accuracy by comparing them with the ones obtained by the conventional CVA-FSMRCCSD.

An efficient algorithm for Cholesky decomposition of electron repulsion integrals.

Approximating the electron repulsion integrals using inner projections is a well-established approach to reduce the computational demands of electronic structure calculations. Here, we present a two-step Cholesky decomposition algorithm where only the elements of the Cholesky basis (the pivots) are determined in the pivoting procedure. This allows for improved screening, significantly reducing memory usage and computational cost. After the pivots have been determined, the Cholesky vectors are constructed using the inner projection formulation. We also propose a partitioned decomposition approach where the Cholesky basis is chosen from a reduced set generated by decomposing diagonal blocks of the matrix. The algorithm extends the application range of the methodology and is well suited for multilevel methods. We apply the algorithm to systems with up to 80 000 atomic orbitals. The accuracy of the integral approximations is demonstrated for a formaldehyde-water system using a new Cholesky-based CCSD implementation.

Efficient Ab Initio Auxiliary-Field Quantum Monte Carlo Calculations in Gaussian Bases via Low-Rank Tensor Decomposition.

We describe an algorithm to reduce the cost of auxiliary-field quantum Monte Carlo (AFQMC) calculations for the electronic structure problem. The technique uses a nested low-rank factorization of the electron repulsion integral (ERI). While the cost of conventional AFQMC calculations in Gaussian bases scales as , where N is the size of the basis, we show that ground-state energies can be computed through tensor decomposition with reduced memory requirements and subquartic scaling. The algorithm is applied to hydrogen chains and square grids, water clusters, and hexagonal BN. In all cases, we observe significant memory savings and, for larger systems, reduced, subquartic simulation time.

The static parallel distribution algorithms for hybrid density-functional calculations in HONPAS package

Hybrid density-functional calculation is one of the most commonly adopted electronic structure theories in computational chemistry and materials science because of its balance between accuracy and computational cost. Recently, we have developed a novel scheme called NAO2GTO to achieve linear scaling (Order-N) calculations for hybrid density-functionals. In our scheme, the most time-consuming step is the calculation of the electron repulsion integrals (ERIs) part, so creating an even distribution of these ERIs in parallel implementation is an issue of particular importance. Here, we present two static scalable distributed algorithms for the ERIs computation. Firstly, the ERIs are distributed over ERIs shell pairs. Secondly, the ERIs are distributed over ERIs shell quartets. In both algorithms, the calculation of ERIs is independent of each other, so the communication time is minimized. We show our speedup results to demonstrate the performance of these static parallel distributed algorithms in the Hefei Order-N packages for ab initio simulations.

Integral partition bounds for fast and effective screening of general one-, two-, and many-electron integrals.

We introduce tight upper bounds for a variety of integrals appearing in electronic structure theories. These include electronic interaction integrals involving any number of electrons and various integral kernels such as the ubiquitous electron repulsion integrals and the three- and four-electron integrals found in explicitly correlated methods. Our bounds are also applicable to the one-electron potential integrals that appear in great number in quantum mechanical (QM), mixed quantum and molecular mechanical (QM/MM), and semi-numerical methods. The bounds are based on a partitioning of the integration space into balls centered around electronic distributions and their complements. Such a partitioning leads directly to equations for rigorous extents, which we solve for shell pair distributions containing shells of Gaussian basis functions of arbitrary angular momentum. The extents are the first general rigorous formulation we are aware of, as previous definitions are based on the inverse distance operator 1/r12 and typically only rigorous for simple spherical Gaussians. We test our bounds for six different integral kernels found throughout quantum chemistry, including exponential, Gaussian, and complementary error function based forms. We compare to previously developed estimates on the basis of significant integral counts and their usage in both explicitly correlated second-order Møller-Plesset theory (MP2-F12) and density functional theory calculations employing screened Hartree-Fock exchange.

Efficient evaluation of the geometrical first derivatives of three-center Coulomb integrals.

The calculation of the geometrical derivatives of three-center electron repulsion integrals (ERIs) over contracted spherical harmonic Gaussians has been optimized. We compared various methods based on the Obara-Saika, McMurchie-Davidson, Gill-Head-Gordon-Pople, and Rys polynomial algorithms using Cartesian, Hermite, and mixed Gaussian integrals for each scheme. The latter ERIs contain both Hermite and Cartesian Gaussians, and they combine the advantageous properties of both types of basis functions. Furthermore, prescreening of the ERI derivatives is discussed, and an efficient approximation of the Cauchy-Schwarz bound for first derivatives is presented. Based on the estimated operation counts, the most promising schemes were implemented by automated code generation, and their relative performances were evaluated. We analyzed the benefits of computing all of the derivatives of a shell triplet simultaneously compared to calculating them just for one degree of freedom at a time, and it was found that the former scheme offers a speedup close to an order of magnitude with a triple-zeta quality basis when appropriate prescreening is applied. In these cases, the Obara-Saika method with Cartesian Gaussians proved to be the best approach, but when derivatives for one degree of freedom are required at a time the mixed Gaussian Obara-Saika and Gill-Head-Gordon-Pople algorithms are predicted to be the best performing ones.

RAQET: Large-scale two-component relativistic quantum chemistry program package.

The Relativistic And Quantum Electronic Theory (RAQET) program is a new software package, which is designed for large‐scale two‐component relativistic quantum chemical (QC) calculations. The package includes several efficient schemes and algorithms for calculations involving large molecules which contain heavy elements in accurate relativistic formalisms. These calculations can be carried out in terms of the two‐component relativistic Hamiltonian, wavefunction theory, density functional theory, core potential scheme, and evaluation of electron repulsion integrals. Furthermore, several techniques, which have frequently been used in non‐relativistic QC calculations, have been customized for relativistic calculations. This article introduces the brief theories and capabilities of RAQET with several calculation examples. © 2018 The Authors Journal of Computational Chemistry Published by Wiley Periodicals, Inc.

Effective-potential theory for time-dependent many-electron wave functions

We derive a variation equation for a time-dependent effective potential that is local both in time and in space and show that this potential generates and propagates spin orbitals with which a many-particle, time-dependent, multiconfigurational wave function $\mathrm{\ensuremath{\Psi}}(t)$ is constructed. Within this approach, the wave function and the effective potential are determined simultaneously in a self-consistent manner both in time-independent and in time-dependent cases. We also derive an equation that determines a real-valued effective potential and show that the equation establishes a relation among the first-order and second-order reduced density matrices and the electron repulsion integrals in the spin-orbital representation. By introducing the first-order density equation, we show that the variation equation for the effective potential can be simplified for an exact wave function. We also show that we can derive an effective potential with which a ground-state wave function that fulfills the Brillouin-Brueckner condition is constructed and that we can derive the effective potential proposed by Slater, with which a ground-state wave function represented by the multiconfiguration expansion is calculated, when an additional constraint is imposed on the varitation equation.

Accuracy of auxiliary density functional theory hybrid calculations for activation and reaction enthalpies of pericyclic reactions

Auxiliary density functional theory (ADFT) hybrid calculations are based on the variational fitting of the Coulomb and Fock potential and, therefore, are free of four-center electron repulsion integrals. So far, ADFT hybrid calculations have been validated successfully for standard enthalpies of formation. In this work the accuracy of ADFT hybrid calculations for the description of pericyclic reactions was quantitatively validated at the B3LYP/6-31G*/GEN-A2* level of theory. Our comparison with conventional Kohn-Sham density functional theory (DFT) results shows that the DFT and ADFT activation and reaction enthalpies are practically indistinguishable. A systematic study of various functionals (PBE, B3LYP, PBE0, CAMB3LYP, CAMPBE0 and HSE06) and basis sets (6-31G*, DZVP-GGA and aug-cc-pVXZ; X = D, T and Q) revealed that the ADFT HSE06/aug-cc-pVTZ/GEN-A2* level of theory yields best balanced accuracy for the activation and reaction enthalpies of the studied pericyclic reactions. With the successfully validate ADFT composite approach consisting of PBE/DZVP-GGA/GEN-A2* structure and transition state optimizations and single-point HSE06/aug-cc-pVTZ/GEN-A2* energy calculations, an accurate, reliable and efficient computational approach for the study of pericyclic reactions in systems at the nanometer scale is proposed.

Analytic Excited State Gradients for the QM/MM Polarizable Embedded Second-Order Algebraic Diagrammatic Construction for the Polarization Propagator PE-ADC(2).

An implementation of a QM/MM embedding in a polarizable environment is presented for second-order Møller-Plesset perturbation theory, MP2, for ground state energies and molecular gradients and for the second-order Algebraic Diagrammatic Construction, ADC(2), for excitation energies and excited state molecular gradients. In this implementation of PE-MP2 and PE-ADC(2), the polarizable embedded Hartree-Fock wave function is used as uncorrelated reference state. The polarization-correlation cross terms for the ground and excited states are included in this model via an approximate coupling density. A Lagrangian formulation is used to derive the relaxed electron densities and molecular gradients. The resolution-of-the-identity approximation speeds up the calculation of four-index electron repulsion integrals in the molecular orbital basis. As a first application, the method is used to study the photophysical properties of host-guest complexes where the accuracy and weaknesses of the model are also critically examined. It is demonstrated that the ground state geometries of the full quantum mechanical calculation for the supermolecule can be well reproduced. For excited state geometries, the deviations from the supermolecular calculation are slightly larger, but still the environment effects are captured qualitatively correctly, and energy gaps between the ground and excited states are obtained with sufficient accuracy.

Double Precision Is Not Needed for Many-Body Calculations: Emergent Conventional Wisdom.

Using single-precision floating-point representation reduces the size of data and computation time by a factor of 2 relative to double precision conventionally used in electronic structure programs. For large-scale calculations, such as those encountered in many-body theories, reduced memory footprint alleviates memory and input/output bottlenecks. Reduced size of data can lead to additional gains due to improved parallel performance on CPUs and various accelerators. However, using single precision can potentially degrade the accuracy of the computed quantities. Here we report an implementation of coupled-cluster and equation-of-motion coupled-cluster methods with single and double excitations in single precision. We consider both standard implementation and one using Cholesky decomposition or resolution-of-the-identity representation of electron-repulsion integrals. Numerical tests illustrate that when single precision is used in correlated calculations, the loss of accuracy is insignificant, and pure single-precision implementation can be used for computing energies, analytic gradients, excited states, and molecular properties. In addition to pure single-precision calculations, our implementation allows one to follow a single-precision calculation by cleanup iterations, fully recovering double-precision results while retaining significant savings.

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.

Local density fitting applied to second order propagator equations for core electron binding energies

The second order electron propagator equations were combined with the local density fitting theory. Our equations require only the calculation of analytical three-center electron repulsion integrals. The same set of auxiliary functions employed for the variational fitting of the Coulomb potential are also used for the evaluation of the self–energy in the propagator calculation. Taking advantage of an efficient Hartree-Fock reference, the resulting implementation has been efficient in the calculation of second order electron propagator energies, especially for core particles. We also show that Almlöf transformation can not be directly applied to propagator equations without further considerations.

Derivative of electron repulsion integral using accompanying coordinate expansion and transferred recurrence relation method for long contraction and high angular momentum

AbstractIn this study, an early‐working algorithm is designed to evaluate derivatives of electron repulsion integrals (DERIs) for heavy‐element systems. The algorithm is constructed to extend the accompanying coordinate expansion and transferred recurrence relation (ACE‐TRR) method, which was developed for rapid evaluation of electron repulsion integrals (ERIs) in our previous article (M. Hayami, J. Seino, and H. Nakai, J. Chem. Phys. 2015, 142, 204110). The algorithm was formulated using the Gaussian derivative rule to decompose a DERI of two ERIs with the same sets of exponents, different sets of contraction coefficients, and different angular momenta. The algorithms designed for segmented and general contraction basis sets are presented as well. Numerical assessments of the central processing unit time of gradients for molecules were conducted to demonstrate the high efficiency of the ACE‐TRR method for systems containing heavy elements. These heavy elements may include a metal complex and metal clusters, whose basis sets contain functions with long contractions and high angular momenta.

Assessment of the overlap metric in the context of RI-MP2 and atomic batched tensor decomposed MP2

The resolution-of-the-identity approximation (RI) is a standard tool to accelerate the evaluation of two electron repulsion integrals. However introducing further approximations on top of for example RI-MP2, makes it important to check again the made choices. Usually the so called Coulomb metric is used. An alternative is the less accurate overlap metric, which has the benefit of a faster decay with distance. We encountered both choices in our atomic batched tensor decomposed MP2 (AB-MP2) and went with the Coulomb metric for the safe choice. In this work we re-investigate the choice of Coulomb and overlap metric for RI-MP2 and AB-MP2.

State-of-the-Art Computations of Vertical Ionization Potentials with the Extended Koopmans' Theorem Integrated with the CCSD(T) Method.

The accurate computation of ionization potentials (IPs), within 0.10 eV, is one of the most challenging problems in modern computational chemistry. The extended Koopmans' theorem (EKT) provides a systematic direct approach to compute IPs from any level of theory. In this study, the EKT approach is integrated with the coupled-cluster singles and doubles with perturbative triples [CCSD(T)] method for the first time. For efficiency, the density-fitting (DF) approximation is employed for electron repulsion integrals. Further, the EKT-CCSD(T) method is applied to a set of 23 molecules, denoted as IP23, for comparison with the experimental ionization potentials. For the IP23 set, the EKT-CCSD(T) method, along with the aug-cc-pV5Z basis set, provides a mean absolute error of 0.05 eV. Hence, our results demonstrate that direct computations of IPs at high-accuracy levels can be achieved with the EKT-CCSD(T) method. We believe that the present study may open new avenues in IP computations.

Libreta: Computerized Optimization and Code Synthesis for Electron Repulsion Integral Evaluation.

A new library called libreta for the evaluation of electron repulsion integrals (ERIs) over segmented and contracted Gaussian functions is developed. Our libreta is optimized from three aspects: (1) The Obara-Saika, Dupuis-Rys-King, and McMurchie-Davidson method are all employed. The recurrence relations involved are optimized by tree-search for each combination of angular momenta, and in the best case, 50% of the intermediates can be eliminated to reduce the computational cost. (2) The optimized codes for recurrence relations are combined with different contraction orders, each of which is suitable for ERIs of different angular momenta and contraction patterns. In practice, libreta will determine and use the best scheme to evaluate each ERI. (3) libreta is also optimized at the coding level. For example, with common subexpression elimination and local memory access, the performance can be increased by about 6% and 20%, respectively. The performance was compared with libint2. For both popular segmented and contracted basis sets, libreta can be faster than libint2 by 7.2-912.0%. For basis sets of heavy elements that contain Gaussian basis functions of large contraction degrees, the performance can be increased 20-30 times. We also tested the performance of libreta in direct self-consistent field (SCF) calculations and compared it with NWChem. In most cases, the average time for one SCF iteration by libreta is less than NWChem by 144.2-495.9%. Finally, we discuss the origin of redundancies occurring in the recurrence relations and derive an upper bound of the least number of intermediates required to be calculated in a McMurchie-Davidson recurrence, which is confirmed by ours as well as previous authors' results. We expect that libreta can become a useful tool for theoretical and computational chemists to develop their own algorithms rapidly.

Gaussian and plane-wave mixed density fitting for periodic systems.

We introduce a mixed density fitting scheme that uses both a Gaussian and a plane-wave fitting basis to accurately evaluate electron repulsion integrals in crystalline systems. We use this scheme to enable efficient all-electron Gaussian based periodic density functional and Hartree-Fock calculations.

An efficient MPI/OpenMP parallelization of the Hartree–Fock–Roothaan method for the first generation of Intel® Xeon Phi™ processor architecture

The Hartree–Fock method in the General Atomic and Molecular Structure System (GAMESS) quantum chemistry package represents one of the most irregular algorithms in computation today. Major steps in the calculation are the irregular computation of electron repulsion integrals and the building of the Fock matrix. These are the central components of the main self consistent field (SCF) loop, the key hot spot in electronic structure codes. By threading the Message Passing Interface (MPI) ranks in the official release of the GAMESS code, we not only speed up the main SCF loop (4× to 6× for large systems) but also achieve a significant ([Formula: see text]×) reduction in the overall memory footprint. These improvements are a direct consequence of memory access optimizations within the MPI ranks. We benchmark our implementation against the official release of the GAMESS code on the Intel® Xeon Phi™ supercomputer. Scaling numbers are reported on up to 7680 cores on Intel Xeon Phi coprocessors.