Evaluation Of Two-electron Integrals Research Articles

Communication: Optical gap in polyacetylene from a simple quantum chemistry exciton model.

A recent virial-theorem-based model of the singlet-triplet splitting in singly excited configurations [A. D. Becke, J. Chem. Phys. 148, 044112 (2018)] offers an accurate and economical tool for the computation of optical gaps in large molecules. Two single-determinant density-functional-theory calculations, one on the (closed-shell) ground state and another on the (restricted open-shell) HOMO → LUMO triplet excited state, followed by a simple two-electron integral evaluation, are all we need. Here the method is used to compute the optical gaps of trans-polyenes containing up to 60 carbon atoms, approaching the polyacetylene limit. Comparisons with previous computations, and experiment, are made. We also explore changes of the exact-exchange mixing fraction in the underlying density functional. Its effect on the optical gap, and also the exciton size, is enormous. Thus we face the vexing, often asked, question: how much exact exchange should be used?

Auxiliary Basis Sets for Density Fitting in Explicitly Correlated Calculations: The Atoms H-Ar.

Auxiliary basis sets specifically matched to the correlation consistent cc-pVnZ-F12 and cc-pCVnZ-F12 orbital basis sets for the elements H-Ar have been optimized at the density-fitted second-order Møller-Plesset perturbation theory level of theory for use in explicitly correlated (F12) methods, which utilize density fitting for the evaluation of two-electron integrals. Calculations of the correlation energy for a test set of small to medium sized molecules indicate that the density fitting error when using these auxiliary sets is 2 to 3 orders of magnitude smaller than the F12 orbital basis set incompleteness error. The error introduced by the use of these fitting sets within the resolution-of-the-identity approximation of the many-electron integrals arising in F12 theory has also been assessed and is demonstrated to be negligible and well-controlled. General guidelines are proposed for the optimization of density fitting auxiliary basis sets for use with F12 methods for other elements.

Accurate localized resolution of identity approach for linear-scaling hybrid density functionals and for many-body perturbation theory

A key component in calculations of exchange and correlation energies is the Coulomb operator, which requires the evaluation of two-electron integrals. For localized basis sets, these four-center integrals are most efficiently evaluated with the resolution of identity (RI) technique, which expands basis-function products in an auxiliary basis. In this work we show the practical applicability of a localized RI-variant (‘RI-LVL’), which expands products of basis functions only in the subset of those auxiliary basis functions which are located at the same atoms as the basis functions. We demonstrate the accuracy of RI-LVL for Hartree–Fock calculations, for the PBE0 hybrid density functional, as well as for RPA and MP2 perturbation theory. Molecular test sets used include the S22 set of weakly interacting molecules, the G3 test set, as well as the G2–1 and BH76 test sets, and heavy elements including titanium dioxide, copper and gold clusters. Our RI-LVL implementation paves the way for linear-scaling RI-based hybrid functional calculations for large systems and for all-electron many-body perturbation theory with significantly reduced computational and memory cost.

Implementation of the electron propagator to second order on GPUs to estimate the ionization potentials of confined atoms

The best way to estimate ionization potentials (I) for confined atoms is by using the same Hamiltonian for the neutral and the corresponding hypothetical ionized atom. For this purpose, we have implemented the electron propagator to second order () by using parallel programming techniques on graphic processing units (GPUs). These programming techniques exploit the GPUs for the evaluation of two-electron integrals, which is required for a self- consistent process and because of the reduction involved in the four-index integral transformation. As an example, we present results for confined helium, beryllium and neon atoms, and these are contrasted with previously reported results. Although Koopmans’ theorem provides good estimates for ionization potentials, it is evident that corrects these estimates. Unfortunately, the correction made by does not reveal a trend for confined atoms because in the case of certain confinement regions overestimates, whereas for other regions, underestimates the ionization potential. The orbital crossing between unoccupied orbitals is responsible for this behavior. In particular, if the lowest unoccupied atomic orbital (LUMO) crosses a virtual orbital, the difference will change its sign. Thus, approximation is required when the ionization potential is estimated for confined atoms.

Mixed-precision evaluation of two-electron integrals by Rys quadrature

A mixed precision implementation of two-electron integrals is demonstrated to have two benefits: (a) computations can be performed reliably in 32-bit precision on architectures for which 32-bit precision is significantly faster than 64-bit precision (e.g. graphical processing units), and (b) numerical results that match those using higher than 64-bit precision can be recovered without a significant penalty associated with performing the entire computation in higher precision. A justification is presented for using mixed precision in the Rys two-electron integral quadrature algorithm, together with timings and numerical results using a variety of floating-point types. The code discussed here presents a systematic way to control the accuracy of the Rys algorithm, regardless of the types and numbers of integrals.

Modelling the Runtime of the Gaussian Computational Chemistry Application and Assessing the Impacts of Microarchitectural Variations

This paper explores the use of a simple linear performance model, that determines execution time based instruction and cache miss counts, for describing the behaviour of two-electron integral evaluation algorithm in the Gaussian computational chemistry package. Four different microarchitecture platforms are considered with a total of seven individual microprocessors. Both Hartree-Fock and hybrid Hartree-Fock/Density Functional Theory electronic structure methods are assessed. In most cases the model is found to be accurate to within 3%. Least agreement is for an Athlon64 system (ranging from 1.8% to 6.5%) and a periodic boundary computation on an Opteron where errors of up to 6.8% are observed. These errors arise as the model does not account for the intricacies of out-of-order execution, on-chip write-back buffers and prefetch techniques that modern microprocessors implement. The parameters from the linear performance model are combined with instruction and cache miss counts obtained from functional cache simulation to predict the effect of cache modification on total execution time. Variations in level 1 and 2 linesize and level 2 total size are considered, we find there is some benefit if linesizes are increased (L1: 8%, L2: 4%). Increasing the level 2 cache size is also predicted to be beneficial, although the cache blocking approach already implemented in the Gaussian integral evaluation code was found to be working well.

Computation of the Hartree-Fock Exchange by the Tensor-Structured Methods

AbstractWe propose a novel numerical method for fast and accurate evaluation of the exchange part of the Fock operator in the Hartree-Fock equation which is a (nonlocal) integral operator. Usually, this challenging computational problem is solved by analytical evaluation of two-electron integrals using the “analytically separable” Galerkin basis functions, like Gaussians. Instead, we employ the agglomerated “grey-box” numerical computation of the corresponding six-dimensional integrals in the tensor-structured format which does not require analytical separability of the basis set. The point of our method is a low-rank tensor representation of arising functions and operators on an n×n×n Cartesian grid and the implementation of the corresponding multi-linear algebraic operations in the tensor product format. Linear scaling of the tensor operations, including the 3D convolution product, with respect to the one-dimension grid size n enables computations on huge 3D Cartesian grids thus providing the required high accuracy. The presented algorithm for evaluation of the exchange operator and a recent tensor method for the computation of the Coulomb matrix are the main building blocks in the numerical solution of the Hartree-Fock equation by the tensor-structured methods. These methods provide a new tool for algebraic optimization of the Galerkin basis in the case of large molecules.

Periodic local-MP2 computational study of crystalline neon

Face-centered-cubic crystalline Neon is taken as a test system to explore the influence of computational parameters on the quality of the MP2 solution provided by the Cryscor program using a local-correlation approach. The effect of the various approximations adopted is analyzed: basis set limitations, finite size of excitation domains, truncation of the tails of the local functions, approximate evaluation of two-electron integrals, estimate (by extrapolation) of long-range contributions are shown to play roles of different importance. The Ne2 dimer is used as an auxiliary test case in order to allow comparison with recent and accurate literature data.

Rational approximants to evaluate four-center electron repulsion integrals for 1s hydrogen Slater type functions

Modelling a problem in quantum chemistry, within the framework of Hartree–Fock approach, requires the evaluation of four-center two-electron integrals. The speed up of the integral evaluation is a key point in order to get a method suitable to deal with large molecules. The most expensive step of the process involves the evaluation of the two-electron integrals. Here we propose a method to compute those involving 1s hydrogen Slater type orbitals, which is based on a expression proposed by Shavitt and Karplus in [J. Chem. Phys. 43 (1965) 398]. After a change of variables performed on a suitable semi-infinite integral given in [J. Chem. Phys. 43 (1965) 398], the integrand is replaced by a rational approximant which is able to be solved analytically using the method of residues. We suggest two alternatives: a pure interpolation and an Hermite interpolant strategy. In both cases, we present evidence of the good numerical behavior of our proposal on some class of examples. Besides, the method requires a computational effort comparable with a two-dimensional numerical integration.

Evaluation of two-electron integrals including the factors r12kexp(− γr212) over Cartesian Gaussian functions

We present a practical scheme for the evaluation of nonstandard two-electron integrals including the factors r 12 k exp(− γr 12 2) which have been appeared recently, where k⩾−1 is an integer. The method used throughout this paper is based on the highly efficient Head-Gordon and Pople (HGP) approach of evaluation of electron repulsion integrals (ERI). Thus only straightforward modifications of existing codes that employ HGP or HGP-PRISM scheme are necessary to implement our approach.

Integral algorithm and density matrix integration scheme for ab initio band structure calculations on polymeric systems.

A new program for band structure calculations of periodic one-dimensional systems has been constructed. It is distinguishable from other codes by the efficient two-electron integral evaluation and the integration schemes of the density matrix in the first Brillouin zone. The computation of polymeric two-electron integrals is based on the McMurchie Davidson algorithm and builds batches of the different cell indices included in the polymeric system. Consequently it presents efficient scaling with respect to the number of unit cells taken into account. Our algorithm takes into account fully the polymeric symmetry rather than the molecular symmetry. A semidirect procedure where only exchange integrals are computed at each SCF cycle is proposed in order to maintain balance between computation time and disk space. In addition, the integration of the density matrix over a large number of cell indices can be performed by different methods, such as Gauss-Legendre, Clenshaw-Curtis, Filon, and Alaylioglu-Evans-Hyslop. This last scheme is able to obtain an accuracy of 10(-13) a.u. on each individual density matrix element for all cell indices with only 48 k-points.

Evaluation of two-electron integrals for explicit r12 theories

We present a practical scheme for the evaluation of nonstandard two-electron molecular integrals that appear in ab initio theories employing explicitly correlated wave functions with linear r12 terms (“linear r12” methods). In contrast with previous efforts, the target integrals are evaluated recursively via intermediates formulated solely in terms of Cartesian Gaussian functions. All working equations fit conveniently the framework of highly efficient Head-Gordon and Pople method of evaluation of electron repulsion integrals. Thus, only straightforward modifications of existing codes that employ HGP or HGP–PRISM scheme are necessary to implement our approach. High potential of the pathway is realized in a robust practical implementation.

Self-consistent treatment of the frequency-independent Breit interaction in Dirac-Fock and MCSCF calculations of atomic structures: I. Theoretical considerations

The self-consistent treatment of the Breit interaction in fully numerical atomic structure calculations is cumbersome due to the computationally demanding evaluation of two-electron integrals as they occur in the original formulation (Grant I P and Pyper N C 1976 J. Phys. B: At. Mol. Phys. 9 761). We present a reformulation of the frequency-independent Breit interaction operator in spherical coordinates and derive the corresponding matrix elements over spinors. With this formulation it becomes possible to compute the matrix elements of the Breit interaction efficiently and analogously to those of the Coulomb interaction: i.e., by determining the corresponding interaction potential functions using Poisson equations. The derived formulae will equally simplify computations using either basis sets or a numerical representation of the orbitals such that the Breit interaction can be included effectively in CI and SCF calculations for atoms and molecules. Of course, the computation of the Breit contribution to the total electronic energy as a first-order perturbation correction is also simplified. Furthermore, the frequency-dependent Breit interaction could be treated analogously.

Variational solution of the Dirac-Coulomb equation using explicitly correlated wavefunctions. Matrix elements and radial integrals

An algorithm for the evaluation of two-electron integrals appearing in an implementation of the Hylleraas-CI method for solving the Dirac-Coulomb eigenvalue problem is presented. All integrals have been expressed analytically, though in some cases as infinite series.

Methods for rapid evaluation of electron repulsion integrals in large-scale LCGO calculations

A method is proposed for efficient use of molecular symmetry in the evaluation of two-electron integrals. This provides a means of avoiding the recalculation of symmetry-redundant integrals, and of symmetry-blocking matrices and supermatrices without the usual time-consuming transformation procedures. Various methods for speeding up the calculation of integrals are also discussed. Integral calculation times are given for some representative molecules.

Removal of the bottleneck in local correlation methods

We present preliminary results from a new hybrid pseudospectral local correlation method in which we combine the pseudospectral approximation to two-electron integral evaluation with local truncation of the virtual space to accelerate singles and double excitation configuration interaction (SDCI). The combination of the two leads even in small molecules to overall speed-ups of factors of 3–5 and reduces the main disk space requirement by factors of 9–20 while retaining at least 98% of the correlation energy of the exact SDCI calculations. These preliminary results bode well for the advantages of further development of pseudospectral local correlation methods.

Some aspects of parallelization of two-electron integrals in molecular orbital programs

Evaluation of two-electron integrals forms a substantial part of the CPU time for any ab initio molecular orbital program. This part of the package, “MICROMOL”, is parallelized. However, this parallelization leads to only sublinear speedups (typically 3 on a 4-node machine). In view of these results, the task of development of an efficient program for two-electron integrals suitable for the parallel environment has been taken up. The program is written in FORTRAN considering specific symmetry features and application of rigorous bounds. This program is further parallelized with a good load balancing strategy. The molecules used as the test cases are: trans-butadiene, benzene, nitrobenzene, naphtalene and cytosine, with 3G and 4–31G basis sets. The results indicate that the parallel version of this program gives a typical speedup of 3.6 for a 3G basis set and approximately 3.4 for a 4–31G basis set for all the molecules tested. The sequential version of this program is ∼1.2 times faster than the sequential version of MICROMOL, whereas the parallel version is ∼1.4 times faster than the parallelized MICROMOL.

The reduced multiplication scheme of the Rys quadrature and new recurrence relations for auxiliary function based two-electron integral evaluation

The reduced multiplication scheme of the Rys quadrature is presented. The method is based on new ways in which the Rys quadrature can be developed if it is implemented together with the transfer equation applied to the contracted integrals. In parallel to the new scheme of the Rys quadrature improvements are suggested to the auxiliary function based algorithms. The two new methods have very favorable theoretical floating point operation (FLOP) counts as compared to other methods. It is noted that the only significant difference in performance of the two new methods is due to the vectorizability of the presented algorithms. In order to exhibit this, both methods were implemented in the integral program seward. Timings are presented for comparisons with other implemenations. Finally, it is demonstrated how the transfer equation in connection with the use of spherical harmonic Gaussians offers a very attractive path to compute the two-electron integrals of such basis functions. It is demonstrated both theoretically and with actual performance that the use of spherical harmonic Gaussians offers a clear advantage over the traditional evaluation of the two-electron integrals in the Cartesian Gaussian basis.

New variations in two-electron integral evaluation in the context of direct SCF procedures

A new horizontal recurrence relation (HRR) is derived which may be used in the generation of two-electron repulsion integrals (ERIs). This work focuses primarily on the application of the new HRR to the 1988 Head-Gordon and Pople method and will be complementary to the more recent Gill, Head-Gordon and Pople procedures applicable to highly contracted basis sets. Special Cartesian coordinate axes may be used in reducing the number of nonzero intermediates dramatically. In direct SCF it may not be desirable to back-transform integrals, but evaluate the contributions to the Fock matrix for many classes of ERIs in the same special axis system, and then back-transform the Fock matrix contribution. Integral derivative evaluation is also discussed. Both the new HRR and the Head-Gordon and Pople HRR may be used in relating ERI derivatives.

Concise, open-ended implementation of Rys polynomial evaluation of two-Electron integrals

A quadrature-point-driven implementation of the standard Rys polynomial method for computing two-electron repulsion integrals of gaussian basis functions has been found to be both concise and openended with respect to the angular momentum of the gaussian functions (i.e., s,p,d,f,g,…). These are important features in certain applications, such as molecular properties and property gradients.