The SHARK integral generation and digestion system

In this paper, the SHARK integral generation and digestion engine is described. In essence, SHARK is based on a reformulation of the popular McMurchie/Davidson approach to molecular integrals. This reformulation leads to an efficient algorithm that is driven by BLAS level 3 operations. The algorithm is particularly efficient for high angular momentum basis functions (up to L = 7 is available by default, but the algorithm is programmed for arbitrary angular momenta). SHARK features a significant number of specific programming constructs that are designed to greatly simplify the workflow in quantum chemical program development and avoid undesirable code duplication to the largest possible extent. SHARK can handle segmented, generally and partially generally contracted basis sets. It can be used to generate a host of one- and two-electron integrals over various kernels including, two-, three-, and four-index repulsion integrals, integrals over Gauge Including Atomic Orbitals (GIAOs), relativistic integrals and integrals featuring a finite nucleus model. SHARK provides routines to evaluate Fock like matrices, generate integral transformations and related tasks. SHARK is the essential engine inside the ORCA package that drives essentially all tasks that are related to integrals over basis functions in version ORCA 5.0 and higher. Since the core of SHARK is based on low-level basic linear algebra (BLAS) operations, it is expected to not only perform well on present day but also on future hardware provided that the hardware manufacturer provides a properly optimized BLAS library for matrix and vector operations. Representative timings and comparisons to the Libint library used by ORCA are reported for Intel i9 and Apple M1 max processors.     

Calculation of two-center overlap integral in molecular coordinate system over Slater type orbital using Löwdin α-radial and Guseinov rotation–angular functions

By use of Löwdin and Guseinov relations for the radial and angular part of two-center overlap integrals, respectively, the computer calculations of overlap integrals over Slater type orbitals (STOs) in molecular coordinate system are performed. The results of calculations are valid for arbitrary principal quantum numbers, screening constants and location of STOs. Excellent agreement with benchmark results and stability of the technique are demonstrated.     

New translation method for STOs and its application to calculation of two-center two-electron integrals

The new translation method for Slater-type orbitals (STOs) previously tested in the case of the overlap integral is extended to the calculation of two-center two-electron molecular integrals. The method is based on the exact translation of the regular solid harmonic part of the orbital followed by the series expansion of the residual spherical part in powers of the radial variable. Fair uniform convergence and stability under wide changes in molecular parameters are obtained for all studied two-center hybrid, Coulomb, and exchange repulsion integrals. Ten-digit accuracy in the final numerical results is achieved through multiple precision arithmetic calculation of common angular coefficients and Gaussian numerical integration of some of the analytical formulas resulting for the radial integrals. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 91–100, 2000     

General expressions for monocenter repulsion integrals in a basis of real atomic orbitals

General expressions for monocenter electron repulsion integrals in a basis of real atomic orbitals are derived in terms of the radial integrals R. The final expressions for these integrals can be classified into five main classes which are characterized by the angular part of the real atomic orbitals. For a basis of real s, p, d, and f AO's the total number of monocenter repulsion integrals is 65536, from which 6652 are different from zero. The nonzero integrals can be classified into 430 groups which contain integrals of equal value.     

Hylleraas method for many‐electron atoms. II. Integrals over wave functions with one interelectronic coordinate

A procedure is proposed to evaluate matrix elements containing r linked with angular functions. Using this procedure, the different types of two‐, three‐, and four‐electron radial and angular integrals that appear in a five‐electron atom, in the case of only one r_ij coordinate per basis function, are written in a compact form, separated in radial coordinates of one electron. The general formulas with which to obtain the integrals for powers ν ≥ 1 are developed, based on the orthogonality of the Legendre polynomials. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Applications of fourier transforms in molecular orbital theory. Calculation of optical properties and tables of two-center one-electron integrals

Fourier transform methods initiated by Geller and Harris are applied to the calculation of optical properties of molecules. Tables of one-electron two-center integrals needed for the accurate computation of molecular absorption and optical activity are calculated by the Fourier transform method. A general theorem is derived which allows the angular part of the integrals to be treated by means of projection operators. The radial parts of the integrals are treated by the methods of Harris. The results are obtained in a simple closed form which avoids the usual transformation to local coordinates. The two-center integrals evaluated include matrix elements of the momentum operator, the dipole moment operator, the tensor operator , the quadrupole moment operator, and the angular momentum operator. These are evaluated between 1s, 2s, and 2p Slater-type atomic orbitals located on different atoms. The results are expressed as functions of the Slater exponents and of the relative coordinates of the two atoms.     

Fourier analysis of the Compton profile: Atoms and molecules

Recent work has shown that the one-dimensional projection of the electron momentum density, the Compton profile, can be usefully interpreted as a position space quantity. This has led to an examination of B(r), the Fourier transform of the momentum density. A number of theoretical results relating to this new observable are given. The wave-mechanical representation with (natural) orbitals is employed, and this forms the basis for the subsequent analysis of B(r). The relationship of B(r) to overlap integrals and more generally to other electron density functions is considered. Atomic wavefunctions for krypton are used to illustrate the potential of this new approach to the analysis of momentum density data. General expressions are derived for atoms and molecules, and the radial and angular dependence of B(r) for various orbitals is displayed. The possibility of extracting accurate bond lengths from B(r) is assessed, and an example is given using some recent theoretical data for the fluorine molecule.     

New recurrence relations for the rapid evaluation of electron repulsion integrals based on the accompanying coordinate expansion formula

We present an algorithm for the rapid computation of electron repulsion integrals (ERIs) over Gaussian basis functions based on the accompanying coordinate expansion (ACE) formula. The present algorithm uses equations termed angular momentum reduced expressions and introduces two types of recurrence relations to ACE formulas. Numerical efficiencies are assessed for (p pmid R:p p) and (sp spmid R:sp sp) ERIs by using the floating-point operation count. The algorithm is suitable for calculating ERIs for the same exponents but different angular momentum functions, such as L shells and derivatives of ERIs. The present algorithm is also capable of calculating ERIs with highly contracted Gaussian basis functions.     

Recursion formula for electron repulsion integrals over Hermite polynomials

A method for computing electron repulsion integrals over contracted Gaussian functions is described in which intermediate integrals over Hermite polynomials are generated by a “pre‐Hermite” recursion (PHR) step before the conversion to regular integrals. This greatly reduces the floating‐point operation counts inside the contraction loops, where only simple “scaling”‐type operations are required, making the method efficient for contracted Gaussians, particularly of high angular momentum. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

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

In 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.     

Density-functional expansion methods: generalization of the auxiliary basis

The formulation of density-functional expansion methods is extended to treat the second and higher-order terms involving the response density and spin densities with an arbitrary single-center auxiliary basis. The two-center atomic orbital products are represented by the auxiliary functions centered about those two atoms, and the mapping coefficients are determined from a local constrained variational procedure. This two-center variational procedure allows the mapping coefficients to be pretabulated and splined as a function of internuclear separation for efficient look up. The splines of mapping coefficients have a range no longer than that of the overlap integrals, and the auxiliary density appears as a single point-multipole expansion to all nonoverlapping atoms, thus allowing for the trivial implementation of a linear-scaling algorithm. The method is tested using Gaussian multipole expansions, and the effect of angular and radial completeness is explored. Several auxiliary basis sets are parametrized and compared to an auxiliary basis analogous to that used in the self-consistent-charge density-functional tight-binding model, and the method is demonstrated to greatly improve the representation of the density response with respect to a reference expansion model that does not use an auxiliary basis.     

Vibration--rotation--tunneling dynamics calculations for the four-dimensional (HCl)2 system: a test of approximate models

Several commonly used approximate methods for the calculation of vibration--rotation--tunneling spectra for (HCl)2 are described. These range from one-dimensional models to an exact coupled four-dimensional treatment of the intermolecular dynamics. Two different potential surfaces were employed--an ab initio and our ES1 experimental surface (determined by imbedding the four-dimensional calculation outlined here in a least-squares loop to fit the experimental data, which is described in the accompanying paper [J. Chem. Phys. 103, 933 (1995)]. The most important conclusion deduced from this work is that the validity of the various approximate models is extremely system specific. All of the approximate methods addressed in this paper were found to be sensitive to the approximate separability of the radial and angular degrees of freedom, wherein exists the primary difference between the two potentials. Of particular importance, the commonly used reversed adiabatic angular approximation was found to be very sensitive to the choice for fixed R; an improper choice would lead to results very much different from the fully coupled results and perhaps to false conclusions concerning the intermolecular potential energy surface.     

Molecular correlation integrals. Two center one electron integrals containing a function of interparticle distance in one center expansion approximation

Two-center one-electron integrals needed in certain molecular correlated wave function calculations, using one-center expansion approximation, have been studied. The form of the basic correlated function used in this study is The parent integral is expressed in terms of an angular integral, and an auxiliary radial integral depending upon the variables r₁, r₂, and r₁₂. Several analytical formulas, and a recursive formula are derived for the auxiliary integral, and other related integrals. All these formulas are given in computationally useful forms. Logical flow charts and FORTRAN programs were constructed for computing the basic integrals discussed in the paper. Numerical values of some integrals, thus obtained, are tabulated for comparisons.     

Molecular integrals by numerical quadrature. I. Radial integration

 This article presents a numerical quadrature intended primarily for evaluating integrals in quantum chemistry programs based on molecular orbital theory, in particular density functional methods. Typically, many integrals must be computed. They are divided up into different classes, on the basis of the required accuracy and spatial extent. Ideally, each batch should be integrated using the minimal set of integration points that at the same time guarantees the required precision. Currently used quadrature schemes are far from optimal in this sense, and we are now developing new algorithms. They are designed to be flexible, such that given the range of functions to be integrated, and the required precision, the integration is performed as economically as possible with error bounds within specification. A standard approach is to partition space into a set of regions, where each region is integrated using a spherically polar grid. This article presents a radial quadrature which allows error control, uniform error distribution and uniform error reduction with increased number of radial grid points. A relative error less than 10⁻¹⁴ for all s-type Gaussian integrands with an exponent range of 14 orders of magnitude is achieved with about 200 grid points. Higher angular l quantum numbers, lower precision or narrower exponent ranges require fewer points. The quadrature also allows controlled pruning of the angular grid in the vicinity of the nuclei. Received: 30 August 2000 / Accepted: 21 December 2000 / Published online: 3 April 2001     

Axial gaussian-lobe orbitals: Optimization of highly angularly dependent functions

Highly angularly dependent axial Gaussian-lobe orbitals (AGLO ) (up to L = 5) are presented. The angular and radial optimizations of these functions have been realized on the ground of theoretical frameworks, previously reported in the literature and somewhat extended here. The numerical difficulties that can appear in the recombination of elementary integrals over lobes are particularly investigated. It is shown that it is necessary to limit the angular accuracy, i.e., the relevant Y_Lo character, in order to preserve the accuracy of atomic integrals. The proposed p, d, f, g, and h AGLOS satisfy this condition, and can be used with confidence in LCAO–MO–SCF calculations. Their advantages, e.g., for the treatment of large symmetrical inorganic systems containing transition metal atoms, are emphasized.     

Singular and nonsingular three-body integrals for exponential wave functions

Integrals which are individually singular, but which may be combined to yield convergent expressions, are needed for computations of relativistic effects and various properties of atomic and quasiatomic systems. As computations become more detailed and precise, more such integrals are required. This paper presents general formulas for the radial parts of the singular and nonsingular (regular) integrals that occur when three-body systems are described using wave functions that include exponentials in all three interparticle coordinates. Our results are compared with those found in the literature for some of the integrals, and are also shown to be consistent with previously reported results for Hylleraas functions (a limiting case in which one of the exponential parameters is set to zero).     

On bound states in two-body systems

The application of the Σ-separation method to the calculation of multicenter two-electron molecular integrals with Slater-type basis functions is reported. The approach is based on the approximation of a scalar component of the two-center atomic density by a two-center expansion over Slater-type functions. A least-squares fit was used to determine the coefficients of the expansion. The angular multipliers of the atomic density were treated exactly. It is shown that this approach can serve as a sufficiently accurate and fast algorithm for the calculation of multicenter two-electron molecular integrals with Slater-type basis functions. © 1995 John Wiley & Sons, Inc.     

Use of binomial coefficients in fast and accurate calculation of Löwdin-<Emphasis Type="Italic">α</Emphasis> radial functions

In the present paper, we introduce a new algorithm for the development of efficient numerical methods for the analytical solution of the Löwdin-α radial function. The final results are expressed through the binomial coefficients, which enable fast and accurate calculation of the Löwdin-α radial function. The comparisons of the obtained results with the literature have shown that the presented approach can be used for the fast calculation of multicenter multielectron integrals over Slater-type orbitals (STOs). The efficiency of the algorithm is discussed and its performance with several examples is demonstrated.     

Evaluation of Hylleraas-CI atomic integrals by integration over the coordinates of one electron. II. Four-electron integrals

An alternative procedure to the classical method for evaluating the four-electron Hylleraas-CI integrals is given. The method consists of direct integration over the r ₁₂ coordinate and integration over the coordinates of one of the electrons, reducing the integrals to lower order. The method based on the earlier work of Calais and L?wdin and of Perkins is extended to the general angular case. In this way it is possible to solve all of the four-electron integrals appearing in the Hylleraas-CI method. The four-electron integrals are expanded in three-electron ones which are in turn expanded in two-electron integrals. Finally the two-electron integrals are expanded into two-electron auxiliary integrals which usually have one negative power. The use of three- and four-electron electron auxiliary integrals is avoided. Some special cases lead to one- and two-electron auxiliary integrals with negative powers which do not converge individually but do converge in combination with others. These relations and their solutions are presented, together with results of various kinds of integrals.