The W algorithm and the D transformation for the numerical evaluation of three‐center nuclear attraction integrals

Three‐center nuclear attraction integrals over exponential‐type functions are required for ab initio molecular structure calculations and density functional theory (DFT). These integrals occur in many millions of terms, even for small molecules, and they require rapid and accurate numerical evaluation. The use of a basis set of B functions to represent atomic orbitals, combined with the Fourier transform method, led to the development of analytic expressions for these molecular integrals. Unfortunately, the numerical evaluation of the analytic expressions obtained turned out to be extremely difficult due to the presence of two‐dimensional integral representations, involving spherical Bessel integral functions. % The present work concerns the development of an extremely accurate and rapid algorithm for the numerical evaluation of these spherical Bessel integrals. This algorithm, which is based on the nonlinear D transformation and the W algorithm of Sidi, can be computed recursively, allowing the control of the degree of accuracy. Numerical analysis tests were performed to further improve the efficiency of our algorithm. The numerical results section demonstrates the efficiency of this new algorithm for the numerical evaluation of three‐center nuclear attraction integrals. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Half-numerical evaluation of pseudopotential integrals

A half-numeric algorithm for the evaluation of effective core potential integrals over Cartesian Gaussian functions is described. Local and semilocal integrals are separated into two-dimensional angular and one-dimensional radial integrals. The angular integrals are evaluated analytically using a general approach that has no limitation for the l-quantum number. The radial integrals are calculated by an adaptive one-dimensional numerical quadrature. For the semilocal radial part a pretabulation scheme is used. This pretabulation simplifies the handling of radial integrals, makes their calculation much faster, and allows their easy reuse for different integrals within a given shell combination. The implementation of this new algorithm is described and its performance is analyzed.     

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.     

Optimized transformation of four center integrals

The n ⁵-order algorithm for the transformation of the quantummechanical four center integrals is analyzed. An optimum implementation of this algorithm for computers with a reasonable amount of direct-access storage is described.     

Study of localized molecular orbitals using group theory methods and its approach to the many-electron correlation problem. IV. The symmetry-adaptation of many-center integrals and hamiltonian matrix elements in MCSCF calculations

Group theoretic methods are presented for the transformations of integrals and the evaluation of matrix elements encountered in multiconfigurational self-consistent field (MCSCF) and configuration interaction (CI) calculations. The method has the advantages of needing only to deal with a symmetry unique set of atomic orbitals (AO) integrals and transformation from unique atomic integrals to unique molecular integrals rather than with all of them. Hamiltonian matrix element is expressed by a linear combination of product terms of many-center unique integrals and geometric factors. The group symmetry localized orbitals as atomic and molecular orbitals are a key feature of this algorithm. The method provides an alternative to traditional method that requires a table of coupling coefficients for products of the irreducible representations of the molecular point group. Geometric factors effectively eliminate these coupling coefficients. The saving of time and space in integral computations and transformations is analyzed. © 1994 by John Wiley & Sons, Inc.     

Numerical calculation of overlap and kinetic integrals in prolate spheroidal coordinates

The efficient algorithm calculating the overlap and the kinetic integrals for the numerical atomic orbitals is presented. The described algorithm exploits the properties of the prolate spheroidal coordinates. The overlap and the kinetic integrals in ?³ are reduced to the integrals over the rectangular domain in ?², what substantially reduces the complexity of the problem. We prove that the integrand over the rectangular domain is continuous and does not have any slope singularities. For calculation of the integral over the rectangle any adaptive algorithm can be applied. The exemplary results were obtained by application of the adaptive Gauss quadrature. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Computation of some new two-electron Gaussian integrals

Summary The evaluation of a new form of two-electron integrals is required if the interelectronic distancer ₁₂ is used as a variable in then-electron functions of electron correlation methods. The McMurchie-Davidson algorithm for the generation of molecular integrals over Gaussian-type functions is ideally suited to this. The new Gaussian integrals are formed from Hermite integrals overr ₁₂ (rather than 1/r ₁₂) by standard techniques. The Hermite integrals overr ₁₂ itself are generated by a simple procedure with negligible computational effort. The key results are discussed in the context of general recursion formulas. On leave from: Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, W-4630 Bochum, Germany     

Two-center overlap integrals,three dimensional adaptive integration,and prolate ellipsoidal coordinates

Numerical, adaptive algorithm evaluating the overlap integrals between the Numerical Type Orbitals (NTO) is presented. The described algorithm exploits the properties of the prolate ellipsoidal coordinates, which are the natural choice for two-center overlap integrals. The algorithm is designed for numerical atomic orbitals with the finite support. Since the cusp singularity of the atomic orbitals vanish in the prolate ellipsoidal coordinate system, the adaptive integration algorithm in generates small number of subdivisions. The efficiency and reliability of the algorithm is demonstrated for the overlap integrals evaluated for the selected pairs of Slater Type Orbitals (STO).     

Calculation of the one-electron two-center integrals with STOS using recurrence-based algorithms

We present a new algorithm for the calculation of two-center one-electron integrals with STOS based on two sets of recurrence relations. The first one enables us to calculate any of these integrals in terms of a few “basic integrals.” Furthermore, these basic integrals are written in terms of certain auxiliary functions which can be obtained through the second set of relations from only two starting values. We give also simple formulas for these starting quantities. Finally, the numerical stability, accuracy, and speed of the different steps of the algorithm are analyzed.     

Two-electron integral evaluation for uncontracted geometrical-type Gaussian functions

A new algorithm for efficient evaluation of two-electron repulsion integrals (ERIs) using uncontracted geometrical-type Gaussian basis functions is presented. Integrals are evaluated by the Habitz and Clementi method. The use of uncontracted geometrical basis sets allows grouping of basis functions into shells (s, sp, spd, or spdf) and processing of integrals in blocks (shell quartets). By utilizing information common to a block of integrals, this method achieves high efficiency. This technique has been incorporated into the KGNMOL molecular interaction program. Representative timings for a number of molecules with different basis sets are presented. The new code is found to be significantly faster than the previous program. For ERIs involving only s and p functions, the new algorithm is a factor of two faster than previously. The new program is also found to be competitive when compared with other standard molecular packages, such as HONDO-8 and Gaussian 86.     

Uniform convergence and computation of the generalized exponential integrals

In this paper, uniform convergence of the generalized exponential (GE) integrals is investigated. We also study the continuity, differentiability, summability and asymptotic behaviour of GE integrals. Then we give an accurate and efficient computation algorithm for these integrals.     

Three‐center Coulomb repulsion integrals with Slater functions

The translation method for products of two Slater functions is improved and combined with the short–long range separation in order to develop a robust and efficient algorithm for the evaluation of three‐center Coulomb integrals with Slater functions. Several tests are carried out showing that the algorithm reported here yields integrals with an absolute error below 10^?12 hartree and a computational cost of a few microseconds per integral. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Nonlinear transformation methods for accelerating the convergence of Coulomb integrals over exponential type functions

It is well known that in any ab initio molecular orbital (MO) calculation, the major task involves the computation of molecular integrals, among which the computation of Coulomb integrals are the most frequently encountered. As the molecular system gets larger, computation of these integrals becomes one of the most laborious and time consuming steps in molecular systems calculation. Improvement of the computational methods of molecular integrals would be indispensable to a further development in computational studies of large molecular systems. The atomic orbital basis functions chosen in the present work are Slater type functions. These functions can be expressed as finite linear combinations of B functions which are suitable to apply the Fourier transform method. The difficulties of the numerical evaluation of the analytic expressions of the integrals of interest arise mainly from the presence of highly oscillatory semi-infinite integrals. In this work, we present a generalized algorithm based on the nonlinear transformation of Sidi, for a precise and fast numerical evaluation of molecular integrals over Slater type functions and over B functions. Numerical results obtained for the three-center two-electron Coulomb and hybrid integrals over B functions and over Slater type functions. Comparisons with numerical results obtained using alternatives approaches and an existing code are listed.     

New program for molecular calculations with Slater‐type orbitals

A new program for computing all the integrals appearing in molecular calculations with Slater‐type orbitals (STO) is reported. This program follows the same philosophy as the reference pogram previously reported but introduces two main changes: Local symmetry is profited to compute all the two‐electron integrals from a minimal set of seed integrals, and a new algorithm recently developed is used for computing the seed integrals. The new code reduces between one and two orders of magnitude the computational cost in most polyatomic systems. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 148–153, 2001     

Evaluation of Hylleraas-CI atomic integrals. III. Two-electron kinetic energy integrals

This paper is the part III of a series about the evaluation of Hylleraas-Configuration Interaction (Hy-CI) integrals by the method of direct integration over the interelectronic coordinates. The two-electron kinetic-energy integrals have been derived using the Hamiltonian in Hylleraas coordinates. We have improved the algorithm used in part II of this series and obtained general expressions. The method used for the two-electron integrals can be used in the same fashion for the evaluation of the three-electron ones. The formulas shown here have been tested in actual Hy-CI calculations of two-electron systems. The two-electron kinetic energy integrals values agree with the ones obtained using the Kolos and Roothaan transformation. The effectiveness of the different methods is discussed.     

Strategies for an efficient implementation of the Gauss-Bessel quadrature for the evaluation of multicenter integral over STFs

In a previous work, a new Gauss quadrature was introduced with a view to evaluate multicenter integrals over Slater-type functions efficiently. The complexity analysis of the new approach, carried out using the three-center nuclear integral as a case study, has shown that for low-order polynomials its efficiency is comparable to the SD. The latter was developed in connection with multi-center integrals evaluated by means of the Fourier transform of B functions. In this work we investigate the numerical properties of the Gauss-Bessel quadrature and devise strategies for an efficient implementation of the numerical algorithms for the evaluation of multi-center integrals in the framework of the Gaussian transform/Gauss-Bessel approach. The success of these strategies are essential to elaborate a fast and reliable algorithm for the evaluation of multi-center integrals over STFs.     

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.     

A full‐pivoting algorithm for the Cholesky decomposition of two‐electron repulsion and spin‐orbit coupling integrals

A significant reduction in the computational effort for the evaluation of the electronic repulsion integrals (ERI) in ab initio quantum chemistry calculations is obtained by using Cholesky decomposition (CD), a numerical procedure that can remove the zero or small eigenvalues of the ERI positive (semi)definite matrix, while avoiding the calculation of the entire matrix. Conversely, due to its antisymmetric character, CD cannot be directly applied to the matrix representation of the spatial part of the two‐electron spin‐orbit coupling (2e‐SOC) integrals. Here, we present a computational strategy to achieve a Cholesky representation of the spatial part of the 2e‐SOC integrals, and propose a new efficient CD algorithm for both ERI and 2e‐SOC integrals. The proposed algorithm differs from previous CD implementations by the extensive use of a full‐pivoting design, which allows a univocal definition of the Cholesky basis, once the CD δ threshold is made explicit. We show that is the upper limit for the errors affecting the reconstructed 2e‐SOC integrals. The proposed strategy was implemented in the ab initio program Computational Emulator of Rare Earth Systems (CERES), and tested for computational performance on both the ERI and 2e‐SOC integrals evaluation. © 2017 Wiley Periodicals, Inc.     

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.     

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.