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.     

Molecular symmetry and ab initio calculations: IV. Symmetry-matrix and symmetry-supermatrix in calculations of two-electron repulsion integrals

The product of two Gaussians having different centers is itself a one-center Gaussian, thus multicenter integrals with a Cartesian Gaussian basis can be reduced to one-center integrals. Recurrence relations for overlap integrals and electron repulsion integrals (ERIs) are derived at these centers. The calculations of overlap integrals and ERIs are carried out step by step from the highest symmetry case (one center) to required cases (different centers) by using the translation of Cartesian Gaussians. Full exploitation of symmetry in calculation processes can result in optimal use of these recurrence relations. Compared with the recently published algorithms, based on the recurrence relations derived by Obara and Saika [J. Chem. Phys., 84 , 3963 (1986)], the floating point operations (FLOPs) for ERI calculations (having four different centers) can be reduced by a factor of ca. 2. A significant extra saving in calculations and storage can be obtained if atoms, linear, or planar molecules are discussed. © 1997 John Wiley & Sons, Inc.     

Gauge invariant gaussian orbitals and the ab initio calculation of diamagnetic susceptibility for molecules

It is shown that gauge terms can be introduced into the Gaussian functions used as the basis functions for an ab initio calculation of the energy of a molecule in the presence of a uniform magnetic field so that all the integrals become independent of the origin of the vector potential. The perturbation treatment of the diamagnetic susceptibility is considered in the molecular orbital approximation. The results show that the susceptibility can be calculated using only the unperturbed orbitals and their first-order corrections. All the integrals that arise can be expressed in terms of known functions.     

Simple yet powerful techniques for optimization of horizontal recursion steps in Gaussian‐type two‐electron integral evaluation algorithms

Simple heuristic rules are given that allow optimization of the performance of horizontal recursion steps present in schemes for calculation of two electron integrals are given. The number of floating point operations and computer timings are compared with the other algorithmic approaches to the problem. It is shown that the presented rules lead to substantial computational savings when compared with the standard implementation and may be also used succesfully instead of the algorithms based on full tree search techniques. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

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.     

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.     

Regularization method and molecular integrals with singular exponential functions

The regularization principle, which is based on the concept of linearly independent singular functions, makes it possible to calculate many important types of molecular matrix elements arising in the variational LCAO-MO-SCF scheme. This is done using a direct approach that employs reduction of these elements to finite sums of convergent and divergent one-electron integrals. A universal algorithm is developed to calculate two-center one-electron molecular integrals involving both singular and ordinary Slater functions. The numerical stability of the algorithms and the accuracy of the integral calculation are analyzed, and numerical estimates are given. V. I. Vernadskii Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences. Translated fromZhurnal Struktunoi Khimii, Vol. 35, No. 2, pp. 3–11, March–April, 1994. Translated by L. Chernomorskaya     

Evaluation of one-electron integrals for arbitrary operators V(r) over Cartesian Gaussians: Application to inverse-square distance and Yukawa operators

A general procedure is presented for generating one-electron integrals over any arbitrary potential operator that is a function of radial distance only. The procedure outlines that for a nucleus centered at point C integrals over Cartesian Gaussians can be written as linear combinations of 1-D integrals. These Cartesian Gaussian functions are expressed in a compact form involving easily computed auxiliary functions. It is well known that integrals over the Coulomb operator can be expressed in terms of F_n(T) integrals, where By means of a substitution for F_n(T) by other simple functions, algorithms that form integrals over an arbitrary function can be generated. Formation of such integrals is accomplished with minor editing of existing code based on the McMurchie–Davidson formalism. Further, the method is applied using the inverse-square distance and Yukawa potential operators V(r) over Cartesian Gaussian functions. Thus, the proposed methodology covers a large class of one-electron integrals necessary for theoretical studies of molecular systems by ab initio calculations. Finally, by virtue of the procedure's recursive nature it provides us with an efficient scheme of computing the proposed class of one-electron integrals. © 1993 John Wiley & Sons, Inc.     

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     

Integrals of the paramagnetic contribution in the relativistic calculation of the shielding tensor

Of the nuclear magnetic resonance (MMR), the nuclear shielding tensor is of a great interest. The relativistic calculation of the nuclear shielding tensor involves extremely challenging integrals of first and second order. Among the first order integrals are paramagnetic contribution integrals, which are extremely difficult to evaluate analytically and numerically, especially when using exponential type functions (ETFs). The main difficulty in the analytical development arises from the presence of 1/r ⁵ in the operators. In the present contribution, we developed the Fourier transform of the operators of the paramagnetic contribution and we used the Fourier integral transformation to derive analytic expressions for the integrals under consideration over ETFs. The main difficulty in the numerical treatment of the obtained analytic expressions arises from the presence of highly oscillatory spherical Bessel integrals. Extrapolation methods and nonlinear transformations are used to develop highly accurate algorithms for the numerical evaluation of the integrals of the paramagnetic contribution in the relativistic calculation of the shielding tensor.     

Transferable integrals in a deformation-density approach to crystal orbital calculations. I

In the usual ab initio method of calculating molecular orbitals, the number of integrals to be evaluated increases as M⁴, where M is the number of basis functions. In this paper, an alternative method is discussed, where the computation time increases much less violently with the number of basis functions. Matrix elements of the deformation potential are evaluated by Fourier transform methods, while matrix elements of the neutral-atom potential are evaluated by means of transferable integrals. The transferable integrals (moments of the neutral-atom potentials) can be evaluated once and for all and incorporated as input data in computer programs. In an appendix to the paper, a general expansion theorem is discussed. This theorem allows an arbitrary spherically symmetric function to be expanded about another center.     

On the evaluation of the characteristic polynomial of a chemical graph

The evaluation of the characteristic polynomial of a chemical graph is considered. It is shown that the operation count of the Le Verrier–Faddeev–Frame method, which is presently considered to be the most efficient method for the calculation of the characteristic polynomial, is of the order n⁴. Here n is the order of the adjacency matrix A or equivalently, the number of vertices in the graph G. Two new algorithms are described which both have the operation count of the order n³. These algorithms are stable, fast, and efficient. A related problem of finding a characteristic polynomial from the known eigenvalues λ_i of the adjacency matrix is also considered. An algorithm is described which requires only n(n ? 1)/2 operations for the solution of this problem.     

Two-electron repulsion integrals over Gaussian s functions

We present an efficient scheme to evaluate the [ 0 ]^(m) integrals that arise in many ab initio quantum chemical two-electron integral algorithms. The total number of floating-point operations (FLOPS ) required by the scheme has been carefully minimized, both for cases where multipole expansions of the integrals are admissable and for cases where this is not so. The algorithm is based on the use of a modified Chebyshev interpolation formula to compute the function exp(?T) and the integral F_m(T) = ∫₀¹u^2mexp(?Tu²) du very cheaply.     

Fluctuation free multivariate integration based logarithmic HDMR in multivariate function representation

This paper focuses on the Logarithmic High Dimensional Model Representation (Logarithmic HDMR) method which is a divide–and–conquer algorithm developed for multivariate function representation in terms of less-variate functions to reduce both the mathematical and the computational complexities. The main purpose of this work is to bypass the evaluation of N–tuple integrations appearing in Logarithmic HDMR by using the features of a new theorem named as Fluctuationlessness Approximation Theorem. This theorem can be used to evaluate the complicated integral structures of any scientific problem whose values can not be easily obtained analytically and it brings an approximation to the values of these integrals with the help of the matrix representation of functions. The Fluctuation Free Multivariate Integration Based Logarithmic HDMR method gives us the ability of reducing the complexity of the scientific problems of chemistry, physics, mathematics and engineering. A number of numerical implementations are also given at the end of the paper to show the performance of this new method.     

The impact of the resolution of the identity approximate integral method on modern ab initio algorithm development

The computation of the two-electron four-center integrals over gaussian basis functions is a significant component of the overall work of many ab initio methods used today. Improvements in the computational efficiency of the base algorithms have provided significant impact. Somewhat overlooked are methods that provide approximations to these integrals and their implementation in application software. A partial review of approximate integral techniques focused on the resolution of the identity (RI) four-center, two-electron integral approximation is given. The past and current uses of the RI algorithms are presented along with possibilities for further exploitation of the technology. Received: 14 January 1997 / Accepted: 11 March 1997     

Slater-orbital molecular integrals by numerical Fourier-transform methods. II. Four-center integrals over is orbitals

The four-center nonplanar electron repulsion integrals over 1s Slater-type atomic orbitals are considered by a numerical Fourier-transform method. It is shown that the highly oscillating integrand appearing in the Fourier inversion formula could be successfully treated by using Tchebyscheff quadrature. The resulting formulas are thoroughly discussed with particular emphasis on their numerical features and convergence properties. It follows that the aforementioned integrals may be calculated with a good accuracy with a moderate amount of computing time.     

Overlap integrals of B functions

Two different methods for the evaluation of overlap integrals of B functions with different scaling parameters are analyzed critically. The first method consists of an infinite series expansion in terms of overlap integrals with equal scaling parameters [14]. The second method consists of an integral representation for the overlap integral which has to be evaluated numerically. Bhattacharya and Dhabal [13] recommend the use of Gauss-Legendre quadrature for this purpose. However, we show that Gauss-Jacobi quadrature gives better results, in particular for larger quantum number. We also show that the convergence of the infinite series can be improved if suitable convergence accelerators are applied. Since an internal error analysis can be done quite easily in the case of an infinite series even if it is accelerated, whereas it is very costly in the case of Gauss quadratures, the infinite series is probably more efficient than the integral representation. Overlap integrals of all commonly occurring exponentially declining basis functions such as Slater-type functions, can be expressed by finite sums of overlap integrals of B functions, because these basis functions can be represented by linear combinations of B functions.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

Compact formulae for three-center nuclear attraction integrals over exponential type functions

In any ab initio molecular orbital calculations, the major task involves the computation of the so-called molecular multi-center integrals. Multi-center integral calculations is a very challenging mathematical problem in nature. Quantum mechanics only determines which integrals we evaluate, but the techniques employed for their evaluations are entirely mathematical. The three-center nuclear attraction integrals occur in a very large number even for small molecules and are among of the most difficult molecular integrals to compute efficiently. In the present contribution, we report analytical expressions for the three-center nuclear attraction integrals over exponential type functions. We describe how to compute the formula to obtain an efficient evaluation in double precision arithmetic. This requires the rational minimax approximants that minimize the maximum error on the interval of evaluation.

     

Techniques for the compression of sequences of integer numbers and real numbers with fixed absolute precision

Algorithms to reduce the space needed to store information either in memory or magnetic media are presented. These algorithms were designed to pack and unpack two common kinds of data types: sequences of sets of integers that change in a regular fashion and real numbers of fixed absolute precision. One typical application of these techniques is in the storage of electron repulsion integrals in ab initio calculations, where the indices of the basis functions are a good example of data of the first type and the integrals of the second type. In this case, savings in storage space of 50% or more can be obtained with reasonable accuracies in the energies. FORTRAN subroutines are presented for packing/unpacking indices and integrals both in the IBM and IEEE 754 64-bit floating point formats. © 1993 John Wiley & Sons, Inc.     

On the use of spatial symmetry in ab initio calculations. Transformation of the two-electron integrals from atomic orbital to localized molecular orbital basis

Anm ⁵-dependent integral transformation procedure from atomic orbital basis to localized molecular orbitals is described for spatially extended systems with some Abelian symmetry groups. It is shown that exploiting spatial symmetry, the number of non-redundant integrals for normal saturated hydrocarbons can be reduced by a factor of 2.5-3.5, depending on the size of the system and on the basis. Starting from a list of integrals over basis functions in canonical order, the number of multiplications of the four-index transformation is reduced by a factor of 2.8-3.5 as compared to that of Diercksen's algorithm. It is pointed out that even larger reduction can be achieved if negligible integrals over localized molecular orbitals are omitted from the transformation in advance.