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     

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).     

An accurate numerical multicenter integration for molecular orbital theory

A powerful and accurate numerical three‐dimensional integration scheme was developed especially for molecular orbital calculations. A multicenter integral is decomposed into the sum of single‐center integrals using nuclear weight functions and calculated using Gaussian quadrature rules. The decomposed single‐center integrands show strong anisotropy. With a careful selection of the Gaussian quadrature rule according to the anisotropy, it is possible to obtain an accuracy of 13 digits with a small number of integration points for the overlap integrals, normalization integrals, and molecular integrals for the hydrogen molecule. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 509–523, 1999     

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.     

Comment on “Analytical evaluation for two-center nuclear attraction integrals over Slater type orbitals by using Fourier transform method” by S. ?zcan and E. ?ztekin (J. Math. Chem. DOI 10.1007/s10910-008-9398-z)

S. ?zcan and E. ?ztekin, (J. Math. Chem. doi:) published formulas for evaluating the two-center nuclear attraction integrals over Slater type orbitals. It is shown that the analytical relations for these integrals through the expansion coefficients of the electron charge density for the one-center case and the overlap integrals presented in Sect. 3 of this work can easily be derived by means of a simple algebra from the formulas published in our papers (I.I. Guseinov, J Mol Struct (Theochem) 417:117, 1997; J Math Chem 42:415, 2007 and B.A. Mamedov, Chin J Chem 22:545, 2004). It should be noted that the formulas of overlap integrals presented by E. ?ztekin et al., in previous paper (E. ?ztekin, M. Yavuz, Ş. Atalay, J Mol Struct (Theochem) 544:69, 2001) for the calculation of two-center nuclear attraction integrals also are obtained from our papers (see Comment: I.I. Guseinov, J Mol Struct (Theochem) 638:235, 2003).     

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.     

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.     

Ab initio program for treatment of related molecules. I. Integral part

Anab initio integral program is described. It utilizes the local symmetries to avoid the redundant computation of integrals over spatially equivalent subsets of the basis. The integrals are grouped in a particular way to facilitate their transfer. The program is very suitable for the treatment of related systems with model geometries. The computing times of different programs are compared and the efficiency of the presented one is demonstrated.     

Evaluation of Two-center One- and Two-electron Integrals over Slater Type Orbitals

A formulation previously presented by the authors for coulomb integrals was generalized to other two-center integrals, except exchange integral. Within this frame, molecular integrals were expressed in terms of some new functions closely related to the well-known incomplete gamma functions and these functions recursively evaluated. Special issues arising in the case of hybrid integrals were addressed, and the results were compared with the ones found in the literature.     

Libcint: An efficient general integral library for Gaussian basis functions

An efficient integral library Libcint was designed to automatically implement general integrals for Gaussian‐type scalar and spinor basis functions. The library is able to evaluate arbitrary integral expressions on top of p, r and σ operators with one‐electron overlap and nuclear attraction, two‐electron Coulomb and Gaunt operators for segmented contracted and/or generated contracted basis in Cartesian, spherical or spinor form. Using a symbolic algebra tool, new integrals are derived and translated to C code programmatically. The generated integrals can be used in various types of molecular properties. To demonstrate the capability of the integral library, we computed the analytical gradients and NMR shielding constants at both nonrelativistic and 4‐component relativistic Hartree–Fock level in this work. Due to the use of kinetically balanced basis and gauge including atomic orbitals, the relativistic analytical gradients and shielding constants requires the integral library to handle the fifth‐order electron repulsion integral derivatives. The generality of the integral library is achieved without losing efficiency. On the modern multi‐CPU platform, Libcint can easily reach the overall throughput being many times of the I/O bandwidth. On a 20‐core node, we are able to achieve an average output 8.3 GB/s for C₆₀ molecule with cc‐pVTZ basis. © 2015 Wiley Periodicals, Inc.     

Recurrence formulas for molecular integrals over Laguerre Gaussian-type functions

Recurrence formulas for overlap, nuclear attraction, and electron-repulsion integrals over Laguerre Gaussian-type functions are presented. They have been derived using compact recurrence relations for homogeneous solid spherical harmonic operators but are rather lengthy as compared to those over Cartesian Gaussian-type functions. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 273–279, 1998     

Overlap Integrals Between Irregular Solid Harmonics and STOs Via the Fourier Transform Methods

In this paper, a unified analytical and numerical treatment of overlap integrals between Slater type orbitals (STOs) and irregular Solid Harmonics (ISH) with different screening parameters is presented via the Fourier transform method. Fourier transform of STOs is probably the simplest to express of overlap integrals. Consequently, it is relatively easy to express the Fourier integral representations of the overlap integrals as finite sums and infinite series of STOs, ISHs, Gegenbauer, and Gaunt coefficients. The another mathematical tools except for Fourier transform have used partial-fraction decomposition and Taylor expansions of rational functions. Our approach leads to considerable simplification of the derivation of the previously known analytical representations for the overlap integrals between STOs and ISHs with different screening parameters. These overlap integrals have also been calculated for extremely large quantum numbers using Gegenbauer, Clebsch-Gordan and Binomial coefficients. The accuracy of the numerical results is quite high for the quantum numbers of Slater functions, irregular solid harmonic functions and for arbitrary values of internuclear distances and screening parameters of atomic orbitals.     

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.     

An integration method of enzyme analysis

The application of an integration method of kinetic analysis to first-order and second-order reactions is discussed with particular emphasis on enzyme analyses. Transducer signals or concentrations of products or substrates are integrated for a Fixed time and the net integral of the increased or decreased signal or concentration is related to the initial substrate or enzyme concentration. Equations are developed describing the dependence of the integrals on enzyme and substrate concentrations for first- and second-order reactions, and examples are presented illustrating different cases. The application of this method to complicated enzymatic systems is discussed.     

Use of Filter-Steinborn B and Guseinov $${Q_{ns}^q}$$ auxiliary functions in evaluation of two-center overlap integrals over Slater type orbitals

An efficient method for computing overlap integral over Slater type orbitals based on the B Filter-Steinborn and Guseinov \({Q_{ns}^q}\) auxiliary functions is presented. The final results are expressed through the binomial coefficients with the help of which the overlap integrals can be evaluated efficiently and accurately. The results of calculation are in good agreement with those obtained by other method for arbitrary principal quantum numbers and different screening constants.     

Fourier transform general formula for systematic potentials

For calculating molecular integrals of systematic potentials, a three‐dimensional (3D) Fourier transform general formula can be derived, by the use of the Abel summation method. The present general formula contains all 3D Fourier transform formulas which are well known as Bethe–Salpeter formulas (Bethe and Salpeter, Handbuch der Physik, Bd. XXXV, 1957) as special cases. It is shown that, in several of the Bethe–Salpeter formulas, the integral does not converge in the meaning of the Riemann integral but converges in the meaning of a hyper function as the Schwartz distribution. For showing an effectiveness of the present general formula, the convergence condition of molecular integrals is derived generally for all of the present potentials. It is found that molecular integrals can be converged in the meaning of the Riemann integral for the present potentials, except for those for extra super singular potentials. It is also found that the convergence condition of molecular integrals over the Slater‐type orbitals is exactly the same as that of the corresponding integrals over the Gaussian‐type orbitals for the present systematic potentials. For showing more effectiveness, the molecular integral over the gauge‐including atomic orbitals is derived for the magnetic dipole‐same‐dipole interaction. © 2012 Wiley Periodicals, Inc.     

Evaluation of two-center overlap integrals using slater type orbitals in terms of bessel type orbitals

Using the definition of STOs in terms of BTOs, we have presented analytical formula for two-center overlap integrals. The obtained formula contains generalized binomial coefficients and Mulliken integrals A_k and B_k. Taking into account the recent advances on the efficient calculation of Mulliken integrals (Harris, Int. J. Quantum Chem., 100 (2004) 142), we have obtained many more satisfactory results for two-center overlap integrals, for arbitrary quantum numbers, scaling parameters, and location of atomic orbitals.PACS No: 31.15.+qAMS Subject Classification: 81V55, 81–08     

对称性轨道的构造

提出1种以配体轨道的旋转性质为基础,并利用群生成元表示矩阵来构造对称性轨道的新方法。讨论了对称性系数之间的相互关系,同时给出群重迭积分的计算过程。     

基于积分渐进展开的气相色谱重叠峰快速解析法

论文提出用积分渐进展开解析气相色谱重叠峰,该方法有3个主要步骤：首先将谷峰或肩峰分成两个积分区域,得到一个子区域的积分方程和一个重叠峰面积的代数方程;然后用数值积分求出这两个方程计算中所需要的峰面积,再用积分渐进公式将积分方程展开成代数方程;最后,将这两个方程与峰高约束方程联立后,得到一个非线性代数方程组,用Gauss-Seidel迭代可以快速求解方程组,方程收敛的最大迭代次数不超过20次。仿真和实验结果表明,解析的峰高和峰面积误差均很小,峰面积最大误差低于6.44%,峰高的最大误差约为6.80%。由于该算法精度高,效率高,所以这个方法可以用于气相色谱重叠峰和一般色谱峰的实时在线解析。     

Molecular integrals of relativistic effects with gaussian-type orbitals

A mathematical analysis is presented of molecular integrals of relativistic interactions in molecules. The integrals are based on Gaussian-type orbitals and include those arising from variation of electron mass with velocity, one-electron Fermi contact interaction, electron spin-same-orbit interaction, electron spin-nuclear spin interaction, electron spin-spin contact interaction, electron spin-other-orbit interaction, electron spin-spin dipolar interaction and electron orbit-orbit interaction. The integrals are expressed in suitable forms for use in computer. It is also pointed out that the integrals are written essentially in terms of the overlap, nuclear attraction, electron repulsion, or field integrals.