Evaluation of bielectronic integrals for 1s Slater orbitals by using averages

A new approach for evaluating the four‐center bielectronic integrals (12|34), involving 1s Slater‐type orbitals, is presented. The method uses the multiplication theorem for Bessel functions. The bielectronic integral is expressed in terms of a finite sum of functions, and a scaling parameter is introduced. In the present work, the scaling parameter used is an average. The results show that the first term in the sum is always the principal contribution, and the remainder has a corrective character. The whole scheme and its numerical trend are understood on the basis of a theorem recently proved. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Numerical integration techniques for quantum chemistry

Korobov theory for multidimensional numerical integration is used to evaluate electronic integrals. This paper shows the important role played by periodization techniques. Singularity (r ₁₂ ^?1 ) in the bielectronic six-dimensional integrals is removed through a twofold three-dimensional integration. Results are presented for atomic integrals involving Slater type atomic orbitals.     

On the role of symmetry in the ab initio hartree-fock linear-combination-of-atomic-orbitals treatment of periodic systems

The symmetry properties of the mono- and bielectronic terms contributing to the Fock matrix in the ab initio Hartree–Fock treatment of periodic systems are discussed. A computational scheme which takes full advantage of the point symmetry is presented; in this respect, it represents a generalization of the scheme proposed in Int. J. Quantum Chem. 17 , 501 (1980). Computational details and numerical examples are reported; it is shown that with respect to two of the bottlenecks of this kind of calculation, namely, computer time and backing storage required for the bielectronic integrals, it is possible to obtain saving factors as large as h and h², respectively, where h is the order of the point group. Preliminary tests are reported which indicate that the study of relatively complicated systems, like quartz or corundum (9 and 10 atoms in the unit cell, respectively) at an ab initio Hartree–Fock level is now within reach.     

Exact-exchange Hartree–Fock calculations for periodic systems. I. Illustration of the method

A scheme is presented for performing linear-combination-of-atomic-orbitals (LCAO ) self-consistent-field (SCF ) ab initio Hartree–Fock calculations of the electronic structure of periodic systems. The main aspects which characterize the present method are (i) a thorough discussion of both translational and local symmetry properties and the derivation of general formulas for the transformation of all the relevant monoelectronic and bielectronic terms under symmetry operators. (ii) The use of general yet powerful criteria for the truncation of infinite sums; in particular, the Coulomb electron–electron interactions are subdivided into terms corresponding to intersecting or nonintersecting charge distributions; the latter are grouped into shell contributions and the interaction is evaluated by multipolar expansions; the exchange interaction may be evaluated with great precision by retaining a relatively small number of two-electron integrals according to a truncation criterion which fully preserves its nonlocal character. (iii) The use of a procedure for performing integrals over k , as needed in the evaluation of the Fermi energy and in the reconstruction of the Fock matrix, which is particularly simple because it employs partially intersecting small spheres as integration subdomains where linear extrapolation is admitted. A comparison is finally made of our fundamental equations in the critical SCF stage with those obtainable by a recent proposal which uses Fourier transforms to express Coulomb and exchange integrals.     

Expansion of atomic orbital products in terms of a complete function set

Summary An alternative method for the evaluation of matrix elements required in an LCAO-SCF calculation is presented. It is based on the use of solutions of the Helmholtz equation within a spherical domain for expanding charge distributions with boundary conditions devised to make the electrostatic-potential integral particularly simple. This method allows the systematic evaluation of bielectronic integrals to be performed for any type of atomic orbitals.Part of a PhD Thesis (J.E.P.) to be presented to the UNRC     

A new algorithm to evaluate bielectronic integrals with 1s Slater‐type orbitals obtained by the integral transforms

This article presents a variation of the integral transform method to evaluate multicenter bielectronic integrals (12|34), with 1s Slater‐type orbitals. It is proved that it is possible to define, out of the expression of (12|34) given by the integral transform method, a function F(q) that has the property of having a unique Q, such that F(Q) = (12|34). Therefore, F(q) may be used to calculate (12|34). It is shown that the evaluation of F(Q) turns out to be simpler than the three‐dimensional integral involved in the calculation of (12|34), and an algorithm is presented to calculate Q. The results show that relative errors on the order of 10^?3 or lower are obtained very efficiently. In addition, it is shown that the proposed algorithm is very stable. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

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.     

Slater orbital molecular integrals with numerical fourier transform methods. I. (coplanar) multicenter exchange integrals over 1s orbitals

Slater orbital r₁₂^?1 integrals are calculated with a numerical Fourier-transform method based on a formulation first given by Bonham, Peacher and Cox. Spherical wave expansions are introduced that decouple the Feynman integrations for the charge distribution Fourier transforms. The Feynman integrals are evaluated semianalytically, and their properties are analyzed in detail. The final computational step involves a numerical integration over charge distribution quantities. Results for (coplanar) multicenter exchange integrals over 1s orbitals are given. As long as the charge distributions are overlapping considerably, the method gives good results, even when these distributions are highly asymmetric. The method as presently implemented fails when highly disconnected charge distributions are involved.     

Evaluation of Hylleraas-CI atomic integrals by integration over the coordinates of one electron. I. Three-electron integrals

A method to evaluate the nonrelativistic electron-repulsion, nuclear attraction and kinetic energy three-electron integrals over Slater orbitals appearing in Hylleraas-CI (Hy-CI) electron structure calculations on atoms is shown. It consists on the direct integration over the interelectronic coordinate r _ij and the sucessive integration over the coordinates of one of the electrons. All the integrals are expressed as linear combinations of basic two-electron integrals. These last are solved in terms of auxiliary two-electron integrals which are easy to compute and have high accuracy. The use of auxiliary three-electron ones is avoided, with great saving of storage memory. Therefore this method can be used for Hy-CI calculations on atoms with number of electrons N ≥ 5. It has been possible to calculate the kinetic energy also in terms of basic two-electron integrals by using the Hamiltonian in Hylleraas coordinates, for this purpose some mathematical aspects like derivatives of the spherical harmonics with respect to the polar angles and recursion relations are treated and some new relations are given.     

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.     

Non‐linear transformations for rapid and efficient evaluation of multicenter bielectronic integrals over B functions

This work presents an extremely efficient nonlinear transformation based on a certain Hankel type transform, originally due to A. Sidi. The approach is applied to evaluating Coulomb integrals in the molecular context. These integrals are bielectronic one, two, three and fourcenter terms arising from the interactions of electron distributions over a Slater type orbital basis. They occur in many millions of terms, even for small molecules, and require rapid and accurate evaluation. The present work shows how we can reduce the order of the linear differential equation required to be satisfied by the integrand considerably. Calculation times as short as 10^-2 ms were obtained for fourcenter terms (the least favorable case) on an IBM RS6000340 workstation. This method represents a considerable advance on previous work on Coulomb integrals.     

Numerical Evaluation of Three-Center Two-Electron Coulomb and Hybrid Integrals Over B Functions Using the HD and H\overline D Methods and Convergence Properties

The difficulties of the numerical evaluation of three-center two-electron Coulomb and hybrid integrals over B functions, arise mainly from the presence of the hypergeometric series and semi-infinite very oscillatory integrals in their analytical expressions, which are obtained using the Fourier transform method.This work presents a general approach for accelerating the convergence of these integrals by first demonstrating that the hypergeometric function, involved in the analytical expressions of the integrals of interest, can be expressed as a finite sum and by applying nonlinear transformations for accelerating the convergence of the semi-infinite oscillatory integrals after reducing the order of the differential equation satisfied by the integrand.The convergence properties of the new approach are analysed to show that from the numerical point of view the method corresponds to the most ideal situation.The numerical results section illustrates the accuracy and unprecedented efficiency of evaluation of these integrals.     

Réductions chimique et électrochimique de 5H-benzopyranno[4,3-d]pyrimidines

The chemical reduction of 5H-[1]benzopyranno[4,3-d]pyrimidines with lithium aluminum hydrrde leads to 3,4-dihydro derivations. The electro-chemical reducation in acidic medium shows two monoelectronic cathodic waves. In netural or basic medium, substituted compounds in 2 position show a single bielectronic wave while two bielectronic waves are observed for unsubstituted compounds In all cases, preparative electrolysis lead to a hydrodimer in the 4,4′-positiom.     

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.     

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.     

Modified all-valence INDO/spd method for ground and excited state properties of isolated molecules and molecular complexes

A new semiempirical all-valence method, GRINDOL (Ghost and Rydberg INDO ), based on the INDO approximation, is described. Linderberg–Seamans relation (extended to the d and Rydberg orbitals) for the resonance integrals and a new semitheoretical expression for the core-core repulsion term and energy correction including basis-set superposition error (intermolecular as well as intramolecular) has been applied. The proposed method enables calculation of ground and excited state properties. The ground state results (including intermolecular interactions) as well as the spectral properties are in reasonable agreement with relevant experimental (or ab initio) studies for isolated molecules, molecular complexes, and transition metal compounds. The method contains only one adjustable parameter, all two-center integrals and terms are only basis-set dependent. The one-center integrals are evaluated from the respective atomic terms.     

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.     

Unified treatment for the evaluation of arbitrary multielectron multicenter molecular integrals over Slater‐type orbitals with noninteger principal quantum numbers

The expansion formula has been presented for Slater‐type orbitals with noninteger principal quantum numbers (noninteger n‐STOs), which involves conventional STOs (integer n‐STOs) with the same center. By the use of this expansion formula, arbitrary multielectron multicenter molecular integrals over noninteger n‐STOs are expressed in terms of counterpart integrals over integer n‐STOs with a combined infinite series formula. The convergence of the method is tested for two‐center overlap, nuclear attraction, and two‐electron one‐center integrals, due to the scarcity of the literature, and fair uniform convergence and great numerical stability under wide changes in molecular parameters is achieved. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

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     

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