Three-center Nuclear Attraction Research Articles

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.

Double exponential transformation for computing three-center nuclear attraction integrals

ABSTRACTThree-center nuclear attraction integrals, which arise in density functional and ab initio calculations, are one of the most time-consuming computations involved in molecular electronic structure calculations. Even for relatively small systems, millions of these laborious calculations need to be executed. Highly efficient and accurate methods for evaluating molecular integrals are therefore all the more vital in order to perform the calculations necessary for large systems. When using a basis set of B functions, an analytical expression for the three-center nuclear attraction integrals can be derived via the Fourier transform method. However, due to the presence of the highly oscillatory semi-infinite spherical Bessel integral, the analytical expression still remains problematic. By applying the S transformation, the spherical Bessel integral can be converted into a much more favorable sine integral. In the present work, we then apply two types of double exponential transformations to the resulting sine integral, which leads to a highly efficient and accurate quadrature formula. This method facilitates the approximation of the molecular integrals to a high predetermined accuracy, while still keeping the calculation times low. The fast convergence properties analyzed in the numerical section illustrate the advantages of the method.

Evaluation of Self-Friction Three-Center Nuclear Attraction Integrals with Integer and Noninteger Principal Quantum Numbers n over Slater Type Orbitals

We have proposed a new approach to evaluate self-friction (SF) three-center nuclear attraction integrals over integer and noninteger Slater type orbitals (STOs) by using Guseinov one-range addition theorem in standard convention. A complete orthonormal set of Guseinov ψα exponential type orbitals (ψα-ETOs, α=2,1,0,-1,-2,…) has been used to obtain the analytical expressions. The overlap integrals with noninteger quantum numbers occurring in SF three-center nuclear attraction integrals have been evaluated using Qnsq auxiliary functions. The accuracy of obtained formulas is satisfactory for arbitrary integer and noninteger principal quantum numbers.

Benchmark values for molecular three-center integrals arising in the Dirac equation.

Previous papers by the authors report that they obtained compact, arbitrarily accurate expressions for two-center, one- and two-electron relativistic molecular integrals expressed over Slater-type orbitals. In the present study, accuracy limits of expressions given are examined for three-center nuclear attraction integrals, which are one-electron, three-center integrals with no analytically closed-form expression. In this work new molecular auxiliary functions are used. They are obtained via Neumann expansion of the Coulomb interaction. The numerical global adaptive method is used to evaluate these integrals for arbitrary values of orbital parameters and quantum numbers. Several methods, such as Laplace expansion of Coulomb interaction, single-center expansion, and the Fourier transformation method, have previously been used to evaluate these integrals considering the values of principal quantum numbers in the set of positive integer numbers. This study of three-center integrals places no restrictions on quantum numbers in all ranges of orbital parameters.

On the fast evaluation of three-center nuclear attraction integrals using one-range addition theorems for Slater functions

Using one-range addition theorems, the three-center nuclear attraction integrals are expressed through the overlap integrals containing χ - and χα -Slater-type orbitals (χ -STOs and χα -STOs), where - ∞ < α ⩽ 2 and . For the fast calculation, the partial summation is utilized for some indices of series expansion relations which correspond to progressively increasing upper limits. The binomial coefficients are stored in the memory of the computer. The convergence and accuracy of series are tested by calculating concrete cases. The best values are obtained for α = 0 .

Accurate and Fast Evaluation of Three-Center Nuclear Attraction Integrals of Coulomb–Yukawa Like Correlated Interaction Potentials and Slater Type Orbitals

Abstract With the use of one-range addition theorems of Slater type orbitals (STOs) introduced by one of the authors, three-center nuclear attraction integrals containing Coulomb–Yukawa like correlated interaction potentials (C-CIPs and Y-CIPs) appearing in the Hartree–Fock–Roothaan (HFR) equations for molecules are evaluated. These integrals are expressed through the overlap integrals which depend on the frictional quantum number α, where −∞ &amp;lt; α ≤ 2. The convergence of the series is tested by calculating three-center nuclear attraction integrals of C-CIPs, Y-CIPs, and STOs for the arbitrary values of potential parameters and locations of orbitals. For rapid calculations of these integrals, we use the partial summations of some indices corresponding to progressively increasing upper limits appearing in the series expansion relations. Additionally, the binomial coefficients arising in the series are stored in the memory of the computer using their recurrence relation. The fast and accurate computation approach suggested in this work is demonstrated.

Analytical evaluation of Coulomb potential generated by multielectron molecule at arbitrary positions in space using one-range addition theorems of Slater type orbitals

By the use of unsymmetrical one-range addition theorems for Slater type orbitals (STO) and Coulomb potential introduced by the author, the analytical formulae in terms of two- and three-center nuclear attraction integrals, and linear combination coefficients of molecular orbitals are derived for the potential produced by the charges of molecule. These formulae can be useful for the study of interaction between atomic-molecular systems containing any number of closed and open shells when the STO are used in the combined Hartree-Fock-Roothaan (HFR) theory suggested by the author. It should be noted that the symmetry of the potential obtained is the same as the symmetry of the molecule. As an example of application, the calculations have been performed for the potential produced by the ground state of BH (3) molecule[Formula: see text].

Calculation of three-center nuclear attraction integral over Slater type orbitals in molecular coordinate system using Löwdin α-radial function and Guseinov’s two-center charge density expansion formulae

Using Lowdin α-radial function and the Guseinov’s charge density expansion formulae, the calculation of the three-center nuclear attraction integrals over Slater type orbitals in molecular coordinate system is performed. The proposed algorithm is especially useful for computation of multicenter-multielectron integrals that arise in the Hartree-Fock-Roothaan approximation, which plays a significant role for the study of electronic structure and electron-nuclei interaction properties of atoms, molecules and solids. The algorithm described in the present work is valid for the arbitrary values of quantum numbers, screening constants and internuclear distances. The calculation results are in good agreement with those obtained using the alternative evaluation procedure.

Bessel, sine and cosine functions and extrapolation methods for computing molecular multi-center integrals

In this work, we present an extremely efficient approach for a fast numerical evaluation of highly oscillatory spherical Bessel integrals occurring in the analytic expressions of the so-called molecular multi-center integrals over exponential type functions. The approach is based on the Slevinsky-Safouhi formulae for higher derivatives applied to spherical Bessel functions and on extrapolation methods combined with practical properties of sine and cosine functions. Recurrence relations are used for computing the approximations of the spherical Bessel integrals, allowing for a control of accuracy and the stability of the algorithm. The computer algebra system Maple was used in our development, mainly to prove the applicability of the extrapolation methods. Among molecular multi-center integrals, the three-center nuclear attraction and four-center two-electron Coulomb and exchange integrals are undoubtedly the most difficult ones to evaluate rapidly to a high pre-determined accuracy. These integrals are required for both density functional and ab initio calculations. Already for small molecules, many millions of them have to be computed. As the molecular system gets larger, the computation of these integrals becomes one of the most laborious and time consuming steps in molecular electronic structure calculation. Improvement of the computational methods of molecular integrals would be indispensable to a further development in computational studies of large molecular systems. Convergence properties are analyzed to show that the approach presented in this work is a valuable contribution to the existing literature on molecular integral calculations as well as on spherical Bessel integral calculations.

Calculation of three-center nuclear attraction integral over Slater type orbitals in molecular coordinate system using Löwdin α-radial function and Guseinov’s two-center charge density expansion formulae

Calculation of three-center nuclear attraction integral over Slater type orbitals in molecular coordinate system using Löwdin α-radial function and Guseinov’s two-center charge density expansion formulae

Tailored Gauss quadratures, a promising route for an efficient evaluation of multicenter integrals over B functions

In the framework of the Fourier integral transform, complicated multicenter integrals, e.g., three-center nuclear attraction and exchange integrals, over B functions involve a multiple integral (double or triple), the innermost of which is a Hankel transform of an exponentially decreasing term. Because of the oscillatory nature of the Hankel transform and the order in which it occurs in the definition of multicenter integrals, i.e., innermost, an efficient evaluation of such a quantity requires highly performant algorithms. In this context, extrapolation techniques emerged, during the past decade, as a possible solution to the problem of evaluating the oscillating semi-infinite integral. With a view to improving the efficiency of future algorithms, this contribution introduces a new technique for the evaluation of the oscillating integral by means of a tailored Gaussian quadrature. Using the case of three-center nuclear attraction integrals as a working example, it is shown that the new approach allows the semi-infinite integral to be evaluated accurately if not exactly. Further, when the roots and weights of the quadrature are available, a complexity analysis of our algorithm shows encouraging results compared to nonlinear extrapolation techniques.

TheW algorithm and theD 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

Analytical expressions of exchange and three-center nuclear attraction integrals over 1s and 2s Slater orbitals with different exponents

Compact analytical formulas are given for exchange integrals and three-center nuclear attraction integrals using nonorthogonal Slater orbitals with equal and different exponents which appear in a Hylleraas-CI basis. Since all exponents in the integrals may be different, they are appropriate to optimize exponents and therefore to shorten the wave function expansions.

Evaluation of Multicenter Nuclear Attraction Integrals by the Use of Translation Formula for Slater Type Orbitals

By the use of translation formula for Slater type orbitals (STOs), three-center nuclear attraction integrals are represented in terms of two-center overlap and nuclear attraction integrals. The computing results for the formula presented here has been tested under wide changes in molecular parameters and good convergence has been obtained with the prior literature.

Computation of molecular integrals over Slater-type orbitals. VII. Calculation of multielectron molecular integrals by single-center expansion method using different translation formulas

Hitherto known formulas for the translation of Slater-type orbitals (STOs) from one center to another encounter serious difficulties in practical applications. In contrast, those derived by the use of arbitrary complete orthonormal sets of exponential-type functions (ETFs) expansion formulas appear to be free of these difficulties. We have examined in this paper the applicability of translation formulas for STOs obtained from the two kinds of ETFs, namely, from the Lambda ( Λ) and Coulomb Sturmian (CS) functions (Int. J. Quant. Chem. 81 (2001) 126) to the quantum-mechanical multicenter problems from the computational point of view and found it very useful. Test calculations on three-center nuclear attraction and four-center electron-repulsion integrals by single-center expansion method are reported. It is shown that these integrals in the case of Coulomb Sturmian ETFs exhibit a faster convergence rate. Therefore, it is recommended to use the expansion formulas for translation of STOs, obtained from the Coulomb Sturmian ETFs, in the calculation of multielectron multicenter molecular integrals.

The HD and HD̄ Methods for Accelerating the Convergence of Three-Center Nuclear Attraction and Four-Center Two-Electron Coulomb Integrals over B Functions and Their Convergence Properties

Three-center nuclear attraction and four-center two-electron Coulomb integrals over Slater-type orbitals are required for ab initio and density functional theory (DFT) molecular structure calculations. They occur in many millions of terms, even for small molecules and require rapid and accurate evaluation. The B functions are used as a basis set of atomic orbitals. These functions are well adapted to the Fourier transform method that allowed analytical expressions for the integrals of interest to be developed. Rapid and accurate evaluation of these analytical expressions is now made possible by applying the HD and HD̄ methods for accelerating the convergence of the semi-infinite oscillatory integrals. The convergence properties of the new methods are analysed. The numerical results section shows the high predetermined accuracy and the substantial gain in the calculation times obtained using the new methods.

Molecular calculations withB functions

A program for molecular calculations with B functions is reported and its performance is analyzed. All the one- and two-center integrals, and the three-center nuclear attraction integrals are computed by direct procedures, using previously developed algorithms. The three- and four-center electron repulsion integrals are computed by means of Gaussian expansions of the B functions. A new procedure for obtaining these expansions is also reported. Some results on full molecular calculations are included to show the capabilities of the program and the quality of the B functions to represent the electronic functions in molecules.

Convergence analysis of the addition theorem of Slater orbitals and its application to three-center nuclear attraction integrals

The mathematical foundation of the methods using addition theorems to evaluate multicenter integrals over Slater-type orbitals is actually well understood. However, many numerical aspects of such approaches still require further investigations. In the framework of these methods, multicenter integrals are generally represented by infinite series which under certain circumstances are very slowly convergent. Accordingly, the determination of the convergence type of such series is of great importance since it allows one to choose adequately the convergence accelerator to be used in the summation procedure. In this work, the convergence of the two-range addition theorem proposed by Barnett and Coulson [Philos. Trans. R. Soc. London, Ser. A 243, 221 (1951)] is analyzed. The results obtained from this study are then applied to study the convergence of three-center nuclear integrals, and most importantly, to discuss the choice of the convergence accelerator to be used in the summation procedure.

Efficient evaluation of integrals for density functional theory: NonlinearD transformations to evaluate three-center nuclear attraction integrals overB functions

Density functional theory requires precise numerical values for three-center nuclear attraction integrals, best obtained over Slater-type orbitals (STOs). Efficient evaluation of three-center nuclear attraction integrals over STOs to predetermined accuracy is made possible by applying the nonlinear D and D transformations. These methods are implemented in Fortran subroutines. This work shows how the conditions of applicability for these transformations are readily proven to be satisfied using the Axiom symbolic computation system. Axiom also provides exact values of the integrals for comparison. Their evaluation by standard numerical quadrature methods (Gauss–Laguerre), proves inadequate: Certain parameters lead to inaccurate values for the integrals, whereas the transformed integrals are highly accurate and rapidly evaluated. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 181–188, 1998

Molecular integrals with Slater basis. IV. Ellipsoidal coordinate methods for three-center nuclear attraction integrals

We develop and check an updated version of the ellipsoidal coordinate methods for the calculation of three-center nuclear attraction integrals with Slater-type orbitals (STOs). A first set of recurrence relations allows us to express these integrals in terms of some basic potentials. The basic potentials are classified in short- and long-range types, and then calculated in specific ways from auxiliary functions obtained by using a new set of both stable and fast recurrence relations. Numerical tests show that the procedure enables to reach very high accuracy in the integrals with a low computational cost.