Nuclear Attraction Integrals Research Articles

Dipole moments and orbital energies from ARCANA: A semi-empirical molecular orbital calculation program

Approximations employed in calculating two-center nuclear attraction and electron repulsion integrals are discussed. A procedure for economical reduction of the number of integrals to be evaluated which satisfies the requirement of rotational invariance is discussed. This calculation technique is incorporated into a semiempirical molecular orbital calculation program and results in a realistic account of the anisotropy of atoms in molecules of low symmetry. Good agreement is found between electric dipole moments and molecular orbital energies determined experimentally and calculated employing ARCANA.

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.

The S and G transformations for computing three‐center nuclear attraction integrals

AbstractIt is now well established that nonlinear transformations can be extremely useful in the case of oscillatory integrals. In previous work, we could show that the G transformation, which is not so well known among those interested in the numerical evaluation of highly oscillatory integrals, works very well for the extremely challenging integral called Twisted Tail. In this work, we demonstrate that these techniques also apply to three‐center nuclear attraction integrals over exponential type functions. The accurate and rapid evaluation of these integrals is required in ab initio molecular structure calculations and density functional theory. Using a basis set of B functions and profiting from their relatively simple Fourier representation, these integrals are formulated as analytical expressions involving highly oscillatory spherical Bessel integral functions. In the present work, we implement two highly accurate algorithms for three‐center nuclear attraction integrals. The first algorithm is based on the G transformation and the second is based on a combination of the S and G transformations. The application of these transformations is largely due to the properties of special functions that allow the computation of higher order derivatives of the integrands with exceptional simplicity. The numerical results illustrate the accuracy of these algorithms applied to three‐center nuclear attraction integrals over exponential type functions with a miscellany of different parameters. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

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

S. Ozcan and E. Oztekin, (J. Math. Chem. doi:10.1007/s10910-008-9398-z) 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. Oztekin et al., in previous paper (E. Oztekin, 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).

Analytical evaluation for two-center nuclear attraction integrals over slater type orbitals by using Fourier transform method

In this study, we shall suggest analytical expressions for two-center nuclear attraction integrals over STO’s with a one-center charge distribution by using Fourier transform method. The derivation is based on partial-fraction decompositions and Taylor expansions of rational functions. Analytical expressions obtained by this method are expressed in terms of Gegenbauer, and binomial coefficients and linear combinations of STO’s. Finally, it is relatively easy to express the Fourier integral representations of two-center nuclear attraction integrals with a one-center charge distribution mentioned above as finite and infinite of series of STO’s and irregular solid harmonics which may be considered to be limiting cases of STO’s.

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.

Analytical evaluation of three-center nuclear-attraction integrals over s-Slater orbitals for asymmetrical conformations of the centers

Analytical formulas for three-center nuclear-attraction integrals over Slater orbitals are given for any location of the three atomic centers. In the mathematical derivations the Neumann expansion has been used and new general auxiliary integrals which depend on the elliptical coordinates of one of the centers are defined. The orbital exponents within the integrals may be different.

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

Reply to “Comment on ‘Calculation of Two-center Nuclear Attraction Integrals over Integer and Noninteger N-slater-type Orbitals in Nonlined-up Coordinate Systems” ’

The comments of Guseinov on our recent paper (T. Özdoğan, S. Gümüş and M. Kara, J. Math. Chem. 33 (2003) 181) are critically analyzed. It is proved that the expansion formula for the product of two normalized associated Legendre functions and the expressions for two-center nuclear attraction integrals over Slater type orbitals have been obtained independently, by the use of basic mathematical rules.

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.

Symbolic programming language in molecular multicenter integral problem

AbstractIt 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 three‐center nuclear attraction and Coulomb integrals is the most frequently encountered. As the molecular system becomes 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 further development in computational studies of large molecular systems. To develop fast and accurate algorithms for the numerical evaluation of these integrals over B functions, we used nonlinear transformations for improving convergence of highly oscillatory integrals. These methods form the basis of new methods for solving various problems that were unsolvable otherwise and have many applications as well. To apply these nonlinear transformations, the integrands should satisfy linear differential equations with coefficients having asymptotic power series in the sense of Poincaré, which in their turn should satisfy some limit conditions. These differential equations are very difficult to obtain explicitly. In the case of molecular integrals, we used a symbolic programming language (MAPLE) to demonstrate that all the conditions required to apply these nonlinear transformation methods are satisfied. Differential equations are obtained explicitly, allowing us to demonstrate that the limit conditions are also satisfied. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Angular momentum in molecular quantum mechanical integral evaluation

Solid-harmonic derivatives of quantum-mechanical integrals over Gaussian transforms of scalar, or radial, atomic basis functions create angular momentum about each center. Generalized Gaunt coefficients limit the amount of cross differentiation for multi-center integrals to ensure that cross differentiation does not affect the total angular momentum. The generalized Gaunt coefficients satisfy a number of other selection rules, which are exploited in a new computer code for computing forces in analytic density-functional theory based on robust and variational fitting of the Kohn–Sham potential. Two-center exponents are defined for four or more solid-harmonic differentiations of matrix elements. Those differentiations can either build up angular momentum about the centers or give forces on molecular potential-energy surfaces, thus generalized Gaunt coefficients of order greater than the number of centers are considered. These 4- j generalized Gaunt coefficients and two-center exponents are used to compute the first derivatives of all integrals involving all the Gaussian exponents on a triplet of centers at once. First all angular factors are contracted with the corresponding part of the linear-combination-of-atomic-orbitals density matrix. This intermediate quantity is then reused for the nuclear attraction integral and the integrals corresponding to each basis function in the analytic fit of the Kohn–Sham potential in the muffin-tin-like, but analytic, Slater–Roothaan method that allows molecules to dissociate into atoms having any desired energy, including the experimental electronic energy. The energy is stationary in all respects and all forces precisely agree with a previous code in tests on small molecules. During geometry optimization of an icosahedral C 720 fullerene computing these angular factors and transforming them via the 4- j generalized Gaunt coefficient takes more than sixty percent of the total computer time. These same angular factors could be used in identical fashion with Gaussian transforms of Slater-type and numerical radial atomic orbitals.

Calculation of two‐center nuclear attraction integrals over slater type orbitals in molecular coordinate system

AbstractA closed analytical relation is derived for the two‐center nuclear attraction integrals over Slater type orbitals (STOs) in terms of binomial coefficients. This formula can be used in highly accurate calculations of the nuclear attraction integrals. The relationships obtained are valid for arbitrary values of quantum numbers and screening constants of STOs and location of nuclei.

Comment on “Calculation of Two-Center Nuclear Attraction Integrals over Integer and Noninteger n-Slater-Type Orbitals in Nonlined-Up Coordinate Systems”

Recently published formulas for the calculation of two-center nuclear attraction integrals (T. Ozdogan, S. Gumus and M. Kara, J. Math. Chem. 33 (2003) 181) are critically analyzed. It is shown that the analytical relations presented in this work are not original and they can easily be derived by means of a simple algebra from the formulas for overlap integrals, their rotation coefficients and expansion of the product of two normalized associated Legendre functions in elliptical coordinates published in our papers (I.I. Guseinov, J. Phys. B 3 (1970) 1399; Phys. Rev. A 32 (1985) 1864; J. Mol. Struct.: Theochem 336 (1995) 17).

Fast and accurate evaluation of three‐center, two‐electron Coulomb, hybrid, and three‐center nuclear attraction integrals over Slater‐type orbitals using the SD transformation

AbstractIn this work, Slater‐type atomic orbitals (STOs) are expanded over B functions to exploit the compact Fourier transform of these functions. The analytic representations of three‐center nuclear attraction integrals and multicenter bielectronic integrals over B functions involve semiinfinite highly oscillatory integrals, which can be transformed into infinite series. These integrals must be evaluated precisely and quickly. This work describes an efficient method for the evaluation of these integrals based on the nonlinear SD transformation, recently developed by Safouhi. The SD method was applied to multicenter integrals over B functions, and it is shown that this method is highly efficient compared with alternatives previously used. The SD approach should lead to a definitive suite of ab initio Slater software. The convergence properties were analyzed and they showed that the approximations obtained using the SD approach converge to the exact values of the semiinfinite integrals without any constraint. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Closed formulae for (1s|1s), Slater two‐center integrals, using the three‐center nuclear attraction integral program in spherical coordinates

AbstractThe strategy of using expansions in spherical harmonics of displaced Slater‐type (exp(−αx)) orbitals to evaluate multicenter molecular integrals is applied to three‐centered nuclear attraction integrals (potential due to the product of separated Slater‐type orbitals). Closed formulae are produced for nine terms of an infinite series for interior region, (inside − ipot) r2 < a, and 14 terms for Exterior Region, (outside − epot) r2 > a. The value a = 2 is chosen for the interior region and r = 14 for the exterior region. Our focus is to obtain a closed formula for the Slater formula. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

An extremely efficient and rapid algorithm for numerical evaluation of three-centre nuclear attraction integrals over Slater-type functions

The present work concerns the development of an extremely accurate and rapid algorithm for the numerical evaluation of the three-centre nuclear attraction integrals over Slater-type functions. These integrals are numerous, they occur in many millions of terms, even for small molecules and they require rapid and accurate evaluation. The new algorithm is based on nonlinear transformation methods, on numerical quadrature and on properties of the sine and Bessel functions. The section with numerical results shows the high accuracy and the substantial gain in calculation time realized using the new algorithm. The complete expressions of the three-centre nuclear attraction integrals over B functions and over Slater-type functions are evaluated for different values of the quantum numbers to show the efficiency of the new approach. Numerical results obtained with linear and nonlinear systems and comparisons with numerical results from the literature are listed.

A new algorithm for accurate and fast numerical evaluation of hybrid and three-centre two-electron Coulomb integrals over Slater-type functions

Recently we developed a new algorithm for a fast and accurate numerical evaluation of three-centre nuclear attraction integrals over Slater-type functions, the results obtained were very satisfactory. Now, it is shown that this new algorithm can also be applied to hybrid and three-centre two-electron Coulomb integrals over Slater-type functions. These integrals, which are numerous, are very difficult to evaluate to a high accuracy, because of the presence of spherical Bessel functions and hypergeometric series in the integrands. We have proved that the integrands that occur in the analytic expressions of the integrals under consideration satisfy all the conditions to apply the approach. The hypergeometric functions which occur in the semi-infinite integrals can be expressed as a finite expansion and the semi-infinite integrals involving the spherical Bessel functions can be transformed into semi-infinite integrals involving the simple sine function.The numerical results obtained with linear and non-linear systems illustrate clearly a further improvement of accuracy and a substantial reduction in calculation times. Comparisons with existing codes, STOP developed by Bouferguene et al (1996 Int. J. Quantum Chem. 57 801) and ADGGSTNGINT developed by Rico et al (1997 Comp. Phys. Commun. 105 216), are listed.

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.

Calculation of Two-Center Nuclear Attraction Integrals over Integer and Noninteger n-Slater Type Orbitals in Nonlined-Up Coordinate Systems

Two-center nuclear attraction integrals over Slater type orbitals with integer and noninteger principal quantum numbers in nonlined up coordinate systems have been calculated by means of formulas in our previous work (T. Ozdogan and M. Orbay, Int. J. Quant. Chem. 87 (2002) 15). The computer results for integer case are in best agreement with the prior literature. On the other hand, the results for noninteger case are not compared with the literature due to the scarcity of the literature, but also compared with the limit of integer case and good agreements are obtained. The proposed algorithm for the calculation of two-center nuclear attraction integrals over Slater type orbitals with noninteger principal quantum numbers in nonlined-up coordinate systems permits to avoid the interpolation procedure used to overcome the difficulty introduced by the presence of noninteger principal quantum numbers. Finally, numerical aspects of the presented formulae are analyzed under wide range of quantum numbers, orbital exponents and internuclear distances.