Untitled

Bouferguène

2006

Multi-center two-electron Coulomb integrals over Slater-type functions are required for any accurate molecular electronic structure calculations. These integrals, which are numerous, are to be evaluated rapidly and accurately. Slater-type functions are expressed in terms of the so-called B functions, which are best suited to apply the Fourier transform method. The Fourier transform method allowed analytic expressions for these integrals to be developed. Unfortunately, the analytic expressions obtained turned out to be extremely difficult to evaluate accurately due to the presence of highly oscillatory spherical Bessel integrals. In this work, we used techniques based on nonlinear transformation and extrapolation methods for improving convergence of these oscillatory spherical Bessel integrals. These techniques, which led to highly efficient and rapid algorithms for the numerical evaluation of three-and four-center two-electron Coulomb and exchange integrals, are now shown to be applicable to the two-center two-electron Coulomb integrals. The numerical results obtained for the molecular integrals under consideration illustrate the efficiency of the algorithm described in the present work compared with algorithms using the epsilon ( ) algorithm of Wynn and Levin's u transform.

Section: Introductionmentioning

confidence: 90%

Extrapolation methods for improving convergence of spherical Bessel integrals for the two‐center Coulomb integrals

Bouferguène

2006

“…We demonstrated previously that ${\cal I}(s)$ satisfies all the conditions to apply D 21. We used a second‐order differential equation, satisfied by the integrand ${\cal F}_s(x)$ 21, 22, to obtain the approximation D

_{italicn}^{(2, italicj)}

for the semi‐infinite integrals ${\cal I}(s)$ 23, which is given by where $g(x) = x^{n_x} {{\hat k}_\nu[R_2 \gamma(s,x)]\over [\gamma(s,x)]^{n_\gamma}}$ and x i for i = 0, 1, 2, …, are the successive positive zeros of j λ ( v x ).…”

Section: W Algorithm and Three‐center Nuclear Attraction Integralsmentioning

confidence: 99%

The W algorithm and the D transformation for the numerical evaluation of three‐center nuclear attraction integrals

Duret

2006

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.

“…μ, m 2 are integers; ν, ν 1 , and ν 2 are half‐integers. The exact definitions of these quantities could be found in 17, 18, 31, 38. v , R 2 , R 21 , and R 34 stand for the modulus of , 2 , 21 , and 34 , respectively.…”

Section: Analytical and Numerical Evaluation Of Molecular Multicentermentioning

confidence: 99%

“…In 38, 39, a second‐order differential equation was used, obtained by Sidi 33, 37, for a function of the form f ( x ) = g ( x ) j λ ( x ), to obtain the approximation D

_{(2)}^{italicn}

for the semi‐infinite integrals, which occur in the analytic expressions of molecular integrals. This approximation D

_{(2)}^{italicn}

is obtained by solving a set of linear equations, which is given by: D

_{(2)}^{italicn}

and β 1, i , i = 0, 1, … , n − 1 are the ( n + 1) unknowns of the above linear system.…”

Section: Analytical and Numerical Evaluation Of Molecular Multicentermentioning

confidence: 99%

Highly accurate numerical results for three‐center nuclear attraction and two‐electron Coulomb and exchange integrals over Slater‐type functions

2004

ABSTRACT:The present work focuses on the recent progress in nonlinear transformation methods for improving the convergence of highly oscillatory integrals and in their applications for an efficient and rapid numerical evaluation of molecular electronic integrals over Slater-type functions (STFs). The nonlinear D transformation, which is probably the most effective general approach for increasing the rate of convergence of semi-infinite oscillatory integrals, is presented. Molecular integrals over STFs are expressed as finite linear combinations of integrals over B functions. The basis set of B functions is suitable to apply the Fourier transform method, which led to analytic expressions, involving highly oscillatory functions, for molecular integrals. Efficient algorithms based on the D transformation are now developed for a numerical evaluation of molecular integrals over STFs. Numerical results that we obtained with linear and nonlinear molecules, and which are in agreement with those obtained using existing codes (Alchemy, STOP, and ADGGSTNGINT), show that the algorithms described in this work are relevant.