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

Safouhi, Hassan

doi:10.1006/jcph.2000.6621

Cited by 25 publications

(20 citation statements)

References 53 publications

(45 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is shown that that the integrand F J (x) ofJ (s, t) satisfies all the conditions for applying the HD method [22,29], and consequently a good approximation of the semi-infinite integralJ (s, t) can be obtained by solving the linear system (28).…”

Section: Thed and Hd Methods For Accelerating Convergence Of Semi-infmentioning

confidence: 99%

“…In [23,29], we showed by using some helpful properties of spherical Bessel, reduced Bessel, and Poincaré series [46] that we can obtain a second-order, linear differential equation of the form required to apply theD-transformation for a function f (x) of the form f (x) = g(x) j λ (x), where j λ (x) stands for the spherical Bessel function and g(x) = h(x)e φ(x) , and where h(x) ∈Ã (γ ) for some γ and φ(x) ∈Ã (k) for k > 0 and α 0 (φ) < 0.…”

Section: Thed and Hd Methods For Accelerating Convergence Of Semi-infmentioning

confidence: 99%

“…Previous work [22,29] focused on the use of some properties of the reduced Bessel and spherical Bessel functions to reduce the order of the differential equation required to apply theD-transformation to 2, keeping all the other conditions fulfilled. This led to the HD method, which greatly simplified the application of theD-transformation.…”

Section: Introductionmentioning

confidence: 99%

“…Following Levin in [25], we can use Cramer's rule, since the zeros of sin(x) are equidistant, to obtain the simple solution which is given by (29) for the unknown SD (2) n . Now let us consider the integrand…”

mentioning

confidence: 99%

See 3 more Smart Citations

Efficient and Rapid Numerical Evaluation of the Two-Electron, Four-Center Coulomb Integrals Using Nonlinear Transformations and Useful Properties of Sine and Bessel Functions

Safouhi

2002

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

Section: Thed and Hd Methods For Accelerating Convergence Of Semi-infmentioning

confidence: 99%

Section: Thed and Hd Methods For Accelerating Convergence Of Semi-infmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Efficient and Rapid Numerical Evaluation of the Two-Electron, Four-Center Coulomb Integrals Using Nonlinear Transformations and Useful Properties of Sine and Bessel Functions

Safouhi

2002

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

“…Many approaches were developed for a fast and accurate numerical evaluation of these Coulomb integrals [41,42,[44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59]. In [60], we used the nonlinear D transformation of Sidi [61][62][63] for a numerical evaluation of the analytic expressions obtained for the overlap-like quantum similarity integrals.…”

Section: Introductionmentioning

confidence: 99%

The Fourier transform method and the SD¯ approach for the analytical and numerical treatment of multicenter overlap-like quantum similarity integrals

Safouhi¹,

Berlu²

2006

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

Mutual conversion of three flavors of Gaussian type orbitals

Mathar

2002

Int J of Quantum Chemistry

View full text Add to dashboard Cite

The set of Gaussian Type Orbitals g(n1, n2, n3) of order (n + 1)(n + 2)/2 and of common n ≡ n1 + n2 + n3 ≤ 7, common center and exponential, is customized to define a set of 2n + 1 linear combinations tn,m (−n ≤ m ≤ n) such that each tn,m depends on the azimuthal and polar angle of the spherical coordinate system like the real or imaginary part of the associated Spherical Harmonic. (Results cover both Hermite and Cartesian Gaussian Type Orbitals.) Overlap, kinetic energy and Coulomb energy matrix elements are presented for generalized basis functions of the type r s tn,m (s = 0, 2, 4 . . .). In addition, normalization integrals |g(n1, n2, n3)|d 3 r are calculated up to n = 7 and normalization integrals |r s tn,m|d 3 r up to n = 5. Gaussian Type Orbitals (GTO's) are widely used construction elements of basis sets in quantum chemistry. They became highly successful owing to fairly simple analytical representations of key integrals 1,2,3,4,5 that pay off in terms of speed when the matrix elements are calculated. This work deals with the conversion of Cartesian or Hermite GTO's which have a product representation in Cartesian coordinates into harmonic oscillator functions which have a product representation in spherical coordinates. The latter are orthogonal with respect to two quantum indices -the total number of integrals to be calculated is reduced by the overlap and kinetic energy integrals over products of orbitals with the same center.Sec. II introduces part of the notation. Sec. III lists those linear combinations of Gaussians defined in Cartesian coordinates that obtain the orthogonal property and are found to be the harmonic oscillator eigenfunctions. To extend coverage of the functional space, the oscillator functions are generalized in Sec. IV. Section V computes two-center overlap, kinetic energy and Coulomb integrals of those without recourse to the expansions in Cartesian coordinates. Sec. VI tabulates some of their products in support of 2-particle Coulomb Integrals. Absolute norms of orbitals play some role in Fitting Function techniques and are given in Sec. VII for the types of Gaussians discussed before.(1) be a primitive Hermite GTO (HGTO) with exponent α and quantum number n ≡ n 1 + n 2 + n 3 centered at the origin. H ni are Hermite Polynomials, and r 2 ≡ x 2 + y 2 + z 2 . It is normalized as 56(n + 1)(n + 2)/2 different g(n 1 , n 2 , n 3 ) exist for a given n. If n ≥ 2, they build an overcomplete set of states compared to only 2n + 1 eigenstates Y m n of the angular momentum operator.shall denote Spherical Harmonics in spherical coordinates, 78 andgeneralized Legendre Polynomials. Their real-valued counterparts areand(0 < m ≤ l), all normalized to unity, with parity (−) l , and orthogonal, The main result of this work is support to use of GTO's in systems with heavy or highly polarized atoms by reduction of the overcomplete sets to sets of 2n + 1 linearly independent combinations of GTO's.

show abstract

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

Cited by 25 publications

References 53 publications

Efficient and Rapid Numerical Evaluation of the Two-Electron, Four-Center Coulomb Integrals Using Nonlinear Transformations and Useful Properties of Sine and Bessel Functions

Efficient and Rapid Numerical Evaluation of the Two-Electron, Four-Center Coulomb Integrals Using Nonlinear Transformations and Useful Properties of Sine and Bessel Functions

The Fourier transform method and the SD¯ approach for the analytical and numerical treatment of multicenter overlap-like quantum similarity integrals

Mutual conversion of three flavors of Gaussian type orbitals

Contact Info

Product

Resources

About