Nuclear Attraction Integrals Research Articles

General expressions of molecular multicenter integrals for Slater‐type orbitals

AbstractEmploying Fourier‐transform technique, this work derives the molecular multicenter integrals for the modified Slater‐type orbitals (STOs) with arbitrary principal and angular quantum numbers, in which the spherical harmonics parts are the homogeneous solutions of original STOs. The integrands include spherical Bessel functions and associated Legendre functions. One‐electron three‐center nuclear attraction integral and two‐electron four‐center Coulomb repulsion integral are presented as numerical integrations in two dimensions and three dimensions, respectively. A simple Python script is provided to run the formulas. The correctness and computational efficiency of the script have been verified.

Realistic nuclear charge distribution model function for analytic nuclear attraction integrals in Gaussian basis functions.

In electronic structure theory, the charge distribution of a nucleus is usually approximated by point charge, Gaussian function, or homogeneously charged sphere, because they have an analytical nuclear attraction integral (NAI) formula. However, these functions do not always provide good approximations for nuclei with large mass number. The two-parameter Fermi (2pF) distribution and more realistic distributions describe well even nuclei with large mass number but do not have analytical NAI formulas. We propose a new function model called augmented Gaussian 12 (AG12), which has sufficient number of parameters and analytical NAI formulas. With the proposed fitting scheme, the AG12 charge distribution model optimally reproduces 2pF and the more realistic charge distributions. Moreover, AG12 fitted to 2pF model reproduces the energy difference of hydrogen-like ions well between the Gaussian distribution and 2pF models. Calculations using AG12 also suggested necessity to use more realistic nuclear charge distributions than 2pF.

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.

Analytical Evaluation of Two-Center Integrals for Slater-Orbitals Type in Hartree-Fock Formalism on Noncommutative Space

In this paper we have examined the influence of noncommutativity geometry on the two-center integrals for Slater-orbitals type in Hartree-Fock Formalism. Within the perturbation theory, we have evaluated analytically the correction due to noncommutativity in nuclear attraction integrals and electronic interaction integrals over Slater-Orbitals type (STO). It is shown that there is no correction due to the noncommutativity in both nuclear integrals for ns-state and interaction integrals between electrons of this quantum state.

Big data analysis of ab Initio molecular integrals in the neglect of diatomic differential overlap approximation.

Most modern semiempirical quantum-chemical (SQC) methods are based on the neglect of diatomic differential overlap (NDDO) approximation to ab initio molecular integrals. Here, we check the validity of this approximation by computing all relevant integrals for 32 typical organic molecules using Gaussian-type orbitals and various basis sets (from valence-only minimal to all-electron triple-ζ basis sets) covering in total more than 15.6 million one-electron (1-e) and 10.3 billion two-electron (2-e) integrals. The integrals are calculated in the nonorthogonal atomic basis and then transformed by symmetric orthogonalization to the Löwdin basis. In the case of the 1-e integrals, we find strong orthogonalization effects that need to be included in SQC models, for example, by strategies such as those adopted in the available OMx methods. For the valence-only minimal basis, we confirm that the 2-e Coulomb integrals in the Löwdin basis are quantitatively close to their counterparts in the atomic basis and that the 2-e exchange integrals can be safely neglected in line with the NDDO approximation. For larger all-electron basis sets, there are strong multishell orthogonalization effects that lead to more irregular patterns in the transformed 2-e integrals and thus cast doubt on the validity of the NDDO approximation for extended basis sets. Focusing on the valence-only minimal basis, we find that some of the NDDO-neglected integrals are reduced but remain sizable after the transformation to the Löwdin basis; this is true for the two-center 2-e hybrid integrals, the three-center 1-e nuclear attraction integrals, and the corresponding three-center 2-e hybrid integrals. We consider a scheme with a valence-only minimal basis that includes such terms as a possible strategy to go beyond the NDDO integral approximation in attempts to improve SQC methods. © 2018 Wiley Periodicals, Inc.

Self-Friction Power Series and Their Use for Construction of One-Range Addition Theorems of Noninteger Slater Functions and Coulomb–Yukawa Like Potentials

Using complete orthogonal \( \mathcal{L}^{{\left( {p_{l}^{*} } \right)}} \)-self-friction polynomials (\( \mathcal{L}^{{\left( {p_{l}^{*} } \right)}} \)-SFPs) in standard convention introduced by the author, the formulas for transformation of noninteger type functions to the power series are presented, where \( p_{l}^{*} = 2l + 2 - \alpha^{*} \) and α * is the integer (\( \alpha^{*} = \alpha ,\; - \infty < \alpha \le 2 \)) or noninteger (\( \alpha^{*} \ne \alpha ,\; - \infty < \alpha^{*} < 3 \)) SF quantum number. With the help of these power series, the one-center one-range addition theorems for χ-Slater type orbitals and V-Coulomb–Yukawa like potentials with noninteger indices (χ-NISTOs and V-NICYPs) are established. As an application, the atomic nuclear attraction integrals of χ-NISTOs and V-NICYPs are investigated.

Self-friction power series and their use for evaluation of atomic nuclear attraction integrals with noninteger indices of Slater orbitals and Coulomb–Yukawa-like potentials

Using complete orthogonal [Formula: see text]-Self-Friction Polynomials ([Formula: see text]-SFPs) introduced by one of the authors, the analytical and power series formulas for SF atomic nuclear attraction integrals over [Formula: see text]-noninteger Slater type orbitals ([Formula: see text]-NISTOs) and [Formula: see text]-noninteger Coulomb–Yukawa-like potentials ([Formula: see text]-NICYPs) are presented, where [Formula: see text] are the integer ([Formula: see text] or noninteger ([Formula: see text] SF quantum numbers and [Formula: see text]. As an application, the computer calculations for dependence of the atomic nuclear attraction integrals over [Formula: see text]-NISTOs and [Formula: see text]-NICYPs functions are presented.

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.

Use of Modified Self-Frictional Laguerre Power Series for Study of Atomic Nuclear Attraction Integrals of Noninteger Slater-Type Orbitals and Coulomb–Yukawa-Like Potentials

Abstract Using complete orthonormal L(α*)-modified self-frictional Laguerre polynomials (L(α*)-MSFLPs) in standard convention introduced by the author, the expressions for transformation of χ-noninteger Slater type orbitals (χ-NISTOs) and V-noninteger Coulomb–Yukawa-like potentials (V-NICYPs) to the MSF Laguerre power series are suggested, where α* are the integer (α* = α, −∞ &amp;lt; α ≤ 2) or noninteger (α* ≠ α, −∞ &amp;lt; α* &amp;lt; 3) SF quantum numbers. By the use of these power series, the atomic nuclear attraction integrals of χ-NISTOs and V-NICYPs are evaluated. The convergence and accuracy of obtained MSF Laguerre power series are tested by calculating concrete cases.

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.

Calculation of two-center nuclear attraction integrals of Slater type orbitals with noninteger principal quantum numbers using Guseinov’s one-center expansion formulas and Löwdin-α radial function

In this work, by the use of Guseinov’s one-center expansion formulas and Löwdin-α radial function, the series expansion relations in molecular coordinate system are established for the two-center nuclear attraction integrals of noninteger n∗ Slater type orbitals in terms of basic two-center nuclear attraction integrals over integer n Slater functions. The Löwdin α-radial function convoluted with the Guseinov’s one-center expansion formulas is one of the most important ingredients for accurate and efficient implementation of electronic structure calculation methods regardless of Hartree–Fock–Roothaan (HFR) method. The proposed algorithm shows better performance in arbitrary quantum numbers, screening constants and location of orbitals leading to significantly reduced run times.

A general formulation for the efficient evaluation of n-electron integrals over products of Gaussian charge distributions with Gaussian geminal functions

In this work, we present a general formulation for the evaluation of many-electron integrals which arise when traditional one particle expansions are augmented with explicitly correlated Gaussian geminal functions. The integrand is expressed as a product of charge distributions, one for each electron, multiplied by one or more Gaussian geminal factors. Our formulation begins by focusing on the quadratic form that arises in the general n-electron integral. Using the Rys polynomial method for the evaluation of potential energy integrals, we derive a general formula for the evaluation of any n-electron integral. This general expression contains four parameters ω, θ, v, and h, which can be evaluated by an examination of the general quadratic form. Our analysis contains general expressions for any n-electron integral over s-type functions as well as the recursion needed to build up arbitrary angular momentum. The general recursion relation requires at most n + 1 terms for any n-electron integral. To illustrate the general method, we develop explicit expressions for the evaluation of two, three, and four particle electron repulsion integrals as well as two and three particle overlap and nuclear attraction integrals. We conclude our exposition with a discussion of a preliminary computational implementation as well as general computational requirements. Implementation on parallel computers is briefly discussed.

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 multicentre nuclear attraction integrals over Slater-type orbitals using unsymmetrical one-range addition theorems

Using the unsymmetrical one-range addition theorems introduced by one of the authors with the help of complete orthonormal sets of Ψα-exponential type orbitals (α = 1, 0, −1, −2, …), this paper presents the sets of series expansion relations for multicentre nuclear attraction integrals over Slater-type orbitals arising in Hartree–Fock–Roothaan equations for molecules. The final results are expressed through multicentre charge density expansion coefficients and basic integrals. The convergence of the series is tested by calculating concrete cases for arbitrary values of parameters of orbitals.

A detailed derivation of Gaussian orbital-based matrix elements in electron structure calculations

A detailed derivation of analytic solutions is presented for overlap, kinetic, nuclear attraction and electron repulsion integrals involving Cartesian Gaussian-type orbitals. It is demonstrated how s-type orbitals can be used to evaluate integrals with higher angular momentum via the properties of Hermite polynomials and differentiation with respect to non-integration variables.

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.

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