Extended method of calculation for two shells of atomic electrons having the same orbital quantum number

An extended method of calculation for two shells of atomic electrons, based upon the use of non-orthogonal radial orbitals, is presented. The rules for going over from the radial integrals of the ordinary method of calculation to the linear combinations of radial integrals of the extended method of calculation are given. The numerical calculations are carried out for the ground states of lithium- and beryllium-like atomic systems.     

Half-numerical evaluation of pseudopotential integrals

A half-numeric algorithm for the evaluation of effective core potential integrals over Cartesian Gaussian functions is described. Local and semilocal integrals are separated into two-dimensional angular and one-dimensional radial integrals. The angular integrals are evaluated analytically using a general approach that has no limitation for the l-quantum number. The radial integrals are calculated by an adaptive one-dimensional numerical quadrature. For the semilocal radial part a pretabulation scheme is used. This pretabulation simplifies the handling of radial integrals, makes their calculation much faster, and allows their easy reuse for different integrals within a given shell combination. The implementation of this new algorithm is described and its performance is analyzed.     

Integrals of the paramagnetic contribution in the relativistic calculation of the shielding tensor

Of the nuclear magnetic resonance (MMR), the nuclear shielding tensor is of a great interest. The relativistic calculation of the nuclear shielding tensor involves extremely challenging integrals of first and second order. Among the first order integrals are paramagnetic contribution integrals, which are extremely difficult to evaluate analytically and numerically, especially when using exponential type functions (ETFs). The main difficulty in the analytical development arises from the presence of 1/r ⁵ in the operators. In the present contribution, we developed the Fourier transform of the operators of the paramagnetic contribution and we used the Fourier integral transformation to derive analytic expressions for the integrals under consideration over ETFs. The main difficulty in the numerical treatment of the obtained analytic expressions arises from the presence of highly oscillatory spherical Bessel integrals. Extrapolation methods and nonlinear transformations are used to develop highly accurate algorithms for the numerical evaluation of the integrals of the paramagnetic contribution in the relativistic calculation of the shielding tensor.     

On the evaluation of overlap integrals with the same screening parameters of Slater-type orbitals using binomial coefficients

A general formula has been established for the overlap integrals with the same screening parameters of Slater-type orbitals in terms of bionomial coefficients. The final results are especially useful for the calculation of these integrals for large quantum numbers, which occur in the multicenter integrals. © 1996 John Wiley & Sons, Inc.     

Calculation of the one-electron two-center integrals with STOS using recurrence-based algorithms

We present a new algorithm for the calculation of two-center one-electron integrals with STOS based on two sets of recurrence relations. The first one enables us to calculate any of these integrals in terms of a few “basic integrals.” Furthermore, these basic integrals are written in terms of certain auxiliary functions which can be obtained through the second set of relations from only two starting values. We give also simple formulas for these starting quantities. Finally, the numerical stability, accuracy, and speed of the different steps of the algorithm are analyzed.     

配位多面体的群重叠积分

本文讨论了配位多面体的群重叠积分计算问题, 给出了这种群重叠积分的一般形式以及它们满足的一些关系, 用第一类点群将群重叠积分的计算化简, 由此定义了有关的几何参数并研究了它们的性质, 以正六面体为例说明了这些参数的具体计算。     

Molecular integrals over the gauge-including atomic orbitals. II. The Breit-Pauli interaction

Each accompanying coordinate expansion (ACE) formula is derived for each of the orbit-orbit interaction, the spin-orbit coupling, the spin-spin coupling, and the contact interaction integrals over the gauge-including atomic orbitals (GIAOs) by the use of the solid harmonic gradient (SHG) operator. Each ACE formula is the general formula derived at the first time for each of the above molecular integrals over GIAOs. These molecular integrals are arising in the Breit-Pauli two-electron interaction for a relativistic calculation. We may conclude that we can derive a certain ACE formula for any kind of molecular integral over solid harmonic Gaussian-type orbitals by using the SHG operator. The present ACE formulas will be useful, for example, for a calculation of a molecule in a uniform magnetic field, for a relativistic calculation, and so on, with the GIAO as a basis function.     

Evaluation of Generalized Exponential Integrals using Multinomial Expansion Theorems

An efficient and reliable method is presented for calculations the generalized exponential (GE) integrals. The basic series expressions of the generalized GE integrals are established. Evaluation of GE integrals for different values of the parameters, show the efficiencies of the new approach. The numerical results illustrate clearly a further reduction in calculation times. The relationships obtained are valid for the arbitrary values parameters and the computation results obtained are in good agreement with the literature. Numerical results obtained and comparisons with numerical results from the literature are listed.     

应用Monte Carlo平均值法计算STO双中心重迭积分

把Monte Carlo方法引进STO双中心重叠积分的计算中,结果表明,它不仅计算简便、快速、很容易在计算机上实现,而且具有较高的精确度,有望推广应用于更复杂的多中心分子积分中.     

Ab initio calculations on large molecules: The multiplicative integral approximation

In the multiplicative integral approximation (MIA), two-electron integrals are evaluated using an expansion of a product of two Gaussians in terms of auxiliary functions. An estimator of the error introduced by the approximation is incorporated in the self-consistent field (SCF) calculations and the integrals for which the error estimate is larger than a preset value are systematically corrected. In this way the results of a MIA-assisted calculation have the same accuracy as a conventional calculation. The full exploitation of the expansion technique while constructing the Fock-matrix allows important time savings. Results are presented for a number of test cases.     

Efficiency of the algorithms for the calculation of Slater molecular integrals in polyatomic molecules

The performances of the algorithms employed in a previously reported program for the calculation of integrals with Slater-type orbitals are examined. The integrals are classified in types and the efficiency (in terms of the ratio accuracy/cost) of the algorithm selected for each type is analyzed. These algorithms yield all the one- and two-center integrals (both one- and two-electron) with an accuracy of at least 12 decimal places and an average computational time of very few microseconds per integral. The algorithms for three- and four-center electron repulsion integrals, based on the discrete Gauss transform, have a computational cost that depends on the local symmetry of the molecule and the accuracy of the integrals, standard efficiency being in the range of eight decimal places in hundreds of microseconds.     

Nonlinear transformation methods for accelerating the convergence of Coulomb integrals over exponential type functions

It 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 Coulomb integrals are the most frequently encountered. As the molecular system gets 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 a further development in computational studies of large molecular systems. The atomic orbital basis functions chosen in the present work are Slater type functions. These functions can be expressed as finite linear combinations of B functions which are suitable to apply the Fourier transform method. The difficulties of the numerical evaluation of the analytic expressions of the integrals of interest arise mainly from the presence of highly oscillatory semi-infinite integrals. In this work, we present a generalized algorithm based on the nonlinear transformation of Sidi, for a precise and fast numerical evaluation of molecular integrals over Slater type functions and over B functions. Numerical results obtained for the three-center two-electron Coulomb and hybrid integrals over B functions and over Slater type functions. Comparisons with numerical results obtained using alternatives approaches and an existing code are listed.     

A new method to evaluate the STO overlap integrals for compilation of an EHMO program containing f orbitals

In this paper, a calculation method to evaluate to STO overlap integrals that have integer or non-integer effective quantum number is reported. Using this method, the overlap integrals of 4f orbitals are calculated and some results are given.     

Ab initio molecular orbital calculations on large lattice cluster models: Use of translational symmetry

Summary Translational symmetry has been shown to be useful in the calculation of electronic structures of large lattice models. The number of unique integrals has been derived for cases of different dimensionality. For the unique integrals zero screening and approximation methods are described. The method has been applied to arrays of hydrogen atoms and to a zincblende surface model. When the size of the system is increased the translationally unique integrals are shown to become either zero or they can be calculated by simple coulombic approximations.     

New translation method for STOs and its application to calculation of two-center two-electron integrals

The new translation method for Slater-type orbitals (STOs) previously tested in the case of the overlap integral is extended to the calculation of two-center two-electron molecular integrals. The method is based on the exact translation of the regular solid harmonic part of the orbital followed by the series expansion of the residual spherical part in powers of the radial variable. Fair uniform convergence and stability under wide changes in molecular parameters are obtained for all studied two-center hybrid, Coulomb, and exchange repulsion integrals. Ten-digit accuracy in the final numerical results is achieved through multiple precision arithmetic calculation of common angular coefficients and Gaussian numerical integration of some of the analytical formulas resulting for the radial integrals. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 91–100, 2000     

An unconventional scf method for calculations on large molecules

An unconventional SCF method for calculations on large molecules with more than 100 basis functions is described. Storage problems which arise in conventional SCF schemes when storing more than 10⁷ integrals are avoided by repeated calculation of integrals. The resulting increase in computational times is kept at a reasonable level by (a) improving the initial guess, (b) accelerating convergence, (c) employing a recursive construction of the Fock matrix, and (d) eliminating insignificant integrals from the calculation by a density-weighted cutoff criterion. Sample calculations show that, compared with conventional SCF calculations, computational times increase by 25%–75% depending on the basis set and the shape of the molecule.     

Non-integer Slater orbital calculations

In order to calculate the one- and two-electron, two-center integrals over non-integer n Slater type orbitals, use is made of elliptical coordinates for the monoelectronic, hybrid, and Coulomb integrals. For the exchange integrals, the atomic orbitals are translated to a common center. The final integration is performed by Gaussian quadrature.As an example, an SCF ab initio calculation is performed for the LiH molecule, both with integer and non-integer principal quantum number.     

Evaluation of screened nuclear attraction and electron repulsion molecular integrals over Gaussian basis functions

Analytical solutions to the Yukawa-like screened Coulomb nuclear attraction and electron repulsion molecular basic integrals, as well as to the basic integrals required to compute the virial coefficient, over Gaussian basis functions, are derived and cast into a practical closed form, suitable to interface with modern codes for the calculation of molecular electronic structure. © 1997 John Wiley & Sons, Inc.     

Calculation of atomic integration data

The calculation of average properties of atoms in molecules and interatomic surfaces is a difficult problem that requires the evaluation of two‐ and three‐dimensional integrals over regions with nontrivial borders. A mathematical formalism is presented that maps these regions onto the whole of ℝ² and/or ℝ³ and allows the construction of efficient and reliable numerical methods for the calculation of these integrals. These methods, which will be part of a forthcoming program package, are described and examples are given. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1040–1048, 2000     

Calculation of three-center nuclear attraction integral over Slater type orbitals in molecular coordinate system using Löwdin <Emphasis Type="Italic">α</Emphasis>-radial function and Guseinov’s two-center charge density expansion formulae

Using Löwdin 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.