Two-center Coulomb Integrals Research Articles

Slater Tipi Orbitaller Bazında Bir- ve İki-Merkezli Coulomb İntegrallerinin Özellikleri

In this study, two-electron one- and two-center Coulomb integrals with the same and different screening parameters are investigated numerically in the real Slater type orbital (STO) basis using Fourier transform method. In momentum space firstly, for atomic, i.e. one-center, Coulomb integrals are calculated, and analytical expressions are obtained in terms of binomial coefficients. Then, the solutions of the two-center Coulomb integrals are made with the modified Bessel function of second kind and the results are expressed in terms of binomial and Gaunt coefficients, irregular solid harmonics, and finite sum of STOs. A computer program is written in the MATHEMATICA language to determine the accuracy of the analytical expressions that are highly suitable for programming. The numerical results obtained from the program are given in the tables, and it is shown that the results agree with the literature.

Expansion of multicenter Coulomb integrals in terms of two-center integrals

Three- and four-center Coulomb integrals in the solid spherical harmonic Gaussian basis are solved by expansion in terms of two-center integrals. The two-electron Gaussian product rule, coupled with the addition theorem for solid spherical harmonics, reduces four-center Coulomb integrals into a linear combination of two-center Coulomb integrals and one-center overlap integrals. With this approach, three- and four-center Coulomb integrals can be reduced to the same form of two-center integrals. Resulting two-center Coulomb integrals can be further simplified into a simpler form, which can be related to the Boys function. Multi-center Coulomb integrals are solved hierarchically: simple two-center Coulomb integrals are used for calculation of more complicated two-center Coulomb integrals, which are used in the calculation of multicenter integrals.

Tight distance-dependent estimators for screening two-center and three-center short-range Coulomb integrals over Gaussian basis functions.

We derive distance-dependent estimators for two-center and three-center electron repulsion integrals over a short-range Coulomb potential, erfc(ωr12)/r12. These estimators are much tighter than the ones based on the Schwarz inequality and can be viewed as a complement to the distance-dependent estimators for four-center short-range Coulomb integrals and for two-center and three-center full Coulomb integrals previously reported. Because the short-range Coulomb potential is commonly used in solid-state calculations, including those with the Heyd-Scuseria-Ernzerhof functional and with our recently introduced range-separated periodic Gaussian density fitting, we test our estimators on a diverse set of periodic systems using a wide range of the range-separation parameter ω. These tests demonstrate the robust tightness of our estimators, which are then used with integral screening to calculate periodic three-center short-range Coulomb integrals with linear scaling in system size.

Fast analytical evaluation of intermolecular electrostatic interaction energies using the pseudoatom representation of the electron density. II. The Fourier transform method.

The Fourier transform method for analytical determination of the two-center Coulomb integrals needed for evaluation of the electrostatic interaction energies between pseudoatom-based charge distributions is presented, and its Fortran-based implementation using the 128-bit floating-point arithmetic in the XDPROP module of the XD software is described. In combination with mathematical libraries included in the Lahey/Fujitsu LF64 Linux compiler, the new implementation outperforms the previously reported Löwdin α-function technique [Nguyen et al. (2018). Acta Cryst. A74, 524-536] in terms of precision of the determined individual Coulomb integrals regardless of whether the latter uses the 64-, 80- or 128-bit precision floating-point format, all the while being only marginally slower. When the Löwdin α-function or Fourier transform method is combined with a multipole moment approximation for large interatomic separations (such a hybrid scheme is called the analytical exact potential and multipole moment method, aEP/MM) the resulting electrostatic interaction energies are evaluated with a precision of ≤5 × 10-5 kJ mol-1 for the current set of benchmark systems composed of H, C, N and O atoms and ranging in size from water-water to dodecapeptide-dodecapeptide dimers. Using a 2012 4.0 GHz AMD FX-8350 computer processor, the two recommended aEP/MM implementations, the 80-bit precision Löwdin α-function and 128-bit precision Fourier transform methods, evaluate the total electrostatic interaction energy between two 225-atom monomers of the benchmark dodecapeptide molecule in 6.0 and 7.9 s, respectively, versus 3.1 s for the previously reported 64-bit Löwdin α-function approach.

Benchmark values for molecular two-electron integrals arising from the Dirac equation.

The two-center two-electron Coulomb and hybrid integrals arising in relativistic and nonrelativistic ab initio calculations on molecules are evaluated. Compact, arbitrarily accurate expressions are obtained. They are expressed through molecular auxiliary functions and evaluated with the numerical Global-adaptive method for arbitrary values of parameters in the noninteger Slater-type orbitals. Highly accurate benchmark values are presented for these integrals. The convergence properties of new molecular auxiliary functions are investigated. The comparison for two-center two-electron integrals is made with results obtained from single center expansions by translation of the wave function to a single center with integer principal quantum numbers and results obtained from the Cuba numerical integration algorithm, respectively. The procedures discussed in this work are capable of yielding highly accurate two-center two-electron integrals for all ranges of orbital parameters.

Reexamination of the calculation of two-center, two-electron integrals over Slater-type orbitals. I. Coulomb and hybrid integrals.

In this paper, which constitutes the first part of the series, we consider calculation of two-center Coulomb and hybrid integrals over Slater-type orbitals. General formulas for these integrals are derived with no restrictions on the values of the quantum numbers and nonlinear parameters. Direct integration over the coordinates of one of the electrons leaves us with the set of overlaplike integrals which are evaluated by using two distinct methods. The first one is based on the transformation to the ellipsoidal coordinates system and the second utilizes a recursive scheme for consecutive increase of the angular momenta in the integrand. In both methods simple one-dimensional numerical integrations are used in order to avoid severe digital erosion connected with the straightforward use of the alternative analytical formulas. It is discussed that the numerical integration does not introduce a large computational overhead since the integrands are well-behaved functions, calculated recursively with decent speed. Special attention is paid to the numerical stability of the algorithms. Applicability of the resulting scheme over a large range of the nonlinear parameters is tested on examples of the most difficult integrals appearing in the actual calculations including, at most, 7i-type functions (l=6).

Efficient algorithm for two-center Coulomb and exchange integrals of electronic prolate spheroidal orbitals

We present a fast algorithm to calculate Coulomb/exchange integrals of prolate spheroidal electronic orbitals, which are the exact solutions of the single-electron, two-center Schrödinger equation for diatomic molecules. Our approach employs Neumann’s expansion of the Coulomb repulsion 1/∣x−y∣, solves the resulting integrals symbolically in closed form and subsequently performs a numeric Taylor expansion for efficiency. Thanks to the general form of the integrals, the obtained coefficients are independent of the particular wavefunctions and can thus be reused later.Key features of our algorithm include complete avoidance of numeric integration, drafting of the individual steps as fast matrix operations and high accuracy due to the exponential convergence of the expansions.Application to the diatomic molecules O2 and CO exemplifies the developed methods, which can be relevant for a quantitative understanding of chemical bonds in general.

Range-Separated Exchange Functionals with Slater-Type Functions.

An implementation of range-separated density functionals utilizing the Yukawa potential and Slater-type functions is described. The density-functional part of the range-separated regime is straightforward. The exact exchange part makes use of established methods for evaluating exchange integrals over Slater-type functions but still requires new one- and two-center integrals. Equations for the one-center integrals are derived. The two-center integrals are evaluated through a combination of new equations and techniques taken from procedures for evaluating two-center Coulomb integrals over Slater-type functions. In a first application, the performance of range-separated functionals in the prediction of transition metal thermochemistry is evaluated using a database of average ligand removal energies. The range-separated functionals perform better than a GGA parent and similarly to commonly used hybrid and meta-hybrid functionals. The results were relatively insensitive to the chosen value of the attenuation parameter.

Evaluation of two-center Coulomb and hybrid integrals over complete orthonormal sets of $$ {\Psi^{\rm{\alpha }}} - {\hbox{ETO}} $$ using auxiliary functions

By the use of ellipsoidal coordinates, the two-center Coulomb and hybrid integrals over complete orthonormal sets of Ψα-ETO exponential type orbitals arising in ab initio calculations of molecules are evaluated, where α = 1,0, -1, -2, ...,. These integrals are expressed through the auxiliary functions Q(ns)(q) and G(-ns)(q). The comparison is made with some values of integrals for Slater type orbitals the computation results of which are in good agreement with those obtained in the literature. The relationships obtained are valid for the arbitrary quantum numbers, screening constants and location of orbitals. Closed form expressions for two-center Coulomb and hybrid integrals for 1s and 2s orbitals with α = 1 are also presented. As an example of application, the Hartree-Fock-Roothaan calculations for the ground state of H(2) molecule are carried out with α = 1 and α = 0.

Evaluation of two-center electric field gradient integrals over STOs using elliptical coordinates

In this study, analytical expressions are obtained for two different types of two-center EFG integrals and the integrals are calculated using these expressions. Analytical expressions for the integrals are given in terms of auxiliary J and K functions defined by Ozmen et al. [Int. J. Quantum Chem. 91, 13 (2003)] for the evaluation of two-center Coulomb integrals using elliptical coordinates. Results are compared and found to be in good agreement for small quantum numbers with the existing results from the literature. Numerical stability of the proposed method is very good and therefore we also present results for higher quantum numbers, however there is no previous results found in the literature to compare.

Contracted auxiliary Gaussian basis integral and derivative evaluation.

The rapid evaluation of two-center Coulomb and overlap integrals between contracted auxiliary solid harmonic Gaussian functions is examined. Integral expressions are derived from the application of Hobson's theorem and Dunlap's product and differentiation rules of the spherical tensor gradient operator. It is shown that inclusion of the primitive normalization constants greatly simplifies the calculation of contracted functions corresponding to a Gaussian multipole expansion of a diffuse charge density. Derivative expressions are presented and it is shown that chain rules are avoided by expressing the derivatives as a linear combination of auxiliary integrals involving no more than five terms. Calculation of integrals and derivatives requires the contraction of a single vector corresponding to the monopolar result and its scalar derivatives. Implementation of the method is discussed and comparison is made with a Cartesian Gaussian-based method. The current method is superior for the evaluation of both integrals and derivatives using either primitive or contracted functions.

Expansion Formulae for Two-center Integer and Noninteger n STO Charge Densities and their Use in Evaluation of Multi-center Integrals

Using complete orthonormal sets of Ψα-exponential type orbitals (Ψα-ETOs, α =1, 0, −1, −2, ...) introduced by the author, the series expansion formulae are derived for the two-center integer and noninteger n STO (ISTO and NISTO) charge densities in terms of integer n STOs at a third center. The expansion coefficients occurring in these relations are presented through the two-center overlap integrals between STOs with integer and noninteger principal quantum numbers. The general formulae obtained for the STO charge densities are utilized for the evaluation of two-center Coulomb and hybrid integrals of NISTOs appearing in the Hartee–Fock–Roothaan approximation. The final results are expressed in terms of both the overlap integrals and the one-center basic integrals over integer n STOs. It should be noted that the result for the multi-center multielectron integrals with two-center noninteger n STO charge densities presented in this paper were not appeared in our past publications.

On The Computation Of Two-Center Coulomb Integrals Over Slater Type Orbitals Using The Poisson Equation

In this paper, a new analytical formula has been derived for the two-center Coulomb integrals over Slater type orbitals using the Poisson equation. The obtained results from constructed computer program for the presented formula have been compared with the available literature and it is seen that the efficiency of the presented algorithm for a wide range of quantum numbers, orbital exponents and internuclear distances is satisfactory.

Comment on ?Computation of Two-Center Coulomb Integrals over Slater-Type Orbitals Using Elliptical Coordinates?

In a recent article in this journal, Ozmen, Karakas, Atav, and Yakar gave formulas for the evaluation of two-center Coulomb integrals over Slater-type orbitals (STOs) in ellipsoidal coordinates Int J Quantum Chem 2003, 91, 13–19. The relation of this work to the rich literature on STO integral evaluation is reviewed, with focus on the important issues of speed of computation and numerical stability. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 93: 332–334, 2003

Coulomb integrals for Gaussian, Slater, Bessel and polynomial-type functions

We present a unified study of two-center Coulomb integrals, involving Gaussian, Slater, Bessel and polynomial radial factors combined with Cartesian and spherical angular factors, performed with the shift operators technique. The master formula for these integrals appears as a linear combination of polynomials of the relative coordinates of the centers. These polynomials are independent of the type of radial factor, and only three families arise: two corresponding to angular factors of the same type and a mixed one. The coefficients depend on the types of both the angular and radial factors. We consider the 30 most important cases and give simple rules for obtaining them in terms of only six reference integrals.

Two-center nonexchange integrals over Slater orbitals

We have used computer algebra (CA) to construct general formulas for the two-center overlap, resonance, Coulomb and hybrid integrals over Slater orbitals (STOs). Individual formulas then were produced for the Coulomb integrals containing all combinations of K, L and M shell orbitals. The numerical evaluation of these formulas is very rapid and allows unrestricted accuracy. Tests give complete agreement with the output of numerical programs written by other workers. Computer algebra eliminates terms of equal magnitude and opposite sign that can cause catastrophic loss of accuracy in a purely numerical computation. Strong parallelization is possible in the construction and evaluation of the formulas. Their availability opens up the prospect of carrying symbolic computation into the eigenvalue finding stage of quantum chemistry. The present calculations also highlight several needs and opportunities in the field of computer algebra.

Computation of molecular integrals over Slater-type orbitals. V. Calculation of multicenter electron-repulsion integrals using auxiliary functions

The multicenter electron-repulsion integrals appearing in the Hartree–Fock–Roothaan (HFR) equations for molecules are expressed in terms of the overlap integrals and the basic two-center Coulomb or hybrid integrals by the use of expansion formulas for the charge-density over Slater-type orbitals (STOs) obtained by the one of authors (I.I. Guseinov, J. Mol. Struct. (Theochem) 417 (1997) 117). For the calculation of basic integrals the new auxiliary functions are introduced. The multicenter electron-repulsion integrals are calculated for extremely large quantum numbers using recurrence relations for the overlap integrals and auxiliary functions.

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. c 2000 John Wiley & Sons, Inc. Int J Quantum Chem 79: 91-100, 2000

Molecular Orbitals Calculation on LiH with Algebraic Treatment of the Integrals

In this work we employ an algebraic computational method to solve the integrals which arise in the study of diatomic molecules. Using the Slater-type orbitals (STO), we obtain analytical solutions for the one-center and two-center hybrid and Coulomb integrals. The exchange integrals are considered, as much as possible, in a similar manner. These results are used to calculate the electronic properties of the ground state of the LiH molecule, by means of a variational calculation using a basis of molecular orbitals. The behavior of the Slater exponents of the atomic orbitals are studied as functions of internuclear distance. We show that the STO undergo a lengthening of their shape when the nuclear separation increases.

Unified treatment of two-center Coulomb and hybrid integrals with Slater-type orbitals using auxiliary functions

Two-center Coulomb and hybrid integrals are combind into a single expression. The final results are expressed in terms of the auxiliary functions.