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.     

Integrals of gaussian and continuum functions for polyatomic molecules. An addition theorem for solid harmonic gaussians

An addition theorem for solid harmonic gaussian functions in terms of surface spherical harmonics is derived and presented in a form particularly useful for evaluating molecular integrals involving gaussian and continuum functions on array processors.     

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.     

Recurrence formulas for molecular integrals over Laguerre Gaussian-type functions

Recurrence formulas for overlap, nuclear attraction, and electron-repulsion integrals over Laguerre Gaussian-type functions are presented. They have been derived using compact recurrence relations for homogeneous solid spherical harmonic operators but are rather lengthy as compared to those over Cartesian Gaussian-type functions. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 273–279, 1998     

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     

Recurrence relations for the evaluation of electron repulsion integrals over spherical Gaussian functions

Recurrence relations are derived for the evaluation of two-electron repulsion integrals (ERIs) over Hermite and spherical Gaussian functions. Through such relations, a generic ERI or ERI derivative may be reduced to “basic” integrals, i.e., true and auxiliary integrals involving only zero angular momentum functions. Extensive use is made of differential operators, in particular, of the spherical tensor gradient ??(?). Spherical Gaussians, being nonseparable in the x, y, and z coordinates, were not included in previous formulations. The advantages of using spherical Gaussians instead of Cartesian or Hermite Gaussians are briefly discussed. © 1993 John Wiley & Sons, Inc.     

Accompanying coordinate expansion formulas derived with the solid harmonic gradient

A series of accompanying coordinate expansion (ACE) formulas for calculating the electron repulsion integral (ERI) over both generally and segmentally contracted solid harmonic (SH) Gaussian-type orbitals (GTOs) can be rederived by the use of the modified operator (called solid harmonic gradient here) of the spherical tensor gradient of Bayman and the reducing solid harmonic gradient defined in this article. The final general formulas contain the reducing mixed solid harmonics defined in a previous article [Ishida, K. J Chem Phys 1999, 111, 4913] and the reducing triply mixed solid harmonics defined previously [Ishida, K. J Chem Phys 2000, 113, 7818]. Each general formula in the series is named ACEb1k1, ACEb2k3, or ACEb3k3. New general algorithm can be obtained inductively from the general formula named ACEb2k3, in addition to the previously developed ACEb1k1 and ACEb3k3. For calculating ERI practically, we select one of these ACE algorithms, as it gives the minimum floating-point operation (FLOP) count. Theoretical assessment by the use of the FLOP count is performed for the (LL/LL) class of ERIs over both generally and segmentally contracted SH-GTOs (L = 1-3). It is found that the present ACE is theoretically the fastest among all rigorous methods in the literature.     

Evaluation of integrals over STOs on different centers and the complementary convergence characteristics of ellipsoidal‐coordinate and zeta‐function expansions

One‐electron integrals over three centers and two‐electron integrals over two centers, involving Slater‐type orbitals (STOs), can be evaluated using either an infinite expansion for 1/r₁₂ within an ellipsoidal‐coordinate system or by employing a one‐center expansion in spherical‐harmonic and zeta‐function products. It is shown that the convergence characteristics of both methods are complimentary and that they must both be used if STOs are to be used as basis functions in ab initio calculations. To date, reports dealing with STO integration strategies have dealt exclusively with one method or the other. While the ellipsoidal method is faster, it does not always converge to a satisfactory degree of precision. The zeta‐function method, however, offers reliability at the expense of speed. Both procedures are described and the results of some sample calculation presented. Possible applications for the procedures are also discussed. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 1–13, 1999     

Use of Cartesian coordinates in evaluation of multicenter multielectron integrals over Slater type orbitals and their derivatives

Using addition theorems for interaction potentials and Slater type orbitals (STOs) obtained by the author, and the Cartesian expressions through the binomial coefficients for complex and real regular solid spherical harmonics (RSSH) and their derivatives presented in this study, the series expansion formulas for multicenter multielectron integrals of arbitrary Coulomb and Yukawa like central and noncentral interaction potentials and their first and second derivatives in Cartesian coordinates were established. These relations are useful for the study of electronic structure and electron-nuclei interaction properties of atoms, molecules, and solids by Hartree–Fock–Roothaan and correlated theories. The formulas obtained are valid for arbitrary principal quantum numbers, screening constants and locations of STOs.     

Analytical calculations of molecular integrals for multielectron R-matrix methods

Closed-form analytical expressions for one- and two-electron integrals between Cartesian Gaussians over a finite spherical region of space are developed for use in ab initio molecular scattering calculations. In contrast with some previous approaches, the necessary integrals are formulated solely in terms of finite summations involving standard functions. The molecular integrals evaluated over the finite region of space are computed by subtracting the contributions outside the region from the integrals over all space. The latter integrals can be efficiently and accurately obtained from existing bound-state algorithms. Our approach incorporates molecular scattering calculations into current quantum chemistry programs and facilitates the unification of bound- and continuum-state calculations for both diatomic and polyatomic molecules. Multidimensional Monte Carlo numerical integrations validate the high accuracy of our closed form results for the two-electron integrals.     

An analytic method for three-center nuclear attraction integrals: A generalization of the Gegenbauer addition theorem

A completely analytic method for evaluating three-center nuclear attraction integrals for STOS is presented. The method exploits a separation of the STO into an ‘evenly loaded’ solid harmonic and a OS STO . The harmonics are translated to the molecular center of mass in closed finite terms. The OS STO is translated using the Gegenbauer addition theorem; 1s STOS are translated using a single parametric differentiation of the OS formula. Explicit formulas for the integrals are presented for arbitrarily located atoms. A numerical example is given to illustrate the method.     

Auxiliary functions for molecular integrals with Slater‐type orbitals. II. Gauss transform methods

The Gauss transform of Slater‐type orbitals is used to express several types of molecular integrals involving these functions in terms of simple auxiliary functions. After reviewing this transform and the way it can be combined with the shift operator technique, a master formula for overlap integrals is derived and used to obtain multipolar moments associated to fragments of two‐center distributions and overlaps of derivatives of Slater functions. Moreover, it is proved that integrals involving two‐center distributions and irregular harmonics placed at arbitrary points (which determine the electrostatic potential, field and field gradient, as well as higher order derivatives of the potential) can be expressed in terms of auxiliary functions of the same type as those appearing in the overlap. The recurrence relations and series expansions of these functions are thoroughly studied, and algorithms for their calculation are presented. The usefulness and efficiency of this procedure are tested by developing two independent codes: one for the derivatives of the overlap integrals with respect to the centers of the functions, and another for derivatives of the potential (electrostatic field, field gradient, and so forth) at arbitrary points. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

General formulae for interacting spherical nanoparticles and fullerenes

Most nanodevices under investigation adopt a computational approach such as molecular dynamics simulations, which gives a numerical value for the potential energy as calculated from the interaction of every atom on one molecule with every atom on a second molecule. Although the simulation only involves short range atom–atom interactions and ignores those interactions at longer distances, the simulation still involves significant computational time. In this paper, we determine analytical formulae for four types of Lennard–Jones interactions: (i) a solid spherical nanoparticle with an atom, (ii) two distinct radii hollow spherical fullerenes, (iii) a solid spherical nanoparticle with a hollow spherical fullerene and (iv) two distinct radii solid spherical nanoparticles. The interaction energy using the 6–12 Lennard–Jones potential for these four situations are determined using the continuum approximation, which assumes that a discrete atomic structure can be replaced by either an average atomic surface density or an average atomic volume density. Using these formulae the computational time for a simulation might be dramatically reduced for those molecular interactions involving spherical nanoparticles or fullerenes. Such formulae might be exploited in hybrid analytical-computational numerical schemes, as well as in metallofullerenes and certain assumed spherical models of molecules such as methane and ammonia. As an illustration of the formulae presented here we determine both the most stable and the maximum radii of a solid spherical nanoparticle inside a fullerene, modelling the centre of a carbon onion or metallofullerenes. We also determine new cut-off formulae for interacting spherical nanoparticles and fullerenes which might be useful in computational schemes.     

Matrix elements for Ĵ2 and Ĵz operators over explicitly correlated Cartesian Gaussian functions

General formalism for evaluation of multiparticle integrals involving J?² and J?_z operators over explicitly correlated Cartesian Gaussian functions is presented. The integrals are expressed in terms of the general overlap integrals. An explicitly correlated Cartesian Gaussian function is a product of spherical orbital Gaussian functions, powers of the Cartesian coordinates of the particle, and exponential Gaussian factors, which depend on interparticular distances. This development is relevant to both adiabatic and nonadiabatic calculations of energy and properties of multiparticle systems. © 1995 John Wiley & Sons, Inc.     

New translation method for STOs and its application to calculation of overlap integrals

A new translation method for Slater‐type orbitals (STOs) is proposed involving 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. The method is positively tested in the case of the overlap integral, showing good rate of convergence and great numerical stability under wide changes in the relevant molecular parameters. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 333–340, 1999     

Gauss' theorem approach to delta function terms in solid spherical harmonic expansions

The nature of the Dirac delta-function singularity in the expansion theorem for an irregular solid spherical harmonic about another centre is discussed for the casel=2. An alternative derivation, motivated by Hobson's derivative expression for solid spherical harmonics and utilizing Gauss' Divergence Theorem, is presented. The orientation dependence is then simply derived from the rotational properties of spherical harmonics.     

Overtone intensities in resonance Raman scattering

A model calculation of resonance Raman scattering tensors has been carried out for a diatomic molecule with a harmonic potential for the ground state and a linear repulsive potential for the excited state. Expressions for scattering tensors have been obtained by using a series of recurrence formulas induced from the Green function of the nuclear hamiltonian of the repulsive potential of the excited state and nuclear wavefunctions for the harmonic potential. The relative scattering intensities of overtones depend on the gradient of the repulsive potential curve and are interpreted in terms of the overlap integrals between nuclear wavefunctions of upper and lower states.     

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

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. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Spherical tensor gradient operator method for integral rotation: a simple, efficient, and extendable alternative to Slater-Koster tables

We present a novel alternative to the use of Slater-Koster tables for the efficient rotation and gradient evaluation of two-center integrals used in tight-binding Hamiltonian models. The method recasts the problem into an exact, yet implicit, basis representation through which the properties of the spherical tensor gradient operator are exploited. These properties provide a factor of 3 to 4 speedup in the evaluation of the integral gradients and afford a compact code structure that easily extends to high angular momentum without loss in efficiency. Thus, the present work is important in improving the performance of tight-binding models in molecular dynamics simulations and has particular use for methods that require the evaluation of two-center integrals that involve high angular momentum basis functions. These advances have a potential impact for the design of new tight-binding models that incorporate polarization or transition metal basis functions and methods based on electron density fitting of molecular fragments.     

Overlap Integrals Between Irregular Solid Harmonics and STOs Via the Fourier Transform Methods

In this paper, a unified analytical and numerical treatment of overlap integrals between Slater type orbitals (STOs) and irregular Solid Harmonics (ISH) with different screening parameters is presented via the Fourier transform method. Fourier transform of STOs is probably the simplest to express of overlap integrals. Consequently, it is relatively easy to express the Fourier integral representations of the overlap integrals as finite sums and infinite series of STOs, ISHs, Gegenbauer, and Gaunt coefficients. The another mathematical tools except for Fourier transform have used partial-fraction decomposition and Taylor expansions of rational functions. Our approach leads to considerable simplification of the derivation of the previously known analytical representations for the overlap integrals between STOs and ISHs with different screening parameters. These overlap integrals have also been calculated for extremely large quantum numbers using Gegenbauer, Clebsch-Gordan and Binomial coefficients. The accuracy of the numerical results is quite high for the quantum numbers of Slater functions, irregular solid harmonic functions and for arbitrary values of internuclear distances and screening parameters of atomic orbitals.