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.     

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.     

Evaluation of multicenter one-electron integrals of noninteger u screened Coulomb type potentials and their derivatives over noninteger n Slater orbitals

Multicenter integrals over noninteger n Slater type orbitals with integer and noninteger values of indices u of screened Coulomb type potentials, f(u)(eta,r)=r(u-1)e(-etar), and their first and second derivatives with respect to Cartesian coordinates of the nuclei of a molecule are described. Using complete orthonormal sets of Psi(alpha) exponential type orbitals and rotation transformation of two-center overlap integrals, these integrals are expressed through the noncentral potential functions depending on the molecular auxiliary functions A(k) and B(k). The series expansion formulas derived for molecular integrals of screened Coulomb potentials and their derivatives are especially useful for the computation of multicenter electronic attraction, electric field, and electric field gradient integrals. The convergence of series is tested for arbitrary values of parameters of potentials and orbitals.     

Extended floating spherical Gaussian basis sets for Molecules. FSGO basis for use in advanced correlated calculations of electronic structures

The transformation of occupied and excited SCF orbitals expressed in Cartesian Gaussian form to a smaller, simpler set of floating spherical Gaussians is described. Illustrative applications at the correlated coupled-cluster level are presented for Lill and H₂O.     

Study of localized molecular orbitals using group theory methods and its approach to the many-electron correlation problem. IV. The symmetry-adaptation of many-center integrals and hamiltonian matrix elements in MCSCF calculations

Group theoretic methods are presented for the transformations of integrals and the evaluation of matrix elements encountered in multiconfigurational self-consistent field (MCSCF) and configuration interaction (CI) calculations. The method has the advantages of needing only to deal with a symmetry unique set of atomic orbitals (AO) integrals and transformation from unique atomic integrals to unique molecular integrals rather than with all of them. Hamiltonian matrix element is expressed by a linear combination of product terms of many-center unique integrals and geometric factors. The group symmetry localized orbitals as atomic and molecular orbitals are a key feature of this algorithm. The method provides an alternative to traditional method that requires a table of coupling coefficients for products of the irreducible representations of the molecular point group. Geometric factors effectively eliminate these coupling coefficients. The saving of time and space in integral computations and transformations is analyzed. © 1994 by John Wiley & Sons, Inc.     

On the use of spatial symmetry in atomic-integral calculations: An efficient permutational approach

The minimal number of independent nonzero atomic integrals that occur over arbitrarily oriented basis orbitals of the form ?(r) · Y_lm(Ω) is theoretically derived. The corresponding method can be easily applied to any point group, including the molecular continuous groups C_∞v and D_∞h. On the basis of this (theoretical) lower bound, the efficiency of the permutational approach in generating sets of independent integrals is discussed. It is proved that lobe orbitals are always more efficient than the familiar Cartesian Gaussians, in the sense that GLO s provide the shortest integral lists. Moreover, it appears that the new axial GLO s often lead to a number of integrals, which is the theoretical lower bound previously defined. With AGLO s, the numbers of two-electron integrals to be computed, stored, and processed are divided by factors 2.9 (NH₃), 4.2 (C₅H₅), and 3.6 (C₆H₆) with reference to the corresponding CGTO s calculations. Remembering that in the permutational approach, atomic integrals are directly computed without any four-indice transformation, it appears that its utilization in connection with AGLO s provides one of the most powerful tools for treating symmetrical species.     

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.     

Recursion formula for electron repulsion integrals over Hermite polynomials

A method for computing electron repulsion integrals over contracted Gaussian functions is described in which intermediate integrals over Hermite polynomials are generated by a “pre‐Hermite” recursion (PHR) step before the conversion to regular integrals. This greatly reduces the floating‐point operation counts inside the contraction loops, where only simple “scaling”‐type operations are required, making the method efficient for contracted Gaussians, particularly of high angular momentum. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

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.     

Evaluation of one-electron integrals for arbitrary operators V(r) over Cartesian Gaussians: Application to inverse-square distance and Yukawa operators

A general procedure is presented for generating one-electron integrals over any arbitrary potential operator that is a function of radial distance only. The procedure outlines that for a nucleus centered at point C integrals over Cartesian Gaussians can be written as linear combinations of 1-D integrals. These Cartesian Gaussian functions are expressed in a compact form involving easily computed auxiliary functions. It is well known that integrals over the Coulomb operator can be expressed in terms of F_n(T) integrals, where By means of a substitution for F_n(T) by other simple functions, algorithms that form integrals over an arbitrary function can be generated. Formation of such integrals is accomplished with minor editing of existing code based on the McMurchie–Davidson formalism. Further, the method is applied using the inverse-square distance and Yukawa potential operators V(r) over Cartesian Gaussian functions. Thus, the proposed methodology covers a large class of one-electron integrals necessary for theoretical studies of molecular systems by ab initio calculations. Finally, by virtue of the procedure's recursive nature it provides us with an efficient scheme of computing the proposed class of one-electron integrals. © 1993 John Wiley & Sons, Inc.     

Fourier transform general formula for systematic potentials

For calculating molecular integrals of systematic potentials, a three‐dimensional (3D) Fourier transform general formula can be derived, by the use of the Abel summation method. The present general formula contains all 3D Fourier transform formulas which are well known as Bethe–Salpeter formulas (Bethe and Salpeter, Handbuch der Physik, Bd. XXXV, 1957) as special cases. It is shown that, in several of the Bethe–Salpeter formulas, the integral does not converge in the meaning of the Riemann integral but converges in the meaning of a hyper function as the Schwartz distribution. For showing an effectiveness of the present general formula, the convergence condition of molecular integrals is derived generally for all of the present potentials. It is found that molecular integrals can be converged in the meaning of the Riemann integral for the present potentials, except for those for extra super singular potentials. It is also found that the convergence condition of molecular integrals over the Slater‐type orbitals is exactly the same as that of the corresponding integrals over the Gaussian‐type orbitals for the present systematic potentials. For showing more effectiveness, the molecular integral over the gauge‐including atomic orbitals is derived for the magnetic dipole‐same‐dipole interaction. © 2012 Wiley Periodicals, Inc.     

Analytical GIAO and hybrid-basis integral derivatives: application to geometry optimization of molecules in strong magnetic fields

Analytical integral evaluation is a central task of modern quantum chemistry. Here we present a general method for evaluating differentiated integrals over standard Gaussian and mixed Gaussian/plane-wave hybrid orbitals. The main idea is to have a representation of basis sets that is flexible enough to enable differentiated integrals to be reinterpreted as standard integrals over modified basis functions. As an illustration of the method, we report a very simple implementation of Hartree-Fock level geometrical derivatives in finite magnetic fields for gauge-origin independent atomic orbitals, within the London program. As a quantum-chemical application, we optimize the structure of helium clusters and some well-known covalently bound molecules (water, ammonia and benzene) subject to strong magnetic fields.     

On the choice and definition of atomic-orbital bases. I. General considerations; promotion and hybridization; electric dipole moments

After a brief discussion of the physical significance of the choice of the basis in molecular calculations, the nature and definition of an atomic-orbital basis for use in limited calculations is discussed, in view of the possibility of replacing, say, ordinary 2s and 2p Slater orbitals by appropriate hybridized-promoted atomic orbitals. It is indicated that, if the orbitals must be defined in connection with a given interpretation scheme for the behavior of molecules, hybridization and promotion may be necessary. The two kinds of conditions one may wish to impose on a restricted atomic-orbital set are explicitly considered. The first is that the atomic orbitals should be hybrids directed along the bonds and at the same time satisfy the maximum overlap criterion; the other is the requirement that the atomic orbitals should be such that the electric dipole moment of a polyatomic molecule described in terms of a semiempirical bond-orbital scheme should be expressed as the dipole moment of the system of bond charges located at the nuclei. The latter condition is treated in detail, showing that it implies a cancellation of atomic and overlap moments. The equations defining the atomic orbitals satisfying the condition in question are given. In the course of the mathematical treatment some general results concerning the expression of the dipole moment of a molecule and the definition of net atomic charges are given, showing that, for systems where overlap integrals are low, the atomic populations can be taken as sums of the squares of the coefficients of orthogonalized atomic orbitals. Applications of the results will be presented in part II.     

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     

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.     

NMR shielding tensors for density fitted local second-order Mo̸ller-Plesset perturbation theory using gauge including atomic orbitals

An efficient method for the calculation of nuclear magnetic resonance (NMR) shielding tensors is presented, which treats electron correlation at the level of second-order Mo?ller-Plesset perturbation theory. It uses spatially localized functions to span occupied and virtual molecular orbital spaces, respectively, which are expanded in a basis of gauge including atomic orbitals (GIAOs or London atomic orbitals). Doubly excited determinants are restricted to local subsets of the virtual space and pair energies with an interorbital distance beyond a certain threshold are omitted. Furthermore, density fitting is employed to factorize the electron repulsion integrals. Ordinary Gaussians are employed as fitting functions. It is shown that the errors in the resulting NMR shielding constant, introduced (i) by the local approximation and (ii) by density fitting, are very small or even negligible. The capabilities of the new program are demonstrated by calculations on some extended molecular systems, such as the cyclobutane pyrimidine dimer photolesion with adjacent nucleobases in the native intrahelical DNA double strand (ATTA sequence). Systems of that size were not accessible to correlated ab initio calculations of NMR spectra before. The presented method thus opens the door to new and interesting applications in this area.     

Symmetry-adapted integral derivatives

A procedure, based on double coset decompositions, is described for reducing formulas for derivatives (with respect to nuclear coordinates) of integrals over symmetry-adapted orbitals to symmetry-distinct integral derivatives over atomic orbitals. The procedure is applicable to any finite point group and to integral derivatives of any order.     

Expansion of atomic orbital products in terms of a complete function set

Summary An alternative method for the evaluation of matrix elements required in an LCAO-SCF calculation is presented. It is based on the use of solutions of the Helmholtz equation within a spherical domain for expanding charge distributions with boundary conditions devised to make the electrostatic-potential integral particularly simple. This method allows the systematic evaluation of bielectronic integrals to be performed for any type of atomic orbitals.Part of a PhD Thesis (J.E.P.) to be presented to the UNRC     

Unified treatment of complex and real rotation-angular functions for two-center overlap integrals over arbitrary atomic orbitals

The new combined formulas have been established for the complex and real rotation-angular functions arising in the evaluation of two-center overlap integrals over arbitrary atomic orbitals in molecular coordinate system. These formulas can be useful in the study of different quantum mechanical problems in both the theory and practice of calculations dealing with atoms, molecules, nuclei and solids when the integer and noninteger n complex and real atomic orbitals basis sets are emploed. This work presented the development of our previous paper (I.I. Guseinov in Phys Rev A 32:1864, 1985).     

Atomic orbital basis sets for molecular interactions

We describe a general approach to the parametrization of linear combinations of Gaussian atomic orbitals, useful for atomic and molecular interactions. We use a Gaussian transform method and Gauss-Legendre quadratures to express hydrogenic atomic orbitals, with varying effective charges, in terms of Gaussian-type orbitals. This procedure provides well-defined rules for calculating exponent factors and combination coefficients of the linear combinations of Gaussians in problems where nuclear distances may vary over large ranges during interactions. © 1994 by John Wiley & Sons, Inc.