Evaluation of two-center,two- and three-electron integrals involving correlation factors over Slater-type orbitals. II. Kinetic and potential energy integrals and examples of numerical results

This article is concerned with the construction of the general algorithm for evaluating two-center, two- and three-electron integrals occurring in matrix elements of one-electron operators in the basis of variational correlated functions. This problem has been solved here in prolate spherical coordinates, using the modified and extended form of the Neumann expansion of the interelectronic distance function r^k_ij derived in Part I of this series for k = ?1, 0, 1, 2. This work expands the method proposed by one of us in the preceding paper for integrals of the types mentioned above. The results of numerical calculations for different types of the two- and three-electron integrals are presented. The problem of convergence of the proposed procedures used is also discussed.     

Calculation of coulomb integrals in molecules over an spd basis of STO

A spherical tensor expansion of 1/r ₁₂ (where r ₁₂ is the separation considered) is used to represent Coulomb integrals in a molecular context for an MNDO method over an spd basis of STO. This is rendered invariant of the space fixed axes chosen using Wigner rotation matrices which transform the integrals from the molecular frame (as distinct from Dewar [1])-This procedure is found to be rigorous only in the long range limit but is satisfactory at separations of the order of most single bond lengths (as Dewar [1]). The pole at R _ab=0 is avoided by adding a constant to the separation, which takes the value that reproduces the Slater-Condon monocentric integrals there. Extension over the whole range is carried out using a unique multiplicative polynomial from the Legendre function expansion of 1/(R _ab ² +A ²) for small R _ab and an exponential decay in R _ab is dictated by symmetry in the overlap region expression, which retains rotational invariance.This calculation results in an easy evaluation of these functions and their first derivatives leading to a very rapid molecular geometry optimisation taking the d-orbitals into account in an MNDO hypothesis.Unité Associée au CNRS no 510 Interactions Moléculaires     

Symmetrie und analytische Struktur der Additionstheoreme räumlicher Funktionen und der Mehrzentren-Molekülintegrale über beliebige Atomfunktionen

Three-dimensional functions f(r) = g(r) · Y _m ^l (, ø), which transform like an irreducible tensor, are transformed simultaneously under rotations and translations. The relationships governing the transformation reveal some general properties. If the addition theorem of a function f(r) can be represented by a one-center expansion in terms of surface spherical harmonics Y _m ^l , each expansion coefficient is given by a Clebsch-Gordan coefficient and a radial function.Because of these properties, addition theorems are especially helpful for the simplification and evaluation of quantum-mechanical matrix elements and multi-center energy integrals in molecular LCAO calculations. The application of addition theorems has two major advantages: First, because addition theorems are equivalent to translation formulas, the number of centers of an integral can be reduced by translation of orbitals and operators. Second, due to the typical analytical structure of the series expansion representing the addition theorem, the dimensionality of a molecular integral can be reduced, because the integration over the angular variables can be executed. Then, a molecular multi-center integral is represented by a series of one-center integrals over functions of the radial variable only.

Herrn Professor Dr. H. Hartmann zum 65. Geburtstag gewidmet.     

Expansion Formulae for Two-center Integer and Noninteger <Emphasis Type="Italic">n</Emphasis> STO Charge Densities and their Use in Evaluation of Multi-center Integrals

Using complete orthonormal sets of Ψ^α-exponential type orbitals (Ψ^α-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.     

Multiconfigurational SCF approach for high-symmetry molecules and its applications to fullerene trianion

A symmetry-adapted multiconfiguration self-consistent field (MC SCF) approach aimed at calculations of high-symmetry molecules is proposed. The self-consistency procedure applicable to the molecular terms of any symmetry and multiplicity is developed. It holds the symmetry transformation properties of varied molecular orbitals, thus taking advantage of the relationships within the set of two-electron integrals through molecular invariants. For orbital optimization, a unified coupling operator is constructed on the basis of the pseudosecular method providing for efficient convergence to energy minimum. Based on the group-theory technique, computer codes have been developed for straightforward determination of the invariant expansions for two-electron integrals and configuration interaction (CI) matrix elements. Calculated in this way, the expansion coefficients are presented for the three-electron states that originate from joint t_1u and t_1g shells of an icosahedral fullerene C₆₀, the case important for the calculations of anion C₆₀³⁻ representing the charge state of the fullerene molecule in the superconducting ionic solids K₃C₆₀ or Rb₃C₆₀. The results of MC SCF calculations for lowest quasi-π-electronic states of C₆₀³⁻ are discussed. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 293–304, 1998     

Computation of overlap integrals over Slater-type orbitals using auxiliary functions

Analytical expressions through the binomial coefficients and recursive relations are derived for the expansion coefficients of overlap integrals in terms of a product of well-known auxiliary functions A_k and B_k. These formulas are especially useful for the calculation of overlap integrals for large quantum numbers. Accuracy of the computer results is satisfactory for the values of quantum numbers up to 50 and for the arbitrary values of screening constants of atomic orbitals and internuclear distances. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 199–204, 1998     

Molecular correlation integrals. Two center one electron integrals containing a function of interparticle distance in one center expansion approximation

Two-center one-electron integrals needed in certain molecular correlated wave function calculations, using one-center expansion approximation, have been studied. The form of the basic correlated function used in this study is The parent integral is expressed in terms of an angular integral, and an auxiliary radial integral depending upon the variables r₁, r₂, and r₁₂. Several analytical formulas, and a recursive formula are derived for the auxiliary integral, and other related integrals. All these formulas are given in computationally useful forms. Logical flow charts and FORTRAN programs were constructed for computing the basic integrals discussed in the paper. Numerical values of some integrals, thus obtained, are tabulated for comparisons.     

Two-electron repulsion integrals over Gaussian s functions

We present an efficient scheme to evaluate the [ 0 ]^(m) integrals that arise in many ab initio quantum chemical two-electron integral algorithms. The total number of floating-point operations (FLOPS ) required by the scheme has been carefully minimized, both for cases where multipole expansions of the integrals are admissable and for cases where this is not so. The algorithm is based on the use of a modified Chebyshev interpolation formula to compute the function exp(?T) and the integral F_m(T) = ∫₀¹u^2mexp(?Tu²) du very cheaply.     

Unified treatment of two-center Coulomb,hybrid, and exchange integrals over Slater-type orbitals

Using Neumann expansion for 1/r₁₂ in elliptical coordinates a combined formula has been obtained for two-center Coulomb, hybrid, and exchange integrals with Slater-type orbitals. © 1995 John Wiley & Sons, Inc.     

Transferable integrals in a deformation-density approach to crystal orbital calculations. I

In the usual ab initio method of calculating molecular orbitals, the number of integrals to be evaluated increases as M⁴, where M is the number of basis functions. In this paper, an alternative method is discussed, where the computation time increases much less violently with the number of basis functions. Matrix elements of the deformation potential are evaluated by Fourier transform methods, while matrix elements of the neutral-atom potential are evaluated by means of transferable integrals. The transferable integrals (moments of the neutral-atom potentials) can be evaluated once and for all and incorporated as input data in computer programs. In an appendix to the paper, a general expansion theorem is discussed. This theorem allows an arbitrary spherically symmetric function to be expanded about another center.     

Accurate nonrelativistic energies for 2Po states of the Li isoelectronic series

With large Hylleraas-configuration interaction (CI) basis sets highly accurate upper bounds for the lowest ²P^o states of the Li isoelectronic series up to Ne are given. The corresponding Hamiltonian H and the operator Σ_{i < j}, ▿_i ▿_j are transformed into nonorthogonal coordinates expressed in interparticle and angular terms. The evaluation of the occurring integrals is reduced to the calculation of well-known auxiliary integrals. Furthermore some expectation values and isotope energies are calculated. The isotope energies are obtained using perturbation theory in first-order approximation. © 1997 John Wiley & Sons, Inc.     

Slater orbital molecular integrals with numerical fourier transform methods. I. (coplanar) multicenter exchange integrals over 1s orbitals

Slater orbital r₁₂^?1 integrals are calculated with a numerical Fourier-transform method based on a formulation first given by Bonham, Peacher and Cox. Spherical wave expansions are introduced that decouple the Feynman integrations for the charge distribution Fourier transforms. The Feynman integrals are evaluated semianalytically, and their properties are analyzed in detail. The final computational step involves a numerical integration over charge distribution quantities. Results for (coplanar) multicenter exchange integrals over 1s orbitals are given. As long as the charge distributions are overlapping considerably, the method gives good results, even when these distributions are highly asymmetric. The method as presently implemented fails when highly disconnected charge distributions are involved.     

Measurements of the thermal neutron cross-sections and resonance integrals for 186W (n,γ) 187W and 98Mo (n,γ) 99Mo reactions

The thermal neutron cross-sections and resonance integrals of the ¹⁸⁶W (n,γ) ¹⁸⁷W and ⁹⁸Mo (n,γ) ⁹⁹Mo reactions in the thermal and 1/E regions, respectively, of a thermal reactor neutron spectrum have been experimentally determined by the activation method using ¹⁹⁷Au (n,γ) ¹⁹⁸Au reaction as a single comparator. The high purity natural W, Mo, and Zr foils; and Au wire diluted in aluminum, were irradiated without Cd shield in two neutron irradiation sites, characterized with different values for the thermal-to-epithermal flux ratios, f at the Second Egyptian Research Reactor (ETRR-2). The induced activities in the samples were measured by high-resolution γ-ray spectrometry with a calibrated germanium detector. Thermal neutron cross-sections for 2200 m/s neutrons and resonance integrals for the ¹⁸⁶W (n,γ) ¹⁸⁷W and ⁹⁸Mo (n,γ) ⁹⁹Mo reactions have been obtained relative to the reference values, σ₀ = 98.65 ± 0.09 b and I ₀ = 1500 ± 28 b for the ¹⁹⁷Au (n,γ) ¹⁹⁸Au reaction. The necessary correction factors for thermal neutron and resonance neutron self-shielding effects, and the epithermal flux index (α) were taken into account in the determinations. The results obtained were: σ₀ = 38.43 ± 0.4 b and I ₀ = 502 ± 65 b for ¹⁸⁶W (n,γ) ¹⁸⁷W, and σ₀ = 0.137 ± 0.014 band I ₀ = 6.47 ± 0.8 for ⁹⁸Mo (n,γ) ⁹⁹Mo. These results are discussed and compared with previous measurements and evaluated data in literature. The traditional method of determining thermal cross-sections and resonance integrals via neutron irradiation with and without Cd shield in one irradiation position was avoided in this work by neutron irradiation without Cd shield in at least two different neutron irradiation positions. This method provides alternative way for determining thermal cross-sections and resonance integrals simultaneously.     

Possible generalization of the optimized diatomics-in molecules theory: Further studies of the diatomic state mixing parameter adjustment

A generalization of optimized diatomics-in-molecules method that provides an accurate treatment of overlap integrals is presented. The possibilities of the method to reproduce potential surfaces of the ⁴B₂, ²B₂, and ²A₁ states of the H₃ molecule are investigated. It is shown that satisfactory results can be achieved only for a single state under consideration, but not for the three states simultaneously.     

New expansion of the Boys function

We propose a new expansion for the Boys function ∫₀¹t^2jexp(−r²t²) dt appearing in the calculation of molecular two-electron matrix elements if Gaussian basis sets are employed. This expansion involves a power series involving the terms C_{i, j}(τ) (r²−R²)ⁱ multiplied by exp(−τr²), where τ is an optimized parameter τ∈[0, 1]. The performances of the introduced expansion are discussed and illustrated by some numerical experiments. It appears that the proposed expansion is considerably shorter than the customary Taylor series, which in turn is the special case for τ=0. This is of some importance, particularly for higher j values. Further, the proposed expansion enables a single expression for calculating erf(x) for the whole range of variable x. The recursive relations for the expansion coefficients are derived and the truncation errors are estimated. A new method for calculating the Boys function by means of asymptotic series is represented too. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 305–315, 1998     

Computation of some new two-electron Gaussian integrals

Summary The evaluation of a new form of two-electron integrals is required if the interelectronic distancer ₁₂ is used as a variable in then-electron functions of electron correlation methods. The McMurchie-Davidson algorithm for the generation of molecular integrals over Gaussian-type functions is ideally suited to this. The new Gaussian integrals are formed from Hermite integrals overr ₁₂ (rather than 1/r ₁₂) by standard techniques. The Hermite integrals overr ₁₂ itself are generated by a simple procedure with negligible computational effort. The key results are discussed in the context of general recursion formulas. On leave from: Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, W-4630 Bochum, Germany     

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.     

Use of unsymmetrical one-range addition theorems of Slater type orbitals in molecular electronic structure determination

In this work, the applicability of the unsymmetrical one-range addition theorems obtained from the use of complete orthonormal sets of Ψ^α-exponential type orbitals (Ψ^α-ETOs, where α = 1, 0, − 1, − 2, ...) to the study of electronic structure of molecules is demonstrated using minimal basis sets of Slater type orbitals (STOs). As an example of application of unsymmetrical one-range addition expansion method to evaluate the multicenter electronic integrals, the calculation has been performed for the ground state of BH ₃ molecule. The results of computer calculations for the orbital and total energies, and linear combination coefficients of symmetrized molecular orbitals are presented.     

Influence of the Excited States of Atomic Nitrogen N(<Superscript>2</Superscript>D), N(<Superscript>2</Superscript>P) and N(R) on the Transport Properties of Nitrogen. Part II: Nitrogen Plasma Properties

In this paper, the calculated values of the viscosity and thermal conductivity of nitrogen plasma are presented taking into account five (e, N, N⁺, N₂ and N₂⁺) or eight (e, N(⁴S), N(²P), N(²D), N(R), N⁺, N₂ and N₂⁺) species. The calculations are based on the supposition that the temperature dependent probability of occupation of the states is given by the Boltzmann factor. The domain for which the calculations are performed, is for p = 1 and 10 atm in the temperature range from 5,000 K to 15,000 K. Classical collision integrals are used in calculating the transport coefficients and we have introduced new averaged collision integrals where the weight associated at each interacting species pair is the probable collision frequency. The influence of the collision integral values and energy transfer between two different species is studied. These results are compared which those of published theoretical studies.     

New approach to the rapid semiempirical calculation of molecular electrostatic potentials based on the am1 wave function: Comparison with ab initio hf/6-31g* results

A new approach to the computation of molecular electrostatic potentials based on the AM1 wave function is described. In contrast to the prevailing philosophy, but consistent with the underlying NDDO approximation, no deorthogonalization of the wave function is carried out. The integrals required for the computation of the electronic contributions to the molecular electrostatic potential are evaluated in a manner similar to that of the AM1 core-electron attraction integrals, while the nuclear contributions are computed using a new semiempirical function—Z_A(S^AS^A, S^pS^p)[1 + exp[ – ω_A(R_Ai – δ_A)]]—where the atomic parameters ω_A and δ_A are obtained by calibration against the results of ab initio HF/6-31G* calculations. Isopotential contour maps for guanine and cytosine obtained with the new method are qualitatively almost indistinguishable from their HF/6-31G* counterparts, while quantitative comparisons for the minima for a wide range of molecules are reproduced with an rms error of 5.2 kcal mol^?1. The locations of the “lone-pair” minima for a wide range of heterosubstituted organic molecules generally fall within 0.02 Å of the corresponding HF/6-31G* minima while those in the π-regions of unsaturated molecules are generally within 0.2 Å. Because of the rapid integral evaluation, the fully semiempirical method described here is extremely economical. For example, for the guanine–cytosine base pair it is >500 times faster than calculations in which the complete integral matrix is computed analytically from the deorthogonalized AM1 wave function. © John Wiley & Sons, Inc.