Two-center Coulomb Integrals Research Articles

Semiempirical calculation of two-electron integrals using bipolar expansion of the Ohno potential

Bipolar expansion of the Ohno potential as a method of calculating two-center Coulomb integrals that appear in the NDDO approximation is generalized to one-center two-electron integrals. A unified semiempirical scheme is suggested for estimating two-electron interactions in molecules. This scheme can be readily extended to arbitrary Slater basis sets (including the s,p,d-orbitals) and involves no a priori data on the valent states of atoms. In this work, the scheme is employed to extend the semiempirical PM3 method to the s,p,d-basis set. The efficiency of the method is proven by test calculations of 24 chromium compounds (π-complexes, carbonyls, isocyanides, etc.).

Asymptotic formulas for Coulomb and hybrid integrals with slater type functions

Asymptotic expressions are obtained for two-center Coulomb and hybrid integrals with Slater functions. Results of numerical test of these expressions are presented. The use of asymptotic formulas in LCAO-MO calculations of electronic molecular structure substantially reduces the computation time.

Combined analytical formula for two-center Coulomb and hybrid integrals in the molecular coordinate system

Combined analytical formula for two-center Coulomb and hybrid integrals in the molecular coordinate system

Analytical evaluation of one- and two-center Coulomb and two-center hybrid integrals for Slater-type orbitals. II

General formulas are derived for the two-center Coulomb and hybrid integrals for real and complex STOs with respect to the common molecular coordinate system. The angular and radial parts of these integrals are expressed through the spherical harmonics and the auxiliary functions ∼(Gq−NN′), respectively.

Parametrized Equations-of-Motion Method at the (1p-1h)+(2h-2p) Level and Its Application to Linear Polyenes

Abstract The previously proposed idea of parametrizing the equations-of-motion method [Int. J. Quantum Chem., 18, 205 (1980)] has been worked out for the π→π* transitions of linear polyenes. The excitation operator consists of not only 1p-1h creation and annihilation operators but also 2p-2h creation and annihilation operators. [Bull. Chem. Soc. Jpn., 61, 1103 (1988)]. The proposed scheme accounts for the effect of σ→σ* mixings in contrast with the traditional pi-electron theories. In the present work, the two-center Coulomb integrals are evaluated by 1/R, and all the one-center integrals on the C atom are assumed to be the same. Values of the three parameters (a, b, ζ) introduced in the scheme are so determined that the computed excitation energies of ethylene and s-trans-butadiene fit the observed values. The scheme has been applied to a series of all-s-trans linear polyenes through dodecahexaene. Some notable results are: (1) The σ→σ* mixings play a significant role in lowering the excitation energies of the allowed transition to 11Bu; (2) the 21Ag state is of Ag− type and almost free from this effect; (3) double excitations (2p-2h states) are dominant in the 21Ag state but practically negligible in the 11Bu state.

Analytic evaluation of two-center molecular integrals

By using the Fourier-transform technique, the explicit expressions for the one-electronndashtwo-center overlap integrals of Slater-type atomic orbitals up to 3d are derived. The final expressions are analytic, simple, and independent of local coordinates. Furthermore, they do not contain the non-closed-form of exponential integrals which were presented in expressions given in earlier work. It is shown that the two-electron--two-center Coulomb integrals, as well as the hybrid integrals, can simply be expressed in terms of these integrals. The numerical instability arising from the situation in which the exponents of the two orbitals are almost equal is discussed, and a solution for this problem based on a Taylor-series expansion of the integral is suggested.

A simple approximation for inverse distance with application to the one-center and two-center Coulomb integrals

A simple approximation for inverse distance with application to the one-center and two-center Coulomb integrals

Influence of the long-range coulomb interaction upon the energy spectrum of infinite polyenes

It is shown that the energies of AMO's (Ek) for an arbitrary homonuclear alternating system can be given with the following expression (identical for AMO with α- and β-spin): $$E_{k\sigma } = \alpha + \tfrac{1}{2}\gamma \pm (\delta ^2 \gamma ^2 + [e_k + \Delta _k (\gamma _{\mu v} )]^2 )^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} {\text{ }}\sigma \in \alpha ,\beta $$ where: ek are the MO energies, obtained by means of the conventional HF method; δ and Δk are correlation corrections depending on the one-center (γ), resp. two-center (γμv) coulomb integrals.

Analytical calculation of two-center coulomb and hybrid integrals with orbitals of the slater type

Analytical calculation of two-center coulomb and hybrid integrals with orbitals of the slater type

Analytical evaluation of three- and four-center electron-repulsion integrals for Slater-type orbitals

A particular method has been introduced for obtaining a spherical harmonic expansion of a Slater-type orbital about a point displaced from the orbital center. Utilizing this method, a general analytical formula has been established for the radial coefficients of the expansion which contains terms only of the product of the overlap integrals and the radial parts of Slater-type orbitals. By the use of the derived expression for a spherical harmonic expansion, three- and four-center electron-repulsion integrals with the arbitrary location of Slater-type orbitals are expressed in terms of overlap integrals and one of the one-center Coulomb, and two-center Coulomb, hybrid and exchange integrals over Slater-type orbitals for the evaluation of which, recently, the analytical formulas have been established by the author. The convergence for the expansion has been tested by calculating concrete cases.

Extremely compact formulas for molecular two-center one-electron integrals and Coulomb integrals over Slater-type atomic orbitals

Extremely compact analytical formulas are derived for molecular two-center one-electron integrals and Coulomb integrals, which occur in linear-combination-of-atomic-orbitals calculations with Slater-type orbitals (STO's). The derivation is based on the connection of the STO's with the reduced Bessel functions (RBF's) and makes use of the convolution theorems of RBF's. The final results are of a surprisingly simple structure and are, therefore, especially useful for practical applications.

Analytical evaluation of one- and two-center Coulomb and two-center hybrid integrals for Slater-type orbitals

General formulas are derived for the two‐center Coulomb and hybrid integrals for real and complex STOs with respect to the common molecular coordinate system. The angular and radial parts of these integrals are expressed through the spherical harmonics and the auxiliary functions ∼(Gq−NN’), respectively.

Analytical calculation of two-center Coulomb and hybrid integrals with orbitals of the Slater type

Analytical calculation of two-center Coulomb and hybrid integrals with orbitals of the Slater type

Analytical calculation of two-center coulomb and hybrid integrals with the correlation factor r 12 k

Analytical calculation of two-center coulomb and hybrid integrals with the correlation factor r 12 k

Unified Treatment of Two-Center, Two-Electron Integrals. I. Electron-Repulsion, Spin-Spin, and Spin-Orbit Integrals Over Slater Functions

Utilizing the Neumann expansions of the electron-repulsion operator, all two-electron two-center spin-spin, spin-orbit, and electron-repulsion, hybrid, Coulomb, and exchange integrals arising in computations of expectation values of the respective operators over wavefunctions written as linear combinations of Slater functions are expressed in terms of a simple formula requiring, effectively, a one-dimensional numerical integration. A general formula for one-center integrals is also derived by utilizing the Laplace expansion of the electron-repulsion operator.

Limited Expansion of Diatomic Overlap (LEDO): A Near-Accurate Approximate Ab Initio LCAO MO Method. I. Theory and Preliminary Investigations

A method is presented for approximating multicenter integrals which arise in LCAO MO calculations utilizing Slater-type orbitals as basis functions. Two-center charge distributions are projected onto one-center functions, allowing all difficult three- and four-center integrals to be expressed as linear combinations of rapidly calculated one- and two-center Coulomb integrals. Test calculations are reported for a wide variety of polyatomic molecules yielding excellent agreement with corresponding ab initio treatments. Integral generation rates are well over an order of magnitude faster than current accurate approaches. The systematic and reliable nature of the approximation is examined and comparison made with the recently reported PDDO (projection of diatomic differential overlap) technique.

Radial Expansion of 3d Orbitals and the Near Degeneracy of the A41g and E4g Excited States of Octahedral Mn(II) Complexes

The relationship between the wavefunctions for mostly 3d molecular orbitals and the energies of the nearly degenerate A41g and E4g excited states of octahedral Mn2+ complexes is investigated from two points of view. First, the expressions relating the energy splitting to the degree of covalency have been extended within the framework of the intermediate neglect of differential overlap (INDO) approximation to include the large but previously neglected two-center Coulomb integrals and all ligand one-center Coulomb and exchange integrals. The results resemble those previously obtained in their inability to account for the spectral assignment of the E4g state below E41g for the MnF64− complex in the salts KMnF3, RbMnF3, and MnF2. Second, the assumption of identical metal-ion repulsion integrals for the eg and t2g molecular orbitals is discarded in a simple analysis of an all-electron molecular SCF calculation for the MnF64− complex. While both the eg and t2g orbitals contain a metal-ion d orbital that is expanded relative to that for the free Mn2+ ion, the slightly greater radial expansion for the t2g orbital is shown to qualitatively account for the observed ordering of the A41g and E4g states.

Unified Treatment of Two-Center Overlap, Coulomb, and Kinetic-Energy Integrals

A single analytical formula is derived which gives the two-center overlap, Coulomb, nuclear-attraction, and kinetic-energy integrals as special cases. There are no difficulties when orbital exponents are nearly equal. Simplification of the derivative operators, which are characteristic of the approach, is discussed in detail. A computational scheme is outlined, and computing times are given.

Triplet-State Zero-Field Splitting Calculations in Aromatic Hydrocarbons and Heterocycles

Zero-field splitting in the lowest triplet states of aromatic hydrocarbons and heterocycles is calculated using several molecular orbital schemes. The most reliable results are obtained using closed-shell SCF molecular orbitals with configuration interaction between all triplet states singly excited relative to the ground state. Slater-type atomic orbitals are used and the two-center Coulomb integrals involving these orbitals are evaluated exactly. Results calculated using best Slater-type orbitals are consistently higher than experimental values, but good agreement with experiment is obtained when two-zeta of four-zeta atomic orbitals are used.

An approximation of the two-center coulomb integrals in the semiempirical LCAO-MO methods

It is shown that the singlet and triplet transition energies of hydrocarbons may be calculated by means of the generalised Mataga-Nishimoto approximation for the two-center Coulomb integrals.