Accurate calculation of Fourier transform of two-center Slater orbital products

A method is presented which leads to accurate Fourier transform values of any 1s?1s Slater-type orbital overlap distribution. The numerical merits are discussed and illustrated by some examples.     

Numerical evaluation of three- and four-center bielectronic integrals using exponential-type atomic orbitals

Three- and four-center Slater orbital bielectronic integrals are evaluated by means of a complete function set. The method provides a series to approximate the bielectronic integrals. Their corresponding partial sums are analyzed in detail for 1s orbitals. The comparison with the Fourier transform–based method brings forth encouraging perspectives for the present approach. © 1995 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.     

Recursion formulae for calculation of overlap integrals

Multi-ζ Slater-type orbitals are frequently used in molecular orbital calculations. Master formulae and numerical tables are available in literature for overlap integrals between s, p, and d atomic orbitals up to principal quantum number (n) = 3 and for some other selected quantum numbers. However, no master formula or numerical table is available for quantum numbers n = 5 and above and involving ? orbitals. In this article recursion formulae have been presented for the calculation of the overlap integral between any two s, p, d, and ? atomic orbitals formed by a linear combination of Slater-type orbitals. These formulae, when expanded, would give rise to all the master formulae reported in the literature as well as formulae hitherto unreported.     

A Closed-Form Relation for Dimension-Dependent Two-Electron Matrix Elements of the Interelectronic Distance

The evaluation of matrix elements of two electron atoms is fundamental for the study of the electronic properties of those systems. We add to this knowledge by presenting an explicit expression for the matrix elements of the inverse of the interelectronic distance of two-electron atoms in any spatial dimension D. The basis functions used are the D-dependent hydrogenic wavefunctions {1s ²,2p ²,3d ²,4f ²,5g ²,...,21y ²,...}, extending and including, in this way, the results of the previous basis set {1s ²,2p ²,3d ²,4f ²}. The methodology used does not employ Fourier integral transforms as in previous works but hypergeometric transformation formulas.     

An alternative computational strategy for direct space hartree–fock calculations on extended model chains

Based on previous analyses of slowly convergent exchange lattice sums entering the configuration space restricted Hartree–Fock–Roothaan scheme for chain systems, an alternative computational strategy is developed. Within the present formalism, the traditionally used finite Fourier transform of k-dependent LCAO density matrices are by-passed and an advantageous computational organization is obtained.     

A new algorithm to evaluate bielectronic integrals with 1s Slater‐type orbitals obtained by the integral transforms

This article presents a variation of the integral transform method to evaluate multicenter bielectronic integrals (12|34), with 1s Slater‐type orbitals. It is proved that it is possible to define, out of the expression of (12|34) given by the integral transform method, a function F(q) that has the property of having a unique Q, such that F(Q) = (12|34). Therefore, F(q) may be used to calculate (12|34). It is shown that the evaluation of F(Q) turns out to be simpler than the three‐dimensional integral involved in the calculation of (12|34), and an algorithm is presented to calculate Q. The results show that relative errors on the order of 10^?3 or lower are obtained very efficiently. In addition, it is shown that the proposed algorithm is very stable. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Applications of fourier transforms in molecular orbital theory. Calculation of optical properties and tables of two-center one-electron integrals

Fourier transform methods initiated by Geller and Harris are applied to the calculation of optical properties of molecules. Tables of one-electron two-center integrals needed for the accurate computation of molecular absorption and optical activity are calculated by the Fourier transform method. A general theorem is derived which allows the angular part of the integrals to be treated by means of projection operators. The radial parts of the integrals are treated by the methods of Harris. The results are obtained in a simple closed form which avoids the usual transformation to local coordinates. The two-center integrals evaluated include matrix elements of the momentum operator, the dipole moment operator, the tensor operator , the quadrupole moment operator, and the angular momentum operator. These are evaluated between 1s, 2s, and 2p Slater-type atomic orbitals located on different atoms. The results are expressed as functions of the Slater exponents and of the relative coordinates of the two atoms.     

Ewald summation of multipolar interactions at an arbitrary order on a two-dimensional lattice

There exist problems in condensed-matter theory that require evaluating infinite Bloch sums of multipolar potential r^?l?1Y_l,m(θ,φ) on a periodic lattice. For an arbitrary multipolar order l, tractable formulas are given for summing such interactions on a two-dimensional Bravais lattice and evaluating their Bloch sums at a point outside as well as inside the plane of the lattice. The approach used is the Ewald method, which consists of separating the original series in rapidly converging sums in reciprocal and real spaces. Computational aspects of the present formulation are briefly reviewed. © 1993 John Wiley & Sons, Inc.     

Transferable integrals in a deformation density approach to crystal orbital calculations. IV. Evaluation of angular integrals by a vector-pairing method

A general method for performing angular integrations is presented. The method depends on the fact that the integral must be invariant under rotations of the coordinate system, and it also makes use of combinatorial analysis. In most cases the method presented is computationally much faster than alternative methods of angular integration using Condon–Shortley coefficients. Applications to charge density analysis and Fourier transforms are discussed, and a general formula for the action of angular momentum projection operators on functions of the Cartesian coordinates is derived. A general angular integration formula for an m-dimensional space is also given.     

A systematic preparation of new contracted Gaussian-type orbitals. IX [54/5], [64/5], [64/6], [74/6], [74/7] and MAXI-1–MAXI-5 from Li to Ne

The new contracted Gaussian-type orbitals (CGTO s) for molecular calculations have been developed from Li to Ne. The CGTO s are minimal type, i.e. composed of two s-type CGTO s, s₁, s₂, and one p-type CGTO , p₁. They are new family of CGTO s given by Tatewaki and Huzinaga, and others. In the previous works three primitive GTO s are used for s₂, which is the main part of the 2s orbital, whereas four primitive GTO s are employed in the present work. The sets generated are [54/5], [64/5], [64/6], [74/6], and [74/7]. In almost all the cases the errors in the 2s and 2p orbital energies are smaller than those of DZ . The resulting 2s orbitals are close to the orbitals of the uncontracted GTO sets, (13/n) and (14/n) of Duijneveldt. It is found that the 2s and 2p orbitals given by [64/6], [74/6], and [74/7] are satisfactorily near to those of Hartree–Fock. The basis sets [54/5], [64/6], and [74/7] are applied to the N₂ molecule in the split valence forms of [5211/311], [6211/3111], and [7211/4111]. Adding the d-type polarization functions from one through three, the quality of the basis sets has been examined. All of the three sets show good behavior and the sets augmented with three d-type polarization functions give almost entirely the same results as the very extended basis set.     

Solution of the Hartree–Fock problem by expansion onto nested bases

A method for solving the Hartree–Fock problem in a finite basis set is derived, which permits each orbital to be expanded in a different basis. If the basis set for each orbital ?_i contains the basis functions for the preceding orbitals, ?_i?1, ?_i?2,… ?₁, then the ?_i form an orthonormal set. One advantage over the standard Hartree–Fock method is that a different long range behavior for each orbital, as for example is required in the Hartree–Fock-Slater method, can be forced. A calculation on the ground state of beryllium is performed using the nested procedure. Very little energy is lost because of nesting, and the node in the 1s orbital disappears.     

Calculation of 4-Center Coulomb Repulsion Matrix Elements in a Basis of Exponential Type Spherical AO Using 9-Dimensional Polyspherical Harmonics

A method for calculating 4-center Coulomb repulsion integrals in a basis of exponential type AO with regular sectorial harmonics as angular terms is proposed. The initial integrals are represented as a partial differentiation operator with respect to the Cartesian coordinates of the centers of AO, acting on the scalar function which is a 4-center integral of s functions. Differentiation is performed by calculating the Fourier transform of this scalar function in 9-dimensional Euclidean space with the help of the sectorial harmonic argument summing theorem. Thus compact representation of quantum-chemical multicenter integrals is obtained in a basis of exponential type functions with arbitrary angular parts.     

On the sudden approximation of cross: computational tests and factorization of cross sections and related scattering phenomena

A sudden approximation recently derived by Cross using a semiclassical treatment of the orbital motion is recast into a form which permits factorization of differential and integral degeneracy averaged cross sections, opacities as a function of final angular momentum quantum number, the scattering amplitude, and the phenomenological cross section which describes spectral line broadening. Calculations are done using an average of initial and final orbital angular momentum quantum numbers for the partial wave parameter for ArN₂, ArTIF, H⁺H₂ and Li⁺H₂. The results indicate that the method is a good approximation for integral cross sections and opacities when the energy sudden approximation is valid and when the coupling of the orbital motion is important.     

Unrestricted CNDO-MO calculations. II. MnO 4 2− and CrO 4 3−

The electronic structures of the tetrahedral molecule ions MnO ₄ ^2– and CrO ₄ ^3– have been investigated within an unrestricted CNDO-MO approximation [Theoret. Chim. Acta (Berl.)20, 317 (1971)]. Calculations assuming the unpaired electron occupies the 3a ₁, 2e, and 4t2 molecular orbitals indicate that the 3a ₁ and2e orbitals have similar orbital energies and that the 4t ₂ orbital is at a higher energy. The experimentally indicated2e orbital for the unpaired electron is obtained with expanded O^1– type atomic orbitals for oxygen and valence metal orbitals of the expanded 3d and plus one ion 4p types. The metal 4s orbitals must be held to the neutral atom type. The optimum valence orbitals above with a slightly contracted 4s type metal orbitals yield the minimum total energy and places the unpaired electron in the 3a ₁ orbital. Since the contracted 4s metal orbital produces results that are not in agreement with experimental data, the method used apparently does not adequately take into account the increased electron-electron repulsions that contracted 4s orbitals produce.     

Picture change error correction in the radial distributions of canonical orbital densities and total electron density of radon atom: the effect of the size of nucleus and the basis set limit

The 2nd order Douglas-Kroll-Hess (DKH2) and the Infinite Order Two Component (IOTC) radial distributions of electron density of canonical Hartree-Fock (HF) orbitals of radon atom are presented. Furthermore, the total electron density is revisited. The picture change error (PCE) correction is investigated by analytical means. The point charge model of nucleus and the Gaussian nucleus model are employed. The basis set is extrapolated by means of including tight s and also p Gaussians within the original triple zeta basis set. It is found that the DKH1 PCE corrected DKH2 total electron and s orbital contact densities are negative for the point charge model of nucleus if tight enough s Gaussians are included in the basis set. It is shown that this failure is caused due to the missing terms of the second order Douglas-Kroll transformation for the DKH2 electron density. PCE is found the most striking in the DKH2/IOTC electron density of s orbitals close to the nucleus. The radial distributions of the 2-component p _1/2 orbital densities are considerably affected by PCE at the nucleus as well. Furthermore, the PCE corrected DKH2/IOTC scalar p orbital densities have a non-zero value of electron density at nucleus and can be considered as an spin-orbit (SO) average of the p _1/2 and p _3/2 orbitals. The d and f orbitals are affected by PCE in the vicinity of the nucleus only little. The PCE corrected DKH2 and IOTC radial distributions of orbital densities are nodeless, which is completely in agreement with the radial distribution of the analytic or numeric DCH orbital densities.     

Refractive index and derived molecular properties of some gases determined by dispersive fourier transform spectroscopy (DFTS) in the visible wave number range

Dispersive Fourier transform spectroscopy (DFTS) is used to determine the dispersion of the real refractive indexn() of the gases CH₄, C₂H₄, C₂H₆, C₃H₈ and cyclo-C₃H₆ in the visible wave number range. Some molecular properties, e.g. oscillator strength sums, are derived from these measurements and compared with available data of literature.     

Applications of fourier transforms in molecular orbital theory

Fourier transform methods introduced by Harris are applied to the evaluation of Frenkel exciton lattice sums. The slowly-convergent direct lattice sum is converted into a rapidly-convergent reciprocal lattice sum which includes all orders in the multipole expansion. A simple example is discussed, and the calculated exciton energy as a function of wave number is compared with the results of the Ewald method.     

Self consistent modified extended Hückel (SC-MEH) calculations on heavy metal systems. I. Platinum(II) tetragonal planar complexes with and without relativistic effects

The Self Consistent Modified Extended Hückel molecular orbital method had been applied to several square planar complexes of platinum (II). Calculations including both the limited 5d, 6s, 6p and extended 5s, 5p, 5d, 6s, 6p starting bases for platinum were made. It is shown that in PtCl ₄ ^2– both the nuclear quadrupole moment and minimum total energy vs. bond distance are calculated to be in good agreement with experiment, only with the extended platinum AO basis.Specific inclusion of relativistic parameters via a pseudo-relativistic approximation are shown to have a significant effect on the energy molecular energy levels, however no meaningful rationalization can be made without the simultaneous inclusion of ligand field parameters as well.Supported in part by a grant made available through the Cancer Association of Greater New Orleans. Use of the facilities of the Computer Research Center of the University of New Orleans is gratefully acknowledged.     

Effect of the exchange integral K2s2p on the 2p-orbitals of the 1P and 3P states of the beryllium isoelectronic sequence arising from the configuration (1s)2(2s)(2p)

The ¹P and ³P states arising from the configuration (1s)²(2s)(2p) of the Be isoelectronic sequence are investigated. In the single configuration approximation, the energies of the two states are expressed as E⁰ + K_2s2p and E⁰ - K_2s2p, respectively. K_2s2p is the exchange integral between the 2s and 2p electrons and E⁰ is the energy of a model in which K_2s2p is deleted. First we calculate the 2s- and 2p-orbitals in this model. Second, by taking account of K_2s2p in this model, effects of this term on the 2p-orbitals in the ^1,3P states are investigated. In this manner, an explanation is given for the following facts which are obtained from a minimal Slater-type orbital set; (1) for Be and B⁺, the 2p-orbital of the ¹P state is broader than that of the ³P state; (2) for C²⁺, the extension of the 2p-orbital in the two states is almost the same; (3) for O⁴⁺ and Ne⁶⁺, in contrast to Be and B⁺, the 2p-orbital of the ¹P state is tighter than that of the ³P state.