One-center Integrals Research Articles

Recurrence relations for two-center harmonic oscillator integrals and their solutions using generating functions

Using the algebraic second quantization formalism, the general recurrence relations for some two-center harmonic oscillator integrals are obtained. The various known special cases for one-center integrals are evident in the formalism. Generating functions for these integrals are obtained and most of them are solved to find explicit expressions for polynomial, exponential, and Gaussian operator integrals. For these cases, all the generating integrals needed for utilizing the recurrence relations are also calculated in their simplest forms.

An intermediate neglect of differential overlap (INDO) technique for lanthanide complexes: studies on lanthanide halides

An intermediate neglect of differential overlap method of use for examining the electronic structure of lanthanide complexes is developed. It is characterized by a basis set obtained from relativistic Dirac-Fock atomic calculations, the inclusion of all one-center two-electron integrals, and a parameter set based on molecular geometry. Lanthanide halides MX2, MX3 and MX4 are studied here, as well as initial results for the twelve coordinate Ce(NO3)6 −2 ion. Geometries obtained are in excellent agreement with experimental values when available. Many MX3 complexes are found to be pyramidal, and EuCl2 and YbCl2 are calculated to be bent even at the SCF level. Models invoking London type forces are therefore not required. Ionization potentials are calculated for the trihalides (δSCF) and are in reasonable agreement with experiment. Contrary to conclusion of others, f-orbital participation, although small, is required — at least in this model — to obtain the spread of metal to halide bond distance observed in these complexes. However f-orbital participation does not seem to be significant even in the twelve coordinate Ce(NO3)6 −2 complex: rather the large coordination number seems to be a consequence of the relatively large size of the lanthanide ion.

Recurrence relations for two-center harmonic oscillator integrals

General recurrence relations for the calculation of two-center harmonic oscillator (HO) integrals are obtained by means of a hypervirial-like-theorem commutator algebra procedure, combined with a second quantization formalism. The method is based on a linear transformation between the creation and annihilation operators of two displaced HO with different frequencies. Ansbacher’s recurrence relations for the calculation of Franck–Condon factors are obtained straightforwardly from the proposed general recurrence relations. The application to polynomial, exponential, and Gaussian operator integrals is shown and new recurrence relations are given. In all cases, the proposed recurrence relations reduce, as particular cases, to the corresponding formulas for the calculation of one-center integrals.

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

The structure of alkoxyfluoroalkyl radicals as studied by M.O. and computer simulation of polycrystalline spectra

A novel approach to the structure of Alkoxyfluoroalkyl radicals (1,2), based on the analysis of the anisotropy of polycrystalline ESR spectra and on M.O. calculations to the INDO level of approximation, is presented. Polymeric primary and secondary perfluoroalkoxyalkyl radicals of type ROCF 2· and ROCF(CF 3) were prepared by UV photolysis at 77 K of peroxidic perfluoropolyethers (2) and their ESR spectra were analyzed by computer simulation with the spin Hamiltonian (3): ▪ A satisfactory computer simulation of ROCF 2· polycrystalline spectrum was obtained by assuming a pyramidal structure with a degree of bending of ∼19° and with an axially symmetric h.f. tensor with principal values: ▪ The ESR spectrum of the ROĊF(CF 3)- radicals was accounted for in terms of a pyramidal radical centre with a degree of bending of about 11° and a locked orientation of the CF 3 group with a Fluorine in the eclipsed position in respect to the half filled orbital. The pertinent h.f. tensors are: ▪ INDO calculations, performed to match the experimental anisotropic components of the h.f. tensor, by using the formula (4) ▪ support the above conclusions and further show that a major contribute (>90%) to the dipolar tensor t ij arises from the one-centre integrals of the F-2p orbitals, followed by (<10%) the two-centre integrals related to the u.s. population at the hybrid orbital of the adjacent carbon atom. The coefficients C(2p)=0.33 and C(2s)=0.086 for the Fluorine orbitals in the LCAO of the u. electron M.O. have been obtained, for radicals ROCF 2·.

Evaluation of one-center coulomb integrals involving s-functions by a programmable calculator

Abstract A calculator program GAMAA is described for evaluation of the one-center Coulomb integrals involving the Slater-type s -functions. The program is partially useful in the complete neglect of differential overlap method to make an estimate of the orbital exponent and to evaluate the atomic energy which is used to calculate the molecular binding energy. The algorithm is presented for a Texas Instruments TI-59 programmable calculator.

Numerical properties of the convolution theorems ofBfunctions

For the computation of convolution integrals of $B$ functions [Eq. (2.14) of E. Filter and E. O. Steinborn, Phys. Rev. A 18, 1 (1978)] which are a special class of exponential-type functions, different analytic representations are available. We have investigated the numerical properties of these convolution formulas and found out that results of sufficient accuracy can be obtained for molecular calculations. We have also investigated the numerical properties of the corresponding one-center integrals. Some new mathematical properties of $B$ functions that are essential for the understanding of the numerical properties of the convolution formulas are derived.

Some CNDO/S parameterised calculations of carbon and nitrogen shieldings including two-centre integrals

Some Sum-Over-States shielding calculations for a number of carbon and nitrogen environments are reported. The CNDO/S parameterised calculations in which only one-centre integrals are included yield rather poor agreement with the experimental results. The inclusion of paramagnetic shielding contributions arising from two-centre integrals produces a significant improvement in the relationship between the calculated and observed nuclear shielding data.

Localized molecular orbital calculations of vibrational circular dichroism. II. CNDO formulation and results for the hydrogen stretching vibrations of (+)-(3R)-methylcyclohexanone

The results of localized molecular orbital (LMO) calculations of vibrational circular dichroism (VCD) for the 12 carbon–hydrogen stretching vibrations in (+)-(3R)-methylcyclohexanone are reported. The VCD is evaluated from the motion of the nuclei and the centroids of LMO’s which are obtained by localization of canonical CNDO molecular orbitals. The detailed expressions used in our CNDO–LMO–VCD procedure are given and the approximation of retaining only the one-center integrals is discussed. Comparison of the CNDO–LMO–VCD calculations to experiment shows good agreement, thus indicating a potential utility of the LMO–VCD calculational method. The results are analyzed in terms of nuclear and orbital vibrational motion and figures depicting these displacements are provided for each of the 12 normal modes.

A CNDO/INDO molecular orbital formalism for the elements H to Br. theory

A CNDO and INDO formalism is presented that can be used for any atom combination up to bromine under inclusion of the first transition metal series. The semiempirical parameters were chosen to reproduce results ofab initio calculations on metalorganic compounds. The calculational results are invariant to rotations of the coordinate system but not to a general transformation into other basis functions. The one-center Coulomb-expressions were selected in order to include intraatomic correlation contributions. Within the CNDO model this could be achieved by the scaled monopole termF0, while in the INDO framework the one-center Coulomb integrals are given as a sum of the monopole-contributionF0 and higher multipole contributions expressed as a linear combination of Slater-Condon parameters. The invariance problem in the case of local rotations within the INDO approximation was solved by considering the combination of one-center Coulomb and exchange integrals as a function ofl but independent ofm. The two-center electron-electron interaction terms were calculated via the Dewar-Sabelli, Ohno-Klopman relation. Penetration effects were treated according to Fischer and Kollmar. For the resonance integralHμvAB parameters are used which carry information related to the directed nature of the chemical bond by using optimized Klondyke functions. The core-core repulsion is constructed as a superposition of a soft potential function, describing polarization effects of the atomic cores, and a hard repulsion function, avoiding the collapse of the atomic cores with decreasing distance.

One-center integrals of the exact effective valence shell Hamiltonian. Empirical analysis for atoms

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTOne-center integrals of the exact effective valence shell Hamiltonian. Empirical analysis for atomsDanny L. Yeager, Maurice G. Sheppard, and Karl F. FreedCite this: J. Am. Chem. Soc. 1980, 102, 4, 1270–1277Publication Date (Print):February 1, 1980Publication History Published online1 May 2002Published inissue 1 February 1980https://doi.org/10.1021/ja00524a009RIGHTS & PERMISSIONSArticle Views34Altmetric-Citations17LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (859 KB) Get e-Alerts Get e-Alerts

Program realizing the INDO method in a basis of up to 232 orbitals with the inclusion of d orbitals

Semiempirical methods of quantum chemistry provide a powerful instrument for studying the structure and properties of complex molecules and molecular ions. Of these methods, the most widely used is the CNDO method, one of the simplest and most reliable [i]. At the same time, the use, in this method, of the averaged values of the one-center Coulomb integrals, together with the underestimation of the role of exchange correlation of the electrons, leads to the energetic equivalence of atomic states of the same configuration with different multiplicities. The underestimation of the exchange correlation is revealed in the incorrect description, in the CNDO method, of the spin density distribution when the unrestricted Hartree--Fock method is used. The INDO method, for which the complexity of the calculations is comparable with that of the CNDO method, does not have these unsatisfactory features [2]. A number of workers in the USSR [3, 4] have now compiled programs which use the CNDO method for elements of the second period in a comparatively small basis.

Analytical calculation of one-center two-electron integrals with orbitals of the slater type

In quantum-mechanical calculations for atoms and molecules, in addition to multicenter integrals, it becomes necessary to calculate one-center two-electron integrals with atomic orbitals of the Slater type, for which no general analytical equations have been published. The present paper gives simple analytical expressions for one-center potential functions and one-center two-electron integrals with orbitals of the Slater type. The one-center two-electron integrals being considered have the form:

Magnetic circular dichroism of molecules in dense media: Spin-forbidden transitions of carbonyl molecules

The purpose of this paper has been to calculate the magnetic rotational strength of the spin-forbidden (1)A(1) --> (3)A(2) transition of the carbonyl group. For this purpose, simple molecular orbitals have been used and only one-center integrals have been considered. The calculated results have been compared with the experimental measurements.

An Investigation of σ and π Contributions to Long-range 1H–1H Coupling Constants in Planar Aromatic Derivatives Using CNDO and INDO Molecular Orbital Calculations

Molecular orbital calculations of coupling constants based on finite perturbation theory in the CNDO/2 and INDO approximations have been carried out for styrene, benzaldehyde, and naphthalene. INDO calculations have also been carried out in which one-center exchange integrals involved in σ–π interaction are omitted (INDO(σ) calculations). Both CNDO and INDO(σ) constants give reasonable estimates of σ contributions to long-range 1H–1H couplings, with the former giving better results when strongly stereospecific σ contributions are present. INDO calculations also give reasonable, but generally overestimated, π contributions to these couplings.

Ab initio calculations on valence-shell molecular orbitals

The following method is proposed for calculating valence electron wave functions and energies in the SCFMO scheme without including inner shells in the basis. (1) The valence orbital bases are Slater functions orthogonalized by the Schmidt procedure to core orbitals. (2) All one-centre integrals are calculated exactly in this basis. The nuclear attraction integrals are multiplied by an effective nuclear charge Zeff which is determined from an atomic calculation to give atomic orbital energies identical to those from an all-electron calculation. (3) All multi-centre two-electron integrals are taken to be equal to the values appropriate to the non-orthogonalized valence basis. (4) Two and three-centre one-electron integrals are calculated from the orthogonalized basis, with the effective nuclear charges equal to the total nuclear charge less the number of core electrons in that atom.This procedure can fit readily into standard SCFMO programs as the core potential is Coulombic for all two and three centre integrals.Calculations on H2O, H2S, CH4 and SiH4 show agreement to within 0.01 a.u. (0.5 eV) of the all-electron calculation for occupied molecular orbital energies. The equilibrium bond angle and bond length for H2O obtained by this model are also in good agreement with those obtained in the all-electron calculation. Calculations on N2, at its equilibrium bond length and greater, show good agreement with the all-electron results but at shorter bond lengths an incorrect ordering of the molecular orbitals is obtained and the total energy is poor.

CNDO investigations of the electronics structures of TiCl 4 and TiCl 6 2−

The electronic structures of TiCl 4 and TiCl 6 2− are investigated on the basis of CNDO procedure. Average energies of electronic configurations of atoms are used to evaluate local core hamiltonian U μμ and the Slater exponents which reproduce one-center repulsion integrals γ μν assuming that they arise from spherically symmetric charge distributions of the valence atomic orbitals. Quite good agreement has been obtained in the interpretation of the experimental data as the photoelectron spectrum of TiCl 4 and the optical spectra of the examined systems.

Description of the radiative properties of unsaturated molecules containing a triple bond based on a quasi π-nodel

In this paper we provide a model for the theoretical calculation of radiative triplet properties of aromatic molecules containing a triple bond. Such molecules have special low-lying electronic states with the symmetry properties of a σπ* -state. These states axe caused by electron promotion from an “in-plane π-system” (quasi π-system), which in highly localized at the triple bond, to the aromatic π-system and vice versa. It is shown that these states lead to one-center spin-orbit integrals and therefore are highly active in radiative triplet deny. Theoretical and spectroscopic investigations of ethynylbenzene show that the radiative properties of the lowest triplet state, such as phosphorescence yield, emission spectrum and degree of polarization can be explained not only qualitatively but also quantitatively by restriction to these states (quasi π-model). The theoretical evaluation of the radiative rate was done with respect to individual promoting modes. It is shown that the first order rate (at the equilibrium position of the molecule), though allowed by symmetry, has nearly no influence on the radiative lifetime. As a consequence the first transition in the phosphorescence emission cannot be assigned as the 0-0 transition.

The intensities of low-energy electronic absorption transitions of some carbonyls and thiocarbonyls

The intensities of low-energy electronic transitions for some carbonyls and thiocarbonyls have been calculated from CNDO wavefunctions. Quite good agreement with experimental results has been obtained, where the latter are available. A satisfactory approximation for calculating intensities employs only one-center integrals. From the calculated trends in oscillator strengths, the absorption of thiophosgene at 4.46 eV can be identified as the π→π* 1A 1←X̃ 1A 1 system. Another, very weak, system of thiophosgene at ≈ 3.9 eV is tentatively assigned to an n→π* 1A 1 ← X̃ 1 A 1 transition, with the n orbital localized on the chlorine atoms.

The inclusion of a correlation function in the calculation of the ZFS of benzene

We calculated the one-center atomic integral of the spin—spin interaction with respect to Slater orbitals where we have included a two-electron correlation function that we used previously to calculate the hyperfine splitting of the 2 3P state of He. The ZFS of the benzene molecule was calculated from this value of the one-center integral and by using the Mulliken approximation combined with a previously introduced approximation for extimating the other integrals. The calculation was performed for two different non-hexagonal configurations I and II, the results are D 1 = 0.132 cm −1 and D II = 0.143 cm −1. The experimental value is D = 0.158 cm −1.