A CNDO/INDO crystal orbital model for transition metal polymers of the 3d series—basis equations

The crystal orbital formalism in the tight-binding approximation is combined with a recently developed CNDO/INDO model for transition metal species of the 3d series in order to allow band structure calculations on the Hartree-Fock (HF) SCF level for one-dimensional (1D) chains with organometallic unit cells. The band structure approach based on the CNDO and INDO approximation can be used for any atom combination up to bromine under the inclusion of the 3d series. The matrix elements for the tight-binding Hamiltonian are derived for an improved CNDO and INDO framework. The total energy of the 1D chain is partitioned into one-center contributions and into two-center increments of the intracell and intercell type. Semiempirical band structure calculations on simple model systems are compared with available ab initio data of high quality.     

Inclusion of relativistic effects into ZDO methods. III. A. Quasi-relativistic INDO/1 version

The quasi-relativistic CNDO /1 method for molecular orbital calculations has been extended to the INDO /1 version where all monocentric repulsion integrals are taken into account. This version has been applied to calculate molecular geometries of MX_n-type molecules, with the central atom belonging to from the second to seventh period. Properties of hypothetical superheavy elements (Z = 110–117) have been predicted. Transition metal complexes of the MCl-type (M = Ni, Pd, Pt, and ePt) have been investigated from the point of view of spin-state and configuration stability.     

Modified all-valence INDO/spd method for ground and excited state properties of isolated molecules and molecular complexes

A new semiempirical all-valence method, GRINDOL (Ghost and Rydberg INDO ), based on the INDO approximation, is described. Linderberg–Seamans relation (extended to the d and Rydberg orbitals) for the resonance integrals and a new semitheoretical expression for the core-core repulsion term and energy correction including basis-set superposition error (intermolecular as well as intramolecular) has been applied. The proposed method enables calculation of ground and excited state properties. The ground state results (including intermolecular interactions) as well as the spectral properties are in reasonable agreement with relevant experimental (or ab initio) studies for isolated molecules, molecular complexes, and transition metal compounds. The method contains only one adjustable parameter, all two-center integrals and terms are only basis-set dependent. The one-center integrals are evaluated from the respective atomic terms.     

Empirical penetration functions and two-center electron repulsion integrals for semiempirical calculations of electronic structure

Explicit functional forms for both the two-electron Coulomb integral, (aa∣bb), and the one-center core–orbital integrals, Z (aa∣Z_A), are derived which permit the penetration integrals to be fully derived and calculated. With these forms the ³Σ of the H₂ molecule is unstable. These forms are generalized so that they are suitable for optimizing semiempirical predictions of experimental one-electron properties.     

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.     

Determination of one-centre core integrals from the average energies of atomic configurations

One-center core integrals for valence orbitals are determined from the experimental average energies of neutral atomic configurations from Li through Zn. These values are compared with those estimated from CNDO /1, “INDO /1”, CNDO /2, “INDO /2” and with theoretical values calculated from a pseudo-potential method. The agreement is good between values obtained from neutral atoms and from the psuedo-potential calculation except for the 3d orbitals of the transition elements where the theoretically calculated integrals over single ξ functions are not realistic. These two methods reproduce both term and average configuration energies for the first two rows of atoms; the semiempirical method reliably reproduces them for the third row. The CNDO /1 and INDO /1 methods underestimate atomic energies, while the CNDO /2 and INDO /2 procedures fail rather poorly. The propriety of using core integrals estimated semiempirically in molecular orbital calculations is discussed.     

CNDO-MO calculation of NnFm molecules

The s-p separation model CNDO -MO method has been introduced and developed for both the open- and closed-shell systems since 1975. This method has two chief advantages over most of the other CNDO methods. (1) The ns and np bases of the same atom may be considered independent to infer different bonding behavior. (2) Pariser–Parr and Nishimoto–Mataga approximations are applied to the Coulombic repulsion integrals, which not only simplifies the calculation and saves computer time but also gives reliable computational results. A series of nitrogen fluoride molecules such as NF, NF₂, NF₃, cis-N₂F₂, trans-N₂F₂, and N₂F₂ have been selected for this MO calculation. In each case, both ionization potential and dipole moment were calculated. The results are closer to the observed values than those reported in other works.     

Simple derivation of the potential energy gradient for an arbitrary electronic wave function

The magnitude of reorganization energies in the photoelectron (PE ) spectra of various transition metal compounds with Mn, Fe, and Ni as 3d center is studied by means of a variable INDO Hamiltonian. The Koopmans defects are analyzed as a function of the one-electron resonance integral β and as function of the one- and two-center electron–electron interaction integrals. β has the property of an inverse coupling constant; reorganization effects are enlarged with reduced β values. In the limit of very small resonance integrals a reduction of the calculated Koopmans defects due to modified localization properties of the orbital wave function is encountered. The two-center electron-electron interaction integrals γ have been calculated via an exponential formula with a variable range parameter. In the limit of long-range potentials with flattened γ; gradients a significant reduction of relaxation and correlation is diagnozed; large defects are predicted in the short-range limit with steep gradients in the repulsion potential. The one-center Coulomb and exchange integrals (γ, K) have been modified by a multiplicative factor. With enlarged one-center integrals enhanced Koopmans defects are encountered. The reorganization energies are determined by means of a Green's function approach with a renormalized approximation for the self-energy part.     

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.     

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.     

Some remarks on the semiempirical one-center electron repulsion integrals

The identification of the stage of ionization for various kinds of one-center electron repulsion integrals, occurring when nonbonding or lone-pair electrons are considered explicitly as well as π-electrons, is discussed for conjugated organic molecules containing heteroatoms N. It is concluded that the value for the negative ions should be used for (π_Cπ_C | π_Cπ_C) in all the states but for (π_Nπ_N | π_Nπ_N) only in the π-π states. In the n-π states, the appropriate value of (π_Nπ_N | π_Nπ_N) is that of the neutral atom if the molecule contains only one N atom. If more than one N atom is involved in the molecule, some weighted mean of the values for the negative ion and for the neutral atom should be used. The value for the neutral atom is most adequate for one-center repulsion integrals other than the (ππ | ππ) type in both the π-π and the n-π states. The method of determining these integrals is also discussed. It is concluded that they are to be determined from the consideration of appropriate electron-transfer reactions except for exchange integrals. The exchange integrals are shown to have to be determined from the Slater–Condon parameters derived from the analysis of the experimental atomic energy levels. Illustrative calculations are given for the lower singlet levels of the formaldehyde, pyrazine, pyridine, and the p-benzoquinone molecule. It is found that the calculated energies of the n-π transitions become much too low unless the (ππ | ππ) values of the heteroatoms in the molecule are chosen differently in the n-π states and in the π-π states as pointed out theoretically in this article.     

On the molecular electrostatic potentials obtained with CNDO and INDO wave functions

Molecular electrostatic potentials computed with CNDO/2 and INDO wave functions are shown to present systematic differences with respect to ab initio potentials in the case of out-of-plane potentials and in-plane vicinal hetero atoms in planar hetero molecules.     

An expansion for four-center integrals over Slater-type orbitals

A new expression is given for the electron repulsion integral over Slater-type orbitals on four different centers. It is based on the asymptotic expansion derived from the bipolar expansion of a previous paper. The expression has the form where q_p = {n_p, l_p, m_p}. Both F and σ are closed expressions. The quantity F is a combination of incomplete gamma functions, Laguerre polynomials and spherical harmonics. It depends upon the relative coordinates of a point P on the AB axis and a point Q on the CD axis. The functions σ_nlm(A, B) depend on the charge distribution (χ_Aχ_B); they have the character of overlap integrals and are of the form      

Theoretical analysis of magnetic and electric hyperfine interactions on 1H and 2H nuclei of bis(acetylacetonato) copper (II)

The INDO method has been used in calculating the tensors of hyperfine structure and the electric field gradient on the ¹H and ²H nuclei in Cu(acac)₂. The procedure used in calculating the magnetic resonance parameters (MRPs) is based on an accounting for one-center and two-center Coulomb integrals of dipole-dipole interaction. The influence of out-of-plane step distortion of the complex on the MRPs is examined critically.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 24, No. 2, pp. 149–155, March–April, 1988.The authors wish to express their appreciation to P. V. Schastnev and S. A. Mustafaev for valuable discussion and for assistance in the work.     

Investigation of the charge state of heteroatoms in organic compounds of third-row elements by x-ray fluorescence spectroscopy 10. Correlations between values of the shifts of the kα lines of sulfur,phosphorus, and chlorine and the values of the effective charges of these atoms determined from semiempirical quantum-chemical calculations

A set of correlation equations which relate the experimental values of the shifts of the E K lines (E=S, P, Cl) for various classes of E-containing compounds to the values of the effective charges of the atoms of E determined from quantum-chemical calculations by the CNDO/S, CNDO/2, INDO, MINDO/3, and MNDO methods in the minimal basis and the CNDO/S method in an expanded basis has been obtained. It has been established that the INDO method in the minimal basis is best for calibrating the values of (E K) on the basis of the values of the effective charges of the atoms, while the CNDO/2 method and the INDO in the expanded basis are practically unsuitable for determining the values of the effective charges of S and P atoms.For previous communication, see [1].Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 2, pp. 403–410, February, 1991.     

Computation of electron repulsion integrals using the rys quadrature method

Following an earlier proposal to evaluate electron repulsion integrals over Gaussian basis functions by a numerical quadrature based on a set of orthogonal polynomials (Rys polynomials), a computational procedure is outlined for efficient evaluation of the two-dimensional integrals I_x, I_y, and I_z. Compact recurrence formulas for the integrals make the method particularly fitted to handle high-angular-momentum basis functions. The technique has been implemented in the HONDO molecular orbital program.     

Inclusion of 3d orbitals in calculations involving second row atoms

Valence-state ionisation potentials, I , and non-empirical one-centre electron repulsion integrals are evaluated for the separate 3s, 3p and 3d orbitals of Na, Mg, Al, Si, P, S, and Cl. These parameters are for use in the application of the CNDO SCMO method to molecules containing second row atoms.

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Zusammenfassung Die Ionisierungspotentiale des Valenzzustands, I und die nicht-empirischen Ein-Zentren-Elektronen-Absto\ungsintegrale werden für die 3s-, 3p- und 3d-Orbitale von Na, Mg, Al, Si, P, S und Cl angegeben. Diese Parameter werden bei der Anwendung der CNDO-SCMO-Methode auf Moleküle, die Atome der zweiten Reihe enthalten, benötigt.

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Résumé On détermine les potentiels d'ionisation de l'état de valence I et les intégrales de répulsion monocentriques pour les orbitales 3s, 3p, 3d de Na, Mg, Al, Si, P, S et Cl. Ces paramètres sont utilisés dans l'application de la méthode CNDO SCMO à des molécules contenant des atomes de la seconde colonne.

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We thank the S.R.C. for a maintenance grant (to K.A.L.).     

The electronic structure,spectra, and reactivity of nitrophenols

The all-valence-electron CNDO /2 calculations were performed for the three isomeric nitrophenols. Using the newly derived σ-core charges and subsequently revising the valence-state ionization potentials and one-center two-electron repulsion integrals, Pariser–Parr–Pople (PPP ) CI calculations were performed on the title compounds following the Nishimoto–Forster scheme. A better agreement between theory and experiment has been observed in spectral assignments compared to the conventional PPP approach. Information from the CNDO /2 calculations was used to obtain useful electronic structural parameters and to get a quantitative insight into the chemical reactivity of these molecules. All the results were compared with the basic compounds, phenol and nitrobenzene. The electronic spectra of these isomers were recorded in both polar and nonpolar solvents.     

On the calculation of arbitrary multielectron molecular integrals over Slater‐type orbitals using recurrence relations for overlap integrals. III. Auxiliary functions \documentclass{article}\pagestyle{empty}\begin{document}$Q_{nn'}^{q}$\end{document} and \documentclass{article}\pagestyle{empty}\begin{document}$G_{-nn'}^{q}$\end{document}

The auxiliary functions $Q_{nn'}^{q}(p,pt)$ and $G_{-nn'}^{q}(p_{a},p,pt)$ which are used in our previous paper [Guseinov, I. I.; Mamedov, B. A. Int J Quantum Chem 2001, 81, 117] for the computation of multicenter electron‐repulsion integrals over Slater‐type orbitals (STOs) are discussed in detail, and the method is given for their numerical computation. The present method is suitable for all values of the parameters p_a, p, and pt. Three‐ and four‐center electron‐repulsion integrals are calculated for extremely large quantum numbers using relations for auxiliary functions obtained in this paper. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Approximate valence-shell electron unrestricted Hartree-Fock calculations with orthogonalized atomic orbitals: Proton hyperfine splittings in benzyl and related radicals

A modified INDO procedure has been used to calculate the proton hyperfine splittings in benzyl and the isoelectronic anilino, phenoxy and 2-azabenzyl as well as 2- and 3-thenyl radicals. The present procedure differentiates between s-, p- and d-orbitals on an atom in estimating various integrals involving them, satisfies the rotational invariance requirements and employs an orthogonalized basis set of atomic orbitals for obtaining core-Hamiltonian matrix elements. The calculations based on using the exponents which depend only on the type of orbital and the nature of atom fail to provide correct relative order of ortho and para proton splittings in benzyl as well as anilino, phenoxy and 2-azabenzyl radicals. On the other hand, use of the exponents which are modified according to the charge densities in various orbitals leads to a high absolute value for para proton splitting compared to that for ortho proton splitting which in case of all these radicals is in agreement with experiment. A spin density calculation on benzyl, anilino and phenoxy radicals considering the variation of one-center one-electron and one-center two-electron integrals for different protons with their charges is found to yield further improvement in the relative order of ortho and para proton splittings in all these radicals. In 2- and 3-thenyl radicals the role of 3d-orbitals on sulfur has also been examined. To our knowledge, no unrestricted INDO calculations including 3d-orbitals on sulfur have been reported in the literature so far.