Determination of α and β parameters in approximate SCF MO theories

The formula proposed by Linderberg for the evaluation of the resonance integral is further investigated. Consistency requirements for the parameters lead to an explanation of the Wolfsberg-Helmholz form for. The simple assumption that is proportional to the overlap integral is improved. The dependency of a and values on the distance in a diatomic molecule in this approximation is presented and discussed.

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Zusammenfassung Linderbergs Formel zur Berechnung des Resonanzintegrals wird weiter untersucht. Konsistenzforderungen für die-Parameter führen zu einer Erklärung der Wolfsberg-Helmholz-Form für. Die einfache Annahme, daß dem Überlappungsintegral proportional ist, wird verbessert. Die Abhängigkeit von- und-Werten vom Atomabstand in einem zweiatomigen Molekül wird in dieser Näherung angegeben und diskutiert.

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Résumé Etude de la formule proposée par Linderberg pour le calcul de l'intégrale de résonance. Des nécessités de cohérence pour les paramètres conduisent à une explication de la formule de Wolfsberg-Helmholz pour. L'hypothèse simple selon laquelle est proportionnel à l'intégrale de recouvrement est améliorée. On présente en la discutant la dépendance à la distance des valeurs de et de dans une molécule diatomique dans cette approximation.

     

Mathematical aspects of the LCAO MO first order density function (5): centroid shifting of MO shape functions basis set, properties and applications

In MO SCF theory and related computational levels, the first order density function (DF) can be considered as a linear combination of the MO shape functions (ShF) set. This work studies the possibility of constructing a centroid function from the ShF set elements using it to perform a ShF set origin shift, while permitting the DF decomposition in two well defined function terms. The properties and consequences of such a simple operation are analyzed in deep.     

Optimization of the basis functions in SCF MO calculations optimized one-center SCF MO basis set for HCL

Equations have been derived which give the first derivatives of the energy, calculated by the expansion SCF MO method, with respect to the parameters contained in the basis functions. The amount of computations needed has a reasonable limit and is particularly reduced if the parameters are the orbital exponents of STO basis functions. Knowledge of such derivatives has been exployted to optimize a one center basis set, comprised of 71 STO functions, for a SCF MO wave function of HCl. A molecular energy of –460.05810 au has been obtained at a bond length of 2.424 au. The resulting wave function, has been used to calculate several observables of interest.

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Zusammenfassung Ausdrücke für die ersten Ableitungen der SCF-Energie nach den in den Basis-Funktionen enthaltenen Parametern werden abgeleitet. Die zusätzlichen Rechnungen sind gering, wenn die Parameter die Orbitalexponenten von Slater-Funktionen sind. Die Kenntnis solcher Ableitungen wird dazu ausgenutzt, eine Einzentren-Basis von Slater-Funktionen für eine SCF-Wellenfunktion des HCl-Moleküls zu optimieren. Die elektronische Molekül-Energie ergab sich zu –460.05810 AE bei einem Kernabstand von 2.424 AE. Mehrere bedeutsame Erwartungswerte wurden berechnet.

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Résumé Des équations ont été établies donnant les dérivées premières de l'énergie par rapport aux paramètres des fonctions de la base utilisée dans la méthode SCF MO. La quantité de calculs nécessaire est raisonnablement bornée; elle est particulièrement réduite s'il s'agit des exposants orbitaux des fonctions de base de Slater. La connaissance de ces dérivées a été utilisée pour optimiser une base à un centre, comprenant 71 orbitales de Slater, pour la fonction d'onde SCF MO de HCl. L'énergie moléculaire obtenue est –460.05810 u.a. pour une longueur de liaison de 2.424 a.u. La fonction d'onde résultante a été utilisée pour le calcul de plusieurs observables intéressantes.

     

A comparative study of approximate SCF MO treatments of hydrides

A comparative study of approximate and exact ab initio treatments of H₂O, NH₃ and CH₄ has been carried out. Three approximate schemes based on the NDDO approach are introduced. The results compare well with best results obtained by other authors.

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Zusammenfassung Die angenäherte und exakte ab initio Behandlung von H₂O, NH₃ and CH₄ wird verglichen. Drei Verfahren, die auf die NDDO-Näherung zurückgehen, werden eingeführt. Die Resultate sind von vergleichbarer Qualität wie die besten Ergebnisse anderer Autoren.

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The work was partly supported by the Institute of Low Temperatures of the Polish Academy of Sciences, Wrocaw.     

Determination of approximate bond dissociation energies by a semi-empirical MO technique

It is suggested that the Extended Hückel Method by Hoffmann may be used for the determination of approximate bond dissociation energies.The author would like to express his sincere gratitude to Miss Etta Rosenzweig for her assistance in this project.     

The use of Politzer's relationship in approximate MO LCAO calculations. I

Conclusion E II approximations in the MO LCAO method, constructed on the basis of Politzer's relationship between the components of the potential energy, are suitable for the approximate calculation of the total energies of molecules. The electron density distributions obtained are unsatisfactory. An attempt to correct the unsatisfactory features of versions I and IA is made in Paper II.Translated from Zhurnal Strukturnoi Khimii, Vol. 23, No. 1, pp. 3–9, January–February, 1982.     

Picture change and calculations of expectation values in approximate relativistic theories

The effect of the so-called picture change on expectation values of one-electron operators in approximate two(one)-component relativistic theories is discussed. This effect is expected to be particularly large for operators which assume large values in the vicinity of heavy nuclei. The numerical results illustrating the picture change effect on electric field gradients at nuclei have been obtained in the spin-free Pauli and Douglas–Kroll approximations. It has been found that the picture change effect lowers the electric field gradient at I in HI by about 1 a.u. Very large picture change effect (−8 a.u.) has been calculated for HAt. It is concluded that in accurate calculations of expectation values of operators involving high inverse powers of the electron–nucleus distance the picture change, which accompanies the transformation of the Dirac (Dirac–Coulomb) equation to approximate two(one)-component relativistic Hamiltonians, must be taken into account. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 159–174, 1998     

Origin of the magic numbers of water clusters with an excess electron

Electron-bound water clusters [e(-)(H(2)O)(n)] show very strong peaks in mass spectra for n=2, 6, 7, and (11), which are called magic numbers. The origin of the magic numbers has been an enigma for the last two decades. Although the magic numbers have often been conjectured to arise from the intrinsic properties of electron-bound water clusters, we attributed them not to their intrinsic properties but to the particularly weak stability of the corresponding neutral water clusters (H(2)O)(n=2,6,7, and (11)). As the cluster size increases; this nonsmooth characteristic feature in stability of neutral water clusters is contrasted to the smooth increase in stability of e(-)-water clusters. As the magic number clusters have significant positive adiabatic electron affinities, their abundant distributions in atmosphere could play a significant role in atmospheric thermodynamics.     

Use of the mixed basis method for ab initio SCF MO calculations

A previously described mixed basis method for performing SCF MO calculations has been applied to the benzene molecule and to the allyl cation. A basis set of Slater type orbitals is used to calculate the one-electron and one-centre two-electron integrals whilst the many-centre two-electron integrals are approximated by small gaussian type orbital expansions of the STO basis. Comparison of the results with all-gaussian basis sets and literature values indicates that the mixed basis 2-GTO approximation is inadequate for molecules of this size because of the consistent underestimation of the electron repulsion integrals. The use of gaussian exponents chosen by a least-squares procedure rather than variationally gives better mixed basis results, but the indications are that a 4-GTO expansion is necessary for reliable mixed basis calculations. A method for more accurate integral evaluation by gaussian expansions of orbital products is suggested.

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Zusammenfassung Eine früher beschriebene Methode mit gemischter Basis zur Durchführung von SCF-MO-Berechnungen wurde auf das Benzolmolekül sowie das Allylkation angewendet. Ein Basissatz aus Slaterorbitalen wird verwendet, um die Einelektronen- und die Einzentrenzweielektronenintegrale zu berechnen, während die Mehrzentrenzweielektronenintegrale mit Hilfe von Entwicklungen der Slaterorbitale nach wenigen Gaußfunktionen angenähert werden. Ein Vergleich mit Ergebnissen von Rechnungen mit Gesamtbasissätzen von Gaußfunktionen und mit Literaturwerten zeigt, daß der gemischte Basissatz 2-GTO ungeeignet für Moleküle von dieser Größe ist, da die Elektronenabstoßungsintegrale durchgehend zu klein berechnet werden. Die Verwendung von Exponenten der Gauß-funktion, die mit Hilfe einer Methode der kleinsten Quadrate gewonnen wurde, ergibt bei der gemischten Basis bessere Ergebnisse als diejenigen, die nach der Variationsmethode gewonnen wurden; es zeigt sich jedoch, daß 4-GTO-Entwicklungen für angemessene Berechnungen mit gemischter Basis notwendig werden. Eine Methode für eine genauere Integralberechnung mit Hilfe der Entwicklung von Orbitalprodukten nach Gaußfunktionen wird vorgeschlagen.

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Résumé Application à la molécule de benzène d'une méthode de base mixte précédemment décrite pour effectuer les calculs SCF MO. Une base d'orbitales de Slater est utilisée pour calculer les intégrales monoélectroniques et les intégrales biélectroniques monocentriques; les intégrales biélectroniques polycentriques sont approchées au moyen d'une expression des orbitales de Slater en orbitales gaussiennes. La comparaison des résultats avec ceux obtenus en bases gaussiennes et avec ceux donnés dans la littérature indique que l'approximation 2-GTO n'est pas adaptée aux molécules de cette taille car elle provoque une sous-estimation constante des intégrales de répulsion électronique. L'emploi d'exposants gaussiens choisis par un procédé de moindres carrés plutôt que par une méthode variationnelle donne de meilleurs résultats en bases mixtes mais il s'avère nécessaire d'utiliser un développement du type 4-GTO. On propose une méthode pour l'évaluation plus précise des intégrales par développement gaussien des produits d'orbitales.

     

Symmetrical overlap transformations of function basis sets: the LCAO MO and quantum similarity practical cases

Quantum chemical computational procedures, like LCAO MO theory and quantum similarity, use non orthogonal function basis sets, which define finite dimensional subspaces of a Hilbert space. Based on the original overlap metric matrices, generated by the chosen finite non orthogonal basis sets, there are several symmetrical overlap and basis set transformations possible. This study tries to find out the general point of view, from where all these procedures can be studied in a clear generalized perspective.     

Mathematical aspects of the LCAO MO first order density function (3): A general localization procedure

A general MO localization process based on Elementary Jacobi Rotations is described. The procedure is connected with Boys, Ruedenberg and Mezey MO localization algorithms.     

Rotational invariance of INDO theories including d-orbitals into the basis set

It is proved that in general the INDO approximation to the full Roothaan theory does not lead to expressions which are invariant under a rotation of local atomic axes. However, when only s- and p-functions are used in the atomic basis set, the equations obtained are invariant due to the special properties of the p-functions. When d-orbitals are included into the basis set, rotational invariance is lost but can be restored if a supplementary approximation is introduced.     

Reconsidering an analytical gradient expression within a divide-and-conquer self-consistent field approach: exact formula and its approximate treatment

An analytical energy gradient formula for the density-matrix-based linear-scaling divide-and-conquer (DC) self-consistent field (SCF) method was proposed in a previous paper by Yang and Lee (YL) [J. Chem. Phys. 103, 5674 (1995)]. Since the formula by YL does not correspond to the exact gradient of the DC-SCF energy, we derive the exact formula by direct differentiation, which requires solving the coupled-perturbed equations while including the inter-subsystem coupling terms. Next, we present an alternative formula for approximately evaluating the DC-SCF energy gradient, assuming the variational condition for the subsystem density matrices. Numerical assessments confirmed that the DC-SCF energy gradient values obtained by the present formula are in reasonable agreement with the conventional SCF values when adopting a reliable buffer region. Furthermore, the performance of the present method was found to be better than that of the YL method.     

Dependence of approximate ab initio molecular loge sizes on the quality of basis functions

Size and shape parameters for the core, bonding, and lone electron pairs of the ten-electron hydrides (CH₄, NH₃, H₂O, HF) were determined from ab initio MO wave functions using various Gaussian basis sets. The fundamental features of approximate electron pair loge representation are somewhat more sensitive to the quality of the basis functions than the molecular total energy. The total size of the molecular electron distribution is less affected by basis set variations than its components: the core, bonding, and lone pair sizes. There is an apparent tendency to “preserve” the total size of molecular distribution.