Academic literature on the topic 'Gaussian-Type atomic orbitals'

A quality test is proposed for SCF atomic orbitals, [Formula: see text] approximated as finite linear combinations of suitable basis functions [Formula: see text] The key is in a function, readily derived from the Hartree–Fock equation [Formula: see text] which is identically zero for true Hartree–Fock spin orbitals and not so for approximate orbitals. In this way, our test measures how closely approximate orbital descriptions approach the true Hartree–Fock limit and thus provides a quality ordering of orbital bases with respect to one another and with respect to that limit, in a scale uniquely defined by the latter. Moreover, this analysis also holds for atomic subspaces of our choice, e.g., the valence region. Examples are offered for representative collections of Slater- and Gaussian-type orbital expansions.

The calculation of the number of atoms of the given dimensional nanoparticle, composed of different type atoms has been researched in this work. The calculations have been carried out for nanoparticles of titanium dioxide. Theoretical visual models have been configured, and quantum – mechanical calculations have been carried out for \((TiO_2 )_6\) nanoparticle. The calculations for titanium dioxide nanoparticle have been carried out on the basis of Gaussian atomic orbitals. Besides, Gaussian functions have been used as atomic orbitals. The numerical values of unknown coefficients of the linear combination of atomic orbitals of the atoms of the titanium nanoparticle have been found from the solution of Hartree–Fock–Roothaan (HFR) equations.The values of orbital energies, ionization potential, and the total electronic energy of titanium dioxide nanoparticles have been determined. The calculations show that ,titanium dioxide nanoparticle is tough, electrophile, and stable dielectric, material. The effective charge of atoms have been calculated, and the theoretical visual mode of titanium dioxide nanopartical have been constructed.

The Hartree–Fock equations for a general open shell atom are described. The matrix equations that result when the single particle orbitals are written in terms of a linear combination of analytic basis functions are derived. Attention is paid to the complexities that occur when open shells are present. The specifics of a working FORTRAN program which is available for public use are described. The program has the flexibility to handle either Slater-type orbitals or Gaussian-type orbitals.

Molecular quantum similarity measurements are based on a quantitative comparison of the one-electron densities of two molecules superposed and aligned to optimize a well-defined similarity function. In most previous work the densities have been related using a Dirac delta leading to the overlap-like quantum similarity function. The densities for the two molecules compared have generally been approximated often with a simple LCAO of s-gaussian functions. In this work, we present a one center two range expansion method for the evaluation of the overlap integrals involved in the overlap-like quantum similarity function over Slater type orbitals (STO). The single center and three types of two-center overlap integrals (involving four atomic orbitals; two in each molecule) have led to finite sums using a single center approach combined with selection rules obtained by analysis of orbital angular momentum (conservation). The three- and four-center integrals are also obtained analytically but involve infinite sums which require further study before leading to a complete set of integral codes for ab-initio quantum similarity.

Molecular orbitals were obtained by X-ray molecular orbital analysis (XMO). The initial molecular orbitals (MOs) of the refinement were calculated by the ab initio self-consistent field (SCF) MO method. Well tempered basis functions were selected since they do not produce cusps at the atomic positions on the residual density maps. X-ray structure factors calculated from the MOs were fitted to observed structure factors by the least-squares method, keeping the orthonormal relationship between MOs. However, the MO coefficients correlate severely with each other, since basis functions are composed of similar Gaussian-type orbitals. Therefore, a method of selecting variables which do not correlate severely with each other in the least-squares refinement was devised. MOs were refined together with the other crystallographic parameters, although the refinement with the atomic positional parameters requires a lot of calculation time. The XMO method was applied to diformohydrazide, (NHCHO)2, without using polarization functions, and the electron-density distributions, including the maxima on the covalent bonds, were represented well. Therefore, from the viewpoint of X-ray diffraction, it is concluded that the MOs averaged by thermal vibrations of the atoms were obtained successfully by XMO analysis. The method of XMO analysis, combined with X-ray atomic orbital (AO) analysis, in principle enables one to obtain MOs or AOs without phase factors from X-ray diffraction experiments on most compounds from organic to rare earth compounds.

La simulation moléculaire est l'un des outils les plus courants de la chimie moderne. Les calculs réalisés au cours de ces simulations présentent souvent des difficultés, qui conduisent à une réduction de leurs performances lorsque les systèmes simulés sont des larges molécules composées de plusieurs atomes. Cette thèse se focalise sur les limitations liées à l'utilisation de fonctions de base centrées sur les atomes des molécules pour la discrétisation des équations de type Schrödinger, qui est un type de discrétisation très populaire en chimie quantique. Nous adoptons une approche d'analyse numérique pour formuler et traiter ces limitations. Le présent travail aborde deux questions fondamentales liées aux éléments de base de type gaussien centrés sur les atomes, à savoir l'évaluation des intégrales moléculaires sur les fonctions de base et la génération des éléments de cette base. Ces deux points ont un impact sur le coût de calcul et les exigences en mémoire des simulations moléculaires. Notre objectif principal est de concevoir des nouvelles méthodes mathématiques ainsi que des nouveaux algorithmes efficaces qui améliorent les simulations moléculaires modernes. Les principales contributions de cette thèse sont les deux suivantes : premièrement, l'accélération de l'évaluation des intégrales moléculaires sur les fonctions de base centrées sur les atomes et, deuxièmement, la proposition d'estimateurs d'erreur a posteriori pour les discrétisations centrées sur les atomes des problèmes linéaires à valeurs propres. Pour le premier objectif, nous avons développé une nouvelle méthode d'ajustement de densité ("Density Fitting" en anglais) pour l’approximation de la densité électronique, complétant les méthodes existantes dans la littérature, qui vise à réduire le coût de calcul en utilisant des approximations de rang faible et creuses, basées sur l’élimination des dépendances linéaires et la décomposition de Cholesky avec pivot. Notre schéma est présenté en utilisant un nouveau formalisme d'optimisation discrète et de recherche du plus court chemin sur les graphes. En outre, nous avons généralisé nos techniques en développant un nouveau schéma d'ajustement de densité indépendant des positions atomiques, en utilisant la méthode des bases réduites. La performance numérique de nos méthodes est démontrée par les résultats numériques des calculs d'énergie d'interaction intermoléculaire basée sur les densités en chimie. Pour le second objectif, notre travail constitue une extension de la théorie de l'estimation de l'erreur a posteriori basée sur les résidus des discrétisations gaussiennes sur des domaines non bornés. Un tel cadre, qui est couramment utilisé en chimie, n'a pas fait l'objet d'études théoriques dans la littérature mathématique jusqu'à présent. Notre contribution dans ce domaine permet la génération adaptative et automatique des bases centrées sur les atomes. Nous présentons des résultats numériques préliminaires des calculs en structure électronique afin d'illustrer un exemple d’application de nos estimateurs d'erreur a posteriori. En résumé, les bases de discrétisation centrées sur les atomes sont largement utilisées dans les simulations moléculaires. Les conclusions de cette thèse contribuent à la compréhension de telles bases du point de vue numérique, tout en proposant des solutions qui permettent l'amélioration des simulations moléculaires en chimie<br>Molecular simulation is among the most common tools in modern chemistry. Suchsimulations often suffer from several computational bottlenecks that reducetheir performance when applied to large systems of molecules or atoms. Thisthesis primarily focuses on the limitations arising from the use ofatom-centered basis functions for the discretization of Schrödinger-typeequations for molecules, which is a popular type of discretization in quantumchemistry applications. We adopt a numerical analysis approach to formulate andtackle such limitations. The present work addresses two of the most impactfulissues related to Gaussian-type atom-centered basis sets, namely, the evaluationof integrals on the basis functions and the generation of such basis sets. Bothissues significantly affect the computational cost and memory requirements ofmolecular simulations. Our main goal is to design novel mathematical methods aswell as new efficient low-complexity algorithms improving modern molecularsimulations. The main contributions of this thesis are twofold: first,accelerating the evaluation of high-dimensional integrals on atom-centered basisfunctions, and, second, establishing a posteriori error estimators foratom-centered discretizations of linear eigenvalue problems. For the firstpurpose, we developed a new density fitting method for approximating theone-electron density, beyond the existing classical and robust density fittingmethods of the literature, achieving tunable cost reduction via sparse low-rankapproximation based on linear dependency elimination and the pivoted Choleskydecomposition. Our scheme is presented using a novel formalism of discreteoptimization and shortest path search on graphs. In addition, we generalizedour main techniques by developing a new atomic-position-independent densityfitting scheme using the reduced basis method. The numerical performance of ourmethods is demonstrated by numerical results of an application to density-basedintermolecular electrostatic interaction energy calculations in chemistry. Forthe second purpose, our work constitutes an extension of residual-based aposteriori error estimation theory to Gaussian discretizations over unboundeddomains. Such a setting, which is routinely used in chemistry, was lackingtheoretical investigation in the mathematical literature up to now. Ourcontribution on this domain paves the way towards adaptive and automaticgeneration of atom-centered basis sets. As numerical evidence, we presentpreliminary numerical results of an application to electronic structure theorycalculations. To sum up, atom-centered Gaussian basis sets are widely used inmolecular simulations. The conclusions of this thesis provide insights to thenumerical analysis as well as to the computational aspects of the use of suchbasis sets in practice, while numerically demonstrating the ability of ourmethodologies to improve realistic simulations in chemistry

Cette thèse contribue aux développements de méthodes numériques utilisées pour reproduire la dynamique électronique induite par des processus à un et plusieurs photons dans les atomes et molécules. Dans le domaine perturbatif, la photoexcitation et la photoionisation ont été étudiées à l'aide de la théorie de la fonctionnelle de la densité à séparation de portée, dans le but de prendre en compte les effets d'interaction électron-électron. De plus, dans le domaine non-perturbatif, les spectres au-delà du seuil d'ionisation et les spectres de génération d'harmoniques d'ordres élevés ont été simulés en utilisant différentes représentations de la fonction d'onde dépendante du temps du système étudié. Cette étude ouvre la possibilité d'explorer des processus matière-rayonnement dans des systèmes plus complexes<br>The present PhD thesis contributes to the development of numerical methods used to reproduce the electron dynamics induced by single and multiphoton processes in atoms and molecules. In the perturbative regime, photoexcitation and photoionization have been studied in atoms with range-separated density-functional theory, in order to take into account the electron-electron interaction effects. Moreover, in the non-perturbative regime, above-threshold ionization and high-harmonic generation spectra have been simulated using different representations for the time-dependent wave function for the purpose of describing the continuum states of the irradiated system. Our studies open the possibility of exploring matter-radiation processes in more complex systems

Abstract This study demonstrates linear scaling with system size in fixednode diffusion QMC calculations with pseudopotentials for carbon fullerenes and for hydrogenated silicon clusters with nearly 1000 valence electrons. The range covered is C20 to C18o and SiH4 to Si211H140, The calculations were carried out with standard pseudopotentials (leaving four electrons per C or Si atom) and Slater Jastrow trial functions with orbitals from LDA density functional calculations. The key elements leading to linear scaling were constructions of sparse determinants and simplification of orbital functions. The LDA orbitals were obtained with a plane-wave basis set which was transformed to yield Wannier functions with maximum localization. These functions were then fitted to cubic spline expressions with a limited number of grid points and cut-off distances matching those of the plane-wave orbitals. The computational effort for evaluating trial functions of this type was clearly seen to behave in a linear fashion. Exploration of the effects of variation of cut-off radius revealed that energies, sensitive only to node structures, increased rapidly for cut-offs less than 7 bohr but were unaffected for cutoffs greater than 7 bohr. The accuracies of the calculations were checked in other ways. These included comparison calculations for SiH3 and Si5H12 using Gaussian, original plane-wave, and Wannier function basis sets which gave close agreement for energies.

Abstract The trial wavefunction for importance sampling in a diffusion QMC calculation for an atom or molecule has most often been assembled from a Slater determinant or a sum of determinants multiplied by a Jastrow function. The determinantal part is readily obtained from easy-to-use quantum chemistry programs which make use of Gaussian basis sets. The programs take advantage of the analytic integrals available for Gaussian exponentials, but these functions fail to give a good fit to realistic one-electron orbitals and produce large fluctuations in the local energy near the electron-nucleus cusps. One solution to the problem has been to fit the Gaussian functions to simple exponentials, adjust them to satisfy the cusp condition, and use them in the determinantsa. In this paper a new scheme, having the advantages of an automatic procedure and avoiding some of the earlier problems is reported. In this scheme a part of each s-type Gaussian function &amp;lt;/J(r) is replaced by a new function ¢ (r) which depends on exp[p(r)] in which p(r) = Do+ a1r + D2r2 + D3r3 + D4r4. Theconstants Di and others are adjusted to fit first and second derivatives to &amp;lt;/J(r), to satisfy the cusp condition, and to optimize the behavior of the local energy. This is accomplished analytically. The scheme was investigated in detail with VQMC and fixednode DQMC for the atom Ne and the molecule H2. Calculations were also made for the 55 molecules of a standard test set. Improvement in the standard deviation in local energy, relative to that for no cusp correction, was dramatic. In the VQMC calculations for the test set the standard deviation in individual local energy was reduced by a factor of 2 to 5 for most molecules. This corresponds to a reduction by a factor of 4 to 25 in calculation effort for a fixed statistical error. Total energies were found to be slightly reduced.

Abstract Results of fixed-node diffusion QMC calculations for 17 small molecules made up of atoms H, C, N, 0, and F were compared in this paper to experimental values for atomization energies, bond lengths, and bond angles, as well as corresponding values from coupled cluster calculations. The QMC calculations were of the Ornstein-Uhlenbeck type,a with trial functions for importance sampling and node locations based on linear combinations of determinants with molecular orbitals composed of floating spherical Gaussian orbitals. Initial geometries were taken from density functional calculations and optimized to find the minima given by the QMC calculations. Zero-point energies were calculated using the energy gradients given by the same calculations. The calculated atomization energies for the 14 species for which experimental data were available were in excellent agreement with the experimental values and with values given by CCSD(T) with a cc-pVQZ basis set. The mean absolute deviation from experimental atomization energies was 0.16 kcal/mol for the QMC and 0.21 kcal/mol for the CCSD(T) calculations. Geometric parameters were in agreement with each other and with experimental values within their uncertainties. Overall, the results provide a solid confirmation of the accuracy of fixed-node diffusion QMC calculations.