Surfaces minimal basis

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These discrepancies result (a) from the harmonic approximation used in all calculations [to,- (theory) > v, (exp)], (b) the known deficiencies of minimal and DZ basis sets to describe three-membered rings [polarization functions are needed to describe small CCC bond angles a>,(DZ + P) > w,(DZ) > to,(minimal basis)] and (c) the need of electron correlated wave functions to correctly describe the curvature of the potential energy surface at a minimum energy point [ [Pg.102]

A minimal basis set of STO (expanded in Gaussian lobe functions) was used, and the least-energy pathway determined, assuming fixed values for some of the parameters. The complete optimization is not practicable for systems of this size and most surface studies make physically reasonable approximations to reduce the number of variables. The computed pathway is a non-least-motion rearrangement. The implications of the results with respect to the stability of F2NOF relative to F3NO were also discussed. 

The focus in the reaction dynamics studies was on the N02 elimination channel, but they also studied the HONO elimination reactions [70]. They based the potential energy surface on experimental data but performed some minimal basis set ab initio calculations to determine geometries, force fields, torsional potentials, and some information about the reaction paths. The representations of the global potential energy surfaces were based on valence force fields for equilibrium structures with arbitrary switching functions operating on the potential parameters to effect smooth and (assumed) proper behavior along the reaction paths. Based on the available experiments [71-73], they assumed that the primary decomposition reaction is simple N-N bond rupture to eliminate N02. 

Between the wall of the cell and any ions (H+, H30+, H502+) forces of supermolecular hydrogen P-bonds and electrostatic y-bonds operate (see Fig. 2). Surfaces of intermolecular potential energy have been calculated by density functional method stated in our paper [6], Necessary data about spatial distributions of electron charge density inside framework of aqua multiparticle had been taken from calculations of aquatic ions and the ring of water (H20)n by using of standard molecular orbital method in the minimal basis set (STO-3G). Results of calculations are shown in Table 1. 

Usually the atoms on the local region include all electrons or a small core pseudopotential whereas the atoms in the outer region include a larger core and most often only the outermost ns electrons necessary to represent the metal conduction band are explicitly included in the quantum mechanical calculation. This is nowadays a standard procedure to model Ni, Cu, Ag and Pt surfaces. " " In addition, a mixed basis scheme is usually employed to further reduce the computational cost while attempting to preserve the quality of the cluster model ab initio calculations. In this scheme, the atoms in the outer region are treated with a rather limited, even minimal, basis set whereas atoms in the local region are treated with a more extended basis set. 

Exact calculations of the potential energy surfaces for complex molecular systems are impossible to carry out from a practical point of view. Such calculations involve the solution of the electronic Schrodinger equation for the system including electron correlation effects and full geometry optimization. However, an estimate of the 8 A lq> value can be obtained in a different way. One can carry out an ab initio calculation in the Hartree-Fock (HF) approximation by using a simple basis set, e.g., (7s, 3p/3s), contracted to a minimal basis set, STO-3G, or 3-21G, etc., with full geom-... 

Other minimal basis sets have been used in S.C.F. calculations for chemical reactions. I must mention a very accurate study of the transition state for the geometrical isomerization of cyclopropane, and a search for nondynamical pathways on the potential surface for this reaction when a minimal basis set of Slater orbitals was used 16 0-162) Calculations with a basis of this kind were also performed for the coplanar decomposition of cyclobutane 163>. Minimal basis sets based on gaussian... 

The surfaces of constant are prolate ellipsoids, those of constant r are hyperbolic sheets. The system is illustrated in Fig. 7.1. If we now take a particular pair of the STO functions comprising the minimal basis for H2O, say Isb and one of the 2po functions we can transform them to the new coordinate system and see what is involved in the integral calculation. 

As with any calculation that involves anions, extended basis sets are required to characterize the diffuse nature of the anion s electron density. At the very least, the basis set should contain a set of diffuse sp functions such as those in the 6-31- -G(d) basis set (see Diffuse Functions). For this reason, methods that are limited to minimal basis sets (i.e., semi-empirical) perform poorly and cannot provide quantitatively meaningful results (see below). Although it may be argued that a favorable cancelation of errors allows minimal basis sets to provide reasonable estimates of relative acidities, this is not necessarily true for proton transfer potential energy surfaces because the localization of the charge changes dramatically in the progression from reactants to products. 

Fig. 5.4. The one- and twc-electron density functions of the bonding. o ) (upper plots) and antibonding 11ct > (lower plots) configurations of the hydrogen molecule on the molecular axis (atomic units). The two-electron densities are represented by surface and contour plots. In the contour plot for the antibonding configuration, the two-electron nodes are represented by dashed lines. The density functions have been calculated in a minimal basis of hydrogenic Is functions with unit exponents.

In this section we characterize the minima of the functional (1) which are triply periodic structures. The essential features of these minima are described by the surface (r) = 0 and its properties. In 1976 Scriven [37] hypothesized that triply periodic minimal surfaces (Table 1) could be used for the description of physical interfaces appearing in ternary mixtures of water, oil, and surfactants. Twenty years later it has been discovered, on the basis of the simple model of microemulsion, that the interface formed by surfactants in the symmetric system (oil-water symmetry) is preferably the minimal surface [14,38,39]. 

For ordered periodic structures of a period A a dimensionless ratio between the two lengths, Ryy = Kyy /, provides an additional characteristic of the structure. On the basis of the results of Ref. 39, we can estimate this ratio for the periodic minimal surfaces. For simple minimal surfaces, P, D, or G [39], we find respectively = 0.306, Ryyi = 0.195, and Ryyi = 0.248. For more complicated periodic structures [39] its value can be even smaller than 0.1 for example, for the surface labeled GX5, Ryy = 0.073. 

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