Density-function overlap

Becke exchange and LYP correlation functional Complete Active Space Self-Consistent Field Configuration Interaction Complete Neglect Differential Overlap Density Functional Theory... 

The method is based on the form of the overlap density function of non-correlating orbitals. 

Helical structure k-k Overlap Density functional theory computations... 

The resulting similarity measures are overlap-like Sa b = J Pxi ) / B(r) Coulomblike, etc. The Carbo similarity coefficient is obtained after geometric-mean normalization Sa,b/ /Sa,a Sb,b (cosine), while the Hodgkin-Richards similarity coefficient uses arithmetic-mean normalization Sa,b/0-5 (Saa+ b b) (Dice). The Cioslowski [18] similarity measure NOEL - Number of Overlapping Electrons (Eq. (10)) - uses reduced first-order density matrices (one-matrices) rather than density functions to characterize A and B. No normalization is necessary, since NOEL has a direct interpretation, at the Hartree-Fodt level of theory. 

Highest occupied molecular orbital Intermediate neglect of differential overlap Linear combination of atomic orbitals Local density approximation Local spin density functional theory Lowest unoccupied molecular orbital Many-body perturbation theory Modified INDO version 3 Modified neglect of diatomic overlap Molecular orbital Moller-Plesset... 

The optimised interlayer distance of a concentric bilayered CNT by density-functional theory treatment was calculated to be 3.39 A [23] compared with the experimental value of 3.4 A [24]. Modification of the electronic structure (especially metallic state) due to the inner tube has been examined for two kinds of models of concentric bilayered CNT, (5, 5)-(10, 10) and (9, 0)-(18, 0), in the framework of the Huckel-type treatment [25]. The stacked layer patterns considered are illustrated in Fig. 8. It has been predicted that metallic property would not change within this stacking mode due to symmetry reason, which is almost similar to the case in the interlayer interaction of two graphene sheets [26]. Moreover, in the three-dimensional graphite, the interlayer distance of which is 3.35 A [27], there is only a slight overlapping (0.03-0.04 eV) of the HO and the LU bands at the Fermi level of a sheet of graphite plane [28,29],... 

FIG. 6 A comparison of the Monte Carlo (points), HHNCl, HNCl, and PYl density profiles. The results are for bulk density 0.403 (lower group of curves) and 0.741 (upper group of curves). The curves at z — 0, are, from the bottom, from HHNCl and HNCl approximations. (Reprinted from A. Trokhymchuk, D. Henderson, S. Sokolowski. Local density of overlapping spheres near a hard wall A density functional approach. Physics Letters A 209, 317-320. 1995, with permission from Elsevier Science.)... 

Davis, J.M. (1997a). Justification of probability density function for resolution in statistical models of overlap. Chromatographia 44, 81. 

Modern theories of electronic structure at a metal surface, which have proved their accuracy for bare metal surfaces, have now been applied to the calculation of electron density profiles in the presence of adsorbed species or other external sources of potential. The spillover of the negative (electronic) charge density from the positive (ionic) background and the overlap of the former with the electrolyte are the crucial effects. Self-consistent calculations, in which the electronic kinetic energy is correctly taken into account, may have to replace the simpler density-functional treatments which have been used most often. The situation for liquid metals, for which the density profile for the positive (ionic) charge density is required, is not as satisfactory as for solid metals, for which the crystal structure is known. 

Figure 5.30 shows the density functions of the estimated y-values (discriminant variable) of the calibration set (left) and the test set (right). The solid lines refer to a PLS model with two components, the dashed lines to a PLS model with one component, and the dashed-dotted lines to a PLS model with three components. The maxima of the density functions are approximately at the values —1 and 1, corresponding to the group codes. The overlap of the density functions indicates wrong group assignments for the calibration set, the overlap is smallest for the model with two components, for the test set two or three components are better than one component. 

The ability to use precisely the same basis set parameters in the relativistic and non-relativistic calculations means that the basis set truncation error in either calculation cancels, to an excellent approximation, when we calculate the relativistic energy correction by taking the difference. The cancellation is not exact, because the relativistic calculation contains additional symmetry-types in the small component basis set, but the small-component overlap density of molecular spinors involving basis functions whose origin of coordinates are located at different centres is so small as to be negligible. The non-relativistic molecular structure calculation is, for all practical purposes, a precise counterpoise correction to the four-component relativistic molecular... 

S v are elements of the overlap matrix. Similar types of expressions may be constructed for density functional and correlated models, as well as for semi-empirical models. The important point is that it is possible to equate the total number of electrons in a molecule to a sum of products of density matrix and overlap matrix elements. ... 

Calculations by the MNDO (modified neglect of diatomic overlap) method with full optimization of geometry were carried out for [Zn, N, C, H]+ ions indicating the possible existence of four stable isomers (Figure 7). According to the same semi-empirical method, [Zn, N, C2, H3]+ ions can form nine stable isomeric stmctnres (Fignre 7) In a separate study by density functional theory, self-assembled helicate architectures have been proposed for ions of the [Zn (CN)2 +i] series. ... 

A variety of empirical rules exist for choosing the exponent(s) for a set of polarization functions. If only a single set is desired, one possible choice is to make the maximum in tlie radial density function, equal to that for the existing valence set (e.g., the 3d functions that best overlap the 2p functions for a first-row atom - note that the radial density is used instead of the actual overlap integral because the latter, by symmetry, must be zero). 

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