Atom-centered indexes

Kier, L.B. and Hall, L.H. (1992a). An Atom-Centered Index for Drug QSAR Models. In Advances in Drug Design (Testa, B., ed.), Academic Press, New York (NY). 

Boys (1950) proposed an alternative to the use of STOs. All that is required for there to be an analytical solution of the general four-index integral formed from such functions is that the radial decay of the STOs be changed from e to e. That is, the AO-like functions are chosen to have the form of a Gaussian function. The general functional form of a normalized Gaussian-type orbital (GTO) in atom-centered Cartesian coordinates is... 

The so-called D-functions concern the descriptors for the n atoms. As described above, in the representation of the electron distribution at any atomic center the localized and delocalized electrons are considered separately. In the atom-descriptor, the number of delocalized electrons is specified together with the number of associated atomic centers to which they belong. Therefore, any D-function acting on the atomic center with index i (1 ig i n) should in principle consist of three components (Aej.l ACj.d Ae z). The first component Ae l e 2 is interpreted as the increase or decrease of the number of localized electrons, the second component Aej.d e 2 as the increase/decrease of the total number of delocalized electrons, and the third component Acj.z e 2 represents the change in the total size of all DE-systems of which A is a member, in terms of the number of participating atoms. 

In all three of these approaches, the similarity indices should be maximized. Both the Carbo index and the Hodgkin-Richards index were used by Boon et al. % whereas in the applications of AIM, the Cioslowski similarity measure was For the atom-centered approach, we can use all... 

There are numerous other applications and extensions of the Fukui function. For instance, we have not included the extension of the Fukui function formalism to conductors (where there are bands of occupied states rather than occupied orbitals) [58, 59]. One can also fruitfully consider condensed Fukui functions [12, 45, 60-64], where one affixes a Fukui index to each atomic center (by partitioning the molecule into regions in either real space or function space and integrating the Fukui function over that region). The relationships between the Fukui function and the grand canonical ensemble of DFT and the relationships between the Fukui function and the softness kernel and local softness are also important [24, 51, 65]. 

This equation is a generalization of the participation ratio (6.19) and gives a more sharp estimate for a number of strongly localized atomic centers (sites). The related index was employed in [92] where it is shown that the index can well distinguish between localized and extended states. From Table 6.6 we see that indeed gives an acceptable average number of the essentially localized unpaired electrons. When using one must keep in mind that this index is informative if that is in the case of a sufficiently sharp EUE localization. 

If one evaluates this product on a grid instead of using atom-centered Gaussian functions for the auxiliary functions Pi r), one then obtains the very similar pseudospectral approximation.Minimizing the self-interaction error, the four-index electron repulsion integrals,... 

Since Vj are short-range potentials, um have non-zero values only in those finite elements located near enough to the corresponding atomic center. This fact can be employed for optimization. The vectors can be treated as sparse by means of the pairs (index, value). Since each set of vectors have the same pattern, this form can be made even more efficient by means of the structure (index, values ). By choosing convenient order of finite elements according to the pertinence to atomic centers, additional optimization could be performed, but, in this case, the benefit would hardly balance the higher computational costs in finite-element mesh generation. 

At the heart of the RI-MP2 method is an expansion of the shell pairs l/rv) in an auxiliary basis of atom-centered Gaussians, fC). This allows the four-index electron repulsion integrals required in MP2 theory to be expressed in terms of three-index integrals ... 

The preceding step to both MP2 and coupled-cluster calculations is to solve the Hartree-Fock equations. The standard approach is, of course, to solve the equations in a basis set expansion (Roothaan-Hall method), using atom-centered basis functions. This set of basis functions is used to expand the molecular orbitals and we will call it orbital basis set (OBS). It spans the computational (finite) orbital space. Occupied spin orbitals will be denoted (pi and virtual (unoccupied) spin orbitals pa- In order to address the terms that miss in a finite OBS expansion, the set of virtual spin orbitals in a formally complete space is introduced, pa- If we exclude from this space all those orbitals which can be represented by the OBS, we obtain the complementary space, with orbitals denoted cp i. The subdivision of the orbital space and the index conventions are summarized in the left part of Fig. 2. 

The spatial one-electron wavefunctions, f/n), are represented as a linear combination of atom-centered functions (i.e. atomic orbitals), called the linear combination of atomic orbitals (LCAO) approximation. The functions (pik constitute a basis set. This is the same approach used for multi-electron atoms and for the molecule. The index k refers to the specific atomic orbital wavefunction, and the index / refers to its contribution to a specific molecular orbital. 

Such simple considerations led Scholten and Konvalinka to confirm the form of the dependence of the reaction velocity on the pressure, as had been observed in their experiments. Taking into account a more realistic situation, on the polycrystalline hydride surface with which a hydrogen molecule is dealing when colliding and subsequently being dissociatively adsorbed, we should assume rather a different probability of an encounter with a hydride center of a /3-phase lattice, an empty octahedral hole, or a free palladium atom—for every kind of crystallite orientation on the surface, even when it is represented, for the sake of simplicity, by only the three low index planes. 

The susceptibility or mixing coefficients, pj and pj , depend upon the position of the substituent (indicated by the index, /) with respect to the reaction (or detector) center, the nature of the measurement at this center, and the conditions of solvent and temperature. It has been held that the p/scale of polar effects has wide general applicability (4), holding for substituents bonded to an sp or sp carbon atom (5) and, perhaps, to other elements (6). The or scale, however, has been thought to be more narrowly defined (7), holding with precision only for systems of analogous pi electronic frameworks (i.e., having a dependence on reaction type and conditions, as well as on position of substitution). 

We now describe a relatively simple MD model of a low-index crystal surface, which was conceived for the purpose of studying the rate of mass transport (8). The effect of temperature on surface transport involves several competing processes. A rough surface structure complicates the trajectories somewhat, and the diffusion of clusters of atoms must be considered. In order to simplify the model as much as possible, but retain the essential dynamics of the mobile atoms, we will consider a model in which the atoms move on a "substrate" represented by an analytic potential energy function that is adjusted to match that of a surface of a (100) face-centered cubic crystal composed of atoms interacting with a Lennard-Jones... 

The group-subgroup relation of the symmetry reduction from diamond to zinc blende is shown in Fig. 18.3. Some comments concerning the terminology have been included. In both structures the atoms have identical coordinates and site symmetries. The unit cell of diamond contains eight C atoms in symmetry-equivalent positions (Wyckoff position 8a). With the symmetry reduction the atomic positions split to two independent positions (4a and 4c) which are occupied in zinc blende by zinc and sulfur atoms. The space groups are translationengleiche the dimensions of the unit cells correspond to each other. The index of the symmetry reduction is 2 exactly half of all symmetry operations is lost. This includes the inversion centers which in diamond are present in the centers of the C-C bonds. 

For an fee lattice a particularly simple surface structure is obtained by cutting the lattice parallel to the sides of a cube that forms a unit cell (see Fig. 4.6a). The resulting surface plane is perpendicular to the vector (1,0,0) so this is called a (100) surface, and one speaks of Ag(100), Au(100), etc., surfaces, and (100) is called the Miller index. Obviously, (100), (010), (001) surfaces have the same structure, a simple square lattice (see Fig. 4.7a), whose lattice constant is a/ /2. Adsorption of particles often takes place at particular surface sites, and some of them are indicated in the figure The position on top of a lattice site is the atop position, fourfold hollow sites are in the center between the surface atoms, and bridge sites (or twofold hollow sites) are in the center of a line joining two neighboring surface atoms. 

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