One-centered approximation

The one-center approximation allows for an extremely rapid evaluation of spin-orbit mean-field integrals if the atomic symmetry is fully exploited.64 Even more efficiency may be gained, if also the spin-independent core-valence interactions are replaced by atom-centered effective core potentials (ECPs). In this case, the inner shells do not even emerge in the molecular orbital optimization step, and the size of the atomic orbital basis set can be kept small. A prerequisite for the use of the all-electron atomic mean-field Hamiltonian in ECP calculations is to find a prescription for setting up a correspondence between the valence orbitals of the all-electron and ECP treatments.65-67... 

In the case of the N2 graphitic slit pore system, w is equal to H - 0.24 nm. Fig. 1 shows potential profiles of a N2 molecule in a slitlike graphite pore as a function of w using the one-center approximation. Here, the molecular position in Fig. 1 is expressed by a distance z from the central plane between two surfaces. 

The SOMF operator was successfully employed in various calculations as implemented in the ORCA [103], MOLPRO [104], and AMFI [105] integral programs. Several additional approximations can be optionally employed in the ORCA program such as a one-center approximation and utilization of atomic densities (in line with the AMFI code) as well as utilization of the RI approximation for the Coulomb-like (second) term of (45) [103]... 

As follows from a brief review of the fundamentals on the kinetics of chain reactions, an elegant one-centered model of the process, which was offered at the begiiming of the development of the chain reaction theory, provides a phenomenological description. Initiation and inhibition of reactions by small additions of compormds, critical phenomena, etc. may serve as examples. However, it should be noted that modeling a chain reaction is usually complicated, if the one-centered approximation is not justified and in the cases when consumption of initial compormds should be taken into accormt, or when the intermediates are participating in the chain process, in other words if one has to deal with chain processes of a more complicated nature. So the efforts aimed at developing the special theoretical approaches that are thought to help for a better orientation in a complex chain chemical process are justified, in particular, rmder the conditions of multi-centered, and consequently multi-routed occurrance. 

Frequently, the basic kinetic laws of chain reactions are described in the one-centered approximation [1]. Certainly such an ideal situation is not of general nature. It is easy to note that even a relatively simple example of a chain chemical reaction between molecular hydrogen and chlorine is realized by means of two active reaction centers, the chain carriers, the H and Cl atoms. 

Performing an expansion in atomic orbitals yj and employing the one-center approximation, that is, neglecting cross-atomic terms, the XES intensity is governed by [6]... 

The atom-in-a-molecule approach is a one-center approximation to the shielding tensor. The calculation is restricted to terms centered on the nucleus in question, which is taken to be the origin. The expression for the shielding tensor components is then unsuitable for accurate prediction of individual shieldings, but is simple enough to be used for discussion of general empirical trends ... 

Going beyond the one-center approximation, the atom-plus-ligand model gives... 

One early approximation, due to Pariser and Parr [12], was to treat the one-center term y A the difference between the ionization potential IP and the electron affinity EA of A (Eq. (51)). 

The ZDO approximation is made only for two-center integrals one-center coulomb Za,b = and exchange... 

PM3, developed by James J.P. Stewart, is a reparameterization of AMI, which is based on the neglect of diatomic differential overlap (NDDO) approximation. NDDO retains all one-center differential overlap terms when Coulomb and exchange integrals are computed. PM3 differs from AMI only in the values of the parameters. The parameters for PM3 were derived by comparing a much larger number and wider variety of experimental versus computed molecular properties. Typically, non-bonded interactions are less repulsive in PM3 than in AMI. PM3 is primarily used for organic molecules, but is also parameterized for many main group elements. 

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In order to obtain nonzero spin densities even on hydrogen atoms in tt radicals, one has to take the one-center exchange repulsion integrals into account in the eigenvalue problem. In other words, a less rough approximation than the complete neglect of differential overlap (CNDO) is required. This implies that in the CNDO/2 approach also, o and n radicals have to be treated separately (98). 

In the limit that nearest-neighbor contributions dominate, such a one-center nb automatically leads to an approximate 21/2-fold increase in overlap and two-fold increase in interaction energy, compared with a two-center 7tb donor. A corresponding enhancement results when the pi-acceptor is reduced from two-center (7tb ) to one-center (nb ) form, i.e., a valence p-type vacancy. Unlike the intrinsically bidirectional character of conjugation between two pi bonds (7ta->-7tb, 7tb 7ta ), the interactions of a pi bond with a nonbonding center are intrinsically mono-directional and lead to uncompensated transfer of pi charge from one moiety to the other. 

In MINDO/2, the one-center integrals are, as usual, determined from spectroscopic data, essentially in the same way as in INDO. The two-center repulsion integrals ii, jj) are given by the Ohno-Klopman 1S approximation ... 

Due to the integral approximations used in the MNDO model, closed-shell Pauli exchange repulsions are not represented in the Hamiltonian, but are only included indirectly, e.g., through the effective atom-pair correction terms to the core-core repulsions [12], To account for Pauli repulsions more properly, the NDDO-based OM1 and OM2 methods [23-25] incorporate orthogonalization terms into the one-center or the one- and two-center one-electron matrix elements, respectively. Similar correction terms have also been used at the INDO level [27-31] and probably contribute to the success of methods such as MSINDO [29-31],... 

Within the Born-Oppenheimer approximation, the nuclei are at rest and have zero momentum. So the electron momentum density is an intrinsically one-center function that can be expressed usefully in spherical polar coordinates and expanded as follows [162,163]... 

Silica makes up 12.6 mass-% of the Earth s crust as crystalline and amorphous forms. It was found that both modifications show similar main luminescence bands, namely a blue one centered at 450 nm ascribed to which substitutes for Si, red centered at 650 nm linked with non-bridge O, and dark-red at 700-730 nm linked with Fe. Time-resolved luminescence of hydrous volcanic glasses with different colors and different Fe, Mn, and H2O contents were measured and interpreted (Zotov et al. 2002). The blue band with a short decay time of 40 ns was connected with T2( D)- Ai ( S) and Ai C G)- Ai ( S) ligand field transitions of Fe " ", the green band with a decay time of approximately 250 ps with a Ti( G)- Ai( S) transition in tetrahedrally coordinated Mn ", while the red band with a much longer decay time of several ms with T1 (4G)- Ai( S) transitions in tetrahedrally coordinated Fe ". We detected Fe " " in the time-resolved luminescence spectrum of black obsidian glass (Fig. 4.43d). 

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