Boys procedure

The main disadvantage in all of the above cases is represented by the fact that the computational complexity of the methods grows in proportion to N5. In the specific case, however, of Boys procedure a fortunate transformation of criterion (7) with (9) leads to the maximization of the equivalent optimum criterion... 

The Boys internal criterion consists essentially in separating as much as possible the so-called centroids of charge of the various molecular orbitals. We have systematically used the Boys procedure for describing the electronic structure of chemical species and for determining the electronic mechanism of chemical reactions. The obtained results will be presented in Sections II and IV. 

Moreover, the Boys procedure may also be used for localizing the singly occupied molecular orbitals obtained by the UHF approach (Pee-ters and Leroy, 1977). Thus, as shown in Sections II and IV, the electronic structure of open-shell systems and the mechanism of free-radical reactions can also be described in terms of the corresponding centroids of charge. 

In conclusion, the Boys localization procedure may be considered as the mathematification of the qualitative Lewis and Linnett theories. As pointed out, in this perspective a formal signification of the symbols used by these authors can be proposed these represent not localized electrons but centroids of charge of localized orbitals. Within this interpretation, the former theories agree much better with the ideas of quantum mechanics such as the wave character of electrons that prevents their localization. Moreover, the results obtained by the Boys procedure allow us to predict the Lewis and (or) Linnett structures of many chemical species without performing explicit calculations. For example, linear and bent 1,3-dipoles are described by the following general formulas ... 

A recently often used practical method is that of proposed by Pipek and Mezey [26], Their intrinsic localization is based on a special mathematical measure of localization. It uses no external criteria to. define a priori orbitals. The method is similar to the Edmiston-Ruedenberg s localization method in the a-n separation of the orbitals while it works as economically as the Boys procedure. For the application of their localization algorithm, the knowledge of atomic overlap integrals is sufficient. This feature allows the adoption of their algorithm for both ab initio and semiempirical methods. The implementation of die procedure in existing program systems is easy, and this property makes the Pipek-Mezey s method very attractive for practical use. 

We have added SIC capabilities to the DFT package in Q-Chem [20]. Initially, this involves localizing the KS canonical orbitals with the Boys procedure [78] and using these to evaluate the self-interaction correction. Thus, the present scheme simply applies the correction perturbatively to the KS energy. Table 9 lists the corrected DFT barriers, evaluated at the re-optimized SIC geometries. 

LMO calculations have been reported for the boron fluorides BF, BHjF, BHF2, BF3, BF2NH2, B4F4, and B2F4. The LMO valence structures obtained by the Boys procedure are typically (33) and (34), where the solid line originates at the atom donating an electron pair, and becomes dotted toward the atom which is electron deficient (indicating bond polarity). These are described as fractional bonds . 

Four different Fock/Kohn-Sham operators have been applied to obtain the orbitals, which are subsequently localized by the standard Foster-Boys procedure. In addition to the local/semi-local functionals LDA and PBE, the range-separated hybrid RSHLDA [37, 56] with a range-separation parameter of /r = 0.5 a.u. as well as the standard restricted Hartree-Fock (RHF) method were used. The notations LDA[M] and LDA[0] refer to the procedure applied to obtain the matrix elanents either by the matrix algebra [M] or by the operator algebra [O] method. All calculations were done with the aug-cc-pVTZ basis set, using the MOLPRO quantum chemical program package [57]. The matrix elements were obtained by the MATROP facility of MOLPRO [57] the Cg coefficients were calculated by Mathematica. 

A molecular orbital located on a certain fragment of a molecular system and spatially separated from other orbitals as much as possible. LMOs are derived from the occupied canonical molecular orbitals by a unitary transformation determined with respect to a physical criterion, e.g., by maximizing the sum of squares of the centroids of occupied molecular orbitals (Foster-Boys procedure) or by minimizing the sum of the exchange (or Coulomb) repulsion integrals between the occupied orbitals (Edmiston-Ruedenberg procedure). 

There is little experience with the von Niessen method, but for most molecules the remaining three schemes tend to give very similar LMOs. The main exception is systems containing both a- and vr-bonds, such as ethylene. The Pipek-Mezey procedure preserves the cr/yr-separation, while the Edmiston-Ruedenberg and Boys schemes produce bent banana bonds. Similarly, for planar molecules which contain lone pairs (like water), the Pipek-Mezey method produces one in-plane cr-type lone pair and one out-of-plane yr-type lone pair, while the Edmiston-Ruedenberg and Boys schemes produce two equivalent rabbit ear lone pairs. 

Although the localization by energy criteria (Edmiston-Ruedenberg) may be considered more fundamental than one based on distance (Boys) or atomic charge (Pipek-Mezey), the difference in computational effort means that the Boys or Pipek-Mezey procedures are often used in practice, especially since there is normally little difference in the shape of the final LMOs. 

Boys, S. F., Bernardi, F., 1970, The Calculation of Small Molecular Interactions by die Differences of Separate Total Energies. Some Procedures With Reduced Errors , Mol. Phys., 19, 553. 

Foster JM, Boys SF (1960) Canonical configurational interaction procedure. Rev Mod Phys 32 300... 

Boys, S. B. and Bemardi, F. 1970. The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors. Mol. Phys. 19 553. 

A single-determinant wave-function of closed shell molecular systems is invariant against any unitary transformation of the molecular orbitals apart from a phase factor. The transformation can be chosen in order to obtain LMOs. Starting from CMOs a number of localization procedures have been proposed to get LMOs the most commonly used methods are as given by the authors of (Edmiston et ah, 1963) and (Boys, 1966), while the procedures provided by (Pipek etal, 1989) and (Saebo etal., 1993) are also of interest. It could be stated that all the methods yield comparable results. Each LMO densities are found to be relatively concentrated in some spatial region. They are, furthermore, expected to be determined mainly by that part of the molecule which occupies that given region and its nearby environment rather than by the whole system. 

As to the localization of occupied orbitals the conventional procedures (Edmiston-Ruedenberg (Edmiston etal., 1963), Boys (Boys, 1966) might not be the most suitable because they do not restrict the magnitude of the off-diagonal Fock matrix elements. Regarding the localization of virtual orbitals, they cannot be localized uniquely into... 

BSSE also affects the shape of the potential energy surface and the energy derivatives. There have been numerous attempts to find a general scheme to eliminate this error, and both a posteriori [2] and a priori [3] schemes are available. The counterpoise approach (CP) by Boys and Bemardi [5] and related methods are the most common a posteriori procedures. Within this method, the monomer electrons are described by the same basis functions as those used in the complex by means of the so... 

The term BSSE was first introduced in 1973 by Liu and McLean (Liu and McLean, 1973). However, as early as 1970 Boys and Bernard (Boys and Bernard , 1970) had proposed the function counterpoise procedure (CP) as a strategy to correct for BSSE. In this procedure the monomer calculations are given the same flexibility that is available to the monomers in the dimer calculation, namely, the monomer energies are evaluated in the complete dimer basis set. The counterpoise-corrected interaction energy then becomes ... 

There have been many attempts to formulate a procedure to avoid it and both a posteriori and a priori schemes are available. The counterpoise approach (CP) (Boys and Bemardi, 1970) and related methods are the most conunon a posteriori procedures. Although this technique represents the most frequently employed a posteriori procedure to estimate this error, several authors have emphasised that the method introduced by Boys and Bemardi does not allow a clear and precise determination of the BSSE. The addition of the partner s functions introduces the "secondary superposition error" a spurious electrostatic contribution due to the modification of the multipole moments and polarizabilities of the monomers. This is particularly important in the case of anisotropic potentials where these errors can contribute to alter the shape of the PES and the resulting physical picture (Xantheas, 1996 and Simon et al., 1996). 

T he radioactive products of a nuclear explosion are said to have under-gone fractionation if their relative proportions in samples taken at various locations differ significantly from their relative proportions as formed. This report describes a study of fractionation in the early fallout from the nuclear cratering shots Danny Boy, Sedan, and Palanquin. Published fallout data for these shots was the basic information used in the study. A normalization procedure was applied to the published data as follows the amount of each radionuclide (or mass chain) of interest measured on a fallout tray is related to the gamma-radiation exposure rate measured at the tray location and to the amount of that radionuclide produced per kiloton of fission by the device. The result is an index... 

Our objective in re-examining the 1446 pieces of Small Boy data has been to extract the maximum amount of information and the minimum amount of misinformation with the least amount of tampering. Our method has turned out to be a loop which we have traveled innumerable times. The first step of the loop was to choose, for a given laboratory, the best substantiated correlation available and select the outliers. We next traced the outliers through every other meaningful correlation to corroborate their spuriousness. The procedure was repeated for the next best substantiated correlation, and so on, as far as we could carry it. The data from the other laboratories were treated similarly in turn. We have thus examined the data exhaustively for mutual consistency. In many cases we were able to show that a datum violated more than one criterion, and we rejected it on that basis. In other cases, data were so far out of line that there was no question as to their abnormality. In still other situations we found that correlations could be established with the data from one laboratory but not with the data from another. We then rejected the irregular data in toto. 

Prior to radiochemical analysis the samples were ashed and separated into size fractions by means of procedures described by Nathans et al. The determinations of the fraction weights and of the mean diameters of the particles in the fractions have also been described extensively in the same paper. An aliquot of each size fraction was dissolved and subjected to a separation procedure to isolate Sr, Ru, Sb, Cs, Ce, Pm, U, and Pu fractions. The procedure is sketched in Figure 1. Further decontamination of Ru and Ce was carried out only with the Johnie Boy sample. The Sb and Pu fractions were set aside for later analysis. After complete analysis of the Cs fractions, anomalies were found in the data for the coral samples. These samples had been ashed at about 475°C. Apparently some Cs had volatilized at this temperature. Such a behavior explained the anomalies, and this was confirmed by Heft by more extensive experimentation (4). Thus, Cs data are reported only for the Johnie Boy sample, which was ashed at low temperature in a Tracerlab low temperature asher. 

The size distribution of the particles in the fractions was determined by measuring the size of about 100 particles in each fraction by optical or electron microscopy, as appropriate for the size range of the fraction. The size of the particles in most of the Johnie Boy samples was measured along a line parallel to the base of the field of view. This procedure extends the size range a little on both ends, because the particles are not all spherical, but otherwise has little effect on the size distribution because the particles are randomly oriented. In some of the Johnie Boy samples, the particles were measured along the longest and the shortest directions, and the averages were taken. 

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