Atoms chemical constraints

The multipole formalism described by Stewart (1976) deviates from Eq. (3.35) in several respects. It is a deformation density formalism in which the deformation from the IAM density is described by multipole functions with Slater-type radial dependence, without the K-type expansion and contraction of the valence shell. While Eq. (3.35) is commonly applied using local atomic coordinate systems to facilitate the introduction of chemical constraints (chapter 4), Stewart s formalism has been encoded using a single crystal-coordinate system. 

If individual atomic coordinate systems are used, as is common when chemical constraints are applied in the least-squares refinement, they must first be rotated to have a common orientation. The transformation of the population parameters under coordinate-system rotation is described in section D.5 of appendix D (Cromer et al. 1976, Su 1993, Su and Coppens 1994). 

Francl et al. (1996) examined the conditioning of the least squares matrix in the fitting procedure, and conclude that the method cannot be used to assign statistically valid charges to all atoms in a given molecule. This problem cannot be alleviated by the selection of more sampling points, and thus may require the introduction of chemical constraints to reduce the number of charges to be determined. 

The principle of maximum symmetry requires that the crystal structure adopted by a given compound be the most symmetric that can satisfy the chemical constraints. We therefore expect to find high-symmetry environments around atoms wherever possible, but such environments are subject to constraints such as the relationship between site symmetry and multiplicity (eqn (10.2)) and the constraint that each atom will inherit certain symmetries from its bonded neighbours. The problems that arise when we try to match the symmetry that is inherent in the bond graph with the symmetry allowed by the different space groups are discussed in Section 11.2.2.4. 

Bond valences can be used in conjunction with other techniques, particularly powder diffraction where, for example, light atoms are difficult to refine in the presence of heavy atoms. Adding the chemical constraints of the bond valence model can stabilize the refinement, particularly in the case of superstructures that have high pseudo-symmetry (Thompson et al. 1999). 

When the number of independent chemical reactions equals C - p, where p is the rank of the atom matrix (mjk), Gibbs free energy is minimized subject to atom balance constraints ... 

One of the biggest problems associated with the use of such flame-heated devices is the constraint imposed by the need for chemical inertness to matrix components, even at flame temperature, and the need to use a metal with a high melting point for the construction of the atomizer. These constraints perhaps constitute one of the major reasons for the relatively rare routine use of such techniques today compared with electrothermal atomization. 

The local axis on each atom is defined by the program s user (see Figure 2) this flexibility is very interesting for big molecules possessing non-crystallographic local symmetry and/or containing chemically equivalent atoms. These symmetry and chemical constraints permit to reduce the number of the k, Pv, and Plm electron density parameters in the least-square process (see applications in ref. 13). For example, all atoms of a benzene ring may be constrained to have the same density parameters and a local symmetry mm2 can be applied to each atom. 

Some of the physical and chemical constraints on the flame atomization process — which usually precluded application to solid samples — were overcome with the advent of flameless atomization, initially accomplished with the pyrolytic coated graphite tube (or carbon rod-type) furnace atomizer. The graphite tube is a confined furnace chamber where pulsed vaporization and subsequent atomization of the sample is achieved by raising the temperature with a programmed sequence of electrical power. A dense population of ground state atoms is produced as a result for an extended interval in relation to the low atom density and short residence time of the flame. The earliest use of furnace devices in analytical atomic spectroscopy is credited to a simultaneous development by Lvov [15] and Massmann [16] however, the first application of one such device to a... 

How is an electron density map interpreted In the case of a conventional small molecule compound, where the electron density map is computed at or near atomic resolution, atoms appear as distinct, well-separated peaks in the map. Chemical constraints, such as reasonable bond lengths and angles, are then applied to the distribution of atoms, and based on the relative peak heights, geometry, and distances of separation, the structure of the molecule deduced. Since there is no explicit information specifying which atoms are directly bonded, connectivity must be implied. In most instances the interpretation is unambiguous and the structure clear. 

Hughes JM, Cameron M (1993) Order/disorder in the apatite anion columns ciystal chemical constraints on F, Cl, QH. Geol Soc Am Ann Meet, Progr Abstr 25 371 Hughes JM, Fransolet AM, Schreyer W (1993) The atomic arrangement of iron-bearing apatite. N Jahrb Mineral Monat 1993 504-510... 

Physticochemical coiiatrainla Further constraints can be imposed on the atoms and bonds of the reaction center, such as those physicochemical factors calculated by the PETRA package (see Section 7.1). For example, the partial charges calculated by the PEOE method can be used to extract the chemically feasible reaction from the two conceivable ones as illustrated in Figure 10.3-11. 

Figure 18.20 The two-dimensional NMR spectrum shown in Figure 18.17 was used to derive a number of distance constraints for different hydrogen atoms along the polypeptide chain of the C-terminal domain of a cellulase. The diagram shows 10 superimposed structures that all satisfy the distance constraints equally well. These structures are all quite similar since a large number of constraints were experimentally obtained. (Courtesy of P. Kraulis, Uppsala, from data published in P. Kraulis et ah. Biochemistry 28 7241-7257, 1989, by copyright permission of the American Chemical Society.)...

More simply, the chemical potentials appear as Legendre multipliers when G is minimised, associated with the constraints on total numbers of atoms, eg. (23). The results are ... 

The distorted sp3 angles at both carbon and sulfur atoms in small ring sulfoxides and sulfones approach their normal size beginning with the thianes. Consequently, the characteristics and chemical behavior of six- and higher-membered sulfoxides and sulfones are expected to be similar to those of the acyclic counterparts. However, in view of the constraints imposed by the cyclic array, three issues deserve study ... 

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