Equivalent atoms

However, it should not be concealed that in some cases the Morgan Algorithm has problems. In some structures the numbering show oscillatoiy behavior (for an example, sec Figure 2-44). Moreover, although equivalent atoms obtain the same extended connectivity value, it cannot always be concluded that atoms with the same extended connectivity value are equivalent. 

As can be seen, this algorithm additionally identifies constitutionally equivalent atoms. These are atoms with the same EC value (3) in the final iteration, such as the two carbon atoms in the ortho positions of the phenyl ring with an EC value of 9 (Figure 2-44). 

In the next step, the neighbor atoms of the second atom (or the current atom plus 1), which are not yet labeled, are assigned in an equivalent manner. This is done for all the atoms arbitrai y decisions are made when numbering the equivalent atoms (Figure 2-45). 

A substructure search algorithm is usually the first step in the implementation of other important topological procedures for the analysis of chemical structures such as identification of equivalent atoms, determination of maximal common substructure, ring detection, calculation of topological indices, etc. 

The chirality code of a molecule is based on atomic properties and on the 3D structure. Examples of atomic properties arc partial atomic charges and polarizabilities, which are easily accessible by fast empirical methods contained in the PETRA package. Other atomic properties, calculated by other methods, can in principle be used. It is convenient, however, if the chosen atomic property discriminates as much as possible between non-equivalent atoms. 3D molecular structures are easily generated by the GORINA software package (see Section 2.13), but other sources of 3D structures can be used as well. 

Chemical equivalence atomic nuclei in the same chemical environment are chemically equivalent and thus show the same ehemieal shift. The 2,2 - and 3,3 -protons of a 1,4-disubstituted benzene ring, for example, are ehemically equivalent because of molecular symmetry. 

An answer was provided in 1931 by Linus Pauling, who showed how an s orbital and three p orbitals on an atom can combine mathematically, or hybridize, to form four equivalent atomic orbitals with tetrahedral orientation. Shown in Figure 1.10, these tetrahedrally oriented orbitals are called sp3 hybrids. Note that the superscript 3 in the name sp3 tells how many of each type of atomic orbital combine to form the hybrid, not how many electrons occupy it. 

Each equivalent atom (the same element, the same number of bonds and lone pairs) has the same formal charge. A check on the calculated formal charges is that their sum is equal to the overall charge of the molecule or ion. For an electrically neutral molecule, the sum of the formal charges is zero. Compare the formal charges of each possible structure. The structure with the lowest formal charges represents the least disturbance of the electronic structures of the atoms and is the most plausible (lowest energy) structure. 

STRATEGY hollow the procedure in. Toolbox 2.2. We need do only one calculation for equivalent atoms, such as the oxygen atoms in the first diagram, because they all have the same arrangement of electrons and hence the same formal charge. 

The coordinate positions for equivalent atoms provided by F 5 are1)... 

The C—C bond in ethane has no polarity because it connects two equivalent atoms. However, the C—C bond in chloroethane is polarized by the presence of the electronegative chlorine atom. This polarization is actually the sum of two effects. In the first of these, the C-1 atom, having been deprived of some of its electron density by the greater electronegativity of Cl, is partially compensated by drawing... 

In the case of malonic acid, CH2(COOH)2 propane, CH2Me2 or any other molecule of the form CH2Y2, if we replace either of the CH2 hydrogens by a group Z, the identical compound results. The two hydrogens are thus equivalent. Equivalent atoms and groups need not, of course, be located on the same carbon atom. For example, all the chlorine atoms of hexachlorobenzene are equivalent as are the two bromine atoms of 1,3-dibromopropane. 

In the following problems the positions of symmetrically equivalent atoms (due to space group symmetry) may have to be considered they are given as coordinate triplets to be calculated from the generating position x,y,z. To obtain positions of adjacent (bonded) atoms, some atomic positions may have to be shifted to a neighboring unit cell. 

The region within which k is considered (—n/a first Brillouin zone. In the coordinate system of k space it is a polyhedron. The faces of the first Brillouin zone are oriented perpendicular to the directions from one atom to the equivalent atoms in the adjacent unit cells. The distance of a face from the origin of the k coordinate system is n/s, s being the distance between the atoms. The first Brillouin zone for a cubic-primitive crystal lattice is shown in Fig. 10.11 the symbols commonly given to certain points of the Brillouin zone are labeled. The Brillouin zone consists of a very large number of small cells, one for each electronic state. 

Translationengleiche subgroups have an unaltered translation lattice, i.e. the translation vectors and therefore the size of the primitive unit cells of group and subgroup coincide. The symmetry reduction in this case is accomplished by the loss of other symmetry operations, for example by the reduction of the multiplicity of symmetry axes. This implies a transition to a different crystal class. The example on the right in Fig. 18.1 shows how a fourfold rotation axis is converted to a twofold rotation axis when four symmetry-equivalent atoms are replaced by two pairs of different atoms the translation vectors are not affected. 

Caesium chloride is not body-centered cubic, but cubic primitive. A structure is body centered only if for every atom in the position x, y, z there is another symmetry-equivalent atom in the position x+ j,y+ j,z+ j in the unit cell. The atoms therefore must be of the same kind. It is unfortunate to call a cluster with an interstitial atom a centered cluster because this causes a confusion of the well-defined term centered with a rather blurred term. Do not say, the 04 tetrahedron of the sulfate ion is centered by the sulfur atom. 

Identical means the very same . Avoid the expression identical atoms when you mean symmetry-equivalent or translationally equivalent atoms. It is impossible to lessen or to increase the term identical nearly identical is nonsense. 

The kinetic energy in this case is the Fermi energy = +(l/35.6)(h2/mR2). So, the total energy of the equivalent atom has the form ... 

Tab. 1.3 Comparison between different sets of atomic point charges for a zwitterionic Gly-Ala dipeptide in aqueous solution. D-RESP electrostatic potential derived charges [12] fitted to all 36 configurations. Hirshfeld average value of the Hirshfeld charges [89c] along the full trajectory, Amber AMBER 1995 force field [86], Gromos GROMOS96 force field [85], The charges of equivalent atoms are imposed to be equal.

---