Equivalence between atoms

In a sense, when considering equivalence between atoms in a molecule, diatomic hydrogen is the simplest situation. However, in reality, just the opposite is found to be true. 

The mass-to-charge ratio, m/n, of those ions that pass through the probe aperture and are analysed in the mass spectrometer is calculated from the equivalence between the potential energy of the atom on the specimen surface at voltage V0, and the kinetic energy that the atom acquires during acceleration to the grounded... 

For example, in a radical with six equivalent carbon atoms (e.g., benzene anion radical), the probabilities are P0 = 0.936, Pi = 0.062, P2 = 0.0017, P3 = 0.00003, etc. In practice, we would probably see only the central line with intensity 0.936 and a pair of satellites with intensity 0.031, corresponding to splitting of the center line by a single 13C (the intensity is distributed between the two resonances). 

Fig. 22 The linkages between triacontahedra and prolate rhombohedra in Tsai-type 2/1 ACs (adapted from Fig. 20) afford some important clues for QC modeling. A trigonal pyramid with particular angles (as marked) and geometric features is formed when equivalent apex atoms of four prolate rhombohedra are linked. The same property exists for Bergman-type 2/1 ACs...

This view was given further support by the discovery of "tautomerism," that is, that some compounds behave as if they have two different structures simultaneously. Peter Laar suggested in 1886 that this can best be explained as the result of continual oscillation of a hydrogen atom between two positions within a single molecule, 109 a hypothesis influenced by Kekule s suggestion that the peculiarities of benzene are the result of an oscillation of atoms in benzene and that "equivalence" or valence is "the relative number of contacts which occur in a unit of time between atoms." 110... 

In the AC-matrices the off-diagonal entries ey are the formal bond orders between atom pairs (Aj, A/). BE-matrices are obtained from the AC-matrices by augmentation with diagonal entries eu indicating the numbers of free valence electrons at the atoms Aj. The indices 1,. .., n can be assigned to the n atoms of a constitution in n different ways. Accordingly there are up to n different but equivalent connectivity lists, or AC- and BE-matrices, respectively. The direct identification and comparison of such representations is essential to Chemical Documentation. Only by uniquely assigning atomic indices can this be accomplished. 

The crystal structure of MPT synthase and the simultaneously determined NMR structure of the MoaD-related ThiS protein involved in thiamine biosynthesis [37] unambiguously demonstrated the evolutionary relationship between a subset of enzymes involved in the biosynthesis of S-containing cofactors (e.g. Moco, thiamine and certain EeS-clusters) and the process of ubiquitin activation. MoaD displays significant structural homology to human ubiquitin (Figure 3.3B and C), resulting in a superposition with a root mean square (rms) deviation of 3.6 A for 68 equivalent Ca atoms out of 76 residues in ubiquitin. The key secondary structure... 

Mulliken draws on the equivalence between two descriptions of a localized double bond, as (a) a tt and a bond superposed and (b) two bent a bonds, as applied, for example, to the case of bonding in ethylene (Hall and Lennard-Jones, 1951). In the present case a pseudo-double bond is visualized involving the tt orbital 4>r oii fhe atom r under attack and a pseudo-77 orbital formed by combining the valence orbitals of the... 

Similar forms can be obtained for coulombic interaction between neutral atoms or between atoms and ions by appropriate consideration of the integrated electron densities of the two species (in the case of interaction between two neutral atoms, the right sides of equations 1.146 and 1.147 are, respectively, equivalent to the nuclear charges of the two species Z and Zb). 

The simple cubic structme, sometimes called the rock salt structure because it is the structme of rock salt (NaCl), is not a close-packed structure (see Figure 1.20). In fact, it contains about 48% void space and as a result, it is not a very dense structure. The large space in the center of the SC structme is called an interstitial site, which is a vacant position between atoms that can be occupied by a small impurity atom or alloying element. In this case, the interstitial site is surrounded by eight atoms. All eight atoms in SC me equivalent and me located at the intersection of eight adjacent unit cells, so that there me 8 x (1/8) = 1 total atoms in the SC unit cell. Notice that... 

Ioffe and Regel (1960) were the first to point out that small values of l such that kl < 1 /2rc are in fact impossible. This point is developed further in the next section. Moreover, when fci l/2rc, which in metals is equivalent to the condition i a where a is the distance between atoms, the Boltzmann approximation becomes a bad one because k is no longer a good quantum number. This occurs particularly... 

The distinction between atomic orbitals and basis functions in molecular calculations has been emphasized several times now. An illustrative example of why the two should not necessarily be thought of as equivalent is offered by ammonia, NH3. The inversion barrier for interconversion between equivalent pyramidal minima in ammonia has been measured to be 5.8 kcal mol However, a HF calculation with the equivalent of an infinite, atom-centered basis set of s and p functions predicts the planar geometry of ammonia to be a minimum-energy structure ... 

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