Atomic equivalence indices

In this section the results obtained from AIMD-B-LYP simulations by energizing three isomers of Li9" clusters will be analyzed [39]. For this purpose three types of quantities will be used. One of them is the root-mean-square (RMS) bond-length fluctuation 8 [25], which is calculated at the end of trajectories. A sharp increase of the 8 value is known as the Lindemann criterion for bulk melting, while in the context of finite-size clusters it can be taken as an indication of transition from solid-like to liquid-like state. We find it particularly instructive to analyze the trajectories in terms of atomic equivalence indexes ,... 

Figure 2.6 Short-time (0.25 ps) averaged kinetic energy per atom (KE) (a) and atomic equivalence indexes a,(r) (b) as functions of time for trajectories initialized by distorting the D4d structures (isomer I) of the cluster. For convenience, the corresponding total energies per atom (in...

The group-subgroup relation of the symmetry reduction from diamond to zinc blende is shown in Fig. 18.3. Some comments concerning the terminology have been included. In both structures the atoms have identical coordinates and site symmetries. The unit cell of diamond contains eight C atoms in symmetry-equivalent positions (Wyckoff position 8a). With the symmetry reduction the atomic positions split to two independent positions (4a and 4c) which are occupied in zinc blende by zinc and sulfur atoms. The space groups are translationengleiche the dimensions of the unit cells correspond to each other. The index of the symmetry reduction is 2 exactly half of all symmetry operations is lost. This includes the inversion centers which in diamond are present in the centers of the C-C bonds. 

Diamond. In this structure (see Chapter 7) all the atoms are equivalent each atom being surrounded by a perfect tetrahedron of four other carbons, forming with each one of them a localized two-electron bond. Diamond has a high density and refraction index and thermal conductivity and the highest melting point ( 4000°C) of any element. 

It is further possible to describe a given constitution by mapping the indexed set of atoms onto the independently indexed graph. The indexed set of atoms must be partially symmetrized with regard to the element equivalence classes of the atoms and the indices of the graph with respect to its symmetry. That constitution is defined as the reference constitution in which the atomic indices a and the graph indices g match. 

Configurations of Ruch s class A occur pairwise, e.g., asymmetric C-atoms. The parities of their permutation descriptors can be used as their -parity descriptors. A permutation containing an even number of transpositionsP has a parity of +1, conversely, odd permutations have parities of — 1. Since a-atom RASI (see Section 5.4.1) correspond in most casses to the CIP sequences, an assignment of +1 to an asymmetric C-atom is equivalent to an R-configuration, and — 1 to the S, if the skeleton is indexed as shown in 25. 

Figure 9 shows a correlation between the heats of atomizations [37] of two families of unbranched hydrocarbons and the Randic connectivity index of their equinumerical (equivalent) caterpillars. 

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