Twelve Atoms

The functions Pcu(c,q) for the fee alloys can be averaged over the concentration c to obtain a probability Pcu(Q) that a Cu atom in any fee alloy will have a charge between q and q+dq. Recall that all of our calculations for fee were carried out with the same lattice constant. We approximate this function by averaging over the five concentrations that we considered, giving equal weight to all of them. It can be seen from the plot of this function in Fig. 5 that the probability is not uniform in q, but has thirteen prominent peeks. Since there are twelve atoms on the nearest-neighbor (nn) shell in a fee lattice, it is reasonable to write Pcu(q) as the sum of conditional probabilities Pcu(ci.q) where ci is the concentration of Cu atoms on the nn-shell. Five of the possible thirteen conditional probabilities are also plotted in... 

Another important polyatomic molecule is benzene, C6f I6, the parent of the aromatic compounds. In the molecular orbital description of benzene, all thirty C2s-, C2p-, and Hls-orbitals contribute to molecular orbitals spreading over all twelve atoms (six C plus six H). The orbitals in the plane of the ring (the C2s-, C2px, and ( 2/ -orbitals on each carbon atom and all six Hls-orbitals) form delocalized o-orbitals that bind the C atoms together and link the H atoms to the C atoms. The six C2pz-orbitals, which are perpendicular to the ring, contribute to six delocalized tt-orbitals that spread all the way around the ring. However, chemists... 

As the ring size is increased above twelve atoms, the number of per-... 

A formulation of structures by layers, such as that represented by the sequence (1) or (2), allows one to derive in a simple way the coordination of the atoms. In the case of perovskite, for example, the atom A of a layer (AX)C is surrounded by twelve atoms X, four located at the corners of the same mesh, and eight at the midpoints of the edges of the (BX2)0 layers above and below (AX)C. The coordination polyhedron is a cuboctahedron. In the case of the rock salt structure, the atom A of a layer (AX)C has coordination six, being surrounded by four atoms X at the corners of the same mesh, and by two atoms X at the center of the meshes (AX)0 above and below (AX)C. In this case the coordination polyhedron is an octahedron. 

The structure is based on a body-centered lattice. At each lattice point there is a small atom (Zn, Al). It is surrounded by an icosahedron of twelve atoms (Fig. 11-14). This group is then surrounded by 20 atoms, at the corners of a pentagonal dodecahedron, each atom lying directly out from the center of one of the 20 faces of the icosahedron. The next 12 atoms lie out from the centers of the pentagonal faces of the dodecahedron this gives a complex of 45 atoms, the outer 32 of which lie at the corners of a rhombic triacontahedron. The next shell consists of 60 atoms, each directly above the center of a... 

Figure 9.7. The twelve atoms of the reaction cage for F2 in Ar corresponding to the fee unit cell. The two cones show the initial orientations of the molecules that result in F atoms separation under photodissociation. (From Alimi et al. [1990].)...

Section 2 reviews the linear-combination-of-atomic-orbitals AIMD forces when the orbital occupations are allowed to vary [23]. Section 3 considers the effect of constraints on orbital occupancy on the dynamics of reaction between two linear 03 molecules [24]. Section 4 considers larger 1-D molecular collisions, the symmetric collision between two twelve-atom collections of van der Waals bonded NO dimers. Section 5 considers the role of bending in organic halide photodissociation using 3-D AIMD. 

Figure 9. The arrangement of the atoms in the / -W structure. There are eight atoms in the primitive cube. Two atoms are centered at (0, 0, 0) and (1/2, 1/2, 1/2). Each of these atoms in surrounded by twelve atoms at the corners of a slightly distorted icosahedron.

Twelve-atom cages are not widespread but play a dominant role in boron chemistry. The most highly symmetrical arrangement is the icosahedron (1-XII), which has... 

STM study which supports this suggestion [54]. The relatively large computing requirements for models involving perhaps twelve atoms per unit cell, and having less than three-fold rotational symmetry, were a barrier to investigating such models in the past, but our current computer resources makes their evaluation feasible, so we plan to test such models soon. 

Among of the systems under study, clusters with three, five, six and twelve atoms possess higher symmetries having more than a twofold rotational axis. As a result doubly degenerate librational states occur in these complexes. In Fig. 10 one of the possible representations of these degenerate states in the HCl(Ar)i2 cluster is displayed, however, any linear combination of these functions is also valid. Three pronounced recurrences shown in Fig. 10 demonstrate repeated collisions of hydrogen with the cluster. The situation becomes dramatically different for photolysis started from the first non-degenerate librationally excited state. In this... 

As a further illustration we will demonstrate here the use of this embedding to resolve a multiplicity case for the symmetry coordinates of a vibrating icosahedron [22]. An icosahedral cage with twelve atoms has 30 internal modes. Since the icosahedron is a deltahedron, the stretchings of the 30 edges form a non-redundant set of internal coordinates. The corresponding symmetry representations are given by ... 

Let us imagine a four-sided prism, of which the eight angles are occupied by eight atoms of carbon, and the centres of the twelve edges by twelve atoms of hydrogen. Let us call this prism the form or fundamental nucleus, and let us represent it by... 

In the presence of a small excess of iodide, Pd gives Pdt which is sparingly soluble in acidic media. When shaken with benzene or DIPE, PdE passes into the organic phase it has been the basis for a sensitive indirect determination method. The suspension of Pdh in the organic solvent is stripped by dilute ammonia. The aqueous solution is acidified, then the iodide is oxidized by bromine to IO3. This iodate reacts with added T to liberate iodine, which is determined as its coloured complex with starch (see Section 25.2.1). One Pd atom in PdE is equivalent to twelve atoms of iodine. 

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