Water reducible representation

For the water molecule, a reasonable set of internal coordinates would be the lengths of the 0-H bonds - let us call them r and V2 - and the H-O-H angle, 9. Displacements of these coordinates form a basis for a reducible representation of C2v that is composed of symmetry species of vibrational motions only. 

In this expression, N is the number of times a particular irreducible representation appears in the representation being reduced, h is the total number of operations in the group, is the character for a particular class of operation, jc, in the reducible representation, is the character of x in the irreducible representation, m is the number of operations in the class, and the summation is taken over all classes. The derivation of reducible representations will be covered in the next section. For now, we can illustrate use of the reduction formula by applying it to the following reducible representation, I-, for the motional degrees of freedom (translation, rotation, and vibration) in the water molecule ... 

Derivation of reducible representation fa degrees of freedom in the water molecule... 

Consider the basis Fh constituted by the two orbitals Ishi ISH2 on the hydrogen atoms in the water molecule. From Table 6.1, we notice that the two orbitals are transformed into themselves by the operations E and ayz (characters equal to 1 +1 = 2), whereas they are interchanged by the operations Cf and (characters equal to 0+0 = 0). The characters associated with the basis Fh, listed in Table 6.7, do not correspond to any of the irreducible representations of the Czv point group, which are all one-dimensional (Table 6.5). This is therefore a basis for a reducible representation. 

The reducible representation for all motions of the water molecule is therefore reduced to... 

Reducible representation for the three translational modes of water. 

When looking for a reducible representation, simply think of a molecule with three vectors (x, y, and z) on every atom, like the water molecule in Figure 7-11. Now, as before when developing your transformation matrices, you need to see what happens to the vectors when doing each symmetry operation. However, there s one key difference You care only about atoms that do not move locations. If an atom appears somewhere different after performing the operation, do not count any of the three vectors. 

Remember, a reducible representation simply means it s composed of irreducible representations for water, every irreducible representation has a value of 1 under E. This means the reducible representation with a 9 under E... 

Before we start our analysis of the quality of the energy approximation by graph invariants, it is reasonable to look at their total numbers for different systems. Table 1 shows these numbers for some ice nanotubes INT . In particular, the dependence of the numbers of distinct graph invariants on the order of the invariant and the indices m and n is revealed. The values in the table can be found numerically by applying the graph invariant definition to a specific water cluster. Alternatively, one can determine these numbers analytically on the basis of symmetry considerations. Transformation of bond variables defines a (reducible) representation T of the symmetry group G of the cluster. The number of the h order invariants is simply the number of the totally symmetric representations in the h symmetric power of T. Thus, only characters of T are necessary to determine numbers in Table 1. 

Schematic representation of the cross section of the surface layer of a metal oxide. , Metal ions O, oxide ions. The metal ions in the surface layer (a) have a reduced coordination number. They thus behave as Lewis acids. In the presence of water the surface metal ions may first tend to coordinate H20 molecules (b). For most of the oxides dissociative chemisorption of water molecules (c) seems energetically favored.

Schematic representation of manganese nodule end-member morphologies. The size of the arrows Indicates the proportion and direction of metal supply, (a) Typical situation In the open ocean with the nodules lying on an oxidized sediment substrate dominant mode of formation Is hydrogenous, (b) Typical situation In nearshore and freshwater environments with nodules lying on a sediment substrate that Is partly reducing In character. Dominant supply of metals Is via Interstitial waters from below the substrate surface. Source From Chester, R. (2003). Marine Geochemistry, 2nd ed. Blackwell, p. 425.

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