Internal displacement coordinate water molecule

The characters Xj for the examples in the previous section were calculated following the method described in Section 8.9, that is, on the basis of Cartesian displacement coordinates. Alternatively, it is often desirable to employ a set of internal coordinates as the basis. However, they must be well chosen so that they are sufficient to describe the vibrational degrees of freedom of the molecule and that they are linearly independent The latter condition is necessary to avoid the problem of redundancy. Even when properly chosen, the internal coordinates still do not usually transform following the symmetry of the molecule. Once again, the water molecule provides a very simple example of this problem. 

Entropy effects. The replacement of the coordination shell of the cation by a multidentate ligand has also the very important effect of decreasing the free energy of the system by the increase in translational entropy of the displaced water molecules. If there were no variation in solvation and internal entropies of the free ligand and of the complexes, the increase in translational entropy would amount to about 8 e.u., where x is the number of displaced solvent molecules minus one (38). This estimate is, however, very inaccurate large deviations are expected, especially in the case of complicated multidentate ligands for which complex formation may produce appreciable internal and solvation entropy changes. 

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. 

Displacements of the internal coordinates of the water molecule, the two O—H distances (dx and d2)y and the HOH angle (A0) are bases for the following irreducible representations ... 

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