Potential Fields and Force Constants

Using Eqs. 11.1 and 12.1, we write the potential energy of a bent XY2 molecule as 

This type of potential field is called a generalized valence force (GVF) field. It consists of stretching and bending force constants, as well as I he interaction force constants between them. When using such a potential field, four force constants are needed to describe the potential energy of a bent XY2 molecule. Since only three vibrations are observed in practice, it is impossible to determine all four force constants simultaneously. One method used to circumvent this difficulty is to calculate the vibrational frequencies of isotopic molecules (e.g., DjO and HDO for H2O), assuming the. same set of force constants.t This method is satisfactory, however, only for simple molecules. As molecules become more complex, the number of interaction force constants in the GVF field becomes too large to allow any reliable evaluation. 

In another approach, Shimanouchi introduced the Urey-Bradley force (UBF) field, which consists of stretching and bending force constants, as well as repulsive force constants between nonbonded atoms. The general form of the potential field is given by 

Here Ar, Aa,-, and A j are the changes in the bond lengths, bond angles, and distances between nonbonded atoms, respectively. The symbols X, XI, 

HU and Fj, F represent the stretching, bending, and repulsive force constants, respectively. Furthermore, n, r and q, are the values of the distances at the equilibrium positions and are inserted to make the force constants dimensionally similar. 

The normal modes of vibration of the tetrahedral XY4 molecule are shown in Eig. 2.12 of Chapter 2. It can be seen that the normal modes, Vi and V3, correspond to the symmetry coordinates and 7 2i,(- 2) of Eq. 1.147, respectively, and that the 

Here Ar Aa,-, and Aqi, are the changes in the bond lengths, bond angles, and distances between nonbonded atoms, respectively. The symbols X, X, Hi, Hi and F,-, F,- represent the stretching, bending, and repulsive force constants, respectively. Furthermore, r r,o , and q, are the values of the distances at the equilibrium positions and are inserted to make the force constants dimensionally similar. Linear force constants (F,-, //, and F) must be included since Ar Aa and Aqt are not completely independent of each other. 

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THEORY OF NORMAL VIBRATION 1-13. POTENTIAL FIELDS AND FORCE CONSTANTS... 

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