Appendix an example of coordinate transformation

In a series of papers dealing with different types of organic families, Jorgensen and his groups have developed the optimized potentials for liquid simulation (OPLS), a force field construction accurately calibrated to reproduce the structure and energetics of organic liquids in Monte Carlo simulations these are reviewed in Section 9.5.2. Another attempt at deriving an empirical force field for molecules in isolation and in condensed phases has been presented by Sun [27]. A compilation of parameters for force field calculations is available [28]. 

Consider a bent triatomic molecule such as water. Fig. 2.1. Each atom is described by a set of three cartesian coordinates, which form an array of nine numbers Xi,i— 1 to 9, and each displacement during a vibration is called 5xi. The atomic mass corresponding to each coordinate is called tm. Defining mass-weighted coordinates qi and writing the total kinetic energy T for the motions during the displacements yields  

The sum of kinetic and potential energy for a classical dynamic system is called the lagrangian, L = T + V, and Newton s equations of motion, one for each coordinate gk, are written as 

This is a system of nine linear equations in nine unknowns, the AjS. It has non-trivial solutions (solutions for which not all AiS are zero) only if the following condition is satisfied  

For the i-th eigenvalue Xi, back substitution into the system 2.20 gives a set of nine coefficients, Ajk. These are the coefficients of the linear combination of the starting mass-weighted cartesian coordinates, giving the normal coordinates Q  

---