Cartesian coordinates, kinetic energy mass-weighted

It is convenient to define a set of 3N mass-weighted Cartesian displacement coordinates q, q2,..., q N such that the first three q s are the components of Qi, the fourth, fifth and sixth q s are the components of Q2, and so on. The kinetic energy T can therefore be written... 

Equation (3.16) presents expressions for the kinetic T and potential V energies in Cartesian mass weighted displacement coordinates x( and corresponding velocities x(... 

In classical terms, if we use the mass-weighted Cartesian displacement coordinates, the kinetic energy of the moving nuclei isf... 

Equation (2.24) shows that, because of the kinetic energy coupling, in internal coordinates the energy is not separable at small displacements. Thus, it is necessary to search for 3N new coordinates (2 called normal mode coordinates, which define a new column vector Q. To begin it will be assumed that the mass-weighted Cartesian coordinates and normal mode coordinates are related by a linear transformation of the form... 

Use of Mass-weighted Coordinates. In this treatment ordinary cartesian displacement coordinates have been used, whereas in the earlier part of the chapter mass-weighted cartesian coordinates were emplo3 ed. If ([i = (mi) i Axi and 52 = (m.2) Aa 2, the kinetic and potential energies become... 

In terms of mass-weighted cartesian coordinates, the kinetic energy is... 

For a spherical penetrant, the energy at each voxel can be used to calculate each rate constant through numerical integration of Eq. (16). The kinetic energy contributions to each partition function are integrated analytically in mass-weighted Cartesian coordinates. 

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 ... 

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