Diagonalization of the internal kinetic energy

It is often useful to transform from simple Cartesian coordinates to other sets of coordinates when we study collision processes including chemical reactions. In a collision process, it is obvious that the relative positions of the reactants are relevant and not the absolute positions as given by the simple Cartesian coordinates. It is therefore customary to change from simple Cartesian coordinates to a set describing the relative motions of the atoms and the overall motion of the atoms. For the latter motion the center-of-mass motion is usually chosen. In the following we will describe a general method of transformation from Cartesian coordinates to internal coordinates and determine its effect on the expression for the kinetic energy. 

R is a column vector with N elements R, r is a column vector with N elements rt, and A is an A x A square matrix with constant elements Ai j. The new momenta Pj may be determined using the definition in Eq. (4.61). From Eq. (D.l) we obtain 

The N x N matrix AmT1 AT will in general not be diagonal, so there will be cross terms of the kind PiPj in the expression for the kinetic energy. This may sometimes be inconvenient, and we shall see in the following how one may choose the matrix A in such a way that the kinetic energy is still diagonal in the new momenta. This leads to the so-called Jacobi coordinates that are often used in reaction dynamics calculations. 

First we want to single out the overall motion of the system, where all atoms move by the same amount, so all distances will be preserved. This is done by introducing the following condition on the matrix elements in A  

If all particles are displaced by the amount u to the position r + u, then coordinates i i. R v are seen from Eq. (D.l) not to change because of the condition in Eq. (D.7), while R is displaced by u. R, . Rn i are thus internal coordinates, whereas Rn describes the overall position of the system. For the momenta, we get 

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