Mass-weighted force-constant matrix

Normal mode analysis is performed by diagonalizing the mass weighted force constant matrix, where V is a matrix representing... 

The relationship between the force constant matrix in Cartesian displacement coordinates Fy, and the force constant matrix for mass weighted Cartesian coordinates F can be written as follows (only the first three rows and columns of the matrices are explicitly shown) ... 

In mass-weighted coordinates, the hessian matrix becomes the harmonic force constant matrix, from which a normal coordinate analysis may be carried out to yield harmonic frequencies and normal modes, essentially a prediction of the fundamental IR transition... 

A normal-mode representation of the Hamiltonian for the reduced system involves the diagonalization of the projected force constant matrix, which in turn generates a reduced-dimension potential-energy surface in terms of the mass-weighted coordinates of the reaction path [64] ... 

In section 2.5, we saw that diagonalization of the force constant matrix gives an eigenvalue matrix whose elements are the force constants of the normal modes, and an eigenvector matrix whose elements are their direction vectors . Mass-weighting the force constants gives the wavenumbers ( frequencies ) of the normal-mode vibrations, and their motions can be identified by using the direction vectors to animate them. So... 

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