Mass-weighted force matrix

The eigenvalues and eigenvectors of the mass-weighted force matrix can be obtained by diagonalizing equation (21.5). Then each eigenvalue corresponds to its normal coordinates, Qj,... 

In Eq. (13), the vector q denotes a set of mass-weighted coordinates in a configuration space of arbitrary dimension N, U(q) is the potential of mean force governing the reaction, T is a symmetric positive-definite friction matrix, and , (/) is a stochastic force that is assumed to represent white noise that is Gaussian distributed with zero mean. The subscript a in Eq. (13) is used to label a particular noise sequence For any given a, there are infinitely many... 

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) ... 

One can simplify Equation 4.95 and obtain a very interesting result. We previously obtained the normal mode vibrational frequencies v by diagonalization of the matrix of the harmonic force constants in mass weighted Cartesian coordinates (Chapter 3). These force constants Fy were obtained from the force constants in Cartesian coordinates fq by using... 

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] ... 

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

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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