tedious vector algebra to derive the corresponding values of the . See for more details on this method, known as the Wilson s-vector method, the book by Wilson et al., or molecular vibration. Now, which can be inverted and put in summation language: Here D is a (3N − 6) × 3N matrix, which is given by (i) the linearization of the internal coordinates s (an algebraic process) and (ii) solution of Wilson's GF equations (a numeric process). Matrices involved in the analysis There are several related coordinate systems commonly used in the GF matrix analysis. These quantities are related by a variety of matrices. For clarity, we provide the coordinate systems and their interrelations here. The relevant coordinates are: Cartesian coordinates for each atom Internal coordinates for each atom Mass-weighted Cartesian coordinates Normal coordinates These different coordinate systems are related to one another by: , i.e. the matrix transforms the Cartesian coordinates to (linearized) internal coordinates. i.e. the mass matrix transforms Cartesian coordinates to mass-weighted Cartesian coordinates. i.e. the matrix transforms the normal coordinates to mass-weighted internal coordinates. i.e. the matrix transforms the normal coordinates to internal coordinates. Note the useful relationship: These matrices allow one to construct the G matrix quite simply as Relation to Eckart conditions From the invariance of the internal coordinates St under overall rotation and translation of the molecule, follows the same for the linearized coordinates stA. It can be shown that this implies that the following 6 conditions are satisfied by the internal coordinates, These conditions follow from the Eckart conditions that hold for the displacement vectors, References Further references Spectroscopy Molecular physics Quantum chemistry