Curvilinear internal coordinates

In order to evaluate these expansion coefficients, one needs only to be able to transform between the curvilinear internal coordinates and the Cartesian coordinates and back to the internal coordinates. We now outline the central steps in this procedure. 

A normal mode analysis can also be performed in curvilinear internal coordinates (Wilson et al., 1955). The approach is the same as that described above for Cartesian coordinates with one major modification. In an internal coordinate normal mode analysis, the internal coordinate cannot be simply scaled by the masses as is done for the Cartesian coordinates, Eq. (2.27), so that the masses become an explicit part of the eigenvalue problem. Thus, in an internal coordinate normal mode analysis, one does not solve Eq. (2.45), but instead solves the eigenvalue equation... 

To express curvilinear internal coordinates in terms of normal coordinafes, bond vectors in terms of normal coordinates are needed. A bond vector pointing from nucleus p to i i,p = 1,2,..., N and / p) in a molecule with N nuclei is given as... 

As was noted earlier, for a geometrical configuration that would correspond to a minimum on the PES to exist, the presence in this minimum of at least one vibration level is required. Therefore, after the verification of necessary conditions, the vibrational spectrum of the molecule can be found from the solution of the classical equations of motion with respect to the curvilinear internal coordinates q [7]. However, in practical calculations it is more convenient to make use of the cartesian coordinates for each atom i(i = 1,2,..., iV) in... 

The kinetic energy of a molecule is most easily expressed in (mass-adjusted) Cartesian coordinates. The potential energy has its most clear physical interpretation in (curvilinear) internal coordinates. When force constants are correlated to spectroscopic observables, mostly (dimensionless) normal... 

From the analysis provided, it is clear that curvilinear internal coordinates are by far the best choice to represent anharmonic force fields. Then the question arises whether one could choose a best set of internal coordinates to represent the stretching and bending motions of molecular systems. 

The calculation of Cartesian coordinates for a molecule distorted by a specified amount along one or some of the curvilinear internal coordinates is not trivial for polyatomic molecules, owing to the nonlinear nature of the transformation between the two coordinate systems (see Section 2.6). 

A simplified method exists" for setting up the required nonlinear coordinate transformations from curvilinear internal coordinates to simple normal coordinates. The transformation coefficients are called the L tensor elements, and the transformation equations can simply be written as... 

Generalized Partition Functions in Curvilinear Internal Coordinates... 

Potential and Vibrational Frequencies in Terms of Curvilinear Internal Coordinates. 

Path Dynamics in Curvilinear Internal Coordinates Including Torsions. 

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