Transformation from Internal to Normal Co-ordinates

Transformation from Internal to Normal Co-ordinates.—It is an important property of the internal co-ordinates, in terms of which we wish to represent the force field, that they should be geometrically defined, i.e. their definition should be made only in terms of the internal distances between the atoms, and should in no way involve the atomic masses. This is necessary to ensure that the force constants are unchanged for different isotopic species. It is clear that 

For harmonic force field calculations it is usual to express the force field in terms of internal co-ordinates Ri defined through a linear transformation from cartesian displacement co-ordinates 8xm (a = x, y, or z) in the molecule-fixed axis system.24 The molecule-fixed axes are located using the Eckart conditions.24 The equations are written in the form 

The internal co-ordinates Rt are also related by a linear transformation to the normal co-ordinates, through the L matrix  

Wilson, J. C. Decius, and P. C. Cross, Molecular Vibrations , McGraw-Hill, New York, 1955. 

In terms of the co-ordinates Rt, 8am, or Q, the kinetic energy is represented by an effective mass matrix whose elements are constants. In a momentum representation this effective mass matrix is the G matrix, a diagonal matrix of the reciprocal atomic masses wm and a unit matrix 1, for the co-ordinates Ri, 8txm, and Qr, respectively. 

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