Internal Coordinates and the FG-Matrix Method

We pointed out in the preceding section that a realistic potential energy function may not be easily expressible in Cartesian coordinates, but may be written more naturally in terms of 3JV generalized coordinates related to the mass-weighted coordinates by a linear transformation (6.40). In fact, only 3N — 6 such coordinates are required to fully specify the potential (3N — 5 in linear molecules) 2K is not sensitive to the center-of-mass position or molecular orientation in space, and a polyatomic molecule exhibits only 3N — 6 (3N — 5) independent bond lengths and bond angles. Such a truncated set of 3N — 6 (iN — 5) generalized coordinates is called an internal coordinate basis, and is commonly denoted S. To illustrate how an internal coordinate basis may be used to evaluate normal modes, we consider the bent H2O molecule in Fig. 6.3. The three internal coordinates are conveniently chosen to be 

This potential assumes that independent Hooke s law restoring forces are 

The force constant matrix need not be diagonal (i.e., more sophisticated potentials may be used). For a bent molecule with three nuclei, an arbitrarily accurate force constant matrix will still be a 3N — 6) x (3N — 6) = 3 x 3 matrix. 

To obtain a secular determinant equation analogous to (6.22) in the internal coordinate basis, both 2Tand 2V must be expressed in terms of S. Since 2T is readily given in terms of mass-weighted coordinates i, we need a transformation of the form 

Note that because S and i have 3N — 6 and 3N elements respectively, D is a rectangular (rather than square) matrix with 3N — 6 rows and 3N columns. The 

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