GF Method - Normal Coordinates in Terms of Cartesian Displacement Coordinates | Normal Coordinates Terms Cartesian Displacement Coordinates

Normal Coordinates in Terms of Cartesian Displacement Coordinates

Often the normal coordinates are expressed as linear combinations of Cartesian displacement coordinates. Let R_A be the position vector of nucleus A and R_A0 the corresponding equilibrium position. Then is by definition the Cartesian displacement coordinate of nucleus A. Wilson's linearizing of the internal curvilinear coordinates q_t expresses the coordinate S_t in terms of the displacement coordinates

$S_t =\sum_{A=1}^N \sum_{i=1}^3 s^t_{Ai} \, d_{Ai}= \sum_{A=1}^N \mathbf{s}^t_{A} \cdot \mathbf{d}_{A}, \quad \mathrm{for}\quad t = 1,\ldots,3N-6,$

where s_At is known as a Wilson s-vector. If we put the into a 3N-6 x 3N matrix B, this equation becomes in matrix language

The actual form of the matrix elements of B can be fairly complicated. Especially for a torsion angle, which involves 4 atoms, it requires 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,

$\mathbf{Q} = \mathbf{L}^{-1} \mathbf{s} = \mathbf{L}^{-1} \mathbf{B} \mathbf{d} \equiv \mathbf{D} \mathbf{d}.$

In summation language:

$Q_k = \sum_{A=1}^N \sum_{i=1}^3 D^k_{Ai}\, d_{Ai} \quad \mathrm{for}\quad k=1,\ldots, 3N-6.$

Here D is a 3N-6 x 3N matrix which is given by (i) the linearization of the internal coordinates q (an algebraic process) and (ii) solution of Wilson's GF equations (a numeric process).

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