GF Method - Normal Coordinates in Terms of 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 RA be the position vector of nucleus A and RA0 the corresponding equilibrium position. Then is by definition the Cartesian displacement coordinate of nucleus A. Wilson's linearizing of the internal curvilinear coordinates qt expresses the coordinate St 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 sAt 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).

Read more about this topic:  GF Method

Famous quotes containing the words normal and/or terms:

    “Perhaps the feelings that we experience when we are in love represent a normal state. Being in love shows a person who he should be.”
    —Anton Pavlovich Chekhov (1860–1904)

    “It is surely a matter of common observation that a man who knows no one thing intimately has no views worth hearing on things in general. The farmer philosophizes in terms of crops, soils, markets, and implements, the mechanic generalizes his experiences of wood and iron, the seaman reaches similar conclusions by his own special road; and if the scholar keeps pace with these it must be by an equally virile productivity.”
    —Charles Horton Cooley (1864–1929)