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Chemistry LibreTexts

Matrix Formulation

Matrix representation of the equations

The matrix representation of the above equations for the kinetic and potential energies is:

where F is the force constant matrix in internal coordinates, G-1 is the transformed inverse mass matrix, and S is a vector of internal coordinates. The normal coordinates are linearly related to the internal coordinates by

S = LQ

In which the transformation coefficients are chosen so that the energies in terms of the normal coordinates have the diagonal forms

Where L is a diagonal matrix whose elements are lk = 4p2n2 and E is the unit matrix. Therefore

The second equation implies that LT = L-1G which can be applied to the left-hand side of the first equation to yield

GFL = LL

which when multiplied on both the right and the left by L-1 gives

L-1GF = L L-1

whose transpose is

FG(L-1)T = (L-1)TL

The condition of compatibility is

|GF - Elk| = 0

This formulation of harmonic analysis problem is the standard matrix formulation.