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Chemistry LibreTexts

Matrix Formulation

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  • 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.