· For a polyatomic molecule, the normal mode frequencies are found from the matrix of second derivatives (the Hessian matrix).

· It is important that the second derivatives are mass-weighted - this is to take into account the fact that a given force will different effects on atoms of different masses.

· For example, the zero-point energy is different for a proton than for a deuteron in the same type of bond, due to the difference in mass. Mass-weighting the Hessian takes care of these effects.

· The mass-weighted Cartesian coordinates (x_m, y_m, z_m) of any atom are related to its Cartesian coordinates (x, y, z) by:

x_m = m^1/2x y_m = m^1/2y z_m = m^1/2z (3)

where m is the mass of the atom.

Next: the mass-weighted Hessian matrix