CCL:RE: Eigenvalues of hessian

> I do know that in the Hessians calculated, there is usually
 > translation
 > and rotation. I want to avoid speaking of frequencies, because we are
 > talking about Hessians which are not necessarily calculated at
 > stationary points.
 Dear Daniel,
 The fact that frequencies are equal to the square
 root of the diagonalised mass-weighted Hessian
 matrix is a result obtained from solving the following
 equations of motions:
 	F = m.a = - H.dx
 where 'm' is a diagonal mass matrix, 'a' the acceleration,
 'H' the Hessian, and 'dx' are the displacement coordinates.
 The 'H.dx' appears if one assumes that (1) the potential energy
 can be written as a harmonic function of the displacement
 coordinates and (2) that the gradient vanishes, i.e.
 	U = U(0) + 1/2 (dx)^T.H.dx
 (1) is called the harmonic approximation and (2) implies
 a stationary point. Hence "frequencies" which are not
 calculated at a stationary point do not necessarily have
 zero translational and rotational components.
 Here is an alternative reasoning. The potential energy
 should not change if the molecule is translated, i.e.
 	dU/dX = 0
 with 'X' are the coordinates of the centre of mass. However,
 	dU/dX = sum (dU/dx) (dx/dX)
 where 'x' are atomic coordinates. More complex expressions
 appear for the second derivatives. Obviously, dX/dx is
 not zero (and hence also dx/dX), so dU/dx must be zero in
 order to have dU/dX = 0, i.e. a stationary point. (or
 if the sum of these terms happens to be exactly zero)
 Hope this helps.
 Groeten
 Jan Dillen