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Subject: CCL:RE: Eigenvalues of hessian
Date: Fri, 30 May 2003 08:28:41 +0200
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Thread-Topic: Eigenvalues of hessian
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From: "Dillen Jan <jlmd:at:sun.ac.za>" <JLMD:at:sun.ac.za>
To: <chemistry:at:ccl.net>
Cc: "Daniel R. Rohr" <rohrd:at:students.uni-marburg.de>
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> 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