[MMTK] Orthogonality of normal modes

Konrad Hinsen research at khinsen.fastmail.net
Mon Apr 23 10:44:54 UTC 2012

Ramon Crehuet writes:

 > Thanks Ralf and Konrad for the explanation,
 > Let me see if I got it right:
 > * EnergeticNormalModes diagonalizes the hessian matrix (K) , in cartesian coordinates.
 > If temperature = None, eigenvectors are normalized, otherwise the norm is
 > temperature-dependent. Therefore dotProduct directly tests for orthogonalisation.

That's correct. You can also use


to get the orthogonal raw mode vectors that are never scaled by the
thermal fluctuation amplitudes.

That was the easy part.

 > * VibrationalNormalModes (or default NormalModes) diagonalizes the mass-weighted hessian
 > matrix : K' = 1/sqrt(M) K 1/sqrt(M))  where M are the masses of the atoms.


 > The resulting eigenvectors are expressed also in mass-weighted
 > cartesian coordinates.

True as well, but note that the modes as returned by MMTK are *not* the eigenvectors.
You can get the raw unmodified eigenvectors through


These mode vectors are in mass-weighted coordinates and not scaled by the thermal
amplitudes, so they are orthonormal.

If you use indexing to extract individual modes, i.e.


you get the eigenvectors converted to unweighted Cartesian
coordinates.  Moreover, unless you set temperature=None, the modes are
also scaled by the thermal fluctuation amplitudes.

 > Therefore one needs to use the massWeighted versions of dotProduct
 > and norm to work with them. In particular, a vector e' in
 > mass-weighted coords is related to the cartesian vector e as:
 > e'=1/sqrt(M)e

As Ralf pointed out, it's just the opposite: e' = sqrt(M) e.

 > So in order to have the eigenvectors of K' in cartesian corrdinates one should do
 > something like (given m1 and m2 as vibrational modes):

The standard (non-raw) modes are already in unweighted coordinates, so you can
just use them directly.

I hope I have not created any further confusion!

Konrad Hinsen
Centre de Biophysique Moléculaire, CNRS Orléans
Synchrotron Soleil - Division Expériences
Saint Aubin - BP 48
91192 Gif sur Yvette Cedex, France
Tel. +33-1 69 35 97 15
E-Mail: research AT khinsen DOT fastmail DOT net

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