Subject: RE: Eigenvalues of hessian

From: "Daniel R. Rohr" <rohrd<<at>>students.uni-marburg.de>
Date: 29 May 2003 11:54:00 +0200

Hi CCLer

 Maybe I should clearify my question:
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. That#s why I talk of the eigenvalues.
In an excat method there should be exactly 3N-6 nonzeroeigenvalues and 6
exact zeros. ANY method I tested (HF, DFT and MP2) with ANY Program I
tested (Gaussian, Turbomole, Jaguar) gives me 3N nonzero Eigenvalues.
The question is now. Why are there too many nonzero Eigenvalues? In my
eyes there are two possible reasons:

1. The method is ok but there are errors made in the derivations of the
method. Maybe neglecting terms in the differentiation. Or maybe
asumptions that hold true only for stationary points.

2. The method is not ok. This means that I do get different Energies for
different orientations of my molecule (As is the case in DFT)

Of course it is also possible that there is a mixture of both. I would
like to know which error is the bigger one? And maybe get a reference
for this question.

Another point which I do not understand:
Some answers state that there is only little translation and rotation
since the frequencies are close to zero. Is the absolute value of an
eigenvalue a measure for the quantity (whaterver that means) of its
eigenvector???? And if that is the case why is there "more" translation
and rotation at nonstationary points? I get "frequencies" (this is
eigenvalues of the Hessian) between 400 and 850 cm^-1 for Water with HF
6-31G** (Gaussian calculation):

Low frequencies --- -410.8545   -0.0007   -0.0006    0.0017  489.4045
853.6142
 Low frequencies --- 1891.8502 3808.4304 3946.8117

These are of course the frequencies calculated BEFORE projection of
translation and rotation.

I hope you can help me on that

Daniel Rohr

Hi Daniel,

Your question is a bit tricky in that the answer depends on the method
you
are using.
 
If you are calculating Hartree-Fock or MP2 hessians then I think you
should
get 6 zero eigenvalues and 3N-6 non-zero eigenvalues at a stationary
point.
Even if the hessian is obtained by numerical differentiation of
gradients
this should come out allright, otherwise the code is broken.

If you are doing a DFT calculation life typically gets a bit more
complicated. If the DFT hessian is implemented as a proper second
derivative
of the energy expression with respect to the nuclear coordinates you
should
still get 6 zero eigenvalues. However this requires that the code takes
the
2nd derivatives of the Becke weights (A.D. Becke, J.Chem.Phys. 88 (1988)
2547-2553) into account as well. Some codes don't bother calculating
these
terms in which case the strict translational invariance of the hessian
is
violated. This typically expresses itself in 3 additional non-zero but
small
valued eigenvalues. 

If you are calculating the vibrational modes by numerical
differentiation of
DFT gradients then there may still be a problem if your molecule has
symmetry. Many quantum chemistry packages will reorientate your molecule
such that the main axes of ineratia coincide with the Cartesian axes of
your
coordinate system. This means that different points in the numerical
differentiation algorithm may be calculated at different orientations.
However, the DFT quadratures are not always invariant for rotations. For
example if you want to do calculations on H2 and you are going to use 6
points per H atom all placed on the Cartesian axes than the energy
you'll
get for the following two orientations will not be the same even if the
bond
distance is:

    H  H
    |   \
    H    H

Again this will lead to additional non-zero but small eigenvalues. In
this
case it may help to switch the use of symmetry off, so that all points
are
calculated in exactly the same orientation. 

As you have given very little details about exactly which calculation
you
are doing I can not say for sure which of the above reasons is the most
like
cause. However I think it is most likely one of these and with some
experimentation and a good look at the documentation for your program
you
should be able to find out what is happening.

Best wishes,

   Huub van Dam

-----Original Message-----
From: Daniel R. Rohr
To: ccl
Sent: 5/28/03 3:20 PM
Subject: CCL:Eigenvalues of hessian

Hi CCLers

 Does anybody of you know, why there are more than 3N-6 nonzero
eigenvalues of the hessian? I am especially interested in the question,
whether this error is due to the errors made by differentiation or
whether they are due to an error within the method itself. I usually get
only 3 zero eigenvalues. No matter if the hessian is calculated on a
stationary point or not.

Thanks for your help

Daniel Rohr

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---

From: "Shobe, Dave" <dshobe{at}sud-chemieinc.com>
Date: Thu, 29 May 2003 09:55:48 -0400

What I said earlier was made on the assumption of being at a stationalry
point.  

I have always heard that a freq calculation is unreliable at non-stationary
points, but I don't remember (and perhaps I never knew) the reason for this
warning.  

--David Shobe
Süd-Chemie Inc.
phone (502) 634-7409
fax     (502) 634-7724
email  dshobe{at}sud-chemieinc.com

Don't bother flaming me: I'm behind a firewall.

-----Original Message-----
From: Daniel R. Rohr [mailto:rohrd{at}students.uni-marburg.de]
Sent: Thursday, May 29, 2003 5:54 AM
To: ccl
Subject: CCL:Eigenvalues of hessian

Hi CCLer

Daniel Rohr

Hi Daniel,

    H  H
    |   \
    H    H

Best wishes,

   Huub van Dam

-----Original Message-----
From: Daniel R. Rohr
To: ccl
Sent: 5/28/03 3:20 PM
Subject: CCL:Eigenvalues of hessian

Hi CCLers

Thanks for your help

Daniel Rohr

 eMail: rohrd{at}stud-mailer.uni-marburg.de        Germany

Send your subscription/unsubscription requests to:
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