Subject: Summary - Scaling of vibrational frequencies

From: Peter Gedeck <[email-masked]>
Date: Tue, 10 Jan 1995 13:42:20 +0100 (NFT)

I received a lot of responses for the following question:

>this time I would like to ask, why vibrational frequencies are usually
>scaled by a factor (0.89-0.9)?
>
>As far as I know, this is independent of the calculational method used
>(ab-initio or semiempirical). So, is it a deficiency of the theoretical
>approach or is it just a question of the accuracy of the calculation?

Thanks a lot to all who replied, 

Peter Gedeck <[email-masked]>

============================================================
Summary - Scaling of vibrational frequencies:
------------------------------------------------------------
Suggested literature:
o J. A. Pople, R. Krishnan, H. B. Schlegel, D. DeFrees, J. S. Binkley,
  M. J. Frisch, R. F. Whiteside, R. F. Hout and W. J. Hehre, Int. J.
  Quantum Chem., Quantum Chem. Symposium, 15 (1981) 269.

o MP2 scaling  D. J. DeFrees, J Comp Chem 82 (1985) 333.

o B.H. Besler, et.al., J. Chem. Phys. 89(1) (1988) 360.

o Possible sources of error in empirical scaling...  C. L. Janssen and
  H. F. Schaefer, J Chem Phys 95 (1991) 5128.

o J.F. Stanton, et.al., J. Chem. Phys. 94(1) (1991) 404.

o M. Flock and M. Ramek, Int J. Quantum Chem., 
  Quantum Chem. Symposium 27 (1993) 331-341.

o There is also a recent (ca. 1993) paper by Pople et al in Israel J Chem
  on scaling MP2 freqs.

o A. P. Scott et.al. Israel J. Chem. 33 (1993) 345. 

o Ab Initio Molecular Orbital Theory - by Hehre, Radom, Schleyer and Pople.
  (Wiley, New York, 1986)

Possible reasons for the deficiency of simple HF-calculations:
Vibrational frequencies are usually calculated from the normal mode 
frequencies using a harmonic osciallator model

 - Zero point energy
 - Anharmonicity in the vibrational potential energy surface 
 - Basis sets are too small 
 - neglect of electron correlation
 - the Hartree-Fock potential is too steep and therefore frequencies 
   too high.

============================================================
Following are all responses (edited):
------------------------------------------------------------
From: Dave Young <[email-masked]>

The answer is that all of these frequencies are being computed with a
harmonic oscilator approximation.  For high frequency modes, the
difference between the harmonic oscilator prediction and the exact or
Morse potential like behavior is about 10% . If you try to look at very
low frequency modes, below a few hundred wave numbers, you will see that
the frequencies calculated are off by a large amount. 

------------------------------------------------------------
From: "David W. Ewing (216) 397-4241" <[email-masked]>

There are three sources of error in the calculation of vibrational 
frequencies via ab initio methods: calculated frequencies are usually 
harmonic, basis sets are too samll, and electron correlation is neglected 
or inadequately treated.  For discussions of the last two factors, see

B.H. Besler, et.al., J. Chem. Phys. 89(1), 360 (1988).

J.F. Stanton, et.al., J. Chem. Phys. 94(1), 404 (1991).

------------------------------------------------------------
From: Anthony P Scott <[email-masked]>

The scaling of calculated harmonic vibrational frequencies to match the 
experimentally determined (anharmonic) vibrational frequencies
is designed to allow for the harmonic approximation that is used in the
theoretically determined values.

Our paper, Israel J. Chem. 1993, 33, 345 is a good place to start when
exploring this.

------------------------------------------------------------
From: Doug Fox <[email-masked]>

  The vibrational frequencies predicted by ab initio and semi empirical
methods are almost all harmonic approximations and limited in some respect
by the dissociation behaviour of the underlying method.  The first
approximation tends to produce values higher than experimental due to the
lack of anharmonic corrections.  The improper dissociation behaviour also
tends toward high estimates because most SCF based methods tend not to
dissociate and single configuration representations tend to be worse away
from equilibrium.

  The remarkable result of the above facts is that for a wide range of
molecules studied at the HF level a scaling of 0.89 or about about 12%
brings the frequencies into good agreement with experiment.  Much better
than attempting to correct the problem with high order correlation treatments.
A different scaling should be used for MP2 or higher order corrected methods
but as you noted often this is not done.

   There is a good bit of discussion of the results in "Ab Initio Molecular
Orbital Theory" by Hehre, Radom, Schleyer and Pople.  There are some recent
papers which are revisiting this issue.  Aue and co-workers have told me they
have one in press using MP2 results.

------------------------------------------------------------
From: "E. Lewars" <[email-masked]>

The general belief seems to be that it's because the calculations (which use the 
eigenvalues of a force constant matrix) assume the vibrations are harmonic.
However, it has been claimed that "this straightfoward looking consideration
is wrong..."; see M. Flock and M. Ramek, Int J. Quantum Chem., Quantum
Chem. Symposium 27, 331-341 (1993).  Some other refs to vib freq scaling are
Possible sources of error in empirical scaling...  C. L. Janssen and
H. F. Schaefer, J Chem Phys 1991 95 5128.
MP2 scaling  D. J. DeFrees, J Comp Chem 1985 82 333.
There is also a recent (ca. 1993) paper by Pople et al in Israel J Chem
on scaling MP2 freqs.

------------------------------------------------------------
From: Per-Ola Norrby <[email-masked]>

        It's a little of both.  If you look at earlier postings to this
list, you can see that the correction factors are different for different
levels of theory, so low level calculations are certainly slightly
deficient in the description of the energy hypersurface.  However,
vibrational frequencies are usually calculated from the Hessian with no
consideration of higher derivatives.  This gives an harmonic approximation,
resulting in a slightly to "hard" system and too high calculated
frequencies.
        Naturally, having one scaling factor for all frequencies at one
level of theory is an oversimplification, but I don't know of anyone who
tried anything more complicated.  If you do, it might not be "ab initio"
anymore...

------------------------------------------------------------
From: Csonka Gabor <[email-masked]>

The IR scaling factor IS method dependent. The 0.9
is for HF. For MP2 you should use different scaling.
For DFT or CCSD(T) you usually get the correct
harmonic frequencies within an error bar, so
no scaling is necessary. The scaling is mainly
for the zero point energy, and it may give wrong
results for individual freqs.

I refer to Hehre et al.: ad Initio MO Theory book
(Wiley, 1986) and the work P. Pulay and G. Fogarasy

------------------------------------------------------------
From: Adel El-Azhary <[email-masked]>

The ab initio frequencies calculated at the Hartree Fock level are usually
overestimated by about 10-20%. This is due to the incompletness of the 
basis set used, neglect of anharmonicity and neglect of the electron 
correlation. Frequencies calculated at the MP2 level of theory are 
overestimated by about 5-10%. This is due to inclusion of the electron 
correlation at the second level but higher excitations in the wave 
funcation are also neglected. You can look at the JPC, 1987, there is 
a paper by R. Amos about furan, pyrrole and thiophene. These is also 
a paper accepted for publication very recently by Petr Bour in the JPC
where frequencies were calculated at the HF, MP2 and MP2 anharmonic also.
This is in addition to the other references mentioned in the e-mail you
received through the CCL.

------------------------------------------------------------
From: Janet Del Bene <@vm.gmd.de:[email-masked]>

    Ab initio calculations of vibrational frequencies at the
Hartree-Fock level are too high compared to experimental frequencies.
The scaling (0.89) is an empirical adjustment that brings the computed
frequencies into better agreement with the experimental. That computed
Hartree-Fock frequencies are too high is a result of two factors:

1) the computed frequencies are based on a parabolic potential and
   are harmonic, whereas the experimental frequencies are anharmonic;
2) the Hartree-Fock potential is too steep and as a result, frequencies
   are too high.

------------------------------------------------------------
From: @uunet.uu.net:[email-masked] (AEleen Frisch)

The reference for the HF frequency scale factor is:

  J. A. Pople, R. Krishnan, H. B. Schlegel, D. DeFrees, J. S. Binkley,
  M. J. Frisch, R. F. Whiteside, R. F. Hout and W. J. Hehre, Int. J.
  Quantum Chem., Quantum Chem. Symposium, 15, 269 (1981).

Scaling is done to account for well-known, systematic errors in Hartree-Fock
frequencies due to its neglect of electron correlation.

------------------------------------------------------------
From: Kui Zhang <[email-masked]>

The fellowing paper and book will answer your question:

J.A. Pople et al, Ab Initio Molecular Orbital Theory (Wiley, New York, 1986).
H.F. Schaefer III et al, J. Chem. Phys. 95, 5128 (1991).

------------------------------------------------------------
From: Eric Bittner <[email-masked]>
 
The scaling looks like a fudge factor to compensate for zero point
energy and anharmonicity in the vibrational potential energy surface.
In most structure calculations, the vbrational frequencies are just
the normal mode frequencies...i.e harmonic classical motion.

------------------------------------------------------------
From: Patrick Bultinck <[email-masked]>

Well, you can't really call it a deficiency in the calculation, but there 
is a deficiency in the model used for the vibrations... it's the harmonic 
approximation. This way you get frequencies that are too big, and that's 
why we use a 0.89 scaling factor. Intensities are even worse, they use 
the double harmonic approximations... 

I think about every book on advanced QC will give some insight (Daudel 
e.g., Pople et al. "Ab Initio MO theory...)

------------------------------------------------------------
From: Earl EVLETH p 74208 <[email-masked]>

The scaling factor 0.89 came from HF 6-31G* level calculations.
Larger basis sets might change that scaling and MP2 level calculations
at the 6-31G* will change it into the 0.95 or 0.96 range. 

However, Pulay showed years ago when he did his modeling
of various structures, scaling factor depends on the type of vibration
one is dealing with, i.e. bond stretching, bond bending, and torsional
modes. He scaled force constants, not the final vibrational frequencies.
Therefore, the 0.89 magic number is just a convenient, practically
off-the-top-of-one's-head factor useable for HF 6-31G* calculations.
The computed low energy torsional modes are largely fiction and scaling
or not scaling them is a technical detail. As for nearly pure bending
and stretching (little coupling) these might be pretty good.  Some people
find that good old semiempirical calculations gives good unscaled results!

------------------------------------------------------------

Peter Gedeck
Inst. f. Physikalische Chemie I
Egerlandstrasse 3
91058 Erlangen
Germany

Tel: ++9131 - 85 7335  Fax: ++9131 - 85 8307
E-Mail: [email-masked]
WWW: http://pctc.chemie.uni-erlangen.de/~gedeck/gedeck.html