Fernando's answer is basically correct, but I feel urged to add a few things, since my PhD research had been centered around the topic of anharmonicity. There is one thing that I am not satisfied with in his answer, and that is that the deviation between the harmonic and anharmonic (true) vibrational band positions is "systematic". I would say that it is very much not.

The anharmonic corrections that need to be applied to a harmonic band prediction are not determined by a single factor, but by multiple constants. In a first approximation, these constants are related to the non-quadraticity of the vibrational potential, and couplings to all other vibrational modes in the system. All of these effects can have different signs and partly cancel out or add to each other, moving the harmonic band around to lower or higher wavenumbers. Where it comes to rest in the end is a question that is very sensitive to these different numbers, and these effects are also non-linear in the relevant vibrational quantum numbers. What you're doing with scaling factors is to approximate this non-linear sum of values with a single scaling factor, and it should be clear that this is a very crude approximation which only seems to work if you don't demand too much from the results. The non-linearity of the true expressions for the vibrational states furthermore means that you need different scaling factors if you want to optimize your scaled harmonic data for different properties of your molecule (zero-point energy, or fundamental vibrations, or perhaps overtone vibrations, ...). You also won't be able to say anything about combination bands, band resonances, or anything like that, because those are all true anharmonic effects which do not appear in harmonic theory. And furthermore, just because you shifted your fundamental frequency around doesn't mean that the harmonic approximation suddenly works any different - it still assumes equidistant vibrational levels, and thermodynamic analyses based on such will still be fundamentally wrong in that regard. And of course, anharmonic effects not only pertain to the band positions, but also to their intensities, and scaling will change absolutely nothing about those.

Coming back to the "non-systematic" aspect of the deviations, consider the fact that different vibrations also have very different anharmonic characteristics: some feel a very large overall impact from it, others don't. There are schemes which apply different scaling factors across the range of your spectrum to capture this, which gives improved results than applying a single factor to all bands. Look into the second piece of literature cited by Fernando for details about that. You will see that for low-frequency vibrations (Table 7), some factors become even larger than unity, meaning that the bands should be shifted towards higher, not lower frequencies, as demanded by more general scaling factors for all fundamental vibrations (Table 2)! This in itself should already tell you how crude the approximation behind scaling factors really is.

As always, it depends a bit on what you really want to do. It's the same thing as anywhere in science - you should know what your approximations do, how well they perform and what they can and can't tell you. I myself am very much opposed to scaling harmonic frequencies, but of course, this is the view of a purist who has really gone into the theory behind anharmonic effects. If you are looking for very rough comparisons to experiment within a limited scope of your spectra, you may be fine. But if your systems allow it, consider anharmonic calculations as a much better and physically sound alternative.

Dear Anthony,

Harmonic spectra have systematic deviation so, in principle, this data can be corrected through scaling parameters. Do not forget we are approximating the potential energy surface to a parabola and it is a source of error (harmonic frequencies of water are 1664.957, 3800.964 and 3914.247 cm^-1, whereas anharmonic frequencies are: 1615.189, 3629.215 and 3726.616 cm^-1, at b3lyp/6-31g(d,p) level, the three harmonic values are greater than the anharmonic ones)

As reference, I recommend you to read:

J. Phys. Chem. 1996, 100, 16502-16513

J. Phys. Chem. A, 2007, 111 (45), pp 11683–11700

and the articles cited in these works.

Hope it helps,

Fernando

Fernando's answer is basically correct, but I feel urged to add a few things, since my PhD research had been centered around the topic of anharmonicity. There is one thing that I am not satisfied with in his answer, and that is that the deviation between the harmonic and anharmonic (true) vibrational band positions is "systematic". I would say that it is very much not.

The anharmonic corrections that need to be applied to a harmonic band prediction are not determined by a single factor, but by multiple constants. In a first approximation, these constants are related to the non-quadraticity of the vibrational potential, and couplings to all other vibrational modes in the system. All of these effects can have different signs and partly cancel out or add to each other, moving the harmonic band around to lower or higher wavenumbers. Where it comes to rest in the end is a question that is very sensitive to these different numbers, and these effects are also non-linear in the relevant vibrational quantum numbers. What you're doing with scaling factors is to approximate this non-linear sum of values with a single scaling factor, and it should be clear that this is a very crude approximation which only seems to work if you don't demand too much from the results. The non-linearity of the true expressions for the vibrational states furthermore means that you need different scaling factors if you want to optimize your scaled harmonic data for different properties of your molecule (zero-point energy, or fundamental vibrations, or perhaps overtone vibrations, ...). You also won't be able to say anything about combination bands, band resonances, or anything like that, because those are all true anharmonic effects which do not appear in harmonic theory. And furthermore, just because you shifted your fundamental frequency around doesn't mean that the harmonic approximation suddenly works any different - it still assumes equidistant vibrational levels, and thermodynamic analyses based on such will still be fundamentally wrong in that regard. And of course, anharmonic effects not only pertain to the band positions, but also to their intensities, and scaling will change absolutely nothing about those.

Coming back to the "non-systematic" aspect of the deviations, consider the fact that different vibrations also have very different anharmonic characteristics: some feel a very large overall impact from it, others don't. There are schemes which apply different scaling factors across the range of your spectrum to capture this, which gives improved results than applying a single factor to all bands. Look into the second piece of literature cited by Fernando for details about that. You will see that for low-frequency vibrations (Table 7), some factors become even larger than unity, meaning that the bands should be shifted towards higher, not lower frequencies, as demanded by more general scaling factors for all fundamental vibrations (Table 2)! This in itself should already tell you how crude the approximation behind scaling factors really is.

As always, it depends a bit on what you really want to do. It's the same thing as anywhere in science - you should know what your approximations do, how well they perform and what they can and can't tell you. I myself am very much opposed to scaling harmonic frequencies, but of course, this is the view of a purist who has really gone into the theory behind anharmonic effects. If you are looking for very rough comparisons to experiment within a limited scope of your spectra, you may be fine. But if your systems allow it, consider anharmonic calculations as a much better and physically sound alternative.

Mr. Nash,

The factor originates from the experiment. For example, 0.8929 is the factor applicable to HF computations of the frequencies.

Yes, scaling is purely empirical proceedure, originating from experiment, and have no any theoretical justification.

Thank you all for your responses. They have been very insightful. 

The vibrational frequencies scaling is simple yet effective procedure to obtain zero point energies or IR/Raman fundamental frequencies. It works in almost all cases, the exceptions are rare. It cannot be just a coincidence between many experimental and theoretical data, it must have a reason behind. For example, Hartree-Fock (HF) method lacks electron correlation, which in turn makes bonds softer and longer. One may correct this behaviour 1) by calculating Hessian matrix for the experimental molecular geometry [1] or 2) just applying scaling factor to all frequencies [2]. Indeed, scaling factors tend to approach 1.0 for the correlated electron methods. Anharmonic potential energy surfaces obtained via HF do not provide better agreement between experimental and calculated frequencies.

I also agree with Matthias Heger, that anharmonic corrections are not determined by a single factor, but by multiple constants and these constants are related to the non-quadraticity of the vibrational potential, and couplings to all other vibrational modes in the system. However, harmonic approximation quality is fine for the zero-point frequency. For example, CCSD(T)/cc-pVTZ calculation for the CO molecule gives value close to 2169.8 cm**-1, while it is not so for IR observed CO frequency 2143 cm**-1. In this case, scaling factor is needed to “convert” zero-point vibrational level to the difference between energy levels. The latter is, certainly, very hard to justify theoretically.

It is possible to make custom DFT functional, where the dynamic electron correlation is somewhat larger than in real life, in order to make harmonic frequencies serve as fundamental frequencies. The scaling factor will be unity, so this will be a specialized DFT for the IR-frequencies prediction, but in that case random error will be large. It is not surprising that one prefers systematic errors rather than random errors, since former can be easily corrected, while the latter can not. That's why people use good-old B3LYP with scaling ca 0.96. It may predict erroneous first vibrational level, second level, but happily random errors cancel each other when we deal with energy level differences, and remaining systematic error are corrected by scaling.

1 Pulay, Peter, et al. "Combination of theoretical ab initio and experimental information to obtain reliable harmonic force constants. Scaled quantum mechanical (QM) force fields for glyoxal, acrolein, butadiene, formaldehyde, and ethylene." Journal of the American Chemical Society 105.24 (1983): 7037-7047

2 Scott, Anthony P., and Leo Radom. "Harmonic vibrational frequencies: an evaluation of Hartree-Fock, Møller-Plesset, quadratic configuration interaction, density functional theory, and semiempirical scale factors." The Journal of Physical Chemistry 100.41 (1996): 16502-16513