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You are here: Calculated > Vibrations > Scale Factors > Why scale vibrations OR Resources > Tutorials > Vibrations > Why scale vibrations

Vibrational scaling factors

The vibrational frequencies produced by ab initio programs are often multiplied by a scale factor (in the range of 0.8 to 1.0) to better match experimental vibrational frequencies. This scaling compensates for two problems:
1) The electronic structure calculation is approximate. Usually less than a relativistic full configuration interaction is performed.
2) The potential energy surface is not harmonic. For bond stretches a better description of the potential energy surface is given by the Morse potential E(x) = D(1-exp(-β( x-x₀ )))² illustrated below. In this equation E is the potential energy, D, β, and x₀ are constants, and x is the interatomic distance.

The programs that predict vibrational frequencies do so by calculating the second derivative of the potential energy surface with respect to the atomic coordinates. This provides the curvature at the bottom (minimum) of the well. For a harmonic potential E(x) = k x ² this is directly related to the vibrational energy level spacing. For a Morse potential (with the same second derivative at the minimum) the anharmonicity causes the vibrational energy levels to be more closely spaced as illustrated in the figure.

Experimentally usually the fundamental (v=0 to v=1 energy) is measured. If enough vibrational levels are observed a harmonic frequency can be estimated with a formula such as G(v) = ω_e(v+1/2) - ω_ex_e(v+1/2)² + ω_ey_e(v+1/2)³ + ... where G(v) is the vibrational energy, v is the vibrational quantum number, ω_e is the harmonic frequency, and ω_ex_e and ω_ey_e are anharmonic constants. For polyatomic molecules usually only the fundamental is experimentally observed.

How we calculate the vibrational scaling factors

How much do the factors change with basis set?

The scaling factors depend weakly on basis set. The uncertainties are about twice as large for the 3-21G basis set as for the other basis sets, all of which include polarization functions. This suggests that polarization functions are important for avoiding markedly poor predictions of vibrational frequencies. For larger basis sets the scaling factors and uncertainties have non-significant changes. Thus the same scaling factors can be used for other basis sets.