Effective Treatment of Fermi Resonances

Let us analyze how the Fermi resonances can be treated within the VPT2 approach, making use of both the variational and perturbative models. In the VPT2 approach, when the harmonic frequencies related to different modes are close to each other, singularities occur in Eqs. 10.16, 10.17, 10.18, and 10.27. Such singularities are usually classified as Fermi type 1 and type 2, corresponding to the conditions 2 or. cOj and -F cOj os. coj, respectively. In the expression of the transition dipole moment for the fundamentals (Eq. 10.27) another type of singularity also occurs when the harmonic frequencies of two fundamental 

The simplest approach to remove the singularities is to define a threshold value for the harmonic frequency differences (usually of the order of 10cm ), evaluate the harmonic frequency differences for all the modes involved in the sums of Eqs. 10.16, 10.17, 10.18, and 10.27, and neglect all the contributions in which the difference is smaller than the threshold. The same approach can be also applied to the treatment of the 1-1 resonances occurring in the transition dipole moment (Eq. 10.27). However, within this approach, the effect of the force constant on the overall resonance is neglected, so that a more refined scheme based on a two-step procedure is often used for the identification of the resonant dyads  

1) The harmonic frequency differences are evaluated for all the possibly resonant modes, and compared with a threshold value, usually of 200cm . If the harmonic frequency difference of a given dyad is lower than the threshold, such a dyad is selected as possibly resonant. 

2) Additional terms, depending on the magnitude of the numerator [249] are evaluated for all the possibly resonant terms identified in the previous step. 

The main drawback of the GVPT2 and DVPT2 treatments is their direct dependence on the semiempirical thresholds, which may lead to inconsistencies for example, when anharmonic frequencies computed using different force fields are compared. It can occur that the harmonic frequency differences and the criteria defined in Eqs. 10.29-10.32 are close to the threshold values, so that different resonant dyads are identified with different methods. Moreover, when reaction paths are considered, it can occur that for two structures on the path, different resonant dyads are found, leading to noncontinuous thermodynamic functions. For these reasons, it is desirable to derive more robust resonance-free models within the VPT2 theory. 

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