Anharmonic frequencies

For very-high-accuracy ah initio calculations, the harmonic oscillator approximation may be the largest source of error. The harmonic oscillator frequencies 

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The first derivative is the gradient g, the second derivative is the force constant (Hessian) H, the third derivative is the anharmonicity K etc. If the Rq geometry is a stationary point (g = 0) the force constant matrix may be used for evaluating harmonic vibrational frequencies and normal coordinates, q, as discussed in Section 13.1. If higher-order terms are included in the expansion, it is possible to determine also anharmonic frequencies and phenomena such as Fermi resonance. 

We also noted already that BLYP results show a better agreement with directly observed, anharmonic frequencies. This is also the case here the BLYP deviations from the fundamental experimental frequencies (vs = 3657 cm4, vas = 3756 enf1) are 2 and 0 cm4, respectively. 

Herein we present calculations [6] for liquid H20 that are similar in spirit but different in detail from those of Buch [71, 110] and Torii [97]. The MD simulations are of the SPC/E model [135]. Local-mode anharmonic frequencies are generated from our most recent map developed for the H0D/D20 system [98], as are our transition dipoles. The relatively small intramolecular coupling fluctuates with molecular environment, and is determined by a separate map parameterized from ab initio calculations on clusters. The form of the intermolecular couplings is transition dipole, which is tested and parameterized from additional ab initio calculations. The effects of motional narrowing are taken into account approximately with the TAA [99]. 

The Impedance Z can Increase to very high values. If this happens, the oscillator prefers to oscillate In resonance with an anharmonic frequency. Sometimes this condition Is met for only a short time and the oscillator oscillation jumps back and forth between a basic and an anharmonic oscillation or It remains as an anharmonic oscillation. This phenomenon Is well known as "mode hopping". In addition to the noise of the rate signal created, this may also lead to Incorrect termination of a coating because of the phase jump. It Is Important here that, nevertheless, the controller frequently continues to work under these conditions. Whether this has occurred can only be ascertained by noting that the coating thickness Is... 

Table 2 Anharmonic frequencies (in cm-1) for the target modes of the different model structures as obtained using Eq. (1) with the one mode potential generated by the DFT/B3LYP method with a 6-31++G(d,p) basis set and the two- and threemode PES obtained by the PM3 approach.

Table 3 Comparison of anharmonic frequencies (in cm-1) for the target modes of the AT-(H20)2 model structures of different two-dimensional calculations and different levels of quantum chemistry (DFT full DFT/B3LYP, 6-31++G(d,p) PM3 full PM3 DUAL DFT one-mode and PM3 two-mode PES).

Table 6 Anharmonic frequencies (in cm-1) for the target modes of the AT-(H20)2 model calculated using the anharmonic expansion, Eq. (2) and the DFT/B3LYP method with a 6-31++G(d,p) basis set. The 4D model is compared with a 6D model which includes in addition the most strongly coupled water stretching and bending modes at the N6-H site whose anharmonic frequencies are 3752 cm-1 and 1588 cm-1, respectively. Also given are results for a 4D model which does not include water molecules [59]. The experimental assignment is shown as well ( /nh2 from Ref. [59], the other modes from Refe. [7-9,14,34-38]).

Comparing to + 2ka to the level spacing [Pg.303]

Figure 12-3. IR-UV double resonance spectrum of GC (structure C) in the mid-IR frequency range (recorded at the FELIX free electron laser facility), compared with three types of ab intio calculations. Harmonic frequencies were obtained at the RI-MP2/cc-pVDZ, RI-MP2/TZVPP, and semiempirical PM3 levels of electronic structure theory. Anharmonic frequencies were obtained by the CC-VSCF method with improved PM3 potential surfaces [30]...

Harmonic frequency for the out-of-plane deformation anharmonic frequencies have been estimated for the other fundamentals (136). 

Case Studies Anharmonic Frequencies of Phenyl and Naphtyl Cation Radicais... 

Table 6.14 with computational results. Harmonic and PT2 anharmonic frequencies have been computed by B3LYP/N07D and CAM-B3LYP/N07D computational models. Since low-temperature experimental frequencies in Argon and Neon are very close, matrix effects have been taken into account by continuum medium... 

Burdett etal " showed that, to fit isotopic spectra accurately, the EFFF needed two corrections a factor that takes into account the particular isotope of CO used and a further correction to ameliorate the effects of using anharmonic frequencies. The effect of the neglect of M-C/C-O conphng was originally explained by Miller in terms of an x-factor, which depended on the particular CO isotope being nsed. Anharmonicity is accounted for by another correction that could be determined from the all- C 0 molecnle and the reduced mass ratio. 

The frequencies that arise from this treatment are listed for the three complexes in Table 3.17. Note first that the anharmonic frequencies of v. (XH in Table 3.17) are quite a bit smaller than the harmonic values in the upper part of the table, differing by about 400 cm . ... 

Table 3.64 Harmonic and anharmonic frequencies (in cm ) of HP and HCN, and the changes resulting from formation of the HCN HF complex. Data calculated with doubly polarized basis set.

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