Scaled frequency

Bond length Bond angle Scaled frequencies Atomization energy Proton affinity EA IP... 

Raw frequency values computed at the Hartree-Fock level contain known systematic errors due to the neglect of electron correlation, resulting in overestimates of about 10%-12%. Therefore, it is usual to scale frequencies predicted at the Hartree-Fock level by an empirical factor of 0.8929. Use of this factor has been demonstrated to produce very good agreement with experiment for a wide range of systems. Our values must be expected to deviate even a bit more from experiment because of our choice of a medium-sized basis set (by around 15% in all). 

System Standard Orientation (Cj, C2) Expected Displacement Scaled Frequency Peak Number... 

System Center Number Scaled Frequency Peak ... 

We ve matched up the predicted to observed frequencies by examining the displacements for each normal mode and determining the type of motion to which it corresponds (just as we did for ground state frequencies). The scaled frequencies are generally in excellent agreement with the observed spectrum. ... 

Solution A geometry optimiiation and frequency calculation (both in solution) are needed for each system (we ran the formaldehyde calculations earlier in this chapter). Here are the resulting scaled frequencies associated with carbonyl stretch for each system, along with the corresponding experimental values ... 

This leads to a remarkable improvement over the uniformly scaled frequencies, and many programs are available for this task. 

Experimental frequencies in cm for C3H2 from Huang et al. 1990 MP4 scaled frequencies... 

To scale frequency to a more convenient range, IR spectroscopists have defined a frequency unit called wave number v given by v = 1 /X, where X is the wavelength in centimeters. The units of v are reciprocal centimeters (cm-1). The wave number is the number of vibrations which occur over a 1-cm distance. Thus the higher the wave number, the more vibrations occur in a 1-cm distance and thus the higher the frequency. Normally IR spectra are recorded between 4000 and 650 cm 1 (2.5 and 15 p,m). 

Results in parentheses use B3LYP/6-31G geometries and 0.96 scaled frequencies. As can be seen, the predicted ionization potentials, electron affinities, proton affinities, and enthalpies are not highly sensitive to the molecular geometries and vibrational frequencies adopted. 

We calculated both d and s for various numbers of feature classes used in features calculations, and chose classes (scale-frequency pairs) characterized by the maximal d. This leads to the iris feature vector of 1152 bits (144 bytes) containing only four feature classes (Figure 7). Simultaneously, for those four selected feature classes we achieved the maximal non-zero separation s. 

Due to these large deviations the direct application of the SQMF method to the B3 molecule may lead to incorrect assignment of the vibrational modes. Therefore we performed a preliminary SQMF calculation on the benzene molecule, where the normal modes are well ascribed [1] and discriminated by symmetry. The extracted scale factors correspond to the set of symmetry-adapted internal coordinates, as introduced by Wilson [1], The same scale factors were then applied to the B3 molecule, which was considered as a system constructed by three benzene molecules. The single C-N bonds in B3 were treated with the same scale factors as the single C-H bonds in the benzene molecule. The scale factors corresponding to the N-H bonds were initially set equal to 1. The as-obtained scaled force-field, after minor adjustment of the scaling factors, was employed to calculate the normal frequencies of B3. Fig. 2(b) shows the corresponding pattern of the calculated scaled frequencies. It can be seen that there... 

Obviously, these structural changes make the transfer of force-constants from the neutral B3 molecule to the B3+ radical inadequate. Instead, we tentatively transferred the scale factors optimized for the neutral molecule to the quantum-mechanical force-field of B3+ and calculated the corresponding scaled normal frequencies. We obtained a clear correspondence between many of the frequencies experimentally observed in Cl doped B3 crystals (Fig. 3(a)) and the calculated scaled frequencies (Fig. 3(b)). We also observed that some of the calculated scaled frequencies in the neutral B3 molecule are present in the spectra of the Cl doped crystals (Fig. 3(c)). This fact tells us that there is some portion of unoxidized B3 molecules in the sample and gives additional proof for the validity of the SQMF calculations performed on the neutral B3 molecule. 

Fig. 10.18. Effects of surface roughness on EHD impedance (amplitude ratio, H(p)IH(p- 0), and phase lag, 9, against scaled frequency, p and comparison with the behaviour of a uniform disc—asymptotic line marked (a) —and an array of UMEs— asymptotic line marked (b). The frequency shift is deduced from the displacement between the two sections of the phase angle diagram where the data superimpose for different n . The modulation frequency, to2, at which the data deviate from that of a uniform electrode, is related to the amplitude of the surface roughness or the spacing between the elements of the UME array. Data from Reference [121], for Fe(CN)i reduction on smooth Pt at 120 rpm 4 240 rpm, and on a rough, Pt-coated silver electrode (roughness scale 5 (im, disc diameter 6 mm) at O 120 rpm + 240 rpm A 500 rpm and x 1000 rpm.

Data Processing derived values, data conversions, scaling, frequencies, periods, calculations, algorithms, validity checks, error correction... 

Fig. 7.6. Numerically computed scaled critical fields (bullets) and experimental 10% threshold fields (squares) (from Moorman and Koch (1992)) as a function of scaled frequency. The full line represents critical fields computed on the basis of Chirikov s overlap criterion. (From Blumel (1994b).)...

Table 4). The antisymmetric, B, CO stretching modes appears at a calculated, scaled frequency of 1535 cm and was not experimentally observed. The symmetric CO stretch of symmetry appears at 1488 cm, 53 cm above the experimentally observed frequency of 1435 cm. The two CC stretching modes of Aj and B. symmetry appear at calculated frequencies of 1643 cm (observed at 1620 cm and 1475 cm (not detected experimentally), respectively. Thus agreement between experimentally measured vibrational frequencies and calculated, scaled vibrational frequencies is moderately good for p-benzosemiquione radical anion and calculated frequencies shift in the direction implied by weaker, longer CO and C=C bonds upon reduction of p-benzoquinone to form its semiquinone anion.

Table 11 lists several scaled calculated vibrational frequencies of UQ and UQ-, along with experimentally measured frequencies.(S/, 89, 90, 99-101) The table focuses on C=0 and C=C stretching and methoxy torsional frequencies because these frequencies have been experimentally measured and are key to inferences concerning the influence of the protein on ubiquinone structure. For neutral UQ, the mode calculated at a scaled frequency of 1683 cm is a stretching vibration concentrated at the C=0 bond located meta to the isoprenoid chain and corresponds most closely to the recently observed UQ-1 C=0... 

Figure 18.2 Impedance data from Figure 18,1 as a function of scaled frequency p = co/Cl for reduction of ferricyanide on a rotating Pt disk electrode a) the real part of the impedance and b) the imaginary part of the impedance.

Remember 18.2 Reduced presentation of impedance data in terms of scaled frequency (given, e.g., in Figures 18.2 and 18.6) provides verification of the origin of the impedance response. 

Calculation The table on the left uses the experimental 0 values to determine the individual terms in the summation of Eq. (2-48) the table on the right uses the scaled frequencies from computational chemistry software and Eq. (2-47) to obtain 0 values and the individual terms in Eq. (2-48). 

Many of the recent studies that examine Raman and infrared spectroscopy have been mentioned in previous sections of this chapter.However, a vibrational spectroscopic smdy by Comerlato and coworkers used HE and B3LYP with the SBKJC EPC for tin to examine IR and Raman spectra of the anionic [NEt4]2[Sn(dmit)3] complex. Comparison of the calculated scaled frequencies to experimental values revealed that the B3LYP method is more accurate than the HE method. The latter method is well known to overestimate frequencies by about 10%. 

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