Experimental fundamentals

The MP2 treatment recovers the majority of the correlation effect, and the CCSD(T) results with the cc-pVQZ basis sets are in good agreement with the experimental values. The remaining discrepancies of 9cm , 13cm and lOcm are mainly due to basis set inadequacies, as indicated by the MP2/cc-pV5Z results. The MP2 values are in respectable agreement with the experimental harmonic frequencies, but of course still overestimate the experimental fundamental ones by the anharmonicity. For this reason, calculated MP2 harmonic frequencies are often scaled by 0.97 for comparison with experimental results. ... 

Vibrational Spectra Many of the papers quoted below deal with the determination of vibrational spectra. The method of choice is B3-LYP density functional theory. In most cases, MP2 vibrational spectra are less accurate. In order to allow for a comparison between computed frequencies within the harmonic approximation and anharmonic experimental fundamentals, calculated frequencies should be scaled by an empirical factor. This procedure accounts for systematic errors and improves the results considerably. The easiest procedure is to scale all frequencies by the same factor, e.g., 0.963 for B3-LYP/6-31G computed frequencies [95JPC3093]. A more sophisticated but still pragmatic approach is the SQM method [83JA7073], in which the underlying force constants (in internal coordinates) are scaled by different scaling factors. 

There are 78 vibrational degrees of freedom for TgHg and it has been shown that the molecule has 33 different fundamental modes under Oh symmetry, 6 are IR active, 13 are Raman active, and 14 vibrations are inactive. The experimental fundamental IR active vibrational frequencies have been assigned as follows 2277 (v Si-H), 1141 (vas Si-O-Si), 881 5 O-Si-H), 566 ( s O-Si-O), 465 (v O-Si-O), and 399 cm ( s O-Si-O). These generally agree well with calculated values The IR spectrum recorded in the solid state shows bands at 2300 and 2293 cm ... 

Table 6-1. Harmonic frequencies and experimental fundamentals for ozone [cm 11. Deviations from the experimental result [%] are given in parentheses.

The equilibrium rotational constants and the theoretical harmonic frequencies calculated by Sherrill and Schaefer424 were in good agreement with the experimental ground-state rotational constants obtained by Leclercq and Dubois420 and in reasonable accord with the experimental fundamental vibrational frequencies obtained by Bengali and Leopold425. 

Experimental fundamental vibrational frequencies of aniline taken from Reference 56. 

The experimental fundamentals for differentiating between metal and porphyrin ring reactions have been given in Section III, thus only the products of the organic redox reactions will be discussed in the following paragraphs. 

The BLYP/6-31G and BP/86 (and probably the pBP/DN ) correction factors are very close to unity. For the frequencies of polycyclic aromatic hydrocarbons calculated by the B3LYP/6-31G method, Bauschlicher multiplied frequencies below 1300cm by 0.980 and frequencies above this by 0.967 [58]. In their paper introducing the modification of Becke s hybrid functional to give the B3LYP functional, Stephens et al. studied the IR and CD spectra of 4-methyl-2-oxetanone and recommended the B3LYP/6-31G as an excellent and cost-effective way to calculate these spectra [49]. With six different functional. Brown et al. obtained an agreement with experimental fundamentals of ca. 4-6%, except for BHLYP [77]. 

We will now comment briefly on the force fields that have been computed for benzene. The first one to incorporate both harmonic and anharmonic force constants (some cubic force constants) was reported by Pulay et al. (146) in 1981. This force field was determined with the 4-21P basis set at the SCF level. The computed harmonic force constants, as expected, were larger than the empirical values, so Pulay et al. developed a scaled quadratic force field. The scale factors were adjusted so that the computed frequencies agreed well with the experimental fundamental frequencies. 

Experimental fundamental vibrational frequencies from Reference 155. 

Percolation theory (the model supposes that asymmetrically structured carbon-black particles are statistically distributed and results in percolation in accordance with probability laws) [3,27,31,32,33,34,35], Although this theory is the most widespread one, it lacks important experimental fundamentals and cannot describe the multitude of factors affecting percolation behaviour,... 

Table 6.1. Computed and experimental fundamental vibrational wavenumbers (cm ) of FjSP S I...

The CH stretching frequencies (v2, v , V13, V20) calculated by the LDA method are close to the experimental fundamental frequencies, but are underestimated by 100 cm" compared to the harmonic frequencies. The large deviation from the experimental harmonic frequencies can be explained by the sensitivity of calculated CH-stretching frequencies to the applied reference geometry. We have shown in the previous section that the error in the reference bond length is the major source of error in the CH-stretching frequencies. 

Ab initio harmonic vibrational frequencies are typically larger than the fundamentals observed experimentally [37]. The overestimation of ab initio harmonic vibrational frequencies is, however, found to be relatively uniform, and as a result generic frequency scaling factors are often applied. The determination of appropriate scale factors for estimating experimental fundamental frequencies from theoretical harmonic frequencies has received considerable attention in the literature [66, 67, 68, 69]. 

Vibrational Frequencies. As noted in Section 15.13, theoretically calculated vibrational frequencies are often multiplied by a scale factor to improve agreement with experiment. For a set of 122 molecules and 1066 vibrational frequencies, scaled theoretical harmonic vibrational frequencies showed the following rms deviations from experimental fundamental (anharmonic) vibrational frequencies also given are the optimum scale factors and the percentages of scaled frequencies with less than 6% error and more than 20% error [A. P. Scott and L. Radom,/ Phys. Chem., 100,16502 (19%)] ... 

Theoretical and experimental fundamental harmonic a.c. cyclic voltammograms for cis-Cr(CO) (dpm) system as described in Figure 8 but with T = 293 K,ki =... 

Experimental fundamentals from the literature were also reported in a collection of vibrational energy levels of polyatomic transient molecules [12]. 

There are two final comments to make in this analysis. Firstly, it it most important that further calculations are performed to gain a greater confidence, with further refinements, on a wider variety of molecules, for which the experimental fundamentals are known. Secondly, the value of such a scheme is its ability to predict, and to this end we must know when the model will fail. 

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