Vibrational scale factor

It is possible to use computational techniques to gain insight into the vibrational motion of molecules. There are a number of computational methods available that have varying degrees of accuracy. These methods can be powerful tools if the user is aware of their strengths and weaknesses. The user is advised to use ah initio or DFT calculations with an appropriate scale factor if at all possible. Anharmonic corrections should be considered only if very-high-accuracy results are necessary. Semiempirical and molecular mechanics methods should be tried cautiously when the molecular system prevents using the other methods mentioned. 

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. 

Extensive comparisons of experimental frequencies with HF, MP2 and DFT results have been reported [7-10]. Calculated harmonic vibrational frequencies generally overestimate the wavenumbers of the fundamental vibrations. Given the systematic nature of the errors, calculated raw frequencies are usually scaled uniformly by a scaling factor for comparison with the experimental data. 

For larger systems, where MP4 calculations are no longer tractable, it is necessary to use scaling procedures. The present results make it possible to derive adapted scaling factors to be applied to the force constant matrix for each level of wave function. They can be determined by comparison of the raw calculated values with the few experimental data, each type of vibration considered as an independent vibrator after a normal mode analysis. 

Rauhut, G., Pulay, P., 1995, Transferable Scaling Factors for Density Functional Derived Vibrational Force Fields , J. Phys. Chem., 99, 3093. 

Within the framework of the scaled quantum mechanical (SQM) procedure, transferable scaling factors (TSFs) were used to compute the vibrational spectra of dibromofuroxan and diiodofuroxan <2002PCA6810>. 

Vibrational frequencies of hexatriene and octatetraene have been reported by several authors21,24-26,36. The increase in the size of these molecules with respect to butadiene limits the use of highly accurate levels of calculation, so that a good choice of scaling factors is necessary to obtain useful results. Kofraneck and coworkers21 have shown that employing scale factors determined from vibrational data for trans structures alone does not give a balanced description of cis and trans structures. 

During recent years DFT methods have been used to reproduce vibrational frequencies and IR intensities (dipole moment derivatives) with high accuracy (scaling factors are close to unity).29,60,61 We therefore used the B3LYP and BLYP functionals to calculate the spectra of la and its isotopomers, and indeed the calculated frequencies, isotopic shifts, and intensities are now in excellent agreement with the experimental values (Fig. 3).62 A careful reexamination... 

The next two steps in the procedure of Leonard and Ashman are the conversion of the diagonal elements from atomic units into force field units and calculation of scaling factors for bond lengths and angles. The calculated force constants had to be scaled down by approximately 25% and 70% to yield force constants comparable in numerical size with those included in MM2. Neither force constants nor scaling factors can be incorporated directly into a different force field. A modification of the described procedure that meets the requirements of CVFF was developed. Fragments with known force field parameters were chosen. After a full geometry optimization (HF/6-31G ) second derivatives and vibrational frequencies were calculated. The force... 

Table 2. Scale factors for ab initio model vibrational frequencies adapted from (Scott and Radom 1996). Please note that these scale factors are determined by comparing model and measured frequencies on a set gas-phase molecules dominated by molecules containing low atomic-number elements (H-Cl). These scale factors may not be appropriate for dissolved species and molecules containing heavier elements, and it is always a good idea to directly compare calculated and measured frequencies for each molecule studied. The root-mean-squared (rms) deviation of scaled model frequencies relative to measured frequencies is also shown, giving an indication of how reliable each scale factor is.

Pople JA, Scott AP, Wong MW, Radom L (1993) Scaling factors for obtaining fundamental vibrational frequencies and zero-point energies from HF/6-31G and MP2/6-31G harmonic frequencies. Israel J Chem 33 345-350... 

Scott AP, Radom L(1996) Harmonic vibrational frequencies An evaluation of Hartree-Fock, Moller-Plesset, quadratic configuration interaction, density functional theory, and semiempirical scale factors. J Phys Chem 100 16502-16513... 

Fractionation factors for Li-HjO clusters are calculated using ab initio vibrational models, in the gas-phase approximation. Vibrational frequencies in this system are largely unknown, and the few that have been measured are contentious. In the absence of reliable experimental constraints, Hartree-Fock model ab initio vibrational frequencies are normalized using a scaling factor of 0.8964. It is generally thought that aqueous lithium is coordinated to four water molecules (Rudolph et al. 1995). The authors speculate that 6-coordinate lithium in adsorbed or solid phases will have lower Li/ Li than coexisting aqueous LF. 

In order to assign more IR signals of 4a, ab initio calculations on Hbdmpza (3b) and 4a were performed. It is well known for the chosen HF/6-31G basis set that calculated harmonical vibrational frequencies are typically overestimated compared to experimental data. These errors arise from the neglecting anharmonicity effects, incomplete incorporation of electron correlation and the use of finite basis sets in the theoretical treatment (89). In order to achieve a correlation with observed spectra a scaling factor (approximately 0.84-0.90) has to be applied (90). The calculations were calibrated on the asymmetric carboxylate Vasym at 1653 cm. We were especially interested in... 

J. P. Merrick, D. Moran, and F. Radom, An Evaluation of Harmonic Vibrational Frequency Scale Factors, J. Phys. Chem. A 111 (2007), 11683. 

The reliable experimental information on the absolute scale and thermal vibrations of beryllium metal made it possible to analyze the effect of the model on the least-squares scale factor, and test for a possible expansion of the 1 s core electron shell. The 0.03 A y-ray structure factors were found to be 0.7% lower than the LH data, when the scale factor from a high-order refinement (sin 6/X) > 0.65 A l) is applied. Larsen and Hansen (1984) conclude that because of the delocalization of the valence electrons, it is doubtful that diffraction data from a metallic substance can be determined reliably by high-order refinement, even with very high sin 0/X cut-off values. This conclusion, while valid for the lighter main-group metals, may not fully apply to metals of the transition elements, which have much heavier cores and show more directional bonding. 

Table 9.3 Scale factors and post-scaling eiTors in vibrational frequencies from different levels of theory ...

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