Harmonic approximation scaling factors

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. 

In order to obtain better agreement between theory and experiment, computed frequencies are usually scaled. Scale factors can be obtained through multiparameter fitting towards experimental frequencies. In addition to limitations on the level of calculation, the discrepancy between computed and experimental frequencies is also due to the fact that experimental frequencies include anharmonicity effects, while theoretical frequencies are computed within the harmonic approximation. These anharmonicity effects are implicitly considered through the scaling procedure. 

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... 

As a rule the quantum-mechanical force-fields and the corresponding normal frequencies are calculated in a harmonic approximation, while the experimentally accessible frequencies are influenced by anharmonic contributions. The Puley s scaling factors are also found to incorporate the relevant empirical corrections for the vibrational anharmonicity. 

The shape of the potential for the proton motion, particularly when the proton is engaged in hydrogen bond formation, is one of the most fascinating problems of molecular physics and chemistry. Because of very low mass the stretching protonic vibrations are characterized by high frequencies and, if independent of additional interactions, they are anharmonic. It is commonly known that expression of quantum-chemical calculations in the harmonic approximation, on various levels of the quantum-mechanical approach, needs the application of some scaling factors [1, 2]. For stretching protonic vibrations this factor is... 

Although the harmonic approximation is satisfactory for small displacements from the equifibrium position, ab initio harmonic force constants and vibrational frequencies are known to be typically overestimated as compared with those experimentally found [86]. Sources of this disagreement are the omission or incomplete incorporation of electron correlation, basis set deficiencies, and the neglect of anharmonicity effects. However, as the overestimation is fairly uniform, the appHcation of appropriate scahng procedures becomes feasible. Due to its simplicity, global scafing (using one uniform scale factor determined by a least-squares fit of the calculated to the experimental vibrational frequencies) has widely been used at different levels of theory [87]. However, for most spectro-... 

Xu are the diagonal anharmonicity constants and Go is the polyatomic counterpart of the small Too Dunham constant [82] in diatomics. Consequently [50, 84, 90], the optimal scaling factor for ZPVEs is almost exactly midway between a 2(co) suitable for harmonic frequencies (as an approximate correction for systematic bias in the calculated frequencies) and a 2(v) suitable for fundamental frequencies (which additionally seeks to approximately corrects for anharmonicity). In fact, Alecu et al. [86] found for a large variety of basis sets and ab initio and DFT methods that 2((o)/2(ZPVE) = 1.014 0.002, which is almost exactly the ratio of 1.0143 found by Perdew and coworkers [87] between harmonic frequencies and ZPVEs derived from experimental anharmonic force fields. Note that the small uncertainty of 0.002 on a ZPVE of 140 kcal/mol still would translate to about 0.3 kcaFmol, and even that is probably optimistic for the uncertainty in an individual... 

For many reactions the calculated structures for potential energy minima are as accurate as those found experimentally. Ab initio and experimental harmonic vibrational frequencies usually agree to within 10-15% at the Hartree-Fock level and 5% at the MP2 level (Hehre et al., 1986). It has been found that Hartree-Fock harmonic frequencies computed with a medium-size basis set ean be scaled by the factor 0.9 to give approximate anharmonic n = 0 — 1 transition frequencies (Hehre et al., 1986). A detailed study has been made of how the computed ab initio frequencies for benzene depend on the size of the basis set and the treatment of electron correlation (Maslen et al., 1992). 

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