Experimental frequencies

Symmetry mode Experimental frequencies (r, cm" ) Theoretical frequencies v. cm" ) Theoretical assignments (207)... 

After scaling, the predicted frequencies are generally within the expected range for carbonyl stretch (-1750 cm ). The table below reproduces our values, published theoretical values using the 6-31+G(d) basis set (this basis set includes diffuse functions), and the experimental values, arranged in order of ascending experimental frequency ... 

Note that the frequency calculation produces many more frequencies than those listed here. We ve matched calculated frequenices to experimental frequencies using symmetry types and analyzing the normal mode displacements. The agreement with experiment is generally good, and follows what might be expected of Hartree-Fock theory in the ground state. ... 

This same istudy can also be done for other, solvents. Here, for example, are the predicted and experimental frequency shifts for cyclohexane drawn from the original study ... 

For comparison with experimental frequencies (which necessarily are anharmonic), there is normally little point in improving the theoretical level beyond MP2 with a TZ(2df,2pd) type basis set unless anharmonicity constants are calculated explicitly. Although anharmonicity can be approximately accounted for by scaling the harmonic frequencies by 0.97, the remaining errors in the harmonic force constants at this level are normally smaller than the corresponding errors due to variations in anharmonicity. 

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. 

Our results indicate that dispersion coefficients obtained from fits of pointwise given frequency-dependent hyperpolarizabilities to low order polynomials can be strongly affected by the inclusion of high-order terms. A and B coefficients derived from a least square fit of experimental frequency-dependent hyperpolarizibility data to a quadratic function in ijf are therefore not strictly comparable to dispersion coefficients calculated by analytical differentiation or from fits to higher-order polynomials. Ab initio calculated dispersion curves should therefore be compared with the original frequency-dependent experimental data. 

The distribution of a data set in the form of a histogram can always be plotted, without reference to theories and hypotheses. Once enough data have accumulated, there is the natural urge to see whether they fit an expected distribution function. To this end, both the experimental frequencies and the theoretical probabilities must be brought to a common scale a very eonvenient... 

Table I. Experimental frequencies of PA and corresponding calculated frequencies of the C12 H14 cluster...

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

But when considered over a wide range of frequencies, the properties of a real electrode do not match those of the equivalent circuits shown in Fig. 12.12 the actual frequency dependence of Z and a does not obey Eq. (12.21) or (12.22). In other words, the actual values of R and or R and are not constant but depend on frequency. In this sense the equivalent circuits described are simplified. In practice they are used only for recording the original experimental data. The values of R and Cj (or R and C ) found experimentally for each frequency are displayed as functions of frequency. In a subsequent analysis of these data, more complex equivalent circuits are explored which might describe the experimental frequency dependence and where the parameters of the individual elements remain constant. It is the task of theory to interpret the circuits obtained and find the physical significance of the individual elements. 

The first-principles calculation of NIS spectra has several important aspects. First of all, they greatly assist the assignment of NIS spectra. Secondly, the elucidation of the vibrational frequencies and normal mode compositions by means of quantum chemical calculations allows for the interpretation of the observed NIS patterns in terms of geometric and electronic structure and consequently provide a means of critically testing proposals for species of unknown structure. The first-principles calculation also provides an unambiguous way to perform consistent quantitative parameterization of experimental NIS data. Finally, there is another methodological aspect concerning the accuracy of the quantum chemically calculated force fields. Such calculations typically use only the experimental frequencies as reference values. However, apart from the frequencies, NIS probes the shapes of the normal modes for which the iron composition factors are a direct quantitative measure. Thus, by comparison with experimental data, one can assess the quality of the calculated normal mode compositions. 

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. 

Surface nitrosyl complexes of TMI have been thoroughly investigated by the computational spectroscopy [22,23,32,33,36,49], and their molecular structure has been ascertained by a remarkable agreement between the theory and experiment of both vibrational (oscillation frequencies and intensities) and magnetic (g and A tensors) parameters. The calculated pNO values for the examined mononitrosyls along with the experimental frequencies are listed in Table 2.6. Analogous collation of the IR data for dinitrosyl species is shown in Table 2.7. 

Scale factors determined from the CH3Br and CH,C1 ab initio and experimental frequencies were used to scale ab initio frequencies for the complexes and central barrier see text... 

At the highest temperature we performed stress relaxation measurements to extend the experimental frequency range. From these relaxation experiments, the corresponding oscillatory data were calculated with the well-known approximate relationships of Schwarzl (13). More details on the preparation of the networks and on the measurements were published previously (14). 

Warshel, Levitt, and Lifson derived a partially optimised consistent force field for amides and lactams (25). It is composed of an alkane part and an amide-part. The former was taken over from analogous earlier calculations for saturated hydrocarbons (17). The potential constants of the amide-part were optimised with the help of a large number of experimental frequencies (taken from TV-methylform amide, acetamide, iV-methylacetamide, and several deuterated species) as well as experimental geometry data for 7V-methylacet-amide. The resulting force field was used for the calculation of vibrational and conformational properties of 2-pyrrolidone, 2-piperidone and e-caprolactam. 

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. 

Figure 8. Computed IR frequencies for the some of the water clusters of Fig. 7. The V symbol indicates the position of experimental frequencies.

It is usually important to get an accurate fit of the frequency response of the model to the experimental frequency only near the critical region where the phase... 

Included in Table III is the comparison of the transition frequencies calculated from the energies obtained in our calculations with the experimental transition frequencies of Dabrowski [125]. To convert theoretical frequencies into wavenumbers, we used the factor of 1 hartree = 219474.63137 cm . For all the frequencies our results are either within or very close to the experimental error bracket of 0.1 cm . We hope that the advances in high-resolution spectroscopy will inspire remeasurements of the vibrational spectrum of H2 with the accuracy lower than 0.1 cm. With such high-precision results, it would be possible to verify whether the larger differences between the calculated and the experimental frequencies for higher excitation levels, which now appear, are due to the relativistic and radiative effects. 

Zero-point energies are obtained from vibrational spectra using experimental frequencies whenever available, while the inactive frequencies are extracted from data calculated by means of an appropriate force-held model. In the harmonic oscillator approximation, the zero-point energy is... 

A comparison of HF, MP2 and density functional methods in a system with Hartree-Fock wave function instabilities, ONO—OM (for M = Li, Na and K), shows that DFT methods are able to avoid the problems that ab initio methods have for this difficult class of molecules. The computed MP2 frequencies and IR intensities were more affected by instabilities than HF. The hybrid B3LYP functional reproduced the experimental frequencies most reliably. The cis,cis conformation of ONO—OM was highly preferred because of electrostatic attraction and was strongest in the case where M = Li. The small Li cation can fit in best in the planar five-membered ring. This is completely different from the nonionic... 

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