Calculated frequencies

Normal modes of water. Experimental and (calculated) frequencies are shown. Theoretical frequencies "ed using a 6-310 basis set. 

SpartanView displays calculated vibrations and frequencies for selected models Calculated frequencies are listed m wavenumbers (cm ) and are consistently larger than observed frequencies (observed frequency = 0 9 X calculated frequency is a good rule of thumb)... 

Near top speed, a fan may operate at a speed that is near or above the natural frequency of the wheel and shaft. Under such conditions, the fan can vibrate badly even when the wheel is clean and properly balanced. Whereas manufacturers often do not check the natural frequency of the wheel and shaft ia standard designs, many have suitable computer programs for such calculations. Frequency calculations should be made on large high speed fans. The first critical wheel and shaft speed of a fan that is subject to wheel deposits or out-of-balance wear should be about 25—50% above the normal operating speed. 

The thiosulfate ion has tetrahedral symmetry and the six fundamental modes are both infrared and Raman active. The calculated frequencies (3) are in good agreement with experimental values (4). 

Unfortunately, particularly for the vendors, the undamped analysis opened somewhat of a Pandora s box. While the calculated frequencies were not quite accurate, they were better than any previous analysis, once the actual bearing parameters were input. 

It is interesting to note that the lowest phonon mode with non-zero frequency at A = 0 is not a nodeless A g mode, but rather an E2g mode with four nodes in which the cross section of the CNT is vibrating with the symmetry described by the basis functions of and xy. The calculated frequency of the E g mode... 

Frequencies computed with methods other than Hartree-Fock are also scaled to similarly eliminate known systematic errors in calculated frequencies. The followng table lists the recommended scale factors for frequencies and for zero-point energies and for use in computing thermal energy corrections (the latter two items are discussed later in this chapter), for several important calculation types ... 

Monomeric neutral SO4 can be obtained by reaction of SO3 and atomic oxygen photolysis of S03/ozone mixtures also yields monomeric SO4, which can be isolated by inert-gas matrix techniques at low temperatures (15-78 K). Vibration spectroscopy indicates either an open peroxo Cj structure or a closed peroxo C2v structure, the former being preferred by the most recent study, on the basis of agreement between observed and calculated frequencies and reasonable values for the force constants ... 

Calculated vibrational frequencies will not exactly match measured frequencies (they are typically too large by about 12%). For example, a measured hequency of 1700 cm would correspond to a calculated frequency of around 1900 cm- . 

Vibrational frequencies Calculated vibrational frequencies are larger than measured values, typically by about 12%. Systematic scaling of calculated frequencies (by 0.88) leads to values which are generally suitable for assignment and interpretation of experimental infrared/Raman spectra. 

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. 

It is clear that the nematic phase exhibits a featureless Rayleigh wing and that several distinct solid phases can be formed depending on cooling rate [79]. This includes an apparently glassy phase. The vertical tick marks indicate the calculated frequencies of vibrational modes as obtained from density functional methods. 

A sum-over-states expression for the coefficient A for the expansion of the diagonal components faaaa was derived by Bishop and De Kee [20] and calculations were reported for the atoms H and He. However, the usual approach to calculate dispersion coefficients for many-electron systems by means of ab initio response methods is still to extract these coefficients from a polynomial fit to pointwise calculated frequency-dependent hyperpolarizabiiities. Despite the inefficiency and the numerical difficulties of such an approach [16,21], no ab initio implementation has yet been reported for analytic dispersion coefficients for frequency-dependent second hyperpolarizabiiities which is applicable to many-electron systems. 

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

The frequencies and relative intensities for the clusters n = 6 to n=12 are given in Table II. As mentioned earlier, the calculated frequencies are 10 to 20 % too high. The corresponding vibrational modes have the character of PA vibrations in so far as the atoms which vibrate most are in the central region of the cluster. 

Normal coordinate analysis of the radical has been carried out and excellent agreement of experimental and calculated frequency values was obtained for the trans structure of HOCO. 

Figure 16. Experimental and calculated IR resonance enhanced photodissociation spectra of Fe" (CH4)3 and Fe" (CH4)4. Experimental spectra were obtained by monitoring loss of CH4. Calculated spectra are based on vibrational frequencies and intensities calculated at the B3LYP/ 6-311+G(d,p) level. Calculated frequencies are scaled by 0.96. The calculated spectra have been convoluted with a 10-cm full width at half-maximum (FWHM) Gaussian. The D2d geometries of Fe (CH4)4 are calculated to have very similar energies, and it appears that both isomers are observed in the experiment.

Table I. Comparison of calculated frequencies and matrix elements m2 (in parentheses) for H2O and NHo on Al(lOO) with experimental values for the NH3-Fe(1107 and H20-Cu(100) systems.

In Table II we compare the calculated frequency and matrix element (In parenthesis) for the D2O-Al(100) hindered rotation with experimental results for D20-Cu(100) (14). The experimental Isotopic factor Is 1.16 for the frequency and =1.6 for the matrix elements,... 

As the main point is the relative positions of these bands with respect to the vibrational origin, the results of the first calculated frequency (v =l->0)can be directly adjusted to the 3.3 0,m observed one(see below). Thus, we find rather regular progressions of about 50 cm- for the ai and 60 cm- for the b2 stretching vibrations due to anharmonicity. 

Figure 2.10 (a) Molecular structure and atomic numbering of adenine, (b) The calculated model of the adenine-silver quadrimer complex, (c) The calculated frequency shifts /Irbm of the Ad-N3 Ag quadrimer and the calculated binding energy as a function of the bond distance for the Ag-N linkage. 

Remaining calculated frequencies and assignments are also given in these references. 

For frequency calculations one usually starts out with a set of approximate existent force constants (e.g. taken over from similar, already treated molecules under the preliminary tentative assumption of transferability), and subsequently varies the force constants in a systematic way by means of a least-squares procedure until the calculated frequencies (square roots of the eigenvalues of Eq. (10)) agree satisfactorily with the experimental values. Clearly, if necessary, the analytical form of the force field is also to be modified in the course of this fitting process. 

At the most negative potentials the gains at 1943 cm-1 and 1828 cm-1 were attributed to the formation of (Dmbpy")Re[CO]3, for which the calculated frequencies are 1967 cm 1836 cm"1 and 1821cm"1 the two lowest frequencies were taken as corresponding to the broad feature near 1828 cm". The large difference between the observed and calculated values for the remaining peak were attributed to the difficulty in calculating the effect of the Dmbpy -. 

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