Harmonization values

The HF vibrational frequencies are too high by about 7% relative to the experimental harmonic values, and by 10-13% relative to the anharmonic values. This overestimation is due to the incorrect dissociation and the corresponding bond lengths being too short (Table 11.1), and is consequently quite general. Vibrational frequencies at the HF level are therefore often scaled by 0.9 to partly compensate for these systematic errors. 

If we use B3LYP/VTZ+1 harmonics scaled by 0.985 for the Ezpv rather than the actual anharmonic values, mean absolute error at the W1 level deteriorates from 0.37 to 0.40 kcal/mol, which most users would regard as insignificant. At the W2 level, however, we see a somewhat more noticeable degradation from 0.23 to 0.30 kcal/mol - if kJ/mol accuracy is required, literally every little bit counts . If one is primarily concerned with keeping the maximum absolute error down, rather than getting sub-kJ/mol accuracy for individual molecules, the use of B3LYP/VTZ+1 harmonic values of Ezpv scaled by 0.985 is an acceptable fallback solution . The same would appear to be true for thermochemical properties to which the Ezpv contribution is smaller than for the TAE (e.g. ionization potentials, electron affinities, proton affinities, and the like). 

Species ) Quantity Anharmonic value Corrected (harmonic) value )... 

Fortunately in a number of cases there is experimental information on these points from broad band pump/probe experiments when the anharmonicity A is larger than the linewidth but much smaller than the bandwidth 8(o of the laser. Then the 0-1 transition is seen as a bleaching signal and the 1-2 (66,67,71) as well as the 2-3 and often higher quantum number transitions (68,95) appear as new absorptions to an extent that depends on the pump intensity. A direct comparison of the total linewidths (1/T2) of these transitions, and the population relaxation times for the v = 1, v = 2 and perhaps higher levels can be obtained from such data. For N3 we found that ratio of the state to state relaxation from v = 2 to v = 1 was 1.8 times that for v = 1 to v = 0, not far from the harmonic value of 2 (50,95). However, the bandwidth of both transitions was roughly the same. 

The matrix codeposition of CsF with HF (31,32) in an excess of argon, up to Ar/HF ratios of 3000, yielded intense bands at 1364 and 1218 cm, within a few wavenumbers of the HF2 band positions in ionic lattices 29]. The deuterium shift, upon formation of DF2, was slightly above the harmonic value of 1.41, indicating that the anion maintained a center of symmetry. Similar results were obtained with other alkali fluoride salts, but the product yield decreased as the radius of the alkali metal cation decreased. These results provided immediate confirmation of the salt/molecule technique, demonstrating that for a known system ion pair formation occurred and that the spectrum of the product anion resembled closely that of the anion in known environments. 

Similar sorts of conclusions apply to the frequencies. A systematic study " found that a DZP basis set yields vibrational frequencies within about 9% of experimental (harmonic) values. The discrepancy diminishes to 4% when correlation is included via CISD and to 2% with a coupled cluster treatment. Another set of calculations confirmed the eost-effec-tiveness of the MP2 treatment of vibrational frequencies, indicating better agreement with experiment than MP3 on some oceasions. Certain types of modes can be more sensitive to the level of theoretical treatment than others. For example, out-of-plane bending motions for it-bonded systems can require triple- plus two sets of polarization functions, as well as a set of/-functions in the basis set . 

The frequencies that arise from this treatment are listed for the three complexes in Table 3.17. Note first that the anharmonic frequencies of v. (XH in Table 3.17) are quite a bit smaller than the harmonic values in the upper part of the table, differing by about 400 cm . ... 

With regard to various levels of theory, it is significant that inclusion of correlation (values in parentheses for ClH-NHj) lowers the frequency but has the opposite effect of increasing v. It is also worth noting that the SCF and Cl frequencies can differ by as much as 300 cm, harmonic or anharmonic. Indeed, anharmonicity and electron correlation effects are additive here the anharmonic Cl v frequency is smaller than the SCF harmonic value by a full 639 cm. ... 

Table 3.18 Vibrational transitions calculated" in units of cm. Harmonic values in parentheses.

Anharmonicity corrections can be quite large in complexes of this type, up to several hundred cm k In FH- NHj and CIH—NH3, for example, the anharmonic v, frequency is lower than the harmonic value by about 400 cm . It is interesting also that whereas correlation lowers the v frequency, it has the opposite effect of increasing v. Nor can one expect cancellation to occur between anharmonic and correlation effects in some cases, the two can be additive. Mechanical anharmonicity leads to a progressively smaller spacing between successive overtones of the v mode as the quantum number increases. 

The calculated Vhh of 3183 cm is too high but this is the harmonic value, whereas the experimental value is an anharmonic value. This is also compounded by the tendency of DFT to overestimate frequencies. Scaling the frequency by the usual value of 0.97 and assuming that the... 

The calculated harmonic vibrational frequencies for N2O are shown in Table 23. The derived experimental values are also quoted ". As expected, the MP2 N—N (2159 cm ) and N—O (1251 cm ) stretches are underestimated, with the triple-bond harmonic mode having the larger calculated-experiment gap. The observed fundamental vibrational frequencies(2224 and 1285 cm respectively) for these two modes are naturally smaller than the derived harmonic values, and are therefore closer to the calculated stretch frequencies. The bending mode is very well calculated, despite the absence of f-type functions from the basis set, as has been shown necessary for acetylene. 

The harmonic vibrational frequencies for acetonitrile, both experimentaP and MP2 calculated, are shown in Table 47. Both the C—H stretch frequencies calculated at 3107 cm and 3197 cm" and the C—C mode (1437 cm ) are close to the experimentally derived harmonic frequencies , while the C—N stretch differs by over 100 cm", as appropriate to the more difficult bond type. The measured fundamental C—H frequencies are about 123 cm below the derived harmonic values, as expected . The CCN bend mode at 360 cm" in Table 47 is particularly well reproduced. This may also represent a cancellation of errors at this... 

The first spectroscopic identification of protonated acetonitrile, CHsCNH", has only very recently been reported, by observation of the N—H fundamental stretch band at 3527 cm. The MP2 calculated harmonic value in Table 47 is 3664 cm If the difference between the observed fundamental and calculated harmonic C—H stretch frequency of ca 123 cm" for acetonitrile is arbitrarily subtracted from the calculated 3664 cm" N—H frequency of CHsCNH, the resulting extrapolated fundamental frequency is 3541 cm, which is close to the measured value. 

The force constants associated with a molecule s potential energy minimum are the harmonic values, which can be found from harmonic normal mode vibrational frequencies. For small polyatomic molecules it is possible (Duncan et al., 1973) to extract harmonic normal mode vibrational frequencies from the experimental anhar-monic n = 0 — 1 normal mode transition frequencies (the harmonic frequencies are usually approximately 5% larger than the anharmonic 0 - 1 transition frequencies). Using a normal mode analysis as described in chapter 2, internal coordinate force constants (e.g., table 2.4) may be determined for the molecule by fitting the harmonic frequencies. 

The computed AEs are found 1.3, 2.6, and 2.9 kJmol" below the ATcT reference values for CH, CH4, and C2H6, respectively. One reason for these discrepancies is that in the present work, we have only included (except for the hindered-rotor treatment) the harmonic ZPVE. For example, the anharmonic correction to the ZPVE of CH amounts to 0.9 kJrnoR at the ae-CCSD(T)/cc-pCVTZ level [107]. Taking this anharmonic correction into account would have produced a theoretical AE for CH of 1209.3 kJrnoR within 0.4 kJ mol of the experimental value. The total ZPVE contribution would have been —77.4 kJrnoR, in good agreement with the value (—77.6 kJmol" ) of Schwenke [113]. For methane, Schwenke s value [114] amounts to —116.1 kJ mol whereas our harmonic value is —117.9 kJ mol Taking Schwenke s value in place of ours would reduce the error in the calculated AE of methane from 2.6 to 0.8 kJ mol k For ethane, an accurate (anharmonic) ZPVE contribution of —194.1 kJmol" is available from benchmark calculations performed by Karton and co-workers [115], This contribution is 2.6 kJmol" smaller in magnitude than our harmonic value, which makes up for almost all of the error of 2.9 kJ mol k Eurthermore, our fc-CCSD(T) value of 2972.5 kJ mol compares well with the value of 2973.7 kJ mol obtained at the W4 level [116[ by these authors. 

Under normal conditions, the fundamental frequency oscillates in the quartz resonator. However, the characteristics of the external network can be utilized to promote oscillation in a higher order overtone mode. The exact overtone frequency is typically not an exact harmonic of the fundamental frequency. However, the overtone is normally close to a harmonic value. The external circuit is tuned to a frequency near the desired overtone frequency. 

The final column of Table II shows matrix isolation spectra from Redington and Hamilll and Andrews and Johnson as reassigned by Redington and Hamill.102 The frequencies appear to be highly perturbed and for some reason are closer to the gas-phase harmonic values than the gas-phase anharmonic ones. 

Since the energy levels of the enharmonic oscillator H in Pd differ from those of an harmonic oscillator, the oscillatory partition function also differs from the harmonic value. Because of the missing coupling terra it can be evaluated as the third power of a one dimensional partition function ... 

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