Anharmonic constants

Vibrational anharmonicity constant Vibrational coordinates Internal coordinates Normal coordinates, dimensionless Mass adjusted Vibrational force constants "eAe A,s get Ri, r 0J, etc. Qr m-i ... 

If all anharmonic constants except cOgXg are neglected, is a linear function of v... 

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. 

These may be explained in the following manner. One band is explained as the anharmonic overtone of the symmetric stretch at 1151cm-1, occurring at 2305 cm-1, with the 3 cm 1 difference attributed to the anharmonic constant. The other three bands may be explained as combination bands. 

Figure 2. Extraction of anharmonicity constants ojexe from the comparison of fundamental OH stretching spectra (center) with overtone spectra (top) and OD spectra (bottom) for the case of jet cooled trifluoroethanol (M) and its most stable dimer conformation, which features a hydrogen bond donor stretching band (Dd) and an acceptor stretching band (Da). The deuteration analysis yields slightly different constants than the overtone approach and underestimates the hydrogen bond effect on donor stretching modes [89].

The anharmonicity of the O—H stretching oscillators changes with cluster size. For the monomer, the anharmonicity constant is on the order of 90 cm 1. A coarse deuteration analysis [16, 88] suggests that it increases by more than 20% upon trimer and tetramer formation [16]. More accurate overtone analyses are possible in a rare gas matrix [88], but the matrix shift complicates a direct comparison to theory. As an example, the overtone-deduced anharmonicity of methanol monomer in a nitrogen matrix [88] is 85 cm whereas in vacuum [16] it is 92 cm The deuteration-estimated anharmonicity is 91 cm 1 for the monomer and 97 cm 1 for the dimer donor in the nitrogen matrix, whereas it is 87 cm 1 for the monomer and 89 cm 1 for the dimer donor in vacuum. Clearly, only a vacuum overtone measurement would be fully conclusive, but as the matrix study [88]... 

The harmonic frequencies and the anharmonic constants may be obtained from experimental vibrational spectra, although their determination becomes difficult as the size of the system increases. In Table 1.10, we have listed experimental harmonic and anharmonic contributions to the AEs. These contributions may also be obtained from electronic-structure calculations of quadratic force fields (for harmonic frequencies) and cubic and quartic force fields (for anharmonic constants). For some of the larger molecules in Table 1.11, we have used ZPVEs calculated at the CCSD(T)/cc-pVTZ level or higher, see Ref. 12. In some cases, both experimental and theoretical ZPVEs are available and agree to within 0.3 kJ/mol [12, 57],... 

Isotope effects on anharmonic corrections to ZPE drop off rapidly with mass and are usually neglected. The ideas presented above obviously carry over to exchange equilibria involving polyatomic molecules. Unfortunately, however, there are very few polyatomics on which spectroscopic vibrational analysis has been carried in enough detail to furnish spectroscopic values for Go and o)exe. For that reason anharmonic corrections to ZPE s of polyatomics have been generally ignored, but see Section 5.6.3.2 for a discussion of an exception also theoretical (quantum package) calculations of anharmonic constants are now practical (see above), and in the future one can expect more attention to anharmonic corrections of ZPE s. 

In Equation 5.34 to is the harmonic frequency, v the vibrational quantum number, and xe and ye the first and second anharmonicity constants (mass dependent, co x /(coxe) = X /X = il/il, l, and i are vibrational reduced masses). The ZPE(v = 0) contribution to RPFR through first order is thus... 

Table 5.4 Gas phase frequencies and frequency shifts on condensation, and anharmonicity constants for CHCI3 and CDCI3 in cm-1 (Jancso, G., Jakli, Gy. and Fetzer, C. Z., Naturforsch. 38a, 184 (1983)) ...

It would seem to us to be extremely interesting to investigate the influence of the cation on the anharmonicity constant of internal vibrations as seen, for example, for K2Cr04 and BaCr04 above. If there is no error in the measurements for chromate, then this influence found for Ba2+ is remarkably large. We hope to measure this effect systematically. 

Once again v is the vibrational quantum number with allowed values of 0, 1, 2,..., and xc and yc are anharmonicity constants characteristic of the molecule. 

If anharmonicity constants were negligible, the band-origin frequencies would be given by the shorthand notation. Give the shorthand designations for each of the following IR bands (a) (000) (202) (b) (010) (011) (c) (110) (300). 

The anharmonicity constant vexe is small compared to ve, but its effect increases as v increases, and the overtones deviate more and more from simple multiples of the fundamental frequency with increasing vSee Fig. 4.7. The infrared region extends from 10 to 14,000 cm-1 (7000 A). Molecular vibrational frequencies run from 100 to 4000 cm-1, so that the fundamental and lower overtones lie in the infrared region. 

Figure 4. Theoretical dependence of rate of IVR as a function of vibrational energy of model molecule CFCl2Br with given anharmonicity constant (from Ref. 1).

The anharmonicities of the potential contribute by the terms involving the constants x, g, y,. .. as well as the energy shifts AEx = 0(h2),. .. and the frequency shifts Aw, = 0(h2),. These anharmonic constants can be calculated by the Van Vleck contact transformations [20] as well as by a semi-classical method based on an h expansion around the equilibrium point [14], which confirms that the Dunham expansion (2.8) is a series in powers of h. Systematic methods have been developed to carry out the Van Vleck contact transformations, as in the algebraic quantization technique by Ezra and Fried [21]. It should be noted that the constants x and g can also be obtained from the classical-mechanical Birkhoff normal forms [22], The energy shifts AEx,... 

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