Harmonicity constant

Even though the system is stable in symmetric form, a vibronic part acts in the inner resorts of the electronic structure. Anticipating, the aromaticity can be presented as the situation when the distortion tendency does not overcome the non-vibronic positive harmonic constant maintaining the high symmetry. The lAg ground state have the aspect of in-phase combination of the Kekule resonance structures, whereas the first excited lB2u state results from their out-of-phase mixing. These states are mixed via the b2u distortion coordinate of bond alternation. Additionally... 

The MM-like model of complexes of nontransition elements requires even less than is given by eq. (4.50) only the first and the second term in the first row. They represent the bare harmonic dependence of the energy on the nuclear shifts and the renormalizations of the respective harmonic constants due to adjustment of the electronic structure to these shifts ... 

Uxed traction (independent of the atom type) of the harmonic constant deteriorates... 

I 1 6 iS A P3 U equilibrium angle = k 6 - 6/6). A fourth-order polynomial enables an independent fit of the barrier to linearity. Such constrained polynomial fittings are rarely done. Instead the bending function is taken to be identical for all atom types, for example a fourth-order polynomial with cubic and quartic constants as a fixed fraction of the harmonic constant. These features are illustrated for H2O in Figure 2.5, where the exact form is taken ... 

The principal progression-forming mode (vj) in the resonance Raman spectrum is almost harmonic up to high quantum number with the an-harmonicity constant = — 0.16cmOther modes show only one or two harmonics. The assumption of a harmonic potential is also reasonable for this molecule. 

The principal progression-forming mode, v, is very close to being a simple harmonic oscillator up to vibrational quantum number 6, with the an-harmonicity constant x,j = —0.11cm . The overtone band wavenumbers derived from the calculation differ from the experimental values by a few cm as the vibrational quantum number increases. Other modes show only one harmonic. The assumption used in the calculation that the potential surfaces are harmonic is reasonable for this molecule. 

Due to increasing accuracy of altimetric data, complete coverage with harmonic constants is required today in order to detide altimeter data in the Baltic Sea, because in some locations the tides may be comparable to other anomalies. 

Table II. Principal Harmonic Constants of the Tide and Tidal Stream for Long Island Sound ... 

Here k and K are the harmonic constants of the longitudinal and hehcoidal springs, respectively. The last term in the Hamiltonian represents a Morse potential approximating the potential of the hydrogen bonds. Eventually, D and a are the depth and the inverse width of the Morse potential well, respectively. 

If higher accuracy is desired (e.g., for computing IR frequencies), a third-order term can be included with an an-harmonicity constant set to be a fraction of... 

Hie tidal range can differ greatly from place to place. Harmonic analysis, in which the tidal osdllations are separated into harmonic components, is frequently used in the calculation of tides. Harmonic constants for the prediction of tides at gauge stations near the selected site may be obtained from the national authorities. 

By force constants, we refer to derivatives of the electronic energy with respect to internal geometrical parameters of a molecule (Table 1). llie harmonic force constants are the second derivatives evaluated at an equilibrium structure, while higher derivatives may be put in the category of anharmonic force constants. From harmonic constants, harmonic frequencies are immediately obtained, and, typically, the harmonic frequencies are at least accurate enough to be used in making zero-point correaions to stabilities. Given harmonic constants and the lowest one or two orders of anharmonic force constants (third and fourth derivatives), transition frequencies of small polyatomics can often be extracted. Usually, this involves a perturbative treatment of the anharmonic parts of the potential in a produa basis of harmonic oscillator functions. 

J. M. L. Martin and P. R. Taylor, Chem. Phys. Lett. 248, 336 (1996). TTie vibrational contributions have been obtained from a CCSD(T) force field, with the harmonic constants extrapolated to the basis-set limit and the anharmonic constants calculated in the cc-pVTZ basis. 

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