Anharmonic potential function

The only types of anharmonic potential function we have encountered so far are the two illustrated in Figure 6.38, both of which show only a single minimum. There are, however, some vibrations whose potential functions do not resemble either of those but show more than one minimum and whose term values are neither harmonic, nor are they given by Equation (6.88) or Equation (6.89). Such vibrations can be separated into various types, which will now be discussed individually. 

The matrix elements connecting 2 = 1/2 and 3/2 states are more complicated, because these states have different sets of vibrational wave functions, and there is no simple expression for the vibrational matrix elements for these highly anharmonic potential functions. These matrix elements are therefore treated as phenomenological spectroscopic parameters, QVtV>, where v and v refer to the 2 = 1/2 and 3/2 states respectively. The addition of centrifugal distortion constants further complicates the analysis [211]. 

It should be noted that our attempt to calculate vibrational relaxation for clusters and complex systems should be regarded as a preliminary attempt because the anharmonic potential function, themselves, are approximate and their performance should be carefully examined by calculating IR spectra in addition to vibrational relaxation. 

Tanaka T, Morino Y (1970) Coriolis interaction and anharmonic potential function of ozone from the microwave spectra in the excited vibiationtil states. J Mol Spectr 33 538-551... 

The question arises whether these series of Q branches could be interpreted on a slightly anharmonic potential function rather than on a cosine function type of pseudo-rotation. For a low barrier ( 7.5 kJ mol ) the contribution to the entropy at 298 K of a hindered pseudo-rotator differs from that of a harmonic oscillator of the same frequency by 6 J K mol , which is certainly measurable. But for a high barrier ( 14.6 kJ mol ) the difference between the entropy contributions is 0.6 J K mol at 298 K and only 1.2 JK mol at 398 K. Therefore only for a low barrier can the two possibilities be distinguished by thermodynamic measurements. For Y-butyrolactone and ethylene carbonate the barriers are calculated to be 46 kJ mol and 75 kJ mol , respectively, which would negate the effect of pseudo-rotation at reasonable temperatures. Thus, in molecules with a sufficiently high barrier the motion can be treated as an ordinary vibration in which the puckering oscillates about a stable configuration. 

D Bermejo, S Montero. Anharmonic potential functions as derived from Raman intensities Methane. J Chem Phys 81 3835-3841, 1984. 

FtG. L13. Hannonic and anharmonic potential functions and energy levels for the C-H stretching vibration in CHCI3. The units for the vertical coordinates are cm" or Elhc. ... 

Fig. 1.26. The equilibrium bond lengths for the v 0 and v = 1 states for HCl and DCl. The solid curve is the anharmonic potential function and the dotted curve is the harmonic potential function for both molecules.

Vr = 53.64 cm ring-puckering vibration, single minimum anharmonic potential function determined ... 

M-L bond deformations are often described with a harmonic potential function, equation (1). This description is suitable for small distortions from equilibrium and it has been demonstrated to provide an accurate and transferable description of M-L bond lengths in many transition metal complexes. When large distortions of M-L bond length occur, a more accurate description of the potential energy surface can be obtained with anharmonic potential functions such as that used in MM3, equation (2). 

Figure 1 Energy surfaces for K-0 bond length distortion computed with an ionic potential function (line), a harmonic potential function (triangles), and the default MM3 anharmonic potential function (circles)...

Fonnulas for evaluating the partial derivatives of internal coordinates with respect to Cartesian displacement coordinates (b i= dKJd [, = x, y, z) are available [162,164], The same coefficients are used in analogous transformation of the anharmonic potential function. 

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