Vibration potentials

Th e vibrational potential in ay be ex pan ded in a Taylor series abon t the etinilibrmni positions of the atoms. 

Notiee that as one moves to higher vf values, the energy spaeing between the states (Eyf-Eyf-i) deereases this, of eourse, refleets the anharmonieity in the exeited state vibrational potential. For the above example, the transition to the vf = 2 state has the largest Franek-Condon faetor. This means that the overlap of the initial state s vibrational wavefunetion Xvi is largest for the final state s Xvf funetion with vf = 2. 

The potential energy of vibration is a function of the coordinates, xj,. .., z hence it is a function of the mass-weighted coordinates, qj,. .., q3N. For a molecule, the vibrational potential energy, U, is given by the sum of the electronic energy and the nuclear repulsion energy ... 

The vibrational potential may be expanded in a Taylor series about the equilibrium positions of the atoms. 

S.4 Vibrational potential functions with more than one minimum... 

The vibrational potential energy U will also be a function of the q s, so I can write it... 

To study processes involving vibration we must add the two diatomic vibrational potentials to stnd we must also generalize... 

By applying Eqs. (4) or (36) we can calculate the average vibrational energy of a molecule with N vibrational modes, and we do this with respect to the zero of the vibrational potentials, implying that we will include all zero-point energies ... 

L. O. Bowen, Nuclear magnetic acoustic resonance and Debye vibration potentials in non-viscous liquids, Proc. Phys. Soc., 1966, 87, 717-720. 

Adiabatic Pseudospectral Methods for Multidimensional Vibrational Potential. 

The vibrational potential energy of subunit A in oxidation state M can be written... 

Fig. 3.1 Born-Oppenheimer vibrational potentials for a diatomic molecule corresponding to the CH fragment. The experimentally realistic anharmonic potential (solid line) is accurately described by the Morse function Vmorse = De[l — exp(a(r — r0)]2 with De = 397kJ/mol, a = 2A and ro = 1.086 A (A = Angstrom = 10 10m). Near the origin the BO potential is adequately approximated by the harmonic oscillator (Hooke s Law) function (dashed line), Vharm osc = f(r — ro)2/2. The harmonic oscillator force constant f = 2a2De...

Note that the first two terms on the r.h.s. of Eq. (2.1) are zero, the first by arbitrary choice, the second by virtue of req being the minimum. If we truncate after the first non-zero term, we have the simplest possible expression for the vibrational potential energy... 

The most extensive potential obtained so far with experimental confirmation is that of Le Roy and Van Kranendonk for the Hj — rare gas complexes 134). These systems have been found to be very amenable to an adiabatic model in which there is an effective X—Hj potential for each vibrational-rotational state of (c.f. the Born Oppenheimer approximation of a vibrational potential for each electronic state). The situation for Ar—Hj is shown in Fig. 14, and it appears that although the levels with = 1) are in the dissociation continuum they nevertheless are quasi bound and give spectroscopically sharp lines. 

Figure 2.11 Addition to a vibration potential (solid curve) of a perturbation of positive slope makes bond stretching more difficult and decreases the equilibrium separation (dashed curve). Reprinted with permission from E. R. Thornton, J. Amer. Chem. Soc., 89, 2915 (1967). Copyright by the American Chemical Society.

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