Vibrational wave functions anharmonic potential

Owing to the effects of mechanical anharmonicity - to which we shall refer in future simply as anharmonicity since we encounter electrical anharmonicity much less frequently -the vibrational wave functions are also modified compared wifh fhose of a harmonic oscillator. Figure 6.6 shows some wave functions and probabilify densify functions (iA A ) for an anharmonic oscillator. The asymmefry in and (iA A ) 5 compared wifh fhe harmonic oscillator wave functions in Figure f.i3, increases fheir magnitude on the shallow side of the potential curve compared with the steep side. 

The relevant question regarding secondary IEs on acidity is the extent to which IEs affect the electronic distribution. How can an inductive effect be reconciled with the Born-Oppenheimer approximation Although the potential-energy function and the electronic wave function are independent of nuclear mass, an anharmonic potential leads to different vibrational wave functions for different masses. Averaging over the ground-state wave function leads to different positions for the nuclei and thus averaged electron densities that vary with isotope. This certainly leads to NMR isotope shifts (IEs on chemical shifts), because nuclear shielding is sensitive to electron density.16... 

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]. 

Combination and difference bands Besides overtones, anharmonicity also leads to the appearance of combination bands and difference bands in the IR spectrum of a polyatomic molecule. In the harmonic case, only one vibration may be excited at a time (the transition dipole moment integral vanishes when the excited state is given by a product of more than one Hermite polynomial corresponding to different excited vibrations). This restriction is relaxed in the anharmonie case and one photon can simultaneously excite two different fundamentals. A weak band appears at a frequency approximately equal to the sum of the fundamentals involved. (Only approximately because the final state is a new one resulting from the anharmonie perturbation to the potential energy mixing the two excited state vibrational wave functions.)... 

To take into account this feature in the molecular models we can make an introducing correction (broadening) of the potential function dependent on the degree of anharmonic transformation of each vibration mode. The value of the corresponding elements of the displacement vector b would seem to be a natural criterion for the selection of the correction magnitude. However, the position of the area of the maximal overlap of the vibrational wave functions in the case of multidimensional displacement turns out to be a more adequate characteristic. This area is characterized by the point X (the top of the potential barrier, see Fig. 3.2) with coordinates x, (x ) in space of the normalized normal coordinates. 

Fig.7.n. The ground-state vibrational wave function i/r of the anharmonic oscillator (of potential energy Vi, taken here as the Morse oscillator potential energy t is asymmetric and shifted toward positive values of the displacement when compared to the wave function i/ro for the harmonic oscillator with the same force constant (the potential energy V2, ) ... 

Anharmonic vibrational energies and wave functions were determined for each of these potentials by solving the vibrational SE. The variational Fourier... 

The pattern of the vibrational levels looks similar to those for the Morse oscillator (p. 192). The low levels are nearly equidistant, reminding us of the results for the harmonic oscillator. The corresponding wave functions also resemble those for the harmonic oscillator. Higher-energy vibrational levels are getting closer and closer, as for the Morse potential. This is a consequence of the anharmonicity of the potential-we are just approaching the dissociation limit where the Uk R) curves differ qualitatively from the harmonic potential. 

As already stated, the Morse potential is our first example of a potential surface that describes a particular motion. The bond vibrates within the constraints imposed by this potential. One may ask, "At any given moment, what is the probability of having a particular bond length " This is similar to questions related to the probability of finding electrons at particular coordinates in space, which we will show in Chapter 14 is related to the square of the wave-function that describes the electron motion. The exact same procedure is used for bond vibrations. We square the wavefunction that describes the wave-like nature of the bond vibration. Let s explore this using the potential surface for a harmonic oscillator (such as with a normal spring), instead of an anharmonic oscillator (Morse potential). For the low energy vibrational states, the harmonic oscillator nicely mimics the anharmonic oscillator. 

H. A localized vibrational wave-packet as a linear combination of delocahzed vibrational states. Figure 1.7 discussed the preparation of a localized vibrational state, a state that vibrates in the potential well in a manner similar to a classical particle. If the well is harmonic the wave function will remain localized indetinitely. Realistic molecular potentials are anharmonic so that after a few oscillations die state will delocahze. Even in the harmonic case, external perturbations such as... 

In general terms, since an exact solution of the vibrational equation in terms of anharmonic wave function is not possible, use is made of the fact at for finite displacements at each step of the potential energy or transitional dipole moment expansion, the higher terms are much smaller than the respective lower terms. A perturbation theoiy treatment becomes, therefore, feasible. The potential energy may be expressed in the form [3]... 

The value of f th depends on the chosen potential function. With AU = 3 kcal, we obtain 3th 2,5 10-s. In reality the value is about 0th 6 10-5 to 8 10 s. This means that the anharmonicity of the thermal vibrations is even greater21). For the potential Eq. (6) (Fig. 4) we need only to calculate the elastic deformation by thermal vibrations in the chosen approximation for one half wave, if B is situated in the neighborhood of A. This leads to... 

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